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2024arXiv240903959D | The stellar masses of galaxies are measured using integrated light via several methods however few of these methods were designed for lowmass Mstarlesssim108rmModot dwarf galaxies whose properties e.g. stochastic star formation low metallicity pose unique challenges for estimating stellar masses. In this work we quantify the precision and accuracy at which stellar masses of lowmass galaxies can be recovered using UVopticalIR photometry. We use mock observations of 469 lowmass galaxies from a variety of models including both semiempirical models GRUMPY UniverseMachineSAGA and cosmological baryonic zoomin simulations MARVELous Dwarfs and FIRE2 to test literature colorMstarL relations and multiwavelength spectral energy distribution SED mass estimators. We identify a list of best practices for measuring stellar masses of lowmass galaxies from integrated photometry. These include updated prescriptions for stellar mass based on gr color and WISE 3.4 mum luminosity which are less systematically biased than literature calibrations and can recover true stellar masses of lowmass galaxies with sim0.1 dex precision. When using SED fitting to estimate stellar mass we find that the form of the assumed star formation history can induce significant biases parametric SFHs can underestimate stellar mass by as much as sim0.4 dex while nonparametric SFHs recover true stellar masses with insignificant offset 0.03pm0.11 dex. However we also caution that noninformative dust attenuation priors may introduce Mstar uncertainties of up to sim0.6 dex. | 2024-09-01T00:00:00Z | ['10.48550/arXiv.2409.03959', '2024arXiv240903959D', 'arXiv:2409.03959'] | ['Astrophysics - Astrophysics of Galaxies'] | Stellar Mass Calibrations for Local LowMass Galaxies | 2,024 | 165 | 0.59 | ['EPRINT_HTML', 'EPRINT_PDF'] | 1 | https://arxiv.org/pdf/2409.03959.pdf | {'Stellar Mass Calibrations for Local Low-Mass Galaxies': 'Mithi A. C. de los Reyes, 1 Yasmeen Asali, 2 Risa H. Wechsler, 3, 4, 5 Marla Geha, 2 Yao-Yuan Mao, 6 Erin Kado-Fong, 2 Ragadeepika Pucha, 6 William Grant, 1 Pratik J. Gandhi, 2, 7 Viraj Manwadkar, 3, 4, 5 Anna Engelhardt, 8 Ferah Munshi, 8 and Yunchong Wang 3, 4, 5 \n1 \n7 8 Department of Physics, George Mason University, 4400 University Dr, Fairfax, VA 22030, USA \nDepartment of Physics & Astronomy, Amherst College, 6 East Drive, Amherst, MA 01002, USA 2 Department of Astronomy, Yale University, New Haven, CT 06520, USA 3 Kavli Institute for Particle Astrophysics & Cosmology, Stanford University, 452 Lomita Mall, Stanford, CA 94305, USA 4 Department of Physics, Stanford University, 382 Via Pueblo Mall, Stanford, CA 94305, USA 5 SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025, USA 6 Department of Physics and Astronomy, University of Utah, Salt Lake City, UT 84112, USA Department of Physics and Astronomy, University of California - Davis, One Shields Avenue, Davis, CA 95616, USA', 'ABSTRACT': "The stellar masses of galaxies are measured using integrated light via several methods - however, few of these methods were designed for low-mass ( M ⋆ ≲ 10 8 M ⊙ ) 'dwarf' galaxies, whose properties (e.g., stochastic star formation, low metallicity) pose unique challenges for estimating stellar masses. In this work, we quantify the precision and accuracy at which stellar masses of low-mass galaxies can be recovered using UV/optical/IR photometry. We use mock observations of 469 low-mass galaxies from a variety of models, including both semi-empirical models (GRUMPY, UniverseMachine-SAGA) and cosmological baryonic zoom-in simulations (MARVELous Dwarfs and FIRE-2), to test literature colorM ⋆ /L relations and multi-wavelength spectral energy distribution (SED) mass estimators. We identify a list of 'best practices' for measuring stellar masses of low-mass galaxies from integrated photometry. These include updated prescriptions for stellar mass based on g -r color and WISE 3.4 µ m luminosity, which are less systematically biased than literature calibrations and can recover true stellar masses of low-mass galaxies with ∼ 0 . 1 dex precision. When using SED fitting to estimate stellar mass, we find that the form of the assumed star formation history can induce significant biases: parametric SFHs can underestimate stellar mass by as much as ∼ 0 . 4 dex, while non-parametric SFHs recover true stellar masses with insignificant offset ( -0 . 03 ± 0 . 11 dex). However, we also caution that non-informative dust attenuation priors may introduce M ⋆ uncertainties of up to ∼ 0 . 6 dex.", '1. INTRODUCTION': "Much of observational astronomy can be distilled into a single exercise: converting the measured light from astrophysical objects to physical properties. The mass of a galaxy is one of its most fundamental properties, and one that heavily influences not only a galaxy's observable properties, but its evolution over cosmic time. The stellar mass ( M ⋆ ) of a galaxy is a particularly useful property that encodes the galaxy's cumulative star formation, merger, and accretion history. As a result, stellar mass is frequently used as a variable in empirical \nrelationships, such as the stellar mass-halo mass relation (e.g., Moster et al. 2010; Wechsler & Tinker 2018; Girelli et al. 2020), the star formation main sequence (e.g., Brinchmann et al. 2004; Popesso et al. 2023), and the mass-metallicity relation (e.g., Tremonti et al. 2004; Maiolino & Mannucci 2019), which are in turn used to probe galaxy evolution processes. \nMeasuring the stellar mass of a galaxy from its integrated light - the only option for most galaxies, which are too distant to resolve individual stars - is a nontrivial exercise. To the first order, a galaxy's stellar mass is correlated with its total luminosity: more stars produce more light. However, determining the exact conversion from light to stellar mass (i.e., the mass-to-light ratio, M ⋆ /L ) is complicated by several factors. \nFor a single coeval stellar population, the most massive stars contribute the majority of light, while the more numerous lower-mass stars contribute the majority of mass; the ratio of high- to low-mass stars is set at birth by the stellar initial mass function (IMF) and modified by stellar evolution. A galaxy consists of multiple such stellar populations with varying ages and metallicities. The light from these populations is also modified by baryonic components like gas and dust located both within the galaxy and along the observed line of sight. To estimate M ⋆ /L , one can model each of these components - the stellar populations and their evolution, nebular emission, and dust attenuation and emission - to obtain a model for a galaxy's observed spectrum. This is the basis of stellar population synthesis (SPS) modeling (Tinsley 1968), the most common method of estimating a galaxy's stellar mass. SPS models are fit to the galaxy's observed light, which are typically in the form of (1) single fluxes or colors, (2) broadband spectral energy distributions (SEDs), or (3) spectra (for a review, see Conroy 2013). Measurements of M ⋆ therefore depend on all of the assumptions implicit in SPS modeling: the observable properties of a single stellar population with a given age and metallicity (which depends on the form of the IMF and stellar evolution models), the stellar populations that comprise a galaxy (which depends on the galaxy's assumed star formation history or SFH), and the impact of gas and dust (which depends on heavily simplified models). \nSeveral works have tested the effect of these assumptions on M ⋆ measurements; in particular, quantifying the uncertainties from SED fitting has become especially relevant in the era of JWST, which has begun to obtain SEDs of galaxies in the early universe. For example, Lower et al. (2020) tested the impact of different SFH assumptions when measuring M ⋆ from SED fitting, and Pacifici et al. (2023) tested the effect of using different SED fitting codes. However, nearly all of these tests have focused on relatively massive galaxies: although Lower et al. (2020)'s sample included galaxies down to 10 7 . 6 M ⊙ , the bulk of their sample was in the higher mass range of 10 8 -12 M ⊙ , and Pacifici et al. (2023) did not consider galaxies below 10 8 M ⊙ at all. \nLow-mass 'dwarf' galaxies ( M ⋆ ≲ 10 8 M ⊙ ) have often been overlooked in these tests, partly because they pose unique problems for SPS modeling. They are thought to have relatively complex, stochastic star formation histories (Emami et al. 2019), making their SEDs and spectra more difficult to fit with SPS models. Furthermore, previous large galaxy surveys have often been incomplete in the low-mass regime, making it natural to prioritize high-mass galaxies. \nHowever, low-mass galaxies are the most numerous type of galaxy in the universe (Schechter 1976). Due to their small gravitational potentials, they are sensitive to both internal processes like stellar feedback (Collins & Read 2022) and external environmental effects (e.g., Dekel et al. 2003; Farouki & Shapiro 1981; Deason et al. 2014), making them excellent testbeds for cosmological and galaxy evolution models. With the advent of a new generation of extragalactic surveys - e.g., the Dark Energy Spectroscopic Instrument (DESI; DESI Collaboration et al. 2016) and DESI Legacy Surveys (Dey et al. 2019), the Legacy Survey of Space and Time (LSST; Ivezi'c et al. 2019), Euclid (Euclid Collaboration et al. 2024), and Roman (Akeson et al. 2019) - we are beginning to observe unprecedentedly large samples of lowmass galaxies across a range of environments. Accurately characterizing the masses of these galaxies is of critical importance: the low-mass end of the galaxy mass function can constrain cosmological models (Sales et al. 2022), and extending scaling relations to low masses can constrain galaxy physics (e.g., Woo et al. 2008). \nIn this work, we test a number of stellar mass measurement techniques on simulated low-mass galaxies ( M ⋆ = 10 4 -8 M ⊙ ). In light of several ongoing and upcoming galaxy surveys, we primarily focus on methods that use UV/optical/IR photometry. We identify some best practices for measuring low-mass galaxy masses, and we develop statistical corrections to extend commonly used empirical calibrations to the low-mass regime. \nThe structure of this paper is as follows. We present the stellar mass calibrations used in this paper in Section 2, and we describe the observed and simulated datasets used to test these stellar mass calibrations in Section 3. We then test the recovery of 'ground-truth' stellar masses from simulated data: using photometric calibrations in Section 4, and using SED fitting in Section 5. We summarize our main results and list 'best practices' for measuring low-mass galaxy masses in Section 6. In Section 7, we discuss potential systematics in our experiment, and we summarize our conclusions in Section 8.", '2. STELLAR MASS CALIBRATIONS': 'Here we describe the stellar mass measurement techniques considered in this work. We emphasize that this is not an exhaustive list, particularly since we only focus on one SED fitting code (Section 2.2). We convert all calibrations to a Chabrier (2003) IMF.', '2.1. Empirical photometric calibrations': "We first consider empirical calibrations based on optical and IR photometry. Such approaches are attractive \nfrom an observational perspective because measuring a galaxy's flux in one or two photometric bands requires far fewer resources than measuring a multi-band SED over a broad wavelength range, let alone a spectrum. Fortunately, several previous studies have found relatively tight correlations between stellar mass (typically measured from full SED fitting) and combinations of different photometric bands. Historically, these have often been parameterized as relationships between M ⋆ /L and a single color (e.g., Bell & de Jong 2001), but a number of studies have developed stellar mass calibrations based on anywhere from one to three photometric bands. In this work, we use the following calibrations.", '2.1.1. Optical M ⋆ /L calibrations': 'While there are many optical-NIR colorM ⋆ /L relations (e.g., Portinari et al. 2004; Zibetti et al. 2009; Gallazzi & Bell 2009; Taylor et al. 2011; Roediger & Courteau 2015; Schombert et al. 2019; Kouroumpatzakis et al. 2023), we have chosen to focus first on the colorM ⋆ /L relations by Bell et al. (2003) and Into & Portinari (2013) because these have been used in a number of recent low-mass galaxy surveys. For example, the Satellites Around Galactic Analogs survey (SAGA; Geha et al. 2017) and other surveys of low-metallicity/lowmass galaxies (Hsyu et al. 2018) use recalibrations of the Bell et al. (2003) relations to measure stellar masses, while the Exploration of Local VolumE Satellites survey (ELVES; Carlsten et al. 2022; Danieli et al. 2023) use a colorM ⋆ /L relation from Into & Portinari (2013). \nIn the following equations in this section, all optical magnitudes are AB magnitudes that have been both corrected for Milky Way extinction and k -corrected to z = 0 using g -r color (Chilingarian et al. 2010). The corrected r - and g -band magnitudes are denoted as M r, 0 and M g, 0 . Unless otherwise noted, we assume absolute solar magnitudes from Willmer (2018): 4.65 for r -band and 5.11 for g -band. \nBell et al. (2003) developed colorM ⋆ /L relations based on optical-NIR SEDs of galaxies with photometry from the Two Micron All Sky Survey (2MASS) and the Sloan Digital Sky Survey (SDSS). We use their g -r relation: \nlog( M ⋆ / M ⊙ ) = 1 . 461 + 1 . 098(g -r) 0 -0 . 4M r , 0 . (1) \nThis relation was calibrated using a sample of galaxies with masses down to ∼ 10 8 . 5 M ⊙ . To fit these galaxies, Bell et al. (2003) constructed a grid of SPS models with \nmetallicities between 10 -2 . 3 < Z/ Z ⊙ < 10 0 . 41 . They assumed exponentially decaying SFHs and did not account for dust attenuation, although they ran some simple tests to quantify the impact of adding star formation bursts and a simple constant dust reddening model. \nInto & Portinari (2013) also proposed colorM ⋆ /L relations using more detailed asymptotic giant branch models. Here we consider the relation between g -r and M ⋆ /L g used by the ELVES survey (Danieli et al. 2023): \nlog( M ⋆ / M ⊙ ) = 1 . 332 + 1 . 774(g -r) 0 -0 . 4M g , 0 (2) \nNote that this equation uses a slightly different absolute solar g -band magnitude of 5.144 (Table 1 of Into & Portinari 2013). Like Equation 1, this relation also assumes an exponentially decaying SFH; however, it was determined from theoretical models, rather than semiempirically from observed photometry. \nThe majority of colorM ⋆ /L relations in the literature, including Equations 1 and 2 above, were calibrated using samples of relatively massive ( M ⋆ > 10 8 -9 M ⊙ ) galaxies. We therefore also consider empirical relations that are specifically based on low-mass galaxies. \nHerrmann et al. (2016) measured colorM ⋆ /L relations for 34 nearby dIrr galaxies from Local Irregulars That Trace Luminosity Extremes, The H I Nearby Galaxy Survey (LITTLE THINGS; Hunter et al. 2012). We consider their g -r relation: \nlog( M ⋆ / M ⊙ ) = 1 . 755 + 0 . 894(g -r) 0 -0 . 4M g , 0 . (3) \nStellar mass density profiles for these galaxies were measured by modeling galaxy SEDs with a multi-component stellar population model. \nKlein et al. (2024) proposed a colorM ⋆ /L relation based on low-mass galaxies in the FIRE-2 simulations (see Section 3.1): \nlog( M ⋆ / M ⊙ ) = 1 . 631 + 1 . 570(g -r) 0 -0 . 4M g , 0 . (4) \nThis relation is based on 20 simulated galaxies with stellar masses between 10 5 -10 M ⊙ . Klein et al. (2024) used the FIRE studio code (Gurvich 2022; Hopkins et al. 2005) to create post-processed images via ray projection from the star particles, then measured synthetic photometry from these mock images. \nFinally, Du et al. (2020) did not develop a new photometric calibration, but instead found corrections for literature colorM ⋆ /L relations in order to make them \ninternally self-consistent when using different photometric bands (McGaugh & Schombert 2014). They used a sample of low surface brightness galaxies to empirically correct several literature relations, including both Bell et al. (2003) and Into & Portinari (2013), listed respectively: \nlog( M ⋆ / M ⊙ ) = 1 . 461 + 1 . 097( g -r ) 0 -0 . 4 M r, 0 (5) \nlog( M ⋆ / M ⊙ ) = 1 . 254 + 1 . 530( g -r ) 0 -0 . 4 M r, 0 (6) \n2.1.2. Infrared (WISE) M ⋆ /L calibrations \nJarrett et al. (2023) computed updated M ⋆ /L calibrations based on near-infrared photometry from the Widefield Infrared Survey Explorer (WISE). They report two stellar mass calibrations, the first based on W1 (3.4 µ m) flux and the second based on W1 -W2 (3.4 µ m-4.6 µ m) color: \nlog( M ⋆ / M ⊙ ) = -12 . 62 + 5 . 00 log( L W1 , 0 ) -0 . 44 log( L W1 , 0 ) 2 +0 . 016 log( L W1 , 0 ) 3 (7) \nlog( M ⋆ / M ⊙ ) = log( L W1 , 0 ) -0 . 376 -1 . 053(W1 -W2) 0 (8) \nHere, L W1 , 0 = 10 -0 . 4( M W1 , 0 -M W1 , ⊙ ) , where the W1 inband solar value is M W1 , ⊙ = 3 . 24 mag (Jarrett et al. 2013). In Equations 7 and 8, all WISE magnitudes and colors are in Vega magnitudes (to convert from AB to Vega, we follow the prescriptions in Table 1 of Jarrett et al. 2011) and are k -corrected following Equations A1 and A2 in Jarrett et al. (2023). These calibrations were based on galaxies in the Galaxy and Mass Assembly (GAMA) survey with WISE fluxes. This sample contained galaxies with stellar masses down to ∼ 10 6 . 5 M ⊙ , but is likely incomplete below ∼ 10 8 M ⊙ . The GAMA DR4 stellar masses (Driver et al. 2022) were computed by fitting a grid of Bruzual & Charlot (2003) SPS models (for more details, see Taylor et al. 2011) and assumed exponential SFHs, uniform metallicities, and Calzetti et al. (2000) dust attenuation.', '2.2. Prospector': "The empirical relations considered in Section 2.1 use only a few photometric bands to estimate stellar masses. Fitting the multi-wavelength SED of a galaxy takes advantage of photometric information across a much broader wavelength range and is frequently considered the 'gold standard' for obtaining galaxy stellar masses from integrated light. Prospector (Leja et al. 2017; Leja et al. 2019; Johnson et al. 2021) is a Bayesian inference code that estimates galaxy properties by forward modeling galaxy SEDs with the Flexible Stellar Population \nSynthesis package (FSPS; Conroy et al. 2009; Conroy & Gunn 2010). \nAs we describe in Section 3, we also use FSPS to produce mock observations of simulated galaxies. Using Prospector to fit these mock observations is therefore an 'apples-to-apples' comparison that directly probes the capabilities of SED fitting rather than the effect of other systematic uncertainties (e.g., stellar model libraries, SED fitting algorithm). We defer tests of these other systematic effects, caused by variations among different SED fitting codes, to future work.", '3. SIMULATED GALAXIES': "We test stellar mass calibrations on mock observations of simulated galaxies, for which the 'true' stellar masses are known. Here we describe the models used and their major assumptions, as well as the method used to produce mock observations from the simulated galaxies.", '3.1. Galaxy models': "3.1.1. GRUMPY \nThe simplest model we consider is GRUMPY (Galaxy formation with RegUlator Model in PYthon; Kravtsov & Manwadkar 2022), a semi-analytic model of the 'regulator' type. In this model, a low-mass galaxy is simulated using a system of differential equations that track the mass conservation of different baryonic components. The key input to the model is an underlying dark matter halo accretion history, which is taken from the halo mass accretion histories from the ELVIS high-resolution simulation suite (Garrison-Kimmel et al. 2014). \nThe gas inflow rate is assumed to be proportional to the halo accretion rate with additional factors describing suppression due to UV heating (i.e., from reionization) and the formation of a hot gaseous halo. The gas outflow rate is assumed to be proportional to the star formation rate with the constant of proportionality (the mass-loading factor) parametrized as a function of stellar mass. The SFH is set by the star formation rate, which assumes a constant molecular hydrogen gas depletion time and instantaneous recycling. Only in-situ formed stars are considered in this model; stellar contribution from galaxy mergers is not accounted for in the model. The production and removal of heavy element abundances in the galaxy's ISM are also parameterized, providing a chemical evolution history for the galaxy. Finally, further stochasticity is added to the SFH by introducing a correlated random perturbation in SFR relative to the mean M ⋆ -SFR relation at each timestep (Pan & Kravtsov 2023). \nIn total, we obtain 69 galaxy SFHs and chemical enrichment histories from GRUMPY (with boosted \nstochasticity). This sample covers a stellar mass range of 10 4 -11 M ⊙ .", '3.1.2. UniverseMachine-SAGA': "The next model we consider is the UniverseMachine (Behroozi et al. 2019), an empirical galaxy-halo connection model. This model assumes that galaxy SFRs depend on host halo properties (halo mass, assembly history, and redshift). It then fits this relationship by predicting galaxy observables (e.g., stellar mass functions, quenched fractions) from a dark-matter-only simulation, and iteratively comparing the predictions with observations over a wide range of galaxy masses and redshifts simultaneously. Wang et al. (2021) extended the DR1 of the UniverseMachine model 2 to lower-mass galaxies by applying it to a joint set of 45 dark-matter-only zoom-in simulations of isolated Milky Way-mass halos from the Symphony compilation (Nadler et al. 2023). \nIn this work, we use SFHs predicted by the updated version of UniverseMachine, UM-SAGA (Wang et al. 2024), which is constrained by SAGA satellites (Geha et al. 2017; Mao et al. 2021, 2024; Geha et al. 2024) and SDSS isolated galaxies (Geha et al. 2012) down to M ⋆ ≳ 10 7 M ⊙ . UM-SAGA provides a better match to observed low-mass galaxies properties than UniverseMachine DR1, and is particularly better at predicting the quenched fraction of low-mass galaxies. We apply UMSAGA to one of the 45 Symphony Milky Way-mass halos and obtain 43 satellite galaxy SFHs. Here, we define a 'satellite' galaxy in UM-SAGA as any subhalo within 5 virial radii of the host, with a virial mass of at least 1 . 2 × 10 8 M ⊙ (corresponding to 300 particle masses). \nOf these 43 simulated galaxies, 35 have stellar masses < 10 7 M ⊙ where the quenched fraction is not well constrained by observations and may lead to additional uncertainties in the SFHs; we include these galaxies in all plots for illustrative purposes but do not include them in our analysis. This model does not track chemical enrichment, so we produce chemical enrichment histories by applying a mass-metallicity relation to the SFHs. For consistency, we obtain this mass-metallicity relation by fitting a simple linear relation 3 to the GRUMPY models (Figure 1): \nlog ( Z ⋆ Z ⊙ ) = 0 . 43 log ( M ⋆ M ⊙ ) -6 . 24 (9)", '2 https://bitbucket.org/pbehroozi/universemachine/src/main/': '3 This and all other fits in this work are, unless otherwise specified, computed with least-squares minimization using a LevenbergMarquardt algorithm. We bootstrap all fits by performing each fit N = 1000 times, sampling with replacement on each iteration; the reported best-fit parameters are the median estimates. \nFigure 1. Relationships between stellar mass and stellar metallicity for the simulations that track chemical enrichment: GRUMPY (dark purple points), MARVEL (pink points), and FIRE-2 (yellow points). These models have roughly consistent mass-metallicity relationships, suggesting that any differences between their derived stellar masses are not due to metallicity. The red dashed line indicates the best linear fit to the GRUMPY data; as described in Section 3.1, this fit is used to compute chemical enrichment histories for the UniverseMachine simulated galaxies. The observed stellar mass-metallicity relations from Kirby et al. (2013) (blue solid line indicates the best fit to Local Group dwarf galaxies; shaded regions indicate the rms about the best-fit line) and Gallazzi et al. (2005) (green solid line indicates the running median; shaded regions indicate the 16th and 84th percentiles) are also shown for comparison. \n<!-- image -->', '3.1.3. MARVELous Dwarfs': "Finally, we use simulated galaxies from two different cosmological baryonic zoom-in simulations. The first is the MARVELous Dwarfs (Bellovary et al. 2021; Munshi et al. 2021; Christensen et al. 2024; Azartash-Namin et al. 2024), a suite of high-resolution zoom-in simulations that use the N-body code ChaNGA (Menon et al. 2015). The MARVELous Dwarfs (hereafter abbreviated as MARVEL) include four simulated volumes of low-mass galaxies in isolated environments using a WMAP3 cosmology (Spergel et al. 2007). These simulations implement gas, initial star, and dark matter particle masses of 1410 M ⊙ , 420 M ⊙ , and 6650 M ⊙ , respectively, and have a force resolution of 60 pc. The high spatial resolution and small particle mass allow MARVEL galaxies to be resolved down to M ∗ ≈ 3 × 10 3 M ⊙ . MARVEL simulates a number of baryonic processes, including: star formation and gas cooling (Christensen et al. 2012), metal line cooling and metal diffusion (Shen et al. 2010), non-equilibrium formation and destruction of molecular hydrogen, photoionization and heating from a cosmological UV field background (Haardt \n& Madau 2012), and a 'blastwave' model of supernova feedback (Stinson et al. 2006) that temporarily shuts off gas cooling during the momentum-conserving 'snowplow phase' (McKee & Ostriker 1977). The combined processes emulate the energy deposited within the interstellar medium by all processes related to young stars, including UV radiation (see Wise et al. 2012; Agertz et al. 2013). The Amiga Halo Finder (AHF; Knollmann & Knebe 2009) is applied to MARVEL to identify dark matter halos, subhalos, and the baryonic content within. We obtain 30 SFHs and chemical enrichment histories of satellite and field galaxies from MARVEL, with a stellar mass range of 10 5 . 7 -10 9 . 2 M ⊙ .", '3.1.4. FIRE-2': "We also consider the Latte (introduced in Wetzel et al. 2016) and ELVIS on FIRE (introduced in GarrisonKimmel et al. 2019a) suites from the FIRE-2 cosmological baryonic zoom-in simulations, part of the Feedback In Realistic Environments project (public data release introduced in Wetzel et al. 2023). The FIRE-2 simulations are run using Gizmo, a Lagrangian Meshless Finite Mass (MFM) hydrodynamics code (Hopkins 2015), and the FIRE-2 physics model (Hopkins et al. 2018). FIRE-2 models the dense, multiphase interstellar medium (ISM) in galaxies and incorporates physically motivated, metallicity-dependent radiative heating and cooling processes for gas. These include free-free, photoionisation and recombination, Compton, photo-electric and dust collisional, cosmic ray, molecular, metal-line, and fine structure processes. The model tracks 11 element species (H, He, C, N, O, Ne, Mg, Si, S, Ca, Fe) across a temperature range of 10 - 10 10 K. FIRE-2 also includes the following time-resolved stellar feedback processes: core-collapse and white dwarf (Type Ia) supernovae, continuous mass loss, radiation pressure, photoionisation, and photo-electric heating. \nIn this analysis, we use a sample of 362 low-mass galaxies around MW/M31-mass galaxies from the z = 0 snapshots from the Latte and ELVIS on FIRE suites, and obtain their SFHs and chemical enrichment histories. The simulations have a dark matter mass resolution of 2 -3 . 5 × 10 4 M ⊙ , and gas and star mass resolution of 3500 -7100 M ⊙ , which allows for well-resolved galaxies down to M ∗ ≈ 10 5 M ⊙ . Identifying dark matter halos, and subhalos is done via the ROCKSTAR halo finder (Behroozi et al. 2013). Importantly, this sample of FIRE-2 galaxies has been benchmarked against the stellar mass functions, radial distance distributions, and star-formation histories of low-mass galaxies in the LG (Wetzel et al. 2016; Garrison-Kimmel et al. 2019a,b; Samuel et al. 2020, 2021). \nIn total, our sample consists of 469 simulated galaxies 4 with a mass range of M ⋆ = 10 4 . 0 -10 . 8 M ⊙ . The majority of our sample (403 galaxies) are low-mass 'dwarf' galaxies, with M ⋆ < 10 8 M ⊙ . The remaining 66 galaxies constitute a relatively small sample that is likely incomplete at the highest masses ( ≳ 10 9 . 5 M ⊙ ); however, this high-mass sample still provides a useful comparison when testing different mass measurement techniques. \nWhile it is not possible to determine whether our simulated galaxies are truly representative of real low-mass galaxies, we can at least check whether the models produce scaling relations that agree with each other and with observations. As shown in Figure 1, both the FIRE-2 and MARVEL galaxies follow roughly the same mass-metallicity relation as the GRUMPY galaxies (and by extension the UM-SAGA galaxies, which by definition follow the red dotted line in Figure 1, given by Equation 9). Figure 1 also shows the observed stellar mass-metallicity relations measured by Kirby et al. (2013) and Gallazzi et al. (2005). The mass-metallicity relation of our sample is somewhat steeper than that measured for Local Group dwarf galaxies by Kirby et al. (2013), who measured a slope of 0.30 compared to our slope of 0.43, but the two relations agree (within the rms about the Kirby et al. 2013 best-fit line) in the mass range 10 6 -8 M ⊙ , which spans most of our lowmass galaxy sample. Where our sample overlaps with the higher mass range of the Gallazzi et al. (2005) massmetallicity relation, our simulated galaxies and simple linear fit are consistent with the Gallazzi et al. (2005) relation within 1 σ .", '3.2. Mock observations': "To produce mock observations of these simulated galaxies, we use the FSPS package (Conroy et al. 2009; Conroy & Gunn 2010) 5 . FSPS takes galaxy SFHs and chemical enrichment histories as input and generates spectra and photometry. For all mock galaxies, we use MIST isochrones (Choi et al. 2016) and the MILES spectral library (Falc'on-Barroso et al. 2011; Vazdekis et al. 2010, 2015) to model stellar populations and spectra. We assume a Calzetti et al. (2000) attenuation curve, normalized by setting the optical depth at 5500 ˚ A to τ V = 0 . 2. We include dust emission using the default FSPS parameters. Finally, to further mimic realistic conditions, we place all simulated galaxies at a redshift \nr \ng \nFigure 2. The g -r color as a function of stellar mass for the GRUMPY (dark purple circles), UM-SAGA (purple triangles), MARVEL (pink squares), and FIRE-2 (yellow contours) models. UM-SAGA SFHs are not well constrained below log( M ⋆ / M ⊙ ) ∼ 7, so these galaxies are marked with small points; for clarity, FIRE-2 galaxies are represented as density contours rather than points. \n<!-- image --> \nTable 1. Photometric filters used to produce mock galaxy SEDs. \nof z = 0 . 01, consistent with the distances probed by ongoing and upcoming surveys of the nearby universe (e.g., the DESI LOWZ survey aims to target low-mass galaxies out to z < 0 . 03; Darragh-Ford et al. 2023). 6 \nUsing the input SFH and chemical evolution history of a galaxy, FSPS calculates the surviving stellar mass of the galaxy. We use this as 'true' stellar mass of the galaxy, rather than the raw output mass measured from GRUMPY, UM-SAGA, MARVEL, or FIRE. This is because the four models assume a variety of different stel- \n6 We note that this is not strictly necessary, since many of the tests described in this work assume that the redshift is exactly known. However, we plan to test the effect of unknown redshift on stellar mass measurements in future works. \nFigure 3. Cumulative SFHs of the simulated galaxies considered in this work. The panels illustrate galaxies of different stellar mass ranges: 10 4 -6 M ⊙ (top), 10 6 -8 M ⊙ (middle), and 10 8 -10 M ⊙ (bottom). UM-SAGA galaxies with M ⋆ < 10 7 M ⊙ are not included in this plot. Solid lines mark the median SFHs of the galaxies from GRUMPY (dark purple), UM-SAGA (purple), MARVEL (pink), and FIRE-2 (yellow). The corresponding filled regions denote the 16thto-84th percentile spread of the SFHs. For comparison, the median SFHs for field dwarf galaxies (blue) and Milky Way satellites (red) measured from resolved color-magnitude diagrams by Weisz et al. (2014) are shown. \n<!-- image --> \nlar physics models, including a range of mass loss rates, and these differences can lead to systematic discrepancies in galaxy M ⋆ measurements. We therefore use FSPS to self-consistently convert between the formed SFH and surviving stellar mass, using the same conversion for all four models. With the MIST isochrones (Choi et al. 2016), this corresponds to a typical differ- \nce of ∼ 40 -50% between the formed and surviving M ⋆ . \nWe also use FSPS to generate noiseless photometry in bands ranging from the UV through IR, listed in Table 1. Our choice of bands is motivated by the existence of wide-area surveys-GALEX in the UV (Martin et al. 2005; Bianchi et al. 2017), SDSS in the optical (Blanton et al. 2017; Doi et al. 2010), and WISE in the near-IR (Wright et al. 2010)-which, combined, provide multi-wavelength photometry for > 700 , 000 galaxies in the low-redshift universe (Salim et al. 2016). We then add realistic noise to the baseline 'noiseless' photometry in order to compare directly with observations. As a reference, we use the third data release of the Satellites Around Galactic Analogs survey (SAGA DR3; Mao et al. 2024), which identified 378 satellite galaxies around 101 Milky Way-mass analogs between 25-40 Mpc away. SAGA used imaging data from several public datasets-in particular, most of the optical grz photometry came from the DESI Imaging Legacy Surveys (Dey et al. 2019). To estimate observational noise for each galaxy in our sample, we first use a k-d tree to identify the SAGA satellite with the nearest r -band magnitude, g -r color, and r -z color. For each photometric band X , we then assign the photometric uncertainty σ X from the nearest-neighbor SAGA satellite to the simulated galaxy, and we perturb the simulated galaxy's noiseless photometry by adding a noise factor δ X randomly drawn from the Gaussian distribution δ X ∼ N (0 , σ X ). \nFigure 2 illustrates the results of this FSPS modeling, showing the distribution of the simulated galaxies in ( g -r )-mass space. In general, the galaxies from all four models (particularly when the poorly-constrained < 10 7 M ⊙ UM-SAGA galaxies are removed) occupy a similar region of optical color space. However, each of the models described in Section 3.1 parameterizes galaxy evolution differently, which leads to some minor variations among the mock observations. For example, the GRUMPY galaxies are systematically redder than the UM-SAGA galaxies, particularly in the mass range 10 6 -8 M ⊙ . This is likely related to differences in the SFHs, as illustrated in Figure 3. When comparing galaxies in the mass range 10 6 -8 M ⊙ , the GRUMPY galaxies typically form more stars at earlier times, whereas the UM-SAGA galaxies have more extended SFHs and higher recent star formation rates, leading to bluer colors. We revisit potential implications of these differences on stellar mass calibrations in Section 7.1.", '4. RESULTS: STELLAR MASS RECOVERY USING PHOTOMETRIC CALIBRATIONS': "We now test how well different stellar mass measurement techniques can recover the 'true' stellar masses from mock observations, beginning with empirical photometric calibrations based on optical and IR fluxes.", '4.1. Literature calibrations': "As discussed in Section 2.1, many studies in the literature have developed optical-NIR colorM ⋆ /L relations to estimate galaxy stellar mass, using either samples of observed galaxies (e.g., Bell et al. 2003; Portinari et al. 2004; Schombert 2006; Taylor et al. 2011; Herrmann et al. 2016; Du et al. 2020) or theoretical SEDs (e.g., Zibetti et al. 2009; Gallazzi & Bell 2009; Into & Portinari 2013; Roediger & Courteau 2015; Kouroumpatzakis et al. 2023; Klein et al. 2024). Figure 4 illustrates how well several of these calibrations work on simulated lowmass ( M ⋆, true < 10 8 M ⊙ ) galaxies. Table 2 summarizes their performance, listing the mean and standard deviation of the residuals log( M ⋆, calib /M ⋆, true ) for low-mass galaxies from each model discussed in Section 3, as well as for all models combined. \nThe optical colorM ⋆ /L relations (top three rows of Figure 4) show two primary features. First, most of the optical calibrations are able to recover M ⋆ of low-mass galaxies ( < 10 8 M ⊙ , left of the vertical red lines in the figure) within ∼ 0 . 2 dex of the true values (dashed horizontal gray lines in the figure). In this low-mass regime, nearly all optical calibrations have average residuals that are consistent with zero within uncertainties, suggesting that they are statistically unbiased. The main exception is the calibration by Klein et al. (2024) (right column, second from top panel in Figure 4), which appears to be overestimate M ⋆ for low-mass galaxies by 0 . 17 ± 0 . 11 dex on average. This is initially somewhat surprising, since Klein et al. (2024) developed this calibration using lowmass galaxies in the FIRE-2 simulation, which also make up the bulk of our sample of mock galaxies. Upon further inspection, this discrepancy is likely due to different assumptions about mass loss prescriptions. Klein et al. (2024) directly measured 'true' stellar masses from counting star particles in simulated FIRE-2 galaxies; in FIRE-2, the typical mass loss (i.e., the percent difference between the formed M ⋆ and the surviving M ⋆ of a galaxy) is ∼ 20 -30%. As described above, we instead obtain M ⋆, true for each simulated galaxy using FSPS with MIST isochrones (Choi et al. 2016), which produce a total mass loss of ∼ 40 -50%. As a result, the surviving M ⋆ of galaxies that Klein et al. (2024) measured directly from the FIRE-2 simulation are higher than the M ⋆, true used in this work, explaining the systematic overestimate. This suggests that uncertain physics in stellar models (i.e., mass loss rates) can lead to uncertainties \nFigure 4. Comparisons between 'true' stellar masses of model galaxies ( M ⋆, true ) and stellar masses measured from mock observations using different empirical photometric calibrations ( M ⋆, calib ). Different colors and shapes represent mock observations obtained from different galaxy models. UM-SAGA (purple) galaxies with log( M ⋆ / M ⊙ ) < 7 are marked with small unfilled points to denote that their SFHs are not well constrained. The dashed horizontal gray lines mark ± 0 . 2 dex errors, which represent typical systematic uncertainties that are assumed for stellar mass measurements. The horizontal blue lines and shaded bands denote the means and standard deviations of the mass residuals for the low-mass ( M ⋆ < 10 8 M ⊙ ) and high-mass ( M ⋆ > 10 8 M ⊙ samples, separated by the vertical red line.) Note the expanded y -axis limits on the bottom two plots. \n<!-- image --> \nof ∼ 0 . 2 dex or more in M ⋆ measured from empirical colorM ⋆ /L relations. \nAdditionally, all optical relations (top three rows of Figure 4) show clear systematic trends in the residuals log( M ⋆, calib /M ⋆, true ) as a function of stellar mass. As shown by the discontinuous blue horizontal lines \nin the figure, the residuals for the low-mass galaxies ( < 10 8 M ⊙ , left of the red vertical lines) are consistently ≳ 0 . 15 dex lower than the residuals for high-mass galaxies. The existence of this trend as a function of mass is perhaps unsurprising, since many of these relations have been calibrated based on samples with limited \nTable 2. Residuals between 'true' and measured masses, log ( M ⋆, calib /M ⋆, true ), for different photometric calibrations. The 'Bands' column lists the various photometric bands used in each calibration, and the bolded band is the reference band (e.g., when calculating M ⋆ /L g , g is the reference band). The values reported are the mean and standard deviation of the residuals for low-mass ( M ⋆, true < 10 8 M ⊙ ) galaxies. \nmass ranges. It may arise because most of the literature relations treat M ⋆ /L as a linear function of optical color, which inherently assumes that M ⋆ ∝ L ; we will revisit this assumption when we develop new calibrations in the next section. \nAs shown in the bottom two panels of Figure 4, the literature calibrations based on near-IR WISE bands perform more poorly than the optical calibrations. The Jarrett et al. (2023) W1 calibration (bottom left panel) has a pronounced downward trend in log( M ⋆, calib /M ⋆, true ) as log M ⋆, true decreases below 10 8 M ⊙ , which causes it to underestimate M ⋆ of low-mass galaxies by 0 . 73 dex on average (blue horizontal line on the left side of the plot). Meanwhile, the W1 -W2 calibration (bottom right panel) produces M ⋆ residuals with a large 1 σ dispersion of 1 . 46 dex for low-mass galaxies (blue shaded region on the left side of the plot). \nThese trends are likely due to a combination of factors. The Jarrett et al. (2023) relation based on W1 is a quadratic function of log( L W1 ), which can produce strong systematic biases when extrapolated. The Jarrett et al. (2023) W1 -W2 calibration, on the other hand, shows dramatic scatter in mass residuals that is almost certainly due to uncertainties in the W2 band. For the low-mass ( < 10 8 M ⊙ ) galaxies in our sample, the assigned uncertainty in W2 magnitude (calculated by finding the nearest-neighbor SAGA galaxy in optical colors; see Section 3.2) is on average 6.2 times larger than the uncertainty in W1 magnitude. For high-mass galaxies, the W2 magnitude is only 3.3 times more uncertain than \nthe W1 band, which explains why the scatter in the Jarrett et al. (2023) W1 -W2 relation decreases significantly above 10 8 M ⊙ . \nWe also note that for all galaxies in our sample, the mock photometric data were produced assuming a single dust attenuation model: the Calzetti et al. (2000) attenuation law with τ V = 0 . 2. Changing the dust model may impact the stellar masses recovered by empirical calibrations. We test this by varying the dust attenuation law in two different ways: we either change the form of the dust law by using an SMC-like attenuation law from Gordon et al. (2003), or we change the total amount of dust by increasing the normalization of the Calzetti et al. (2000) law from τ V = 0 . 2 to up to τ V = 1 . 8. In both cases, we find that while altering the dust model may affect the average stellar mass residuals log( M ⋆, calib /M ⋆, true ) (i.e., the blue horizontal lines in Figure 4), these changes are typically smaller than the 1 σ dispersion in the residuals. Furthermore, changing the dust law has the same effect on both low- and high-mass galaxies, so the systematic trends in log( M ⋆, calib /M ⋆, true ) as a function of stellar mass remain unchanged.", '4.2. Updated stellar mass calibrations': 'As discussed in the previous section, many literature optical and NIR calibrations for M ⋆ /L appear to be systematically biased as a function of stellar mass. We can use our mock dataset to define new M ⋆ /L calibrations that avoid these biases. \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 5. Top panels: linear fits to stellar mass residuals as a function of absolute g -band magnitude (left) and M ⋆ /L W1 as a function of W1 luminosity (right), which are used to produce updated stellar mass calibrations (Equations 11 and 12). Note that the most massive galaxies ( M ⋆ > 10 10 M ⊙ ; open circles) are not included in these fits. Bottom panels: stellar mass residuals as a function of mass for the updated mass calibrations; axes, colors, and symbols are the same as in Figure 4. \n<!-- image --> \nFor the optical colors, we begin by attempting to fit M ⋆ /L g as a linear function of g -r color, as has previously been done in the literature; from this best fit, we calculate an initial estimate for stellar mass M ⋆, linear : \nlog( M ⋆, linear / M ⊙ ) = 1 . 613+1 . 433(g -r) -0 . 4M g , 0 (10) \nHowever, when we compute the residuals between the true stellar mass and this initial estimate (expressed as log( M ⋆, linear /M ⋆, true )), we find that these residuals are a strong function of g -band luminosity. As shown in the top left panel of Figure 5, we fit the residuals with a quadratic function; we find that this functional form performs better than other forms, including arctangent and piecewise linear functions. Note that we do not include the most massive galaxies ( M ⋆ > 10 10 M ⊙ ) in this fit, since their small sample size might strongly bias the fit. This yields a new calibration for stellar mass: \nlog( M ⋆ / M ⊙ ) = 1 . 433(g -r) +0 . 00153 M 2 g, 0 -0 . 335 M g, 0 +2 . 072 (11) \nHere, as before, all magnitudes have been k -corrected using g -r color (Chilingarian et al. 2010), and an abso- \nlute solar g -band magnitude of 5.11 is assumed (Willmer 2018). The bottom left panel of Figure 5 shows the result of this updated calibration. This stellar mass calibration does not appear to have significant systematic residual trends as a function of mass; the galaxies below and above 10 8 M ⊙ have residuals that are consistent within 1 σ dispersion. \nBy fitting for M ⋆ as a function of both g -r color and absolute g -band magnitude M g, 0 , we inherently break the assumption that M ⋆ ∝ L g . Future studies examining M ⋆ /L relations may want to consider this multivariable approach, which seems to avoid systematic biases in M ⋆ . However, we caution that given the small size of our simulated sample at low and high masses, this relation is likely to suffer from extrapolation errors outside of the magnitude range -17 < M g, 0 < -5 . 5 (corresponding to an approximate mass range of M ⋆ ∼ 10 5 . 5 -9 M ⊙ ). \nFor the NIR colors, we focus on the W1 band, since relatively large uncertainties in the W2 band make it less useful as a mass indicator. We plot M ⋆ /L W1 as a function of L W1 and find that it is reasonably well fit by a linear function. This fit, shown in the top right panel \nof Figure 5, yields an equation for stellar mass: \nlog( M ⋆ / M ⊙ ) = 0 . 867 log(L W1 ) + 0 . 705 (12) \nThis equation uses the same equations for L W1 and k -correction as described in the previous section for the Jarrett et al. (2023) calibrations. The result of this updated calibration are shown in the bottom right panel of Figure 5. While this NIR calibration has a slightly larger dispersion in stellar mass residuals ( -0 . 05 ± 0 . 11 dex for low-mass galaxies) than our updated optical calibration, it does not suffer from the same systematic biases as the NIR calibrations considered in the previous section. As before, we note that this calibration might perform poorly outside of the mass range M ⋆ ∼ 10 5 . 5 -9 M ⊙ , but a larger sample of simulated galaxies in this high-mass regime is needed to confirm this. \nWe briefly comment on the impact of the assumed dust attenuation model on our new photometric calibrations. As with the literature calibrations, we find that if we vary the form or normalization of the dust attenuation law in the simulated galaxies, the average stellar mass residuals vary slightly. However, this effect is again statistically insignificant; regardless of the dust model assumed in the mock photometry, both Equations 11 and 12 produce stellar mass residuals that are approximately constant as a function of stellar mass.', '5. RESULTS: STELLAR MASS RECOVERY USING SED FITTING': "We now consider stellar masses estimated from SED fitting. In a Bayesian framework, the assumptions about prior distributions can have significant effects on the best-fit parameters, and SED fitting is no exception. In this work, we focus on four primary assumptions that may impact the recovery of M ⋆ for low-mass galaxies: (1) the form of the SFH, (2) the dust model, (3) the IMF, and (4) the choice of photometric bands. These choices are particularly impactful in the low-mass regime, since low-mass galaxies have stochastic SFHs that may not be described well by classical SFH models and may lead to stochastic sampling of the IMF. Furthermore, the (sometimes extreme) low-metallicity conditions in lowmass galaxies may affect dust production and evolution in ways that are not captured well by the dust models built for and largely calibrated on more massive galaxies. Many other assumptions are involved in SED modeling (e.g., the stellar model library used, the fitting method), but we defer discussion of these to a later work. \nThe full priors used in all tests in this section are described in Appendix A. Table 3 summarizes the results of all tests in this section, including the mean and standard deviation of the residuals log( M ⋆, calib /M ⋆, true ) for \nlow-mass galaxies from each model, as well as for all models combined. \nAs with any fitting procedure, the best-fit model identified by Prospector may not actually fit the data well. To test the goodness-of-fit, we use the reduced chisquared statistic χ 2 red to compare the true synthetic SED with the SED prediction for the highest probability sample. We identify fits with χ 2 red > 10 as 'poor' fits, marked with x's in all plots in this section. 7 Table 3 also lists the relative fraction of poor fits for each test, since our goal is to identify which prior assumptions most successfully recover M ⋆ ; however, the 'poor' fits are not considered in any further analysis (including the other summary statistics listed in Table 3).", '5.1. Parametric vs. nonparametric SFHs': "We first test the effect of different assumptions about the shape of the SFH on stellar mass measurements. We modify some of the default values in Prospector to accommodate the lower masses and metallicities expected for low-mass galaxies (see Appendix A). For this test, we fix the dust attenuation, dust emission, and IMF to be the same as the input SEDs-we will modify these in the following sections. We also use all 7 photometric bands listed in Table 1, which tests the 'best-case' scenario for lowz galaxies in the SAGA and DESI surveys; we will investigate some of the effects caused by missing photometric coverage later in Section 5.4. \nThe left column of Figure 6 presents a comparison of Prospector's parametric SFH templates: constant SFH, delayedτ SFH, and delayedτ with an additional instantaneous burst of star formation. We find that the assumption of a constant SFH (top left panel of Figure 6) leads to significant offsets for galaxies with M ⋆ < 10 8 M ⊙ : using a constant SFH model, Prospector can underestimate true stellar masses by up to ∼ 4 dex. This is not unexpected; Lower et al. (2020) also showed that a constant SFH can lead to similarly large offsets in M ⋆ for more massive galaxies. Additionally, we find that a constant SFH produces mostly 'poor' fits: ∼ 97% of the fits produce χ 2 red > 10. This is not surprising, since z ∼ 0 galaxies like the ones in our sample are thought to have quenched over time (e.g., Searle et al. 1973). A constant SFH is more appropriate for high-redshift starforming galaxies that are young enough to have maintained the same SFR over their short lifetimes. \nFor lowz galaxies whose SFR has likely declined over time, a more realistic parametric SFH model is the \nFigure 6. Same as Figure 4, but M ⋆, calib represents stellar masses measured using Prospector with various SFH priors. The left column shows the results of different parametric SFH priors: a constant SFH (top), a delayedτ ( ∼ t exp -t/τ ) SFH (middle), and a delayedτ SFH with a δ -burst added (bottom). The right column shows the results of non-parametric SFH priors: a Log M SFH (top), a continuity SFH (middle), and a Dirichlet SFH with a Dirichlet hyperparameter α = 0 . 7 (bottom). Bad fits ( χ 2 red > 10) are marked with x's. Note the expanded y -axis limits on the top plots. \n<!-- image --> \ndelayedτ model (left middle panel of Figure 6), in which SFR is described as a delayed declining exponential: \nSFR( t ) ∝ 0 t < T 0 ( t -T 0 ) e -t -T 0 τ t ≥ T 0 (13) \nwhere T 0 is the delay time of the SFR (or the maximum stellar age of the galaxy) and τ is the e -folding time. For more massive galaxies ( M ⋆ > 10 8 M ⊙ ), the delayedτ model underpredicts the true stellar mass by ∼ 0 . 2 dex, in agreement with Lower et al. (2020). However, for lower-mass galaxies the systematic bias is much less dramatic; on average, after removing poor fits (which constitute only 3% of our sample), stellar masses are only underpredicted by -0 . 05 dex on average, with a dispersion of 0 . 14 dex. As shown in the bottom left panel of Figure 6, increasing the flexibility of the model by adding an instantaneous δ -function burst of star formation (characterized by the time of the burst and the frac- \ntellar mass produced in the burst) further improves the recovery of M ⋆ for low-mass galaxies (the percentage of poor fits drops even further, to 1.7%) and reduces the dispersion in stellar mass residuals to 0 . 09 dex. Neither model shows a strong trend in residuals as a function of stellar mass below M ⋆ < 10 8 M ⊙ . \nWe now consider 'nonparametric' SFH models, which are described not with analytic functional forms but with hyperparameters-e.g., some number of time bins (which are each assumed to have constant SFR), the bin widths, and other hyperparameters describing the relationships between the SFRs in the bins. The right column of Figure 6 shows the performance of three of the SFH models implemented in Prospector (Leja et al. 2017). For this test, we fix the number of bins to N bin = 10 for all SFH models (see Section 7.2 for further discussion of this choice). Following the procedure of Lower et al. (2020), the bins are logarithmically spaced \nexcept for the two youngest bins, which in this work are fixed to span lookback times of 0 -10 Myr and 10 -100 Myr. \nThe Log M model is the simplest nonparametric model, in which the free parameters are the masses formed in each of the fixed time bins. This SFH model is extremely flexible and can reproduce multiple bursts of star formation and quenching. However, as shown in the top right panel of Figure 6, the Log M model struggles to accurately recover stellar masses. Only 39.5% of the fits are 'good' ( χ 2 red < 10); of these 'good' fits, stellar mass is underestimated by -0 . 24 dex on average, with a dispersion of 0 . 32 dex. \nNonparametric SFH templates with more constrained functional forms perform better than the Log M model. The Continuity model (right middle panel of Figure 6) fits directly for the difference in SFR between adjacent time bins, but explicitly weights against sharp transitions in SFR. Although low-mass galaxy SFHs are thought to be highly stochastic and bursty, the Continuity model performs remarkably well, with an average stellar mass residual of 0 . 01 dex and a dispersion of 0 . 09 dex. The Dirichlet model (bottom panel), like the Continuity model, also constrains the relationship between SFRs in each of the bins. In this model, the fractional sSFR for each time bin follows a Dirichlet distribution (Betancourt 2012), set by a 'concentration' parameter α : low α values weight toward bursty SFHs, while higher α weights toward smooth SFHs. We assume α = 0 . 7 following Lower et al. (2020), although we discuss the effect of changing α in the following section. The Dirichlet template is comparable to the Continuity model in terms of stellar mass recovery, with an average offset of -0 . 03 dex and a dispersion of 0 . 11 dex. Both the Continuity and the Dirichlet models return the highest number of 'good' fits of all the SFH models (with only 1.28% of all fits being discarded as 'poor' fits), and neither template produces a strong trend in the residuals as a function of stellar mass. \nThe overall success of the Continuity and Dirichlet SFH models suggests that the additional flexibility provided by non-parametric SFHs improves stellar mass measurements of low-mass galaxies. However, too much flexibility (as in the Log M model) may under-constrain the fit, leading to poor SED fits and unreliable M ⋆ measurements. For the remainder of this work, we use the Dirichlet template as our fiducial SFH model; although the Continuity template performed similarly well, the Dirichlet model provides a more direct comparison to the work by Lower et al. (2020), as discussed in the next section.", '5.1.1. Comparison with literature results': "These results are a direct extension of the work on higher-mass galaxies by Lower et al. (2020), who demonstrated that for galaxies with M ⋆ ≳ 10 8 M ⊙ , assuming simple parametric SFHs can lead to systematic biases in measured stellar mass of up to ∼ 0 . 4 dex. Like this work, Lower et al. (2020) also used FSPS to generate mock photometry of simulated galaxies, then compared the 'true' masses with Prospector's estimates. \nThere are some differences between the studies-most notably, Lower et al. (2020) used galaxies from a single large-volume hydrodynamic simulation (SIMBA; Dav'e et al. 2019), while we used several different models that all specifically aim to model lower-mass galaxies (including zoom-in simulations that have much finer mass resolution than SIMBA). Rather than using FSPS to apply a simple dust attenuation law while generating mock SEDs, as we have done here, Lower et al. (2020) used the more complex 3D radiative transfer code POWDERDAY to model a dust screen surrounding all stars. Finally, while Lower et al. (2020) considered the effect of applying 3% uncertainties to their mock photometry, we used realistic observational uncertainties that vary as a function of wavelength and for different galaxies. \nDespite these slight differences in methods, many of our results are qualitatively in agreement. We find that parametric SFHs generally perform worse than nonparametric SFHs, often underestimating the true stellar mass. This is particularly apparent for the Constant SFH template: we find an large stellar mass offset of -1 . 09 ± 0 . 80 dex for low-mass galaxies (when poor fits are included), which is consistent with the offset of -0 . 48 ± 0 . 61 dex reported by Lower et al. (2020). Lower et al. (2020) also find that other parametric SFHs (Delayedτ , Delayedτ +burst) tend to underestimate M ⋆ , which is qualitatively consistent with the high-mass ( M ⋆ > 10 8 M ⊙ ) galaxies in our sample (right of the red vertical lines in the left middle and left bottom panels of Figure 6). \nHowever, we find that for low-mass galaxies, parametric SFHs can still perform quite well: the Delayedτ and Delayedτ +burst models produce stellar mass residuals that are, on average, consistent with zero within their 1 σ dispersions. Additionally, Lower et al. (2020) find that the non-parametric Continuity SFH template overestimates stellar masses by 0 . 24 ± 0 . 15 dex, while we find that the Continuity template recovers masses that are much closer to the true mass (0 . 01 ± 0 . 09 dex for the low-mass galaxies). This is somewhat surprising: the parametric and Continuity templates prefer smoothly changing SFHs, but low-mass galaxies are expected to have burstier SFHs than massive galaxies. \nThe exact reason for this discrepancy is unclear. One option is that short-duration bursts of star formation simply do not contribute significantly to the overall stellar mass of a low-mass galaxy. As a result, parametric and smooth SFHs may still recover a reasonably accurate stellar mass even if Prospector's SFH models are unable to capture the full behavior of the SFH. We defer a full test of this hypothesis to future work. In the meantime, we can at least confirm, as Lower et al. (2020) did, that non-parametric SFH models with some constraints (e.g., the Dirichlet and Continuity models) are the best among the Prospector SFH templates tested for stellar mass recovery. Finally, because the Dirichlet and Continuity models performed comparably for our sample, for convenience we choose the Dirichlet model as our fiducial SFH model, since Lower et al. (2020) also adopted the Dirichlet model as their default non-parametric SFH.", '5.1.2. Non-parametric SFH hyperparameters': "As described above, the non-parametric SFH models defined by Prospector are not truly 'non-parametric,' in that they must still be described by hyperparameters such as the number and width of time bins ( N bins ) of constant SFH. Ocvirk et al. (2006) suggest using age bins spaced logarithmically in time, and that no more than eight characteristic star-forming episodes can be recovered with high-quality optical spectra. We therefore test the effect of changing N bins for our fiducial SFH model (Dirichlet SFH with α = 0 . 7). We find that if N bins is too small, many of the fits are poor quality ( χ 2 red > 10); for example, with N bins = 3, 70% of the galaxies have poor fits. However, the 'good' fits are still able to recover stellar masses that are on average consistent with the true stellar masses. The number of poor fits can be reduced by increasing the number of bins, and the results for N bins = 6 , 10 , 12 are essentially the same. \nWe also test the effect of varying the Dirichlet α parameter, which determines the burstiness of the SFH. Again, we find that if α is too small, the number of poor fits increases - if α = 0 . 2, 41 . 8% of the galaxies have poor fits - but the 'good' fits are still strong predictors of the true stellar mass. Increasing α reduces the number of poor fits, and α = 0 . 7 (the fiducial value) and α = 1 . 0 again return near-identical results. \nThese tests suggest that there are some minimum values for N bins and α , below which Prospector begins to struggle to fit many low-mass galaxy SFHs. While determining these exact values is beyond the scope of this work (and is likely to depend on the sample of galaxies used - or in this case, the details of the simulations used to model our galaxies), we can at least safely as- \nume that the fiducial hyperparameters discussed in the previous section N bins = 10 and α = 0 . 7 are reasonable assumptions.", '5.2. Dust prescription': "In the previous sections, we tested SFH assumptions by using idealized models in which the dust attenuation and emission exactly matched the input SEDs. Of course, this is not the case in reality. Having identified a fiducial SFH model, we can now address the impact of dust on our SED fits. \nDust both attenuates and emits light from a galaxy, generally leading to a reddening of galaxy spectra. While significant work has been done to progress our understanding of both of these effects - see, e.g., Salim & Narayanan (2020) for a recent review of the dust attenuation law - the impact of dust on galaxy observations remains an open question, particularly in low-mass and low-metallicity galaxies. This is both because the physical properties of dust in low-metallicity environments are not well understood (e.g., Nanni et al. 2020; Galliano et al. 2021), and because dust, metallicity, and age have inherently degenerate effects on a galaxy's SED (and this degeneracy is also a complex function of galaxy mass and metallicity; e.g., Nagaraj et al. 2022). \nIn this work, we do not attempt to choose the most physically correct global dust prescription for low-mass galaxies. Our goal is simply to test the robustness of Prospector's ability to recover M ⋆ of low-mass galaxies, given different assumptions about dust models. To do this, we vary the dust model used to produce the mock observations, then run Prospector with different parameterizations of the dust attenuation model. For these tests, we fix the SFH to our fiducial nonparametric Dirichlet SFH with α = 0 . 7.", '5.2.1. Dust attenuation': "We first test the impact of the assumed dust attenuation model. We consider both (1) the overall normalization of the attenuation law, and (2) the shape of the attenuation law. \nAs described in Section 3.2, our primary set of mock observations are produced by assuming a Calzetti et al. (2000) attenuation law normalized to τ V = 0 . 2. To test how strongly the normalization of the dust attenuation curve affects stellar mass measurements, we synthesize additional sets of observations using a Calzetti et al. (2000) law normalized to τ V = 1 . 0 and 1 . 8. We then fit these mock SEDs using the Calzetti et al. (2000) law, but where τ V is allowed to vary as a free parameter. This parameterization of the dust attenuation law is a common assumption when fitting the SEDs of low-mass galaxies (e.g., Pandya et al. 2018; Greco et al. 2018), \nFigure 7. Same as Figure 4, but M ⋆, calib represents stellar masses measured using Prospector with different dust attenuation laws. The rows represent mock observations with varying amounts of dust, produced by using the Calzetti et al. (2000) dust law with different normalization parameters τ V, true : 0.2 (top), 1.0 (middle), and 1.8 (bottom). The left column shows the results when assuming a Calzetti et al. (2000) dust law with the normalization parameter τ V allowed to vary as a free parameter. The right column shows the results when assuming a SMC bar attenuation law (Gordon et al. 2003). Bad fits ( χ 2 red > 10) are marked with x's. \n<!-- image --> \nalthough the exact form of the prior on τ V may vary; in this case, we assume a uniform prior for 0 < τ V < 2. \nThe results of changing the dust normalization are shown in the left column of Figure 7, where the true value of τ V (i.e., the input value used to produce the mock observations) increases from the top row to the bottom row. Two primary features stand out: first, regardless of the true τ V of the mock observations, there is a increase in scatter in the recovered stellar mass residuals below M ⋆, true < 10 8 M ⊙ . This is visibly apparent in the left column of Figure 7; the blue horizontal bands to the left of the red vertical lines are larger than the blue horizontal bands to the right of the vertical lines. This increase in scatter is the direct result of allowing the dust normalization to vary: for low-mass galaxies, the dispersion in stellar mass increases from 0 . 09 dex when the dust prescription exactly matches the input dust law (bottom right panel of Figure 6) to [0 . 19 , 0 . 23 , 0 . 15] dex \nwhen τ V is allowed to vary (for τ V, true = [0 . 2 , 1 . 0 , 1 . 8], respectively). \nAdditionally, the average residuals for low-mass galaxies (the blue horizontal bands left of the red vertical lines in Figure 7) appear to be a function of τ V, true . The freeτ V models typically overestimate stellar masses (mean offset of 0 . 21 dex) when τ V, true = 0 . 2 and underestimate stellar masses (mean offset of -0 . 17 dex) when τ V, true = 1 . 8. This makes sense, since we stipulated in our prior assumptions that 0 < τ V < 2. As a result, if τ V, true = 0 . 2 ( τ V, true = 1 . 8), Prospector is more likely to overestimate (underestimate) τ V , and overestimating the total amount of dust will lead to an overestimate (underestimate) of the true stellar mass. To check this hypothesis, Figure 8 plots the residuals in stellar mass as a function of the residuals in τ V (i.e., the difference between the measured and 'true' τ V values). As expected, there is a clear trend between the two residuals: \nFigure 8. Difference between measured and actual stellar mass log( M ⋆, calib /M ⋆, true ) as a function of the difference between measured dust normalization τ V, calib and actual dust normalization τ V, true . The dashed horizontal gray lines mark ± 0 . 2 dex uncertainties in stellar mass. UM-SAGA galaxies with log( M ⋆ / M ⊙ ) < 7 are marked with small unfilled points to denote that their SFHs are poorly constrained. \n<!-- image --> \noverestimating τ V ( τ V, calib -τ V, true > 0) is correlated with overestimating M ⋆ , and underestimating τ V is correlated with underestimating M ⋆ . \nWe can also test the effect of changing the shape of the dust attenuation law by fitting our mock SEDs with the extinction law measured by Gordon et al. (2003) for the SMC bar, which has a much steeper UV-optical slope than the Calzetti et al. (2000) law. Rather than a single normalization parameter, as was used with the Calzetti et al. (2000) law, we now assume a dust screen geometry in which young stars are more strongly attenuated by their birth clouds, while older stellar populations are less strongly attenuated by diffuse cirrus dust. Prospector uses the Charlot & Fall (2000) parameterization to model how opacity varies as a function of stellar age t . We assume a power-law dust screen for young stars (with ages ≤ 10 7 y), while both young and old stars are attenuated by diffuse dust: \nτ young = τ 1 ( λ/ 5500 ˚ A) -δ CF (14) \nτ diff = τ 2 ( A ( λ ) /A ( V )) (15) \nFollowing Conroy et al. (2010) and Charlot & Fall (2000), we set δ CF = 1 . 0, τ 1 = 1 . 0, and τ 2 = 0 . 3. Here, the diffuse dust attenuation A ( λ ) /A ( V ) is set by the SMC bar extinction law (Table 4 of Gordon et al. 2003). \nThe results of changing the dust attenuation law are shown in the right column of Figure 7, where the SMC dust law is applied to mock SEDs produced using the Calzetti et al. (2000) law with varying τ V, true . As before, τ V, true traces the overall amount of dust. For galaxies with low amounts of dust ( τ V, true = 0 . 2; top row), the \nexact form of the dust attenuation law does not significantly change the output; assuming the SMC dust law produces stellar masses that are as accurate as if the 'true' dust law was used (0 . 00 ± 0 . 09 dex, compared to -0 . 03 ± 0 . 09 dex when the dust law is fixed to match the input). However, as the total amount of dust increases, using the incorrect dust law more dramatically impacts the recovered stellar masses. For both τ V, true = 1 . 0 and τ V, true = 1 . 8, the SMC dust law underpredicts the stellar masses of M ⋆ < 10 8 M ⊙ galaxies by ∼ -0 . 3 dex on average. Furthermore, as the true value of τ V increases, the number of 'poor' fits increases, particularly for the most massive galaxies in our sample.", '5.2.2. Dust emission': "We also check the effect of the dust emission model on stellar mass measurements. We again fix the SFH to a Dirichlet prior, this time with the dust attenuation law fixed to be exactly the same as the input SEDs (Calzetti et al. 2000, with τ V = 0 . 2). In this test, we allow the dust emission parameters in the Draine & Li (2007) dust model-the minimum radiation field U min , the fraction of dust heated by starlight γ , and the PAH mass fraction q PAH -to vary. Compared to the same model in which these parameters were fixed (right bottom panel of Figure 6), this additional flexibility improves the fits for some galaxies, particularly those with stellar masses ≳ 10 8 M ⊙ ; the percentage of 'poor' fits decreases to 0.64%. However, for low-mass galaxies with M ⋆, true < 10 8 M ⊙ , varying the dust emission parameters only slightly changes the recovered stellar masses, and the overall mean and standard deviation of the residuals do not change at all. \nTo summarize, the assumed dust model has a strong impact on both the systematic offset and the scatter in stellar masses recovered from SED fitting, and these effects are particularly apparent for low-mass ( M ⋆ < 10 8 M ⊙ ) galaxies: \n- · Assumptions about the overall normalization of the attenuation law (i.e., the total amount of dust) affect the recovered stellar mass in two ways.\n- 1. Allowing dust normalization to vary increases the population-level scatter in recovered masses for low-mass galaxies.\n- 2. Any offset in the prior on dust normalization will lead to systematic offsets in the recovered stellar masses, because overestimating (underestimating) the dust normalization will lead to overestimating (underestimating) M ⋆ . Since most low-mass galaxies are likely \nto have low metallicities and correspondingly low dust masses, a wide prior on the dust normalization is more likely to overestimate the amount of dust and consequently overestimate M ⋆ . \n- · The shape of the dust attenuation law can also have an effect on the recovered stellar masses, but this only becomes a significant issue for dusty galaxies with τ V > 0 . 2; again, since most low-mass galaxies may have low dust content, this may not be a major issue in this mass regime.\n- · Assumptions about dust emission do not appear to significantly impact the results of SED fitting to UV through near-IR bands, although we note that it may be more relevant when mid- and far-IR photometry is included.", '5.3. Stellar initial mass function': 'We briefly consider the role of the stellar IMF in SED fitting. The IMF directly sets M ⋆ /L , so assuming the wrong IMF should simply add a constant offset to log M ⋆ . Fitting models with a Kroupa (2000) IMF to mock observations produced using a Chabrier (2003) IMF should underestimate stellar masses by ∼ 0 . 03 dex, while fitting a Salpeter (1955) IMF to a Chabrier (2003) IMF should overestimate stellar masses by ∼ 0 . 24 dex. We check this intuition by applying different IMFs in Prospector, assuming the fiducial Dirichlet SFH and dust parameters fixed to match the input SEDs. \nAs expected, we find that changing the IMF leads to constant shifts in stellar mass residuals (see Table 3): the average offset shifts, but the 1 σ dispersion does not change. The magnitudes of these offsets are also smaller than expected. Compared to the true Chabrier (2003)IMF, assuming a Kroupa (2000) IMF changes the average offset in stellar mass by only 0 . 1 dex, while assuming a Salpeter (1955) IMF increases the stellar mass residuals by 0 . 14 dex on average.', '5.4. Photometric coverage': "In the previous subsections, we considered only the results of fitting to all 7 photometric bands in Table 1. In real lowz galaxy surveys, incomplete photometric coverage is common. We now investigate how well Prospector can recover stellar masses with realistic subsets of the 7 photometric bands. For this test, we run Prospector on our fiducial set of mock observations (produced using a Chabrier 2003 IMF and a Calzetti et al. 2000 attenuation law with τ V = 0 . 2). We aim to mimic a more 'realistic' usage of Prospector, in which the true dust attenuation law is unknown and must be described by \none or more free parameters. However, some basic assumptions can be made: due to low-mass galaxies' low metallicities, one might expect these galaxies to have relatively little dust - indeed, recent observations suggest that low-mass galaxies in the nearby universe have 0 ≲ τ V ≲ 0 . 4 (e.g., Geha et al. 2024; Greco et al. 2018; Pandya et al. 2018). As discussed in Section 5.2, for galaxies with such low dust content, the exact form of the assumed dust attenuation law does not strongly impact the recovered stellar mass. We therefore assume a Calzetti et al. (2000) attenuation law with a more restricted uniform prior on the normalization parameter τ V : U (0 , 0 . 4). \nWe then fit Prospector to three different sets of bands: (1) only optical grz bands, (2) optical grz with near-IR coverage (WISE W1 and W2), and (3) optical grz with limited near-UV and near-IR coverage (GALEX NUV and WISE W1). The first case is motivated by the DESI Imaging Legacy Surveys (Dey et al. 2019), an extremely wide-field ( ≈ 14 , 000 deg 2 ) grz survey. The second case is motivated by the fact that while the WISE satellite imaged the entire sky in four near-to-mid-IR bands at 3.4, 4.6, 12, and 22 µ m (W1, W2, W3 and W4; Wright et al. 2010; Cutri & et al. 2012), substantially deeper coverage is available for the two bluest bands, W1 and W2. Finally, the third case is motivated by the SAGA survey (Mao et al. 2024; Geha et al. 2024), which uses GALEX NUV fluxes to measure star formation rates. \nThe results of this test are shown in Figure 9. As illustrated in the top panel, the optical grz bands alone are able to recover much of the stellar mass information for low-mass galaxies. While the grz bands produce a systematic trend in stellar mass residuals as a function of M ⋆ , similar to the results of the empirical g -r calibrations (Figure 4), on average the low-mass residuals are consistent with zero ( -0 . 13 ± 0 . 13 dex). Adding more bands slightly improves the M ⋆ estimates for low-mass galaxies: the average residuals decrease to -0 . 11 ± 0 . 12 dex if W1 and W2 are added, and to -0 . 05 ± 0 . 09 dex if NUV and W1 are added. The NUV+ grz +W1 test shows no systematic trend in residuals as a function of stellar mass. However, it is worth noting that the inclusion of the W2 band produces 'poor' ( χ 2 red > 10) fits, likely due to the large W2 photometric uncertainties in our sample. We recommend that future studies exercise caution in using uncertain photometric measurements when fitting low-mass galaxy SEDs. \n6. BEST PRACTICES FOR MEASURING THE STELLAR MASSES OF LOW-MASS GALAXIES \nTable 3. Residuals between 'true' and measured masses, log ( M ⋆, calib /M ⋆, true ), for different SED fitting tests. The values reported are the mean and standard deviation of the residuals for low-mass ( M ⋆, true < 10 8 M ⊙ ) galaxies. Only 'good' fits ( χ 2 red < 10) are included when computing the mean and 1 σ residuals; the percentage of 'poor' fits ( χ 2 red > 10) out of 645 total fits is also listed. \nFigure 9. Same as Figure 4, but M ⋆, calib represents stellar masses measured using Prospector with different sets of photometric bands: optical grz only (top), optical grz and near-IR (middle), and optical grz with limited near-UV and near-IR coverage (bottom). \n<!-- image --> \nBased on our investigation, we now outline recommendations for measuring the stellar masses of low-mass galaxies (5 < log( M ⋆ / M ⊙ ) < 8). \n- 1. If multi-wavelength SED fitting is not computationally or observationally feasible, we recommend using Equations 11 or 12 to calculate stellar masses from optical or NIR fluxes, respectively (Figure 5). Based on our simulated sample, these calibrations are less systematically biased as a function of mass than literature relations (Figure 4), and will recover stellar masses with 1 σ errors of ∼ 0 . 1 dex. This is comparable to the best results from multi-wavelength SED fitting, and may significantly outperform SED fitting with certain choices of assumptions (see below); however, we note that these calibrations should only be used in the magnitude range -17 < M g, 0 < -5 . 5 ( M ⋆ ∼ 10 5 . 5 -9 M ⊙ ) to avoid extrapolation errors.\n- 2. If multi-wavelength SED fitting is used, we recommend assuming a non-parametric SFH, rather than a parametric SFH (Figure 6). In a best-case scenario, SED fitting with a non-parametric SFH can produce stellar mass estimates with 1 σ -level dispersions of ∼ 0 . 1 dex.\n- 3. For SED fitting with a non-parametric SFH, we recommend using at least six logarithmically spaced age bins, and applying some restrictions to prevent extremely large variations between SFH bins (i.e., using a Dirichlet SFH template with α ≥ 0 . 7 or a Continuity SFH template).\n- 4. Allowing dust attenuation parameters to vary in an SED fit further increases the dispersion in M ⋆, calib -M ⋆, true (Figure 7) and may potentially introduce uncertainties in individual stellar mass estimates of up to ∼ 0 . 6 dex (Figure 8), depending on how badly the total dust content is overor under-estimated. For low-mass galaxies that are not likely to be very dusty, we therefore recommend using narrow prior distributions (e.g., a Calzetti et al. (2000) dust attenuation law with normalization 0 < τ V < 0 . 4) to avoid significantly overestimating the amount of dust.\n- 5. Some SED fitting assumptions do not appear to strongly impact stellar mass recovery. For example, dust emission parameters do not significantly affect M ⋆ measurements based on optical and near-IR photometry. Changing the assumed stellar IMF may introduce constant offsets to M ⋆ measurements, but these are relatively small ( < 0 . 2 dex) for low-mass galaxies.\n- 6. If attempting to choose a limited subset of photometric bands to observe, the optical ( grz ) bands alone provide the most information about M ⋆ for low-mass galaxies. While adding more bands can further improve stellar mass estimates, any bands with large observational uncertainties (e.g., WISE W2 in this study) may actually increase the uncertainty in M ⋆ .", '7.1. Differences between models': "A fundamental flaw of the tests presented in Sections 4 and 5 is our inherent assumption that simulated galaxies accurately represent real low-mass galaxies. We have attempted to mitigate this by using several different galaxy models that use a variety of approaches and physical assumptions, as described in Section 3. Here \nwe discuss whether our results depend strongly on these models. \nFor each of the empirical photometric calibrations discussed in Section 2.1, the mean residuals computed for the low-mass galaxies from each of the four models (GRUMPY, UM-SAGA, MARVEL, FIRE) typically agree within 1 σ (as seen in each row of Table 2). The only exceptions are the UM-SAGA sample, which sometimes has systematically higher stellar mass residuals than the other models (e.g., for the Klein et al. (2024) calibration, the UM-SAGA sample has an average stellar mass residual of 0 . 33 ± 0 . 03 dex, compared to 0 . 18 ± 0 . 08 dex for the GRUMPY sample and 0 . 16 ± 0 . 11 dex for the FIRE sample). This is because only the UM-SAGA galaxies with well-constrained SFHs (and M ⋆ > 10 7 M ⊙ ) are included when calculating average residuals. Not only is this a small sample of galaxies ( N = 7), it also means the UM-SAGA sample is biased to higher stellar masses than the samples from the other models. Because of the upward trend in log( M ⋆, calib /M ⋆, true ) as a function of stellar mass for the optical calibrations (and the Jarrett et al. 2023 W1 relation), the average residuals for the UM-SAGA sample are also higher. \nA similar effect is apparent when more complex SED fitting is used. As shown in Table 3, the average stellar mass residuals log( M ⋆, calib /M ⋆, true ) are typically highest for the UM-SAGA galaxies and lowest for the GRUMPY galaxies. However, as with the photometric calibrations described above, this difference is not statistically significant (i.e., the average UM-SAGA and GRUMPY residuals are consistent within 1 σ errorbars) for many of the SED fits we performed. For the cases where this difference is > 1 σ , the discrepancy may be due to a number of non-physical factors, including the small sample size and bias to higher M ⋆ of the wellconstrained UM-SAGA sample. Alternatively, as noted in Section 3.2, the UM-SAGA galaxies are systematically bluer than the GRUMPY galaxies at a given mass due to more extended SFHs (Figures 2 and 3). Because more recent bursts of star formation 'outshine' older stellar populations (Narayanan et al. 2024; Haskell et al. 2024), Prospector may overestimate the stellar mass contributed by the older populations in the UMSAGA galaxies with higher recent SFRs. Larger sample sizes are needed to distinguish between these scenarios. \nHowever, we emphasize that differences between the four models are relatively minor. Qualitatively, the trends in the stellar mass residuals log( M ⋆, calib /M ⋆, true ) for photometric calibrations (Figure 4) and SED fitting (Figures 6-9) are consistent among all four models. This overall agreement implies that our key conclusions - in \nparticular, the best practices outlined in Section 6 are robust to model-specific assumptions.", '7.2. Additional systematics': 'We now discuss systematic effects that may impact the applicability of our results. Some of these involve physics that have been simplified or ignored in modeling low-mass galaxies, while other systematic effects are due to our experimental setup (i.e., fixing certain variables, choice of SED fitting code).', '7.2.1. Physical factors': "For simplicity, we have ignored a number of physical factors in our modeling of low-mass galaxies. \nFirst, we have overlooked the potential contribution of active galactic nuclei (AGN) to low-mass galaxies (see, e.g., the recent review of Reines 2022). None of the model galaxies used in this work include AGN, and we fixed the AGN contribution to zero when performing SED fits using Prospector. In real galaxy surveys, the AGN contribution may not be exactly zero. This can lead to systematic offsets in M ⋆ measurements, since ignoring the contribution of an AGN to a galaxy's light can lead to an overestimate of the stellar mass. The magnitude of this effect is unclear and highly dependent on how AGN are modeled: some studies find that AGN in the local universe do not strongly impact M ⋆ measurements from SED fitting (e.g., Leja et al. 2018; Thorne et al. 2022), while others suggest that some AGN models (particularly for high-luminosity AGN) may lead to M ⋆ uncertainties of up to ∼ 0 . 5 dex (e.g., Buchner et al. 2024; Ciesla et al. 2015). \nMost of this literature has focused on higher-mass ( M ⋆ > 10 8 M ⊙ ) galaxies, so it is even less clear how AGN contributions might affect M ⋆ measurements in the low-mass regime. Fortunately, the fraction of lowmass galaxies in the nearby universe with AGN is expected to be quite low. Recent estimates suggest that the AGN occupation fraction is ≲ 0 . 5% in galaxies with M ⋆ ≲ 10 9 out to redshifts z < 0 . 1 (e.g., Birchall et al. 2020; Mezcua et al. 2018), so the results outlined in this work are likely valid for the vast majority of local lowmass galaxies. However, it is difficult to overstate just how little we know about the effects of AGN on lowmass galaxies. Additional modeling of AGN in low-mass galaxies is therefore needed to better constrain uncertainties on measurements of physical properties. \nAnother physical phenomenon that may impact lowmass galaxies is quenching from reionization. We briefly consider how different assumptions about reionization might affect the stellar masses of the mock galaxies in our sample. The galaxy models described in Section 3 parameterize reionization in a variety of ways. In \nthe semi-analytic model GRUMPY, UV heating from reionization is parameterized as a suppression factor on gas inflow, while in the zoom-in simulations MARVEL and FIRE, reionization is directly implemented as a photoionizing UV background (described respectively in Haardt & Madau 2012; Faucher-Gigu'ere et al. 2009). 8 While a full discussion of all the differences among these models is beyond the scope of this paper, we focus on one major difference in the timing of reionization in each model. The redshift of reionization is set to z reioniz = 6 in GRUMPY, z reioniz ∼ 15 in MARVEL, and z reioniz ∼ 10 in FIRE. This may have a direct effect on stellar mass: to first order, reionization occurring later leaves more time to form stars before quenching, leading to higher M ⋆ . \nTo check this, we would ideally look at the galaxies in our sample with M ⋆ ≲ 10 6 M ⊙ , since these are thought to be the most strongly impacted by reionization (e.g., Wheeler et al. 2019). However, our sample in this mass regime is limited - both GRUMPY and MARVEL suffer from small sample sizes in the 10 4 -6 M ⊙ mass range ( N = 5 and N = 1, respectively, as shown in the top panel of Figure 3), and only a few GRUMPY galaxies have stellar masses < 10 5 M ⊙ (Figure 2) - making it difficult to identify any differences among simulations. A larger sample of model galaxies is needed to fully test whether stellar mass recovery depends on z reioniz . Additionally, we note that all models in this work assume uniform reionization for all galaxies. A potentially more realistic model of 'patchy' reionization (e.g., Pentericci et al. 2014) would imply that low-mass galaxies in denser environments should undergo reionization quenching first; in other words, z reioniz may also be a function of a galaxy's environment. As described in Section 3, nearly all of the model galaxies in our sample are satellite galaxies. Testing the effect of uniform vs. patchy reionization would require not only a larger sample of < 10 6 M ⊙ galaxies, but also a sample that spans a wide range of environments (satellite, field, and isolated).", '7.2.2. Experimental setup': "Other systematic effects may be due not to physical factors, but our choices when setting up our tests of stellar mass measurement techniques. \nFor example, throughout this work, we assumed the redshifts of our mock galaxies were exactly known. While this is a reasonable assumption for galaxy surveys \nthat obtain precise spectroscopic redshifts, it may have a larger impact on stellar mass measurements obtained from purely photometric surveys of low-mass galaxies. Over- or underestimating a galaxy's redshift will produce systematic offsets in stellar mass. The magnitude of this effect can be estimated with simple scaling relations: for z ≪ 1, z ∝ d and luminosity L ∝ Fd 2 (where d is luminosity distance and F is observed flux). Assuming M ⋆ ∝ L (which is generally reasonable, as shown by the empirical calibrations in Section 2), this leads to log( M ⋆, calib /M ⋆, true ) ≈ 2 log( z calib /z true ). For a galaxy with z true ≈ 0 . 1, typical photometric redshift errors of ∼ 0 . 03 (Salvato et al. 2019) can produce stellar mass residuals of ∼ 0 . 3 dex. This is an appreciable uncertainty, particularly when compared to the 1 σ dispersions of ∼ 0 . 1 dex that we found for our optical/NIR photometric calibrations and SED fitting (Section 6). 9 \nFurthermore, for most of the SED fitting tests in Section 2.2, redshift was not the only parameter we held fixed. We treated the SFH prior, dust attenuation, dust extinction, and IMF as independent assumptions. By only varying the assumption under investigation and fixing all other parameters, we may have ignored covariant effects. The true uncertainties in stellar mass likely depend on how assumptions about the SFH, dust, and IMF are combined. \nFinally, as discussed in Section 2.2, we chose Prospector as our SED fitting code because it uses FSPS to create galaxy models. Since our mock observations were produced using FSPS, our experiment is therefore a direct 'apples-to-apples' comparison that tests, to first order, the SED fitting mechanism rather than other systematic effects that may depend on the exact choice of SED fitting code (e.g., stellar model libraries, dust parameterization, SED fitting algorithm). However, when fitting the SEDs of real galaxies, there is no way to select a code that exactly matches the input SED, so it is important to understand the systematic uncertainties introduced by different SED fitting codes. Pacifici et al. (2023) recently investigated the effects of using 14 different SED fitting codes on high-mass ( > 10 8 M ⊙ ) galaxies at z ∼ 1 and z ∼ 3 and found that all codes return stellar mass distributions that are qualitatively similar. We plan to extend this test to the low-mass regime in a future study.", '8. SUMMARY': "As we enter an era of unprecedentedly large extragalactic surveys, we are beginning to observe statistical samples of low-mass galaxies across a range of environments. Accurately estimating the physical properties of these galaxies is critical if we want to understand the physics that drives their evolution. However, many of the methods used to measure galaxy properties from integrated light have primarily been developed for and tested on massive ( > 10 8 M ⊙ ) galaxies. In this study, we have attempted to verify several of these methods for low-mass galaxies. \nIn particular, we have tested how well different stellar mass measurement techniques can recover the masses of low-mass galaxies from integrated UV/optical/nearIR photometry. Our test sample included 469 simulated galaxies from four different models. Using multiple types of simulations - semi-analytic models, empirical halo-galaxy connection models, and two different hydrodynamic simulations with varying resolutions and prescriptions for feedback - makes our results more robust to variations among different simulations. For each simulated galaxy, we generated mock photometry using the stellar population synthesis code FSPS. \nWe then tested empirical colorM ⋆ /L calibrations and showed that many literature relations produce systematic trends in stellar mass residuals that are only apparent at the low-mass end. This is particularly true for calibrations based only on near-IR color, which can produce stellar mass errors of > 1 dex for galaxies below 10 8 M ⊙ . We have provided updated prescriptions for stellar mass based on g -r color (Equation 11) and WISE 3.4 µ mluminosity (Equation 12) that reduce these systematic biases. These new calibrations can recover stellar masses with minimal offsets (0 . 04 ± 0 . 08 dex and -0 . 05 ± 0 . 11 dex, respectively). \nWe also tested different assumptions that go into multi-wavelength SED fitting. In agreement with previous studies of high-mass galaxies (Lower et al. 2020), we found that non-parametric SFH models generally perform better than parametric SFH models when measuring stellar masses of low-mass galaxies. While parametric SFHs can underestimate stellar mass by as much as ∼ 0 . 4 dex, under ideal conditions non-parametric SFH templates can recover M ⋆ of low-mass galaxies with offsets of -0 . 3 ± 0 . 11 dex. Assumptions about dust attenuation parameters introduce larger uncertainties in M ⋆ of low-mass galaxies: over- or under-estimating total dust content can lead to significant ( ∼ 0 . 6 dex) overor under-estimates of stellar mass. The shape of the dust attenuation law may also impact stellar mass estimates, though more work is needed to better understand these effects; in this study we only compared two simple atten- \ndels with different UV-optical slopes, and we did not consider other features in the attenuation curve (e.g., the UV bump; Salim & Narayanan 2020). Our results are summarized in Section 6, in which we have laid out recommendations for measuring M ⋆ of low-mass galaxies from integrated photometry. \nThis study represents an initial step towards a full re-evaluation of measurement techniques based on the integrated light of low-mass galaxies. Much remains to be done, including further tests of SED fitting. As discussed in Section 7.2, this work aimed to test the assumptions that go into a single SED fitting code (Prospector, which uses the same SPS models that were used to generate our mock observations). This work also focused on stellar mass as a 'zeroth-order' property that traces a galaxy's integrated history. In the future, we plan to compare other SED and spectral fitting codes, as well as to investigate other galaxy properties-star formation rates and histories, metallicities, AGN properties-that can reveal even more information about the physics underlying low-mass galaxy evolution. \nMAdlR acknowledges the financial support of the Stanford Science Fellowship while writing this paper. This research has made use of NASA's Astrophysics Data System Bibliographic Services. \nThere are many communities without whom this work would not have been possible. We acknowledge that this work is rooted in Western scientific practices and is the material product of a long and complex history of settler-colonialism. MAdlR wishes to recognize her status as a settler on the unceded homelands of the Pocumtuc Nation. We hope to work toward a scientific practice guided by pono and a future in which we all honor the land. \nThis research used data from the SAGA Survey (Satellites Around Galactic Analogs; sagasurvey.org). The SAGA Survey is a galaxy redshift survey with spectroscopic data obtained by the SAGA Survey team with the Anglo-Australian Telescope, MMT Observatory, Palomar Observatory, W. M. Keck Observatory, and the South African Astronomical Observatory (SAAO). The SAGA Survey also made use of many public data sets, including: imaging data from the Sloan Digital Sky Survey (SDSS), the Dark Energy Survey (DES), the GALEX Survey, and the Dark Energy Spectroscopic Instrument (DESI) Legacy Imaging Surveys, which includes the Dark Energy Camera Legacy Survey (DECaLS), the Beijing-Arizona Sky Survey (BASS), and the Mayall z-band Legacy Survey (MzLS); redshift catalogs from SDSS, DESI, the Galaxy And Mass Assem- \nbly (GAMA) Survey, the Prism Multi-object Survey (PRIMUS), the VIMOS Public Extragalactic Redshift Survey (VIPERS), the WiggleZ Dark Energy Survey (WiggleZ), the 2dF Galaxy Redshift Survey (2dFGRS), the HectoMAP Redshift Survey, the HETDEX Source Catalog, the 6dF Galaxy Survey (6dFGS), the Hectospec Cluster Survey (HeCS), the Australian Dark Energy Survey (OzDES), the 2-degree Field Lensing Survey (2dFLenS), and the Las Campanas Redshift Survey (LCRS); HI data from the Arecibo Legacy Fast ALFA Survey (ALFALFA), the FAST all sky HI Survey (FASHI), and HI Parkes All-Sky Survey (HIPASS); and compiled data from the NASA-Sloan Atlas (NSA), \nthe Siena Galaxy Atlas (SGA), the HyperLeda database, and the Extragalactic Distance Database (EDD). The SAGA Survey was supported in part by NSF collaborative grants AST-1517148 and AST-1517422 and Heising-Simons Foundation grant 2019-1402. SAGA Survey's full acknowledgments can be found at https:// sagasurvey.org/ack/. \nSoftware: Prospector (Johnson et al. 2021; Leja et al. 2019), python-FSPS (Johnson et al. 2023), FSPS (Conroy et al. 2009; Conroy & Gunn 2010), Matplotlib (Hunter 2007), Seaborn (Waskom 2021), Astropy (Robitaille et al. 2013), Scipy (Jones et al. 2001)", 'A. PROSPECTOR PRIORS': 'For the sake of reproducibility, the full priors for all Prospector runs used in this study are described in Table 4.', 'REFERENCES': 'Agertz, O., Kravtsov, A. V., Leitner, S. N., & Gnedin, N. Y. 2013, ApJ, 770, 25 \nBruzual, G., & Charlot, S. 2003, MNRAS, 344, 1000 \nAkeson, R., Armus, L., Bachelet, E., et al. 2019, arXiv e-prints, arXiv:1902.05569 \nAzartash-Namin, B., Engelhardt, A., Munshi, F., et al. 2024, ApJ, 970, 40 \nBehroozi, P., Wechsler, R. H., Hearin, A. 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2024arXiv240908003S | Global expansion of the Event Horizon Telescope EHT will see the strategic addition of antennas at new geographical locations transforming the sensitivity and imaging fidelity of the lambda sim 1mm EHT array. A possible South African EHT station would leverage a strong geographical advantage local infrastructure and radio astronomy expertise and have strong synergies with the Africa Millimetre Telescope in Namibia. We assessed three South African candidate millimetre sites using climatological simulations and antenna sensitivity estimates and found at least two promising sites. These sites are comparable to some existing EHT stations during the typical April EHT observing window and outperform them during most of the year especially the southern hemisphere winter. The results suggest that a strategically placed South African EHT station will have a sizable positive impact on nextgeneration EHT objectives and the resulting black hole imaging science. | 2024-09-01T00:00:00Z | ['2024arXiv240908003S', '10.48550/arXiv.2409.08003', 'arXiv:2409.08003'] | ['Astrophysics - Instrumentation and Methods for Astrophysics'] | Evaluation of South African Candidate Sites for an Expanded Event Horizon Telescope | 2,024 | 165 | 0.43 | ['EPRINT_HTML', 'EPRINT_PDF'] | 0 | https://arxiv.org/pdf/2409.08003.pdf | {'Evaluation of South African Candidate Sites for an Expanded Event Horizon Telescope': '1 \nSenkhosi Simelane 1 , Roger Deane 1 , 2 , Athol Kemball 3 , 1 , Roelf Botha 4 , Roufurd Julie 4 , Keitumetse Molamu 4 , Adrian Tiplady 4 , and Aletha de Witt 5 Wits Centre for Astrophysics, School of Physics, University of the Witwatersrand, Braamfontein, Johannesburg, 2017, South Africa 2 Department of Physics, University of Pretoria, Hatfield, Pretoria, 0028, South Africa 3 Department of Astronomy, University of Illinois Urbana-Champaign, Urbana, 61801, USA 4 South African Radio Astronomy Observatory, River Park Liesbeek Parkway, Settlers Way, Cape Town, 7705, South Africa 5 Department of Science, Technology and Innovation, 671 Meiring Naudé Road, Pretoria, 0184, South Africa \nAbstract - Global expansion of the Event Horizon Telescope (EHT) will see the strategic addition of antennas at new geographical locations, transforming the sensitivity and imaging fidelity of the λ ∼ 1 mm EHT array. A possible South African EHT station would leverage a strong geographical advantage, local infrastructure, and radio astronomy expertise, and have strong synergies with the Africa Millimetre Telescope in Namibia. We assessed three South African candidate millimetre sites using climatological simulations and antenna sensitivity estimates, and found at least two promising sites. These sites are comparable to some existing EHT stations during the typical April EHT observing window and outperform them during most of the year, especially the southern hemisphere winter. The results suggest that a strategically placed South African EHT station will have a sizable, positive impact on next-generation EHT objectives and the resulting black hole imaging science.', '1. Introduction': "A host of novel black hole (BH) science goals will be enabled by the expansion of the Event Horizon Telescope (EHT) [1, 2], including the reconstruction of highfidelity movies of the Messier 87 ∗ (M87 ∗ ) and Sagittarius A ∗ (Sgr A ∗ ) supermassive BHs (SMBHs). Both sources display dynamic behaviour, with timescales of ∼ 20 days [3] and ∼ 20 minutes [4-6], respectively. However, the typical one-week duration of EHT observing campaigns is too short to capture the full dynamical evolution of M87 ∗ [7] and limited snapshot ( u , v ) -coverage has meant that the rapid time-variability of Sgr A ∗ remains poorly constrained [8]. These limitations have precluded experiments that allow dynamical imaging of SMBHs. New and strategically placed stations, especially several on the African continent, would extend the annual monitoring window for M87 ∗ and also significantly improve the instantaneous and full-track ( u , v ) -coverage towards Sgr A ∗ [9, 10], while prompting significant strides in African Very Long Baseline Interferometry (VLBI) development, particularly with intra-African \nVLBI baselines. \nSouthern Africa holds a strong geographical advantage for imaging southern-hemisphere sources, such as Sgr A ∗ . Simulations by [10] showed that adding the Africa Millimetre Telescope (AMT) in Namibia and a Canary Islands station to the EHT improves Sgr A ∗ movie fidelity through better constraints on source extent and a factor ≳ 4 increase in observation length with adequate snapshot ( u , v ) -coverage for dynamical imaging. Static Sgr A ∗ imaging from [9] showed that an EHT array augmented with the AMT and a South African station achieves significantly improved ( u , v ) -coverage, especially in the eastern sub-array. Through long northsouth and east-west baselines to NOEMA and ALMA, respectively, a South African EHT station (SA-EHT hereafter) would further enhance dynamical reconstructions of Sgr A ∗ and M87 ∗ and potentially enable polarimetric photon ring experiments, as proposed by [11]. Additionally, a short ( ∼ 1000 km) SA-EHT-AMT baseline would enable the measurement of low-spatial-frequency emission critical for linking jet launch with horizon-scale dynamics. The added short-baseline constraints would also improve the gain calibration solutions and hence the achieved image fidelity [12] and dynamic range, particularly with Maximum Entropy Method imaging [13]. \nThe prospective array performance improvements offered by the addition of SA-EHT to the EHT strongly motivate an investigation of feasible millimetre (mm) sites in South Africa. To this end, three sites were identified for preliminary evaluation: Ben Macdhui (BMAC) in South Africa's Eastern Cape, Sutherland (STL) in the Northern Cape, and Matjiesfontein (MATJ) in the Western Cape. Details of the sites, selected for their relatively high altitude and existing infrastructure, are summarised in Table 1. The pre-selection criterion for sites to have existing infrastructure is motivated by a more rapid and cost-effective SA-EHT project delivery. \nTropospheric water vapour is the chief cause of atmospheric absorption and scattering of mm waves, necessitating high-altitude, low-humidity sites. The frequencydependent absorption, quantified by the optical depth, \nTable 1: Geographical information of the three SA-EHT candidate sites: Ben Macdhui (BMAC), Sutherland (STL) and Matjiesfontein (MATJ). \nτ ( ν ) , attenuates incoming mm radiation as e -τ ( ν ) and a fast-evolving water vapour distribution above a mm station causes rapid phase fluctuations. The coherence time, defined as the mean time interval over which the troposphere induces a 1-radian phase shift, characterises the rate of these fluctuations. Thus, the columnintegrated precipitable water vapour (PWV) and τ ( ν ) are key parameters in mm site evaluations. For the nextgeneration EHT (ngEHT), the 86GHz, 230GHz, and 345GHz zenith optical depths are of the most interest as these align with the proposed receiver bands [2]. Wind speeds are also critical, as high wind speeds degrade coherence times, disrupt dish pointing and tracking and may even necessitate stowing of the dish. This work evaluated these climatological proxies, among others, and used them to estimate antenna sensitivity for various dish sizes via the System Equivalent Flux Density (SEFD). This analysis marks a first step towards identifying an SAEHT dish size that offers the best value proposition while delivering significant scientific impact. \nThe work is presented as follows: the methods employed in the climatological site characterisation and antenna sensitivity estimation study are presented in Sections 2 and 3. The results are presented and discussed in Section 4. Finally, Section 5 summarises the key results and gives an outlook on the next steps and outstanding questions.", '2. Climatological Modelling': 'To characterise the candidate SA-EHT sites, we employed a meteorological modelling approach akin to [14-17]. Using data from Modern-Era Retrospective Analysis for Research and Applications, version 2 (MERRA-2) [18], a global database of atmospheric measurements on a 0 . 5 · latitude by 0 . 625 · longitude (or ∼ 55 km latitude by ∼ 70 km) grid from 1980 to date, we generated realistic model atmospheres. MERRA-2 has a three-hour temporal resolution, but a temporal resolution of 1 month was used for this analysis as it suffices for the goal of shortlisting sites for onsite testing. We used daily MERRA-2 data files from 01-01-2009 to 31-12-2022 to calculate monthly means of the quantities needed for model atmosphere generation, which we then used to create the model atmosphere. \nIn the absence of on-site air pressure measurements, we devised a surface pressure estimation approach requiring only the site latitude, S lat , longitude, S lon and altitude (height above sea level) S alt as inputs. The method employs the MERRA-2 vertical pressure levels, lev , and the mean edge heights, H , the mean altitudes to which they correspond, from the surrounding grid points. For each grid point, the closest mean edge heights above and below S alt are determined, the corresponding pressure levels identified and a linear interpolation between the pair of points is carried out to estimate the pressure at the height S alt . The estimated pressures at S alt from the four grid points are then interpolated bilinearly to estimate the surface pressure, P surf , at the site location, (S lat , S lon , S alt ). Finally, the mean vertical profiles of other quantities (e.g. Figure 1 of [17]) are determined by bilinearly interpolating the mean MERRA-2 profiles of the surrounding grid points to (S lat , S lon ), truncating the profiles at the closest vertical pressure level above S alt and using P surf to logarithmically extrapolate them to (S lat , S lon , S alt ). To avoid negative wind speed predictions, wind speeds were instead estimated by linearly interpolating between the values at the pressure levels closest to P surf above and below. \nFollowing atmospheric model generation, we used the am atmospheric model [19] to solve the radiative transfer equation for the models, effectively estimating ground-based radiometer measurements by simulating the effect of atmospheric absorption, emission, and scattering on radio-frequency signals passing through the model atmospheres. We chose a frequency range of 80 GHz to 700 GHz with 500-MHz resolution. Estimated quantities include the mean precipitable water vapour (PWV), zenith optical depth τ ( ν ) , zenith brightness temperature T b ( ν ) , and wind speed v wind .', '3. Sensitivity Estimation': "Equipped with the τ ( ν ) and T b ( ν ) values, we estimated the effective sensitivity of the antenna and receiver at 230 GHz through single-dish SEFDs for different dish diameters using the definition [20] \nSEFD ≡ 2 k B T sys η A geo . \nIn the equation above, k B is the Boltzmann constant, T sys the absorption-corrected system temperature, A geo the geometric area of the dish and η an efficiency encapsulating all relevant effects. Assuming a perfectly sidebandseparating receiver system, we calculated T sys as \nT sys ( ν ) = e τ ( ν ) [ T rec ( ν ) +( 1 -e -τ ( ν ) ) T atm ( ν )] , \nwhere T rec ( ν ) is the receiver temperature and T atm ( ν ) , the effective atmospheric temperature, is \nT atm ( ν ) = T b ( ν ) -T CMB e -τ ( ν ) 1 -e -τ ( ν ) . \nT b ( ν ) is the zenith atmospheric brightness temperature. We assumed the ALMA 230-GHz receiver temperature of T rec = 40 K (see Table 5 of [17]) and T CMB = 2 . 725 K [21]. Our η estimates considered only the forward efficiency and aperture efficiency: \nη ( ν ) = η F ( ν ) η ap ( ν ) . \nWe chose η F = 0 . 9 for all sites, which is similar to values adopted in other EHT expansion analyses [17]. The aperture efficiency, η ap, was calculated using Ruze's formula [22], assuming a focus offset of 10 µ m and a commercially available dish with an RMS surface accuracy of 40 µ m. \nThese calculations underpin the 230-GHz noise performance analysis of hypothetical dishes at the candidate sites in Section 4, incorporating dish and receiver specifications and the site-specific atmospheric conditions calculated in Section 2. Three indicative dish diameters (9 m, 13 m and 18 m) were included based on ngEHT and next-generation Very Large Array designs.", '4. Results and Discussion': 'We validated the climatological characterisation results through comparison with BMAC weather data from the ngehtsim package, which was the only South African candidate site whose weather data was included in [15] and ngehtsim . The results were consistent, yielding mean relative differences of ∼ 4 %, ∼ 9 % and ∼ 17 % for the mean 230-GHz zenith optical depth, mean PWV and mean wind speed, respectively, which are lower than the RMS of each parameter. With the validation completed, the analysis of all three candidate SA-EHT sites was undertaken.', '4.1. Nominal EHT observing window comparison': "Over the decade analysed, BMAC generally exhibited the lowest mean PWV values among the three South African sites. As an EHT site located at a relatively low altitude of 1900 m, Kitt Peak (KP) has some of the least favourable mm observing conditions in the array. Thus, we used it as a benchmark non-premier site against which to evaluate the SA-EHT candidate sites. In April, the customary EHT observing window, the mean PWV and the 230-GHz optical depths and brightness temperatures above BMAC are comparable to those above KP (see middle column of Figure 1), a pattern also observed at 86 GHz and 345 GHz. While STL's mean PWV, τ ( ν ) \nBMAC (3000 m) \nSTL (1800 m) \nMATJ (1340 m) \nALMA (5070 m) \nKP (1900 m) \nFigure 1: Monthly mean PWV (top row), 230-GHz zenith opacity (second row) and brightness temperature (third row), and wind speed (bottom row) above BMAC, STL, and MATJ for March, April, and May during 2013-2022. Predictions for BMAC, STL, and MATJ are shown by blue, orange, and green curves, respectively, with shaded regions indicating ± σ . Means plus σ above KP (dashed) and means minus σ above ALMA (dotted) contextualise the SA-EHT values within the typical range of atmospheric conditions in the array. The dashed magenta line (bottom row) marks a wind speed of 15 m/s. \n<!-- image --> \nYear \nand T b ( ν ) curves approach those of BMAC during certain years, the STL and MATJ curves generally exceed the upper bound of the KP one-standard deviation ( σ ) range. In May, BMAC's curves fall well below KP + σ , while those of STL remain close to them. The ∼ 10-mm conditions at BMAC and STL in March decline to ∼ 5 mm (STL) and < 5 mm (BMAC) from April through August, representing an important trend as the ngEHT project seeks to extend the observing window to ∼ 3 months in Phase 1 [2]. \nSimilar wind speeds to STL and MATJ during the first half of the year and significantly higher values during the latter half make BMAC the windiest site of the three. Wind-induced sensitivity degradation depends on antenna specifications but at wind speeds < 15 m/s, degradation is typically insubstantial [17]. The mean wind speeds at all three candidate SA-EHT sites rarely reach this mark. We note, however, that the MERRA-2 wind speed data lack the time-resolution to capture gusting. This is left for future analysis of on-site measure- \nFigure 2: Mean SEFD estimates for different dish sizes during the month of April from 2013 to 2022. Thicker line widths correspond to large dish diameters at two South African sites, BMAC (purple) and STL (green). A black, dashed horizontal line at the mean value of KP over the decade is added for context. Values generally showed a standard deviation of ∼ 3000 Jy. \n<!-- image --> \nFigure 3: Winter monthly mean PWV above BMAC, STL, MATJ and KP during 2013-2022. Predictions for BMAC, STL, MATJ and KP are shown by blue, orange, green and black curves, respectively, with shaded regions indicating ± σ . The dashed black line marks a PWV of 10 mm. \n<!-- image --> \nments. \nWe explored three dish diameters in our estimated sensitivity analysis: 9 m, 13 m and 18 m. The hypothetical 13-m dish at BMAC and 18-m dish at STL showed mean SEFDs similar to those of the KP antenna in April (see Figure 2). Specifically, KP marginally outperforms a BMAC 13-m dish and is marginally outperformed by an STL 18-m dish, reflecting KP's slightly superior April 230-GHz zenith opacities and brightness temperatures. The noise performance difference between BMAC and STL diminishes with increasing dish diameter. While an 18-m dish at BMAC outperforms the same dish at STL, the difference is minimal during the latter half of the decade analysed, partly due to an apparent uptrend in April brightness temperatures and zenith opacities at BMAC. This trend warrants further monitoring with a MERRA-2 data subset including 2023 and 2024.", '4.2. Winter-winter comparison': "The goal of achieving year-round observations in Phase 2 of the ngEHT project [2] relaxes the requirement for sub-10 mm PWV conditions specifically around April, allowing a broader assessment of site quality across seasons. This enables us to remove the effect of opposing seasonal patterns between the southern and northern hemispheres by comparing the quality indicators of the candidate SA-EHT sites and KP during equivalent seasonal periods. Since winter observing conditions are generally more favourable than summer conditions, we focused on the meteorological winters: 1 June to 31 August in the southern hemisphere and 1 December to 28/29 February in the northern hemisphere. Figure 3 demonstrates that STL and KP offer similar site quality for mm observations, while the higher-altitude BMAC site exhibits conditions superior to both. Typically falling below the 10 mm mark in the same figure, all three SA-EHT candidate sites become competitive mm sites in the southern hemisphere winter. Notably, only in March and the period plotted in Figure 3 do KP's mean PWVs and zenith spectral opacities consistently suggest more favourable conditions than BMAC and STL. Furthermore, observing conditions at the South African sites are relatively stable throughout the year. For instance, PWV values above KP deteriorate by a factor > 6 (to ∼ 30mm in July and August) in the northern hemisphere summer compared to April, while even the MATJ PWV values seldom exceed 15 mm. This stability is significant for year-round observing and opens the door to southernwinter VLBI observations with the AMT and Atacama Desert stations (ALMA and APEX), particularly with Frequency Phase Transfer (FPT)-enabled stations.", '5. Conclusion': "Among the three candidate SA-EHT sites selected for desktop evaluation, BMAC showed the most promising proxies for mm site quality, comparable to KP during the nominal April EHT observing window and outperforming it during most of the year. In its most favourable years, however, STL's weather conditions are similar to those of BMAC in this window, and it is comparable to KP when the sites' weather conditions are compared for similar meteorological seasons. In our view, BMAC and STL justify the financial investment of in situ characterisation. \nThe positive mm site quality and antenna sensitivity indicators emphasise the importance of the next steps. FPT will very likely play a pivotal role in future EHT operations, particularly at non-premier sites. It is, therefore, also important to incorporate this technique in an extension of this study to dynamical imaging, especially \ncharacterising its performance as a function of sensitivity to help inform the recommended SA-EHT dish size.", '6. Acknowledgements': 'We thank Lindy Blackburn, Iniyan Natarajan, Daniel Palumbo and Dominic Pesce for insightful discussions that greatly benefitted this work. We are especially grateful to Dominic Pesce for adding our candidate sites to ngehtsim , which significantly streamlined the analysis. We also appreciate the constructive feedback from two anonymous referees.', '7. References': '- 1. M. D. Johnson, K. Akiyama, L. Blackburn, K. L. Bouman, A. E. Broderick et al., \'Key Science Goals for the NextGeneration Event Horizon Telescope,\' Galaxies , 11 , 3, April 2023, p. 61.\n- 2. S. S. Doeleman, J. Barrett, L. Blackburn, K. L. Bouman, A. E. 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2024arXiv240900197K | Globular clusters GCs are sites of extremely efficient star formation and recent studies suggest they significantly contributed to the early Milky Ways stellar mass buildup. Although their role has since diminished GCs impact on the Galaxys initial evolution can be traced today by identifying their most chemically unique starsthose with anomalous nitrogen and aluminum overabundances and oxygen depletion. While they are a perfect tracer of clusters be it intact or fully dissolved these highNO highAlFe GCorigin stars are extremely rare within the current Galaxy. To address the scarcity of these unusual precious former GC members we train a neural network NN to identify highNO highAlFe stars using lowresolution Gaia BPRP spectra. Our NN achieves a classification accuracy of approximately approx99 and a false positive rate of around approx7 identifying 878 new candidates in the Galactic field. We validate our results with several physicallymotivated sanity checks showing for example that the incidence of selected stars in Galactic GCs is significantly higher than in the field. Moreover we find that most of our GCorigin candidates reside in the inner Galaxy having likely formed in the protoMilky Way consistent with previous research. The fraction of GC candidates in the field drops at a metallicity of FeHapprox1 approximately coinciding with the completion of spinup i.e. the formation of the Galactic stellar disk. | 2024-08-01T00:00:00Z | ['arXiv:2409.00197', '10.48550/arXiv.2409.00197', '2024arXiv240900197K'] | ['Astrophysics - Astrophysics of Galaxies'] | The Ones That Got Away Chemical Tagging of Globular ClusterOrigin Stars with Gaia BPRP Spectra | 2,024 | 165 | 0.57 | ['EPRINT_HTML', 'EPRINT_PDF'] | 2 | https://arxiv.org/pdf/2409.00197.pdf | {'The Ones That Got Away: Chemical Tagging of Globular Cluster-Origin Stars with Gaia BP/RP Spectra': 'Sarah G. Kane, 1 ★ Vasily Belokurov, 1 Miles Cranmer, 1 , 2 , 3 Stephanie Monty, 1 Hanyuan Zhang, 1 Anke Ardern-Arentsen, 1 and Elana Kane 1 \n- 1 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK\n- 2 Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK\n- 3 Kavli Institute for Cosmology, Cambridge, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK \nAccepted XXX. Received YYY; in original form ZZZ', 'ABSTRACT': "Globular clusters (GCs) are sites of extremely efficient star formation, and recent studies suggest they significantly contributed to the early Milky Way's stellar mass build-up. Although their role has since diminished, GCs' impact on the Galaxy's initial evolution can be traced today by identifying their most chemically unique stars-those with anomalous nitrogen and aluminum overabundances and oxygen depletion. While they are a perfect tracer of clusters, be it intact or fully dissolved, these high-[N/O], high-[Al/Fe] GC-origin stars are extremely rare within the current Galaxy. To address the scarcity of these unusual, precious former GC members, we train a neural network (NN) to identify high-[N/O], high-[Al/Fe] stars using low-resolution Gaia BP/RP spectra. Our NN achieves a classification accuracy of approximately ≈ 99% and a false positive rate of around ≈ 7%, identifying 878 new candidates in the Galactic field. We validate our results with several physically-motivated sanity checks, showing, for example, that the incidence of selected stars in Galactic GCs is significantly higher than in the field. Moreover, we find that most of our GC-origin candidates reside in the inner Galaxy, having likely formed in the proto-Milky Way, consistent with previous research. The fraction of GC candidates in the field drops at a metallicity of [Fe/H] ≈ -1, approximately coinciding with the completion of spin-up , i.e. the formation of the Galactic stellar disk. \nKey words: stars: abundances - Galaxy: abundances - globular clusters: general - Galaxy: formation - Galaxy: halo", '1 INTRODUCTION': 'Globular clusters (GCs) are sites of extremely efficient star formation. Comprised of 10 5 -10 7 of densely-packed stars that form at approximately the same time, these objects are suggested to the primary contributors to galaxy formation at early times (e.g. as nuclear star clusters, building blocks of stellar halos, and of chemically diverse discs and spheroids through their association with star forming clumps; Tremaine et al. 1975; Gnedin et al. 2014; Martell et al. 2011; Schiavon et al. 2017; Belokurov & Kravtsov 2023; Clarke et al. 2019; Debattista et al. 2023). Observations at high redshift are now beginning to suggest that this may indeed be the case. For instance, Mowla et al. (2024) have identified in the JWST imaging a 𝑧 = 8 . 3 galaxy with approximately half of its mass locked in massive star-forming clusters. The interpretation of these results is of ongoing community interest, with Rusta et al. (2024) suggesting that these bright, unresolved stellar clumps may instead be "building block galaxies" themselves. \nNearby GCs, resolved into individual stars, ubiquitously exhibit a light-element anti-correlation indicative of multiple stellar populations (early evidence of this phenomena emerged through measurements of the molecule CN in MW GCs, e.g. Freeman & Rodgers 1975; Norris & Bessell 1975; Bessell & Norris 1976; Cottrell & Da \nCosta 1981). Stars in the first population (1P), so called because they are thought to have formed first, display light element abundances consistent with MW field stars at similar metallicities. By contrast, members of the second population (2P) are exceptionally enhanced in nitrogen, aluminum, and sodium and depleted in oxygen, carbon, and magnesium relative to field stars (Gratton et al. 2004; Carretta et al. 2009b,a, 2010; Milone & Marino 2022a). The precise origin of the anomalous chemistry of the 2P stars in GCs is not fully understood, with theories invoking enrichment from supermassive stars (Denissenkov & Hartwick 2014; Gieles et al. 2018) among the 1P, winds from AGB stars (e.g. Ventura & D\'Antona 2009; D\'Antona et al. 2016) in the dense cluster environment and massive binary interactions (for a more complete summary, see Bastian & Lardo 2018; Gratton et al. 2019; Milone & Marino 2022a). Nonetheless, the light-element anticorrelation is distinctive to and universal across GCs (Bastian & Lardo 2018; Gratton et al. 2019; Milone & Marino 2022b). By comparison, within almost any given cluster members exhibit spreads in metallicity on the order of the abundance uncertainty, regardless of whether those stars are in the first or second population (Gratton et al. 2004). \nThe anomalous chemistry of the 2P GC stars is so distinct relative to MW field stars that we can tag field stars with this pattern of overabundances and depletions as having originated in a GC; nitrogen overabundance in particular has been used consistently for this form of chemical tagging (Horta et al. 2021; Schiavon et al. \n2017; Belokurov & Kravtsov 2023; Martell & Grebel 2010; Phillips et al. 2022). Belokurov & Kravtsov (2023) use APOGEE data (Abdurro\'uf et al. 2022) to identify MW field stars with high [N/O] ratio and show that the majority of these likely GC 2P members are part of Aurora , the old ([Fe/H] ≲ -1 . 3), in-situ MW halo (Belokurov & Kravtsov 2022; Rix et al. 2022). Moreover, Belokurov & Kravtsov (2023) find that the fraction of high-[N/O] field stars in the Galactic halo drops rapidly with increasing metallicity around [Fe/H] ≃ -1 . 0. This metallicity dependence of the high-[N/O] fraction indicates that in the early MW a significant portion of star formation was locked in GCs but the clusters\' relative contribution to the Galaxy\'s stellar mass buildup began to decline around the time of the emergence of the MW thick disk, or at the time of \'spin-up\' (see also Belokurov & Kravtsov 2022; Chandra et al. 2023; Belokurov & Kravtsov 2024). In spite of their hypothesized importance for tracing the early history of the Galaxy, high-[N/O] stars are exceptionally rare in the present-day Galactic field, constituting only 2 -3% of the inner halo and even lower further out from the galactic centre (Martell et al. 2016; Horta et al. 2021; Schiavon et al. 2017; Belokurov & Kravtsov 2023). The scarcity of these chemically anomalous stars poses a real challenge for investigating the history of GCs in the MW. \nThe anomalous nitrogen enhancement of 2P GC stars has become particularly relevant within the context extragalactic astronomy given the recently discovered nitrogen-rich chemistry of GN-z11, a 𝑧 = 10 . 6 galaxy discovered first with Hubble and then observed with JWST (Bunker et al. 2023; Cameron et al. 2023). The extreme nitrogen enrichment of GN-z11 and in several other galaxies at high redshift (e.g. Ji et al. 2024; Topping et al. 2024; Yanagisawa et al. 2024) along with the mounting evidence that GCs contribute significantly to star formation in the early MW (Belokurov & Kravtsov 2023) has spurred theories that the source of nitrogen enrichment in 2P GC stars and in high-z Universe may be one and the same. One of the more popular hypotheses is that in GN-z11 supermassive stars are responsible for the elevated N abundance (Charbonnel et al. 2023; Denissenkov & Hartwick 2014; Nandal et al. 2024). Alternative theories have also been proposed, such as an intermittent star formation history (Kobayashi & Ferrara 2024). Nonetheless, this is an active area of research that makes the study of clusters extremely relevant. \nThe scarcity of high-[N/O] stars in the Galactic field poses a challenge for using these stars as a tracer of the history of globular clusters in the MW. High resolution spectroscopic surveys are both limited in how many stars they can observe as well as the difficulty in measuring N at all wavelengths, thus making the number of known 2P GC-type stars outside of clusters limited. The European Space Agency\'s Gaia mission (Gaia Collaboration et al. 2016) has recently revolutionized our understanding of the Milky Way by providing astronometric, photometric, and spectroscopic data for hundreds of millions of stars. Among the products from the most recent data release (Data Release 3; DR3) are low-resolution (R ≈ 30-100) Blue Photometer/Red Photometer (BP/RP) spectra for approximately 220 million stars (Gaia Collaboration et al. 2023; De Angeli et al. 2023; Montegriffo et al. 2023). Combined, the full BP/RP spectra have a resolution of 𝑅 ∼ 60 and cover wavelengths spanning from 330 nm to 1050 nm, which includes a CN band at 388 nm, a second CN feature at 421.5 nm for stars with [Fe/H] ∼ -1, and a NH band at the edge of the wavelength range at 336 nm. Usefully, the resolution of the Blue Photometer is highest at lower wavelengths, where the nitrogen features are located (Carrasco et al. 2021). Thus, we propose to leverage the large quantity of low resolution Gaia BP/RP spectra to identify new high-[N/O] candidates. \nThe usefulness of the Gaia BP/RP spectra to derive various stellar \nparameters and abundances has been of broad interest to the community (see e.g. Witten et al. 2022), and a wealth of papers on the topic have been published in the past few years. Many of these recent works rely on machine learning methods to extract information from the BP/RP spectra, as the low resolution nature of the data makes more traditional abundance line analysis challenging. Among these works, multiple groups have derived stellar effective temperatures, surface gravities, and metallicities from the BP/RP spectra with relatively high accuracy (e.g. Andrae et al. 2023; Liu et al. 2012; Yao et al. 2024; Khalatyan et al. 2024). Other works have extracted 𝛼 -element abundances from the spectra as well as the three fundamental stellar parameters (e.g. Li et al. 2024; Hattori 2024). Some groups have also used the BP/RP spectra to identify chemically peculiar stars. For instance, Lucey et al. (2023) identified carbon-enhanced metal-poor star candidates from the BP/RP spectra; while Sanders & Matsunaga (2023) used BP/RP to pick out C-rich long periodic variables. Recently, Fallows & Sanders (2024) used a neural network to derive data-driven 𝑇 eff , log 𝑔 , metallicity, [ 𝛼 /Fe], [C/Fe], and [N/Fe] abundances from the BP/RP spectra. \nIn this work, we leverage the vast quantity of Gaia BP/RP spectra to detect new high-[N/O] stars in the MW halo. Thanks to Gaia , we are able to increase the number of known halo stars with 2Ptype chemistry by a factor of as compared to APOGEE. We perform heteroscedastic regression with a simple neural network to derive our own estimates of 𝑇 eff , log 𝑔 , [Fe/H], [N/O] and [Al/Fe] from the BP/RP spectra, and from these predictions, we classify stars as high-[N/O], high-[Al/Fe] candidates. With these classifications, we identify 878 new high-[N/O] candidates in the MW halo. Using these new high-[N/O] field stars, we are able to use their properties and distribution to study the contribution of GCs to the Galaxy at an unprecedented scope. \nOur paper is organised as follows. Section 2 describes our datadriven approach to locating new high-[N/O] candidates in three subsections. In Section 2.1, we discuss our training dataset, and in Section 2.2, we outline the neural network we use to infer stellar parameters and [N/O] and [Al/Fe] abundances. Section 2.3 details the sample of red giant stars upon which we use our trained neural network to identify new high-[N/O] stars. We validate the results of our network performance in Section 3.1. The rest of Section 3 details our newly identified high-[N/O] candidates, with Section 3.3 detailing additional tests we used to verify the reliability of our selection. Section 4 discusses the properties of our candidates and their distribution within the Galaxy. We provide a summary of our analysis in Section 5.', '2.1 Training and Validation Data': "Because our goal is to predict [N/O] and [Al/Fe] abundances from the Gaia BP/RP coefficients, we require stars with both BP/RP observations in Gaia DR3 and high-quality abundance measurements with which to train our neural network. To this effect, the core of our training data is comprised of cross-matches between Gaia BP/RP observations and stars with data from the Apache Point Observatory Galactic Evolution Experiment (Abdurro'uf et al. 2022, APOGEE). APOGEE is unique among spectroscopic surveys and particularly well-suited to our goal of having training labels for [N/O] abundances because APOGEE spectra contain many N lines (see Schiavon et al. 2017). The robust nitrogen abundances provided by APOGEE have prompted its use in previous studies of nitrogen-rich stars (e.g., Schiavon et al. 2017; Horta et al. 2021; Belokurov & Kravtsov 2023), \n<!-- image --> \nFigure 1. Left: The color-magnitude diagram (CMD) of our training and validation dataset of APOGEE and Gaia BP/RP sources (colored 2D histogram) overlaid on the CMD of 32 459 252 randomly selected BP/RP sources for comparison (grey 2D histogram). The extension of the RGB into redder colors is caused by dust effects. Right: The distribution of APOGEE [N/O] abundances in our training and validation dataset. The purple histogram shows the normalized distribution of [N/O] abundances in the dataset overall, and the transparent black histogram shows the distribution of [N/O] abundances of stars in our training and validation dataset tagged as GC members by Vasiliev & Baumgardt (2021). The vertical, dashed red line marks the nominal cutoff of high-[N/O] stars are [N/O] > 0.55. The GC members are included in the overall training and validation datasets; note that almost all high-[N/O] giants from APOGEE are tagged as GC members. \n<!-- image --> \nwhich has the added benefit of providing some consistency between our data and that of previous works. Despite known biases in the absolute value of O in APOGEE (Carrillo et al. 2022), it has been shown to trace other 𝛼 elements to high fidelity. From these crossmatches, we restrict our training data to giants by selecting stars with 𝑇 eff < 5200 and log 𝑔 < 3 . 0. We further apply the following quality cuts to the abundances and stellar parameters from APOGEE: \n- · Bitmasks to remove telluric contamination and duplicate targets ( EXTRATARG TELLURIC & DUPLICATE ) 1\n- · ASPCAP flags: STAR\\_BAD , TEFF\\_BAD , & LOGG\\_BAD < 0\n- · SNR > 20\n- · Errors for [Fe/H], [Mg/Fe], [N/Fe], [O/Fe], & [Al/Fe] < 0 . 1 \nWe also apply several cuts based on the Gaia catalog to ensure the exclusion of potentially problematic BP/RP spectra from our training data. \n- · phot\\_g\\_mean\\_mag < 16 . 0\n- · parallax / parallax\\_error > 3 . 0 & parallax > 0\n- · RUWE < 1 . 4 (to remove clear binary contamination per Belokurov et al. 2020)\n- · bp\\_chi\\_squared < 10 . 5 × bp\\_degrees\\_offreedom \nAfter applying these cuts, our set of BP/RP-APOGEE crossmatches contains 199 078 stars, of which we use 80% for training and withhold a randomly selected 20% subset upon which to validate our network. The color-magnitude diagram (CMD) of our full training and validation dataset is shown in the left panel Fig. 1, wherein our selected stars clearly populate the red giant branch (RGB). The right panel of Fig. 1 depicts the normalized distribution of [N/O] abundances in our training data and the distribution of [N/O] abundances from the subset of our training data that are identified as likely GC members in Vasiliev & Baumgardt (2021). \nAs is consistent with the presence of a chemically anomalous 2P population in GCs, the GC members have a much higher fraction of high-[N/O] stars as compared to the training dataset overall. Within the APOGEE data, we identify true high-[N/O] stars as those with [N/O] -[N/O] error > 0 . 55 and [Al/Fe] > -0 . 1. The cuts at [N/O] > 0 . 55 and [Al/Fe] > -0 . 1 are consistent with the criteria used in Belokurov & Kravtsov (2023) to select a sample of 2P stars in the field without contamination (e.g., from nitrogen-rich but not oxygen-depleted or aluminum enhanced stars that thus do not display the full pattern of light element anticorrelations typical of 2P GCorigin stars). Using this selection, 1 755 stars among the combined training and validation data are high-[N/O], high-[Al/Fe] stars. \nThe Gaia BP/RP spectra are provided as a set of 110 coefficients of a set of spectral basis functions, with 55 corresponding to the Blue Photometer (BP) and 55 to the Red Photometer (RP) (De Angeli et al. 2023). The coefficient form of the spectra can itself in some sense be considered a form of data compression and feature extraction; although the Gaia team provides software to convert the coefficients into mean spectra ( GaiaXPY 2 ), these converted spectra contain no more information than the coefficients themselves (i.e., because they are converted from the coefficients and their errors). For this reason, we use the coefficients as the input to our neural network, as has most previous work to extract stellar parameters and abundances from the BP/RP spectra. The values of the BP and RP coefficients vary with apparent magnitude, which we do not want to effect the predictions of our network. Thus, we scale each BP and RP coefficient to the first BP or RP coefficient, respectively, associated with that star, as the first coefficients contain the information regarding the overall shape and scaling of the spectrum: \nBP i , scaled = BP i , original / BP 1 , original \nand likewise for the RP coefficients. After this scaling, the values of the first BP and RP coefficients are 1 for all stars, so we remove \nthem from the array of BP/RP coefficients. Thus, the input predictors for our neural network are a set of 108, rather than the original 110, coefficients. Fallows & Sanders (2024) employ a similar procedure to normalize the BP/RP coefficients; however, unlike their work, we do not include photometric colors (ex. from 2MASS or WISE) in the vector of our predictors. \nWe further normalize the coefficients using the 1st and 99th percentiles of the coefficients, i.e.: \nBP i , final = BP i -BP i , 1% BP i , 1% -BP i , 99% \nwherein BP 𝑖, 1% is the 1st percentile and BP 𝑖, 99% is the 99th percentile of all of the 𝑖 th BP coefficients in the training data. We perform the same procedure for the RP coefficients. \nIn summary, the training data for our neural network are comprised of a prediction features vector of the 108 standardized BP and RP coefficients (with the first coefficients removed after scaling) and labels in the form of a five-dimensional vector of APOGEE measurements of 𝑇 eff , log 𝑔 , [Fe/H], [N/O], and [Al/Fe] for the star corresponding the BP/RP coefficients in the features. \nWe believe that it is possible to gauge nitrogen abundances from the BP/RP spectra due to the presence of the two CN bands within the wavelength range of the Gaia Blue Photometer and also potentially via the NH band near the lower wavelength limit of the Blue Photometer. The recent work of Fallows & Sanders (2024) to estimate nitrogen abundances from the BP/RP spectra also supports this assertion. To illustrate the nitrogen information in the BP/RP spectra, we provide sample spectra in Fig. 2 converted from the coefficients with GaiaXPY . The stars share similar stellar parameters but differ in their [N/O] abundances, with a high-[N/O] star in purple and a nitrogen-typical star in gray. The CN and NH bands are indicated with arrows and are magnified in the right panel of Fig. 2 and are visibly more prominent in the spectrum of the nitrogen-rich star as compared to the nitrogen-typical star. We further include the spectrum of a carbon-enhanced star to show that high-[N/O] and high-[C/Fe] stars can be distinguished from their BP/RP spectra using the CH and C 2 bands, which is especially important given that we believe that most of the nitrogen information in the BP/RP spectra comes from the CN bands. The flux in the blue wavelengths of the spectrum of the carbon-rich star is noticeably lower than that of the two carbon-typical stars, making the distinction between nitrogen- and carbon-enhanced stars possible by eye from their BP/RP spectra. As we discuss in more detail in Section 3.1, we do not find carbon enhancement to be a confounding factor in our selection of high-[N/O] candidates. \nThe results from Ting et al. (2018) suggest that these same CN, CH, and C 2 molecular features in the BP/RP spectra may contain the oxygen information which is part of our [N/O] predictions. The oxygen information in the carbon molecular features arises from the fact that most of the oxygen in stellar atmospheres exists in CO; thus, the oxygen abundance directly influences the abundance of other carbon-bearing molecules. Ting et al. (2017) leveraged the oxygen information inherent in the carbon features to determine carbon abundances from optical spectra without oxygen lines, although notably the 𝑅 ≈ 1800 LAMOST spectra they used are nonetheless much higher resolution than the BP/RP spectra. Ting et al. (2018) note that the effect of oxygen on the molecular carbon features is non-degenerate with the carbon and nitrogen abundance, which is particularly important given that we seek [N/O] predictions as a classifier of 2P stars. We tested predicting [N/Fe] and [O/Fe] as separate output features rather than in a combined [N/O] prediction and found a negligible difference in performance. \nRegarding aluminum, which is especially difficult to measure from stellar spectra, we discuss in Section 3.3 that the [Al/Fe] predictions of our network may in fact be closely linked to Na abundances. Hattori (2024) produces 𝛼 -element predictions from BP/RP spectra which they find arise from information near the Na D absorption lines at 589 and 589.6 nm. They further indicate that some 𝛼 -element information arises from the Mg I line at 516 nm, which is consistent with our findings that our [Al/Fe] predictions are distinct from [ 𝛼 /Fe] values (discussed in Section 3.3). As Na is also enhanced in 2P GC stars (for example, see Milone & Marino 2022a; Carretta et al. 2009b,a), Na information in our [Al/Fe] predictions would nonetheless be useful for identifying new 2P candidates. We cannot discount the possibility that our [Al/Fe] predictions may in part arise from a series of complicated correlations with other abundances and spectral features; we address this possibility throughout the paper.", '2.2 Data-Driven Identification of High-[N/O] Stars': 'Our machine learning model is a multi-layer perceptron (MLP) a type of neural network - which we parameterise using PyTorch (Paszke et al. 2019). Our MLP has an input layer with 108 nodes, each corresponding to one of the BP/RP coefficients, two hidden layers of 128 nodes each with rectified linear unit (ReLU) activation functions, followed by an output layer. Each hidden layer has a dropout fraction of 0.2 initialized, wherein 20% of the nodes are randomly not used in each forward pass. Dropout (Hinton et al. 2012) is a common technique used to prevent overfitting to the training data for neural networks, and our dropout rate of 0.2 was selected after testing rates within the range 0.05 to 0.2 and examining the training and validation loss curves during network training to prevent overfitting. Per Gal & Ghahramani (2015), we use dropout during evaluation as well as training and make 100 predictions per BP/RP spectrum. These individual predictions, which can be thought of as samples from the distribution of predictions, are then averaged for the final inferred abundances and variances used to classify 2P-type candidate stars. This technique helps address uncertainty arising from the initialization of the MLP itself and also has been shown to improve network performance. Fallows & Sanders (2024) employ the same technique and provide a thorough discussion of model variance in their work. We use the Adam optimizer (Kingma & Ba 2014) with a learning rate of 0.001. All other Adam hyperparameters are set to the PyTorch defaults. We train with a batch size of 128. The learning rate and batch size were set following testing to minimize the loss on the validation set (with additional checks to ensure the false negative and false positive rates were not excessively high, see Section 3.1). \nHeteroscedastic regression is a form of regression that takes into account non-uniform variance among the data (Section 5.3.4 of Prince 2023); we elect to use this form of regression given that we naturally expect stellar parameters and abundances to have nonuniform variance (ex. for abundance measurements to have different variances based on 𝑇 eff , metallicity, etc.). To perform heteroscedastic regression, we use a loss proportional to the negative log-likelihood of a parameterised Gaussian, wherein for each batch in the training the loss is: \nloss = ( 𝜇 -𝑦 true ) 2 𝜎 2 + log 𝜎 2 \nIn the loss function above, 𝜇 is the vector of the network\'s predicted values for each of the five parameters and abundances used as labels, log 𝜎 2 is the corresponding logarithm of the variances on each of those predictions, and 𝑦 true is the true value of each parameter or abundance. Note that the variances are inferred by the network and \nFigure 2. Sample Gaia BP/RP spectra, with [N/O]-normal ([N/O]=-0.066, grey), high-[N/O] ([N/O]=1.098, purple) and, [N/O] normal, high-[C/Fe] ([C/Fe]=1.3, dark green) stars with comparable 𝑇 eff and [Fe/H]. The Gaia IDs are 1454785210768079744, 2259240659843862016, and 2982933097213087616, respectively, with 𝑇 eff ≈ 5000 K, log 𝑔 ≈ 2 . 4, and [Fe/H] ≈ -1 . 35. The left panel shows the whole BP/RP spectra as converted from the basis coefficients with GaiXPy , and the right panel shows the magnified region of the high- and normal-[N/O] spectra containing the CN bands. The abundances of the carbon-rich star are from Yoon et al. (2016), while the other two were selected from APOGEE abundances. \n<!-- image --> \nnot given as part of the training labels. From here forward, we refer to the network\'s mean prediction of a parameter or abundance by 𝜇 value and correspondingly to the network\'s predicted variance on the value as 𝜎 2 value . For instance, we refer to the predicted [N/O] abundance as 𝜇 [N/O] and the corresponding variance as 𝜎 2 [N/O] .The loss function is such that when the residual is large, it dominates the loss; when the residual is low or the variance is particularly high, the log 𝜎 2 term dominates the loss. The consideration of the variance in the data within the loss function helps to decrease bias in predictions (i.e. a "regression towards the mean" wherein low values are over-predicted and high values are under-predicted). Moreover, we use the network\'s predicted variances to help determine which 𝜇 value predictions are reliable (see Section 3.1). Fallows & Sanders (2024) stellar parameter predictions from the BP/RP spectra utilized a comparable loss function. \nWe make note of the fact that a machine learning model\'s estimations of abundances and stellar parameters are different from actually \'measuring\' these values, as the models are sufficiently complex to learn complicated correlations between various features (i.e., a general correlation of N with metallicity or 𝛼 -element abundance, among many other features). This capability of machine learning approaches to learn correlations between features in the data may itself be a strength in the context of real relationships between abundances (i.e. the relationship between O and carbon molecular features discussed above), particularly given the low resolution of the BP/RP spectra with limited visible atomic and molecular features.', '2.3 Red Giant Sample': 'Because our model is trained solely on red giants from APOGEE, we likewise can only produce predictions for giants. Andrae et al. (2023) produce a reliable selection of RGB stars (see Table 2 of the referenced paper) to which we apply our neural network to make predictions of [N/O] and [Al/Fe] as well as new predictions of log 𝑔 , 𝑇 eff , and [Fe/H]. The Andrae et al. (2023) catalog also includes predictions for log 𝑔 , 𝑇 eff , and [Fe/H], but we include our own new \npredictions to maintain consistency among our data. This consistency is especially relevant given that Andrae et al. (2023) uses a very different machine learning model ( XGBoost , Chen & Guestrin 2016) as compared to our neural network. \nFor the MLP to provide reasonable predictions, it is essential that the data to which we apply the network is well represented in our training data. For that reason, we apply the same cuts from the Gaia catalog that were applied to our training and validation data in Section 2.1 and remove any stars that were in our training or validation data. Note, however, that Andrae et al. (2023) make the cut parallax / parallax\\_error > 4, which is slightly more restrictive than the cut we use in our training and validation data. We apply an additional cut on the log 𝑔 prediction produced by Andrae et al. (2023), log 𝑔 XGBOOST < 3 . 0, again with the goal of maintaining consistency with our training data from APOGEE. Andrae et al. (2023) already apply a cut of 𝑇 eff,XGBOOST < 5200 K, which is consistent with the cut we made on 𝑇 eff,APOGEE in the training and validation data. Our ability to rely on the XGBOOST predictions of 𝑇 eff and log 𝑔 to ensure consistency with our training data motivates our use of the vetted Andrae et al. (2023) RGB catalog for our selection of candidates. For additional cuts applied to the vetted RGB catalog, the reader is referred to Andrae et al. (2023). Finally, we apply an extra cut to remove spectra with high extinction with ebv < 1 (from Schlegel et al. 1998); although this cut is not included in the training data to maximize the number of samples we train with, stars with ebv < 1 are nonetheless well represented within the training and validation data. By contrast, high-extinction sources are relatively less well-represented in the training data, and to ensure high quality predictions, we thus remove these stars. Perhaps even more importantly, high extinction significantly impacts sensitivity in blue wavelengths, with E ( B -V ) = 1 . 0 corresponding to a decrease of about 3.5 magnitudes around 430 nm, near the CN features. For this reason, we construct a high-reliability sample with ebv < 0 . 2 in addition to the full catalog. With the exclusion of stars in our APOGEE training and validation data, we are left with 6 878 665 RGB stars in the full catalog to which we apply our MLP.', '3 NETWORK PERFORMANCE': 'In this section, we describe several tests that validate the results of our neural network. These tests include the performance of our network on the 20% of the APOGEE-BP/RP data that we withheld from training and performance checks on the selection of new candidates from the Andrae et al. (2023) RGB catalog and from a catalog of giants from GALAH (Buder et al. 2021; De Silva et al. 2015).', '3.1 Validation Data Performance': "Our test data is comprised of 39 816 stars with both BP/RP spectra and APOGEE abundances. We show the validation performance of our network in Fig. 3, separated by each of the five stellar parameters or abundances we predict. As is evident in the first three panels of Fig. 3, the predictions of log 𝑔 , 𝑇 eff , and [Fe/H] appear to well follow the 1:1 line with the true APOGEE values; this result is consistent with many previous works which find these fundamental stellar parameters to be well-predicted from the BP/RP coefficients through various data-driven approaches (e.g. Andrae et al. 2023; Fallows & Sanders 2024, among others). We note a small population of stars with over-predicted surface gravities at log 𝑔 ≈ 2, which could be related to the red clump. There is also a slight systematic overprediction of metallicities at [Fe/H] ≲ -1, with an average residual of + 0 . 07 dex for stars with [Fe/H] ≲ -1. The root mean squared error (RMSE) of 𝜇 𝑇 eff , 𝜇 log 𝑔 , and 𝜇 [Fe/H] are 62.4 K, 0.162, and 0.096, respectively. \nThe [N/O] and [Al/Fe] predictions also approximately follow the 1:1 lines with the APOGEE values well. However, examination of the residuals of these predictions in the bottom panels in Fig. 3 do reveal a bias towards the median (i.e. over-prediction of low values and under-prediction of high values). This behavior is consistent with the trends in [N/Fe] prediction discussed in Fallows & Sanders (2024). This trend is a natural behavior of many machine learning models and is unsurprising given the unbalanced nature of our training data wherein we have relatively few high-[N/O] stars. However, we argue that this bias is not exceedingly problematic for our use case given that we use the 𝜇 [N/O] and 𝜇 [Al/Fe] predictions only to produce a binary classification of stars as 2P-type or not 2P-type rather than directly as abundances, as we will discuss in more detail shortly. It is worth noting that the bias appears to be the most significant for the 𝜇 [Al/Fe] estimates, which may be related to the relative difficulty of identifying Al (or Na) information in the BP/RP spectra as compared to N or O (as discussed in Section 2.1). The RMSE of 𝜇 [N/O] is 0.096, and the RMSE of 𝜇 [Al/Fe] is 0.076. \nIn Fig. 4, we examine the MLP's output standard deviation of the [N/O] prediction, 𝜎 [N/O] , in the validation data as a function of 𝑇 eff , log 𝑔 , [Fe/H], [N/O], and the 𝜇 [N/O] residual. There are several trends in the value of 𝜎 [N/O] worth noting. First, 𝜎 [N/O] increases noticeably at higher temperatures ( 𝑇 eff ≳ 4700 K. This trend is likely physically motivated given that the CN bands grow weaker in the spectra of hotter stars. Second, 𝜎 [N/O] is also higher at [Fe/H] ≲ -1. We believe this behavior may be caused both by the fact that there are fewer metal-poor than metal-rich stars in our training data and also because the CN bands become less prominent in stars at lower metallicities (e.g., as recently shown in Carretta & Bragaglia 2024). 𝜎 [N/O] is consistently high for stars with [Fe/H] ≈ -2. Interestingly, the trend of 𝜎 [N/O] with the prediction residual in 𝜇 [N/O] is somewhat less distinct than the trends with temperature and metallicity, though stars with higher magnitudes of prediction residuals do tend to have slightly higher values of predicted 𝜎 [N/O] as well. This trend coincides with the higher values of 𝜎 [N/O] in stars with true [N/O] abundances that \nare well above or below the median value of [N/O] in our training and validation data. The behavior is typical of machine learning algorithms given that these values of [N/O] are comparably less represented in our training data as compared to stars with [N/O] ≈ 0 . 1, which drives both the residuals and uncertainties for these less common stars to be higher. Finally, there is a useful linear correlation between 𝜎 [N/O] and the reported [N/O] error from APOGEE, which is marked by the black line and which we utilize to make our selection of high-[N/O] candidates (as described in Section 3.2). \nInferred standard deviations of the 𝑇 eff , log 𝑔 , [Fe/H], and [Al/Fe] are included in Appendix B.", '3.2 High-[N/O] Candidate Selection': 'We use the following criteria to select high-[N/O] candidate stars using our network predictions: \n- · 𝜇 [N/O] -0 . 19 × 𝜎 [N/O] > 0 . 65\n- · 𝜇 [Al/Fe] -𝜎 [Al/Fe] > 0 .\n- · 𝜇 𝑇 eff < 5000\n- · 𝜇 [Fe/H] > -2 . 0 \nThe cut on predicted temperature has two motivations, one physical and one empirical. First, it is known that the CN bands grow weak or disappear altogether at 𝑇 eff ≳ 5000 K, and since we believe that most of the nitrogen information in the BP/RP spectra comes from those CN bands, there is good reason to exclude stars with higher temperatures from our sample of high-[N/O] candidates. Second, our network predicts increasing 𝜎 [N/O] values at high 𝑇 eff (Fig. 4), suggesting greater uncertainties in the [N/O] predictions at higher temperatures. This result is unsurprising given the behavior of the CNbandsat higher temperatures and further justifies the exclusion of warm giants from our high-[N/O] candidates. The cut on metallicity as part of our criteria for high-[N/O] candidates is motivated by the metallicity distribution of the training APOGEE-BP/RP cross-match, which contains very few stars with [Fe/H] < 2 . 0. As compared to the rest of the stars in the training data, proportionally more of these very metal-poor stars were high-[N/O] stars (as is also noted in Belokurov &Kravtsov 2023), biasing the neural network to predict high 𝜇 [N/O] values for very metal-poor stars. We felt for this reason that our [N/O] predictions for stars with [Fe/H] < -2 . 0 may be unreliable and thus exclude them from our sample of high-[N/O] candidates. \nThe [N/O] selection at 𝜇 [N/O] -0 . 19 × 𝜎 [N/O] > 0 . 65 takes advantage of the observed correlation between 𝜎 [N/O] and the APOGEE [N/O] error in Fig. 4. By approximating the relation between 𝜎 [N/O] and the APOGEE [N/O] error as linear, we take the slope of the best-fit line (shown in Fig. 4), 0.19, as a factor for 𝜎 [N/O] in our cut; thus, 𝜇 [N/O] -0 . 19 × 𝜎 [N/O] is intended to be approximately analogous to APOGEE [N/O] -[N/O] error. One high outlier in 𝜎 [N/O] was excluded in the calculation of the relation between 𝜎 [N/O] and the APOGEE [N/O] error. We note that the thresholds in 𝜇 [N/O] and 𝜇 [Al/Fe] that we use to select high-[N/O] candidates are slightly higher than the cuts we use to make the true classification from APOGEE abundances (as described in 2.1). These higher thresholds are motivated by the fact that our [N/O] and [Al/Fe] predictions, although robust, do sometimes have errors of a few tenths of a dex, and our goal is to construct a pure, reliable sample of new high[N/O] candidates. Thus, to maintain a low false positive rate in our selection of new candidates, we use more stringent cuts on 𝜇 [N/O] and 𝜇 [Al/Fe] to select stars with true abundances of [N/O] > 0 . 55 and [Al/Fe] > -0 . 1. These cuts were selected after testing with our validation data to maintain a balance of a low contamination rate without losing most or all of the true 2P stars from our classification. \nFigure 3. Moving from left to right and top down, the validation performance of the network predictions for 𝜇 𝑇 eff , 𝜇 log 𝑔 , 𝜇 [Fe/H], 𝜇 [N/O], and 𝜇 [Al/Fe]. The large upper panel of each subplot depicts the prediction versus the true value from APOGEE with the 1:1 line marked in black. The narrow lower panel is the prediction residual (i.e., 𝜇 𝑣𝑎𝑙𝑢𝑒 -𝑣𝑎𝑙𝑢𝑒 𝐴𝑃𝑂𝐺𝐸𝐸 ) with the horizontal black line marking a residual of zero. The colorbar is shared for all panels and marks the color-mapping of the histograms as the log-scaled number of stars per pixel. In the panels depicting the [N/O] and [Al/Fe] predictions, the vertical lines depict the true cutoff values we use from APOGEE abundances to classify stars as high-[N/O] at [N/O] = 0 . 55 and [Al/Fe] = -0 . 1, respectively. The horizontal red dashed lines represent the cutoffs we use in our predicted values of 𝜇 [N/O] and 𝜇 [Al/Fe] at 0.65 and 0.0, respectively, as is described in Section 3.1; note that these lines do not account for 𝜎 [N/O] and 𝜎 [Al/Fe], which are also used in our selection cuts. \n<!-- image --> \nFigure 4. From top down, the MLP standard deviation prediction of [N/O] ( 𝜎 [N/O]) versus APOGEE values for 𝑇 eff, log 𝑔 , [Fe/H], [N/O], the residual of the 𝜇 [N/O] prediction in the validation dataset, and ebv .from Schlegel et al. (1998) The colorbar is shared for all panels and marks the color-mapping of the histograms as the log-scaled number of stars per pixel. The black line in the fourth row depicts the best-fit line relating the APOGEE [N/O] error to 𝜎 [N/O], APOGEE [N/O] error = 0 . 19 × 𝜎 [N/O] + 0 . 013. \n<!-- image --> \nFigure 5. The confusion matrix of classifications of stars in the validation dataset, with true labels on the vertical axis generated from APOGEE abundances (as described in Section 2.1) and the predicted labels made from the network predictions as described in Section 3.2. The 2P-type label refers to high-[N/O], high-[Al/Fe] stars consistent with having formed in the second generation of GCs, while the not 2P-type label refers to all other stars. \n<!-- image --> \nApplying these criteria to our validation data, we classify stars as either "2P-type" or "not 2P-type" with over 99% accuracy. The confusion matrix showing the results of our classification in the validation data is presented in Fig. 5. 103 stars are classified as having 2P chemistry with a 6.8% false positive rate. We note, however, that due to the rarity of 2P stars in the Galaxy and thus within our validation data, this false positive rate corresponds to only 7 out of 39 468 non-2P-type stars being misclassified as stars with 2P [N/O] and [Al/Fe] abundances. Inspection of these 7 false positive classifications in the validation data reveal that 3 have APOGEE [N/O] > 0 . 55 but have errors sufficiently high such that they do not satisfy the condition [N/O] -[N/O] error > 0 . 55. The remaining 4 false positives all have [N/O] abundances close to the threshold, with all having reported APOGEE [N/O] > 0 . 49. It appears that these stars are very nitrogen-rich but not sufficiently oxygen-depleted to satisfy the [N/O] requirement. All false positives clear the [Al/Fe] threshold at -0.1. We note that carbon overabundance does not appear to be a source of false positives, as the mean [C/Fe] of the false positive classifications in the validation data is -0.16. This result is reassuring given that we believe the main source of nitrogen information in the BP/RP spectra comes from the CN bands, with the NH band barely being in the range of the Gaia Blue Photometer. \nNotably, our classification of 2P-type stars in the validation data has a somewhat substantial false negative rate, with 72% of true 2P-type stars being misclassified as not 2P-type. Some of these misclassifications arise not only from 𝜇 [N/O] and 𝜇 [Al/Fe] predictions inconsistent with our selection criteria but also from the exclusion of warm or very metal-poor giants from our classification of 2P-type stars; naturally, some true 2P-type stars will be hotter than 5000 K or more metal-poor than [Fe/H] = -2 . 0. Our false negative rate is approximately consistent with that of the identification of carbonenhanced metal-poor (CEMP) stars from the BP/RP spectra in Lucey et al. (2023). The false negative rate is a necessary side effect of maintaining high purity in our candidate selection, and given the rarity of known 2P-type stars in the Galactic field, even with this false negative rate our sample of candidates is multiple times larger than the comparable sample that can be compiled from APOGEE data (see Section 3). Thus, since we nonetheless construct a large sample \nFigure 6. The figure shows the sodium versus oxygen abundances of stars in our GALAH dataset. The background gray histogram represents the overall distribution of [Na/Fe]-[O/Fe] abundances among GALAH stars with [Fe/H] < -0 . 5, with metal-poor stars selected for a more realistic comparison with GCs, which tend to be metal-poor. The grey contours marking 30%, 50%, and 90% contours. The small orange triangles mark members of GCs in the GALAH data (Vasiliev & Baumgardt 2021). The purple circles represent stars that we identify as high-[N/O] candidates, and the large purple triangles represent stars that are both high-[N/O] candidates and GC members. \n<!-- image --> \nof candidates, we find having a high purity in our selection to be a worthwhile trade-off for the false negative rate. However, we will publish our full catalog of predictions, enabling the use of less stringent candidate selections by the community if desired. The catalog of our predictions is detailed in Appendix A.', '3.3 Additional Tests of High-[N/O] Candidates': "In addition to testing our neural network on validation data from APOGEE, we perform several additional tests to confirm the reliability of the sample of our new high-[N/O] candidates. First, we identify stars from Andrae et al. (2023) that are in GCs according to Vasiliev & Baumgardt (2021) and compare the high-[N/O] fraction among these cluster members to the full set of field giants. We find that 9.82% of GC members are classified by our algorithm as high[N/O] stars as compared to just 0.16% of field giants in the halo-a difference of well over a magnitude. This result is consistent with the expectation that GCs should contain many more 2P stars (ex. Milone et al. 2017), which are very rare in the field. These fractions are likely to be artificially low due to the relatively high false positive rate in our classification; approximately 72% of true 2P stars in our validation data were misclassified as not high-[N/O]. For comparison, if we make classifications for 2P-type stars with APOGEE abundances in our combined training and validation data, we find that 31.5% of giants in GCs are classified as 2P-type as compared to just 0.43% in the halo. \nOur second test of the predictive power and accuracy of our network utilizes the GALactic Archaeology with HERMES (GALAH, Buder et al. 2021; De Silva et al. 2015; Kos et al. 2017; Zwitter et al. 2021) survey, which provides abundances for several light elements \nassociated with the GC abundance anti-correlations, including Na, O, Al, Mg, and C. To further validate our candidate selection, we select stars from GALAH that have been identified as GC members in Vasiliev & Baumgardt (2021) and explore their light element abundances. We apply several cuts to the data, beginning by selecting giants by using the cuts 𝑇 eff < 5200 K and log 𝑔 < 3. Stars with potentially erroneous stellar parameters are removed via the cut flag\\_sp = 0, problematic metallicity measurements removed with flag\\_fe\\_h = 0, and high signal-to-noise observations are selected with snr\\_c3\\_iraf > 30. When other element abundances are used, we also apply a cut for flag\\_X\\_h = 0. Each of these criteria is used at the recommendation of the GALAH collaboration's best practices 3 . We also apply the same cuts from the Gaia that we describe in Section 2.1, including a cut for ebv < 0 . 2 for consistency with our high-reliability sample of candidates, and by necessity, our sample is limited to stars with BP/RP spectra available from Gaia DR3. Stars that are present in our training or validation data are removed. After these cuts, we have 30 604 stars with both GALAH data and BP/RP spectra. \nIn Fig. 6, we show the [Na/Fe]-[O/Fe] abundance patterns of stars in our GALAH sample. The distribution of the overall population of giants is depicted in the 2D histogram in the background with GC membergiants (Vasiliev & Baumgardt 2021) over-plotted on top. The light element anti-correlation of GC members is evident, with a group of GC member(1G)stars falling within the regime of typical GALAH stars. Another group of GC member stars (2P) is clearly outside the limits of the typical GALAH stars in the [Na/Fe]-[O/Fe] plane, with evident Na-enhancement and O-depletion relative to GC 1G stars, as is expected for GC 2P stars (e.g., as described in the review in Milone &Marino 2022a). We then use the BP/RP spectra of all stars in this sample to identify high-[N/O] candidates from our GALAH sample using the same procedure outlined in Section 3.1. We note that our candidates generally populate the Na-enhanced, O-depleted region of the abundance space, again as one would expect of second generation GCstars. The fraction of GC members in GALAH identified as high[N/O] candidates, 12.8%, is much higher than the overall fraction of high-[N/O] candidates that we identify in GALAH stars that are not GC members in Vasiliev & Baumgardt (2021), just 0.046%. As in our test of GC members above, this result is consistent with the fact that GCs ought to contain more high-[N/O] and high-[Al/Fe] stars than the MW field. We recognize that there are many 2P GC stars that our algorithm does not tag as high-[N/O], which is consistent with our recovery of high-[N/O] stars having a false negative rate of approximately 72%. When we relax the cuts for candidate selection for this sample to 𝜇 [N/O] > 0 . 55 and 𝜇 [Al/Fe] > -0 . 1, we find that we recover most of the apparent 2P stars from this figure as well as many more field giants in that region of the [Na/Fe]-[O/Fe] plane. However, with those relaxed selection cuts we also tag a small but noticeable number of field stars outside the Na-enriched regime of the abundance space which we assume are false positives, which is consistent with our tests in Section 3. We find our original cuts at 𝜇 [N/O] > 0 . 65 and 𝜇 [Al/Fe] > 0 . 1 to thus be well-suited to this test, too, although this result may suggest that many more 2G stars can be found in the field from our predicted abundances if one relaxes the selection cuts from those we use here. \nFinally, Al abundances in the optical rely on only three lines, one of which (669.8 nm) can appear quite weak in metal-poor stars. Thus, because aluminum abundances are notoriously difficult to derive from spectra and because Gaia BP/RP spectra are so low resolution, \nFigure 7. Moving from left to right and beginning with the top row, GALAH [Al/Fe], [Fe/H], [ 𝛼 /Fe], [Na/Fe], [Mg/Fe], and [Ca/Fe] versus 𝜇 [Al/Fe]. The GALAH [Al/Fe] versus 𝜇 [Al/Fe] has the 1:1 line depicted in red, and the GALAH [Na/Fe] versus 𝜇 [Al/Fe] has a vertical red line at 𝜇 [Al/Fe] = -0 . 04 to depict the cutoff between low- and high-Na stars in the abundance plane. Panels are log-scaled in density and share a colorbar. \n<!-- image --> \nFigure 8. GALAH [Na/Fe] versus 𝜇 [Al/Fe] for GALAH [Fe/H] < -0 . 5 (left) and[Fe/H] > -0 . 5 (right). Both panels are log-scaled in density and share a colorbar. The red dashed line in the left panel is the best-fit line relating 𝜇 [Al/Fe] and GALAH [Na/Fe]. \n<!-- image --> \nthe 𝜇 [Al/Fe] predictions demand further examination. We again use GALAH abundances to further explore the nature of the 𝜇 [Al/Fe] predictions; in particular, we are looking at whether the aluminum predictions of our neural network are correlated with other similar elements, in spite of the relatively good agreement between our 𝜇 [Al/Fe] and the APOGEE [Al/Fe] measurements in Fig. 3. In Fig 7, we examine 𝜇 [Al/Fe] as a function of [Al/Fe], [Fe/H], [ 𝛼 /Fe], [Na/Fe], [Mg/Fe], and [Ca/Fe] from GALAH. The correlation between 𝜇 [Al/Fe] clearly does not follow the 1:1 line with the GALAH [Al/Fe] measurement as well as is seen in our validation data, though this behavior can be explained given that GALAH aluminum abundances are known to differ from those reported by APOGEE (see Buder et al. 2022). \nAs is apparent in the top right panel of Fig 7, there is a correlation between 𝜇 [Al/Fe] and [ 𝛼 /Fe], and this is again likely to be expected as aluminum is predominantly produced in core-collapse supernovae (Kobayashi et al. 2020) and thus can be expected to generally correlate with 𝛼 -element abundances. Likewise, there is some relation between metallicity and our 𝜇 [Al/Fe] prediction, but the nonmonotonicity of that relation suggests that our neural network is not purely correlating one with the other. Interestingly, there are clearly multiple tracks in 𝜇 [Al/Fe] as a function of metallicity. \nThe correlation of 𝜇 [Al/Fe] with [Mg/Fe] and [Ca/Fe] is present but less distinct, which is again perhaps a result of the fact that all of these elements share a nucleosynthetic source in Type II supernovae. However, in the [Na/Fe]𝜇 [Al/Fe] plane (bottom left panel of Fig 7), there is a distinct and valuable pattern. By placing a cut at 𝜇 [Al/Fe] = 0, as we illustrate with the vertical red line, stars are very clearly divided into low- and high-Na groups. This clean separation is not possible with any of the other elements that we examine with GALAH and suggests that there may in fact be information on [Na/Fe] in the 𝜇 [Al/Fe] values. We further examine the relation between 𝜇 [Al/Fe] and [Na/Fe] for metal-poor stars ([Fe/H] < -0 . 5) and metal-rich stars ([Fe/H] > -0 . 5) in Fig. 8. We find that 𝜇 [Al/Fe] particularly wellpredicts Na abundance for metal-poor stars, though the relationship is less strong for metal-rich stars; given that most 2P-type stars, including our candidates, tend to be metal-poor, this is a positive result. Moreover, if the cut on 𝜇 [Al/Fe] is removed for candidates, three more candidates, seemingly false positives, appear in Fig. 6; two have [Na/Fe] < 0, further suggesting that there is Na information in the 𝜇 [Al/Fe] predictions. It has been previously observed that Al and Na abundances trace each other in GC light element anticorrelations (e.g. Carretta et al. 2010, among many others), indicating that this may be a useful and sensible result. \nIn short, we believe that there is likely information on several elements, especially 𝛼 -elements, in our 𝜇 [Al/Fe] predictions. In particular, we find that a cut at 𝜇 [Al/Fe] = 0 cleanly separates stars into low- and high-[Na/Fe] populations in data from GALAH. The cor- \nbetween 𝜇 [Al/Fe] and GALAH [Na/Fe] is especially evident at [Fe/H] < -0 . 5. Given that most of our high-[N/O] stars exist at lower metallicities and that Na-enhancement is also present in 2P stars, it is perhaps unsurprising that our cut for high 𝜇 [Al/Fe] values helps to select 2P stars. In validation testing, including the cut of 𝜇 [Al/Fe] -𝜎 [Al/Fe] > 0 . removes an additional 6 false positive stars, and with numbers of high-[N/O] stars so low, this removal of 6 false positives is relevant to our contamination rate. Moreover, as we discuss in Section 4.1, these 𝜇 [Al/Fe] predictions even produces a sensible, if imperfect, separation between in-situ and accreted stars.", '4 PROPERTIES OF GALACTIC HIGH-[N/O] CANDIDATES': "In this section, we examine the properties of our high-[N/O] candidates. We are particularly interested in the distribution and kinematics of our high-[N/O] candidates within the Galactic halo, which allows us to explore the contribution of GCs to the early Galaxy (Belokurov & Kravtsov 2022; Rix et al. 2022; Conroy et al. 2022). To remove thin disk stars, which are more likely to be N-enhanced from sources other than an origin in GCs (Belokurov & Kravtsov 2023), we convert Gaia astrometric data into spherical Galactocentric positions and velocities, and we apply a cut at 𝑣 𝜙 < 160 km/s. Radial velocities are from Gaia , and the distances are calculated from parallaxes. We use Astropy 's (Astropy Collaboration et al. 2013, 2018, 2022) default Galactocentric reference frame, which has a Solar distance of 8.122 kpc from the Galactic center and a Solar velocity of (12.9, 245.6, 7.78) km/s in Cartesian Galactocentric coordinates. After applying the selection for halo stars and a cut at ebv < 0 . 2 in addition to the selection criteria for high-[N/O] candidates that we outline in Section 3.1, we identify 878 new high-[N/O], high-[Al/Fe] candidates in the Galactic halo that are not associated with a GC in Vasiliev & Baumgardt (2021). This low extinction selection constitutes our high-reliability sample and is shown in Fig. 9. If all 6 878 665 RGB stars with ebv < 1 from (Andrae et al. 2023) are used, we identify 1 432 new high-[N/O], high-[Al/Fe] candidates in the Galactic halo. We publish this full catalog but encourage caution regarding predictions for highly extincted stars due to the decreased sensitivity at blue wavelengths, and for the remainder of this paper we use our high-reliability, low extinction sample. By contrast to our 878 new candidates in the low extinction sample, our selection of APOGEE giants has only 133 high-[N/O] stars with 𝑣 𝜙 < 160 km/s that are not associated with a GC (using the cuts [N/O] -[N/O] error > 0 . 55 and [Al/Fe] > -0 . 1, as specified in Section 2.1). Thus, our candidate sample is over six times larger than the existing comparable catalog of known high-[N/O] stars from APOGEE.", '4.1 Spatial and Kinematic Distribution of High-[N/O] Candidates': 'In the left panel of Fig. 10, we explore the distribution of our high[N/O] candidates in the total orbital energy and vertical component of the angular momentum, 𝐸 -𝐿 𝑧 space. Energies are calculated using the MilkyWayPotential from gala (Price-Whelan 2017; Price-Whelan et al. 2020; Bovy 2015). [Al/Fe] abundances have previously reliably been used to separate stars that formed in situ in the Galaxy from those that were accreted with other structures (Hawkins et al. 2015), with the MW\'s satellites having consistently lower aluminum abundances than in-situ stars (Hasselquist et al. 2021). With this in mind, we attempt to use our 𝜇 [Al/Fe] predictions to separate accreted and in-situ stars. We impose rather strict cuts on 𝜇 [Al/Fe] to attempt a clean selection, motivated by the higher bias \nFigure 9. In red points, the distribution of high-[N/O] and high-[Al/Fe] candidates in Galactic latitude and longitude. The grey 2D histogram is the log-scaled distribution of all stars not in GCs that pass the cut at 𝑣 𝜙 < 160 km/s. Note that as a result of this cut and the extinction cut, low Galactic latitutdes are excluded. \n<!-- image --> \nin the 𝜇 [Al/Fe] prediction noted in Section 3.1 and the notable but imperfect trend with [Na/Fe] observed in Fig. 8. Thus, low-aluminum "accreted" stars are selected via 𝜇 [Al/Fe] < -0 . 2, and high-aluminum " in-situ stars" are selected to have 𝜇 [Al/Fe] > 0 . 2. We also impose a cut on metallicity, -1 . 4 < 𝜇 [Fe/H] < -1 . 1, as this is the metallicity range in which accreted and in-situ stars can be best separated by aluminum abundances (Belokurov & Kravtsov 2023). We maintain the cut at 𝜇 𝑇 eff < 5000 for both samples and make the same velocity cuts that we applied to our high-[N/O] candidates to remove thin disk stars. The black dashed line marks the empirical separation of accreted and in-situ ( Aurora ) stars developed in Belokurov & Kravtsov (2023), which is: \n𝐸 = -1 . 316 : 𝐿 𝑧 < -0 . 58 𝐸 = -1 . 416 + 0 . 3 𝐿 2 𝑧 : -0 . 58 < 𝐿 𝑧 < 0 . 58 𝐸 = -1 . 341 + 0 . 075 𝐿 2 𝑧 : 𝐿 𝑧 < 0 . 58 \nwith 𝐿 𝑧 in units of 10 -3 kpc km / s and 𝐸 in units of 10 -5 km 2 / s 2 . The energy values are shifted by -0 . 016 × 10 -5 km 2 / s 2 to account for the different energies of gala \'s MilkyWayPotential and the Milky Way model used in Belokurov & Kravtsov (2023). This separation of accreted and in-situ stars was developed using the observed differences in APOGEE [Al/Fe] abundances between accreted and in-situ stars. Interestingly, other chemical elements also show strong differentiation across this 𝐸 -𝐿 𝑧 divide. For example, Monty et al. (2024) demonstrate that both field stars and MW Globular Clusters display distinct levels of [Si/Fe], [Eu/Fe] and [Eu/Si] on either side of the boundary. We primarily use this orbital separation as a means with which to judge the efficacy of our identification of accreted and in-situ stars using 𝜇 [Al/Fe] . \nNotably, our selection of stars with high 𝜇 [Al/Fe] predictions appears to be a fairly clean sample of in-situ stars, as the contours of this sample lie almost entirely within the kinematic selection in the 𝐸 -𝐿 𝑧 space. This behavior is especially remarkable given the aforementioned difficulty of measuring aluminum spectroscopically and reinforces the suggested [Al/Fe] and [Na/Fe] information in the 𝜇 [Al/Fe] , both of which are useful to distinguish accreted and insitu stars (Hawkins et al. 2015; Nissen & Schuster 2010; Das et al. 2020). By contrast, the selection of low 𝜇 [Al/Fe] "accreted" stars is \nFigure 10. Left: The separation of low𝜇 [Al/Fe] accreted and high𝜇 [Al/Fe] in-situ ( Aurora ) field stars from our sample in energy𝐿 𝑧 space. The orange and blue contours depict the 20%, 50%, and 80% contours of the low𝜇 [Al/Fe] and high𝜇 [Al/Fe], and the black dashed line marks the kinematic separation adopted from Belokurov & Kravtsov (2023). The grey 2D histogram is the distribution of high-[N/O] candidates in the E𝐿 𝑧 space. Metallicities are restricted to -1 . 4 < 𝜇 [Fe/H] < -1 . 1, where separation of accreted and in-situ stars based on [Al/Fe] is most effective. Right: The frequency of high-[N/O] candidates (black solid line), high𝜇 [Al/Fe] (blue dashed line), and low𝜇 [Al/Fe] (orange dash-dotted line) stars in the halo as a function of orbital energy. The same [Fe/H] restrictions are used. \n<!-- image --> \nFigure 11. Ratio of high-[N/O] candidates to total number of stars in the halo (black line), in-situ or Aurora stars (dashed blue line), and accreted stars (dashed orange line) as a function of distance from the Galactic center in spherical Galactocentric coordinates. The shaded regions indicate the statistical uncertainties. We maintain our selection for -1 . 4 < 𝜇 [Fe/H] < -1 . 1. \n<!-- image --> \nnotably more contaminated than the high 𝜇 [Al/Fe] selection, although the contours do fall above the line of separation more than the insitu stars. We conclude that the 𝜇 [Al/Fe] values contain information of abundances that are useful to separate accreted and in-situ stars, albeit imperfectly. It is especially reassuring that the in-situ selection appears to be pure, as this result suggests that the stars identified as high-[Al/Fe] by our network are actually high-[Al/Fe], as is typical of in-situ MWstars. Given that we use a selection for high-[Al/Fe] to identify 2P candidates, this result can be taken as yet another form of validation of our selection. The apparent contamination of the "low[Al/Fe], accreted" stars is less concerning in this context as it will not affect the purity of our selection of high-[Al/Fe] 2P candidates; the fact that high-[Al/Fe] stars are sometimes tagged as low-[Al/Fe] may be related to the false negative rate discussed in Section 3.2. \nFinally, the distribution of high-[N/O] candidates, shown by the grey 2D histogram, clearly falls mostly among the in-situ regime of the 𝐸 -𝐿 𝑧 space defined both by the kinematic separation and the high 𝜇 [Al/Fe] contours. This result is consistent with the previous findings of Belokurov & Kravtsov (2023) that most high-[N/O] field stars from APOGEE are members of Aurora . The connection between the high-[N/O] candidates and selected in-situ stars can be further observed in the right panel of Fig. 10, which shows the energy distributions of the high-[N/O], in-situ , and accreted candidates. There is a clear difference in the energy distribution of the in-situ and accreted candidates, with the high 𝜇 [Al/Fe] stars tending to have lower energies than the low 𝜇 [Al/Fe] stars, which is consistent with the expected behavior from in-situ and accreted stars, respectively. Notably, the high-[N/O] candidate energy distribution closely traces that of the in-situ candidates, with many of the candidates having lower orbital energies than the accreted stars. This trend is again indicative of the in-situ and high-[N/O] populations being linked \nand is also in accordance with results from Belokurov & Kravtsov (2023). We perform a 2-sample Kolmogorov-Smirnov (K-S) test to compare the energy distributions of the high-[N/O] candidates with the in-situ and accreted distributions and find that the high-[N/O] and in-situ energy distributions are approximately consistent with having been drawn from the same populations, producing a p-value of 0.455 and a test statistic of 0.037. By contrast, the K-S test results of the high-[N/O] and accreted energy distributions indicate that these two groups differ significantly, producing a p-value of 1 . 5 × 10 -101 and a test statistic of 0.380. The results of the K-S tests further suggest a close association between the high-[N/O] candidates and in-situ stars. \nWe examine the radial distribution of the high-[N/O] candidates within the Galaxy in Fig. 11. We see the expected increase in the fraction of high-[N/O] stars with decreasing Galactocentric radius in the halo as a whole as noted in Horta et al. (2021); Schiavon et al. (2017); Martell et al. (2011); Belokurov & Kravtsov (2023). Our false negative rate results in systematically lower fractions of high-[N/O] stars as compared to those quoted in Horta et al. (2021); regardless, the general trend recovered is similar. The central concentration of the 2P-type candidates is also evident in the sky distribution in Fig. 9. \nFig. 11 also shows the ratio of high-[N/O] candidates to accreted and in-situ stars, which were classified according to the chemical criteria described above. We notice an increasing trend in the fraction of high-[N/O] candidates to accreted stars, with the fraction growing by a factor of ≈ 10 from 𝑟 ≈ 10 kpc to 𝑟 ≈ 1 . 25 kpc. The large increase in this fraction at small Galactocentric radii is in agreement with the results from Belokurov & Kravtsov (2023), although they found that the fraction increased by a factor of ≈ 40. We believe that the rate of the increase in the fraction of high-[N/O] stars relative to the accreted candidates approaching the Galactic center is more similar to the fraction relative to the halo overall due to the considerable contamination in our selection of accreted stars, as is evident in the right panel of Fig. 10. This contamination makes our sample of accreted candidates more similar to the halo as a whole rather than being a distinct subgroup, as is the case with the accreted stars selected via APOGEE abundances in Belokurov & Kravtsov (2023). As a result, the trend of 2P-type candidates to accreted halo stars does closely resemble that of the ratio relative to the halo overall. We recover a flatter trend in the ratio of high-[N/O] candidates to in-situ stars, as noted by Belokurov & Kravtsov (2023). The more similar radial distribution of 2P-type candidates and in-situ halo stars further indicates that these candidates are predominately members of Aurora . As we noted in Fig. 10, the in-situ selection with 𝜇 [Al/Fe] appears mostly pure, suggesting that this result is particularly robust. \nWith regards to all radial distributions, our selection criteria for 2Ptype, in-situ , and accreted stars differ from those used by Belokurov & Kravtsov (2023) by necessity to maintain relatively reliable selections. For these reasons, the fractions of high-[N/O] stars that we recover relative to the halo, it-situ halo, and accreted halo cannot be directly compared to the results of that work, although the trends are nonetheless very similar.', '4.2 Metallicity Distribution of the High-[N/O] Candidates': 'We now discuss the metallicities ( 𝜇 [Fe/H] ) of our high-[N/O] field giants. We show the metallicity distribution of our candidates in the right panel of Fig. 12 and the corresponding high-[N/O] fraction as a function of metallicity in the left panel. The high-[N/O] candidates are comprised of a larger fraction of metal-poor stars as compared to the overall population of RGB stars. As a consequence, the high-[N/O] fraction drops with increasing metallicity, \nwhile remaining at a low rate by [Fe/H] ≈ -0 . 8. For comparison, we also show the high-[N/O] fraction as a function of metallicity using APOGEE abundances for stars in our training and validation data. We note that using Gaia BP/RP spectrophotometry we recover a trend extremely similar to that in Belokurov & Kravtsov (2023), who also use APOGEE data. "Spin-up," the formation of the MW disk (Belokurov & Kravtsov 2022), is marked with the horizontal gray line in Fig. 12. By the metallicity of spin-up ([Fe/H] ≈ -0 . 9, the high-[N/O] fraction is much lower than it was at lower metallicities. \nWe echo the interpretation of Belokurov & Kravtsov (2023) regarding these results. Namely, the production of high-[N/O] stars, which currently consensus suggests is a yet-unknown process taking place between 1G and 2P stars in GCs, was much more prevalent in star formation at lower metallicities. Thus, at lower metallicities and hence further in the MW\'s past, GCs contributed a much higher fraction of stellar production in the Galaxy. Around the time of spinup, that contribution, traced by the high-[N/O] fraction, dropped precipitously.', '5 CONCLUSIONS': 'We use a multi-layer perceptron neural network to perform heteroscedastic regression on the Gaia BP/RP spectra. Our MLP takes only the BP/RP coefficients as input and predicts stellar log 𝑔 , 𝑇 eff , [Fe/H], [N/O], and [Al/Fe]. We use the 𝑇 eff , log 𝑔 , [N/O], and [Al/Fe] predictions to classify candidates with chemistry typical of GC second generation stars (e.g. nitrogen and aluminum overabundance and oxygen depletion, or "high-[N/O] stars" in our phrasing). We show that our predictions of 𝜇 𝑇 eff , 𝜇 log 𝑔 , 𝜇 [Fe/H] , 𝜇 [N/O] , and 𝜇 [Al/Fe] are robust as compared to baseline APOGEE values in our validation data (Fig. 3) and that by selecting stars with 𝜇 [N/O] -0 . 19 × 𝜎 [N/O] > 0 . 65, 𝜇 [Al/Fe] -𝜎 [Al/Fe] > 0 . , 𝜇 𝑇 eff < 5000, and 𝜇 [Fe/H] > -2 . 0, we can produce a pure sample of high-[N/O] candidates from their BP/RP spectra. We further validate this selection using GALAH DR3 data by comparing our predicted high-[N/O] fraction in the field to the high[N/O] fraction in GCs, finding a much higher fraction of high-[N/O] stars among cluster members, as expected. \nFrom our selection, we identify 878 new field stars in the Galactic halo as high-[N/O] candidates, constituting a sample over six times larger than can be made from APOGEE. We use these new candidates to study the properties of high-[N/O] stars in unprecedented detail. We summarize our findings as follows: \n- (i) Our high-[N/O] candidates are present among known GC members at a higher rate than in the Galactic field (Section 3.2), and examination of these candidates within the [Na/Fe]-[O/Fe] plane constructed from GALAH abundances shows that they generally fall within the Na-enhanced, O-depleted regime, as is consistent for 2P stars. These results serve to validate our candidate selection.\n- (ii) Exploration of the 𝜇 [Al/Fe] prediction with GALAH abundances reveal that this prediction can be used to separate stars into low- and high-Na groups. Al has previously been found to trace Na in GCs (Carretta et al. 2010), with both being useful to separate stars into accreted and in-situ (Hawkins et al. 2015; Nissen & Schuster 2010; Das et al. 2020). We thus use 𝜇 [Al/Fe] to make this separation in the 𝐸 -𝐿 𝑧 plane in Fig. 10, finding that a relatively clean separation is possible between the accreted, low𝜇 [Al/Fe] and in-situ , high𝜇 [Al/Fe] stars. The selection of in-situ stars appears to be particularly pure.\n- (iii) Within the 𝐸 -𝐿 𝑧 plane, the majority of the high-[N/O] candidates clearly appear to be associated with the in-situ population, which can further be confirmed via examination of their orbital energy and Galactocentric radial distributions (Fig. 10, 11). We also \n<!-- image --> \nFigure 12. Left: The fraction of high-[N/O] field stars in the halo as a function of metallicity. The orange dashed line shows the data from APOGEE using APOGEE metallicities, and the black line is constructed from our sample of candidates with 𝜇 [Fe/H] as the metallicity. The high-[N/O] fraction from our candidates is multiplied by a factor of 3.6, which is used to represent our projected false positive rate of 72% from the validation data. The shaded regions surrounding the trend lines indicate the statistical uncertainties; note that the uncertainties from our candidate data set are much smaller due to the larger sample size compared to APOGEE. The completion of spin-up, the formation of the Galactic disk, is marked by the vertical gray shading at -1 . 0 ≲ [Fe/H] ≳ -0 . 9 (Belokurov & Kravtsov 2022). Right: The normalized metallicity distribution using 𝜇 [Fe/H] of all giants in our sample (purple) and the candidate high-[N/O] stars (black outline). \n<!-- image --> \nfind the increasing trend in the high-[N/O] fraction in the halo with decreasing Galactocentric radius found by Horta et al. (2021); Martell et al. (2011); Schiavon et al. (2017). \n(iv) The fraction of high-[N/O] candidates in the halo drops above a metallicity of [Fe/H] ≈ -1, which is approximately concurrent with spin-up and is consistent with the findings of Belokurov & Kravtsov (2023) using APOGEE data. \nOur sample of high-[N/O], high-[Al/Fe] candidates represents the current largest collection of stars with chemistries consistent with the second generation of GCs in the Galactic field. Moreover, our approach adds to the mounting community evidence that the Gaia BP/RP spectra contain valuable information regarding stellar parameters and can be used to identify chemically peculiar candidates in spite of their low resolution. We intend to leverage the large number of candidates to tag high-[N/O] stars to their GC of origin using their orbital properties, perhaps enabling the discovery of new GC stellar streams.', 'ACKNOWLEDGEMENTS': "SGK thanks Tom Hehir for his useful discussions regarding neural networks. SGK acknowledges PhD funding from the Marshall Scholarship, supported by the UK government and Trinity College, Cambridge. HZ thanks the Science and Technology Facilities Council (STFC) for a PhD studentship. AAA acknowledges support from the Herchel Smith Fellowship at the University of Cambridge and a Fitzwilliam College research fellowship supported by the Isaac Newton Trust. \nThis work made extensive use of the Python packages Numpy (Harris et al. 2020), Scipy (Virtanen et al. 2020), Matplotlib (Hunter 2007), Scikit-learn (Pedregosa et al. 2011), and Gala (De Silva et al. 2015; Price-Whelan et al. 2020). This work made use \nof Astropy : 4 a community-developed core Python package and an ecosystem of tools and resources for astronomy (Astropy Collaboration et al. 2013, 2018, 2022). \nThis work has made use of data from the European Space Agency (ESA) mission Gaia ( https://www.cosmos.esa.int/ gaia ), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/ consortium ). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. \nThis work made use of data from the Apache Point Observatory Galactic Evolution Experiment (APOGEE Abdurro'uf et al. 2022). Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. \nSDSS-IV acknowledges support and resources from the Center for High Performance Computing at the University of Utah. The SDSS website is www.sdss4.org. \nSDSS-IV is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, Center for Astrophysics | Harvard & Smithsonian, the Chilean Participation Group, the French Participation Group, Instituto de Astrofísica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU) / University of Tokyo, the Korean Participation Group, Lawrence Berkeley National Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck-Institut für Extraterrestrische Physik (MPE), National Astronomical Observatories of China, New Mexico \nState University, New York University, University of Notre Dame, Observatário Nacional / MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Autónoma de México, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University. \nThis work made use of the Third Data Release of the GALAH Survey (Buder et al. 2021). The GALAH Survey is based on data acquired through the Australian Astronomical Observatory, under programs: A/2013B/13 (The GALAH pilot survey); A/2014A/25, A/2015A/19, A2017A/18 (The GALAH survey phase 1); A2018A/18 (Open clusters with HERMES); A2019A/1 (Hierarchical star formation in Ori OB1); A2019A/15 (The GALAH survey phase 2); A/2015B/19, A/2016A/22, A/2016B/10, A/2017B/16, A/2018B/15 (The HERMES-TESS program); and A/2015A/3, A/2015B/1, and A/2015B/19, and A/2016A/22, and A/2016B/12, and A/2017A/14 (The HERMES K2-follow-up program). We acknowledge the traditional owners of the land on which the AAT stands, the Gamilaraay people, and pay our respects to elders past and present. This paper includes data that has been provided by AAO Data Central (datacentral.org.au). \nThis paper made used of the Whole Sky Database (wsdb) created by Sergey Koposov and maintained at the Institute of Astronomy, Cambridgewithfinancialsupport from the Science & Technology Facilities Council (STFC) and the European Research Council (ERC).", 'DATA AVAILABILITY': "This paper relies on publicly available data from Gaia DR3 (Gaia Collaboration et al. 2023), APOGEE (Abdurro'uf et al. 2022), and GALAH (De Silva et al. 2015). The catalog of GC members is publicly available from Vasiliev & Baumgardt (2021). Our catalog of new high-[N/O] candidates will be made publicly available upon publication of this article. A description of the columns in the catalog is included in Appendix A.", 'REFERENCES': "Carretta E., Bragaglia A., Gratton R., Lucatello S., Bellazzini M., D'Orazi V., 2010, ApJ, 712, L21 \nPhillips S. G., et al., 2022, MNRAS, 510, 3727 \nPrice-Whelan A. 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J., 1975, ApJ, 196, 407 Vasiliev E., Baumgardt H., 2021, MNRAS, 505, 5978 Ventura P., D'Antona F., 2009, A&A, 499, 835 Virtanen P., et al., 2020, Nature Methods, 17, 261 Witten C. E. C., et al., 2022, MNRAS, 516, 3254 Yanagisawa H., et al., 2024, arXiv e-prints, p. arXiv:2405.01823 Yao Y., Ji A. P., Koposov S. E., Limberg G., 2024, MNRAS, 527, 10937 Yoon J., et al., 2016, ApJ, 833, 20 Zwitter T., et al., 2021, MNRAS, 508, 4202", 'APPENDIX A: CATALOG OF PREDICTIONS': 'Our catalog of inferred abundances and variances has 12 columns: \n- (i) Gaia source\\_id\n- (ii) teff , logg , feh , no , and alfe , which correspond to 𝜇 𝑇 eff , 𝜇 log 𝑔 , 𝜇 [Fe/H] , 𝜇 [N/O] , and 𝜇 [Al/Fe] , respectively \n(iii) teff\\_stdev , logg\\_stdev , feh\\_stdev , no\\_stdev , and alfe\\_stdev , which correspond to 𝜎 𝑇 eff , 𝜎 log 𝑔 , 𝜎 [Fe/H] , 𝜎 [N/O] , and 𝜎 [Al/Fe] , respectively \n(iv) ebv from Schlegel et al. (1998), for convenient selection of our high-reliability, low extinction sample \nWe will also provide as a separate table the variances of the 100 iterations of network predictions for each inferred value (which is described in Sec. 2.2', 'APPENDIX B: ADDITIONAL INFERRED VARIANCES': 'Below we present the values of 𝜎 𝑇 eff , 𝜎 log 𝑔 , 𝜎 [Fe/H] , and 𝜎 [Al/Fe] for our validation data. \nThis paper has been typeset from a T E X/L A T E X file prepared by the author. \nFigure B1. From top down, the MLP standard deviation prediction of 𝑇 eff ( 𝜎 𝑇 eff < 5000) versus APOGEE values for 𝑇 eff, log 𝑔 , [Fe/H], [N/O], the residual of the 𝜇 𝑇 eff prediction in the validation dataset, and ebv .from Schlegel et al. (1998). The colorbar is shared for all panels and marks the color-mapping of the histograms as the log-scaled number of stars per pixel. \n<!-- image --> \nFigure B3. From top down, the MLP standard deviation prediction of [Fe/H] ( 𝜎 [Fe/H]) versus APOGEE values for 𝑇 eff, log 𝑔 , [Fe/H], [N/O], the residual of the 𝜇 [Fe/H] prediction in the validation dataset, and ebv from Schlegel et al. (1998). The colorbar is shared for all panels and marks the color-mapping of the histograms as the log-scaled number of stars per pixel. \n<!-- image --> \nFigure B2. From top down, the MLP standard deviation prediction of log 𝑔 ( 𝜎 log 𝑔 ) versus APOGEE values for 𝑇 eff, log 𝑔 , [Fe/H], [N/O], the residual of the 𝜇 log 𝑔 prediction in the validation dataset, and ebv .from Schlegel et al. (1998). The colorbar is shared for all panels and marks the color-mapping of the histograms as the log-scaled number of stars per pixel. \n<!-- image --> \nN \nFigure B4. From top down, the MLP standard deviation prediction of [Al/Fe] ( 𝜎 [Al/Fe]) versus APOGEE values for 𝑇 eff, log 𝑔 , [Fe/H], [N/O], [Al/Fe], the residual of the 𝜇 [Al/Fe] prediction in the validation dataset, and ebv .from Schlegel et al. (1998) The colorbar is shared for all panels and marks the color-mapping of the histograms as the log-scaled number of stars per pixel. \n<!-- image -->'} |
2024arXiv240803189K | We study from both a theoretical and observational perspective the physical origin and spectroscopic impact of extreme nebular emission in highredshift galaxies. The nebular continuum which can appear during extreme starbursts is of particular importance as it tends to redden UV slopes and has a significant contribution to the UV luminosities of galaxies. Furthermore its shape can be used to infer the gas density and temperature of the ISM. First we provide a theoretical background showing how different stellar populations SPS models IMFs and stellar temperatures and nebular conditions impact observed galaxy spectra. We demonstrate that for systems with strong nebular continuum emission 1 UV fluxes can increase by up to 0.7magnitudes or more in the case of hotmassive stars above the stellar continuum which may help reconcile the surprising abundance of bright highredshift galaxies and the elevated UV luminosity density at zgt10 2 at high gas densities UV slopes can redden from betalesssim2.5 to betasim1 3 observational measurements of xiion are grossly underestimated and 4 UV downturns from twophoton emission can masquerade as DLAs. Second we present a dataset of 58 galaxies observed with NIRSpec on JWST at 2.5ltzlt9.0 that are selected to have strong nebular continuum emission via the detection of the Balmer jump. Five of the 58 spectra are consistent with being dominated by nebular emission exhibiting both a Balmer jump and a UV downturn consistent with twophoton emission. For some galaxies this may imply the presence of hot massive stars and a topheavy IMF. We conclude by exploring the properties of spectroscopically confirmed zgt10 galaxies finding that UV slopes and UV downturns are in some cases redder or steeper than expected from SPS models which may hint at more exotic e.g. hottermore massive stars or AGN ionizing sources. | 2024-08-01T00:00:00Z | ['arXiv:2408.03189', '10.48550/arXiv.2408.03189', '2024arXiv240803189K'] | ['Astrophysics - Astrophysics of Galaxies', 'Astrophysics - Cosmology and Nongalactic Astrophysics'] | 21 Balmer Jump Street The Nebular Continuum at High Redshift and Implications for the Bright Galaxy Problem UV Continuum Slopes and Early Stellar Populations | 2,024 | 166 | 0.67 | ['EPRINT_HTML', 'EPRINT_PDF'] | 22 | https://arxiv.org/pdf/2408.03189.pdf | {'21 BALMER JUMP STREET: THE NEBULAR CONTINUUM AT HIGH REDSHIFT AND IMPLICATIONS FOR THE BRIGHT GALAXY PROBLEM, UV CONTINUUM SLOPES, AND EARLY STELLAR POPULATIONS': 'Harley Katz 1 , Alex J. Cameron 2 , Aayush Saxena 2 , Laia Barrufet 3 , Nicholas Choustikov 2 , Nikko J. Cleri 4 , 5 , 6 , Anna de Graaff 7 , Richard S. Ellis 8 , Robert A. E. Fosbury 9 , Kasper E. Heintz 10 , 11 , 12 , Michael Maseda 13 , Jorryt Matthee 14 , Ian McConachie 13 , 15 , Pascal A. Oesch 12 , 10 , 11 \n- 1 Department of Astronomy & Astrophysics, University of Chicago, 5640 S Ellis Avenue, Chicago, IL 60637, USA 2 Department of Physics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford, OX1 3RH, UK 3 Institute for Astronomy, University of Edinburgh, Royal Observatory, Edinburgh, EH9 3HJ 4 Department of Astronomy and Astrophysics, The Pennsylvania State University, University Park, PA 16802, USA 5 Institute for Computational and Data Sciences, The Pennsylvania State University, University Park, PA 16802, USA 6 Institute for Gravitation and the Cosmos, The Pennsylvania State University, University Park, PA 16802, USA 7 Max-Planck-Institut fur Astronomie, Konigstuhl 17, D-69117, Heidelberg, Germany 8 Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK 9 European Southern Observatory, Karl-Schwarzschild-Strasse 2, 85748 Garching, Germany 10 Cosmic Dawn Center (DAWN), Denmark 11 Niels Bohr Institute, University of Copenhagen, Jagtvej 128, DK-2200 Copenhagen N, Denmark. 12 Department of Astronomy, University of Geneva, Chemin Pegasi 51, 1290 Versoix, Switzerland. 13 Department of Astronomy, University of Wisconsin-Madison, 475 N. Charter St., Madison, WI 53706 USA 14 Institute of Science and Technology Austria (ISTA), Am Campus 1, 3400 Klosterneuburg, Austria and 15\n- Department of Physics and Astronomy, University of California, Riverside, 900 University Avenue, Riverside, CA 92521, USA Version August 8, 2024', 'ABSTRACT': 'We study, from both a theoretical and observational perspective, the physical origin and spectroscopic impact of extreme nebular emission in high-redshift galaxies. The nebular continuum, which can appear during an extreme starburst, is of particular importance as it tends to redden UV slopes and has a significant contribution to the UV luminosities of galaxies. Furthermore, its shape can be used to infer the gas density and temperature of the interstellar medium. First, we provide a theoretical background, showing how different stellar populations (SPS models, initial mass functions (IMFs), and stellar temperatures) and nebular conditions impact observed galaxy spectra. We demonstrate that, for systems with strong nebular continuum emission, 1) UV fluxes can increase by up to 0.7 magnitudes (or more in the case of hot/massive stars) above the stellar continuum, which may help reconcile the surprising abundance of bright high-redshift galaxies and the elevated UV luminosity density at z ≳ 10, 2) at high gas densities, UV slopes can redden from β ≲ -2 . 5 to β ∼ -1, 3) observational measurements of ξ ion are grossly underestimated, and 4) UV downturns from two-photon emission can masquerade as damped Ly α systems. Second, we present a dataset of 58 galaxies observed with NIRSpec on JWST at 2 . 5 < z < 9 . 0 that are selected to have strong nebular continuum emission via the detection of the Balmer jump. Five of the 58 spectra are consistent with being dominated by nebular emission, exhibiting both a Balmer jump and a UV downturn consistent with two-photon emission. For some galaxies, this may imply the presence of hot massive stars and a top-heavy IMF. We conclude by exploring the properties of spectroscopically confirmed z > 10 galaxies, finding that UV slopes and UV downturns are in some cases redder or steeper than expected from SPS models, which may hint at more exotic (e.g. hotter/more massive stars or AGN) ionizing sources.', '1. INTRODUCTION': 'The successful launch of JWST (Gardner et al. 2006) has opened a new frontier to study galaxy formation at the earliest cosmic epochs. Understanding the physical properties of these early galaxies, how they differ from those in the low-redshift Universe, and how they impacted subsequent galaxy evolution remain key open goals of modern extragalactic astronomy. One of the primary advantages of JWST compared to HST is its spectroscopic capabilities in the rest-frame UV and optical at high-redshift. This tool can be used not only to confirm the redshifts estimated from photometry, but also to obtain deep insights into the stellar populations, star formation histories, and the conditions of the interstellar medium (ISM) of galaxies in the early Universe (e.g. Sanders et al. 2023; Cameron et al. 2023b). \nNumerous galaxies have now been spectroscopically \nconfirmed at z > 10 (e.g. Curtis-Lake et al. 2023; Wang et al. 2023; Arrabal Haro et al. 2023; Carniani et al. 2024) and have been shown to exhibit a diverse set of properties. For example GNz11 (Bunker et al. 2023b) and GHZ2 (Castellano et al. 2024; Zavala et al. 2024) are characterized by extreme UV emission lines, including those rarely seen at lower redshifts (e.g. N IV ] λ 1486), while others exhibit much lower EW (and undetected) UV lines (e.g. Carniani et al. 2024; Curtis-Lake et al. 2023). Compilations of high-redshift galaxies (e.g. Heintz et al. 2024b; Roberts-Borsani et al. 2024) show that UV slopes can range from close to -3, which would be consistent with the intrinsic values of massive stars, to redder than -2, which is typically reconciled by assuming dust attenuation, clearly indicating a difference in ISM properties or stellar populations. With JWST, we are seeing unprecedented spectral features at high-redshift. \nMore generally, the photometric detection and spectroscopic confirmation of numerous bright galaxies at z ≳ 10 as well as the elevated UV luminosity density at similar redshifts is in tension with most theoretical models of high-redshift galaxy formation (e.g. Finkelstein et al. 2023; Harikane et al. 2024; Leung et al. 2023; Chemerynska et al. 2023, although c.f. Willott et al. 2023). Various physical mechanisms have been proposed to explain this overabundance of bright (M UV ≳ -19) galaxies such as: \n- 1. an increased scatter in the relation between dark matter halo mass and UV luminosity compared to the lower-redshift Universe, possibly driven by bursty star formation (e.g. Ren, Trenti & Mason 2019; Mason, Trenti & Treu 2023; Shen et al. 2023; Sun et al. 2023; Kravtsov & Belokurov 2024),\n- 2. an increased star formation efficiency at high redshift (e.g. McCaffrey et al. 2023), potentially caused by weak feedback (e.g. Dekel et al. 2023),\n- 3. a lack of dust in massive galaxies due to strong radiation pressure (Ferrara, Pallottini & Dayal 2023),\n- 4. or a change in the mass-to-light ratio of early stellar populations, possibly due to a top-heavy IMF (e.g. Yung et al. 2024, although c.f. Cueto et al. 2024). \nThese solutions address the problem by either changing the observed mass-to-light ratio or forming stars more efficiently. The true origin of the bright galaxy problem and excess UV luminosity density is likely a combination of many physical effects. For example, nearly all high-resolution numerical simulations of high-redshift galaxies that model cold gas in the ISM predict that star-formation is bursty (e.g. Ma et al. 2018; Katz et al. 2023; Pallottini et al. 2022). However, as this scatter is mass-dependent (e.g. Gelli, Mason & Hayward 2024; Kravtsov & Belokurov 2024) it remains an open question whether the level of SFR burstiness generically expected in galaxy formation models is sufficient to produce the level of UV luminosity variability required to match observations (e.g. Pallottini & Ferrara 2023). \nNumerous 1D (e.g. Omukai et al. 2005) and 3D models (e.g. Chon et al. 2022) predict that the stellar initial mass function (IMF) becomes more top-heavy at lower metallicities and in the early Universe when the cosmic microwave background (CMB) temperature is higher. A more top-heavy IMF is a generic prediction in the extreme limit of metal-free star formation at high redshift (e.g. Bromm, Coppi & Larson 2002; Abel, Bryan & Norman 2002; Hirano et al. 2014). The mass-to-light ratio is lower at young stellar ages for IMFs with flatter high-mass slopes, which would lead to an increase in UV luminosity at fixed star formation rate. \nWhat has perhaps been considered less in the literature is the role of the nebular continuum, which is observed to become important in lower mass, low-metallicity environments (e.g. Izotov, Guseva & Thuan 2011). When ionizing photons are absorbed by gas they can be reprocessed into continuum emission through three different processes: free-free, free-bound, and two-photon 1 . For \n1 Note that throughout this work we will only consider the nebular continuum contribution as we will analyze spectroscopic data. For photometric data, line emission can also contribute to the inferred luminosity integrated over a photometric passband. \nFig. 1.Spectral shape of the three components of the nebular continuum as a function of wavelength. We show the shape of the nebular continuum for gas at 2 × 10 4 K and an electron density of 10 2 cm -3 assuming Case B recombination. Annotated are the locations of the Balmer jump caused by free-bound emission and the UV downturn from two-photon emission. \n<!-- image --> \na typical star-forming galaxy, nebular continuum emission is generally subdominant at rest-frame wavelengths of 1500 ˚ A, where UV luminosities are typically measured. However, strong nebular continuum emission has now been detected in both low (e.g. Guseva, Izotov & Thuan 2006; Guseva et al. 2007) and high-redshift galaxies (e.g. Cameron et al. 2023a; Roberts-Borsani et al. 2024; Welch et al. 2024) undergoing bursts of star formation. \nFree-bound emission is arguably the most easily recognizable due to the Balmer jump, which is a sharp discontinuity at rest-frame 3645 ˚ A (see the pink line in Figure 1). Two-photon emission, particularly the 2 s → 1 s transition of neutral hydrogen peaks very close to 1500 ˚ A (in f λ ) and smoothly drops towards both shorter and longer wavelengths, going to zero at 1216 ˚ A (see the green line in Figure 1). In addition to being emitted by H II regions, two-photon emission can also arise from warm cooling flows that may be present in the high-redshift Universe (e.g. Dijkstra 2009). Two-photon emission is rarely directly detected because for Population II star formation, the power-law stellar continuum increases towards the blue end of the spectrum and typically outshines the nebula at short wavelengths. However, in low-metallicity environments where H II region temperatures are elevated, Case B departures can occur which increases both the two-photon and Ly α emission via collisional excitation (e.g. Raiter, Schaerer & Fosbury 2010; Mas-Ribas, Dijkstra & Forero-Romero 2016). \nIn contrast to two-photon emission, free-bound emission increases towards longer wavelengths and can thus outshine the stellar emission. Hence it can be directly detected in the rest-frame optical. The key aspect of the nebular continuum is that it should only become important when galaxies are undergoing a burst of star formation. The natural UV magnitude fluctuations driven by deviations from the star-forming main-sequence can be further enhanced by continuum emission from the gas. Identifying a sample of high-redshift galaxies with strong \nnebular continuum emission may provide an ideal testbed for elucidating the physical mechanisms that drive extreme bursts of star formation, which may explain the over-abundance of bright high-redshift galaxies and the excess UV luminosity density. \nUnfortunately, a sample of high-redshift galaxies with detected nebular continuum has yet to be compiled. Only one individual galaxy with a Balmer jump has been reported at z > 5 (Cameron et al. 2023a) and this same feature has appeared in a stack of z ∼ 6 Ly α emitters (Roberts-Borsani et al. 2024). \nThe lack of reported Balmer jumps is perhaps surprising because high-resolution simulations of early galaxies generically predict bursty star formation histories such that the SFRs can fluctuate by multiple orders of magnitude on tens of Myr timescales (e.g. Ma et al. 2018; Katz et al. 2023; Pallottini et al. 2022), which is exactly the scenario where the nebular continuum is expected to be strong. SED fitting of the photometry of highredshift galaxy candidates often predicts extreme nebular emission (e.g. Endsley et al. 2023; Bradaˇc et al. 2024). Moreover, the flux-limited nature of high-redshift surveys preferentially biases observations towards galaxies that are highly star-forming, which is when the nebular continuum should begin to appear. The spectral resolution of the NIRSpec prism and sensitivity of the grating may also impact our ability to detect the Balmer jump if present. As the prevalence and characteristics of the nebular continuum at high-redshift are poorly known, its importance for the luminosities of early galaxies remains to be quantified. \nIn this work, we aim to better understand the physics of galaxies undergoing extreme bursts of star formation by focusing on the spectral impact of extreme nebular continuum emission and the underlying physics that may be driving its observability. We begin in Section 2 by providing a theoretical background for how nebular emission modifies the observational properties of starbursting galaxies, focusing on changes to the properties of the underlying stellar populations and ISM conditions. In Section 3, we test the theoretical results in the context of a newly compiled sample of 58 spectroscopically confirmed high-redshift galaxies that show Balmer jumps. Finally, in Sections 4 and 5, we discuss the implications our findings in the context of z > 10 galaxies and present our conclusions.', '2. THEORETICAL BACKGROUND': 'We begin by providing a theoretical overview of the expected nebular continuum contribution of different stellar populations to the observed spectra of star-bursting galaxies and how this emission impacts our ability to infer the underlying properties of these same galaxies.', '2.1. Spectral synthesis modeling': 'SEDs of galaxies including the nebular continuum are computed with CLOUDY version C23 (Chatzikos et al. 2023). Unless otherwise specified, models assume a slab configuration with a dimensionless ionization parameter of log 10 U = -2 . 0, a metallicity of 1% solar by scaling down abundance ratios from Asplund et al. (2009), and a hydrogen gas density of 10 3 cm -3 , consistent with that of galaxies in the early Universe (e.g. Isobe et al. \n2023). Calculations are stopped when the electron fraction reaches 1% 2 . The exact details of the models have little impact on our conclusions. We consider five different stellar population synthesis (SPS) models, Starburst99 (Leitherer et al. 1999) 3 , BPASS v2.2.1 (Stanway & Eldridge 2018) 4 , FSPS (Conroy, Gunn & White 2009; Conroy & Gunn 2010) 5 , C&B 2019 models 6 , and individual, metal-poor, massive star models of Larkin, Gerasimov & Burgasser (2023) 7 . Within each subset of models, we choose the SEDs that have the closest metallicity to the fiducial value of 1% that we adopt for the gas 8 . All models assume an instantaneous burst of star formation and we consider stellar ages up to 20 Myr. Beyond this, the ionizing output is so low that the nebular emission is negligible. Note that the upper mass limits for all of our chosen SPS models differ. The fiducial upper mass cutoffs are 120 M ⊙ , 300 M ⊙ , and 100 M ⊙ for Starburst99 , BPASS , and FSPS , respectively. For the C&B19 models, we consider three different upper mass limits of 100 M ⊙ , 300 M ⊙ , and 600 M ⊙ . \nThe output from these models are the transmitted stellar emission as well as the continuum and line emission from the gas. It should be noted that in real galaxies, the turbulent nature of H II regions can modify the gas emission from expectations of spherical models (e.g. Jin, Kewley & Sutherland 2022) and interpretation biases can arise if multiple H II regions with different properties combine to produce the spectrum of a galaxy (e.g. Cameron, Katz & Rey 2023). Nevertheless, the 1D models used here provide both quantitative and qualitative insights into how galaxy spectra may change under different conditions under a restrictive set of assumptions.', '2.2. How strong can the nebular continuum be?': 'We first address the question of how much the nebular continuum can contribute to the UV luminosity of a galaxy.', '2.2.1. Standard SPS models': 'In Figure 2 we show the fraction of the total 1500 ˚ A or 3640 ˚ A continuum emission from the galaxy comes from the nebula as a function of stellar age for four SPS models. For young stellar ages, ∼ 1 Myr, all models predict that the nebula contributes at least 30% to the 1500 ˚ Aluminosity, which corresponds to brightening the galaxy by ∼ 0 . 4 magnitudes. The nebular continuum contribution can be > 70% in the rest-frame optical. All SPS models predict that after ∼ 10 Myr, the nebular contribution at 1500 ˚ A is negligible, a consequence of the massive stars \n2 We consider this to be the edge of an ionization bounded nebula. Adopting a density bounded model would reduce the contribution of nebular emission with respect to the stellar continuum as the escape fraction will be non-zero. Moreover, extending the calculation beyond the Stromgren sphere will not impact our results unless the gas is hot enough so that it can be collisionally excited. This possibility is further discussed below. \n- 3 https://www.stsci.edu/science/starburst99/docs/default.htm\n- 4 https://bpass.auckland.ac.nz/9.html\n- 5 https://github.com/cconroy20/fsps\n- 6 http://www.bruzual.org\n- 7 https://atmos.ucsd.edu \n8 Note that we do not assume any α enhancement in stellar SEDs which is expected both at low metallicities and at high redshifts (e.g. Kobayashi, Karakas & Lugaro 2020; Cullen et al. 2021). \nFig. 2.Nebular continuum fraction of the total SED at either 1500 ˚ A (magenta), where the two-photon continuum becomes important, or at 3640 ˚ A (cyan), where the free-bound emission reaches its maximum. We show results for four different SPS models as a function of stellar age. For the C&B19 models, different line styles represent different upper mass limits for the IMF. \n<!-- image --> \nevolving off the main sequence and the lower ionizing photon production efficiencies of less massive stars. The timescale over which the nebular continuum remains important is slightly longer for the free-bound emission due to the fact that it is stronger with respect to the stellar continuum compared to two-photon emission. We note that burst models maximize the nebular contribution to each SSP and the nebular continuum contribution for realistic SFHs will be explored in Section 2.3.1. \nWhile the general trends between the SPS curves are similar, in detail the shapes of the curves and the maximum nebular contributions are different. For example the 1500 ˚ A nebular contribution of the BPASS models peak at ∼ 40% at 1 Myr, much higher than the Starburst99 models, due to the increased ionizing photon production efficiencies caused by binary stellar evolution. The C&B19 models with maximum stellar masses of 300 M ⊙ and 600 M ⊙ can reach a 1500 ˚ A nebular contribution as high as 50% at similarly young stellar ages, which corresponds to a brightening the UV luminosity by 0.75 magnitudes. In contrast to the other three SPS models, the nebular contribution of the FSPS models peaks between 3 -5 Myr at nearly 45%, which results from different assumptions regarding Wolf-Rayet stars.', '2.2.2. A top-heavy IMF': "Numerous theoretical models predict that the stellar IMF becomes more top-heavy at high redshift due to lower gas metallicities and an elevated CMB temperature (e.g. Chon et al. 2022; Bate 2023). This has two important effects. First, as the upper-mass IMF slope becomes \nflatter 9 , at young stellar ages, more UV luminosity is produced per unit star formation, which would help explain the bright galaxy problem by decreasing the mass-tolight ratio at young ages. Second, the ionizing photon production per unit star formation also increases as the IMF becomes more biased towards high-mass stars. To test the impact of a varying IMF, we adopt Starburst99 models 10 , where we have systematically varied the uppermass slope, α , from -2 . 6 (i.e. more bottom heavy than the canonical -2 . 3) to -0 . 5 (where nearly all of the mass is locked in massive stars). We define the IMF such that: \nξ ( m ) ∝ { m -1 . 3 m ≤ 0 . 5 M ⊙ m α m> 0 . 5 M ⊙ (1) \nThe lower and upper mass limits of the IMF are set to 0.08 M ⊙ and 120 M ⊙ , respectively. \nIn the top panel of Figure 3, we show the fractional nebular contribution at 1500 ˚ A and 3640 ˚ A as a function of stellar age. The general trend of the nebular contribution is the same as above - the peak in nebular emission comes at young stellar ages and begins to subside after a few Myr, with a small secondary peak from Wolf-Rayet stars. Unsurprisingly, a more top-heavy IMF \n9 Here we only consider modifications to the IMF where the power-law slope of the high-mass end is varied. However, the shape of the IMF may be fundamentally different and deviate from a power-law at high-redshift (e.g. Chon et al. 2022). For our work, the important parameter is the fraction of mass in high-mass stars with respect to that at lower masses rather than the exact shape. \n10 Starburst99 models were adopted because it is trivial to vary the properties of the IMF, in contrast to some of the other SPS models. Our primary conclusions likely generalize to the other SPS models. \nFig. 3.Contribution of the nebular continuum to the total SED at either 1500 ˚ A (solid), where the two-photon continuum becomes important, or at 3640 ˚ A (dashed), where the free-bound emission reaches its maximum. We show results for Starburst99 models assuming a gas density of 10 3 cm -3 with different stellar initial mass functions where the upper-mass slope is varied as listed in the legend (top) as well as for low-metallicity massive star models of different stellar temperatures (bottom) where we vary the gas density of the nebula. For models with varying stellar temperature, we only consider the zero-age main-sequence spectra. \n<!-- image --> \n1 \n. \n0 \n0 \n. \n8 \n0 \n. \n6 \n0 \n. \n4 \n0 \n. \n2 \n0 \n. \n0 \n40000 \n60000 \n80000 \n100000 \nStellar Temperature [K] \nresults in an increased fractional nebular contribution to the 1500 ˚ A luminosity. While the fiducial model predicts that ∼ 30% of the 1500 ˚ A luminosity at an age of 1 Myr comes from nebular emission, by flattening the upper-mass IMF slope to -0 . 5, the nebular contribution increases to ∼ 38% at the same stellar age. This increase is rather modest, especially compared to the total increase in UV luminosity which is more than a factor of ten for an upper-mass IMF slope to -0 . 5 (see below). \nIt is straightforward to understand why extreme IMF variations have only a minimal impact on the nebular continuum fraction at 1500 ˚ A. A massive O star has temperature of ∼ 40 , 000 K (Pecaut & Mamajek 2013), which according to Wien's law has a spectral peak at ∼ 725 ˚ A. For a standard IMF, these stars dominate the ionizing \n. \nneb \nf \n2 \n10 \n3 \n10 \n4 \n10 \n5 \n10 \ncm \n- \ncm \n- \ncm \n- \ncm \n- \n3 \n3 \n3 \n3 \noutput as going slightly lower in temperature pushes the peak redward of the hydrogen ionizing wavelength. Since hotter blackbodies are brighter at all wavelengths, the most massive stars also contribute significantly to the 1500 ˚ A UV luminosity, despite both their lower numbers and their spectral peak at shorter wavelengths. The absolute maximum nebular contribution is then that which comes from the most massive star in the IMF. Since it is these stars that provide both the ionizing photons and a significant amount of the 1500 ˚ A luminosity, flattening the upper-mass slope of the IMF (i.e. making it more 'top-heavy') does not substantially increase the fractional nebular contribution because the upper limit to this value is set by the most massive star in the IMF.", '2.2.3. Hotter and more massive stars': 'We have shown that for an IMF with a fixed upper mass, the fractional nebular contribution at 1500 ˚ A does not increase substantially if the upper-mass IMF slope flattens from -2 . 3 to -0 . 5. However, increasing the maximum mass sampled by the IMF represents a possible alternative. This was already hinted at in Section 2.2.1 where we showed for the C&B19 models that increasing the maximum mass from 100 M ⊙ to 600 M ⊙ increased the fractional nebular contribution at both 1500 ˚ A and 3640 ˚ A. \nTo explore this effect further, we adopt the lowmetallicity, massive star models of Larkin, Gerasimov & Burgasser (2023) which are available for a range of stellar temperatures. We assume that these are generally representative of low-metallicity massive stars in the early Universe for which there is limited public data of stellar atmospheres. The effect on nebular contribution strength due to very massive stars was recently discussed in Schaerer et al. (2024). Our work differs from Schaerer et al. (2024) in that they chose a very specific model for massive stars at a metallicity that is higher than typically measured for high-redshift galaxies and hence the effective temperatures of those models are lower than considered here. Moreover, they apply a more simplistic model for the nebular continuum by assuming a constant temperature, which both ignores the temperature structure of the nebula and neglects potential Case B departures that may become important at low metallicity (e.g. Raiter, Schaerer & Fosbury 2010; Mas-Ribas, Dijkstra & Forero-Romero 2016). \nIn the bottom panel of Figure 3 we show the fractional nebular contribution at 1500 ˚ A and 3640 ˚ A as a function of stellar temperature. While at a temperature of 40,000 K, the nebular fraction at 1500 ˚ A is only ∼ 14%, for stars with T = 100 , 000 K, nebular emission represents 80% of the total 1500 ˚ A UV luminosity. For these hot stars, the UV luminosity could then be brightened by 1.75 magnitudes, purely due to nebular emission. \nThis prediction is not unique to our work. In fact, a generic characteristic of hot star models is that they have nebular dominated spectra across nearly the entire UV and optical (e.g. Panagia 2002; Schaerer 2003; Trussler et al. 2023), and Raiter, Schaerer & Fosbury (2010) have already shown that the nebular continuum can dominate the 1500 ˚ A UV luminosity in the context of Pop. III stars under various assumptions of IMF. Raiter, Schaerer & Fosbury (2010) also pointed out that at low metallici- \nies, one expects departures from standard Case B assumptions due to the fact that the 2 s and 2 p states can be collisionally populated. Case B departures have the effect of enhancing the two-photon emission (and Ly α ), both due to the fact that the collisionally excited atom can fluoresce down to the ground state and that in the excited state, lower energy photons are needed to ionize the gas, although this latter effect is likely subdominant (Mas-Ribas, Dijkstra & Forero-Romero 2016). Case B departures due to collisional excitation will also increase the intrinsic ratio of H α /H β to values > 2 . 86.', '2.2.4. Other physics that impacts the nebular contribution': "Cooling Flows -Just as collisions impact the level populations within H II regions, gas accreting onto haloes can undergo the same process as long as the virial temperature of the halo is high enough ( ≳ 10 4 K) (e.g. Fardal et al. 2001; Faucher-Gigu'ere et al. 2010). This process then cools the gas and is particularly efficient in the temperature range 15 , 000 -20 , 000 K. Numerical simulations predict that cold-mode accretion, where cooling is strongest is much more prevalent at highredshift compared to the local Universe (Kereˇs et al. 2005). A nebular-only spectrum dominated by Ly α and two-photon emission is then expected when cooling is sufficiently strong (e.g. Dijkstra 2009). This process can boost the nebular contribution, particularly Ly α and two-photon emission, which has the effect of decreasing the overall EWs of UV emission lines, reducing the perceived ionizing photon production efficiency, and increasing the UV magnitude of a galaxy. Cooling flows have been used, for example, to explain the presence of Ly α blobs at intermediate redshift (e.g. Goerdt et al. 2010; Faucher-Gigu'ere et al. 2010; Rosdahl & Blaizot 2012) and the relative contribution of cooling radiation to the total Ly α (and also two-photon) emission is predicted to increase substantially at higher redshift (e.g. Dayal, Ferrara & Saro 2010; Yajima et al. 2012). Given the fact that catastrophic cooling is required to drive a starburst it is reasonable to expect that cooling radiation occurs simultaneously with star formation, although the fractional contribution of cooling radiation is likely maximized immediately preceding or just after the starburst (e.g. Smith et al. 2019). The exact details of how much cooling radiation can contribute to the total nebular continuum warrants further investigation under the conditions likely present in early galaxies and under conditions that may increase the electron fraction in the cooling gas (e.g. X-ray radiation). \nHigh Density Gas -While we have primarily focused on the physical effects that can increase the nebular contribution to the UV luminosity, we also highlight the fact that certain nebular conditions can inhibit two-photon emission. For example, at high gas densities, l -changing collisions can shift electrons between the 2 s to 2 p states. However, because the Einstein coefficient is orders of magnitude larger for Ly α compared to two-photon emission, at high densities, Ly α is enhanced at the expense of two-photon emission. This is demonstrated in the bottom panel of Figure 3 where we show the fractional contribution of nebular emission at 1500 ˚ A at four densities from 10 2 -10 5 cm -3 . Between 10 3 cm -3 and 10 4 cm -3 the nebular contribution begins to decrease and at a den- \ny of 10 5 cm -3 the decrease is much more significant. Nevertheless, at high stellar temperatures, even at extreme densities of 10 5 cm -3 , which have only been observed in a select few high-redshift galaxies (e.g. Topping et al. 2024c,b; Senchyna et al. 2024), the nebular continuum can still increase the UV luminosity of the galaxy by a factor of two. The electron densities of typical galaxies in the reionization epoch are such that collisional de-excitation is not expected to be very important (e.g. Isobe et al. 2023).", '2.3. Observational implications': 'It is clear that during a starburst with ages ≲ 5 Myr, the nebular continuum can represent a significant fraction of the observed SED of a galaxy. Under specific circumstances, e.g. the presence of very hot massive stars, the nebular continuum can even dominate over the stellar emission at all wavelengths observable by NIRSpec on JWST. Here we explore the observational implications for galaxies with strong nebular continuum emission.', '2.3.1. UV luminosities and implications for the bright galaxy problem': "As we have shown in Section 2.2.1, in some of the SPS models, e.g. the C&B19 models with maximum stellar masses of 300 M ⊙ and 600 M ⊙ , the nebular contribution at 1500 ˚ A can reach as high as 50%. This corresponds to a UV luminosity increase of 0.75 magnitudes. Even in the optimistic scenario of the C&B19 models, the excess UV luminosity cannot explain the full UV variability needed to match the z > 10 UV luminosity functions as estimated by Kravtsov & Belokurov (2024). However, the inclusion of the nebular emission can bring down the implied 'burstiness' of star formation. \nOne important caveat is that our models assume an instantaneous burst of star formation rather than a realistic star formation history. Older stars can contribute to the UV luminosity while providing no additional ionizing photons that excite the nebular continuum. The general impact of including a realistic star formation history would be to decrease the fractional contribution of the nebular continuum in the UV. To better understand this effect, we perform a simple experiment and apply the C&B19 model with a maximum mass of 600 M ⊙ (our most optimistic model) to the realistic star formation histories of simulated galaxies from the SPHINX public data release (Katz et al. 2023). The stellar masses of these galaxies range between ≳ 10 7 M ⊙ to ≲ 10 10 M ⊙ , and all have 10 Myr-averaged SFRs > 0 . 3 M ⊙ yr -1 . We then account for the nebular contribution from the young stellar populations to the total SED following our fiducial photoionization model. Figure 4 shows a histogram of the change in UV magnitude from including the nebular continuum for > 1 , 400 star-forming galaxies from the SPHINX simulation between 4 . 6 ≤ z ≤ 10. While the typical magnitude increase is only ∼ 0 . 2 magnitudes, galaxies undergoing particularly strong bursts of star formation see much more substantial increases ( > 0 . 5 magnitudes) in UV luminosity due to the inclusion of the nebular continuum. Hence, during the burst, the UV magnitude is enhanced both due to the typical magnitude of the galaxy due to both the increase in SFR and the gaseous emission. We emphasize that the effect of \nFig. 4.Histogram of the change in 1500 ˚ A UV magnitude of galaxies from the SPHINX simulation when accounting for the nebular contribution to the SED. In all cases, the nebular continuum increases the UV luminosity for galaxies with realistic star formation histories, in some extreme examples by more than 0.5 magnitudes. \n<!-- image --> \nthe nebular continuum is unavoidable unless the gas is at extreme densities or the escape fraction is very high. \nModifying the upper IMF slope of the maximum stellar mass of the IMF can similarly increase the UV luminosity of a galaxy. We demonstrate this in Figure 5. In the top panel, we show the 1500 ˚ A luminosity of the Starburst99 models with different upper-mass IMF slopes as a function of stellar age, normalized by the model with an upper-mass slope of -2 . 3. A modest flattening of the slope from -2 . 3 to -1 . 7 increases the UV luminosity by a factor of five (1.75 magnitudes) at young stellar ages. Recall that under such circumstances, the nebular continuum contribution was not substantially increased and thus it is clearly the modification of the stellar component that leads to such a substantial increase in UV luminosity. \nIn the bottom panel of Figure 5 we show the ratio of 1500 ˚ A luminosity of stars of different temperatures normalized to a 40,000 K star. Here we find that hotter stars tend to increase the UV luminosity; however, in contrast to the IMF, it is the nebular continuum that drives the substantial increase. At high gas densities, where twophoton emission is strongly suppressed, a 100,000 K star is not even twice as bright as a 40,000 K star. However, at low gas densities, when the nebular continuum is strong in the UV, the brightness can increase by a factor of five. Hence, we emphasize that there is a clear difference between simply assuming a more top-heavy IMF by modifying the upper-mass power-law slope and changing the maximum mass star in the IMF as the physical effects that lead to a luminosity increase are different. Nevertheless, the combination of both a top-heavy IMF and more massive stars can lead to substantial changes in the UV luminosity of early galaxies, especially when the nebular continuum is strong. \nFig. 5.(Top) 1500 ˚ A UV luminosity as a function of stellar age for Starburst99 models using a density of 10 3 cm -3 with different upper-mass IMF slopes, normalized to the model with a slope of -2 . 3. (Bottom) 1500 ˚ A UV luminosity as a function of stellar temperature for zero-age main-sequence metal-poor massive star models, normalized to a star with a surface temperature of 40,000 K. All models include both the stellar and nebular continuum. In the case of IMF variations, the number of high mass stars drives the UV luminosity increase but in the case of stellar temperature, the nebular continuum is the primary factor that leads to an increase in UV luminosity. \n<!-- image -->", '2.3.2. UV continuum slopes, β': 'The most common discussion on the implications of a strong nebular continuum is the impact on UV continuum slopes (e.g. Schaerer 2002; Bouwens et al. 2010; Dunlop et al. 2013; Cullen et al. 2024; Topping et al. 2024a). In particular, the nebular continuum acts to redden the observed spectrum. This is demonstrated in Figure 6 where we show the intrinsic (i.e. stellar only) and observed (stellar + nebular) UV slopes for all SPS models. Here we have measured UV slopes in the wavelength range 1400 ˚ A to 2600 ˚ A. For young stellar ages, the intrinsic UV slopes are all bluer than -3 . 0. After the nebular continuum is added to the spectrum, the observed slope reddens to ∼ -2 . 5. This is consistent with other works that have performed similar calculations (e.g. Schaerer 2002; Bouwens et al. 2010; Dunlop et al. 2013; Cullen et al. 2024; Topping et al. 2024a). \nMaking the IMF more top-heavy only marginally changes the UV slope. The top panel of Figure 6 shows that as the IMF upper mass slope increases from -2 . 6 \n<!-- image --> \nStellar Age [yr] \nFig. 6.UV slope as a function of stellar age for various SPS models (top) or stellar temperature for the individual metal-poor star models (bottom). Solid lines represent the UV slope as measured from total (stellar+nebular) emission while dashed lines represent the UV slope of only the stellar component. The dotted red lines indicate a slope of -2 . 3 for reference in all panels. For the Starburst99 models, colors represent different upper-mass IMF slopes (see legend in top panel), while for the C&B19 models, colors indicate different upper-mass limits (see legend in the third panel). For the individual massive star models, colors indicate the gas density of the nebula. \n<!-- image --> \nto the extreme value of -0 . 5, β only increases by 0.15 at ages prior to the main sequence lifetime of massive stars. The reason the change is so mild is that the massive stars already dominate both the UV luminosity and the ionizing photon production. Adding more massive stars increases the contribution of these stars to the 1500 ˚ A luminosity, but this effect quickly saturates and the intrinsic β slope only becomes slightly bluer. At the same time, the nebular contribution is rising faster which reddens the slopes. Hence the net effect for a top-heavy IMF is that the UV continuum slope is redder than in the scenario with a more standard IMF (see also discussion in Cullen et al. 2024). We emphasize that these results are relatively independent of the chosen SPS model. Although the exact value of β can change slightly, the trends with IMF will be similar. \nTo make β even redder than in models with a top-heavy IMF, hotter stars are needed. This is shown in the bottom panel of Figure 6 where UV slope is plotted as a function of stellar temperature. For typical ISM densities of 10 2 -10 3 cm -3 , β saturates at a slope of ∼ -2 . 0 consistent with earlier work that assumed Pop. III SEDs (e.g. Trussler et al. 2023). By increasing gas density, twophoton emission is suppressed and the free-bound emission dominates, which leads to extremely red UV slopes approaching -1. This is extremely important because we do not yet know the ISM conditions around Pop. III stars and selection criteria should not necessarily remove redder objects as the ISM could simply be at high density. Hence we encourage photometric selection criteria to allow for these exotic scenarios. \nOne important caveat to these calculations is that we have computed the UV slopes without dust attenuation. Many high-redshift galaxies have Balmer decrements and UV slopes that are consistent with a dust-free scenario (e.g. Cullen et al. 2024; Sandles et al. 2023) and dust will only act to redden the SEDs. High quality spectra will be required to differentiate reddening due to dust versus the nebular continuum; however, it should be noted that β is not constant with wavelength when the nebular continuum is strong. \nThe modification of β due to the nebular continuum has important implications for interpreting spectra. For example, Chisholm et al. (2022) recently proposed using the UV slope as an indicator for LyC escape. In their sample of galaxies, β is largely set by dust attenuation and the galaxies with less dust are more likely to be LyC leakers. Applying their model to our dust-free CLOUDY simulations where β can be as low at ∼ -2 . 55, one would predict an escape fraction of ∼ 17% at β = -2 . 55. Therefore, in the zero-dust limit, this relation should be revised or not applied (see also Choustikov et al. 2024) while at higher metallicities, when dust is present, the relation likely remains applicable. \nAnother key result is that under normal nebular conditions and for typical SPS models, the UV slope should be bluer than approximately -2 . 3 for star-bursting galaxies 11 . This is indicated as the dotted red line on each \n11 We emphasize that here we are discussing star-bursting galaxies. For older stellar populations, the UV slope will naturally redden. Based on the realistic star formation histories in the SPHINX simulation (Katz et al. 2023), even for galaxies with mass-weighted ages of 100 Myr, the UV slope measured from stellar+nebular emission remains ∼ -2 . 5. \npanel of Figure 6 and represents the maximum slope that one obtains for standard SPS models. Redder slopes imply either dust or an increased nebular fraction (e.g. due to hotter/more massive stars), the former of which can be tested using the ratio of Balmer lines.', '2.3.3. Ionizing photon production efficiencies, ξ ion': 'The ionizing photon production efficiency ξ ion , is defined as \nξ ion = Q/L ν, 1500 ˚ A erg -1 Hz , (2) \nwhere Q is the number of ionizing photons produced per second and L ν, 1500 ˚ A is the 1500 ˚ A monochromatic luminosity. Note that throughout this work, we often refer to log 10 ( ξ ion ). As Q cannot be directly measured observationally, it is often inferred from emission lines (e.g. Balmer lines) using a conversion factor based on recombination theory (see Appendix B or e.g. Osterbrock & Ferland 2006). \nBecause the nebular continuum can significantly impact the UV luminosity at 1500 ˚ A, measurements of ξ ion , which often normalize a Balmer line (e.g. H α ) to the 1500 ˚ A luminosity can also be impacted. More specifically, the ionizing photon production efficiency is intended to be a measurement that traces the intrinsic properties of the underlying stellar (or more generally photoionizing) population. However, if the nebular continuum represents a large fraction of the UV luminosity, these measurements will be systematically biased low. \nThis is demonstrated in the left panel of Figure 7. For each SPS model, we adopt the fiducial CLOUDY model and measure the intrinsic ξ ion using the number of emitted ionizing photons ( Q ) by the intrinsic stellar continuum at 1500 ˚ A as well as the observed ξ ion where we add the nebular continuum to L UV . Independent of the chosen SPS model, the trend is nearly identical. For log 10 ( ξ ion / erg -1 Hz) ≲ 25 . 5 the intrinsic and observed ξ ion agree. This is a result of the fact that when ξ ion is low, there are not enough ionizing photons to create a significant nebular continuum. However, at log 10 ( ξ ion / erg -1 Hz) ≳ 25 . 5 the observed ξ ion deviates low compared to the intrinsic value. For the highest ξ ion values among all of our SPS models, this deviation can be nearly 1 dex. The primary conclusion here is that, while the different SPS models vary in terms of maximum intrinsic ξ ion , at high ionizing photon production efficiencies, the ξ ion value one would measure deviates from the true value in the exact same way in all cases. \nThe reason why the underlying SPS models have very little impact on the relation between intrinsic and observed ξ ion is because the contribution of the nebular continuum to the 1500 ˚ A UV luminosity is primarily sensitive to the properties of the nebula. In the right panel of Figure 7 we adopt the hot star models of Larkin, Gerasimov & Burgasser (2023) 12 and vary the details of the CLOUDY models such that gas density is sampled in the range 10 2 -10 4 cm -3 , the log of the ionization parameter is varied from between -2 . 5 and -1 . 5, while \nmetallicity ranges between 10 -3 -10 -1 Z ⊙ , all of which are conditions that are thought to be generally representative of the high-redshift Universe. The models show almost no sensitivity to ionization parameter (compare line styles), while there is a weak dependence on metallicity (compare line opacities). The primary impact of metallicity is that at lower Z , gas temperature is higher. At higher gas temperature, two-photon emission becomes stronger partially due to the departures from Case B (Raiter, Schaerer & Fosbury 2010). This effect is particularly noticeable when the metallicity of the nebula drops to 10 -3 Z ⊙ . Non-solar abundance patterns may modify where this effect matters, particularly if carbon or other strong coolants are removed from the gas. \nThe largest impact on the ξ ion discrepancy arises with gas density (compare line colors). As already explained above, this is due to l -changing collisions being able to shift electrons between the 2 s and 2 p states which then preferentially decay as Ly α due to the large difference in Einstein coefficients. Nevertheless, in all cases, when log 10 ( ξ ion ) ≳ 25 . 5, the true intrinsic value must be higher that what is observed. \nWe conclude from this experiment that under typical ISM conditions, if the gas in galaxies is being irradiated by a normal stellar population, log 10 ( ξ ion ) should not be observed to be ≳ 25 . 8. This is because the intrinsic log 10 ( ξ ion ) of even the optimistic SPS models peaks at ∼ 26 . 1 which corresponds to an observed log 10 ( ξ ion ) of 25 . 8. Measurements higher than this value would require any one of the following: \n- · An incorrect dust correction. For dusty galaxies UV luminosity is suppressed more than the Balmer lines which would lead to an over-prediction of ξ ion in the case where dust acts as a screen.\n- · High gas densities such that the two-photon emission is suppressed and the true ξ ion can be measured.\n- · An exotic stellar population that intrinsically has a much higher ξ ion than that predicted by normal SPS models.\n- · Extreme nebular conditions (e.g. incredibly high temperatures) or geometries, or non-stellar ionizing sources. \nWe discuss these options further below in the context of JWST galaxies. \nFinally, we note that the bias in ξ ion measurements when the nebular continuum is strong is unlikely to impact inferences on reionization history. This is because in such calculations, ξ ion is multiplied by UV luminosity density (e.g. Robertson et al. 2013) and even though ξ ion is biased low, this is compensated by the fact that observed UV luminosity density is increased by the nebular continuum.', '2.3.4. UV turnovers masquerading as DLAs': 'A further implication of a strong nebular continuum is that as the ratio of the two-photon emission to the stellar continuum increases, a downturn in the UV emission just redward of Ly α begins to appear (see e.g. Cameron et al. 2023a). This downturn may be perceived to be a damped \n<!-- image --> \nFig. 7.(Left) Observed log 10 ( ξ ion ) as a function of intrinsic log 10 ( ξ ion ) for various SPS models for our fiducial photoionization model parameters. All models follow the same trend despite differences in the underlying spectra. (Right) Observed log 10 ( ξ ion ) as a function of intrinsic log 10 ( ξ ion ) for metal-poor, hot star models for various assumptions of gas density (colors), ionization parameters (line styles), and metallicities (line opacities). This demonstrates that the properties of the nebula dictate the observed ξ ion . Note that each intrinsic ξ ion value in the right panel corresponds to a single stellar temperature for the massive star models while in the left panel, the correspondence is with stellar age. In both panels, the diagonal dotted red line shows the one-to-one relation while the horizontal line represents log 10 ( ξ ion ) = 26 . 0. \n<!-- image --> \nLy α system (DLA). The deficit in UV luminosity near Ly α would raise its EW; however, at high-redshift, an increasingly neutral IGM may absorb emission close to rest-frame 1216 ˚ A. For this reason, high-redshift galaxies with significant two-photon emission may be perceived to have strong absorption from neutral hydrogen (i.e. a DLA). \nTo demonstrate this, we fit the hot star SPS models in the wavelength interval (1250 ˚ A -2000 ˚ A) using a power law + DLA model. For this simple experiment, we assume that the galaxy is at z = 9, account for IGM absorption using the transmission curves from Garel et al. (2021), as well as the JWST/NIRSpec Prism/Clear line spread function 13 . In Figure 8 we show the inferred DLA column density as a function of stellar temperature. A clear trend emerges such that as stellar temperature increases, so does the inferred DLA column. At T ∼ 100 , 000 K, the inferred DLA column is ∼ 10 22 . 5 cm -2 , which is very high compared to known DLAs (e.g. Tanvir et al. 2019; Hu et al. 2023), and consistent with some of the extreme systems observed at high-redshift with JWST (Heintz et al. 2024c; Umeda et al. 2023). \nWe emphasize two key points. First, the exact value of the inferred DLA column is highly subject to the assumed intrinsic SED. For our experiment, we have fit a power law to the continuum at wavelengths < 2000 ˚ A. However, the SED is not a perfect power law, noise can modify the inferred slope, and the wavelength baseline matters for the inferred UV slope. Flatter slopes lead to lower inferred column densities, while steeper slopes require more absorption. The exact shape of the underlying SED also contributes to the inferred DLA properties. \nFig. 8.Strength of the UV downturn in terms of the inferred (masquerading) DLA column density as a function of stellar temperature. The UV downturn strength increases with stellar temperature as the two-photon emission becomes more visible. \n<!-- image --> \nHence Figure 8 should be taken more qualitatively than quantitatively (although in all cases the inferred DLA column will still be exceptionally high for high stellar temperatures). Second, the shape of a spectrum modulated by a DLA differs from the characteristic profile of two-photon emission. The DLA fits are thus not perfect representations of the true spectrum. This effect is likely diminished when noise is present and is subject to the redshift of the object as the spectral resolution of NIRSpec changes with wavelength. It should also be emphasized that strong nebular emission is not mutually exclusive with the presence of a DLA and both may occur simultaneously in the same galaxy. Nevertheless, a clear prediction of a nebula being irradiated by hot massive stars is a steep UV downturn that may masquerade as an extreme DLA. \n<!-- image --> \nFig. 9.(Left) 1500 ˚ A nebular fraction as a function of observed log 10 ( ξ ion ) for various SPS models for our fiducial photoionization model parameters. All models follow the same trend despite differences in the underlying spectra. (Right) Observed log 10 ( ξ ion ) as a function of intrinsic log 10 ( ξ ion ) for metal-poor, hot star models for various assumptions of gas density (colors), ionization parameters (line styles), and metallicities (line opacities). \n<!-- image -->', '2.4. Inferring a high nebular contribution': 'As we have shown in Figure 2, for typical SPS models, the nebular continuum contribution does not exceed 40% unless more massive/hotter stars are included. Here we summarize the key characteristics of a galaxy that may have a high nebular contribution. \n- 1. The presence of a Balmer jump. The only way to avoid a Balmer jump in systems with strong nebular continuum is to increase the temperature to extreme values, well beyond what are typically associated with H II regions. In this case, the strength of the jump becomes much smaller with respect to line emission and can be dominated by other continuum processes that may make detection/identification difficult.\n- 2. A UV slope redder than ∼ -2 . 7. The nebular continuum reddens the intrinsic stellar population and this reddening is dominated by the free-bound emission. In the extreme scenario where the continuum is dominated by two-photon emission, a lower limit of β ∼ -2 . 7 will be measured 14 . Assuming the galaxy is undergoing a starburst, UV slopes redder than ∼ -2 . 3 may require more extreme ionizing sources than those typically assumed in standard SPS models or dust attenuation.\n- 3. A high ξ ion as measured via a Balmer line. If ξ ion is measured to be high, ionizing photons are both being produced at a high rate and being efficiently absorbed by the gas (i.e. not leaking such that f esc ∼ 0). log 10 ( ξ ion ) values above ∼ 25 . 7 -25 . 8 may require more extreme ionizing sources than those typically assumed by standard SPS models. \n14 Note that this value is slightly sensitive to how the wavelength range over which β is measured is set. \n- 4. Strong UV downturns. The appearance of a UV downturn, which can easily be confused with a high column density DLA, scales very strongly with nebular contribution due to two-photon emission. When a UV downturn appears simultaneously with bright Ly α emission, this is an even cleaner signature of strong nebular emission because DLAs are optically thick to Ly α . The presence of Ly α is however not a necessary characteristic as both the IGM, dust, and geometry can absorb Ly α even if the nebular continuum is strong.\n- 5. High H α (or H β ) EWs. For instantaneous bursts of star formation, the H α EW can reach as high as ∼ 3 , 200 ˚ A across all of the different SPS models. Higher values require more extreme ionizing sources. We note that high H α EW is not required for extreme nebular emission as there are numerous scenarios where it can be lower than predicted for the instantaneous burst. For example, the presence of an older stellar population can impact the rest frame optical while having minimal contribution in the UV. Moreover, if the two-photon emission is particularly strong, the H α and H β EW would also be reduced compared to predictions from both nebular-only models that assume Case B recombination or SPS models where the temperature is set by the SPS model. \nAny combination of the above spectral characteristics may indicate a galaxy is a candidate for having strong nebular continuum emission. \nMore specifically, one can even quantify the exact nebular contribution if certain properties of the gas are known. Because nebular emission is only weakly sensitive to the shape of the input ionizing spectra and strongly sensitive to the properties of the nebular gas, if the underlying physical properties of the nebula are \nknown, the nebular contribution at any wavelength 15 can be uniquely determined. Hence, all one needs to infer the nebular fraction is the observed ξ ion and a measure of gas density and temperature. We demonstrate this in the left panel of Figure 9 where we show the nebular contribution at 1500 ˚ A as a function of observed ξ ion (i.e. that which can be measured from the spectrum) for a metallicity of 1% Z ⊙ and a gas density of 100 cm -3 . The shape and normalization of the trend is relatively independent of the SPS model. The largest spread occurs between FSPS and Starburst99 where at an observed log 10 ( ξ ion ) of 25.71, the nebular contribution to the total spectrum differs by 6%. \nIn the right panel of Figure 9 we show the 1500 ˚ A nebular fraction as a function of observed ξ ion for a selection of the metal-poor massive star models while varying gas density, ionization parameter, and metallicity. At typical ISM densities below 10 3 cm -3 the nebular fraction is only sensitive to metallicity (via gas temperature). Above this critical density, gas density becomes the dominant factor as discussed in the previous section. For all gas conditions, ionization parameter is a subdominant factor. \nWe emphasize that there are important caveats to this analysis. First, the gas temperature in the photoionization models is determined via the gas metallicity and the underlying shape of the spectrum. Additional heating terms can modulate this relation such that lower observed ξ ion means a higher nebular fraction. A similar behavior can be achieved by assuming non-solar abundance patterns that decrease cooling (i.e. fixing the oxygen abundance but decreasing other metal species). This is because the two-photon emission is enhanced with respect to free-bound and Balmer emission in this scenario. Likewise, any other physics, e.g. collisional excitation, cooling flows, harder radiation, that increases two-photon emission without impacting the recombination lines will mean that f neb . is under-predicted for a given observed ξ ion . \nThe photoionization models also assume that the nebula is ionization bounded. In the case of density bounded nebulae, the trends can differ. Finally, there are nuances to measuring ξ ion . A fixed conversion between H α (or another Balmer line) and ionizing luminosity is often assumed; however, this conversion depends on temperature. Measuring the Balmer line correctly is often difficult in JWST data where the spectrum can be very noisy in the continuum near H α . Finally, dust can significantly impact 1500 ˚ A luminosity and due to the large wavelength difference between the Balmer lines and 1500 ˚ A, an accurate dust correction is paramount.', '3. A SAMPLE OF HIGH-REDSHIFT JWST GALAXIES WITH STRONG NEBULAR CONTINUUM': "While the theory behind the nebular continuum is well understood, there are very few spectroscopically confirmed examples of high-redshift galaxies with a prominent nebular continuum contributions. The Balmer jump at rest-frame 3465 ˚ A is the most easily identifiable feature and this has only been reported for a select few high-redshift JWST galaxies (e.g. Cameron et al. 2023a; Welch et al. 2024), or in a stack of strong Ly α emit- \nrs (Roberts-Borsani et al. 2024). The lack of reported Balmer jumps is perhaps surprising because SED fits to NIRCam photometry often predict strong nebular continuum emission (e.g. Endsley et al. 2023; Bradaˇc et al. 2024) for galaxies with extreme emission lines. For this reason, we have undertaken a survey of public JWST data to assemble a catalog of galaxies with strong nebular continuum emission. \nMore specifically, we visually inspected all prism spectra in v2 of the DAWN JWST Archive (DJA, Heintz et al. 2024c) at z ≳ 2 . 5 for a possible spectral discontinuity at the location of the Balmer jump 16 . All spectra in this database were reduced using msaexp 17 and full details of the reduction can be found in Heintz et al. (2024c). No other sample selection criteria were applied except for the removal of galaxies with obvious artifacts in the 2D spectra that coincidentally overlapped with the location of the Balmer jump. This means no objects were removed for obvious broad features (that may indicate the presence of an AGN) or any other reason. Our broad selection criteria leads to an extreme diversity in the types of objects that are included. Figure 10 shows example spectra and RGB images of 20 galaxies in the data set. Our sample contains data from 14 different JWST programs as listed in Table 1, with the most coming from JADES GTO (IDs: 1180, 1181, 1210, 3215, PIs: Eisenstein, Leutzgendorf, Bunker et al. 2023a; D'Eugenio et al. 2024), RUBIES (ID: 4233, PI: de Graaff, de Graaff et al. 2024), and UNCOVER (ID: 2561, PI: Labb'e, Bezanson et al. 2022). We emphasize that because we adopt visual inspection, the sample is unlikely to be both complete and pure. The strength of spectral break that we are sensitive to is highly dependent on the signal-to-noise ratio of the continuum and we empirically find that most breaks in the sample have a > 20% reduction in flux between rest-frame 3500 ˚ A, and 4200 ˚ A. \nThe final sample contains 58 objects in the redshift interval 2 . 5 ≲ z ≲ 9. Redshifts are calculated by fitting templates for emission lines to each prism spectrum or grating spectrum when available. The distribution of redshifts is shown in the top panel of Figure 11, with most of the objects falling in the redshift interval 4 ≲ z ≲ 6. The bias towards lower redshifts is likely a reflection of sample size and the ease of detecting the continuum at high enough signal-to-noise to identify the Balmer jump rather than strong nebular emission being more common at lower redshift. The data set spans more than four magnitudes in UV luminosity from M UV > -18 to M UV < -22 as shown in the bottom panel of Figure 11 18 . \nFor each galaxy we measure various emission lines and continuum fluxes. We mask out strong emission lines and fit the remaining data with a seven parameter model using templates for the nebular continuum at various temperatures generated with PyNeb (Luridiana, Morisset & Shaw 2015) and for stellar spectra (Stanway & Eldridge 2018) including both young and old stellar populations. We use an MCMC sampler (Foreman-Mackey et al. 2013) \n16 This possibly omits galaxies where the nebula is at such high temperatures that two-photon emission completely dominates over the Balmer jump. We leave such a search to future work. \nFig. 10.A gallery of example Balmer jump galaxies. We show the observed spectrum of each galaxy as well as an RGB image combining JWST NIRCam filters F444W, F200W, and F150W in the red, green, and blue channels, respectively. Each image is 1' × 1'. \n<!-- image --> \n5000 \nFig. 11.Histograms of redshift (top), 1500 ˚ A UV magnitude (middle), and H α or [O III]+H β EW (bottom) for the 58 galaxies in our dataset. For comparison, we show the [O III]+H β EW distribution from the JWST primal database (Heintz et al. 2024b) of well-detected spectroscopically confirmed galaxies at z > 5 . 5. We have renormalized the JWST primal histogram by the ratio of Balmer jump galaxies to the Primal galaxies (i.e. 58/494). \n<!-- image -->", 'TABLE 1': 'Table of program name, pi, program id, and the number of \ngalaxies from each program that are part of our Balmer jump sample. Programs are sorted descending by the number of galaxies (and then alphabetically by program name). \nto measure the posterior distribution on the contributions of each component the continuum as well as the gas temperature and ages of each stellar component. To measure emission line luminosities, we randomly sample the continuum posterior and subtract it from the observed spectrum. The residuals are then fit with Gaussians using specutils 19 , accounting for the error in the residuals. Line fluxes and their uncertainties are taken as the median and standard deviation of the Monte Carlo samples. \nTypical rest-frame equivalent widths (EWs) for H α in our sample of galaxies is ≳ 1000 ˚ A. This is significantly higher than the 500 -800 ˚ A H α EWs found for the general population of high-redshift galaxies from Roberts-Borsani et al. (2024). A similar trend is seen for [O III]+H β EW where the typical galaxy in our sample has an EW a few times higher than that found in Roberts-Borsani et al. (2024) and similarly that estimated for photometric samples (e.g. Endsley et al. 2023). More quantitatively, comparing to the JWST Primal database (Heintz et al. 2024b), the median [O III]+H β EWis 2.3 × higher in our dataset compared to the typical high-redshift galaxy and the distributions are very different (see Figure 11). Some galaxies in our sample exhibit incredibly high EWs, approaching 5000 ˚ A in [O III]+H β .', '3.1. Star Formation Histories of Balmer Jump Galaxies': 'We have argued that the Balmer jump sample consists of galaxies with bursty star formation histories due to the visibility of the nebular continuum. To demonstrate this further, we have estimated the star formation history of each galaxy using SED fitting. \nWe employ the Dense Basis code (Iyer & Gawiser 2017; Iyer et al. 2019), which reconstructs the star formation history of the galaxy using Gaussian Processes. The star \nFig. 12.Normalized stacked star formation histories of the Balmer jump galaxies. The star formation history of each galaxy is normalized so that the SFR at the present time is 1. The black line and gray shaded region show the median star formation history and the 1 σ scatter. \n<!-- image --> \nformation history is split into five bins of equal stellar mass. We adopt a Jeffreys prior on the shape parameters. We first construct a sparse atlas using a wide range of final stellar mass, SFR, metallicity, and A V . Rather than fitting the spectrum directly (as we find that many SED fitting codes struggle to simultaneously fit the Balmer jump strength and the emission line properties), we measure photometry for all JWST NIRCam wide and medium bands from the spectrum and fit the photometry. This assumes that the spectra are properly flux calibrated as we do not rescale to NIRCam fluxes apart from what was applied at the time of reduction in the DJA, i.e. no additional slit loss corrections are applied. Uncertainties on the photometry are calculated by Monte Carlo resampling the observed spectrum from the error distribution on each pixel. Because the redshift is kept fixed in the fit to that measured from the spectrum, we consider only filters that are redward of (and do not include) Ly α . From the sparse atlas, we select the best fit model and create a refined atlas adopting priors that are 2 σ deviations around the best model. Our final best model comes from the refined atlas. \nIn Figure 12, we show the normalized star formation history stack of all of the Balmer jump galaxies over the past 200 Myr. For each galaxy, we normalize the star formation history by the SFR at the current time. The histories are then interpolated onto the same time axis and we show the median and 1 σ deviations. \nThe star formation histories appears to be nearly a delta function at the present time. The median history prefers almost no star formation beyond ∼ 35 Myr and the present day SFR of the median history is > 200 × that at 20 Myr. This exercise clearly demonstrates that the Balmer jump sample represents a unique class of galaxies undergoing extreme bursts of star formation, which potentially makes them analogues of the extreme highredshift galaxy population.', '3.2. The Characteristic Spectrum of Strong Nebular Emission': 'We begin by stacking the galaxies to increase the signal-to-noise. Each prism spectrum is de-redshifted to the rest-frame and we use the FluxConservingResampler from specutils to resample the spectra onto a common wavelength grid. Resampled spectra are then normalized to have a flux density of unity at 3200 ˚ A. Mean and median stacks are shown in Figure 13. Scatter in the median stack is calculated from 10,000 bootstrap resamples. Apart from exhibiting very strong emission lines, both in the rest-frame UV and optical, the most notable feature is the strong spectral discontinuity at 3645 ˚ A which demonstrates that these galaxies exhibit a very clear Balmer jump. \nOnly minor differences are apparent between the mean and median stacks. Both exhibit remarkably little variation in UV flux as a function of wavelength between 1500 ˚ Aand 3000 ˚ A, indicating a spectral slope of β ∼ -2. The strength of the Balmer jump is also similar and both stacks exhibit a downturn in the UV, just redward of Ly α . Emission line fluxes tend to be slightly higher for the mean stack. This is particularly true for C IV λλ 1550 and Ly α , where we find that fewer than half of the Balmer jump galaxies are strong Ly α emitters. \nThe stack exhibits strong rest-frame UV and optical emission lines, consistent with the individual EWs measured in our sample. There is also evidence for slightly rarer emission lines such as the combination of He II λ 4686 and the nearby [Ar IV ] λλ 4711,4740 doublet. The [O II ] quadruplet auroral line feature at rest-frame 7320-7330 ˚ A is clearly visible as was also recently seen in the lower-redshift stack of Strom et al. (2023). Numerous He I lines are present as well as weak [Ar III ] at 7136 ˚ A.', '3.3. The nebular contribution in Balmer jump galaxies': 'Now considering individual galaxy spectra, one of the key predictions of the SPS models is that the nebular contribution at 1500 ˚ A maximizes at ∼ 40% unless hotter/more massive stars are included. Having a sample of galaxies with bursty star formation histories and clear nebular continuum contributions in the optical, we can test whether this ∼ 40% upper limit holds for real galaxies. \nInferring the nebular contribution is not straightforward unless the detailed properties of the nebula are known. Robust temperatures, densities, and ξ ion values are not available for the vast majority of galaxies in our sample and thus we resort to an alternative technique. In principle, the shape of the continuum encodes the contributions of stellar and nebular emission. However, the key confounding factor occurs in the UV where downturns can occur both due to neutral hydrogen absorption and two-photon emission. We thus fit the galaxies twice, once using our seven parameter MCMC fitting code and a second time allowing for a DLA in the fit. We then focus on sources where the nebular continuum is predicted to be strong independent of the model.', '3.3.1. Nebular dominated galaxies': "Based on this procedure, a few galaxies immediately appear to require an unusually strong nebular contribution. The first galaxy is 1210 13176 at z = 5 . 943 (also \nFig. 14.Comparison of 1210 5217 at z = 4 . 890 with potential nebular-dominated galaxies 2189 7807 ( z = 5 . 387) and GS 9422 ( z = 5 . 946). All three show a clear Balmer jump and a very similar UV downturn. Spectral modeling predicts a very high nebular fraction in each of these objects. \n<!-- image --> \n- \nFig. 13.Mean (cyan) or median (magenta) stack of all 58 galaxies in our sample. We highlight the spectral discontinuity at the location of the Balmer jump. The shaded region around the median stack shows the 5th-95th percentile distribution calculated from 10,000 bootstrap resamples. This demonstrates that no single galaxy has an extreme impact on the stack. \n<!-- image --> \n- \nreferred to as NDG 9422 in Cameron et al. 2023a), the spectrum of which is shown as the blue line in Figure 14 and the spectral properties can be found in Table 2. Our template fitting indicates a 1 σ lower limit of 80% for the nebular contribution to the 1500 ˚ A luminosity. This galaxy is discussed at length in Cameron et al. (2023a) and the spectrum was first described in Saxena et al. (2024), so here we only highlight the key features. \nWhat makes 1210 13176 interesting is that not only does it exhibit a very strong downturn in the UV, which if interpreted as a DLA would imply an extreme neutral column density of ∼ 10 23 cm -2 , it also has strong Ly α emission. If a such a high column density DLA is present, it remains unclear why such a large fraction ( > 25%, Saxena et al. 2024) of the Ly α escapes, especially considering that the galaxy is very compact with no visible companion that could otherwise account for the Ly α emission. Interestingly, Cameron et al. (2023a) demonstrated that the shape of the continuum is consistent with the expectations of the nebular continuum at a gas temperature similar to that measured from [O III ] λ 4363/[O III ] λ 5007. The log 10 ( ξ ion / erg -1 Hz) for this galaxy is 25.8, indicating that the nebular contribution in the UV could be very strong. While Cameron et al. (2023a) present numerous physical mechanisms to explain the shape of the spectrum, hot stars remain a natural explanation for the continuum shape of this galaxy. \nMore recently, Schaerer et al. (2024) have argued, based on Case B recombination, that the H β equivalent width is too low for the spectrum to be fully nebular dominated. We argue that the discussion presented in Schaerer et al. (2024) does not necessarily apply to this object. It is clear from the Balmer decrement that the Case B assumption is not valid for this galaxy (e.g. Yanagisawa et al. 2024). Moreover, the temperature inferred from both the [O iii ] auroral line ratio and the Balmer jump is T e > 18 , 000 K, much higher than the typically assumed 10 4 K. At these temperatures, the expected Case B H β EW is 860 ˚ A (somewhat lower than the ∼ 1,000 ˚ A suggested in Schaerer et al. 2024, under the assumption of T e = 15 , 000 K). \nMore importantly, as shown in Appendix A, the twophoton emission still contributes significantly in the restframe optical. Collisional excitation to the n = 2 state can significantly enhance the two-photon continuum, further lowering the observed H β EW. This has, in fact, been predicted for hot and metal-poor environments Raiter, Schaerer & Fosbury (2010); Mas-Ribas, Dijkstra &Forero-Romero (2016). The best-fit model of Cameron et al. (2023a) predicts such an enhancement which naturally explains both the observed H β and H α EW. Finally this system shows clear oxygen, carbon, and neon emission, which indicates that there would have been a previous generation of stars. This would further lower the Balmer EWs without impacting the fit in the UV. Thus, as presented in Cameron et al. (2023a), we stress that the spectrum of 1210 13176 is fully consistent with having a dominant contribution from the nebular continuum in the UV. \nOther studies have suggested alternative explanations for the observed UV downturn and concomitant Lyα detection. Based on the fact that several other galaxies in \nthe same field as 1210 13176 have DLAs, Heintz et al. (2024a); Terp et al. (2024) argue that a lower-redshift DLA at z ∼ 5 . 4 in a foreground galaxy cluster would offset the Ly α absorption trough, which could allow Ly α to leak and reconcile the differences in shape between the observed UV downturn and a DLA. However, this solution would imply an even larger gas column density. Likewise, Tacchella et al. (2024); Li et al. (2024) argue that an obscured black hole with an offset narrow line region and a large DLA column could provide a similar spectral shape. Because of the UV downturn and strong Ly α , any solution to this puzzle that invokes a DLA requires a 'fine-tuned' geometry. Therefore we argue that, for a single object these could be reasonable explanations; however, if more galaxies like 1210 13176 are identified, such models become less favorable. \nInterestingly, a second galaxy in our sample, 2198 7807 at z = 5 . 387 from the Cycle 1 program: 'Quiescent or Dusty? Unveiling the Nature of Red Galaxies at z > 3' (Program ID: 2198; PIs: L. Barrufet & P. Oesch; Barrufet et al. 2024) shows a very similar UV downturn and strong Ly α emission as 1210 13176 (see the orange line in Figure 14. This galaxy is remarkably similar to 1210 13176 in UV magnitude, β slope, and ξ ion , and has similarly high emission line EWs (see Table 2). There is no indication of dust in the object and our 1 σ lower-limit on the 1500 ˚ A nebular fraction of 60%, much higher than that of typical SPS models (see Figure 2). As these two spectra are so similar, we refrain from a more detailed discussion since the same arguments presented for 1210 13176 in Cameron et al. (2023a) apply to 2198 7807. However, we briefly highlight a few key differences. The C IV λλ 1550 emission is much weaker in 2198 7807 and there is no evidence for He II λ 1640, which suggests the emission is even less likely to be powered by a black hole 20 . [O II ] λλ 3727 is also stronger in 2198 7807 compared to 1210 13176, suggesting it is less likely that the emitting H II regions are density bounded. Because 2198 7807 is at a redshift consistent with or lower than the presumed proto-cluster that was used in Heintz et al. (2024a); Terp et al. (2024) to explain both the UV downturn and the Ly α emission in 1210 13176, the same argument cannot apply to 2198 7807, unless there is another foreground cluster at even lower redshift. This would be a remarkable coincidence given that the shapes of the UV spectra are so similar. Finally, fine-tuned models that invoke strong DLAs (e.g. Tacchella et al. 2024; Li et al. 2024) are less appealing given that there is more than one spectra with an almost identical UV shape. \nA third galaxy that is predicted to have a high 1500 ˚ A nebular fraction is 1210 5217 at z = 4 . 89 (green line in Figure 14). This galaxy was presented in Boyett et al. (2024) where they noted an extremely high log 10 ( ξ ion ) value of 25 . 9 ± 0 . 1 and no dust. Such a value is much higher than expected for standard SPS models when the nebular continuum is taken into account (see above). This value is also consistent with that measured in our work (26 . 04 +0 . 03 -0 . 03 ) to ∼ 1 σ . This UV luminosity of this galaxy is much fainter by two magnitudes compared to the previous two discussed in this section, but it has a \n20 Although we emphasize that for 1210 13176, nebular diagnostics suggest that this object is not AGN dominated or at best a composite (Cameron et al. 2023a). \nProperties of five galaxies predicted to have nebular contributions far exceeding that predicted from SPS models. We list the galaxy ID, redshift, 1500 ˚ A UV magnitude, UV slope, Balmer decrement, EWs of H α and [O III]+H β , H II region temperature predicted from the MCMC fitting of the spectra, ionizing photon production efficiency, 1 σ lower limit on the estimated nebular contribution at 1500 ˚ A, and finally the DLA gas column density or 1 σ upper limit. For each galaxy we list two values of ξ ion computed using either H α (top) or H β (bottom). The first three galaxies may host a population of massive stars while the bottom two spectra of different regions of a gravitationally lensed arc appear to be consistent with purely Case B recombination. \nTABLE 2 \nvery similar UV slope of -1 . 99 +0 . 07 -0 . 09 . As this is much redder than that expected from SPS models for a single burst and given the lack of dust in this object, a high UV nebular fraction is a likely explanation. Moreover, this galaxy has extreme emission line equivalent widths. The equivalent widths of H α and [O III ] + H β are both ≫ 3 , 000 ˚ A, the highest H α EWamong the entire Balmer jump sample. No He II λ 1640 or λ 4686 is detected in the prism or gratings, despite a clear grating detection of O III λ 1666, and strong line ratios indicate that this source is not a candidate AGN (Scholtz et al. 2023; Curti et al. 2024). \nWhile 1210 5217 is not a Ly α emitter, it also exhibits a strong UV downturn that has a remarkably similar shape to 1210 13176 and 2198 7807 (Figure 14). Even allowing for a DLA in our MCMC fit, the shape of the UV downturn prefers a nebular solution and hence we find a 1 σ lower limit of 63% for the nebular fraction at 1500 ˚ A. Like the other two galaxies, the gas temperature is predicted to be much higher than local H II regions, which could enhance the nebular contribution in the UV. We emphasize that a lack of Ly α emission does not preclude 1210 5217 from being nebular dominated. The IGM is not completely transparent at these redshifts (e.g. Inoue et al. 2014) and the actual emission profile of Ly α is highly viewing-angle dependent, even for galaxies without DLAs (e.g. Blaizot et al. 2023). The presence of massive stars is a natural explanation for the UV downturn and the relatively red β slope given the high ξ ion . If such stars are common at high redshift, we may expect more to look like 1210 5217 than either 1210 13176 or 2198 7807 due to the sight-line variability of Ly α and the fainter UV magnitude.", '3.3.2. No young stars in the shutter?': "One of the key features of the three galaxies in the previous section is that their Balmer decrements indicate that they are dust-free but their UV slopes are redder than that of typical SPS models. Redder slopes can be achieved if the intrinsic ionizing photon production efficiency is higher than the SPS models which enhances the nebular contribution at all wavelengths. Such is the case when hot/massive stars are present (see Section 2.2.3 and \n- \nalso Schaerer et al. 2024). However, even in the case of hot/massive stars, if the gas density is ≲ 10 4 cm -3 , the UV slope for a young stellar population cannot be redder than β ∼ -1 . 7. If a Balmer jump is present in the spectrum and the Balmer decrement indicates that there is no dust in the system, a β redder than ∼ -1 . 7 would either imply that the gas is either exceptionally dense, suppressing the two-photon continuum, or there is no underlying young stellar continuum and we are observing pure nebular emission in the UV. As we show in Appendix A, in the latter scenario, β from pure Case B recombination can approach values of ∼ -1 . 3. \nThe latter scenario where there is little or no young stellar population can occur if the NIRSpec slit only covers the nebula. We find two example spectra where this may be occurring. In Figure 15 we show an image of a gravitationally lensed arc (The Cosmic Gummy Worm) being lensed by the Abell 2744 cluster. Two separate JWST programs (2561, PI: Labb'e and 2756, PI: Egami) placed shutters on different parts of the arc and the spectra are shown in the bottom two panels of Figure 15. This source has already been discussed extensively in the literature (Vanzella et al. 2022; Lin et al. 2023) and is thought to host proto-globular clusters. Most notably, Lin et al. (2023) provide a map of UV slope across the arc and find values ranging from -2 . 5 to -1 . 5 with the bluest values centered on the location of where there are thought to be young stars. Both previous analyses conclude that this system likely hosts a young, metal-poor stellar population with relatively little dust. \nThe key features of these spectra (see Table 2) are that their Balmer decrements are consistent with Case B values indicating that little or no dust is present in the object and the log 10 ( ξ ion ) values are very high at 25 . 97 or 26 . 03. The UV slopes are very red with values of -1 . 58 and -1 . 33, which is much redder than can be generated by even the hot star models, unless extreme densities are assumed. Both spectra exhibit exceptionally high [O III ]+H β EWs > 3 , 000 ˚ A. \nWhat is unique about our MCMC fits to these spectra is that not only do they require a very high nebular fraction at 1,500 ˚ A that is significantly higher than \nwhat can be produced with typical SPS models, especially 2756 301 where the model prefers a fully nebular solution 21 , but the model also simultaneously infers a very strong DLA column between 10 22 . 50 -10 22 . 75 cm -2 . These two spectra are the only ones in our sample that exhibit such behavior. The continuum shapes of these spectra are consistent with being nebular continuum from Case B recombination with a DLA. \nIn fact, this system is already known to host a DLA based on shorter wavelength data from MUSE that covers Ly β (Lin et al. 2023), with those authors inferring a DLA column of N H ∼ 10 21 . 8 cm -2 . The DLA column likely varies across the system which may explain why our measurement, which only covers part of the arc, finds a higher column density. \nAs speculated by Lin et al. (2023), our results support a scenario in which the red portions of the arc are dominated by the nebular continuum. Spectra such as 2561 17467 and 2756 301 are extreme examples where gravitational lensing can help distinguish the gas from the underlying continuum source and hence the spectra may appear as being nebular dominated. More such examples exist in the DJA; however, none are as exemplary as those shown here. The high measured ξ ion values are thus not necessarily reflective of the underlying ionizing source population, but rather a manifestation of separating a continuum source from the nebula. \nIt should be emphasized that this system is not a Ly α emitter. This is a clear example of a galaxy with a very high, spatially varying DLA column density, and yet the spectrum looks completely different from the objects above with strong UV downturns and Ly α emission. If the ISM of this object has any comparison to 1210 13176 and 2198 7807, it makes it even more peculiar that if a high column density DLA was the explanation for their spectrum, Ly α emission can leak significantly from those objects but not for 2561 17467 and 2756 301.", '4. IMPLICATIONS FOR REDSHIFT > 10 GALAXIES': "Both the theoretical and observational exercises presented here provide insight into the expected properties and physical nature of galaxies that are undergoing extreme bursts of star formation. Nearly all numerical simulations that attempt to resolve the ISM predict that star formation becomes increasingly bursty at high-redshift (e.g. Ma et al. 2018; Katz et al. 2023; Pallottini et al. 2022). Moreover, the leading explanation for the highredshift bright galaxy problem is that surveys are sampling the extreme 'tip-of-the-iceberg' of bursty star formation (e.g. Ren, Trenti & Mason 2019; Mason, Trenti & Treu 2023; Shen et al. 2023; Sun et al. 2023; Kravtsov & Belokurov 2024). For both of these reasons, the nebular continuum is thus expected to become increasingly important at z > 10. \nIn this Section, we discuss some of the physical properties of known z > 10 galaxies in the context of the models we have presented for the nebular continuum which may help elucidate their physical origin. We emphasize that in order to explain the high-redshift bright galaxy problem, only a subset of the known objects would require \n21 Note that the fit for 2756 301 requires a temperature extrapolation for the nebular continuum and there is a systematic uncertainty, particularly related to temperature associated with this fit. \n<!-- image --> \nFig. 15.(Top) RGB image of the Cosmic Gummy Worm combining JWST NIRCam filters F444W, F200W, and F150W in the red, green, and blue channels, respectively. Overlaid in magenta are the locations of the MSA shutters. (Bottom) Prism spectra of two regions of the gravitationally lensed arc at z = 3 . 99, both showing red continua and a Balmer jump. \n<!-- image --> \ndeviations from standard galaxy formation physics. The expectation is that many objects will be in agreement with standard SPS models of bursty galaxy formation, while others may require a different interpretation.", '4.1. Do the bright high-redshift galaxies have strong nebular continuum emission?': 'Except for extreme examples like GNz11 where the continuum is detected and a Balmer jump detection is not conclusive 22 , the signal-to-noise of the spectra of known z > 10 galaxies at rest-frame 3,645 ˚ A is too low to see a Balmer jump. Stacking all publicly available spectra in the DJA with z > 10 also leads to a too low signal to detect a Balmer jump. Thus we discuss these high-redshift galaxies using more indirect indicators of strong nebular emission. \nUnfortunately, very few properties can be measured for most z > 10 galaxies apart from their UV magnitudes, continuum slopes, and an occasional emission line. Thus in Figure 16, we plot β as a function of M UV for all galaxies in the JWST Primal sample (Heintz et al. 2024b). The region shaded in cyan show the expected \n22 In one pointing, a spectral discontinuity appears at the location of the Balmer jump. Since not all observations show this, the presence of a Balmer jump is inconclusive. \nFig. 16.UV continuum slope, β , as a function of M UV for all z > 10 galaxies in the JWST Primal database (Heintz et al. 2024b) are shown as black points. The region shaded in cyan show the expected UV slopes from all of the different SPS models that we test. Slopes bluer than this (shaded in black) would require a non-zero escape fraction, while redder slopes (magenta), would indicate the presence of either hotter stars, the combination of hotter stars and high gas density, or dust. Red points represent the three galaxies in our Balmer jump sample that appear to have a very strong nebular contribution at 1500 ˚ A. \n<!-- image --> \nUV slopes from all of the different SPS models that we test. Slopes bluer than this (shaded in black) would require a non-zero escape fraction, while redder slopes (magenta), would indicate the presence of either hotter stars, the combination of hotter stars and high gas density, or dust. \nMost of the z > 10 galaxies fall close to the cyan region indicating that vanilla nebular continuum presents a reasonable explanation. Recall that this does not rule out the presence of a flatter upper-mass slope, as long as the upper mass does not increase much above ∼ 100 -300 M ⊙ . Three galaxies have β < -2 . 55 which would imply a weaker nebular continuum contribution. This could be caused by a non-zero escape fraction, or if the massive stars output fewer ionizing photons than expected compared to the SPS models that we consider. Finally, 3/17 galaxies have β > -2 . 2, although within the uncertainties, two are consistent with the cyan region. The outlier is the z = 11 . 5 23 galaxy (CEERS2 588) that has β = -1 . 79 ± 0 . 13, more than 3 σ away from the vanilla prediction. \nPerhaps the simplest explanation for this galaxy is that it is dusty. However, in the context of theoretical models, this explanation is unsatisfying because most pre-JWST predictions of the high-redshift bright galaxy number counts from full-box simulations catastrophically fail in reproducing observations (e.g. Finkelstein et al. 2023) except for those that adopt a top-heavy IMF (Cowley et al. \n23 Note that this is the same galaxy as reported in Arrabal Haro et al. (2023) with MSA ID 10 (CEERS2 588). The Lyman break and possible detection of [O II] λλ 3727 provide different redshifts. Following the JWST primal database, we adopt the redshift from the break. \n2018; Lu et al. 2024) or those with enhanced efficiency of star formation (e.g. McCaffrey et al. 2023; Katz et al. 2023). Since dust lowers the observed UV luminosity, if obscuration was the explanation for this galaxy, the intrinsic M UV would be much greater, exacerbating the tension between theoretical models and observations. In contrast, appealing to hotter stars with T ≳ 80 , 000 K and high gas densities of ∼ 10 4 cm -3 (or even cooler stars at higher gas density) would also naturally explain this object. Moreover, the inclusion of hotter stars would also help reconcile why the object is so bright. \nHowever, as discussed previously, the hot/massive star solution implies other behavior. First, we might expect a small UV downturn. At the stellar temperatures considered here, the inferred DLA column density would be ≳ 10 21 . 8 cm -2 . Second, depending on the exact shape of the ionizing spectrum, one might expect high EW UV lines. The metal lines can easily be weakened by appealing to low metallicities; however, the key uncertainty is the strength of He II λ 1640. However, the regions of the spectrum at wavelengths below the He + ionizing threshold is arguably highly uncertain (e.g. Kewley, Nicholls & Sutherland 2019). For this reason, it is unclear how much flexibility there is in the He II λ 1640 predictions. Nevertheless, CEERS2 588 remains a very viable candidate for hosting hot, metal-poor stars at a high gas density. This same explanation can also be applied to the recently-discovered z > 14 candidate that has a UV slope of ∼ -2 . 2 ± 0 . 7 and very low EW UV lines (Carniani et al. 2024) and also GS-z12-0 from Curtis-Lake et al. (2023) at z = 12 . 6 with a measured β = -1 . 84 ± 0 . 19. \nIn summary, all but two z > 10 galaxies are consistent with having strong nebular continuum emission. Most galaxies do not appear to require hot massive stars in very high density gas, but with limited information, we cannot rule out massive stars with normal ISM densities, a top-heavy IMF, or contributions from an extended star formation history.', '4.2. UV downturns and β slopes at z > 10 ?': "Akey signature of the presence of a strong nebular continuum is the appearance of downturns in the spectrum in the UV, just redward of Ly α , or more generally deviations from a power-law slope produced by the stellar continuum. Recently, Heintz et al. (2024b) defined the 'Ly α damping parameter' that encapsulates UV downturns when the value is strongly positive. While Ly α emission can impact this metric, at z ≳ 10, the IGM is likely to absorb the vast majority of Ly α emission and thus, the difference between the measured damping that is expected from the IGM provides a clear probe of the spectral deviation from a pure power-law. \nHeintz et al. (2024b) show that the fraction of galaxies with perceived absorption that would imply a neutral column density in excess of 10 21 cm -2 mildly increases from ∼ 60% at z ∼ 6 to 65% -90% at z > 8. Moreover, Umeda et al. (2023) show that all of their z > 10 galaxies have UV downturns corresponding to absorption by column densities > 10 22 cm -2 . Hence, there is tentative evidence that UV downturns, in particular strong ones, are more common at high-redshift. \nIn Figure 17, we plot the Ly α damping parameter ( D Ly α ) as a function of UV slope for the various SPS models (left) and the individual metal-poor star models \n<!-- image --> \nFig. 17.Ly α damping parameter, D Ly α , as a function of UV slope compared to various SPS models (left) and for hot metal-poor star models at various gas densities (right). Black data points with error bars represent individual z > 10 from the JWST Primal database (Heintz et al. 2024b). Red points represent the three galaxies in our Balmer jump sample that appear to have a very strong nebular contribution at 1500 ˚ A. \n<!-- image --> \nat various ISM densities in the range 10 2 -10 5 cm -3 (right). Note that for this calculation, we have recalculated β to match the definition used in Heintz et al. (2024b), although this results in only a marginal change. We also apply IGM absorption assuming the highest redshift curve from Garel et al. (2021). \nWhile many of the objects are consistent with the SPS models, two z > 10 galaxies seem to be outliers for having high D Ly α ∼ 90 ˚ A and a relatively flat UV slope, slightly bluer than -2 . 0, consistent with the galaxies in our Balmer jump sample that seem to exhibit large nebular contribution at 1500 ˚ A 24 . The standard SPS models, even with IGM absorption, cannot reproduce these galaxies and hence very dense DLAs would be required as was the interpretation in Heintz et al. (2024b); Umeda et al. (2023). However, the right panel of Figure 17 shows that these galaxies are very consistent with the hot star models and a low-density ISM. Since our CLOUDY calculations were stopped when the electron fraction reached 1%, no additional absorption is required as the perceived damping is entirely due to H I two-photon emission. These galaxies are a z = 12 . 5 galaxy from JADES (Curtis-Lake et al. 2023) and a z = 11 . 4 galaxy from CEERS (Arrabal Haro et al. 2023). \nAnother cluster of high-redshift galaxies appears to have much lower D Ly α and redder β than predicted by the SPS models. As we show in the right panel of Figure 17, these can be reconciled with increasing gas density up to 10 5 cm -3 (although not necessarily hot massive stars). In this case, two-photon emission is suppressed which reduces the amount of perceived absorption. This can also be reconciled with Ly α emission, though this is rare at these redshifts (although c.f. Bunker et al. 2023b). \nFinally, there is one object (1210 14220) that has \nD Ly α ∼ 80 ˚ A and β ∼ -3 which cannot be explained by any of our models. This would imply a dust-free DLA as to not redden the β , but the DLA must be far enough from the continuum source to keep the nebular emission low enough as to not redden the UV slope. The physical nature of this particular galaxy thus remains uncertain.", '4.3. Considerations for Finding Population III Stars at High-Redshift': 'The stellar spectra of hot massive stars are extremely blue (e.g. Schaerer 2003) and without a nebular contribution, we expect Pop. III galaxies to have β < -3 . 0. However, the nebular emission is clearly non-negligible and the search for Pop. III galaxies has primarily consisted of looking for blue ( β ≲ -2) galaxies that show strong He II λ 1640 emission (e.g. Trussler et al. 2023). However, as discussed in Cameron et al. (2023a) and shown again here (see also Trussler et al. 2023; Zackrisson, Inoue & Jensen 2013; Inoue 2011), the two-photon emission is necessarily dominant if the stellar population consists of stars with T ≳ 100 , 000 K. Hence if strong He II emission is found but no UV downturn is present (e.g. Saxena et al. 2020), either the ISM of the galaxy is at very high density (which suppresses the two-photon emission), or the galaxy is not hosting hot massive stars. The latter does not rule out the galaxy from being Pop. III; however, this would imply that not all Pop. III stars are extremely hot and might populate a lower range of masses, in agreement with some numerical simulations (e.g. Stacy, Bromm & Lee 2016). In the case of highdensities, the strength of the UV downturn is correlated with the UV slope such that weaker downturns imply redder spectra. We re-emphasize here that the ISM density surrounding Pop. III stars is unknown. ISM densities surrounding massive stars are typically regulated by a combination of photoheating, radiation pressure, and stellar winds. However in the Pop. III scenario, winds are much weaker due to the lower opacities of the stellar at- \ne (e.g. Vink, de Koter & Lamers 2001), radiation pressure is caused by UV absorption and Ly α scattering rather than IR photons scattering on dust (e.g. Kimm et al. 2018), and finally, the ISM is at much higher pressure due to inefficient cooling (e.g. Omukai et al. 2005), which lowers the pressure gradient between the H II region and the PDR. For these reasons, one may reasonably expect that the gas densities around Pop. III stars may be considerably higher than in the local ISM. We therefore stress that selection criteria for high-redshift Pop. III galaxies should not rule out systems with red β slopes approaching values of -1.', '5. CONCLUSIONS': "Understanding the physical nature of galaxies at cosmic dawn remains a primary goal of modern cosmology and galaxy formation. Arguably one of the biggest surprises of early high-redshift JWST observations is the number density of both photometrically selected and spectroscopically confirmed objects at z > 10 as their number counts are significantly higher that expected from many pre-JWST models (e.g. Finkelstein et al. 2023; Harikane et al. 2024; Leung et al. 2023; Chemerynska et al. 2023). One of the leading explanations for this anomaly is UV variability driven by extremely bursty star formation histories (e.g. Ren, Trenti & Mason 2019; Mason, Trenti & Treu 2023; Shen et al. 2023; Sun et al. 2023; Kravtsov & Belokurov 2024). While most numerical simulations with a resolved ISM predict that star formation histories at early epochs are much burstier than in the low-redshift Universe (e.g. Ma et al. 2018; Katz et al. 2023), there remains debate on whether this is enough to reconcile theory with observations (Pallottini & Ferrara 2023; Gelli, Mason & Hayward 2024). Hence a detailed study of galaxies with extremely bursty star formation histories is warranted. \nOne method for probing galaxies with extremely bursty star formation histories is strong nebular emission, and specifically, the visibility of the nebular continuum. More generally, the nebular continuum can significantly modify the observed spectrum of a galaxy, by reddening its spectral slope (e.g. Bouwens et al. 2010; Dunlop et al. 2013; Cullen et al. 2024; Topping et al. 2024a), contributing to the 1500 ˚ A luminosity of the galaxy, and causing a downturn in the UV spectrum, just redward of Ly α (e.g. Fosbury et al. 2003; Raiter, Schaerer & Fosbury 2010; Cameron et al. 2023a). For this reason, we began with a theoretical exercise, adopting five different stellar population synthesis models to quantify how changes in the underlying stellar population synthesis model, IMF slope, stellar temperature, and nebular conditions impact the observed spectra of galaxies under the assumption of an extreme burst of star formation. The conclusions from our theoretical exercise can be summarized as follows: \n- 1. 1500 ˚ A UV luminosities can increase by up to ∼ 0 . 7 magnitudes in the presence of extreme bursts of star formation for published stellar population synthesis models at young stellar ages ( ≲ 5 Myr). This result holds even when considering realistic star formation histories from state-of-the-art numerical simulations. If common at high-redshift, the nebular continuum contribution at 1500 ˚ A has the potential to significantly reduce the amount of\n- 'burstiness' required to reproduce the measured abundance of bright high-redshift galaxies.\n- 2. Assuming a top-heavy IMF only marginally increases the nebular contribution to the 1500 ˚ A UV luminosity and only slightly reddens the SED as long as the upper-mass limit of the IMF remains fixed. In contrast, assuming hotter (more massive) stars can redden UV slopes to values of β > -2 . 0. When the gas density is allowed to increase above the critical density of H I two-photon emission, β slopes around hot, massive stars can approach ∼ -1. Hence, searches for Population III galaxies should not rule out such red objects.\n- 3. When the nebular continuum is strong, twophoton emission can masquerade as DLA absorption with apparent column densities reaching up to ∼ 10 23 cm -2 . When UV downturns are observed at high-redshift, the DLA solution should be compared with predictions from strong nebular continuum emission.\n- 4. Because ξ ion measurements adopt the observed 1500 ˚ A UV luminosity, observationally inferred ξ ion values deviate from intrinsic ξ ion at values > 10 25 . 5 erg -1 Hz. Assuming normal nebular conditions, published SPS models predict that measured values of ξ ion should never be ≳ 10 25 . 81 erg -1 Hz. When the effects of dust can be rules out, values above this limit would imply either 1) non-standard ISM conditions, 2) an intrinsic ξ ion ≳ 10 26 . 2 erg -1 Hz which would point to objects like hot, massive stars or an AGN, or 3) that the spectrum does not fully cover both the continuum source and the nebular emission. \nWe then proceeded to study bursty star formation observationally by compiling a sample of 58 galaxies from the Dawn JWST Archive (DJA, Heintz et al. 2024c) in the redshift interval 2 . 5 < z < 9 that have clear Balmer jumps in their R ∼ 100 JWST NIRSpec prism spectra. This sample increases the number of reported highredshift JWST Balmer jumps by a factor of ∼ 30 × , confirming some expectations from SED fitting (e.g. Endsley et al. 2023). The conclusions from our observational exercise can be summarized as follows: \n- 1. SED fitting using non-parametric star formation histories indicates that nearly all sources in the sample are well described by an extreme recent burst of star formation, consistent with expectations for when the Balmer jump becomes visible.\n- 2. Five spectra in the sample are consistent with having a nebular dominated spectrum - significantly more nebular emission is preferred in the continuum fit to the observed spectrum compared to expectations from SPS models. Two of these spectra come from the same strongly gravitationally lensed galaxy at z ∼ 4 and have red UV slopes, but have Balmer decrements consistent with Case B recombination expectations. We argue that in these spectra, the MSA slit preferentially covers the nebular gas and not the underlying ionizing source due to \nthe high effective spatial resolution enabled by lensing. In contrast, three other spectra have β ∼ -2, two of which have Ly α emission in addition to a Balmer jump, and the third has an H α EW of ∼ 3 , 300 ˚ A. All three have a remarkably similar shape for their UV downturn that is consistent with being two-photon emission from H I . The leading explanation is that these galaxies host hot, massive stars as the similarities across multiple galaxies seem to disfavor fine-tuned geometric scenarios that require a DLA. \nFinally, we discussed some of the recently detected z > 10 galaxies in the context of our nebular continuum models. While many galaxies are consistent with expectations for ordinary bursty star formation combined with nebular emission, some seem to deviate by either having excessive downturns in the UV and redder UV slopes. Some of these galaxies are consistent with hot massive star models, which may help explain some of the highredshift bright galaxy problem. \nBecause the nebular continuum is sensitive to both ISM density and temperature, a sample such as this provides a unique opportunity to constrain the properties of the high-redshift ISM using emission that has different sensitivities to temperature and density compared to existing strong-line methods. Ideally one could constrain the abundance of Balmer jump galaxies at high-redshift; \nhowever, the visual selection and ill-constrained selection functions of the various observational programs used in this work currently inhibits accurate estimates for this quantity. Topics such as this will be explored in future work.", 'ACKNOWLEDGMENTS': "HK thanks Andrey Kravtsov for insightful comments and thoughtful discussions. We sincerely thank the PIs and Co-Is of the JWST programs where spectral data was made publicly available on the DJA. We refer interested readers to the following papers for survey descriptions regarding the spectral data: Bunker et al. (2023a); D'Eugenio et al. (2024); Bezanson et al. (2022); Barrufet et al. (2024); de Graaff et al. (2024); Finkelstein et al. (2024); Glazebrook et al. (2024); Pierel et al. (2024); Siebert et al. (2024); Maseda et al. (2024). This work is based in part on observations made with the NASA/ESA/CSA James Webb Space Telescope. The data were obtained from the Mikulski Archive for Space Telescopes at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-03127 for JWST. These observations are associated with programs listed in Table 1. 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J. et al., 2023, arXiv e-prints, arXiv:2311.12234 Yajima H., Li Y., Zhu Q., Abel T., 2012, MNRAS, 424, 884 Yanagisawa H. et al., 2024, arXiv e-prints, arXiv:2403.20118 Yung L. Y. A., Somerville R. S., Finkelstein S. L., Wilkins S. M., Gardner J. P., 2024, MNRAS, 527, 5929 Zackrisson E., Inoue A. K., Jensen H., 2013, ApJ, 777, 39 \n- Zavala J. A. et al., 2024, arXiv e-prints, arXiv:2403.10491 \n<!-- image --> \n<!-- image --> \nFig. 18.(Left) UV slope, β as a function of gas temperature for the nebular continuum assuming Case B recombination. (Center) Equivalent widths of H α and H β as a function of temperature. Solid and dashed lines show the results with and without two-photon emission, respectively. (Right) log 10 ( ξ ion ) as a function of gas temperature for the nebular continuum. \n<!-- image -->", 'Balmer Equivalent Widths': 'In the middle panel of Figure 18 we show the equivalent widths (EWs) of H α and H β as a function of temperature. These quantities are strong functions of temperature such that at higher temperatures, the predicted EWs drop significantly. For example, at a temperature of 10 , 000 K, the H β EW is nearly 1400 ˚ A, but at 30 , 000 K, it drops to 658 ˚ A. One of the key features of the Balmer line EWs is that they are also sensitive to the presence of two-photon emission. This is perhaps counter-intuitive as the two-photon emission falls off steeply at wavelengths longer than ∼ 1450 ˚ A (in f λ ). The dashed lines in the middle panel of Figure 18 shows the Balmer line EWs when the two-photon emission has been removed and they are clearly significantly higher in the absence of two-photon emission. \nIt is important to consider that these calculations all assume Case B recombination. However, as discussed in Raiter, Schaerer & Fosbury (2010); Mas-Ribas, Dijkstra & Forero-Romero (2016), in low-metallicity environments, Case B departures are expected because the gas is hotter and collisional excitation can preferentially enhance both the Ly α emission and the two-photon emission. The Balmer lines are not enhanced nearly as much due to their higher energies and thus lower collisional excitation coefficients. The effect of these Case B departures is that the Balmer EWs would then further decrease in low-metallicity environments.', 'Ionizing photon production efficiencies': 'In the right panel of Figure 18 we show the observed ξ ion that would be inferred from H α and L UV , 1500 ˚ A as a function of gas temperature for the nebular continuum. This quantity again varies as a function of temperature such that the measured ξ ion decreases with increasing gas temperature. This reflects the decrease in line emissivity with respect to the continuum at high-temperatures. Note that the inferred ξ ion values from a nebular-only spectrum are not too dissimilar from the intrinsic values of the SPS models. We again emphasize that any Case B departures or cooling radiation would systematically decrease the observed ξ ion from the Case B values presented here.', 'FITTING FORMULAE FOR ξ ION MEASUREMENTS': 'Measuring the ionizing photon production efficiency, ξ ion requires converting between an observed emission line (e.g. a Balmer line such as H α or H β ) and the total number of ionizing photons. Typically a single value is used for this conversion that assumes Case B recombination and a fixed temperature of, for example, 10 4 K (e.g. Bouwens et al. 2016; Matthee et al. 2017; Maseda et al. 2020). Under these assumptions the value adopted is ∼ 7 . 3 × 10 11 erg -1 (Osterbrock & Ferland 2006). However, as we measure the full posterior distribution on electron temperature from our galaxy fits, we can adopt a temperature-dependent conversion rate. For convenience, we provide fitting formulae to convert H α and H β luminosity into ionizing photon luminosity such that: \nQ [s -1 ] = 10 -0 . 007358 log 10 ( T ) 2 +0 . 1729 log 10 ( T )+11 . 289 [erg -1 ] × L H α [erg s -1 ] (B1) \nand \nQ [s -1 ] = 10 0 . 0309 log 10 ( T ) 2 -0 . 212 log 10 ( T )+12 . 679 [erg -1 ] × L H β [erg s -1 ] . (B2) \nThese equations assume that the escape fraction is zero and are based on atomic data from Pequignot, Petitjean & Boisson (1991). Note that the ratio of the two conversion factors provides the H α /H β ratio as a function of temperature. Because H II regions in the early Universe tend to be hotter than their low-redshift counterparts, likely due to lower metallicity (e.g. Laseter et al. 2024; Morishita et al. 2024), the effect of using a temperature-dependent conversion factor is that the measured ξ ion values can be elevated by ≳ 10%. \nOne final effect to consider is that these conversions assume that all Lyman continuum photons emitted by a source population interact with hydrogen. In practice other species such as helium (and possibly dust) will be present which \nreduces the ionizing photon budget for hydrogen by ∼ 10% (e.g. Tacchella et al. 2022). In this situation, the Q measured from a Balmer line will be an underestimate of the true value. \nThis paper was built using the Open Journal of Astrophysics L A T E X template. The OJA is a journal which provides fast and easy peer review for new papers in the astro-ph section of the arXiv, making the reviewing process simpler for authors and referees alike. Learn more at http://astro.theoj.org .'} |
2024ApJ...976...69H | We present a comprehensive analysis of the broadband spectral energy distribution SED of the extreme highenergy peaked BL Lac source 1ES 0229200. Our study utilizes nearsimultaneous data collected at various epochs between 2017 September and 2021 August MJD 5811959365 from different instruments including AstroSatUVIT Soft Xray focusing Telescope LAXPC SwiftUVOT FermiLAT and MAGIC. We investigate the onezone synchrotron and synchrotron selfCompton SSC model employing diverse particle distributions such as the log parabola broken power law power law with a maximum electron energy energydependent diffusion EDD and energydependent acceleration EDA models to fit the broadband SED of the source. Our findings indicate that both peaks in the SED are well described by the onezone SSC model across all particle distribution models. We estimate the jet power for different particle distributions. The estimated jet power for broken power law particle distributions is found to be on the order of 10SUP47SUP 10SUP44SUP erg sSUP1SUP for a minimum electron energy inlineformula mmlmath overflowscrollmmlmsubmmlmrowmmlmimmlmimmlmrowmmlmrowmmlmiminmmlmimmlmrowmmlmsubmmlmath inlineformula 10 10SUP4SUP. However for intrinsically curved particle energy distributions e.g. log parabola EDD and EDA models the estimated jet power is 10SUP44SUP erg sSUP1SUP. The SED fitting at five epochs enables us to explore the correlation between the derived spectral parameters of various particle distribution models. Notably the observed correlations are inconsistent with the predictions in the powerlaw with a maximum model although the EDD and EDA models yield the correlations as expected. Moreover the estimated physical parameter values are consistent with the model assumptions. | 2024-11-01T00:00:00Z | ['2024arXiv240912827H', '10.48550/arXiv.2409.12827', '2024ApJ...976...69H', '10.3847/1538-4357/ad8085', 'arXiv:2409.12827'] | ['BL Lacertae objects', 'Active galaxies', 'Blazars', '158', '17', '164', 'Astrophysics - High Energy Astrophysical Phenomena'] | Multiwavelength Study of Extreme Highenergy Peaked BL Lac Source 1ES 0229200 Using Ultraviolet XRay and Ray Observations | 2,024 | 166 | 0.51 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF'] | 0 | https://arxiv.org/pdf/2409.12827.pdf | {'Multi-wavelength study of extreme high-energy peaked BL Lac (EHBL) source 1ES 0229 + 200 using ultraviolet, X-ray and gamma-ray observations': 'J/y.pc/o.pc/t.pc/i.pc/s.pc/h.pc/r.pc/e.pc/e.pc H/o.pc/t.pc/a.pc, 1 R/u.pc/k.pc/a.pc/i.pc/y.pc/a.pc K/h.pc/a.pc/t.pc/o.pc/o.pc/n.pc, 2 R/a.pc/n.pc/j.pc/e.pc/e.pc/v.pc M/i.pc/s.pc/r.pc/a.pc, 3 /a.pc/n.pc/d.pc A/n.pc/a.pc/n.pc/t.pc/a.pc C. P/r.pc/a.pc/d.pc/h.pc/a.pc/n.pc 1 \n1 Department of Physics and Astronomy, National Institute of Technology, Rourkela - 769008, India \n2 Centre for Space Research, North-West University, Potchefstroom - 2531, South Africa \n3 Inter-University Center for Astronomy and Astrophysics, Post Bag 4, Ganeshkhind, Pune - 411007, India', 'ABSTRACT': 'Wepresent a comprehensive analysis of the broadband spectral energy distribution (SED) of the extreme highenergy peaked BL Lac (EHBL) source, 1ES 0229 + 200. Our study utilizes near-simultaneous data collected at various epochs between September 2017 and August 2021 (MJD: 58119 -59365) from different instruments, including AstroSat -UVIT, SXT, LAXPC, Swift -UVOT, Fermi -LAT, and MAGIC. We investigate the one-zone synchrotron and synchrotron self-Compton (SSC) model, employing diverse particle distributions such as the log parabola, broken power law, power law with a maximum electron energy 𝛾 , energy-dependent diffusion (EDD), and energy-dependent acceleration (EDA) models to fit the broadband SED of the source. Our findings indicate that both peaks in the SED are well described by the one-zone SSC model across all particle distribution models. We estimate the jet power for different particle distributions. The estimated jet power for broken power law particle distributions is found to be on the order of 10 47 (10 44 ) erg s -1 for a minimum electron energy 𝛾 𝑚𝑖𝑛 ∼ 10 (10 4 ). However, for intrinsically curved particle energy distributions (e.g., log parabola, EDD, and EDA models), the estimated jet power is ∼ 10 44 erg s -1 . The SED fitting at five epochs enables us to explore the correlation between the derived spectral parameters of various particle distribution models. Notably, the observed correlations are inconsistent with the predictions in the power-law with a maximum 𝛾 model, although the EDD and EDA models yield the correlations as expected. Moreover, the estimated physical parameter values are consistent with the model assumptions. \nKeywords: galaxies: active - BL Lacertae objects: general - BL Lacertae objects: individual: 1ES 0229 + 200 - acceleration of particles - diffusion - X-rays: galaxies', '1. INTRODUCTION': "Blazars are a sub-class of active galactic nuclei (AGN) whose relativistic jet points in the direction of the observer's line of sight (Urry & Padovani 1995; Blandford et al. 2019). Thekeycharacteristics of blazars are their non-thermal broadband spectra, strong radio and optical polarization, and fast variability (Sambruna 2000; Fan et al. 2008). The spectral energy distribution (SED) of blazars shows a double bump that ranges from radio to very high energy (VHE) 𝛾 -ray (Fossati et al. 1998). According to the leptonic model, the first bump, which peaks in the optical to X-ray bands, is attributable to synchrotron emission while the second bump, which peaks in 𝛾 -ray energies, is explained by inverse Compton (IC) scatter- \[email protected], [email protected] \nlow-energy photons (Urry & Mushotzky 1982; Ghisellini et al. 1985; Begelman et al. 1987; Blandford & Levinson 1995; Bloom & Marscher 1996; Sokolov et al. 2004). \nThe low-energy photons can be either synchrotron photons or photons that are external to the jet. When synchrotron photons serve as the target for IC scattering, it is called synchrotron self-Compton (SSC) (Jones et al. 1974; Maraschi et al. 1992; Ghisellini et al. 1993). On the other hand, the scattering of external photons is termed as the external-Compton process (Dermer et al. 1992; Sikora et al. 1994; B la˙zejowski et al. 2000; Shah et al. 2017). Furthermore, the high energy emission can be explained through hadronic processes such as the proton synchrotron process and pion production process (Mannheim & Biermann 1992; Mucke & Protheroe 2001). \nBased on their spectral line in the optical bands, blazars are commonly classified into two types, such as flat spectrum radio quasars (FSRQs) and BL Lac objects. Further, depending upon their low-energy peak position, BL Lacs are categorized \ninto three groups: low-energy peak (LBL), intermediate energy peak (IBL), and high-energy peak (HBL) (Padovani & Giommi 1995). \nIn addition to the above classification, a new class of highenergy peak BL Lac sources, called extreme HBL (EHBL) sources, exhibit ambiguous spectral properties in high-energy emission. These sources show either a low-energy peak, a high-energy peak, or both, with peak frequencies surpassing 10 17 Hzfor the synchrotron peak and above one TeV for the IC peak (Costamante, L. et al. 2001; Biteau et al. 2020). As predicted by the blazar sequence (Ghisellini & Tavecchio 2008; Ghisellini et al. 2017), these objects are located at the upper edge of the peak frequency and are at the lowest luminosity end. Over the last decade, the exceptional operational results of the Imaging Atmospheric Cherenkov Telescopes such as H.E.S.S., MAGIC, and VERITAS have identified 14 sources at VHE (E > 100 GeV) (Foffano et al. 2019; MAGIC Collaboration et al. 2019a). Among these sources, Costamante et al. (2018) and MAGIC Collaboration et al. (2019a) have detected seven objects in TeV energies and categorized them as hard-TeV blazars (high-energy peak located above one TeV) and the remaining seven EHBL sources are categorized as soft-TeV blazars by Foffano et al. (2019). Remarkably, it is noticed that two more sources, Mrk 501 and 1ES 1959 + 650, also show EHBL behaviour (hard-TeV spectra) during their flaring states (Pian et al. 1997; MAGIC Collaboration et al. 2018). \nThe EHBL source, 1ES 0229 + 200 (RA = 38.202562 · , DEC: 20.28819 · ), is located at a considerable distance with a redshift of 𝑧 = 0 . 14, (Woo et al. 2005; Aharonian et al. 2007a; Tavecchio et al. 2009)). This blazar is known for emitting VHE gamma rays in the hard TeV range (Aliu et al. 2014). Initially discovered during the Einstein IPC Slew Survey (Elvis et al. 1992), it was later identified as a highfrequency peaked BL Lac source based on the location of its synchrotron peak (Ackermann et al. 2011). H.E.S.S. detected the hard spectrum of this source in VHE emission, reaching up to 10 TeV, classifying it as a hard-TeV EHBL (Aharonian et al. 2007b). The source is also included in the third catalogue of Fermi -LAT, having been detected by Fermi -LAT after integrating four years of exposure time (Acero et al. 2015; Vovk et al. 2012). 1ES 0229 + 200 stands out as an ideal object for studying the extragalactic background light (EBL; Aharonian et al. 2007b; Kneiske & Dole 2010) and the intergalactic magnetic field (IGMF; Neronov & Vovk 2010; Tavecchio et al. 2010; Acciari et al. 2023) due to its distant and hard-spectrum nature as a blazar. \nModelling of hard-TeV spectra of a source is a challenging task. Various one-zone leptonic SSC models have been developed to explain the VHE emission of the source 1ES 0229 + 200 (Abdo et al. 2011; Aleksi'c et al. 2012; Aliu et al. 2014; Tanaka et al. 2014; Costamante et al. 2018; Foffano et al. \n2019; Prandini et al. 2019; Diwan et al. 2023). Recently, a few more models have also been developed to explain the hardTeV blazars. Zech & Lemoine (2021) developed a new onezone lepto-hadronic emission model for extreme-TeV blazars and showed that the acceleration in a single standing shock is capable of reproducing the jet emission. However, the re-acceleration on a second shock is required for the source like 1ES0229 + 200 (having hardest 𝛾 -ray spectra). Li et al. (2022) investigated the one-zone hadronuclear (pp) model and found that the hard-TeV spectrum of 1ES0229 + 200 is reproduced by the 𝛾 -ray emission from the 𝜋 0 decay in the p -p interactions. A lepto-hadronic model proposed by AguilarRuiz et al. (2022) suggests that the two-zone emission can reproduce the broadband SEDs of the hard-TeV blazars. The two-zone lepto-hadronic model was also able to relax several parameters required by the one-zone models. \nTo understand the hard-TeV systems, one has to find the energy-generating mechanism responsible for the high jet power ( 𝑃 𝑗𝑒𝑡 ) in such blazars. There are two mechanisms, Blandford-Znajek process (Blandford & Znajek 1977) and Blandford-Payne process (Blandford & Payne 1982), proposed for the jet formation in the hard-TeV blazars. The Blandford-Znajek model suggests that the blazar jet has an origin from a spinning black hole, and the 𝑃 𝑗𝑒𝑡 is associated with the spin and mass of the black hole with the magnetic field at its horizon, whereas the Blandford-Payne model suggests that the jet is originated from the black hole accretion disk. A recent study by Zhang et al. (2022) suggests that the relativistic jets are probably dominated by the Blandford-Znajek process for both FSRQs and BL Lacs. The 𝑃 𝑗𝑒𝑡 estimation in the one-zone SSC model is primarily governed by the electron energy distribution in the blob, which is characterized mostly by the broken power law with a minimum electron Lorentz factor ( 𝛾 𝑚𝑖𝑛 ). Xue et al. (2019) and Zech &Lemoine (2021) suggested a large 𝛾 𝑚𝑖𝑛 value ( ∼ 10 4 ) is required to fit the SED in the TeV range spectra of the hard-TeV systems under one-zone SSC model. Zech & Lemoine (2021) proposed a standard one-zone lepto-hadronic emission model to study the broadband SED of TeV blazar 1ES 0229 + 200, and they calculated the total jet power 𝑃 𝑗𝑒𝑡 in the order of ∼ 10 44 ergs/sec. Acciari et al. (2020) applied the single-zone SSC model, spine-layer model, and Proton-Synchrotron Scenario (PSS) to fit the spectra of various extreme blazars and estimated their total jet powers. The jet power calculated using the spine-jet scenario (10 42 erg 𝑠𝑒𝑐 -1 ) consistently appears more than one order of magnitude lower than those predicted by the SSC model (10 44 erg 𝑠𝑒𝑐 -1 ). In contrast, the 𝑃 𝑗𝑒𝑡 estimated with the PSS is giving a higher value ∼ 0 . 15 -45 . 6 × 10 46 erg 𝑠𝑒𝑐 -1 and this is because PSS requires a rather large power in the protons responsible for the emission, often larger than the Eddington luminosity of the black hole powering the AGN (Zdziarski & Bottcher 2015). \nThe 𝑃 𝑗𝑒𝑡 estimates mentioned above are primarily derived by assuming that the underlying electron energy distribution follows either a broken power law or a single power law with an exponential cutoff (Acciari et al. 2020). Nonetheless, there is evidence that suggests the underlying electron distribution, which may resemble a log-parabola distribution, has a considerable curvature in the observed spectrum (Massaro et al. 2004; Tramacere et al. 2007). The curvature in the electron distribution is attributed to the energy dependency of particle acceleration/escape time-scales (Sinha et al. 2017; Goswami et al. 2018; Hota et al. 2021; Khatoon et al. 2022). Additionally, physically motivated models have been employed, including distributions where the curvature of the spectrum can be attributed to the energy dependency of the diffusion time-scale (EDD) or due to the energy dependency of the acceleration time-scale (EDA), or with a high energy cutoff attributable to radiative losses ( 𝛾 -max models) (Hota et al. 2021; Khatoon et al. 2022). \nThis study presents a standard SSC model for the broadband spectra of the hard-TeV blazar 1ES 0229 + 200. Various electron energy distribution models, such as the broken powerlaw model, log-parabola model, power-law particle distribution with maximum electron energy (PL with 𝛾 𝑚𝑎𝑥 ), energydependent diffusion (EDD), and energy-dependent acceleration (EDA) models, have been considered. These models have previously been verified for the X-ray spectral curvature of the HBL source, Mkn 421 (Hota et al. 2021; Khatoon et al. 2022). Furthermore, a study on the broadband emissions of a HBL source Mkn501 has been conducted (Bora et al. 2024). \nWehave applied these particle distribution models to investigate the hard-TeV blazar source, extending the model for a broadband SED analysis using various observations over five epochs in a wide energy range, from ultraviolet (UV; 0.01 KeV) to VHE 𝛾 -ray ( ∼ 10 TeV) bands. \nThe paper is organized as follows: In Section 2, we provide observations and data analysis procedure details. In Section 3, we present details of the 𝛾 -ray and VHE 𝛾 -ray analysis of the observed data. In Section 4, we discuss the broadband SED modelling of the source with synchrotron and SSC emission processes. The results are presented in Subsection 4.1 and 4.2. In Section 5, we summarized the results and their implications are discussed. \nThroughout the paper, the following cosmological parameters are assumed: 𝐻 0 = 70 𝑘𝑚 𝑠 -1 𝑀𝑝𝑐 -1 , Ω 𝑀 = 0.3, Ω Λ = 0.7.", '2. MULTI-WAVELENGTH OBSERVATIONS AND DATA ANALYSIS': 'AstroSat is the first multi-wavelength space observatory of India, carrying five major payloads with energies ranging from UV to hard X-ray (Agrawal 2006; Singh et al. 2014; Rao et al. 2016). AstroSat onboard instruments are \nUltra-Violet Imaging Telescope (UVIT: 130 -300 nm; Tandon et al. 2017a,b), Soft X-ray focusing Telescope (SXT: 0.3 -8.0 keV; Singh et al. 2017, 2016), Large Area X-ray Proportional Counter (LAXPC: 3 -80 keV; Yadav et al. 2016), Cadmium Zinc Telluride Imager (CZTI: 10 -100 keV; Rao et al. 2017) and Scanning Sky Monitor (SSM). SXT was the primary instrument used to observe the blazar, 1ES0229 + 200 at five different epochs between 2017 and 2022. LAXPC and UVIT were also used to observe the source simultaneously. The AstroSat observation details of the source are listed in Table 1. Furthermore, for the LAXPC and SXT observations conducted during August 8-12, 2021, no simultaneous UVIT observations are available. Hence, we have used UV data from Swift/ultraviolet optical telescope (UVOT) observations for the above duration. \nIn this study, we have used the quasi-simultaneous observations from UVIT and UVOT for UV data, SXT, and LAXPC for X-ray data, Fermi -LAT for 𝛾 -ray data, and VHE spectral data from the MAGIC observations (MAGIC Collaboration et al. 2019b, 2020). The details of the data reduction of UVIT, UVOT, SXT, LAXPC, and Fermi -LAT observations are as follows.', '2.1. UVIT': 'UVIT is an imaging telescope onboard AstroSat consisting of three channels: FUV (1300 - 1800 ˚ A), NUV (2000 - 3000 ˚ A), and Visible (3200 - 5500 ˚ A). The data reduction of the UVIT observations was performed with the software package CCDLAB (Postma & Leahy 2017). CCDLAB converts the Level 1 (L1) data into science-ready astronomical images. We extracted the L1 files of the UVIT observations and removed all the duplicate observations and short exposures (less than 120 seconds). The extracted L1 files were then supplied into CCDLABfor various corrections, viz., FPN, CPU distortion, flat-field, drift corrections, etc. The drift-corrected frames were then co-added to get the orbit-wise science-ready images for each observed filter. The images were co-aligned across all the orbits and merged into the final science images. Astrometric corrections were performed using the Gaia EDR3 catalogue. We applied the Source Extractor (SExtractor) tool (Bertin & Arnouts 1996) to extract the count rates in UVIT fits images corresponding to each filter. Then, the flux of the blazar source was calculated in all the UVIT filters using the unit conversion factor suggested by Tandon et al. (2017b). The flux values were then corrected for reddening using E(B -V) = 0.1192 values mentioned in Schlafly & Finkbeiner (2011).', '2.2. UVOT': "The UVOT (170-650 nm; Roming et al. 2005) onboard Swift uses three optical (U, B, V) and three UV (W1, M2, W2) filters to cover both the optical and UV portions of the \nTable 1. Details of the AstroSat observations with LAXPC, SXT, and UVIT at five epochs. \nspectrum. We have used Swift-UVOT observations to compliment the AstroSat SXT and LAXPC observations during 8 -12 August, 2021 (at the fifth epoch) in the UV band. We have used observations in the W1, M2, and W2 filters of UVOT. We used the /u.pc/v.pc/o.pc/t.pc/i.pc/m.pc/s.pc/u.pc/m.pc command to combine all the observed images in each UVOT filter. The task /u.pc/v.pc/o.pc/t.pc/s.pc/o.pc/u.pc/r.pc/c.pc/e.pc was used to extract the magnitudes from the combined images. The source and background regions were selected with radii 5 '' and 10 '' , respectively, as input parameters in the /u.pc/v.pc/o.pc/t.pc/s.pc/o.pc/u.pc/r.pc/c.pc/e.pc command. The observed magnitudes were then corrected for Galactic extinction using E(B -V) = 0.1192 mag and 𝑅 𝑉 = 3 . 1 (Schlafly & Finkbeiner 2011). We converted the UVOT magnitudes to flux units using the photometric zero-points from Breeveld et al. (2011) and the conversion factors from Giommi et al. (2006).", '2.3. SXT': "SXT is a focusing telescope with a field of view (FOV) of around ∼ 40 ' diameter and operates in the soft X-ray energy range of 0.3 - 8.0 keV for X-ray imaging and spectroscopy (Singh et al. 2017). The SXT observed 1ES 0229 + 200 in the Photon Counting (PC) mode for all the epoch and sxtpipeline /one.sup (AS1SXTLevel2, version 1.4b) was used to reduce the Level \n1 data and obtain the cleaned Level 2 event files from different orbits. The standard Julia script developed by the instrument team was used to merge the cleaned event files from all orbits into a single file to avoid the problem of time-overlapping event files from successive orbits. The XSELECT (V2.4d) package built-in HEASOFT(V6.28) was used to extract the source spectrum from the processed Level-2 cleaned event files within a circular region of 15 ' centred on the source. An off-axis auxiliary response file (ARF) was generated by using the SXT ARF generation tool /two.sup with the help of the on-axis ARF (sxt pc excl00 v04 20190608.arf) as input. Further, we used 'SkyBkg comb EL3p5 Cl Rd16p0 v01.pha' for a background spectrum and 'sxt pc mat g0to12.rmf' for response matrix file (RMF; given by the SXT team). The SXT spectrum was re-binned using the /g.pc/r.pc/p.pc/p.pc/h.pc/a.pc tool. For the spectral analysis, we used the energy range between 0.5 -7.0 keV. To modify the gain of the response file, the gainfit tool in XSPEC was used with a fixed slope of value one and the offset as a free parameter. To account for the Galactic absorption, we have utilized the TBabs model (Wilms et al. 2000) available in the XSPEC for the spectral fit. The equivalent hydrogen column density ( 𝑁 𝐻 ) was fixed at 7 . 9 × 10 20 cm -2 for \nthe X-ray observation, which was estimated from the online tool /three.sup , by the LAB survey group (Kalberla et al. 2005).", '2.4. LAXPC': 'The LAXPC X-ray proportional counter has a high time resolution ( ∼ 10 𝜇 s) and covers the energy range of 3 -80 keV (Yadav et al. 2016; Antia et al. 2017; Agrawal et al. 2017; Misra et al. 2017). It comprises three identical co-aligned proportional counter units, named LAXPC 10, LAXPC 20, and LAXPC 30, with the effective area of each detector as ∼ 2000 𝑐𝑚 2 . All the Level 1 data were downloaded from the AstroSat archive. The Level 1 raw data were processed by the LAXPCSOFT package (version as of 2022 August 15; which is recommended by the instrument team and accessible at the website of the AstroSat Science Support Cell /four.sup ). LAXPCcommand laxpc make event were used to merge different orbits. To avoid the Earth occultation and the South Atlantic Anomaly, good time intervals (GTI files) were created using the command laxpc make stdgti. Finally, the source and background spectra were extracted by using the command laxpc make spectra. We have adopted the faint source method to extract the background spectrum and lightcurve (Misra et al. 2021). Out of the three detectors, LAXPC 30 wasswitched off due to a gain instability issue arising from the gas leakage in the detector, and the LAXPC10 was working at low gain during the observation. Therefore, in our analysis, we have used only the LAXPC20 detector and limited the spectral analysis energy range to 4 -18 keV.', '2.5. FERMI': "The Fermi satellite carries the 𝛾 -ray instrument Large Area Telescope (LAT; Atwood et al. 2009), which is sensitive in the energy range 20 MeV to 300 GeV. The Fermi -LAT continuously monitored more than 5000 extragalactic 𝛾 -ray sources (4FGL; Abdollahi et al. 2020) between 2008 and 2018, out of which more than 3000 sources are blazars, suggesting that the 𝛾 -ray sky is heavily inhabited by relativistic jets. In this analysis, we have collected the Fermi -LAT data of the source, 1ES 0229 + 200 from 2008 -08 -04 (MJD54682.6) to 2022 -10 -30 (MJD 59882). The analysis was carried out in the energy band 100 MeV -300 GeV using the latest version of fermipy -v0.17.4 /five.sup and fermitools1 -v1.2.23 /six.sup . We selected a 15 · region of interest (ROI) around the source to extract the photon events with evclass=128 and evtype=3, as recommended by the Fermi -LAT team in the fermitools documentation. The source model file was created using the Fermi 4FGL catalogue (Abdollahi et al. 2020), and the background \nFigure1. Spectral fitting with a simple power law model for the combined 𝛾 -ray (blue) and VHE 𝛾 -ray (green) data for 1ES 0229+200. \n<!-- image --> \n𝛾 -ray emission and isotropic background emission were handled using 'gll iem v07.fits' and 'iso P8R3 SOURCE V3 v1.txt' files, respectively. Additionally, a 90 · zenith angle was chosen to prevent contamination from the earth limb. While extracting within the ROI, the source parameters were left thawed while those outside were frozen to their 4FGL catalogue values. The test statistics (TS) defined as TS = 2logL, where L is the likelihood parameter of the analysis (Mattox et al. 1996) and is used to evaluate the detection significance of each source in the ROI.", '3. GAMMA-RAY ANALYSIS': 'This Section presents the 𝛾 -ray spectral analysis of 1ES 0229 + 200 using quasi-simultaneous observations from Fermi -LAT and MAGIC. EHBLs are generally considered relatively faint sources in the high-energy 𝛾 -ray domain, owing to their low average brightness and the shifts in the IC peak location at higher energies. The Fermi -LAT data considered in this work covers a substantial period from 2008 to 2022 ( Section 2.5). \nFor the VHE 𝛾 -ray observations, we used the TeV spectra obtained from the MAGIC observations reported by MAGIC Collaboration et al. (2020), with a total exposure time of 117.46 hours and spanning a duration from 2013 to 2017. The VHEspectra has been corrected for extragalactic background light (EBL) absorption given by Franceschini et al. (2008). \nWe undertake a joint fitting of the two, using a simple power law and the best fit was found to be with an index Γ ∼ 1.78 ± 0.3 and the fit resulted in a 𝜒 2 / 𝑑𝑜 𝑓 = 4 . 28 / 10. The best-fit spectra with residuals are shown in Figure 1.', '4. SED MODELING': "Broadband SED modelling of blazars is used to understand the underlying physical processes driving the broadband emission in both high-flux and low-flux states. We carried out broadband SED modelling of 1ES 0229 + 200byusing the data \nFigure 2. Broadband spectral energy distribution (SED) of 1ES 0229 + 200 for data sets of epoch-4. Different panels show the particle distribution models (mentioned at the top left corner of each plot) used to construct the SEDs. The solid lines represent the best-fit model without Galactic absorption. The dotted lines represent the best-fit model spectra for the SXT data with Galactic absorption. \n<!-- image --> \nfrom UV to VHE 𝛾 -ray bands. The time intervals selected for the broadband spectral study were determined based on the availability of simultaneous observations in UV and Xray energies. We generated simultaneous multi-wavelength SEDs for all the five epochs of the observations of the source 1ES 0229 + 200 with AstroSat . Details of the observations from AstroSat instruments are provided in Table 1. \nThe one-zone SSC model provides an explanation for the broadband emission of EHBL sources like 1ES 0229 + 200. Leptonic models assume that interactions between relativistic electrons and the magnetic field in the emission zone produce the first hump of the SED in the energy range covering from radio to soft X-rays. Whereas the second hump of the SED is produced by IC scattering of a photon population, either the \nsynchrotron photons themselves (SSC) and/or the photon field external to the jet (EC). Based on SSC models (Ghisellini et al. 1993), synchrotron photons generated by relativistic electrons in the magnetic field will be up-scattered. We have employed a one-zone SSC leptonic model to explain the broadband SED fitting at each epoch. The model assumes a non-thermal emission from the blazar jet that arises from a spherical blob of radius R, filled with a tangled, homogeneous magnetic field (B), and isotropic electron distribution 𝑛 ( 𝛾 ) moving with the relativistic velocity along the jet. The blazer jet is moving with a bulk Lorentz factor, Γ , at an angle 𝜃 with respect to the observer's direction, affecting emission region by the beaming factor 𝛿 = 1 / Γ ( 1 -𝛽 cos 𝜃 ) . \nIf the electron Lorentz factor, 𝛾 , is represented in terms of 𝜉 in such a way that 𝜉 = 𝛾 √ C , where C = 1 . 36 × 10 -11 𝛿𝐵 1 + 𝑧 with z being the redshift of source and 𝛿 is the jet Doppler factor, the synchrotron flux that the observer receives at energy 𝜖 will be (Begelman et al. 1984), \n𝐹 syn ( 𝜖 ) = 𝛿 3 ( 1 + 𝑧 ) 𝑑 2 𝐿 𝑉 A ∫ 𝜉 𝑚𝑎𝑥 𝜉 𝑚𝑖𝑛 𝑓 ( 𝜖 / 𝜉 2 ) 𝑛 ( 𝜉 ) 𝑑𝜉 (1) \nhere, V is the volume of the emission region, 𝑑 𝐿 is the luminosity distance, A = √ 3 𝜋𝑒 3 𝐵 16 𝑚 𝑒 𝑐 2 √ C and f(x) is the synchrotron emissivity function (Rybicki & Lightman 1986). Rather than employing the electron's Lorentz factor, 𝛾 , the particle energy distribution is expressed by 𝑛 ( 𝜉 ) ). Note that 𝜉 represents 𝛾 , and 𝜉 2 has dimension of keV. Hota et al. (2021) solved Equation 1 numerically and included it as a local convolution model, 𝑠𝑦𝑛𝑐𝑜𝑛𝑣 ⊗ 𝑛 ( 𝜉 ) in XSPEC (Arnaud 1996). In this work, we have extended the numerical codes presented in Hota et al. (2021) to estimate the emissivities and to calculate the observed fluxes corresponding to the synchrotron and SSC emission processes. We have included it as a local convolution model, 𝑠𝑠𝑐𝑖𝑐𝑜𝑛 ⊗ 𝑛 ( 𝜉 ) in XSPEC to fit the broadband SED of the source where 𝑛 ( 𝜉 ) is the particle distribution. \nFor this study, we utilize empirical models, such as log parabola (LP) and broken power law (BPL), and we input the particle distributions to the 𝑠𝑠𝑐𝑖𝑐𝑜𝑛 ⊗ 𝑛 ( 𝜉 ) model. Additionally, we incorporate physical models, including powerlaw with maximum energy due to radiative cooling (PL with 𝛾 𝑚𝑎𝑥 ), EDD, and EDA in order to fit the observed spectrum with the SSC emission. The goal of this analysis is to comprehensively explore and understand the observed spectrum through the application of both empirical and physical models. \n- · Log parabola (LP) : The underlying particle distribution in this scenario is described as \n𝑛 ( 𝜉 ) = 𝐾 ( 𝜉 / 𝜉 𝑟 ) -𝛼 -𝛽 log ( 𝜉 / 𝜉 𝑟 ) (2) \nhere, particle spectral index is denoted as 𝛼 at the reference energy 𝜉 2 = 𝜉 2 𝑟 , the spectral curvature is represented by 𝛽 , and K is the normalization of the particle density. The reference energy 𝜉 2 𝑟 kept constant at one keV throughout the spectral fit, whereas the parameters 𝛼 , 𝛽 , and norm K remained free. \n- · Broken power-law (BPL) : This scenario of particle distribution is given by \n𝑛 ( 𝜉 ) = 𝐾 ( 𝜉 / 1 √ 𝑘𝑒𝑉 ) -𝑝 for 𝜉 < 𝜉 𝑏𝑟𝑒𝑎𝑘 𝐾𝜉 𝑞 -𝑝 break ( 𝜉 / 1 √ 𝑘𝑒𝑉 ) -𝑞 for 𝜉 > 𝜉 𝑏𝑟𝑒𝑎𝑘 (3) \nWhere K represents the normalization, 𝑝 and 𝑞 represent the low and high energy photon indices, respectively, \nwhile the Lorentz factor associated with the break energy is 𝜉 𝑏𝑟𝑒𝑎𝑘 and the transformation is represented by 𝜉 = 𝛾 √ C . The distributions is non-zero for 𝜉 𝑚𝑖𝑛 < 𝜉 < 𝜉 𝑚𝑎𝑥 , which correspond to 𝛾 𝑚𝑖𝑛 and 𝛾 𝑚𝑎𝑥 such that 𝜉 𝑚𝑖𝑛 = 𝛾 𝑚𝑖𝑛 √ C and 𝜉 𝑚𝑎𝑥 = 𝛾 𝑚𝑎𝑥 √ C . \n- · Power-law particle distribution with maximum electron energy (PL with 𝛾 𝑚𝑎𝑥 ) : In this case, we have considered that the particles accelerated by a shock, and then subsequently, these accelerated particles lose energy through radiative processes. \nIn such a case, the steady-state particle density is given by \n𝜕 𝜕𝛾 GLYPH<20> GLYPH<18> 𝛾 𝜏 𝑎𝑐𝑐 -𝛽 𝑠 𝛾 2 GLYPH<19> 𝑛 𝑎 GLYPH<21> + 𝑛 𝑎 𝜏 𝑒𝑠𝑐 = 𝑄𝛿 ( 𝛾 -𝛾 0 ) (4) \nwhere 𝜏 𝑎𝑐𝑐 and 𝜏 𝑒𝑠𝑐 represent the acceleration and escape time scales, respectively, and the radiative loss term includes 𝛽 𝑠 = 4 3 𝜎 𝑇 𝐵 2 8 𝜋𝑚 𝑒 𝑐 , B is the magnetic field, 𝜎 𝑇 is the Thomson cross-section and 𝑚 𝑒 the electron mass. We consider that a mono-energetic injection of electrons Q at energy 𝛾 0. The solution of the steady-state equation is described as \n𝑛 ( 𝜉 ) = 𝐾𝜉 -𝑝 GLYPH<18> 1 -𝜉 𝜉 𝑚𝑎𝑥 GLYPH<19> ( 𝑝 -2 ) (5) \nWhere, 𝐾 = 𝑄 0 𝜏 𝑎 𝛾 𝑝 -1 0 C 𝑝 / 2 , 𝑝 = 𝜏 𝑎𝑐𝑐 / 𝜏 𝑒𝑠𝑐 + 1 is the particle spectral index, 𝜉 𝑚𝑎𝑥 = 𝛾 𝑚𝑎𝑥 √ C with 𝛾 𝑚𝑎𝑥 = 1 /( 𝛽 𝑠 𝜏 𝑎𝑐𝑐 ) is the maximum Lorentz factor that an electron can attain before losing energy. The free parameters are, 𝜉 𝑚𝑎𝑥 , 𝑝 , and the normalization N defined as \nN = 𝛿 3 ( 1 + 𝑧 ) 𝑑 2 𝐿 𝑉 A 𝑄 0 𝜏 𝑎𝑐𝑐 𝛾 𝑝 -1 0 C 𝑝 / 2 (6) \nFor 𝛾 𝑚𝑎𝑥 model, the distribution is only for 𝜉>𝜉 𝑚𝑖𝑛 such that 𝜉 𝑚𝑖𝑛 = 𝛾 𝑚𝑖𝑛 √ C . \n- · Energy dependent diffusion model (EDD) : In this case, weassume that the diffusion takes place in a region consisting of the tangled magnetic field, which may cause the diffusion coefficient dependent on the gyration radius. Consequently, escape time scale energy dependent or the diffusion coefficient energy is dependent as 𝜏 𝑒𝑠𝑐 ( 𝛾 ) is given by 𝜏 𝑒𝑠𝑐 = 𝜏 𝑒𝑠𝑐,𝑅 GLYPH<16> 𝛾 𝛾 𝑅 GLYPH<17> -𝜅 . By ignoring the synchrotron energy loss and considering this escape time-scale dependence, the solution to Equation 4 will be (for detailed derivation see Hota et al. 2021; Khatoon et al. 2022) \n𝑛 ( 𝜉 ) = 𝑄 𝑜 𝜏 𝑎𝑐𝑐 √ C 𝜉 -1 exp GLYPH<20> -𝜂 R 𝜅 GLYPH<18> GLYPH<18> 𝜉 𝜉 R GLYPH<19> 𝜅 -GLYPH<18> 𝜉 0 𝜉 R GLYPH<19> 𝜅 GLYPH<19> GLYPH<21> (7) \nwhere 𝜉 𝑅 = √ C 𝛾 𝑅 , 𝜉 0 = √ C 𝛾 0 and 𝜂 𝑅 ≡ 𝜏 𝑎𝑐𝑐 / 𝜏 𝑒𝑠𝑐,𝑅 . \nTable 2. The best-fitted spectral parameter values of the log-parabola, broken power law, PL with 𝛾 𝑚𝑎𝑥 , EDD, and EDA models. The broadband SED fitting for the various models is constructed using observations from UV to gamma rays at each epoch. \nTable 2. (Continued...) The best-fitted spectral parameter values of the log-parabola, broken power law, PL with 𝛾 𝑚𝑎𝑥 , EDD, and EDA models. The broadband SED fitting for the various models is constructed using observations from UV to gamma rays at each epoch. \nNotes : For broadband analysis, the size of the emission region we have considered is 𝑅 = 10 17 cm and bulk Lorentz factor Γ = 20. Jet Power 𝑃 𝑗𝑒𝑡 is in logarithmic scale with the units of erg 𝑠 -1 . \nTable 2. (Continued...) The best-fitted spectral parameter values of the log-parabola, broken power law, PL with 𝛾 𝑚𝑎𝑥 , EDD, and EDA models. The broadband SED fitting for the various models is constructed using observations from UV to gamma rays at each epoch. \nNotes : For broadband analysis, the size of the emission region we have considered is R= 10 17 cm and bulk Lorentz factor, Γ = 20. Jet Power 𝑃 𝑗𝑒𝑡 is in logarithmic scale with the units of erg 𝑠 -1 . 𝑓 𝑎𝑐𝑡𝑜𝑟 𝑠𝑥𝑡 is the relative cross-normalization constant between X-ray instrument SXT and LAXPC. This factor was kept frozen at 1 for LAXPC, whereas it was kept free for the SXT instrument. \nAfter removing the degenerate parameters, we can use the updated Equation as follows \n𝑛 ( 𝜉 ) = 𝐾 ' 𝜉 -1 exp GLYPH<20> -𝜓 𝜅 𝜉 𝜅 GLYPH<21> (8) \nwhere 𝜓 = 𝜂 𝑅 GLYPH<0> C 𝛾 2 𝑅 GLYPH<1> -𝜅 / 2 = 𝜂 𝑅 𝜉 -𝜅 𝑅 , and the normalization parameter ( 𝐾 ' ) is given as \n𝐾 ' = 𝑄 0 𝜏 𝑎𝑐𝑐 √ C exp GLYPH<20> 𝜂 R 𝜅 GLYPH<18> 𝛾 0 𝛾 R GLYPH<19> 𝜅 GLYPH<21> (9) \nand further modified into N as \nN = 𝛿 3 ( 1 + 𝑧 ) 𝑑 2 𝐿 𝑉 A 𝐾 ' (10) \nWe have considered 𝜓 , 𝜅 , and N as the free parameters for the above model. \n- · Energy dependent acceleration model (EDA) : We next consider a case where the energy dependence of acceleration time scale as 𝜏 𝑎𝑐𝑐 = 𝜏 𝑎𝑐𝑐,𝑅 GLYPH<16> 𝛾 𝛾 𝑅 GLYPH<17> 𝜅 . So considering the above dependency 𝜏 𝑎𝑐𝑐 , the solution to Equation 4 will be (defined in Hota et al. 2021; Khatoon et al. 2022) \n𝑛 ( 𝜉 ) = 𝑄 0 𝜏 𝑎𝑐𝑐,𝑅 √ C 𝜉 -𝜅 𝑅 𝜉 𝜅 -1 exp GLYPH<20> -𝜂 𝑅 𝜅 GLYPH<18> GLYPH<18> 𝜉 𝜉 𝑅 GLYPH<19> 𝜅 -GLYPH<18> 𝜉 0 𝜉 𝑅 GLYPH<19> 𝜅 GLYPH<19> GLYPH<21> (11) √ √ \nwhere 𝜉 0 = C 𝛾 0, 𝜉 𝑅 = C 𝛾 𝑅 and 𝜂 𝑅 ≡ 𝜏 𝑎𝑐𝑐,𝑅 / 𝜏 𝑒𝑠𝑐 . \nWe can recast the distribution as \n𝑛 ( 𝜉 ) = 𝐾 ' 𝜉 𝜅 -1 exp GLYPH<20> -𝜓 𝜅 𝜉 𝜅 GLYPH<21> (12) \nwhere 𝜓 = 𝜂 𝑅 GLYPH<0> C 𝛾 2 𝑅 GLYPH<1> -𝜅 / 2 = 𝜂 𝑅 𝜉 -𝜅 𝑅 , and the normalization ( 𝐾 ' ) is defined in Equation 13 and further modified as N (Equation 14). \n𝐾 ' = 𝑄 0 𝜏 𝑎𝑐𝑐,𝑅 √ C 𝜉 -𝜅 𝑅 exp GLYPH<20> 𝜂 R 𝜅 GLYPH<18> 𝜉 0 𝜉 R GLYPH<19> 𝜅 GLYPH<21> (13) \nN = 𝛿 3 ( 1 + 𝑧 ) 𝑑 2 𝐿 𝑉 A 𝐾 ' (14) \nWe have considered 𝜓 , 𝜅 , and N as the free parameters for the above model. \nTo consider Galactic absorption while fitting the broadband SED of the source, we used the TBabs model (Wilms et al. 2000) available in the XSPEC with the equivalent hydrogen column density ( 𝑁 𝐻 ) set at 7 . 9 × 10 20 cm -2 as determined by the online tool /seven.sup created by the LAB survey group (Kalberla et al. 2005). \nFor the SED spectral fitting, we fixed the size R, the bulk Lorentz factor Γ , and the opening angle 𝜃 , at 10 17 cm, 20, and 0 · , respectively, while keeping other parameters free. However, we have discussed later the impact of variations of these parameters. For the broken power-law model, we further fix the minimum and maximum Lorentz factors, 𝛾 𝑚𝑖𝑛 and 𝛾 𝑚𝑎𝑥 to 10 and 10 8 , respectively. The best-fit parameters with errors are listed in Table 2 along with 𝜒 2 for each of the epochs and different particle distributions. Figure 2 shows the model along with the data for a representative epoch. The solid lines in Figure 2 represent the best-fit model without Galactic absorption. The dotted lines represent the best-fit \nFigure 3. Jet power estimated for all five models as indicated in the legend at five different epochs. \n<!-- image --> \nmodel spectra for the SXT data with Galactic absorption. Note that to take into account the calibration uncertainties between LAXPC and SXT instruments a constant factor has been multiplied to the SXT model, which causes the dotted lines to be shifted compared to the solid ones. The constant factor was fixed for LAXPC, whereas it was free to vary for the SXT ( 𝑓 𝑎𝑐𝑡𝑜𝑟 𝑠𝑥𝑡 ) instrument. The constant factor for SXT ( 𝑓 𝑎𝑐𝑡𝑜𝑟 𝑠𝑥𝑡 ) for each of the epochs is listed in Table 2.", '4.1. Jet Power': "We have calculated the total jet power ( 𝑃 𝑗𝑒𝑡 ), or the total power carried by electrons, Poynting flux, radiation, and protons (Celotti & Ghisellini 2008) using Equation 15 as given below, \n𝑃 𝑗𝑒𝑡 = 2 𝜋 2 𝑅 2 Γ 2 𝛽 𝑐 𝑢 ' 𝑘 (15) \nHere, factor 2 refers to two-sided jets, Γ represents the bulk Lorentz factor, R is the size of the emission region, and 𝑢 ' 𝑘 are the energy densities in the co-moving jet's frame of the magnetic field (k = mag), relativistic electrons (k = ele), and cold protons (k = kin). \nWe have calculated the 𝑃 𝑗𝑒𝑡 values for all the five particle distribution models used for the broadband SED fitting described in Section 4. Since we have performed SED fitting by freezing R and Γ values at 10 17 cm and 20, respectively, 𝑃 𝑗𝑒𝑡 values are calculated for the fixed R and Γ for all the particle distribution models and at all the five epoch observations, given in Table 2. In Figure 3, we present the variation of jet power at different epochs for all the models. The maximum jet power values are obtained for the PL with 𝛾 𝑚𝑎𝑥 model and the broken power law model i.e., approximately two orders of magnitude higher compared to the log parabola, EDD, and EDA models. Notably, the values of jet power remain nearly constant for the log parabola, EDD, and EDA models, while for the broken power law and PL with 𝛾 𝑚𝑎𝑥 models, the jet power values exhibit variability across all five epochs. \nFurthermore, to examine the variation in the jet power of the blazar source 1ES 0229+200 at different values of Γ and R, we calculated the jet power by systematically varying Γ and \nR for both the log parabola and broken power law models. The top left panel of Figure 4 illustrates the variation of Γ with 𝜒 2 (top) and jet power (bottom). The jet power shows a slight increase (by a factor of one or two) as the Γ values vary from 10 to 35, becoming approximately constant at Γ ≥ 35. In the top right panel of Figure 4, the variation of R with its respective 𝜒 2 (top) and jet power (bottom) values is depicted. The plot indicates a slight decrease (by a factor of 2(1)) in jet power for the log parabola model (broken power law), and the 𝜒 2 remains stable with decreasing R values. However, 𝜒 2 values increases below R = 10 15 (10 16 ) for the broken power-law (log parabola model). Therefore, the general 𝑃 𝑗𝑒𝑡 estimates with Γ = 20 and size R = 10 17 cm are insensitive to their presumptions. \nAdditionally, the bottom panel of Figure 4 displays the variation in jet power (bottom) and 𝜒 2 (top) of the source 1ES 0229+200 with respect to 𝛾 𝑚𝑖𝑛 for the broken power-law particle distribution. The jet power decreases with increasing 𝛾 𝑚𝑖𝑛 for large values, while the 𝜒 2 remains constant up to 10 4 and increases beyond that threshold. We find that the 𝑃 𝑗𝑒𝑡 for the broken power-law model is ∼ 10 44 ergs/sec for a value as large as 𝛾 𝑚𝑖𝑛 = 10 4 .", '4.2. Correlation study of spectral parameter': "The study of correlation among various spectral parameters is important to understand the dependence between the fit parameters and the observed properties. We obtained spectral parameters for log-parabola, Broken power law, PL with 𝛾 𝑚𝑎𝑥 , EDD, and EDA models. Then, we determined Spearman's rank correlation coefficient ( 𝑟 𝑠 ) and null hypothesis probability ( 𝑃 𝑟𝑠 ) for all the derived parameters in each model. \nThe scatter plots between spectral parameters of the logparabola are shown in Figure 5. A significant anti-correlation is observed between fit parameters, 𝛼 vs. 𝛽 and normalization each with 𝑟 𝑠 = -0 . 99 ( 𝑃 𝑟𝑠 = 1 . 4 × 10 -24 ) . A positive correlation is found between 𝛽 vs. normalization and 𝛼 vs. B, each with 𝑟 𝑠 = 0.99 ( 𝑃 𝑟𝑠 = 1 . 4 × 10 -24 ) . Furthermore, a significant anti-correlation is observed between fit parameter magnetic field vs. 𝛽 and normalisation, each having 𝑟 𝑠 = -0.99 ( 𝑃 𝑟𝑠 = 1 . 4 × 10 -24 ) . \nThe scatter plots for the broken power law model parameters are presented in Figure 6. A strong anti-correlation is observed between index 'p' vs. normalization and the magnetic field 'B' vs. normalization, both with 𝑟 𝑠 = -0 . 99 ( 𝑃 𝑟𝑠 = 1 . 4 × 10 -24 ) . On the contrary, a strong positive correlation is found between index p and magnetic field B with 𝑟 𝑠 = 0 . 99 ( 𝑃 𝑟𝑠 = 1 . 4 × 10 -24 ) . \nFigure 7 illustrates the scatter plot for the PL with 𝛾 𝑚𝑎𝑥 model parameters. A pronounced anti-correlation is noted in both index 'p' vs. normalization and magnetic field 'B' vs. normalization, each with 𝑟 𝑠 = -0 . 99 ( 𝑃 𝑟𝑠 = 1 . 4 × 10 -24 ) . The observed strong anti-correlation between index p and \nFigure 4. Top left panel represents the variation of Γ with 𝜒 2 and 𝑃 𝑗𝑒𝑡 whereas the top right panel represents the variation of size of the region (R) with 𝜒 2 and 𝑃 𝑗𝑒𝑡 for both the Broken power law (blue) and Log parabola (red) models for data epoch-4. The bottom panel shows the variation of 𝛾 𝑚𝑖𝑛 with 𝜒 2 and 𝑃 𝑗𝑒𝑡 only for the Broken power law model. \n<!-- image --> \nnormalization contradicts the theoretical form of N ( 𝑙𝑜𝑔 N ∝ 𝑝 ; Equation 6). Additionally, a strong positive correlation is also noted between index 'p' and the magnetic field; however, this conflicts with the theoretical relation, B ∝ ( 𝑝 -1 ) -1 / 𝑛 (Khatoon et al. 2022), which predicts a negative correlation between 'p' and 'B'. \nFigure 8 shows the scatter plot between EDD model fit parameters. It is expected that log 10 ( 𝜓 ) will be inversely and linearly proportional to 𝜅 since 𝜓 = 𝜂 𝑅 𝜉 -𝜅 𝑅 , and as expected we found a strong anti-correlation in 𝜓 vs 𝜅 . \nThe above equation can be further expressed as \nlog 10 ( 𝜓 ) = log 10 ( 𝜂 𝑅 ) -𝜅 × log 10 ( 𝜉 𝑅 ) (16) \nEquation 16 was fitted using the values of 𝜅 and 𝜓 obtained from the broadband SED fitting. The fitted line in the left panel of Figure 8 on the 𝜅 vs. log 10 ( 𝜓 ) plot is having a slope \n(log 10 ( 𝜉 𝑅 ) ) of 2.54 and an intercept (log 10 ( 𝜂 𝑅 ) ) of 0.51. This implies that the observed photon energy 𝜉 2 𝑅 = 122 MeV falls within the spectral coverage of our broadband spectra. We can estimate the 𝛾 𝑅 ( 𝜉 𝑅 / √ C ) and 𝜂 𝑅 to be 3 . 6 × 10 8 and 3.24, respectively. \nAdditionally, injection energy can be estimated using the correlation between the best-fit parameters, 𝜅 and N . The relation between 𝜅 and N is given in Equation 10 which can be further expressed as \nln ( N ) = 𝜂 𝑅 𝜅 𝐴 𝜅 + 𝐵 (17) \nEquation 17 was fitted with the 𝜅 and N values obtained from the broadband SED fitting. \nUsing 𝜂 𝑅 = 3.24 from above, we fitted the 𝜅 vs. ln ( N ) plot shown in Figure 8, to obtain 𝐴 = 9 . 66 × 10 -04 , 𝐵 = -23 . 4. Since, 𝛾 0 ∼ 𝐴𝛾 𝑅 , we can estimate the injection energy, 𝛾 0 \nFigure 9 shows the scatter plot between the EDA model fit parameters. The free parameters 𝜅 , 𝜓 , and N in the EDA model are similar to the EDD model. The 𝜅 and 𝜓 parameters \n<!-- image --> \n× \nFigure 5. Scatter plots between the derived parameters obtained for log parabola model at five epochs. Top left panel: 𝛼 vs. 𝛽 , top middle panel: normalisation vs. 𝛼 , top right panel: 𝛼 vs. magnetic field value, bottom left panel: 𝛽 vs. normalisation, bottom middle panel: 𝛽 vs. magnetic field, and bottom right panel: normalisation vs. magnetic field. \n<!-- image --> \n2 \n. \n3 \n2 \n. \n4 \n2 \n. \n5 \np \nN ( \n8 \n9 \n10 \n11 \n12 \n× \n10 \n- \n12 \n) \nFigure 6. Scatter plots between the derived parameters with broken power law model at five epochs. Left panel: 𝑝 vs. normalisation, middle panel: 𝑝 vs. magnetic field, and right panel: normalisation vs. magnetic field. \n∼ 3 . 5 × 10 5 , which is significantly smaller than 𝛾 𝑅 . Such a high value of minimum Lorentz factor with a standard onezone SSC model for EHBLs sources has also been reported by several authors (Kaufmann et al. 2011; Zech & Lemoine 2021; Goswami et al. 2024). \nof the EDA model follow a similar trend as in Equation 16. However, the 𝜅 and N parameters of the EDA model have a relation given in Equation 14 which can be further expressed as \nln ( N ) = 𝜂 𝑅 𝜅 𝐴 𝜅 -𝜅 ln ( 𝜉 𝑅 ) + 𝐵 (18) \nWe fitted both Equation 16 and Equation 18 with the 𝜅 , 𝜓 , and N values of the EDA model. Equation 17 was applied to the 𝜅 vs. log 10 ( 𝜓 ) plot (left panel of Figure 9), while \n3 \n. \n5 \n3 \n. \n0 \n2 \n. \n5 \nGauss) \n3 \n- \n10 \n× \n( \nB \n2 \n. \n0 \n1 \n. \n5 \n3 \n. \n5 \n3 \n. \n0 \n2 \n. \n5 \nGauss) \n3 \n- \n10 \n× \n( \nB \n2 \n. \n0 \n1 \n. \n5", 'H/o.pc/t.pc/a.pc /e.pc/t.pc /a.pc/l.pc.': '<!-- image --> \n<!-- image --> \n<!-- image --> \n× \nFigure 7. Scatter plots between the derived parameters with PL with 𝛾 𝑚𝑎𝑥 model at five epochs. Left panel: 𝑝 vs. normalisation, middle panel: 𝑝 vs. magnetic field, and right panel: normalisation vs. magnetic field. \n<!-- image --> \nFigure 8. Scatter plots between the free parameters of the EDD model at five epochs. Left panel : 𝜅 vs. log 10 𝜓 ; right panel: ln 𝑁 vs. 𝜅 . A solid curve in each panel is the best-fitted function as described in the text. \n<!-- image --> \nEquation 18 was applied to the 𝜅 vs. ln ( N ) (right panel of Figure 9). From the fitted equations of the EDA model, we obtained 𝜉 2 𝑅 = 66 MeV, 𝛾 𝑅 = 2 . 66 × 10 8 , 𝜂 𝑅 = 3 . 21, 𝐴 = 7 . 5 × 10 -04 , and 𝐵 = -21 . 8. With the relation, 𝛾 0 ∼ 7 . 5 × 10 -04 𝛾 𝑅 , we find 𝛾 0 = 2 × 10 5 . Thus the results obtained from the EDA model are qualitatively similar to those obtained for the EDD model.', '5. SUMMARY AND DISCUSSION': "In this work, we have performed a detailed broadband SED analysis of an EHBL source, 1ES 0229 + 200, using simultaneous multi-wavelength observation taken at different epochs from September 2017 to August 2021 (MJD 58119 -59365) \nusing AstroSat -LAXPC, SXT, and UVIT. We have also included the 𝛾 -ray data from Fermi -LAT observed from August 2008 to October 2022 (MJD 54682.6 -59882) and VHE 𝛾 -rays data of MAGIC observed from 2013 to 2017 (MJD 56293 -58118). We used the one-zone synchrotron and SSC model ( 𝑠𝑠𝑐𝑖𝑐𝑜𝑛 ⊗ 𝑛 ( 𝜉 ) ) with various particle distributions viz. log parabola, broken power law, power law with maximum gamma ( 𝛾 𝑚𝑎𝑥 ), energy-dependent diffusion (EDD) and energy-dependent acceleration (EDA) model to fit the broadband SED. \nAccording to Costamante et al. (2018), the broadband SED modelling in hard-TeV blazars can be explained by the onezone SSC model with a smooth broken power-law particle \n<!-- image --> \nFigure 9. Scatter plots between the free parameters of the EDA model at five epochs. Left panel : 𝜅 vs. log 10 𝜓 ; right panel: ln 𝑁 vs. 𝜅 . A solid curve in each panel is the best-fitted function as described in the text. \n<!-- image --> \ndistribution. They estimated the break energy of the electrons, 𝛾 𝑏𝑟𝑒𝑎𝑘 ∼ 10 6 and magnetic field strengths in the range of a few mG for six hard-TeV blazars. In this work, for the TeV blazar 1ES 0229 + 200, we also find the magnetic field strength in the range of a few mG and the break energy of the electrons in the order of ∼ 10 6 . \nWecomputed the jet power, 𝑃 jet, for various particle energy distributions at five different epochs. In the case of broken power-law and PL with 𝛾 max models, we find 𝑃 jet ∼ 10 47 ergs/sec with a minimum Lorentz factor ( 𝛾 min) set to 10. This value decreases to approximately 10 44 ergs/sec when 𝛾 min is increased to 10 4 . However, for other particle energy distributions with intrinsic curvature, the calculated 𝑃 jet remains around 10 44 ergs/sec, irrespective of 𝛾 min. Interestingly, we found that the estimated 𝑃 jet is nearly independent of the bulk Lorentz factor ( Γ ) and size (R). In the case of intrinsically curved particle energy distributions, such as the log parabola, EDD, and EDA models, the 𝑃 jet ( ≈ 10 44 ergs/sec) represents only a small fraction of the Eddington luminosity (1.26 × 10 47 ergs/sec) of the blazar's black hole mass (10 9 𝑀 ⊙ ; Meyer et al. 2012), suggesting that accretion processes might be driving the jet. \nFor the source 1ES 0229 + 200, Acciari et al. (2020) estimated 𝑃 jet ( ≈ 10 44 ergs/sec) applying the one zone SSC model assuming a power law with exponential cutoff particle distribution which agrees well with our estimation. Consistent findings have been reported recently by Bora et al. (2024). They calculated the jet power for the HBL source Mrk 501 using the same particle distributions. They found that the \nestimated jet power with a broken power-law distribution was around 10 47 (10 44 ) ergs/sec with a minimum electron energy of 𝛾 𝑚𝑖𝑛 = 10 (10 3 ). However, the estimated jet power was found to be considerably lower than a few times (10 42 ergs/sec) for electron energy distributions with intrinsic curvature (such as the log-parabola form). \nCorrelation studies among the best-fit model parameters provide significant insights regarding the consistency and adequacy of the model in describing the observed broadband SED. For the power-law with a maximum gamma ( 𝛾 𝑚𝑎𝑥 ) model, we find a strong anti-correlation between the index p and normalisation N and a positive correlation between index p and magnetic field B. As discussed in Hota et al. (2021), these correlations are against expectations. This is because, in this model, the change in normalization (Equation 6), Δ N / N = 𝑙𝑜𝑔 ( 𝜉 2 0 ) Δ 𝑝 / 2, where 𝜉 2 𝑜 ≡ 𝛾 2 0 C , occurs for a change in the index, Δ 𝑝 . Since 𝛾 𝑜 is much smaller than the 𝛾 required to produce X-ray photons, 𝑙𝑜𝑔 ( 𝜉 2 0 ) should be large; the normalization should vary significantly with a positive correlation when the index changes. However, we find that the normalisation varies with the change in the index with a strong anti-correlation. The observed anti-correlation could be explained by a more sophisticated model in which the acceleration time scale and magnetic field are associated. \nWe showed that the spectral curvature may also be reproduced by the energy-dependent electron diffusion (EDD) model, where we consider the escape or diffusion time scale to be energy-dependent. As expected by the predictions of the model, we found a strong anti-correlation between the \ntwo model parameter normalization and 𝜓 with the third parameter 𝜅 . The compatibility of the mentioned correlation with the predicted one, for Γ = 20, allows us to calculate the reference photon energy 𝜉 2 𝑅 = 122 MeV, arising from an electron with energy 𝛾 𝑅 = 3 . 6 × 10 8 , and the energy of the electrons that are injected into the acceleration region, as 𝛾 0 = 3 . 5 × 10 5 . Similarly, we consider the spectral curvature may also be reproduced by the energy-dependent electron acceleration (EDA) model, where the acceleration time scale is energy-dependent. For the EDA model, the calculated values are approximate as 𝜉 2 𝑅 = 66 MeV, 𝛾 𝑅 = 2 . 66 × 10 8 , and 𝛾 0 = 2 × 10 5 . \nIn a previous investigation, Hota et al. (2021) utilized the EDD and EDA models to analyze the X-ray observations of the HBL source, Mkn 421, during its flaring phase, deriving respective spectral parameters. They identified a different range of spectral parameters, with 𝜅 falling between 0.3 and 1.0, and 𝜓 between 1.12 and 1.96. These findings resulted in a slope of log 10 𝜉 𝑅 = 0.38 for the 𝜅 vs. log 𝜓 anti-correlation. However, in our analysis of EHBL source 1ES 0229+200 (present study), we observe 𝜅 and 𝜓 values within the ranges of 0.11 -0.16 and 1.30 -1.65, respectively. The correlation between these parameters yields a steeper slope, log 10 𝜉 𝑅 = 2.54. Consequently, we obtain a significantly higher value of 𝜉 2 𝑅 (10 4 times larger than the HBL source). Notably, the calculated values of 𝜉 2 𝑅 in both the EDD and EDA models predominantly fall within the MeV range, surpassing the Synchrotron spectral energy range under consideration. This suggests favourable conditions for these models to fit the broadband spectrum of EHBL sources effectively. In contrast, for Mkn 421 during its flaring phase, the corresponding photon energy ( 𝜉 2 𝑅 ∼ 5.6 keV) remains within the considered energy range (0.5 -18 keV) in both the EDD and EDA models, contradicting the spectral fitting of HBL sources with these models. Additionally, we get a relationship 𝛾 0 = 9.6 × 10 -04 𝛾 𝑅 when fitting the correlation between 𝜅 and ln N , which is more likely appropriate in the physical framework than it is observed for the HBL source (i.e 𝛾 0 ∼ 0.2 𝛾 𝑅 ) (Hota et al. 2021). Moreover, we can now get the actual values of 𝛾 𝑅 and, more importantly, 𝛾 0 while using the SSC model (current study). \nThe intrinsically curved particle distribution models (EDD and EDA models) considered in this work are simple and have \nanalytical solutions; the physical situation may really be more complicated. Furthermore, considering both escape and acceleration time scales to be energy-dependent would be more physically plausible. Compared to the power-law form used in this work, the energy dependency of various time scales may differ. Eventually, in order to provide a more complete picture, the analysis must be expanded to include some other EHBL blazars. Moving forward, it would be valuable to explore additional aspects, such as temporal variations and extended datasets, to further refine our understanding of the jet dynamics and uncover any nuanced behaviours that might emerge over an extended observation period. Additionally, comparative analyses with other blazar sources could provide a comprehensive perspective on the universality of the observed trends in jet power and its dependence on particle energy distributions.", 'ACKNOWLEDGEMENTS': "We thank the anonymous referee for insightful comments and constructive suggestions. The authors JH and ACP would like to acknowledge Inter-University Centre for Astronomy and Astrophysics (IUCAA), Pune, India, for providing facilities to carry out this work. This publication uses data from the Astrosat mission of the Indian Space Research Organisation(ISRO), archived at the Indian Space Science Data Centre(ISSDC). This work has used the data from the Soft X-ray Telescope (SXT) developed at TIFR, Mumbai. LaxpcSoft software is used for analysis of the LAXPC data and we acknowledge the LAXPC Payload Operation Center (TIFR, Mumbai). This research has made use of data, software and/or web tools obtained from the High Energy Astrophysics Science Archive Research Center (HEASARC), a service of the Astrophysics Science Division at NASA/GSFC and of the Smithsonian Astrophysical Observatory's High Energy Astrophysics Division. \nFacilities: Fermi(LAT), AstroSat (UVIT, SXT AND LAXPC), Swift(UVOT) \nSoftware: fermipy -v0.17.4 (Fermipywebpage:https: //fermipy.readthedocs.io/en/latest/), XSPEC (https://heasarc.gsfc.nasa.gov/xanadu/xspec/), HEASARC(https://heasarc.gsfc.nasa.gov/docs/software/ heasoft/)", 'REFERENCES': 'Abdo, A. 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2024JCAP...10..075L | The gravitational lensing of supermassive black holes surrounded by dark matter halo has attracted a great number of interests in recent years. However many studies employed simplified dark matter density models which makes it very hard to give a precise prediction on the dark matter effects in real astrophysical galaxies. In this work to more accurately describe the distribution of dark matter in real astrophysical galaxies we study the gravitational lensing of black holes in astrophysical dark matter halo models Beta Burkert Brownstein and Moore. The deflection angle is obtained using a generalized GibbonsWerner approach. The visual angular positions and the Einstein rings are also calculated by adopting the gravitational lens equation. Specifically we choose the supermassive black holes in Milky Way Galaxy Andromeda galaxy M31 Virgo galaxy M87 and ESO138G014 galaxy as examples including the corresponding fitted value of dark matter halos. The results suggest that the dark matter halo described by the Beta model has nonnegligible influences on the gravitational deflection angle and gravitational lensing observations. However the Burkert Brownstein and Moore models have relatively small influences on angular position of images and the Einstein ring. | 2024-10-01T00:00:00Z | ['10.1088/1475-7516/2024/10/075', '10.48550/arXiv.2312.15760', '2024JCAP...10..075L', 'arXiv:2312.15760', '2023arXiv231215760L'] | ['GR black holes', 'weak gravitational lensing', 'massive black holes', 'rotation curves of galaxies', 'General Relativity and Quantum Cosmology', 'Astrophysics - Cosmology and Nongalactic Astrophysics', 'Astrophysics - Astrophysics of Galaxies'] | Gravitational lensing of spherically symmetric black holes in dark matter halos | 2,024 | 166 | 0.39 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML'] | 4 | https://arxiv.org/pdf/2312.15760.pdf | {'Yi-Gao Liu, a Chen-Kai Qiao, b Jun Tao a': 'a College of Physics, Sichuan University, Chengdu, 610065, China \nb \nCollege of Science, Chongqing University of Technology, Chongqing, 400054, China \nE-mail: [email protected], [email protected], [email protected] \nAbstract. The gravitational lensing of supermassive black holes surrounded by dark matter halo has attracted a great number of interests in recent years. However, many studies employed simplified dark matter density models, which makes it very hard to give a precise prediction on the dark matter effects in real astrophysical galaxies. In this work, to more accurately describe the distribution of dark matter in real astrophysical galaxies, we study the gravitational lensing of black holes in astrophysical dark matter halo models (Beta, Burkert, Brownstein, and Moore). The deflection angle is obtained using a generalized Gibbons-Werner approach. The visual angular positions and the Einstein rings are also calculated by adopting the gravitational lens equation. Specifically, we choose the supermassive black holes in Milky Way Galaxy, Andromeda galaxy (M31), Virgo galaxy (M87), and ESO138-G014 galaxy as examples, including the corresponding fitted value of dark matter halos. The results suggest that the dark matter halo described by the Beta model has non-negligible influences on the gravitational deflection angle and gravitational lensing observations. However, the Burkert, Brownstein, and Moore models have relatively small influences on angular position of images and the Einstein ring.', '1 Introduction': "In the past years, abundant astrophysical observations revealed that our universe is dominated by dark matter and dark energy. The cosmic microwave background observation shows that 26.8% of our universe has consisted of dark matter, and 68.3% of our universe is made up of dark energy, while only 4.9% of universe is composed of baryonic matter [1-4]. The dark matter has non-negligible influences on the galactic and cosmological predictions. Especially, the large-scale structures formation of the universe [5-8], bullet clusters [9, 10], star motions in galaxies (galactic rotational curves [11-13]), and gravitational lensing can hardly get desired explanations without assuming the dark matter. Furthermore, in past few years, a number of X-ray and matter-antimatter satellites and detectors (such as Chandra, Fermi-LAT, AMS and DAMPE) reported anomalous excess of X-rays and other cosmic-rays [14-17]. The dark matter near the galaxy center provides a viable explanation on such excess. There are a number of dark matter candidates proposed in the past few decades, and most of them were given by some unknown non-baryonic particles generated in theories beyond the Standard Model [17, 18]. In recent years, the weakly-interactive massive particles (WIMPs) and axions have attracted considerable interest in dark matter detection experiments [19-25]. \nThe dark matter vastly influences the behavior of galaxies / galaxy clusters, for it could produce extremely strong gravitational potential, heavily affecting the particles and stars motions in galaxies / galaxy clusters. The dark matter usually forms halo structures in the galaxy and galaxy clusters [26-28]. The range and mass of dark matter halos are very large [28]. The supermassive black hole in a galaxy center is usually surrounded by a dark matter halo with a typical length scale h ∼ 10 kpc. The dark matter halo structure strongly affects the galactic rotational curves [11-13, 29-31], the motions of matter in Bullet cluster collision observations [10], the gravitational lensing of massive objects [32-34], the particle motions and chaos [27, 35], the black hole shadow and its polarized image [37-40]. Through simulation studies and astrophysical observations, many dark matter halo models have been proposed, and most of them suggest that the dark matter distribution is spherically symmetric. The most popular halo models include the Navarro-Frenk-White (NFW) model [41, 42], Einasto model [44-46], Beta model [47, 48], Burkert model [49, 50], Brownstein model [51], and Moore model [52]. These models have become extremely valuable in phenomenological studies, which restrain the indirect and direct dark matter detections [53-56]. \nThe gravitational lensing is one of the most significant methods to extract knowledge and properties from massive objects in the galaxy and our universe, especially in the presence of dark matter. The enormous amount of dark matter and luminous matter in galaxies and galaxy clusters makes the light rays deflected and distorted during the propagation, producing single or multiple images for distant light sources. Essentially, the gravitational lensing is nothing but relevant to the particle motions. It is natural to infer that dark matter, which produce strong gravitational potential, may have notable influences on gravitational lensing observations in the galactic scales. The anomalous excess of X-rays and other cosmic-rays reported by Chandra, Fermi-LAT, AMS and DAMPE [14-17] indicates the large amount dark matter near the galaxy center around the supermassive black holes (which may generate potentially observable effects on the gravitational lensing). In the last few decades, gravitational lensings have often been used to constrain non-luminous matters, such as dark matter distributions, and other massive compact objects [57-59]. In astrophysical observations, the gravitational deflection angle of light, gravitational lens equation, and Einstein ring are crucial quantities, closely connected with the distributions and properties of gravitational sources. These have been extensively researched in many gravitational systems [60-82]. \nBased on the aforementioned reasons, it is extremely important to study the dark matter effects on gravitational lensing observations in galaxies, especially for the gravitational lensing of supermassive black hole in the galaxy center. Recently, the gravitational lensing of black holes surrounded by dark matter has been extensively studied in literature [38, 83-93]. These studies suggested that dark matter has a significant effect on gravitational deflection and gravitational lensing. However, many studies employed simplified dark matter density models [85-89], from which it is hard to give a precise dark matter mass distribution (as well as its influences on central black holes) in real galaxies. Therefore, using precise and authentic dark matter models rather than over-simplifying ones in the gravitational lensing of central black holes is necessary. Until very recently, Óvgün et al first considered the gravitational lensing of black hole within the cold dark matter halo model with an NFW density profile [92], where a real galactic dark matter distribution goes beyond the over-simplified model is included in analytical studies. \nInspired by Óvgün's work, we are interested in the gravitational lensing of black holes interplayed with dark matter medium in galaxy centres. Using several astrophysical dark matter halo models (Beta, Burkert, Brownstein, and Moore), we are committed to give a \nprecise description on the galactic dark matter halo. In this work, we study the gravitational deflection of spherically symmetric black holes in dark matter halos. The dark matter distributions in galaxy and halo structures are given by several phenomenological dark matter density profiles: the Beta, Burket, Brownstein, and Moore models. Furthermore, using a generalized Gibbons and Werner (GW) approach [94-98], the gravitational deflection angles of light are derived and calculated. Additionally, we also study the dark matter halo effects on gravitational lensing observations. The important observables in gravitational lensing are the positions of luminous sources and the lensed images, and they are constrained by gravitational lens equations [99-104]. One of the most important observables in gravitational lensing, the Einstein ring, is mainly focused in the present work. To directly connect with the astrophysical gravitational lensing observations, it would be helpful to select some real supermassive black holes and dark matter halos in galaxies and galaxy clusters to carry out calculations. We consider the supermassive black holes in Milky Way Galaxy, Andromeda galaxy (M31), Virgo galaxy (M87), and ESO138-G014 galaxy, and the fitted value of dark matter halos in these galaxies are taken into calculations. \nThis paper is organized as follows. In section 2, we derived the effective spacetime metrics for spherically symmetric black holes in dark matter halos. The theoretical treatment (the generalized GW method to gravitational deflection angle, the gravitational lens equation to angular positions of images and Einstein ring) is introduced in section 3. Section 4 gives the analytical results of the gravitational deflection angle for black holes surrounded by dark matter halos. We derive the gravitation deflection angle for both the receiver and the light source at infinity. In section 5, the angular positions of lensed images and Einstein ring are investigated by solving the gravitational lens equation. We conclude our works and give perspectives in section 6. Appendix A compares the effects of dark matter and luminous matter on gravitational deflection angles. Throughout the paper, we adopt the convention G = c = 1 .", '2 The Spherically Symmetric Black Hole Surrounded by a Dark Matter Halo': "In this section, we give the descriptions of the spacetime metric for black holes surrounded by a dark matter halo. The dark matter distributions in our galaxies and other spiral galaxies are usually modeled by several astrophysical phenomenological models, such as NFW, Einasto, Beta, Burkert, Brownstein, and Moore models [31, 42, 47-52, 105]. For most galaxies, the dark matter halo can be effectively described by a spherically symmetric distribution. For simplicity, we shall focus on the non-rotating black holes surrounded by dark matter halos whose spacetime metric can be expressed by, \nd s 2 = -f ( r ) dt 2 + f ( r ) -1 dr 2 + r 2 d θ 2 + r 2 sin 2 θ d ϕ 2 . (2.1) \nThe effects of dark matter halo in gravitational lensing are through its mass profile and mass density. The dark matter halo mass profile is defined as \nM DM ( r ) = 4 π ∫ r 0 ρ ( r ' ) r ' 2 dr ' , (2.2) \nwhere ρ ( r ) is the density of dark matter distributions, and the detailed expressions are given by astrophysical phenomenological models. Using the mass profile, the tangential velocity \nof the test particle in dark matter halo is calculated easily by v tg 2 ( r ) = M ( r ) /r . For a test particle in spherical symmetric spacetime, its rotation velocity in the equatorial plane is determined by the metric function f ( r ) as [36] \nv tg 2 ( r ) = r √ f ( r ) · d √ f ( r ) dr = r ( dln √ f ( r )) dr . (2.3) \nAccording to the rotational velocity in Eq. (2.3), the metric function of the dark matter halo can be derived by solving the ordinary differential equation Eq. (2.3) \nf DM ( r ) = exp[2 ∫ v tg 2 ( r ) r d r ] . (2.4) \nIn this work, we use several famous spherically symmetric dark matter profiles (Beta, Burkert, Brownstein, Moore). These models were proposed and developed for many years. Cavaliere and Fusco-Femiano proposed the isothermal Beta model in 1976 when studying the distribution of matter in galaxies [48]. Burkert proposed an empirical profile that successfully fitted the halo rotation curves of four dark matter-dominated dwarf galaxies in 1995 [49, 50]. In 1999, Moore et al. simulated that the profile has a cusp proportional to r -1 . 5 in both galaxy-sized and cluster-sized halos [52]. Brownstein showed that the core-modified profile with a constant central density fits excellently well to the rotation curves of both high- and low-surface brightness galaxies in 2009 [51]. These dark matter profiles have been widely used in theoretical predictions and numerical simulations in physics and astronomy. The dark matter distributions of the Beta, Burkert, Brownstein, and Moore models can be expressed as \nρ Beta ( x ) = ρ 0 Beta (1 + x 2 ) 3 / 2 , (2.5a) \nρ Bur ( x ) = ρ 0 Bur (1 + x )(1 + x 2 ) , (2.5b) \nρ Bro ( x ) = ρ 0 Bro 1 + x 3 , (2.5c) \nρ Moo ( x ) = ρ 0 Moo x 3 / 2 (1 + x 3 / 2 ) , (2.5d) \nwhere x = r/h , ρ 0 and h are the characteristic density and radius of dark matter halos respectively. Among these dark matter distribution, the Beta, Burkert, and Brownstein models are cored halo models with a smooth density profile near the galaxy center, and the Moore model is a cusp halo model with a rapidly increased density profile near the galaxy center [31]. \nTo get the spacetime metric of black holes surrounded by dark matter halos, one can combine the dark matter density and the central black hole mass M to the energy-momentum tensor T µν in the Einstein field equation. It turns out that the spacetime metric can be decomposed into f ( r ) = f DM ( r ) -2 M r , where f DM ( r ) in Eq. (2.4) describes the dark matter halo and -2 M r describes the effects of supermassive black hole in the galaxy center, seeing Ref. [106] for details. Eventually, we get the metric function of the black hole in each dark \nmatter halo model: \nf Beta ( r ) = e -8 πk r sinh -1 x -2 M r , (2.6a) \nf Bur ( r ) = e 4 πk r (1+ x ) arctan x (1 + x ) -4 πk r (1+ x ) (1 + x 2 ) 2 πk r ( x -1) -2 M r , (2.6b) \nf Bro ( r ) = e 4 πkx 2 h 2 F 1 ( 2 3 , 1; 5 3 ; -r 3 h 3 ) (1 + x 3 ) -8 πk 3 r -2 M r , (2.6c) \nf Moo ( r ) = e 16 πk √ 3 h arctan 2 √ x -1 √ 3 (1 + x 3 / 2 ) -16 πk 3 r ( 1 + x -√ x 1 + x +2 √ x ) -8 πk 3 h -2 M r , (2.6d) \nwhere M is the mass of the supermassive black hole, 2 F 1 represents the hypergeometric function and k = h 3 ρ 0 can be used to give an estimation of the dark matter mass. \nThe various matter fields in astrophysical galaxies around the supermassive black holes also influence the gravitational lensing, and a comprehensive study including dark matter halo and all the baryonic matter in galaxies is extremely complicated. Apart from dark matter halos, there are nearby luminous stars, interstellar gas, dust, and plasma mediums. In this paper, the effect of luminous matter on the deflection of light in the Milky Way and Andromeda galaxies is roughly investigated in Appendix A. The gravitational effects of other matters are also of interest and will be explored in future studies.", '3 Theoretical Treatment': 'In this section, we use the generalized Gibbons and Werner (GW) method to calculate the gravitational deflection angle in asymptotically flat spacetimes. At the same time, we obtain the angular positions of lensed images and the Einstein ring angular radius by solving the famous lens equation. Notably, we assume the horizon of a supermassive black hole, impact parameter, and dark matter halo scale satisfies r H ∼ M ≪ b ≪ h . The gravitational deflection of light under this assumption is illustrated in Fig. 1.', '3.1 Gravitational deflection angle': "According to Einstein's general theory of relativity, gravitational interaction is a geometric effect, so analyzing gravitational bending and gravitational lensing through new techniques of geometry and topology is possible. The Gauss-Bonnet theorem was first proposed by G. W. Gibbons and M. C. Werner to calculate the gravitational deflection angle of particles [94]. In their work, the gravitational deflection angle is calculated by a surface integral in the optical geometry \nα = -∫ ∫ D KdS. (3.1) \nwith K to be the Gaussian curvature of optical geometry. \nRecently, this approach has been applied to large numbers of static and stationary gravitational systems, and several improved schemes are proposed [95-97, 107-109]. Notably, Y. Huang and Z. Cao proposed a generalized Gibbons and Werner (GM) method that works well for asymptotically flat and nonflat spacetimes, and it deeply simplify the calculation of the deflection angle, seeing Ref. [96, 97] for details. In this paper, we will take this approach to carry out the calculation. Based on Huang's derivation, the gravitational deflection angle is reduced from Eq. (3.1) as \nα = ∫ ϕ R ϕ S [1 + H ( r γ )] dϕ, (3.2) \nFigure 1 . This figure illustrates the gravitational deflection of light in the presence of dark matter. The dark matter in the galaxy forms a halo structure, which encloses the supermassive black hole in the Galactic center. The b labels the impact parameter in the gravitational lensing, and h is the characteristic scale of the dark matter halo. In this work, we assume the horizon of a supermassive black hole, impact parameter, and dark matter halo scale satisfies r H ∼ M ≪ b ≪ h . \n<!-- image --> \nwith the function H ( r ) defined by \nH ( r ) = -1 2 √ g ∂g OP ϕϕ ∂r . (3.3) \nThe above expressions are derived by using the Gauss-Bonnet theorem in a two-dimensional manifold M ( g 2) , corresponding to the equatorial plane of optical space for static and spherically symmetric spacetimes(SSS) and ϕ R , ϕ S in Eq. (3.2) are the azimuthal angle for observer and light source. The two-dimensional optical space metric is usually constructed from the SSS spacetime metric by constraining ds 2 = 0 and θ = π/ 2 \ndt 2 = g OP rr ( r ) dr 2 + g OP ϕϕ ( r ) dϕ 2 = 1 f ( r ) 2 dr 2 + r 2 f ( r ) dϕ 2 , (3.4) \nwhere g in Eq. (3.3) is the determinant of the above optical metric. The r γ denotes the radial coordinate of photon orbit from the light source to the observer, which can be determined by the orbital equation \n( du dϕ ) 2 = 1 b 2 -u 2 F ( u ) , (3.5) \nwhere the variable u is defined as u = 1 /r , and F ( u ) = f (1 /u ) is the metric of SSS spacetime. \nApplying the famous Gauss-Bonnet theorem in the two-dimensional optical space of black holes, the Gibbons and Werner's method and its generalizations have achieved a great success in a number of gravitational systems, such as the rotational black holes and wormholes [95, 110-113], the finite distance deflections [107, 114-116], the spacetimes in the presence of plasma and dark matter [83, 92, 117-119], and asymptotically non-flat spacetimes [96, 108, 109, 120, 121]. The results obtained from these methods are consistent with traditional approaches in the weak deflection cases [122-126].", '3.2 The lens equation and einstein ring': "In astrophysical observations, light beams emitted from remote luminous sources can be deflected and converged due to the central supermassive black hole, which plays the role of a gravitational lens. Sometimes, luminous light sources may exhibit multiple images after a gravitational lens, and the Einstein ring is such example. The angular radius of the Einstein ring is usually achieved by solving the lens equation. Many gravitational lens equations have been proposed in astrophysical studies [99-103]. In our work, we choose a famous lens equation is given by V. Bozza [104], \nD OS · tan β = D OL · sin θ S -D LS · sin ( α -θ S ) cos ( α -θ S ) . (3.6) \nHere, D OS is the distance between the observer and the source plane, D OL is the distance between the observer and the lens plane, D LS is the distance between the lens plane and the source plane, angle β represents the precise angular position of the luminous source, and angle θ S is the visual angular position of the lens image as seen by a distant observer. The actual position of a luminous source and its visual images produced by a gravitational lens are shown in figure 2. In many cases, the lensed astrophysical luminous sources could have multiple images. The figure illustrated the most general two-image cases in gravitational lensing observations. In the source plane, the image S 1 and the light source are located on the same side, while the image S 2 are located on the other side of the light source (they are separated by the central supermassive black hole). We adopt the following convention for angle α , β , and θ S : the angles in the counter-clockwise direction to be positive, and the angles in the clockwise direction to be negative. Under such convention, the observables in the gravitational lensing can be applied to the same lens equation (3.6), regardless of which direction the light beams are deflected during the propagation. \nIn most of the gravitational lensing, the light beams are emitted from distinct sources, which makes the gravitational deflection angle very small and the weak gravitational deflection limit available. In the weak gravitational deflection limit, the approximations tanβ ≈ β , sinθ S ≈ θ S , and sin ( α -θ S ) ≈ α -θ S can be made. And the relation D OS = D OL + D LS reduces the gravitational lens Eq. (3.6) to \nβ = θ S -D LS D OS · α. (3.7) \nIn theory, given any value of β , we can find the corresponding θ S . However, we are of great interest to the Einstein ring angular radius. By taking β = 0 to calculate the angular radius of Einstein ring, the formula is obtained \nθ E = D LS D OS · α. (3.8) \nIn addition, in the weak deflection cases, the impact parameter b is satisfied \nb ≈ D OL · sin θ E . (3.9) \nThe angular radius θ E of Einstein's ring can be solved by Eqs. (3.8) and (3.9) as long as the gravitational deflection angle α is known. \nGLYPH<dc7a> \nGLYPH<dc73> \nFigure 2 . This figure illustrates the light propagation in the gravitational lensing. The locations of the observer, luminous light source, and the central supermassive black hole (which acts as a gravitational lens) are labeled respectively. The gravitational deflection angle of light α , angular position of the light source β , angular positions of the lensed images θ S 1 , θ S 2 , and impact parameter b have been shown in the figure. Notably, the angles θ S 1 , α 1 , β in the clockwise direction are assigned to be positive, and the angles θ S 2 , α 2 in the clockwise direction are assigned to be negative. \n<!-- image --> \nGLYPH<d835> \nGLYPH<d835>GLYPH<dc73> \nGLYPH<d835> \nGLYPH<d835>GLYPH<d835>", '4 Calculation of Gravitational Deflection Angles': 'In this section, we present the results and discussions on the gravitational deflection angle for black hole surrounded by a dark matter halo. Besides, we mainly focused on the weak gravitational deflection limit, so that the gravitational deflection angles are dominantly contributed by the leading-order M/b and k/b terms. The subsections specify the gravitational deflection angles for the dark matter distributions given by Beta, Burkert, Brownstein, and Moore models.', '4.1 Beta model': 'In this subsection, we calculate the gravitational deflection angle of black hole surrounded by Beta model dark matter halo. The metric of M ( g 2) corresponding to the equatorial plane of optical space for dark matter distribution is \nd l 2 = 1 f 2 ( r ) dr 2 + r 2 f ( r ) dϕ 2 . (4.1) \nAccording to the generalized GW method described in subsection 3.1, the gravitational deflection angle can be calculated through the integration of function H ( r ) . Substituting Eqs. (2.6a) and (4.1) into Eq. (3.3) leads to \nH ( r ) = -1 + 2 M r + 4 πk r [ 2arcsinh r h -1 √ 1 + ( h r ) 2 ] + O ( M 2 , k 2 , kM ) . (4.2) \nConsidering the motion on the equatorial plane, the photon orbit equation can be represented by Eq. (3.5) \n( du dϕ ) 2 = 1 b 2 -u 2 +2 u 3 ( M -4 πk ln hu 2 ) + O ( M 2 , k 2 , kM ) , (4.3) \nwith a new variable u = 1 /r defined in photon orbit. Thus, we can get the perturbation solution \nu = sin ϕ b + M ( 1 + cos 2 ϕ ) b 2 + 2 πk b 2 [ 1 + 3 ln 2 b csc ϕ h +(1 + ln 2 b csc ϕ h ) cos 2 ϕ +4cos ϕ ln tan ϕ 2 ] + O ( M 2 , k 2 , kM ) . (4.4) \nFor both the light source and the receiver are at infinity, u = 1 /r → 0 , the azimuthal angles ϕ R and ϕ S can be approximated by ϕ R ≈ π + α ≈ π and ϕ S ≈ 0 . We calculate the gravitational deflection angle through Eq. (3.2) \nα = ∫ π 0 [1 + H ( r γ )] dϕ = 4 M b + 4 πk h ( π -z 3 ) + O ( M 2 , k 2 , kM ) , (4.5) \nwhere z = b/h and the r γ = 1 /u have been used with Eq. (4.4). The higher-order terms are not presented here. With the gravitational deflection angle, we can solve the lens equation to obtain the Einstein ring angular radius, which is studied in section 5.', '4.2 Burkert model': 'We present calculation of the gravitational deflection angle for a black hole surrounded by the Burkert model dark matter halo in this subsection. Substitute Eqs. (2.6b) and (4.1) into Eq. (3.3) to get \nH ( r ) = -1 + 2 M r + πk r [ (2 -r h ) ln(1 + r 2 h 2 ) -2(2 + r h )(arctan r h -ln(1 + r h )) ] + O ( M 2 , k 2 , kM ) . (4.6) \nSince the motion is restricted on the equatorial plane, the photon orbit equation can be represented \n( du dϕ ) 2 = 1 b 2 -u 2 +2 Mu 3 -2 πku 2 h [ π + hu ( π -4 + 4 ln hu ) ] + O ( M 2 , k 2 , kM ) . (4.7) \nThe perturbation solution is \nu = sin ϕ b + M (1 + cos 2 ϕ ) b 2 -πk 2 b 2 [ 3 π -16 + 2 π b h sin ϕ +12ln h sin ϕ b +( π -8 + 4 ln h sin ϕ b ) cos 2 ϕ +( π 2 b h -2 πϕ b h +16lncot ϕ 2 ) cos ϕ ] + O ( M 2 , k 2 , kM ) . (4.8) \nWhen the light source and the receiver are at infinity, having employed approximations of ϕ R ≈ π + α ≈ π and ϕ S ≈ 0 , we obtain the gravitational deflection angle through Eq. (3.2) \nα = ∫ π 0 [1 + H ( r γ )] dϕ = 4 M b + πkz 4 18 b (5 -3 π +12ln z 2 ) + O ( M 2 , k 2 , kM ) , (4.9) \nwhere the r γ = 1 /u has been used with Eq. (4.8).', '4.3 Brownstein model': 'For a black hole surrounded by the Brownstein model dark matter halo, the gravitational deflection angle can be calculated using the same scheme in the previous subsections. The function H ( r ) can be obtained through substituting Eqs. (2.6c) and (4.1) into Eq. (3.3) \nH ( r ) = -1 + 2 M r + 2 πk r [ 4 3 ln(1 + r 3 h 3 ) -r 3 h 3 2 F 1 ( 2 3 , 1; 5 3 ; -r 3 h 3 ) ] + O ( M 2 , k 2 , kM ) . (4.10) \nThe photon orbit equation on the equatorial plane can be expressed \n( du dϕ ) 2 = 1 b 2 +2 Mu 3 -u 2 (1 + 16 √ 3 π 2 k 9 h )(1 -8 πku +8 πku ln hu ) + O ( M 2 , k 2 , kM ) . (4.11) \nThe perturbation solution can be gained \nu = sin ϕ b + M (1 + cos 2 ϕ ) b 2 + 2 πk 9 b 2 [ -2 cos ϕ ( √ 3 π ( π -2 ϕ ) b h +18lncot ϕ 2 ) +18(2 + cos 2 ϕ ) -9(cos 2 ϕ +3)ln h sin ϕ b -4 √ 3 π b h sin ϕ ] + O ( M 2 , k 2 , kM ) . (4.12) \nFinally, the gravitational deflection angle is calculated from Eq. (3.2) with the function H ( r ) and photon orbit specified by Eqs. (4.11) and (4.12) \nα = ∫ π 0 [1 + H ( r γ )] dϕ = 4 M b + 2 πk 9 h (8 √ 3 πz -9 C 1 ( z )) + O ( M 2 , k 2 , kM ) , (4.13) \nwhere r γ = 1 /u has been used and C 1 ( z ) = z 2 ∫ π 0 csc 2 ϕ 2 F 1 ( 2 3 , 1; 5 3 ; -z 3 csc 3 ϕ ) d ϕ is a function of z = b/h , which could be calculated numerically. The detailed behavior of function C 1 ( z ) can be seen in Appendix C.', '4.4 Moore model': 'For the calculation of black hole surrounded by Moore model dark matter halo, through substituting Eqs. (2.6d) and (4.1) into Eq. (3.3), one can get \nH ( r ) = -1 + 2 M r + 4 πk 3 r [ 4 ln(1 + ( r h ) 3 2 ) -2 r h ( √ 3 arctan 2 √ r h -1 √ 3 +arctanh 3 1 + 2 √ r h +2 √ h r ) ] + O ( M 2 , k 2 , kM ) . (4.14) \nOn the equatorial plane, the photon orbit equation can be represented \n( du dϕ ) 2 = 1 b 2 + u 2 [ 2 u ( M +4 πk ) -8 √ 3 π 2 k 3 h -1 ] -8 πku 3 ln hu + O ( M 2 , k 2 , kM ) . (4.15) \nWe can get the perturbation solution \nu = sin ϕ b + M (1 + cos 2 ϕ ) b 2 -2 πk 3 b 2 [ ( √ 3 π ( π -2 ϕ ) b h +12lncot ϕ 2 ) cos ϕ -6(2 + cos 2 ϕ ) + 3(3 + cos 2 ϕ ) ln h sin ϕ b +2 √ 3 π b h sin ϕ ] + O ( M 2 , k 2 , kM ) . (4.16) \nFigure 3 . The gravitational deflection angles of black holes surrounded by Beta, Burkert, Brownstein, and Moore dark matter halos with different b/M . In this figure, we take the parameter k = M , and the characteristic radius of the dark matter halo is h = 100 b . The green line is the Beta model result, the red square dotted line is the Burkert model result, the orange line is the Brownstein model result, and the blue line is the Moore model result. Since the Burkert and Brownstein models results overlap, the Burkert model result is distinguished by a red square dotted line. \n<!-- image --> \nThe r γ = 1 /u has been used with Eq. (4.16). Obtaining the above integration function H ( r γ ) and photon orbit, the gravitational deflection angle can be calculated through Eq. (3.2) \nα = ∫ π 0 [1 + H ( r γ )] dϕ = 4 M b + 8 √ 3 πk 9 b [ 2 πz 3 / 2 ( π 3 / 2 Γ( 3 4 ) Γ( 7 12 ) 2 Γ( 11 12 ) 2 Γ( 5 4 ) -2 √ z ) -3 C 2 ( z ) + √ 3 C 3 ( z ) ] + O ( M 2 , k 2 , kM ) , (4.17) \nwhere C 2 ( z ) = z ∫ π 0 arctan 2 √ z csc ϕ -1 √ 3 d ϕ and C 3 ( z ) = z ∫ π 0 arctanh 3 √ z csc ϕ 2+2 z csc ϕ + √ z csc ϕ d ϕ can be calculated numerically. The details of functions C 2 ( z ) and C 3 ( z ) are shown in Appendix C.', '4.5 Comparison of numerical results': 'To study the dark matter effects on gravitational deflection of black holes surrounded by dark matter halos, it would be necessary to give Comparisons between the gravitational deflection results using the four dark matter halo models. In the actual astrophysical gravitational lensing observations, the observer is usually very far from the lensed black holes. Therefore, the assumption of infinite distance observer and light source in the calculations of gravitational deflection angles displayed in subsections 4.1-4.4 is reasonable. We plot the gravitational deflection angle for infinite distance observers and light sources in Figs. 3 and 4. \nAt different b/M for both the light source and the receiver at infinity, the gravitational deflection angles of black holes surrounded by Beta, Burkert, Brownstein, and Moore dark matter halos are shown in Fig. 3. We take the parameter k = M , and the characteristic radius of the dark matter halo h = 100 b in this figure. We can see from the figure that the gravitational deflection angle decreases with the increase of the b/M . In other words, when the dark matter halo parameters are fixed, with the increase of the impact parameter, the influence of the central supermassive black hole on the light decreases, resulting in the \nFigure 4 . The gravitational deflection angles of black holes surrounded by Beta, Burkert, Brownstein, and Moore dark matter halos with different values of h/b . In this figure, we also take the parameter k = M , and the impact parameter is selected as b = 100 M . The thumbnail is an enlarged image of the Brownstein model. The green, orange, blue lines correspond to Beta, Brownstein, Moore results, and the Burkert model result is distinguished by a red square dotted line, which is the same as in Fig. 3 \n<!-- image --> \nreduction of the gravitational deflection angle. Furthermore, for the same dark matter mass, dark matter halo scale, and impact parameters, the Beta dark matter halo model results in the largest gravitational deflection angle, followed by the Moore model. The deflection angle in the Burkert and Brownstein models are the smallest, and the curves of these two models are overlapped in the figure. \nAt different h/b for both the light source and the receiver at infinity, the gravitational deflection angles of black holes surrounded by Beta, Burkert, Brownstein, and Moore dark matter halos sre shown in Fig. 4. We also take the parameter k = M , and the impact parameter is elected as b = 100 M in this figure. With the increase of h/b , the gravitational deflection angles in Beta, Burkert, Brownstein, and Moore models decrease and converge to M/b = 0 . 04 . In other words, when the impact parameter is determined, for the Beta, Burkert, and Moore models, as the dark matter halo characteristic radius increases, the influence of the dark matter halo on gravitational deflection angle decreases. This is because, for a dark matter halo with a larger scale and constant mass, the dark matter is more sparsely distributed and the central black hole is more weakly affected by the dark matter medium. In the h → ∞ limit, the gravitational deflection angles for different halo models converge to the 4 M/b term (the gravitational deflection angle without the dark matter). However, the Burkert and Brownstein model curves are nearly straight lines, indicating that the dark matter halo has little effect on their gravitational deflection angles. Furthermore, for the same dark matter mass, dark matter halo scale, and impact parameters, the Beta dark matter halo model results in the largest gravitational deflection angle, followed by the Moore model. The Burkert and Brownstein models result in the smallest deflection angles, and their curves are overlapped in the figure.', '5 Angular Position Of Lensed Images and Einstein Ring': 'In this section, we discuss some valuable observables in gravitational lensing observations. Through the gravitational lensing, the visual angular position of lensed images θ S could be \nproduced. Especially, the Einstein ring, a special case of gravitationally lensed images with β = 0 , is of more importance and deserves a detailed discussion in our work. \nWe mainly discuss the gravitational lensing from the supermassive black holes in our Galaxy (Sgr*A in the Milky Way), Andromeda galaxy(M31), Virgo galaxy(M87), and ESO138G104 galaxy in the presence of dark matter halos. These black holes act as gravitational lenses and could provide typical examples in astrophysical gravitational lensing observations. Since the light sources in these gravitational lensing observations are distant from us, one can use the gravitational lens equation and the analytical gravitational deflection angle derived from the weak deflection limit to calculate the Einstein ring angular radius. The position of the lensed image is calculated numerically by using Eq. (3.7). Further, setting β = 0 , the Einstein ring angular radius of the above dark matter models can be calculated with θ E = D LS D OS · α , where the gravitational deflection angle α of the four models can be expressed as \nα Beta = 4 M b + 4 πk h ( π -z 3 ) , (5.1a) \nα Bur = 4 M b + πkz 4 18 b (5 -3 π +12ln z 2 ) , (5.1b) \nα Bro = 4 M b + 2 πk 9 h (8 √ 3 πz -9 C 1 ( z )) , (5.1c) \nα Moo = 4 M b + 8 πk 9 b [ 2 πz 3 / 2 ( π 3 / 2 Γ( 3 4 ) Γ( 7 12 ) 2 Γ( 11 12 ) 2 Γ( 5 4 ) -2 √ z ) -3 C 2 ( z ) + √ 3 C 3 ( z ) ] . (5.1d) \nNote that the Einstein ring angular radius θ E is usually very small in the astrophysical gravitational lensing observations. The impact parameter b in gravitational lensing satisfies Eq. (3.9). Finally, the Einstein ring angular radius of a lensed luminous object can be calculated by solving Eqs. (5.1) and (3.9). \nTo present the numerical calculations of observables in the gravitational lensing of black holes in dark matter halos, we first consider the supermassive black hole Sgr*A in the Milky Way Galaxy. According to the recent works in the Milky Way [128-130], the mass of the supermassive black hole Sgr*A is M = 4 . 3 × 10 6 M ⊙ , the distance between the observer and the lens plane is D OL = 8 . 3 kpc, the characteristic radius of the dark matter halo is about h = 10 . 94 kpc, and the dark matter halos densities corresponding to models Beta, Burkert, Brownstein, and Moore are shown in Table 1. The angular radius of the Einstein ring and the angular positions of lensed images for this situation are drown in Fig. 5. The actual angular positions of the light source are chosen to be β = 0 and β = 1 arcsec to show the most general images configurations in the astrophysical gravitational lensing observations. The Einstein ring image is generated at β = 0 . At β = 1 arcsec, it can produce the double images S 1 and S 2 , as illustrated in Fig 2. The Fig. 5(a) shows, as the D LS /D OS increases, the Einstein ring angular radius of the Beta model increases the fastest, while the Burkert, Brownstein, and Moore models increase slower. The results show that under the same conditions, the Einstein ring angular radius of the Beta model is the largest, and the Burkert, Brownstein, and Moore models are smaller. At β = 1 arcsec, as the Fig. 5(b) shows, with the D LS /D OS increases, the angular positions of lensed images have the similar tendency with the β = 0 case. Note that for β = 0 and β = 1 arcsec, the results of Burkert, Brownstein, and Moore models almost coincide with the case of a pure black hole without a dark matter halo. It shows that the dark matter halos in Burkert, Brownstein, and Moore models contribute little to the visual angular positions of lensed images for the supermassive black hole Sgr*A in the center of the Milky Way. Furthermore, we compare the effects of luminous matter and dark \nFigure 5 . The angular radius of the Einstein ring at β = 0 ( LEFT ) and the angular positions of lensed images at β = 1 arcsec ( RIGHT ) for the supermassive black hole Sgr*A in the center of the Milky Way surrounded by a dark matter halo (described by the Beta, Burkert, Brownstein, and Moore halo models). The horizontal axis labels the ratio of D LS /D OS , and the vertical axis labels the Einstein ring angular radius θ E and the angular positions of lensed images θ S in units of arc-second. The curve label by "BH" represents the gravitational lensing of a central black hole without dark matter. Here, the mass of the supermassive black hole is M = 4 . 3 × 10 6 M ⊙ , the characteristic radius of the dark matter halo is h = 10 . 94 kpc, and the distance between the observer and the lens plane is D OL = 8 . 3 kpc. The density ρ 0 of dark matter halos in the Milky Way are presented in Table. 1 \n<!-- image --> \nTable 1 . The dark matter halo density ρ 0 for the Beta, Burkert, Brownstein, and Moore models in the Milky Way. The numerical values listed here are selected according to the fitted values in [31, 129]. \nmatter in the Milky Way in Appendix A, and the results show that the luminous matter has a nonnegligible contribution to gravitational deflection angle compared with dark matter. In particular, Appendix B shows that the local random fluctuations of gravitational field in the Milky Way Galaxy affect the gravitational lensing variables in the order of 10 microarcseconds ( µ as), which can be neglected in our study. \nBesides the Sgr*A in the Milky Way Galaxy, it is also important to see the gravitational lensing of supermassive black hole in Andromeda galaxy (M31), which could provide an interesting example. According to the recent works in the Andromeda galaxy (M31) [131133], the mass of the supermassive black hole is M = 1 . 4 × 10 8 M ⊙ , the distance between the observer and the lens plane is D OL = 785 kpc, which is exactly the distance between the earth and Andromeda galaxy (M31). The characteristic radius h and density ρ 0 of dark matter halos corresponding to the Beta, Burkert, Brownstein, and Moore models are given \nFigure 6 . The angular radius of the Einstein ring at β = 0 ( LEFT ) and the angular positions of lensed images at β = 1 arcsec ( RIGHT ) for the supermassive black hole in the center of the Andromeda galaxy (M31) surrounded by a dark matter halo (described by the Beta, Burkert, Brownstein, and Moore halo models). The horizontal axis labels the ratio of D LS /D OS , and the vertical axis labels the Einstein ring angular radius θ E and the angular positions of lensed images θ S in units of arc-second. The curve label by "BH" represents the gravitational lensing of a central black hole without dark matter. Here, the mass of the supermassive black hole is M = 1 . 4 × 10 8 M ⊙ , and the distance between the observer and the lens plane is D OL = 785 kpc. The characteristic radius h and density ρ 0 of dark matter halos in Andromeda galaxy (M31) are presented in Table 2. \n<!-- image --> \nTable 2 . The dark matter halo characteristic radius h and density ρ 0 for the Beta, Burkert, Brownstein, and Moore models in the Andromeda galaxy (M31). The numerical values listed here are selected according to the fitted values in [132]. \nin Table 2. Based on the gravitational deflection angle in Eq. (5.1), the angular positions of lensed images for the supermassive black hole surrounded by a dark matter halo in the center of the Andromeda galaxy (M31) are shown in Fig. 6. The actual angular positions of the light source are chosen to be β = 0 and β = 1 arcsec to show the most general images configurations in the astrophysical gravitational lensing observations. As the Fig. 6(a) shows, with the D LS /D OS increases, the Einstein ring angular radius of the Beta model increases the fastest, the Moore model result model the second, while the result in Burkert and Brownstein models the slowest. The results show that under the same conditions, the Einstein ring angular radius of the Beta model is the largest. At β = 1 arcsec, as the Fig. 6(b) shows, with the D LS /D OS increases, the angular positions of lensed images have a similar tendency with \nTable 3 . Calculated data of the Einstein ring angular radius within the Burkert model in the gravitational lensing of Milky Way, Andromeda galaxy(M31), Virgo galaxy(M87), and ESO138-G014 galaxy. We have taken D LS /D OS as 0 . 2 , 0 . 5 , and 0 . 8 respectively. \nthe β = 0 case. Note that for β = 0 and β = 1 arcsec, the results of Burkert and Brownstein models almost coincide with the case of a black hole without dark matter. It shows that the dark matter halos in Burkert and Brownstein models contribute little to the visual angular positions of lensed images for the supermassive black hole in the center of the Andromeda galaxy (M31). As can be seen from Figs. 5 and 6, to see a larger image of the Einstein ring, the distance between the lens plane and the source plane D LS should be as large as possible. In addition, we also compare the effects of luminous matter and dark matter in the Andromeda galaxy in Appendix A. Unlike the case of the Milky Way galaxy, the dark matter halo has greater influences than luminous matter in the gravitational deflection angle. \nFurthermore, it is also worthy to see the magnitude of Einstein rings in the gravitational lensing of supermassive black holes in different galaxies in the presence of dark matter halos. Here, we restrict ourselves to the Burkert halo model to show the Einstein ring angular radius for gravitational lensing of the Milky Way Galaxy, Andromeda galaxy(M31), Virgo galaxy(M87), and ESO138-G014 galaxy. In the Virgo galaxy(M87), the mass of the supermassive black hole is M = 6 . 5 × 10 9 M ⊙ , the distance between the observer and the lens plane is D OL = 16 . 8 Mpc, the characteristic radius and density of the dark matter halo are h = 91 . 2 kpc, ρ 0 = 6 . 9 × 10 6 M ⊙ /kpc 3 [38, 134, 135]. In the ESO138-G104 galaxy, the total mass of Hydrogen Intensity in the 21kpc range is M = 4 . 6 × 10 9 M ⊙ , the distance between the observer and the lens plane is D OL = 18 . 57 Mpc, the characteristic radius and density of the dark matter halo are h = 7 . 5 kpc, ρ 0 = 1 . 3 × 10 7 M ⊙ / kpc 3 [136, 137]. As shown in Table 3, by taking D LS /D OS as 0 . 2 , 0 . 5 , and 0 . 8 respectively, we can obtain the angular radius values of Einstein rings in gravitational lensing of these galaxies. And we can observe that the gravitational lensing of Milky Way results in the largest angular radius of the Einstein ring, followed by those for Virgo galaxy(M87), the ESO138-G104 galaxy\'s results are smaller, and the Einstein rings for gravitational lensing of Andromeda galaxy(M31) are smallest. In other words, we can observe the largest Einstein ring images from the gravitational lensing observations of Milky Way Galaxy (the Sgr*A gravitational lensing observations).', '6 Conclusion and Perspectives': 'In this work, we study the gravitational lensing of central black holes surrounded by dark matter halos. Instead of treating the dark matter effects within simplified density models, we choose several widely adopted phenomenological dark matter halo models (which are the Beta, Burkert, Brownstein, and Moore models) in the current study. These phenomenological dark matter models can give precise dark matter distributions in astrophysical galaxies and galaxy clusters. The gravitational deflection angles of light for black holes in such dark matter halos \nare derived analytically using a generalized Gibbons-Werner approach, and some important observables (including the visual angular positions of lensed images and the Einstein ring) in gravitational lensing observations are calculated numerically using the gravitational lens equation. Particularly, in the calculations of visual angular positions and Einstein rings, the supermassive black holes in our Galaxy (the Sgr*A in Milky Way), Andromeda galaxy (M31), Virgo galaxy (M87), and ESO138-G014 galaxy have been chosen as typical examples, with the fitted values of dark matter halos taken into numerical calculations. \nThe analytical results show that the dark matter halos described by four models all contribute to the gravitational deflection angle as leading order effects, as seen by our expansion. The dark matter contributions to the gravitational deflection angle are proportional to the dark matter halo characteristic mass k = h 3 ρ 0 , and they decrease for larger dark matter halo characteristic radius. In the h →∞ limit, the gravitational deflection angles for different halo models all converge to the gravitational deflection angle without dark matter. The numerical results show that, the Beta dark matter model has non-negligible influences in gravitational deflection angle, the visual angular position of images and the Einstein ring. Under the same conditions, the visual angular position and Einstein ring angular radius of the Beta model are the largest, followed by the Moore model, and results in the Burkert and Brownstein models are smaller. The dark matter halos described by the Burkert and Brownstein models only have very small influences on the angular position of images and Einstein ring angular radius. To compare the dark matter effects and luminous matter effects on gravitational lensing, a simple analysis is also carried out in the Appendix based on the gravitational potential of various matter fields, which shows that the luminous matter in the Milky Way has greater influences than the Andromeda galaxy M31. A more detailed comparison of these effects deserves a subsequent study. \nWe have concentrated on one of the important effects on galactic supermassive black holes in gravitational lensing observations - the dark matter halo effects. The interactions coming from other matter fields are omitted for simplicity. However, the various matter fields in astrophysical galaxies around the supermassive black holes are extremely complicated. Apart from dark matter halos, there are nearby luminous stars, as well as interstellar gas, dust, and plasma mediums. They are all interplayed with the central supermassive black holes in the gravitational lensing observations, which also deserve extensive studies. In principle, our approach (which utilizes the effective spacetime metric produced by the central black hole and the surrounding matter fields and employs the generalized Gibbons and Werner approach on gravitational deflection angles) could effectively treat these effects in the gravitational lensing 1 , as long as the matter distributions have a spherical symmetry. We hope that studies in this direction could inspire more interesting works in the near future.', 'A Comparing the galactic dark matter and luminous matter effects on deflection angles': "In addition to dark matter, there are a number of luminous matter in galaxies that also contribute to the gravitational deflection of photons. This raises an intriguing question, which has a greater influence on the gravitational deflection of photons, the dark matter or luminous matter? To give a qualitative understanding of the influences from dark matter and luminous matter, we begin with the analysis of the deflection angle for photon orbits through \ngravitational potential Φ 2 . The deflection angle is closely connected to the gravitational potential via \nα = 2 ∫ ∇ ⊥ Φ ds. (A.1) \nwhere ∇ ⊥ Φ is the gradient of gravitational potential in the transverse direction (the direction transverse to the photon propagation). By calculating the gravitational potential produced by dark matter and luminous matter, their contributions to gravitational deflection angles can be compared indirectly. Using Poisson's equation, the gravitational potentials of the Beta, Burkert, Brownstein, and Moore dark matter models are expressed as \nΦ Beta ( r ) = 4 πkG h (arctanh √ 1 + x 2 -arccoth 1 √ 1 + x 2 + 1 x arcsinh x ) , (A.2a) \nΦ Bur ( r ) = πkG r [( x -1) ln(1 + x 2 ) + 2( x +1)(arctan x -ln( x +1))] , (A.2b) \nΦ Bro ( r ) = 2 πkG 3 r [3 x 3 2 F 1 ( 2 3 , 1; 5 3 ; -x 3 ) -2 ln(1 + x 3 )] , (A.2c) √ \nΦ Moo ( r ) = 4 πkG 3 r [2 ln(1 + x 3 / 2 ) -2 x (ln(1 + √ x ) + √ 3arctan 2 x -1 √ 3 )] . (A.2d) \nIn this appendix, we give a comparison about the dark matter and luminous matter effects on gravitational deflections in Milky Way and Andromeda galaxies. For the luminous matter around the Milky Way and Andromeda galaxies, the bulge and disk make up a large proportion of the total luminous matter mass. It is reasonable that we only consider the gravitational potential of these two parts and neglect other luminous parts. Noticeably, the potentials produced by the bulge part can also be assumed as spherically symmetric. However, in most spiral galaxies, the luminous stars are usually accumulated in the galaxy disk, which shows that the disk part cannot be treated in the spherical symmetrical formulation as the bulge part and dark matter halo. These luminous stars result in a larger gravitational potential | Φ | in the disk plane. For simplicity, we restrict ourselves to the potential produced by luminous matter (the bulge and disk parts) in the disk plane in this appendix. This enables us to give an upper estimation on the influences of galactic luminous matter on gravitational deflection angle, because the mass of luminous matter and their gravitational potential outside the disk plane could be much smaller compared with those in the disk plane (even if at the same distance r to the central black hole). From Ref. [142], the gravitational potentials of the bulge and disk of the Milky Way are modeled as \nΦ b ( r ) = GM b r + c 0 , (A.3a) \nΦ d ( r ) = GM d r (1 -e -r/b d ) , (A.3b) \nwhere M b ≃ 10 10 M ⊙ is the total mass of the bulge, c 0 ≃ 0 . 6kpc is its scale length, M d ≃ 5 × 10 11 M ⊙ is the mass of the disk, and b d ≃ 3kpc is the disk scale length. For Andromeda galaxy, the bulge has the same gravitational potential model as the Milky Way, and the disk's \ngravitational potential is modeled as [143] \nΦ ' d ( r ) = 2 πG Σ 0 R 2 d r (1 -e -r/R d ) , (A.4) \nwhere M b = 3 . 3 × 10 10 M ⊙ is the total mass of the bulge, c 0 = 0 . 61kpc is its scale radius, Σ 0 = 4 . 6 × 10 8 M ⊙ / kpc 2 is the disk central surface density, and R d = 5 . 4kpc is the disk scale radius. In this work, the natural unit G = 1 has been set. \nTo study the influence form bulge and disk on gravitational deflection angle, we calculate and observe how the gradient of gravitational potential changes with respect to r . Fig. 7 and Fig. 8 show the results for the Milky Way Galaxy and Andromeda galaxy, respectively. The numerical results of |∇ Φ | produced by Beta, Burkert, Brownstein, Moore dark matter halo models, as well as the bulge and disk distributions for luminous matter are given in these figures. The region of distance is chosen according to the scope of galaxies as well as the distance D OL , D LS in the gravitational lensing. In the Milky Way Galaxy, the distance r is chosen no more than 10kpc (although the dark matter distribution scale in the Milky Way is much larger, the D OL ∼ D LS ∼ 10kpc restrict the r in the gravitational lensing of Milky Way). For the Andromeda Galaxy M31, the dark matter distribution could exist up to 100kpc, namely the entire dark matter halo contributes to the gravitational deflection (because D OL ∼ 785kpc ≫ 100kpc). Therefore, we should pick a larger distance range of r for the Andromeda Galaxy. From Fig. 7, the results show that the Milky Way disk's gravitational potential gradient is the largest, while the Burkert dark matter model's gradient is the smallest. In this case, it can be roughly inferred that the luminous matter in the Milky Way has a greater influence on the gravitational deflection of photons than the dark matter. On the other hand, in the Andromeda galaxy, the results in Fig. 8 show that the Moore model has the largest gravitational potential gradient, the bulge has the smallest, and the sum of the gravitational potential gradient of luminous matter (bulge plus disk) is smaller than that of dark matter described by four models. It is safe to concluded that luminous matter (bulge plus disk) has less effect on the deflection of photons than dark matter in Andromeda galaxy. In summary, combining the results in Fig. 7 and 8, when studying the gravitational deflection of photons near central supermassive black holes, it is necessary to consider the effect of luminous matter in the Milky Way galaxy, while the effect of luminous matter in the Andromeda galaxy (if very high precision is not required) can be neglected. \nHowever, it should be emphasized that the above comparisons only serve as a rough estimation of the luminous matter contributions to gravitational deflection. There are two main reasons. Firstly, our calculation for the disk part of luminous matter is restricted in the disk plane, while the luminous matter distribution and their gravitational potential outside the disk plane are less than those in the disk plane (which means we may overestimate the effects of luminous matter in the above comparisons). Secondly, for simplicity, we give numerical calculations on the total gradient of gravitational potential, rather than the transverse gradient which is more closely related to the deflection angle. Therefore, this appendix gives rough quantitative conclusions rather than high-precision predictions. A more detailed and comprehensive analysis of the luminous matter effects on gravitational lensing of supermassive black hole in the galaxy center deserve an additional study. \nFigure 7 . The gradient of gravitational potential |∇ Φ | produced by dark matter and luminous matter in the Milky Way. This figure present the results for Beta, Burkert, Brownstein, Moore dark matter halo models, as well as the bulge and disk distributions for luminous matter. The horizontal axis is distance r in unit of kpc , and the vertical axis is the gravitational potential gradient in unit of 10 9 M ⊙ / kpc . \n<!-- image --> \nFigure 8 . The gradient of gravitational potential |∇ Φ | produced by dark matter and luminous matter in the Andromeda galaxy. This figure present the results for Beta, Burkert, Brownstein, Moore dark matter halo models, the bulge and disk distributions for luminous matter. The horizontal axis labels r in unit of kpc , and the vertical axis is the gravitational potential gradient in unit of 10 8 M ⊙ / kpc . \n<!-- image -->", 'B Effect of random fluctuations of gravitational field on gravitational lensing observable': "In an astrophysical galaxy, due to the complex (and stochastic) motions of stars and other ultra-compact objects, the gravitational field in the galaxy is not perfectly static. The local fluctuations of the gravitational field may have influences on gravitational lensing observations. This appendix provides a concise discussion of whether they have non-negligible contributions to the lensing observables calculated in our present work. \nFrom a pioneering work given by S. Chandrasekhar [144], the local variations of the gravitational field in time caused by the galactic objects (or the fluctuations of the galactic matter density) can be treated as a stochastic process. Inspired by this work, many studies investigate their effects on gravitational lensing [145-149]. Recent studies suggested that local fluctuations of gravitational field in galaxies could lead to a 'jitter' effect for apparent positions of light sources in gravitational lensing observations [147, 148]. T. I. Larchenkova et al. found that the galactic gravitational random fluctuation affects the apparent position of the light source outside the Milky River. The standard deviation of the deflection angle can reach tens of microarcseconds ( µas ) in the galactic center direction, while it drops to 4-6 µas in the high silver latitude direction [147]. They also found that the jitter effect of apparent celestial positions of distant sources due to local fluctuations of the galaxy gravitational field can be detected when the accuracy of differential astrometric observations is around 10 µas [148]. Other studies [149] also concluded that the influence coming from the random fluctuations of the gravitational field is at the order of microarcseconds ( µas ), which is not negligible in the study of microlensing [150-152]. However, the gravitational lensing observables calculated in our work (the angular radius of lensed images and Einstein ring in Sec. 5) are at the order of arcseconds ( as ). One can safely infer that the effects of the stochastic influence of stars / local fluctuations of gravitational field on our gravitational lensing results are small.", 'C Numerical calculations on C 1 ( z ) , C 2 ( z ) and C 3 ( z )': 'In this appendix, we give the numerical results on the several defined functions C 1 ( z ) , C 2 ( z ) and C 3 ( z ) in the analytical expressions of the gravitational deflection angle. The detailed \nFigure 9 . The numerical results of C 1 ( z ) , C 2 ( z ) , and C 3 ( z ) with respect to z . \n<!-- image --> \nexpressions of these functions are: \nC 1 ( z ) = z 2 ∫ π 0 csc 2 ϕ 2 F 1 ( 2 3 , 1; 5 3 ; -z 3 csc 3 ϕ ) d ϕ, (C.1a) \nC 3 ( z ) = z ∫ π 0 arctanh 3 √ z csc ϕ 2 + 2 z csc ϕ + √ z csc ϕ d ϕ. (C.1c) \nC 2 ( z ) = z ∫ π 0 arctan 2 √ z csc ϕ -1 √ 3 d ϕ, (C.1b) \nThe numerical results of C 1 ( z ) , C 2 ( z ) , and C 3 ( z ) with respect to z are shown Fig. 9. As can be seen from the figure, C 1 ( z ) also increases and presents a linear relationship with the increase of z , while C 2 ( z ) first increases and then decreases, resulting in a local minimum, and C 3 ( z ) also increases with z but has a non-linear behavior.', 'Acknowledgments': "The authors are grateful to Peng Wang, Aoyun He, Yang Huang, and Yadong Xue for useful discussions. The authors would like to thank the anonymous referee for helpful comments and suggestions, which helped to improve the quality of this paper. This work is supported by the National Natural Science Foundation of China (Grant No. 12147207, No. 12175212, and No. 12275184), the 'zhitongche' program for doctors from Chongqing Science and Technology Committee (Grant No. CSTB2022BSXM-JCX0100), the Natural Science Foundation of Chongqing Municipality (Grant No. CSTB2022NSCQ-MSX0932), the Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant No. KJQN202201126).", 'References': '- [1] P. A. R. Ade et al. [Planck], Planck 2013 results. XVI. Cosmological parameters, Astron. Astrophys. 571 , A16 (2014). arXiv:1303.5076[astro-ph.CO].\n- [2] P. A. R. Ade et al. [Planck], Planck 2013 results. I. Overview of products and scientific results, Astron. Astrophys. 571 , A1 (2014). arXiv:1303.5062[astro-ph.CO].\n- [3] N. Jarosik et al. [WMAP], Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Sky Maps, Systematic Errors, and Basic Results, Astrophys. J. Suppl. 192 , 14 (2011). arXiv:1001.4744[astro-ph.CO].\n- [4] N. Aghanim et al. [Planck], Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys. 641 , A6 (2020); erratum: Astron. Astrophys. 652 , C4 (2021). arXiv:1807.06209[astro-ph.CO].\n- [5] G. R. Blumenthal, S. M. Faber, J. R. Primack and M. J. Rees, Formation of Galaxies and Large Scale Structure with Cold Dark Matter, Nature 311 , 517-525 (1984).\n- [6] M. Davis, G. Efstathiou, C. S. Frenk and S. D. M. White, The Evolution of Large Scale Structure in a Universe Dominated by Cold Dark Matter, Astrophys. J. 292 , 371-394 (1985).\n- [7] R. M. Reddick, R. H. Wechsler, J. L. Tinker and P. S. Behroozi, The Connection between Galaxies and Dark Matter Structures in the Local Universe, Astrophys. J. 771 , 30 (2013).\n- [8] M. Rocha, A. H. G. Peter, J. S. Bullock, M. Kaplinghat, S. Garrison-Kimmel, J. Onorbe and L. A. Moustakas, Cosmological Simulations with Self-Interacting Dark Matter I: Constant Density Cores and Substructure, Mon. Not. Roy. Astron. Soc. 430 , 81-104 (2013). arXiv:1208.3025[astro-ph.CO].'} |
2024LRSP...21....1K | Magnetic storms on stars manifest as remarkable randomly occurring changes of the luminosity over durations that are tiny in comparison to the normal evolution of stars. These stellar flares are bursts of electromagnetic radiation from Xray to radio wavelengths and they occur on most stars with outer convection zones. They are analogous to the events on the Sun known as solar flares which impact our everyday life and modern technological society. Stellar flares however can attain much greater energies than those on the Sun. Despite this we think that these phenomena are rather similar in origin to solar flares which result from a catastrophic conversion of latent magnetic field energy into atmospheric heating within a region that is relatively small in comparison to normal stellar sizes. We review the last several decades of stellar flare research. We summarize multiwavelength observational results and the associated thermal and nonthermal processes in flaring stellar atmospheres. Static and hydrodynamic models are reviewed with an emphasis on recent progress in radiationhydrodynamics and the physical diagnostics in flare spectra. Thanks to their effects on the space weather of exoplanetary systems and thus in our search for life elsewhere in the universe and their preponderance in Kepler mission data whitelight stellar flares have reemerged in the last decade as a widelyimpactful area of study within astrophysics. Yet there is still much we do not understand both empirically and theoretically about the spectrum of flare radiation its origin and its time evolution. We conclude with several bigpicture questions that are fundamental in our pursuit toward a greater understanding of these enigmatic stellar phenomena and by extension those on the Sun. | 2024-12-01T00:00:00Z | ['2024arXiv240207885K', '2024LRSP...21....1K', '10.48550/arXiv.2402.07885', 'arXiv:2402.07885', '10.1007/s41116-024-00039-4'] | ['Stellar flares', 'Solar flares', 'Optical flares', 'Stellar atmospheres', 'Astrophysics - Solar and Stellar Astrophysics'] | Stellar flares | 2,024 | 166 | 0.68 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML'] | 20 | https://arxiv.org/pdf/2402.07885.pdf | {'Adam F. Kowalski 1,2,3': '1 Department of Astrophysical and Planetary Sciences, University of Colorado, 2000 Colorado Ave, Boulder, 80305, CO, United States. 2 National Solar Observatory, University of Colorado, 3665 Discovery Drive, Boulder, 80303, CO, United States. 3 Laboratory for Atmospheric and Space Sciences, University of Colorado, 3665 Discovery Drive, Boulder, 80303, CO, United States. \nContributing authors: [email protected];', 'Abstract': 'Magnetic storms on stars manifest as remarkable, randomly occurring changes of the luminosity over durations that are tiny in comparison to the normal evolution of stars. These stellar flares are bursts of electromagnetic radiation from X-ray to radio wavelengths, and they occur on most stars with outer convection zones. They are analogous to the events on the Sun known as solar flares, which impact our everyday life and modern technological society. Stellar flares, however, can attain much greater energies than those on the Sun. Despite this, we think that these phenomena are rather similar in origin to solar flares, which result from a catastrophic conversion of latent magnetic field energy into atmospheric heating within a region that is relatively small in comparison to normal stellar sizes. We review the last several decades of stellar flare research. We summarize multi-wavelength observational results and the associated thermal and nonthermal processes in flaring stellar atmospheres. Static and hydrodynamic models are reviewed with an emphasis on recent progress in radiation-hydrodynamics and the physical diagnostics in flare spectra. Thanks to their effects on the space weather of exoplanetary systems (and thus in our search for life elsewhere in the universe) and their preponderance in Kepler mission data, white-light stellar flares have re-emerged in the last decade as a widely-impactful area of study within astrophysics. Yet, there is still much we do not understand, both empirically and theoretically, about the spectrum of flare radiation, its origin, and its time evolution. We conclude with several big-picture questions that are fundamental in our pursuit toward a greater understanding of these enigmatic stellar phonemena and, by extension, those on the Sun. \nKeywords: Solar-stellar connection', '1 Introduction': "Stars produce energetic bursts of electromagnetic radiation that follows a sudden magnetic energy release into their atmospheres. These electromagnetic bursts are called flares, which occur on a very wide range of timescales, from seconds to days. Stellar flares are observed in all spectral windows, from the high-energy X-rays through the long wavelength radio waves. However, all wavelength regimes do not respond equally in energy and simultaneously in time. Some wavelengths (e.g., microwave) are dominated by nonthermal radiation, while others (optical, low-energy X-rays) are thought to result from primarily thermal radiative processes. The flare energy - both thermal and nonthermal - ultimately originates in the energy that is released from magnetic fields in stellar coronae. These magnetic fields, in turn, originate from turbulent convective energy transport and shear in rotating stellar envelopes, beneath the visible photosphere. The Sun is the best studied flare star due to its proximity to Earth and a fleet of multi-messenger instruments that continuously provide spatially resolved observations. \nFrom Earth, solar and stellar flares are not detectable by the unaided human eye. Stars like the Sun, and other active stars like Algol that are visible to the naked eye, \nhave too much background glare for our eyes to sense the flare brightness increases. In all-sky surveys that are sensitive to the apparent magnitudes of low-mass stars (fainter than a Johnson V -band magnitude of ∼ 9), however, stellar flares are the most dramatic source of variability. By most dramatic , we mean that flares produce the largest changes in their apparent magnitudes per unit time when compared against nearly all other astrophysical phenomena 1 . \nIn data from the upcoming Vera C. Rubin Observatory's ten-year Legacy Survey of Space and Time (LSST), stellar flares will constitute a major source of variability (Hawley et al, 2016). Further, many of the largest events that are observed serendipitously (and those from very faint but populous low-mass stars) will be bona-fide transients, whereby the changes in brightness occur from sources that have not been detected in quiescence. Stellar flares are much shorter in duration and much less luminous than extra-galactic transient phenomena, such as supernovae, tidal disruption events, and optical counterparts to neutron star mergers. Most stellar flares can in principle be readily distinguished from these events given enough empirical information, despite some striking spectral similarities (Chang et al, 2020) in the optical spectra of kilonovae (Shappee et al, 2017). Due to their proximity to the solar system, the largest stellar flare events can trigger gamma ray observatories; these events are known as 'superflares' in X-rays. Stellar superflares are factors of ≈ 10 2 -10 4 more energetic and luminous in comparison to the largest solar flares, which have bolometric energies of E bol ≈ 3 -6 × 10 32 erg (Woods et al, 2004, 2006; Cliver et al, 2022b; Hayakawa et al, 2023a). Such superflares were likely common when the Sun was very young and rotating much more rapidly than today (Maehara et al, 2012). Understanding the physical processes in stellar flares thus provides insight into the heliospheric conditions in the early history of our solar system (Ribas et al, 2005). \nIt is timely for a review of recent observations and models of stellar flares. The next sunspot cycle maximum is approaching in late 2024 (Upton and Hathaway, 2023) 2 , and with it a deluge of flares and eruptions from the Sun. New and upcoming solar observatories, such as the Daniel K. Inouye Solar Observatory (Rimmele et al, 2020), the Expanded Owens Solar Valley Array (Gary et al, 2018), and the Interface Region Imaging Spectrograph (De Pontieu et al, 2014), will provide new avenues for solarstellar comparisons. The immaculate precision of Kepler (Koch et al, 2010), K2 (Howell et al, 2014), and the Transiting Exoplanet Survey Satellite (TESS; Ricker et al, 2015) observations has recently facilitated a resurgence in the study of optical broadband stellar flares. These missions provide access to stellar magnetic activity over long time baselines previously not feasible to observe from ground-based campaigns, and over a much larger variety of stars. Models of energy transport, atmospheric response, and emission line broadening have increased in accuracy and sophistication over a large range of heating parameters. As the community looks to next-generation modeling paths, analysis methods, and observational capabilities (such as new space missions and ground-based instruments), synthesizing recent findings and outstanding problems could help to steer these into productive directions. Additionally, stellar flare radiation is now considered an important factor in assessing exoplanet habitability \nand photochemistry (Shields et al, 2016; Segura, 2018), and general interest within the solar and astrophysics communities has grown over the last decade.", '1.1 Overview of this review': "There have been several reviews on stellar flares prior to ca. 1990 (Pettersen, 1989; Byrne, 1989; Haisch et al, 1991). The current review supplements these with results from the past three decades. This review features many results from flare studies of low-mass, M-dwarf (dM / MV) stars. Because of their low background glare from nonflaring regions, large contributions to Galactic populations, and inherently high flare rates, the M dwarfs tend to be the most commonly studied flare stars. Flares from active binary systems have also been very well-observed across the electromagnetic spectrum. Results from studies of post main-sequence single stars and young solar-type stars are covered as well. Surprisingly, certain wavelength regimes in M-dwarf flares have been more thoroughly observed than solar flares. Solar flares are not reviewed in any detail here, but a general overview is provided, since the interpretation of and modeling approach to stellar flares are dependent on knowledge of the particle acceleration, magnetic fields, and spatial resolution from the study of the Sun. \nThe primary purpose of this review is to serve as a compendium of references and to facilitate research in multi-wavelength observations and models of stellar flares. The target audience consists of beginning PhD students or interested scientists in other areas of astrophysics or solar physics. Every current topic in the study of stellar flares is not included here: for example, I leave all results from the TESS mission to the next iteration of this Living Review . Unfortunately, many important results and references cannot be discussed extensively in order to keep this review a reasonable length, while also allowing growth with future addenda. For the same reasons, every important result from the references cited herein regrettably cannot be included at this time. There are several comprehensive introductions to sub-topics in the stellar flare literature; these are indicated throughout instead of being repeated. Broader applications of flare research beyond stellar astrophysics (e.g., exoplanet habitability and exoplanet atmospheric photochemistry) are outside the scope of this review and are not covered. The study of stellar coronal mass ejections and their associated energetic particles is a vast and newly emerging field; these topics are only very briefly alluded to. This review focuses on the electromagnetic response of flaring stellar atmospheres and detailed modeling of the associated physical processes. \nThis review is organized as follows. We begin with a brief tour of the flare star next door, Proxima Centauri, which is about the same age as the Sun but is otherwise a much different star (Sect. 2). Then we take a detour for a brief introduction to solar flare terminology and phenomenology (Sect. 3). A general overview of all known types of stars that tend to flare is summarized in Sect. 4. The bulk of the review contains a synthesis of stellar flare observations across the electromagnetic spectrum with a focus on the near-ultraviolet (NUV) and optical response, which has not been the subject of most other reviews. Observational studies can be separated into two general topics: flare rates, which are primarily diagnosed through single-band photometry (Sect. 5), and multi-wavelength analyses, which are primarily accomplished through spectroscopy (Sect. 7). The final third of the review summarizes stellar flare modeling \napproaches (slab, semi-empirical, and radiative-hydrodynamic), beginning in Sect. 8. We go into detail in several areas within stellar flare modeling, focusing on key elements of radiation-hydrodynamics (Sect. 9) and the interpretation of chromospheric spectral line broadening in flares (Sect. 10). A comprehensive analysis of an 'ideal' multi-wavelength data set is discussed in Sect. 11, and inferences of the stellar flare geometries are reviewed in Sect. 12. We conclude in Sect. 13 with six questions that we think are currently central in stellar flare research (excluding questions directly related to exoplanets). Throughout, we use the cgs (Gaussian) units system except for wavelengths in ˚ A or in cases that are otherwise noted. In the appendices, we include a visual guide to common filters used in optical studies of stellar flares (Appendix A), some additional observational results from the literature are merged (Appendix B), several common slab modeling practices and assumptions are reviewed (Appendix C), and the basic microphysical processes that symmetrically broaden hydrogen lines in flares is summarized (Appendix D).", '2 Our flare star neighbor: Proxima Centauri': "The nearest star outside the solar system is Proxima Centauri, which is ≈ 1.3 parsecs, or 268,000 astronomical units (au), from the Earth, and is a low-mass ( ≈ 0 . 12 M Sun ) M dwarf star of spectral type M5.5Ve (dM5.5e) (Hawley et al, 1996). The hydrogen Balmer α line (6562.81 ˚ A) is in emission (e) 3 in quiescence outside of detectable flaring events, which occur very frequently. This star is a relatively old ( ≈ 5 -6 Gyr) main sequence star, but it is still very magnetically active, consistent with the long magnetic activity lifetimes of low mass stars of its spectral type (West et al, 2008). We describe some of the flare properties of this star from a night of monitoring during an observing run at La Silla Observatory in 2010, while also introducing basic pointsource photometry measurements and terminology that are common to almost all analyses of flare star data. \nA NUV light curve of Prox Cen (Kowalski et al, 2016) is shown in Fig. 1. The photometry was measured through a custom filter with a full-width-at-half-maximum (FWHM) of 100 ˚ A centered at λ cen = 3500 ˚ A. This is a wavelength range over which the star has a very faint luminosity during non-flaring ('quiescent', denoted by q ) intervals. In less than seven hours of continuous 3 s exposures (and 1.5 s exposures at the end of the night), the star produced at least 16 flares at > 3 σ (1 σ ≈ 6%) above the mean, which is indicated by a flux enhancement equal to 1.0 in the light curve. In quiescence, the star is very 'red' since it has a very cool photospheric temperature ( < 3000 K), but the flare light causes the star's total flux at Earth to become much 'bluer'. In addition to the time-integrated flare energy (fluence), the peak flux enhancement is sometimes used to characterize the size of a flare. The peak magnitude change of a flare is related to the peak flux enhancement by \n∆mag peak = -2 . 5 log 10 ( C rel ( t peak ) /C rel ,q ) = -2 . 5 log 10 ( I f ( t peak ) + 1 ) (1) \nFig. 1 A light curve of Proxima Centauri in a narrowband (filter FWHM of 100 ˚ A) centered in the near-ultraviolet, U -band wavelength regime. These data are from Kowalski et al (2016) and can be obtained from the Zenodo repository at https://doi.org/10.5281/zenodo.45878. The horizontal dashed lines indicate the mean flux enhancement (1.0) and 3 σ above the mean (where σ is the standard deviation outside of flaring). The inset shows the flare peaking at 2010 May 25 01:31:02 UT, which is referred to as the 'IF10' event in Kowalski et al (2016). The inset shows time relative to peak. \n<!-- image --> \nwhere C rel represents the measured counts per exposure through a bandpass and is calculated relative to the counts in that same exposure from a nearby comparison (nonvariable) star or set of stars. The count flux enhancement, C rel ( t ) /C rel , q , is normalized to a time interval over which the flare star is not clearly varying, usually taken right before a flare. The count flux enhancement has traditionally been denoted as I f +1 (Gershberg, 1972). Because the star is spatially unresolved, the values of I f ( t ) + 1 during a flare consist of the flux from the non-flaring regions on a flare star in addition to the flaring source; if the flare source has an optical thickness τ ≳ 1 at wavelengths in this bandpass, then some or all of the pre-flare flux from the flare area may be diminished at time t (Hawley et al, 1995). \nNone of the flares in Fig. 1 are all that energetic. The flare with the largest peak flux enhancement of five would correspond to an energy of 3 × 10 29 erg in the Johnson U bandpass, which is traditionally the optimal broadband filter ( λ peak ∼ 3700 ˚ A, a full-width-at-half maximum of ∼ 700 ˚ A; Moffett, 1974; Bessell and Murphy, 2012) 4 for stellar flare observations at ground-based observatories. This energy is nearly four times larger than the average U -band flare energy from Proxima Centauri (Walker, 1981) and a factor of 100 smaller than the largest event on the Sun that has been observed at similar wavelengths (Neidig et al, 1994). Multi-wavelength scaling relations (Hawley and Pettersen, 1991; Kretzschmar, 2011; Schrijver et al, 2012; Osten and Wolk, 2015; Namekata et al, 2017; Cliver et al, 2022b) facilitate comparison to the standard classification of solar flares according to their peak flux at Earth over the \nλ = 1 -8 ˚ A band, as measured by the X-ray Sensor (XRS) on the Geostationary Operational Environmental Satellite (GOES). The A1.0, B1.0, C1.0, M1.0, and X1.0class flares correspond to a range of peak X-ray irradiances logarithmically spanning 10 -8 to 10 -4 Wm -2 , while those at ≥ 10 -3 Wm -2 are sometimes clipped (Hudson et al, 2023) and are given designations beginning with X10.0. The scaling relations imply that the large event on Proxima Centauri would correspond to approximately a GOES 1 -8 ˚ A C-class flare if it had occurred on the Sun. However, Prox Cen occasionally erupts with energies that are comparable to, or even much greater than, typical GOES X-class flares from the Sun (Gudel et al, 2002a; Howard et al, 2018). Prox Cen also hosts a near Earth-mass planet at ∼ 0 . 05 au (Anglada-Escud'e et al, 2016), and unlike our relatively safe distance from solar flares at 1 au, equivalent flare energies from this star would bathe the surrounding planet in 400 x the flux of highenergy radiations. Higher-mass, M3-M4 stars are known to emit even higher-energy flares than Prox Cen. For example, the most luminous event known resulted in a remarkably fast ∆ t ≈ 35 s rise to a peak flux change of ∆ V = -5 mags ( L V, peak ≈ 1 . 7 × 10 32 erg s -1 ) from a young M4+M4 binary DG CVn (Caballero-Garc'ıa et al, 2015), for which the estimated values of the U -band energy and GOES class are 4 . 7 × 10 34 erg and X600,000, respectively (Osten et al, 2016; Youngblood et al, 2017). Pettersen (2016) describes a remarkable EV Lac flare with a peak magnitude change in the U -band of -7 . 2 mags ( L U, peak = 4 . 6 × 10 31 erg s -1 ) and a U -band energy of 7 . 23 × 10 33 erg. Howard et al (2019) highlight several flares with extreme amplitudes (∆ g ' ≤ -3 mag) and energies ( E ≈ 10 35 -10 36 erg) on M1-M4 stars in their optical survey. \nThe most common qualitative description of a flare light curve is a 'FRED': a fast-rise, exponential-decay. In Fig. 1, the flares exhibit a simple FRED shape at low time-resolution; however, at high-time resolution, much more variation in the temporal morphology is apparent, including two periods of the rise phase (1a, 1b), an extended peak (1c), two intervals of fast decay (2a, 2c), a stall between these two intervals (2b), and a gradual decay phase (3). These features are clear in some other events too (e.g., Figure 2 of Kowalski et al, 2013, which is reproduced in the middle, right of Fig. 8 here), though the respective phases may have different durations, relative amplitudes, and integrated energies. High-time resolution observations of stellar flares were actually routine in the 1970s and 1980s using photometers, which have been superseded by high efficiency CCDs, culminating in the unprecedented precision from Kepler , K2, and TESS. The short timescale variations in stellar flares have actually long been recognized. The seminal study of Bopp and Moffett (1973) aptly summarized their findings as follows: 'As the time resolution of observations has improved, the great complexity of the flare phenomenon has been revealed. The classical definition of a flare (i.e., rapid rise to maximum followed by a slower quasi-exponential decay) appears to be a gross oversimplification of the complex structures observed'.", '3 Solar Flares and the Standard Flare Model': "Only the very largest solar flare events could be detected in the optical if the Sun was at a similar distance as other stars. Recently, Kepler has provided the precision to \ndetect a signal of 0 . 0170% -0 . 0270% in optical enhancements that have been observed in Sun-as-a-star data (Woods et al, 2004; Kretzschmar, 2011; Moore et al, 2014) and from slowly rotating G dwarfs (Maehara et al, 2012; Notsu et al, 2019; Okamoto et al, 2021). Nonetheless, it is generally thought that flares from other stars originate from the same or similar processes as solar flares. This is supported by the empirical Neupert effect (Neupert, 1968), which was first reported in stellar flares in Hawley et al (1995) and Gudel et al (1996) (Sect. 7.7). In this section, we briefly review the standard solar flare model paradigm, whose phenomenology (e.g., footpoints, loops) and fundamental physical processes are widely adopted in the interpretation and analyses of stellar flares - either through direct application or through some sort of physical scaling to higher densities, magnetic fields, accelerated particle fluxes, etc... This section synthesizes decades of observations and theoretical work, and it presents our own (rather highly simplified) viewpoint of the entire process. Thus we give a nonexhaustive reference list. For more extensive reviews of solar flares, see Svestka (1976), Hudson (2007), Hudson (2011), Benz (2017), Hudson (2021), the entries within the Space Science Reviews Volume (Dennis et al, 2011), the books by Aschwanden (2004a) and Tandberg-Hanssen and Emslie (2009), and the review by Shibata and Magara (2011). For modern, comprehensive reviews of the observations and modeling of solar flares, see Fletcher et al (2011), Reeves (2022), Kerr (2022), and Kerr (2023). \nIn the standard flare paradigm, magnetic potential energy is transferred to the atmosphere, which responds by radiating away this energy as the flare. There are generally two categories of solar flares (Thalmann et al, 2019; Kazachenko, 2023). Mass and magnetic field are ejected away from the Sun during eruptive flares. In contrast, magnetized plasma is not ejected in confined, or compact, flares or the eruption may be undetectable. The collective process of the mass eruption and the electromagnetic flare is called a 'solar eruptive event' (SEE). The left panel of Figure 2 illustrates the geometry of eruptive magnetic field above compact flare loops, with reconnection of magnetic fields in between (Shibata et al, 1995). This framework underlies most modern generalizations of flare-productive magnetic field topologies (e.g., Figure 1 of Kazachenko et al, 2022), which we now describe in further detail. \nAn SEE begins in active regions where there are colliding, non-conjugate 5 sunspots of opposite polarity (Chintzoglou et al, 2019; Toriumi and Wang, 2019; Rempel et al, 2023). The demarcation that separates the bulk of the north and south polarities in an active region is termed the polarity inversion line (PIL). At the PIL, mixed negative and positive magnetic polarities appear as 'salt-and-pepper' patterns in magnetograms where there is newly emerging magnetic flux. Ongoing magnetic flux emergence adds to a twisted, cool filament (flux rope) that is parallel to the PIL, and it is thus driven to an instability (Lin et al, 2001). The start of the SEE occurs when this filament becomes unstable and erupts through overarching magnetic field into interplanetary space, developing into a coronal mass ejection (CME). The CME produces a shock that accelerates a large number of solar energetic particles (SEPs; Reames, 2021) away from the Sun; these particles may reach Earth within ≈ 10 minutes of the X-ray flare, peak about ≈ 6 -12 hours after, and persist for days. Another class of SEPs are prompt or impulsive-type SEPs, which originate from the \nflare site. The arrival of the CME itself within ≈ 18 -50 hours of the flare disturbs the terrestrial magnetic field (if the orientation of the CME magnetic field is opposite that of the Earth's magnetic field upon arrival), induces DC electric fields and currents in the ground, and increases the particle flux into the poles and radiation belts. The ground-induced currents can damage transformers if power is not diverted away from high-load regions. \nReturning to the Sun, the solar magnetic field lines that initially arch over a filament current channel (which is relatively low-lying in the corona; Sun et al, 2012; Rempel et al, 2023) are 'stretched' during the eruption. Oppositely-directed magnetic field lines pinch together in a 'current sheet' in the wake of the erupting filament (a simplified analogy to magnetic tension and retraction is often made to a rubber band building up elastic tension and releasing it by snapping back). The fields undergo magnetic reconnection at many so-called X-points, and the rapid speed of this process is thought to be facilitated by the tearing mode/plasmoid instability and MHD turbulence. The magnetic potential energy in the stretched field line is released as they shorten and retract (Section 4 of Longcope et al, 2018). \nThe fundamental physics of the conversion of magnetic energy into particle kinetic energy is described in the overview by Dahlin (2020). As the retraction of field proceeds to a more relaxed state, an Alfven-ish-speed outflow ('jet'/'exhaust') is directed from the reconnection region in the direction toward the stellar surface. Slow-mode MHD ('Petschek') shocks (e.g., Longcope et al, 2016) and parallel electric fields (Egedal et al, 2015) form in the just-reconnected, bent field lines, enhancing the temperature and density in the reconnection outflow. Thus, the magnetic potential energy initially stored in the contorted magnetic field is converted into kinetic energy of bulk flows, the thermal energy of the plasma, and the kinetic energy of particles out of the ambient/thermal distribution. \nIn large solar flares, a sizable fraction of the magnetic energy that is released seems to go into accelerating ambient particles (electrons, protons, ions) to mildly relativistic energies (Lin et al, 2003; Emslie et al, 2012; Warmuth and Mann, 2016, 2020). The nonthermal ('beam') distribution is a power-law extending from ∼ 10 keV to tens of MeV for electrons and ∼ 1 -1000 MeV for protons. Recent observations have shown that the Sun is a prolific particle accelerator: ≈ 1 -100% of the pre-flare coronal electrons are accelerated (White et al, 2003; Kundu et al, 2009; Krucker et al, 2010; Oka et al, 2013; Krucker and Battaglia, 2014; Fleishman et al, 2022; Kontar et al, 2023). The inferred nonthermal particle flux density into the lower atmosphere can attain a correspondingly extreme range, ∼ 10 12 -10 13 erg cm -2 s -1 , at some locations in solar flares (McClymont and Canfield, 1986; Canfield et al, 1991; Wulser et al, 1992; Neidig et al, 1993a; Krucker et al, 2011). \nDetermining how and where the bulk of nonthermal particles is accelerated in the solar atmosphere is actually a very active research topic (for an overview, see Zharkova et al, 2011). Generally speaking, there are three classes of particle acceleration that may operate in the solar atmosphere (Aschwanden, 2004a, and see also Knuth and Glesener 2020 and Chen et al 2020 for succinct overviews of the more modern classifications): first-order Fermi and shock (including shock-drift and diffusive shock), stochastic ('AC', or resonant interaction with a spectrum of magnetic turbulence), \nFig. 2 ( Left ) Illustration of several salient features and processes in the standard model of solar flares, reproduced from Shibata et al (1995) with permission. ( Right) Generalization of the chromospheric evaporation and condensation phenomena in radiative-hydrodynamic models of high-flux electronbeam heating at the footpoints of newly-reconnected flare loops (adapted from Kowalski and Allred, 2018). Here, we also include an illustration of the radiative backwarming of the photosphere, which is predicted by some 1D models (Allred et al, 2006) of flare chromospheres that are optically thin at Balmer and Paschen continuum wavelengths (see Fisher et al, 2012, for the expected geometry of 3D radiative backwarming of the photosphere). \n<!-- image --> \nand electric field ('DC') acceleration. Many theoretical frameworks roughly fall under one of these types in one way or another, and all may operate to some degree over the volume and time-evolution of a flare. \nIn and around the volume of reconnection and its outflow jets, there have been exciting advances in producing power-laws from multiple first-order Fermi reflections (through curvature-drift motions of particles) among volume-filling plasmoids, which are contracting and coalescing magnetic 'islands' (Shibata and Tanuma, 2001; Drake et al, 2006; Oka et al, 2010; Karpen et al, 2012; Drake et al, 2013; Dahlin et al, 2014; Guidoni et al, 2016). These models have recently been extended to 3D (Dahlin et al, 2015) and expanded to much larger physical sizes (Drake et al, 2019; Arnold et al, 2021) than particle-in-cell simulations, demonstrating efficient acceleration of highenergy electrons into power-law tails that extend over many orders of magnitude. The Alfvenic-speed reconnection exhaust may develop turbulence, and stochastic acceleration within turbulence (Miller et al, 1996; Petrosian and Liu, 2004; Petrosian, 2012; Bian et al, 2012) in the presence of Coulomb collisions (Bian et al, 2014) is thought to be one of the possible acceleration mechanisms 6 . Sub-Driecer (e.g., Holman, 1985; Benka and Holman, 1992; Alaoui et al, 2021) electric fields have also been investigated in accelerating a small fraction of the ambient particles, and super-Driecer electric fields (accelerating the entire ambient population) that result from the observed decay of magnetic field over a large loop-like volume have also been discussed (Fleishman \net al, 2022), possibly related to a type of hybrid particle acceleration involving electric fields and turbulent reconnection (Vlahos and Isliker, 2019). Despite the apparent lack of consensus among these impressive modeling and theoretical directions, a commonality is that a significant amount of energy is transferred to the atmosphere through shortening of magnetic field lines (facilitated by reconnection), and particles are efficiently accelerated over a large volume. For reviews of magnetic reconnection and particle acceleration, see also Uzdensky (2007); Cassak et al (2008); Priest (2014); Cassak et al (2017); Pontin and Priest (2022); Arber et al (2015); Lazarian et al (2020); Nishikawa et al (2021); Ji et al (2022). \nWe should mention that acceleration at a coronal termination shock where the reconnection outflow/exhaust collides with the lower-lying (previously reconnected) magnetic loops could also be important (e.g., Somov and Kosugi, 1997), and plasmoid interaction with a fast-mode MHD shock at the collision interface could contribute to the accelerated particle flux into the lower atmosphere (Shibata and Tanuma, 2001). Acceleration based on terrestrial aurorae or magnetotail sub-storm processes are also considered (Birn et al, 2017; Haerendel, 2018), and some models argue for a mechanism where the bulk of electron acceleration takes place close to (Simnett and Haines, 1990; Tsiklauri, 2017) or within (Fletcher and Hudson, 2008) the chromosphere. Whatever the case or cases may be, acceleration models should attempt to reconcile one way (e.g., Brown et al, 1998, and see also Section 3.6 in Aschwanden et al 1996a) or another (Cheng et al, 2012) with energy-dependent time delays of hard X-rays (Aschwanden et al, 1995, 1996b,a; Aschwanden, 1996; Qiu et al, 2012; Altyntsev et al, 2019; Knuth and Glesener, 2020) and the general agreement between inferred path lengths and direct imaging measurements of flare loop sizes. Other critical tests are matching the properties of the white-light continuum radiation (Fletcher et al, 2007), reproducing the energy content in accelerated protons (e.g., E ≈ 7 × 10 32 erg above a proton kinetic energy of 30 MeV; Murphy et al, 1997), and generating strong nonthermal electron radio signatures at coronal heights (Chen et al, 2020). \nIn order to consider the rest of the flare process, let us assume that the nonthermal particle beams are injected into a region around the apex of a just-reconnectedand-retracted coronal loop. Initially, the particles are injected in relatively low-lying magnetic fields ( ≲ 10 Mm), resulting in the onset of the flare impulsive phase. Over time, larger (and weaker) magnetic fields reconnect and energize particle beams. The particle beams are injected into the newly reconnected loops with a distribution of pitch angles with respect to the magnetic field orientation. The E ≳ 100 keV electrons in the beams produce gyrosynchrotron radiation as they spiral (with Larmor radii of ∼ cm) in the magnetic fields. If the magnetic flux density [G] increases along the beam path (e.g., if the cross-sectional area of the loop correspondingly converges into the lower atmosphere), then the particles with small pitch angles with respect to the magnetic field direction will freely stream ('precipitate') into the chromosphere. Those with larger initial pitch angles will gradually and adiabatically increase their pitch angle, which is a result of Faraday's Law of Induction in the frame of the gyrating electrons. These particles can eventually reflect, or 'mirror', at the footpoints (which could \noccur in the transition region, high chromosphere, low chromosphere, or low corona) 7 . After some time, the mirrored particles too will scatter off enough ambient particles in the corona and precipitate out of the magnetic 'trap' into the chromosphere. This is the so-called 'trap+precipitation' paradigm (Melrose and Brown, 1976). The highest energy particles take the longest to scatter and thus remain trapped for a time proportional to E 1 . 5 n e where E is the kinetic energy of the nonthermal electrons and n e is the ambient/thermal electron density that the beam heats (under the dilute beam assumption: n e ≫ n e -beam ). These times are rather short ( < 10 s) for typical conditions and energies. In solar flares, proton/ion beams can certainly contribute to atmospheric heating (Proch'azka et al, 2018; Allred et al, 2020), nuclear excitation, a gamma-ray continuum spectrum, the 2.2 MeV neutron-capture Deuterium-formation line, the 511 keV positron-annihilation line, and neutrons that are directly detected at Earth (Murphy et al, 1997; Vilmer et al, 2011). For the sake of brevity, we further consider the effects of only accelerated electrons. \nWhen the electron beam particles stream into the chromosphere, they encounter a wall of dense, partially ionized gas to which they rapidly lose their energy through Coulomb collisions with ambient electrons. At the same time, free-free radiative transitions in collisions with ambient protons occur in the chromosphere, producing hard X-ray ( E ≳ 25 keV), nonthermal bremsstrahlung radiation that exhibits a power-law distribution. Hard X-ray light curves define the impulsive phase in solar flares. The standard model of hard X-ray emissivity that accounts for Coulomb energy loss as the electrons radiate bremsstrahlung is called the collisional (cold) thick target model (CTTM; Brown, 1971, for reviews, see Brown et al 2003; Holman et al 2011; Kontar et al 2011). Electron beam power-laws that are inferred from the CTTM and injected into radiative-hydrodynamic simulations are typically characterized by powerlaw indices of δ ≈ 5, a low-energy cutoff of E c ≈ 15 -25 keV, and energy flux densities in the range of ≈ 10 10 to ≈ 10 11 erg cm -2 s -1 (Carlsson et al, 2023). Often, the hard X-ray light curves consist of gradual variations with superimposed short bursts of O (0 . 5 -5) s. In some interpretations, these timescales are sensibly related to the processes of prompt and delayed precipitation (described in the previous paragraph) of nonthermal electrons. \nThe relative displacement of the accelerated electrons from the slower ions in the corona results in a 'return current electric field' (e.g., van den Oord, 1990; Siversky and Zharkova, 2009); this electric field drains energy from the beam and transfers it as a drifting velocity component to the ambient Maxwellian distribution of electrons. The drift is towards the loop apex. In steady state, the flux densities ( J ; el s -1 cm 2 ) of the beam and return current drifting electrons are equal and opposite, thus preventing pulsar-strength, B ≈ 10 9 G, magnetic fields from forming in the solar atmosphere. The resistivity of the background results in 'Joule heating' of the coronal plasma ( ≈ ηe 2 J 2 beam ; Holman, 2012; Allred et al, 2020; Alaoui and Holman, 2017). For nondilute beams, the density of the electron beam is comparable to the ambient electron density, and interactions of plasma wave turbulence and parallel electric fields (e.g., Langmuir waves, electrostatic double layers) with the beam are additionally expected. \nThe nonthermal electron beam kinetic energy is thermalized in the low corona and mid-to-upper chromosphere; see right side of Figure 2. Initially hydrogen primarily provides the radiative cooling that balances the chromospheric temperature increase. As the chromosphere increases from T ≈ 8000 K to T ≈ 30 , 000 K, hydrogen becomes fully ionized, and then helium I and helium II take over the radiative cooling in the range of T ≈ 30 , 000 -80 , 000 K. Thus, it is thought that non-equilibrium rates of helium and detailed radiative transfer are important for calculating accurate evolutionary states of the atmosphere. After helium is fully ionized, oxygen, carbon, and neon regulate the cooling at T ≈ 10 5 K, which is the peak of the optically thin cooling curve (e.g., Cox and Tucker, 1969; Rosner et al, 1978; Dere et al, 1997; Dorfi, 1998). If enough heat is deposited to fully ionize these elements, then a temperature runaway, or explosion, ensues because the optically thin cooling decreases as a function of temperature up to T ≈ 20 MK. Thermal conduction transports energy away from the region of maximum beam deposition. Thus, the temperature 'bubble' expands, resembling a one-dimensional blast wave (which is more or less confined to the magnetic field direction) in an exponential atmosphere with pervasive beam heating. The increased pressures and their gradients result in a multitude of mass motions. Additionally, an increase in temperature of the very low corona (through thermal heat conduction) can also drive mass motions on top of the beam-generated mass motions. The sources and sinks of the equation for energy conservation are discussed further in Sect. 9. \nThe impulsively-generated upflows and downflows have been studied numerically as part of the standard solar flare paradigm for many decades (Livshits et al, 1981; Fisher et al, 1985c,b,a; Fisher, 1989). They are respectively known as explosive chromospheric evaporation and chromospheric condensation (Figure 2, right) 8 . The condensations and evaporations manifest as redshifts and blueshifts in cool and hot lines, respectively, with a transition temperature that has been constrained in solar flares (Milligan and Dennis, 2009). The high-speed upflows fill the loops with T ≈ 10 -30 MK plasma, while the higher density downflows accrue/accrete mass over time and radiatively cool through T ≈ 10 , 000 K (i.e., this is a non-adiabatic process). The chromosphere is compressed as in a snow plow effect while also experiencing continuous energy deposition directly by the beam. The condensations are a result of shock phenomena because they increase in density up to 10x the ambient density (whereas sound waves would ostensibly smooth out the density perturbations). The closest physical process to the temperature bubble and condensation that we have found (e.g., Kowalski et al, 2022) is that of a thermal wave (Section 6.6, pp 671-672 of Zel'dovich and Raizer, 1967, and see also Atzeni and Meyer-ter Vehn 2004 for similarities to supersonic ablative heating waves in laboratory implosion experiments), which are described as a 'secondkind temperature wave' in Livshits et al (1981). \nThe representative properties of the condensation and evaporation flows at any time in the evolution follow from a simple momentum balance equation: \n( vρ ∆ z ) cond ≈ -( vρ ∆ z ) evap , (2) \nwhere ∆ z is a physical depth range over which gas with a mass density of ρ has a bulk velocity of v . \nHere, we ignore the downward momentum of proton and electron beam particles (Ichimoto and Kurokawa, 1984; Allred et al, 2015). Typical values of the condensation are ∆ z ≈ 30 km (which scales inversely with surface gravity; Kowalski and Allred, 2018) and v ≈ -50 km s -1 . Typical evaporation flows are fully ionized and have n e ≈ 10 11 cm -3 ( ρ ≈ 3 × 10 -13 g cm -3 ) with flow speeds of v ≈ 100 -500 km s -1 . The density of the condensation increases over time, which is compensated by an increase in the ∆ z of the evaporation as chromospheric/transition region mass ablates into and fills the newly-reconnected magnetic loop. After the flows fill up a ∆ z ≈ 10 Mm loop, then Equation 2 implies a gas density of 10 -9 g cm -3 in the condensation, in agreement with numerical simulations 9 . The coronal loops thus emit in soft X-rays, while the footpoints emit in optical and UV radiation. The condensation manifests as red-shifts in broad, chromospheric lines (Fe II, Mg II, H α , Si II) that exhibit a red-wing asymmetry (RWA). The RWA evolves in velocity and intensity over ≈ 30 s (Ichimoto and Kurokawa, 1984; Zarro and Canfield, 1989; Wulser et al, 1992; Graham et al, 2020). \nFisher (1989) calculated the analytic relationship between the beam energy flux deposited above the flare transition region ( F evap ), the peak downflow (condensation) speed ( v peak ) just below the flare transition region, and the pre-flare chromospheric mass density ( ρ chrom ) at which the flare transition region forms: \nv peak ≈ 0 . 6 ( F evap ρ chrom ) 1 / 3 (3) \nTypical upflows of 100 -500 km s -1 imply durations up to ≈ 30 -90s to fill a flare loop, at which time the confined, field-aligned flows from conjugate footpoints collide. Superhot (lower volume and lower density) T ≈ 30 -50 MK sources can be produced above the looptops of the cooler, T ≈ 20 MK, plasma that is thought to be due to chromospheric evaporation; these are thought to be due to heating from slowmode (Petschek) shocks in reconnection, as demonstrated in recent multi-dimensional MHD simulations (e.g., Longcope et al, 2016). However, some of such sources that are consistent with ultrahot 100 -200 MK thermal spectra that appear early on are generally more consistent with nonthermal power-law spectra. \nIn the impulsive phase, bright sources are observed from the footpoints (Figure 2, right) of flare loops in cooler emission lines and in the UV, optical, and infrared continuum radiation. A variety of source morphologies appear and could be affected by temporal and spatial resolution. However, one can typically separate the geometry into elongated 'ribbons' and compact, brighter circular 'kernels' (see, for example, the remarkable H α images in Kawate et al, 2016). Within a relatively short time, the footpoints light up quasi-sequentially along the PIL, forming flare ribbons (e.g., Qiu et al, 2017; Kazachenko et al, 2022). One ribbon typically develops in plage 10 , and another ribbon develops in a penumbral or umbral region of sunspot. As the \nflare progresses, the distance between the two ribbons increases, which is the so-called perpendicular apparent motion. This is thought to be a signature of the reconnection occurring higher and higher in the corona, thus releasing energy from larger and larger loops, which is a hallmark of the famous 'CSHKP' model of two-ribbon flares (Carmichael, 1964; Sturrock, 1966; Hirayama, 1974; Kopp and Pneuman, 1976). As a result, the brightest emission is generally seen at the 'leading bright edge' of the ribbons as they spread apart (Figure 2 left). It is thought that excitation from above, e.g. through particle beams, changes location rather than through cross-field thermal heat transport / diffusion in the low atmosphere. However, radiative backwarming from the X-ray, EUV, and FUV lines may play an important role in heating the surrounding atmosphere away from the brightest kernels (such as in the so-called 'core-halo' morphology; Neidig et al, 1993a; Allred et al, 2006; Isobe et al, 2007; Fisher et al, 2012; Namekata et al, 2022a). After the excitation front passes, the impulsively-generated chromospheric condensation continues to propagate even in the absence of electron beam heating, but it eventually reaches pressure equilibrium with the lower atmosphere. The loops cool from tens of MK to 1 MK, first primarily through thermal conduction, then primarily through radiation, resulting in the appearance of the famous 'post-flare' loops as seen in filtergrams such as AIA 171 ˚ A (which are, really, post-impulsive phase loops; see Fig. 3 and Liu et al 2013a). In cooler lines such as H α or He II 304, the very late phase of each flare loop is sometimes associated with the phenomena of coronal rain, which is a second phase of 'condensation'. In this phase, dense material that was chromospheric before the flare drains out of the corona. New flare loops are continuously formed through the gradual decay phase of GOES soft X-ray solar flares (Warren, 2006), but the details of the partition among various possible heating sources in the low atmosphere (thermal heat flux 11 , electron beams, proton beams, direct heating by waves) remain poorly understood, especially during stellar flares. We note that the origin of the long cooling times of post-flare loops on the Sun is still a very active area of research (e.g., Ryan et al, 2013; Liu et al, 2013b; Qiu and Longcope, 2016; Bian et al, 2018; Zhu et al, 2018; Kerr et al, 2020; Reep et al, 2022; Allred et al, 2022; Ashfield and Longcope, 2023). \nSeveral notable and well-studied solar flares are the SOL1992-Jan-13T17:25 M2.0 flare ('the Masuda flare'; Masuda et al, 1994), the SOL2000-Jul-14T10:00 X5.7 flare ('the Bastille Day flare' 12 , e.g., Qiu et al, 2010), the SOL2003-Oct-28T11:30 X17 flare (one among the 'the Halloween storms', e.g., Woods et al, 2004), the SOL2002Jul-23T00:30 X4.8 ( 'double power-law'; Holman et al, 2003) flare, the SOL2014Mar-29T17:48 X1.0 flare (NASA's 'best observed X-class flare'; e.g., Kleint et al, 2016), the SOL2014-Sep-10T17:45 X1.6 flare (e.g., Graham and Cauzzi, 2015), and the SOL2017-Sep-10T16:06 X8.2 flare (e.g., Chen et al, 2020).", '4 A Survey of Flare Stars': 'The characteristics that lead to flaring in stellar atmospheres are generally thought to be some combination of the following: rapid rotation, an outer convective zone, and \nFig. 3 SDO/AIA images around the EUV wavelength of 171 ˚ A Lemen et al (2012), obtained from Helioviewer.org, showing thermal emission from flare loops at two times during a large X-class flare on the solar limb. The top panel shows the compact, low-lying thermal flare arcade (indicated by the arrow) during what is usually the peak of the nonthermal hard X-ray (not shown) luminosity from the footpoints. Later in the flare, the bright arcade volume exhibits an expansion horizontally and vertically - the expansion is, however, apparent in the sense that newly reconnected, larger magnetic loops participate in the flare as time progresses. The large arcades have traditionally been called \'post-flare\' loops, which shine brightly much longer after the end of the main hard X-ray impulsive phase (they are perhaps more accurately called \'post-nonthermal-X-ray-footpoint\' loops). It is generally thought that the flare loops in these images are the result of magnetic structures that have cooled down from much higher temperatures to T ≈ 1 MK (e.g. Aschwanden and Alexander, 2001). Note that a cusp-like (fork-shaped) dark region just above the bright loops in the top panel is where many models and observations suggest that the bulk of magnetic reconnection and particle acceleration occur (see Longcope et al, 2018; Chen et al, 2020). For illustrative comparisons of flare and active region loops, see Gudel (2004). \n<!-- image --> \ndisorganized surface magnetic fields. The ratio of stellar rotation to the convective turnover time 13 is called the Rossby number, or the inverse Coriolis number, which may be indicative of how much internal shear, differential rotation, and turbulent amplification of kinetic energy into magnetic energy contribute to the dynamo. The dynamo may operate in the shear flows around the tachocline interface, and/or throughout the \nFig. 4 Quiescent R X = L X /L bol vs. the Rossby number for a large sample of stars, reproduced from Wright et al (2011) with permission. R X is \'saturated\' at ≈ 10 -3 for stars with Rossby numbers ≲ 0 . 1, and R X decreases according to a power-law for lower-activity stars. \n<!-- image --> \nturbulent convection zone, and/or in the near-surface shear layers. Stars with small Rossby numbers are found in the so-called \'saturated regime\' of quiescent magnetic activity (outside of large flares). The saturated regime is empirically characterized by nearly constant luminosity ratios of L X /L bol ≈ 10 -3 (Figure 4) and L Hα /L bol ≈ 10 -4 (e.g., Pizzolato et al, 2003; Wright et al, 2011; Reiners et al, 2014; West et al, 2015; Wright and Drake, 2016; Newton et al, 2017; Brun and Browning, 2017; Wright et al, 2018), where L bol is the quiescent bolometric luminosity, and the numerator is a luminosity (e.g., the soft X-ray luminosity, L X ; Schmitt and Liefke, 2004; Wright et al, 2011) that is a proxy for magnetic heating in the corona or chromosphere 14 . There is a rather large spread about the saturated value from star to star, and the Sun varies from L X /L bol ≈ 5 × 10 -8 to ≈ 5 × 10 -7 over its 11 year sunspot cycle (Judge et al, 2003; Ayres, 2015a). For comparison, the least active M dwarfs have ratios of 10 -6 (Wright and Drake, 2016; France et al, 2020; Brown et al, 2023; Engle, 2023). At very small Rossby numbers, a super-saturated regime may occur, while stellar activity proxies follow a power-law with increasing Rossby numbers ≳ 0 . 1, relatively independent of spectral type 15 For an introduction to dynamo theory in the stars and the Sun, see Brandenburg (2005), Brandenburg and Subramanian (2005), Charbonneau (2013), Brun and Browning (2017), and Schrijver et al (2019). For a review of the literature, see Brun and Browning (2017) and Kapyla et al (2023). Much recent progress has been made in modeling stellar dynamos in low-mass stars just below (Bice and Toomre, 2020, 2022) and above (Browning, 2008; Brown et al, 2020; Kapyla, 2021) the fully convective transition, and with especially high-resolution simulations of the solar magnetic field and differential rotation (Hotta et al, 2022). \nMagnetic fields buoyantly rise through the stellar photosphere; once in the corona, they undergo destabilization and reconnection to release energy into the atmosphere to produce flares. Flaring occurs at different times through stellar evolution as convective envelopes come and go with changes in internal structure, core fusion processes, and external factors such as tidal locking and synchronous rotation. An overview of the myriad of flare stars is best summarized in a color-magnitude diagram (CMD). Prolific and/or representative flare stars are shown on a Gaia CMD in Fig. 5. The data were obtained from Gaia Data Release 2 (Gaia Collaboration et al, 2018), except for a few cases that are only available through the Early Data Release 3 or through common filter transformations (van Leeuwen, 2007) for some of the brightest stars. The background data are the Gaia photometry for a collection of stars compiled for the Palomar/Michigan State University spectroscopic survey (PMSU; Reid et al, 1995; Hawley et al, 1996; Gizis et al, 2002; Reid et al, 2002; Reid and Hawley, 2005), downloaded from Neill Reid\'s website 16 . Many of these stars are discussed in the recent 10 pc Gaia sample (Reyl\'e et al, 2021) and the CNS5 catalog (Golovin et al, 2023). Isochrones from the PARSEC v1.2S (Bressan et al, 2012) stellar evolution models are shown for solar age and metallicity and the age of the β Pic moving group (Mamajek and Bell, 2014). Characteristics 17 of selected flare stars in the CMD are summarized in Tables 1 - 6, and in some cases a reference to an example of a flare study featuring the respective star. \n17 We have tried to update properties with some \'standard\' values as best as possible within the time frame of writing this review but we have certainly missed some notable flare stars. Spectral types of lowmass stars are determined to only about ± 0.5 spectral types using broadband flux spectral-typing facilities (Covey et al, 2007). For example, Davison et al (2015) quote a spectral type of M4.0Ve for YZ CMi, whereas spectral typing with the Hammer facility can result in determinations closer to M5 (and similar differences result for the M0/M1 star AU Mic). Reyl\'e et al (2021) discuss the evolutionary status of YZ CMi, a notorious flare star, as a candidate pre-main sequence star with an age of around 24 Myr. Gyrochronology suggests an age range of 810 ± 60 56 Myr (Engle, 2023); the large intrinsic age spread at a given rotation period, especially at young ages for low-mass stars, is widely-recognized (e.g., Irwin et al, 2011; Engle and Guinan, 2023). There is a correspondingly large range of radius estimates for this star found in the literature (Morin et al, 2008; Baroch et al, 2020). Aside from coordinates, parallaxes, and proper motions, one should use caution for some fundamental stellar parameters (binarity, surface gravity, evolutionary status) obtained from large catalog databases such as SIMBAD and Gaia. For example, the Gaia DR3 archive returns an inaccurate surface gravity of log g /(cm s -2 ) = 4 . 0 for the notorious flare star AD Leo (as discussed by Hawley et al (1999) and Davison et al (2015), surface gravities determined from spectra of active low mass stars are unreliable). \nβ \nBoo \nFig. 5 Gaia color-magnitude diagram of absolute magnitude vs. color for notable flare stars (outside of flaring). Note that corrections for extinction and reddening from interstellar dust are not applied because most of these stars are in close proximity (within ∼ 20 pc) of Earth. The linecolors and linestyles for the annotations are arbitrary. Open circles signify that the data were transformed from Hipparcos magnitudes and/or parallaxes. Several solar-metallicity isochrones from the PARSEC stellar evolutionary models are shown for reference. \n<!-- image --> \n- \nstars. \nflare \nnotable \nand \ncommon \nof \nable \nT \n1 \nable \nT \n) \n<!-- image --> \n... \nued \ntin \nCon \n2 \nable \nT \n<!-- image --> \nd Stellar pa rameters extensiv el y review e d and discussed in F eiden and Chab o y er ( 201 3 ); e Most flares in the SDSS Strip e 82 flare sample. f 2. \'\' 3 from Gl 65A (BL Ceti), whic h is a dM5.5e BY Dra system. \'EB\' indicates an eclipsing binary . \'S82\' refers to a SDSS 82 flare star with v ery large mag nitude enhancemen ts during the observ ed ev en ts. g Largest magnitude enhancemen t in the SDSS Strip e 82 flare sample. \n) \n<!-- image --> \n) \n<!-- image --> \n) \n<!-- image --> \n26 \n<!-- image --> \nStellar classification according to binarity, evolutionary status, and spectral type is important for understanding the origin and release of magnetic energy. We describe general classifications of the types of stars that flare across the CMD (Fig. 5). In the following \'early\' spectral types refer to those of hotter effective temperatures (e.g., M0-M2), and \'later\' spectral types refer to the cooler effective temperatures (e.g., M7M9); this jargon has no direct correspondence to time or age. Mid-type M dwarfs span the fully convective transition and roughly correspond to stars with spectral types ≈ M3-M6, though there is over an order of magnitude range of bolometric luminosities just within this sub-grouping. \n- · RS CVn . Though rapid rotation is often associated with stellar youth, tidal locking of binaries can lead to prolonged magnetic activity and enhanced flare rates for billions of years (Osten et al, 2012) in systems that would not normally produce energetic events. RS CVn systems are synchronously rotating, detached binaries consisting of an early-type main sequence star and an evolved, cooler G/K subgiant or giant companion. In RS CVn systems, the flares are thought to originate from the cooler component. The cool star component has a large convection zone, which combined with very rapid rotation for its size, leads to large flares with rise and decay times on the order of days. The flares of RS CVns are among the highest energy (10 36 -10 37 erg) and longest-lasting stellar flares. Flare rates from the EUV and X-rays indicate about one flare per day (Osten and Brown, 1999), but optical enhancements are relatively rare due to the enormous background glare from the non-flaring source; peak U -band changes (Equation 1) are typically much less than a magnitude for energies of E U > 10 33 erg (rates of ≈ 0 . 17 hour -1 ; Mathioudakis et al, 1992); the largest flares increase the U -band flux by a factor of ≈ 2 (Doyle et al, 1990b). The best-studied RS CVn system is arguably HR 1099 (V711 Tau), which consists of a primary K1 IV (sub-giant) and a G5 V (main sequence) star with orbital separation of ≈ 3 R Sun and a 2.8 day synchronized orbital and rotation period (Fekel, 1983). The most extensive multi-wavelength study of the flaring and quiescence of RS CVns is presented in Osten and Brown (1999); Osten et al (2000, 2002, 2003, 2004). A large catalog of the non-flaring properties of these stars is contained in Strassmeier et al (1993).\n- · Algol-type flare stars are semi-detached binaries with a history of significant mass transfer from a cool evolved sub-giant onto a main sequence B-type star. Algol (B8V + G8III/K1IV/K2III) consists of eclipsing binary (EB) stars separated by 14.14 R Sun , whose components have radii of 2.9 and 3.5 R Sun with a 2.87-day orbital period. Algol produces giant flares on occasion (van den Oord et al, 1989; Favata and Schmitt, 1999), and serendipitous eclipses of the flares constrain the geometries and locations of the events on the cooler, less-massive subgiant component, Algol B (Schmitt and Favata, 1999; Schmitt et al, 2003), which is also the source of the coronal radio emission (Lestrade et al, 1993). Algol B is one of the few stars besides the Sun whose magnetosphere has been spatially resolved in radio observations (Peterson et al, 2010). Detailed comparisons of Algol-type and RS CVn systems have been presented in Singh et al (1996); Sarna et al (1998); Drake (2003). They are otherwise quite similar, except the total optical (non-flaring) light from the system is dominated by the main sequence B star in Algol-like systems, whereas RS \nCVns consist of two cool stars (spectral type G or later) with one subgiant or giant star that dominates the total system light 18 . The different locations on the CMD (Fig. 5) are clear. \n- · BY Dra -type stars are, here, informally referred to as closely orbiting (EB or nonEB), detached binaries consisting of two main-sequence, typically late K or early M, stars. Arguably, the best known BY Dra flare star is the eclipsing 0.8 day period binary YY Gem, which consists of nearly identical, and probably rotationally synchronized, dM0e stars with radii of ≈ 0 . 6 R Sun and masses of ≈ 0 . 6 M Sun separated by 3.83 R Sun (Torres and Ribas, 2002; Feiden and Chaboyer, 2013; Butler et al, 2015; Kochukhov and Shulyak, 2019). Other eclipsing BY Dra flare star systems are CM Dra and CU Cnc. BY Dra itself is composed of two active main-sequence K stars with semi-major axes of 7 . 4 -8 . 4 × R Sun and pseudo-synchronous rotation with a period of 5.9 d (Heglyph[suppress]lminiak et al, 2012). Formally, BY Dra is a term that describes the class of magnetically active spotted stars that exhibit periodic variation in their optical light curves, due to starspots and rotational modulation (General Catalogue of Variable Stars: Version GCVS 5.1; Samus\' et al, 2017) 19 ); however, the BY Dra classification more uniquely describes (e.g., Reid and Hawley, 2005) detached active binary systems that consist of main sequence late-type (K or M) stars that orbit very closely together 20 , like BY Dra itself.\n- · Pre-Main Sequence (PMS) stars, such as Young stellar objects (YSO\'s) and T Tauri stars exhibit flaring emission (e.g., Flaccomio et al, 2018) in addition to the emission from accretion: the flow, shocks, and hotspots at the stellar surface all generate optical, UV, IR, and X-ray continuum and emission line radiation (e.g. Herczeg and Hillenbrand, 2008) that can be both steady-state and transient. The optical, NUV, and FUV flares on T Tauri stars tend to be very energetic, ≳ 10 35 erg (Tofflemire et al, 2017; Hinton et al, 2022; Getman et al, 2023). Tofflemire et al (2017) conducted a large optical photometric monitoring campaign on the T Tauri binary star DQ Tau and compared the color differences between accretion and flare variations. Getman et al (2011) performed an extensive study of X-ray giant flares in the DQ Tau system. The Chandra Orion Ultradeep Project (COUP; Getman et al, 2005) detected many X-ray superflares from YSOs; the flares produced X-ray hardness ratios consistent with coronal temperatures of T > 10 8 K (Getman et al, 2008). M-type stars take longer to contract onto the main sequence after losing their gaseous disks and are thus not actively accreting, but may exhibit inflated radii for tens of Myr (e.g., AU Mic) as they are still contracting. These are still PMS stars but are also often considered as dwarf, UV Ceti-type stars.\n- · UV Ceti -type stars are low-mass, M-type main sequence (dwarf) flare stars that exhibit frequent flaring and H α in emission in quiescence (Gershberg et al, 1999). There is a high correspondence between the dMe status and high flare rates (Kowalski et al, 2009). These stars are traditionally denoted as \'dMe\' or \'MVe\' stars, \nthey exhibit optical rotational modulation (with periods typically in the 0 . 2 -5 day range, but see West et al 2015 and Newton et al 2017), and many but not all are in the saturated regime ( L x /L bol ≈ 10 -3 ) of quiescent X-ray luminosity. Many of the UV Cet/dMe-type stars are in the early stages of their main sequence evolution and are possibly only several hundred Myr or less in age. UV Ceti itself is a triple star system with two dM5.5e stars (A and B components) separated by 5.3 au (Benz et al, 1998); UV Ceti B is a binary BY Dra-type system (using our adopted definition above). This category includes some non-accreting PMS stars (AU Mic, AT Mic). A catalog of UV Ceti stars and their quiescent properties was compiled by Gershberg et al (1999) and is provided on VizieR 21 . Though the M-dwarfs are the most populous stars in the Galaxy (e.g., Reid and Hawley, 2005), none are visible to the naked eye ( V ≲ 6) from Earth due to their low quiescent luminosities. \n- · Very Low-Mass (VLM) Stars, Ultracool dwarfs, Brown dwarfs. The spectral types M7 through early L span the edge of the core-hydrogen burning regime (i.e., the star/brown-dwarf boundary). The exoplanet host star TRAPPIST-1 is now a well-known, M8 flare star, due to its system of planets in or around the traditional habitable zone (Gillon et al, 2017; Agol et al, 2021). VB 10 is another prolific VLM flare star (Herbig, 1956; Linsky et al, 1995; Fleming et al, 2000; Kanodia et al, 2022). The G \' \'udel-Benz relationship (Gudel and Benz, 1993) is a correlation between the quiescent radio and soft X-ray fluxes that holds over many orders of magnitudes but breaks down in the VLM regime (with the VLMs being radio overluminous compared to the X-rays). Nonetheless, flares from VLMs and L-type stars have shown dramatic optical continuum enhancements and broad and bright H α lines (Liebert et al, 1999; Schmidt et al, 2007; Gizis et al, 2013) that otherwise resemble the very energetic flares from the earlier M dwarf spectral types, as noted by Reid and Hawley (2005). White-light flare rates and magnitude variations have been rather well-characterized recently in time-domain surveys (e.g., ASASSN, Kepler, K2, NGTS; Schmidt et al, 2014, 2016; Gizis et al, 2017; Vida et al, 2017; Paudel et al, 2018, 2019; Jackman et al, 2019a; Paudel et al, 2020). The L-dwarf flare luminosities are remarkably super-bolometric and exceed magnitude changes of -9 in the V -band (Schmidt et al, 2014). X-ray superflares have been reported from L dwarfs (e.g., De Luca et al, 2020) as well.\n- · Weakly active and \'inactive\' M dwarfs (dM or MV) stars are \'optically inactive\' stars without H α in emission during quiescence. The inactive dM\'s also flare on occasion (Paulson et al, 2006; Kowalski et al, 2009). These are thought to be descendants of MVe stars, which are inferred to hold on to their high levels of activity for several billion years (Hawley et al, 2000; West et al, 2008; Kiman et al, 2021). After several Gyr on the main sequence, the M-dwarf stars maintain a lower level of chromospheric emission in Ca II H & K (with sometimes H α showing a deeper absorption profile), Mg II h & k emission lines, transition region emission (e.g., C II), and faint X-rays. One notable example of a low-activity M dwarf is the ≈ 10 Gyr-old, slowly rotating ( P ≈ 150 d) star Gl 699 (Barnard\'s star; Walkowicz and Hawley, 2009; Fontenla et al, 2016; Toledo-Padr\'on et al, 2019; France et al, 2020). Optical flares are rare (Hilton, 2011; Hawley et al, 2014) on optically inactive stars, \nwhich do not exhibit much if any rotational modulation in Kepler . The visibility (contrast) of flares at shorter wavelengths is much greater, however, and thus the FUV and X-ray flares are rather prolific (Loyd and France, 2014; France et al, 2016; Loyd et al, 2018a; Froning et al, 2019; France et al, 2020; Brown et al, 2023). \n- · Solar-type stars Flares from solar-type, G-dwarf stars have been observed rather infrequently, and mostly through serendipitous means in the FUV (Ayres et al, 1994), X-ray (Getman et al, 2008; Pye et al, 2015), and optical (Schaefer et al, 2000; Maehara et al, 2012). The detectable flares above the spatially integrated, background luminosity are very energetic, and in almost all cases they exceed broadband optical energies of E > 10 33 erg. Detailed flare rates in the EUV have been studied for the rapidly rotating, young, single G-type stars, EK Dra, 47 Cas, and κ Ceti (Audard et al, 2000). \nThe Kepler and K2 missions have transformed our knowledge about the white-light flare rates of G-type stars. The literature has been summarized in detail by Cliver et al (2022b) and Hayakawa et al (2023b), and we refer the reader to these works. Here, we provide only a brief synopsis. After the discovery of G-dwarf white-light superflares in Kepler (Maehara et al, 2012), Shibayama et al (2013) and Notsu et al (2013b) performed a comprehensive flare rate analysis of the 30-minute cadence data (with some comparisons to the 1-minute cadence data), and Candelaresi et al (2014) extended analyses to lower mass K and M stars (see also Walkowicz et al, 2011, for analyses of 30-minute cadence data of M star flares). Yang et al (2018) also compares short and long cadence Kepler data. Notsu et al (2013a), Nogami et al (2014), and Karoff et al (2016) began the detailed spectroscopic characterization of the chromospheric and photospheric properties of the solar-type superflare stars reported in Maehara et al (2012) and Shibayama et al (2013). Notsu et al (2015a) and Notsu et al (2015b) presented spectroscopic follow-up of 46 of the flare stars with 30-minute cadence data in Shibayama et al (2013). Maehara et al (2015) analyzed 1-minute cadence data of 23 solar-type superflare stars. The most prolific flare star (KIC 11551430; Table 4) was confirmed as a visual and spectroscopic binary in Notsu et al (2019), who presented spectroscopic follow-up of 18 of these 23 stars, leveraging Gaia DR2 data as well to constrain evolutionary statuses on the subgiant branch or main sequence. Okamoto et al (2021) re-analyzed all solar-type Kepler data and presented up-to-date flare rates. As a result of the group\'s detailed follow-up, the calculated superflare occurrence rates on Sun-like stars ( T eff = 5600 -6000 K, P rot = 20 -40 d) has decreased by ≈ an order of magnitude since Shibayama et al (2013). These refined constraints provide interesting comparisons to extrapolated, multiwavelength scaling relations from smaller solar flare energies and from large solar energetic particle events (see also, e.g., Figure 33 of Usoskin, 2023). Wu et al (2015) presented flare rates and power-law indices for 77 individual G stars that produce superflares, including KIC 10422252 which is the prolific flare star that is discussed in Shibayama et al (2013). Davenport (2016), Van Doorsselaere et al (2017), and Yang and Liu (2019) present Kepler flare catalogs, where the latter includes all spectral types including A-type stars (see below) and only long-cadence data. Lawson et al (2019) discussed source contamination in the Davenport (2016) catalog in the higher mass stellar range. Recently, Aschwanden and Gudel (2021) analyzed the power-law \ndistributions of the Kepler long-cadence data flare catalog of Yang and Liu (2019) and tested against self-organized criticality theory (Aschwanden, 2014). \n- · Single Active Giants (Simon and Drake, 1989) are classified into one of two general categories (see Ayres et al, 1998). The first type is helium core burning (post Helium flash) red clump active giants like β Ceti. Giant flare events with rise times of ∼ 1 day are observed from these stars: an impressive EUV light curve of β Ceti is shown in Fig. 6 from Ayres et al (2001). The active clump stars are rotating slower than evolutionary models predict, yet they have detectable magnetic fields (unlike the rest of this stellar population). The origin of the magnetic activity in these stars is not well understood, but it has been hypothesized that swallowing a companion planet or brown dwarf increases the shear in the stellar interior (Siess and Livio, 1999), resulting in \'magnetic rejuvenation\' in old age. An alternative idea is that these stars, like some other active giant stars (Stepien, 1993), are descendants of Ap stars with strong fossil fields (Tsvetkova et al, 2013). The second type of active giants occur in the Hertzsprung Gap, which is a short-lived phase of stellar evolution after rapidly rotating A and B-type stars leave the main sequence before rotationally breaking down and ascending the red giant branch. White-light flares have been widely reported in Kepler data of subgiants and giants (Notsu et al, 2019; Okamoto et al, 2021; Katsova et al, 2018; K"ov\'ari et al, 2020; Ol\'ah et al, 2022), and some exceed 10 38 erg from KIC 2852961, whose binary status is not yet clear.\n- · B5-F5 single stars have convective cores and radiative exteriors and are not expected to produce complex surface magnetic fields and flares. B stars and Ap stars are magnetic but the magnetism is thought to be a remnant of fossil fields, which are not disorganized enough to facilitate reconnection and impulsive energy release. The F2 star HR 120 (Mullan and Mathioudakis, 2000) is a surprisingly early-type flare star that has a detailed EUV flare rate presented in Audard et al (2000). White-light superflares from A-type stars have been reported in Kepler data (Balona, 2012, 2019), and spectroscopic follow-up by Pedersen et al (2017) found evidence for binarity in most of these sources. If originating from the companion, Balona (2012) argues that the energy requirements are yet rather large and unreasonable (see also Mullan, 2009). The A7 star Altair is magnetically active, and its dynamo may operate locally in an equatorial zone that is cool enough for convection to occur due to the star\'s very fast rotation (Robrade and Schmitt, 2009). But to our knowledge, this \'backyard star\' is not known to produce white-light superflares. Algol-like systems consist of a B- or A-type star, but the flares originate from the cooler, evolved star.', '5 Flare Rates and Flare Frequency Distributions': "The rates at which stars flare have been measured through continuous ground-based observations, continuous space-based observations, and coarse sampling with serendipitous detections. The coarse sampling may result in just one data point per flare; thus an accurate characterization of the flaring source (Sect. 4) is often desired through follow-up spectroscopy. The study of flare rates is important for a wide variety of wider applications, including coronal heating physics (Hudson, 1991), characterizing \nFig. 6 A small fraction of single, red clump giant stars like β Ceti exhibit continuous variability that consists of week-long, giant EUV (60 -180 ˚ A) flares. A flare on µ Velorum (also shown) is even more unexpected in its evolutionary stage. Figure reproduced from Ayres et al (2001) with permission. \n<!-- image --> \nserendipitous variability in time-domain and exoplanet surveys (Becker et al, 2004; Welsh et al, 2007; Hilton et al, 2010; Berger et al, 2013; Gezari et al, 2013; Hawley et al, 2016; Fuhrmeister et al, 2018; Mondrik et al, 2019; Howard et al, 2019; Chang et al, 2020; Jackman et al, 2021; Rodr'ıguez Mart'ınez et al, 2020; Koller et al, 2021; Webb et al, 2021; Wu et al, 2022), and studying the effects on exoplanet environments (e.g., Howard et al, 2018; Tilley et al, 2019). Much knowledge of stellar flares from low-mass stars is a result of monitoring a handful of nearby dMe stars, which reliably flare at a high rate. Large etendue 22 time-domain capabilities of wide-field surveys such as the Sloan Digital Sky Survey, 2MASS, and ASAS-SN have been leveraged to characterize flare rates of inactive and early-type M dwarfs Kowalski et al (2009), at infrared wavelengths (Davenport et al, 2012), and of ultracool dwarfs (Schmidt et al, 2014, 2016, 2019).", '5.1 Basic Methods and Early Studies': "Hilton et al (2010) distinguish among flare rates , flare duty cycles (flaring fractions), and flare frequency distributions . Here, we focus on flare frequency distributions (FFDs), which refer to the energy dependence of the flare rates from continuous monitoring. Flaring fractions in sparsely sampled data are discussed in Section 5.3. \nThe seminal study of Lacy et al (1976) presented an extensive flare rate analysis of several nearby active M dwarf (dMe) stars from hundreds of hours of groundbased monitoring in typical Johnson bandpasses. The high-time resolution data and observational details, along with some fundamental measurements such as equivalent durations, timescales, and peak amplitudes, can be found in Moffett (1974). In this section, we first present several basic measurements that are derived from flare light curves. The most fundamental calculation is the equivalent duration of a flare through a bandpass. The equivalent duration of a flare is the time that a star must spend in quiescence to produce the same energy as in the flare. It is a proxy of the flare energy, which is spectral type and luminosity class dependent, but can be readily obtained from relative photometry. The equivalent duration is (Gershberg, 1972): \nED = ∫ t end t start I ( t ) -I q I q dt = ∫ t end t start I f ( t ) dt (4) \nwhich has units of time and requires establishing flare start and end times. A small equivalent duration can result from a small flare energy and/or a large quiescent count flux ( I q ) at Earth. Note that the symbol for intensity, I , is traditionally used, though this quantity is actually the spatially unresolved instrumental ( I ) count flux from the star 23 . The equivalent duration is multiplied by quiescent luminosity through this bandpass to obtain the absolute flare energy. 24 \nThe integrand of the equivalent duration is known as the 'flare visibility' or 'flare contrast' and is denoted as I f ( t ). The flux enhancement relative to the flare star's quiescence is known as I f ( t ) + 1 (Sect. 2). The numerator in the integrand is the 'flare excess', which can be denoted using a prime symbol to indicate a change. If the average flux is calculated from a continuum or pseudo-continuum region of a spectrum, then the flare-only flux is denoted as C λ ' c , where λ c is a representative wavelength within the spectral window. The I quantity is formally the count flux of the flare star normalized to the count flux from a nearby non-variable star ( I ( t ) = C rel ( t ) in Eq. 1) in the same exposure and same aperture radius. For a photon-counting instrument, such as a CCD, the theoretical value of I is \nI ( t ) = ∫ T ( λ ) f λ ( λ, t ) λdλ ∫ T ( λ ) f λ, 0 ( λ ) λdλ (5) \nwhere T ( λ ) is the total system response or effective area, including filter, atmospheric transmission (if applicable), and the detector gain; f λ, 0 is the flux spectrum of a comparison (non-variable) star or stars, and f λ is the flux spectrum of the target star in \nunits 25 in erg cm -2 s -1 ˚ A -1 . The ED is then multiplied by the bandpass ( T ) quiescent luminosity ( L q,T ) to find the bandpass-integrated flare energy, E T = ED × L q,T . For known and well-characterized bandpasses, these can be determined using zeropoint fluxes, f ZP (Willmer, 2018) and published apparent magnitudes 26 , m T , at low-levels of flare activity: \nm T = -2 . 5 log 10 ⟨ f q,λ ⟩ T f ZP ,T (6) \nwhere the zeropoint flux in the Vega mag system is determined by the numerical integration of a bandpass over the spectrum of the A0 V spectrophotometric standard star, Vega. If a flux-calibrated quiescent spectrum of the flare star is available, numerical integration over the total system response, T ( λ ), including bandpass, alternatively yields the following: \nL q,T = ⟨ f q,λ ⟩ T ∆ λ 4 πd 2 = ∫ T ( λ ) f q,λ ( λ ) λdλ ∫ T ( λ ) λdλ ∆ λ 4 πd 2 , (7) \nwhere ⟨ f q,λ ⟩ T is the system-weighted flux (Sirianni et al, 2005), f q,λ ( λ ) is the quiescent stellar spectrum at Earth in units of erg cm -2 s -1 ˚ A -1 , and d is the distance to the star. The width of the bandpass, ∆ λ , is usually taken to be the FWHM or the effective width, which vary from ∆ λ ≈ 600 ˚ A to 4000 ˚ A, for the SDSS u -band and Kepler bands, respectively. In recent years, quasi 27 -bolometric conversions from white-light filters have assumed a blackbody function for all flares, and all phases of flares (e.g., Shibayama et al, 2013). Other studies assume constant spectral energy distributions, with energies reported over the limits of the bandpass (e.g., Hawley et al, 2014). Several quiescent luminosities used for M dwarfs are given in Table 16 by Moffett (1974), but it should be noted that stars, especially active M dwarfs, have a rather large spread of U -band magnitudes reported in the literature (e.g., reported quiescent apparent U -band magnitudes for the flare star YZ CMi range between 13 . 70 -13 . 85). Total system response characteristics through U bandpasses are also inherently uncertain (e.g. Ma'ız Apell'aniz, 2006) and to some extent, intractable. Thus, non-negligible systematic uncertainties in absolute energies are to be expected from a combination of these various issues (see Hilton, 2011, for empirical comparisons of flare energies in SDSS u and Johnson U from two different telescopes). In Appendix A, we show several of the broadband and narrowband filters that are commonly employed in optical stellar flare studies. \nFFDs are presented as either downward cumulative, Q ( > E ), or differential, n ( E ), distributions in log 10 -log 10 space (see Audard et al, 2000, for comparisons of these methods). The most common practice is to fit a power-law (pareto) model to the unbinned, downward cumulative FFD: \nQ ( > E ) = N ( E E 0 ) β (8) \nwith power-law index, β , and N total flares greater than or equal to a fiducial lowenergy limit, E 0 (e.g., the detection completeness limit) observed within a monitoring duration of ∆ t (hr). The usual convention is that β is negative. The differential FFD (# of flares per unit energy) is then \nn ( E ) = -dQ dE = N α -1 E 0 ( E E 0 ) -α (9) \nwhere β = 1 -α (again, the convention is that β is negative and α is positive). Note that if A ∫ ∞ E 0 n ( E ) dE = 1 is solved for A , the converted differential distribution A n ( E ) is a probability density function (PDF): given a flare occurred, A n ( E ) dE is the probability for a flare to have an energy between E and E + dE . \nConvenient analytic forms exist for the power-law index and its statistical uncertainty from a direct (i.e., unbinned) maximum likelihood (ML) analysis (Wall and Jenkins, 2003; Clauset et al, 2009, Appendix B): \nˆ β ML = N ∑ N i =1 ln E i E 0 (10) ˆ √ . \nAlternatively, weighted least-squares fits of the power-law index (slope) and intercept of a line in log-log space has been sometimes employed in stellar flare studies. For example, Lacy et al (1976) express the power-law for each flare star in their sample of active M dwarfs as \nfrom which the result follows that σ ˆ β ML ≈ β ML / N \nlog 10 ν ( > E ) = a + β log 10 E (11) \nwhere ν ( > E ) = Q ( > E ) / ∆ t is the number of flares per hour greater than or equal to energy E (expressed in erg) through an optical bandpass T . (To increase statistical confidence, flares that are observed in several bandpasses were multiply-counted using empirical bandpass energy conversions with the effective observing time adjusted accordingly.) Of course, the counts in even well-populated cumulative distributions have statistical uncertainties that are asymmetric in logarithmic space, and the values in cumulative distributions are non-independent. Nevertheless, this method is employed as a sufficient approximation in some cases given the statistics that are possible with the relatively low number of flares in ground-based observing campaigns. \nPower-laws are typically fit to FFDs over an intermediate range of energy that is carefully chosen (e.g., Silverberg et al, 2016) to exclude the high-energy and lowenergy ends, which may have incomplete sampling that can affect the fits. Note that Clauset et al (2009) present more sophisticated methods for uncertainty estimation for power-laws and have been employed in some more recent flare rate studies (Medina et al, 2020). The Markov Chain Monte Carlo approach is described in Davenport et al (2016), and Davenport et al (2019) use an appropriate uncertainty analysis for the counts (Gehrels, 1986). Modifications to the basic power-law form (Eq. 11) and completeness functions have been used to make adjustments to the the low-energy end (Rosner and Vaiana, 1978; Aschwanden, 2015; Davenport et al, 2012; Medina et al, 2020; Okamoto et al, 2021) and high-energy tail (Aschwanden and Gudel, 2021) of FFDs. For reference, solar flare frequency distributions (Cliver et al, 2022b) exhibit \npower-law indices in (nonthermal) hard X-ray energy of ≈ 1 . 5 and peak luminosity of ≈ 1 . 7 (Crosby et al, 1993), whereas the thermal soft X-ray peak flux exhibits a larger power-law index of 2.0 (Veronig et al, 2002a; Aschwanden and Freeland, 2012; Hudson et al, 2023). \nWe summarize the general range of values of α calculated from FFDs in the X-ray and the optical regimes. Ground-based optical and space-based XEUV (0 . 01 -10 keV, which corresponds to 1 . 24 -1240 ˚ A, and where only EUV data, such as DS count rates, are available, a two-temperature, T = 6+23 MK, optically thin free-free model extrapolates to the full XEUV range to calculate energies) flare rates have generally given values from α ≈ 1 . 4 -2 . 2 with statistical uncertainties of ≳ 0 . 1 (Audard et al, 2000). RS CVn systems exhibit power-law indices in the EUV on the lower end of this distribution (Osten and Brown, 1999). Wu et al (2015) summarize the power-laws from 77 G-type stars. Hawley et al (2014) and Ilin et al (2021a) summarize values of α for low-mass stars in the optical, and Loyd et al (2018a) present power-laws for lowmass stars of different activity levels in the FUV. A useful document that summarizes flare rates from low mass M stars of various spectral types is Osten (2016) 28 . Among the active M dwarfs, the higher luminosity stars (in quiescence) exhibit flatter powerlaws (Pettersen et al, 1984), and thus the sum of the energy in all flares is dominated by the occurrence of the highest-energy flares (Lacy et al, 1976). Alternatives to the power-law functional form (Eq 9) are summarized in Aschwanden (2011), Rosner and Vaiana (1978), and Sakurai (2022). \nA few stars have had power-law indices measured independently several times; these stars show some interesting systematic differences in the reported power-law properties. In the M dwarf study of Lacy et al (1976), the famous flare star AD Leo appeared to be an outlier from the others. Due to the relatively short observing time in the Lacy et al (1976), its FFD was recalculated after more data were collected. Pettersen et al (1984) found a value of α = 1 . 62 ± 0 . 09 from 85 U -band flare events (with 115 individual flare peaks) in 111 hr of monitoring. The most energetic flare was 10 33 erg. The U -band FFD of Proxima Centauri has been quantified by Walker (1981) for flare energies between 5 × 10 27 erg and 10 30 erg, giving a similar value of α ≈ 1 . 7 to AD Leo but flatter than other stars of the same quiescent luminosity, such as CN Leo. Recently, Davenport et al (2016) doubled the number of flares in the statistics for Prox Cen using white-light optical data from the MOST satellite and calculated a similar power-law slope to Walker (1981). Howard et al (2018) found a steeper powerlaw for Proxima Centauri from the Evryscope survey and investigated the effect on the power-law index with and without an extreme superflare outlier. \nLacy et al (1976) and Pettersen et al (1984) found trends in power-law indices with effective temperature or quiescent U -band luminosity: larger luminosity (earlier-type) main-sequence stars exhibiting flatter power-laws (i.e., smaller α , | β | ). However, the study of Pettersen et al (1984) did not include the results of Walker (1981) for the interesting case of Proxima Centauri, which shows a flatter power-law than expected; evidence for and against Proxima Centauri being less magnetically active than others of the same sub-type, such as CN Leo, are summarized in Reiners and Basri (2008). \nAudard et al (2000) suggested that α derived from the XEUV may exhibit an opposite trend with spectral type, though this is reported with low confidence. They find an increasing rate of E > 10 32 erg flares corresponds with greater quiescent X-ray luminosity. \nOther interesting quantities that have been calculated from flare star monitoring are the energy per flare (average flare energy) and the radiated flare energy per unit time (average luminosity due to flaring). These quantities are usually calculated through the U band. They correlate with the non-flaring, bandpass quiescent luminosity (Lacy et al, 1976) or non-flaring, bolometric luminosity of the star over several orders of magnitude. It should be noted that the number of detectable flares is anti-correlated with the quiescent luminosity, an effect largely due to brighter stellar background that raises the detection limit (e.g., Davenport et al, 2012). Nonetheless, including many small, unobserved flares in the total energy calculation does not change the trends. From modern space-based data, it seems that these relationships generally extend to active binaries, which comprise much higher luminosity stellar components (Osten et al, 2012). An extrapolation of these relationships to the Sun, however, dramatically over-predicts the flare energy release rate in optical solar flares by many orders of magnitude, as first noted by Lacy et al (1976). From Kepler data (Sect. 5.2), a somewhat similar quantity of L f vs. L Kp has been constructed by Lurie et al (2015) and is making for powerful comparisons for normalized flaring efficiencies among stars of different spectral types (e.g., Davenport et al, 2019). Davenport (2016) investigates evidence for saturation in the relative flare luminosity given by L f /L Kp at small Rossby numbers, analogous to the quiescent L X /L bol saturation (Section 4). \nThere has been a severe paucity of low-activity M dwarf flare rates until Hawley et al (2014) presented white-light 1-minute cadence data from Kepler and some previously unpublished ground-based U -band data from the PhD work within Hilton (2011). These studies revealed that stars that are optically inactive in quiescence (Sect. 4) produce broadband optical flares, a result that had been quantified in sparsely sampled data from the Sloan Digital Sky Survey (Kowalski et al, 2009) and from serendipitous spectral detections (Paulson et al, 2006). Loyd et al (2018a) compared FUV flare energies among stars with comparable quiescent bolometric luminosities and found that FFDs are shifted according to the quiescent flux in the FUV. \nThe different apparent behaviors at the high-energy ends of stellar flare FFDs are still largely unexplained. Does the power-law that is well-fit to a middle range of energies continue on indefinitely, or does it turn over at the high-energy end and follow a steeper power-law index (Doyle and Mathioudakis, 1990; Osten et al, 2012) or another type of roll-over (e.g., Osten et al, 2004; Dal, 2020; Aschwanden and Gudel, 2021; Cliver et al, 2022b; Sakurai, 2022) for some stars? What physical parameters determine the upper limits of flare energies (and peak luminosities) that are possible on a given star? Extending the power-law that is fit to the FFD of YZ CMi predicts a U -band flare to occur about once per month with an energy that is a factor of 5000 -10 , 000 larger than its mean flare energy (Lacy et al, 1976; Kowalski et al, 2010). Dal and Evren (2011b) discuss an upper limit to the energy released in dMe flares is approached as the flares become longer in total duration. Shibata et al (2013) and Aulanier et al (2013) present theoretical grounds and semi-empirical relations for \nupper limits of flare energies. Empirical constraints from very long monitoring times have only been possible recently using Kepler data.", '5.2 White-Light Flare Rates & Kepler': "The release of Kepler data (Borucki et al, 2010) has transformed knowledge of flare rate statistics in several important ways. First, the high precision and long monitoring times facilitate detection of relatively rare white-light flares from older, less active low-mass stars, from much larger luminosity stars such as solar-type stars, and from higher mass K and early-type M dwarf stars, and evolved stars. Second, it provided many more hours of monitoring per star: ground-based monitoring efforts resulted in ≈ 30 -100 hours per star, while the nominal baseline for Kepler was two contiguous months of observations per star. The flare statistics correspondingly improved, but the high precision also revealed comparable amounts of background flux variation due to starspots and rotation. Nonetheless, the Kepler field began as a relatively poorly known star field (due to its relatively faint magnitude limits), and systematic contamination of the flare sources from binaries, subgiants, and other non-flare sources was an issue until targeted, ground-based spectroscopic (and asteroseismic) analyses caught up and Gaia DR2 parallaxes became available. Additionally, the vast majority of data from the Kepler mission is at 30-minute cadence, and most superflares span just two to four data points. The 1-minute cadence data were made possible for select stars through Guest Observer programs (e.g., Hawley et al, 2014). \nThe dM4e star GJ 1243 has been observed for 11 months at 1-minute cadence with Kepler , making it the longest observed flare star in the sky besides the Sun. The FFDs and flare properties of this star have been studied extensively by Ramsay et al (2013); Hawley et al (2014); Davenport et al (2014); Davenport (2016); Silverberg et al (2016). From the database of over 6000 flares, Silverberg et al (2016) obtained a value of α = 2 . 008 ± 0 . 002, and Davenport et al (2020) inferred α = 1 . 942 ± 0 . 001. From the first two months of data, Hawley et al (2014) found α = 2 . 01. They also noted that the turnover in the FFD at the low-energy end, which has often been attributed to a detection limit effect, may be astrophysical at some level since flares of these energies should be readily detectable above the precision of Kepler . The FFD of GJ 1243 is shown in Fig. 7(top) in comparison to several other M dwarfs spanning different spectral types and levels of quiescent magnetic activity (dM v. dMe). Comparable diagrams with compilations of FFDs of many individual stars are shown in Shakhovskaia (1989), Ramsay et al (2013), and Dal (2020). Lurie et al (2015) presented a detailed analysis of the GJ 1243 AB system, where they separated the Kepler light into the two stellar components and found values of α ≈ 2 for both stars, but a higher flare rate occurs on the slower rotator. The bottom panel of Fig. 7 shows the Kepler energies converted to energies in the U -bandpass. The flare rates of 'less-active' M3-M5 stars are closer to the FFD of GJ 1243, whereas the very-active M3-M5 stars include the dMe stars from Lacy et al (1976), such as YZ CMi, EV Lac, and AD Leo. A continuum of flare rates is clearly evident as the vertical offsets of the power-laws, while different power-law slopes may be a result of vast differences in statistics from ground and space-based monitoring (Hawley et al, 2014). \nFig. 7 Flare frequency distributions of several main-sequence M stars in the Kepler band (top) and in the U -band (bottom). Figure reproduced from Hawley et al (2014) with permission. Note the spread in flare rates in the bottom panel among the very active and less active stars of the same spectral types. \n<!-- image --> \nThe spread in flare rates among active M-dwarfs of similar spectral types (Fig. 7) contrasts with the findings of Kowalski et al (2009), who found that the average M4VeM5Ve flare frequency distribution from sparsely sampled data of field stars in the Sloan Digital Sky Survey (SDSS) Stripe 82 is sensibly in line with that of the well-known flare star YZ CMi (which is one of those among the 'more active' stars in Fig. 7). Hilton (2011) began to investigate the chance alignment of FFDs in Kowalski et al (2009) using Monte Carlo sampling of simulated flares at the SDSS riuzg cadence. The resolution of this issue (which is a non-trivial adjustment to the statistical rate calculation; E. Hilton, priv. communication 2013) will be important for accurately mapping flare rates of the Galaxy using sparsely sampled data from the LSST (Ivezi'c et al, 2019). \nThe immaculate flare statistics of Kepler facilitated new timing analyses of flare occurrence and correlation. The flare occurrence of GJ 1243 was investigated as a function of the phase of the longer-timescale, flux modulation that exhibits a ≈ 1% amplitude and is attributed to stellar rotation. No significant correlation was found at either maximum or minimum flux levels. The relationship between flare occurrence and phase was further investigated in detail with larger samples of flares on GJ 1243 (Silverberg et al, 2016) and for a large sample of M dwarfs (Lurie et al, 2015; Doyle et al, 2018). Morris et al (2018) reported a larger flare occurrence when the very lowmass (M8V) star TRAPPIST-1 is slightly brighter than average and when it goes through the increasing flux phase in the Kepler light curve. Dal and Evren (2011a) investigated the occurrence of different types of flares with phase from a large sample of ground-based observations of flares. \nHawley et al (2014) finds that the intervals between successive flares ('waiting times') on an active M dwarf (GJ 1243) over many rotation periods is consistent with a single exponential probability distribution, p (∆ t next ; τ 0 ) ∝ 1 τ 0 e -∆ t next /τ 0 (where p is the probability density and τ 0 is the average interval between consecutive flares). Equivalently, the number of flares occurring in fixed time intervals, ∆ t , is Poissondistributed, p ( N flares ; r, ∆ t ) = ( r ∆ t ) N flares e -r ∆ t / ( N flares !) (e.g., Lacy et al, 1976). Stellar flares from active stars thus occur independently and continuously at a constant rate, r = 1 /τ 0 (note that some studies use λ instead of r for the rate), except possibly on some short time-intervals (∆ t ≲ 30 min) within complex events (Hawley et al, 2014; Davenport et al, 2014). Correlations between waiting time and flare energy, as may be expected from magnetic energy buildup and release (e.g., Rosner and Vaiana, 1978) in a single region on the star, are not clearly consistent with the stellar data (Hawley et al, 2014). Efforts to understand the flare occurrence as a function of light-curve phase are ongoing in the era of TESS data (Ikuta et al, 2023). \nDavenport et al (2014) constructed a canonical flare template from the normalized, classical (single-peaked) flares of GJ 1243 in short-cadence Kepler data. They modeled the decay phase with a sum of two exponentials and the rise phase with a polynomial. The template reflects a remarkable self-similarity in many flares on one-minute integrations. Further analysis in this work demonstrated that many complex flares can be decomposed into linear superpositions of several templates with adjustable parameters. They then injected synthetic flares over a long time baseline and simulated flare statistics, concluding that the number of complex flares observed in the data is not fully consistent with a chance superposition of many uncorrelated classical flares. This is tantalizing, quantitative evidence for a solar-like 'sympathetic' (e.g., Lynch et al, 2016) flaring property of stars. \nThe rates of flares in the high-energy regime have been investigated in detail with Kepler data. Silverberg et al (2016) discuss a deviation from a single power-law between E Kp = 10 32 . 5 -10 34 erg flares in the GJ 1243 sample. (In the following, flare energies now refer to the energies integrated over a T = 9000 K blackbody function, as reported in the respective studies, rather than an energy over a particular bandpass). The upper limit flare energy for white-light flares has been pretty well established from the long monitoring times from Kepler to be E max ≈ 10 37 erg for subgiants, \nE max ≈ 10 36 erg for rapidly rotating G-dwarfs, and E max ≈ 10 35 erg for slower rotating G dwarfs (Notsu et al, 2019; Okamoto et al, 2021). For G, K, and M stars in the Kepler field, Candelaresi et al (2014) discuss a correlation between a star's superflare ( E > 5 × 10 34 erg) rate and its light-curve amplitude modulation due to starspots. The ground-based Evryscope network has also provided new insights into g ' -band superflare rates: Howard et al (2019) find that the rate of E ≥ 10 33 erg superflares (by extrapolating the best-fit power-laws down over 0 . 5 -1 . 5 orders of magnitude in energy for the K5-M2 stars) decrease from the averaged active K5-K7 stars to the averaged active M4 stars. Similar to many previous studies (Section 5.1), they report a larger average flare energy for earlier-type stars, which they attribute to the 'size of the stellar convective region'. Pettersen (1989) review and analyze the time-averaged luminosity due to flaring (and quiescent magnetic activity proxies) in a sample of dKe and dMe stars, and they correlate it with the convective envelope volume. Recent stellar evolution models of dM flare stars in eclipsing binary systems (Feiden and Chaboyer, 2013, 2014) and dK/dG stars (Matt et al, 2011) also show that early type stars, which have larger mean flare energies, have larger convective zone volumes.", '5.3 Timing on Long Scales: Flare Rates and Stellar Age': "By characterizing the FFDs across many types of stars in different evolutionary stages, astronomers begin to piece together how flare rates change over billions of years. With age-dating techniques such as open cluster membership / association and gyrochronology, quantitative evolutionary tracks in the flare history of stars are possible. At the youngest stellar ages, high flare rates and energies have been reported from PMS stars in nearby open clusters, such as Orion, in the X-ray and optical (Getman et al, 2005; Jackman et al, 2020). Signatures of accretion (e.g., in T Tauri systems) indicate a very young age range for some flare stars (e.g., Tofflemire et al, 2017). Membership among moving groups, such as the well-known β Pic moving group (with the flare star members AU Mic and AT Mic AB) probe ages around ∼ 20 Myr for stars that are no longer actively accreting gaseous material from a circumstellar disk. At older ages ( ≳ 100 Myr), flare rates and properties have been reported in open clusters, such as the Pleiades (Stelzer et al, 2000; Ilin et al, 2021b). At the oldest ages, globular cluster and the Galactic bulge membership facilitates calibration of the flare-rate-age relationship, but such stars are located at very large distances. As stars spin down due to angular momentum loss over their main sequence evolution, flare rates are expected to correspondingly decrease. This correlation is supported by detections of stellar superflares from main sequence, rapidly rotating (periods < 5 d) G-type stars. However, binarity plays an important role in maintaining flaring among a population stars to old ages, compared to the Galactic disk populations. This may occur through a co-evolutionary process in M dwarf-white dwarf binaries (Morgan et al, 2016) or rotational tidal synchronization in RS CVn-like systems (Osten et al, 2012). Binarity may also directly affect flare occurrences and energies (e.g., Doyle and Mathioudakis, 1990; Gao et al, 2008; Adams et al, 2011, and see references in Section 7.5), but it is in general a difficult property to assess without adaptive optics (e.g., Clarke et al, 2018) and high-resolution spectroscopy (e.g., Notsu et al, 2013a), which are observational techniques that are limited to nearby stars. \nFollowing the statistical inference of quiescent magnetic activity lifetime analyses of West et al (2008), the vertical distances above or below the Galactic plane have been used as proxies of stellar age for M dwarf stars in the field. The basic idea is that over time, stars experience repeated gravitational scatterings in the midplane of the Galaxy and are consequently more likely to be found at larger vertical displacements. Large time-domain surveys, such as the SDSS and soon the LSST, allow characterization of flare rates in multiple populations of stars in sparsely sampled data sets. The stellar flaring fraction and the flare fraction (duty cycle) are two quantities that are useful in mapping the flare rate of a galaxy and the star's vertical distance from the galactic plane. A stellar flaring fraction is the fraction of stars that produce some number of flares, while flaring fraction , or duty cycle, is the fraction of sparsely sampled epochs that flare. UV and optical surveys have been leveraged to map these quantities with increasing vertical distances (Welsh et al, 2007; Kowalski et al, 2009; Hilton et al, 2010; Walkowicz et al, 2011). There is suggestive evidence that flaring fraction from sparsely sampled data indicates that flaring occurrence among a population decreases faster with stellar age than other measures of quiescent magnetic activity (Kowalski et al, 2009; Hilton et al, 2010). One may be tempted to speculate that this is one possible source for the spread in flare rates among the mid-type dMe stars in Fig. 7, but more investigation is needed. The effects of solar-like activity cycles (Crosby et al, 1993) might also sensibly contribute to the variation of flare rates within a population of stars of the same spectral type (but see Pettersen et al, 1984; Davenport et al, 2020). \nMost recently, the improved methods and data samples for flare statistics, asteroseismology, gyrochronology, and open cluster calibrations using Kepler and K2 data have facilitated much more quantitative characterization of the flare rate as a function of stellar age (e.g., Ilin et al, 2021a,b). Davenport et al (2019) used Kepler field stars and gyrochronology (Mamajek and Hillenbrand, 2008) to present the first stellar flaring fraction as a function of age and g -i color, which maps to spectral type (Covey et al, 2007). Davenport et al (2019) used MCMC non-linear least squares to constrain α ( t ) in an equation similar to Eq. 11 over a range of stellar masses, from G to M dwarfs, and ages older than t ≥ 10 Myr. They found that α ( t ) remains approximately constant, flare rates indeed decrease over time for the M dwarfs, but that the flare rate decreases much less than for the G dwarfs. Notsu et al (2019) and Okamoto et al (2021) characterized the evolution of differential flare frequency of G-type stars in the Kepler field by splitting up into different rotational period bins from < 5 d to 20 -40 d. They further divided the < 5 d sample into smaller bins and found that flare frequency is approximately constant for periods < 3 d, for which the corresponding ages are ≲ 500 Myr but are not as well constrained from gyrochronology calibrations. Johnstone et al (2021) combine several state-of-the-art techniques to calculate the quiescent L X as a function of age and mass, and then they leverage a scaling relation (discussed in Sect. 5.1) from Audard et al (2000) to map the rate of high-energy XEUV flares on GKM stars over ages spanning 2 Myr to 5 Gyr.", '6 Light Curve Analyses': 'In addition to integrated flare energy in a particular bandpass (Sect. 5), a number of other strictly empirical quantities can be calculated from single-bandpass data with a moderately fast cadence. A useful decay timescale measurement is the time it takes for a light curve to reach 1 /e of its maximum flux (e.g., Maehara et al, 2015; Namekata et al, 2017), which prevents ambiguities in determining the bona-fide end of a flare. This measure also avoids model-dependent uncertainties in fitting an exponential function to flares with events in the decay phase, which may also show lengthening timescales (Osten et al, 2005; Davenport et al, 2014). Here, we summarize a few relationships between decay times and other quantities. The total rise and total decay times of flares are not well-correlated (Moffett, 1974; Lacy et al, 1976; Hawley et al, 2014), which possibly suggests some loss of \'memory\' about the details of the rise phase after the peak. Integrated energy and total duration exhibit a tighter correlation. Larger peak-flare amplitudes typically correspond to longer decay times and larger energy flares, but there is significant scatter. Namekata et al (2017) find that the 1 /e decay duration scales with total white-light energy to the 1 / 3 power and magnetic field to the -5 / 3 power for flares on the Sun, rapidly rotating G stars (see also Maehara et al, 2015), and M stars (see also Howard et al, 2019). K"ov\'ari et al (2020) found that the 1 / 3 power scaling between duration and energy also extends to a latetype giant. For similar-duration ( ≈ 5 -10 min) solar flares and white-light flares on solar-type stars, the energies of the latter are several orders of magnitude larger ( cf Figures 8-9 of Namekata et al, 2017).', '6.1 Classification of Flares by Optical Broadband Evolution': "Several classification schemes of flares have been developed based on optical broadband durations and light curve shapes ('morphologies'). Bopp and Moffett (1973) and Moffett (1974) established some of the basic descriptive terminology of flare classification, such as complex , spike , slow , typical , or multipeak (a type of complex flare). More recently, a condensed scheme denotes flares as either classical or complex (Davenport et al, 2014); several classical and complex events are evident throughout a small section of the Kepler data of GJ 1243 in Figure 8(top) panel. Kowalski et al (2019b) argued that flares with several events that comprise the rise phase can appear as relatively simple, 'classical' flares if degraded to 1-minute time-integrations. A classification of solar flares into 'gradual', 'impulsive', and 'secondary' was proposed by Cliver et al (1986) based on extremely high-time resolution, hard X-ray data. Hard X-ray data is rarely available for stellar flares, and impulsive and gradual classifications have thus been determined from broadband U - and optical light curves. \nA common classification utilizes a measure of the impulsiveness (also referred to as 'impulse' or 'impulsivity'), of a flare. Hawley and Pettersen (1991) quantified the impulsiveness as the fraction of energy emitted in the impulsive phase, which ends after the fast-decay phase(s) in broadband optical radiation. They found that about 70% of the energy was emitted in the impulsive phase during two 29 flares that are separated by ∼ 2 orders of magnitude in total energy. These were both considered very impulsive \nevents. Kowalski et al (2011) classified three events as 'gradual', 'impulsive', and 'traditional' in ultra-high-cadence (0.16 s) photometry data. Kowalski et al (2013) calculated the impulsiveness index ( I ) as the peak value of I f divided by the FWHM ( t 1 / 2 ) of the broadband ( U ) light curve to classify events into 'impulsive flares (IF)', 'hybrid flares' (HF), and 'gradual flares' (GF). Examples of high-energy, E U > 10 32 erg, GF and IF events are shown in Figure 8(middle). Lower-energy and loweramplitude IF, HF, and GF events are shown on the same axes in the bottom panel of Figure 8. Dal and Evren (2010) divide a large sample of dMe flares into fast and slow flares and suggest the differences are related to the position on the stellar disk. This hypothesis could be verified with spectral properties that are thought to change with position on the solar disk (Neidig et al, 1993b). \nThe Kowalski et al (2013) IF/HF/GF impulsiveness index is primarily a diagnostic of the broadband evolution during the impulsive phase of a flare. Thus, it is smaller for flares with a gradual decay phase that starts at a flux level that is a significant fraction of the peak flux. Among the IF-type events, there is a large range, 30 -70%, in the percentage of the total energy that is emitted in the impulsive phase. Several spectral quantities (Balmer jump ratios, Balmer line-to-continuum ratios; Sect. 7.2) at the time of peak broadband flux correlate with this index; one possible interpretation is that the average impulsive-phase heating rate varies among events. If the heating rate is attributed to the flux of electron beams, then the inter-flare variation of these optical spectral quantities suggests (Kowalski et al, 2016) an important connection between large-scale flare evolution and electron acceleration, which is sometimes discussed in the context of nonthermal hard X-ray properties of solar flares (McClymont and Canfield, 1986; Holman et al, 2011). This expands upon the foundational paradigm that stellar flare scaling relations (e.g., H γ v. U -band energies), which extend over many orders of magnitude, mean that differences from flare to flare primarily result from the flaring surface area (e.g., Butler et al, 1988; Hawley and Pettersen, 1991). The empirical differences in IF/HF/GF spectra thus eventually motivated the semiempirical modeling work in Kowalski (2022), which suggests that differences among flares further result from scaling the relative areas of higher and lower concentrations of particle beams in the stellar atmosphere. Recent analyses of H α line profiles from different locations within solar flare ribbons have begun to quantitatively explore the implications for spatially unresolved stellar data (Namekata et al, 2022a). \nThe fraction of flares within each of the IF, HF, and GF types is not yet wellestablished. The reported numbers of flares that are IF/HF/GF (Kowalski et al, 2013) are biased toward the IF type because the larger signal-to-noise at peak facilitates spectral characterization. Note that rigorous distinctions cannot be made between an HF and GF classification for certain events (Kowalski et al, 2019b). It is also not clear how many IF events produce larger Balmer jumps (Kowalski et al, 2016) than expected according to the original Kowalski et al (2013) IF/HF/GF classification scheme. GF events in the decay phase of highly-impulsive events also have shown some of the smallest Balmer jumps (in absorption). \nNonetheless, it is rather clear that the most gradual-type events in the light-curve evolution exhibit the largest Balmer jumps ratios and ratios of H γ / C4170 ' at the peak time in broadband optical flux, whereas the most impulsive-type events show \nFig. 8 Representative high-cadence optical light curves of M Ve flares. (Top) A randomly chosen section of the 1-minute cadence Kepler light curve of GJ 1243 from Davenport et al (2014) showing a variety of classical and complex flares superimposed on the modulation, which is due to starspots (Davenport et al, 2015). The data were obtained from https://github.com/jradavenport/ GJ1243-Flares. Note, there was a single flare event (not shown) that was observed simultaneously in both Kepler and in the U -band (Hawley et al, 2014). For reference, this flare had a relative peak contrast of ∆ F/F median = 0 . 014 in the Kepler band, a peak contrast of I f = 0 . 9 in the U band, and a total U -band energy of 10 31 erg. (Middle left) The impulsive and gradual decay phases of a highenergy gradual flare (GF) event on EV Lac; reproduced with permission from Kowalski et al (2013). The impulsive phase and gradual decay phases are indicated. At the peak (vertical black dashed line), GF-type events like this one have large Balmer jumps, which are still much smaller than predicted by optically thin hydrogen recombination theory alone (e.g., Sect. 7.2.1). The t 1 / 2 = 13 min is the FWHM of the light curve, and the U -band impulsiveness index, I = I f /t 1 / 2 , is 0.5 for this flare. (Middle right) An example high-energy IF event ( t 1 / 2 = 2 . 5 min, I = 8 . 3) on EQ Peg A, reproduced with permission from Kowalski et al (2013). (Bottom) Examples of lower energy flares within the flare IF/HF/GF classification scheme based on the impulsiveness in the U -band or a narrow-band filter (NBF) with λ c = 3500 ˚ A. The HF and IF events have similar peak contrasts ( I f = 2 . 7 -3 . 7) but differ significantly in the values of t 1 / 2 (14 s and 120 s). The IF and GF events are referred to as IF4 and GF2 in Kowalski et al (2016); the HF event is referred as the HST-2 flare in Kowalski et al (2019b). Note, the flare-only impulsive-phase spectrum of the IF event is shown in comparison to common photometry filters in Appendix A. 45 \n<!-- image --> \nthe smallest Balmer jumps in emission and the largest fraction of optical energy that is radiated in the continuum (Sect. 7.1). This is possibly an interesting connection to the Type I and Type II white-light flare classification on the Sun (Sect. 7.2), but the stellar events that would, ostensibly, be most analogous to Type II white-light flares in the spectral properties are actually the stellar events that are most impulsive. \nWe should clarify that an impulsive-type flare exhibits an impulsive phase and a distinct gradual decay phase, while a gradual-type flare can show a gradual decay phase that is clearly distinct from its impulsive phase. Figure 8 (middle left) shows an energetic, gradual-type flare event that exhibits several faster impulsive events superimposed on a more gradual rise phase; the impulsive phase of this GF event ends around t = 2 . 3 hrs and transitions to a more gradually decaying phase.", '6.2 Quasi-Periodic Pulsations': 'High-time resolution data of stellar flares facilitate searches for periodicity over the light curve evolution. Some events have shown fluctuations in the broadband flux having periods with a modulated amplitude or an evolving period over a cycle or two; these are called quasi-periodic pulsations (QPPs). This topic is of significant interest within the solar and stellar communities (Vievering et al, 2023), but an exhaustive discussion and reference list are outside the scope of this review. I summarize a few selected results. For detailed discussions and recent reviews, see Broomhall et al (2019a) and Zimovets et al (2021). \nQPPs in stellar flares are reported on a variety of timescales, from 10 s to 30 minutes, and they occur in the impulsive and gradual phases. Mathioudakis et al (2006) report a 10 s period during the u -band flare in the peak phase using a wavelet analysis (Torrence and Compo, 1998) and discuss different interpretations, including one in which the reconnection and particle acceleration are modulated through MHD oscillations in a nearby large loop (Nakariakov et al, 2006). Anfinogentov et al (2013) detect a damped 32 minute oscillation during the decay phase of a U -band megaflare (Kowalski et al, 2010) on YZ CMi using Lomb-Scargle and autocorrelation analyses of the detrended light curve. Doyle et al (2022) discuss shorter, 1 -4 minute, QPPs in two giant ( E ≈ 5 -10 × 10 33 erg) flares observed on YZ CMi with 0 . 25 -0 . 6 s sampling. The AFINO method (Inglis et al, 2015, 2016) is another powerful technique for evaluating the statistical significance of QPPs. Inglis et al (2015) apply the AFINO method to the stellar event in Anfinogentov et al (2013) and Kowalski et al (2010) using a Fourier power-spectrum analysis that includes the component of the flare signal that is detrended in the wavelet technique. This approach accounts for the fact that some types of power-spectra may naturally give rise to types of bursty light curve behavior in the temporal domain. \nStellar flare QPPs are reported in data from the optical to the X-ray regimes on a wide variety of stars. Cho et al (2016) investigate a tight correlation between the damping times and periods of QPPs in a sample of X-ray stellar flares. White-light QPPs in Kepler data are presented in Balona et al (2015). Mathioudakis et al (2003) report remarkable oscillations with a 240 s period during the peak phase of a U -band flare on the RS CVn II Peg, and Mitra-Kraev et al (2005b) discuss several physical origins in the damped 750 s period in the X-ray flare on the lower-mass system AT \nMic. For an analysis of multi-band data during the decay phase of an X-ray flare on the young G star EK Dra, see Broomhall et al (2019b). QPPs of 320 s and 660 s periods were reported over the peak phase of a remarkably energetic flare on a PMS M3 star (Jackman et al, 2019b).', '7 Multi-Wavelength Spectral Observations': 'Multi-wavelength observations of stellar flares are grouped into categories according to the regions of the atmosphere producing the electromagnetic response. A predominantly thermal response in the optical and near-ultraviolet originates from the cool, T ≈ 10 4 K, dense lower atmosphere (Sect. 7.1 - 7.2), while the far-ultraviolet emission lines probe the rapid response of transition region temperatures around T ≈ 10 5 K (Section 7.3). Nonthermal radiation originates from accelerated particles gyrating in coronal magnetic fields (Sect. 7.4 - 7.5), and a thermal response results from the hot, T ≳ 10 7 K, tenuous upper atmosphere (Sect. 7.6). The temporal correlations (Sect. 7.7) among these radiative responses justify a solar-like modeling paradigm with nonthermal electron beams and chromospheric evaporation and condensation processes (Sects. 3 and 8.2).', '7.1 NUV and Optical: The Thermal Radiative Response of the Footpoints to Impulsive Nonthermal Heating from Above': 'The optical and NUV response is one of the most enigmatic and energetic aspects of stellar flares. This phenomenon is known for its dramatic impulsive phase, but it also exhibits a long-duration gradual decay phase that can extend for many hours after a bright peak flux phase. Long-duration continuum flare radiation may persist after relatively small peaks as well (see the GF1 event in Kowalski et al 2013 reproduced in the middle left panel of Fig. 8 and the event studied in Hawley et al, 1995). The optical and NUV radiation is thought to be thermal radiation that originates from photospheric and/or chromospheric heights. For reference, Table 7 summarizes several of the largest broadband optical flux enhancements in well-studied MVe flares with multi-band photometry data that include the U - or SDSS u -band.', '7.1.1 Overview of Emission Line Properties': "An overview of the temporal response of various emission lines and continuum fluxes in the U band and optical regimes throughout a high-energy flare ( E u > 10 33 erg; Table 7) is shown in Fig. 9. The temporal sequence in the figure is representative of many stellar flares, although quantitative differences among events occur (Houdebine et al, 1991; Houdebine, 2003; Kowalski et al, 2013, 2019b). The line fluxes respond rapidly in the impulsive phase, except that Ca II K (and H) peak well into the gradual decay phase of the optical flare continuum flux. The late peak of Ca II K is a remarkable phenomenon in stellar flares (Hawley and Pettersen, 1991; Houdebine et al, 1993a; Houdebine, 2003; Garc'ıa-Alvarez et al, 2002; Kowalski et al, 2013). The blue continuum flux is the fastest to rise to its maximum and then begin its decay, followed by the highest order Balmer lines (Hawley and Pettersen, 1991). The Balmer H α line is the \nT able 7 Flux enhancemen ts at p eak phase for sev eral e sp ecially large MV e ev en ts with sim ultaneous m ulti-band photometry . The flux enhancemen t v alues ( I f + 1; Sect. 2 ) are rep orted relativ e to the pre-flare or quiescence in eac h band. The flux enhancemen ts are related to magnitude c hanges through Equation 1 . \n† \n6010 \n† \n4170 \n† \n3500 \ni \nI \nR \nr \nV \ng \nB \nu \nU \n(erg) \n/u \nU \nE \nSpT \nr \nSta \nw) \n(ro \n1.7 \n7.6 \n30.9 \n33 \n10 \n× \n4 \n≈ \ne \nM4.5V \nCMi \nYZ \n(1a) \n8 \n. \n3 \n28.7 \n106 \n33 \n10 \n× \n4 \n≈ \ne \nM4.5V \nCMi \nYZ \n(1b) \n1.35 \n2.6 \n6.3 \n78.2 \n33 \n10 \n× \n85 \n. \n1 \ne \nM4.5V \nCMi \nYZ \n(2) \n1.13 \n3.4 \n33 \n10 \n× \n6 \n. \n1 \nM1(V)e \nMic \nU \nA \n(3) \n1.6 \n3.4 \n11 \n70 \n33 \n10 \n× \n5 \n. \n6 \ne \nM3V \nLeo \nAD \n(4) \n4 \n40 \n40 \n32 \n10 \n× \n9 \n. \n3 \ne \nM3.5V \nLac \nEV \n(5) \n1.18 \n1.4 \n7.4 \n8.0 \n32 \n10 \n× \n2 \n. \n6 \ne \nM3.5V \nLac \nEV \n(6) \n2.0 \n8.6 \n16.9 \n158 \n... \ne \nM6V \nJ001309 \nSDSS \n(7) \nen \ngiv \nis \nenergy \n-band \nU \nestimated \nbined, \ncom \nthe \n); \n2016 \n( \nal \net \nalski \nw \nKo \nfrom \n, \nely \nctiv \ne \nresp \nflares, \nIF3 \nand \nIF1 \nthe \ner \nv \no \nmaxima \nThe \n(1b): \nand \n(1a) \n(4): \n); \n2023 \n( \nal \net \nristan \nT \nfrom \n23", '': "Flare \n(3): \n); \n2013 \n( \nal \net \nalski \nw \nKo \nfrom \nt \nen \nev \nIF3 \nThe \n(2): \nt. \nen \nev \n'Ultraflare' \nsame \nthe \nf \no \npart \nare \nthey \nuse \na \nec \nb \neled \nlab \n(although \nt \nen \nev \ne \nsiv \nimpul \nmost \nthe \nand \n) \n2012 \n( \nal \net \nt \nd \nhmi \nSc \nfrom \nflare \nbiggest \nThe \n(5): \n); \n1991 \n( \nettersen \nP \nand \nwley \nHa \nfrom \nFlare \nGreat \nThe \n. \n8 \nFigure \nof \npanel \nleft \nmiddle \nthe \nin \nwn \nsho \nis \nthat \n) \n2013 \n( \nal \net \nalski \nw \nKo \nfrom \nt \nen \nev \nGF1 \nThe \n(6): \n); \n2013 \n( \nal \net \nalski \nw \nKo \nof \nsample \nthe \nin \nIF10) \nas \n552 \nJ001309.33-002 \nSDSS \n(7) \n(NBF). \nfilters \nw-band \nnarro \nM \nCA \nTRA \nUL \nthe \nof \nelengths \nv \na \nw \ntral \nen \nc \nthe \nto \nrefer \ncolumns \n6010 \nand \n4170, \n00, \n35 \nThe \n† \n. \n1 \nable \nT \nin \nand \n) \n2009 \n, \nal \net \nalski \nw \nKo \n( \nsample \nstar \nflare \n82 \ne \nStrip \nSDSS \nthe \nfrom \nslowest to decay relative to its peak (see also Namekata et al, 2020). In some flares, the Balmer line peaks occur significantly after the peak of the continuum, which may be empirically related to secondary events (e.g., Hawley and Pettersen, 1991; Garc'ıaAlvarez et al, 2002, and see also the IF4/F2 event in Kowalski et al 2016). The neutral helium lines exhibit a very wide range of decay timescales: The blue He I lines decay quickly (Hawley and Pettersen, 1991), but He I 10830 ˚ A (studied in a different MVe flare) exhibits one of the most gradually evolving light curves (Schmidt et al, 2012). \nQuantitative assessments of the time-scales of emission line and continuum fluxes have been investigated with a time-decrement (Kowalski et al, 2013, 2019b). The timedecrement is defined as the empirical trend of the full-width-at-half-maximum ( t 1 / 2 ) of each light curve, plotted as a function of the wavelength of the transition. Are the relative decay timescales in Fig. 9 consistent with a simple model consisting of a slowly decreasing average temperature over the flare footpoints on the star (Gurzadian, 1984; Houdebine et al, 1991)? This explanation would be analogous to the cooling thermal loop sequence in solar flares (Aschwanden and Alexander, 2001) that is attributed to the temporal offsets in the maximum of optically thin, XEUV light curves. The average temperature properties of the flare region are undoubtedly important in understanding stellar flare light curve evolution in Fig. 9. However, the ratios of the Balmer line fluxes (which is called the Balmer flux decrement, or just 'Balmer decrement') and line-tocontinuum flux ratios in the observations suggest that chromospheric optical depths (Kunkel, 1970; Drake and Ulrich, 1980), vertical and transverse spatial inhomogeneities (Kowalski et al, 2015; Kowalski, 2022), and non-LTE radiative transfer (Hawley and Fisher, 1992; Allred et al, 2006) are probably also critical to interpret the sustained chromospheric emission line fluxes (and the relative evolution of the optical continuum fluxes) in stellar flares. In summary, there are not yet any comprehensive physical models that explain the absolute and relative timescales of the various optical spectral variations in Fig. 9.", '7.2 Spectral Properties of the Optical and NUV Continuum Radiation': "The continuum fluxes through the NUV and optical are clearly much more impulsive than the emission lines in Fig. 9 (see also Hawley and Pettersen (1991); Garc'ıa-Alvarez et al (2002); Kowalski et al (2019b)). The observed flare continuum radiation from the NUV through the optical is collectively referred to as the white-light 30 . In this \nIn reference to stellar flare radiation, white-light has traditionally meant 'detected in broadband optical filters'; before the immaculate precision provided by Kepler, this unambiguously referred to flare continuum radiation. In solar physics, a common anecdotal use of 'white-light flare' refers to an enhancement of photospheric continuum radiation and thus the excitation of the solar atmosphere in very deep layers. Adding to the ambiguity, there may be two types of white-light solar flares, Type I and Type II, which are classified according to their optical and Balmer jump spectral properties (see, e.g., Proch'azka et al, 2018). The introduction of Kowalski et al (2019a) provides further discussion about the ambiguities in terminology surrounding the use of 'white-light' in solar physics and astrophysics. Some researchers colloquially refer to the white-light ('flare' implied) as an increase in the optical continuum radiation only (excluding shorter \nFig. 9 Representative time-evolution of optical spectral quantities over a giant flare on the M4.5Ve star, YZ CMi. The peak optical broadband magnitude enhancements are listed in row (2) of Table 7. The emission line fluxes are integrated over wavelength, and all light curves are normalized to their respective peak fluxes. The C4170 ' light curve is a proxy for the flare-only blue continuum flux averaged over λ = 4155 -4185 ˚ A and is the fastest to decay after its peak flux is attained. The H α and Ca II K emission lines are the slowest. Figure reproduced from Kowalski et al (2013) with permission. \n<!-- image --> \nreview, we adopt the empirical definition (without regard to its origin in the stellar atmosphere) that a white-light flare is a broad-wavelength increase in the observed NUV, U -band, and optical continuum stellar flare radiation that sometimes extends into the FUV and NIR. Thus, the white light would contribute a majority of the integrated flux in optical broadband filters (e.g., UBVR , Kepler, TESS). \nIn this section, we review observational properties of optical and NUV flare continuum radiation, as revealed by spectroscopic measurements. We separately discuss peak/impulsive phase spectra (Sect. 7.2.1) and gradual decay phase spectra (Sect. 7.2.2). In Sect. 7.2.4 and Sect. 8.2, we review some closely-related results from analyses of colors in broadband photometry ('colorimetry'). The M-dwarf flare spectral observations, combined with detailed modeling, suggest that there are unexpected amounts of heating over a large (deep) column mass density (hereafter, just 'column mass' or m c ; g cm -2 ). Whether the λ = 1300 -9000 ˚ A continuum radiation in stellar flares is caused by heating the photosphere to bona-fide incandescence (i.e., isotropic blackbody radiation or a blackbody-like spectral intensity) is an open question that is discussed in the next section and further in Sect. 8.2. The data discussed in this review come mostly from dMe flares, which produce large contrasts against their non-flaring photospheric radiative fluxes at blue and optical wavelengths. This property facilitates \nwavelength U -band and NUV continuum radiation). Note that bona-fide 'white light' is defined according to the CIE D65 white-light standard illuminant, which is noon sunlight. See Cranmer (2021) for human color vision synthesis of various stellar colors. Note, we have synthesized the 'true' colors in Fig. 31 from representative spectra of each source that is illustrated. \nisolating 'flare-only' spectra, which can be readily compared to models without the ambiguities of subtraction artifacts.", '7.2.1 Rise and Peak Phase': "A commonly reported empirical property of the optical, λ ≥ 4000 ˚ A, flare-only continuum radiation in the spectra of the impulsive phase of stellar flares is a color temperature matching a T ≈ 8 , 000 -14 , 000 K blackbody (Mochnacki and Zirin, 1980; Hawley and Pettersen, 1991; Katsova et al, 1991; Paulson et al, 2006; Kowalski et al, 2013; Gizis et al, 2013; Lalitha et al, 2013; Kowalski et al, 2016). The blackbody color temperatures might be purely phenomenological, in which case the color temperature values are parametrizations of the spectral shapes rather than analogs to in situ 'thermometer readings' (Appendix C.1). A temperature value is simply easier to relate to than continuum flux ratios, which are unique to specific spectral windows for a given color temperature. The continuum radiation that is consistent with T ≈ 10 4 K optical blackbody color temperatures is thus referred to as 'hot, blackbody-like'. As we discuss further in Sect. 8.2, there are possible physical explanations from optically thick hydrogen recombination radiation. A peak-phase spectrum is shown in Fig. 10, which was taken at the peak of the event whose evolution is showcased in Fig. 9. Single-component blackbody function models in the range of T ≈ 9000 -12 , 000 K can be fit to this flare spectrum at λ ≥ 4000 ˚ A outside of the major emission lines (Kowalski et al, 2013). Multi-component blackbody curves with higher and lower temperatures are fit to the entire λ ≳ 4150 ˚ A wavelength range and are compared to a single blackbody fit to the blue-optical, λ = 4000 -4800 ˚ A, range in Fig. 10. \nThe broadband photometry observations from Hawley and Pettersen (1991) established the 9500 K blackbody hypothesis for the continuum in an energetic E ∼ 10 34 erg flare from the M3Ve star AD Leo (see Sect. 8.2 for additional discussion), which was verified in Kowalski et al (2013) by fitting a blackbody temperature of T ≈ 11 , 600 K to the optical range in the spectra of this flare. Further spectral investigation at resolving powers of R ∼ 500 -1000 in other events has revealed that a T BB ≈ 8500 -14 , 000 K blackbody color temperature in the blue-optical, λ = 4000 -4800 ˚ A, is not a unique property of extremely energetic, large amplitude flares: for example, similar color temperatures were calculated in the impulsive phase flare spectrum (Appendix A) of the IF event in Fig. 8(bottom), which has an energy of E U ≈ 1 . 6 × 10 31 erg (Kowalski et al, 2016). Hot blackbody-like continuum radiation in the blue-optical regime tends to be more prominent, or more often reported, in the rise and peak phases of impulsivetype (IF; Sect. 6.1) events. The flare event in Fig. 10 is a highly impulsive event. We refer the reader also to the discussions in Kowalski et al (2016) and Kowalski (2023) pertaining to the variation in the optical blackbody color temperatures on ∆ t = 3 s cadences in the large, IF-type events listed in rows (1a) and (1b) in Table 7. \nIn echelle flare spectra with high-resolving power, Schmitt et al (2008) and Fuhrmeister et al (2008) have shown that the continuum radiation within the U band at λ = 3250 -3860 ˚ A also exhibits hot blackbody color temperatures of ≈ 11 , 300 K in the impulsive phase (Fig. 11a). At lower resolving power, similarly blue spectral trends at wavelengths shortward of the Balmer limit have been noted in Kowalski et al (2013) for a large sample of flares. Though spectral observations within the U -band \nFig. 10 Flare spectrum at the peak of the large, impulsive-type (IF) event on YZ CMi in Fig. 9. Several blackbody fits to optical ( λ ≥ 4000 ˚ A) continuum regions are shown. The flare spectrum (black) has the pre-flare (dotted) spectrum subtracted. Figure reproduced from Kowalski et al (2013) with permission. \n<!-- image --> \nbenefit from increased flare contrast, especially afforded by M dwarfs, accurate measurements are generally very difficult due to the systematic uncertainties of the flux calibration near the terrestrial atmospheric limit. \nSpectral observations with high signal-to-noise (S/N) at λ > 3600 ˚ A reveal a very important clue about the origin of the white-light continuum radiation and its hot blackbody-like property. An extrapolation of a blackbody function that is fit to λ = 4000 -4800 ˚ A (thus giving T BB ), or an extrapolation of a blackbody that is fit to the continuum flux ratio, f ' 4170 ˚ A /f ' 6010 ˚ A =C4170 ' / C6010 ' (thus giving T FcolorR ), fails \nto account for excess continuum flux at wavelengths shorter than λ ≲ 3700 ˚ A. In other words, there is a Balmer jump between the blue-optical continuum flux and continuum flux at shorter wavelengths within the U band. An isothermal blackbody function thus does not comprehensively explain the panchromatic continuum flux properties. A Balmer jump was tentatively noted in Kunkel (1970) and Moffett (1974), and it was clearly evident and extensively characterized in impulsive-phase flare spectra using modern instrumentation and high-time resolution in Kowalski et al (2013, 2016). An impulsive phase spectrum during a E U ≈ 10 31 erg, gradual/hybrid-type flare on GJ 1243 is shown in Fig. 11(b) over a broader spectral range than in panel (a). This is one of two flares analyzed in Kowalski et al (2019b) that produced similar decreases in the continuum flux into the NUV range at λ < 3200 ˚ A, as constrained by Hubble Space Telescope data (not shown here). The Balmer jump in this event is the largest that has been detected spectroscopically over the impulsive phase of an M dwarf flare. In flares with very prominent hot blackbody-like continuum radiation at optical wavelengths, \nFig. 11 (a) VLT/UVES echelle ( R ∼ 40 , 000) spectral observations at λ < 3860 ˚ A that nearly extend to the terrestrial atmospheric cutoff. These spectra show the evolution of a giant whitelight flare on the M6Ve star CN Leo, reproduced from Fuhrmeister et al (2008) with permission. A T ∼ 11 , 300 K blackbody is fit in the impulsive phase (2nd from top), and the gradual phase spectrum is fit with a T ∼ 9100 K blackbody in the third panel. A catalog of the emission lines in the spectrum in the middle panel, extending to λ = 3060 ˚ A, is available through VizieR. (b) Other MVe flares in the impulsive phase show a large Balmer jump and a flatter U -band continuum spectrum that decreases into the NUV at shorter wavelengths than shown here (Kowalski et al, 2019b) are not satisfactorily explained by a T ≈ 9000 K blackbody fit to the blue-optical continuum wavelengths. Here, the two blue-colored spectra correspond to the rise/peak phase and early gradual decay phase, respectively, which are averaged into the black spectrum. The red-colored spectrum is an optically thin hydrogen recombination continuum model from a RHD simulation that invokes a solar-type electron beam heating function; the model is scaled to the observed spectrum at λ ≈ 3615 ˚ A. The spectra have a low-resolving power, R ≈ 600, which is clearly sufficient for robust wavelength-integrated emission line fluxes. Figure reproduced with permission from Kowalski et al (2019b). \n<!-- image --> \nsuch as the Great Flare of AD Leo and the flare in Fig. 10, the Balmer jumps are evident as smaller flux excesses above the blackbody extrapolations to λ ≲ 3700 ˚ A. Impulsive-phase continuum flux ratios, which are 'colors' if they are converted to magnitude differences, are calculated from flare spectra in Kowalski et al (2013), Kowalski et al (2016), and Kowalski et al (2019b) and are shown in Fig. 12. Flux ratios have been calculated from isothermal static, slab models for τ ( λ ) ≫ 1 (blackbody) and τ = 0 (continuous hydrogen LTE emissivity; Appendix C) and are shown in Fig. 12 for comparison. The flare peak 'color-color' diagrams from the spectra demonstrate that the Balmer jump strengths ( f ' 3615 / f ' 4170 =C3615 ' / C4170 ' ) imply that moderate-to-large continuum optical depths develop in stellar flare atmospheres. Further quantitative understanding about the observed offsets from the blackbody line \nin Fig. 12 requires detailed RHD model spectra, which are represented in the figure by the thick dashed line. These RHD model calculations include wavelength-dependent opacities and optical depths over column mass/depth-dependent temperatures and electron densities (Sect. 8.2), which are not included in blackbody or other simple slab models. \nIn Appendix B, we discuss comparable color-color diagrams using narrow-band (∆ λ = 50 -100 ˚ A) filters around λ = 3500 ˚ A, 4170 ˚ A, and 6010 ˚ A, which were designed specifically to avoid emission lines in flares and to measure the Balmer jump with the ULTRACAM instrument (Dhillon et al, 2007). These narrow-band filters provide high-cadence (∆ t ≈ 0 . 3 -3 s), simultaneous constraints of continuum variations while avoiding degeneracies in model fits to broadband filters (Sect. 7.2.4). ULTRACAM count-flux ratios from a large sample of MVe flares were analyzed and converted into flare-only continuum flux ratios in Kowalski et al (2016). The flare color from relative photometry is the ratio of two values of \nI f ( t ) ⟨ f q,λ ⟩ T (12) \nin the notation of Equation 7. Several flares were observed simultaneously with spectra over the impulsive to gradual decay phases, which facilitated unambiguous interpretation of the filter ratios ( cf Figure 3 in Kowalski et al 2016). A relationship between color temperatures fit to blue-optical spectra and to blue-to-red optical flux ratios is characterized by an offset of ∆ T ≈ 1900 K (see Table 5 in Kowalski et al 2016), in line with the various blackbody fits to the peak flare spectrum in Fig. 10.", '7.2.2 The Decay Phase and The Evolution of Optical Spectral Properties': "The T ≈ 10 , 000 K blackbody-like optical color temperatures do not last for long during MVe flares (e.g., Kowalski et al, 2013; Kowalski, 2023). After the peak of a broadband (e.g., U -band) light curve, a gradual decay phase almost always follows a shorter period of faster decay. A characteristic blue-optical, blackbody color temperature value at the start of the gradual decay phase is T BB ≈ 8000 K (Kowalski et al, 2013). For equal λ ≈ 3615 ˚ A flare-only flux levels during the fast rise and fast decay phases, the fast decay phase is more than ∆ T ≈ 1000 K cooler. Mochnacki and Zirin (1980) observed similar trends for optical color temperatures using continuum data during several MVe flares at λ = 4200 -6900 ˚ A (e.g., see their Figure 3). At red-optical wavelengths, the continuum becomes very flat in the gradual decay phase, with even cooler temperatures. This was referred to as a red 'conundruum' in Kowalski et al (2013); the cooler blackbody fits were linked to the increasing fraction of energy that was previously reported in the redder bands of photometry in the decay phase of the Great Flare of AD Leo (see Figure 10 and accompanying discussion in Hawley and Pettersen, 1991) and retrospectively to the deviations from blackbody fits to the photometry of that flare (bottom right panel of Figure 11 in Hawley and Fisher 1992). See Figure 31 of Kowalski et al (2013) for examples of the stellar flare gradual phase continuum spectra that exhibit cool T ≲ 5000 K blackbody color temperatures in large \nFig. 12 Compilation of impulsive-phase continuum flux ratios (color-color diagrams) from spectra (Kowalski et al, 2013, 2016, 2019b). The y-axis is a measure of the Balmer jump ratio, and the flux ratios on the x-axis correspond to optical color temperatures. Note that the x and y uncertainties are correlated, but only marginal error bars are shown. The prime symbols indicate that a pre-flare flux has been subtracted. The optical continuum colors from the Great Flare of AD Leo (Hawley and Pettersen, 1991) are estimated from extrapolations of blackbody curve fits over the λ = 3800 -4440 ˚ A range (Kowalski et al, 2013). The wavelength range of C4170 ' can be seen in the right panel of Fig. 11. The flux ratio values at the peaks of flares indicate optical depths between blackbody radiation and optically thin hydrogen recombination radiation. The general trend is well-reproduced by approximations (KA18 CC models; Kowalski and Allred, 2018) to the dynamic atmospheres in RHD simulations (Sect. 8.2). In the right margin plot, flares with Balmer jump measurements only are shown. The flares are color-coded to their U -band impulsiveness (IF/HF/GF) classification. Other flare-peak flux ratios from narrowband ULTRACAM photometry (Kowalski et al, 2016) are summarized in Appendix B. The color temperatures corresponding to the blue-to-red flux ratios in this figure are typically ≳ 1500 K less than that obtained from spectral fitting over the λ = 4000 -4800 ˚ A range. All data in this figure are provided in supplementary online material hosted on Zenodo. \n<!-- image --> \ndMe events. In smaller flares, it is difficult to accurately subtract the preflare spectrum at λ ≳ 7500 ˚ A (Kowalski et al, 2019b) and thus determine the NIR continuum properties. \nIt is not clear how to explain such persistent flat optical continua with relatively moderate-sized Balmer jump ratios; optically thin (bound-free) Paschen continua and photospheric backwarming predict cooler color temperatures but also much larger Balmer jumps, which are not observed - hence the conundrum. A related phenomenon during the impulsive phase may be the cause of the lower color temperatures that \nare calculated over a very wide optical wavelength range (such as those on the xaxis in Fig. 12) that spans the blue-optical and red-optical regimes. On the other hand, Fuhrmeister et al (2008) report hotter blackbody temperatures over the redwavelengths (not shown) in the flare in Fig. 11a. Clearly, further characterization of the red-optical and near-IR continuum properties is warranted. There may be important contributions to Kepler flare energetics in the gradual decay phase (Hawley et al, 2014). \nAt shorter wavelengths within the U -band, the Balmer jump ratio increases in the gradual decay phase of a flare. The fraction of the wavelength-integrated energy in emission lines (primarily the hydrogen Balmer series in the optical and within the U -band wavelength range) also becomes larger. The energy partition evolution is shown in Fig. 13 for a large sample of dMe flares. It was found that the impulsive flare (IF) events were more continuum-dominated than the gradual flare (GF)-type events in both the impulsive phases and gradual phases of each event. Thus, an empirical connection among the broadband impulsiveness, fraction of energy in the Balmer component, Balmer jump strength (right panel of Fig. 13), and Balmer line-to-continuum ratios (namely, the H γ /C4170 ' ratios - see Appendix B) was established from simultaneous spectral and broadband photometry observations (see also Kowalski et al, 2016, 2019b). \nStellar flares in the NUV generally exhibit a rapid time evolution (Brasseur et al, 2019). However, the spectral properties in the NUV ( λ = 2000 -3200 ˚ A) are not nearly as well-observed as in the optical, in either impulsive- or gradual-type flares. The NUV spectral observations and model predictions are compiled and reviewed in Brasseur et al (2023). Kowalski et al (2019b) investigated the NUV properties in two HF-type events (that show some optical spectral properties that are more in line with other GF-type events) with shorter-wavelength spectra ( λ = 2440 -2840 ˚ A) from the Hubble Space Telescope / Cosmic Origins Spectrograph. They concluded that a Balmer continuum enhancement upon a blackbody-like optical continuum component is necessary to account for the continuum flux down to at least λ ≈ 2500 ˚ A. International Ultraviolet Explorer (IUE) FUV spectra covering λ = 1900 -3100 ˚ A have been reported in the decaying phases of two large events on MVe stars (Hawley and Pettersen, 1991; Robinson et al, 1995). For a review of the timing of the IUE/NUV and IUE/FUV spectral integrations within the AD Leo Great Flare, see Kowalski (2022). Of the few (relatively small) flares with NUV spectra that have been analyzed through their peak and decay phases, the fractions of energy in the continuum were calculated in the large range of ≈ 50 -90% (Hawley et al, 2007; Kowalski et al, 2019b), which is much larger than in quiescence (35%). The evolution of the λ ≈ 3300 -3800 ˚ A continuum-to-line energy fractionation is also noticeable after the peak phase in the M5.5/6Ve event in Fig. 11(a). \nFig. 13 Energy of the hydrogen Balmer component vs. the optical continuum energy evolution in a large sample of MVe flares (reproduced from Kowalski et al 2013 with permission). The Balmer jump ratios ( χ flare ) increase into the gradual decay phase and are also generally correlated with the IF/HF/GF classification scheme. The hydrogen Balmer component includes an estimate for the Balmer continuum flux above an optical continuum model extrapolation to λ < 3650 ˚ A. Several more flares with comparable spectroscopy (Kowalski et al, 2016, 2019b) since this result have been consistent with these trends. \n<!-- image -->", '7.2.3 Empirically-determined, Broadband Radiated Energy Budgets Relative to the U band': "Empirical radiative energy fractionation from the X-ray to the radio provides modelindependent 31 characterization of stellar flares. The energy in the optical and NUV continuum provides a large contribution to the fractionation. To date, Osten and Wolk (2015) is the most comprehensive analysis of the multi-wavelength energy budget in archival observations of stellar flares across a wide variety of spectral types and quiescent activity levels. At λ = 1200 -8000 ˚ A, 96% of the peak flare luminosity can originate from the continuum in the impulsive phase (Hawley and Pettersen, 1991). In the gradual decay phase, the continuum accounts for as much as 83% when the emission lines remain highly elevated for longer durations (e.g., Fig. 9). The energy radiated in soft X-rays (0.04 - 2 keV) is estimated to be ∼ 11x the H γ energy (Butler et al, 1988) and to be comparable to that in the U -band, which is roughly 1/6 of the integrated 1200 -8000 ˚ Awhite-light energy (Hawley and Pettersen, 1991; Hawley et al, 1995), and a factor of 1/10 of the bolometric flare energy (Osten and Wolk, 2015). Hawley and Pettersen (1991) constrain E U = 0 . 64 E 2000 -3260 ˚ A , however the bandpassaveraged energy [erg ˚ A -1 ] in the U -band is reported as a factor of 1.2 greater. Several \nestimates of the U -band to Kepler-band (Kp) energy ratios have been empirically \ndetermined to be E U = (0 . 4 -0 . 65) E Kp (Hawley et al, 2014). Lacy et al (1976) determine E U = 1 . 2 E B = 1 . 8 E V . Line-integrated flare energies in H α are typically factors of ≳ 0 . 04 -0 . 08 smaller than the U -band energies, and the H γ energies are factors of ≈ 0 . 08 of the U -band energies over several orders of magnitude in total flare energy (see the large compilation in Figure 4.2 in Kowalski, 2012). \nThe relative energy in soft X-rays ( E ≈ 0 . 1 -5 keV) is difficult to compare among flares. This difficulty arises due to different temporal (e.g., Hawley et al, 1995) and wavelength integration limits (e.g., by including a significant portion of the EUV range) among datasets. The X-ray and optical thermal radiation occur on much different time-scales (Sect. 7.7), and a limiting assumption (e.g. Audard et al, 1999, 2000) is sometimes that the broad-band flare energy is accurately accounted for with a scaled 2-temperature quiescent coronal model (Gudel et al, 1997). Nonetheless, many interesting empirical constraints on the X-ray to optical flare energies are reported in the literature. The ratio of the soft X-ray (and/or EUV) to the U -band energy can take on a large range of values, from essentially no detectable response in the soft X-rays within a ± 10 -20 minute window (Doyle et al, 1988a; Osten et al, 2005)), down to 1 / 3 (Hawley et al, 1995), to as large as a factor of ≳ 10 (Gudel et al, 2002a; Osten et al, 2016). Namekata et al (2020) study the X-ray properties (Sect. 11) of several X-ray flares with NICER spectra. The E = 0 . 5 -10 keV soft X-ray energies fall in the range of 0 . 3 -1 × 10 32 erg, but the event with complete multi-wavelength coverage in their study showed an H α response without a detectable g ' -band (white-light) response. In general, using the energy fractionation relation of E U = 1 / 3 E 0 . 01 -10keV = 1 / 3 E XEUV in Osten and Wolk (2015), one obtains consistent cumulative flare rates in the XEUV and in the U -band (e.g., ∼ 1 per day greater than 10 32 erg in U Pettersen et al, 1984) corresponding to 1 flare per day with E XEUV ≳ 3 × 10 32 erg (Audard et al, 2000) for non-simultaneous flares on AD Leo. Tristan et al (2023) calculate the energy ratios of six simultaneous U -band and soft X-ray ( E = 0 . 2 -12 keV) flares to be E SXR /E U ≈ 1 . 5. Schmitt et al (2019) extend the comprehensive, multi-wavelength analysis of a E ≈ 10 34 erg flare on the K0Ve star AB Dor from Lalitha et al (2013); they also report comparable energies released in soft X-rays and in an NUV bandpass. This is consistent with a paradigm that a majority (Osten and Wolk, 2015; Kuznetsov and Kolotkov, 2021) of the total bolometric response in many (but not all) white-light stellar flares occurs from energy deposition in the dense, chromospheric footpoints (Fig. 2(right)). \nThe peak radio (3.6 cm) specific luminosity (erg s -1 Hz -1 ) is ∼ 1 / 4 of the U -band, but the bandpass-integrated energy is a factor of O (10 4 ) smaller (Osten et al, 2005). New broadband observations have suggested much larger energy releases than expected 32 in the NUV regime relative to the optical (Brasseur et al, 2023), while others are rather well-matched by current RHD models that invoke large heating rates (Osten et al, 2016; Kowalski et al, 2019b). Studies of high-cadence, simultaneous Galex/FUV and Galex/NUV photometry have reported much larger peak luminosities in the broadband FUV than in the broadband NUV (Robinson et al, 2005) (but see the discussion of bright Galex sources in Sect. 7.3). In the NIR, Davenport et al (2012) \npresent model predictions for the magnitude enhancements and energies relative to the U band. Tofflemire et al (2012) used high-cadence, high-precision monitoring of ∆ U = -1 . 5 mag flares on mid-type M dwarfs to constrain the broadband NIR response to | ∆ NIR | ≲ 5 -12 milli-magnitudes.", '7.2.4 Optical Broadband Flare Colorimetry': 'Optical flare colorimetry is the study of the temporal tracks of broadband flare colors or absolute fluxes. For example, the U -B vs. B -V colors of flares on the dM3.5e star EV Lac were compared to regimes of optically thick and optically thin hydrogen spectra in Zhilyaev et al (2007). They conclude that the flare continuum becomes optically thick (and more blackbody-like) in the peak phase of flares, but in the decay phase, the optical depths decrease. Hawley et al (2003) find a best-fit blackbody temperature of T ∼ 8500 K by fitting to the fluxes in UBVR filters for a sample of eight moderatesized flares on AD Leo. Colorimetry with the traditional Johnson/Bessell bandpasses is a powerful and convenient method to test a wide variety of flare continuum models (Kunkel, 1970; Mullan, 1976; Mullan and Tarter, 1977; Cram and Woods, 1982; Doyle et al, 1989; Hawley and Fisher, 1992; Maas et al, 2022; Namekata et al, 2022a; Brasseur et al, 2023), and similar methods using broadband filters in the NUV and FUV have inferred a wide range of even hotter blackbody temperatures, T ≈ 20 , 000 -50 , 000 K (Robinson et al, 2005; Getman et al, 2023). \nHowever, there are several important assumptions in broadband colorimetry that warrant spectroscopic verification or calibration (e.g., Kowalski et al, 2016). We discuss a few here. First, a correction to the contribution from emission lines in the optical bandpasses must be considered (Hawley and Pettersen, 1991; Hawley and Fisher, 1992). The V band is relatively free of major flare emission lines, but the contributions from the Balmer lines vary from flare to flare in the B band (e.g., Fig. 13, Fig. A1 in Appendix A). The U band integrates over the Balmer jump, a pseudo-continuum of blended Balmer lines, and to a lesser extent Ca II K and H emission. Allred et al (2006) demonstrated that a RHD model spectrum with a large Balmer jump and a discontinuity at λ = 3646 ˚ A exhibits colors that are consistent with a T ∼ 9000 K blackbody when convolved with the UBVR filters - but a blackbody function has no Balmer discontinuity or jump! Kowalski et al (2019b) convolved an observed flare spectrum with broadband filters and showed that very large blackbody temperatures ≈ 15 , 000 -22 , 000 K could be inferred if the Balmer jump and high-order Balmer lines are not included in a model for the U -band and/or a constraint on the NUV continuum flux is not available (see their Figure 16). They showed that the full optical continuum regime may be flatter than a hot blackbody fit to the blue-optical wavelengths in such flares. The analysis of Hawley et al (2003) included a bona-fide continuum flux constraint from HST spectra in the FUV, which may have otherwise resulted in much hotter peak blackbody temperatures, as in Zhilyaev et al (2007). \nThe results from broadband flare colorimetry have largely motivated spectroscopic investigations (Giampapa, 1983; Hawley and Pettersen, 1991; Kowalski et al, 2013) of flares and detailed modeling (Hawley and Fisher, 1992), which will be discussed in Sect. 8.2.', '7.3 FUV Observations: The Rapidly Evolving Flare Transition Region (TR)': "The FUV continuum and emission line response may be a complex mixture of thermal, nonthermal, and nonlocal radiative processes in the flare transition region ( T ≳ 50 , 000 K) and the deeper atmosphere at T ≲ 10 , 000 K (Hawley and Pettersen, 1991; Phillips et al, 1992; Hawley et al, 2003; Ayres, 2015b; Froning et al, 2019; Kowalski, 2022; Doyle et al, 2012, 2013). Many questions remain about the origin of the spectral and temporal properties in stellar flares in this wavelength regime. A few observational properties are summarized in this section. \nHawley et al (2003) presented a comprehensive, high-resolving power ( R = 70 , 000) study of eight moderate-sized M dwarf flares in the FUV with contemporaneous optical spectroscopy and broadband photometry. The energy budgets and timing properties in the brightest spectral lines and continua were compared in detail. In the biggest flare, the peak line fluxes in C II λ 1335, Si IV λ 1394, C IV λ 1548, and C III λ 1176 were calculated in the ratio of ≈ 25:35:50:100, respectively (note that the lineshifts in this study are discussed in Sect. 10). The study also presented a novel timing analysis of the light curves of Si IV, C IV, N V, and the FUV continuum region (averaged over λ ≈ 1266 -1295 , 1420 -1452 , 1675 -1710 ˚ A) compared to the U -band by fitting b and m in the relation log 10 ( F X ) = b + m log 10 ( F U ). For C IV 1551 and Si IV 1403, the slopes are rather consistent with a value of ≈ 1, but the value of m FUVcont ≈ 1 . 7 indicates a significantly more rapid evolution of the FUV continuum compared to the U -band, an effect that has also been noted in a much larger event on AD Leo (Hawley and Pettersen, 1991). The power-law relationship between the FUV continuum and the U band is not yet explained by physical models. A linear scaling among neutral silicon FUV continua, C IV, and C II luminosities among a sample of dMe and RS CVn flares was interpreted in Phillips et al (1992) to be consistent with radiative backheating of the temperature minimum region. \nRemarkable M-star FUV flare spectra were analyzed in Hawley and Pettersen (1991), Redfield et al (2002), Loyd et al (2018b), Froning et al (2019), MacGregor et al (2021), and Feinstein et al (2022). In nearly all cases, the continuum spectra are either very flat (in units of erg cm -2 s -1 ˚ A -1 ; see also the spectra compiled in Butler et al, 1981; Bromage et al, 1986; Byrne et al, 1990; Phillips et al, 1992) or an isothermal blackbody continuum fit to the regions between emission lines results in T ≳ 15 , 000 K, up to T ≈ 30 , 000 -40 , 000 K. Of these, Hawley and Pettersen (1991) and MacGregor et al (2021) had simultaneous multi-wavelength observations that were analyzed in detail, and Froning et al (2019) and Loyd et al (2018b) discuss redshifted line asymmetries in the FUV. Ayres (2015b) discuss interesting FUV continuum bursts during the gradual decay phase of a Si IV light curve in an energetic event from the young solar analog EK Dra (Fig. 14); similar responses during the FUV bursts do not show up in higher temperature lines. These continuum-only bursts were interpreted as possible stellar analogs of 'Type II' white-light solar flares (Fang and Ding, 1995). \nBroadband comparisons of FUV and NUV responses during flares have been possible due to the time-tagged photon capability of the Galex mission. Robinson et al (2005) analyzed a giant flare in both Galex bands, and Welsh et al (2006) showed similar trends in other flares: very large ratios of the FUV to NUV fluxes that are \nFig. 14 Figure 7 from Ayres (2015b), reproduced with permission, showing various FUV spectral quantities from HST/COS spectra during the decay phase of a large flare on the young solar analog, EK Dra. The bottom x -axis is elapsed time (hr) and top x -axis is rotational phase. Note the bursts that only occur in the FUV continuum ( λ = 1435 ± 25 ˚ A, 1610 ± 25 ˚ A). Later in the decay phase, shorter FUV wavelengths were observed by HST/COS, providing constraints on the continuum at λ = 1344 . 5 ± 6 ˚ A and λ = 1381 . 0 ± 8 . 0 ˚ A, the forbidden coronal line [Fe XXI] λ 1354 ( T form ≈ 10 7 K), and C II λ 1335. \n<!-- image --> \nfar larger than expected from any known continuum model constrained by optical spectroscopy (Sect. 7.2). It has been pointed out that non-linearity corrections are important considerations at these large count rates in Galex (Morrissey et al, 2007; Fleming et al, 2022). More recently, the gPhoton tool (Million et al, 2016) has been employed to study events in the NUV without such concerns at large peak count rates, \nrevealing remarkably fast time variations on a wide variety of stars (Brasseur et al, 2019). Simultaneous constraints from Kepler and Galex observations favor very large flux ratios that are, again, far larger than RHD models of optically thick continuum radiation from high-energy electron-beam heating models (Brasseur et al, 2023). \nPotential signatures of proton beams are found in both the FUV and in the gamma rays. Loh et al (2017) and Song and Paglione (2020) searched for gamma-ray stellar flares, which result from the higher energy ( E ≳ 30 MeV / nucleon) ions and protons interacting with the stellar chromosphere. Lower-energy protons ( E ≪ 1 MeV) in the beams are expected to undergo charge-exchange with the neutral hydrogen in the chromosphere before it explodes in temperature. The nonthermal hydrogen beams then emit red-shifted Lyman α photons as a captured electron falls to the ground state. This is called the Orrall-Zirker effect (Orrall and Zirker, 1976; Kerr, 2023). (Woodgate et al, 1992) reported a broad satellite component that was highly redshifted from Ly α and was attributed to low-energy proton beams in the early phase of a stellar flare in AU Mic. Follow-up observations with high-time resolution spectroscopy of the Ly α red and blue wings have not resulted in similar detections (Robinson et al, 1993, 2001; Feinstein et al, 2022).", '7.4 Centimeter (Microwave/Radio) Observations: Nonthermal Gyrosynchrotron Radiation from Mildly Relativistic Electrons': "The radio/microwave regime at cm wavelengths (GHz frequencies) is a probe of nonthermal emission from accelerated electrons in flare coronae. Lower frequencies, ≲ 2 GHz, tend to be highly circularly polarized, shorter in duration, and narrowband, indicating plasma or electron-cyclotron maser emission, while higher frequencies are largely unpolarized, broadband, and longer in duration. The unpolarized higher frequency radiation is consistent with gyrosynchrotron radiation from a power-law distribution of mildly relativistic electrons trapped in the magnetic fields of the flare region. The peak frequency of the gyrosynchrotron spectrum determines the demarcation at which the spectrum transitions from optically thin (at frequencies higher than the peak frequency) to optically thick (at frequencies lower than the peak frequency). For a homogeneously emitting source, the power-law index of accelerated electrons is directly related to the power-law index of the optically thin gyrosynchrotron flux spectrum at Earth; see Dulk (1985) and Sect. 11 for more discussion and references. \nRadio stellar flares have typically been observed in two bands, such as 6 cm (4.9 GHz) and 3.6 cm (8.4 GHz), which are close to or at lower frequencies than typical peak frequencies ( ≈ 10 GHz) (note that some solar flares have peak frequencies constrained to at least ≳ 30 GHz; White et al, 2003). A large multi-wavelength flare from the dM3.5e star EV Lac was studied in Osten et al (2005) and a light curve is shown Fig. 15. The peak radio luminosity at 8.4 GHz is 1 . 9 × 10 15 erg s -1 Hz -1 , but the U -band (not reproduced here; L U, peak ≈ 10 30 erg s -1 , E U ≈ 5 × 10 31 erg) peaks about 54 s earlier and decays much more rapidly. The spectral index ( α ; S ν ∝ ν α ) evolution is discussed in Osten et al (2005) and indicates rapidly varying optical depths at these frequencies over the flare; in the decay phase, the higher frequency likely becomes optically thin, while in the peak phase, a reasonable range of expected power-law \nFig. 15 A remarkable radio flare on the dM3.5e EV Lac from Osten et al (2005) with the spectral indices calculated from observations at the two observed frequencies. Reproduced with permission. \n<!-- image --> \nindices for the nonthermal electrons cannot explain the values of α for a homogeneous optically thick source. The lengthening decay timescales were interpreted in terms of the 'trap+precipitation' model (Section 3). Source parameters were derived during the long decay, indicating very large loops on comparable scales to the stellar radius with small trapped nonthermal electron densities (10 4 -10 6 cm -3 ), and large magnetic fields (100 G) increasing over time and decreasing in source size as the higher frequency becomes more optically thin (R. Osten, priv. communication 2011). \nFig. 16 shows three radio flares from the comprehensive multi-wavelength study of the RS CVn system HR 1099 (Osten et al, 2004). The data show low polarization during the flares and a similar spectral index evolution patterns among these longduration flares. Gudel et al (1996) and Smith et al (2005) present radio analyses of stellar flares that will be discussed below (Sect. 7.7, Sect. 11) in multi-wavelength contexts. \nThe emission at lower frequencies results from a plethora of possible phenomena. Decimetric flares 33 can be very luminous and highly circularly polarized indicating coherent emission from beam-plasma instabilities ('plasma emission') or electron cyclotron maser (Lang et al, 1983; Bastian et al, 1990; Osten and Bastian, 2008; Villadsen and Hallinan, 2019, see references and discussion in Bastian 1990; Gudel et al 1996; Osten et al 2005). Coherent bursts can occur at higher frequencies too, and a remarkable example is the 100% left-hand circularly polarized 30 mJy peak-flux, 3-minute \nFig. 16 Flux (top row), polarization (middle row), and spectral indices (bottom row) during three gyrosynchrotron radio flares from the RS CVn HR 1099, reproduced from Osten et al (2004) with permission. \n<!-- image --> \nburst at 5 GHz discussed in Smith et al (2005) from the dM3e AD Leo. Assuming plasma emission, they derive a coronal electron density of ≈ 3 × 10 11 cm -3 ; assuming electron cyclotron, they derive a magnetic field of B ≈ 1 . 8 kG. Between 1-2 GHz, stellar analogies to Type III decimetric bursts 34 have been reported in very high timeresolution data from Arecibo (Osten and Bastian, 2006). A surprising lack of Type II radio bursts has been constrained in the context of extensive ground-based monitoring of optical flaring (Crosley and Osten, 2018). Many recent numerical, theoretical, and observational efforts have sought to explain the lack of Type II radio bursts from other stars (Alvarado-G'omez et al, 2018; Mullan and Paudel, 2019; Alvarado-G'omez et al, 2020; Wood et al, 2021, see also Cliver et al 2022a).", '7.5 Millimeter Observations from ALMA: Nonthermal Radiation with Linear Polarization': "Bright nonthermal flare radiation occurs at much higher frequencies around ν ≈ 230 GHz ( λ ≈ 1 . 3 mm) too. Millimeter flares have been reported in ALMA data from AU Mic and Proxima Centauri (MacGregor et al, 2018, 2020). Particularly remarkable is the extremely impulsive flare from Proxima Centauri with multi-wavelength observations (MacGregor et al, 2021). The dMe mm flares are short in duration (2 -30 s), have rather symmetric (Gaussian-like) impulsive-phase time profiles, and exhibit only a very weak gradually decaying tail. They note that these properties are rather similar to the FUV continuum response in the same flares (see Sect. 7.3). The spectral \nindices, α ( S ν ∝ ν α ), are negative with a significant amount of linear polarization that 'flip-flops' between negative and positive values during the peak phase. \nThe millimeter flare emission is interpreted as either an optically thin extension to the gyrosynchrotron radiation component that peaks around ν = 10 GHz (see the previous section), or as optically thin synchrotron radiation that undergoes some type of depolarization to the observed levels of ± 20% (see MacGregor et al, 2020). The radiation sources are relativistic or ultra-relativistic electrons, but the magnetic field strengths are largely different between these hypotheses. Being in the optically thin regime of either interpretation, these observations have facilitated important inferences of the power-law indices of the accelerated electrons in the range of δ r ≈ 2 . 8 -5 . 2, indicating rather hard distributions of accelerated particles. At much higher frequencies, Beskin et al (2017) reported short (FWHM durations of ≲ 1 s) synchrotron bursts in the optical U -band superimposed on a giant, unpolarized dMe event from UV Ceti. Only lower limits on the linear polarization were possible, but the analyses of the energetics of the bursts favored very hard, δ < 3 . 4, power-laws of ultra-relativistic electrons. \nFlares at ν ≈ 100 GHz ( λ ≈ 3 mm) have been reported from PMS stars, RS CVn's, and eccentric binaries with colliding magnetospheres (Phillips et al, 1996; Bower et al, 2003; Salter et al, 2008, 2010; Brown and Brown, 2006; Adams et al, 2011). In the case of the remarkable periastron event in the binary T Tauri V773 A (Massi et al, 2006), the flare was interpreted as synchrotron radiation. There is a notable observation of an RS CVn event at ν ≤ 100 GHz showing a break and a rise toward higher frequencies (Beasley and Bastian, 1998): i.e., an opposite spectral index to the dMe events reported in ALMA, and one that is more similar to some reports in solar flares (Kaufmann et al, 2004). Colliding magnetospheres at periastron are also attributed to some flare-like optical and NUV variability in the PMS DQ Tau system (Salter et al, 2010; Getman et al, 2023), while the optical variability properties at periastron at other times follow the expectations of accretion alone (Mathieu et al, 1997; Tofflemire et al, 2017).", '7.6 Soft X-rays and EUV Observations: Loops Filled with Evaporated, Previously-Chromospheric/TR, T > 10 7 K Thermal Plasma': "Stellar flares have been studied in detail in soft X-rays ( E ≈ 0 . 1 -10 keV) for many decades through XMM-Newton, Chandra, Swift, and NICER spectroscopy and photometry. X-ray flare spectra are a composite of many optically thin, highly-ionized emission lines and free-free bremsstrahlung continua that originate from the flaring coronal volume. The bulk of the radiation is thought to originate in the chromospheric gas that has been heated to millions of K and ablated into the magnetic loops, which confine the flows (Sect. 3). One of the highest signal-to-noise flare spectra during a stellar flare, which occurred in an RS CVn system, is showcased in Fig. 17. \nAn extensive review of stellar X-ray flares and analysis methods (prior to and including 2004) are provided by Gudel (2004) and Gudel (2006). A general review of X-ray spectroscopy of stars is contained in Gudel and Naz'e (2009). Since 2004, there have been many important X-ray flare studies (Osten et al, 2005; Robrade and Schmitt, 2005; Stelzer et al, 2006; Fuhrmeister et al, 2007; Nordon and Behar, 2007, 2008; \nFig. 17 X-ray spectra from Chandra during quiescence (top) and flaring (bottom) in the active binary σ 2 Coronae Borealis, reproduced from Osten et al (2003) with permission. The authors note the enhanced continuum flux increasing towards shorter wavelengths in the flare spectrum. \n<!-- image --> \nPandey and Singh, 2008; Huenemoerder et al, 2010; Liefke et al, 2010; Fuhrmeister et al, 2011; Pandey and Singh, 2012; Lalitha et al, 2013; Pye et al, 2015; Namekata et al, 2020; Paudel et al, 2021; Karmakar et al, 2022). In this section, we focus on a few key results from the last several decades pertaining to abundance changes and the stellar flare Neupert effect. Several other results are included throughout this review in other contexts where they are especially complementary (e.g., multi-wavelength analyses).", '7.7 The Neupert Effect in Stellar Flares': "The backbone of the solar-stellar flare connection is the Neupert effect (Neupert, 1968). This was originally established in three solar flares as a correlation between the cumulative time-integral of the nonthermal microwave emission around 2.7 GHz and the thermal soft X-ray flux at 1.87 ˚ A, attributed to the Fe XXV line emission (1s 2 -1s2p, which is sensitive to T ≈ 30 MK). It has since been the subject of many follow-up studies in solar flares using either the hard X-rays or the derivative of the GOES 1-8 ˚ A soft X-rays as a proxy for the nonthermal particle heating in the chromosphere (e.g., Dennis and Zarro, 1993; McTiernan et al, 1999; Veronig et al, 2002b, 2005; Warmuth et al, 2009). Alternatively, the empirical Neupert effect is expressed as a relationship between integral of the hard X-ray flux and the luminosity of the \nGOES soft X-rays. The relationship between the impulsive, early nonthermal radiation and the gradual, later-peaking thermal flux is thought to be linked by the physics of impulsive nonthermal particle propagation and bombardment of the chromosphere, rapid evaporation/ablation of T ≈ 10 -50 MK chromospheric mass into the coronal loops, and the eventual cooling to pre-flare coronal temperatures (Sect. 3). \nThe Neupert effect is also reported in multi-wavelength observations of stellar flares. Because stellar flare hard X-ray ( E ≫ 10 keV) emission is almost always too faint to detect, and radio observations are relatively difficult to plan and secure, the (thermal) response in the U -band is utilized as a proxy. This approach is justified by the close temporal and spatial correspondence between hard X-ray and white-light emission on the Sun (e.g., Kane et al, 1985; Neidig, 1989; Hudson et al, 1992; Neidig and Kane, 1993; Fletcher et al, 2007; Krucker et al, 2011, 2015; Kleint et al, 2016). A remarkable example of the stellar Neupert effect is shown for a flare from Proxima Centauri (Gudel et al, 2002a) in Fig. 18. \nA stellar flare Neupert effect was first reported in Hawley et al (1995) using the correlation between the extreme-ultraviolet (65-190 ˚ A) flux and the integrated U -band energy during a long-duration, 3.5-hour rise phase of a flare from the dM3e AD Leo. This was followed shortly thereafter in Gudel et al (1996) using a correlation between the E = 0 . 5 -2 keV soft X-rays and 3.6 cm radio emission during a flare on the dM5.5e UV Ceti A. The Neupert effect has subsequently been reported in stellar flares from RS CVn systems (Gudel et al, 2002b; Osten et al, 2004), dMe stars (Gudel et al, 2002a, 2004; Mitra-Kraev et al, 2005a; Wargelin et al, 2008; Fuhrmeister et al, 2011; Caballero-Garc'ıa et al, 2015; Tristan et al, 2023), dK-stars (Lalitha et al, 2013), T Tauri stars (Audard et al, 2007). There are also notable exceptions to the expected multi-wavelength relations from the Neupert effect in stellar flares (Doyle et al, 1988a; Osten et al, 2005); see these papers for discussions of possible explanations, which include deep heating from ultra-relativistic electrons or MeV protons. On the Sun, a clear violation of the Neupert effect in a 'late impulsive [hard X-ray] burst' (adopting the terminology from Xia et al, 2021) has been modeled with a large, low-energy cutoff E ≳ 100 keV in the nonthermal electron beam (Warmuth et al, 2009), which would reasonably generate small amounts of collisional heating within and chromospheric evaporation into the corona (however, the explanation is still being investigated; Alaoui and Holman, 2017). \nThe Neupert effect in Flare D in Gudel et al (1996) was investigated in detail and was compared to a similar multi-wavelength relationship in a solar flare from Dennis and Zarro (1993). Intriguingly, the microwave response of flare D returned to pre-flare levels well before the maximum of the soft X-ray light curve, which is not in strict agreement with the expected Neupert effect. It was shown that similar deviations in the solar flare could be explained by a temperature sensitivity on the Neupert effect, such that radiation from hotter plasma more strongly followed the expectations (see also McTiernan et al, 1999). Gudel et al (1996) thus separately derived a generalized Neupert effect for the light curve and a generalized Neupert effect , following energy conservation over the flaring coronal volume, V : \nd dt ( 3 n e k B TV ) = αf radio ( t ) -n 2 e V ψ ( T ) (13) \nwhere n e is the ambient/thermal electron density of the heated plasma at a temperature T , k B is Boltzmann's constant, f radio is the flare radio flux as a function of time t , α is a proportionality constant, and ψ ( T ) is the optically thin radiative loss function in units of erg cm 3 s -1 . The left hand side of Equation 13 is the change of the thermal energy of the corona due to chromospheric mass evaporation, and the right hand side is the balance between radio/ microwave flux at Earth (assumed to be proportional to the kinetic energy of nonthermal electrons causing the chromospheric evaporation) and the optically thin radiative cooling of the coronal volume (assuming conductive cooling is overall small; see also Fisher and Hawley, 1990). Thus, accounting for the total plasma energetics and the bolometric luminosity evolution more naturally explains the apparent deviations from the Neupert effect assessed in a specific X-ray bandpass. An additional finding from the solar-stellar flare comparison was that the M dwarf flare was 'radio-overluminous', possibly suggesting a greater efficiency of acceleration of microwave-emitting ( E ≳ 100 keV) electrons (Sect. 7.4) and/or a lower efficiency of chromospheric evaporation, among several other possibilities (see Gudel et al, 1996). \nThe long rise times of Ca II K and H in dMe flares are suggestive of a Neupertlike effect, which could provide insight into their origin (e.g., through XEUV radiative backwarming; Hawley and Fisher, 1992). Kowalski et al (2013) investigated the relationship between the time-integral of the blue continuum flux around λ = 4170 ˚ A and the Ca II K line flux for a large sample of dMe flares. They found a Neupert-like relationship for a wide variety of flare types, both impulsive and gradual-type events, but none of these observations were complemented with X-ray data.", '7.8 Abundance Changes': 'Impulsive heating in the chromosphere evaporates gas into the coronal loops, but all chemical elements are not evaporated equally. This results in relative abundance changes in the hot, coronal flare plasma such that elements with low first ionization potential (FIP) of < 10 eV, like Fe, Si, and Mg, become preferentially enhanced above their photospheric values. The fractionation may be explained by ponderomotive forces (Laming, 2004) that selectively act on singly-ionized species in the upper pre-flare chromosphere. \nOptically thin, coronal equilibrium modeling of X-ray and EUV spectra has been used to search for the so-called FIP effect (or its inverse, the IFIP effect) in stellar flares. Detections of changes in low-FIP abundances have been suggestive but not highly statistically significant (Gudel et al, 1999), as have relative changes in highFIP abundances such as Ne (Osten et al, 2005). An example of an abundance analysis of an RS CVn flare in Fig. 17 is shown in Fig. 19 where an abundance change in Fe was seen, but other expected FIP dependencies were not. Recently, Paudel et al (2021) searched for the (I)FIP effect in individual flares from the dM3.5e star EV Lac; they discuss the importance of field geometry in the context of the Laming theory. A homogeneous Sun-as-a-star (i.e., stellar) EUV spectral analysis of a large sample of solar flare Fe lines revealed coronal abundances that were remarkably close to photospheric, thus suggesting that chromospheric heating and evaporation occur deeper in the atmosphere than the standard models of electron beam heating and elemental fractionation (Warren, 2014), but Laming (2021) briefly contests the interpretation. \nFig. 18 A remarkable example of the Neupert effect in a stellar flare in Proxima Centauri; figure reproduced with permission, from Gudel et al (2002a). The U -band light curve is a (thermal) proxy for the nonthermal, impulsive energy deposition into the lower atmosphere. This flare was subsequently studied and modeled in Gudel et al (2004) and Reale et al (2004). \n<!-- image --> \nNonetheless, these solar results provide tantalizing connections to the deep heating rates (Sect. 8.2) that are inferred and hydrogen line blueshifts (Sect. 10) that are reported from optical observations of M dwarf flares.', '8 Atmosphere Modeling of Stellar Flares: Overview of Results': 'The physics of stellar flares is a complicated (and therefore interesting!) intersection of particle and nuclear physics, radiative transfer, plasma processes, gas dynamics and shock phenomena. All of the physical processes are inter-dependent and occur in a highly magnetized, coupled atmosphere, which is probably also highly turbulent. Flares are transient non-equilibrium departures of the stellar luminosity, and all layers of the atmosphere are thought to produce a response to some degree. In this section, \nFig. 19 Abundances relative to hydrogen vs. the first ionization potential (FIP) in quiescent (black) and flaring (red) intervals in Chandra spectra of the RS CVn σ 2 Coronae Borealis. Figure reproduced from Osten et al (2003) with permission. They note that there is a slight enhancement of Mg, Si, O, and Ne during the flare, and Fe increases by nearly twice. \n<!-- image --> \nwe review different stellar flare modeling techniques. We focus on efforts toward understanding chromospheric flare processes in the context of the response of the entire stellar atmosphere. In contrast to purely magneto-hydrodynamic models, radiative models include detailed, time-dependent spectral predictions, which are critical for connecting to observations. \nStellar flare modeling can be sub-divided into several types of techniques. Isothermal, static slab modeling and atmospheric modeling are the two most general categories. Atmospheric modeling can be further divided into a static semi-empirical approach (e.g., Cram and Woods, 1982, wherein the atmospheric structure is adjusted by hand to match observations) and radiative-hydrodynamic (RHD) forward modeling, whereby the input heating function is data-constrained and is not fine-tuned by the modeler to match observations (e.g., Allred et al, 2006). A hybrid, semi-empirical RHD approach is also possible, whereby the injected flare heating function is varied by hand over a large parameter space to find a time-dependent spectral response that matches the observations; thus, the atmospheric response and depth-dependent heating rates are calculated in a fully self-consistent manner (e.g., Kowalski et al, 2015). Though atmospheric modeling accounts for the vertical heterogeneity at a location in a flare source, additional techniques are required to model the lateral heterogeneity of the flare source and the source geometries (to be discussed further in Sect. 12).', '8.1 Slab Models': "Uniform, static slab modeling techniques include the following: fitting a Planck function to spectra or multi-band photometry (Hawley et al, 1995, 2003; Zhilyaev et al, 2007; Kowalski et al, 2013), calculating LTE continuum spectra for a range of optical depths τ ( λ ) = [0 , ∞ ) (Kunkel, 1970; Eason et al, 1992), and modeling the NLTE Balmer line decrements, optical depths, and continuum flux ratios (Drake and Ulrich, 1980; Jevremovic et al, 1998a; Garc'ıa-Alvarez et al, 2002; Morchenko et al, 2015). The pioneering study of Kunkel (1970) found evidence for a prominent Balmer jump \nin their spectral observations of M dwarf flares. Their models of optically thin hydrogen (emissivity) continuum calculations, however, were largely inconsistent with the colors of M dwarf flares. They concluded that an additional moderately heated photospheric flare component could better account for observations (it was not until the radiative-hydrodynamic models of Allred et al (2006) that essentially produced this scenario through a chromospheric hydrogen recombination spectrum and a radiatively backheated photospheric spectrum; see below). In Appendix C, we update the LTE hydrogen continuum emissivity calculations from Kunkel (1970) with non-ideal occupational probabilities around the Balmer limit and extend them to higher temperatures and shorter wavelengths. These are useful for verifying for oneself how poorly first-principle simplifications hold up against continuum observations of actual stellar flares 35 . \nThe higher quality spectral observations of the Great Flare of AD Leo (Hawley and Pettersen, 1991) were found to (apparently) exhibit little to no Balmer jump in the flux, but there was unprecedented power in the optical continuum radiation and highly broadened hydrogen line wings. The broadband colors from the FUV through the optical ( R band) were tested against all of the models available at the time (Hawley and Fisher, 1992), and a T ≈ 8500 -9500 K blackbody was clearly favored over a hydrogen recombination model and a hot T = 10 7 K free-free slab model interpretation; the former model has a slope that is too red, and the latter interpretation is inconsistent with X-ray to optical energy fractionations (see Section 4.3.2 of Hawley and Fisher 1992, Osten and Wolk, 2015, and Sect. 7.2.3 here) and representative volume emission measures (Sect. 11). The various interpretations (blackbody vs. optically thin Balmer continuum) and reported characteristics of the properties at the Balmer limit eventually motivated additional spectral observations (Kowalski et al, 2013). More immediately, they paved the way toward advances in the realism of models of stellar flares. It became essential to consider the comprehensive atmospheric response that includes the complexities of depth-dependent heterogeneities of opacities, temperatures, and densities. Then, self-consistent non-equilibrium calculations of velocity fields, mass advection, radiative cooling, and flare heating became possible. We summarize these efforts next.", '8.2 Atmospheric Models': "A traditional approach to modeling stellar flare spectra is to start with a hydrostatic equilibrium model atmosphere and semi-empirically adjust the temperature gradients of the chromosphere, photosphere, and the location of a transition region. Cram and Woods (1982) explored the results of this technique using six atmospheric variations, including two very extreme adjustments where the chromosphere and transition region are placed at large column mass depths. They successfully reproduced a T ≈ 14 , 000 K blackbody-like continuum spectral property in one of these models (their model #5), which also had very broad hydrogen lines and deep central reversals. Similar approaches have been widely adopted to model EUV, optical, and infrared stellar flare spectra (Houdebine, 1992; Mauas and Falchi, 1996; Jevremovic et al, 1998b; Christian \nFig. 20 Demonstration of semi-empirical temperature adjustments used to model the red- and blue-wavelength optical spectra of a giant flare from CN Leo. Figure from Fuhrmeister et al (2010) reproduced with permission from the author. See also Fuhrmeister et al (2011) and Fuhrmeister et al (2005) for similar approaches for modeling flares from Proxima Centauri and the dM6 star AZ Cnc, respectively. \n<!-- image --> \net al, 2003; Fuhrmeister et al, 2010; Schmidt et al, 2012). An example of this modeling technique from Fuhrmeister et al (2010) is shown in Fig. 20, where the chromospheric temperature adjustments result in satisfactory fits to the emission lines in the giant CN Leo flare (Fig. 11(a)) from Fuhrmeister et al (2008). Further, a depth-dependent filling factor and missing photospheric heating were inferred through this technique. \nHow to create these deep transition region locations and chromospheres selfconsistently with a flare heating function, Q flare ? One early hypothesis (Mullan and Tarter, 1977) was that intense irradiation of soft, thermal X-rays from the flare corona penetrated to the low atmosphere, causing the optical emission lines and continuum in the gradual decay phase (whereas, the impulsive phase optical emission was assumed to be caused by particle beams, as was known to be consistent with hard X-ray and whitelight timing in solar flares). Hawley and Fisher (1992) made a major advancement in stellar flare X-ray backheating models by calculating the full atmosphere, including the coronal structure. The apex temperatures were set and all else were determined selfconsistently through energy balance (X-ray heating and photoionization v. radiative cooling) and hydrostatic equilibrium. Although the models produced hydrogen lines that were too narrow, and the optical and NUV continuum radiation was too weak compared to the observations (Hawley and Pettersen, 1991), a large amount of broad, Ca II K line flux was predicted (perhaps offering clues to understanding its very slow temporal response). Updates to XEUV irradiation modeling (e.g., the atomic physics and geometrical assumptions therein) were employed in evolving atmospheres (Hawley and Fisher, 1994; Abbett, 1998; Allred et al, 2005, 2006, 2015) that facilitated comparisons to the role of nonthermal electron heating. The role of reprocessing much larger X-ray emissivities into lower atmosphere optical continuum radiation in superflares has been recently reconsidered in static atmosphere calculations (Nizamov, 2019). \nSignificant advances were made in modeling the temporal evolution of the atmospheric response in gas-dynamic simulations, starting in the 1980s (Livshits et al, 1981; McClymont and Canfield, 1983; Fisher et al, 1985c,b,a; Emslie et al, 1992). An accurate treatment of energy deposition in a partially ionized, nonuniform chromosphere due to an injected nonthermal electron distribution was developed (Hawley and Fisher, 1994) based on analytic formulae for Coulomb collisions (Emslie, 1978, 1981a). With the inclusion of radiative-transfer, this effort resulted in a series of papers on radiative 36 -hydrodynamic modeling of solar flares (Abbett and Hawley, 1999; Allred et al, 2005) using a version of the RADYN code (Carlsson and Stein, 1992, 1995, 1997) that was configured for a larger surface gravity, a cooler photosphere, and a hotter and denser corona of an M dwarf in Allred et al (2006). The RHD simulations are currently limited to one spatial dimension, which facilitates resolving short-timescale shock phenomena, non-equilibrium ionization rates of hydrogen and helium, and detailed radiative transport. \nThe Allred et al. stellar flare models studied the atmospheric response to electron beam heating inputs from the first widely studied solar flare (SOL2002-07-23T00:30 GOES class X4.8) of the RHESSI (Lin et al, 2002) hard X-ray spectroscopic imaging era (Dennis et al, 2022). A double-power law distribution with a low-energy cutoff of E c = 37 keV was obtained from the hard X-ray modeling of Holman et al (2003), and several injected fluxes were investigated. These models reproduced the broad Balmer lines for the first time and bright optical and NUV continuum radiation. However, the Balmer jump was even more conspicuous than in the energy equilibrium models with X-ray flare irradiation (Hawley and Fisher, 1992) and semi-empirical models (Fuhrmeister et al, 2010). The dominant mode of radiative backheating in the RHD models takes place through irradiation of the photosphere by the Paschen and Balmer continuum fluxes of optically thin thermal radiation in response to the nonthermal electron beam energy deposition in the mid-to-upper chromosphere (the beam electrons do not penetrate to the photosphere; see Sect. 9). The detailed RHD spectrum provided an alternative explanation to the T ≈ 9000 K blackbody interpretation of broadband photometry reported in earlier observations (e.g., Hawley et al, 2003), and the general shape of the Balmer continuum in the U -band was later found to be a satisfactory model in the decay phase spectra of a megaflare from YZ CMi (Kowalski et al, 2010). The lower flux model was run for several minutes in Allred et al (2006), and it well-reproduced several transition region flare lines from Hawley et al (2003). Thus, the Allred et al. models provided a comprehensive solar-flare modeling framework for interpreting many of the observed properties of M dwarf flares. \nYet, re-analyses of archival spectra and new flare spectra and narrowband ULTRACAM photometry still did not match the shape and relative strength of the optical continuum radiation predicted by the highest-beam flux model in the Allred et al (2006) simulations. Some important details of the line broadening and blending at the Balmer limit remained unexplained as well (Sect. 10.2.1). The M dwarf gas-dynamic models of Livshits et al (1981) produced a chromospheric condensation using a large nonthermal electron flux in an initially isothermal atmosphere and suggested that \na large optical depth in the continuum could be produced well above the photosphere (see also Katsova et al, 1997). Gan et al (1992) investigated chromospheric condensations as a source of white-light in solar flares. Impulsive-phase chromospheric condensations have been thought to be important in generating emission line spectral features in solar flares, such as the red-wing asymmetries in H α (Ichimoto and Kurokawa, 1984; Kowalski et al, 2022; Namekata et al, 2022a) and in more optically thin lines such as Fe II in IRIS NUV spectra (Kowalski et al, 2017b; Graham et al, 2020). However, stellar radiative-hydrodynamic models did not produce chromospheric condensations that were dense enough to explain the observed continuum radiation until recently when the effects of very large electron beam fluxes were explored in RADYN (Kowalski et al, 2015) and the RH code (Uitenbroek, 2001). (A realistic M dwarf atmosphere has a dense chromosphere and very efficient radiative cooling that can make the onset of complete ionization and explosive hydrodynamics a rather energetic threshold to attain, especially for a beam with a moderately large low-energy cutoff of E c = 37 keV, as traditionally employed in the RADYN dMe flare models.) The large electron beam fluxes were actually motivated by scaling the radiative backwarming fluxes from the Allred et al (2006) models, but when the large beam fluxes were simulated, the lower chromosphere become too optically thick for the NUV and optical continuum radiation to penetrate to the pre-flare photosphere. Instead, the τ ( λ ) surfaces shift to a dense chromospheric condensation and the beam-heated layers just below the condensation (Fig. 2(right)). \nA snapshot of the lower flare atmosphere in a RADYN model from Kowalski et al (2015) is shown in Fig. 21. The electron beam flux that is injected at the model loop apex is very large, 10 13 erg cm -2 s -1 (which is hereafter referred to as a beam flux of 'F13', whereas 'F12' refers to an injected beam flux of 10 12 erg cm -2 s -1 , and so on). By the time shown ( t = 2 . 2 s), the chromospheric condensation has cooled and accrued mass into a narrow ∆ z ≈ 15 km region with a maximum electron density of n e ≈ 5 × 10 15 cm -3 . This was an exciting development because the model produced a small Balmer jump ratio and an optical color temperature that was in line with the impulsive phase spectral observations of dMe flares. In these models, the emergent optical radiative flux spectrum is a result of thermal response to the nonthermal electron beam energy deposition. The hydrogen recombination rates are determined by the Maxwellian velocity distribution of recombining ambient electrons, which are rapidly equilibrated as the beam kinetic energy is transferred to them (Sect. 9). Non-thermal ionization rates of hydrogen by the beam electrons (and subsequent nonthermal recombination) (Hudson, 1972; Ricchiazzi and Canfield, 1983; Aboudarham and Henoux, 1986; Fang et al, 1993) are much less than the thermal rates in these atmospheres at all times except for the first small fraction of a second. Optical and NUV spectral calculations with dominant non-thermal ionization rates (Zharkova and Kobylinskii, 1993) do not seem to predict the observed blue continuum properties and Balmer jump strengths (e.g., Fig. 10). \nThe formation of the optical and NUV continuum radiation in a F13 electron beam model is illustrated in Fig. 21. The cumulative contribution functions are shown for λ = 3550 ˚ A, 4300 ˚ A, and 6690 ˚ A. The detailed analyses of emissivity sources and optical depths explain the emergent spectrum as Balmer and Paschen recombination \nradiation from optically thick layers (see also Kowalski et al, 2016; Kowalski, 2016, for further analyses of these models). The λ = 3550 ˚ A and 6690 ˚ A radiative fluxes are very optically thick and escape from the top of the condensation; the blue ( ∼ 4300 ˚ A) continuum flux is less optically thick and emerges from the condensation and deeper layers that have a lower electron density of n e ≈ 10 15 cm -3 and a lower gas temperature. The excitation by the highest energy electrons in the beam, and to a lesser extent, Balmer continuum backwarming, heat these deep layers and lead to an emergent spectrum that departs significantly from an optically thin hydrogen recombination spectrum from a homogeneous slab. Kowalski and Allred (2018) reported self-similar patterns in chromospheric condensation models (using a basic HI 37 pattern recognition algorithm) and parameterized the RHD atmospheres at especially interesting times in their evolution. The two main parameters are a reference column mass ( m ref ) and a reference temperature ( T ref ), which are indicated in the cartoon in Fig. 2(right). These parameterized models facilitate LTE estimates of the Balmer jump and optical continuum ratios over a large range of condensation column masses and chromospheric temperature gradients. A prediction from Kowalski and Allred (2018) is shown in Fig. 12 for T ref = 11 , 000 K and a range of m ref . The chromospheric condensation models can account for the trend over most of the impulsive-phase, flare-only Balmer jump ratios and optical continuum flux ratios in spectral observations. \nThere has been follow-up work by Kowalski et al. on the modeling of chromospheric condensations to further evaluate their role as the origin of cooler line and continuum radiation in flares. Here, we summarize some of the challenges and successes with the hypothesis. First, these model atmospheres evolve very quickly, and the emergent flux spectra around the Balmer jump rapidly change on short timescales. Thus, an exposure-time averaged model spectrum does not necessarily exhibit properties that are consistent with the observational constraints. An average condensation model that produces a continuum spectrum that resembles observations requires a very hard, δ ≈ 3, electron beam distribution (e.g., Kowalski et al, 2016, 2017b). In other cases, an average model spectrum has satisfactorily produced the broadband continuum shape and Balmer jump strength in two moderate-sized dMe flares with HST/COS observations (Kowalski et al, 2019b), and success has been achieved in an even broaderwavelength comparisons to a superflare event (Osten et al, 2016). There are also important theoretical issues with the huge current density and charge displacement due to an electron beam with a flux of 10 13 erg cm -2 s -1 that have not yet been addressed. \nA further severe challenge has been encountered in dense chromospheric condensation modeling of dMe flares, as detailed in Kowalski et al (2017b) and Kowalski (2022). Updated hydrogen emission line broadening treatments (Sect. 10) reveal that the optical depths and the ambient (thermal) charge densities in the model condensations are far too large to be consistent with the symmetric broadening in archival dMe flare observations. Namekata et al (2020) modeled the broad H α emission lines in a E > 10 33 erg, ∆ g ≈ -1 . 35 mag superflare from AD Leo. They used updated hydrogen broadening profiles in RADYN simulations and concluded that lower electron beam fluxes with hard power-law distributions produce optical depths and electron densities \nFig. 21 RADYN model of the low flare atmosphere after 2.2 s of heating by a high-flux, F13 (10 13 erg cm -2 s -1 ), double power-law electron beam (Kowalski et al, 2015). (top panel) The cumulative of the contribution function to the emergent intensity, I ( λ = 3550 ˚ A), shows that the chromospheric condensation becomes optically thick in the Balmer continuum. The emergent blue-optical ( λ = 4300 ˚ A) continuum intensity has larger contribution from the layers below the condensation, which is the large increase in gas density in the bottom panel. (Bottom panel) Several trajectories of nonthermal electrons are shown on top of the gas dynamic quantities (the initial kinetic energies are indicated, and the kinetic energy variations, E ( z ), are normalized from 0 to 1 on a linear scale). The initial temperatures and gas densities are shown as black dashed lines. A movie is available online. \n<!-- image --> \nin the low atmosphere that explain the broadening. The models with energy transport through only thermal conduction were largely discrepant (Fig. 22). \nSimilarly to Namekata et al (2020), the approaches in Kowalski et al (2017b), Kowalski (2022), and Brasseur et al (2023) employ semi-empirical forward modeling with the RADYN and RH codes. These studies use large low-energy cutoffs ( E c ≥ 85 keV), hard power-law indices ( δ = 3), and high energy fluxes (10 12 -10 13 erg cm -2 s -1 ) to heat the low chromosphere at log 10 m c / [g cm -2 ] ≳ -2 to gas temperatures hotter than 10 , 000 K (see also Kowalski, 2023). This approach results in optically thick continuum radiation at NUV and optical wavelengths and broad Balmer wings in the early impulsive and peak phase. However, an additional model component that is attributed to lateral spatial heterogeneity in the flare heating is required to simultaneously account for the narrow and broad components of the hydrogen Balmer emission line fluxes. The slow-rising Ca II K emission line flux is not yet explained with this approach. Large, low-energy electron beam cutoffs or some other energy transport mechanism (e.g., Kontar et al, 2012) that similarly results in a large fraction of the beam heating over log 10 m c = -2 . 2 to -1 . 2 (Kowalski, 2023) rather than in the upper chromosphere (log 10 m c ≲ -3) seems to be a promising avenue for explaining some of \nFig. 22 (Top) Model spectra of H α that were calculated with the RADYN code and analyzed in Namekata et al (2020). The larger electron beam heating rates reproduce the broad, symmetric Balmer lines in an AD Leo flare. Figure reproduced with permission from Namekata et al (2020). (Bottom) Echelle resolving power, flare-only spectra of the H α line during the decay of a large flare on YZ CMi. This figure demonstrates the phenomenological Voigt profile + Gaussian modeling that is used to isolate and characterize the asymmetries in the wings of the Balmer lines. In this spectrum, a Gaussian model is fit to the broad blue asymmetry; reproduced from Fig. 42(a) in Notsu et al (2023), with permission. \n<!-- image --> \nthe more challenging M dwarf flare observations. For example, a model with 2 × 10 12 erg cm -2 s -1 and E c = 500 keV (i.e., a high-flux, fully relativistic electron beam) adequately reproduces the spectra of the secondary flare events in the decay phase of the YZ CMi megaflare (Kowalski et al, 2010, 2013). These events produced an 'A star on an M star', such that the flare-only spectra resembled the main sequence A0 V star, Vega. The Balmer lines and Balmer continuum are 'in absorption'. The colors from two secondary events are shown in Fig. 12 as star symbols. Alternative explanations are summarized in Anfinogentov et al (2013). Kowalski et al (2013) claimed that some flares show a narrowing of the hydrogen Balmer lines in their impulsive-phase peaks that is consistent with a spatially unresolved component on the star that has the hydrogen Balmer lines in absorption. \nRelatively little attention has been given to the coronal and late phase predictions in the recent RADYN models of stellar flares. The RADYN models are generally viewed as simulations of the short, pulsed beam-heating events that sequentially light up along a two ribbon flare arcade, as in typical solar flare geometries (see Sect. 12). Ostensibly, the very hot T ≳ 50 MK coronal temperatures produced within a short time are expected to eventually cool down and shine in emission lines that probe cooler \ntemperatures (Hawley and Fisher, 1992; Aschwanden and Alexander, 2001). However, it is imprudent to speculate further on these issues without X-ray spectral synthesis because of the vast range of electron densities over the hot temperatures in the model corona and transition region. \n<!-- image --> \nFig. 23 ( Left ) Hydrodynamic modeling of a coronal explosion on CN Leo. The panel shows temperature, number density, and the emission measure weighted cooling function. The solid lines show the evolution at t = 0 , 2 , 4 , 8 , 10 s when a flare heating function was applied, while the dashed lines correspond to t = 13 , 16, and 19 s. ( Right ) Modeled X-ray light curve over the first 20 s of the model compared to XMM-Newton data of the early phase of the giant X-ray flare. Figures reproduced from Schmitt et al (2008) with permission. \n<!-- image --> \nSchmitt et al (2008) used the Palermo-Harvard adaptive regridding code (Peres et al, 1982; Betta et al, 1997) to model the hydrodynamic response of the giant Xray CN Leo flare in Fig. 11(a) over the first 20 s. They assumed a Gaussian heating function centered in the low corona and extended into the chromosphere. The atmospheric evolution is shown in Fig. 23. The flare transition region moves downward, and there is an increase in density that resembles a chromospheric condensation, but low-temperature spectral synthesis was not the goal of these modeling efforts (see Fuhrmeister et al, 2010, and Fig. 20 here). They successfully reproduced the X-ray luminosity evolution by multiplying the model by an area of 10 19 cm 2 . The model X-ray luminosity is shown in the right panel of Fig. 23. They explain the early phase X-ray light curve as a result of heating within the bottom parts of a flaring loop during the first 10 s, followed by a v ∼ 1000 km s -1 evaporation upflow that dominates the coronal emission after that. Compressional heating at the apex occurs later on as \nwell (see their dashed temperature profiles) and persists after the applied flare heating function is turned off at 10 s. The Palermo-Harvard code has modeled longer heating durations to simulate X-ray flares from the M dwarf Proxima Centauri (Reale et al, 2004) and the active giant HR 9024 as well (Testa et al, 2007; Argiroffi et al, 2019). Reale et al (2004) model the second X-ray flare in the decay of the event in Fig. 18 here as the first in a series of loops. They develop a two-phase heating model whereby the total energy input consists of energy deposition near the loop apex and near the low-altitude coronal footpoints.", '9.1 An Introduction to Radiative-Hydrodynamic Flare Modeling': "Radiative-hydrodynamic flare modeling consists of solving the coupled, non-linear conservation equations for mass, momentum, energy, charge, radiative transfer, level populations, and grid motion. The atmosphere begins in a steady-state, and flare heating is introduced through a term in the energy equation, Q flare . In this section, I follow the general framework and modeling assumptions employed in the RADYN flare code, which has unique capabilities for stellar flare applications (e.g., radiative transfer and non-solar gravity). \nA conservation/continuity equation for a volumetric quantity w ( w hatever cm -3 ), which is a function of space and time, can be written (in laboratory frame coordinates) as \n∂w ∂t + ∇· ( w v ) = 0 (14) \nin three spatial dimensions (if the time-derivative is 0, then the equation leads to the steady-state solution), where v is the macroscopic gas velocity vector, w v is the flux 38 of the quantity in w , and we have excluded a diffusion term, -D ∇ w (Weinberg, 2021), in the parentheses. The second term is the advection term which is a gradient of the flux (while in more than one spatial dimension, it is the divergence of the flux). From here on, we discuss the 1D form of Eq. 14 and consider the spatial coordinate z ('height') along a hypothetical magnetic loop extending from below the photosphere to the loop apex in the corona. At the loop apex, the z direction is parallel to the stellar surface, and the surface gravity in the momentum equation decreases accordingly. In 1D flare modeling, one assumes that flows and energy transfer are aligned with the magnetic field, meaning that the magnetic pressure is assumed to be much larger than the gas pressure everywhere at all times (which may be appropriate for the center of a strong magnetic flux tube). Note also that the typical 1D form of the continuity equation in Eq. 14 usually assumes a constant flux tube cross-sectional area; for a non-variable magnetic field along the flow direction, the adjustments that must be made to account for funneling effects are written in Emslie et al (1992) and Reep et al (2022). \nThe conservation equations are solved simultaneously with an adaptive grid equation (Dorfi and Drury, 1987) using a semi-implicit numerical scheme that allows for timesteps that are much longer than the Courant step, which is very small in the transition region during flares. Implicit radiation-hydrodynamics essentially consists of linearization and iteration (i.e., multi-dimensional Newton-Raphson) of the conservation equations with a five-point stencil monotonic upstream scheme (van Leer, 1977, 1979) to ensure numerical stability of advected quantities around steep gradients, such as shocks. The spatial derivatives are written as finite differences and are centered in time between the current and next time (semi-implicitly) during the linearization of the Taylor-expansion about the iterative guesses. We do not discuss the details here and instead direct the interested reader to Kneer and Nakagawa (1976), McClymont and Canfield (1983), Carlsson (1998), Abbett (1998) for general overviews 39 . We especially recommend the lucid and comprehensive lecture on the topic in Dorfi (1998). \nThe Eulerian (fixed space) forms of the conservation equations that are solved in RADYN flare modeling are written in Allred et al (2015). Here, I highlight the conservation of internal energy density, e , which is the first law of thermodynamics (i.e., dU + PdV = Q ) connecting several of the critical ingredients in flare RHD: \n∂e ∂t + ∂ ( ev z ) ∂z +( P + q ) ∂v z ∂z = -∂ ( F c + F r ) ∂z + Q flare + ∑ n Q n (15) \nwhere P is the thermal gas pressure from all species s ( P = k B T ∑ s n s ) and enters into the PdV work term, v z is the gas velocity along the loop axis ( z direction), q is a viscous stress term added to aid in numerical stability, F c is the thermal conduction flux (temperature diffusion), F r is the radiative flux, and Q n can be any number of external heating terms, which can be flare-related (e.g., beam-generated electric fields) or non-radiative sources to keep the photosphere and corona hot in the preflare equilibrium state. The form of thermal conduction is taken from Smith and Auer (1980) and Fisher et al (1985c). The internal energy density is the sum of thermal and excitation energy densities: \ne = 3 2 k B T ∑ s n s + ∑ s,i n s,i χ s,i (16) \nwhere n s,i is the population level density of species s and level i and χ represents atomic excitation energies ( χ = 0 for the ground state). Equation 15 is derived by starting from the continuity equation (setting Eq. 14 equal to all Q terms, instead of 0) for the energy density, ε , which is the usual (e.g., Thorne and Blandford, 2017): \nε = ρ ( 1 2 v 2 z + e ' +Φ ) (17) \nwhere e ' is the internal energy per unit mass ( e = ρe ' ), Φ is the gravitational potential energy, gz . Magnetic energy density is neglected. The energy flux is \nF = ρ ( 1 2 v 2 z + h +Φ ) v z (18) \nwhere h is the enthalpy per unit mass ('specific' enthalpy; h = e ' + P ρ ). After substituting the continuity equations for mass density and linear momentum density, some algebra and reduction, the form of Equation 15 is readily obtained 40 . \nThe equation of radiative transfer is solved simultaneously with the other equations through complete linearization and a low-order Feautrier method that is stable in the presence of steep gradients and shocks (M. Carlsson, priv. communication 2022). The monochromatic specific radiative intensity, I ν ( z, ν, µ ), at frequency ν , at an angle θ = cos -1 µ from the atmospheric normal, at a height z , is assumed to be in steadystate ( ∂I ν ( z,ν,µ ) /c ∂t = 0 where c is the speed of light; cf. Section 11.2 of Hubeny and Mihalas 2014) within each hydrodynamic time-step. The solution to the intensity at each depth point is integrated over angle using a Gaussian quadrature numerical weighting (Chandrasekhar, 1960), and then it is integrated over frequency to give the net radiative flux. The gradient in the energy equation (Eq. 15) thus gives the radiative heating or cooling rate at each time-step in the flare simulation. Due to this coupling, the time-history of the atomic level populations, ionization, rates, and radiative transfer are critical in the self-consistent energy balance, pressure gradients, and hydrodynamics in a flare simulation.", '9.2 The Flare Heating Term': "The flare heating term, Q flare , in the energy equation (Eq. 15) may be a sum from a large number of possible sources: nonthermal electrons, nonthermal protons, Joule heating, shocks in the reconnection process, and Alfven waves. Here, we assume the major source of heating in stellar flares is through nonthermal electrons, as is widely accepted in and borrowed from the solar flare analogy. The radio gyrosynchrotron radiation from mildly relativistic electrons in stellar flares also supports this approach (Sect. 11). The following is a synopsis of several relevant points from the collisional energy loss theory of Emslie (1978) and a unified model calculation presented in Allred et al (2015). Additional formalism about the theory of energy loss on ionized species and neutrals is contained in Trubnikov (1965), Chambe and Henoux (1979), Emslie (1981a), Fang et al (1993), Hawley and Fisher (1994), Mott and Massey (1949), Snyder and Scott (1949). We consider particle velocities with magnitude, v , and three directional components (whereas in the previous section, one-dimensional bulk velocities of the ambient/thermalized gas were denoted as v z ). \nA nonthermal (beam) electron penetrating a plasma deposits heat thus affecting the internal energy of the plasma by increasing the temperature, excitation, and ionization of the plasma constituents. A beam particle is assumed to begin with a kinetic energy that is much greater than the average kinetic energy of the free electrons in the background gas or plasma (thus, the background plasma is a 'cold target'); the beam particle is then followed until it decreases to an energy that is close ( ∼ 2 . 5 k B T ) to the energy of the thermal distribution, at which point it joins the background/thermal pool of particles. Energy loss from the beam electron occurs through Coulomb collisions, and the interactions with all particles in the plasma require an integral that, when evaluated (with somewhat artificially imposed distance limits), is known as a \nCoulomb logarithm (e.g., Benz, 2002). The Coulomb logarithms give a sense of the relative importance of distant to nearby interactions with the beam particle. A beam particle loses energy to ambient (thermal) electrons and protons, with an associated Coulomb logarithm 41 Λ. The integral that describes collisions with ambient neutral atoms also has an associated Coulomb logarithm, Λ ' , for each neutral species. For a typical beam electron, values for Λ and Λ ' are ≈ 20 and 8.5, respectively. Including the ionization fraction of a hydrogen gas, x , the fraction of energy loss that goes into the neutrals is (Ricchiazzi and Canfield, 1983) 1 -x x Λ+(1 -x )Λ ' . For a 50% ionized gas, the energy loss is about a factor of three more efficient on the ionized component (i.e., the ambient thermal electrons). \nThe pitch angle of the beam particle changes through scattering, which can be elastic (for collisions with ambient protons, electrons, and neutrals) or inelastic (for collisions with neutrals). The momentum changes in the direction parallel to the trajectory of the beam particle must include an additional Coulomb logarithm, Λ '' , for elastic scattering off of neutrals (e.g., collisions with bound electrons that do not undergo atomic transitions). Distribution-averaged approximations to the Coulomb logarithms are given in Emslie (1978) and Hawley and Fisher (1994); Allred et al (2015, 2020) gives the energy-dependent, relativistic formulae. Then, the total differential kinetic energy change per unit path length [keV km -1 ] of an electron beam particle interacting with hydrogen and ambient electrons in a partially ionized gas is, following from Eq. 24a of Emslie (1978), \n( dE dz ) hyd = ( dE dz ) hyd , I + ( dE dz ) hyd , N (19) \n= -10 5 2 πe 4 µE × a 2 ( x Λ ee +(1 -x )Λ ' ) γn hyd (20) \nwhich is the sum of the ionized (I) and neutral (N) components of energy loss in a hydrogen gas. We ignore the ∼ 10 3 less energy loss on ambient protons, E is the kinetic energy of the electron beam particle in keV, a = 1 . 602 × 10 -9 keV erg -1 , µ specifies its pitch angle ( µ = 1 for beamed along hypothetical magnetic field), x is the ionization fraction of the hydrogen target, n hyd is the number density of neutral hydrogen atoms plus ambient protons in the target, γ is the relativistic gamma factor of the beam electron, and all of the rest of the units are in cgs . As an example, if x = 0 . 75, n hyd = 1 . 7 × 10 13 cm -3 , T ≈ 10 4 K, which are representative conditions at the top of a model M dwarf chromosphere, then dE dx = -0 . 1 keV km -1 for E = 40 keV. Additional energy losses on helium and other gas constituents can be included. \nAssuming a fully ionized, uniform hydrogen plasma, Eq. 19 can be integrated from the electron's initial energy E = E 0 to E = 0 to find the stopping length of an electron, L stop ∝ E 2 0 n -1 e , which defines an effective absorption cross section, σ , according to σn e L stop = 1. Higher energy particles penetrate deeper because they have more energy to lose and because they spend less time losing energy per collision. The energy loss \nFig. 24 A pre-flare and flare model (from the RADYN code) of the temperature distribution in the low atmosphere compared to Coulomb stopping depths of beam particles. For reference the (preflare) photosphere and upper photosphere corresponds to log 10 m c > 1 on this plot ( m c is the column mass density in units of g cm -2 ). The collisional stopping depths of mono-energetic beam particles are shown on the right using Equation 19. The kinetic energy decrease of an electron with injected energy of 50 keV is shown for each state of the atmosphere; these two curves are on a linear scale and normalized to 1.0 at the top of the loop (where they are injected) outside the range of the plot on the right. An electron gets stopped at higher altitudes when the chromosphere has experienced a large amount of hydrogen ionization; as the atmosphere further evolves, the lower energy electrons in the beam can be stopped in the evaporation flows ( cf. Fig. 21), and all the rest of the beam particles except for the highest energy electrons are stopped in the chromospheric condensation (Fig. 2(right)). Note, these 'cold-target' stopping depth approximations for the protons are not accurate significantly below ≲ 1 MeV at which the warm-target transport treatment becomes essential (Allred et al, 2020). \n<!-- image --> \nrate per electron is, dE dz dz dt = dE/dt = -2 πe 4 n e v Λ /E which can be integrated over a uniform plasma to give a timescale for energy loss, τ ∝ E 1 . 5 /n e at non-relativistic energies. Note, this is the same proportionality in the estimate for scattering timescales out of a magnetic trap (Sect. 3). \nThe heating from a power-law distribution of electrons and protons is analytically solved in Emslie (1978) following Brown (1973) and Lin and Hudson (1976) in an integral from, Q beam ( m c ), which is a function of column mass over an atmosphere with arbitrary, constant ionization fraction. Hawley and Fisher (1994) extended the formulae to atmospheres with non-uniform ionization fractions. The analytic formulae \nfor Q beam were used in RADYN simulations of stellar flares up through the models in Kowalski et al (2015, 2016). \nAn alternative method is to numerically solve the distribution function, f ( z, µ, E, t ), of the nonthermal electrons (Leach and Petrosian, 1981; McTiernan and Petrosian, 1990), and evaluate the spatial gradient/divergence. This is the FokkerPlanck solution (Rosenbluth et al, 1957), which tracks the phase space evolution due to systematic decrease (drag) and diffusion of the kinetic energies, E , and of the cosines of the beam pitch angles, µ . Generally speaking, the Fokker-Planck solution is required when some or all of the following are important: time dependence of the beam propagation, diffusion of pitch angle and energy, external forces (e.g., magnetic mirror, synchrotron losses), and beam feedback effects such as return currents and plasma waves (Hamilton and Petrosian, 1987; Mauas and G'omez, 1997; Zharkova and Gordovskyy, 2006; Allred et al, 2015). We refer the reader to the comprehensive description of the collisional Boltzmann transport equation in Trubnikov (1965) and the application of the Fokker-Planck solution to flare modeling in Allred et al (2020). Other helpful references are Thorne and Blandford (2017) and Boyd and Sanderson (2003). In current RHD simulations, numerical limitations necessitate a steady-state solution ( ∂f ∂t = 0) at each hydrodynamic time-step, thus giving the flare heating rate as: \nQ flare ( z ) = Q beam ( z ) = d dz ∫ µ ∫ E µvEf ( z, µ, E ) dEdµ (21) \nwhich is the spatial gradient of the beam flux at height z in the atmosphere.", '10 Models of Chromospheric & Transition Region Flare Emission Line Broadening (& Asymmetries)': 'The emission line properties (strength and shapes) during flares encode much of the evolution of the atmospheric physics, which may be quite dissimilar to the density, optical depth, and velocity regimes in the atmosphere before a flare 42 . The interpretation of flare emission lines, especially in the chromosphere, in general requires forward modeling with non-equilibrium physics. Thus, there is still much we do not know about the origin of certain well-observed changes in the spectral line profile shapes during flares. The efforts to make headway in this area have been stymied because, further, not all flares show similar behavior in the same emission line ( cf. in solar flares, H α shows a rather wide range of empirical properties; Canfield et al 1990). Detailed line shape measurements are further obfuscated by the heterogeneity of the flare source and long exposure times, which are generally required of echelle observations. \nThis section is divided into two parts. First, we categorize the line broadening mechanisms that are thought to be important in the interpretation of transition-region and chromospheric line widths and asymmetries during stellar flares (Sect. 10.1). Then, \nin Sect. 10.2, we summarize the symmetric pressure broadening of hydrogen lines in more detail. Observations and modeling are reviewed together in this section.', '10.1 An Overview of Broadening Sources': "There are many types of possible sources of chromospheric and transition region emission line broadening in stellar flare spectral observations. Microscopic broadening occurs over scales that are much smaller than the optical depth, and each process is generally assumed to be statistically independent. Thus, the total microscopic broadening enters the line opacity after a convolution of all the respective probability density functions. Microscopic broadening is largely a source of symmetric broadening (to first order). Macroscopic broadening sources can affect the line opacity though the mean intensity, J ν , but this term typically refers to the broadening from spatially unresolved, inhomogeneous sources in the plane transverse to the observer. \nHere, we summarize the most important sources of micro- and macroscopic broadening that are relevant to stellar flare chromospheric, transition region, and photospheric lines. We exclude rotational and Zeeman broadening, which affect the broadening of lines in active stars in their quiescent states. Note that microscopic broadening sources are modeled with an ambient, thermal distribution of perturbers even when they are considered separately from thermal Doppler broadening. In Fig. 25 we show several profiles for the hydrogen Balmer γ and α lines, which complement the descriptions throughout this section. For each type of broadening, we state the probability distribution that most closely describes the perturbation magnitudes. \n- · H I + p + /e -: 'Stark broadening', or electric pressure broadening of hydrogen, hydrogen-like ions, or Rydberg atoms by ambient charged particles (electrons, protons, ions) in the flare chromosphere, which is partially ionized by definition. The perturbations are caused by both electron collisional 'damping' with a Lorentzian probability distribution and quasi-static splitting of energy levels from ambient positive charged protons (and ions) with a Holtsmark-like 43 probability distribution. The splitting of energy levels is due to the first-order/linear Stark effect which is discussed in more detail in Appendix D. Optically thin emission line profiles from a homogeneous slab exhibit a Holtsmark power-law slope ( log 10 I ∝ log 10 | ∆ λ | -5 / 2 ) in the far wings , if non-ideal effects are not important.\n- · LEPT : Oks and Gershberg (2016) discuss the role of broadening of hydrogen lines due to chromospheric plasma turbulence. Power-law electron beams drive a cospatial drifting return current, whose speed may exceed the threshold for the ion-acoustic / current instability and therefore lead to the development of ion-acoustic turbulence in the chromosphere (E. Oks, priv. communication 2022). The low-frequency electrostatic plasma turbulence (LEPT) is broadband, and it is quasi-static compared to radiative timescales. LEPT is treated as a statistically independent broadening process to the ionic microfield broadening and is therefore convolved with the Holtsmark or Hooper distribution of field strengths that enters into the nominal hydrogen line broadening (above). Oks and Gershberg (2016) analyzed several archival dMe \nFig. 25 Optically thin spectral line profiles from the TB09+HM88 pressure broadening theory for Balmer H γ (top two panels) and H α (bottom two panels) at n e ≈ 10 15 cm -3 . These are identical to the Vidal et al (1973) (VCS73) calculations at this density, and they do not include thermal Doppler convolution in this figure. The 2nd and 4th panels show the slopes, η , calculated for each of the profiles in 1st and 3rd panels. Lorentzian profiles with scaled damping widths are overplotted for comparison. The slope of the H α pressure broadening profile is between the Holstmark and Lorentzian because of the higher importance of collisional broadening from electrons for this transition. The higher order lines, including H γ , are more obviously dominated by the quasistatic (Holtsmark-distributed) perturbations ( cf. Fig 3 of Vidal et al, 1971), are less optically thick, are closer to LTE because of less scattering in this line, and are less affected by thermal broadening (the FWHM of H α is affected by thermal Doppler broadening even at such high densities; 1 /e Doppler widths are indicated by horizontal red lines). The 'self-broadening' of hydrogen is relatively less in H γ and in the higher order lines, but it is important when the atmosphere is largely neutral (Barklem et al, 2000). The Balmer H γ line is also less affected by thermal, T = 10 4 K ion-dynamic effects Stehle (1994): we show an area-normalized (rather than peak-normalized) profile from Stehl'e and Hutcheon (1999), which includes additional Lorentzian damping near line-center from non-quasistatic components of the perturbing ionic field. Also shown are emergent spectral intensities (dashed-dotted black), with Doppler broadening included, from slabs with a physical depth of dl = 500 m to demonstrate the large differences that result from curve-of-growth-like optical depth effects. 86 \n<!-- image --> \n- \nflare observations of the Balmer series. Without detailed wing shapes, the values of FWHM/ λ 0 from the data are degenerate with either electron densities of > 10 15 cm -3 or rms LEPT fields of 10 -30 kV cm -1 . Lower electron densities combined with LEPT development in the chromosphere are favored by considering the Balmer flux decrements or the last resolved Balmer line. Profiles dominated by LEPT have an exponential dependence in the far wings, I ∝ exp ( -k | ∆ λ | γ ) , γ ≈ 1 . \n- · non-H + e -, non-H + H : Collisional broadening of non-hydrogenic lines due to collisions with ambient electrons (the quadratic Stark effect) or with neutral hydrogen atoms (van der Waals broadening or the more complete ABO theory; Anstee and O'Mara, 1995; Barklem and O'Mara, 1998). The quadratic (second order) electron impact damping of resonance ion lines, such as Ca II H and K, causes a small wavelength shift and a broadening that is typically 10 3 smaller than the linear effect in hydrogen. The passage of H atoms is relatively fast and so this is a damping. nonH + non-H : Collisions among like atoms or ions results in resonance broadening. This is an exchange and thus described as a lifetime broadening effect (Bohm 1951). Optically thin emission line profiles have a Lorentzian shape (or Voigt if a Doppler broadening is included).\n- · HI + HI : Collisional damping of hydrogen lines from transient perturbations from other hydrogen atoms. van der Waals and resonance forces, collectively known as self-broadening, drop off rapidly with distance and so this interaction has a short duration. The neutral fraction of hydrogen is too small, even at ∼ 50%, in the flare chromosphere for self-broadening to be significant in comparison to broadening by charged particles, unlike in the quiet Sun photosphere (Barklem et al, 2000). Optically thin emission line profiles have a Lorentzian shape (Voigt if thermal Doppler included).\n- · Thermal Doppler broadening is important for line opacity near λ o but decreases precipitously according to the Gaussian distribution. Note that one Doppler width is ≈ 0.2 ˚ A for H γ at T = 10 , 000 K. Thus thermal Doppler broadening cannot explain broad wing emission unless there are very high temperatures T ≫ 10 , 000 K. Optically thin emission line profiles exhibit a Gaussian profile.\n- · Microturbulence 44 , or nonthermal Gaussian Doppler broadening. An ad hoc microturbulence parameter is often used to account for missing broadening sources when all other known broadening sources have been exhausted. This parameter enters into the opacity (extinction coefficient) through a convolution of a Gaussian with the thermal Doppler core. (Certain values of this parameter may propagate through decades of stellar models with little critical reconsideration of their physical meaning or necessity.) For recent use in solar flare modeling of Fe II and Mg II lines, see Kowalski et al (2017b) and Zhu et al (2019), respectively. Non-Gaussian microturbulence is discussed often in the interpretation of solar flare transition region lines (Dud'ık et al, 2017), but the densities are rather large and the optical depths are non-negligible even in transition region lines of quiescent state of active stars (e.g., Mathioudakis et al, 1999). Osten et al (2005) report nonthermal broadening \nvelocities of several tens of km s -1 in transition region flare lines with a temperature dependence, such that N V ( T > 10 5 K) does not exhibit excess broadening. \nDoes the microturbulence parameter represent bona-fide gas-dynamic turbulence (eddies, vortices, random gas motions) in the footpoints of a flare loop (e.g., Bornmann, 1987), and if so, through which fluid instabilities (e.g., Kelvin-Helmoltz, Rayleigh-Taylor) does the flare turbulence occur? The turbulence hypothesis can be investigated by considering the energetics inferred through spectral lines and the emission line flaring area on the star (see Sect. 12). Hawley et al (2007) reported very broad, symmetric wings of Mg II h and k emission that had a small, bulk blueshift during a NUV flare observed by the Hubble Space Telescope/STIS. The wings would require electron densities as high as 10 17 cm -3 if due to collisional broadening. A Gaussian fit to the broad wings gives the line-of-sight velocity component root-mean-square (rms), σ los , and the kinetic energy of the chromospheric turbulence is \nKE = 3 2 σ 2 los ρ chrom ∆ z chrom A flare (22) \nwhere A flare is the projected area on the star (Section 12) that produces the emission line flux at Earth. In this flare, the turbulence would be highly supersonic (200 km s -1 ) and its kinetic energy would be several orders of magnitude larger than the radiated NUV energy. \n- · Opacity broadening, optical depth effects : It has long been known that simultaneous modeling of electron density and optical depth is required to interpret the hydrogen lines properly in stellar and solar flares (e.g., Drake and Ulrich, 1980). Optical depth effects cause a type of macroscopic broadening in chromospheric flare lines in the emergent spectral lines from stellar atmospheres. Sometimes this is loosely referred to as opacity broadening, which is described in Wood et al (1996), Rathore and Carlsson (2015), and Hansteen et al (2023). Essentially, opacity broadening follows from the Eddington-Barbier relation for a depth-dependent source function over optically thick wavelengths of a line 45 . If the source function decreases in the outer layers a central reversal can form due to an absorbing layer for very optically thick lines; this is also sometimes called 'self-absorption'. An optical depth broadening effect that is closely related to opacity broadening occurs in the enhancement of the emergent intensity at wavelengths that correspond to the transition from a formation over very large optical depths (where there is opacity broadening, proper) to smaller optical depths in the far wings. This is similar to the broadening in textbook illustrations of the curve-of-growth of a spectral line, whereby the Doppler core saturates at the local source function, and the Lorentzian or Holtsmark wings are very prominently in emission but are not close to saturation. The core saturation and wing enhancements of H γ and H α are demonstrated in Fig. 25 for a slab thickness of dl = 500 m and electron density n e = 10 15 cm -3 . An analysis \n- of these effects in a heterogeneous, non-equilibrium model chromospheric condensation is presented in Kowalski et al (2022). Namekata et al (2020) analyze in detail the self-absorption and pressure broadening in stellar flare H α emission lines.\n- · Red-wing and blue-wing asymmetries cause an increase in the breadth of a spectral line, especially if the line is spectrally and/or spatially unresolved. In stellar flares, red- and blue-shifted components of H α have maximum line-of-sight velocity extents of ≈ 100 -300 km s -1 from the rest wavelength (Vida et al, 2019). Reports of line asymmetries in the literature attribute the asymmetries to a variety of phenomena that involve chromospheric mass motions: beam-generated impulsive phase chromospheric condensations, coronal rain (a 'coronal condensation') from post-flare loop plasma draining and cooling to transition region and chromospheric temperatures, cool filament eruptions of plasma, absorbing downflows, and chromospheric evaporation of cool material into magnetic loops (see Wu et al, 2022, for an overview). Many stellar flare observations, however, with echelle resolving power have rather long exposure times ( t exp ≈ 15 minutes), and some are serendipitous single-exposure observations from exoplanet radial velocity monitoring. Additionally, optical depth effects and other microscopic broadening sources are important for interpretation. Optically thin emission lines can exhibit a velocity 'smearing' in the emergent spectra, simply due to an integration of the emissivity over the velocity gradient along the line-of-sight, and optically thick lines are complicated by opacity broadening (Namekata et al, 2020; Wu et al, 2022). \nThere are many reports of line asymmetries in hydrogen, helium I, and metallic lines throughout the UV, optical, and NIR that have been interpreted with these phenomena. The spectral observations consist of a bright emission line component around the rest wavelength and a fainter asymmetric wing component, or more than one wing component. Most modeling of these lines uses linear superpositions of Gaussians to characterize the shifts and widths of each component. Wu et al (2022), Namekata et al (2022b), and Namizaki et al (2023) use Voigt/Lorentzian profiles to isolate a Gaussian fit to a red-shifted components in large stellar flares. The bottom panel of Fig. 22 demonstrates this method, which effectively isolates a ≈ -100 km s -1 blueshifted component in the H α line flux during the decay phase of a whitelight flare (Notsu et al, 2023). Eason et al (1992) and Fuhrmeister et al (2008) report broad H α lines that are blueshifted during the rise and peak spectra during flares on UV Ceti and CN Leo, respectively. In the event on CN Leo, the blueshifts were -10 to -20 km s -1 and the Gaussian FWHM values were 4.8 ˚ A; the event on UV Ceti produced a similar width but a larger blueshift of -70 km/s. Fuhrmeister et al (2011) presents detailed line profiles at high-time resolution during flares on Proxima Centauri with VLT/UVES. Vida et al (2016) and Vida et al (2019) discuss blueshifts observed across the Balmer series with some temporal resolution. Notsu et al (2023) present a large survey of echelle spectral observations of the H α and H β lines in M dwarf flares; the spectra had relatively short exposure times and were complemented with simultaneous optical broadband photometry, allowing the changes in the profile shapes to be characterized in the context of the impulsive and gradual phases of the white-light response (e.g., Fig. 22(bottom)). Honda et al (2018) report on high-time cadence monitoring of EV Lac and present persistent \nblue-wing asymmetries through the entire duration of a flare (see also Maehara et al, 2021). Namekata et al (2022c) discuss a transient blueshifted H α absorption feature, while Namekata et al (2022b) analyze broad H α emission lines with redshifted components, in flares on the young, solar-type flare star EK Dra, and Lalitha et al (2013) analyze the redshift evolution of a broad H α line component in a flare on the K0Ve star AB Dor. Gunn et al (1994b) discuss blueshifted emission components in AT Mic. Of course, the location on the stellar disk is generally unconstrained in stellar flares, and the inferred velocities are not corrected for line-of-sight effects for field-confined flows, contrary to what is possible for solar flare observations (Brosius and Inglis, 2018). \nFlux-weighted line-center redshifts in Si IV and C IV during and after the impulsive phase of several dMe flares were reported in Hawley et al (2003). Chromospheric-line, red-wing emission in H α , Na I D, and He I 5876 ˚ A was described in Fuhrmeister et al (2005). These observations were tested against scaling relations (e.g., Eq. 3) from analytic models of impulsive-phase chromospheric condensations (Fisher, 1989). Late-phase redshifts, on the other hand, are usually interpreted as condensations that are analogous to the draining of cool material from the post-flare loops on the Sun. Wu et al (2022) discuss evidence from the time-evolution of the broadening of the H α line that could favor beam-generated condensations in the late-phase instead (which we note would be more in line with the multi-thread RHD modeling of the continuum radiation in the decay phase of large flares; Sect. 12, Fig. 30). Fuhrmeister et al (2008) and Kanodia et al (2022) report on red wing enhancements in He I 7065 ˚ A and 10830 ˚ A emission lines, respectively, in two flares. Fuhrmeister et al (2020) show remarkably broad He I 10830 ˚ A wings during a flare on the dM3e star EV Lac; correspondingly broad H α profiles were analyzed in Fuhrmeister et al (2018) and consist of a spectrally unresolved red wing satellite component. Loyd et al (2018b), Froning et al (2019), and France et al (2020) report redshifts in other transition region lines, such as C II, from low-mass stellar flares. Linsky and Wood (1994) and Ayres (2015b) analyze the profile variations in FUV lines during flares on AU Mic and EK Dra, respectively. The optical Balmer lines have been decomposed into 'separate' broad and narrow components, which exhibit clearly different Doppler shifts in some flares (Houdebine, 1992; Houdebine et al, 1993b; Fuhrmeister et al, 2008; Kowalski, 2022). Extreme blueshifted (Houdebine et al, 1990) and redshifted (Woodgate et al, 1992; Bookbinder et al, 1992) emission has been reported out to several 1000 km s -1 in chromospheric and transition region lines. Koller et al (2021) discuss blueshifts and redshifts in a large sample of galactic field stars at larger distances and over a wide range of spectral types. \n- · Macroturbulence is a term that refers to a superposition of several Doppler-shifted profiles that originate from sequentially-ignited, neighboring flare loop footpoints, as in a solar arcade. The superposition is intended to emulate spatially unresolved flows, which are termed 'directed mass motions' in the literature. The unresolved flows may be preferentially blueshifted, preferentially redshifted (Rubio da Costa and Kleint, 2017), or equally blueshifted and redshifted (Doyle et al, 1988a). To generate symmetric wings, the emergent radiation from downflows (e.g., a condensation) would have to precisely balance the emergent radiation, and thus the optical \ndepths, from the upflows (e.g., evaporation) in neighboring loops. This precarious balance would have to be maintained over long durations, as for typical exposure times of 60 -300s. The lineshapes may take on a variety of symmetric or asymmetric forms that depart from Holtsmark and Lorentzian wing profiles.", '10.2 Symmetric Wing Broadening of H Lines': "The symmetric (or very nearly symmetric) wing broadening of hydrogen Balmer lines is a remarkable property of stellar flares. Full widths at 10% maximum in emission line spectra are as large as ≈ 15 -20 ˚ A (Hawley and Pettersen, 1991; Namekata et al, 2020). The source of the symmetric broadening has been debated as a superposition of directed mass motions (macroturbulence), gas turbulence (as represented by a microturbulence parameter), and pressure broadening due to the Stark effect (Sect. 10.1). Doyle et al (1988a) and Eason et al (1992) argue against the Stark effect in the large dMe flares that they analyze by comparing to the optically thin wing predictions of Holtsmark profiles and the Vidal et al (1973) profiles for electron densities of n e ≈ 10 15 cm -3 (see also Houdebine et al, 1993a,b; Gunn et al, 1994a). The better fits with multiple Gaussian components in these studies support the hypothesis of some type 46 of supersonic laminar- or turbulent-emitting flows. The latter corresponds to a microturbulence parameter of 50 km s -1 around line center and 150 -600 km s -1 in the wings. Montes et al (1999) present similar two-component Gaussian fits to broad and narrow components in H α and H β in flares from the K dwarf LQ Hya; the broad components show relative shifts with respect to the narrow components and are interpreted in terms of various types of unresolved mass motions. \nThe alternative hypothesis for the source of the broad wings of hydrogen lines is pressure (linear Stark) broadening from ambient charged particles, which imprints unique signatures in the observations. Namely, the pressure broadening of hydrogen produces larger widths for larger n j within a series (e.g., within the Balmer, Paschen, or Lyman series) in the absence of optical depth effects and heterogeneities (here, j refers to the upper level of a transition). However, the smoking gun signature is generally thought to be broader hydrogen emission line wings compared to nearby resonance lines of metallic ions, such as Mg II and Ca II, which experience a much smaller amount of broadening due to collisions with charged particles (Dimitrijevi'c and Sahal-Br'echot, 1992; Dimitrijevic and Sahal-Brechot, 1993). Hawley and Pettersen (1991) and Garc'ıa-Alvarez et al (2002) present much broader Balmer series lines in comparison to the Ca II K emission line (see Fig. 4 of Garc'ıa-Alvarez et al, 2002) in the impulsive phases of two large dMe flares (see also Lalitha et al, 2013). \nWe show two remarkable examples of the hydrogen broadening in stellar flare spectra in Fig. 26. The spectral broadening and emission line shape differences among Ca II K, H, and Balmer H ϵ ( n j = 7 → n i = 2) are clearly evident in these echelle spectral observations. The top panel shows a serendipitous flare spectrum ( t exp = 30 min) from the inactive dM3.5-4 star Gl 6l 699 (Barnard's star) that was studied in detail and modeled with the RADYN and MULTI codes in Paulson et al (2006). The bottom panel displays a sequence of spectra during a flare from the dM4.5e YZ CMi \nat the same times 47 as shown for the H γ spectra in Fig. A.1, top panel, of Vida et al (2019). The inactive M-star flare shows some qualitative similarities to the line broadening in the spectrum of the decay phase of the continuum in the YZ CMi flare. These similarities are remarkable given that the stars have very different ages ( > 7 Gyr vs. < 500 Myr, respectively) and quiescent magnetic activity levels. The impressive FUVflaring activity of Barnard's star has been reported recently in France et al (2020). A detailed analysis of the H9 Balmer line was presented for the large flare on CN Leo in Fuhrmeister et al (2008). Broadened Paschen lines have been reported in the decay phase of a large flare from the very low mass star VB10 (Kanodia et al, 2022). Similar comparisons are, in principle, possible with the broadening of the high-order Paschen series and the Ca II infrared triplets (Neidig and Wiborg, 1984), for which opacity broadening is perhaps less of a concern in comparisons to Ca II H and K.", '10.2.1 Recombination Edge Modeling': "Accurate interpretation of stellar spectra from EUV through NIR wavelengths depends on modeling how the wings of the hydrogen lines overlap and merge near bound-free ionization limits (Hubeny et al, 1994). Accurate modeling, which should lack any sharp edge feature, opens up diagnostic power from the rich spectral region at wavelengths just longward of a recombination limit. For example, the Inglis-Teller relation is (Inglis and Teller, 1939; Rutten, 2003) \nlog 10 n e = 23 . 2 -7 . 5 log 10 n Balmer max (23) \nwhich has commonly been used to determine electron density from the last discernible Balmer emission line near the Balmer recombination limit (Kurochka and Maslennikova, 1970; Hawley and Pettersen, 1991). Besides stellar flares, many types of astrophysical and laboratory phenomena also lack discontinuities in spectra around the expected location of hydrogen recombination edges (Herczeg and Hillenbrand, 2008; Esteban et al, 2004; Meigs et al, 2013; Shull et al, 2012). In solar flares, features such as a 'blue continuum bump', a shift of the edge to redder wavelengths, or a completely flat continuum have been reported in observations (Neidig, 1983; Donati-Falchi et al, 1985). The complete lack of any type of edge feature, and a lack of a jump in flux across the expected edge location, is ostensibly an indication of blackbody radiation. \nHowever, the lack of an edge is also expected from recombination theory that includes non-ideal, level dissolution. The fundamental question pertaining to stellar flares is thus if there is enough instrumental blending of the wings and/or overlapping, linearly-superposed wing opacities to account for the lack of the observed discontinuities. A remarkable example of stellar flare spectra with no edge feature is shown in Fig. 11(a). Instead, these spectra (and others) show a series of faint, broad hydrogen lines that fade into a continuum. Other spectra, albeit at much lower resolving power, show some noisy features here, however (Hawley and Pettersen, 1991). Ostensibly, the conservation of oscillator strength density should result in a smooth transition of observed flux from a pseudo-continuum of a confluence of high-order, highly broadened \n<!-- image -->", 'YZ CMi (dM4.Se)': "Fig. 26 Dramatic broadening differences between hydrogen and metallic emission lines, such as the Ca II resonance lines, in flare spectra are considered evidence for the important role of pressure broadening of hydrogen. The pressure broadening of hydrogen constrains the charge densities (and optical depths) in the flare chromosphere. This conclusion is supported by detailed spectral synthesis with, e.g., the RADYN, RH, MULTI codes. The top figure shows these effects during a flare on the dM4 star Gl 699, reproduced from Paulson et al (2006), and the bottom panel shows the coarse temporal evolution of the broadening differences during a flare (Vida et al, 2019) on the dM4.5e star YZ CMi. \n<!-- image --> \nand blended emission lines to the onset of recombination radiation. A phenomenological description of the transfer of bound-bound opacity to bound-free opacity was developed by Dappen et al (1987) using the occupational probability ( w n ) formalism in the non-ideal equation-of-state of Hummer and Mihalas (1988). The 'dissolvedlevel' bound-free Balmer continuum opacity longward of the Balmer edge has been recently incorporated into spectral models of stellar flares (Kowalski et al, 2017b; Kowalski, 2022). However, self-consistent modeling of the pressure broadening of the bound-bound Balmer lines (Tremblay and Bergeron, 2009) was not also included in \nthe first models (Kowalski et al, 2015). This incorrectly supported the extremely dense chromospheric condensations in 1D RADYN models (Sect. 8.2) as realistic models of flare density enhancements at cool temperatures (Kowalski et al, 2016). Otherwise, the dissolved-level continuum opacities have largely reconciled observations with models by explaining the lack of an edge around λ = 3646 -3700 ˚ A and the range of observed pseudo-continuum properties in between the longer-wavelength, high-order Balmer lines at λ = 3700 -4000 ˚ A. The non-discontinuous, transition of dissolved-level opacities at T ≈ 10 4 K have apparently reconciled the hydrogen recombination (Balmer jump) theory of the NUV with the blackbody-like theory of the optical continuum radiation. \nFig. 27 Direct comparisons of two methods (Donati-Falchi et al, 1985; Kowalski et al, 2017b) for calculating the Balmer jump spectral region in solar and stellar flares. The emergent LTE intensity is calculated from a slab with n e = 3 . 9 × 10 13 cm -3 . The method from Donati-Falchi et al (1985) predicts a 'blue continuum bump', whereas the current treatment with occupational probabilities predicts an extension of the Balmer continuum intensity to wavelengths longward of the 'ideal' recombination limit (indicated by an ∞ symbol) at λ = 3646 ˚ A. \n<!-- image --> \nThe physical cause of the continuous, dissolved-level opacity that was parameterized in Dappen et al (1987) for hydrogen is still not well understood. Experiments with Rydberg atoms support the idea that Landau-Zener transitions (e.g., Zwiebach, \n2022) at avoided level crossings (Zimmerman et al, 1979; Rubbmark et al, 1981; Pillet et al, 1984) facilitate a cascading ionization process. In this picture, an electron is photo-excited to a high-lying level n that is pressure-broadened to overlap with the broader, next higher-energy level n +1, thus effectively 'dissolving' the level n . This description apparently explains the Inglis-Teller relationship through dynamic microfield ionization (Hummer and Mihalas, 1988). There are actually many types of calculations (Littman et al, 1978; Damburg and Kolosov, 1979; Bergeman, 1984; Seidel et al, 1995; Benenti et al, 1999; Fisher and Maron, 2002, 2003; Cho et al, 2022, and see the review in Sections 86.2.2 - 86.2.3 in Drake 2006, Weisheit and Murillo 2006) 48 that may contribute to 'lowering the ionization potential' or 'continuum lowering' in a hydrogen plasma with microfield fluctuations and collisional perturbations 49 . \nDue to level-dissolution in dense regions of flare atmospheres that are not very optically thick in the Balmer continuum, applying a simple relationship, such as the Inglis-Teller, that connects the last-visible hydrogen line to a single electron density may be misleading. Kowalski et al (2022) discuss a scenario wherein level-dissolution of hydrogen in a chromospheric condensation can generate an approximately grey opacity window at λ ≈ 3700 -3750 ˚ A. Thus, the Balmer lines that are in emission in the emergent spectra actually originate from deeper layers with smaller ambient (thermal) electron densities and no apparent gas velocities. Due to the heterogeneous vertical stratification of the flare atmosphere, the high-order hydrogen Balmer series in emergent intensity spectra can be as broad as or less broad than the low-order Balmer lines (H γ , H β , H α ), which are very optically thick in some of the more recent RHD flare models. Some models of stellar flare spectra at the Balmer limit also incorporate spatial and temporal heterogeneity within the flare ribbons (Kowalski, 2022). These seek to explain the inconsistencies in large n Balmer max ≳ 15 (Eq. 23) being observed (indicative of low electron densities, n e = 10 13 -10 14 cm -3 ) with the necessity to also produce the observed continuum properties from a location on the star with much larger electron density ( n e ≳ 10 15 cm -3 ; n Balmer max ≈ 11). Note that in occupational probability theory, however, there is not one single maximum upper level for all atoms in a plasma. \nFig. 27 directly compares two different modeling techniques within the Balmer jump spectral region. The emergent intensity is calculated in LTE from a static slab with dl = 400 km, T = 10 4 K, and ρ = 10 -10 g cm -3 ( n e = 3 . 9 × 10 13 cm -3 ). Donati-Falchi et al (1985) modeled several solar flare spectra (with high resolving power) using Voigt profiles up to n max = 2 × 10 4 T 0 . 25 n -0 . 25 e ≈ 65 -85; this is a saddle-point estimate with Debye-screening of the proton field (Mihalas, 1978). This method convolves the Balmer edge with the Voigt function corresponding to the line with n max . The 'current (2022)' method uses the TB09+HM88 profiles (Tremblay and Bergeron, 2009) and the dissolved level bound-free continuum opacity. Many other \nFig. 28 A variety of approximations to opacity line profile functions for the Balmer series have been used in stellar flare modeling codes over many decades. This figure summarizes the differences in models of the line profile function, ϕ λ , for the H11 Balmer line (see also Johns-Krull et al, 1997, for a similar figure for the H10 line). The calculations use n e = n p = 3 . 16 × 10 14 cm -3 , or about 10x larger than in Fig. 27, and thermal Doppler broadening at T = 10 4 K is included. The 'ideal' profiles, ϕ V CS , are the VCS73 calculations from Lemke (1997), and do not account for line narrowing due to occupational probability / level dissolution. Several Voigt profiles are also plotted for a range of Lorentzian FWHM values, Γ. Currently, the most accurate theory corresponds to the ϕ TB09+HM88 (solid black) curve. \n<!-- image --> \n- \nvarious approximations for the pressure broadening of hydrogen have been used in models of solar and stellar flares. The ambiguities were discussed extensively in JohnsKrull et al (1997) and were resolved only in recent years. In Fig. 28, we summarize the wide variety of broadening prescriptions for the Balmer H11 line, whose integrated flux is sensitive to the amount of level dissolution in the densities of flare atmosphere models (Kowalski et al, 2015). For a description of each method presented in the figure, we refer the reader to Kowalski et al (2022) where similar comparisons are shown for the Balmer H γ line.", '11 A Comprehensive Multi-Wavelength Stellar Flare Dataset': 'What fundamental physical parameters can one infer using a multi-wavelength dataset of a stellar (super)flare? A comprehensive analysis of a series of two consecutive superflares from the binary dM4e+dM4e system DG CVn was presented in Osten et al (2016), who summarize the plasma and geometric properties that can be constrained in a flaring atmosphere. This culminates an extensive history of stellar flare research, and we briefly review the salient points to the algorithm.', '11.1 Inferring Nonthermal Electron Properties from Stellar Data': "The nonthermal radiation from power-law electrons in stellar flares occurs as hard X-rays ( E ≳ 10 keV) and as incoherent gyrosynchrotron radiation in the radio/microwaves at cm wavelengths. The radiative processes are thought to be similar to solar flares, in which the hard X-rays are emitted through collisional bremsstrahlung in the chromosphere, whereas the radio flux originates from predominantly higher energy electrons ( E ≈ 100 -300 keV) trapped in magnetic fields; see the review in White et al (2011). The distribution of electrons is parameterized in the hard X-rays and radio using two power-law indices, δ x and δ r , respectively. From hard X-rays, the particle flux distribution is \nF ( E ) = F o ( E E c ) -δ x (24) \nwhere E is the electron kinetic energy in keV, E c is the low-energy cutoff, and F o is the differential (specific) electron flux density (el s -1 cm -2 keV -1 ) around the cutoff energy. Similarly, a nonthermal electron beam density distribution determined from the radio is written as \nn ( E ) = n o ( E E c ) -δ r (25) \nwhere n o is the differential number density (el cm -3 keV -1 ) of electrons around energy E c . If the same population of electrons is responsible for both X-rays and radio, then δ x ≈ δ r is expected for electrons with velocity v very close to the speed of light c ; otherwise, δ x ≈ δ r -0 . 5. The different forms ( i.e., , flux density vs. number density) of these equations is because the injected power-law flux distribution is inferred through the collisional thick target model of the hard X-ray spectrum, whereas a power-law density distribution is determined through observations at optically thin microwave frequencies (see below). \nNote that the integral of Eq. 24 multiplied by E gives the total beam energy flux density (erg cm -2 s -1 ), which is an important input for flare modeling (e.g., F10, F13, etc...). Integrating Eq. 25 over energy gives the total beam density, N , where N is the quantity that is used in the gyrosynchrotron expressions in Dulk (1985) if E c is set to 10 keV. In the studies of Smith et al (2005); Osten et al (2016); Dulk \nand Marsh (1982); Dulk (1985), Eq. 25 is integrated from E c to ∞ to replace our n o with the prefactor N ( t ) δ r -1 E c where N ( t ) is the total electron density (el cm -3 ) above the cutoff (and the same is usually done for Eq. 24). Eq. 24 can be multiplied by a probability density function for the angular distribution, M ( θ ), to specify the directivity of particles with respect to the magnetic field (Allred et al, 2020). A highlycollimated Gaussian distribution is usually chosen for M ( θ ). Then, a '1.5D' solution to the Fokker-Planck equation (Sect. 9.2) accounts for the changes in the pitch angles of the particles. \nThe best diagnostic of accelerated particles in stellar flares is optically thin frequencies in the mm or radio. If the observations are in the radio at GHz frequencies, one must ensure that these frequencies are above the peak frequency ( ≈ 10 GHz) so as to be in the optically thin part of the gyrosynchrotron spectrum. An expression from Dulk (1985) converts from a power-law index in the optically thin flux spectrum, S ν ∝ ν α r , to the power-law index in the electron distribution, δ r : \nα r = 1 . 22 -0 . 90 δ r (26) \nThe power-law index in the optically thick part of the spectrum is much less sensitive to the power-law index of nonthermal electrons. \nFollowing Smith et al (2005), the kinetic energy of nonthermal particles can be constrained from the radio spectrum as follows. The power per volume in nonthermal electrons above the assumed low-energy cutoff value, E c , is given by Eq. 9 of Osten et al (2016). In this equation, the total number of nonthermal electrons, N ( t ) V ( t ), over the flaring volume V ( t ) is an unknown, but it is directly proportional to the optically thin spectral radio luminosity, L ν . By constraining the value of δ r in this part of the spectrum and invoking some prior knowledge or assumption about the peak frequency, an integral over the entire radio spectrum can be calculated - including the optically thick part - to give the time-integrated energy of nonthermal electrons in the flare. This leaves one unknown, the magnetic field, B , in the gyrosynchrotron emissivity formula for the optically thin part of the spectrum (see below for further constraints on the magnetic field, but it must be assumed that the B derived from the radio originates from the same B that is derived from the soft X-rays). If the radio or mm data are available at only one frequency, and thus a spectral index cannot be determined from the data, then contours of nonthermal energies and plausible ranges of magnetic field strengths and spectral indices are reported, as in Fig. 29, which is reproduced from Osten et al (2016) (see also MacGregor et al, 2021, where a spectral index was derived from mm observations). \nX-ray and optical data can constrain the physical parameters of flares further. From time-resolved X-ray spectra, one obtains the volume emission measure, VEM . The VEM is ∫ n 2 e dV , which is the amount of optically thin emitting plasma at a fixed X-ray temperature, T X . Typical values of the VEM (in M-dwarf flares) are 10 51 -10 52 cm -3 . In a log 10 T X vs. log 10 √ VEM diagram, the peak T X may clearly occur before the peak VEM (Favata et al, 2000; Aschwanden et al, 2008; Liefke et al, 2010). Over the decay phase evolution of a flare, the slope in this diagram provides one of three inputs for the widely employed flare loop modeling of Reale et al (1997). Combined \nFig. 29 Arange of possible nonthermal electron beam power-law indices and magnetic field strengths that can be constrained from gyrosynchrotron data of stellar flares; reproduced from Figure 6 in Osten et al (2016) with permission. See also Smith et al (2005) and MacGregor et al (2021). Contours of constant electron beam kinetic energies add a third constraint. Cross sectional footpoint areas from optical data can constrain the beam flux density for forward modeling inputs (Osten et al, 2010, 2016). \n<!-- image --> \nwith an exponential decay constant from the X-ray light curve and the maximum Xray temperature, a semi-circular loop's half-length, l , can be inferred. The loop sizes in giant X-ray flares are typically reported to be on the order of the stellar radius (e.g., Favata et al, 2000; Osten et al, 2010; Lalitha et al, 2013; Osten et al, 2016). Note, large 50 -200 Mm loop lengths are also inferred under some interpretations of QPPs (Section 6.2) in optical broadband light curves (Anfinogentov et al, 2013; Doyle et al, 2022). \nOptical and/or NUV data provide data to relax the geometrical assumption of a single, giant monolithic loop (Sect. 12), while also providing constraints on the electron density and the magnetic field strength in the flare corona. The optical and NUV probe the footpoint radiation, from which a dimensionless ratio of R foot / (2 l ) can be calculated, where R foot is an equivalent radius of the area of an optical flare footpoint: A flare , opt = 2 πR 2 foot . By combining with an alternative expression for the VEM in terms of these quantities and using the value of the VEM from X-ray spectral fitting, a representative value of the coronal n e is readily obtained that is independent of the number of identical loops in a hypothetical stellar arcade (Eq. 25 of Osten et al, 2016). One also can assume that the magnetic pressure must be at least equal to the gas pressure of the evaporated MK plasma for it to be confined to the loops instead of cooling through expansion, as in a scenario analogous to a filament eruption or coronal mass ejection (Cully et al, 1994). The magnetic field confinement condition places a lower-limit on the coronal magnetic field of the flaring loops. \nMullan et al (2006) homogeneously analyzed a large sample of many sizes of Xray stellar flares from various spectral types and evolutionary stages using 'Haisch's simplified analysis' (HSA; Haisch, 1983), which involve scaling relations (see also van den Oord and Mewe, 1989) that infer coronal temperature, electron density, loop length, and confining magnetic field strength from an emission measure and light \ncurve decay timescale. Typical X-ray temperatures derived from multi-component fits to E = 0 . 5 -10 keV spectra are similar to those in solar flares, T X ≈ 15 -30 MK (e.g., Osten et al, 2005; Liefke et al, 2010; Namekata et al, 2020). At the low coronal temperature end, T ≈ 2 -15 MK, electron density variations, or lack thereof, are constrained with X-ray spectra of the f (forbidden) and i (intercombination) components of helium-like triplets (e.g., bottom right spectra in Fig. 18; Gudel et al, 2002a; Osten et al, 2003; Testa et al, 2004; Osten et al, 2005; Liefke et al, 2010). When significantly detected, the flare-enhanced electron densities are reported in the range of 10 11 -10 12 cm -3 at these cooler coronal temperatures. \nThe peak stellar flare values of VEM and T X generally compare well with scaling relations over many orders of magnitude, from superflares to solar flares, in the famous 'Hertzsprung-Russell (H-R) diagram of solar and stellar flares' (Yokoyama and Shibata, 1998). The H-R diagram of solar and stellar flares was first theoretically explained using MHD scaling relations in Shibata and Yokoyama (1999). This diagram was investigated further in Aschwanden et al (2008), and the tightest correlation was reported in the empirical relationship \nOsten et al (2016) summarize the physical parameters in three of the largest dMe flares that have been observed with X-ray spectra ( cf. their Table 5). The inferred temperatures are T ≈ 50, 140, 290 MK; the electron densities are 3 × 10 11 , 3 × 10 12 3 × 10 11 cm -3 ; the confining magnetic field flux densities are 230, 1100, 580 G, respectively; and VEM ranges between 10 54 -10 55 cm -3 . Generally similar numbers are found in the largest RS CVn flares, which produce integrated X-ray energies and luminosities one to two orders of magnitude larger (Osten et al, 2007; Karmakar et al, 2023). YSO's also exhibit such extreme volume emission measures and temperature (40 -80 MK; Vievering et al, 2019). These superhot temperatures correspond with X-ray luminosities that are close to or in excess of the bolometric luminosity of the star 50 . Note that in an X-ray superflare from the dM3.5e star EV Lac with comparable temperatures ( T X, peak ≈ 70 MK) that were derived from the hydrodynamic modeling analysis of Reale et al (1997), Favata et al (2000) argue that accounting for the E > 10 34 erg of energy release during the flare places further constraints on coronal magnetic fields in the ≈ 4 kG range before the flare. \nVEM ( T p ) = 10 50 . 8 ( T p 10 MK ) 4 . 5 ± 0 . 4 [ cm -3 ] (27) \nwhere T p is the peak flare X-ray temperature. Their relation for decay time and peak temperature is, in principle, useful for inferring physical parameters if only the decay phase is caught in the observations (Ayres, 2015b); however, the empirical scatter about the trends is an order of magnitude in both quantities. Getman et al (2008) and Osten et al (2016) suggest that the hottest ( T X > 100 MK) stellar flares exhibit a 'saturated'-like volume emission measure of ≈ 10 55 cm -3 and do not follow a solar extrapolation in the flare H-R diagram. Shibata and Yokoyama (1999) include two additional parameters (either loop length or magnetic field and ambient thermal \nelectron density) in their scaling relation analogue (see also further alternatives summarized in Heinzel and Shibata, 2018) to Equation 27 to explain the apparent offset in volume emission measure from the solar extrapolation. Howard et al (2022) include a much smaller flare from an active M-dwarf flare in the context of stellar superflares and solar flares. Osten et al (2016) report on a stellar superflare (their 'F2' event) with T X ≈ 50 MK that is within a factor of 2.5 in the predicted VEM given by Equation 27. \nLarge stellar flares occasionally trigger Swift 's Burst Alert Telescope (BAT trigger numbers 172969, 310125, 331592, 596958, 625898), which is intended to automatically slew to gamma-ray bursts and other extra-galactic transients 51 . The combination of X-ray data from the BAT and at softer X-ray energies from the Swift 's XRT facilitates classical model hypothesis testing among thermal and nonthermal models out to E ≈ 100 keV. In superflares from II Peg and DG CVn, nonthermal bremsstrahlung models were found to be equally statistically robust fits as a superhot thermal model fit. However, in the superflare event from DG CVn (Osten et al, 2016) that the BAT spectra triggered on, the T X = 290 MK thermal model was favored over a nonthermal model because the time-evolution of the X-rays was delayed with respect to the optical V -band data (in line with the thermal, empirical Neupert effect; CaballeroGarc'ıa et al, 2015), and a footpoint area from the optical gives (even by stellar flare standards) unrealistic collisional thick target electron beam energy and flux into the lower atmosphere. The collisional relaxation times were estimated to be rather short compared to the (classical) thermal conduction times, thus further supporting the thermal interpretation. Additional evidence for thermal interpretation of superflare X-ray emission from YSO's is discussed in Vievering et al (2019). \nIn soft X-ray flare spectra, features around the Fe 6.4 keV K α emission line and the 6.7 keV Fe XXV thermal emission line complex are also reported. These features have been interpreted as either electron beam excitation (Emslie et al, 1986) or nonLTE fluorescence of a larger-area photospheric 'halo' (Osten et al, 2007; Drake et al, 2008; Osten et al, 2010). However, there are some calibration issues in Swift data around these features that have been reported only recently (Pagani et al, 2011). The recalibrations greatly diminish the 6.4 keV feature while enhancing the 6.7 keV complex. Fe K α fluorescence was studied in a flare from an active giant star using a different observatory (Chandra) without instrumental problems and higher resolving power (Testa et al, 2008); see Sect. 12.2. \nTable 8 highlights a few important multi-wavelength diagnostics in stellar flares that have been discussed in this review. Of course, the interpretations of these diagnostics are complicated by issues pertaining to heterogeneities of the unresolved nature of stellar flare sources (in time, and in all three spatial directions), and thus are generally model-dependent. In the final section of this review, we discuss assumptions for and inferences of stellar flare geometries and the associated implications for heterogeneity. \nT able 8 A summary of flare diagno sti cs \n<!-- image --> \n102 \n'NT' is an abbreviation for n on thermal.", '12 Stellar Flare Geometries': "Stellar flares are spatially unresolved point sources. Yet, from spectral data, model fitting, and sufficient time-resolution, there are many properties of the spatial dimensions of flaring sources that can be inferred. These reveal similarities to solar flare phenomenology at similar wavelengths and strongly reinforce the solar-stellar connectivity of flare physics. In this section, we review the methods and results of stellar flare geometry calculations. \nOptical and NUV observations constrain the chromospheric/photospheric flare areas. A blackbody temperature, T flare , and the fraction ('filling factor') of the visible stellar hemisphere that is flaring, X flare = R 2 flare /R 2 star , can be fit to spectral data or multi-band photometry through the following equation (Hawley et al, 1995, 2003): \nf ' λ, Model = ( πB λ ( T flare ) -S λ,q ) X flare R 2 star d 2 (28) \nwhere B λ ( T flare ) is the Planck function intensity model of the flare, S λ,q is the surface flux of the quiescent star, and f ' λ, Model is the 'flare-only' model flux at Earth above the atmosphere that is to be fit to the observed, calibrated [erg cm -2 s -1 ˚ A -1 ] flareonly flux, f ' λ . The ratio of flare-only fluxes at two wavelengths is a flare color and is the minimum information that is needed to solve for T flare in Eq. 28 (through linearization and iteration or a lookup table). An m -component, linear superposition of RHD surface flux flare spectra, S λ, RHD ,m , or a multi-thermal blackbody model are typically required to reproduce the optical and NUV continuum observations over a wavelength range spanning more than ∆ λ ≈ 1000 ˚ A. Additional terms are thus included in Equation 28. For an m -component RHD model superposition, the solution is a linear least squares fit to an observed flare-only spectrum, f ' λ , resulting in the estimated parameters, ˆ X flare ,m (see Osten et al, 2016; Kowalski et al, 2017b; Kowalski, 2022). For further discussion, see Appendix C. \nConstraints on X flare from blackbody fitting to optical flare-only spectra result in inferences that a small fraction of the visible stellar hemisphere flares for even the largest, E ≈ 10 34 erg, dMe events. For example, the broadband photometry in the Great Flare of AD Leo (Hawley and Pettersen, 1991) constrain X flare to be 0.5% (Hawley and Fisher, 1992) in the first impulsive peak. Fits to the flare-only, optical continuum in the rise/peak spectrum of this event are consistently small (0.2%) but with a correspondingly larger blue-optical blackbody color temperature of T flare ≈ T BB = 11 , 600 K (Kowalski et al, 2013). An m = 2 component RHD model of this spectrum gives a similar filling factor for a model with large electron-beam heating fluxes, which are combined with the second RHD model component with filling factors that are ≈ 2 -10x larger in order to account for most of the narrow emission line flux (Kowalski, 2022, see also Cram and Woods 1982; Hawley and Fisher 1992; Kowalski et al 2010). The corresponding projected area at the star is A flare = X flare πR 2 star , which may instead consist of a series of K circular flare kernels. Inferences in smaller dMe flares from broadband photometry (Hawley et al, 2003) and optical spectroscopy (Kowalski et al, 2016) 52 are consistent with very small areas ≈ 5 -10 × 10 16 cm 2 , \nthus reinforcing the analogy to compact white-light kernels that are observed at the footpoints of impulsively heated loops in solar flares (e.g., Fletcher et al, 2007). During the most energetic dMe events, however, peak-phase flare areas are inferred to be several orders of magnitude larger than solar flare white-light and NUV areas (Krucker et al, 2011; Kowalski et al, 2017a). \nIn the gradual decay phase, and in the impulsive phase of gradual-type dMe flare events with large Balmer jump ratios, blackbody fits to broadband photometry may be misleading (Sect. 7.2.4). Blackbody fits to the gradual phase continuum give a rather inconsistent picture of the post-peak footpoint areal evolution from event to event; perhaps, heterogeneous RHD model superpositions in the gradual decay phase are much more important for accurate areal inferences than in the 10 4 K blackbody-like rise and peak phase. A superposition of RHD model components in the gradual decay phase of a megaflare on the dM4.5e star YZ CMi is shown in Fig. 30 from Kowalski et al (2017b). This model was previously developed to explain the NUV to V -band ratio in the DG CVn superflare event reported in Osten et al (2016), and it also adequately modeled the NUV continua and Balmer jumps in much smaller, gradual-type events (Kowalski et al, 2019b). This 'DG CVn superflare multi-thread (F13) model' consists of several RHD model spectra over the evolution of the atmosphere that is shown in Fig.21, thus superposing snapshots from a range of optical depths, temperatures, and densities (see Sect. 12.1) in order to explain the gradual decay phase. \nVarious size scales in solar and stellar flares are shown in Fig. 31.", '12.1 A Dominant Loop or an Arcade?': "There are two general approaches to modeling spatially unresolved flare light curves and spectra (Liefke et al, 2010). The first approach is summarized in Karmakar et al (2023), and references therein, where it is most recently applied to an RS CVn X-ray superflare. This modeling follows the analysis of Reale et al (1997) and describes the rise and decay of a flare as result of a dominant monolithic loop (or a superposition of a few dominant loops that are heated and cooled simultaneously). The heating and cooling timescales in the loop are on the order of tens of minutes to hours (see also Reale et al, 2004; Testa et al, 2007; Argiroffi et al, 2019). In Reale et al (2004), the initial X-ray event in the same flare in Fig. 18 was modeled with this method, and the subsequent X-ray flare in the event was modeled as an arcade of ≈ five loops that were sequentially heated in a nearby region. Coincidentally, a solar analogy inspired a similar hypothetical explanation for the NUV and optical footpoint ignition during the decay phase of a different M dwarf flare in Kowalski et al 2017b. Longitudinal MHD oscillations in a monolithic, giant (200 Mm) loop have been proposed as an explanation for the triggering of quasi-periodic, U -band, light-curve oscillations in the decay phase of this megaflare event (Anfinogentov et al, 2013). The comparisons of particularly dense snapshots (e.g., t = 2 . 2 s; Kowalski et al, 2015) from RHD models to observations also inherently make the assumption that a single dominant kernel at any one time contributes to the bulk of the flux received at Earth (but see Namekata et al, 2022a, for recent tests of this assumption). \nAn alternative hypothesis is known as 'multi-thread modeling', which was pioneered by Hori et al (1997), Reeves and Warren (2002), and Warren (2006) to explain \nFig. 30 Gradual-decay phase, flare-only spectrum of a megaflare on the MV4.5e star YZ CMi (Kowalski et al, 2010). Multi-thread RHD modeling of another superflare on DG CVn (Osten et al, 2016) adequately reproduces the Balmer jump, optical continuum shape, and some of the hydrogen Balmer emission line properties (but some important shortcomings remain). Each RHD model component is calculated self-consistently, but the linear superposition of the model components is semi-empirical; see discussion in Kowalski et al (2017b) and Kowalski et al (2019a). \n<!-- image --> \nSun-as-a-star soft X-ray solar flare light curves. These works addressed the large discrepancy between the observations of long decay times in GOES soft X-rays and the very short decay times in models of individual loops; multithread techniques have further been used to model a variety of solar flare phenomena (e.g., Reeves et al, 2007; Rubio da Costa et al, 2016; Reep and Toriumi, 2017). Elements of this approach and of observations of sequential footpoint brightenings in the hard X-ray and optical wavelengths in solar flare ribbons (e.g., Kosovichev and Zharkova, 2001; Qiu et al, 2010) have inspired stellar flare RHD model superpositions (Osten et al, 2016; Kowalski et al, 2017b) for more direct comparisons to spatially unresolved observations. Getman et al (2011) argue that the rise phase of a giant X-ray flare from the PMS star DQ Tau consists of a multi-threaded source; see also Schmitt et al (2008) in reference to the modeling approach showcased in Fig. 23. Doyle et al (2022) suggest that a series of tens of loops contribute to the geometrical dimensions, as inferred from their QPP (Section 6.2) analysis, in two high-energy flares. Mathioudakis et al (2006) discuss the loop triggering scenario of Emslie (1981b) as an alternative hypothesis for \nFig. 31 Sizes to scale. An image of the Sun at λ = 6173 ˚ A from the Helioseismic and Magnetic Imager (HMI; Scherrer et al, 2012) on SDO, a representative size of an active mid-type (MV3eMV4.5e) M star ( R star = 0 . 3 R Sun ), and a circular flare area ( ≈ 5 × 10 18 cm 2 ) that corresponds to about 0.5% of the visible hemisphere of this hypothetical M star. For comparison, Earth and Jupiter are shown to scale. The insets each have a field-of-view of ≈ 10 19 cm 2 , and they show an image of the solar active region in the NW quadrant during the hard X-ray impulsive phase of the SOL2014-0329T17:48 X1.0 flare (total intensity left, difference image right). For a detailed analysis of the HMI data of this solar flare, we refer the reader to Kleint et al (2016). Jupiter image credit: NASA, ESA, and J. Nichols; Earth image credit: GOES/NOAA/NESDIS. \n<!-- image --> \nshort-duration QPPs in white-light. Osten et al (2005) hypothesize that an arcade of heterogeneous gyrosynchrotron-emitting loops may explain the microwave spectral indices in a large dMe flare. \nMulti-thread modeling approaches seek to be more representative of a solar flare arcade geometry, following arguments previously made in stellar (Hawley and Fisher, 1992) and solar modeling (Hawley and Fisher, 1994). However, ad hoc geometrical assumptions and simplifications must be imposed due to the absence of appropriate \n(i.e., broad spectral coverage, high-time resolution, unsaturated data at comparable wavelengths) constraints from solar data. Recent analyses of IRIS solar data at hightime resolution, such as the 'new area' calculations of Graham et al (2020), and spatially resolved spectral line analyses (Namekata et al, 2022a), may be able to provide clearer guidance to stellar models of sequential footpoint ignition. Nonetheless, there may be important differences between solar and stellar flares that complicate a direct application of the thermal multi-thread modeling of Warren (2006): shorterduring footpoint pulses (e.g., Mathioudakis et al, 2006), greater importance of the gradual-phase white-light continuum radiation (Hawley and Pettersen, 1991; Heinzel and Shibata, 2018) and particle acceleration (Osten et al, 2005), and larger apparent loop height growth over the flare evolution ( τ c ∝ l 2 ; e.g., Osten et al, 2005, 2016) all are examples of properties that distinguish stellar flares. \nNovel stellar flare applications of the two-ribbon solar flare reconnection model of Kopp and Poletto (1984) are described in Poletto et al (1988) and Gudel et al 1999 (see also Haisch, 1983). Gudel et al (1999) model the ∆ t ≈ 8000 s rising phase of a flare on the RS CVn UX Ari in this framework. This flare exhibited a remarkable peak X-ray luminosity of > 10 32 erg s -1 , and they infer the evolution of the reconnection heights and electron densities (which evolve through ≈ 10 11 cm -3 at peak). A longduration decay phase is predicted by the model, and the paper discusses how the lack of nonthermal particles in the early rise phase is a potential limitation of the original assumptions.", '12.2 Stellar Core-Halo': "Multi-dimensional geometrical analysis of flare sources is possible by combining hydrodynamic modeling of loop lengths and high resolution spectral observations at Fe K α . The technique is described in Testa et al (2008), who infer an X-ray source height that irradiates a large area of the photosphere, from which the Fe K α is observed at a viewing angle ϕ . The geometry is shown in Fig. 32. The combined constraints from the Palermo-Harvard code and the MOCASSIN 3D Monte Carlo radiation code (Ercolano et al, 2003, 2008; Drake and Ercolano, 2007) are reproduced from their paper on the right of Fig. 32. Recently, Werner band H 2 flare emission has been reported in France et al (2020) and interpreted as fluorescence due to O VI λ = 1032 ˚ A irradiation of the temperature minimum region ( T ≤ 1500 K). \nIn 1D radiative-hydrodynamic modeling, XEUV backwarming of the photosphere is approximated using the treatments of Gan and Fang (1990), Hawley and Fisher (1992), Hawley and Fisher (1994) and Abbett and Hawley (1999) through exponential integrals and detailed radiative transfer (Allred et al, 2015; Wahlstrom and Carlsson, 1994). A multi-dimensional extension of UV and optical radiative backwarming is illustrated in a cartoon of Figure 4 of Fisher et al (2012), and the geometrical dilution of chromospheric radiative flux impinging on the photosphere is given by their Eq. 25. \nThe remarkable, state-of-the-art 3D MHD codes that have been recently developed with the primary goal to study quiet Sun magnetism are not ready to address some of the most pressing current problems in stellar flare physics. Flare models require accurate treatments of non-equilibrium physics and detailed radiative transfer for comparisons to chromospheric observations. An approximation to nonthermal \nFig. 32 Geometrical inference from spectral analysis of the 6.4 keV (1.94 ˚ A) Fe K α emission line during a flare on the giant star HR 9024, reproduced from Testa et al (2008) with permission. The X-rays from hot flare loops fluoresce the neutral iron in the surrounding photosphere. The modeling leverages hydrodynamic modeling from Testa et al (2007). \n<!-- image --> \nparticle heating can in principle be included as Q flare in the energy equation in 3D simulations of small flares (Frogner et al, 2020). The geometrical grid spacing in 3D MHD models, however, is immobile and too coarse (e.g., see Sect. 8.2) to resolve the gradients and contribution function features for the optically thick spectral lines and continua. Section 4.6.1 of Bjørgen et al (2019) discusses the issues facing a 3D MHD solar flare simulation that predict chromospheric emission lines in response to thermally conducted fluxes. The RADYN code (for example) was originally a 'quiet' Sun code, but it had implemented the 1D adaptive grid scheme of Dorfi and Drury (1987) and was flexible enough to simulate particle beam heating in a non-solar gravity. These features proved fortuitous in applications to stellar flare models. The stellar community is in urgent need of collaborative development of multi-dimensional extensions of Carlsson's and Dorfi's works, such as Stokl and Dorfi (2007), in order to understand the comprehensive, radiative-hydrodynamic response of stellar atmospheres to the impulsive release of magnetic energy.", '13 Conclusions and Future Outlook': "In conclusion, we summarize six, non-exhaustive big-picture questions that are (in our view) at the forefront of stellar flare research. \n- · Stellar flares routinely attain energies and peak luminosities that are factors of 10 2 -10 4 greater than the largest solar flares. How are the flare energies connected to photospheric magnetic flux densities/geometries, particle acceleration properties, flaring footpoint areas, and energy transport processes in the atmospheres of other stars\n- · Stellar superflares exceed the quiescent bolometric stellar luminosity. The rate of superflares, however, is not characterized with high statistical significance. What are the rates of events in the superflare regime for solar analogs and other low-mass cool stars as a function of stellar age?\n- · On the Sun, the brightness of chromospheric flaring sources is transient at any single location for (arguably) ≲ 10 -20 s. A comprehensive, self-consistent model (e.g., within frameworks similar to Reeves and Forbes 2005 and Rempel et al 2023 pertaining to solar flare soft X-rays) has not yet been developed for optical or radio \n- stellar flare light curves that last for tens of minutes to hours. What are the principal physical parameters that drive the observed temporal scales of stellar flare light curves of optically thick radiation in the rise and decay phases?\n- · Blue asymmetries in Balmer lines are often interpreted as evidence of mass eruptions from a star. Are there direct analogues (e.g., as in Mg II; Tei et al, 2018) on the Sun, and does chromospheric evaporation in stellar flares contribute to blueshifted spectral features in cool lines?\n- · Turbulent mass motions and other non-equilibrium macroscale processes (e.g., UV and X-ray radiative backwarming) in stellar flares remain poorly understood in three spatial dimensions. If gas-dynamic turbulence is important for explaining the observed spectral line shapes, what processes generate large Reynold's numbers in flare atmospheres and, more specifically, in flare chromospheres?\n- · Optical and NUV stellar flare spectra are inconsistent with optically thin hydrogen recombination theory. Electron-beam and thermally-conducted heating fluxes that are within our expectations of the standard solar flare model paradigm do not explain the observed strength and spectral shape of the continuum radiation at NUV and optical wavelengths in M-dwarf flares. What is the source of stellar flare white-light continuum radiation, which is prominent in almost all of the Kepler (and TESS) broadband flare data? \nSupplementary information. Additional data are provided in supplementary online material hosted on Zenodo at https://doi.org/10.5281/zenodo.10641273. \nAcknowledgments. I thank Dr. Yuta Notsu for providing helpful comments and corrections. I gratefully acknowledge an anonymous referee for valuable comments and suggestions. I am grateful to Dr. Suzanne Hawley for much guidance and many stimulating discussions about stellar flares over the years. I thank Dr. Mihalas Mathioudakis and the solar physics group at Queen's University Belfast for many productive discussions. I thank Dr. Rachel Osten for many discussions about stellar flares, radio and X-ray observations, and about energy partitions. I thank Dr. John Wisniewski for many conversations about near-ultraviolet stellar flares. I thank Dr. Thomas Gomez for helpful discussions about line broadening theory, Dr. Eugene Oks for discussions about plasma line broadening, Dr. Jim Drake and Dr. Joel Dahlin for discussions about PIC simulations and related plasma physics topics, Dr. Joel Allred for extremely helpful guidance in radiative-hydrodynamics and assistance with flare modeling, Dr. Mats Carlsson with invaluable assistance with flare modeling, Dr. Han Uitenbroek for helpful conversations about stellar atmosphere codes, Dr. Lyndsay Fletcher, Dr. Gianna Cauzzi, Dr. Dana Longcope, Dr. Eduard Kontar, and Dr. Markus Aschwanden for stimulating discussions on solar flare topics, Dr. Gregory Fleishman and Dr. Stephen White for discussions about solar flare radio observations, Dr. Andy Inglis for helpful discussion about quasi-periodic pulsations in solar and stellar flares, Dr. Matthias Rempel and Cole Tamburri for conversations about 3D flare modeling, and Dr. Manuel Gudel for discussions about the Neupert effect and stellar flares on Proxima Centauri. I acknowledge the International Space Science Institute (ISSI) for my participation in several solar-stellar workshops that were organized and lead by Dr. Lyndsay Fletcher, Dr. Sven Wedemeyer, Dr. Louise Harra, Dr. Graham Kerr, Dr. Vanessa Polito, and \nDr. Paola Testa. Lastly, I thank all of the researchers in this field for developing such an engrossing area of study within astrophysics.", 'Appendix A Optical Filter Reference': 'We include several appendices in this Living Review. This appendix, Appendix A, serves as a quick-reference for optical broadband filter curves. Appendix B shows supplementary spectral constraints from optical dMe flare data, Appendix C reviews two continuum modeling approaches with slab geometries (blackbody fitting and optically thin hydrogen emissivity spectra) and the continuum modeling approach with parameterized chromospheric condensations (Kowalski and Allred, 2018), and Appendix D reviews the salient physics (through a mostly qualitative exposition) of the pressure broadening of the spectral lines of hydrogen. \nFig. A1 Quantum efficiency curves of common optical filters, normalized to peak transmission. An example of a terrestrial atmospheric transmission curve from the Sloan Digital Sky Survey is shown for an airmass of 1.0 at the Apache Point Observatory. One should note that the quantum efficiency of the CCD, telescope primary mirror, and other optical elements are not included. For reference, the impulsive phase flare spectrum from the IF4 event in Kowalski et al (2016) is shown. The wavelengths that are affected by the dichroic are in a lighter shade of red. The ULTRACAM photometry (NBF3500, NBF4170, NBF6010) at the peak of this flare were calibrated independently, and they are scaled to account for the factor of ten shorter integration time. The flare data were obtained from the Zenodo repository at: https://doi.org/10.5281/zenodo.45878. \n<!-- image --> \nIn Fig. A1, we show the optical broadband and narrowband filters that are commonly used in stellar flare studies. There are numerous databases that provide these, such as the SVO Filter Profile Service at http://svo2.cab.inta-csic.es/theory/fps/, and some are published alongside photometry calibration surveys (Bessell and Murphy, 2012). For comparison, we show the low-resolving power, flare-only spectrum (from \nFigure 3 of Kowalski et al, 2016) over the impulsive phase of the IF event from the bottom panel of Fig. 8. The spectral shape from the calibrated ULTRACAM narrowband filters at the flare peak time is in agreement with the spectrum.', 'Appendix B A Summary of Supplemental Model Constraints in the Optical': "In this appendix, we supplement the impulsive phase flare colors from spectra, shown in Fig. 12, with the peak colors calculated from narrow-band continuum filters. Fig. B2(top) is a summary of the peak-phase ULTRACAM filter (Appendix A) ratios from Kowalski et al (2016). These flare colors were calculated at higher time resolution than from the spectra that were discussed in the main text (Sect. 7.2). Figure B2(middle) shows the evolution of the ULTRACAM filter ratios in the large IF1 event described in row (1a) of Table 7; see also Kowalski (2023). Average fluxes in the ULTRACAM narrowband filters are calculated from the spectra of the energetic IF3 event (also on YZ CMi) from Kowalski et al (2013) and are shown for comparison in Fig. B2(middle). Note, the peak flare spectrum for this event is showcased in Fig. 10. In the bottom panel, we show Balmer jump ratios against the ratios of the H γ line flux to the flare-only continuum flux around λ = 4170 ˚ A at the time corresponding to the optical continuum peak flux. (The H γ line flux is continuum-subtracted, integrated over the wavelengths of the line, and pre-flare subtracted; C4170 ' is the average the flare-only flux over λ = 4155 -4185 ˚ A). Fig. B2(bottom) combines the impulsive-phase spectral measurements from Kowalski et al (2013) with the same quantities obtained in the four flares (indicated on the top axis) with low-resolving-power, optical spectra that were analyzed in Kowalski et al (2016) and Kowalski et al (2019b). It is our hope that as more flares are observed, these figures will be updated to provide compelling and comprehensive constraints for future multi-dimensional flare models. \n<!-- image --> \n<!-- image --> \nFig. B2 ( Top ) Flare-peak colors from ULTRACAM narrowband filter ratios; similar to Fig. 12 and see text for details. ( Middle ) Evolution of flare colors in two large flares on YZ CMi. Time increases over the tracks in a clockwise direction. These data have small statistical uncertainties (which are shown) but non-negligible systematic errors. The uncertainties are correlated, which we show for representative 5% systematic absolute color calibration uncertainties in both flare colors located at ( x, y ) ≈ (2 , 2). The error-ellipses indicate representative joint probabilities: the outer is the joint 68% confidence interval, while the inner translates to the marginal 68% uncertainties. ( Bottom ) Flare-only, line-to-continuum flux ratios from the same peak spectra in Fig. 12 in the main text. \n<!-- image -->", 'C.1 Blackbody Fitting to Stellar Flare Optical Data': "In this appendix, we summarize the assumptions and techniques employed by fitting isothermal blackbody functions to spectra, broadband photometry, and narrowband photometry of flares. A blackbody 'color' temperature refers to the temperature of a blackbody that is fit to a ratio of two fluxes (e.g., a flare color calculated from a ratio of two flare-only continuum fluxes; Sect. 7.2.1), where color is equivalently a difference of magnitudes in two bandpasses. The term also refers to the temperature of a blackbody that is fit to the absolute value of more than two continuum flux values (outside of prominent emission lines) over a limited spectral range (∆ λ ≈ 1000 -2000 ˚ A). Generally, a color temperature contains rather limited information about the actual nature of the radiation, such as whether it is optically thick, thin, an intermediate optical thickness, or which atomic processes contribute to the emergent radiative flux. A color temperature does not unambiguously constrain how the spectrum of continuum radiation extends to wavelengths outside of the range over which it is calculated. Within the total wavelength span that a flare color is calculated (e.g., the ratio of flare fluxes in narrowband filters around 4170 ˚ A and 6010 ˚ A), there may be emission lines that affect the calculated value, and the actual spectrum between two wavelengths may be best represented by a more complex continuum model. Therefore, flare color temperatures are equivalent color temperatures of blackbody functions. \nOther measures of temperature include gas temperature, radiation (or brightness) temperature, and effective temperature (in addition to the color temperatures of optically thin recombination and bremsstrahlung spectra). None of these are possible to robustly measure directly from optical stellar flare observations and must be inferred from detailed modeling. In principle, flare effective temperatures could be measured, but there has not yet been a more complete broadband spectral observation of a flare than analyzed in Hawley and Pettersen 1991. To demonstrate the ambiguities in these measures of temperature, it is interesting to note that for an optically thin hydrogen recombination and free-free spectrum calculated from LTE level populations for a slab with a gas temperature of T gas = 10 , 000 K (Appendix C.2, Fig. C5), the radiation temperature at λ = 4170 ˚ A is T rad = 3600 K. The emergent intensities are calculated for a slab thickness of dl = 0 . 1 km to ensure that τ ( λ ) < 0 . 005 across the optical regime. The color temperature measured from the ratio of intensities at λ = 4170 ˚ A to 6010 ˚ A is T FcolorR = 5300 K. Thus, T gas > T FcolorR > T rad in this example. \nAnother interesting ambiguity in inferred gas temperatures occurs when the emergent continuum flux is formed over significant ( τ ≈ 1) optical depth in dynamic, non-LTE, non-isothermal flare atmospheres. In Fig. C3, we summarize the argument in Kowalski (2023), which is also closely related to the analyses in the appendices of Kowalski et al (2017b). Two Planck functions with T = 13 , 000 K and T = 14 , 000 K are shown. If wavelength-dependent, continuum optical depths are large over an inhomogeneous temperature structure, then one could possibly, for example, measure radiation temperatures of T rad = 13 , 000 K at λ = 4170 ˚ A and T rad = 14 , 000 K at λ = 6010 ˚ A. The ratio of the corresponding emergent intensities gives a color temperature that is only T FcolorR = 10 , 500 K. Thus, T rad > T FcolorR in this example. For \nFig. C3 Demonstration showing the relationship between a color temperature and wavelengthdependent radiation (brightness) temperatures at λ = 4170 ˚ A and λ = 6010 ˚ A. \n<!-- image --> \nsemi-infinite atmospheres that exhibit a linear dependence between the optical depth and the (LTE or non-LTE) source function, the Eddington-Barbier approximation relates the radiation temperature to the gas temperature at the depth corresponding to τ ( λ ) = 1; however, dynamic flare atmosphere calculations are usually much more complicated over the large physical depth ranges of continuum formation (e.g., in chromospheric condensations and the stationary layers below; cf. Fig. 2(right) and Fig. 21), which results in differences between the radiation temperature and the gas temperatures over which the emergent continuum intensity forms (see Kowalski, 2023, for details). \nThe temperatures and filling factors calculated from optical and NUV flare data are in general obtained through two methods: \n- 1. From a ratio of flare-only fluxes in two bandpasses, one can solve for the temperature and filling factor ( T flare , X flare ) of a Planck function following Hawley et al 1995 (i.e., by equating ratios of the RHS of Eq. 28 in Sect. 12 to the observed flare-only ratio). The ratio equation is a transcendental equation for T flare and can be solved numerically using linearization and iteration (i.e., Newton-Raphson) or through a simple lookup table. The result for T flare is used to infer X flare . \nAs discussed in Kleint et al (2016) and Castellanos Dur'an and Kleint (2020), by not accounting for the fraction of stellar flux ( X flare ) that no longer shines through with the quiescent brightness, such as in a blackbody flare (see Hawley et al, 1995) or a flare with a transient overlying absorbing layer, it is possible to calculate a color temperature of ≈ 8500 K from flare-only irradiance observations of solar flares, as reported in Kretzschmar (2011). In Fig. C4, we demonstrate what \nFig. C4 Lines of constant blackbody flare temperatures ( T flare ) and lines of constant blackbody color temperatures that are calculated after subtracting a quiescent blackbody surface flux spectrum, S λ,q = B λ ( T q ), from the flare model spectrum, S λ, flare = B λ ( T flare ). The color temperature calculations, here shown for λ 1 = 4170 ˚ A and λ 2 = 6010 ˚ A, from stellar flare data usually account for or consider these issues in M dwarf flares (Hawley et al, 1995; Kowalski et al, 2016), but hotter flare color temperatures in higher-mass stars could be a result of 'oversubtraction' effects. \n<!-- image --> \nhappens to the calculated color temperature of an optically thick ( τ ≫ 1) flare source if one does not use the proper equation in Hawley et al (1995). For example, a 5800 K Planck function subtracted from a 7800 K Planck function at λ 1 = 4170 ˚ A and λ 2 = 6010 ˚ A gives a blackbody color temperature of 9000 K ( c.f. , where the 9000 K solid line and the 7500 K dashed line approximately intersect). However, for certain conditions, S λ, flare ≫ S λ,q where S λ, flare is the model flare surface flux spectrum, one may ignore the background subtraction errors (in which case, the problem must still be solved numerically or through a simple lookup table). Using the actual quiet Sun photospheric intensity, which is larger than a Planck function at these wavelengths, a 7500 K Planck function is sufficient to produce a 9000 K blackbody shape in the subtraction residuals. The surface fluxes of M dwarfs are much fainter than on G dwarfs. M-dwarf surface fluxes are lower than blackbody functions given by their effective temperatures, and the errors indicated in Fig. C4 are consequently smaller. For optically thick emission lines, similar issues arise if the model surface fluxes are comparable to the pre-flare surface fluxes (thus, it is fortuitous that large electron beam flux models produce spectra that are extremely bright compared to the pre-flare surface flux spectrum!). \n- 2. Second, a color temperature can be fit to a spectrum of observed fluxes. A Planck function is typically fit to the excess (flare-only) spectrum, or to flare-only multiband photometry data (Hawley et al, 2003), with nλ points ( n > 2) and Gaussian \nuncertainties, thus deriving the maximum likelihood parameters ˆ T flare , ˆ X flare . This can be accomplished with non-linear least squares through linearization and iteration for unconstrained parameter fits and with the Levenberg-Marquardt algorithm (e.g., IDL's mpcurvefit ) for constrained parameter fits. The color temperatures (e.g., T BB ) that were quoted from blackbody fitting in Kowalski et al (2013) were obtained by fitting Planck functions directly to the observed flare-only spectral fluxes (i.e., assuming B λ ( T flare ) ≫ S λ,q ). Also in that paper, the excess RHD model flux spectra were similarly compared to a Planck function temperature without subtracting a pre-flare from the Planck function. In this approach, the color temperature is just a convenient parametrization for comparing models to flare-only spectral observations, since temperatures are easier to compare than color indices in different filters or at different wavelengths (as noted in the main text). Whatever one does to the data, one should also do analogously to the model, whether the model is a blackbody or an RHD spectrum, irrespective of spectral type of the star (from RHD models of M dwarf flares that are optically thin in the chromospheric continuum radiation, it is actually desirable to isolate the chromospheric flare radiation by subtracting the pre-flare surface flux because the model photospheric spectrum, which does not have proper molecular opacities, otherwise shines through and obfuscates comparisons to observations). \nThe main purpose of subtracting a pre-flare spectrum from the flare observations is to remove the relatively large, wavelength-dependent flux from the area on the star that is not flaring so that the flare emission is more accurately characterized. A method for determining an accurate subtraction of the preflare spectrum based on molecular bandpass veiling by the flare continuum is described in Kowalski et al (2013). Alternatively, a very wide slit may be employed, but spectral purity and wavelength resolution are sacrificed; see Kowalski et al (2016). \nIn summary, the first method uses two filters and assumes that the flare spectrum is inherently blackbody radiation emitting at a single temperature T flare , and the second method is used for calculating color temperatures ( T BB , T FcolorR ) directly from a spectrum of flare-only fluxes assuming that the effects in Figure C3 are not important (although this assumption can readily be relaxed in the second method).", 'C.2 Hydrogen Continuum Emissivity Models': "Kunkel (1970) calculated a grid of optically thin, LTE hydrogen recombination models with free-free contributions. These are often used as a starting place for comparisons to flare observations (e.g., Fig. 12). The equations are not included in Kunkel (1970) but were restated in Kowalski (2012). Here, we update the calculations with freefree Gaunt factors from CHIANTI and include the dissolved-level Balmer continuum components (longward of the Balmer edge at λ = 3646 ˚ A) in the spectra. These are shown in Fig. C5 for a range of temperatures and a gas density of ρ = 10 -8 g cm -3 . The Balmer jump ratios are calculated from these spectra and are shown in Fig. C6. All spectra are available as FITS tables for a range of gas densities in the supplemental online material hosted on Zenodo. \nFig. C5 Hydrogen LTE emissivity calculations with bound-free (recombination) emissivity from H I and H -and free-free (thermal bremsstrahlung) emissivity from H II. The spectral shapes here are identical to the shapes of intensity spectra, I λ , that emerge from optically thin, homogeneous slabs. A blackbody function and a mono-energetic synchrotron spectrum is shown for comparison. The smooth transitions from the Balmer jump to the optical wavelengths are caused by the 'dissolved level Balmer continuum' (Dappen et al, 1987), which extends to redder wavelengths for larger electron densities. \n<!-- image --> \nFig. C6 Balmer jump ratios calculated from the LTE emissivity models in Fig. C5 and from the same calculations of the LTE emissivity in a hydrogen gas with a density of ρ = 10 -9 g cm -3 . \n<!-- image --> \nAt constant slab temperature, a large range of flare colors result from adjusting the continuum optical depths (Morchenko et al, 2015). In Fig. C7, we show the LTE continuum spectra from RHD model parameterizations (Kowalski and Allred, 2018); the colors of these spectra are calculated and are shown in Fig. 12 as the 'KA18 CC Models'. A range of reference column masses ( m ref ) were chosen to increase the continuum optical depths within the model chromospheric condensations. Occupational probabilities ( w n ) for the dissolved level Balmer continuum opacities are included as in Figure C5. \nFig. D8 shows the splitting of the energy levels of hydrogen in a static electric field. The energy shifts were calculated using Equation D1 and the quadratic (secondorder) and cubic (third-order) perturbation terms for hydrogen (Condon and Shortley, \n<!-- image --> \nFig. C7 (Left) LTE continuum flux spectra calculated from the parameterized evolution of chromospheric condensations from RHD models. (Right) Examples of two of the parameterized atmospheres. \n<!-- image -->", 'Appendix D Pressure Broadening of Hydrogen Lines in Flares': "Electric fields split the degenerate energy levels of hydrogen atoms, Rydberg atoms, and hydrogen-like ions by an amount directly proportional to the field magnitude ( ∝ E 1 ; e.g., Condon and Shortley, 1963; Bethe and Salpeter, 1957; Gallagher, 2006; Goldman and Cassar, 2006), which is commonly known as the linear Stark, or StarkLo Surdo 53 effect. The splitting increases for larger n , and n = 1 experiences only second-order energy shifts, which are small. Within a partially ionized gas, the electric fields from ambient, thermal charges (electrons and protons) within a Debye radius broaden spectral lines. This explains the broad hydrogen absorption profiles in main sequence A stars and white dwarfs (combined with, of course, large optical depths in their photospheric layers). \nThe symmetric, atomic energy-level shifts that are linearly proportional to an external, uniform electric field E are described by the quantum numbers nqm , according to the following absolute value (in cgs units): \n| ∆ E (1) | = | 3 a 0 e 2 Z nq E| (D1) \nIn Eq. D1, n is the principal quantum number, q is the difference of two parabolic quantum numbers, k 1 -k 2 , that satisfy n = k 1 + k 2 + | m | +1, m = m l is the magnetic quantum number, a 0 is the Bohr radius, e is the electron charge, and Z is the nuclear charge (where Z = 1 for hydrogen). \n1963; Bethe and Salpeter, 1957; Drake, 2006; Goldman and Cassar, 2006). The secondorder and third-order terms create asymmetric splittings but are only important for hydrogen at very high electric field strengths. The maximum possible wavelength shift for an allowed Balmer transition ( n lower = 2) and ∆ m ± 1 , 0 follows from Eq. D1 and is proportional to (van Dien, 1949): \n( n 2 j n 2 j -4 ) 2 ( n j ( n j -1) + 2) n 2 / 3 e (D2) \nwhere ( n 2 j n 2 j -4 ) 2 is the appropriate conversion from dE to dλ , and n j is the upper level of the Balmer transition. This formula, which goes as n 2 j in the limit of large n j , is sometimes used as an equivalent Doppler width that is added in quadrature with the true Doppler width in a Voigt profile. However, this approximation, among others (Sutton, 1978), predicts inaccurate scalings for the line shapes and widths within a series. Thus, we recommend that these approximations not be used in place of proper line broadening calculations, which we briefly summarize next. \nIn a partially ionized gas, the electric microfield amplitude and direction vary from atom to atom and are not constant in time even at one atom because the charge distribution changes. The microfield amplitude probability is described by a probability distribution function (e.g., in the top panel of Fig. D8). A brief phenomenological summary of the microphysics of the broadening of hydrogen spectral lines follows. See Barklem (2016), Gigosos (2014), Gomez et al (2022), Hubeny and Mihalas (2014) for more comprehensive and rigorous reviews 54 . The densities of perturbing, ambient (thermal) protons ( n p ) and electrons ( n e ) affect the total electric pressure broadening of radiative (dipole) transitions in hydrogen differently. The slow-moving protons and ions produce quasi-static (meaning that the microfield changes slowly compared to radiative/orbital timescales; see Griem, 1974; Hummer and Mihalas, 1988) electric microfields that split the energy levels of hydrogen into sub-states (Fig. D8), which are superpositions of the various orbital angular momentum states, l , within each n ('manifold'). The microfield magnitude is given by a Hooper probability distribution (which is a modified Holtsmark distribution that accounts for Debye screening and plasma correlations; Nayfonov et al, 1999). The collisional perturbations from ambient electrons are more complicated. Classically, the individual elastic electron impacts on each atom are randomly occurring and well-separated in time, thus causing a damping/Lorentzian profile function with broadening FWHM Γ that is proportional to n e . However, on very short timescales, which correspond to large frequency displacements from line center (in the far line wings) in Fourier-conjugate space, the surrounding electrons are nearly still and thus produce a quasi-static (Holtsmark) broadening proportional to n 2 / 3 e . The transition in electron broadening from its impact to its quasistatic regimes occurs at wavelength shifts, of order | λ -λ rest | ≈ 20 ˚ A (for the H γ line). Thus, large optical depths can make the effects of the quasi-static limit of \n<!-- image --> \nE \nn=4 n=4 n=4 n=4 n=4 n=4 n=4 n=4 n=4 Fig. D8 Energy level splitting for hydrogen in a static external electric field as a function of the electric field (in SI units). A principal level n is split into roughly 2 n -1 (including the unshifted) sub-states. For even n , there is an unshifted component. Every split sub-state ( k 1 , k 2 , m ) is a mix of all the l -states for that value of n . The maximum splitting of the n = 5 level shown here corresponds to 0.016 eV, which would correspond to a maximum wavelength displacement of ∆ λ ≈ 24 ˚ A for the H γ transition. Only levels that can result in a dipole transition with n = 2 have been plotted. The transitions with ∆ m = 0 are known as the π components, and ∆ m ± 1 are the σ components. The field strength that forms a 'saddle point' (potential energy maximum) and destroys all levels > n in hydrogen is shown as a function of n (if there is a saddle-point in the potential, then tunneling is allowed as well; Bethe and Salpeter, 1957). The field ionization of hydrogen has been observed (see the photograph from Rausch v. Traubenberg et al (1930) that is reproduced in Montgomery et al 2022). The Inglis-Teller relation corresponds to the critical field that causes the 'bluest' sub-state of n to intersect with the 'reddest' sub-state of n +1, which occurs at a much smaller electric field for a given n than the saddle point estimate (the ratio of the critical field to the saddle-point field is given by the factor known as K n in the literature). Holtsmark probability distributions of electric field magnitudes are shown in the top panel to illustrate an occupational probability ( w n ) calculation for n = 11 and n e = 10 15 cm -3 . A 'normal field', F 0 , is the electric field magnitude at an average interparticle separation; the critical fields for n e = 10 14 and 10 15 cm -3 are indicated by vertical dotted lines in the top panel. \nelectrons important in the wings. Additionally, lifetime broadening occurs if there are inelastic effects, such as collisional transitions to states of different principle quantum \nn=3 \nn=3 \nn=3 \nn=3 \nn=3 \nn=3 \nnumber n (e.g., Hummer and Mihalas, 1988; Bransden and Joachain, 2000), but these are generally small for well-separated levels (and are ignored in the 'no-quenching' assumption in the VCS73 theory). \nThe energy-level perturbations from protons and from electron collisions are not independent random variables, and the line profiles thus cannot be modeled as convolutions of two probability distribution functions, such as a Lorentzian convolved with a Holtsmark (or Hooper) distribution. This is because the electron collisions with hydrogen are transient (short duration) and non-adiabatic 55 (Smith et al, 1969; Sobel'Man et al, 1995; Gigosos, 2014; Hubeny and Mihalas, 2014): they can cause transitions and redistribution among the microfield-split sub-states of the same n . Quantum calculations (Eq 10 of Tremblay and Bergeron, 2009, which is reproduced from Vidal et al 1970) that account for the perturbing electrons in both of their limiting regimes are crucial. If the dynamical aspects (Stehle, 1994) of the ions are ignored, the normalized line profile functions are then the ionic microfield averages of the electronic broadening profiles calculated at each value of the quasi-static (QS), ionic microfield E QS . When the spacing between the upper levels n j becomes small and the broadening causes the states to have a significant probability of overlapping, level dissolution occurs and the bound-bound transitions become narrower (Seaton, 1990; Tremblay and Bergeron, 2009) and less Holtsmark-like in the wings. The extensions of the VCS73 profiles that account for the effects of level dissolution through occupational probabilities (Sect. 10.2.1) are known as the TB09+HM88 profiles. Cho et al (2022) discuss comparisons of high-density analytic profiles to new numerical calculations without level dissolution effects (see also Stehle and Jacquemot, 1993; Gomez, 2017). To our knowledge, the TB09+HM88 profiles are yet the only quantum calculations extended to large n j that also include a framework, albeit phenomenological, for ionization-limit lowering consistently with the bound-bound opacities and line shapes. 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2024PSJ.....5..222M | NearEarth Object NEO Surveyor will detect asteroids and comets using midinfrared thermal emission however groundbased followup resources will require knowledge of the expected visible light brightness in order to plan characterization observations. Here we describe the range of visualtoinfrared colors that the NEOs detected by Surveyor will span and demonstrate that for objects that have no previously reported Visual band observations estimates of the Johnson Visualband brightness based on infrared flux alone will have significant uncertainty. Incidental or targeted photometric followup of objects discovered by Surveyor enables predictions of the fraction of reflected light visible and nearinfrared wavelengths supporting additional detailed characterization. | 2024-10-01T00:00:00Z | ['10.3847/PSJ/ad7859', 'arXiv:2409.05753', '2024arXiv240905753M', '2024PSJ.....5..222M', '10.48550/arXiv.2409.05753'] | ['Asteroids', 'Albedo', 'Near-Earth objects', 'Surveys', '72', '2321', '1092', '1671', 'Astrophysics - Earth and Planetary Astrophysics', 'Astrophysics - Instrumentation and Methods for Astrophysics'] | Visualband Brightnesses of NearEarth Objects that will be Discovered in the Infrared by NEO Surveyor | 2,024 | 166 | 0.48 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF'] | 0 | https://arxiv.org/pdf/2409.05753.pdf | {'Visual-band brightnesses of Near Earth Objects that will be discovered in the infrared by NEO Surveyor': 'Joseph R. Masiero 1 , Tyler Linder 2 , Amy Mainzer 3 , Dar W. Dahlen 1 , Yuna G. Kwon 1', 'ABSTRACT': 'NEO Surveyor will detect asteroids and comets using mid-infrared thermal emission, however ground-based followup resources will require knowledge of the expected visible light brightness in order to plan characterization observations. Here we describe the range of visual-to-infrared colors that the NEOs detected by Surveyor will span, and demonstrate that for objects that have no previously reported Visual band observations, estimates of the Johnson Visual-band brightness based on infrared flux alone will have significant uncertainty. Incidental or targeted photometric followup of objects discovered by Surveyor enables predictions of the fraction of reflected light visible and near-infrared wavelengths, supporting additional detailed characterization.', '1. Introduction': "The Near-Earth Object (NEO) Surveyor mission will survey the Solar system at two thermal infrared wavelengths (4 -5 . 2 µ m and 6 -10 µ m) with the singular goal of detecting the majority of objects that pose a regional hazard to Earth. Over the course of the 5-year required survey period, the mission is expected to detect ∼ 100 , 000 -200 , 000 near-earth asteroids larger than 25 m in size, and will report these observations to the MPC within 3 days of data acquisition. Mainzer et al. (2023) provide an overview of the mission and the design of survey simulation tools that will be used for survey completeness validation. \nThe survey strategy employed by the NEO Surveyor mission is designed to provide sufficient self-followup to establish the orbit and effective spherical diameter of the NEOs observed. Diameters can be determined from NEO Surveyor data even if visible light flux is not available (Mainzer et al. 2015), as was done for the objects discovered by the NEOWISE survey that did not receive ground based followup (e.g. Mainzer et al. 2011; Masiero et al. 2011). This is possible because the total thermal emission is only weakly dependent on albedo, and so the added uncertainty due to the unknown albedo is only a small component of the total error. \nThe mission also has a Targeted Followup Observation (TFO) mode for high-priority objects that can be used for improving orbital knowledge, obtaining high-cadence light curve observations (individual Exposures are taken every 30 s), and improving thermal modeling constraints. The TFO mode, however, takes time away from normal survey operations and so will only be used sparingly for the most critical objects. Additional characterization such as spectroscopy would need to employ ground- or space-based followup telescopes, the vast majority of which observe at visible or near-infrared wavelengths. In order to support asteroid and comet followup observations by the community, the NEO Surveyor mission will provide estimated Johnson Visual magnitude brightnesses for discovered objects to enable these investigations. \nThe recent International Asteroid Warning Network (IAWN 1 ) observing campaigns present a model for the types of followup that could be obtained for a potential impactor. IAWN is a global collection of amateur and professional astronomers dedicated to studying and monitoring the nearEarth asteroid population. IAWN to date has held six asteroid campaigns, two of which were dedicated to the observational timing assessments of fast-moving asteroids and four were intended to assess the ability of the community to rapidly recover and characterize NEOs. Those latter four campaigns in particular demonstrated how additional characterization data would be used to help assess the hazard posed by an impactor, and how this knowledge would be disseminated among the various stakeholders. The types of characterization data that could possibly be obtained and the timing for when it would be available can have significant impacts on the responses that are available. Therefore, it is important to understand the resources that would be able to access new NEO discoveries. \nThe most recent campaign was a rapid response exercise following the discovery of NEO 2023 DZ2, whose impact probability quickly rose with follow-up observations before measurements from precovery data retired the impact risk (Reddy et al. 2024). This object was discovered at a V = 20 . 3 mag ten days prior to close approach, and rose to a peak brightness of V ∼ 13 mag, allowing a wide range of characterization tools to be used. Visible and near-infrared photometry, spectroscopy, and polarimetry all provided an immediate estimate of the albedo of this NEO that was fed into the hazard model. \nPrevious characterization campaigns focused on recovery and tracking of 2012 TC4 (Reddy et al. 2019), spectral characterization of (66391) Moshup (Reddy et al. 2022a), and simulated discovery and hazard assessment of (99942) Apophis (Reddy et al. 2022b). The characterization campaign for 2012 TC4 was able to obtain light curve properties while the object was at V ∼ 22 . 5 mag, however these rotational observations were at the practical limit for the 4 -5 m-class telescopes used. In each case, ground-based followup allowed the impact hazard model to be refined and improved. \nTo assess the potential future capacity for followup of NEO Surveyor discoveries, we take the \nsynthetic population of near-Earth objects that were created to verify the mission's Level 1 survey requirements and extend this population down to diameters of D = 25 m. This population is assigned both orbital and physical properties. Using the current best knowledge of the survey rules and system sensitivities for NEO Surveyor, we determine the Visual and mid-infrared fluxes of each NEO at the time when it would first be discovered by the survey. We present the results of this analysis and related discussion below.", '2. Population Model and Survey Simulation': "Weuse the survey simulation tools recently developed for the NEO Surveyor mission to generate a synthetic population of near-Earth objects down to sizes below the current completeness limit. We simulate objects with diameters of D ≥ 25 m following the size-frequency distribution described in Mainzer et al. (2023), with orbital and albedo distributions based on the nearly-complete regime of the currently known NEO population (see Stokes et al. 2017, for more details about the NEO size distribution). Our survey simulator calculates the position and expected flux of each object for every time it would be in the field of view of a survey pointing. Dynamical and flux calculation routines were validated against standard software tools as well as the NEOWISE data, as described in Masiero et al. (2023). \nPredicted fluxes for NEOs are compared to the current best estimate of the system sensitivity and the predicted zodiacal light background flux for each observation to determine if the synthetic object would be detected. A fraction of these potential detections are then discarded based on a signal-to-noise-dependent factor to account for incompleteness in the data extraction due to e.g. confusion, bad pixels, etc. The remaining detections are then assembled into tracklets following the same rules that will be used when processing the flight data. This produces a set of data analogous to what the NEO Surveyor will be submitting to the Minor Planet Center once the survey commences. \nBecause physical properties are generated for each object, it is possible to determine the Visual band flux from reflected light as well as the thermal infrared flux from emission at the time each object would be detected. While thermal emission is determined using the Near-Earth Asteroid Thermal Model (NEATM Harris et al. 1998), Visual band flux is determined using the H-G asteroid photometric system (Bowell et al. 1989) and the orbital geometry at the time of observation. Using these two fluxes for each detection, we can determine the apparent magnitude difference from V band to the NC2 bandpass centered at 8 µ m (the band where Surveyor will be most sensitive to NEOs). In this implementation, this difference doesn't strictly indicate a spectral reflectance gradient, however we will still refer to it here as the V -NC 2 color. \nApproximately 40% of the NEOs larger than 140 m in diameter are expected to be known at the time of launch based on current survey capabilities (Grav et al. 2023) and thus will already have visible light measurements available to use for estimating visible brightnesses. The Vera Rubin \ntelescope is scheduled to begin survey prior to the launch of NEO Surveyor, and so will improve the catalog of reflected light measurements with observations of a large number of Solar system objects. Wagg et al. (2024) show that Rubin will typically observe NEOs at V ∼ 22 -23 (see their Figure 7), but will observe few NEOs with V > 24 mag. The remaining NEOs detected by Surveyor will have unknown visible light properties (in particular an unknown Visual band albedo) and as such would be expected to be drawn randomly from the population below. Quantification of this uncertainty is needed to properly plan for any followup observations.", '3. Survey Simulation Results': "We show in Figure 1 the distribution of Johnson V-band magnitudes for all NEOs at the first time that they would be detected during the survey. We use the time of the initial detection, rather than the peak brightness of each object over the 5-year survey period, as it is more representative of the brightness at time when immediate characterization followup would be most desirable for an object with non-zero impact probability, and when the predicted on-sky position will be best known prior to followup. We mark the location of V = 22 mag in this plot to show the approximate limits for the most common NEO survey and followup assets in use today (e.g. Larson et al. 2003; Denneau et al. 2013; Bellm et al. 2019; Devog'ele et al. 2019). As demonstrated in this plot, the vast majority of NEOs detected by Surveyor will be magnitudes fainter than what is currently submitted to the MPC and published for followup on the NEO Confirmation Page 2 . This is consistent with the previous findings of Mainzer et al. (2015). \nFigure 2 shows the spatial histogram of NEO V magnitudes compared to the elongation (onsky angular distance from the Sun) at the time of first detection by NEO Surveyor. The magnitude distribution is consistent across most elongations, with a ∼ 1 mag brightening at the lowest elongations. However, low elongations are the hardest for ground-based telescopes to access as they are only above 2 airmasses in the twilight sky or for a short period of time outside twilight. Incidental visible light photometry will likely be available for objects at larger elongations from ground-based surveys such as the Vera C. Rubin telescope (Jones et al. 2018). The trend of increasing numbers of objects at larger elongations is due primarily to the large population of amor-class NEOs, which spend all their orbit outside the Earth's, and thus are most accessible at high elongations. Amors tend to have higher minimum orbital intersection distances (MOID) with the Earth's orbit, and thus are less likely to be categorized as potentially hazardous objects which are defined as objects with MOID < 0 . 05 AU. \nThe planning of any desired characterization followup of NEO Surveyor discoveries beyond mission-obtained TFO characterization will be difficult for objects that do not receive any incidental visible light observations, and thus have unknown brightnesses at reflected light wavelengths. \nFig. 1.- V-band magnitude distribution of all simulated NEOs at the time they would be first detected by NEO Surveyor. The vertical dotted line indicates V= 22 mag, the approximate limiting magnitude of current NEO surveys and followup. \n<!-- image --> \nBecause of the wide range of albedos that have been observed for NEOs (Mainzer et al. 2011; Wright et al. 2016) and the albedo distribution that is expected for the population observed by NEO Surveyor (Mainzer et al. 2023), the predicted reflected light brightness for an object of a given infrared flux will be drawn from a large potential range. Figure 3 shows the distribution of V-NC2 colors for all NEOs in our simulation at the time of first detection. While the distribution is peaked at V -NC 2 ∼ 10, the full-width at half-maximum of this distribution spans ∼ 3 mags. The figure also shows the median V -NC 2 color (10.2) as well as the 2 . 5 th and 97 . 5 th percentile ranges ( -2 . 2 and +2 . 3, respectively.) which span the 2-sigma region of the distribution. \nOther detection parameters available at the time of observation, such as elongation or signalto-noise in the NEO Surveyor bandpasses, do not provide constraints on the expected V -NC 2 color. As shown in Figures 4, the color distribution does not show significant structure with respect to brightness at the NC2 wavelengths. Parameters that can be derived from NEO Surveyor data after self-followup and orbit fitting such as diameter also show large spreads in V -NC 2 color, as shown in Figure 5. Larger objects do tend to show bluer colors, as they are more likely to be detected at larger heliocentric distances where their temperatures are lower, however these are also the most likely to be already recorded in the current catalog. \nFig. 2.- V-band magnitude as a function of elongation for all simulated NEOs at the time they would be first detected by NEO Surveyor. The horizontal dotted line indicates V= 22 mag, the approximate limiting magnitude of current NEO surveys and followup. Spatial bins in this plot represent a linear count of objects. \n<!-- image -->", '4. Discussion': "Our simulations show that there is a wide dispersion in possible V band magnitudes for infrared-detected NEOs. It is not possible to make a precise prediction of these objects' expected Visual magnitudes using only the data obtained by NEO Surveyor. This means that targeted followup of individual objects of interest will need to account for a wide range of possible brightnesses, where only part of the possible range may be detectable with the followup instrumentation available, e.g. spectroscopy. \nA complete analysis of the capabilities of follow-up assets currently available is challenging due to the large number of variables. The location of the telescope, observing time availability, and specific instrument parameters such as image scale and readout time make a direct general comparison difficult. Instead, we discuss a few specific cases commonly needed for astrometric followup. \nThe most critical type of astrometric observation will be if NEO Surveyor discovers a virtual impactor. Selecting the correct ground-based assets is crucial to successful recovery. For example, the first IAWN campaign recovered 2012 TC4 using VLT at V ∼ 27 mag which is the practical \nFig. 3.- V-NC2 color distribution of all simulated NEOs at the time they would be first detected by NEO Surveyor. The cross indicates the median value of the distribution, while the whiskers show the range spanned by the 2 . 5 th and 97 . 5 th percentiles. \n<!-- image --> \nupper limit of recovery using an 8 meter ground-based telescope. The NEO community will likely be limited by observing time constraints to a handful of recoveries at these brightnesses per telescope per year. Considering all the 8 meter telescope time that could be even potentially be available for recovery, less than 50 asteroids per year could receive astrometric follow-up fainter than V = 26 mag. \nThe typical limiting magnitude of 2-4m class telescopes is V ∼ 25 mag. A 2 meter-class telescope optimized for moving object astrometric recovery can outperform a 4 meter telescopes with a more generalized setup and instrument set. However, the availability of facilities for asteroid recovery between V ∼ 23 -25 mag will be limited to a theoretical maximum of hundreds of asteroids per night if all potential facilities were used in a dedicated fashion. Objects brighter than V ∼ 23 mag are in the range of the current NEO survey and followup programs, and so could be expected to receive followup based on the current operational program, however the number of potential targets will likely saturate the available time for the currently operational programs. \nPhotometric and spectroscopic characterization observations are limited to V ∼ 22 mag currently, which means that it will be difficult to obtain further analysis for many members of the population discovered by NEO Surveyor. This is especially true for objects where the possible Visual band brightness is predicted to straddle this sensitivity threshold. One method to address this challenge is for ground-based telescopes to report negative detections and the limiting magnitude \nFig. 4.- V-NC2 color vs signal-to-noise (SNR) of the NC2 detection for simulated NEOs at the time of first detection. Spatial bins in this plot represent a logarithmic count of objects; the vast majority of detections are at SNR ∼ 5. Due to the expected photometric systematic uncertainties, sources have an effective upper limit to their SNR causing the apparent increase in source density at SNR ∼ 34 for all objects brighter than this limit. \n<!-- image --> \nthat was searched. This would allow for setting a limit on the possible H V magnitude and rule out the possibility of characterization for some objects. \nIt is important for the NEO Surveyor project to report the minor planet observations in a way that provides as much utility as possible for the followup observer community. Experience from NEOWISE, where the team reported estimated V magnitudes for most objects to aid in followup, found that predicted V magnitudes could often be 2 mags fainter than the recovery magnitude, as NEOWISE preferentially discovered low-albedo NEOs (Mainzer et al. 2011). Some NEOWISE-discovered objects had no reported estimated visible brightness at all, resulting in objects in the orbit catalog without H V magnitudes, which significantly hampered followup planning. The NEOWISE reported V magnitudes were estimates based on the infrared flux, not visible light measurements, but this fact is not explicitly indicated as such in the MPC observation archive. Users of the data would need to refer to the NEOWISE documentation (e.g. Cutri et al. 2012, 2015) to learn the source of these values. \nTo resolve this situation, we propose to use the new reporting fields enabled by the Astrom- \nFig. 5.- V-NC2 color at the time of first detection vs the size of the simulated NEOs. Spatial bins in this plot represent a logarithmic count of objects. Larger objects are skewed bluer as they are more likely to be detected further from the Sun, but they still span a significant range of colors. \n<!-- image --> \netry Data Exchange Standard (ADES 3 ) format, the preferred format for submitting data to the Minor Planet Center. Specifically, the mission can report estimated V magnitudes as was done for NEOWISE, but now can include an uncertainty on that value that will indicate it should only be considered a weak constraint on the actual expected V mag. Based on the distribution shown in Fig 3, and the range spanned by the 2-sigma region, the value to report would be \nV estimate ≈ NC 2 + 10 ± 2 mag \nwhich would cover the majority of the expected V mag distribution. \nThe benefit of a large uncertainty value is that any real photometric measurement that follows the NEO Surveyor observations would quickly dominate the H V calculation carried out during orbit fitting (assuming magnitude uncertainties are also reported for those measurements). Expansion of the MPC's NEO Confirmation Page to include the uncertainty on both the predicted V and H V values will enable observers to understand the provenance of these predicted values. \nBeyond NEO Surveyor, estimated V magnitudes may be useful for observation reports from other infrared observatories such as the James Webb Space Telescope (JWST). As described by \nMuller et al. (2023), during calibration observations the JWST Mid-Infrared Instrument (MIRI) detected a previously unknown asteroid, and the authors estimate that every MIRI observation near the ecliptic will likely contain a few unknown asteroids. These MIRI data alone were sufficient to constrain the size of this unknown object, despite the lack of orbital arc coverage needed to fit an orbit. While astrometric recovery and followup of these short-arc objects is nearly impossible with current resources, the ability to report visual magnitude estimates for infrared detections will potentially help future efforts to link these detections to later observations.", '5. Conclusions': "We use the survey simulation tools developed for NEO Surveyor to investigate the expected brightnesses of near-Earth objects that the mission will discover. We find that the vast majority of objects will be significantly fainter than the current depth reachable with most reflected light followup assets. Ground-based targeted astrometric followup will be difficult to achieve on a large scale, although the mission's survey is designed to provide astrometric self-followup. In order to obtain optical photometry and spectral characterization of any virtual impactors discovered by NEO Surveyor it will be necessary to invest in more sensitive followup resources and exercise these systems. \nEstimating the Visual brightness of NEO Surveyor discoveries is complicated by the wide range of possible V -NC 2 colors that detected objects are predicted to have. Reports of these estimates will use the photometric uncertainty field to identify to followup observers the large range of brightnesses that are possible. Propagation of these uncertainties to the predicted apparent and absolute magnitudes provided on the MPC's NEO Confirmation Page will allow followup observers to plan for a range of activities based on the recovered brightness. This will fulfill the needs of the followup community for observation planning purposes, as well as avoid empty entries in the orbit catalogs, all while making it clear to data users that this value is only an estimate of the actual brightness of the object.", 'Acknowledgments': "We thank the referees for their helpful comments that improved this manuscript. This publication makes use of software and data products from the NEO Surveyor, which is a joint project of the University of California, Los Angeles and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. This research has made use of data and services provided by the International Astronomical Union's Minor Planet Center. This research has made use of NASA's Astrophysics Data System Bibliographic Services. This research has made use of the neospy , numpy , scipy , astropy , and matplotlib Python packages.", 'REFERENCES': "Stokes, G.H., Barbee, B.W., Bottke Jr., W.F. et al. , 2017, NASA Report of the Near-Earth Object Science Definition Team, 'Update to Determine the Feasibility of Enhancing the Search and Characterization of NEOs', https://cneos.jpl.nasa.gov/doc/2017 neo sdt final e-version.pdf \nWagg, T., Juric, M., Yoachim, P., et al. 2024, arXiv:2408.12517. doi:10.48550/arXiv.2408.12517 \nWright, E. L., Mainzer, A., Masiero, J., et al. , 2016, AJ, 152, 79. doi:10.3847/0004-6256/152/4/79"} |
2018ApJ...869L..41A | We introduce the Disk Substructures at High Angular Resolution Project DSHARP one of the initial Large Programs conducted with the Atacama Large Millimetersubmillimeter Array ALMA. The primary goal of DSHARP is to find and characterize substructures in the spatial distributions of solid particles for a sample of 20 nearby protoplanetary disks using very high resolution 0.035 or 5 au FWHM observations of their 240 GHz 1.25 mm continuum emission. These data provide a first homogeneous look at the smallscale features in disks that are directly relevant to the planet formation process quantifying their prevalence morphologies spatial scales spacings symmetry and amplitudes for targets with a variety of disk and stellar host properties. We find that these substructures are ubiquitous in this sample of large bright disks. They are most frequently manifested as concentric narrow emission rings and depleted gaps although largescale spiral patterns and small arcshaped azimuthal asymmetries are also present in some cases. These substructures are found at a wide range of disk radii from a few astronomical units to more than 100 au are usually compact 10 au and show a wide range of amplitudes brightness contrasts. Here we discuss the motivation for the project describe the survey design and the sample properties detail the observations and data calibration highlight some basic results and provide a general overview of the key conclusions that are presented in more detail in a series of accompanying articles. The DSHARP dataincluding visibilities images calibration scripts and moreare released for community use at A hrefhttpsalmascience.orgalmadatalpDSHARPhttpsalmascience.orgalmadatalpDSHARPA. | 2018-12-01T00:00:00Z | ['2018ApJ...869L..41A', 'arXiv:1812.04040', '10.48550/arXiv.1812.04040', '2018arXiv181204040A', '10.3847/2041-8213/aaf741'] | ['circumstellar matter', 'planets and satellites: formation', 'protoplanetary disks', 'Astrophysics - Solar and Stellar Astrophysics', 'Astrophysics - Earth and Planetary Astrophysics'] | The Disk Substructures at High Angular Resolution Project DSHARP. I. Motivation Sample Calibration and Overview | 2,018 | 166 | 0.73 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF'] | 901 | https://arxiv.org/pdf/1812.04040.pdf | {'No Header': "The Disk Substructures at High Angular Resolution Project (DSHARP): I. Motivation, Sample, Calibration, and Overview \nSean M. Andrews, 1 Jane Huang, 1 Laura M. P'erez, 2 Andrea Isella, 3 Cornelis P. Dullemond, 4 Nicol'as T. Kurtovic, 2 Viviana V. Guzm'an, 5, 6 John M. Carpenter, 5 David J. Wilner, 1 Shangjia Zhang, 7 Zhaohuan Zhu, 7 Tilman Birnstiel, 8 Xue-Ning Bai, 9 Myriam Benisty, 10, 11 A. Meredith Hughes, 12 Karin I. Oberg, 1 and Luca Ricci 13 \n1 Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA 2 Departamento de Astronom'ıa, Universidad de Chile, Camino El Observatorio 1515, Las Condes, Santiago, Chile 3 Department of Physics and Astronomy, Rice University, 6100 Main Street, Houston, TX 77005, USA 4 Zentrum fur Astronomie, Heidelberg University, Albert Ueberle Str. 2, 69120 Heidelberg, Germany 5 Joint ALMA Observatory, Avenida Alonso de C'ordova 3107, Vitacura, Santiago, Chile 6 Instituto de Astrof'ısica, Pontificia Universidad Cat'olica de Chile, Av. Vicu˜na Mackenna 4860, 7820436 Macul, Santiago, Chile 7 Department of Physics and Astronomy, University of Nevada, Las Vegas, 4505 S. Maryland Pkwy, Las Vegas, NV 89154, USA 8 University Observatory, Faculty of Physics, Ludwig-Maximilians-Universitat Munchen, Scheinerstr. 1, 81679 Munich, Germany 9 Institute for Advanced Study and Tsinghua Center for Astrophysics, Tsinghua University, Beijing 100084, China 10 Unidad Mixta Internacional Franco-Chilena de Astronom'ıa, CNRS/INSU UMI 3386, Departamento de Astronom'ıa, Universidad de Chile, Camino El Observatorio 1515, Las Condes, Santiago, Chile 11 Univ. Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France 12 Department of Astronomy, Van Vleck Observatory, Wesleyan University, 96 Foss Hill Drive, Middletown, CT 06459, USA 13 Department of Physics and Astronomy, California State University Northridge, 18111 Nordhoff Street, Northridge, CA 91130, USA", 'ABSTRACT': "We introduce the Disk Substructures at High Angular Resolution Project (DSHARP), one of the initial Large Programs conducted with the Atacama Large Millimeter/submillimeter Array (ALMA). The primary goal of DSHARP is to find and characterize substructures in the spatial distributions of solid particles for a sample of 20 nearby protoplanetary disks, using very high resolution ( ∼ 0 . '' 035, or 5 au, FWHM) observations of their 240 GHz (1.25 mm) continuum emission. These data provide a first homogeneous look at the small-scale features in disks that are directly relevant to the planet formation process, quantifying their prevalence, morphologies, spatial scales, spacings, symmetry, and amplitudes, for targets with a variety of disk and stellar host properties. We find that these substructures are ubiquitous in this sample of large, bright disks. They are most frequently manifested as concentric, narrow emission rings and depleted gaps, although large-scale spiral patterns and small arc-shaped azimuthal asymmetries are also present in some cases. These substructures are found at a wide range of disk radii (from a few au to more than 100 au), are usually compact ( glyph[lessorsimilar] 10 au), and show a wide range of amplitudes (brightness contrasts). Here we discuss the motivation for the project, describe the survey design and the sample properties, detail the observations and data calibration, highlight some basic results, and provide a general overview of the key conclusions that are presented in more detail in a series of accompanying articles. The DSHARP data - including visibilities, images, calibration scripts, and more - are released for community use at https://almascience.org/alma-data/lp/DSHARP. \nKeywords: protoplanetary disks - circumstellar matter - planets and satellites: formation", '1. INTRODUCTION AND MOTIVATION': "There is a long-standing desire to link the properties of circumstellar disks with the initial conditions of planetary systems. The theoretical aspiration in the field is to develop a deterministic framework that takes a set of measured disk properties (e.g., the spatial distribution of densities and temperatures; Andrews et al. 2009, 2010; Isella et al. 2009, 2010) and predicts the key characteristics of the exoplanet population (e.g., masses, orbital architectures, atmosphere compositions; Ida & Lin \n2004, 2008; Alibert et al. 2005; Mordasini et al. 2009). A quality reproduction in this population synthesis context requires the tuning of increasingly sophisticated models for the formation of planetary systems, their interactions with disk material, and their subsequent long-term dynamical evolution (see Benz et al. 2014). \nThe most crucial obstacle in the planet formation process is the assembly of planetesimals (see Johansen et al. 2014). The formation of terrestrial planets and giant planet cores hinges on the rapid agglomeration of small \nparticles into these much larger ( glyph[greaterorsimilar] km-sized) bodies. Astronomers have worked on this topic and its pitfalls for more than 50 years, although without much observational guidance. Fortunately, that is changing. Resolved observations of the continuum emission from mm/cmsized particles in disks measure how the solids are distributed. Resolved variations in the continuum spectrum shape have been interpreted as radial gradients in the particle size distributions (larger solids closer to the star; Isella et al. 2010; Guilloteau et al. 2011; P'erez et al. 2012, 2015; Menu et al. 2014; Tazzari et al. 2016; Tripathi et al. 2018). Pronounced discrepancies between the spatial distributions of continuum and spectral line emission have led to suggestions that the mass ratio of solids relative to gas also varies with radius (higher closer to the star; Pani'c et al. 2009; Andrews et al. 2012; de Gregorio-Monsalvo et al. 2013; Rosenfeld et al. 2013; Zhang et al. 2014; Facchini et al. 2017; Ansdell et al. 2018). Those results provide strong qualitative support for evolutionary models of solids early in the planetesimal assembly process (e.g., Birnstiel & Andrews 2014; Testi et al. 2014; Birnstiel et al. 2016). \nDespite that progress, there is still considerable tension regarding planetesimal formation timescales for the default assumption of a smooth gas disk (with pressure, P , decreasing monotonically with radius, r ). This tension is associated with radial drift, the inward migration of solids toward the global P maximum that occurs when they decouple from the sub-Keplerian gas flow (Adachi et al. 1976; Weidenschilling 1977; Nakagawa et al. 1986). The predicted drift rates for mm/cm solids located tens of au from the host star are fast enough to severely limit planetesimal growth (Takeuchi & Lin 2002, 2005; Brauer et al. 2007, 2008) and are in conflict with routine observations of emission from those particles at r ≈ 10-100 au (e.g., Tripathi et al. 2017; Tazzari et al. 2017; Barenfeld et al. 2017; Andrews et al. 2018). \nThis contradiction indicates that the P ( r ) profiles in disks are likely not smooth. Localized P modulations can slow or trap drifting solids (Whipple 1972; Pinilla et al. 2012a), perhaps concentrating them enough to trigger gravitational and/or streaming instabilities that rapidly convert pebbles to planetesimals (e.g., Youdin & Shu 2002; Youdin & Goodman 2005; Johansen et al. 2009). Such particle traps or other migration bottlenecks could be produced by the dynamics associated with how gas, dust, and magnetic fields are coupled (e.g., Dzyurkevich et al. 2013; Bai & Stone 2014; Flock et al. 2015; Lyra et al. 2015; Dipierro et al. 2015; B'ethune et al. 2017; Dullemond & Penzlin 2018; Suriano et al. 2018) or by strong gradients in material properties (e.g., Okuzumi et al. 2012; Estrada et al. 2016; Armitage et al. 2016; Stammler et al. 2017; Pinilla et al. 2017; but see van Terwisga et al. 2018; Long et al. 2018). These small-scale material concentrations substructures - are largely absent in contemporary models of planet forma- \nion, but they would likely play fundamental roles in nearly all aspects of the formation process. \nIf such substructures were prominent in disks at early evolution stages, it is possible that planetesimals and even entire planetary systems were created much more efficiently than is expected in the traditional models (e.g., Greaves & Rice 2010; Najita & Kenyon 2014; Nixon et al. 2018). In that scenario, the typical ∼ Myrold disk may harbor a 'second generation' of substructures created by the dynamical interactions between young planets and their nascent disk material (see Lin & Papaloizou 1993; Kley & Nelson 2012), which in turn can affect the orbital architectures of those burgeoning planetary systems (e.g., Coleman & Nelson 2016). \nIn any case, observations of disk substructures are essential. Direct constraints on small-scale gas pressure variations in disks based on high resolution measurements of molecular line emission are a formidable challenge. However, the particle trapping capabilities of even modest pressure maxima should substantially amplify the associated local mm/cm-sized particle density (e.g., Paardekooper & Mellema 2006; Rice et al. 2006; Pinilla et al. 2012b; Zhu et al. 2012), generating a bright signature in the broadband (sub-)mm continuum that is much easier to measure on the smallest scales. \nThe initial foray into such work came from the 'transition' disks (Strom et al. 1989; Skrutskie et al. 1990; Calvet et al. 2002), which show dense particle rings at r ≈ tens of au, outside depleted central cavities (e.g., Andrews et al. 2011; van der Marel et al. 2018; Pinilla et al. 2018). Observations with sufficient resolution reveal that these particle traps exhibit complex substructures, including azimuthal asymmetries (Casassus et al. 2013; van der Marel et al. 2013; Isella et al. 2013; P'erez et al. 2014), additional rings (Fedele et al. 2017; van der Plas et al. 2017), warped geometries (and/or radial inflows; Rosenfeld et al. 2012, 2014; Marino et al. 2015; Casassus et al. 2018), and spiral arms (Christiaens et al. 2014; Boehler et al. 2018; Dong et al. 2018). Similar features have been identified from the IR starlight scattered off the disk atmospheres (e.g., Muto et al. 2012; Grady et al. 2013; Quanz et al. 2013; Avenhaus et al. 2014; Rapson et al. 2015; Benisty et al. 2015; de Boer et al. 2016; Ginski et al. 2016; Akiyama et al. 2016). \nSome serendipitous discoveries at modest ( ∼ 10-20 au) resolution hint that the more general disk population frequently exhibits substructures in the forms of rings/gaps (Zhang et al. 2016; Isella et al. 2016; Cieza et al. 2016, 2017; Loomis et al. 2017; Huang et al. 2017; Cox et al. 2017; Dipierro et al. 2018; Fedele et al. 2018; van Terwisga et al. 2018) and spirals (P'erez et al. 2016). The richness of these substructures becomes clear for the few individual cases that have had their continuum emission probed at resolutions of only a few au (HL Tau, ALMA Partnership et al. 2015; TW Hya, Andrews et al. 2016; MWC 758, Dong et al. 2018). Again, similar conclusions are being drawn from complementary mea- \nements of scattered light from small dust grains (e.g., van Boekel et al. 2017; Avenhaus et al. 2018). \nAll of these observations suggest that substructures are common, and therefore are likely significant factors in many disk evolution and planet formation processes. Moreover, they demonstrate a tremendous opportunity: high resolution mm continuum measurements can quantify the forms, prevalence, and diversity (e.g., in scales, locations, amplitudes) of disk substructures, and thereby help develop a more robust theoretical framework for characterizing the early evolution of planetary systems. The next step along that path is to move from a serendipitous discovery-space to a principled survey specifically designed to study these features. \nIn this article, we introduce a new survey that moves in this direction. The Disk Substructures at High Angular Resolution Project (DSHARP) was conducted as one of the first ALMA Large Programs. DSHARP measures the 240 GHz continuum emission at ∼ 35 mas (5 au) resolution for 20 disks, to help better understand the evolution of solid particles during the planet formation process. Having motivated the project, this article also describes the DSHARP survey design and sample (Section 2), the ALMA observations (Section 3) and their calibration (Section 4), along with some basic observational results and the DSHARP data release (Section 5). We conclude with an overview of the highlights from a series of accompanying articles (Section 6).", '2. SURVEY DESIGN AND SAMPLE': "The DSHARP survey was designed to optimize the spatial resolution and contrast sensitivity to continuum emission substructures. Secondarily, measurements of CO line emission were also of interest as a preliminary opportunity to identify corresponding gas structures and infer other relevant bulk disk properties (e.g., geometry). We defined two criteria to guide the survey design, based on previous observations and theoretical expectations for the origins of disk substructures. \nThe first criterion was access to a wide range of spatial scales down to a FWHM resolution of ∼ 5 au. Such high resolution was essential for identifying the disk substructures in the sharpest ALMA continuum images available to date (ALMA Partnership et al. 2015; Andrews et al. 2016). Moreover, it is comparable to the (disk-averaged) pressure scale height, h P (where h P /r ≈ 0 . 1; Kenyon & Hartmann 1987), a benchmark size that is directly related to the P deviations generated by turbulent zonal flows (e.g., Johansen et al. 2009), vortices (e.g., Barge & Sommeria 1995), or planetary gaps (e.g., Bryden et al. 1999). At 5 au resolution, h P -sized features in radius or azimuth are resolved in the outer disk, and detectable down to r ≈ 10 au (for sufficient contrast). \nThe second criterion was the ability to detect a ∼ 10% contrast out to Solar System size-scales ( r ≈ 40 au). This is roughly the contrast measured for the weaker substructures in the HL Tau and TW Hya disks (e.g., \nAkiyama et al. 2016; Huang et al. 2018). It is also sufficient to detect the continuum emission that (indirectly) traces the ∼ 20% pressure variations produced by glyph[greaterorsimilar] 0 . 1 M Jup planets (Fung et al. 2014), zonal flows (Simon & Armitage 2014), or weak vortices (e.g., Goodman et al. 1987), even if (contrary to expectations) there is no accompanying amplification in the concentration of the solids (presuming the emission is optically thin). \nThe combination of these criteria and ALMA technical restrictions meant that the optimal observing frequency was in the vicinity of 240 GHz (Band 6). Higher frequency observations at comparable (or better) resolution were not permitted for Cycle 4 Large Programs, and the resolution and sensitivity options at lower frequencies were both insufficient for our goals. \nThe resolution criterion drove planning for the survey sample. The Cycle 4 configuration schedule was set to provide the requisite resolution (with baseline lengths out to 6.8-12.6 km) during 2017 June and July. We targeted disks that are nearby enough to give the required spatial resolution for those configurations, and that transit at high elevations at night during this period. This limited the sample pool to the Oph (Wilking et al. 2008), Lup (Comer'on 2008), and Upper Sco (Preibisch & Mamajek 2008) regions, plus a few isolated targets. The field was narrowed to focus on Class II sources to avoid confusion with envelope emission. We excluded 'transition' disks, since they are already known to exhibit substructures (by definition). \nThose criteria leave ∼ 200 viable targets. A more severe cut was then made to meet the contrast criterion. A general framing of that criterion is somewhat arbitrary, but we chose some fiducial numbers as a guide. For a target at 140 pc and with a synthesized beam FWHM of 35 mas, we aimed to measure a 10% deviation from an otherwise smooth brightness profile (at SNR ≥ 2 per beam) out at r = 40 au ( ∼ 0 . '' 3). This metric requires previous continuum observations at modest (0 . '' 3) resolution for selection (Andrews et al. 2009, 2010; Ansdell et al. 2016; Barenfeld et al. 2016). For reasonable assumptions about the shape of the brightness profile, 1 this criterion can be met with a cut on the 0 . '' 3 peak brightness. Experimentation with simulated data suggested a peak brightness cut at 20 mJy per 0 . '' 3 beam (4.8 K) is appropriate, implying an objective noise level of 17 µ Jy per 35 mas beam (0.3 K). 2 The caveat is that much of the available data at 0 . '' 3 resolution were taken at 340 GHz; in the applicable cases, we assumed that I ν ∝ ν 2 . 5 (cf., Andrews & Williams 2005). \nWhile that brightness cut substantially reduces the pool, the sample size was ultimately set by ALMA re- \nTable 1. DSHARP Sample: Host Star Properties \nReferences -In Col. (11), the references for the quoted SpT, { T eff , L ∗ } , and accretion luminosity measurements, respectively: 1 = Alcal'a et al. (2017), 2 = Fairlamb et al. (2015), 3 = Luhman & Mamajek (2012), 4 = Barenfeld et al. (2016), 5 = Rigliaco et al. (2015), 6 = Salyk et al. (2013), 7 = Luhman & Rieke (1999), 8 = Andrews et al. (2010), 9 = Natta et al. (2006), 10 = Andrews et al. (2009), 11 = Wilking et al. (2005), 12 = Muzerolle et al. (1998), 13 = Bouvier & Appenzeller (1992), 14 = Eisner et al. (2005), 15 = Herbig & Bell (1988). \nstrictions. Only ∼ 30 hours in the LST ranges of interest were set aside for Large Programs in each of the two relevant array configurations. The desired noise could be reached in ∼ 1 hour of integration per target, but the factor of three overhead costs meant that the sample size was limited to 10 targets per configuration. We selected 10 targets (mostly) in Oph for the more compact of the two configurations (C40-8, ≤ 6 . 8 km baselines; 50 mas resolution), based on their nominally closer distances (125 pc; de Geus et al. 1989; Loinard et al. 2008). 3 Ten \nmore targets (primarily in Lup) were chosen for C40-9 ( ≤ 12 . 6 km baselines; 35 mas resolution). \nThe resulting sample and its stellar host properties are compiled in Table 1. Target distances ( d ) were derived from Gaia DR2 parallax measurements (Gaia Collaboration et al. 2018), following Astraatmadja & Bailer-Jones (2016) for a flat d prior. Literature estimates of the effective temperatures ( T eff ) and luminosities ( L ∗ ; re-scaled for the appropriate d ) were adopted to derive masses ( M ∗ ) and ages ( t ∗ ) based on the MIST models (Choi et al. 2016), following the methodology described by Andrews et al. (2018). Accretion rates ( ˙ M ∗ ) were calculated from those host parameters and literature measurements of accretion luminosities (scaled for d ; see Table 1). The sample hosts exhibit a range of young star properties, with M ∗ ≈ 0 . 2-2 M glyph[circledot] and nearly two decades spanned \nFigure 1. Broadband SEDs for the DSHARP targets. The ordinate is L ν = 4 πd 2 νF ν in L glyph[circledot] units. These SEDs have been de-reddened using the extinction values quoted by the references in Col. (11) of Table 1 (second entries) and the prescription described by Andrews et al. (2013). Blue curves show the Nextgen / BT-settl photosphere models (Allard et al. 2003, 2011) corresponding to the stellar parameters listed in Table 1. Red curves show the Spitzer IRS spectra. Note that the SEDs for HT Lup and AS 205 include contributions from multiple components. Optical photometry was collected from a range of sources (Vrba et al. 1993; Herbst et al. 1994; Hughes et al. 1994; Oudmaijer et al. 2001; Wilking et al. 2005; Gras-Vel'azquez & Ray 2005; Eisner et al. 2005; Padgett et al. 2006; Grankin et al. 2007; Mer'ın et al. 2008; Mendigut'ıa et al. 2012); infrared data were culled from 2MASS (Skrutskie et al. 2006), WISE (Wright et al. 2010), Spitzer imaging surveys (Carpenter et al. 2008; Evans et al. 2009), AKARI (Ishihara et al. 2010), and Herschel (IRSA); (sub-)mm data come from various sources (Andre & Montmerle 1994; Mannings & Emerson 1994; Sylvester et al. 1996; Nuernberger et al. 1997; Mannings & Sargent 1997; Dent et al. 1998; Henning et al. 1998; Natta et al. 2004; Stanke et al. 2006; Andrews & Williams 2007; Lommen et al. 2007, 2009; Roccatagliata et al. 2009; Andrews et al. 2009; Isella et al. 2007; Pinte et al. 2008; Isella et al. 2009; Ricci et al. 2010; Sandell et al. 2011; Oberg et al. 2011; P'erez et al. 2012, 2015; Qi et al. 2015; Ansdell et al. 2016; Cleeves et al. 2016; Barenfeld et al. 2016; Ubach et al. 2017; Huang et al. 2017; Tripathi et al. 2017; Cox et al. 2017; Andrews et al. 2018). These SEDs are available in the DSHARP data release. \n<!-- image --> \nin both L ∗ and ˙ M ∗ . The mean age is 1 Myr, although with considerable individual uncertainties (and various untreated systematics; see Soderblom et al. 2014). \nThe broadband spectral energy distributions (SEDs) for the sample are shown together in Figure 1. Relative to the median SED (normalized at 1.5 µ m) of larger samples of Class II targets (e.g., Ribas et al. 2017), RU Lup, AS 205, and AS 209 are in the top quartile (i.e., are over-luminous); the SEDs for HD 143006, SR 4, and DoAr 25 are relatively low in the near-infrared and high in the far-infrared (similar to, though not nearly as pronounced as, the typical transition disk SED); and the SEDs for DoAr 33 and WaOph 6 are in the bottom quartile. This diversity in the SEDs is one potential basis for future explorations of how the resolved emission distributions vary with relevant 'bulk' parameters (e.g., the amount of dust settling toward the disk midplane). \nWhile these sample targets do cover a range of properties, it is worth emphasizing that this range is not representative of the general population. The sample hosts tend to have earlier spectral types, and are accordingly more massive, luminous, and accreting more vigorously than stars at the peak of the initial mass function. This host bias enters implicitly with the sensitivity criterion, since we required a previous resolved measurement. The studies that provided those data are biased toward brighter continuum sources, which permeates to the host properties since the continuum luminosity scales steeply with M ∗ (Andrews et al. 2013; Mohanty et al. 2013) and ˙ M ∗ (Manara et al. 2016; Mulders et al. 2017). The same is true for multiple star systems: these were not explicitly excluded, but the selection criteria bias against them because close companions tend \nto reduce the system continuum emission (e.g., Jensen et al. 1994; Harris et al. 2012). \nThe bias in favor of targets with brighter continuum emission also translates into a preferential selection of larger disks, given the observed size-luminosity correlation (Tripathi et al. 2017; Tazzari et al. 2017; Andrews et al. 2018). This corresponding size bias is decidedly beneficial for achieving the DSHARP goals discussed in Section 1. As we noted above, the general theoretical predictions for substructure sizes are comparable to the gas pressure scale height ( h P ), which increases roughly linearly with disk radius. For a fixed resolution, it should be easier to identify and characterize the larger substructures expected at larger disk radii. \nTo roughly quantify these biases, we can make a comparison between targets that are more representative of the general disk population and the average member of the DSHARP sample. A 'typical' target has a host star mass near the peak of the mass function ( M ∗ ≈ 0 . 3 M glyph[circledot] , or spectral type M3-M4) and continuum emission from its disk that is both relatively faint ( F ν ≈ 10-15 mJy; Ansdell et al. 2016; Cieza et al. 2019) and compact ( R eff ≈ 10-20 au, with the effective radius defined by Tripathi et al. 2017; see also Andrews et al. 2018). The DSHARP averages are M ∗ ≈ 0 . 8 M glyph[circledot] (spectral type K7), F ν ≈ 150 mJy, and R eff ≈ 50 au; only the sample extremes stretch down toward 'typical' values. \nThese biases are difficult to mitigate for studies focused on finding and characterizing disk substructures, presuming their size scales are usually comparable to h P . The 'typical' disk is compact enough that h P for the radii where there is still continuum emission is smaller than the best resolutions available with ALMA. If this is the case, then we could be left probing only the extreme large end of substructures in 'typical' disks, making any assessments of prevalence difficult (i.e., failed searches for substructures would still permit plenty of h P -sized features to be present on sub-resolution scales). One option is to push to higher frequencies and thereby better resolution, but then high optical depths would limit the discovery-space to substructures in the form of dramatic depletions (e.g., very deep gaps) only.", '3. OBSERVATIONS': "The DSHARP ALMA observations were conducted in 2017 May-November as part of program 2016.1.00484.L. All measurements used the Band 6 receivers and correlated data from four spectral windows (SPWs) in dual polarization mode. The continuum was sampled in three SPWs, centered at 232.6, 245.0, and 246.9 GHz, each with 128 channels spanning 1.875 GHz (31.25 MHz per channel). The remaining SPW was centered at the 12 CO J =2 -1 rest frequency (230.538 GHz) and covered a bandwidth of 938 MHz in 3840 channels (488 kHz channel spacing, 0.64 km s -1 velocity resolution). The plan was to observe each target briefly in the C405 (hereafter 'compact') configuration, and also for ∼ 1 \nhour in the C40-8 or C40-9 (hereafter 'extended') configurations. The compact observations are necessary to recover emission on the larger angular scales that are not sampled in the extended configurations. The actual observing log is provided in Table 2. \nThe compact observations used an array with baseline lengths from 15 m to 1.1 km (a resolution of ∼ 0 . '' 25). The FWHM continuum and CO (per channel) emission scales are glyph[lessorsimilar] 2 '' , so spatial filtering should be negligible (cf., Wilner & Welch 1994). These observations cycled between nearby targets and totaled ∼ 12 minutes of integration time per source. A nearby phase calibrator was observed every 6 minutes; an additional 'check' calibrator (to assess the quality of phase transfer) was observed every 30 minutes. A bandpass and amplitude calibrator (sometimes the same quasar) were observed during each observing block. The log in Table 2 includes information about the observing conditions and calibrators. \nWe relied on archival ALMA observations of 5 targets (IM Lup, HD 142666, Elias 24, Elias 27, HD 163296) instead of obtaining new compact data, and folded in archival data for 3 other targets (HD 143006, AS 205, AS 209). Information about these datasets are compiled in Table 3. The setups, observing strategies, and weather conditions are described in the listed references. \nDue to a long stretch of inclement weather, the extended configuration observations were delayed until 2017 September, and continued through November. Despite the non-optimal scheduling for the DSHARP sample, nearly all of the targets were observed for two executions (often in different configurations) in good conditions, each with ∼ 35 minutes of on-source integration time. Sz 114, AS 205, and DoAr 25 each had only a single successful execution. The spectral setup was the same as for the compact datasets. Observations cycled between a single target and a nearby phase calibrator on 1 minute intervals, with a 'check' calibrator visited every 30 minutes. Bandpass and amplitude calibrators were observed in each execution block (see Table 2).", '4. CALIBRATION AND IMAGING': 'Since the DSHARP survey is among the first to collect a large volume of ALMA data on such long baselines, a substantial effort was made to explore various calibration strategies to enhance the data quality. The standard methodology we adopted is described here. The specific details on the calibration of datasets for individual targets (i.e., calibration scripts) are available in the DSHARP data release (see Section 5). All calibration tasks are performed with the CASA package (McMullin et al. 2007) and a small supplement of python tasks.', '4.1. Pipeline Calibration': "The first step was a standard ALMA pipeline calibration. This procedure was performed by ALMA staff separately for the compact and extended data, using CASA v4.7.2 or v5.1.1 for datasets that were processed be- \nTable 2. DSHARP Observing Log (ALMA Program 2016.1.00484.L) \nNote -Col. (1) Target name. Col. (2) UTC date and time at the start of the observations. Col. (3) ALMA configuration. Col. (4) Minimum and maximum baseline lengths. Col. (5) Number of antennas available. Col. (6) Target elevation range. Col. (7) Range of precipitable water vapor levels. Col. (8) From left to right, the quasars observed for calibrating the bandpass, amplitude scale, phase variations, and checking the phase transfer. Additional archival observations used in our analysis are compiled in Table 3. Table 2 is published in its entirety in the electronic edition of the journal. A portion is shown here for guidance regarding its form and content. \nTable 3. Archival ALMA Datasets Used by DSHARP \nNote -Col. (1) Target name. Col. (2) UTC date and time at the start of the observations. Col. (3) ALMA configuration. Col. (4) Range of baseline lengths. Col. (5) Number of antennas available. Col. (6) From left to right, the quasars observed for calibrating the bandpass, amplitude scale, phase variations, and checking the phase transfer. An entry of ' . . . ' indicates no calibrator was observed for checking the phase transfer. Col. (7) ALMA program ID. Col. (8) Original references for these datasets. Table 3 is published in its entirety in the electronic edition of the journal. A portion is shown here for guidance regarding its form and content. \nReferences -1 = Oberg et al. (2015), 2 = Cleeves et al. (2017), 3 = Pinte et al. (2018), 4 = Salyk et al. (2014), 5 = Dipierro et al. (2018), 6 = P'erez et al. (2016), 7 = Huang et al. (2016), 8 = Fedele et al. (2018), 9 = Flaherty et al. (2015), 10 = Isella et al. (2016). \nfore or after 2017 November, respectively. The pipeline imports the raw data and flags problematic scans, channels, or antennas. It then derives a table of system temperatures ( T sys ). Most of the DSHARP data have T sys ≈ 60-80 K; in the poorest conditions it reached 130 K, and in the best cases it was 50 K. Next, the pipeline adjusts the visibility phases according to water vapor radiometer (WVR) measurements. For the extended data, the WVR corrections improved the median RMS phase variations by a factor of ∼ 1.7, although individual datasets saw improvements between 1.2-3. The corrected RMS phase variations (far from the reference antenna) were typically 30 · (with a range ∼ 15-50 · ). The compact observations saw similar improvement factors (1.5-3.0) and RMS phase variations ( ∼ 10 · ). \nThe pipeline then performs a bandpass calibration, using the first quasar in the calibrator list in Table 2. It continues by setting the amplitude scale, using measurements of the second quasar in the Table 2 list. The flux density in each SPW for that quasar is determined from a power-law spectral model based on bi-monthly monitoring in ALMA Bands 3 and 7 ( ∼ 100 and 340 GHz) \nthat is tied to primary calibrators (planets or moons). Finally, the gain variations with time are corrected, with reference to repeated measurements of the nearby quasar listed third in the Table 2 calibrator list.", '4.2. Self-Calibration': "We next performed some substantial post-processing, with particular emphasis on combining datasets (from different array configurations and observations) and selfcalibrating the visibilities. We generally followed the homogenized strategy described below, using CASA v5.1.1 and a set of custom python routines. \nThe procedure started with the compact data. A pseudo-continuum dataset was created by flagging data within ± 25 km s -1 from the CO J =2 -1 line center and averaging into 125 MHz channels. The visibilities corresponding to each individual observation were imaged (Section 4.3) and checked to ensure consistent astrometric registration and flux calibration (if necessary, they are corrected; see Section 4.4). The individual datasets were then re-combined. Next, we performed a series of phase-only self-calibration iterations, stepping down the \nsolution interval (60, 30, 18, and 6 s). Reference antennas were selected based on data quality and proximity to the array center. When possible, we avoided combining SPWs (or scans) to correct for SPW-dependent gain variations. After each iteration, the data were imaged. A noise estimate was made in an annular region within a 4 . '' 25-radius circle centered on the target but excluding the image mask. This self-calibration sequence is stopped after reaching a solution interval on the record length (6 s) or if the peak SNR does not increase by > 5% from the previous iteration. Finally, we performed one iteration of amplitude self-calibration (for each SPW independently) on a scan interval ( ∼ 6 minutes). The (phase + amplitude) self-calibration provided a dramatic improvement in quality. The typical peak SNR increased by a factor of 3; the resulting noise was 30 µ Jy beam -1 (10 mK) for a ∼ 0 . '' 25 beam. The same procedure was applied to archival datasets. \nNext, we prepared the extended data as was described above, with an additional time-averaging to 6 s integrations (from the original 2 s records). The data for each individual extended observation were imaged and checked for misalignments and flux discrepancies. Once those are corrected (if necessary; see Section 4.4), the compact (already self-calibrated) and extended datasets were combined. The phases for this combined dataset were iteratively self-calibrated on solution intervals of { 900, 360, 180, 60, 30 s } (usually only the latter 3 are necessary). The SPWs were combined in this case to enhance the SNR on longer baselines. For antenna pairings with SNR ≤ 1 . 5 on these intervals, the selfcalibration solutions were not applied but the corresponding data were not flagged ( applymode='calonly' in the applycal task). The sequence was stopped when the peak SNR does not increase by > 5% and the map quality does not visually improve. One iteration of amplitude self-calibration was attempted on the starting interval of the phase self-calibration sequence. \nThis self-calibration of the combined datasets resulted in a typical improvement of 40% in the peak SNR, although there is a large range in benefits across the sample. The improvements are generally smaller here because the compact data were already self-calibrated and the extended data were taken in excellent conditions. The typical noise measured in the combined, selfcalibrated datasets is 10-20 µ Jy beam -1 (0.1-0.5 K). \nOnce the continuum self-calibration was satisfactory, the same gain tables are applied to the non-spectrallyaveraged visibilities (after any required astrometric and flux calibration adjustments) to obtain a corresponding calibrated measurement set for the region of the spectrum around the CO J =2 -1 emission line.", '4.3. Imaging During Self-Calibration': "Self-calibration uses continuum emission models assembled from the 'clean' components derived from interferometric imaging. We adopted a set of imaging stan- \ndards to homogenize that process. These were informed by considerable experimentation with the associated parameter choices. We explored alternative sets of deconvolution scales, clean thresholds, masks, and pixel sizes and found that reasonable other options had negligible influence on the end products of self-calibration. \nAll imaging was performed with the tclean task. For the compact data, we imaged out to the primary beam FWHM (26 '' ) with 30 mas pixels ( ∼ 10 per synthesized beam FWHM, θ b ) to check for problematic background sources. Finding nothing of concern, we used 9 '' -wide images with 3 mas pixels (again, ∼ 10 pixels per θ b ) for the combined datasets. We used the multi-scale, multifrequency synthesis (assuming a flat spectrum) deconvolution mode (Cornwell 2008) with a Briggs robust =0.5 weighting scheme. Elliptical masks were designed to reflect the target geometry (aspect ratio, position angle) and pad the outer reaches of the emission distribution. The adopted (Gaussian) deconvolution scales are targetdependent, but always include a point-like contribution and scales comparable to θ b and 2-3 × θ b ; additional scales (increasing by factors of 2-3) could be selected up to the mask radius. The algorithm was halted on thresholds; 3 × the noise early in the self-calibration sequence, and 2 × the noise for the last phase-only step and the amplitude self-calibration. \nSpecial effort was made to verify that sidelobes in the point spread function (PSF, or 'dirty' beam) do not corrupt the self-calibration. The extended ALMA configurations place antennas along three distinct arms (set by the site topography). The corresponding spatial frequency coverage generates complicated PSF features, with sidelobes up to ∼ 30%. Figure 2 illustrates the impact, showing the connections between the sampling function ( u , v coverage), PSF, and image for different configurations and weighting schemes. We vetted the effects of those PSF features on self-calibration by repeating the process for different combinations of weighting schemes and tapers. Coupling lower robust values with tapers can mitigate PSF artifacts while maintaining resolution, but at a substantial SNR cost. Direct comparisons (of both visibilities and images) between these variants and the standard methodology outlined above demonstrated that the PSF features had negligible impact on the self-calibration. 4 \nWhile the effects on self-calibration are minimal, the resulting images can still exhibit PSF-related artifacts. One of the more interesting is the imprint of a hexagonal structure on emission rings (e.g., second image from left, bottom row of Figure 2), produced by convolution with a 'spoked' PSF (a consequence of the the extended \n4 The HD 163296 disk is the one exception (albeit a quite modest one): we find ∼ 10% SNR improvements (relative to the standard) when self-calibration is conducted for images with robust =-0.5, due to the combination of the target emission distribution and the unusual spatial frequency coverage from the archival data. \nFigure 2. Illustration of the effects of spatial frequency coverage and visibility weighting on PSF structure and image morphology, for the RU Lup disk. The top panels show the observed u , v coverage. The middle panels show the corresponding PSF structures, annotated with the corresponding robust weighting parameter. The bottom panels show the corresponding images, on the same T b scale, created from subsets of the end product of the self-calibration. FWHM beam dimensions are marked in the lower left corners of each image. \n<!-- image --> \nALMA configuration arms). As demonstrated in the bottom right panel, this can usually be minimized with an appropriate visibility weighting and/or tapering.", '4.4. Astrometric and Flux Scale Alignment': "Half the sample targets show clear spatial offsets between their emission centers in different observations. For the larger of these shifts ( ∼ 100 mas), the cause is proper motion (especially when using archival data); in other cases, smaller (10-30 mas) mismatches might instead be attributed to instrumental or atmospheric artifacts. Combining these datasets without correcting these shifts creates blurred (or even double) images, which is problematic when they are used as initial self-calibration models. The solution is to simply adjust the visibility phases to shift into alignment. We measure emission centroid positions with Gaussian fits in the image plane for each individual observation and calculate the offsets relative to the highest quality extended dataset. The fixvis task then implements the appropriate phase adjustments. In cases where the ob- \nrvations have different pointing centers, we manually reconcile them with the fixplanets task. \nWe also routinely found mismatches in the amplitude scales among different observations of a target. Some experimentation showed that noticeably improved self-calibration results were obtained if the relative flux scales between observations were consistent within 5%. To quantify any mismatches, we inspected the deprojected (according to the Gaussian fit geometries noted above), azimuthally-averaged visibilities from different datasets on 200-500 k λ baseline lengths (at lower spatial frequencies, the extended configuration data are too sparse, and at higher frequencies the averages are more strongly affected by low SNR and phase noise). \nThese mismatches are caused by inaccurate flux calibration. The claimed calibration accuracy is ∼ 10%, although the adopted methodology for estimating calibrator fluxes (interpolation in time and frequency) can lead to some added uncertainty. About a third of the sample had 5-10% mismatches, but the majority exhibited 15-25% discrepancies for at least one dataset. In some \nTable 4. DSHARP Fiducial Continuum Image Properties \nNote -Col. (1) Target name. Col. (2) Mean frequency. Col. (3) Synthesized beam FWHM and position angle. Col. (4) RMS noise in the map, as described in Section 4.3. Col. (5) Peak intensity in the map. Note that noise and peak brightness temperatures are calculated assuming the Rayleigh-Jeans limit. Col. (6) Integrated flux density inside the image mask. Col. (7) Briggs robust value. Col. (8) FWHM and position angle of the taper (if applicable). \nReferences -II = Huang et al. (2018a), III = Huang et al. (2018b), IV = Kurtovic et al. (2018), VIII = Guzm'an et al. (2018), IX = Isella et al. (2018), X = P'erez et al. (2018). \ncases, these were tracked down to a bookkeeping issue: the data were pipeline-processed before a relevant calibrator catalog update. Some 2017 November datasets that used J1427-4206 as the calibrator were problematic. There is no obvious error in the calibrator catalog, so the issue must be with the interpolation: perhaps this quasar flared or changed its spectrum between catalog entries. Regardless of the cause, these misalignments were rectified. We selected a reference dataset and used the gaincal task to re-scale the outlier datasets.", '4.5. Fiducial Images': 'After the calibration was complete, we synthesized a set of fiducial images for further analysis. The continuum imaging followed the methodology outlined in Section 4.3, but was tailored to individual sources with the aim of minimizing PSF artifacts. In many cases, this involved adopting a visibility weighting scheme that traded SNR for resolution, as well as a visibility taper to improve the PSF symmetry. Table 4 lists the basic parameters and resulting properties of these fiducial images. A gallery of the continuum images are shown in Figure 3. Small-scale substructures are notable in all of the DSHARP targets, often with compact (FWHM glyph[lessorsimilar] 10 au) dimensions. Figure 4 emphasizes the utility of push- \ning the ALMA resolution for recovering such features in one particularly illustrative example. \nWe also synthesized channel maps of the CO J =2 -1 emission following the basic steps outlined above. The self-calibrated CO visibilities were continuumsubtracted and imaged in LSRK velocity channels at roughly the native channel spacing (0.35 km s -1 ; the actual velocity resolution is about two channels, due to Hanning smoothing in the ALMA correlator). The DSHARP data are generally not sensitive enough to reconstruct useful channel maps of the emission line at the best available resolution. We compromised by increasing the relative weight of shorter baselines and employing a modest taper. Table 5 lists the imaging parameters, and Figure 5 shows the channel maps. For many of the targets, the CO channel maps exhibit partially recovered large-scale emission structures from the ambient cloud material. These are noted in Table 5 to prevent confusion in the interpretation of extended emission features in some cases (e.g., Elias 24 and WSB 52 are particularly problematic cases).', '5. DATA RELEASE': "One key inspiration for conducting the DSHARP survey was to provide a set of resources to the com- \nFigure 3. A gallery of 240 GHz (1.25 mm) continuum emission images for the disks in the DSHARP sample. Beam sizes and 10 au scalebars are shown in the lower left and right corners of each panel, respectively. All images are shown with an asinh stretch to reduce the dynamic range (accentuate fainter details without over-saturating the bright emission peaks). For more quantitative details regarding the image dimensions and intensity scales, see Huang et al. (2018a) and Kurtovic et al. (2018). \n<!-- image --> \nFigure 4. The deprojected, azimuthally-averaged radial brightness temperature profile for the 240 GHz continuum emission from the AS 209 disk (see Huang et al. 2018a; Guzm'an et al. 2018, for more details). The corresponding image is shown in the bottom row of Figure 3, second from right. The PSF profile (resolution) is marked in black in the upper right corner, along with a gray Gaussian profile that has FWHM = 10 au, to illustrate that the disk substructures typically have compact dimensions. \n<!-- image --> \nmunity that can seed and develop a range of related work. To that end, we have released a suite of data products that go beyond the standard contents in the ALMA archive. This release is available online at https://almascience.org/alma-data/lp/DSHARP. It includes: (1) CASA scripts and associated python modules used to calibrate and image the data; (2) fully calibrated continuum and CO measurement sets (visibility datafiles); (3) continuum images and CO channel maps; and (4) some secondary products (radial intensity profiles, SED data). With this data release and the standard ALMA archive products, the community has the access needed to both reproduce and expand on the efforts detailed in the initial series of DSHARP articles.", '6. OVERVIEW: INITIAL DSHARP RESULTS': "This article has detailed the scientific motivations behind DSHARP, introduced the survey strategy and sample, described the observations and calibration process, and presented the resulting products as part of our data release. It is also the first in a series of articles that explore and analyze the data in more detail. The principal DSHARP conclusions can be summarized as follows: \n- · Continuum substructures are ubiquitous in this sample, as can be deduced from Figure 3. Small-scale emission features are found at effectively any disk radius, from 5 au out to more than 150 au.\n- · The most common form of these substructures are concentric bright rings and dark gaps. There are no obvious patterns in their distributions or connections to the stellar host properties. There are hints of ring/gap \nsubstructures that are obfuscated due to their smaller size scales (relative to the DSHARP resolution) and/or their modest amplitudes with respect to an optically thick background in the inner disk. Measurements of the rings and gaps, as well as a more detailed exploration of their potential origins and associated issues, are presented by Huang et al. (2018a). \n- · While less common, the spiral morphologies identified for a subset of disks in the DSHARP sample are striking. For the cases with apparently single host stars (IM Lup, Elias 27, and WaOph 6), the spiral patterns are complex and appear to be superposed with rings and gaps. Their emission distributions and potential origins are characterized by Huang et al. (2018b).\n- · For the two known multiple star systems in the DSHARP sample, HT Lup and AS 205, the disks around the primary stars show clear two-armed spirals and complicated CO distributions that are indicative of strong dynamical interactions. The circumstellar material in these systems is studied by Kurtovic et al. (2018).\n- · Azimuthal asymmetries are rare in this sample. Substantial deviations from axisymmetry (or point symmetry for the spirals) are only identified in two cases. The disks around HD 143006 and HD 163296 show small, arc-shaped features in otherwise emission-depleted regions (i.e., beyond the continuum disk edge and in a gap, respectively). The properties and potential origins of these special cases are scrutinized by P'erez et al. (2018) and Isella et al. (2018), respectively.\n- · In some cases, the continuum emission can be decomposed into only small-scale substructures. The AS 209 disk is a particularly compelling example. Guzm'an et al. (2018) quantify its substructures and highlight an important point: there are analogous features lurking in the gas (even as traced by optically thick 12 CO), at radii well beyond the extent of the continuum emission.\n- · The ring substructure sizes and amplitudes suggest that these features can be understood as dust trapped in axisymmetric gas pressure bumps. Dullemond et al. (2018) demonstrate this conclusion and derive a lower limit on the strength of the turbulence in the disk. These and other related analyses are guided by a fiducial dust model developed by Birnstiel et al. (2018).\n- · A new suite of hydrodynamics simulations by Zhang et al. (2018) suggest that dynamical interactions between low-mass (sub-Jupiter) planets and their local disk material are plausible explanations of the observed ring/gap substructures. Assuming this is the case, those simulations are used to reconstruct the associated planet population in the mass - semimajor axis plane. \nThere is, of course, much more to learn from the DSHARP dataset. Our hope is that this preliminary foray not only provides useful results and motivation for \nTable 5. DSHARP Fiducial CO Datacube Properties \nNote -Col. (1) Target name. Col. (2) Synthesized beam FWHM and position angle. Col. (3) RMS noise per channel, measured as described in Section 4.3. HD 143006 and HD 163296 are imaged with 0.32 km s -1 channels; for all other targets, we used 0.35 km s -1 channels. Col. (4) Peak intensity. Note that noise and peak brightness temperatures are calculated assuming the Rayleigh-Jeans limit. Col. (5) Briggs robust value. Col. (6) FWHM and position angle of the adopted taper (if applicable). Col. (7) Comments on issues with the channel maps, including degree of contamination from the ambient molecular cloud and the presence of non-disk features. \nReferences -III Huang et al. (2018b). IV Kurtovic et al. (2018). VIII Guzm'an et al. (2018). IX Isella et al. (2018). X P'erez et al. (2018). \nmany other studies, but also lays some technical groundwork for designing and calibrating future ALMA surveys of disks at very high angular resolution. \nWe thank Erica Keller and Tony Remijan for their pipeline calibration efforts, the ALMA staff for pushing through the observations even after the re-configuration delays, Todd Hunter and Crystal Brogan for some useful technical consultations, Ian Czekala for his advice on measuring stellar parameters, and an anonymous reviewer for thoughtful suggestions. S.A. and J.H. acknowledge support from the National Aeronautics and Space Administration under grant No. 17XRP17 2-0012 issued through the Exoplanets Research Program. J.H. acknowledges support from the National Science Foundation Graduate Research Fellowship under Grant No. DGE-1144152. L.P. acknowledges support from CONICYT project Basal AFB-170002 and from FCFM/U. de Chile Fondo de Instalaci'on Acad'emica. A.I. acknowledges support from the National Aeronautics and Space Administration under grant No. NNX15AB06G issued through the Origins of Solar Systems program, and from the National Science Foundation under grant No. AST-1715719. C.P.D. acknowledges support by the German Science Foundation \n(DFG) Research Unit FOR 2634, grants DU 414/22-1 and DU 414/23-1. V.V.G. and J.C acknowledge support from the National Aeronautics and Space Administration under grant No. 15XRP15 20140 issued through the Exoplanets Research Program. T.B. acknowledges funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme under grant agreement No. 714769. M.B. acknowledges funding from ANR of France under contract number ANR-16-CE31-0013 (Planet Forming disks). Z.Z. and S.Z. acknowledge support from the National Aeronautics and Space Administration through the Astrophysics Theory Program with Grant No. NNX17AK40G and the Sloan Research Fellowship. L.R. acknowledges support from the ngVLA Community Studies program, coordinated by the National Radio Astronomy Observatory, which is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. This paper makes use of the following ALMA data: ADS/JAO.ALMA #2016.1.00484.L, ADS/JAO.ALMA #2011.0.00531.S, ADS/JAO.ALMA #2012.1.00694.S, ADS/JAO.ALMA #2013.1.00226.S, ADS/JAO.ALMA #2013.1.00366.S, ADS/JAO.ALMA #2013.1.00498.S, ADS/JAO.ALMA #2013.1.00631.S, ADS/JAO.ALMA #2013.1.00798.S, \nADS/JAO.ALMA #2015.1.00486.S, ADS/JAO.ALMA #2015.1.00964.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. This work presents results from the European Space Agency (ESA) space mission Gaia . Gaia data are being processed by the Gaia Data Processing and Analysis Consortium (DPAC). Funding for the DPAC is provided by national institutions, in particular the institutions participating in the Gaia MultiLateral Agreement (MLA). The Gaia mission website is https://www.cosmos.esa.int/gaia. The Gaia archive website is https://archives.esac.esa.int/gaia. \nFacilities: ALMA \nSoftware: CASA (McMullinetal.2007), Numpy (VanDer Waltetal.2011), Matplotlib (Hunter2007), Astropy (AstropyCollaborationetal.2018), ScottiePippen (Czekala et al. 2016, https://github.com/iancze/ScottiePippen). \nFig. Set 5. DSHARP CO J =2 -1 Channel Maps \nFigure 5.3. Channel maps of the 12 CO J =2 -1 line emission from the IM Lup disk. \n<!-- image --> \nFigure 5.1. Channel maps of the 12 CO J =2 -1 line emission from the HT Lup disk. \n<!-- image --> \n<!-- image --> \nFigure 5.2. Channel maps of the 12 CO J =2 -1 line emission from the GW Lup disk. \n<!-- image --> \nFigure 5.4. Channel maps of the 12 CO J =2 -1 line emission from the RU Lup disk. \nFigure 5.7. Channel maps of the 12 CO J =2 -1 line emission from the MY Lup disk. \n<!-- image --> \nFigure 5.5. Channel maps of the 12 CO J =2 -1 line emission from the Sz 114 disk. \n<!-- image --> \n<!-- image --> \nFigure 5.6. Channel maps of the 12 CO J =2 -1 line emission from the Sz 129 disk. \n<!-- image --> \nFigure 5.8. Channel maps of the 12 CO J =2 -1 line emission from the HD 142666 disk. \nFigure 5.9. Channel maps of the 12 CO J =2 -1 line emission from the HD 143006 disk. \n<!-- image --> \nFigure 5.10. Channel maps of the 12 CO J =2 -1 line emission from the AS 205 disk. \n<!-- image --> \nFigure 5.11. Channel maps of the 12 CO J =2 -1 line emission from the SR 4 disk. \n<!-- image --> \nFigure 5.12. Channel maps of the 12 CO J =2 -1 line emission from the Elias 20 disk. \n<!-- image --> \nFigure 5.13. Channel maps of the 12 CO J =2 -1 line emission from the DoAr 25 disk. \n<!-- image --> \nFigure 5.16. Channel maps of the 12 CO J =2 -1 line emission from the DoAr 33 disk. \n<!-- image --> \nFigure 5.14. Channel maps of the 12 CO J =2 -1 line emission from the Elias 24 disk. \n<!-- image --> \nFigure 5.15. Channel maps of the 12 CO J =2 -1 line emission from the Elias 27 disk. \n<!-- image --> \nFigure 5.19. Channel maps of the 12 CO J =2 -1 line emission from the AS 209 disk. \n<!-- image --> \nFigure 5.17. Channel maps of the 12 CO J =2 -1 line emission from the WSB 52 disk. \n<!-- image --> \n<!-- image --> \nFigure 5.18. Channel maps of the 12 CO J =2 -1 line emission from the WaOph 6 disk. \n<!-- image --> \nFigure 5.20. Channel maps of the 12 CO J =2 -1 line emission from the HD 163296 disk.", 'REFERENCES': "Tripathi, A., Andrews, S. M., Birnstiel, T., et al. 2018, ApJ, 861, 64, doi: 10.3847/1538-4357/aac5d6 Ubach, C., Maddison, S. T., Wright, C. M., et al. 2017, MNRAS, 466, 4083, doi: 10.1093/mnras/stx012 van Boekel, R., Henning, T., Menu, J., et al. 2017, ApJ, 837, 132, doi: 10.3847/1538-4357/aa5d68 van der Marel, N., van Dishoeck, E. 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J. 1994, ApJ, 427, 898, doi: 10.1086/174195 Wright, E. L., Eisenhardt, P. R. M., Mainzer, A. K., et al. 2010, AJ, 140, 1868, doi: 10.1088/0004-6256/140/6/1868 Youdin, A. N., & Goodman, J. 2005, ApJ, 620, 459, doi: 10.1086/426895 Youdin, A. N., & Shu, F. H. 2002, ApJ, 580, 494, doi: 10.1086/343109 Zhang, K., Bergin, E. A., Blake, G. A., et al. 2016, ApJL, 818, L16, doi: 10.3847/2041-8205/818/1/L16 Zhang, K., Isella, A., Carpenter, J. M., & Blake, G. A. 2014, ApJ, 791, 42, doi: 10.1088/0004-637X/791/1/42 Zhang et al. 2018, ApJL, in press (DSHARP VII) Zhu, Z., Nelson, R. P., Dong, R., Espaillat, C., & Hartmann, L. 2012, ApJ, 755, 6, doi: 10.1088/0004-637X/755/1/6 \nTable 2 . DSHARP Observing Log (ALMA Program 2016.1.00484.L) \nTable 2 (continued) \nNote -Basic information from the individual execution blocks conducted as part of ALMA Program 2016.1.00484.L. Col. (1) Target name. Col. (2) UTC date and time for the start of the execution block. Col. (3) ALMA configuration. Col. (4) Minimum and maximum baseline lengths. Col. (5) Number of antennas available. Col. (6) Target elevation range. Col. (7) Range of precipitable water vapor levels. Col. (8) From left to right, the quasars observed for calibrating the bandpass, amplitude scale, phase variations, and checking the phase transfer. Additional archival observations used in our analysis are compiled in Table 3. \nTable 3. Archival ALMA Datasets Used by DSHARP \nNote -Col. (1) Target name. Col. (2) UTC date and time at the start of the observations. Col. (3) ALMA configuration. Col. (4) Range of baseline lengths. Col. (5) Number of antennas available. Col. (6) From left to right, the quasars observed for calibrating the bandpass, amplitude scale, phase variations, and checking the phase transfer. An entry of ' . . . ' indicates no calibrator was observed for checking the phase transfer. Col. (7) ALMA program ID. Col. (8) Original references for these datasets. Table 3 is published in its entirety in the electronic edition of the journal. A portion is shown here for guidance regarding its form and content. \nReferences -1 = Oberg et al. (2015), 2 = Cleeves et al. (2017), 3 = Pinte et al. (2018), 4 = Salyk et al. (2014), 5 = Dipierro et al. (2018), 6 = P'erez et al. (2016), 7 = Huang et al. (2016), 8 = Fedele et al. (2018), 9 = Flaherty et al. (2015), 10 = Isella et al. (2016)."} |
2024ApJ...972...92J | To understand how galaxies reionized the Universe we must determine how the escape fraction of Lyman continuum LyC photons f SUBescSUB depends on galaxy properties. Using the z 0.3 Lowredshift Lyman Continuum Survey LzLCS we develop and analyze new multivariate predictors of f SUBescSUB. These predictions use the Cox proportional hazards model a survival analysis technique that incorporates both detections and upper limits. Our best model predicts the LzLCS f SUBescSUB detections with an rms scatter of 0.31 dex better than singlevariable correlations. According to ranking techniques the most important predictors of f SUBescSUB are the equivalent width EW of Lymanseries absorption lines and the UV dust attenuation which track lineofsight absorption due to H I and dust. The H I absorption EW is uniquely crucial for predicting f SUBescSUB for the strongest LyC emitters which show properties similar to weaker LyC emitters and whose high f SUBescSUB may therefore result from favorable orientation. In the absence of H I information star formation rate surface density SUBSFRSUB and O IIIO II ratio are the most predictive variables and highlight the connection between feedback and f SUBescSUB. We generate a model suitable for z gt 6 which uses only the UV slope SUBSFRSUB and O IIIO II. We find that SUBSFRSUB is more important in predicting f SUBescSUB at higher stellar masses whereas O IIIO II plays a greater role at lower masses. We also analyze predictions for other parameters such as the ionizingtononionizing flux ratio and Ly escape fraction. These multivariate models represent a promising tool for predicting f SUBescSUB at high redshift. SUPSUP Based on observations made with the NASAESA Hubble Space Telescope obtained at the Space Telescope Science Institute which is operated by the Association of Universities for Research in Astronomy Inc. under NASA contract NAS 526555. These observations are associated with programs GO15626 GO13744 GO14635 GO15341 and GO15639. | 2024-09-01T00:00:00Z | ['10.48550/arXiv.2406.10171', '2024arXiv240610171J', '10.3847/1538-4357/ad58b9', 'arXiv:2406.10171', '2024ApJ...972...92J'] | ['Astrostatistics', 'Reionization', 'High-redshift galaxies', 'Starburst galaxies', 'Interstellar medium', 'Ultraviolet astronomy', 'Radiative transfer', '1882', '1383', '734', '1570', '847', '1736', '1335', 'Astrophysics - Astrophysics of Galaxies'] | Multivariate Predictors of Lyman Continuum Escape. I. A Survival Analysis of the Lowredshift Lyman Continuum Survey | 2,024 | 167 | 0.59 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF'] | 14 | https://arxiv.org/pdf/2406.10171.pdf | {'Multivariate Predictors of LyC Escape I: A Survival Analysis of the Low-redshift Lyman Continuum Survey ∗': "Anne E. Jaskot, 1 Anneliese C. Silveyra, 1, 2 Anna Plantinga, 3 Sophia R. Flury, 4 Matthew Hayes, 5 John Chisholm, 6 Timothy Heckman, 7 Laura Pentericci, 8 Daniel Schaerer, 9 Maxime Trebitsch, 10 Anne Verhamme, 9, 11 Cody Carr, 12, 13 Henry C. Ferguson, 14 Zhiyuan Ji, 15 Mauro Giavalisco, 4 Alaina Henry, 14 Rui Marques-Chaves, 9 Goran Ostlin, 5 Alberto Saldana-Lopez, 5 Claudia Scarlata, 16 G'abor Worseck, 17 and Xinfeng Xu 18 \n1 Department of Astronomy, Williams College, Williamstown, MA 01267, USA \n2 Department of Physics, University of Nevada, Reno, NV 89557, USA \n3 Department of Mathematics & Statistics, Williams College, Williamstown, MA 01267, USA \n4 Department of Astronomy, University of Massachusetts Amherst, Amherst, MA 01002, USA \n5 Department of Astronomy, Oskar Klein Centre, Stockholm University, AlbaNova, SE-10691 Stockholm, Sweden \n6 Department of Astronomy, University of Texas at Austin, Austin, TX 78712, USA \n7 Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA \n8 INAF - Osservatorio Astronomico di Roma, via Frascati 33, 00078, Monteporzio Catone, Italy \n9 Observatoire de Gen'eve, Universit'e de Gen'eve, Chemin Pegasi 51, 1290 Versoix, Switzerland \n10 Astronomy, Kapteyn Astronomical Institute, Landleven 12, 9747 AD Groningen, The Netherlands \n- Univ. Lyon, Univ. Lyon 1, ENS de Lyon, CNRS, Centre de Recherche Astrophysique de Lyon UMR5574, 69230 Saint-Genis-Laval, France \n11 \n12 Center for Cosmology and Computational Astrophysics, Institute for Advanced Study in Physics, Zhejiang University, Hangzhou 310058, China \n13 Institute of Astronomy, School of Physics, Zhejiang University, Hangzhou 310058, China \n14 Space Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD 21218, USA \n15 Steward Observatory, University of Arizona, Tucson, AZ 85721, USA \n16 Minnesota Institute for Astrophysics, School of Physics and Astronomy, University of Minnesota, 316 Church St. SE, Minneapolis, MN 55455, USA \n17 VDI/VDE Innovation+Technik, Berlin, Germany \n18 Center for Interdisciplinary Exploration and Research in Astrophysics, Northwestern University, Evanston, IL 60201, USA", 'ABSTRACT': 'To understand how galaxies reionized the universe, we must determine how the escape fraction of Lyman Continuum (LyC) photons ( f esc ) depends on galaxy properties. Using the z ∼ 0 . 3 Low-redshift Lyman Continuum Survey (LzLCS), we develop and analyze new multivariate predictors of f esc . These predictions use the Cox proportional hazards model, a survival analysis technique that incorporates both detections and upper limits. Our best model predicts the LzLCS f esc detections with a rootmean-square (RMS) scatter of 0.31 dex, better than single-variable correlations. According to ranking techniques, the most important predictors of f esc are the equivalent width (EW) of Lyman-series absorption lines and the UV dust attenuation, which track line-of-sight absorption due to H i and dust. The H i absorption EW is uniquely crucial for predicting f esc for the strongest LyC emitters, which show properties similar to weaker LyC emitters and whose high f esc may therefore result from favorable orientation. In the absence of H i information, star formation rate surface density (Σ SFR ) and [O iii ]/[O ii ] ratio are the most predictive variables and highlight the connection between feedback and f esc . We generate a model suitable for z > 6, which uses only the UV slope, Σ SFR , and [O iii ]/[O ii ]. We find that Σ SFR is more important in predicting f esc at higher stellar masses, whereas [O iii ]/[O ii ] plays a greater role at lower masses. We also analyze predictions for other parameters, such as the \nionizing-to-non ionizing flux ratio and Ly α escape fraction. These multivariate models represent a promising tool for predicting f esc at high redshift.', '1. INTRODUCTION': "Star-forming galaxies likely caused one of the most significant transformations in cosmic history: the reionization of hydrogen in the intergalactic medium (IGM) at z ≳ 6. Current constraints on the galaxy and quasar luminosity functions at z > 6 and on galaxies' ionizing photon production efficiencies favor stars as the dominant source of ionizing, Lyman continuum (LyC) photons (e.g., Bouwens et al. 2015, 2016; Finkelstein et al. 2015, 2019; Robertson et al. 2015; Ricci et al. 2017; Shen et al. 2020; Faucher-Gigu'ere 2020; De Barros et al. 2019; Endsley et al. 2021). Exactly which star-forming galaxies contribute these LyC photons remains unclear, however. Studies disagree as to whether low-, intermediate-, or high-luminosity galaxies dominate reionization (e.g., Razoumov & Sommer-Larsen 2010; Wise et al. 2014; Paardekooper et al. 2015; Finkelstein et al. 2019; Cain et al. 2021; Begley et al. 2022; Rosdahl et al. 2022; Saldana-Lopez et al. 2023; Wyithe & Loeb 2013; Naidu et al. 2020; Izotov et al. 2021; Ma et al. 2020; Matthee et al. 2022) or whether other properties such as concentrated star formation, nebular ionization, or starburst age demarcate the LyC-emitting galaxy population (e.g., Heckman et al. 2001; Clarke & Oey 2002; Alexandroff et al. 2015; Sharma et al. 2016; Marchi et al. 2018; Jaskot & Oey 2013; Nakajima & Ouchi 2014; Izotov et al. 2018b; Zastrow et al. 2013; Ma et al. 2015; Trebitsch et al. 2017; Naidu et al. 2022). \nObservations with the James Webb Space Telescope (JWST) are revealing that proposed LyC-emitting galaxy populations exist at high redshift (e.g., Schaerer et al. 2022a; Williams et al. 2023; Endsley et al. 2023; Fujimoto et al. 2023; Tang et al. 2023; Mascia et al. 2024; Atek et al. 2024), although it remains to be seen whether they are sufficiently numerous at z > 6. However, identifying the galaxies responsible for reionization also requires knowing f esc , the fraction of LyC photons that escape into the IGM. The value of f esc for z > 6 galaxies is not known, and its physical connection with galaxy properties may be complex. LyC escape can depend on a galaxy's interstellar gas geometry, dust content, stellar feedback, and gravitational potential and may vary with time (e.g., Heckman et al. 2001, 2011; Wise et al. 2014; Ma et al. 2015; Sharma et al. 2016; Trebitsch et al. 2017; Chisholm et al. 2018; Barrow et al. 2020; Gazagnes et al. 2020; Mauerhofer et al. 2021; Saldana-Lopez et al. 2022). \nLyC escape is an inherently multi-parameter problem. \nBecause of attenuation in the IGM, detecting the LyC flux from galaxies becomes unlikely above z ∼ 4 (e.g., \nInoue et al. 2014). As a result, the astronomy community has relied on lower-redshift samples in order to investigate f esc and its dependence on galaxy properties. Studies of z ∼ 2 -4 galaxies find that f esc may be enhanced in galaxies with lower UV luminosities, lower dust attenuation, higher Ly α equivalent widths (EWs), strong [O iii ] λ 5007 emission, and/or compact sizes (e.g., Marchi et al. 2018; Steidel et al. 2018; Bassett et al. 2019; Fletcher et al. 2019; Nakajima et al. 2020; Begley et al. 2022; Saxena et al. 2022). More local samples, at z ∼ 0 . 3, likewise identify strong Ly α emission, elevated [O iii ]/[O ii ] ratios, high star formation rate surface densities (Σ SFR ), and low dust attenuation as characteristics of LyC emitters (LCEs) (e.g., Borthakur et al. 2014; Izotov et al. 2016b, 2018b; Verhamme et al. 2017; Chisholm et al. 2018). Nevertheless, f esc can have a range of values, even for galaxies with similar properties (e.g., Izotov et al. 2018b; Vanzella et al. 2016; Rutkowski et al. 2017; Fletcher et al. 2019; Bian & Fan 2020; Marques-Chaves et al. 2021). \nThe recently completed Low-redshift Lyman Continuum Survey (LzLCS; Flury et al. 2022a) offers a new opportunity to investigate all these physical properties simultaneously in a large statistical sample of LCEs and non-emitters. Through the LzLCS and archival programs (Izotov et al. 2016a,b, 2018a,b, 2021; Wang et al. 2019), 89 galaxies at z ∼ 0 . 3 have LyC observations from the Hubble Space Telescope (HST) Cosmic Origins Spectrograph (COS), and 50 of these galaxies are detected in the LyC. This combined sample, hereafter the LzLCS+, covers a wide range of luminosities, metallicities, H β and Ly α EWs, [O iii ]/[O ii ], and Σ SFR (Flury et al. 2022a), which enables it to more clearly reveal the trends and scatter between f esc and galaxy properties. By quantifying the relationship between f esc and observables, the LzLCS+ can generate predictions for f esc in z > 6 galaxies, where direct LyC detections are inaccessible. \nThe results from the LzLCS+ confirm that f esc correlates with a variety of galaxy properties, from line-ofsight measurements such as H i covering fraction, dust attenuation, and Ly α escape fraction to global properties such as [O iii ]/[O ii ] and Σ SFR (e.g., Saldana-Lopez et al. 2022; Flury et al. 2022b; Chisholm et al. 2022; Xu et al. 2023). The latter properties hint at the possible role of mechanical or radiative feedback in creating low optical depth sightlines that allow ionizing photons to escape. Nevertheless, all correlations between f esc and observable properties show significant scatter \n(Wang et al. 2021; Saldana-Lopez et al. 2022; Flury et al. 2022b; Chisholm et al. 2022; Xu et al. 2023). \nA combination of properties might more accurately predict f esc than a single variable alone. For instance, several studies have generated multivariate predictions of Ly α properties for low- and high-redshift galaxies (e.g., Yang et al. 2017; Trainor et al. 2019; Runnholm et al. 2020). Runnholm et al. (2020) present such an analysis for Ly α luminosity using galaxies in the Lyman Alpha Reference Sample (LARS; Hayes et al. 2013; Ostlin et al. 2014). By applying multivariate regression using physical and observable galaxy properties, they predict the Ly α luminosities of the LARS galaxies with a root-mean-square residual scatter of 0.2-0.3 dex. \nSimilarly, Maji et al. (2022) and Choustikov et al. (2024) perform multivariate linear fits for f esc using simulated galaxies from the SPHINX cosmological simulations (Rosdahl et al. 2018). Maji et al. (2022) find that a combination of four variables (escaping Ly α luminosity, gas mass, gas metallicity, and recent star formation rate) can explain 85% of the variance in escaping LyC luminosity, while three variables (escaping Ly α luminosity, recent star formation rate, and gas mass) explain 66% of the variance in f esc . However, some of the relevant variables (e.g., Ly α luminosity and gas mass) will be difficult or impossible to obtain for many galaxies at z > 6. In contrast, Choustikov et al. (2024) focus on multivariate f esc predictions using observable properties, such as [O iii ]/[O ii ], UV magnitude, and UV slope. While promising, the Maji et al. (2022) and Choustikov et al. (2024) simulation results require further testing and confirmation using observational samples. \nOther recent studies have taken a more empirical approach by using the z ∼ 0 . 3 LyC observations to derive relationships between f esc and observable properties. Lin et al. (2024) fit for the probability of a galaxy having detectable vs. undetected f esc as a function of UVmagnitude, UV slope, and [O iii ]/[O ii ] and use these results to assess the likelihood of LyC escape from galaxies at z > 6. Mascia et al. (2023) develop a model that predicts f esc based on some of the strongest correlating variables in the LzLCS: β 1550 , UV half-light radius, and [O iii ]/[O ii ]. \nIn this paper, we investigate a variety of empirical multivariate predictions for f esc using the LzLCS+, the largest observational sample of LyC measurements at low redshift, which enables both reliable individual f esc measurements and comprehensive rest-frame UV and optical ancillary data. Whereas the Ly α data used by Runnholm et al. (2020) includes only Ly α -emitters and the simulated LyC data used by Maji et al. (2022) and Choustikov et al. (2024) include measurements down to \nf esc = 0, the LzLCS+ sample includes 39 galaxies with non-detected LyC corresponding to f esc upper limits of 0.03-5.9%. Standard multivariate regression models do not account for upper limits, but a valid statistical analysis of the full LzLCS+ dataset should include information from both detections and non-detections (e.g., Isobe et al. 1986). Hence, we turn to the statistical approach of survival analysis, which can handle censored data like that of the LzLCS. We apply the Cox proportional hazards model (Cox 1972), a semi-parametric survival analysis method, to the full LzLCS+ dataset to derive multivariate predictions of f esc . By testing different sets of variables and assessing their predictive ability, we explore which physical properties combine to set a galaxy's f esc . We adopt a cosmology of H 0 = 70 km s -1 Mpc -1 , Ω m = 0 . 3, and Ω Λ = 0 . 7.", '2.1. Sample: The Low-redshift Lyman Continuum Survey': "To derive multivariate predictors of f esc , we use the Low-redshift Lyman Continuum Survey (LzLCS; Flury et al. 2022a), the largest sample of LCEs at low redshift. Flury et al. (2022a) describe the survey sample, observations, data processing, and measurements in full, but we review some of the key details here. The LzLCS is a 134-orbit Cycle 26 HST program (GO15626; PI Jaskot) that obtained far-ultraviolet spectra, including the rest-frame LyC, with the COS G140L grating for 66 star-forming galaxies at z ∼ 0 . 3. The targeted galaxies each fulfill one or more hypothesized selection criteria for LCEs: high nebular ionization (O32 = [O iii ] λ 5007/[O ii ] λ 3727 ≥ 3), blue UV slopes (power law index β < -2), and/or concentrated star formation (Σ SFR > 0 . 1 M ⊙ yr -1 kpc -2 ). We combine the LzLCS with archival COS observations from Izotov et al. (2016a,b, 2018a,b, 2021) and Wang et al. (2019). We exclude one galaxy, J1333+6246 (Izotov et al. 2016b), whose [O iii ] λλ 5007,4959, H α , and H β line fluxes may be inaccurate; these lines appear truncated in the SDSS spectrum and the galaxy's H α /H β and H β /H γ line ratios are unphysical. Excluding this problematic galaxy, the combined LzLCS and archival galaxies (hereafter the LzLCS+ sample) consists of 88 low-redshift galaxies with LyC observations. The sample spans a range of 10 8 -10 10 M ⊙ in stellar mass, 12 + log 10 (O / H) = 7 . 5 -8 . 5, and -21.5 to -18.3 in observed (not corrected for internal reddening) 1500 ˚ A absolute magnitude. \nWe process the raw COS spectra using the calcos pipeline (v3.3.9) and the FaintCOS software routines (Worseck et al. 2016; Makan et al. 2021) to reduce and calibrate the spectra and account for the backgrounds \ndue to dark current and scattered geocoronal Ly α . We correct all spectra for Milky Way attenuation using the Green et al. (2018) dust maps and Fitzpatrick (1999) attenuation law. The LyC flux measurements represent the flux in a 20 ˚ A-wide region near rest-frame 900 ˚ A; we exclude wavelengths above an observed wavelength of 1180 ˚ A because of telluric contamination. For consistency, we re-process and re-measure the archival observations using this same methodology. Following the criteria in Flury et al. (2022a), we define LyC detections as observations that have a probability < 0 . 02275 of originating from background counts. By this definition, 49 of the 88 total galaxies have detected LyC. As in Flury et al. (2022a), we adopt the 84th percentile of the background count distribution as the upper limit for non-detections. \nUsing these LyC fluxes, we then calculate the absolute f esc from different estimates of the intrinsic LyC, as described in Flury et al. (2022a); for this paper, we adopt the f esc values from Saldana-Lopez et al. (2022), derived from Starburst99 (Leitherer et al. 2011, 2014) spectral energy distribution (SED) fits to the UV continuum, following the methods outlined in Chisholm et al. (2019). As discussed in Flury et al. (2022a), the UV SED-fitting method is more reliable than H β -derived f esc estimates for the diverse LzLCS sample, because the presence of underlying, older ( > 10 Myr old) stellar populations that emit optical emission can bias the H β f esc results. The H β estimates of intrinsic LyC also do not account for LyC photons absorbed by dust and assume isotropic escape. With our preferred UV-based f esc estimate, the LzLCS+ LyC detections cover f esc = 0 . 4 -88 . 9% and the non-detections have upper limits ranging from 0.035.9%. We also consider an alternative, purely empirical measure of LyC escape, the F λ LyC /F λ 1100 flux ratio in Section § 4.1. \nIn addition to LyC measurements, the LzLCS+ dataset contains a wealth of ancillary measurements, described in full in Flury et al. (2022a) and SaldanaLopez et al. (2022). Each galaxy in the survey has rest-frame optical photometry and spectroscopy through the Sloan Digital Sky Survey (SDSS; Blanton et al. 2017) and UV photometry via the Galaxy Evolution Explorer ( GALEX ; Martin et al. 2003). We derive stellar masses ( M ∗ ) using Prospector (Leja et al. 2017; Johnson et al. 2019) fits to the SDSS and GALEX photometry (Flury et al. 2022a, Ji et al. in prep.). We measure nebular line fluxes by fitting the SDSS spectra with multiple Gaussian profiles and then iteratively derive the electron temperature ( T e ), electron density, stellar absorption, and nebular E ( B -V ) from the [O iii ] λλ 5007,4959,4363, [S ii ] λλ 6716,6731, and Balmer \nlines. Where the [S ii ] doublet or [O iii ] λ 4363 are undetected, we adopt n e = 100 cm -3 and the estimated [O iii ] λ 4363 flux from the Pilyugin et al. (2006) 'ffrelation', respectively. We then derive oxygen abundances via the direct method as implemented in pyneb (Luridiana et al. 2015). To estimate the star formation rate (SFR), we use the dust-corrected H β luminosities, Case B H α /H β ratio (Storey & Hummer 1995), and Kennicutt & Evans (2012) SFR calibration. \nThe COS data provide estimates of additional physical parameters. We use the COS near-UV acquisition images to calculate the UV half-light radius. The sources all appear to be very compact within the central 1 '' diameter of the COS aperture, so that vignetting will not strongly affect our radius measurements. From the measured radius, we also derive Σ SFR as \nΣ SFR = SFR 2 πr 2 50 , NUV . (1) \nFrom the Starburst99 SED fits to the UV spectra, we fit for the dust excess E(B-V). We denote the E(BV) derived from the UV spectral fits as E(B-V) UV to distinguish it from E(B-V) neb derived from the Balmer line ratios. We also extrapolate the UV SED fits to infer the 'observed' (non-extinction corrected) absolute magnitude at 1500 ˚ A ( M 1500 ) and the power law index slope at 1550 ˚ A ( β 1550 ; Saldana-Lopez et al. 2022). \nThe G140L spectra also cover Ly α . For our Ly α measurements, we use spectra extracted using a slightly wider spatial aperture (30 pixel vs. 25 pixel), because the scattered Ly α emission may be more spatially extended than the continuum light (Flury et al. 2022a). After masking the Si iii λ 1206 and N v λ 1240 regions, we linearly fit the continuum within 100 ˚ A of Ly α using an iterative sigma clipping algorithm and integrate the Ly α fluxes relative to this continuum level. This linear continuum fit typically agrees within < 5% with the continuum estimated from SED fits to the 25-pixel aperture extracted spectra (Saldana-Lopez et al. 2022). We do not correct for the small contribution of stellar Ly α absorption, but we conservatively adopt a 25% uncertainty on the Ly α continuum estimate (Flury et al. 2022a). From these fluxes, we derive Ly α luminosities, equivalent widths (EWs), and escape fractions ( f esc , Ly α ); we use the dust-corrected H β flux and the galaxies' measured electron temperatures and densities to infer the intrinsic Ly α flux. The reported Ly α measurements represent the sum of both absorption and emission along the line of sight. Even galaxies with net Ly α emission may have underlying Ly α absorption troughs, which are commonly detected in samples at z < 0 . 1 (e.g. McKinney et al. 2019; Hu et al. 2023). At the higher redshifts \nof our sample, z ∼ 0 . 3, the COS aperture covers a larger physical area, and these troughs may experience more infilling due to scattered Ly α emission. Because the radii of our Ly α spectral apertures are ∼ 2.6 times larger than the UV continuum half-light radius (Flury et al. 2022a), we expect our Ly α measurements to capture most of the scattered emission (e.g. Hayes et al. 2013); future Ly α imaging of LzLCS+ galaxies will test this assumption (HST GO-17069, P.I. Hayes). Nine galaxies in our sample have significant Ly α absorption that overlaps with the Si iii λ 1206 absorption feature. To account for the uncertain strength of Si iii λ 1206 contamination in these galaxies, we re-measure the Ly α fluxes and EWs, excluding wavelengths within 500 km s -1 of Si iii λ 1206, and we increase our Ly α uncertainties to account for this possible flux difference. We note that this effect is minor, changing EWs by < 3 ˚ A and f esc , Ly α by ≤ 0 . 017, with a median change of only 0.001 in f esc , Ly α for the nine affected galaxies. \nLastly, we use the G140L spectra to measure the EWs and residual fluxes near line center ( R l ) for a variety of low-ionization absorption lines, as described in SaldanaLopez et al. (2022). The UV absorption line measurements use the same (25 pixel aperture) spectral extraction as the LyC measurements, and the SED fits are used to estimate the continuum level. For optically thick gas, well-resolved absorption lines, and assuming a uniform dust screen the residual flux is related to the gas covering fraction ( C f ) by \nR ( λ i ) = 1 -C f ( λ i ) (2) \n(see e.g., Gazagnes et al. 2018 for a discussion of geometry effects on R ). This equation assumes that all lines are saturated; while this assumption appears to hold for most of the H i lines, the observational uncertainties are too high to draw conclusions about saturation for the metal lines (see Saldana-Lopez et al. 2022 for more information). In this work, we consider some of the strongest low-ionization metal absorption lines (Si ii λ 1260, C ii λ 1334, H i Ly β +O i ). To improve the S/N of the measurements, we also consider the inversevariance weighted average of the EWs and R l values for a given galaxy, calculated for the available H i Lymanseries lines (Ly β -Ly6) or for the available low-ionization metal (LIS) lines (Si ii λλ 1190 , 1193, Si ii λ 1260, [O i ] λ 1302, C ii λ 1334; see Saldana-Lopez et al. 2022 for details). We refer to these average absorption line measurements as EW(H i ,abs), R l (H i ,abs), EW(LIS), and R l (LIS). Because the measured residual intensity depends on resolution, we caution that the measured R l will not perfectly correspond to the true R l of the system; for the LzLCS+, this systematic error is of order \n10-20% and is within the reported uncertainty (SaldanaLopez et al. 2022). While trends between R l and f esc will still be apparent (e.g., Saldana-Lopez et al. 2022), future investigations should use care when applying models derived from the LzLCS+ R l data to spectroscopic data with a different resolution.", '2.2. Survival Analysis': "As described above, one of the strengths of the LzLCS+ dataset is its large and diverse sample, spanning a wide range of measured f esc . The full sample of 88 galaxies includes 39 non-detections with f esc upper limits of only a few percent. These non-detections can convey essential information about which properties do and do not distinguish LCEs from non-LCEs, and omitting non-detections can bias fitted relations (c.f., Isobe et al. 1986). Standard multivariate linear regression does not account for censored data, that is, data with upper limits. Therefore, we instead apply survival analysis techniques to the LzLCS+ sample, as these techniques properly treat censored data. \nAs implied in the name, survival analysis originated in the field of medicine (see reviews by Clark et al. 2003 and Bradburn et al. 2003, and see Feigelson & Nelson 1985 and Isobe et al. 1986 for applications to astronomy). In a medical context, the censored data often consist of known and unknown lifetimes for individuals participating in a medical study. People who are alive at the end of the study have an unknown lifetime; all we know is that they will live longer than their current age. Survival analysis techniques ultimately seek to describe the probability of a particular lifetime for a population or to compare how the survival probability changes between different populations, based on data that include some known lifetimes and some limits. \nTo set up an analogous scenario for predicting the probability of a particular f esc , we do the following. Instead of a 'death' or known lifetime, we have a detection. Most survival analyses involve lower limits ('rightcensored' data). For simplicity in following common techniques, we therefore adopt the absorbed fraction of LyC ( f abs = 1 -f esc ) and its corresponding lower limits rather than f esc and its upper limits. With this setup, increasing f abs (decreasing f esc ) is equivalent to increasing the 'age' of the study participant. A non-detection, i.e., a lower limit to f abs (and upper limit on f esc ), is analogous to a lifetime greater than some threshold. Instead of starting at time = 0 and proceeding until we record a death or the study ends, we can imagine trying to observe a galaxy with f esc = 1 ( f abs = 0) and proceeding with deeper and deeper observations until we have a detection or cease our efforts. Just as the \nthreshold between recording a measured lifetime vs. a limit will depend on factors such as the length of the study and age of the participant, the threshold between detected f esc and an upper limit will depend on aspects of our study such as the observation depth and galaxy brightness.", '2.2.1. The Cox Proportional Hazards Model': "The Cox proportional hazards regression model (Cox 1972) describes the probability of an event in an infinitesimally small window of time, given a combination of independent variables and assuming the event did not already occur. In the Cox model, this event probability (or 'hazard function') at time t for a set of variables x is modeled as \nh ( t | x ) = h 0 ( t ) exp[ n ∑ i =1 b i ( x i -¯ x i )] , (3) \nwhere b i are fitted coefficients for each variable x i and where h 0 ( t ) represents the baseline hazard, the probability of an event given average values of all variables (¯ x i ). An increase in variable x i relative to that variable's average value in the sample results in an exponential increase or decrease in the detection probability, depending on the sign of the coefficient b i . In our case, instead of modeling the probability of an event at time t , the hazard function represents the probability of a LyC detection in an infinitesimally small window of f esc for a set of independent variables, assuming the galaxy was not already detected at a higher value of f esc . \nThe Cox model is semi-parametric. One of its strengths is that it does not assume a particular functional form for the baseline hazard function but rather estimates it non-parametrically. In other words, the event times or detection values do not have to obey a known statistical distribution (Bradburn et al. 2003). Because the Cox model predicts the detection probability at each possible f esc value, rather than predicting the value of f esc itself, the model results are identical for f esc and for log 10 ( f esc ). In other words, the non-parametric estimate of h 0 ( f abs ) for an input array of f esc values would be identical to h 0 (log 10 ( f abs )) for the equivalent input array of log 10 ( f esc ) values. \nYet, the parametric nature of the hazard function equation (Equation 3) with its reported fitted coefficients makes it straightforward to analyze the relative importance of variables and generate predictions for future datasets. On the other hand, this fixed functional form implicitly assumes that the effect of the independent variables is multiplicative and does not depend on t , in our case, f abs . This functional form would not be appropriate, for example, if one variable tended to introduce a step function in f esc , while the others had more of \na smooth correlation. However, the adopted functional form has some flexibility in that independent variables are allowed to take any functional form (e.g., x i can be replaced with log 10 ( x i ), x 2 i , etc). Another limitation of the Cox model is that, while it handles censored dependent variable values, it cannot handle censored data in the independent variables. Consequently, we have to limit our analysis to input variables, x i , that are available for all (or nearly all) of the sample galaxies, as we explain below. In summary, the Cox proportional hazards model serves as a multivariate regression model for describing and predicting dependent variables with upper limits. Many other common statistical methods either do not treat upper limits (e.g., multivariate linear regression, principal component analysis), are univariate analyses (e.g., Kaplan-Meier analysis), or are fully parametric, less-flexible models (e.g., the Weibull model). \nIn fitting the Cox model to the LzLCS+, we experiment with different combinations of independent variables from the parent list given in Table 1. Where possible, we take the base 10 logarithm of the relevant variable. We cannot take the logarithm for variables that contain both positive and negative values, such as the UV LIS and Ly α lines; these lines range from absorption to emission within the sample. Putting variables on a logarithmic scale serves two purposes. First, it puts variables on a consistent scale of similar order of magnitude. In addition, the coefficients in Equation 3 have a simple interpretation; increasing a variable by an order of magnitude leads to a factor of e b i increase in the probability h ( f abs | x ). Exploring alternative functional forms for the input variables is outside the scope of this paper, but we test the effects of switching between logarithmic and linear scalings in the following subsection ( § 2.2.2). While some statistical methods require putting all variables on an identical scale, such as scaling from 0 to 1, this further rescaling is not necessary in the Cox model. Equation 3 effectively rescales variables; the sample mean is subtracted from each input variable measurement, and the b i parameter translates a linear change in variable x i to a change in the probability of measuring a given f esc . Any normalization would change the derived values of b i but not the statistical significance of input variables nor their ultimate effect on f esc . \nIn selecting a subset of variables to use in our fits, we include multiple variables from each category in Table 1, but we avoid any variables that are closely related to each other and which may therefore be highly collinear. Collinearity can result in multiple best-fit solutions and can cause the Cox model to fail to converge. Consequently, we typically consider either Σ SFR alone or SFR and r 50 , NUV , but not all three variables at once. Sim- \nlarly, we choose only one variable that traces nebular ionization, one variable to trace UV color, and limited subsets of the UV absorption line variables. We discuss the effect of selecting different variables in Section § 3.2. As mentioned above, the Cox proportional hazard model cannot account for missing or censored independent variable data. Hence, we exclude a galaxy from our fits if it is missing any of the required variables. Numbers in brackets in Table 1 indicate the number of galaxies missing a given measurement. Most of the models discussed in this paper exclude only one galaxy, J1046+5827, a non-leaker that lacks a reported Σ SFR and r 50 , NUV . We apply the Cox model to f esc as our primary dependent variable, but we also perform and discuss model fits to the F λ LyC /F λ 1100 ratio, ( § 4.1), LyC luminosity ( § 4.1), f esc , Ly α , and Ly α luminosity ( § 4.2). \nWe apply the Cox proportional hazards model to our data using the python package lifelines (DavidsonPilon 2019). The lifelines CoxPHFitter routine returns the best-fit coefficients for our selected independent variable set, their p-values, and various goodnessof-fit statistics. As shown by Equation 3, the Cox proportional hazards model gives the probability of observing a particular f esc value given a set of physical or observable properties. Instead, our goal is to predict the expected value of f esc given a set of properties. As described below, to find the expected f esc , we find the median of the probability distribution, where the true f esc has an equal probability of lying above or below this adopted value (e.g., Bradburn et al. 2003; DavidsonPilon 2019); f esc is predicted to be above the median 50% of the time and below it the other 50% of the time. \nThe median f esc = 1 f abs represents the f esc value where the survival function S ( f abs ), the probability that there is no detection at f abs , detect < f abs , reaches 0.5. This probability is equivalent to the probability that f esc , detect is detected at f esc , detect < f esc . The survival function is calculated as (e.g., Cox 1972; Bradburn et al. 2003; Davidson-Pilon 2019; McLernon et al. 2023) \nS ( f abs ) = exp[ -HF 0 ( f abs ) · ph( x )] , (4) \nwhere HF 0 is the baseline cumulative hazard function: \nHF 0 ( f abs ) = ∫ f abs 0 h 0 ( f ) df (5) \nand ph( x ) is the partial hazard function for a set of variables x: \nph( x ) = exp[ n ∑ i =1 b i ( x i -¯ x i )] . (6) \nOccasionally, S ( f abs ) for a set of parameters will never reach 0.5 and the predicted median f esc is indeterminate. This situation corresponds to an arbitrarily small predicted f esc ∼ 0. \nThe best fit coefficients for the Cox proportional hazards model are found by maximizing the partial likelihood, which compares ph( x ) for each detection with the sum of ph( x ) for all galaxies with higher f abs (lower f esc ), including both detections and non-detections. The resulting coefficients b i are those that best sort f abs in order. Once the coefficients are determined, Breslow's estimator (Breslow 1972) determines the baseline cumulative hazard function, HF 0 ( f abs ), using the number of detections with values lower than each f abs (i.e., detections with higher values of f esc ) plus the ph( x ) values for all detections and limits higher than f abs (i.e., with lower f esc ). The model fits HF 0 ( f abs ) non-parametrically, reporting a value of HF 0 for each of the input f abs values of the LzLCS+ sample. \nOne can also use the survival function to evaluate the values at which S reaches 0.159 and 0.841, the bounds corresponding to the Normal-theory 1 σ uncertainty of the predicted f esc . These probabilities account for the scatter in the correlations between f esc and the independent variables. The scatter may arise from observational uncertainty in the measurements as well as inherent variation among the population. In addition to this method, we have also used a Monte Carlo (MC) method to explore how the f esc predictions change if we vary each variable within its uncertainties. We randomly resample each independent variable measurement and each dependent variable detection using their uncertainties, re-run the Cox fit, and obtain new estimates of the median predicted f esc for each galaxy. Using the distribution of predicted f esc , we then calculate the 15.9 and 84.1 percentiles. For nearly all galaxies, the uncertainties estimated using the survival function are greater than or equal to the uncertainties determined from this MC method, which indicates that the scatter in the correlations is the dominant effect. We conclude that the survival function sufficiently represents the uncertainty in predicted f esc in most cases.", '2.2.2. Interpreting Cox Model f esc Predictions': 'As shown by Equation 4, the Cox model does not represent an equation that shows how f esc itself depends on particular variables. Rather, it describes how the probability of observing a given f esc changes for different sets of galaxy properties. The baseline cumulative hazard function HF 0 ( f abs ) describes the expected probability distribution for galaxies that have the average properties of the LzLCS+ dataset. In this case, x i = ¯ x i for all vari- \nTable 1. Complete List of Independent Variables \nNote -Numbers within brackets denote the number of galaxies within the LzLCS+ sample missing these measurements. Positive values of EW(H β ) and EW(Ly α ) denote net emission; positive values of the UV absorption lines denote net absorption. \nables, and the reference probability that the observed f esc , detect is less than a particular value of f esc is simply S ref ( f esc ) = exp[ -HF 0 ( f esc )]. If we increase one variable by an increment of 1, x i -¯ x i = 1, ph( x ) = exp( b i ), and the new probability S ( f esc ) = S ref ( f esc ) exp( b i ) . Because the probability S ref ( f esc ) < 1, raising it to the power of exp( b i ) decreases the probability, which means that low values of f esc , detect < f esc are less likely. The probability distribution therefore shifts such that the probability corresponding to S =0.5 occurs at higher f esc , resulting in a larger predicted median f esc . Changing a second variable, x j , by an increment of 1 changes the new probability again by a power of exp( b j ), such that S ( f esc ) = [ S ref ( f esc ) exp( b i ) ] exp( b j ) ]. \nWe illustrate an example of the survival function probabilities, the associated median f esc , and the dependence of these quantities on input variables in Figure 1. In the left panel, we show a simple Cox model with three input variables: β 1550 , log 10 (Σ SFR ), and log 10 (O32). Dots represent the predicted median f esc where S ( f esc )= 0 . 5 for each value of O32. For log 10 (O32)= 1, S ( f esc )= 0 . 5 at f esc = 0 . 005. Changing only the log 10 (O32) value by 1, from O32 = 1, the blue dotted line, to O32 = 10, the magenta long-dashed line, shifts S ( f esc = 0 . 005) from 0.5 down to 0 . 5 exp(0 . 996) = 0 . 153, where 0.996 is the best-fit coefficient b i for the log 10 (O32) variable. Thus, according to this model, high f esc values are more common among galaxies with high O32. In this example, we changed only one parameter, but changing the other variables at same time could either further shift the probability distribution to higher f esc or counteract the change caused by increasing O32. We note that, unlike an equation for f esc as a function of these input variables, which could conceivably reach unphysical values of f esc > 1 for extreme parameters, the probability dis- \nibution can shift toward higher values of f esc , but it never extends beyond f esc = 1. \nFigure 1 also shows how the predicted probability responds to linear vs. logarithmic variables. As outlined above, the probabilities shift by a power of exp( b i ) for a step size of 1 in x i , which corresponds to an order of magnitude increase if x i represents the base 10 logarithm of a variable. In the left panel, the model uses log 10 (O32) as the input variable, and in the right panel, we show the effect of changing the form of the input variable from log 10 (O32) to linear O32. The differences are most apparent where a large linear spacing (e.g., changing O32 from 10 to 25) is not equivalent to a large logarithmic spacing and vice versa (e.g., O32 from 0.1 to 1). However, the change in the predicted median f esc values is largely minor. In changing to a linear scale, f esc changes from 0 to 0.006 for O32 = 0 . 1 and from 0.027 to 0.059 for O32 = 25. For the other plotted values, the change in f esc is < 0 . 002. The best-fit coefficients b i and HF 0 ( f esc ) values also differ between the two models and are optimized to match the observed distribution of f esc in the LzLCS+. Consequently, although the choice of variable form does affect predictions for galaxies at the extremes of the input parameter space, our main results, including the overall quality of our model predictions, are not highly sensitive to the functional form of the input variables. \nWe do not thoroughly explore functional forms here, but we experiment with changing the ISM absorption measurements (EW(H i ,abs) and R l (H i ,abs)) and Ly α variables (EW(Ly α ), L (Ly α ), f esc , Ly α ) to logarithmic forms. For the logarithmic Ly α variables, we exclude all galaxies with net Ly α absorption from the dataset. We find the same variables are statistically significant in our models, although their exact coefficients and pvalues necessarily change. The goodness-of-fit metrics \n<!-- image --> \nFigure 1. Examples of the survival function probabilities predicted from the Cox model. The left panel uses β 1550 , log 10 (Σ SFR ), and log 10 (O32) as input variables, and the right panel uses β 1550 , log 10 (Σ SFR ), and O32 instead of log 10 (O32). Lines show the probability that the detected f esc , detect is less than a particular value of f esc or equivalently, the probability that f abs , detect is not less than a given f abs . The lines are plotted at f esc values corresponding to the measured f esc for the LzLCS+ dataset, from which the baseline cumulative hazard function is estimated non-parametrically. Data points show the median predicted f esc , where S ( f esc )= 0 . 5. The different lines show predictions for different input values of O32. As O32 increases, the probability distribution shifts to higher values of f esc , and the extent of the shift depends on whether the variable is linear or logarithmic. The black solid line in the left panel (log 10 (O32)= -1) shows an example where S never reaches 0.5, indicating an inferred median f esc ∼ 0. \n<!-- image --> \n( § 2.3) are also similar, typically changing by ≤ 0 . 02. The change in variable form also does not affect our main conclusions regarding the most important ranked variables ( § 2.4), although it can result in minor changes in ranked variable order. \nObservationally, for most variables, log 10 ( f esc ) does seem to change significantly with the logarithm of a variable (Flury et al. 2022b). These trends could indicate that f esc has a power law dependence on many variables, although, as noted above, such relationships could only apply over the range where the predicted f esc ≤ 1. One exception to this logarithmic dependence is the UV absorption line measurements, which do show a dependence of log 10 ( f esc ) on the linear form of the variables (Saldana-Lopez et al. 2022). This functional form may indicate a more complex dependence of f esc on the gas geometry. We find that our predictions using the linear EW(H i ,abs) better match the observed f esc by every metric compared to models using log 10 (EW(H i ,abs)), and we therefore choose to keep EW(H i ,abs) in the linear form. Models using the linear or logarithmic form of R l (H i ,abs) perform comparably well, but with the linear form better matching the observed f esc for the strongest LCEs. Given their comparable or improved performance and for consistency with the LIS line measurements, which can reach negative values, we therefore \nkeep all absorption line variables in their linear form. We also choose to keep the Ly α measurements in a linear form so as to not bias our dataset by excluding nonLy α emitters. We emphasize that our fit quality and main results are not sensitive to these choices regarding variable form. \nIn summary, the Cox model predicts the probability of observing particular f esc values for different galaxy populations and can identify which variables most affect that probability. Statistically significant variables signify that the f esc probability is highly responsive to an incremental change in that variable: either a linear or logarithmic increase, depending on the variable form. While this work identifies these significant variables, future theoretical or observational programs could endeavor to derive the exact functional dependence of f esc on these parameters.', '2.3. Model Assessment': "We assess the goodness-of-fit for our models in a few different ways. Our primary method is the concordance index (Harrell et al. 1982), which applies to censored data. The concordance index considers all possible pairs of data points and how the observed rank order of f esc compares to the order predicted by the model. The model classifies each pair as concordant if the data point \nwith a higher observed f esc also has a higher predicted f esc and discordant if it does not. Pairs can also be tied, if their predicted f esc values are identical. For some pairs with upper limits, their rank is ambiguous. The concordance index is \nC = n c +0 . 5 n t n c + n d + n t , (7) \nwhere n c is the number of concordant pairs, n t is the number of tied pairs, and n d is the number of discordant pairs. The concordance index ranges from 0 (perfect disagreement) to 0.5 (perfectly random) to 1.0 (perfect concordance; e.g., Davidson-Pilon 2019). 1 We also calculate the R 2 statistic for the galaxies with f esc detections as \nR 2 = 1 -∑ i ( y i -f i ) 2 ∑ i ( y i -¯ y ) 2 , (8) \nwhere y i are the observed values of log 10 ( f esc ), ¯ y is their mean value, and f i are the model-predicted log 10 ( f esc ) values. Following Maji et al. (2022), we also calculate the adjusted R 2 : \nR 2 adj = 1 -(1 -R 2 ) n -1 n -p -1 , (9) \nwhere n is the number of data points and p is the number of independent variables in the model. The R 2 adj parameter measures whether additional variables improve the model fit more than would be expected by random chance. Lastly, we also calculate the root-mean-square (RMS) dispersion about the model predictions: \nRMS = √ ∑ i ( y i -f i ) 2 n . (10) \nWe can only calculate these three quantities, R 2 , R 2 adj , and RMS, for the detections in our sample. Although they cannot provide a complete picture of model performance, they can tell us how well our predictions work for the galaxies with known f esc and can quantify our model performance at the high f esc end. Each of these metrics correlates strongly with the others and with the concordance index, and our main results are insensitive to the exact metric used. \nWe choose to evaluate both R 2 and RMS using log 10 ( f esc ) rather than the linear f esc , because the scatter in the predicted f esc values is approximately constant in logarithmic space over the full observed f esc \nrange. A single RMS value is therefore representative of both low and high f esc galaxies. In contrast, in linear space, the RMS changes systematically from low to high observed f esc . For instance, for our fiducial model ( § 3.1), the RMS is 0.02 for galaxies with f esc in the lowest third of the sample, 0.16 for the next third, and 0.22 for the final third, whereas the logarithmic RMS is ∼ 0.3-0.4 dex across the full sample range. Because R 2 and RMS use log 10 ( f esc ), these statistics only include galaxies with both detected f esc and non-zero predicted f esc . In contrast, the concordance index C incorporates the full dataset, including galaxies with observed upper limits and those with predicted f esc =0. \nWe also assess the predictive ability of our models using cross validation, applying the model to data that was not used in generating the model itself. We perform a k -fold cross validation, in which we randomly split the dataset into k groups. We combine k -1 of the groups to become our training set, which we use to fit the model. We then test the model on the remaining group using the C , R 2 , and RMS metrics described above. ( R 2 adj is often undefined, due to runs where n for the small test set is equal to p ). We then repeat this process until each group has been used as the test group with the remaining groups used as the training set. Following Runnholm et al. (2020), we adopt k = 3, repeat the group selection 100 times, and average the final results. A value of k = 3 gives us sufficient galaxies in the test set to calculate statistics, and the repetition of the process ensures our results are not sensitive to the exact group selected (Runnholm et al. 2020). We provide the results of these tests in our evaluation of the models and distinguish the metrics derived from this cross-validation analysis by the subscript 'CV'.", '2.4. Variable Selection': 'We explore different subsets of independent variables and discuss the resulting model fits in Section § 3.2. With these different selections, we test the effect of switching the absorption line probed, of including or excluding Ly α , and of limiting the variable set to observables accessible to JWST (Section § 3.4). However, we also wish to analyze the relative importance of each independent variable and identify the variable subsets that most accurately predict f esc . To evaluate the importance of each variable, we turn to the tools of forward and backward selection (e.g., Runnholm et al. 2020; Maji et al. 2022). \nIn forward selection, we run the Cox model with each independent variable individually in turn and determine which variable provides a fit with the highest concordance. We select this variable and then combine it with each remaining variable in turn to determine which \ncombination of two variables provides the highest concordance. We proceed in this manner until we have a ranked order of the most significant variables. Conversely, with backward selection, we start with the full set of independent variables, and remove each one in turn. The combination of variables that gives the highest concordance identifies the least important excluded variable. We exclude it and proceed with the remaining variables, identifying and removing the next-least important variable each time to obtain a rank-ordered list. Importantly, each round of forward or backward selection only ranks variables with respect to the current best-performing model. Hence, some variables may be poorly ranked if they provide similar information as an already-selected variable. Variable rankings can reveal the most important predictors of f esc , but a poor ranking does not necessarily imply that a variable is uncorrelated with f esc . \nWhen we perform our forward and backward selections, we do not use the full list of variables in Table 1 but limit it in the following ways. Firstly, we include only the EW(H i ,abs), EW(LIS), R l (H i ,abs), and R l (LIS) measurements, which are averages of multiple lines, rather than the measurements of individual absorption lines. This choice avoids overly limiting our sample sizes, since 22 galaxies are missing one or more individual absorption line measurements. Secondly, we avoid highly collinear variables and variables that measure very similar properties. We include only Σ SFR rather than SFR and r 50 , NUV , we use only one ionization-sensitive line ratio (O32 = [O iii ]/[O ii ]), and we use only one measure of UV dust attenuation ( β 1550 or E(B-V) UV ). When included separately, alternative measures of these properties end up with the same or nearly the same ranks, which suggests that they are indeed largely interchangeable. We explore three different sets of variables in our rankings: one set includes β 1550 and all remaining variables, one set corresponds to the variables in our best-performing Cox model, and one set is limited to variables accessible at high redshift. We also test our final variable rankings using an MC method. We re-run the forward and backward selection processes after randomly sampling the independent and dependent variables 100 times using their uncertainties. We present the results of the forward and backward selection methods in Section § 3.3.', '3.1. Fiducial Model': "For our fiducial model, we choose the following variables: log 10 ( M ∗ ), M 1500 , log 10 (EW(H β )), E(B-V) neb , 12+log 10 (O/H), log 10 (O32), log 10 (Σ SFR ), E(B-V) UV , \nf esc , Ly α , EW(LIS). We present the fitted coefficients and significance for these variables in Table 2 and compare the model-predicted median f esc with the observations in Figure 2. We list the goodness-of-fit metrics for this model in Table 3. Statistically significant variables, with p -values ≤ 0 . 05 are f esc , Ly α , E(B-V) UV , log 10 (Σ SFR ), E(B-V) neb , and log 10 (O32). Not surprisingly, these same variables individually correlate or anticorrelate with f esc , as discussed in Flury et al. (2022b) and Saldana-Lopez et al. (2022). We choose to include both O32 and EW(H β ) in the fiducial model, even though they correlate with each other (e.g., Flury et al. 2022b), because they may have subtly different relationships with f esc ; a young starburst may have both high EW(H β ) and high O32, but high global f esc will tend to increase O32, while decreasing EW( Hβ ) (e.g., Zackrisson et al. 2013; Nakajima & Ouchi 2014; Jaskot & Ravindranath 2016). However, we find that EW(H β ) is not significant and its coefficient is near zero, which indicates that EW(H β ) does not contribute meaningful information to the fit beyond what O32 already provides. For the fiducial model, we choose EW(LIS) instead of EW(H i ), because EW(LIS) is less affected by IGM and circumgalactic medium opacity and is therefore a viable indirect indicator across a wider redshift range. We consider the effect of substituting EW(H i ) or other measurements of absorption line strength in § 3.2. \nAlthough most variables show the behavior expected from their individual relationship with f esc , one exception is the nebular attenuation, E(B-V) neb , which correlates with f esc in the Cox model instead of anticorrelating as might be expected. However, the model already contains the UV E(B-V) and f esc , Ly α , two parameters that show strong trends with E(B-V) neb . The nebular E(B-V) therefore essentially operates as a second-order effect, where the nebular attenuation would increase f esc at a fixed value of line-of-sight UV attenuation or f esc , Ly α . This extra nebular E(B-V) correlation could indicate a trend with some other physical property not included in the model, such as starburst age, global gas content, or gas clumpiness. In the case of f esc , Ly α , E(B-V) neb may help quantify how much of the Ly α escape is due to a low H i optical depth. At fixed f esc , Ly α , a higher E(B-V) neb implies a greater contribution of dust and weaker contribution of H i to the Ly α absorption. As expected, if we remove both Ly α and E(B-V) UV from the model, the correlation with E(BV) neb disappears. \nAs discussed in § 2.2.2, the best-fit coefficients in Table 2 quantify how the probability of observing a given f esc responds to an incremental change in each variable. \nThe probability S of observing f esc , detect <f esc changes from the original probability S ref by S = S exp( b i ∆ x i ) ref for a change in variable x i . The coefficients in Table 2 indicate that changing f esc , Ly α by 0.1 results in raising the probability to the power of exp(0 . 7) = 2 . 0, leading to a lower probability of low f esc values (see § 2.2.2. Conversely, increasing E(B-V) UV by 0.1 increases the probability of low f esc values by an even greater factor; the probability is raised to exp( -1 . 49) ∼ 1 / 5. For the other statistically significant variables, a 0.1 change in E(BV) neb also raises the probability to the power of ∼ 2, while 0.1 changes in either log 10 (Σ SFR ) or log 10 (O32) raise the probability to powers of 1.2 and 1.3, respectively. \nAs illustrated in Figure 2 and Table 3, the fiducial model predicts the observed f esc values for detections with an RMS scatter of 0.36 dex for both the full sample and when the dataset is split into test and training sets (RMS CV = 0.36; see § 2.3). The model's high concordance, C = 0 . 89 ( C CV = 0 . 85), shows that it successfully ranks galaxies on the basis of f esc . The R 2 , R 2 adj , and R 2 CV metrics are 0.60, 0.49, and 0.55 respectively. This successful fit shows that including multiple variables results in a substantial improvement over predictions using a single variable. Even for some of the best single-variable correlations, f esc spans 2 dex at a given galaxy property (e.g., Flury et al. 2022b; Chisholm et al. 2022). \nThe improved accuracy of the fiducial multivariate model is driven by only a handful of variables. Table 2 shows that only four variables are statistically significant in the fiducial model, and limiting the model to only these four variables achieves comparable accuracy, with the R 2 , RMS, and C metrics changing by 0.010.02. As can be seen in Table 2, highly insignificant variables tend to have best-fit coefficients near 0, which means the variable plays almost no role in the prediction. Excluding these variables therefore has little effect on the model, and because these extra parameters have negligible effects, overfitting is not a major concern. \nDespite some overall improvement, the scatter in the fiducial Cox model is still significant, which demonstrates the difficulty in predicting the line-of-sight f esc even given a large amount of information. The residuals in the bottom panel of Figure 2 also show that the fiducial model systematically underpredicts the value of f esc for the strongest LCEs, even though it does typically identify them as having f esc > 0 . 1. Mascia et al. (2023) find a similar result and suggest that it may arise from the limited number of strong LCEs in the LzLCS+. We \nsuggest that this under-prediction arises from a different effect, where the strongest LCEs represent a subset of galaxies with favorably oriented low-column density channels but whose global f esc may actually be substantially lower. We elaborate on this possibility in §§ 3.2 and 3.6. Because LyC may escape from narrow channels, the chance orientation of a galaxy can introduce scatter in relationships between global galaxy properties and the measured line-of-sight f esc (e.g., Cen & Kimm 2015; Flury et al. 2022b; Seive et al. 2022). Although the fiducial model explicitly includes variables that trace line-ofsight properties, such as EW(LIS) and E(B-V) UV , the LIS metals may imperfectly trace the LyC-absorbing H i gas. \nOther effects may also contribute to the observed scatter. Simulations show that f esc may fluctuate in time, which may introduce scatter between f esc and observables that are sensitive to a different timescale (e.g., Trebitsch et al. 2017; Barrow et al. 2020). Other possible causes of the model scatter and underprediction of f esc are uncertainties in the observed f esc values or a disconnect between the global properties we measure and the local properties of the LyC-emitting region (e.g., Martin et al. 2015; Kim et al. 2023). Rivera-Thorsen et al. (2022) point out that a small number of massive stars can potentially dominate the escaping LyC emission; the properties of this LyC source region may differ from both the global galaxy properties and the properties inferred from the non-ionizing UV spectrum. Finally, because different properties may regulate LyC escape in different types of galaxies (e.g., Flury et al. 2022b), one relation may not suffice to predict f esc in all types of galaxies. We explore this possibility further in Section § 3.5.", '3.2. Modifications to the Fiducial Model': "Modifications to the fiducial model demonstrate that properties sensitive to dust and H i column density are the most useful predictors of f esc . We summarize the performance of the fiducial model and compare it with different modifications in Table 3. We do not test all possible combinations of the variables in Table 1, but we experiment with modifying the fiducial model by swapping alternative measures of some properties and by dropping each statistically significant variable. \nWe first substitute different absorption line tracers (Table 1) for EW(LIS). Of these different tracers, only EW(Ly β ), EW(H i ,abs), and R l (H i ,abs) give statistically significant coefficients with p < 0 . 05. Because the sample size changes slightly for the other low-ionization metal absorption line selections (see Table 1), we cannot compare the metrics for the other models in detail. However, the fit quality appears comparable to the fidu- \nTable 2. Fiducial Model Coefficients \nNote -Variables are listed in order of their p -value significance. Positive values of EW(LIS) represent net absorption; positive values of EW(H β ) represent net emission. \ncial model (RMS = 0 . 33 -0 . 39 and C = 0 . 88 -0 . 89). Only EW(H i ,abs) clearly improves the model fit, with RMS = 0 . 31 and C = 0 . 91. \nThe fit with EW(H i ,abs) (Figure 3) is also our bestperforming model, and it scores better than the fiducial model on all metrics (Table 3). Moreover, this model more accurately predicts f esc for strong LCEs (Figure 3). Hence, the f esc underprediction in the fiducial model arises, at least in part, from the fact that EW(LIS) and the other variables imperfectly trace the line-of-sight H i gas. The Cox model is predicting the median f esc for a given set of parameters, and the strongest LCEs may not be noticeably distinct from more moderate LCEs in the fiducial model's set of parameters. In contrast, the low EW(H i ,abs) values of the strongest leakers do distinguish them from almost all of the moderate LCEs (e.g., Saldana-Lopez et al. 2022). The exact conversion from EW(LIS) to H i absorption and f esc depends on metallicity and gas geometry, since the trace metals will only show up in absorption for a sufficiently large H i column. While a range of physical conditions could cause weak LIS absorption, weak H i absorption more directly indicates LyC escape. Lastly, the observational uncertainty in EW(LIS) is also higher than in EW(H i ,abs), and these uncertainties will be the most problematic for the strongest LCEs with the weakest absorption lines. In conclusion, while LIS lines can identify LCEs, H i absorption lines appear necessary to accurately determine their line-of-sight f esc . \nLike EW(H i ,abs), Ly α measurements can contribute key information about H i column density. Indeed, Maji et al. (2022) find that Ly α luminosity, L (Ly α ), is the \nmost important variable in predicting the LyC f esc from simulations. In our fiducial model, f esc , Ly α was the most statistically significant coefficient, consistent with observational and theoretical studies that find a connection between LyC and Ly α (e.g., Dijkstra et al. 2016; Verhamme et al. 2017; Steidel et al. 2018; Izotov et al. 2020). In the fiducial model, f esc , Ly α serves as the best estimate of the H i optical depth. When EW(H i ,abs) is substituted in the model, f esc , Ly α remains statistically significant, but its p-value drops from 6E-9 to 3E-3 and its best-fit coefficient declines from 7 to 4. Because it is less sensitive to scattered emission, EW(H i ,abs) more directly traces the line-of-sight H i optical depth, which reduces the model's reliance on f esc , Ly α . Without EW(H i ,abs), however, f esc , Ly α provides crucial information in the fiducial model. If we exclude it, the fit quality worsens by all metrics; the RMS rises from 0.36 to 0.46, C drops from 0.89 to 0.84, and R 2 drops from 0.60 to 0.35. Substituting L (Ly α ) for f esc , Ly α also decreases the quality of the fit (RMS = 0.44, C = 0.85, R 2 = 0.41), whereas EW(Ly α ) performs almost as well as f esc , Ly α (RMS = 0.39, C = 0.88, R 2 = 0.54). Interestingly, when included together, both EW(Ly α ) and f esc , Ly α show up as significant coefficients. The two variables may provide slightly different information relevant to LyC escape. While f esc , Ly α may be more directly linked to the fraction of escaping LyC, the EW provides additional information about starburst properties such as the intrinsic Ly α production and underlying continuum. However, including both EW(Ly α ) and f esc , Ly α results in marginal, if any, improvement in the fit. All metrics change by only 0-0.01. \nOf all the variables in the fiducial model, excluding E(B-V) UV or f esc , Ly α has the most detrimental effect \nFigure 2. The f esc predictions from the fiducial Cox model compared with the observed f esc values for the LzLCS+ sample. Red circles indicate detections, and blue triangles represent the 1 σ upper limit on non-detections. The dashed line shows a one-to-one correspondence. The black cross in the upper left indicates the median size of the uncertainties in the observed and predicted f esc . The uncertainties in the predicted f esc represent the 15.9 and 84.1 percentiles of the f esc probability distribution. We plot galaxies with predicted f esc ∼ 0 at the bottom of the figure with their predicted f esc set to 10 -3 . The bottom panel shows a plot of the residuals on a linear scale. This model predicts f esc for the LyC detections with an RMS scatter of 0.36 dex. \n<!-- image --> \non the model fit. Without information on the line-ofsight dust attenuation from E(B-V) UV , R 2 drops to 0.30 and the RMS scatter rises by ∼ 0.1 dex to 0.48. The concordance also drops slightly to C = 0 . 86. Excluding f esc , Ly α has the strongest negative impact on the concordance ( C = 0 . 84) and the second largest impact on R 2 and RMS. The concordance metric includes nondetections, whereas the R 2 and RMS parameters do not. Hence, the larger effect of f esc , Ly α on concordance may illustrate that even without information on dust, a lack of Ly α emission is generally sufficient to identify nonleakers in the LzLCS+, even if the presence of Ly α does not necessarily imply LyC escape. Conversely, predicting accurate f esc for LCEs requires measurements of the line-of-sight dust attenuation. \nSubstituting alternate measures of dust attenuation or ionization has little effect on the fiducial model. Using β 1550 instead of E(B-V) UV changes the goodness-offit metrics by only 0-0.03, consistent with the observed \ntight correlation between β 1550 and E(B-V) UV in the LzLCS+ (Chisholm et al. 2022). However, the β 1200 parameter is not as successful at tracing dust content (e.g., Chisholm et al. 2022) and worsens the fit quality by all metrics. Likewise, alternate measures of ionization such as [Ne iii ]/[O ii ] or [O ii ]/H β work as well as O32, with no change in C and minor ( ≤ 0.03) changes in R 2 and RMS. Of the three measures of ionization, [O ii ]/H β performs best in all metrics, but this improvement is marginal.", '3.3. Most Important Variables': "As discussed in Section § 2.4, we perform forward and backward selection to determine which variables have the greatest effect on the model fit quality. In Table 4, we show a ranked ordering of variables based on forward and backward selection for three representative models. The 'Full Model' includes β 1550 instead of E(B-V) UV and includes some variables that measure similar but not highly collinear properties, in order to \nFigure 3. The f esc predictions from the Cox model improve when we include EW(H i ,abs) instead of EW(LIS). Symbols are the same as in Figure 2. \n<!-- image --> \ntest which ones perform best. For instance, we include f esc , Ly α , EW(Ly α ), and L (Ly α ), which are related but may have different relationships with f esc . Our second set of variables is the more limited list from the Fiducial+HI model, our best-performing Cox model. This model also differs from the Full Model by using E(BV) UV . The third model, the 'JWST Model' excludes absorption line measurements, which are difficult or impossible to measure for most galaxies at z > 6, and Ly α measurements, which are heavily affected by the IGM at z > 6; we discuss this model in Section § 3.4. We also list the mean MC ranks for each variable obtained by sampling the observational uncertainties and rerunning the ranking process 100 times. We plot the distribution of these ranks in Figure 4. \nTable 3. Goodness-of-Fit for Cox f esc Models \nT able 4 . V ariables b y Rank ed Order \n<!-- image --> \na After this v ariable, add ing v ariables impro v es C b y < 0 . 01. \nNote -V ariables in rank order from 1 (most imp ortan t) to 15 (least imp ortan t) b y forw ard or bac kw ard selection. Num b ers in brac k ets indicate the mean of the v ariable's ranks in the 100 MC runs. W e plot the full distribution of ranks in the MC runs in Figure 4 . The F ull Mo del includes most v ariables in T able 1 aside from those that are highly collinear. The Fiducial +HI Mo de l includes the v ariables in the b est-p erforming Co x Mo del. The JWST Mo del includes only v ariables acce ssible at z > 6. \nWe find that the most important factors for predicting f esc are almost always the strength of H i absorption (EW(H i ,abs)) and dust attenuation (E(B-V) UV or β 1550 ). These factors are naturally connected to the lineof-sight f esc , since the two sources of LyC absorption are H i atoms and dust. Previous studies have used the combination of H i absorption lines and dust to derive predictions for f esc (e.g., Reddy et al. 2016; Chisholm et al. 2018; Saldana-Lopez et al. 2022), and the β 1550 parameter alone predicts much of the variation in f esc (Chisholm et al. 2022). Importantly, unlike other tracers of H i content and dust such as Ly α and the nebular E(BV), the H i absorption lines and UV measures of dust attenuation are more direct tracers of the H i and dust absorption along the line of sight to the UV-emitting stars and can thus substantially aid the prediction of the lineof-sight f esc . The combination of EW(H i ,abs) and one of the UV dust measures alone achieves a concordance C = 0 . 84. \nThe MC method also finds that these two parameters, H i absorption and UV dust attenuation, have the top mean ranking. The MC mean ranks for the other variables are not necessarily consistent with their rank order in Table 4, which reflects the fact that the ranked order varies considerably among different MC runs (Figure 4). Since C barely changes after adding the first three to five variables, the order of most variables may be largely random, consistent with the broad ranking distribution in Figure 4. As also noted in § 3.1, only the most significant variables dominate the model predictions, and limiting the model to this smaller set of variables gives comparable results to a model generated using a larger variable set. Consequently, we recommend using only the most statistically significant or top-ranked variables when using Cox models to predict f esc . \nTable 4 and Figure 4 show that Ly α measurements are often some of the more important variables, although generally not as important as EW(H i ,abs). For all rankings that involve Ly α , including the MC rankings, a Ly α measurement, either EW(Ly α ), L (Ly α ), or f esc , Ly α , achieves a rank between 1-5. In the MC runs, for the Fiducial+HI model, f esc , Ly α is the third most important variable after EW(H i ,abs) and E(B-V) UV . In Section 3.2, we found that excluding f esc , Ly α had one of the most detrimental effects on the fiducial model. Crucially, the fiducial model lacks EW(H i ,abs), so f esc , Ly α provided essential information on the H i optical depth in its place. With EW(H i ,abs) included, f esc , Ly α plays a lesser role but is still ranked among the top variables. \nThe rankings for the Full Model in Table 4 show that EW(H i ,abs) is a better predictor of f esc in the LzLCS+ sample than the residual intensity R l (H i ,abs). This \nranking may result from the low resolution of the FUV spectra. Low spectral resolution, which depends on the spatial size of the source, will artificially increase the observed residual intensity, R l (H i ,abs). In contrast, the EW(H i ,abs) is less sensitive to spectral resolution and may serve as a more accurate measure of the lineof-sight gas. Indeed, Saldana-Lopez et al. (2022) find that at a fixed R l , stronger LCEs have lower EWs. At the same time, the observational uncertainties are also higher in R l , such that f esc trends may appear more readily with EW. To investigate the effect of observational uncertainty on the rankings, we compare each variable's median uncertainty with standard deviation of that variable in the LzLCS+ sample. If the measurement uncertainty is comparable to the standard deviation, we may not be able to discern trends with that variable across the LzLCS+. For most variables, the median uncertainty is much lower, ≲ 20% of the standard deviation. These variables span the full range of possible ranks and MC ranks, which suggests that we can distinguish variables that do and do not affect the f esc predictions. However, a few variables ( R l (H i ,abs), log 10 ( M ∗ ), EW(LIS), and R l (LIS)) have higher ratios of median uncertainty to standard deviation (0.50-1.03). These variables all have high MC ranks in the Full Model (6.73-10.81) and any genuine trends with f esc may be hidden by the uncertainties in their measurements. As a test, we insert log 10 ( f esc ) as a dummy variable, as it should correlate perfectly with itself. Its MC rank begins to deviate from 1.00 when we give it an uncertainty ≥ 0 . 6 times the standard deviation. To further test the effect of uncertainty, we re-run the MC rankings after doubling the uncertainty in the EW(H i ,abs) measurements, such that the EW(H i ,abs) and R l (H i ,abs) variables have similar ratios of uncertainty to standard deviation. The EW(H i ,abs) MC ranks of 1.21-4.90 increase to 2.6-7.2 when the uncertainty is doubled. We conclude that the higher uncertainty in R l (H i ,abs), log 10 ( M ∗ ), EW(LIS), and R l (LIS) may prevent us from determining the importance of these variables. Higher resolution data are necessary to test whether EW(H i ,abs) or R l (H i ,abs) better predicts f esc . \nIn their multivariate analysis of f esc from cosmological simulations, Maji et al. (2022) also use forward and backward selection to rank the importance of variables. They find that the three most important predictors of f esc are L (Ly α ), the SFR, and galaxy gas mass. Our rankings share some broad similarities with these simulation results. We also find that gas content (as measured by EW(H i ,abs)), SFR (in the form of the UV luminosity), and Ly α emerge as important parameters. However, our sample and the Maji et al. (2022) sim- \nForward MC Rank Full Model \n<!-- image --> \n<!-- image --> \nForward MC Rank Fiducial+HI \n<!-- image --> \n<!-- image --> \nForward MC Rank JWST Model \n<!-- image --> \n<!-- image --> \nBackward MC Rank Full Model \n<!-- image --> \n<!-- image --> \n<!-- image -->", 'Backward MC Rank Fiducial+HI': '<!-- image --> \nBackward MC Rank JWST Model \n<!-- image --> \nFigure 4. The distribution of variable rankings from 100 forward selection (left) and backward selection (right) MC runs, after resampling each variable using its observational uncertainties. Variable combinations are the same as in Table 4. Thick solid lines indicate the four variables with the lowest (best) mean MC ranks. \n<!-- image --> \nulated galaxies have some crucial differences. First, we measure the line-of-sight f esc , whereas Maji et al. (2022) measure the total global f esc through all sightlines; consequently, the relevant parameter for our analysis is the line-of-sight H i rather than the global H i . Secondly, the galaxies in the Maji et al. (2022) sample are much less massive than the LzLCS+ sample, with a median stellar mass of log 10 ( M ∗ / M ⊙ )=6.41 vs. 8.8. Dust may play a greater role in determining f esc in the higher mass, more enriched galaxies in our sample, which would account for its higher importance in our predictions.', '3.4. Predictions at z > 6 with JWST': "Although the fiducial model and the models using the top-ranked variables predict f esc reasonably well, we cannot apply these models to JWST observations of galaxies in the epoch of reionization. The GunnPeterson trough and Lyman-series IGM absorption at lower redshifts will prevent H i absorption line measurements, and the partially neutral IGM at z > 6 can suppress Ly α emission (e.g., Stark et al. 2011; Schenker et al. 2014). Measuring LIS absorption lines instead requires high signal-to-noise observations of the restUV continuum, which will be difficult for faint galaxies. Consequently, we explore alternative models that use parameters that can be derived from z > 6 JWST observations. \nWe create a JWST model by modifying the fiducial model to exclude EW(LIS) and f esc , Ly α . We also choose β 1550 instead of E(B-V) UV as it is more easily inferred from observations without any required stellar population modeling (e.g., Chisholm et al. 2022) or assumptions about the dust-attenuation law. The JWST model thus includes the following variables: log 10 ( M ∗ ), M 1500 , log 10 (EW(H β )), E(B-V) neb , 12+log 10 (O/H), log 10 (O32), log 10 (Σ SFR ), β 1550 . We show the resulting Cox model fit in Figure 5a and list the goodness-of-fit metrics in Table 3. The scatter is noticeably higher by 0.1 dex than in the fiducial model, with RMS=0.47 dex. Like the fiducial model, the JWST model also tends to under-predict f esc in several of the strongest LCEs with f esc > 0 . 2. This reduced fit quality shows that information about the line-of-sight H i content from absorption lines and Ly α is essential to precisely predict f esc . Nevertheless, other observable properties can still provide rough f esc estimates and distinguish strong LCEs from non-leakers. Since the JWST model relies more on global properties rather than line-of-sight H i to predict f esc , the underprediction for the most extreme LCEs suggests that their global properties are not distinct from more moderate LCEs. The model-predicted f esc may indicate the typical f esc value for this combination \nof parameters. Moreover, if the most extreme f esc occurs only along favorable, nearly transparent sightlines, the f esc predicted from global parameters could be a better estimate of these galaxies' global average f esc . The strongest LCEs in the LzLCS+ sample do tend to have high nebular EWs (EW(H β ≳ 200 ˚ A; Flury et al. 2022b), which shows that they cannot be devoid of absorbing gas in all directions. \nOnly three variables are statistically significant coefficients in the JWST model: β 1550 , log 10 (Σ SFR ), and log 10 (O32). These same three parameters are ranked as the most important in forward and backward selection and have the lowest mean ranks from forward and backward selection with MC sampling (Table 4, Figure 4). When we limit the model to just these three variables, the fit quality is similar to or better than the JWST model with the full set of variables (Table 3). We also find that the C CV for the cross-validation analysis is closer to the C derived from the full sample; with fewer variables in the fit, even a smaller training sample can generate a reliable model. We display the predicted f esc from this model in Figure 5b. Using [Ne iii ]/[O ii ] instead of O32 results in a similar quality fit (Table 3) and may be more suitable for observations covering a limited spectral range, observations at z ≳ 11 where [O iii ] is redshifted out of the NIRSpec wavelength range, or observations with uncertain nebular dust attenuation (Levesque & Richardson 2014). These fits demonstrate that the multivariate Cox model can predict f esc for high-redshift galaxies using JWST observables. We apply Cox models to high-redshift galaxy samples in a forthcoming paper (Jaskot et al., in prep.), where we also provide all parameters needed to apply these models to future samples. \nOur JWST model includes similar variables as the multivariate f esc predictions derived by Choustikov et al. (2024) from the SPHINX simulations, but we find a different dependence on some of these variables. Choustikov et al. (2024) use the following observables: UV slope β , E(B-V) neb , H β luminosity, EW(H β ), UV magnitude, R23=([O iii ] λλ 5007,4959+[O ii ] λ 3727)/H β , O32, and half-light radius. Like the Choustikov et al. (2024) model, we find that β 1550 is statistically significant and that high f esc is associated with blue UV slopes. We likewise find that O32 is statistically significant. However, in the LzLCS+ sample, O32 correlates with f esc , whereas the SPHINX simulations find an anti-correlation. Strong LCEs with high O32 do not appear within the SPHINX galaxy population (Choustikov et al. 2024). This disagreement may result from the different properties of the observed vs. simulated galaxy populations, such as the stellar mass range. Alterna- \n<!-- image --> \nFigure 5. The f esc predictions from JWST models with the full set of variables (a) and top-ranked variables only (b). Symbols are the same as in Figure 2. The scatter in these models is higher than in the fiducial model, but they still reproduce the observed f esc with an RMS scatter of 0.46-0.47 dex. \n<!-- image --> \nit may indicate the need for more efficient radiative feedback and/or resolving smaller-scale turbulent gas structure, to allow LyC escape from younger clusters (e.g., Kimm et al. 2019; Kakiichi & Gronke 2021; Choustikov et al. 2024). \nOur model also disagrees with Choustikov et al. (2024) regarding the role of Σ SFR . Whereas we find that Σ SFR is one of the most important predictors of f esc in the JWST model and is statistically significant, Choustikov et al. (2024) find that compactness is insignificant and shows little relationship with f esc . One of main differences between the LzLCS+ and SPHINX galaxy datasets is the mass range probed; the LzLCS+ sample has a median mass of log 10 ( M ∗ / M ⊙ )= 8 . 8 and extends up to log 10 ( M ∗ / M ⊙ )= 10 . 8, whereas the majority of the SPHINX galaxies have masses log 10 ( M ∗ / M ⊙ ) < 9. At even lower masses, log 10 ( M ∗ / M ⊙ ) < 7, simulations suggest that LyC escape may occur in a more extended mode, driven by star clusters in low-column density regions in the outskirts of galaxies (Kostyuk et al. 2023). The LzLCS+ likewise hints that the link between Σ SFR and f esc may be mass dependent. Several of the LCEs with high Σ SFR in the LzLCS+ sample have high masses (log 10 ( M ∗ / M ⊙ )= 9 . 7 -10 . 5), and as we discuss in the following section, Σ SFR may be a particularly significant predictor of f esc in high-mass galaxies.", '3.5. Variations with Galaxy Properties': "Flury et al. (2022b) discuss the possibility that the LyC escape process may vary among different types of galaxies. For example, the dominant feedback mecha- \nnism may shift from radiative feedback to mechanical supernova feedback as a starburst ages or in galaxies with different masses or metallicities (e.g., Jaskot et al. 2019; Kimm et al. 2019; Jecmen & Oey 2023). In addition, the strongest LCEs could have a nearly density-bounded gas geometry, whereas LyC photons may escape along narrow channels in weaker LCEs (e.g., Gazagnes et al. 2020; Flury et al. 2022b). We therefore explore whether the Cox model predictions change if we limit our analysis to different subpopulations of the larger LzLCS+ sample. We first divide our galaxy sample into two bins, above and below the median value of stellar mass, log 10 ( M ∗ / M ⊙ )= 8 . 8. \nDifferent variables appear important for the low- vs. high-mass sample fits. For the fiducial model, Σ SFR is statistically significant in the high-mass fit only, with a coefficient value ∼ 5 × higher than that found in the low-mass fit. Conversely, M 1500 and O32 are only statistically significant in the low-mass fit, with coefficients ∼ 8 and ∼ 5 × stronger than in the high-mass fit. The same pattern occurs with the JWST models, with Σ SFR significant in the high-mass fits and M 1500 and O32 significant in the low-mass fits. \nThe variable ranks by forward and backward selection (Tables 5-6, Figures 6-7) likewise suggest that different variables are important for the two subsamples. As with the full sample (Table 4), variables tracing dust attenuation, H i absorption, and Ly α emission are some of the most important variables for both the high- and low-mass subsamples. However, for the high-mass subsample (Table 5), log 10 (Σ SFR ) consistently ranks as the \nmost important variable in all models and in its mean MC rank values. In the low-mass subsample (Table 6), log 10 (O32) actually appears at the bottom of the ranked list for the full set of variables (rank 15), but by mean MC rank, log 10 (O32) is one of the four most important variables in all models and is the most important variable in the JWST model runs. This seemingly discrepant rank may suggest that O32 provides similar information as some of the other top-ranked variables. Figure 7 shows that O32 replaces EW(H i ,abs) as the top-ranked variable ∼ 10-25% of the time, and high O32 may generally imply weak H i absorption. In the JWST model, when EW(H i ,abs) is not included, O32 takes over as the most important variable in a majority of the MC runs. To test whether the smaller sample size of the high-mass and low-mass subsets affects these results, we re-run the MC variable ranking using bootstrap resampling to change the selected sample. The resulting ranking distributions are similar to those in Figures 6-7 with only minor changes to the top four ranked variables. Our main conclusions remain unchanged. Σ SFR consistently holds the top MC rank for every model for the high-mass subset. Dust, H i absorption, Ly α and O32 are important for the low-mass subset, and O32 is the top-ranked variable for the low-mass JWST model. \nThe separate high- and low-mass models do a slightly better job at reproducing the observed f esc values in LCEs than the full fiducial model with all galaxies included. The separate models show a slightly higher R 2 = 0 . 64 -0 . 70 and lower RMS= 0 . 28 -0 . 32. However, the R 2 adj values of 0.31-0.43 are lower than the full model R 2 adj = 0 . 49, which suggests the improvement mainly comes from fitting a smaller sample. One exception is the JWST Model, where the high-mass subsample predictions are substantially improved compared to the full model; R 2 rises to 0.54 from 0.29, R 2 adj is 0.23 instead of 0.14, the RMS drops from 0.47 to 0.34, and C increases from 0.83 to 0.90. The Σ SFR variable takes on greater importance in the JWST model, since it lacks H i and Ly α information. The high-mass subsample f esc depends strongly on Σ SFR , and the best-fit coefficient for Σ SFR increases by a factor of four compared to the full sample JWST model. Evidently Σ SFR is key for accurately predicting f esc in the higher mass galaxies within the LzLCS+. \nInterestingly, the importance of Σ SFR vs. O32 for different subsamples only emerges when we split the sample by stellar mass, not by UV luminosity. Almost all the highly ionized galaxies, with O32 ≳ 5 are in the lower stellar mass bin. Some of these galaxies can be quite luminous for their mass, so they are not as distinguishable by luminosity. The galaxies with the highest O32 ratios \nalso have high nebular emission line EWs, suggestive of young ages. Lower-mass galaxies may have more bursty star formation histories in general (e.g., Lee et al. 2007). Among the low-mass galaxies in the LzLCS+, the high O32, high EW galaxies may represent currently bursting galaxies with extremely young average ages for their UVemitting stellar populations (e.g., Izotov et al. 2011). \nThe noticeably different roles of the Σ SFR and O32 variables in the high- vs. low-mass samples suggest that the causes of LyC escape may indeed differ between these galaxy populations. In the low-mass sample, radiative feedback in young starbursts may drive high LyC escape over wide opening angles (e.g., Flury et al. 2022b). Galaxies with high luminosities and highly ionized gas would have higher f esc . At the same time, if the youngest stellar population dominates the UV light output, the measured line-of-sight dust attenuation and H i absorption would also be representative of the conditions near the LyC-emitting stars and would be sufficient to constrain f esc . In contrast, escape in higher-mass galaxies, with potentially longer-duration starburst episodes may rely on the cumulative effect of supernova feedback punching holes in the interstellar medium (ISM; e.g., Flury et al. 2022b). The ability to drive these required outflows may be linked to the galaxy's Σ SFR (e.g., Heckman et al. 2001, 2011; Kim et al. 2020). If the size of these holes is sufficiently small, the measured dust attenuation and H i EW across the full starburst may not reflect the conditions at the LyC escape site, especially if the youngest LyC-emitting populations do not dominate the total UV light. This disconnect could account for the reduced importance of β 1550 and EW(H i ,abs) and enhanced importance of Σ SFR in the high-mass models. \nT able 5 . Most Imp ortan t V ariables for High-Mass Sample (log 10 ( M ∗ ) ≥ 8 . 8) \n7] \n[1.1 \n) \nSFR \n(Σ \n10 \nlog \n[1.10] \n) \nSFR \n(Σ \n10 \nlog \n] \n[2.98 \n) \nSFR \n(Σ \n10 \nlog \n[1.79] \n) \nSFR \n(Σ \n10 \nlog \n[4.35] \n) \nSFR \n(Σ \n10 \nlog \n24] \n[3. \n) \nSFR \n(Σ \n10 \nlog \n[4.61] \n(O32) \n10 \nlog \n[5.06] \n) \n∗ \nM \n( \n10 \nlog \n[4.37] \nα \nLy \n, \nesc \nf \n[3.68] \nUV \nE(B-V) \n[7.17] \n) \nα \n(Ly \nL \n[5.38] \n,abs) \ni \nEW(H", 'JWST Mo del': "del \nMo \nFiducial+HI \ndel \nMo \null \nF \nRank \nard \nkw \nBac \nard \norw \nF \nard \nkw \nBac \nard \norw \nF \nard \nkw \nBac \nard \norw \nF \nT op-Rank ed V ariables \nOrder \nRank \nMC \ny \nb \nariables \nV \nop \nT \n[2.93] \n1550 \nβ \n[2.64] \n1550 \nβ \n[4.33] \n,abs) \ni \nEW(H \n[3.53] \nα \nLy \n, \nesc \nf \n5] \n[6.7 \n(O32) \n10 \nlog \n[5.34] \n1550 \nβ \n[5.14] \nneb \nE(B-V) \n[5.36] \n)) \nβ \n(EW(H \n10 \nlog \n[4.76] \nUV \n) \nE(B-V \n3.99] \n[ \n,abs) \ni \nEW(H \n[7.41] \n1550 \nβ \n[6.18] \nα \nLy \n, \nesc \nf \nNote -Num b ers in brac k ets indicate the mean of the v ariable's ranks in the 100 MC runs. \nForward MC Rank Full Model High Mass \n<!-- image --> \n<!-- image --> \nForward MC Rank Fiducial+HI High Mass \n<!-- image --> \n<!-- image --> \nForward MC Rank JWST Model High Mass \n<!-- image --> \n<!-- image --> \nBackward MC Rank Full Model High Mass \n<!-- image --> \n<!-- image --> \n<!-- image -->", 'Backward MC Rank Fiducial+HI High Mass': '<!-- image --> \n<!-- image -->', 'Backward MC Rank JWST Model High Mass': 'Figure 6. The distribution of MC variable rankings from forward selection (left) and backward selection (right) for galaxies with log 10 ( M ∗ / M ⊙ ) ≥ 8 . 8. Variable combinations are the same as in Table 4. Thick solid lines indicate the four variables with the lowest (best) mean MC ranks. \n<!-- image -->', 'T able 6 . Most Imp ortan t V ariables for Lo w-Mass Sample (log 10 ( M ∗ ) < 8 . 8)': "bles \naria \nV \ned \nop-Rank \nT \nOrder \nRank \nMC \ny \nb \nariables \nV \nop \nT \nNote -Num b ers in brac k ets indicate the mean of the v ariable's ranks in the 100 MC runs. \nTo explore variations related to feedback, we also split the sample based on the median values of log 10 (Σ SFR )=0.77 and O32=3.4. While splitting the LzLCS+ sample on mass resulted in slightly different variable selections, the high O32 and high Σ SFR subsample fiducial fits more closely resemble the full model, likely because they contain nearly the same population of strong LCEs; the high Σ SFR and high O32 subsets each include all but two of the strong LCEs with f esc > 0 . 1. The two most important predictors of f esc remain H i absorption (either R l (H i ,abs) or EW(H i ,abs)) and dust attenuation. Interestingly, Σ SFR is no longer a statistically significant variable in the fiducial and JWST models for the high Σ SFR and high O32 subsamples. Σ SFR may be more useful for identifying non-leakers than in actually quantifying f esc . Indeed, LCEs in the LzLCS+ sample nearly all lie above a threshold value of Σ SFR = 10 M ⊙ yr -1 kpc -2 , but above this value, no discernible relationship exists between f esc and Σ SFR (Flury et al. 2022b). \nFor the low Σ SFR subsample, only f esc , Ly α and E(BV) UV are statistically significant in the fiducial model, while these same variables plus E(B-V) neb are significant for the low O32 subset. As usual, in the variable rankings, EW(H i ,abs) and dust measurements are typically important variables. However, Ly α measurements also appear to be relevant predictors. For the Full Model variable set, L (Ly α ) is the top-ranked variable for the low O32 subsample. Similarly, for the low Σ SFR subsample, f esc , Ly α is the top-ranked variable for the Fiducial+HI model and the top-ranked variable in forward modeling for the Full Model. Because these subsamples contain few strong LCEs, this result emphasizes that Ly α emission may be useful in distinguishing weak LCEs from non-leakers.", '3.6. Outliers in f esc Trends': "Many studies have identified relationships between f esc and physical or observable parameters (e.g., Verhamme et al. 2017; Izotov et al. 2018b; Flury et al. 2022b; Saldana-Lopez et al. 2022; Chisholm et al. 2022), yet even the best trends show significant scatter, with individual galaxies showing f esc significantly above or below the expected value. Schaerer et al. (2022b) highlight one outlier galaxy in particular, J1248+4259. J1248+4259's high nebular ionization and strong Ly α emission resemble those of other strong low-redshift LCEs, yet its f esc is ≤ 1.3%. Here, we investigate whether the Cox model improves f esc predictions for J1248+4259 and other outlier galaxies. \nWe select outliers based on the variables that show the strongest trends with f esc in the Flury et al. \n(2022b) analysis of the LzLCS+ sample: r 50 , NUV , O32, Σ SFR / M ∗ , EW(Ly α ), Σ SFR , and f esc , Ly α . From the strong LCEs ( f esc ≥ 0 . 05) in the LzLCS+ sample, we choose the three with the highest r 50 , NUV , the three lowest in O32, the three lowest in Σ SFR / M ∗ , the three lowest in EW(Ly α ), the three lowest in Σ SFR , and the three lowest in f esc , Ly α . Conversely, we choose three weak or non-LCEs ( f esc < 0 . 05) with the lowest r 50 , NUV , highest O32, highest Σ SFR / M ∗ , highest EW(Ly α ), highest Σ SFR , or highest f esc , Ly α . This selection gives us a list of 10 strong LCEs and 13 weak or non-LCEs that are outliers by one or more selection. In other words, applying a simple, single-variable trend as in Flury et al. (2022b) would incorrectly identify these galaxies as strong LCEs or as non-leakers. \nFigure 8a shows that the fiducial model modified to include EW(H i ,abs) predicts f esc for these 'outliers' quite well. In fact, the outlier predictions are generally no more or less accurate than the predictions for other galaxies in the LzLCS+ sample. For example, the outlier galaxy J1248+4259, which 'should' be a strong LCE based on its high O32 ratio and high EW(Ly α ), has a predicted f esc of 0.006 according to the model, consistent with its observed limit of f esc ≤ 0.013. \nCrucially, the fiducial+EW(H i ,abs) model includes information about the optical depth along the line of sight, which is necessary to predict the f esc observed along the same line of sight. As discussed in Sections 3.2 and 3.3, EW(H i ,abs), f esc , Ly α , and E(B-V) UV , all of which are sightline dependent, have some of the largest effects on the accuracy of the model predictions. If we remove these and other sightline-dependent parameters (E(B-V) neb , M 1500 ) from the fit (Figure 8b), we can no longer reproduce the line-of-sight f esc of many galaxies. Figure 8b indicates that we would expect a different f esc for these galaxies based on their global properties. Hence, if the only cause of the scatter in Figure 8b is chance galaxy orientation, the predicted f esc should represent the global average across all sightlines. By this model, J1248+4259 should indeed be an LCE, although not an exceptionally strong one, with f esc = 0 . 051. \nAt high f esc , the model without line-of-sight information seems to systematically under-predict the observed f esc , which may suggest that the global properties of the strongest LCEs are not distinct from those of weaker LCEs. The observed properties of the strongest LCEs support this interpretation. For example, the strongest LCEs all tend to have high O32 (Flury et al. 2022b), yet the median observed f esc for the 11 galaxies with O32 ≥ 10 is only 0.05. For these same 11 galaxies, the model without line-of-sight information finds the median of their predicted f esc values to be 0.04, compa- \nForward MC Rank Full Model Low Mass \n<!-- image --> \n<!-- image --> \nForward MC Rank Fiducial+HI Low Mass \n<!-- image --> \n<!-- image --> \nForward MC Rank JWST Model Low Mass \n<!-- image --> \n<!-- image --> \nBackward MC Rank Full Model Low Mass \n<!-- image --> \n<!-- image --> \nBackward MC Rank Fiducial+HI Low Mass \n<!-- image --> \n<!-- image --> \nBackward MC Rank JWST Model Low Mass \n<!-- image --> \nFigure 7. The distribution of MC variable rankings from forward selection (left) and backward selection (right) for galaxies with log 10 ( M ∗ / M ⊙ ) < 8 . 8. Variable combinations are the same as in Table 4. Thick solid lines indicate the four variables with the lowest (best) mean MC ranks. \n<!-- image --> \n<!-- image --> \nFigure 8. The f esc predictions from the fiducial model (a) modified to use EW(H i ,abs) and (b) modified to exclude information about the line of sight from Ly α , absorption lines, dust attenuation, or the observed UV luminosity. Black crosses show strong LCEs that are outliers by one or more selection criteria; gray circles and triangles show weak and non-LCEs that are outliers by one or more selection criteria. Small red circles and blue triangles represent the other detections and non-detections in the LzLCS+ sample. The fiducial+EW(H i ,abs) model reproduces the f esc of the outliers, whereas the model without line-of-sight information does not. \n<!-- image --> \nrable to the observations. The only model that successfully reproduces the f esc of the strongest LCEs is the fiducial+EW(H i ,abs) model. Hence, of the variables we consider, only EW(H i ,abs) distinguishes the strongest LCEs from other galaxies. Either the strongest LCEs differ in some other unknown global property or their high f esc results from a favorable orientation compared to weaker LCEs. The high O32 galaxies may have f esc ∼ 0 . 04 -0 . 05 along most sightlines but may also contain several extremely optically thin channels, which lead to much higher observed f esc , ≳ 0 . 5 in some cases. \nAlthough orientation bias likely causes some of the scatter in f esc predictions, the Cox modeling results also suggest that some of the 'outlier' galaxies are not outliers at all, at least not when multiple global variables are considered. For example, J0919+4906 and J02320426 have high O32 ratios of 12.7 and 9.9 but f esc of 0.049 and < 0 . 047, respectively. However, both galaxies also have low stellar masses (log 10 ( M ∗ )= 7 . 5 and 7.4), such that their O32 values are not extreme compared to other galaxies of similar mass (see Figure 23 in Flury et al. 2022b). Furthermore, other criteria, such as their moderate or low Σ SFR values (log 10 (Σ SFR )= 0 . 66 and -0.39) would argue against a high f esc . The combination of multiple properties leads to a moderate predicted f esc for J0919+4906 ( f esc =0.048) and little escaping LyC ( f esc =0.005) for J0232-0426, even when the model includes only global, and not sightline-dependent, properties. Similarly, several galaxies have concentrated star formation (J0910+6105, J1209+3053, J1248+1234, J1349+5631, J1503+3644, J1517+3705) or strong Ly α (J0122+0520, J0811+4141, J0901+2119, J1648+4957) but f esc < 0 . 05; these galaxies' other properties yield consistent predictions of f esc < 0 . 05, even in models that neglect line-of-sight information. \nThe Cox model's success at reproducing f esc even for these outliers demonstrates that accurately predicting f esc requires multiple input variables. Moreover, because the observed f esc depends on our line of sight, the best models also require sightline-specific information about optical depth from absorption lines, UV dust attenuation, and Ly α . Although Ly α and H i absorption lines are inaccessible for most z > 6 galaxies, highredshift f esc predictions can include information about UV dust attenuation as in the JWST models discussed in Section 3.4. Information about dust attenuation is critical to predict f esc , and large sample sizes can probe multiple random orientations to estimate the average f esc for a galaxy population (e.g., Cen & Kimm 2015; Saldana-Lopez et al. 2023).", '4.1. Alternative Measures of LyC: Predicting F λ LyC /F λ 1100 and L (LyC)': "As shown above, multivariate Cox models can successfully predict f esc ; we now consider whether they can predict alternative measurements of the escaping LyC. To model how galaxies reionized the universe, we need to know the input rate of LyC photons into the IGM from different galaxy populations. Deriving this quantity from f esc requires knowing the production rate of LyC photons in each galaxy, which is typically derived from SED fitting or Balmer line observations. To avoid this process and its associated systematics, we could instead predict a more directly observable quantity, such as the F λ LyC /F λ 1100 ratio or the total escaping LyC luminosity log 10 ( L (LyC)). The F λ LyC /F λ 1100 flux ratio measures the ratio of observed LyC flux to the observed flux at 1100 ˚ A. As such, it depends on observed quantities and does not require an assumed stellar population model or dust attenuation law. To predict F λ LyC /F λ 1100 using the Cox proportional hazards model, we follow the same procedure as for f esc and convert the F λ LyC /F λ 1100 measurements to right-censored data (i.e., detections and lower limits) by predicting 1F λ LyC /F λ 1100 . Unlike f esc and F λ LyC /F λ 1100 , which cannot be higher than 1, L (LyC) has no obvious maximum allowed value. To convert it to a right-censored format, we therefore adopt a maximum of log 10 ( L (LyC))= 41, greater than the maximum luminosity in the sample of log 10 ( L (LyC))=40.6, and predict 41-log 10 ( L (LyC). We present the goodnessof-fit metrics for the fiducial and JWST model predictions of F λ LyC /F λ 1100 , L (LyC), and Ly α measurements in Table 7. \nThe F λ LyC /F λ 1100 models perform worse than the f esc models for every variable combination we consider (see Figure 9) and by all three metrics ( R 2 , RMS, C ). Although F λ LyC /F λ 1100 is simpler to measure, f esc appears simpler to predict. It may be more directly linked to galaxy physical properties such as optical depth, porosity, and feedback. Furthermore, while measuring F λ LyC /F λ 1100 does not depend on assumptions about the galaxy SED, predicting it does. The galaxy SED shape changes with age and metallicity, which affects the ratio of F λ LyC /F λ 1100 even for f esc = 1 (see e.g., Chisholm et al. 2019). \nAlthough we derive both E(B-V) UV and f esc from the COS UV spectra, the greater accuracy of the f esc Cox models does not appear to result from this interdependency. First, we see the same improvement in accuracy for f esc vs. F λ LyC /F λ 1100 when we use β 1550 instead of E(B-V) UV ; β 1550 is directly related to the observed spectral slope and is not sensitive to the choice of stellar population model or star formation history (Chisholm et al. \n2022). Secondly, the f esc models have higher R 2 and C than the F λ LyC /F λ 1100 predictions, even for the models that contain no UV spectral information at all. We conclude that although F λ LyC /F λ 1100 is a useful quantity, accurately predicting it is difficult given its dependence on the intrinsic SED shape. We list and discuss the topranked variables for the F λ LyC /F λ 1100 predictions in the Appendix. \nThe F λ LyC /F λ 1100 predictions rely on much the same variables as the f esc predictions, except for a weaker dependence on the UV dust attenuation, as expected. For the fiducial model, the same variables are statistically significant except for E(B-V) UV , and excluding E(B-V) UV from the fiducial model changes R 2 , RMS, and C by only ≤ 0 . 03. However, when we substitute EW(H i ,abs) in the fiducial model, E(B-V) UV does appear statistically significant, indicating that its influence is not always negligible. Table 8 in the Appendix lists the top-ranked variables and likewise indicates a reduced role for dust attenuation in predicting F λ LyC /F λ 1100 compared to its role in predicting f esc . \nLike F λ LyC /F λ 1100 , the predictions for L (LyC) (Figure 10, Table 7) are also less accurate than for the f esc predictions. The concordance drops for every model, and R 2 is negative in all cases, meaning that simply adopting the mean L (LyC) for the detections would result in a better fit than using the model predictions. The negative R 2 arises from the fact that the R 2 calculation (Equation 8) uses only the detections and their mean, whereas the Cox model fits to the full set of galaxies; R 2 becomes positive if we use the median L (LyC) from the full sample of limits and detections, instead of the mean L (LyC) of the detections alone. Not surprisingly, other measures of luminosity show up as important variables for the L (LyC) models. M 1500 is statistically significant in the fiducial model and is one of top three ranked parameters by forward and backward selection, while L (Ly α ) holds the top place in forward selection for the full variable list (see Table 9 in the Appendix for a list of top-ranked variables). However, the L (LyC) models' poor performance indicates that key information is missing. Specifically, a lack of detailed information about the SED likely prevents an estimate of the intrinsic L (LyC) from M 1500 . The escaping L (Ly α ) also imperfectly traces the escaping L (LyC) because of scattering. In addition, the production of the Ly α recombination line requires LyC absorption, which could cause L (Ly α ) to drop for the few extreme LCEs in the LzLCS+ sample (e.g., Nakajima & Ouchi 2014). In conclusion, neither the L (LyC) nor the F λ LyC /F λ 1100 mod- \nide satisfactory alternatives to the f esc predictions.", '4.2. Predicting f esc , Ly α': 'Although closely related to the H i optical depth, the escape of Ly α may depend on different parameters than the LyC escape fraction does. Here, we apply the Cox model to the f esc , Ly α data for the LzLCS+ sample and assess the differences between predictions for f esc , Ly α vs. f esc . We use the same input variables, with one exception: we exclude any variables containing Ly α measurements ( f esc , Ly α , EW(Ly α ), L (Ly α )) from the models. Also, unlike our f esc data, the f esc , Ly α measurements include no upper limits, since Ly α can appear in absorption or emission. For the negative values of f esc , Ly α , representing net absorption, we cannot calculate log 10 ( f esc , Ly α ) and do not include these galaxies in the reported R 2 or RMS; they are included in the concordance. \nThe fiducial model is not very successful at predicting f esc , Ly α (Figure 11a, Table 7), with R 2 = 0 . 33, R 2 adj = 0 . 24, RMS=0.32, C = 0 . 75. The lack of upper limits and need to precisely rank galaxies with even very low f esc , Ly α may explain the reduced C values for the f esc , Ly α model. Including EW(H i ,abs) improves the f esc , Ly α predictions ( R 2 = 0 . 44, RMS=0.30, C = 0 . 78; Figure 11b). As with the f esc predictions, the JWST f esc , Ly α model performs worse than the fiducial model (Figure 11c), which demonstrates that the observed f esc , Ly α does depend on line-of-sight H i properties such as the neutral gas covering fraction or H i column density. Unlike the f esc models, for f esc , Ly α , the fiducial model without E(B-V) UV performs comparably to the fiducial model ( R 2 = 0 . 37, RMS=0.29, C = 0 . 75), because Ly α is more sensitive to the nebular attenuation rather than the stellar attenuation. \nWe also investigate L (Ly α ), using a similar method as for L (LyC) and predicting 43.5-log 10 ( L (Ly α )). We exclude all galaxies with net Ly α absorption and predict only the luminosities of Ly α Emitters. Whereas we found that L (LyC) was harder to predict than f esc , for L (Ly α ), the situation is reversed: the L (Ly α ) models are actually slightly better than the f esc , Ly α models (Table 7 and Figure 12). The L (LyC) predictions suffered from the large uncertainty in estimating the intrinsic LyC production from the FUV continuum. The available Balmer line measurements make estimating the intrinsic Ly α luminosity less of a concern. Another explanation for the better performance of the L (Ly α ) predictions may be mundane; since it depends on predicting the intrinsic Ly α , f esc , Ly α is a more modeldependent measurement than L (Ly α ) and consequently \n<!-- image --> \n(a) \nFigure 9. Examples of F λ LyC /F λ 1100 model predictions using the variable sets for the fiducial model (a) and the JWST model (b). Symbols are the same as in Figure 2. Predictions of F λ LyC /F λ 1100 are not as accurate as the model predictions for f esc . \n<!-- image --> \nTable 7. Goodness-of-Fit Metrics for Different Dependent Variables \na The f esc , Ly α variable is not included in the fiducial model for the f esc , Ly α and L (Ly α ) predictions. \nhas a greater uncertainty (26% on average for f esc , Ly α vs 21% for the Ly α flux alone). Again the L (Ly α ) JWST model (Table 7 and Figure 12b) has slightly worse R 2 and RMS metrics than the fiducial model because of the lack of information about line-of-sight gas. \nTables 10 and 11 in the Appendix list the topranked variables for predicting f esc , Ly α and L (Ly α ). For f esc , Ly α , we find that H i absorption strength and nebular dust attenuation are two of the most important \npredictive variables. In contrast, the UV dust attenuation is less important, always ranking after E(B-V) neb in forward and backward selection. As a nebular emission line, Ly α escape is more closely linked to the nebular attenuation rather than the attenuation experienced by the stellar light. \nFor L (Ly α ), the most important variables include Σ SFR , M 1500 , E(B-V) neb , EW(H β ), and EW(H i ,abs). These five variables are also statistically signifi- \n<!-- image --> \n(a) \n<!-- image --> \n(b) \nFigure 10. Examples of L (LyC) model predictions using the variable sets for the fiducial model (a) and the JWST model (b). We plot galaxies with predicted L (LyC) ∼ 0 at the bottom of each panel. Symbols are the same as in Figure 2. The models fail to accurately predict L (LyC). \n<!-- image --> \n<!-- image --> \nFigure 11. Examples of f esc , Ly α model predictions using the variable sets for the fiducial model (a), the fiducial model with EW(H i ,abs) (b) and the JWST model (c). Symbols are the same as in Figure 2. We plot galaxies with negative f esc , Ly α at f esc , Ly α =2E-3. Of the three models, the fiducial model with EW(H i ,abs) predicts f esc , Ly α with the lowest RMS scatter. \n<!-- image --> \ncant in either the fiducial model (Σ SFR ), the fiducial+EW(H i ,abs) model (EW(H β ), EW(H i ,abs)), or both (E(B-V) neb and M 1500 ). The dependence on UV luminosity reflects the fact that higher SFRs enhance Ly α production, while the dependence on E(B-V) neb and EW(H i ,abs) arises because lower dust attenuation and lower H i column densities enhance Ly α escape. As a recombination line, the H β EW is also directly linked to nebular Ly α production. While the dependence on Σ SFR may reflect the role of feedback in Ly α escape (e.g. Kim et al. 2021), it may also serve to quantify aperture losses, as more extended galaxies may have more of their Ly α emission scatter outside the COS aperture. \nRunnholm et al. (2020) also assess the most important variables in predicting L (Ly α ) using the Lymanα \nReference Sample (LARS). They find that the top five ranked variables by both forward and backward selection are SFR, E(B-V) neb , M ∗ , r 50 , and the gas covering fraction. If using observables, the top variables are the FUV luminosity and the UV size. Despite the different methodologies between the LARS Ly α measurements and the LzLCS+ Ly α measurements, our results generally agree with their findings, as Σ SFR , E(B-V) neb , and the galaxy FUV luminosity also appear among our most important predictors.', '5. DISCUSSION': "Both theoretical and observational studies have long recognized that LyC escape is likely anisotropic due to the inherently inhomogeneous structure of the ISM (e.g., Gnedin et al. 2008; Wise & Cen 2009; Kim et al. 2013; \n<!-- image --> \n(a) \nFigure 12. Examples of L (Ly α ) model predictions using the variable sets for the fiducial model (a) and the JWST model (b). Symbols are the same as in Figure 2. We exclude galaxies with negative L (Ly α ). The models do a better job at predicting L (Ly α ) compared to f esc , Ly α . \n<!-- image --> \nZastrow et al. 2013; Paardekooper et al. 2015; Cen & Kimm 2015; Trebitsch et al. 2017; Keenan et al. 2017; Rivera-Thorsen et al. 2017, 2019; Kim et al. 2023). LyC may escape in a 'picket-fence' geometry, emerging only through holes in the ISM (e.g., Heckman et al. 2001), and simulations find that supernova feedback may carve these low-density paths in the ISM (e.g., Wise & Cen 2009; Cen & Kimm 2015; Paardekooper et al. 2015; Trebitsch et al. 2017). Observationally, the detection of residual flux from saturated low-ionization absorption lines (e.g., Heckman et al. 2001, 2011; Gazagnes et al. 2018) and the co-existence of narrow Ly α emission with underlying broad absorption (e.g., McKinney et al. 2019) provide evidence for optical depth variations within galaxies. Highly ionized channels observed in some low-redshift galaxies (e.g., Zastrow et al. 2013; Bik et al. 2018) and spatially resolved LyC observations in the lensed Sunburst Arc at z = 2 . 37 (Rivera-Thorsen et al. 2019; Kim et al. 2023) likewise imply that LyC may escape through narrow opening angles, perhaps driven by outflows (e.g., Amor'ın et al. 2024). Even when f esc is large, with escape occurring in most or all directions, H i column density variations will still lead to variations in the observed f esc along different lines of sight (e.g., Gazagnes et al. 2020). \nBecause of the anisotropy of LyC escape, estimating global f esc values from line-of-sight LyC detections is difficult, particularly since the observed f esc depends both on galaxy properties and on orientation. The multivariate Cox models present new evidence for the anisotropy of LyC escape by showing that the global properties of \nthe strongest LCEs ( f esc ≳ 0 . 4) do not distinguish them from weaker LCEs ( f esc ∼ 0 . 04; see § 3.6). At z ∼ 3, Nakajima et al. (2020) reach a similar conclusion, finding that high O32 galaxies can have wide variations in f esc , despite sharing comparable spectral properties. We find that the only variable that successfully predicts the high observed f esc of the strongest LCEs is EW(H i ,abs), which traces the line-of-sight H i absorption. This result supports the picture described by Gazagnes et al. (2020), where galaxies with high O32 may have sufficient ionizing radiation to ionize most lines of sight, but where the highest f esc occurs in only the most diffuse channels. \nMultivariate survival analysis models such as those presented here may offer a tool for quantifying the line-of-sight variation in f esc among galaxy populations. Given a large reference sample of galaxies with measurements of relevant global properties, the Cox models generate the probability distribution of f esc for a given set of variables. If the variation in the observed f esc for similar galaxies results only from orientation effects, the derived probability distributions should reflect the distribution of f esc across all sightlines. \nAlthough the outsized importance of EW(H i ,abs) in accurately predicting high observed f esc suggests that favorable orientation accounts for the strongest LCEs in the LzLCS+, orientation alone may not fully explain the scatter between predicted and observed f esc across the entire LzLCS+ sample. LyC escape will depend on local properties, which global variables may not accurately capture (e.g., Rivera-Thorsen et al. 2019). Observational studies at higher spatial resolution and sim- \nulations that track the evolution of f esc with high spatial resolution will help reveal the origin of f esc variations in otherwise similar galaxies. \nThe variable rankings in § 3.3 show that the two most essential variables in predicting the line-of-sight f esc are variables sensitive to the line-of-sight H i gas (EW(H i ,abs)) and the line-of-sight dust (E(B-V) UV or β 1550 ). Although z > 6 observations will not be able to measure the former, β 1550 is easily accessible. JWST observations of the epoch of reionization are rapidly expanding the available measurements of β 1550 at high redshift (e.g., Schaerer et al. 2022a; Mascia et al. 2023; Saxena et al. 2024; Morishita et al. 2024), and this variable alone can provide useful constraints on f esc (Chisholm et al. 2022). By examining the spread of β 1550 values for a fixed intrinsic M 1500 or set of galaxy properties, high-redshift studies could gain insight into the possible variation in the line-of-sight f esc among a population, at least the variation due to changing levels of dust attenuation. To understand the H i component of the LyC absorption and its variation, high-redshift studies will need to rely on simulations and lower-redshift observations. \nOur models provide some clues as to which physical properties most affect a galaxy's H i distribution and resulting f esc . In the absence of EW(H i ,abs) and Ly α measurements, which more directly trace H i , the models find that Σ SFR and O32 are the most important variables in predicting f esc , given constraints on the dust attenuation from β 1550 ( § 3.4). These variables suggest that feedback is key in determining a galaxy's average H i column density. Interestingly, we find in § 3.5 that O32 may be more important in lower mass galaxies (log 10 ( M ∗ / M ⊙ ) < 8 . 8), whereas Σ SFR takes on greater importance in the higher mass LzLCS+ galaxies. Previous studies have suggested that the mechanism responsible for LyC escape may be different in different galaxy populations (e.g., Flury et al. 2022b; Katz et al. 2023). Here, we find that differences in the multivariate f esc models are specifically associated with galaxy mass, rather than luminosity. We hypothesize that the typical star formation histories of the low- and high-mass LzLCS+ galaxies may differ, with the lower-mass starbursts exhibiting burstier star formation (e.g., Lee et al. 2007). Radiative feedback in extremely young, strong starbursts may drive LyC escape over wide opening angles in these low-mass galaxies, whereas the cumulative effect of supernovae in longer-lived concentrated starbursts could regulate LyC escape at higher masses (e.g., Bremer & Dayal 2023). \nWhile the same physical processes, such as bursty star formation with radiative feedback or an extended star \nformation episode with supernova feedback, may occur at high redshift, they may not occur in exactly the same type of galaxies as the LzLCS+. In particular, at a given stellar mass, a high-redshift galaxy could have a different gas mass, total mass, or star formation history compared to the same stellar mass galaxy at low redshift. Nevertheless, the distinct models for the lowvs. high-mass subsets in the LzLCS+ demonstrate that different parameters and physical mechanisms may regulate LyC escape in different types of galaxies. These differences could explain why some studies disagree regarding the properties of LCEs (e.g., Alexandroff et al. 2015; Rutkowski et al. 2017; Izotov et al. 2018b; Naidu et al. 2018). Depending on the sample probed, high O32 or high Σ SFR may or may not be a relevant predictor of f esc . Understanding the origin of the relationships between O32, Σ SFR , and f esc in low- vs. high-mass LzLCS+ galaxies will require more observational and theoretical studies examining the interplay between star formation history, geometry, and feedback in these types of galaxies.", '6. SUMMARY': "LyC escape depends on a range of galaxy properties, from gas and dust distributions in the ISM to galaxy star formation histories and the resulting stellar feedback. Because of this complexity, f esc correlates observationally with a variety of parameters, and correlations with individual variables show high scatter. Using the results of the LzLCS and other z ∼ 0 . 3 LyC observations from the literature, we have generated new multivariate models to predict f esc . We use the statistical technique of survival analysis, specifically the Cox proportional hazards model (Cox 1972), which accounts for data from both LyC detections and f esc upper limits from nondetections. With these models, we explore which combinations of variables best predict f esc and discuss the possible physical basis for these trends. \nWe summarize our main findings below. \n- 1. The f esc predictions from our fiducial model match the observed f esc of z ∼ 0 . 3 LCEs with an RMS of 0.36 dex, and our best-performing model, which includes H i absorption lines, reaches an RMS of 0.31 dex. These multivariate models offer a substantial improvement over correlations with single variables, which can span ≳ 2 dex in f esc for a given galaxy property (e.g., Wang et al. 2021; Flury et al. 2022b; Saldana-Lopez et al. 2022; Chisholm et al. 2022). ( §§ 3.1-3.2).\n- 2. Our best-performing model accurately predicts f esc even for galaxies that are outliers in a singlevariable selection. This model, which includes H i \n- absorption line measurements, is the only model that reproduces the high observed f esc values of the strongest LCEs in the LzLCS+. Among the variables we consider, weak H i absorption is the only distinctive property of these strong LCEs, which suggests that these high f esc values may occur only along fortuitous sightlines. ( § 3.6).\n- 3. The most important variables in the models, as determined by forward and backward selection, are the EW of the H i Lyman series absorption lines and either E(B-V) UV or the UV slope β 1550 . These properties trace the line-of-sight H i absorption and UV dust attenuation, which determine the line-of-sight f esc that we measure (e.g., SaldanaLopez et al. 2022). In our fiducial model, E(BV) UV is a statistically significant variable, along with f esc , Ly α , Σ SFR , E(B-V) neb , and O32. Excluding E(B-V) UV or f esc , Ly α has the greatest adverse effects on model performance, and including f esc , Ly α appears to be particularly important for identifying non-LyC emitters. These analyses show which variables are the most important to include for accurate predictions; models using only the statistically significant variables or the 3-5 topranked variables achieve comparable accuracy to models with a longer list of inputs ( §§ 3.2-3.3).\n- 4. We generate a 'JWST model' for predicting f esc , which can apply to z > 6 observations. The model excludes Ly α and absorption line measurements, which will be inaccessible for most z > 6 galaxies. When compared with the z ∼ 0 . 3 LCEs, this model has a higher scatter (RMS= 0 . 47 dex) than the fiducial model, but it can distinguish LCEs from non-emitters. Three variables in this model are statistically significant: β 1550 , Σ SFR , and O32. These three variables alone can predict f esc with an RMS of 0.46 dex in the z ∼ 0 . 3 sample. ( § 3.4).\n- 5. We find that f esc may depend on different variables for high- vs. low-mass galaxies. Σ SFR is statistically significant for the high-mass subsample but not the low-mass subsample and is the top-ranked variable for the high-mass subsample by forward and backward selection. In contrast, M 1500 and O32 are statistically significant for the low-mass subsample but not for the high-mass subsample. These differences suggest that the physical causes of LyC escape and the feedback mechanisms involved could differ across the galaxy population, with ionization and radiative feedback potentially playing a greater role in low-mass galaxies. ( § 3.5).\n- 6. The multivariate Cox models are less successful at accurately predicting alternative measures of LyC escape, such as F λ LyC /F λ 1100 and the LyC luminosity. We hypothesize that accurately predicting these properties requires information about the unattenuated SED shape, which is not included in our set of variables. ( § 4.1).\n- 7. We also create Cox models to predict Ly α , specifically the f esc , Ly α and Ly α luminosity. Unlike the LyC f esc predictions, H i absorption EW and E(BV) neb , rather than E(B-V) UV , are two of the most important variables for predicting f esc , Ly α . For L (Ly α ), top-ranked and statistically significant variables are the H i absorption EW, E(B-V) neb , M 1500 , EW(H β ), and Σ SFR , which are similar to results found for the LARS sample (Runnholm et al. 2020). ( § 4.2). \nOur models highlight the complexity of LyC escape and the dependence of f esc on multiple variables. Multivariate Cox models represent a new tool to investigate the relationships between f esc and galaxy properties and to predict f esc in the epoch of reionization. These models offer a substantial improvement over f esc predictions based on a single variable, and the models presented here can be refined and expanded to incorporate future observations. Our results show which variables have the greatest effect on f esc predictions for the LzLCS+ sample and for galaxies in different mass ranges. By exploring these same variables, future simulations could try to reproducing these scalings and determine their physical origin. These flexible multivariate models also offer an empirical approach to predicting f esc at high redshift. In a follow-up paper (Paper II, Jaskot et al., in prep.), we apply Cox models derived from the LzLCS+ to z ∼ 3 and z > 6 samples and provide the best-fit models to the community for application to future observations. 2 \n- 2 We have also created a python script, available at https://github. com/sflury/LyCsurv, that generates Cox model f esc predictions using any desired combination of variables from the LzLCS+. The LzLCS+ dataset used in this paper is available at the same page, and version 0.1.0 of the code is archived in Zenodo (Flury et al. 2024). \nWe thank Kate Follette for interesting discussions that led to this project, and we thank the anonymous referee for comments that improved the clarity of this work. AEJ and SRF acknowledge support from NASA through grant HST-GO-15626. STScI is operated by AURA under NASA contract NAS-5-26555. ASL acknowledges support from Knut and Alice Wallenberg Foundation. MT acknowledges support from the NWO grant 0.16.VIDI.189.162 ('ODIN').", 'APPENDIX': "Tables 8-11 list the four top-ranked variables by forward selection, backward selection, and an MC rank ordering for the F λ LyC /F λ 1100 , L (LyC), f esc , Ly α , and L (Ly α ) predictions. We describe these ranking methods in § 2.4 and discuss the results in §§ 4.1 and 4.2. The 'Full Model' uses β 1550 as a measure of UV dust attenuation and includes multiple variables that trace Ly α or UV absorption line strength. The 'Fiducial+HI Model' uses E(B-V) UV to quantify dust attenuation, f esc , Ly α as a measure of Ly α , and EW(H i ,abs) as the only absorption line variable. The 'JWST Model' excludes absorption line and Ly α variables.", 'T able 8 . Most Imp ortan t V ariables for F λ LyC /F λ 1100 Predictions': "ariables \nV \ned \nop-Rank \nT \nOrder \nRank \nMC \ny \nb \nariables \nV \nop \nT \n] \n[1.09 \n) \nSFR \n(Σ \n10 \nlog \n3] \n[1.1 \n) \nSFR \n(Σ \n10 \nlog \n[1.94] \n,abs) \ni \n(H \nEW \n.00] \n[1 \n,abs) \ni \nEW(H \n[2.87] \n,abs) \ni \nW(H \nE \n] \n[1.01 \n,abs) \ni \nEW(H \n2.14] \n[ \n(O32) \n10 \nlog \n[3.22] \n(O32) \n10 \nlog \n[3.64] \nα \nLy \n, \nesc \nf \n[3.57] \nα \nLy \n, \nesc \nf \n[5.83] \n,abs) \ni \n(H \nl \nR \n.86] \n[2 \n,abs) \ni \n(H \nl \nR \n[3.11] \n)) \nβ \n(EW(H \n10 \nlog \n[5.45] \n)) \nβ \n(EW(H \n10 \nlog \n[4.58] \n) \nSFR \n(Σ \n10 \nlog \n[3.79] \n1500 \nM \n95] \n[5. \n)) \nβ \n(EW(H \n10 \nlog \n[7.20] \nα \nLy \n, \nesc \nf \n[4.73] \n1550 \nβ \n[3.97] \n1550 \nβ \n.85] \n[4 \n(O32) \n10 \nlog \n[4.21] \n) \nSFR \n(Σ \n10 \nlog \n[6.68] \n) \nα \nEW(Ly \n[7.50] \n) \nSFR \n(Σ \n10 \nlog \nor \n) \nα \nEW(Ly \nNote -Num b ers in brac k ets indicate the mean of the v ariable's ranks in the 100 MC runs. \n.71] \n[1 \n(O32) \n10 \nlog \n[1.17] \n) \nSFR \n(Σ \n10 \nlog \n[2.95] \n1500 \nM \n[2.61] \n1500 \nM \n[3.05] \n1500 \nM \n[1.00] \n) \nα \n(Ly \nL \n[2.87] \n) \nSFR \n(Σ \n10 \nlog \n[3.53] \n1500 \nM \n[4.27] \nα \nLy \n, \nesc \nf \n] \n[3.30 \n) \nSFR \n(Σ \n10 \nlog \n6.08] \n[ \n,abs) \ni \nEW(H \n[6.21] \nneb \nE(B-V) \n[4.58] \n)) \nβ \n(EW(H \n10 \nlog \n.38] \n[4 \n(O32) \n10 \nlog \n] \n[4.54 \n(O32) \n10 \nlog \n[3.32] \nα \nLy \n, \nesc \nf \n52] \n[6. \n(O32) \n10 \nog \nl \n[6.96] \n) \nα \nEW(Ly", 'T able 9 . Most Imp ortan t V ariables for L (LyC) Predi ctions': "l \nde \nMo \nST \nJW \ndel \nMo \ncial+HI \nFidu \ndel \nMo \null \nF \nRank \nard \nkw \nBac \nard \norw \nF \nard \nkw \nBac \nard \norw \nF \nard \nkw \nBac \nard \norw \nF \nariables \nV \ned \nop-Rank \nT \nOrder \nRank \nMC \ny \nb \nariables \nV \nop \nT \n[2.55] \n1500 \nM \n[3.40] \n1550 \nβ \n9] \n[3.3 \n,abs) \ni \nEW(H \n[3.28] \n,abs) \ni \nEW(H \n[4.78] \n) \nα \nEW(Ly \n[4.43] \n1500 \nM \nNote -Num b ers in brac k ets indicate the mean of the v ariable's ranks in the 100 MC runs.", 'T able 10 . Most Imp ortan t V ariables for f esc , Ly α Predictions a': 'del \nMo \nJWST', 'Fiducial+HI Mo del': "del \nMo \null \nF \nRank \nard \nkw \nBac \nard \norw \nF \nard \nkw \nBac \nard \norw \nF \nard \nkw \nc \na \nB \nard \norw \nF \nT op-Rank ed V ariable s \n)) \nβ \n(EW(H \n10 \nlog \n) \nSFR \n(Σ \n10 \nlog \n)) \nβ \n(EW(H \n10 \nlog \n) \nSFR \n(Σ \n10 \nlog \n1500 \nM \n) \nSFR \n(Σ \n10 \nlog \n1 \n1500 \nM \nneb \nE(B-V) \n1500 \nM \nneb \nE(B-V) \n)) \nβ \n(EW(H \n10 \nlog \nneb \nE(B-V) \n2 \n(O/H) \n10 \n12+log \n1500 \nM \n,abs) \ni \nEW(H \n1500 \nM \n,abs) \ni \nEW(H \n1500 \nM \n3 \n) \nSFR \n(Σ \n10 \nlog \n(O/H) \n10 \n12+log \nUV \nE(B-V) \n(O/H) \n10 \n12+log \nneb \nE(B-V) \n(O/H) \n10 \n12+log \n4 \nOrder \nRank \nMC \ny \nb \nariables \nV \nop \nT \n[1.94] \n1500 \nM \n0] \n[1.0 \n) \nSFR \n(Σ \n10 \nlog \n[1.98] \n1500 \nM \n0] \n[2.1 \n) \nSFR \n(Σ \n10 \nlog \n[1.93] \n1500 \nM \n73] \n[1. \n) \nSFR \n(Σ \n10 \nlog \nruns. \nMC \n100 \nthe \nin \nranks \nariable's \nv \nthe \nof \nmean \nthe \nindicate \nets \nk \nbrac \nin \ners \nb \n-Num \nNote \na V ariable lists exclude an y measures of Ly α .", 'T able 11 . Most Imp ortan t V ariables for L (Ly α ) Predic tions a': 'del \nMo \nJWST', 'REFERENCES': 'Worseck, G., Prochaska, J. X., Hennawi, J. F., & McQuinn, \nM. 2016, ApJ, 825, 144, \ndoi: 10.3847/0004-637X/825/2/144 \nWyithe, J. S. B., & Loeb, A. 2013, MNRAS, 428, 2741, \ndoi: 10.1093/mnras/sts242 \nXu, X., Henry, A., Heckman, T., et al. 2023, ApJ, 943, 94, \ndoi: 10.3847/1538-4357/aca89a Yang, H., Malhotra, S., Gronke, M., et al. 2017, ApJ, 844, 171, doi: 10.3847/1538-4357/aa7d4d Zackrisson, E., Inoue, A. K., & Jensen, H. 2013, ApJ, 777, \n39, doi: 10.1088/0004-637X/777/1/39 \nZastrow, J., Oey, M. S., Veilleux, S., & McDonald, M. 2013, ApJ, 779, 76, doi: 10.1088/0004-637X/779/1/76'} |
2024arXiv240910919P | Extending the standard LambdaCDM model by considering dissipative effects within a causal viscous framework and obtaining an analytical solution for the Hubble parameter remains a challenge in the literature. In this work we resolve this dilemma by deriving a complete and original solution for the Hubble parameter by introducing a novel form for the bulk viscous coefficient associated with bulk viscous dark matter vDM. A thorough analysis of the model is conducted by deriving theoretical constraints on the parameters and comparing the model with the latest observational data sets. Intriguingly we find that the model predicts a signswitching bulk viscous pressure which facilitates both the early decelerated expansion and the late accelerated expansion of the universe. Also the redshift at which the viscous pressure switches sign is found to be strongly correlated with the relaxation time parameter of the viscous fluid. Thermodynamic analysis revealed that the model satisfies both the covariant and generalized second law of thermodynamics as well as the convexity condition for entropy. Additionally we reconstructed the model by unifying viscous dark matter and dark energy into a single unified dark matter UDM component and found that this unified model predicts identical dynamical evolution for the Universe while satisfying the necessary nearequilibrium condition throughout that evolution both in early and late phases. | 2024-09-01T00:00:00Z | ['2024arXiv240910919P', '10.48550/arXiv.2409.10919', 'arXiv:2409.10919'] | ['Astrophysics - Cosmology and Nongalactic Astrophysics', 'General Relativity and Quantum Cosmology'] | Dissipative LambdaCDM model with causal signswitching bulk viscous pressure | 2,024 | 167 | 0.34 | ['EPRINT_HTML', 'EPRINT_PDF'] | 0 | https://arxiv.org/pdf/2409.10919.pdf | {'Dissipative Λ CDM model with causal sign-switching bulk viscous pressure.': 'Vishnu A Pai 1 , ∗ , Sarath N 2 , † and Titus K Mathew 1 , ‡ 1 Department of Physics, CUSAT, Kalamasserry, Kochi 682022, Kerala, India 2 Indian Institute of Technology, Kanpur 208016, India [email protected] ∗ , [email protected] † , [email protected] ‡ and \nOctober 29, 2024', 'Abstract': 'Extending the standard Λ CDMmodel by considering dissipative effects within a causal viscous framework, and obtaining an analytical solution for the Hubble parameter remains a challenge in the literature. In this work, we resolve this dilemma by deriving a complete and original solution for the Hubble parameter by introducing a novel form for the bulk viscous coefficient associated with bulk viscous dark matter (vDM). A thorough analysis of the model is conducted by deriving theoretical constraints on the parameters and comparing the model with the latest observational data sets. Intriguingly, we find that the model predicts a sign-switching bulk viscous pressure, which facilitates both the early decelerated expansion and the late accelerated expansion of the universe. Also, the redshift at which the viscous pressure switches sign is found to be strongly correlated with the relaxation time parameter of the viscous fluid. Thermodynamic analysis revealed that, the model satisfies both the covariant and generalized second law of thermodynamics as well as the convexity condition for entropy. Additionally, we reconstructed the model by unifying viscous dark matter and dark energy into a single unified dark matter (UDM) component, and found that this unified model predicts identical dynamical evolution for the Universe, while satisfying the necessary near-equilibrium condition throughout that evolution (both in early and late phases).', '1 Introduction': 'Standard Λ CDM model in cosmology was proposed to explain the observed recent accelerated expansion of the universe. It is derived by reintroducing the cosmological constant ( Λ ) into Einstein\'s field equations and modeling dark matter as an ideal pressure-less fluid known as Cold Dark Matter (CDM). In spite of its agreement with with observational data [1, 50], Λ CDM model faces significant challenges, such as the cosmic coincidence problem [1, 50], the cosmological constant problem [55, 11], Hubble tension [33], S8 tension [54, 13] etc.. These limitations have led researchers to explore extended versions of the standard model by incorporating new physics, or derive entirely different cosmological models via alternative cosmological frameworks. Such modifications often involve altering the gravitational part of the Einstein-Hilbert action to induce changes in the dynamics of the universe [9, 21, 23, 52] or adjusting the matter stress-energy tensor, either by introducing exotic varying dark energy components [20, 25], or by modifying the properties of already hypothesized dark matter sector [40]. \nThe perfect fluid assumption considered in the Λ CDM model could only represent an approximation to reality as the fluids found in nature has some amount of dissipation associated with them. Hence, an obvious extension of the standard Λ CDM model is to consider dissipative effects in the dark matter sector. In literature, majority of such extension of the standard model are done by utilising the Eckart\'s formalism to model bulk viscosity [34, 49, 16]. However, being a first order theory, this formalism violates causality [35, 14] and provides only unstable final equilibrium states [31]. This motivates authors to consider IsraelStewart theory (FIS theory) for incorporating bulk viscosity in the dark sector, as it may provide causal & stable solutions. However, the incorporation of cosmological constant in such cases, introduces non-linearity \ninto the differential equations governing the dynamics [15]. This occurrence of non-linearity completely negates the possibility of obtaining an analytical solution for the Hubble parameter, which in turn prohibits one\'s efforts to understand the dynamics of the Universe in a transparent way. Furthermore, the numerical analysis carried out using those differential equations, in the context of dissipative Λ CDM model, predict abnormally large value for the bulk viscous coefficient, which even turns negative for a particular case [15]. It is therefore desirable to find an extended version of the Λ CDM model, which not only accounts for bulk dissipation in the dark matter sector, but also provides analytical solution for Hubble parameter. In the present work we aim to find such an extension of the standard model. This will then enable us to better constrain the dissipative parameters in the model and gain a better understanding about the dark sector and associated physical processes. \nOne of the crucial ingredients in dissipative cosmological models is the form of bulk viscous coefficient. This choice not only determines the magnitude and evolution of the bulk viscous pressure but also governs the dynamics of both the early and the late Universe. Hence, the feasibility of a dissipative cosmological model mainly depends on the suitable choice of this very coefficient. However, owing to the lack of knowledge about the exact origin of bulk viscous dissipation in the cosmological context, the bulk viscosity coefficient is often chosen phenomenologically. In most cases, this coefficient is assumed to depend on factors such as the energy density of the cosmic fluid, the expansion rate of the Universe, or both. However, one must note that such choices are discretionary rather than a rule. In this work, we propose that since bulk viscosity is a transport phenomenon linked to entropic flux and heat exchange within a system, the coefficient of bulk viscosity should depend on the enthalpy density [19] of the dissipative fluid rather than its energy density. Building on this idea, we postulate a bulk viscosity coefficient that depends on enthalpy density rather than just its energy density. This approach introduces an essential equation of state dependence to the bulk viscosity coefficient, a feature absent in previous models discussed in the literature. \nIn this article, we adopt Truncated Israel-Stewart (TIS) theory to model bulk viscosity [35, 57], primarily because it is a relatively simple second-order framework compared to the Full Israel-Stewart (FIS) formalism. Also, like FIS theory, TIS formalism is also causal and stable in the linear regime [47, 30, 42], and is sometimes preferred over FIS theory. Moreover, there is significant evidence favoring the TIS theory over the FIS formalism [4, 36]. Also, some studies suggest that TIS can either be seen as a more constrained version of FIS or as a distinct phenomenological extension [44]. Hence, by using the TIS formalism to incorporate viscosity, along with a novel "enthalpy density"-dependent bulk viscous coefficient, we demonstrate that it is possible to obtain an analytical solution for the Hubble parameter even in the presence of a cosmological constant and other independently evolving cosmic components. \nThe article is organized as follows: In Sec. 2, we will introduce bulk viscosity in the standard Λ CDM model and formulate the Λ vDM model of the Universe. In Sec. 3, we impose theoretical and observational constraints on the model parameters and estimate their best-fit values. Following this, we investigate the evolution of relevant cosmological observables in Sec. 4 and conduct a comprehensive thermodynamic analysis of the model in Sec. 5. Subsequently, in Sec. 6, we develop a Unified Dark Matter (UDM) interpretation for this model and hence show that, under the said UDM interpretation, the present model satisfies the \'near equilibrium condition\' defined in TIS theory in Sec. 7. Finally, in Sec. 8, we conclude the study by highlighting the critical results from this investigation.', '2 FLRWUniverse with causal viscous matter and cosmological constant': "In Einstein's gravity, the Friedmann equations describing the evolution of a flat, homogeneous and isotropic Universe with bulk viscous dark matter (vDM) and cosmological constant ( Λ ), as the cosmic components takes the form, \n1 3 H 2 = ρ m +Λ (1) \n2 ˙ H +3 H 2 = -( p m +Π -Λ) . (2) \nHere, H = ˙ a/a represents the Hubble parameter of the universe and an over dot signifies derivative with respect to cosmic time. Also, ρ m denotes energy density, p m represents kinetic pressure and Π corresponds \nto the bulk viscous pressure of vDM component. Conservation equation associated with vDM component is then given by, \n˙ ρ m +3 H ( ρ m + p m +Π) = 0 (3) \nIn order to obtain the solution to equation (3) one must first specify the equation of state of vDM component as well as the evolution of bulk viscous pressure. Following the conventional approach we consider an equilibrium barotropic pressure of the form p m = ω 0 ρ m , where ω 0 is the constant barotropic equation of state parameter, and model bulk viscosity using TIS theory. The evolution of the bulk viscous pressure of vDM is then, \nΠ+ τ ˙ Π = -3 ζH. (4) \nHere, τ is the relaxation time and ζ is the bulk viscous coefficient. According to this equation, gaining an exact understanding about the evolution of bulk viscous pressure necessitates one to know the exact dependence of τ & ζ on the local cosmological/thermodynamic variables. By perturbation the equation (4) by demanding causality and stability of solutions, Marteens derived a relation connecting these variables as [39], \nτ = ζ ϵ 0 ( ρ m + p m ) (1 -c 2 s ) . (5) \nwhere, c 2 s = ∂p m /∂ρ m , is the adiabatic speed of sound in the medium and ϵ 0 is a constant free parameter, which characterizes the contrast between the speed of sound and speed of the propagation of viscous perturbations through the fluid, and it have value within the domain (0 , 1] . Then, by using (5) in (4), and taking into account the barotropic nature of equilibrium pressure, one obtains the evolution equation for bulk viscous pressure as, \nΠ+ ζ ϵ 0 (1 -ω 2 0 ) ρ m ˙ Π = -3 ζH. (6) \nFinally, to close the set of equations, we must suitably postulate the form of bulk viscous coefficient. Even though literature consider the ansatz, ζ ∝ √ ρ . But still there is room for better choice for ζ , as one is not yet entirely sure what the exact form of ζ is. In the present study, we postulate a novel form for the bulk viscous coefficient based on two assumptions. Firstly, since bulk viscosity is a transport phenomenon directly related to entropy flux [58], which in turn depends on total heat content in the system, we assume that bulk viscous coefficient depends on enthalpy density of the fluid, h = ρ m + p m , rather than just on its energy density 1 . This formulation basically provides an 'equation of state' dependence to the bulk viscous coefficient ζ , which is absent in the conventional considerations 2 . Secondly, one also expect that the bulk viscous coefficient to depend on the expansion rate of the fluid, that is the Hubble parameter of the Universe. With these two basic considerations, the simplest ansatz that we propose for the bulk viscous coefficient ζ (which has the dimension as ζ ) is of the form; \nζ ∝ ρ m + p m H = ⇒ ζ = ζ 0 [ ρ m + p m H ] . (7) \nHere, ζ 0 is a dimensionless proportionality constant. \nAn interesting feature of the above choice is that, in the case of CDM, which has pressure p m = 0 , we get ζ = ζ 0 ρ m /H , which coincidentally aligns with an earlier proposal [28]. Also, for dark energy dominated epoch, with equation of state, p de = -ρ de , equation (7) can be analogously recast by replacing ρ m and p m by ρ de and p de respectively, which then implies that ζ = 0 . Which means, the de-Sitter epoch cannot be dissipative. This is exactly what one might expect, because a de-Sitter phase corresponds to a stable equilibrium state of the universe, and therefore any out-of-equilibrium contributions such as Π should vanish.", '2.1 Evolution of Hubble parameter': "Now will determine the analytical solution for Hubble parameter for the viscous universe by using the equation for the viscous coefficient given in equation (7). We will start by substituting p m = ω 0 ρ m in the \nconservation equation (3) and then apply a change of variable from ' t ' to ' x = ln( a ) '. Henceforth, we obtain the evolution equation for bulk viscous pressure as, \nΠ = -[ ρ ' m 3 +(1 + ω 0 ) ρ m ] . (8) \nHere, ' prime ' denotes a derivative with respect to ' x '. On further differentiating (8) with respect to x , and combining it with (6) & (7), we obtain the second order linear differential equation for ρ m as, \nαρ '' m + βρ ' m + γρ m = 0 , (9) \nHere the coefficients are of the form, \nα = ζ 0 [ ϵ 0 (1 -ω 0 )] -1 , (10) \nβ = 1 + 3 α (1 + ω 0 ) , (11) \nγ = 3(1 + ω 0 )(1 -3 ζ 0 ) . (12) \nOn solving this equation, we get the evolution of vDM density as, \nρ m = C 1 a -N 1 + C 2 a -N 2 (13) \nHere, N 1 and N 2 are constants give by, \nN 1 = n 1 -n 2 & N 2 = n 1 + n 2 , (14) \nwith n 1 and n 2 defined as, \nWhile C 1 and C 2 are integration constants, the values of which are to be determined by applying suitable boundary conditions. Imposing the present condition 3 , H → H 0 and Π → Π 0 , as a → a 0 = 1 , in equations (3) and (13), we obtain the two independent equations connecting these coefficients which upon solving gives, \nn 1 = β 2 α & n 2 = √ β 2 -4 αγ 2 α . (15) \n˜ C 1 = [ N 2 -3(1 + ω 0 ) 2 n 2 ] ( 1 -Ω 0 Λ ) -3Ω 0 Π 2 n 2 , (16) \n˜ C 2 = 1 -Ω 0 Λ -˜ C 1 . (17) \nHere, ˜ C 1 = C 1 / 3 H 2 0 , ˜ C 2 = C 2 / 3 H 2 0 , Ω 0 Λ = Λ / 3 H 2 0 , Ω 0 Π = Π 0 / 3 H 2 0 . Now following equation (1), we can write the equation for the Hubble parameter as, \nH = H 0 √ Ω 0 Λ + ˜ C 1 a -N 1 + [ 1 -Ω 0 Λ -˜ C 1 ] a -N 2 (18) \nH ≊ H 0 √ ˜ Ω 0 Λ +(1 -˜ Ω 0 Λ ) a -2 n 1 ; { ˜ Ω 0 Λ = Ω 0 Λ + ˜ C 1 (19) \nOne of the intriguing possibility observed in this model is that, if n 1 ≈ n 2 , (which can happen if ζ ≈ 1 / 3 or 4 αγ/β 2 ≪ 1 ), then the above solution reduces to, \nand if n 1 ≈ 3 / 2 , model becomes equivalent to Λ CDM model, but with an effective cosmological constant ˜ Ω 0 Λ .", '3 Constraints on model parameters': 'Introducing bulk viscous effect in the standard Λ CDM model has introduced 4 additional free parameters in the model, all of which are related to vDM component, i.e, [ Ω 0 Π , ζ 0 , ϵ 0 , ω o ]. Determining their exact value is essential for gaining a robust understanding about the dark sector. Hence, in this section, we will constrain the parametric space of these new variables based on certain theoretical requirements, and then determine their best fit values by comparing the model with observational data sets.', '3.1 Theoretical Constraints': 'In this subsection we will constrain the parameters by demanding that this model (i) predicts a lateaccelerated expansion of the Universe, (ii) obeys the covariant second law of thermodynamics (CSLT), (iii) satisfies the weak energy condition ( ρ m ≥ 0 ), (iv) maintains causality and (v) satisfies null-energy condition ( ρ m + p m ≥ 0 ) thereby avoiding phantom behaviors. \nTo safeguard the model from phantom behaviors and violations in null energy condition, we must develop further constraints on model parameters by analyzing the behavior of effective equation of state ( ω e ) and deceleration parameter ( q ). By combining (1), (2) and (18), we obtain the effective equation of state and the deceleration parameter as, \nFirstly, notice that the only acceptable value of ω 0 is within the interval [0 , 1) . This is because, for any value of ω 0 > 1 in (5) the relaxation time becomes negative, which is unacceptable, and for any ω 0 < 0 we will have a negative value for squared speed of sound ( c 2 s = ω 0 ) in the medium. Secondly, to satisfy CSLT 4 as given in equation (2.5) in [39], it is essential to constrain the parameters such that the bulk viscous coefficient is always greater than zero, and from (7) this implies the condition ζ 0 > 0 . Thirdly, for satisfying weak-energy condition (given as ρ m ≥ 0 ) throughout the evolution of the Universe, it is essential to constrain the parameters such that, ˜ C 1 ≥ 0 and ˜ C 2 ≥ 0 . Finally, for maintaining causality, the values of ϵ 0 must be restricted within the domain 5 ϵ 0 ∈ (0 , 1] . \nω e = ˜ C 1 N 1 a -N 1 + [ 1 -Ω 0 Λ -˜ C 1 ] N 2 a -N 2 3 Ω 0 Λ + ˜ 1 a -N 1 + 1 Ω 0 Λ ˜ 1 a -N 2 -1 \n[ C [ --C ] ] (20) q = ˜ C 1 N 1 a -N 1 + [ 1 -Ω 0 Λ -˜ C 1 ] N 2 a -N 2 2 [ Ω 0 Λ + ˜ C 1 a -N 1 + [ 1 -Ω 0 Λ -˜ C 1 ] a -N 2 ] -1 (21) \nFrom (18), it is clear that, due to the presence of Λ , this model can surely predict an accelerated expansion for the universe. However, according to (21), for this model to have an initial deceleration era, one must at least have either, N 1 > 2 or N 2 > 2 . From (20) and (21), it is also clear that, the model shows far future phantom behaviors for the universe if N 1 < 0 or N 2 < 0 . Hence, for the model to be devoid of phantomness, we must constrain the parameters such that, N 1 ≥ 0 and N 2 ≥ 0 . Subsequently, using relations (10), (11), (12), (15) and considering the prior values of ω 0 , ϵ 0 and ζ 0 stated in the earlier paragraph, we notice that both n 1 and n 2 cannot be negative. Hence, combining this knowledge with the inequality N 1 ≥ 0 , we obtain a single refined constraint on n 1 and n 2 as, 0 < n 2 ≤ n 1 . This single constraint will then ensure that the model is free from any phantom behavior. Additionally, notice that this refined constraint in which we have both n 1 > 0 and n 2 > 0 , also satisfies the requirement for having an early decelerated expansion (i.e, n 1 + n 2 > 2 ). \nFinally, the constraint 0 < n 2 ≤ n 1 , together with equation (15), implies the condition that γ ≥ 0 . And from (12), this is achievable only if ζ 0 ≤ 1 / 3 . Thus, the exact value of bulk viscous coefficient ζ 0 is crucial in determining whether the model predicts a phantom era in the far future. Specifically, for values of ζ 0 within the range [0 , 1 / 3] , the condition 0 < n 2 < n 1 is met, resulting in a model devoid of phantom behavior and predicting a universe that asymptotically approaches a de Sitter phase in its late epochs. while for ζ 0 > 1 / 3 the model exhibits phantom behavior in the distant future. In this study, since we require vDM to satisfy both weak energy and null energy conditions, we will constrain the bulk viscous coefficient to the range ζ 0 ∈ [0 , 1 / 3] .', '3.2 Observational constraints': "We will now compare the model with observational datasets by performing a χ 2 analysis, whilst incorporating the theoretical constraints and priors established in the previous section. For this analysis, we will use the \nTable 1: Best estimated value of model parameters for different data combinations. \n- \n- \nlatest Observational Hubble Data (OHD), Pantheon+ data, Baryon Acoustic Oscillation (BAO) data, and the shift parameter observed in the Cosmic Microwave Background (CMB). \nOHD: For our analysis we consider the data set [51, 45] which includes 76 Hubble parameter values within the redshift range of 0 . 07 ≤ z ≤ 2 . 36 . The χ 2 analysis between the model and this data set is a direct comparison using the relation, \nχ 2 OHD = n =76 ∑ i =1 [ H ( H 0 , Ω 0 Λ , Ω 0 Π , ω, ζ 0 , ϵ 0 , z i ) -H i ] 2 σ 2 H i (22) \nHere, H i is the value of Hubble parameter observed at a redshift z i with σ 2 H i variance. \nPantheon+ data: Recently released Pantheon+ data sample contains 1701 light curves of 1550 distinct Type Ia supernovae (SNe Ia) spanning a redshift range of 0 . 001 ≤ z ≤ 2 . 26 [7]. Unlike OHD data set, this data sample consist of 1701 apparent magnitude data points, each observed at a particular redshift. For comparing the model with this observational data, one must first relate the Hubble parameter to apparent magnitude ( µ ) using the expression, \nµ ( H 0 , ..., z i ) = 5 log 10 [ d L ( H 0 , ..., z i ) Mpc ] +25 . (23) \nHere, d L is the luminosity distance of i th supernovae at a redshift of z i which is defined as, \nd L ( H 0 , ..., z i ) = c (1 + z i ) ∫ z i 0 dz H ( H 0 , ..., z i ) (24) \nThen, the χ 2 analysis is done by minimizing the function, \nχ 2 PAN + = ⃗ Q T · ( C stat + sys ) -1 · ⃗ Q (25) \nHere, ⃗ Q is a 1701-dimensional vector defined using the relation ⃗ Q = µ i -µ ( H 0 , ..., z i ) -M, and C stat + sys is the covariance matrix of Pantheon+ data sample. Also, µ i is the observed apparent magnitude of supernovae and M is the absolute magnitude of supernovae which is often treated as a nuisance parameter. \nCMB shift parameter: Usage of shift parameter ( R ) associated with the Cosmic Microwave Background (CMB) temperature anisotropies is one of the simplest ways of constraining the free parameters in the model with data obtained from the early universe. However, one must note that the calculation of R from CMB is done by assuming Λ CDM model as the background [12, 22], and hence one must be cautious while constraining arbitrary dark energy models using R . Nevertheless, in the present case, we expect this model to deviate only slightly from the standard model, and hence utilize shift parameter to constrain the model. The relation connecting R and Hubble parameter for a flat homogeneous and isotropic universe is given as, \nHere, z r is the recombination redshift and h represents the dimensionless Hubble parameter h = H/H 0 . From Planck 18 analysis [12] we have R o = 1 . 7502 ± 0 . 0046 at z r = 1089 . 92 . The χ 2 function to be minimized in this case is then, \nR = √ Ω 0 m ∫ z r 0 1 h (Ω 0 Λ , Ω 0 Π , ω, ζ 0 , ϵ 0 , z ) dz. (26) \nχ 2 CMB = ( R-R o ) 2 σ 2 R o . (27) \nFigure 1: Corner plot of 2D posterior contours with 1 σ (68%), 2 σ (95%) and 3 σ (99.7%) confidence level and 1D marginalized posteriors of all model parameters plotted using [5] for different datasets given in Table. \n<!-- image --> \nBAO data: BAOs corresponds to the preferred length-scale imprinted in the distribution of photons and baryons by propagating sound waves in the primordial plasma before decoupling occurred. In [46], authors found that using BAO data in association with CMB shift parameter can improve the accuracy of estimation. Hence, in the present study, we follow that approach and constrain the model using BAO data reported in [2]. This data set encompasses two distinct parameters; the transverse comoving distance D M (which is equivalent to line of sight comoving distance D c in a spatially flat universe (i.e. D M = D c )) defined as [17], \nD M ( z ) = D c ( z ) = c H 0 ∫ z 0 dz ' h (Ω 0 Λ , ..., z ' ) (28) \nand the volume average angular diameter distance D v given as, \nD V ( z ) = [ cz H 0 D 2 M ( z ) h (Ω 0 Λ , ..., z ) ] 1 / 3 . (29) \nThe χ 2 function to be minimized in this case is then, \nχ 2 BAO = n =6 ∑ i =1 ( D-D o ) 2 σ 2 z i . (30) \nHere, D and D o are theoretical and observed values of BAO parameters, either D M ( z i ) or D V ( z i ) , at redshift z i , and σ 2 D i is the variance in those observed values. \nFor comparing the present model with observations, we consider the two combinations of above mentioned data sets. The 'PAN + +OHD+BAO' data combination and the 'PAN + +OHD+BAO+CMB' data combination. Former set constrains the parameters based on data gathered from the late phase, whereas, the latter constraints the model with CMB data also. The overall χ 2 function to be minimized in each of these cases is then, χ 2 D = ∑ N i =1 χ 2 i . Here, χ 2 i represents chi-squared function of i th data set and N denotes number of data sets involved. \nData Analysis: The χ 2 minimization is conducted using the Markov Chain Monte Carlo (MCMC) technique, implemented via the emcee Python library [24]. The estimated values and variances of the model parameters are provided in Table 1, and the corresponding contour plot is displayed in Figure 1. \nFigure 2: Evolution of deceleration parameter and effective equation of state with redshift. \n<!-- image --> \nFrom the results obtained, the first noticeable feature is the consistency in the observed value of Hubble parameter despite comparing the model with different sets of observational data. That is, the model consistently predicts the present value of the Hubble parameter to be around 72 km/s/Mpc for both datasets. A careful analysis and comparison with the values reported in [36] reveal no significant tension between the H 0 values estimated from the two data sets. Therefore, by introducing a cosmological constant, we have significantly reduced the tension in the H 0 values found in the pure bulk viscous scenario examined in [36]. \nSecondly, it is important to note that the equation of state parameter associated with the adiabatic speed of sound in the medium, ω 0 is very close to zero. This suggests that the matter component behaves almost like viscous cold dark matter (vCDM), having a value of ω 0 that is much lower than those found in [36]. Additionally, ω 0 has an even smaller value when constrained using the D2 dataset, which includes contributions from both the early and late universe. \nFinally, as seen in Fig. 1, the present value of the reduced bulk viscous pressure, Ω 0 Π cannot be constrained using late-time data alone (i.e., the D1 dataset). To achieve this, early-time data, such as the CMB shift parameter, is also required. Once contributions from early-time data are included, the value of ω 0 Π becomes significantly more constrained, as shown in Fig. 1. Therefore, we conclude that the parametric space of all model parameters related to the macroscopic properties of the fluid (such as ϵ 0 , ω 0 , Ω 0 Π and ζ 0 ) can be more tightly constrained by incorporating additional early-universe data.", '4 Evolution of Cosmological Observables': "Using the best estimated value of model parameters obtained in previous sections (D2 data set), we will now investigate the evolution of some relevant cosmological observables. \nAge of the universe: Age of the Universe is determined using the equation Age = ∫ 1 0 da/ ( aH ) . By integrating this expression by substituting Eqn. (18) and the best fit value of model parameters, we found the age predicted by the model to be approximately 13 . 14 Gyrs, which is very close to Λ CDM prediction but slightly lower. Nevertheless, this estimate is in agreement with age deduced from both CMB anisotropy data [53] and from the oldest globular clusters [10]. \nDeceleration parameter ( q ) and Effective equation of state ( ω e ): Evolution of deceleration parameter and the effective equation of state, defined by relations (21) and (20) respectively, are as plotted in Fig. (2). Accordingly, we learn that the model predicts the observed late-time accelerated expansion. It is to be noted that the effective equation of state parameter stabilizes around -1 in the far future, implying that universe will approach a de Sitter epoch at the asymptotic late phase of the evolution. Hence, the constraints developed in Sec. 3.1 are successful in prohibiting phantom behavior from this model. Also, according to the present analysis, the transition from prior deceleration to late acceleration occurs at a redshift of z T = 0 . 69 , and the current value of deceleration parameter to be q 0 = -0 . 55 , which are in conformity with observations. \nFigure 3: Evolution of dissipative variables with the evolution of the Universe. In Fig. (3b), ' ζ p ' represents the present value of bulk viscous coefficient, which is in agreement with bounds proposed in [6, 48, 41]. \n<!-- image --> \n<!-- image --> \na \n(b) Coefficient of bulk viscosity. \nDissipative variables: The evolution of bulk viscous coefficient and the associated dissipative pressure are obtained by plotting Eqns. (7) and (8) respectively. Accordingly, from the plot of bulk viscous pressure, i.e, from Fig. (3a), we see that it evolves from a positive value in the early universe, decreases gradually to zero at a redshift z s (the sign-switching redshift), and then continues to decrease further to negative values. But, at a later stage, it achieves a minimum negative value from which it then starts increasing to finally approach the value zero at asymptotic conditions. On the other hand, from Fig. (3b), we see that the bulk viscosity coefficient remains positive throughout the evolution. This means, a positive bulk viscosity coefficient does not necessarily imply a negative bulk viscous pressure. This is an interesting behavior as compared the conventional Eckart formalism of dissipative evolution, in which a positive viscous coefficient always implies a negative bulk viscous pressure. Hence, in the present case, owing to this sign-switching behavior, bulk viscous pressure can aid both, the prior decelerated expansion, and after the sign change, the late-time accelerated expansion. Thus, in addition to the cosmological constant, bulk viscous pressure also plays a significant role in initiating the transition from the earlier decelerated phase to the current accelerated expansion. \nThe asymptotic vanishing nature of viscous pressure is not fully apparent in the plot. However, it can be verified by analyzing the expression of reduced viscous pressure obtained by substituting Eqn. (13) in Eqn. (8) as, \nΠ 3 H 2 0 = -{ ˜ C 1 3 [3(1 + ω 0 ) -N 1 ] a -N 1 + ˜ C 2 3 [3(1 + ω 0 ) -N 2 ] a -N 2 } . (31) \nUsing the best estimates of model parameters (D2) we obtain, N 1 ≈ 0 . 0003 and N 2 ≈ 3 . Using these values in the above expression, we find that the first term on the right hand side of the above equation behaves almost \nFigure 4: Evolution of ln( | Π | ) with redshift for different value of relaxation time parameter ' ϵ 0 '. \n<!-- image --> \nFigure 5: Evolution of specific entropy and total entropy production rates (CSLT) with scale factor of the Universe. \n<!-- image --> \nlike a constant, while the second term evolves as a -3 . Consequently, it can be easily concluded that, in the late stages of the evolution the first term has the dominating behavior, while in the early epochs contribution from the second term outweighs that of the first. However, it may be noted that, since the exponent N 1 is not exactly zero, the bulk viscous pressure is not a perfect constant in the late universe, but is only a slowly decaying function of time. And in the asymptotic limit, i.e. as z → -1 , viscous pressure vanishes and the Universe enters de Sitter epoch driven by cosmological constant. In addition to providing the exact evolution characteristics of bulk viscous pressure, Eqn. (31) also provides insights into the reason behind the sign-switching behavior of viscous pressure. Considering the constraints imposed on model parameters, i.e, ˜ C 1 & ˜ C 2 are greater than zero, and the resulting values of N 1 and N 2 , we obtain 3(1 + ω 0 ) - N 1 > 0 and 3(1 + ω 0 ) -N 2 < 0 . Taking account of these, the above equation then implies that, the relative viscous pressure was positive in the prior epochs and becomes negative in the later epochs. \nFrom our analysis, we also found a strong correlation between the sign-switching z s , and the relaxation time parameter ' ϵ 0 '. This can be clearly visualized from Fig. (4) which depicts the evolution of ' ln( | Π | ) ' with redshift of the Universe, for different values of ' ϵ 0 '. The troughs seen in this surface plot represents different possible combinations of ( z s , ϵ 0 ) , which corresponds to points where bulk viscous pressure undergoes sign switching. Consequently, we see that for fixed values of other parameters, a decrease in the value of relaxation time parameter causes an increase in the value of sign-switching redshift.", '5 Thermodynamic analysis': 'In this section we will show that, the present model exhibits thermal evolution in conformity with the conventional laws of thermodynamics which includes; the covariant second laws of thermodynamics (CSLT), the generalized second laws of thermodynamics (GSLT) and also the convexity condition of entropy, such that the total entropy of the system maximizes at the end epoch of the evolution.', '5.1 Covariant Second Laws of Thermodynamics': "It is to be noted that, the evolution of total entropy production rate and specific entropy rate associated with the dissipative fluid differs in Israel-Stewart theory in contrast to the Eckart's model, where the evolution of these two factors coincide. In TIS theory the specific entropy rate evolves as [32], \n˙ σ = -3Π H nT (32) \nand the total entropy production rate evolves as, \nS µ ; µ = Π 2 ζT . (33) \nHere n = n 0 a -3 represents the particle number density, with n 0 as the number density at the present epoch, and T corresponds to temperature evolution of vDM which is given as [32], \nT = T 0 [ ρ m ρ 0 m ] ω 0 / (1+ ω 0 ) . (34) \nHere, T 0 is the present value of temperature which we set to one. By analyzing Eqns. (32) and (33), by taking into account the sign-switching behavior of bulk viscous pressure, we see that the specific entropy rate evolves from negative values in the past to positive in the future (since ˙ σ ∝ -Π ), whereas the total entropy production rate remains positive through out the evolution (as it behaves, S µ ; µ ∝ Π 2 ). Physically, this behavior can result from entropic flux leaving the comoving volume in the early universe, causing a decrease in total entropy production rate. However, since particle number density within the comoving volume doesn't change ( n µ ; µ = 0 ), a decrease in entropy production rate in the comoving volume leads to decreasing entropy per particle in the comoving volume and hence a negative specific entropy rate. Furthermore, as universe undergoes accelerated expansion, entropic flux enters the comoving volume causing an increase in S µ ; µ and hence an increase in entropy per particle. \nWe plotted the evolution of specific entropy rate and entropy production rate in Fig. (5a) & Fig. (5b) respectively, by using Eqns. (7), (8) and (18) in the above expressions. From the resulting figures we see that, as expected the specific entropy evolves form negative values, becomes zero at z s and becomes positive when viscous pressure switches its sign. On the other hand, the total entropy production rate decreases from a large positive value (which diverges at big-bang singularity), becomes zero at the sign switching redshift and then appears to be increasing towards a constant maximum value in the late phase. However, by evaluating lim z →-1 S µ ; µ ( z ) by substituting best fit value of model parameters, one can easily verify that S µ ; µ ( z ) does go to zero as z →-1 .", '5.2 Generalized Second Law of Thermodynamics': "According to generalized second law of thermodynamics (GSLT), the total change in entropy of the Universe, which is the sum of entropy change occurring in the bulk ( S ' m ) and entropy change occurring on the horizon ( S ' H ), must be an increasing function in time [26]. That is, \nS ' H + S ' m ≥ 0 . (35) \nHere, ' prime ' symbolizes a derivative with respect to a suitable cosmological variable. In the present case, the most suited choice for this variable would be the scale factor of the Universe. The equation for total \nFigure 6: Evolution of total entropy and convexity condition with scale factor of the Universe. \n<!-- image --> \nentropy stored in a horizon surface of area A = 4 π ˜ r 2 A is S H = k B A/ 4 l 2 p [18]. And for a flat FLRW universe, the apparent horizon radius becomes ˜ r A = c/H . Substituting this in the above equation one obtains the Horizon entropy as, \nS H = [ πc 2 k B l 2 p ] 1 H 2 . (36) \nHence, for an expanding universe, the horizon entropy is always at an increase and in its asymptotic far future, it reaches its maximum when Universe enters the end de Sitter stage. \nTo determine the evolution of matter entropy inside the Hubble horizon, we use the Gibbs equation, \nTdS m = c 2 V dρ m + ( c 2 ρ m + p m +Π ) dV. (37) \nUsing Eqns. (1) and (3), along with the expression V = (4 π ˜ r 3 A ) / 3 , we simplify the above relation into, \nS ' m = 4 πc 5 3 TH 3 ρ ' m [ 1 + aH ' H ] = -c 5 H ' q TGH 2 . (38) \nHere, q is the deceleration parameter of the Universe. Adding on this, the corresponding derivative of (36), gives the rate of change of total entropy as, \nS ' T = S ' H + S ' m = H ' H 2 { c 5 q TG -2 πc 2 k B l 2 p H } . (39) \nAccording to Eqn. (38), the matter entropy rate evolves from a positive value in the early decelerating epoch, to a negative value in the late accelerating epoch. In the far future of it's evolution, it asymptotically approaches zero from the negative side (since H ' → 0 as a → ∞ ). However, analyzing the derivative of Eqn. (36), we find that Horizon entropy rate is positive throughout the evolution. And, by comparing Eqns. (36) and (38), we can see that the magnitude of entropy change happening in the bulk is inconsequential compared to that occurring on the Hubble horizon. Hence, the total entropy change in the system mimics the behavior of horizon entropy, and is clear from Fig. (6). Consequently, the model satisfies GSLT throughout the evolution.", '5.3 Convexity Condition': "In addition to satisfying the law of thermodynamics, one also expects the model to evolve towards a stable thermal equilibrium. In a closed thermodynamic system, a stable equilibrium is said to be achieved only when the entropy of that system gets maximized. The minimum requirement for this is to have the entropy evolution that satisfies the constraints S ' T ≥ 0 and S '' T < 0 , at-least in the end stage of that evolution [29, 8]. The requirement S '' T < 0 means that the rate of increase of entropy in the universe should be decreasing, which suggest that the system is gradually approaching a stable equilibrium state. In the cosmological context, the first condition corresponds to GSLT, while the second represents the convexity condition. \nTo determine whether the model complies with the convexity condition, we differentiate Eqn. (39) with respect to the scale factor of the Universe and plot its evolution as shown in Fig. (6). We see that S '' T initially increases towards a maximum positive value until the transition redshift, which is then quickly followed by a rapid decrease towards the negative side at the present epoch. From there, it evolves towards zero from the negative side in an asymptotic fashion. This evolution confirms the fact that the present cosmological model is evolving towards a stable thermodynamic equilibrium which in turn implies its agreement with the convexity condition.", '6 Reconstructing the model as a dissipative UDM model with constant adiabatic speed of sound': "Until now we have considered Λ and vDM as separate independent and non-interacting dark components, with the former representing dark energy and the latter signifying viscous dark matter. However, there is an interesting approach in which the cosmological constant is hypothesized to arise from equation of state of dark matter component having constant adiabatic speed of sound [3, 56, 37]. Here, one considers dark matter and dark energy as two faces of a single dark component called the unified dark matter (UDM). It is then interesting to check the possibility of re-interpreting the present model as a UDM model, and analyze the results based on both interpretations. Hence, in this section, we provide an alternate interpretation for the present model by considering the UDM modeling approach proposed in [3]. In the upcoming section we will see that, adopting this UDM interpretation enables the model to satisfy the much required 'near equilibrium condition' (NEC) through out the evolution of the universe, without changing the background dynamics predicted by the model. In contrast, under the 'two fluid' Λ vDM interpretation, the near equilibrium condition is satisfied only in the early epoch of accelerated expansion (see the later section for more discussion). \nFor introducing the description of a UDM fluid, we will follow the approach made in literature [3, 56]. First, we define the equilibrium energy density of the unified dark fluid as ρ eq and it's equilibrium pressure as p eq . With reference to the standard model, it is possible to take, ρ eq = ρ m +Λ , (in units of 8 πG = 1 ) and p eq = ω e ρ eq [3]. Here, ω e ( ρ eq ) represents barotropic equation of state of UDM fluid. As a result the Friedmann equations are, \n3 H 2 = ρ eq ≡ ρ m +Λ (40) \n2 ˙ H +3 H 2 = -ω e ρ eq . (41) \nThen, by assuming the adiabatic speed of sound c 2 s = ∂p eq /∂ρ eq , to be a constant, we obtain, \nc 2 s = ω e + dω e dρ eq ρ eq = ¯ ω (42) \nHere, c 2 s = ¯ ω = constant . One can then integrate the above equation to obtain the expression for effective equation of state of the unified fluid as, \nω e = ¯ ω + I ρ eq . (43) \nHere, I is an integration constant whose value can be determined by imposing required initial conditions. For instance, if we demand the unified dark matter to depict a cosmological constant like behavior in the asymptotic late phase, we can consider ρ eq → Λ & ω e →-1 as a →∞ , and obtain the value of integration constant as, I = -Λ(1 + ¯ ω ) . If we then substitute this value of I in above equation we can obtain the equilibrium pressure of the unified fluid as, \np eq = ω e ρ eq = ¯ ωρ eq -Λ(1 + ¯ ω ) . (44) \nFeeding the the general form of ρ eq in the right side of above equation we retain the pressure term, p eq = ¯ ωρ m -Λ , where ρ m = ρ eq -Λ . This means that, the dynamics of the Universe predicted by the unified dark matter model is identical to a two-fluid Λ -WDM model of the universe, with warm dark matter (WDM) \nFigure 7: Evolution of energy densities in the Λ vDM and v Λ ¯ ω DM interpretations. \n<!-- image --> \nFigure 8: Evolution of NEC with expansion of the Universe in Λ vDM and v Λ¯ ω DM interpretations for the best estimated value of model parameters. Gray regions in graph corresponds to regimes where NEC is violated. \n<!-- image --> \nhaving a constant barotropic equation of state parameter ' ¯ ω '. Also, note that when ¯ ω ≈ 0 , the model becomes identical to Λ CDM case [37]. \nTo extend this UDM formalism in the context of the present model, we introduce bulk viscosity in the Friedmann equations obtained in previous paragraph as, \n3 H 2 = ρ eq ≡ ρ m +Λ (45) \n2 ˙ H +3 H 2 = -( p eq +Π) ≡ -( ω 0 ρ m -Λ+Π) . (46) \nHere, we have considered ¯ ω = ω 0 , since ω 0 represent constant adiabatic speed of sound in the fluid medium in the presence of bulk viscosity. We will call this the 'v Λ ¯ ω DM' model of the Universe. However, note that, in the v Λ¯ ω DM case, the viscous pressure is associated with the effective fluid as a whole, rather than with dark matter alone. \nIn both cases, i.e, Λ vDM and v Λ ¯ ω DM, Friedmann equations describing the dynamics of the Universe are exactly identical, as is clear from Eqns. (1) & (2) and (45) & (46). Also, if we replace ρ m and p m in Eqns. (7) and (5), with the new equilibrium variables, ρ eq and p eq , we find that the relations for viscous coefficient and relaxation time are also identical in both cases. Hence, the evolution of bulk viscous pressure associated with the UDM model matches with the expressions defined in (6) and (8). Consequently, the three independent equations, i.e., (1), (3) & (6) that define the dynamics of the Universe, are the same in both Λ vDM and v Λ ¯ ω DM cases. Therefore, the only difference between these two interpretations lies in the way in which one interprets the dark components, and owing to this difference, the local equilibrium variables associated with viscous fluid changes from ρ m & p m in Λ vDM case, to ρ eq & p eq , in v Λ¯ ω DM case. \nIn Fig. (7) we have compared the evolution of unified dark matter density in v Λ¯ ω DM ( ρ eq ), with the evolution of ρ m and ρ Λ components in Λ vDM model. In the latter case, the early Universe is dominated by vDM component having a net positive pressure while the late universe is dominated by dark energy density ( Λ ) having a constant negative pressure. As a result, there exists a redshift ( z c ) at which the energy densities of these cosmic components coincide, following which, the energy density of dark energy dominates over dark matter, thereby causing a late-accelerated expansion. While in v Λ ¯ ω DM case, the entire dynamics of the Universe is governed only by a single cosmic component which drives both, the past deceleration as well as the late acceleration of the Universe. According to Fig. (7), the energy density of this component (denoted as ρ udm = ρ eq ) evolves from a singular value in the the early Universe towards a constant positive value in the late Universe. Consequently, the late acceleration of the Universe is achieved via the effective equation of state of UDM component, which is almost zero in the early phase and varies as -(Λ(1 + 3¯ ω )) /ρ eq in the late universe (enabling the model to behave like of dark energy). Consequently, under UDM interpretation, the model not only satisfies NEC throughout the evolution (which is shown in the next section), but also evades the cosmic coincidence problem [43].", '7 Near Equilibrium Condition': 'Relativistic dissipative theories such as FIS and TIS theories are derived by defining a local equilibrium state for the fluid using equilibrium thermodynamic variables and then considering minute deviation from that equilibrium state. Hence, such viscous theories are valid only in near-equilibrium regimes where deviations from equilibrium states are small. In these cases, the bulk viscous pressure arises as a response to the perturbations in the dissipative medium, which drives the system back to its equilibrium state. Thus, the bulk viscous pressure is expected to die out when the fluid achieves equilibrium. These dissipative theories define viscous pressure as a deviation from the equilibrium pressure of the cosmic fluid, and as a result, the difference between the relative magnitude of bulk viscous pressure compared to that of the equilibrium pressure of the fluid acts as the measure of the deviation of the fluid from its equilibrium state. Consequently, for the fluid to remain in near-equilibrium regimes (minimal deviations from equilibrium), one must have, \n∣ ∣ ∣ Π p eq ∣ ∣ ∣ ∣ < 1 . (47) \n∣ This constraint is often quoted in literature as the near equilibrium condition. Yet, while studying viscousdriven accelerated expansion of the Universe, one hypothesizes the validity of these dissipative theories to cases where the fluid is far from equilibrium (where NEC is violated) [38]. However, under such circumstances, the applicability of either TIS or FIS is questionable. Hence, having a cosmological model that abides by NEC is always preferential. In the present model, we analyze the status of NEC by adopting both Λ vDM and v Λ ¯ ω DM interpretations. This is because, even though both models predict identical dynamics for the Universe, the NEC evolves differently in each case due to the differences in the definition of equilibrium thermodynamic variables. \nEvolution of NEC in Λ vDM model: From Fig. (8a), it is clear that the vDM component in this case evolves from a barely near-equilibrium state in the early epoch (since NEC is very close to but strictly less than one), fluctuates at sign-switching redshift (sudden dip in cure in Fig. (8a) that arises because at z = z s , Π = 0 ) and turns far-from-equilibrium in the late accelerating phase. However, in the far future of its evolution, i.e, as a →∞ , both p eq and Π must vanish according to Eqn. (31), and thus vDM eventually attains a global equilibrium state. \nEvolution of NEC in v Λ¯ ω DM model: Evolution of NEC under UDM interpretation can be analyzed from Fig. (8b). It is then clear that, under this interpretation the present model satisfies NEC not only in the early decelerating phase but also during the late accelerating epoch of the Universe. The NEC evolves from a near unit value (but less than one) in the early Universe, undergoes two fluctuations in the recent past and vanishes in the asymptotic far future as seen in Fig. (8b). In this case, the fluctuation seen in Fig. (8a), arises because of two different reasons. The initial spike in the curve where NEC grows rapidly (and goes singular) occurs due to the vanishing of equilibrium pressure, causing NEC to blow up briefly in the early Universe. Meanwhile, the second spike, which rapidly decays to zero, occurs due to the vanishing of viscous pressure at the sign-switching redshift. Furthermore, since NEC is a measure of deviation of the \nfluid from its equilibrium state, we infer that in this interpretation, the dissipative UDM fluid depicts an evolution from an out-of-equilibrium (but not far from equilibrium) to an equilibrium state.', '8 Results and Conclusion': 'Several authors have attempted to extend the Λ CDM model by introducing causal dissipative effects in the dark sector, and thereby propose a more generalized model of the universe. However, none of these models have succeeded in obtaining an analytical solution for the Hubble parameter in the presence of causal viscous dissipation. Such a solution is highly desirable, as it would provide stringent constraints on model parameters and deepen our understanding about the dark sector. In this article, we reconciled this longstanding challenge by obtaining an analytical solution to extended Λ CDM model in which causal viscous dissipation is accounted using Truncated Israel-Stewart (TIS) theory. Specifically, we replaced the ideal dark matter component in the standard Λ CDM model with causal bulk viscous dark matter, with a novel enthalpy-density dependent coefficient, and derived an analytical solution for the Hubble parameter. The obtained solution reveals a universe characterized by an early decelerated expansion phase followed by a late accelerated expansion phase with a pre-quintessence phase which asymptotes to a future de Sitter era, or future de Sitter epoch followed by a far-future phantom evolution. \nObtained analytical solution for the Hubble parameter predicts three possible late-time evolutionary scenarios depending on the value of the bulk viscous coefficient ( ζ 0 ), which in-turn determines the nature of vDM in the late phase. If ζ 0 > 1 / 3 , the bulk viscous dark matter (vDM) leads to a phantom evolution. For ζ 0 < 1 / 3 , the universe enters a quintessence phase. And notably, when ζ 0 ≊ 1 / 3 both the energy density and bulk viscous pressure of the dissipative matter stabilize, mimicking the behavior of a cosmological constant in the late universe. We found that, to avoid violation of null energy condition (phantom evolution) and covariant second law of thermodynamics (CSLT), ζ 0 must be constrained within the range ζ 0 ∈ (0 , 1 / 3] . \nDespite the sign-switching nature of the bulk viscous pressure in this model, the coefficient of bulk viscosity remains positive throughout the evolution of the universe, hence ensuring a positive entropy production rate in the dissipative fluid and safeguarding the validity of the CSLT. This is a novel feature compared to conventional viscous models where a positive bulk viscous pressure often leads to a negative entropy production rate, thereby violating the CSLT. Interestingly, we found that during periods when the bulk viscous pressure is positive, the dissipative fluid exhibits a negative specific entropy rate in the instantaneous comoving frame. Since the model satisfies the CSLT during this era, we infer that this behavior results from a decaying entropy production rate while maintaining a constant number of particles in the comoving volume. In addition to preserving the CSLT, the model also adheres to the generalized second law of thermodynamics (GSLT) and the convexity condition, confirming that the universe evolves toward a stable equilibrium state of maximum entropy. \nA key feature of the present model is the sign-switching behavior of the bulk viscous pressure. Initially, it starts with a large positive value, singular at the big-bang, and transitions to negative values around a specific redshift, z s known as sign-switching redshift. After reaching a minimum, the pressure approaches zero asymptotically. We found that the redshift at which this sign switching occurs depends on the relaxation time parameter ϵ 0 . A decrease in ϵ 0 results in an increase in z s . Based on theoretical constraints, we estimated the optimal value of ϵ 0 to be ϵ 0 ≈ 2 . 4 × 10 -4 , corresponding to z s = 1 . 178 . \nThese findings have significant implications in viscous cosmology. For instance, in [27] sign-switching bulk viscous pressure was considered in the acausal model to study the effect of bulk dissipation induced in the energy spectra of relic gravitons. It will be interesting to see the consequence of a more general, causal sign switching bulk viscous pressure in the context discussed in that work, which we leave as a future scope. Investigating modifications to structure formation rates in the early universe in the presence of a positive viscous pressure ia yet another intriguing possibility. \nFinally, we reconsidered our model, by treating the cosmic fluids, the viscous dark matter and dark energy, as a single unified dissipative dark fluid (v Λ¯ ω DM)and showed that the resultant model i.e, v Λ¯ ω DM model, is dynamically similar to Λ vDM model. Despite the similarity in the evolutionary dynamics, the near equilibrium condition (NEC) associated with background dissipative theory evolves differently in each case. Notably, under v Λ¯ ω DM interpretation, the model satisfies NEC both in the early and late Universe. While in Λ vDM NEC is violated in the late accelerating epoch. In addition to this, in v Λ¯ ω DM interpretation, \nevolution of NEC depicts a viscous fluid evolving from a out of equilibrium (not far) to a stable equilibrium state. Consequently, we conclude that it is better to follow v Λ¯ ω DM model interpretation for this model, in which the cosmological constant arises as an inherent part of the equation of state of the viscous fluid itself, rather than as an independent dark energy component.', 'Acknowledgments': 'The authors of the manuscript are thankful to the Indian Institute of Technology Kanpur for providing the Param Sanganak high-performance computational facility for faster execution of python program. Vishnu A Pai is thankful to Cochin University of Science and Technology for providing Senior Research Fellowship. Sarath Nelleri is thankful to the Indian Institute of Technology Kanpur for providing the Institute Postdoctoral Fellowship.', 'References': '- [1] Nabila Aghanim, Yashar Akrami, Mark Ashdown, J Aumont, C Baccigalupi, M Ballardini, AJ Banday, RB Barreiro, N Bartolo, S Basak, et al. Planck 2018 results-vi. cosmological parameters. Astronomy & Astrophysics , 641:A6, 2020.\n- [2] Shadab Alam et al. The clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: cosmological analysis of the DR12 galaxy sample. Mon. Not. Roy. Astron. Soc. , 470(3):2617-2652, 2017.\n- [3] Amedeo Balbi, Marco Bruni, and Claudia Quercellini. Λ α DM : Observational constraints on unified dark matter with constant speed of sound. Phys. Rev. D , 76:103519, Nov 2007.\n- [4] Fábio S. Bemfica, Marcelo M. Disconzi, and Jorge Noronha. Causality of the einstein-israel-stewart theory with bulk viscosity. Phys. Rev. Lett. , 122:221602, Jun 2019.\n- [5] Sebastian Bocquet and Faustin W. Carter. pygtc: beautiful parameter covariance plots (aka. giant triangle confusograms). Journal of Open Source Software , 1(6):46, 2016.\n- [6] Iver Brevik. Temperature Variation in the Dark Cosmic Fluid in the Late Universe. Mod. Phys. Lett. A , 31(08):1650050, 2016.\n- [7] Dillon Brout et al. The Pantheon+ Analysis: Cosmological Constraints. Astrophys. J. , 938(2):110, 2022.\n- [8] Herbert B Callen. Thermodynamics and an Introduction to Thermostatistics . John wiley & sons, 1991.\n- [9] Salvatore Capozziello. Curvature quintessence. Int. J. Mod. Phys. D , 11:483-492, 2002.\n- [10] Eugenio Carretta, Raffaele G. Gratton, Gisella Clementini, and Flavio Fusi Pecci. Distances, ages, and epoch of formation of globular clusters*. The Astrophysical Journal , 533(1):215, apr 2000.\n- [11] Sean M. Carroll. The Cosmological constant. Living Rev. Rel. , 4:1, 2001.\n- [12] Lu Chen, Qing-Guo Huang, and Ke Wang. Distance priors from planck final release. Journal of Cosmology and Astroparticle Physics , 2019(02):028, feb 2019.\n- [13] Steven J. Clark, Kyriakos Vattis, JiJi Fan, and Savvas M. Koushiappas. H 0 and S 8 tensions necessitate early and late time changes to ΛCDM . Phys. Rev. D , 107:083527, Apr 2023.\n- [14] A. A. Coley and R. J. van den Hoogen. 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Finkbeiner, Bruce Gillespie, Karl Glazebrook, Gregory S. Hennessy, David W. Hogg, Željko Ivezić, Bhuvnesh Jain, David Johnston, Stephen Kent, Donald Q. Lamb, Brian C. Lee, Huan Lin, Jon Loveday, Robert H. Lupton, Jeffrey A. Munn, Kaike Pan, Changbom Park, John Peoples, Jeffrey R. Pier, Adrian Pope, Michael Richmond, Constance Rockosi, Ryan Scranton, Ravi K. Sheth, Albert Stebbins, Christopher Stoughton, István Szapudi, Douglas L. Tucker, Daniel E. Vanden Berk, Brian Yanny, and Donald G. York. Cosmological constraints from the sdss luminous red galaxies. Phys. Rev. D , 74:123507, Dec 2006. \n- [54] Licia Verde, Tommaso Treu, and Adam G Riess. Tensions between the early and late universe. Nature Astronomy , 3(10):891-895, 2019.\n- [55] Steven Weinberg. The cosmological constant problem. Rev. Mod. Phys. , 61:1-23, Jan 1989.\n- [56] Lixin Xu, Yuting Wang, and Hyerim Noh. Unified dark fluid with constant adiabatic sound speed and cosmic constraints. Phys. Rev. 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2024A&A...691A.124A | We present a kinematic analysis based on the large integral field spectroscopy IFS dataset of SDSSIV MaNGA Sloan Digital Sky SurveyMapping Nearby Galaxies at Apache Point Observatory 10 000 galaxies. We have compiled a diverse sample of 594 unique active galactic nuclei AGNs identified through a variety of independent selection techniques encompassing radio 1.4 GHz observations optical emissionline diagnostics BPT broad Balmer emission lines midinfrared colors and hard Xray emission. We investigated how ionized gas kinematics behave in these different AGN populations through stacked radial profiles of the O III 5007 emissionline width across each AGN population. We contrasted AGN populations against each other and nonAGN galaxies by matching samples by stellar mass O III 5007 luminosity morphology and redshift. We find similar kinematics between AGNs selected by BPT diagnostics compared to broadlineselected AGNs. We also identify a population of nonAGNs with similar radial profiles as AGNs indicative of the presence of remnant outflows or fossil outflows of past AGN activity. We find that purely radioselected AGNs display enhanced ionized gas line widths across all radii. This suggests that our radioselection technique is sensitive to a population in which AGNdriven kinematic perturbations have been active for longer durations potentially due to recurrent activity than in purely optically selected AGNs. This connection between radio activity and extended ionized gas outflow signatures is consistent with recent evidence that suggests radio emission expected to be diffuse originated due to shocks from outflows. We conclude that different selection techniques can trace different AGN populations not only in terms of energetics but also in terms of AGN evolutionary stages. Our results are important in the context of the AGN duty cycle and highlight integral field unit datas potential to deepen our knowledge of AGNs and galaxy evolution. | 2024-11-01T00:00:00Z | ['10.48550/arXiv.2408.16831', '2024A&A...691A.124A', 'arXiv:2408.16831', '2024arXiv240816831A', '10.1051/0004-6361/202451738'] | ['galaxies: active', 'galaxies: evolution', 'quasars: supermassive black holes', 'Astrophysics - Astrophysics of Galaxies'] | Mapping AGN winds A connection between radiomode AGNs and the AGN feedback cycle | 2,024 | 167 | 0.66 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF'] | 0 | https://arxiv.org/pdf/2408.16831.pdf | {'Mapping AGN winds: a connection between radio-mode AGN and the AGN feedback cycle': 'M. Albán 1 , D. Wylezalek 1 , J. M. Comerford 2 , J. E. Greene 3 , and R. A. Ri ff el 4 , 5 \n- 1 Zentrum für Astronomie der Universität Heidelberg, Astronomisches Rechen-Institut, Mönchhofstr, 12-14 69120 Heidelberg, Germany\n- 2 University of Colorado Boulder, 2000 Colorado Avenue, Boulder, CO 80309, USA\n- 3 Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08544, USA\n- 4 Departamento de Física, Centro de Ciências Naturais e Exatas, Universidade Federal de Santa Maria, 97105-900, Santa Maria, RS, Brazil\n- 5 Laboratorio Interinstitucional de e-Astronomia - LIneA, Rua Gal. José Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil \nReceived Month xx, 202x; Month Xxxxxx xx, 202x', 'ABSTRACT': "We present a kinematic analysis based on the large Integral Field Spectroscopy (IFS) dataset of SDSS-IV MaNGA (Sloan Digital Sky Survey / Mapping Nearby Galaxies at Apache Point Observatory; ∼ 10.000 galaxies). We have compiled a diverse sample of 594 unique Active Galactic Nuclei (AGN), identified through a variety of independent selection techniques, encompassing radio (1.4 GHz) observations, optical emission line diagnostics (BPT), broad Balmer emission lines, mid-infrared colors, and hard X-ray emission. We investigate how ionized gas kinematics behave in these di ff erent AGN populations through stacked radial profiles of the [OIII] 5007 emission-line width across each AGN population. We contrast AGN populations against each other (and non-AGN galaxies) by matching samples by stellar mass, [OIII] 5007 luminosity, morphology, and redshift. We find similar kinematics between AGN selected by BPT diagnostics compared to broad-line selected AGN. We also identify a population of non-AGN with similar radial profiles as AGN, indicative of the presence of remnant outflows (or fossil outflows) of a past AGN activity. We find that purely radio-selected AGN display enhanced ionized gas line widths across all radii. This suggests that our radio-selection technique is sensitive to a population where AGN-driven kinematic perturbations have been active for longer durations (potentially due to recurrent activity) than in purely optically selected AGN. This connection between radio activity and extended ionized gas outflow signatures is consistent with recent evidence that suggests radio emission (expected to be di ff use) originated due to shocks from outflows. We conclude that di ff erent selection techniques can trace di ff erent AGN populations not only in terms of energetics but also in terms of AGN evolutionary stages. Our results are important in the context of AGN duty cycle and highlight integral field unit (IFU) data's potential to deepen our knowledge of AGN and galaxy evolution. \nKey words. Catalogs - galaxies: active", '1. Introduction': "Active galactic nuclei (AGN) have become a common element in galaxy evolution studies (Heckman & Best 2014) and a fundamental engine for supermassive black hole (SMBH) growth (Alexander & Hickox 2012). Observational studies have suggested the connection between supermassive black holes and their host galaxies, finding significant empirical correlations between them (Kormendy & Ho 2013). Specifically, the mass of the SMBH has been seen to correlate with fundamental galaxy properties such as the bulge luminosity (Ferrarese & Merritt 2000) and the bulge velocity dispersion (Marconi & Hunt 2003). Further evidence has shown that star formation rate history in galaxies peaks at z ∼ 2, exactly where the black hole accretion history (related to AGN activity) is at its height (Madau & Dickinson 2014; Aird et al. 2015). This suggests an interaction (and coevolution) between the AGN and the interstellar medium (ISM) of its host galaxy (Fabian 2012; Morganti 2017b), known as AGN feedback. Indeed, the released energy required for such a massive black hole to have grown is comparable to or greater than the binding energy of the host galaxy itself (Silk & Rees \n1998), placing AGN in the spotlight as relevant for understanding galaxy evolution (see also Hopkins et al. 2006). \nA common property of galaxies hosting an AGN is the presence of strong winds or outflows (e.g., Mullaney et al. 2013; Harrison et al. 2014; Cheung et al. 2016; Wylezalek et al. 2020) in the ionized gas. Such outflows can be deployed in the form of collimated jets (Worrall & Birkinshaw 2006) or as radiativelydriven winds (Netzer 2006) where gas can be ejected and transferred into the host galaxy (see King & Pounds 2015, for a review). This ubiquitous characteristic is a popular mechanism to explain how AGN feedback works and has been a key parameter introduced to solve theoretical problems faced in cosmological simulations (Somerville & Davé 2015; Naab & Ostriker 2017). For example, one notable application is helping to explain the regulation of star formation in massive galaxies (see also Harrison 2017). These phenomena (winds or outflows) have been observed in multiple gas phases (e.g., Aalto et al. 2012; Fiore et al. 2017; Herrera-Camus et al. 2020; Baron et al. 2021; Riffel et al. 2023), from extremely broad X-ray outflow features (reaching fractions of the speed of light; Tombesi et al. 2012) to cold-molecular gas winds (e.g., Cicone et al. 2014). \nEven when focusing on one specific phase, outflow signatures can turn out to be very complex (e.g., Zakamska et al. 2016a). In the ionized gas (the main subject of this paper), for example, such outflows display non-gravitational winds with velocity dispersion (FWHM > 500 km s -1 ) that cannot be explained by the intrinsic rotation of the host galaxy or its dynamical equilibrium (Karouzos et al. 2016). Outflows usually appear in the spectra as secondary spectral components that accompany the main spectral lines (e.g., Heckman et al. 1981; Mullaney et al. 2013). Therefore, the shape of these spectral lines can acquire complex features that a single Gaussian profile cannot model. Instead, multi-component fitting procedures have been widely used to characterize outflow signatures (e.g., Förster Schreiber et al. 2014). A widely used tracer to study these signatures is the [O III] λ 5007 emission line. This forbidden emission line is restricted to low-density environments (such as the narrow line region) and can be produced as a result of shocks or photoionization (Osterbrock 1989). \nMuch of what has been learned from AGN has been through the study of their ionized gas kinematics. For example, in a large sample of optically selected Type-II AGN, Woo et al. (2016) found that the velocity dispersion of the outflow as well as the fraction of emission-line ([O III]) shapes exhibiting multiple components both tend to escalate with an increase in [O III] luminosity ( L [O III]). This is relevant because the L [O III] has been shown to be a good indicator of an AGN's bolometric luminosity (L bol ; Heckman et al. 2004; LaMassa et al. 2010), which is an important parameter to understand the involved energy injection of the AGN's supermassive black hole (Heckman & Best 2014) to the host galaxy. For example, Fiore et al. (2017) found that the wind mass outflow rate correlates with L bol . \nIonized outflows have been routinely found in AGN selected from Infrared (e.g., DiPompeo et al. 2018), X-ray (e.g., Rojas et al. 2019) and optical surveys (e.g., Wylezalek et al. 2020), to mention a few examples. However, none of the multiple AGN selection techniques today o ff er an ultimate clean AGN population (Padovani 2017) by itself. Attempts to create more complete AGN samples have shown that di ff erent selection techniques can find AGN candidates that other single selection techniques would miss (e.g., Alberts et al. 2020). Statistical analysis of AGN selected based on techniques that are limited to a certain wavelength window can su ff er from important biases such as obscuration or data coverage. This is not a simple task and di ff erent selection techniques (using various wavelengths) find di ff erent AGN populations, even with contrasting host-galaxy properties (e.g., Hickox et al. 2009; Comerford et al. 2020; Ji et al. 2022). \nConsequently, the estimated outflow properties and, therefore, AGN feedback studies can be compromised by the way the AGNpopulation is selected. For example, Mullaney et al. (2013) found that the most extreme [O III] kinematics arise from AGN with moderate radio luminosities (10 23 W Hz -1 > L1 . 4 GHz > 10 25 WHz -1 ), finding evidence of compact radio cores being responsible for driving the most broadened profiles (see also Jarvis et al. 2019; Molyneux et al. 2019; Jarvis et al. 2021). Baron & Netzer (2019) found that AGN that present outflows (using the [O III] emission line) exhibit an excess in the mid-infrared spectral energy distribution component, suggesting that outflows are carrying dust. Di ff erent selection techniques can also be sensitive to di ff erent AGN powering mechanisms or stages of the current AGNduty cycle. The latter has been suggested by comparing directly between optically selected AGN candidates against midinfrared radio-detected AGN candidates (see Kau ff mann 2018), finding the former to dominate black hole growth in lower mass systems. \nAn additional complication is that ionized outflows can extend from sub-kpc (e.g. Singha et al. 2022) to kpc scales (e.g., Liu et al. 2010; Sun et al. 2017). Due to the limitations of the instruments, most of the studies mentioned above base their results on single-fiber observations. Integral field spectroscopy (IFS) provides a valuable technique to study the spatial distribution of outflows in more detail (e.g., Wylezalek et al. 2018; Luo et al. 2021; Singha et al. 2022). One of the latest pioneering IFS surveys is the MaNGA (Mapping Nearby Galaxies at Apache Point Observatory) survey (Bundy et al. 2015), providing 10 010 unique galaxies with spatially resolved spectra. Hence, our primary objective is to investigate how outflow properties vary, not only spatially but also based on the selection technique employed. The responsiveness of our selection methods to outflow characteristics can potentially shed light on their driving mechanisms and a connection to the AGN duty cycle. \nThis paper is organized as follows. In Section 2, we describe our data and some available catalogs about them that are relevant to this study to assemble a multi-wavelength AGN catalog. The methods employed to study our sample are described in Section 3, with a description of the host galaxy properties of our sample. The results are explained in Section 4, and we present a discussion in Section 5. Lastly, we summarize our conclusions in Section 6. The cosmological assumptions used in this study are H0 = 72 km s -1 Mpc -1 , Ω M = 0.3 and ΩΛ = 0 . 7.", '2.1. The MaNGA Survey': "In this study, we use the ∼ 10,000 galaxies (0 . 01 < z < 0 . 15) observed in the SDSS-IV / MaNGA survey (Sloan Digital Sky Survey / Mapping Nearby Galaxies at Apache Point Observatory). MaNGA is an integral field unit (IFU) survey, providing 2D mapping of optical spectra at 3622-10354 Å at a resolution of R ∼ 2000. It's field-of-view ranges from 12 '' to 32 '' in diameter. Data reduction has been performed by MaNGA's Data Reduction Pipeline (DRP, Law et al. 2015). Complete spectral fitting is provided by MaNGA's Data Analysis Pipeline (DAP, Westfall et al. 2019). The DAP fits models for multiple spectral components (e.g., stellar continuum, emission lines) to the entire spectra. Throughout this paper, we are using the spectra (reduced by the DRP) after subtracting their stellar continuum (i.e., emission line-only spectra provided by the DAP; see details in Section 3). \nAdditionally, Sánchez et al. (2022) presents a comprehensive catalog reporting multiple characteristics and integrated host galaxy properties based on a full spectral analysis with the pyPipe3D pipeline (Lacerda et al. 2022). Most of the galaxy properties used in our study are taken from this catalog (e.g. stellar mass, star formation rates). Other galaxy properties, such as emission-line ratios, H α equivalent widths (EW(H α )), are taken from (Albán & Wylezalek 2023). In this paper, we furthermore compute additional parameters as described in Section 3 (e.g., L[ O III ]).", '2.2. AGN catalogs': 'This paper aims to assess the behavior of spatially resolved ionized gas kinematics in AGN samples selected through various selection methods 1 . We use the following set of MaNGA-AGN catalogs that we further describe in the following subsections: \n- -An optical emission line-based catalog from Albán & Wylezalek (2023) (using the 2 kpc aperture).\n- -A broad-line based AGN catalog from Fu et al. (2023).\n- -A mid-infrared selected AGN catalog from Comerford et al. (2024).\n- -A hard X-ray-selected AGN catalog from Comerford et al. (2024).\n- -A catalog of radio-selected AGN that we construct in this paper (see section 2.2.4). \nThe full MaNGA sample contains a small number of repeated observations, most of which can be identified through their MaNGA-IDs (although there are exceptions; see more in Appendix A). We exclude duplicate sources in our final statistics, tables, and figures. In the following sections, we describe the individual AGN catalogs and the respective selection criteria in more detail. The sky coverage of the di ff erent surveys used for the classifications described below overlaps with MaNGA.', '2.2.1. DR17 optical AGN catalog in flexible apertures from Albán & Wylezalek': "Albán & Wylezalek (2023) present galaxy classifications based on optical emission line diagnostics (Baldwin et al. 1981; Veilleux & Osterbrock 1987) measured within apertures of varying size for the entire MaNGA survey. Galaxies are classified into Star-forming (SF), Composite, Seyfert, LINER (lowionization emission-line region, Halpern & Steiner 1983), or Ambiguous galaxies (e.g., if a galaxy received two di ff erent classifications based on di ff erent line ratio diagnostics). The final AGN sample is then defined based on the galaxies in the Seyfert and LINER classes with an additional cut on H α equivalent width > 3 Å 2 . This additional cut minimized the contamination of faint 'fake' AGN (Cid Fernandes et al. 2010). \nIn this paper, we use the 399 AGN candidates from the catalog based on a 2 kpc aperture 3 . The aperture is chosen to keep a balance between MaNGA's spatial resolution limit ( ∼ 1 . 37 kpc, Wake et al. 2017) and the physical extent of gas ionized by an AGN (known as the narrow line region (NLR), Bennert et al. 2006; Netzer 2015).", '2.2.2. Broad-line AGN catalog': "Some active galaxies present broad Balmer emission lines (known as Type-I AGN; e.g., Oh et al. 2015). This is attributed to Doppler broadening due to high-velocity ionized gas surrounding the SMBH (Peterson 2006). Comerford et al. (2020) presented a crossmatch between the MaNGA survey and Oh et al. (2015)'s Type I classification which is based on SDSS DR7 data single-fiber spectroscopic observations with a size of 3'. More recently, Fu et al. (2023) has carried out an analysis to identify broad-line AGN and double-peaked emission line signatures for the total MaNGA sample using the DR17 data release. MaNGA not only uses smaller fibers (2') but also provides additional spatial information. \nFor each galaxy, Fu et al. (2023) use DAP flux residuals to compare them to the original flux in specific spectral regions \n(with a size of 20 Å) corresponding to the location of H α and [O III] emission lines to assess the quality of the DAP's fitting procedure. They arrange the sample in 20 S / N bins (G-band S / N from the DAP) and select galaxies with residuals > 1 σ of the residual distribution at each S / N bin (see details on Fu et al. 2023). They then perform a spectral fitting on this sample of 1652 galaxies, allowing multiple components to be fitted to emission lines. \nThey ultimately select broad-line AGN as galaxies where the emission line width ( σ ) of the broad component is at least 600 km s -1 larger than the emission line width of the narrow component and present a catalog of 139 broad-line AGN (TypeI) candidates. We find a few duplicate galaxies in this catalog observations (see Appendix A), which reduces the sample to 135 targets. The work by Fu et al. (2023) almost doubles the number of broad-line selected galaxies presented Comerford et al. (2020); on the other hand, 21 galaxies presented in Comerford et al. (2020) are not found in Fu et al. (2023). Discrepancies in the latter context can be related to the di ff erence in FWHM( H α ) constraints and possibly e ff ects from changing look AGN (see Ricci & Trakhtenbrot 2023, for a review).", '2.2.3. Mid-IR and X-ray AGN catalogs of Comerford et al.': 'Comerford et al. (2024) cross-match MaNGA galaxies with known AGN candidates from multi-wavelength surveys (as in Comerford et al. 2020). For this study, we will use the following catalogs: \n- -Mid-Infrared AGN catalog based on observations with Wide-field Infrared Survey Explorer (Wright et al. 2010, WISE): 123 AGN.\n- -X-Ray selected catalog based on observations with the Burst Alert Telescope (Barthelmy et al. 2005, BAT): 29 AGN. \nComerford et al. (2024) also provides a radio-AGN catalog and a broad-line (Type-I) AGN catalog, which we choose not to use due to our science goals (see 2.2.4 and Section 2.2.2, respectively). Due to repeated observations or critical flags (from the MaNGA DRP; see the details in Appendix A) we exclude 7 galaxies from the Mid-Infrared-selected (3 were repeated) and 1 from the X-ray-selected (1 has a critical flag).', '2.2.4. Selection of radio AGN': "In this section, we present a catalog of AGN candidates solely based on radio data and independent of any radio-loud / radioquiet classification often used in the literature. Broadly speaking, independent of having a jet or not (see the details in some comprehensive reviews; e.g., Heckman & Best 2014; Padovani 2016; Panessa et al. 2019). For example, Best & Heckman (2012) used a selection technique combining both optical and radio signatures, and this is the catalog that Comerford et al. (2024) presents as the radio-selected MaNGA AGN population (see Section 2.2.3). However, this classification prioritizes sources with f 1 . 4 GHz > 5 mJy as its emphasis lies on the radio-loud population of AGN (see Urry & Padovani 1995; Padovani et al. 2017, for a review). Instead, we use a di ff erent approach and first crossmatch MaNGA SDSS-IV galaxies with data from the NRAO Very Large Array Sky Survey (Condon et al. 1998, NVSS) and the Faint Images of the Radio Sky at Twenty centimeters (Becker et al. 1995, FIRST) radio surveys and adopt a less strict flux cut of > 1 mJy. We note that this threshold is close to the sensitivity \nFig. 1. Definition of the radio-selected AGN candidates. We plot in the y-axis the expected SFR that one would measure, assuming that all the radio luminosity can be attributed to star formation processes (SFR(Lrad; see Section 2.2.4). Similarly, in the x-axis, the SFR is expected from H α luminosity (SFR(H α )). The black line corresponds to the location where SFR( L rad) = SFR(H α ). The other colored lines (orange, red, and blue) correspond to our SFR excess definition in SFR(H α ) steps of 0.5, 1.0, and 1.5 dex (log( xi )). We define each sample of AGN-selected candidates by selecting the targets whose values (as well as their error bars) are above the corresponding line (following: SFR( L rad) / SFR(H α ) = xi ) and we color them according to the colored lines (except for the xi = 0 . 0 dex, which corresponds to the yellow ones). We also show targets that did not satisfy any SFR-excess criteria (gray without marker) or did not pass the S / Ncriteria (gray with marker) for our kinematic analysis (See section 3.2). The contours (dashed-blue, light-blue, and teal) represent the density where specific galaxy populations gather (Star-forming, Composite, Non-radio selected AGN). The top and right-hand plots show the individual parameter distribution of these three galaxy populations. \n<!-- image --> \nof the surveys, and there is likely a population of AGN emitting even below this limit (e.g., White et al. 2015). \nPrevious studies have shown that finding genuine associations between targets of two di ff erent surveys comes with a trade-o ff between completeness and reliability (e.g., Best et al. 2005; Ivezi'c et al. 2002). Choosing a larger o ff set for searching counterparts can lead to high completeness but increases the number of false associations. To ensure precise spatial alignment between the optical and radio sources, we search for the closest galaxy within a 1.5 arcsecond aperture for the FIRST survey, which ensures 85% of completeness and 97% reliability (see Ivezi'c et al. 2002), and we use a 5.0 arcseconds aperture for the NVSS survey. For comparison, Best et al. (2005) estimates 90% of completeness and 6% of contamination from random targets using a 10.0 arcseconds aperture. This choice of apertures aims to maximize completeness and reliability. We find 936 and 1035 cross-matches for the FIRST and NVSS survey, respectively. In total, 1383 unique targets were found, with 588 coincident targets between FIRST and NVSS. \nOur aim is to develop a radio-selected sample as independent as possible of known optical diagnostics (e.g., BPT diagrams) or \n<!-- image --> \nFig. 2. Optical diagnostic diagrams for MaNGA galaxies. The black circles in the scatterplot show the emission line ratio values for all MaNGA galaxies. In colored shapes, we feature the di ff erent AGN-selected candidates (see the legend). We take the flux ratio values from the ones measured around their central 2 kpc region (Albán & Wylezalek 2023). In the top panel, the gray line corresponds to the demarcation line from Kau ff mann et al. (2003), and the black line (both in the top and bottom plot) corresponds to the ones from Kewley et al. (2001). \n<!-- image --> \nother selection criteria. Two significant contributors to the extragalactic radio sky are star formation processes and nuclear activity in galaxies (Padovani 2016). AGN host galaxies have been shown to span several decades in radio luminosity, often so faint, raising the question of whether their radio emission is dominated by star formation processes (Panessa et al. 2019) rather than the AGN event. Zakamska et al. (2016b) studied whether the radio signatures (1.4 GHz flux density) of confirmed AGN can be explained purely by star formation processes when comparing it with a variety of star formation rate tracers. Independent of the used SFR tracer, they conclude that AGN had a systematic excess in radio luminosity not consistent with star formation, which might be attributed to the activity in the nucleus. Similarly, Kau ff mann et al. (2008) used the H α luminosity and compared it with the 1.4 GHz flux densities of a sample of SDSS galaxies (cross-matched with radio surveys; FIRST and NVSS) and found that SF galaxies form a tight correlation between both \nparameters and that their AGN candidates systematically exceed this tight relation towards higher 1.4 GHz flux densities. \nBased on the findings described above, we construct a radioselected AGN sample similar to the 'LH α versus L rad ' method used in Best & Heckman (2012). Specifically, we identify AGN activity based on the excess in the SFR estimated based on the radio luminosity compared to the H α -based SFR as reported in the PIPE3D value-added catalog of Sánchez et al. (2022); i.e., values that are above the expected 1-to-1 relation. We use the extension named log\\_SFR\\_SF, meaning that only the spaxels that were consistent with star formation regions were used to measure the SFR. Additional ways to minimize or correct for the contribution of the AGN during the SFR measurement have been shown in De Mellos in prep. We note that di ff erent methods to estimate the SFR from optical spectra (e.g., SSP-method, Sánchez et al. (2022)) do not change our results significantly (as has also been seen in Zakamska et al. 2016b). \nIn Figure 1, we show the relation between the H α -based SFR and the radio-based SFR for the radio-detected MaNGA galaxies described above, assuming all radio emission is related to SF processes. We also show the density contours of di ff erent galaxy sub-classes based on optical diagnostics presented in Albán & Wylezalek (2023) and described in Section 2.2.1. Pure SF galaxies agglomerate close to the 1-to-1 line (blue dashed contours), in agreement with the findings presented in Kau ff mann et al. (2008). Composite galaxies (see Section 2.2.1) occupy values consistent with a radio excess (cyan solid contours). Indeed, the emission in composite galaxies is expected to be a mix of star formation and AGN processes. We also show the location of AGN candidates that have been selected by any of the described selection techniques (apart from a radio selection), i.e., using mid-infrared, hard X-rays (see Section 2.2.3), broad lines (see Section 2.2.2) and optical diagnostics (see Section 2.2.1) and label this sample as 'AGN (no radio)' in the figure (teal-colored solid line). We show that this AGN population gathers preferentially in the excess region of the plot, consistent with the findings of Zakamska et al. (2016b) and Kau ff mann et al. (2008). On the top and right-hand borders of the plot, we show the distribution of the individual SFR values using a smooth histogram. \nUsing o ff sets from the 1-to-1 line, following SFR( L rad) / SFR(H α ) = xi , we have colored with yellow, orange, red and maroon the galaxy populations with excesses from log ( xi ) = 0 . 0 to log ( xi ) = 1 . 5 dex. The gray-colored targets in the plot represent the galaxies that we exclude from our analysis due to one or two of two reasons: they were not above the 1-to-1 relation, or their signal-to-noise (S / N) from the [OIII] 5007 emission line had a bad quality to be accepted for our kinematic analysis (see Section 3.2). \nWe define our radio-AGN sample using the galaxies whose L rad plus associated flux uncertainties were at least 0.5 dex above the 1-to-1 SFR relation. We find 642 galaxies that satisfy this criterion, while 28 of those are either duplicate or critical targets (see Appendix A). We note that only 5% of the SF classified galaxies are above the 0.5 dex line. Employing larger cuto ff s risks excluding low-luminosity AGNs. Notably, 25% of targets identified as AGN by alternative methods, i.e., excluding radio observations, were found below the 0.5 dex line. Taking into account additional quality criteria necessary for our emission line analysis (see Section 3.2), we will work with a sample of 288 radio-AGN, which we will refer to as the radio-selected AGN sample in the remaining parts of this paper. \nFurthermore, at the moment of writing this paper, Suresh & Blanton (2024) studied a radio-AGN sample (selected from MaNGA in a very similar way as in this paper) and their Ed- \ndington ratio to estimate their radio activity. They find that the Eddington ratio distribution within their AGN sample exhibits a significant dependency on stellar mass, whereas it shows no correlation with the specific star formation rate (sSFR) of the host galaxies. This led them to conclude that, at a fixed stellar mass, SFRs of host galaxies do not influence the radio-AGN selection.", '2.2.5. Overlap and discrepancy between the AGN catalogs': "In Table 1, we compile the number of galaxies identified as AGN using the various selection methods discussed above, noting that some galaxies were selected as AGN by multiple techniques. In total, we identify a sample of 970 galaxies that are classified as AGN by at least one method. \nSince the work of this paper focuses on the ionized kinematics as traced by the [O III] 5007 emission line in AGN, we require additional S / N cuts on emission line fluxes (e.g., S / N < 7; see the details in 3.2), which reduces the sample we continue to work with. Table 2 lists 594 individual AGN candidates (621 if repetitions or critical targets are not taken into account; see Appendix A) that will be used in our kinematic analysis, indicating most AGN selections remain largely una ff ected by our S / N criteria, except for the radio-selected sample. We further discuss this in Section 3.3. \nIt is largely known that no selection technique is free from limitations. For example, optical selection techniques are mostly biased towards unobscured AGN. This spectral window is significantly impacted by absorption and scattering due to the presence of dust and gas that can obscure the central regions of an AGN (where most of its energetic input occurs). Some contaminants to optical selection techniques can be associated with galaxies dominated by post-asymptotic giant branch stars (e.g., Singh et al. 2013). Furthermore, dilution from the host galaxy can also play a role in missing AGN emission. Given that opacity due to dust is less e ff ective at longer wavelengths, Mid-infrared selection techniques are less a ff ected by dust attenuation. Most of its critical contaminants dominate at larger redshifts; with MaNGA we work with sources at z < 0 . 5. Finally, radio selection techniques are also less a ff ected by obscuration. However, low-luminosity AGN can be di ffi cult to distinguish from starforming processes. Padovani et al. (2017) provides a broad and comprehensive overview of this topic. \nIn Figure 2, we display the emission-line ratio diagrams highlighting our AGN samples, except for optically selected sample. AGN do not show any preferred location on the diagrams (or a specific side of the demarcation lines; see Kewley et al. 2001; Kau ff mann et al. 2003). Using AGN that were classified di ff erently than optical techniques, only 4% of the SF galaxies are AGN, 14% in the case of Composite, and 7% for Ambiguous. This is an excellent example of the well-known problem that using only one single criterion is insu ffi cient to obtain a complete picture of the AGN population. \nWenote that the hard X-ray-selected AGN candidates are the smallest sample. This is not surprising, as the BAT's integration time is kept short to fulfill its scientific goals (Barthelmy et al. 2005). It is also the sample with the largest overlap with the other AGNsamples; all X-ray-selected AGN are also selected as AGN in at least one other selection technique reported in this paper. Indeed, x-ray emission appears to be universal in AGN and the emission is not significantly contaminated by its host galaxy (see a detailed discussion in Padovani et al. 2017). Hence, the optical, infrared, broad-line, and radio selection techniques make up the four biggest AGN sub-samples in our study, with the largest number of independently selected candidates. Additionally, we \nTable 1. Coincident targets between the di ff erent AGN-selection techniques in the full MaNGA sample. \nNotes. Here, we have already excluded targets that were repeated observations or targets that had a critical flag (see Appendix A). \nTable 2. Coincident targets between the di ff erent AGN-selection techniques in a sample of MaNGA targets limited by S / N. \nNotes. Here, we only use targets that satisfy the quality criteria described in Section 3.2. \ndefine a sample of non-AGN galaxies that will be used in the discussion section. Our non-AGN sample contains all MaNGA galaxies that were not selected as AGN by any method used in this paper.", '3. Analysis': 'Hereinafter, when referring to kinematics, we specifically refer to the ionized gas (traced by the [O III] 5007 emission line).', '3.1. Fitting procedure': "Spectra from regions with kinematics dominated by winds can display complex emission line profiles (e.g., Liu et al. 2013). This is not taken into account by the DAP emission-line fitting routine. Therefore, we develop a fitting procedure to account for up to two Gaussian components for each emission line. Our fitting method is based on a least-squares Python program using the documentation from Non-Linear Least Squares Minimization (LMFIT, Newville et al. 2016) and it follows standard fitting procedure techniques (e.g., Liu et al. 2013; Wylezalek et al. 2020). In summary, for all spectra in each MaNGA galaxy, our procedure operates in the rest-frame stellar-subtracted region where the [O III] 5007 emission line is (see the details in Appendix B). From the maps of the best-fit parameters, we create a non-parametric emission line width W80 map. The W80 parameter is the most essential value we extract from our fitting procedure and will be the most relevant for the discussion throughout this paper. Whenever we refer to it, we refer to the W80 value obtained from [O III] 5007. \nTo study the spatial distribution of this parameter, we construct radial profiles for all galaxies from elliptical annuli in steps of 0.25 e ff ective radius (R e f f ). To obtain the parameters for the elliptical apertures for each galaxy, we use the b / a axis ratio and position angle (PA) from PIPE3D's value-added catalog from Sánchez et al. (2022), as well as the e ff ective radius. These parameters are adopted from the NASA-Sloan Atlas catalog (Blanton et al. 2011). They use the Petrosian system (see Petrosian 1976; Blanton et al. 2001) applied to the SDSS r-band imaging of galaxies using elliptical apertures. Here R e f f is defined as the major axis containing 50% of the flux inside 2 Petrosian radii, and b / a , and PA are obtained from the elliptical aperture (see \nthe details in Wake et al. 2017). To perform a weighted average on each annulus, we capture the fraction of each pixel enclosed by an annulus so that we avoid average properties over a set of discrete pixels and recover a smooth distribution. Specifically, we follow the pixel-weighted average procedure used in Albán &Wylezalek (2023) but using ellipses. \nIn a sample of ∼ 160 000 normal SDSS (SF BPT selected) galaxies ( z < 0 . 7, with 8 < log ( M ∗ / M ⊙ ) < 11 . 5 and -3 < log ( S FR / M ⊙ yr -1 ) < 2), Cicone et al. (2016) find that the gas velocity dispersion ( σ ) hardly exceeds 150 km s -1 . The latter corresponds to a W80 of ∼ 380 km s -1 ( W 80 = 2 . 56 σ ). Furthermore, Gatto et al. (2024) conclude a lower cut, of ∼ 315 km s -1 when studying the W80 in a control sample of nonAGN (matched to optically-selected AGN in stellar mass, morphology, inclination and redshift). Therefore, W80 values greater than this threshold suggest the presence of non-gravitational motion of gas, such as outflows.", '3.2. Galaxies selected for the kinematic analysis based on S/N quality criteria': "One crucial factor to consider is the impact of the S / N on measuring the line width W80. As S / N decreases, the W80 measurements tend to get underestimated (see Liu et al. 2013), especially if there is indeed a (faint) broad component present in the line profile (Zakamska & Greene 2014). To ensure the accuracy of our analysis and avoid incorrect W80 measurements, we exclude all spaxels with an A / N < 7 (amplitude over noise) before we perform the spectral line fitting. Given the tight relation between S / Nand A / Nseen in Belfiore et al. (2019), we will refer to A / Nas simply S / N. To ensure that each individual galaxy retains enough high S / N spaxels, we furthermore apply the following criteria: \nWe only include galaxies in our final sample which satisfy the following criteria: \n- -More than 10 spaxels with a S / N > 7.\n- -There are at least two annuli (for the radial profile derivation) where the area covered by the spaxels with a S / N > 7 is at least 10% of each annulus's total area. \nThe thresholds for the S / N and pixel fractions are chosen to minimize the number of excluded galaxies while retaining a suit- \nFig. 3. Stellar mass versus star formation rates of MaNGA host galaxies. The left panel shows the impact of our quality criteria (see Section 3.2), excluding galaxies with higher stellar mass and lower star formation rates (represented by the black dots in the scatter plot and black hatched distribution in the top and right-hand diagrams). In red, orange, and yellow, we show the distribution of the AGN candidates selected by radio (using 0.0, 0.5, and 1.0 dex of excess; see Section 2.2.4) that satisfy the S / N quality criteria. To understand the properties of the excluded radio-selected hosts, we encourage the reader to look at Figure 1. It can be seen that a long tail of deficient SFR hosts are excluded ( -2 > log(SFR(H α )) > -4). On the right, we show only the galaxies chosen after the quality criteria and their corresponding AGN classification. We have provided labels for each color in a panel between both plots. \n<!-- image --> \nFig. 4. Smooth histograms of multiple host galaxy properties. On top (left to right): the b / a axis ratio, log( M ⋆ ), and log(SFR) from Pipe3D. On the bottom (left and middle plot): log( L [O III]) and log(EW(H α )), both extracted from an aperture of 1 R e ff , and (to the right) a complementary illustration for Table 2 employing Venn diagrams. In order to maintain visual clarity, the hard X-ray selected AGN sample was intentionally omitted from the diagrams. The gray shaded histogram shows the distribution of all MaNGA that pass our S / N criteria (see Section 3.2). \n<!-- image --> \nFig. 5. W80 stacked radial profiles for various galaxy groups. In the upper central panel, colors represent these groups: black, gray, and blue for all MaNGA galaxies, non-AGN, and SF galaxies (selected by BPT), while the remaining colors denote AGN-selected candidates. Each line in the plots represents the median value at each R e ff ring, with shaded areas indicating the 14th to 86th percentiles. The leftmost plot displays unshaded profiles for easier comparison. Di ff erent line styles are used for visual clarity. \n<!-- image --> \nable quality for the analysis. These quality cuts introduce a bias (driven by the S / N ) that rejects more likely low star-forming and some high-mass galaxies, respectively (e.g., Brinchmann et al. 2004; Albán & Wylezalek 2023). Our final sample contains 5696 targets (see the left plot on Figure 3). Table 1 and Table 2 show the cross-matches between the di ff erent AGN populations and how the subsample sizes decrease after applying the S / N and quality cuts. Radio-selected AGN candidates are significantly impacted by the quality criteria (see Figure 1). In contrast, the other AGN samples remain relatively una ff ected. \nAccording to the optical classification from Albán & Wylezalek (2023), more than 90% of the radio-selected-AGN that were excluded from the final sample due to the quality criteria are LINERs ( ∼ 30%, with EW(H α < 3)) or 'lineless' ( ∼ 60%, galaxies that could not be classified by optical analysis, with S / N < 3; see the details in Albán & Wylezalek 2023), and around 5% are classified as ambiguous.", '3.3. Typical properties of AGN-selected host galaxies': "Focusing on the final sample of galaxies that fulfill the S / N and quality criteria, Figure 4 shows host galaxy properties for our AGN populations, including a Venn 4 diagram (following to Table 2). Comparing Table 1 with Table 2 (samples before and after the quality cuts) reveals that the AGN samples do not experience a significant cut, with the exception of radio-selected AGN where a notable fraction of massive galaxies, with low SFR and low EW(H α ) got excluded (see 3.2). However, the distribution of b / a and L[O III] for radio-selected AGN remains less a ff ected by the S / N cut. \nMost AGN are found in host galaxies with high stellar masses ( M ⋆ ; see the top-middle plot of Figure 4), regardless of the AGN selection technique. This is a ubiquitous trend that has been found in various AGN samples from di ff erent studies (see e.g. Kau ff mann et al. 2003; Powell et al. 2018; Barrows et al. 2021; Best et al. 2005). Our di ff erent AGN subsamples all have similar stellar mass distributions. This is an important fact to notice, given that more massive galaxies are expected to have larger emission-line widths (e.g., Chae 2011; Zahid et al. 2016; Cappellari 2016, see also Appendix C). \nFigure 4 reveals that the di ff erent AGN samples probe di ff erent distributions of their host SFR, L[ O III ], and EW(H α ), showcasing the biases of each selection technique. The SFR di ff erences between the samples are reflected in Figure 3, where our AGN candidates tend to gather below the star formation main sequence (SMFS). AGN studies for samples in the local universe have found similar results, where AGN are found in the so-called transition zone or the green valley (e.g., Schawinski et al. 2007; Salim 2014; Leslie et al. 2016). At slightly higher redshifts (0 . 25 < z < 0 . 8), Hickox et al. (2009) find that midinfrared selected AGN have bluer colors and are found preferentially in the blue cloud, while radio-selected AGN gather more likely in the red sequence, suggesting the latter are relevant for understanding the evolutionary transition of host galaxies from actively star-forming to more quiescent states. \nWylezalek et al. (2018) find that AGN-selected MaNGA (DR14) targets have mostly low to intermediate luminosities ( L [O III] ∼ 10 40 erg s -1 ) for an optically-selected AGN sample. We observe here the same behavior for our optically-selected AGN in MaNGA-DR17. However, for AGN selected via infrared, hard X-rays, or broad Balmer lines, we typically observe higher L[ O III ], with distributions peaking at ∼ L[O III] ∼ 10 41 . 6 erg s -1 . On the other hand, radio-selected AGN candidates show some \nFigure 5 (see the left panel) reveals that the median W80 radial profiles of AGN-selected populations are significantly di ff erent. In Section 3.3, we have shown that the host galaxies of the different AGN samples are similar with respect to some properties (e.g., stellar mass, or b / a axis-ratio) but significantly di ff erent with respect to other properties (e.g., L [O III], or SFR(H α )). We note that samples with higher M ⋆ will systematically select galaxies with higher L [O III] and vice versa (see also Appendix C). At the same time, samples with higher M ⋆ and L [O III] will systematically select galaxies with higher W80 (see discussion in Section 3.3). Therefore, in this Section, we investigate if and how the di ff erences in the kinematics persist or change when we carefully match the AGN samples so that they have the same host galaxy properties. \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFig. 6. Comparison of stacked W80 profiles and host galaxy properties between non-AGN and Optical AGN. In each plot, blue corresponds to the behavior of a specific parameter for Optically selected AGN and black for non-AGN. We show the W80 stacked radial profile, the log( M ⋆ ) distribution, the log( L [O III]) distribution, and the log(SFR(H α )) distribution. For this comparison, only log( M ⋆ ), redshift, and morphology were controlled. \n<!-- image --> \nlower L[O III] values. Interestingly, L [O III] is known to correlate with the AGN's bolometric luminosity (Heckman et al. 2004; LaMassa et al. 2010; Pennell et al. 2017), which in turn is correlated with AGN-driven wind velocities (e.g., Fiore et al. 2017). \nFrom the 288 radio-selected AGN used for this analysis, 52 of them were optically classified as LINERs with AGN (see Section 2.2.1), and 55 were not classified as AGN as they did not meet the minimum H α equivalent width of 3 Å. The 52 LINER galaxies have a median EW(H α ) of ∼ 1.75 Å. Other authors have used less strict EW(H α ) constraints (Sánchez et al. 2018, e.g., 1.5 Å) to include fainter AGN in optically selected samples. However, this might introduce some LINER-like galaxies with no AGN but dominated by a population of post-AGB stars (Singh et al. 2013) that can mimic AGN-like ionization in typical optical classification. This emphasizes the importance of a multi-wavelength AGN selection technique for a more complete population census. Quite remarkably, AGN that were selected by MIR, broad-lines, or X-ray observations show clearly EW(H α ) > 3 . 0 Å (only three galaxies have EW(H α ) < 3 . 0 Å ∼ 2.0 Å), while radio-selected AGN show lower EW(H α ) ∼ 2.0Å. We recall that optically-selected AGN were required to have EW(H α ) > 3 . 0 Å (Albán & Wylezalek 2023). \nTherefore, we will investigate whether the di ff erences in the W80 radial profiles (see Section 3.4) can be attributed to any differences in host-galaxy properties.", '3.4. Radial profiles of ionized gas kinematics': 'Several studies have investigated the overall AGN kinematic properties across many di ff erent AGN samples (e.g., Mullaney et al. 2013; Zakamska & Greene 2014; Baron & Netzer 2019; Rojas et al. 2019, among others). However, most of these studies have used single-fiber data and have, therefore, been limited in assessing spatial dependencies. Consequently, comprehensive studies with large spatially resolved spectral samples are crucial to assessing the impact of selection techniques. We investigate the radial profiles of ionized gas kinematics in the various AGN samples we have defined above. To do so, we first stack the W80 profiles (see Section 3.1) of galaxies within each individual subsample and use the median value at each annulus. \nThe resulting profiles are shown in Figure 5. The shaded regions around the profiles represent the 14 th and 86 th percentiles of the W80 distribution at each specific annulus. The median W80 profiles reveal distinct behaviors between the AGN samples. Visual inspection indicates variations not only in the magnitude but also in the slope of these profiles. Notably, regardless of AGN selection technique, there is a systematic behavior of an \nenhanced W80 profile in the AGN population compared to the overall MaNGA sample. This characteristic continues out to 2 e ff ective radii. Similarly, if we focus only on galaxies that were not selected as AGN by any of our selection techniques (the nonAGN, see Section 2.2.5), we find the same trend. Moreover, SFclassified galaxies exhibit less pronounced profiles, with minimal enhancements near the center. The subsequent section will explore potential explanations for these observations.', '4. What drives the differences in the observed kinematics?': 'We create control samples based on a M ⋆ and L [O III] parameter space. Given that the number of galaxies per each M ⋆ and L [O III] bin becomes limited, controlling for redshift and morphology becomes challenging. Therefore, we select the galaxy that is closest in redshift and in morphology. The morphology is used as a number (obtained from Sánchez et al. 2022). We also note that the radial profiles take the R e f f of each galaxy into account by using it as a step for the average W80 at each annulus (see Section 3.4).', '4.1. AGN versus non-AGN': 'In a recent study, Gatto et al. (2024) used a catalog of opticallyselected AGN (selected through emission-line diagnostics) and created a control sample matching properties to the AGN hosts similarly as in this paper, except for the L [O III]. When looking at all spaxels, they find that AGN have greater W 80 values than the control galaxies, attributing the ionized gas kinematic disturbances to the presence of the AGN. We obtain similar results as in Gatto et al. (2024) (see Figure 1) when using only optically selected AGN and a similar control sample. In Figure 6, we present \nFig. 7. Comparison of stacked W80 profiles and host galaxy properties between non-AGN and Sample A. In each plot, blue corresponds to the behavior of a specific parameter for Sample A (see Section 4.1) and black for non-AGN. For each column of plots, from left to right, we show the W80 stacked radial profile, the log( M ⋆ ) distribution, the log( L [O III]) distribution, and the log(SFR(H α )) distribution. The plots in the bottom row show how both samples behave after they are matched to have the same log( M ⋆ ) and log( L [O III]). The plots in the top row show how both samples behave if the matched is done only for the log( M ⋆ ). The Venn diagrams shown in each W80 plot represent Sample A. The discrepancy in the Venn diagram numbers in Sample A arises from the incorporation of L [O III] into the control. \n<!-- image --> \nFig. 8. Comparison of stacked W80 profiles and host galaxy properties between non-AGN and Sample B. Same as Figure 7 but comparing non-AGN (black) to Sample B (blue). \n<!-- image --> \nthe results as radial profiles for this comparison 5 , finding that optically selected AGN have larger velocity widths than non-AGN of similar masses. We find similar results when selecting AGN through other techniques (see below). \nGatto et al. (2024) find that both, AGN and control galaxies, have L [O III] values that correlate positively with their average W 80. We see that their control sample has a median L [O III] that is \n∼ 1 dex lower than the ones from AGN. Therefore, in our analysis, we ask whether the kinematic di ff erences are still present between non-AGN and AGN if taking the L [O III] into account during the control (this removes the most luminous AGN). Under these conditions, we observe that AGN and controls are more alike in terms of their W 80 values (see the discussion in Section 5). \nWith this approach, we aim to assess the question: is the current nuclear activity responsible for the enhanced radial profiles in the AGN populations? And if so, to what extent are these \nFig. 9. Comparison of stacked W80 profiles and host galaxy properties between di ff erent AGN samples. The plots in the top and bottom rows show how both samples behave after they are matched to have the same log( M ⋆ ) and log( L [O III]). The Venn diagrams shown in each W80 plot represent the AGN sample in the label. On top: blue for optically-selected (without broad-line-selected) and gray for the opposite. On the bottom: blue for Sample B and gray for Sample A. For each column of plots, from left to right, we show the W80 stacked radial profile, the log( M ⋆ ) distribution, the log( L [O III]) distribution, and the log(SFR(H α )) distribution. \n<!-- image --> \nperturbations spreading? We also seek to investigate how the kinematics may be dependent on the AGN selection technique. Therefore, we look at the W 80 radial profiles comparing nonAGN (see Figure 7 and Figure 8) with the following samples: \n- -Sample A: targets selected via optical or broad lines, excluding the ones selected via radio.\n- -Sample B: targets selected via radio but not via optical or broad lines. \nWe do not observe a significant di ff erence (in terms of ionized gas kinematics) between optically selected and broad-line selected AGN (see the details in Section 4.2). This is the main reason why we merge them when defining Sample A and Sample B for the purpose of comparing them with radio-selected AGN. \nWe present a comparison between non-AGN and AGN, using as a control sample non-AGN first matched only in stellar mass, morphology, and redshift, and, later, including L [O III] in the parameter match (as mentioned at the beginning of this section). Figure 7 shows two rows of plots. The top row shows the W80 radial profiles as well as histograms of host galaxy properties of Sample A and non-AGN without including L [O III] during the match. The bottom row shows the comparison considering the L [O III] during the match. Note that the latter matching procedure leads to the removal of the most extreme AGN in Sample A exhibiting the highest L [O III]. The shaded areas in the radial profiles show the 14th and 86th percentiles as in Figure 4, while the histograms report the M ⋆ and L [O III] (both parameters used during the match) and the SFR( H α ) (same SFR used in Figure 1), which is not used for the match. \nThe upper-left plot shows that Sample A has greater W80 values at all annuli than non-AGN when only the M ⋆ is considered during the match. We can see that the L [O III] is systematically lower for non-AGN. As mentioned above, Gatto et al. (2024) find that their non-AGN control sample has [O III] linewidths that correlate with [O III] luminosity. If we include the L [O III] \n(bottom left plot of Figure 7), the median W80 value at all annuli of Sample A and the non-AGN sample behaves similarly. While there might be a small di ff erence at small radii R e ff < 0 . 5, at large radii, there is no di ff erence between both samples. This suggests that the stacked W80 radial profiles of optically selected AGN together with broad-line selected AGN can be easily reproduced by non-AGN hosts with the same distribution of mass M ⋆ and L [O III]. \nIn contrast to Sample A, Sample B (see Figure 8) shows higher W80 values compared to non-AGN. Remarkably, the most significant di ff erence between non-AGN and sample B is seen at the largest annuli, where high W80 values are achieved by Sample B. Excluding optically (and broad-line) selected samples from our radio-selected AGN systematically removes AGN with low W80 at larger annuli. For Sample A, it appears that excluding radio-selected AGN from optically (or broad-line) selected samples consistently removes the W80 kinematic excess at larger annuli.', '4.2. AGN versus AGN': 'In Figure 4, we find that our broad-line selected AGN tend to have larger L [O III] than optically-selected AGN galaxies. Therefore, in the top panel of Figure 9, we match the two samples within 39 . 4 < log( L [O III]) < 42 . 1 and 10 . 4 < log( M ⋆ ) < 11 . 7 (limits within which the parameter distribution can be matched) and compare the two samples: optically-selected AGN excluding broad-line selected AGN and vice-versa. We find that the median W80 radial profile of broad-line-only AGN is similar to the optically-selected AGN (excluding broad-line AGN) when both samples are matched in host galaxy properties, with a small excess in the center for the broad-line selected AGN (see Figure 9). Gatto et al. (2024) arrive at similar conclusions for their broad-line and optically-selected AGN, reporting no di ff erence when comparing their W 80 distributions. The results remain un- \nanged if we also control for inclination ( b / a ), ruling out possible orientation e ff ects. Therefore, we combine these samples when setting up Sample A and Sample B (in Section 4.1). \nWe now proceed with comparing Sample A with Sample B directly. As before, we also match the samples in M ⋆ and L [O III], redshift and morphology. Note that, when excluding radio-selected sources from the optically-selected AGN catalog (Sample A), we do not claim that the remaining opticallyselected AGN have no AGN-related radio emission, but we rather aim to investigate kinematic properties of a sample that would not have been detected as AGN through radio techniques and vice versa. \nIn the bottom-left panel of Fig. 9, we present the W80 radial profiles of Sample A and Sample B. Sample A is forced to match Sample B within 39 . 2 < log( L [O III]) < 41 . 5 and 10 . 1 < log( M ⋆ ) < 11 . 7. Sample B (represented in blue) shows elevated W80 values across all annuli, notably at large R e ff , aligning with the findings in Section 4.1. This comparison suggests that while optical and broad-line selection methods can identify AGN hosts with perturbed kinematics extending to large galacto-centric distances, the absence (or exclusion) of radioselected AGN (as in Sample A, shown in gray) results in a population characterized by systematically reduced kinematic disturbances at these distances. Conversely, when optically and broadline selected AGN are removed from a radio-selected sample (as in Sample B), the remaining AGN hosts predominantly exhibit significant kinematic perturbations, especially at extended R e ff scales. This suggests that AGN radio-selection techniques are sensitive to finding AGN hosts with disturbed kinematics over larger galacto-centric distances. \nA key takeaway message is that the selection technique is sensitive to the kinematics found in AGN galaxies and might also be sensitive to the evolutionary stage of AGN (see the Discussion section). We point out that employing alternative cuto ff lines in our radio selection technique (e.g. 1.5 dex; see Section 2.2.4 and Figure 1) produces similar outcomes. As illustrated in Figure 4, a larger cuto ff would also result in a cut on stellar mass. However, this also would substantially reduce the number of targets. Our results concerning radio AGN are very similar if we consider a di ff erent SFR estimator (as mentioned in Section 2.2.4) when selecting our radio sample.', '4.3. SF galaxies vs AGN': 'We perform a similar comparison between AGN and (see Section 2.2.1) SF galaxies (as classified by BPT diagnostics). We find that SF galaxies have a lower median W80 radial profile compared to any of the AGN-selected samples (AGN were excluded from the SF galaxy sample, although only 15 AGN overlap with it). Furthermore, SF galaxies have indeed higher SFRs than our selected AGN and even higher SFRs when controlling for M ⋆ and L [O III] to a specific AGN selected population (they also have younger D4000 ages, and higher H α equivalent widths). This suggests that SF galaxies (at least, BPTclassified) in MaNGA do not seem responsible for driving significant ionized gas outflow signatures, even when having significantly higher SFRs. \nLastly, in Section 3.2, we described that we only use galaxies with at least two available annuli where at least 10% of their spaxels have S / N > 7. This quality criteria results in some galaxies having no W80 values in some of their annuli. To control for a possible impact of this e ff ect, we study the same comparisons described in this Section with two samples: one where we use all the galaxies and all their spaxels that have a S / N > 3 (low \nS / N), and the other one where we only use galaxies that have at least six available annuli where at least 10% of their spaxels have S / N > 7 (a high S / N constraint). Using the latter samples, we confirm that the behavior described in this Section is still present for all the comparisons.', '5.1. AGN selection and their integrated host galaxy properties': "In this paper, we find that di ff erent AGN selection techniques select AGN samples that hardly overlap in more than 50% of their targets. Similar results have been found in Oh et al. (2022) at z < 0 . 2 when comparing X-ray selected to optically-selected AGN. Additionally, for higher redshifts (0 . 25 < z < 0 . 8), in a sample of mid-infrared, radio, and X-ray selected AGN, Hickox et al. (2009) find AGN candidates hardly overlapping (their radio selection is at L1 . 4 GHz > 10 23 . 8 W Hz -1 ). These findings impose a clear challenge in AGN studies, since AGN found by the di ff erent selection techniques do not always trace the same host galaxy properties and / or AGN accretion state (Hickox & Alexander 2018). \nWe find that our radio-selected AGN are typically found below the main sequence of SF galaxies (SFMS). Similar results have been found specifically for MaNGA (Comerford et al. 2020; Mulcahey et al. 2022), and other low redshift studies (e.g. Smolˇci'c 2009). Accordingly, Sánchez et al. (2022) find that optically-selected AGN lie below the SFMS in the Green Valley. Additionally, Schawinski et al. (2007) find that SF galaxies, composite and AGN (all optically selected) seem to follow an evolutionary sequence in the star formation and stellar mass plane, with SF galaxies having bluer colors and AGN found more in the transition zone. Our composite-selected targets are also found between SF and AGN-selected galaxies in the M ⋆ versus SFR plane. Hickox et al. (2009) find similar results, also in agreement with our mid-infrared selected targets, which they find to be more likely found in slightly more star-forming hosts. They propose an interpretation suggesting an evolutionary picture where, as star formation decreases, AGN accretion changes from optical or infrared-bright to optically faint radio sources. These findings suggest that AGN selection techniques are sensitive not only to the physical processes powering them but also to the stage of their duty cycle. We discuss this further in Section 5.3.", '5.2. Spatially resolved ionized gas kinematics': 'We first investigated the radial properties of the [OIII] ionized gas kinematics of unmatched AGN and non-AGN samples, showcasing a diverse range of ionized gas kinematics (this was done before controlling for host galaxy properties; see Figure 5). Non-AGN and SF galaxies exhibit less disturbed kinematics compared to all AGN samples (lower W80 radial profiles). \nWhen comparing AGN samples matched in M ⋆ and L[O III], intrinsic distinct kinematic behaviors emerge. Specifically, the exclusion of radio-selected AGN from an optical and broad-line selected AGN sample (Sample A) results in lower W80 values at greater galacto-centric distances, suggesting that much of the kinematic disturbances within an optically-selected sample are linked to the radio emission in AGN (see more discussion on the connection of outflows and radio emission in the next Section). \nThe analysis of Sample A also reveals that there is a population of non-AGN galaxies that can easily produce AGN-like W80 \nprofiles when controlling for host galaxy properties (see bottomleft panel of Figure 7). Simulations suggest that kiloparsec-scale AGN-driven outflows can outlast the AGN activity phase, extending from a few to several orders of magnitude longer in duration (a few Myr, King et al. 2011; Zubovas 2018). For example, Zubovas et al. (2022) predicts that fossil outflows (outflows living after the AGN switches o ff ) could actually be more common than finding an outflow and an AGN in a galaxy simultaneously. Consequently, MaNGA non-AGN galaxies may include some galaxies showing fossil outflows. The possible presence of fossil outflows in MaNGA galaxies will be discussed in a future paper.', '5.3. Radio-selected AGN as tracers of the final phases of AGN evolution': "We find that AGN identified through radio techniques alone (Sample B) show notably stronger kinematics at larger R e f f than any other AGN sample. The presence of AGN-related radio emission in AGN may, therefore, seem to trace AGN with more spatially extended outflows. \nOne explanation for this behavior may be that sources with AGN-related radio emission trace host galaxies that have been experiencing AGN activity (or activities; see below) for a longer time. However, kinematic perturbances up to kpc scales would typically imply an active (AGN) phase longer than the duration of a typical AGN duty cycle. For example, King et al. (2011) uses analytical models to study the outflow propagation during an AGN event. They show that an outflow with an initial velocity of a couple of hundred km s -1 in an AGN episode lasting about ∼ 1 Myr can last up to ten times more than the AGN itself, reaching several kpc. \nAlternatively, radio-selected AGN may be sensitive to AGN that have gone through multiple cycles of AGN activity. Indeed, some galaxies show evidence of past and recurring AGN events (e.g., Schawinski et al. 2015; Shulevski et al. 2015; Rao et al. 2023). Recent studies have used low-frequency (MHz) radio spectra combined with high-frequency spectra (GHz) to trace back emissions from previous activities (e.g., Jurlin et al. 2020). A younger AGN phase is characterized by a peaked spectrum in the center, while a remnant from past events displays a more spread-di ff use emission. Therefore, if a combination of the latter is observed in one target, it can suggest the target is a strong candidate for a restarted AGN phase. In this context, Kukreti et al. (2023) find that around 6% of the targets in their sample (at 10 23 WHz -1 > L1 . 4 GHz > 10 26 WHz -1 , 0 . 02 < z < 0 . 23) are peaked sources classified as compact in GHz frequencies but have extended emission at MHz frequencies, suggesting them as restarted AGN candidates. \nAlso, the simulations from Zubovas & Maskeli¯unas (2023) show that fossil outflows in gas-poor systems tend to last longer than in gas-rich hosts. Radio-selected AGN, indeed, are preferentially found in gas-poor galaxies. \nWith respect to an AGN's impact on its host galaxy, a recent review by Harrison et al. (2023) discusses that simulations predict that feedback that leads to galaxy quenching does not come from a single AGN event but is rather a cumulative e ff ect of multiple AGN episodes (see also Piotrowska et al. 2022). Thus, given the findings in our analysis, AGN selected through radio observations may preferentially trace galaxies that have experienced episodic AGN events (Morganti 2017a). Sample B indeed contains host galaxies with older stellar populations (compared to Sample A), as traced by average D4000 measurements from the PiPE3D catalog. \nIn a sample of radio-selected AGN from MaNGA MPL-8, Comerford et al. (2020) find that radio-mode AGN host galaxies reside preferentially in elliptical galaxies have more negative stellar age gradients with galacto-centric distance. The authors suggest that radio-mode AGNs may represent a final phase in the evolution of AGN. In addition, Hickox et al. (2009) propose a scenario where radio-AGN are key to the late stages of galaxy evolution, with them being, in general, more passive and low Eddington ratios than their infrared and optical counterparts. In fact, radio-selected AGN typically have larger black hole masses (Best et al. 2005; Hickox et al. 2009). Interestingly, the latter parameter is found to be a strong predictor for galaxy quenching (Piotrowska et al. 2022). These results are in line with the here presented work on the spatially resolved ionized gas kinematics in radio-selected AGN, which suggest that radio selection methods may be used to identify AGNs at a more advanced stage of their activity (and feedback) cycle. Lastly, we note that most of the removed radio-selected AGN (see Section 3.2 and Section 3.3) are massive galaxies with low SFRs (see Figure 1), located near the red sequence, suggesting an even later evolutionary phase.", '5.4. The connection between radio-emission and outflow activity in AGN': 'Our results discussed above raise the question of what mechanisms are responsible for the observed radio emissions. The possible origins of radio emission in low-luminosity radio AGN is reviewed in Panessa et al. (2019). The review discusses several mechanisms such as by jets, winds, accretion disk corona, and star formation. For the context of our work, winds are discussed as a mechanism in which a shock is driven by the wind (e.g., Ri ff el et al. 2021) and produce radio emission due to the acceleration of relativistic electrons on sub-kpc scales. Similarly, in a small sample of AGN ( z < 0 . 07), Mizumoto et al. (2024) found that NLR-scale shocks (traced by [Fe II] / [P II]; see Oliva et al. 2001) are likely triggered by ionized outflows (traced by [S III] in Mizumoto et al.). \nNotably, in a sample of galaxies at z < 0 . 8, Zakamska & Greene (2014) show that the radio luminosity in formally radioquiet AGN correlates with the [O III] velocity width, consistent with our findings. Zakamska & Greene (2014) propose two scenarios: one where radio emission is produced by accelerated particles as a result of shock fronts due to outflows (extended and di ff use radio emission), and another one where an unresolved radio jet (unresolved in FIRST / NVSS data) is launching an outflow (expected to be compact). We argue that both scenarios could simultaneously be present in one system (e.g., if the galaxy had more than one recent AGN event). High spatial resolution radio observations would be needed to distinguish between them. \nCalistro Rivera et al. (2023) arrives at a similar conclusions by analyzing the CIV and [O III] velocities (in a sample of ∼ 100 AGN). They discover minimal or no correlation between the CIV velocities and radio luminosity, in contrast to a connection between the [OIII] velocity width and radio luminosity. Given that CIV emission originate from within the broad-line region (subpc scales), and [O III] emission traces ionized gas on galactic / kpc scales, Calistro Rivera et al. (2023) conclude that the interplay between winds and radio luminosity predominantly occurs on these circumnuclear scales. Similarly, Liao et al. (2024) not only shows that [O III] velocity widths of AGN (in their sample: z < 1 . 0, and a median log(L[O III]) ∼ 42 . 1) correlate with radio emission but also that the conversion e ffi ciencies align with those needed to account for the observed radio luminosities in \ngalaxies exhibiting large [O III] velocity widths. Their results also support the idea that AGN-driven outflows contribute to the radio emission in AGN. \nWhile the results discussed above suggest a connection between the radio emission and the ionized gas kinematics in AGN, we note that they were done predominantly using single fiber spectra and investigating higher redshift galaxies ( z ≲ 0 . 8), averaging the gas kinematics over larger areas. Our work adds to the picture using a spatially resolved kinematic analysis and while we cannot exclude the presence of jetted radio-AGN in our radio-selected sample, our results also suggest that there is a strong connection between radio activity and ionized gas outflows in AGN.', '5.5. SF galaxies': 'We find that SF galaxies show less enhanced kinematic profiles when compared to AGN candidates, even when controlling for M ⋆ , L[O III], morphology and redshift. A detailed comparison between Sample A and SF galaxies highlights that Sample A demonstrates significantly higher W80 values within its central regions. In contrast, such di ff erences fade at larger e ff ective radii (R e f f ), where the kinematic behaviors of both populations (Sample A and the matched star-formation sample) align closely. Moreover, the matched SF galaxy sample exhibits higher star formation rates than Sample A (and higher than the SF sample before matching). In a larger sample ( > 50000) of local (0 . 05 < z < 0 . 1) SF galaxies, Yu et al. (2022) studied the ionized gas kinematics of these galaxies, finding that they can indeed present outflow signatures. But the authors also show that the star-forming sample hardly ever reaches σ > 150 km s -1 , i.e., W80 > 375 km s -1 . This is consistent with our findings. Therefore, we infer that the enhanced W80 values in our AGNselected population are likely driven by AGN and do not expect star-forming processes to play a significant role. \nHowever, the most massive ( M ⋆ > 10 11 M ⊙ ) SF galaxies in our sample reveal remarkably high W80 values (although not as high as AGN). Sabater et al. (2019) found that 100% of the galaxies with masses above this limit (10 11 M ⊙ ) host radio AGN even though sometimes with radio luminosities (L150 MHz > 10 21 . 5 WHz -1 , or L1 . 4 GHz ≳ 10 21 WHz -1 ; most of our galaxies are above this limit). Indeed, > 50% of massive SF galaxies in our sample have radio detections, while only ∼ 10% of lower mass SF galaxies ( < 10 11 M ⊙ ) have radio detections. This suggests that some of our massive SF galaxy populations may be AGN as well.', '6. Summary and conclusions': 'We have assembled a multi-wavelength AGN-selected sample for the SDSS-IV MaNGA-DR17, comprising 594 unique AGN identified through optical, hard X-ray, radio, infrared, and broadline selection techniques. We seek to explore the extent to which ionized gas kinematics, as quantified by W80 of [O III] λ 5007, is influenced by the diversity in AGN selection methods, thereby o ff ering insights into feedback processes and the duty cycle of AGN activity. To do so, we fit up to two Gaussian components to the [O III] λ 5007 emission line region in all spaxels ( S / N > 7; see Section 3.2) of each galaxy and derive the W80 velocity widths (see Section 4). We then map the spatial distribution of this parameter for each galaxy. Furthermore, we create W80 radial profiles and stack them according to each defined AGN subsample. Our findings are summarized as follows: \n- -We find that di ff erent AGN selection techniques do not completely overlap with each other. Overlap ranges from ∼ 34% (e.g., between radio and optical selection) up to ∼ 80% (the latter percentage only achieved by the X-ray selection, although it is the smallest sample).\n- -The di ff erent AGN populations are found in galaxies with di ff erent host galaxy properties. The most significant differences are found in the distribution of L[O III], EW( H α ), D4000, and the W 80 radial profiles.\n- -Regarding AGN vs. non-AGN: Regardless of the selection technique, all AGN populations show more perturbed ionized gas kinematics (traced by W 80) at all annuli when compared to non-AGN of similar M ⋆ of the host, redshift, and morphology. These kinematic di ff erences become less pronounced when L[O III] is taken into account in the non-AGN control sample. Remarkably, the di ff erences between AGN and non-AGN disappear when we compare pure optical (BPT and broad-line, but exclude radio-detect AGN: Sample A) AGN to non-AGN (see Section 4.1). We suggest that some non-AGN may host fossil outflows (i.e., relic outflows of a past AGN phase), which may outnumber outflows in currently active AGN (Zubovas et al. 2022).\n- -Regarding AGN vs AGN: Our di ff erent AGN samples display not only hosts with di ff erent properties but also hosts with di ff erences in the stacked radial profiles of their kinematic signatures. Interestingly, when controlling for host galaxy properties, we find that removing radio-selectedAGN from optically selected candidates leaves a sample (Sample A) of galaxies that lack significantly high W 80 at high R e f f , suggesting that much of the kinematic disturbances within an optically-selected sample are linked to the radio emission in AGN. In addition, radio-selected AGN show more enhanced ionized gas kinematics at all radii and their hosts show evidence of older stellar populations. Our results support a scenario in which radio selection methods may be used to identify AGNs at a more advanced stage of their activity (and feedback) cycle.\n- -AGNvs star-forming: SF galaxies in our sample do not show significant kinematic signatures in the ionized gas compared to AGN (regardless of the selection technique; see Section 2). We highlight that when controlling for L[O III] and M ⋆ when comparing AGN to non-AGN, SF galaxies tend to have significantly larger SFR(H α )) than AGN. We conclude that in our sample, the main driver of the enhanced kinematic signatures in AGN cannot be accounted for by star formation processes alone. \nOur study shows that a given AGN selection technique can impact what sort of ionized kinematic signatures are found in their host galaxies. Our results are tested in low-redshift ( z < 0 . 1) galaxies with low- to intermediate luminosities. The impact of AGN selection techniques could be more significant at higher redshift. Moreover, our results highlight the importance and utility of spatially resolved spectroscopy. \nAcknowledgements. D.W. acknowledges support through an Emmy Noether Grant of the German Research Foundation, a stipend by the Daimler and Benz Foundation and a Verbundforschung grant by the German Space Agency. M.A. extends gratitude to the GALENA research group for their invaluable discussions, which have significantly shaped the ideas presented in this paper. J.M.C. is supported by NSF AST-1714503 and NSF AST- 1847938. RAR acknowledges the support from Conselho Nacional de Desenvolvi20232023-7), Fundação de Amparo à pesquisa do Estado do Rio 2551-0002018-0), and CAPES (Proj. \nmento Científico e Tecnológico (CNPq; Proj. 303450 / 2022-3, 403398 / 1, & 441722 / Grande do Sul (FAPERGS; Proj. 21 / 88887.894973 / 2023-00). \nThis project makes use of the MaNGA-Pipe3D dataproducts. We thank the IAUNAM MaNGA team for creating this catalogue, and the Conacyt Project CB285080 for supporting them. \nFunding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy O ffi ce of Science, and the Participating Institutions. SDSS-IV acknowledges support and resources from the Center for HighPerformance Computing at the University of Utah. The SDSS web site is www.sdss.org. \nSDSS-IV is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, the Chilean Participation Group, the French Participation Group, HarvardSmithsonian Center for Astrophysics, Instituto de Astrofísica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU) / University of Tokyo, the Korean Participation Group, Lawrence Berkeley National Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg), MaxPlanck-Institut für Astrophysik (MPA Garching), Max-Planck-Institut für Extraterrestrische Physik (MPE), National Astronomical Observatories of China, New Mexico State University, New York University, University of Notre Dame, Observatario Nacional / MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Autonoma de México, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University', 'References': "Aalto, S., Garcia-Burillo, S., Muller, S., et al. 2012, A&A, 537, A44 \nAird, J., Coil, A. 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D., Laor, A., et al. 2019, Nature Astronomy, 3, 387 \nPennell, A., Runnoe, J. C., & Brotherton, M. S. 2017, MNRAS, 468, 1433 \n- Peterson, B. M. 2006, in Physics of Active Galactic Nuclei at all Scales, ed. D. Alloin, Vol. 693, 77\n- Petrosian, V. 1976, ApJ, 210, L53\n- Piotrowska, J. M., Bluck, A. F. L., Maiolino, R., & Peng, Y. 2022, MNRAS, 512, 1052\n- Powell, M. C., Cappelluti, N., Urry, C. M., et al. 2018, The Astrophysical Journal, 858, 110 \nRao, V. V., Kharb, P., Rubinur, K., et al. 2023, MNRAS, 524, 1615", 'Appendix A: Targets in our sample': "Due to the amount of data, and catalogs available for MaNGA, this Section details our sample. We intend to remove duplicate observations to be specific about which targets we use since we require specific host-galaxy properties from the galaxies. \nTable A.1. Plate-IFU pairs of repeated observations. \nWe start defining the sample with all the targets present on Sánchez et al. (2022) value-added catalog, which starts with a sample of 10 220 galaxies. We use the plate-ifu as the main identifier of our targets, given that MaNGA has a number of repeated observations (some with the same MaNGAID). We follow MaNGA's steps to mask the sample for unique galaxies 6 , reducing the sample to 9 995 targets. The sample gets reduced to 9 992 galaxies because three of them had no data stored on the public website of the data reduction pipeline: 11939-1901, 11949-1901, and 8626-9102 (also reported in a list of targets that failed to be analyzed by the DAP 7 . \nIn Sánchez et al. (2022), a table showing duplicate observations is reported, and similarly, a table of duplicates is also reported on the latter website (warning about duplicate galaxies with di ff erent MaNGA IDs). We note that some targets present in the MaNGA's duplicate table are not present in Sanchez's table. Therefore, we merge both repeated target tables and select from each pair the plate-ifu of which had more available annuli with higher S / N when measuring their W80. This removes twenty more galaxies, leaving our sample with 9 972. We further double-checked for duplicate observations matching targets by MaNGA-ID and ensuring that the coordinates were consistent with each other and found more repetitions in the sample. We show these duplicate observations in Table A.1, while some observations are repeated more than two times, as shown in Table A.2 (most of the targets in both tables come from cluster ancillary programs discussed in the drpall website mentioned above). As before, we remove these, keeping the one that o ff ers a better quality of W80. With the latter, we end up with 9 853 targets. Finally, we remove targets flagged by the MANGA\\_DRP3QUAL as CRITICAL by the DRP. This leaves us with a final sample of 9777 galaxies. \nTable A.2. Repeated observations with more than two elements. \n<!-- image --> \nNotes. the last row is a target repeated five times. \nThe quality criteria used in our analysis remove a number of extra galaxies from the study (see Section 3.2). We do not analyze separately additional galaxies (e.g., if more than one galaxy was found in a specific plate-ifu, Pan et al. 2019) found in the same IFU, and do not include any special treatment where this happens.", 'Appendix B: Fitting procedure details': "Our pipeline starts by subtracting the stellar continuum (provided by the DAP Westfall et al. 2019) from all spectra and moving each to its rest frame. We focus on the 4920-5080 Å region and subtract an additional continuum component from a 1D polynomial using two spectral windows (the first between 4870-4900 Å and the second between 5040-5100 Å). We execute the fitting two times: the first using a single Gaussian for each emission line and the second allowing two Gaussians for each emission line to account for possible asymmetries in the line profiles. \nBelow, we list the constraints used during the fitting procedure. The model with just one Gaussian profile has three free parameters to be fitted: amplitude, width, and systemic velocity, denoted by A , σ , and µ , respectively. The details are given below: \n- -The [O III] 4959,5007 Å doublet is fixed to the theoretical flux ratio of 2.98 ( λ 5007 / λ 4959; Storey & Zeippen 2000; Laker et al. 2022).\n- -The velocity dispersion ( σ ) and systemic velocity ( µ ) of both [O III] 4959 Å and 5007 Å are tied to the same value, which will be a free parameter on the fitting procedure.\n- -We limit velocity values to 0 < σ < 1000 km s -1 , and -1000 < µ < 1000 km s -1 \nIn the two-Gaussian model, the fitting procedure has six free parameters. The first Gaussian component with A , σ , and µ , and similarly, the second with Aw , σ w , and µ w (for the amplitude, width and o ff set of the 'wing' component). We adopt the same considerations as listed above, and we add the following to the second Gaussian component: \n- -The amplitude Aw is a fraction of the main Gaussian amplitude ( A ), constrained between 0 and 1.\n- -The velocity dispersion σ w is forced to be higher and up to 1500 km s -1 to avoid fitting noise.\n- -The systemic velocity µ w can be blue or redshifted up to 1000 km s -1 from the main Gaussian's o ff set µ . \nA visual inspection of many of our results motivated us to add an extra condition to prevent the second component from fitting noise. To do this, we impose an additional condition to decide whether to use one or two Gaussians for the emission line. The second Gaussian component (after fitted) should have at least a S / N [ O III ] > 3. If this S / N requirement is not satisfied, the emission line is kept fitted with only one Gaussian. \nWe store each fitted parameter in maps (for the single- and double-Gaussian fitting procedures), including the reduced-chisquare provided by LMFIT. From the resulting maps, we construct the L[ O III ] map (the sum of both components' fluxes in the case of the double-Gaussian model) and a non-parametric emission-line width map. To capture the emission-line width of a complex profile (e.g., a mixture of two Gaussian profiles) and reduce being influenced by the criteria of our fitting procedure, non-parametric measurements are routinely adopted (e.g., Zakamska & Greene 2014; Wylezalek et al. 2020). Specifically, we use the width that encloses the 80% of the total flux, known as the non-parametric W80 parameter (see the details in Liu et al. 2013). This parameter aims to prevent discarding information from the additional components of a profile composed of multiple components. \nFinally, the decision to keep one fitting procedure from both models is based on the best-reduced chi-square (the one closer to \nFig. B.1. Output for MaNGA plate-IFU: 8244-3702. Final W80 map (bottom plot) combined from the W80 map of each model (top plots) based on the best χ 2 red mask (middle plot) and an additional S / N cut on the second Gaussian component (see Section 3.1). All these Figure's W80 maps have the same contrast colored following the same colorbar (on the middle-left) \n<!-- image --> \n. \n1; Andrae et al. 2010). With the latter, we construct best-model mask maps (see Figure B.1), which are used to combine the results from the two fitting techniques into one containing the results of the models that fitted the spectral region the best (in the Figure, we show this for the [O III] λ 5007 W80). The same bestmodel map creates the combined L[ O III ] map for each galaxy. From these two maps, we extract the following parameters: \n- -W80 radial profiles for each map: average W80 at elliptical ring apertures with a step of 0.25 R e f f from the center of each target (see below).\n- -L[ O III ] averaged at a radius of 0.5 R e f f .", 'Appendix C: Binning parameters': "We show how the ionized gas kinematics from the [O III] emission line (traced by the central W80 averaged over an aperture of 0.5 R e f f ) changes (for di ff erent galaxy populations) when observed in di ff erent parameter spaces of host galaxy properties. This is shown in Figure A1.C.2 and Figure A1.C.1. When looking at SF galaxies, in most cases, there is no significant evolution in their W80. Conversely, in the case of AGN-selected galaxies, stronger W80 values are found as we move to a specific direction of the parameter spaces. This is visually represented in Figure A1.C.1, where minimal gradients are observed for SF galaxies, whereas AGN-selected galaxies show not only larger gradients but also a distinct trend towards increased stellar mass ( M ⋆ ) and L[O III] luminosity. \nFig. C.1. Average W80 binned on a plane of M ⋆ vs. L[O III]. The bins have a size of 0.3 dex in each parameter, colored by the strength of the W80. The scatter dots show the distribution of a specific galaxy population and the line shows a 1D polynomial fitted to the location of the SF galaxies. The green arrows in the plots illustrate the gradient change of W80 in the parameter space, with the arrowhead indicating the direction and the arrow's size representing the magnitude. \n<!-- image --> \n3.2 \nFig. C.2. Average W80 binned on a plane of star formation rate measured from L rad and L H α . The bins have a size of 0.3 dex in each parameter, colored by the strength of the W80. The scatter dots show the distribution of a specific galaxy population and the line shows the 1-to-1 relation if both SFR tracers are equal. \n<!-- image -->"} |
2024ApJ...973L..31R | Several recurrent Xclass flares from Active Region AR 13664 triggered a severe G5class geomagnetic storm between 2024 May 10 and 11. The morphology and compactness of this AR closely resemble the AR responsible for the famous Carrington Event of 1859. Although the induced geomagnetic currents produced a value of the Dst index probably 1 order of magnitude weaker than that of the Carrington Event the characteristics of AR 13664 warrant special attention. Understanding the mechanisms of magnetic field emergence and transformation in the solar atmosphere that lead to the formation of such an extensive compact and complex AR is crucial. Our analysis of the emerging flux and horizontal motions of the magnetic structures observed in the photosphere reveals the fundamental role of a sequence of emerging bipoles at the same latitude and longitude followed by converging and shear motions. This temporal order of processes frequently invoked in magnetohydrodynamic modelsemergence converging motions and shear motionsis critical for the storage of magnetic energy preceding strong solar eruptions that under the right timing location and direction conditions can trigger severe space weather events on Earth. | 2024-09-01T00:00:00Z | ['2024arXiv240904408R', '10.48550/arXiv.2409.04408', '2024ApJ...973L..31R', '10.3847/2041-8213/ad77cb', 'arXiv:2409.04408'] | ['Sunspot groups', '1651', 'Astrophysics - Solar and Stellar Astrophysics', 'Physics - Space Physics'] | Analyzing the Sequence of Phases Leading to the Formation of the Active Region 13664 with Potential Carringtonlike Characteristics | 2,024 | 167 | 0.53 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF'] | 2 | https://arxiv.org/pdf/2409.04408.pdf | {'Analyzing the Sequence of Phases Leading to the Formation of the Active Region 13664, with Potential Carrington-like Characteristics': 'P. Romano 1 , A. Elmhamdi 2 , A. Marassi 3 and L. Contarino 1 \[email protected] \nReceived \n; \naccepted', 'ABSTRACT': 'Several recurrent X-class flares from Active Region (AR) 13664 have triggered a severe G5-class geomagnetic storm between May 10 and 11, 2024. The morphology and compactness of this AR closely resemble the active region responsible for the famous Carrington Event of 1859. Although the induced geomagnetic currents produced a value of the Dst index, probably, an order of magnitude weaker than that of the Carrington Event, the characteristics of AR 13664 warrant special attention. Understanding the mechanisms of magnetic field emergence and transformation in the solar atmosphere that lead to the formation of such an extensive, compact and complex AR is crucial. Our analysis of the emerging flux and horizontal motions of the magnetic structures observed in the photosphere reveals the fundamental role of a sequence of emerging bipoles at the same latitude and longitude, followed by converging and shear motions. This temporal order of processes frequently invoked in magnetohydrodynamic models - emergence, converging motions, and shear motions - is critical for the storage of magnetic energy preceding strong solar eruptions that, under the right timing, location and direction conditions, can trigger severe space weather events at Earth. \nSubject headings: Sun: photosphere - Sun: ARs photospheric dynamics - Sun: flares - Sun: geomagnetic effects', '1. Introduction': "The solar atmosphere is an inherently dynamic system where magnetic fields play a crucial role in the manifestation of various solar phenomena. Among these, active regions (ARs) are particularly notable for their intense magnetic activity, which can lead to solar flares and coronal mass ejections (CMEs). These events are significant not only for their impact on space weather but also for their potential effects on Earth's magnetosphere. \nARs usually appear in narrow latitudinal belts between +35 ° and -35 ° , approaching the equator as the 11-year solar cycle advances. Preferred longitudes of sunspot formation, known as active longitudes, have been identified (e.g., Bogart 1982; Balthasar & Schuessler 1983), although there are inconsistencies regarding their number, lifetime, location, and rotation rate (Berdyugina & Usoskin 2003). Persistent active longitudes separated by about 180 ° have also been detected on different types of cool active stars (e.g., Rodon'o et al. 2000; Berdyugina et al. 2002; Korhonen et al. 2002). \nThe most powerful solar eruptions typically occur in ARs characterized by significant and unusual emerging magnetic flux, as observed, for instance, in AR 12673, which produced the most intense flare of Solar Cycle 24. In that case, the flux emergence rate reached an extraordinary peak value of approximately 1.12 × 10 21 Mx hr -1 . However, it is also important to consider that strong horizontal displacement of sunspots may play a crucial role in storing magnetic free energy (e.g., Romano et al. 2018). Romano et al. (2019) specifically demonstrated that shear motions observed along the polarity inversion line (PIL) of the main sunspots, when combined with intense magnetic fields exceeding 4000 G, can serve as a reference for studying not only the most powerful solar flares but also flares on other stars characterized by higher orders of magnitude. \nOne of the most famous events associated with a solar AR is the Carrington Event of March 1859, with a peak Dst-index reaching extraordinary value of ∼ -589 nT, the most \npowerful geomagnetic storm on record (see Cliver & Dietrich 2013; Saiz et al. 2016; Boteler 2019 and references therein). This AR was characterized by strong magnetic field concentrations and a particular compactness (Carrington 1859). Understanding the conditions and processes leading to such peculiar morphology and extreme events is essential for improving space weather prediction and mitigating its effects. \nRecently, AR 13664, during its passage over the solar disc from May 2 to May 14, 2024, showed impressive activity with many recurrent high-intensity flares. Notably, two X-class flares occurred on May 9 and 10, each associated with a corresponding CME. These events triggered an extreme G5-level storm (K p =9) on May 10-11, with geomagnetic indices, commonly used to assess the severity of a geomagnetic storm at Earth, attaining impressive values: solar wind speed exceeding ∼ 750 km s -1 up to ∼ 1000 km s -1 the following days, the north-south magnetic field component B z almost going down to -50 nT, and Dst-index about -412 nT around 03:00UT on 11 th of May. AR 13664 presents a unique opportunity for studying the emergence and evolution of prolific ARs. Initially appearing at the East limb as a relatively ordinary AR - comprising fewer than 10 sunspots, each typically between 5,000 and 20,000 km in diameter, with a total area of less than 300 millionths of the solar hemisphere, a magnetic field strength between 1,000 and 3,000 G, and a relatively simple magnetic configuration, typically α or β , with few polarity inversions - it rapidly evolved within 3-4 days. The AR developed characteristics reminiscent of the one responsible for the Carrington Event, growing to over 100 sunspots often grouped in complex formations, with the largest spots exceeding 30,000 km in diameter, a total area approaching 1,000 millionths of the solar hemisphere ( µ Hem), magnetic field strengths surpassing 4,000 G, and a very complex βγδ configuration with numerous close polarities. \nThis work focuses on the unusual magnetic flux emergence and peculiar horizontal motions during its development. These motions are investigated to understand their role in the \nrapid intensification and complexity of the magnetic field within the region. By analyzing high-resolution solar observation data, we aim to provide new insights into the mechanisms behind the formation of compact active regions and their potential for extreme solar activity. \nIn the next section, we describe the dataset used and the evolution of AR 13664 during its passage across the solar disc before the onset of the flares responsible for the G5 geomagnetic storm on May 10-11, 2024. The method of analysis and the results are described in Section 3, while the main conclusions are highlighted in Section 4.", '2. Evolution of AR 13364': "For this study, we utilized high-resolution data from Helioseismic and Magnetic Imager (HMI; Scherrer et al. 2012) onboard the Solar Dynamics Observatory spacecraft (SDO; Pesnell et al. 2012). The HMI provides comprehensive observations of the solar photosphere, enabling detailed analysis of magnetic field configurations and motions within active regions. \nWe specifically used Space-weather HMI Active Region Patches (SHARPs) data products, based on vector magnetograms data series with a cadence of 12 minutes and a spatial resolution of 1 arcsecond per pixel (Bobra et al. 2014). These data were collected continuously from May 2, 2024, at 00:00 UT to May 10, 2024, at 23:48 UT, with an interruption on May 8 from 16:36 UT to 23:59 UT. These 1012 magnetograms allowed us to capture the entire passage of the AR, including the formation phase, the buildup of magnetic complexity, and the eruption phases associated with the X-class flares. \nTo gain a more comprehensive understanding of the photospheric features associated with AR 13664, we complemented the magnetogram data with continuum intensity images from HMI. These images provided valuable context for the magnetic field observations, aiding \nin the identification of sunspot structures and other photospheric phenomena associated with the active region's evolution. \nThe global evolution of AR 13664 was marked by significant changes as it traversed the solar disk, culminating in a series of powerful X-class flares that triggered the geomagnetic storm on May 10, 2024. AR 13664 first appeared on the eastern limb of the Sun on May 1, 2024. Initially, the region was relatively small and magnetically simple (a β -class until late third of May, panel (a) of Fig. 1). However, as it progressed across the solar disk, it rapidly developed in size and complexity, evolving into a βγδ -class (see Hale et al. (1919) for a complete description of the complexity classes of ARs). \nBy May 5, 2024, the active region exhibited significant growth in both its magnetic field strength and area, with numerous sunspots forming and expanding (see panel (b) of Fig. 1). This period was characterized by intense magnetic flux emergence, where new magnetic fields bubbled up from beneath the solar surface (see panel (d) of Fig. 1). The region's complexity increased as these new fields interacted with the existing magnetic structures, leading to the formation of complex magnetic configurations with components significant proper motions and shearing episodes accompanied by the onset of a notable flux growth phase (see panels (c) and (d) of Fig. 1 and discussions in next paragraphs). Panel (d) of Fig. 1 illustrates the temporal evolution of the retrieved SHARP magnetic parameters 'Unsigned Flux' and 'Area'. Note that the area is calculated using the HMI line-of-sight magnetic field, while the unsigned flux is derived from the vector magnetic field components. Both parameters are based on the SHARP automated tracking system, which identifies magnetically strong-field structures within the AR (known as active pixels) after applying geometric corrections (refer to Bobra et al. 2014; Hoeksema et al. 2014; Bobra et al. 2015 for more details). \nThe rate of the total unsigned flux emergence increased dramatically just prior to the flares responsible for the severe geomagnetic disturbances. We quantified the growth rates \nFig. 1.- In the left panels a sequence of three HMI continuum images describing the main evolution of the AR 13664 before the X8.7 flare occurrence has been reported. The blue and red contours correspond to -500 G and +500 G, respectively. The panel (d) displays the temporal evolution of the computed total unsigned magnetic flux (green) and the area in µ Hem covered by the sunspots forming the AR (dark-yellow dashed line). The central meridian passage (CMP) is highlighted by vertical short-dashed line. The curves start at 18:00 UT on May 2, 2024. The three arrows indicate the times when the images reported in panels (a), (b) and (c) have been acquired. Best linear least-squares fits are drawn, depicting the significant growth phases (see text for details). \n<!-- image --> \nfor both area and flux by performing linear least-squares fitting over the period from May 7 to May 9. The resulting best-fit lines are shown in Fig. 1 (d). The flux exhibited a significant increase, transitioning from a slow growth phase (until mid May 6; 0.348 ± 0.001 × 10 22 Mx/day) to a rapid growth phase (starting around May 7th; 2.202 ± 0.012 × 10 22 Mx/day). A similar trend was observed for the area, although less pronounced compared to the flux, which increased from 353.47 ± 3.25 µ Hem/day to a higher rate of 670.44 ± 5.26 µ Hem/day. Interestingly, and in particular on May 9th AR 13664 reached the peak of the area covered by its sunspots, i.e., ∼ 3494 µ Hem, although the peak of the total unsigned magnetic flux ( ∼ 13.53 × 10 22 Mx) has been reached about two days later. Remarkably, this large magnetic flux rate in AR 13664 appears to be among the fastest flux emergence rates observed so far, exceeding even that of AR 12673 (August 2017) that hosted the most intense flare of SC24, and AR 12192 (October 2014) the largest AR in SC24 (refer to Figure 2 of Sun et al. 2024). \nOn May 9 AR 13664 showed pronounced shear motions indicative of significant magnetic stress building up within the region, usually a key precursor to solar flares. Indeed, the magnetogram analysis revealed strong horizontal flows and significant shearing along the PIL, where the positive and negative magnetic fields met and interacted. AR 13664 had become one of the most magnetically complex and active regions observed in the 25 th solar cycle. We argue that, consequently, the magnetic field was most likely highly sheared, creating a highly unstable configuration prone to eruptions. This set the stage for the series of X-class flares. The first of these flares occurred at 9:13 UT on May 9, rapidly followed by several more intense flares within the subsequent 24 hours. \nThe flaring activity culminated in a major X3.98-class flare on May 10, 2024, which was directly responsible for the ensuing G5-class geomagnetic storm. The rapid release of magnetic energy during these flares was probably facilitated by the highly sheared magnetic \nfields that have been observed over the preceding days and that allowed to reach the enormous value of the flare index (Li et al. 2004; Romano & Zuccarello 2007) of ∼ 5280 1 , quantifying the flare productivity of the active region during its solar disk presence: 39 C-class, 61 Mclass and 11 X-class flares. This sequence of events highlights the critical role of magnetic evolution and horizontal motions in driving extreme solar activity.", '3. Results': "The HMI observations were processed using standard SolarSoft routines to remove instrumental effects and correct for projection. We employed the Differential Affine Velocity Estimator for Vector Magnetograms (DAVE4VM; Schuck 2005, 2006) technique to analyze the horizontal motions of magnetic elements within the photosphere, using a window size of the apodizing window of 11 pixels (5 '' .5) which balances the need for capturing smallscale features while minimizing noise in the velocity estimations. The temporal cadence of the magnetograms was maintained at 12 minutes to ensure a high temporal resolution that captures the rapid evolution of magnetic structures. We used the standard configuration of DAVE4VM with default settings for the noise threshold and other algorithm-specific parameters, which have been optimized for HMI data in previous studies (e.g., Romano et al. (2014)). This method provides a robust estimation of the velocity field from the time series of vector magnetograms, allowing us to quantify the shearing and converging motions critical to understanding the magnetic energy buildup and release processes in AR 13664 . \nOur analysis identifies four distinct and key evolutionary stages in the AR's development: the early phase, the emergence phase, the compaction phase and the shear phase. Despite \n1 AR 13664 associated flare index is the highest for ARs of SCs 24 and 25 so far; AR 12673 the most active AR in SC24 had FI ∼ 3000. \n<!-- image --> \nFig. 2.- Horizontal velocity maps obtained by DAVE4VM using SHARP HMI vector magnetograms, describing the phase of the maximum rate of magnetic flux emergence, from May 5 at 10:00 UT to May 7 at 10:00 UT. We used the color green to indicate region where no data are available in the SHARP magnetograms. A complete animation covering this phase from May 5 at 10:00 UT to May 7 at 10:00 UT is also available online. In the animation, the emergence of two main bipoles is observed on the eastern side of the preexisting magnetic features, with their opposite polarities gradually moving apart from each other. \n<!-- image --> \n<!-- image --> \nFig. 3.- Same of Figure 2 but describing the compaction of the AR 13664, from May 7 at 10:00 UT to May 8 at 10:00 UT. We used the color green to indicate region where no data are available in the SHARP magnetograms. A complete animation covering this phase from May 7 at 10:00 UT to May 8 at 10:00 UT is also available. In the animation, we observe a change in the direction of the previously emerged structures, which no longer move apart but begin to converge and compact. \n<!-- image --> \nFig. 4.- Same of Figure 2 but describing the shearing motions, from May 8 at 10:00 UT to the end of the dataset. We used the color green to indicate region where no data are available in the SHARP magnetograms. The green box highlights the region where the shearing motions are located. A complete animation covering this phase from May 8 at 10:00 UT to May 10 at 24:00 UT is also available. In the animation, shearing motions can be observed along most of the main PIL of the AR, where the positive polarity predominantly moves toward the northeast, while the negative polarity moves toward the southwest. This pattern is consistently visible throughout the entire sequence. \n<!-- image --> \n<!-- image --> \nsome overlap between these stages, we were able to clearly distinguish their sequential onset. \nThe early phase. AR 13664 appeared at the East limb at a latitude of 18 ° S as a common bipolar region, oriented along the East-West direction with the preceding negative polarity in the direction of the solar rotation. This phase, corresponding to the first 3-4 days of the AR passage across the solar disc as shown in Fig. 1, does not show any significant evolution in terms of magnetic flux variations and area covered by the AR sunspots. Only the projection effects show some variation in the extent of AR 13664, while the magnetic flux remains just below 4 × 10 22 Mx, which is average for active regions at this phase of the solar cycle (Shanmugaraju et al. 2023). \nThe emergence phase. This phase started on May 5, when several magnetic bipoles began to emerge on the eastern side of the pre-existing sunspot system. The first new bipole appeared about 200 '' to the east of the pre-existing field and was oriented along the EastWest direction, with the negative polarity leading. In the horizontal velocity maps, this emergence is manifested by the progressive separation of the two polarities, with velocities of a few hundred meters per second oriented parallel to the equator (see the region at x=500 '' , y=-220 '' in the top panel of Fig. 2). On May 6, a second bipole emerges between the initial AR location and the previously emerged bipole (see the region at x=-50 '' , y=-260 '' in the bottom panel of Fig. 2). This bipole, also characterized by a leading negative polarity in the direction of rotation, is initially oriented along the Southeast-Northwest direction. This new bipole appears to emerge with greater intensity compared to the previous one. This is evident not only in terms of the new magnetic flux that appears on the surface between May 6 and 7, but also from the separation velocities of its polarities, which reach up to 0.5 km s -1 . We note that the peak emergence rate of 2.202 ± 0.012 × 10 22 Mx/day was reached on May 7, although this phase continued in the following days, overlapping with the two subsequent phases. Before the AR reached the western limb, where projection effects become \nsignificant, the total unsigned magnetic flux was approximately ∼ 13.53 × 10 22 Mx. \nThe compaction phase. During this phase the emerging fluxes expanded horizontally, contributing to the overall growth of the active region. The horizontal velocity magnitudes in the southern part of the AR range from 0.5 to 1.0 km s -1 (see the region at x=150 '' , y=330 '' in the top panel of Fig. 3), consistent with the vigorous flux emergence seen in highly ARs (Romano et al. 2018). In particular, we note that the combined process of new flux emergence and its westward propagation leads to the formation of a particularly compact active region. This region's morphological characteristics are reminiscent of the active region responsible for the Carrington Event. At the end of this phase, we observe a significant decrease in the separation velocity of the two polarities along the south-north direction. Additionally, a broad area characterized by a sea-serpent pattern of the magnetic field extends between them (see the bottom panel of Fig. 3). These variations in the orientation of horizontal motions, which shifted from separating opposite polarities (as typically occurs during the emergence of individual flux tubes) to converging different magnetic structures within the AR, resulted in a reduction in sunspot fragmentation and the formation of exceptionally large sunspots, with the largest spots exceeding 30,000 km in diameter. Additionally, these compaction processes resulted in a decrease in the number of individual sunspots within the AR, from around 81 to approximately 43, despite the continued emergence of new magnetic flux. \nThe shear phase. Based on the horizontal velocity maps, we observe a correlation between magnetic flux complexity and intricate velocity patterns (Fig. 4). Specifically, after the onset of the emergence and compaction phases, we identify strong shearing motions along the PIL, where the southern negative portion of the active region exhibits westward displacement, while the positive flux on the opposite side of the PIL moves eastward (see the green box around the region at x=400 '' , y=-300 '' in the top panel of Fig. 4). This \nshear persists for several hours, extending at least until the AR reaches the western limb, and affects the magnetic flux concentrations at the eastern edge of the AR (refer to the region at x=340 '' , y=250 '' in the bottom panel of Fig. 4). The peculiarity of these motions lies in their strength (up to 1.0 km s -1 ), duration (over 2 days), and extent (about 200 '' in latitude, i.e., from x=300 '' to x=550 '' in the bottom panel of Fig. 4). These shear motions, probably driven also by differential rotation and convective flows, seems to be crucial in building up magnetic stress within the region, before the trigger of the main flares occurred in the AR. Alongside shearing, converging flows were also observed near the PIL. Specifically, the velocities of the two polarities of the second emerged bipole transitioned from diverging to converging (compare the arrows direction of the central part of the AR in the bottom panel of Fig. 2 with the bottom panel of Fig. 4), thereby reducing the latitudinal extent of the AR in its central portion. These converging motions, reaching velocities up to 0.5 km s -1 , facilitated the accumulation of magnetic flux and likely intensified the magnetic field gradient. \nInterestingly, the time evolution maps of the average latitudinal (top panel of Fig. 5) and longitudinal (bottom panel of Fig. 5) components of the horizontal velocities along the AR clearly confirm and emphasize the four phases described earlier. Initially, until May 5, the average velocities of both components do not exhibit significant variations. Specifically, in the average latitudinal components, there is a prevalence of northward and southward velocities above and below y=-250 '' , respectively, which can be interpreted as the initial expansion of the AR in latitude. The emergence phase becomes evident in both maps starting from 12:00 UT on May 5. The emergence of the second bipole between the initial AR location and the previously emerged bipole is characterized by significantly higher velocities reaching up to 0.5 km s -1 , which influence the average velocities with a much more intense signal compared to the previous phase. We note the effects of the emergence of this bipole in the prevalence of northward and southward directed velocities above and below y=-260 '' , respectively (see the \n<!-- image --> \nFig. 5.- Average of the latitudinal (top panel) and longitudinal (bottom panel) components of the horizontal velocities along the latitudinal extension of the AR over time. Positive (white) and negative (black) values correspond to northward and southward directions, respectively, for the latitudinal component. For the longitudinal component, positive (white) and negative (black) values represent westward and eastward directions, respectively. The red box highlights the emergence of the second main bipole. The gray vertical bars indicate periods of SHARP data gaps. \n<!-- image --> \nred box in Fig. 5). This corresponds to the location of the neutral line of the new magnetic system overlapping the pre-existing one. \nThe converging phase becomes clearly evident starting from May 7, when the northward component at y=-250 '' progressively shifts to a southward component in the average latitudinal velocity map. \nGiven that the PIL is oriented longitudinally, the shear is noticeable in the strong eastward component above y=-300 '' , countered by a similarly significant westward component below it.", '4. Discussion': "Our detailed analysis of AR 13664, spanning from its initial appearance on May 1, 2024, to its peak activity on May 10, 2024, reveals a complex evolutionary path marked by substantial quantitative changes. Initially classified as a β -class region with a total unsigned magnetic flux of just below 4 × 10 22 Mx, AR 13664 rapidly evolved in size and complexity. By May 5, the active region expanded significantly, reaching a magnetic flux growth rate of 2.202 ± 0.012 × 10 22 Mx/day, a notable increase from the previous slower growth rate of 0.348 ± 0.001 × 10 22 Mx/day. \nDuring the emergence phase, the total unsigned magnetic flux increased to approximately 13.53 × 10 22 Mx before the region reached the western limb. In comparison, historical reconstructions suggest that the active region responsible for the Carrington Event in 1859 had a total unsigned magnetic flux of around 5 × 10 22 Mx, although this estimate carries significant uncertainties due to the limitations of data from that period (see Cliver & Dietrich 2013). \nThe compaction phase saw horizontal velocity magnitudes ranging from 0.5 to 1.0 km/s \nin the southern part of the region. This phase resulted in a reduction in the number of sunspots from around 81 to approximately 43, despite ongoing flux emergence. The largest sunspots in AR 13664 exceeded 30,000 km in diameter, whereas historical records indicate that the Carrington AR had sunspots possibly larger than 50,000 km (Hayakawa et al. 2019). \nThe shear phase, marked by strong shearing motions along the PIL, exhibited velocities up to 1.0 km/s. This shearing persisted for over 2 days and covered a latitudinal extent of approximately 200 '' . The analysis of horizontal velocities confirmed these phases with detailed maps showing shifts from diverging to converging motions and significant increases in the latitudinal and longitudinal components of velocity. \nThe culmination of these processes led to a series of intense X-class flares, with the flare index reaching approximately 5280, the highest recorded for ARs in solar cycles 24 and 25. This intense activity ultimately triggered a major geomagnetic storm on May 10, 2024. For context, the Carrington Event produced the most intense geomagnetic storm on record, with estimated Dst indices exceeding -1600 nT, which is several times stronger than typical modern geomagnetic storms (Siscoe et al. 2006). This comparison underscores the critical role of magnetic flux emergence, shearing, and horizontal motions in driving extreme solar activity and its impact on the solar and geomagnetic environment. \nAR 13664 exhibited notable similarities and differences when compared to the active region responsible for the Carrington Event of 1859. Both regions were characterized by a compact and highly complex magnetic configuration, with strong magnetic fields and multiple sunspots, indicative of intense magnetic activity. This compactness and complexity were key features that contributed to their significant flare activity, as both regions unleashed multiple X-class flares. \nHowever, despite these similarities, there were also important differences between the two regions. AR 13664, while exhibiting intense activity, was smaller in size compared to the \nCarrington Event active region, which was notably larger and more extensive (see Hodgson 1859). This difference in scale had significant implications for the geomagnetic impact of each region. The geomagnetic storm induced by AR 13664, while significant and classified as a G5 storm, was less intense than the Carrington Event, which produced a superstorm with extreme global geomagnetic disturbances (see Tsurutani et al. 2003; Cliver & Dietrich 2013). \nAdditionally, the duration of active phases between AR 13664 and the Carrington AR also showed notable differences. The emergence phase of AR 13664, characterized by a rapid increase in flux, spanned approximately 4 days, whereas the emergence phase of the Carrington AR, based on historical records, was more abrupt and occurred over a shorter timescale. The shear phase in AR 13664, marked by significant shearing motions along the PIL, persisted for over 2 days, covering a latitudinal extent of approximately 200'. The Carrington AR, most apparently, exhibited a shorter but more intense shearing phase, which played a crucial role in the rapid release of magnetic energy, ultimately leading to the subsequent extreme geomagnetic storm (Hayakawa et al. 2019)). \nThese differences in the duration and nature of the active phases highlight the variability in the lifecycle of solar active regions, even among those capable of producing extreme space weather events. While AR 13664 exhibited a more prolonged and gradual buildup of activity, the Carrington AR's rapid and intense phase transitions were key factors in its historical significance.", '5. Conclusions': "Our investigation of AR 13664 offers valuable and key insights into the characteristics of solar ARs, particularly when compared to other super ARs documented in the literature. On \nthe one hand, the peak of the unsigned magnetic flux reached by AR 13664, of approximately 13.53 × 10 22 Mx, is comparable to other super ARs such as AR 12192, which exhibited a peak flux of about 2 × 10 23 Mx (Sun et al. 2015). On the other hand, and in terms of flux emergence rate, AR 13664 had a peak rate significantly higher than that observed in some other large ARs, such as AR 12192 (peak rate of approximately 3.4 × 10 22 Mx/day) or AR 12673 (peak rate of approximately 2.9 × 10 22 Mx/day) (Sun & Norton 2017). \nThe largest sunspots in AR 13664 had diameters of up to 30,000 km. For comparison, the largest sunspots observed in AR 12192 were about 40,000 km in diameter (Jain et al. 2017). Instead, during the compaction phase the number of sunspots in AR 13664 decreased from about 81 to 43, within the range reported for largest super ARs observed since 1996 up to now, such as AR 12192, AR 10486, AR 9393, where the number of sunspots varied between 60 and 108 during their peak activity. \nWe believe that the identified sequence and order of evolutionary phases in this AR are likely fundamental to the formation and development of such complex solar ARs. Indeed, the subsequent emergence of magnetic field concentrations one after another at the same latitude and, importantly, at the same longitude, followed by predominant converging motions among the various photospheric magnetic structures, enables the strengthening of the magnetic field and the concentration of alternating magnetic polarities. Only subsequently vigorous shear motions along the PIL contribute significantly to energy buildup within the active region. These motions facilitate the accumulation and redistribution of magnetic flux, leading to the development of complex magnetic configurations. On the contrary, a different sequence/order in these phases could result in an earlier release of accumulated energy and the consequent diffusion of the magnetic field, preventing the formation of a compact AR like the AR 13664 and the one responsible for the Carrington event. This result is further validated by the similar temporal evolution of the total magnetic flux observed in other super ARs as \nreported in the literature (e.g., Romano & Zuccarello 2007; Smyrli et al. 2010). \nWe also remark that the driving mechanisms behind the observed horizontal motions in AR 13664 could be linked to differential rotation, convective flows, and magnetic instabilities in the solar photosphere. Differential rotation could be one of the causes that contribute to shear motions along the PIL, while convective motions could drive converging flows and facilitate the transport of magnetic flux. The interaction of emerging magnetic fields with pre-existing structures also could play a significant role in shaping the dynamics of solar active regions. In particular, in AR 13664, we observe the appearance of two subsequent bipoles, corresponding to additional flux tubes at the same latitude and longitude as the preexisting field. This behavior can be attributed to the higher rate of magnetic buoyancy and emergence during the solar cycle's maximum phases. When preferred longitudes of sunspot formation also coincide in latitude, we observe this succession of recurrent, intense flares attributable to the interaction between emerging and pre-existing magnetic fields (Zhang et al. 2007). This interaction is accompanied by a significant accumulation of energy due to the field's intensity and shear. These processes are essential for understanding the initiation of solar flares and CMEs from active regions like AR 13664, as well as similar activity observed in young sun-like stars (e.g., Lanza et al. 2009). Our study specifically highlights how horizontal motions and shearing contribute to the magnetic evolution of solar active regions, which in turn influences their potential to generate significant space weather events. By deepening our understanding of these dynamics, we can enhance predictive models for both solar and stellar activity, thereby improving our ability to anticipate and mitigate the impacts of extreme events on Earth's technological systems. \nWe are thankful to the editorial board and anonymous reviewers for their valuable and constructive suggestions, that have significantly enhanced this paper. This work was supported by INAF (Bando per il finanziamento della Ricerca Fondamentale 2022 - Study of \nthe correlation between the solar activity and the geomagnetically induced currents in gas \npipelines systems- and Bando per il finanziamento della Ricerca Fondamentale 2023 - IDEASWproject), by ASI under contract with INAF no. 2021-12-HH.0 'Missione Solar-C EUVST - Supporto scientifico di Fase B/C/D' and no. 2022-29-HH.0 'MUSE'. The research work of A. Elmhamdi in this project was supported by King Saud University's Deanship of Scientific Research and College of Science Research Center in Saudi Arabia. \nWe aknowledge the use of the different facilities, databases and tools appearing in our paper: SDO (AIA, HMI).", 'REFERENCES': "Antiochos S. K., DeVore C. R. and Klimchuk J. A. 1999, ApJ, 510, 485. doi:10.1086/306563 Balthasar, H. & Schuessler, M. 1983, Sol. Phys., 87, 23. doi:10.1007/BF00151156 \nBerdyugina, S. 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2024arXiv240816816C | Fifth force and equivalence principle tests search for new interactions by precisely measuring forces between macroscopic collections of atoms and molecules and their properties under free fall. In contrast the early Universe plasma probes these interactions at a more fundamental level. In this paper we consider the case of a scalar mediating a fifth force and show that the effects of dimensional transmutation spontaneous symmetry breaking and the running of the gauge couplings cause the scalars lowenergy interactions to mix leading to nearly universal dynamics at early times. We use known expressions for the pressure of the Standard Model during its various epochs to compute the scalar effective potential and find that the cosmological dynamics of this scalar are very sensitive to the reheat temperature of the Universe. Given the unknown reheat temperature we show that scalar couplings to matter larger than sim 106mphirm eV14 relative to gravity produce the correct dark matter abundance motivating new physics searches in this part of parameter space. | 2024-08-01T00:00:00Z | ['10.48550/arXiv.2408.16816', 'arXiv:2408.16816', '2024arXiv240816816C'] | ['High Energy Physics - Phenomenology', 'Astrophysics - Cosmology and Nongalactic Astrophysics', 'High Energy Physics - Experiment', 'High Energy Physics - Theory'] | Minimal targets for dilaton direct detection | 2,024 | 167 | 0.3 | ['EPRINT_HTML', 'EPRINT_PDF'] | 1 | https://arxiv.org/pdf/2408.16816.pdf | {'Minimal targets for dilaton direct detection': "David Cyncynates /orcid 1, /coffee and Olivier Simon /orcid 2, /rebel \n1 Department of Physics, University of Washington, Seattle, WA 98195, U.S.A. 2 Princeton Center for Theoretical Science, Princeton University, Princeton, NJ 08544, U.S.A. (Dated: September 2, 2024) \nFifth force and equivalence principle tests search for new interactions by precisely measuring forces between macroscopic collections of atoms and molecules and their properties under free fall. In contrast, the early Universe plasma probes these interactions at a more fundamental level. In this paper, we consider the case of a scalar mediating a fifth force, and show that the effects of dimensional transmutation, spontaneous symmetry breaking, and the running of the gauge couplings cause the scalar's low-energy interactions to mix, leading to nearly universal dynamics at early times. We use known expressions for the pressure of the Standard Model during its various epochs to compute the scalar effective potential, and find that the cosmological dynamics of this scalar are very sensitive to the reheat temperature of the Universe. Given the unknown reheat temperature, we show that scalar couplings to matter larger than ∼ 10 -6 ( m ϕ / eV) -1 / 4 relative to gravity produce the correct dark matter abundance, motivating new physics searches in this part of parameter space.", 'I. INTRODUCTION': "The Standard Model (SM) of particle physics describes the Universe in exquisite detail over a tremendous range of scales, yet the unexplained nature of quantum gravity, the cosmological constant, the cosmological dark matter, and the numerous unnatural hierarchies that apparently plague the SM suggest physics beyond the current paradigm. A goal of modern particle phenomenology is then to identify how the resolution to any of these outstanding problems may lead to experimental signatures. One category of signatures predicted by Beyond the SM (BSM) physics is the Fifth Force, where a new boson mediates additional interactions between SM particles. Fifth forces are a common feature in String Theories [1, 2], in proposed resolutions of the unnatural hierarchies in the Higgs and gravitational sectors [3-9], and particle models for the dark matter [10, 11] and other extensions of the Standard Model [12-14]. Numerous experiments have been devised to look for deviations from known force laws [15-19], including equivalence principle tests which measure deviations from the universality of free fall [20-22]. So far, these probes have all confirmed the predictions of the SM to the O (10 -3 ) level over scales larger than 50 µ m [23]. Nevertheless, fifth forces with range below 50 µ m are only weakly constrained, leaving open the possibility that shorter-range forces may still be discovered. \nThe simplest particle physics model of a fifth force is the addition of a singlet scalar to the SM, whose exchange among SM particles leads to the emergence of a Yukawa potential between them, with a range set by the scalar's mass [15, 24-29]. Scalars are well known to emerge from ultraviolet (UV) completions of the SM [30-32], although their couplings to matter can be sensitive to the details of the particular UV completion, making it is useful to take \n/coffee \nthe perspective of Effective Field Theory (EFT). Because the scalar is not associated with any conserved charges, it is expected to couple to all SM operators, leading to new interactions between all the known particles. While the particular nature and strengths of these couplings vary, their low energy experimental signatures can be degenerate. For instance, a scalar interacting exclusively with any one of gluons, quark masses, or photons, all lead to the exchange of scalars between the nucleons. On the other hand, dense astrophysical environments and the early Universe access much higher energies than Earthbound experiments, and therefore probe the particular nature of the interaction. \nThe dynamics of a scalar field in the early Universe are known to depend on the thermodynamics of the SM particles [33, 34]. A scalar can be interpreted as modulating the fundamental constants of the Standard Model [2, 29, 32, 35-38], and so an entropic force [39-41] will coherently pull the scalar towards field values that maximize entropy of the SM as a function of the fundamental constants [2, 42]. In the case that the scalar mediates a fifth force, maximum entropy is achieved for field values generically displaced from the scalar's Vacuum Expectation Value (VEV). Some relic abundance of the scalar will therefore result, in a process called thermal misalignment [43-45]. This relic abundance leads to new signatures, since the background scalar field will lead to spacetime dependent fundamental constants. Many experiments have searched for such variations [46-64], and many others may do so in the future [65-74]. \nIn this paper, we demonstrate that the strength of the entropic forces in the early Universe are largely insensitive to the nature of the microscopic interactions responsible for the macroscopic fifth force between composite states of matter at low energies, owing both to the running of the Standard Model gauge couplings and to the changing degrees of freedom as the Universe undergoes its various phase transitions. A key consequence is that the scalar relic abundance depends sensitively on the reheat temperature of the Universe. Precisely because of this UV \nsensitivity, we are able to show that thermal misalignment can be an effective production mechanism in previously unrecognized parameter space, motivating new searches for varying fundamental constants over a large but finite range of couplings and masses. \nThe paper is organized as follows. In Sec. II, we review scalar interactions with matter and obtain the thermal effective potential of the scalar in terms of the pressure of the cosmological SM thermal bath during the epochs of QED plasma, quark gluon plasma, and electroweak plasma. We stress the critical role played by the mass dimension of the operator to which the scalar couples to in determining the behavior at high temperatures. In Sec. III, we explain how to precisely compute the resulting scalar relic abundance by solving the sourced scalar equation of motion in the cosmological background. In Sec. IV we discuss the results of the relic abundance computations under different assumptions for the low-energy, experimentally accessible interactions of ϕ to composite matter. In Sec. V, we discuss the extent to which our results are sensitive to uncontrolled higher order terms, with respect to both scalar-SM and scalar-scalar interactions. In Sec. VI, we compute cosmological constraints on the available scalar-mass parameter space under the assumption that the scalar constitutes an O (1) fraction of the dark matter. We conclude in Sec. VII. In App. A we provide detailed expressions for the relativistic degrees of freedom used in our numerical calculations. In App. B we provide additional details regarding the joint Higgs-scalar potential. Finally, in App. C we provide simple asymptotic expressions useful for understanding the behavior of the scalar field.", 'II. SCALAR INTERACTIONS': 'In this paper, we consider a new, real scalar field ϕ described by the Lagrangian \nL ϕ = 1 2 ∂ µ ϕ∂ µ ϕ -V ϕ ( ϕ ) + L int . , (1) \nwhere V ϕ ( ϕ ) is a bare potential for ϕ , and L int . represents interactions with the SM particles. In the cosmological context, it is useful to define the dimensionless scalar field \nφ = √ 4 πGϕ = ϕ √ 2 M pl . (2) \nThis section provides extensive setup for our computation of the cosmological relic abundance of ϕ under different assumptions. Our objective is first to determine how ϕ interacts with the cosmological SM bath as the Universe evolves. 1 To do so, we propose the perspective \nthat the evolving Universe is most conveniently described as a succession of EFTs. First, in Sec. II A, we recapitulate how ϕ can be coupled to an existing EFT through its fundamental constants. In Sec. II B, we review how this framework applies to the present-day laboratory contexts. Sec. II C expands this perspective to the cosmological context. Finally, Secs. II D to II G explain how the dynamics of ϕ can be obtained from the dependence of the pressure and vacuum energy density of the EFT on the fundamental constants of the theory and applies this formalism to different phases of the SM.', 'A. Couplings of ϕ through the fundamental constants of existing theories': "Whether they are fundamental or effective, field theories must be specified by set of number-valued so-called input parameters in order to become predictive. Together, they form the fundamental constants of the theory within its range of applicability. For example, the Standard Model of particle physics (with massless neutrinos) contains 19 input parameters which must be experimentally measured (at a given renormalization energy scale); only then does the SM become predictive [75-77]. \nSystematically deriving low-energy physics from the SM often proves intractable, and the modern perspective of effective field theory tells us that studying physical systems in terms of SM degrees of freedom (DOF) and parameters is in fact often not appropriate, useful or necessary. Low-energy systems are often better viewed as characterized on their own, through a new set of input parameters and particle DOF, without regard for a high-energy theory, even if one exists. 2 For example, at exceedingly low energies, the SM reduces to quantum electrodynamics (QED), narrowly defined as the theory of interaction between electrons and light. The set of fundamental constants is then greatly reduced to { α EM , m e } , namely the electromagnetic fine-structure constant and the mass of the electron. It is well known that QED on its own suffices to describe a large array of atomic phenomena. \nThis EFT perspective is particularly useful when considering the way ϕ may couple to the degrees of freedom at a certain scale. Consider an EFT consisting of particles { X } (not including ϕ ) and input parameters { ζ } . This EFT is assumed to aptly describe physics at some energy scale. Because ϕ has no internal or spacetime symmetries, a generic theory containing both { X } and ϕ is simply the original theory of { X } where the fundamental parameters { ζ } are replaced by generic functions of ϕ . In other words, ϕ couples in a manner that effectively generates space and time dependence of the parameters of the original theory. \nThe field-dependence of a parameter ζ is then described via the scalar response function d ' ζ , defined as [38] \nd ' ζ ( φ ) ≡ 1 ζ ( φ ) dζ ( φ ) dφ . (3) \nIntegrating Eq. 3 and using the freedom to define d ζ to vanish at φ = 0 gives \nζ [ φ ] = ζ (0) e d ζ ( φ ) ≈ ζ (0) [1 + d ζ ( φ )] , (4) \nwhere d ζ ( φ ) = ∫ φ 0 d ' ζ ( φ ' ) dφ ' and the second equality holds at leading order in small d λ ( φ ). We will often make use of the Taylor expansion of d ζ ( φ ) in the form \nd ζ ( φ ) = ∑ n =1 d ( n ) ζ φ n n ! = d (1) ζ φ + d (2) ζ 2 φ 2 + . . . (5) \nExamples of theories where new scalars enter via preexisting fundamental parameters are numerous. In scalartensor theories of gravity, the magnitude of the gravitational constant G ( ϕ ) effectively becomes a function of dynamical scalar field. Many extra-dimensional theories, such as string theory, contain new scalar dilaton fields associated with the sizes of the extra dimensions, which, at low energies, appear to modulate the fundamental constants of IR theories. In this work, we will colloquially refer to any scalar fields which modulate the magnitude of the fundamental constants of a theory are as a dilaton, regardless of its presumed origin in the UV. In this colloquial sense, the Higgs field itself may be viewed as a dilaton [78], since it promotes the masses of SM fermions to scalar functions of space and time. The QCD axion is also an example, to the extent that it comes about via the promotion of the θ QCD parameter to a function of space and time. \nThis EFT framework for ϕ admits a Lagrangian description, which consists of promoting the Lagrangian parameters of a theory to functions of ϕ . For example, a spin 1 / 2 species ψ couples to ϕ through its mass parameter m ψ as \nL ϕψψ = -m ψ [ φ ] ¯ ψψ, (6) \nwhile a spin 0 scalar species s couples through its mass parameter m s as \nL ϕss = -1 2 m s [ φ ] 2 s 2 . (7) \nCoupling ϕ to the dimensionless strength g of a gauge interaction mediated by the field strength tensor F A µν can be done through \nL ϕgg = -1 4 g [ φ ] 2 F A µν F µν A , (8) \nwhere A is the gauge group index and we used the YangMills normalization of the gauge field.", 'B. Laboratory couplings of ϕ': "The EFT perspective on the coupling of ϕ through the input parameters of a theory has the advantage of allowing for ϕ to couple to low energy degrees of freedom while maintaining a level of agnosticism about the underlying high energy theory. This is useful because, outside high energy collider contexts, laboratory experiments are performed with composite states (e.g. a small or macroscopic number of atoms, molecules or ions). The MICROSCOPE satellite experiment [21], for example, looks for a new scalar-mediated force between the Earth (a mixture of silica SiO 2 and iron Fe) and two test masses of metallic alloys (Pt-Rh and Ti-Al-V). \nThe effective framework for such experiments really is an EFT of neutral atoms and molecules. The set of input parameters (fundamental constants) { ζ } of this low-energy EFT is then the set of atomic and molecular masses, along with the strength of the longrange gravitational interaction between them { ζ } = { m SiO 2 , m Fe , m Pt-Rh , . . . } ∪ { G } . A Lagrangian can in principle be built for such a theory, where field operators create and annihilate neutral atoms, although it is cumbersome to do so because neutral atoms do not have definite spin representations. \nLow energy experiments can therefore only probe the coupling of ϕ to the neutral atomic or molecular species used in the experiments. Within the narrow context of this low-energy EFT, the different response functions d ' ζ ( φ ) may be viewed as a priori independent. For example, in the extreme case, ϕ could, in principle, modulate the mass of iron atoms, but not the mass of platinum atoms. The other extreme is the universal case, where all atomic masses are modulated according to the same function of ϕ . We expect, however, that the large number of a priori free parameters of the atomic EFT will be determined by a smaller number of parameters in the UV. \nOnce a UV theory is specified from which the EFT descends, the coupling functions of the IR theory can be related to those of the UV theory. If the EFT contains, as effective degrees of freedom, composite states of the UV theory, then, in the most naive sense, a composite species should inherit its coupling to ϕ from the 'sum of its constituents.' \nTo a very good approximation, the progenitor theory of atomic and molecular EFTs is the combined theory of electromagnetism and chromodynamics with two light quarks. It is convenient to parameterize the input parameters of this theory by the set { ζ } = { α EM ≡ e 2 / 4 π, Λ QCD , m e , ˆ m ≡ ( m u + m d ) / 2 , δm ≡ m u -m d } , where α EM is the electromagnetic fine-structure constant, e the elementary electric charge, Λ QCD is the confinement scale of chromodynamics, m e is the mass of the election, and m u and m d are the masses of the up and down quarks. Making this set of input parameters ϕ -dependent corresponds, at the Lagrangian level, to coupling ϕ to the electromagnetic strength tensor F µν F µν , the gluon strength tensor G A µν G µν A , and the light fermion mass op- \n¯ dd [38]. \nBased on semi-empirical nuclear mass formulas, Damour and Donoghue [38] established formulas for the elements Q i [ Z, A ] of the transfer matrix \nd ' m [ Z,A ] ( φ ) = ∑ α EM , Λ QCD , ˆ m,δm Q i [ Z, A ] d ' i ( φ ) , (9) \nwhich relates the response function an atomic species with nuclear charge Z , atomic mass number A and atomic mass m [ Z, A ] to the response function of the five input parameters of the progenitor theory 3 \nd ' α EM = 1 α EM ∂α EM ∂φ = 2 d ' e , (10a) \nd ' Λ QCD = 1 Λ QCD ∂ Λ QCD ∂φ , (10b) \nd ' m e = 1 m e ∂ m e ∂φ , (10c) \nd ' ˆ m = 1 ˆ m (Λ QCD ) ∂ ˆ m (Λ QCD ) ∂φ , (10d) \nd ' δm = 1 δm (Λ QCD ) ∂δm (Λ QCD ) ∂φ . (10e) \nThe matrix element Q i [ Z, A ] is called the dilatonic charge of atomic species A Z X ; it encapsulates the species' sensitivity to the couplings of ϕ in the progenitor theory. \nIt is constraints on these 'experimentally accessible' couplings Eq. 10 which are often displayed in the literature, often under the assumption that all but one of them is nonzero [46, 69]. Alternatively, one can look at the sensitivity of a given experiment to particular linear combinations of couplings [59, 79]. This is especially relevant if one adopts the top-down perspective that a theory defined at still higher energies predicts a particular relationship between the numerical values of the five couplings Eq. 10 (in much the same way as the couplings Eq. 10 predict a particular relationship between the coupling functions of atoms). In particular, there exists a scenario of 'universal' couplings d ' m e = d ' Λ QCD = d ' m u = d ' m d and d ' e = 0, in which case all atomic d ' m X are equal and no EP violation can be measured in non-relativistic systems [38, 80]. \nIn this work, we will generally adopt the bottom-up perspective. We imagine measuring the experimentally accessible couplings Eq. 10 at laboratory scales (through e.g. a fifth force experiment) and study the implication for the cosmological history of ϕ . We further address the top-down perspective in the cosmological context in Sec. IV C.", 'C. Couplings of ϕ in cosmology': "We reviewed how the EFT perspective on the couplings of ϕ through the input parameters (fundamental constants) of different theories helps link high-energy fundamental physics to low-energy laboratory observables. In this section, we argue that, because the physical history of the Universe covers a broad range of energy scales, the EFT perspective is particularly apt for studying the interactions of ϕ with the cosmological bath of SM matter as it evolves. In particular, we elucidate the set of SM fundamental constants through which ϕ can couple during each epoch of cosmological evolution. \nEven if one were to believe that the SM (i.e. SU(3) c × SU(2) L × U(1) Y theory) is the fundamental theory of the Universe, it is hardly the most efficient framework to describe it across its ∼ 13 . 8 billion years of history. As the Universe cooled, it underwent phase transitions, which entailed a change in the effective DOF. We know the Universe went through, at the very least, primordial nucleosynthesis (BBN), recombination, and late-stage cosmic chemical evolution (in which heavy atoms and molecules formed in stars and the interstellar medium), each of which introduced new composite particles coupled to ϕ . \nThe highest temperature reached by the cosmological bath in the very early Universe, which we call the 'reheat temperature,' T RH , however, remains unknown. 4 The most conservative assumption is that the Universe was only ever hot enough to undergo BBN, corresponding to a lower bound T RH ≳ 4 MeV [81]. Depending on the reheat temperature, the cosmological bath might have gone through the electroweak and QCD phase transitions, and may or may not have contained heavy leptons and quarks. \nThe fundamental constants of the particular EFT that applies at a period of cosmological evolution dictate the coupling of ϕ to SM matter at that time, and therefore its cosmological production. It is therefore useful to think of the Universe as a succession of EFTs whose input parameters (and therefore the coupling of their respective DOF to ϕ ) are related. Partitioning cosmological history into different EFTs is of course inherently ambiguous. We nevertheless propose dividing cosmological particle physics history into at least four EFT eras based on the the effective content and temperature of the SM bath. Each EFT is specified by a set of particles { X } (not including ϕ ) and input parameters { ζ ( ϕ ) } which we take to be ϕ -dependent. We indicate when the theory requires an additional input parameter which we do not make ϕ -dependent in this work.", '0. Atomic and molecular era ( T ≲ eV):': 'As discussed above, in a laboratory context, ϕ couples to atomic and molecular degrees of freedom and modulates the set of fundamental constants associated with them. The set of particle DOF in this EFT is therefore, heuristically, \n{ X } = { Fe , Pt , SiO 2 , . . . , H , γ, ν e , ν µ , ν τ } , (11) \nwhere we have highlighted a few elements relevant to experimental tests. The . . . designate the rest of the atoms and molecules. We have also included the most cosmologically abundant species, namely hydrogen, photons and the three species of neutrinos. \nThe set of possibly ϕ -dependent fundamental constants associated with this energy scale is the set of all atomic and molecular masses, as well as an energy scale Λ 0 associated with the vacuum: \n{ ζ ( ϕ ) } = { m Fe , m Pt , m SiO 2 , . . . , Λ 0 } . (12) \nLong range interactions are dominated by gravity, whose coupling strength G is also an input parameter of the theory, which we take to be independent of ϕ , which is always possible through a field redefinition - the Einstein frame [82-85]. We have also neglected the set of effective coupling parameters associated with short-range chemical interactions between atomic and molecular species. \nAs we will discuss, one may define a ϕ -dependent vacuum energy M 2 pl Λ 0 [ ϕ ] such that ∼ ∂ ln Λ 0 /∂φ dictates the matter-independent part of the dynamics of ϕ . In this way, all the dynamics of ϕ , both vacuum dynamics and those induced by couplings to matter, can be obtained by promoting input parameters of the theory without ϕ to functions of ϕ . Following changes in the vacuum energy M 2 pl Λ 0 across EFTs helps in accounting for the effects of the Higgs and QCD chiral condensates in the cosmological dynamics of ϕ . \nAs our notation suggests, one may adopt (e.g. [86]) the perspective that Λ 0 is the measured cosmological constant Λ CC ≈ (10 -33 eV) 2 . For us, it only need be a tool to track changes in the energy of the vacuum across phase transitions involving spontaneous symmetry breaking. Elucidating the relationship between the vacuum energy parameter of the particle physics theory Λ 0 and the measured value of the cosmological constant Λ CC is the notorious cosmological constant problem which has yet to be conclusively resolved [87-89]. \n- 1. Nuclear and electron pairs era (eV ≲ T ≲ Λ QCD ): Below the QCD phase transition, the appropriate non-leptonic degrees of freedom are hadrons. Only the lightest hadronic states however are ever abundant in the cosmological bath. The particle degrees \nof freedom during this period are therefore those included in canonical treatments of BBN and the ionized Universe after BBN, namely the light leptons, nucleons, light nuclei, photons and neutrinos. Given that hydrogen-1 and helium-4 constitute ≳ 99 . 99% of nuclei after BBN, other light nuclei may be neglected in a first approximation. Therefore, \n{ X } = { γ, e -, p, n, 4 He 2+ , ν e , ν µ , ν τ , π 0 , π + , { ¯ X } , Λ I } , (13) \ncorresponding to the photon, electron, proton, neutron, helium-4 nucleus, the three neutrinos, the neutral and charged pions, all their anti-particles { ¯ X } , and the vacuum energy scale of that EFT Λ I . The ϕ -dependent input parameters are then \n{ ζ ( ϕ ) } = { e, m e , m p , m n , m π 0 , m π ± , m He } , (14) \nnamely the electromagnetic coupling strength e , and the masses of the electron, proton, neutron, neutral and charged pions, and light nuclei. The gravitational strength G , the strength of short-range nucleon-nucleon interactions, as well anomalous baryon magnetic moments remain additional parameters of this theory independent of ϕ . \nWe will find that relativistic states of matter by far dominate the cosmological production of ϕ . Nucleons and nuclei are only ever present in the cosmological bath while non-relativistic and are therefore included in our set only for completeness and to connect with the previous, lower-energy EFT. \nMoreover, because no spontaneous symmetry breaking takes places between the two epochs, Λ I = Λ 0 . \n- 2. Plasmas era (Λ EW ≳ T ≳ Λ QCD ≃ 150 MeV): Below the electroweak phase transition, the SM bath is well described by combined quark-gluon and electrodynamics plasmas with massive fermions. Then \n{ X } = { u, d, s, c, b, e -, µ, τ, γ, G A µ ν e , ν µ , ν τ , { ¯ X }} , (15) \ncorresponding respectively to the massive up, down, strange, charm and bottom quarks, the electron, muon and tau leptons, the massless photon, the ( A = 1 , . . . , 8) massless gluons, three massless neutrinos, and the set of anti-particles. Species are only abundant in the equilibrium thermal bath when the temperature is above their mass threshold. \nThe set of input parameters which we allow to be ϕ -dependent are \n{ ζ ( ϕ ) } = { m u , m d , m s , m c , m b , m e , m µ , m τ , e, Λ QCD , Λ II } , (16) \ncorresponding respectively to the masses of the up, down, strange, charm, and bottom quarks, the \nmasses of the electron, muon and tau leptons, the coupling strength of electromagnetism, the confinement scale of chromodynamics, and the vacuum energy scale. \nAdditional input parameters which we do not take to be ϕ -dependent are the three CKM mixing angles θ 12 , θ 13 , θ 23 , the CKM CP-violating angle δ CP , the theta parameter of QCD θ QCD , and the gravitational coupling strength G . Note that promoting θ QCD to a field dependent quantity amounts to introducing a QCD axion in the theory [90-93].', '3. Electroweak era ( T ≳ Λ EW ≃ 125 GeV):': 'Above the electroweak phase transition, the SM is described by the full SU(3) c × SU(2) L × U(1) Y theory with massless leptons and quarks. Then, \n{ X } = { G A µ , W A µ , B µ , H, Q i , L i , u i R , d i R , e i R , { ¯ X }} , (17) \ncorresponding respectively to the gluons ( A = 1 , . . . , 8), the SU(2) L gauge mediators ( A = 1 , 2 , 3), the U(1) Y mediator, the complex Higgs doublet, the 3 generations ( i = 1 , 2 , 3) of left-handed quark doublets, left-handed lepton doublets, right-handed up-type quarks, right-handed down-type quarks, and right-handed leptons, and their anti-particles. Then, \n{ ζ ( ϕ ) } = { g 1 , g 2 , g 3 , ν 2 H , λ H , y 2 t , y 2 b , y 2 e , y 2 µ , y 2 τ , y 2 c , y 2 u , y 2 d , y 2 s , Λ III } , (18) \nwhere g 1 is the U(1) Y gauge coupling strength, g 2 is the SU(2) L gauge coupling strength, g 3 is the strong gauge coupling strength, y 2 q,ℓ are the singular values (squared) of the matrices of Yukawa couplings of quarks and leptons, λ H and ν 2 H the strength of the quartic self-interaction and the mass squared parameter of the Higgs respectively, and Λ III the scale of vacuum energy in the SM. \nAgain, additional input parameters to the SM which we do not take to be ϕ -dependent are θ 12 , θ 13 , θ 23 , δ CP , θ QCD and G .', 'D. Finite temperature potential for ϕ': "So far, we have emphasized the way ϕ may modulate the fundamental parameters of the SM and its IR descendants. Conversely, an abundance of SM species will modulate the dynamics of any scalar field coupled to it. Assuming Standard Model species X are each (approximately) described by thermal equilibrium distribution functions defined by a finite temperature T X and/or a finite a chemical potential µ X , they generate dynamics for ϕ that can in general be computed in thermal field theory. If ϕ moreover varies slowly in space and time relative to the energy scales of the thermal bath (i.e. SM \nmasses or temperatures, whichever is largest), this potential can be computed using the background field methods [33, 34, 94]. In the background field method, the thermal path integral over SM species (corresponding to a sum of SM bubble diagrams) is evaluated with constant ϕ , and the spacetime dependence ϕ → ϕ ( t, ⃗x ) is re-introduced only in the final result. 5 \nFurther, because ϕ enters as modulations of the fundamental constants, the effective potential is especially simple. In a homogeneous Universe, at fixed values of the theory parameters { ζ } , the thermal path integral (i.e. the sum of bubble diagrams) evaluated over the SM species at finite temperature T X and/or finite chemical potential µ X corresponds simply to the thermodynamic pressure of the SM sector. The effective potential for ϕ is then the ϕ -dependent pressure of the system where the fundamental constants obtain their ϕ -dependent values locally [33, 34]: \nV eff ( ϕ ) = -P ( { ζ ( φ ) } ) , (19) \nwhere the thermodynamic pressure P ( { ζ } ) as a function of the input parameters of the theory can be evaluated as the finite { T X } and/or finite { µ X } part of bubble diagrams in thermal field theory at one-loop level and beyond. \nIn a loop expansion, the roles played by the one-loop and higher-loop diagrams have different physical significance. The one-loop contribution amounts to approximating the SM bath as a collection of independent, ideal, non-interacting gases. This is the leading order contribution to the energy densities and pressures of the Universe which drive its evolution according to the cosmological Friedmann equations. We call this total ideal gas contribution P non-int. ( ϕ ). At two-loops and beyond, interactions (and self-interactions) between species are accounted for. This contribution is suppressed relative to the ideal gas contribution by the various coupling strengths and is therefore generally neglected from the Friedmann equations outside of precision computations. When coupling strengths depend on ϕ however, this 'interacting' contribution P int. ( ϕ ) to the pressure can contain the leading dependence of P ( ϕ ) on ϕ and must be taken into account when computing the dynamics of ϕ . \nAll in all then, at any given stage in the evolution of the Universe, \nP ( φ ) = P non-int. ( φ ) + P int. ( φ ) . (20) \nIn Sec. II E and Sec. II F, we will discuss the terms in Eq. 20 in detail.", 'E. Finite temperature at one-loop order: Coupling to masses': "The total ideal gas pressure P non-int. is a sum over the individual pressures P non-int. X of each species. Because no couplings are involved at this level of approximation, ϕ only enters through the ϕ -dependent masses: \nP ( ϕ ) ⊃ P non-int. ( ϕ ) = ∑ X P non-int. X ( m X [ ϕ ]) . (21) \nThe quantity which enters into the ϕ equation of motion is the 'differential pressure,' \n( ∂ ∂φ P ( ϕ ) ) { µ X ,T X } ⊃ ( ∂ ∂φ P non-int. X ( ϕ ) ) µ X ,T X = -∑ X d ' m X ( φ ) θ non-int. X ( m X [ φ ]) , (22) \nwhere \nθ non-int. X ( m X [ φ ]) = ρ non-int. X ( m X [ φ ]) -3 P non-int. X ( m X [ φ ]) , (23) \nand ρ non-int. X ( m X [ φ ]) is the (non-interacting) energy density of the species X . Note that θ non-int. X ( φ ) is the φ -dependent trace of the stress energy of the species (in a non-interacting theory). In the ultra-relativistic limit [43] m X [ φ ] ≪ T X , \nθ non-int. X → (3 ∓ 1) g X 48 m X [ φ ] 2 T 2 X , (24) \nwhere the upper sign is for fermionic species, the lower sign for bosonic species and g X is the number of internal degrees of freedom. In the non-relativistic m X [ φ ] ≫ T X limit \nθ non-int. X → m X [ φ ] n X . (25) \nIn all the expressions above, particles and anti-particles must be counted separately. \nDuring Standard Model radiation-domination, a species in kinetic equilibrium with the radiation bath has temperature that is directly related to the Hubble parameter H ( a ): \nT X ( a ) 2 = T γ ( a ) 2 = 3 √ 5 π 1 √ g ∗ ( a ) H ( a ) √ 4 πG , (26) \nwhere a is the cosmological scale factor, and g ∗ ( T ) is the total number of relativistic degrees of freedom and T γ ( a ) is the temperature of bath of SM photons (see App. A). \nMost species in the SM are unstable (so that | µ X | ≪ m X ). Their equilibrium number density is therefore strongly suppressed (i.e. n X ≈ 0) once T X ≪ m X [ φ ]. This fact, in combination with the observed baryon asymmetry of the Universe, means that the non-relativistic limit is only relevant to stable SM matter species, namely baryons \nand electrons (and potentially massive neutrinos). The production of ϕ by the cosmological non-relativistic thermal bath however is generally greatly suppressed (roughly because non-relativistic states only make up negligible fraction of the energy budget of the Universe in RD) and will therefore not be further considered.", 'F. Finite temperature at two-loop order and beyond: Plasma effects': 'In thermal field theory, the interactions between particle species are accounted for at two-loop order and beyond. Let T denote the common temperature of the interacting plasma and µ X = -µ ¯ X be the set of chemical potentials. Then when T ≫{ m X , | µ X |} is much larger than all other mass scales in the system, the pressure of interactions is [95] \nP int. g ∝ [ O ( g 2 ) + O ( g 3 ) + . . . ] T 4 , (27) \nwhere g 2 is a dimensionless number characterizing the strength of 2 × 2 collisions. Comparing this expression to the high temperature limit Eq. 24, one can see that the differential pressure of interactions ∂P int /∂φ dominates the total differential pressure ∂P/∂φ . As a result, the cosmological production of scalars which modulate dimensionless parameters is particularly sensitive to the reheat temperature of the Universe. \nWithin the SM, the plenitude of mass scales in the IR descend from the combination of a large number of dimensionless couplings with only two apparently fundamental mass scales in the UV: the (negative) Higgs mass squared parameter ν 2 and the confinement scale of quantum chromodynamics Λ QCD . One can surmise that this trend continues in UV completions of the SM; as energy increases, dimensionful scales are traded for dimensionless couplings. \nThe precise determination of the proportionality between P int and T 4 at high temperatures in a given EFT is generally very challenging. Indeed, a large literature is devoted to precision computations of pressures in the Standard Model and its IR realizations. Fortunately, we are interested in the differential pressure, and only at large temperatures, where the calculation can be simply performed at leading order in the coupling. We will now compute the various contributions to the differential pressure throughout the different phases of the early Universe.', '1. QED plasma': "Below the electroweak phase transition, quantum electrodynamics with a single massless photon is a good approximation. The O ( e 2 ) correction to the pressure of a QED plasma of interacting spin 1 / 2 fermions at high temperature is well known [95, 96]. Accounting for all \nFIG. 1. The number of relativistic charged particles weighted by their electromagnetic charge squared, normalized so that the electron positron plasma contributes a factor of 1. Above the electroweak phase transition, we smoothly connect to the effective g C defined in Eq. 40, and we have neglected the charged pions. \n<!-- image --> \ncharged leptons and quarks, \nP int QED,high T ( φ ) = -5 288 g C ( T ) e 2 [2 πT,φ ] T 4 P + O ( e 2 µ 2 X T 2 ) + O ( e 3 T 4 ) , (28) \nwhere \ng C = ∑ X =rel. fermions g X Q 2 X 4 , (29) \nwith Q X the U (1) EM charge of the species in the broken electroweak phase, and the factor of 4 is a normalization chosen so that g C = 1 when electrons and positrons are the only relativistic charged species. 6 We plot g C ( T ) in Fig. 1, and write explicit expressions for it in App. A. We neglect the contribution of the charged pions, since their impact is limited to a very narrow temperature range between the pion mass and the QCD phase transition. \nEq. 28 is the 'Renormalization Group (RG) improved' leading order result which uses the running value of the coupling constant e evaluated at the scale ¯ µ = 2 πT , where the running of the fine structure constant is given by the β -function, \n1 e 2 [¯ µ, φ ] = 1 e 2 [0 , φ ] -1 12 π 2 m f < ¯ µ ∑ f Q 2 f log [ ¯ µ 2 m 2 f [ φ ] ] . (30) \nThis choice minimizes the impact of dropping terms in Eq. 28 that are formally of higher order in the coupling [95, 96]. \nNote that if the scalar couples to any charged particle's mass, it will acquire a coupling to P int . QED due to the running of the fine structure constant. While this effect is second order in α EM , it is the leading term in the effective potential in the case that the scalar couples only to the electron's mass today. \nDifferentiating Eq. 30 with respect to φ , we find \n2 d ' e (¯ µ, φ ) = 2 d ' e (0 , φ ) + 2 α EM (¯ µ, φ ) 3 π m f < ¯ µ ∑ f Q 2 f d ' m f (0 , φ ) . (31) \nDifferentiating Eq. 28 with respect to φ gives \n( ∂ ln P int QED,high T ∂φ ) T,µ X = 2 d ' e (2 πT,φ ) , (32) \nwhere the subscripts on the parentheses indicate that T and µ X are held constant.", '2. QCD plasma': "At temperatures above the QCD phase transition, the colored sector of the Standard model is a quark-gluon plasma. Similarly to QED plasmas, the pressure of a QCD plasma with colored fermions has been computed perturbatively in the strong coupling constant g 3 to high precision (at zero chemical potential) [95-97]. The result is, \nP int . QCD , high T = -g 2 3 [2 πT,φ ] 6 ( 1 + 5 12 N q ) T 4 + O ( g 3 3 (2 πT )) , (33) \nwhere N q is the number of effectively massless quarks (see App. A). Again, this is the RG improved leading order result which uses the running coupling constant g 3 evaluated at the scale ¯ µ = 2 πT , where \n1 g 2 3 (¯ µ, φ ) ≈ -β 0 ln [ ¯ µ Λ QCD ( φ ) ] + 2 27 m q < ¯ µ ∑ q = c,b,t ln [ ¯ µ m q [ φ ] ] , (34) \nwith β 0 = -18 / (4 π ) 2 . \nAt temperatures T ≫ Λ QCD , this choice minimizes the impact of dropping terms that are formally of higher order in the coupling. Even so, QCD becomes non-perturbative as T → Λ QCD and analytic control is poor. This is perhaps most evident in Ref. [97], where it is illustrated that, close to the QCD phase transition, higher order terms do not agree on whether the QGP interactions are net attractive or repulsive. Nevertheless, the magnitude of the interaction pressure remains roughly comparable among all the terms, and so we take the leading g 2 3 term as representative of the true interaction pressure. \nIn Eq. 34, we allow Λ QCD ( φ ) to depend on φ , and define Λ QCD ( φ ) to be the value at which 1 /g 2 3 = 0 in that background φ . We also allow the masses of heavy quarks to depend on φ . Differentiating Eq. 34 yields a relationship between the scalar coupling functions for the (running) coupling g 3 and that of Λ QCD : \nd ' g 3 (¯ µ, φ ) ≡ ∂ ln g 3 (¯ µ, φ ) ∂φ = β 0 2 g 2 3 (¯ µ, φ ) × \n d ' Λ QCD , ¯ µ < m c [ φ ] , d ' Λ QCD + 2 27 d ' m c , m c [ φ ] < ¯ µ < m b [ φ ] , d ' Λ QCD + 2 27 ( d ' m c + d ' m b ) , m b [ φ ] < ¯ µ < m t [ φ ] , d ' Λ QCD + 2 27 ( d ' m c + d ' m b + d ' m t ) , m t [ φ ] < ¯ µ. (35) \nDifferentiating Eq. 33, \n( ∂ ln P int QCD,high T ∂φ ) T,µ P = 2 d ' g 3 (2 πT,φ ) . (36) \nIt is important to recognize that Eq. 35 expresses the gluon coupling at a particular scale ¯ µ in terms of the coupling to Λ QCD and the heavy quarks as measured in vacuum . For example, if at zero temperature only the charm quark mass depends on ϕ , then above the charm mass the scalar must inherit a coupling to gluons for consistency.", '3. Electroweak plasma': "The interaction pressure of the electroweak plasma, excluding the pure QCD contribution, above the electroweak phase transition is, at leading order in the couplings [98102] \nP int . EW,high T = P SU(2) × U(1) + P y + P λ + 1 6 T 2 ν 2 [ φ ] + P int . QCD , highT + O ( g 3 SM ) , (37a) \nwhere electroweak gauge interactions yield the partial interaction pressure \nP int . SU(2) × U(1) = -T 4 24 ( 43 8 g 2 2 [ φ ] + 55 24 g 2 1 [ φ ] ) , (37b) \nand Higgs-fermion Yukawa interactions yield \nP int . y = -5 T 4 288 ∑ ℓ = e,µ,τ y 2 ℓ [ φ ] + 3 ∑ q = u,c,t,d,s,b y 2 q [ φ ] , (37c) \nwhere we have extended the usual results, which include only the top quark contribution, to include all the Stan- \ndard Model Yukawa couplings. Finally, the pressure contributed by Higgs self-interactions is \nP int . λ H = -λ H [ φ ] T 4 24 . (37d) \nIn these expressions, g 2 SM ∈ { g 2 1 , g 2 2 , g 2 3 , y 2 q , y 2 ℓ , λ H } . \nNear the electroweak phase transition, ν 2 ∼ g 2 SM T 2 and the leading order terms explicitly given in Eq. 37 are subdominant to the O ( g 3 SM ) terms. Accounting for O ( g 3 SM ) terms is therefore crucial if one studies the behavior of the pressure itself [102] or the symmetry breaking potential [103-106] across the electroweak phase transition, as if often the case in the literature. The same cancellation however will not generally occur for the differential pressure ∂P int . EW,high T /∂φ , which is of interest to us, unless ∂P int . EW,high T /∂φ ∝ P int . EW,high T , which happens only if all coupling functions of the partial pressures in Eq. 37 are equal. We assume this is not the case. \nIn terms of the electromagnetic charge: \ne [ φ ] = g 1 [ φ ] g 2 [ φ ] √ g 2 1 [ φ ] + g 2 2 [ φ ] , g 1 [ φ ] g 2 [ φ ] = tan θ W [ φ ] , (38) \nwhere tan θ W [ φ ] is the ( φ -depdendent) tangent of the Weinberg angle, and all coupling constants are evaluated at the electroweak scale. As a result, we find the following relationship between the coupling functions { d ' e ( φ ) , d ' tan θ W ( φ ) } and { d ' g 1 ( φ ) , d ' g 2 ( φ ) } : \nd ' e = d ' g 1 cos 2 θ W + d ' g 2 sin 2 θ W , ≈ 0 . 77695 d ' g 1 +0 . 22305 d ' g 2 , d ' tan θ W = d ' g 1 -d ' g 2 . (39) \nConversely, we may write \nP SU(2) × U(1) = -e 2 [ φ ] T 4 576 ( 129 csc 2 θ W +55sec 2 θ W ) , ≈ -64 . 9135 × 5 288 e 2 [ φ ] T 4 . (40) \nComparing Eq. 40 with Eq. 28, we can define an effective value for g C above the electroweak phase transition g C ( T ≳ 150 GeV) ≈ 64 . 9135. When calculating the relic density of the scalar coupled to electromagnetism, we smoothly match g C below the electroweak phase transition to its value above the electroweak phase transition with an exponential cutoff, as illustrated in Fig. 1 and written in App. A. \nBelow the electroweak phase transition, fermions acquire masses \nm f = y f ν √ 2 λ H , (41) \nfrom which it follows that \nd ' m f = d ' y f + d ' ν -d ' λ H 2 (42) \nWhile contributions form d ' ν and d ' λ are universal across SM fermions, the different d ' y f are in principle independent. If only the low energy couplings d ' m f can be probed, one cannot uniquely determine the UV couplings.", 'G. Zero-temperature potential and coupling to condensates': "We have examined how SM species at finite temperatures and/or chemical potentials generate an effective potential for ϕ . This formally corresponds to the finite T/µ parts of the resummation of the leading-order bubble diagrams of the SM and its EFTs. Similarly, we expect the vacuum dynamics of ϕ to be generated by the T = µ = 0 contributions of the bubbles diagrams. Those formally correspond to the vacuum energy of the (effective) theory. In Sec. II C, we have quantified the vacuum energy M 2 pl Λ j of each theory via an input energy scale Λ j to be included in the EFT Lagrangian. As a result, in each EFT, the vacuum dynamics are accounted for by adding \n-∂ ∂ϕ M 2 pl Λ j [ φ ] , (43) \nto the Euler-Lagrange equation for ϕ . In other words, M 2 pl Λ j [ ϕ ] can be thought of as the bare potential for ϕ [86]. \nWe are interested in the case where the vacuum potential for ϕ today has a minimum, which we designate without loss of generality as ϕ = 0. Expanding around this minimum, one has Λ 0 [ φ ] ≈ Λ 0 [0] and d ' Λ ( φ ) ≈ d (2) Λ φ , and therefore \n-∂ ∂ϕ M 2 pl Λ 0 [ φ ] = -1 √ 2 d ' Λ ( φ )Λ 0 [ φ ] M pl = -m 2 ϕ ϕ + . . . , (44) \nwhere m 2 ϕ ≡ d (2) Λ Λ 0 [0] / 2 defines the vacuum mass of the scalar today.", '1. Coupling to condensates': "As the Universe cools from high temperatures, the SM bath goes through electroweak symmetry breaking and chiral symmetry breaking of the QCD sector. At each of these stages, condensates form - the Higgs and chiral quark condensates respectively - resulting in a shift to the vacuum energy density. Taking into account the energy densities of the Higgs and chiral QCD condensates [94], we find the approximate relations \nΛ 0 ≈ Λ I ≈ Λ II -ˆ m Λ 3 QCD M 2 pl , ≈ Λ III -ˆ m Λ 3 QCD M 2 pl -ν 4 4 λ H M 2 pl , (45) \nwhere we recall that the subscripts 0,I,II,III refer to the present-day, QED, QCD, and electroweak epochs respectively. Recall also that both the Higgs and chiral QCD condensate contribute negatively to the vacuum energy. Λ 0 therefore contains the sum contribution of each of the condensates that formed at earlier epochs and encodes their contribution to the vacuum properties of ϕ today. At earlier times, we use Eq. 45 to remove these contributions as the SM bath heats up and crosses phase transitions. Thus, above the QCD phase transition, the non-thermal contribution to the equation of motion of ϕ is \n-∂ ∂ϕ M 2 pl Λ II [ ϕ ] = -∂ ∂ϕ M 2 pl Λ 0 [ ϕ ] , -( d ' ˆ m ( φ ) + 3 d ' Λ QCD ( φ ) ) ˆ m [ ϕ ]Λ 3 QCD [ ϕ ] √ 2 M pl . (46) \nAbove the electroweak phase transition, \n-∂ ∂ϕ M 2 pl Λ III [ ϕ ] = -∂ ∂ϕ M 2 pl Λ II [ ϕ ] , -( d ' ν ( φ ) -1 4 d ' λ H ( φ ) ) ν [ φ ] 4 √ 2 M pl λ H [ φ ] . (47)", '2. Naturalness': 'As mentioned, the generating functional M 2 pl Λ 0 [ φ ] for the vacuum dynamics of ϕ formally corresponds to the T = µ = 0 part of the sum of bubble diagrams of the SM, where input parameters are then promoted to functions of ϕ . The simplest estimate is that M 2 pl Λ 0 [0] ∼ Λ 4 cutoff , where Λ cutoff is the cutoff energy scale of the SM. If M 2 pl Λ 0 [ φ ] depends on φ through the SM couplings, we also expect d (2) Λ ∼ ( d (1) g SM ) 2 g 2 SM . This results in a mass for ϕ that is naturally of order \nm 2 ϕ ≃ ( d (1) g SM ) 2 g 2 SM Λ 4 cutoff M 2 pl . (48) \nThis is the well-known sensitivity of scalars to large radiative corrections [107-109]. Regions of the parameter space for which m ϕ is smaller than that dictated by the mass-to-coupling relation Eq. 48 require fine-tuning. \nThe cosmological production of ϕ from the SM bath, which we discuss in the next sections, is sensitive to the highest temperature at which the scalar is coupled to the SM. Taking this to be T RH requires Λ cutoff > T RH , in which case naturalness requires \nm 2 ϕ ≳ ( d (1) g SM ) 2 g 2 SM H 2 RH , (49) \nwhere H RH is the Hubble parameter at reheating. \nOn the other hand, first principles estimates of M 2 pl Λ 0 based only on SM physics are famously discordant with the measured value of the gravitating vacuum energy M 2 pl Λ CC by several orders of magnitude (anywhere from 55 to 122 orders of magnitude [88, 89]). Yet-undiscovered physics \npresumably reconciles these values. Another perspective then is to take Λ 0 as a input parameter that must agree with measurements and set Λ 0 [0] ≃ Λ CC . In this case, \nm 2 ϕ ≃ d (2) Λ Λ CC . (50) \nFrom this perspective, a scalar mass m ϕ ≫ 10 -33 eV is tantamount to d (2) Λ ≫ 1 . Of course, d (2) Λ is now no longer tied to SM physics exclusively and d (2) Λ ≫ ( d (1) g SM ) 2 g 2 SM may be realized. Note that because then \nM 2 pl Λ 0 [ φ ] = M 2 pl Λ CC + 1 2 m 2 ϕ ϕ 2 + . . . (51) \nthe requirement that the ϕ -dependence of Λ 0 [ ϕ ] not affect well established features of background ΛCDM cosmology does not constrain d (2) Λ beyond the requirement that ϕ be no more abundant then the totality of the cosmological dark matter.', 'III. COSMOLOGICAL PARTICLE PRODUCTION': "The Euler-Lagrange equation of motion for ϕ in the cosmological spacetime is \n¨ ϕ +3 H ˙ ϕ + ⃗ ∇ 2 ϕ a 2 = -∂ ∂ϕ M 2 pl Λ j [ φ ] + ∂P [ φ ] ∂ϕ , (52) \nwhere H is the Hubble parameter. We expand this equation around the minimum (i.e. ϕ = 0) of the vacuum potential at late-times. Furthermore, if we assume that φ ≪ 1 at all points in cosmic evolution, then, in the formulae for P ( ϕ ), all coupling functions are linearized around the minimum of ϕ 's vacuum potential: d ' ζ ( φ ) ≈ d ' ζ (0) ≡ d (1) ζ , while SM fundamental constants obtain ζ [ φ ] ≈ ζ [0] ≈ ζ 0 . 7 In that case, the spatially homogeneous component of ϕ obeys \n¨ ϕ ( t ) + 3 H ˙ ϕ ( t ) + m 2 ϕ ϕ ( t ) = -∆ V ' (0) , (53) \nwhere \n-∆ V ' (0) = ∂P [ φ ] ∂ϕ ∣ ∣ ∣ φ =0 -M 2 pl ∂ ∂ϕ (Λ j [ φ ] -Λ 0 [ φ ]) ∣ ∣ ∣ φ =0 . (54) \nAs the Universe expands and cools, its pressure P [ φ ] decreases and therefore acts as a time-dependent source term for the production of the field ϕ ( t ). \nDuring radiation domination (RD), H = 1 / 2 t . The method of (retarded) Green's functions for the homogeneous operator ∂ 2 t +3 H∂ t + m 2 ϕ = 0 then yields that the \nsolution to Eq. 53 is \nϕ ( t ) = M pl ( H eq 2 m ϕ ) 1 / 4 √ 3 π 2 ξ 1 / 4 ( A ( ξ ) Y 1 / 4 ( ξ ) -B ( ξ ) J 1 / 4 ( ξ ) ) , (55) \nwhere ξ ≡ m ϕ t , J 1 / 4 and Y 1 / 4 are the Bessel functions of order 1 / 4, and \nA ( ξ ) = A i +∆ A ( ξ ) , (56) \nB ( ξ ) = B i +∆ B ( ξ ) , (57) \nwhere A i and B i represent the initial conditions of the scalar field when the Universe is at a temperature T RH and are related to ϕ ( t i ) and ˙ ϕ ( t i ) via Eq. C1, and where \n∆ A ( ξ ) = m ϕ M pl √ π 3 ( 2 m ϕ H eq ) 1 / 4 × 1 m 3 ϕ ∫ ξ ξ i dξ ' ξ ' 5 / 4 J 1 / 4 ( ξ ' )( -∆ V ' (0)) , (58a) \n∆ B ( ξ ) = m ϕ M pl √ π 3 ( 2 m ϕ H eq ) 1 / 4 × 1 m 3 ϕ ∫ ξ ξ i dξ ' ξ ' 5 / 4 Y 1 / 4 ( ξ ' )( -∆ V ' (0)) , (58b) \naccount for the influence of the Standard Model plasma. Here, t i is the time corresponding to when the Universe was at a temperature T RH , where we assume instantaneous reheating, and ξ i ≡ m ϕ t i is the dimensionless time of reheating. \nAs we will discuss later in Sec. VI C, the expression with the initial conditions set to zero is a meaningful representation of the irreducible effect of the Standard Model plasma on the relic abundance of the scalar. Therefore, we will take A i = B i = 0 for the remainder of this discussion. \nAs long as ∆ V ' (0) decays in time faster than ∼ t -7 / 4 ∝ a -7 / 2 , then for ξ → ∞ , the coefficients A ( ξ ) and B ( ξ ) tend to some finite values in RD, which we denote A 0 and B 0 . These corresponds to a net, finite production of ϕ particles. Long after the scalar has started oscillating and the source terms have effectively decoupled, the energy density 1 2 ˙ ϕ 2 + 1 2 m 2 ϕ ϕ 2 of ϕ at late times (i.e. ξ →∞ ) is then \nρ ϕ ( a ) = 3 2 M 2 pl H 2 eq ( a eq a ) 3 ( A 2 0 + B 2 0 ) . (59) \nIn other words, the condensate ϕ behaves as cold dark matter. The latter expression extends to subsequent periods of matter and Λ -domination. With our normalization, the complete cosmological dark matter density is obtained for A 2 0 + B 2 0 = 1. \nWe compare this expression to the relic abundance of a non-interacting scalar produced through the so-called 'misalignment' mechanism, which is often parametrized \nin terms of the initial value of the misalignment angle φ i = ϕ i / √ 2 M pl . When m ϕ ≳ H eq , \nρ free ϕ ( a ) = M 2 pl H 2 eq φ 2 i ( m ϕ H eq ) 1 / 2 ( a eq a ) 3 . (60) \nComparing this to the form of Eq. 59, we see that we can define an effective 'initial' misalignment angle \nφ i, eff = √ 3 2 √ A 2 0 + B 2 0 ( H eq m ϕ ) 1 / 4 . (61)", 'A. Integration over sources': "The total pressure is a sum over different components of the Standard Model pressure. These contributions can be viewed as independent source terms which turn on and off at different stages in cosmological history. Because, the integrals Eq. 58 for ∆ A and ∆ B are linear in V ' eff (0), each contribution adds independently to the total production vector (∆ A, ∆ B ). Expressions Eq. 58 fit the usual paradigm for particle production from a source: the efficiency of the source goes as the overlap between that source and one of two solutions to the homogeneous differential equation of motion. In practice, we numerically evaluate the integrals in Eq. 58 for each contribution to the total time-dependent source V ' eff (0). Nonetheless, it is instructive to get some approximate analytical handles on these expressions. This is possible because there are only a few types of time dependencies in V ' eff (0) to consider. In this section, we enumerate the various source terms relevant to the computation of the ϕ relic density, and we present approximate expressions for the effective misalignment angle in the presence of each of these terms, reserving more details for App. C.", '1. Relativistic massive species': "In RD, terms corresponding to ideal gas contributions from a relativistic massive species go as \nd (1) m X θ X ( m X [ φ ]) ∼ d (1) m X m 2 X, 0 2 t , (62) \nafter reheating and prior to pair annihilation. Given one such massive species, there is therefore a natural inflection point to the production depending on whether the source turns off well before or after the dynamical timescale of the scalar field ξ ∼ 1. In the limit where A i , B i ≪ ∆ A 0 , ∆ B 0 , the contribution of such species to the effective misalignment is \nφ i, eff ∝ d (1) m X × 1 , m ϕ ≲ m 2 X, 0 M pl , m 2 X, 0 M pl m ϕ , m 2 X, 0 M pl ≲ m ϕ ≲ T 2 X,i M pl , m 2 X, 0 T 2 X,i ( T 2 X,i M pl m ϕ ) 5 / 4 , T 2 X,i M pl ≲ m ϕ . (63) \nHere, T X,i is the temperature at which the source 'turns on.' It is the minimum of the reheating temperature and the UV cutoff of the theory in which X is an effective massive degree of freedom (e.g. the electroweak phase transition, above which particle masses are replaced by dimensionless Yukawa couplings to the Higgs).", '2. Relativistic plasma effects': "Interactions in relativistic gases (i.e. plasmas) contribute to ∆ V ' (0) as \n∼ d (1) g g [0] 2 √ 4 πG 1 4 t 2 , (64) \nwhere g is the associated dimensionless coupling strength. In strongly running theories, both d (1) g and g [0] evaluated around φ = 0 inherit dependence on the temperature due to the RG flow. For the purpose of illustration, we neglect this additional time-dependence here only, in addition to the time-dependence of g ∗ ( a ). Suppose then that the source turns on when the Standard Model bath is at a temperature T g,i : \nφ i, eff ∝ d (1) g g 2 0 × log ( T 2 g,i M pl m ϕ ) , m ϕ ≲ T 2 g,i M pl , ( T 2 g,i M pl m ϕ ) 5 / 4 , m ϕ ≳ T 2 g,i M pl . (65) \nIt is worth noting that in the case m ϕ ≳ T 2 g,i /M pl , the effective initial misalignment is much larger than the maximum displacement of the scalar φ max ∼ d (1) g g 2 0 ( T 2 g,i /M pl m ϕ ) 2 . 8 This is because the effective misalignment addresses the question 'what would the scalar's expectation value have been at H = m ϕ to match its observed relic abundance,' but if m ϕ ≫ T 2 g,i /M pl , the initial Hubble rate is much smaller than m ϕ , so it has undergone less dilution than the effective misalignment anticipates.", '3. Condensation': "Finally, we comment on the scalar production that occurs when it couples to a field undergoing spontaneous symmetry breaking. Condensation is an event that takes place at a particular temperature, and typically involves an order-one fraction of the energy density of the Universe at that time. Thus, as far as the scalar is concerned, a coupling to a condensing field is much like a coupling to a fermion mass, and we should expect similar scalings, at least if the Universe's temperature was ever above the \nsymmetry restoration scale. The condensate contribution to ∆ V ' (0) is \n∼ d (1) Λ M pl Λ j Θ( t PT -t ) , (66) \nwhere Θ is the Heaviside function, leading to the effective misalignment (assuming T PT ≪ T RH ) \nφ i, eff ∝ d (1) Λ PT × M 2 pl Λ j T 4 PT m ϕ ≲ T 2 PT M pl , M 2 pl Λ j T 4 PT ( T 2 PT M pl m ϕ ) 5 / 4 m ϕ ≳ T 2 PT M pl , (67) \nwhere j = 0 , I , II , III. One can see that the effective misalignment scales with m ϕ in the same way as it does for a coupling to a massive fermion Eq. 63, as expected. Further, the size of the effective misalignment is proportional to the fraction of the critical density contained in the condensate M 2 pl Λ j /T 4 PT , analogous to the factor m 2 X, 0 /T 2 X,i in Eq. 63, which represents the fraction of the critical density proportional to m 2 X, 0 in relativistic fermions when the source 'turns on.' We note that an analogous factor is also present in Eq. 65, although it is O (1) in this case, since the part of the critical density proportional to g 2 scales as T 4 .", 'IV. COSMOLOGICAL RELIC ABUNDANCE': "To this point in our discussion, we have reviewed how the cosmological Standard Model thermal bath, if it was coupled to a scalar particle that modulates the size of its fundamental constants, would have generated an abundance of this scalar in the form of a coherent, oscillating condensate whose energy density would eventually come to dilute like that of cold dark matter. The amount of ϕ produced is linked to the dependence of the SM pressure on ϕ through the fundamental constants, and is sensitive to the hottest temperature T RH achieved in the early Universe. The UV sensitivity of the scalar relic abundance and the insensitivity of this abundance to the nature of the scalar's interactions are our main results, and owe to the scale-dependence of the Standard Model, of its couplings, and of the effective degrees of freedom to which ϕ couples. We now discuss our results for the amount of cosmological production under different assumptions.", 'A. Experimentally accessible couplings': "In the present-day, five canonical couplings of the scalar to the SM, introduced in [38] and reviewed in Eq. 10, are experimentally accessible in the context of fifth force experiments with macroscopic source and test masses. These couplings are therefore measured today at the low energies characteristic of the experimental atomic and molecular systems. Constraints on the low-energy couplings are often presented under the assumption that only one of the couplings is nonzero at the time [46, 69]. In \nkeeping with this convention, we imagine that one of { d (1) e , d (1) Λ QCD , d (1) m e } is non-zero at present-day laboratory energies, and consider the implied cosmological production of ϕ by integrating Eq. 58, under different assumptions for T RH . While { d (1) m u , d (1) m d } are also experimentally accessible, we do not treat the couplings to light quark masses because properly accounting for them requires calculating the dependence of the strong gauge coupling on the light quark masses down to ¯ µ ∼ Λ QCD . The low-energy couplings relevant for experiments today are related to their values at higher energies relevant to cosmological production through Eqs. 31, 35 and 42. For each of { d (1) e , d (1) Λ QCD , d (1) m e } , we display (Figs. 2 to 5) the values of the coupling for which the amount of ϕ produced accounts for the totality of the measured cosmological dark matter under different assumptions for T RH . At fixed T RH , scalar DM is overproduced with the given assumptions in models living in the parameter space above the colored contour (although the limits of our analysis at large couplings are discussed in Sec. VI). \nWe discuss each coupling individually below. Before starting, we stress the remarkable feature, visible in Figs. 2 to 5, that, at high enough T RH , the universal scaling d (1) ζ ∝ m -1 / 4 ϕ applies to all couplings up to masses m ϕ ≲ H RH . This is the scaling law associated with the presence of ϕ -dependent dimensionless fundamental constants, such as gauge coupling strengths, and corresponds to an effective initial misalignment angle independent of the mass. This can be thought of, in the 'bottom-up' perspective, as a consequence of the fact that dimensionful parameters in the IR tend to feed into dimensionless parameters in the UV, whether through RG evolution, dimensional transmutation, or spontaneous symmetry restoration (Eqs. 31, 35 and 42). In fact, there is a choice of coupling in the UV which does not have UV sensitivity, namely the superrenormalizeable Higgs portal ϕH † H [44]. On the other hand, running our argument in reverse shows that all the low energy couplings will be non-zero, and will be related to one another in a particular way. We comment on this possibility at the end of the section.", '1. Low energy coupling to photons d (1) e': 'The blue-to-red lines in Fig. 2 show the values of the coupling d (1) e for which the relic abundance of ϕ constitutes the totality of the cosmological dark matter when d (1) Λ QCD = d (1) m e = 0 at laboratory energies. This is the most straightforward case, because the dominant contribution to the SM pressure proportional to α EM is simply accounted for through the number of fermionic relativistic charged species weighted by their charge g C (Eq. 29). To the left of m ϕ ≃ H RH , the effective initial misalignment angle is independent of the mass and has a mild (logarithmic) dependence on H RH Eq. 65, such that the total DM abundance is obtained for d (1) e ∝ m -1 / 4 ϕ , per Eq. 60. \nFIG. 2. The available parameter space for a scalar coupled to the photon at low energies accessible to experiments. The colored lines indicate the parameters for which the scalar relic abundance is that of the dark matter for various reheat temperatures corresponding to the Hubble rates H RH in the legend to the right of the plot. The dotted part of the contours correspond to where inflationary vacuum misalignment is expected to overproduce the scalar (see Sec. VI C), while the dashing indicates where isocurvature perturbations are too large to be consistent with observations of the CMB [110]. We avoid plotting additional bounds that depend on the scale of inflation or reheating since they would clutter the figure, and instead refer the reader to Fig. 6 for a representative example of the various constrains considered throughout the text. The prospective sensitivity curves of various experiments are plotted in red [65-70], while current observational bounds are in gray [15, 46-62, 111-114]. \n<!-- image --> \nFIG. 3. The available parameter space for a scalar coupled to the QCD scale at low energies. At scales above the Λ QCD , the scalar is coupled to gluons, and as a result its effective potential scales as T 4 in the early Universe, leading to similar contours as those in Fig. 2. Because QCD is asymptotically free, the scalar relic density is UV finite, which is visible as the near perfect overlap of the colored curves at small m ϕ . At very small reheat temperatures near BBN, the scalar couples to composite states which are falling out of equilibrium, and consequently a larger coupling is necessary to achieve the correct relic abundance for H RH = 10 -13 eV. The prospective sensitivity curves of various experiments are plotted in red [67, 71], while current observational bounds are in gray [21-23, 115-127]. We refer the reader to Fig. 6 for more details. \n<!-- image --> \nFIG. 4. The available parameter space for a scalar coupled to the electron mass at low energies via a coupling to the Higgs potential. This plot represents the particular case of a coupling to the Higgs mass parameter, though a coupling to the Higgs quartic leads to a plot which is almost visibly indistinguishable up to the decay rate for m ϕ ≫ m higgs (the faster quartic decay rate is indicated in lighter gray). Although the coupling to the Higgs mass parameter would naively only lead to an effective potential scaling as T 2 at high temperatures, the running of the strong gauge coupling with the heavy quark masses ultimately leads to a coupling between the scalar and the gluons at high temperatures, again leading to the ubiquitous m -1 / 4 ϕ contours. Prospective sensitivity curves are plotted in red [65-68, 70-74]. Observational constraints are in gray [15, 21, 28, 46, 47, 49, 50, 52, 54-59, 61, 63, 64, 111, 124, 126]. See Fig. 6 for more details. \n<!-- image --> \nFIG. 5. In contrast to Fig. 4, here we plot the available parameter space for a scalar coupled to the electron mass at low energies by a coupling only to the Higgs-electron Yukawa interaction. Generically, one would expect the scalar to couple to any number of particle masses through the Standard Model Yukawa interactions, so this example represents an edge case. Above the electroweak scale, the scalar interaction with the electron Yukawa leads to an effective potential proportional to y 2 e T 4 , although the smallness of the electron Yukawa means this contribution is highly suppressed. Ultimately, the high temperature behavior of the scalar effective potential is dominated by the running of the fine structure constant Eq. 30, since y 2 e ≪ α 2 EM . Nonetheless, α 2 EM is a small number, and the tree-level coupling dominates scalar production for small m ϕ . \n<!-- image --> \nTo the right, the coupling necessary to produce a dark matter abundance of ϕ scales as d (1) e ∝ m ϕ , illustrating the fact that scalars with masses larger than the Hubble rate at reheating decouple.', '2. Low energy coupling to gluons d (1) Λ QCD': "Fig. 3 shows the values of d (1) Λ QCD for which the relic abundance of ϕ produced by the SM cosmological thermal bath equals the totality of the cosmological dark matter when d (1) e = d (1) m e = 0 at laboratory energies. At low reheat temperatures m π ≲ T RH ≲ Λ QCD , production is dominated by the relativistic composite pions which eventually fall out of equilibrium. We approximate this regime with a free pion gas (one-loop, ideal gas limit) and use the Gell-Mann-Oakes-Renner relation [94] m π ∝ ˆ m 1 / 2 Λ 3 / 2 QCD which implies d ' m π = 3 2 d ' Λ QCD . Production in this regime is much suppressed and requires d (1) Λ QCD ∼ 1, which is currently excluded (under the assumption that all other low energy couplings Eq. 10 are vanishing). \nWhen T RH ≳ Λ QCD , the coupling through the dimensionful Λ QCD is promoted to the dimensionless strong coupling g 3 through dimensional transmutation (Eq. 35). Scalar production is then very similar to the preceding case of a coupling to photons. Because QCD is asymptotically free, the QCD pressure actually becomes smaller at very large temperatures thus canceling out the logarithmic dependence on H RH at small masses, in contrast to the case of a coupling to photons. Note that Fig. 3 assumes that the scalar coupling function d ' m q of heavy quarks is zero when applying Eq. 35.", '3. Low energy coupling to electrons d (1) m e': "Figs. 4 and 5 show the values of d (1) m e for which the relic abundance of ϕ produced by the SM cosmological thermal bath equals the totality of the cosmological dark matter when d (1) e = d (1) Λ QCD = 0 at laboratory energies, assuming two possible UV origins for the coupling d (1) m e , as we explain below. At the lowest scalar masses, production is dominated by ideal gas contribution of relativistic electrons. If this were the only source term in the scalar equation of motion, the parameter space where the scalar abundance would match that of the dark matter would be entirely excluded, as pointed out in [43]. However, at larger masses, the dominant source depends on the assumptions one makes about the UV origin of the d (1) m e coupling. Per Eq. 42, at tree-level, the dependence of the electron mass on ϕ descends from the ϕ -dependence of the electron Yukawa parameter y e and/or the dependence of the Higgs's potential parameters. \nIf the dependence occurs through ϕ -dependent Higgs parameters, then it is passed on to all massive fermions. \nIn particular, the masses of heavy quarks will depend on ϕ . Then, even if d (1) Λ QCD = 0, as is already assumed, the strong coupling constant g 3 inherits a dependence on ϕ at temperatures above the masses of the heavy quarks whose masses depend on ϕ (Eq. 35). In such a theory, ϕ production is therefore dominated by the quark-gluon plasma at energies above e.g. the mass of the charm quark. If the dependence of fermion masses on ϕ is realized by a ϕ -dependent λ H , then there is also significant production from the pressure term ∼ λ H [ φ ] T 4 at a level comparable to that from the quark gluon plasma. \nIf, on the other hand, the varying electron mass occurs through the ϕ -dependence of the electron Yukawa, it need not be passed on to all other massive fermions. In this case, there is production through the y 2 e T 4 term in the electroweak pressure, but it is greatly suppressed by the smallness of y e . But a significant source for production nevertheless persists. This is because, if d (1) e = 0 at atomic energies, but d ' m e is non-zero, the running electromagnetic coupling e at high energies inherits a dependence on ϕ through its dependence on the electron mass threshold at order e 2 (Eq. 31). While this coupling is suppressed, the production from QED effects in the UV nevertheless has the characteristic scaling ∝ m 1 / 2 ϕ , and therefore dominates at larger m ϕ if T RH is large enough.", 'B. Couplings to heavy leptons, heavy quarks': "Fewer laboratory constraints exist on fifth forces in the heavy leptons and heavy (and strange) quarks sectors. By the same token however, they hold less potential for experimental discovery (although the muon is a notable exception [64, 128]). Nevertheless, we stress that the cosmological production of a ϕ particle coupled through any of their masses will share many of the features explored for the experimentally accessible couplings above, especially when T RH is high. \nThe case of a scalar coupled to µ or τ leptons is similar to that of scalar coupled to the electron (see also [43]), with the additional condition that T RH must at least the lepton's mass, or else the scalar's displacement will be exponentially suppressed. Also similarly to the electron, at reheat temperatures above the lepton mass, the scalar will necessarily inherit a coupling to the photon at order e 2 , which will dominate the relic abundance of heavier scalars at very large reheat temperatures. \nAlong similar lines, the production of a scalar coupled to the mass of any of the three heavy quarks will be dominated by a coupling to gluons at large-enough reheat temperatures. Again, however, the reheat temperature must be at least the heavy quark's mass, or else the scalar will feel only an exponentially small force.", 'C. Bottom-up versus top-down': "In this paper, we have taken the bottom up perspective that the set of scalar couplings is only restricted by observational limits, and found that a general set of scalar couplings at low energy implies a coupling to dimensionless parameters at higher energies. We adopted the bottom-up perspective partly because it is the one consistent with the frequent way of displaying experimental limits on dilatonic couplings, that is, by assuming that only one coupling at the time is non-zero at laboratory scales [29, 46]. \nOn the other hand, one can specify the set of scalar couplings in the UV, in which case the couplings at low energies will be determined by running the argument of Sec. II backwards. This top-down approach is, of course, in principle exactly equivalent to the bottom-up approach, since the full renormalization group equations are reversible. One subtlety of the top-down perspective is whether the high energy scale at which the UV theory is defined (sometimes taken to be the scale of new physics) is itself a function of ϕ . The distinction can be important in determining, for example, whether the baryonic sector exhibits equivalence principle violation or not (see the opposing perspectives of Refs. [38, 80]). In contrast, in the bottom-up approach, there is no such ambiguity, because the defining scales are by definition associated with the low energy scales being measured by experiments (as is evident in Eqs. 30 and 35). \n̸ \nFor the particle content of the Standard Model, there is a choice of scalar coupling in the UV electroweak theory which avoids the steep temperature scaling ∝ T 4 , namely the superrenormalizable Higgs coupling [44], which in our notation corresponds to d (1) ν = 0 with all other coupling functions set to zero above the electroweak phase transition (Eq. 37a). At tree level, the superrenormalizable Higgs coupling implies d (1) m X = d (1) ν . At loop level, the Standard Model gauge couplings run with the masses of the fermions, implying that the scalar will acquire a coupling to the QCD scale and the fine structure constant at laboratory energies. For instance, in the case of QCD, one fixes d ' g 3 (¯ µ, φ ) = 0 for ¯ µ larger than the electroweak scale in Eq. 35, which in turn implies d (1) Λ QCD = -(6 / 27) d (1) m X . Similarly, the photon coupling can be determined from Eq. 31, yielding d (1) e = -(16 α EM / 3 π ) d (1) m X . Ultimately, when performing searches for or placing constraints on any model specified in the UV, one must be careful to compute signatures in terms of the relevant linear combination of couplings in the IR. We leave a detailed treatment of this scenario to future work.", 'V. HIGHER-ORDER INTERACTIONS AND LIMITS OF PERTURBATIVE RESULTS': "In this section, we assess the limits of the dynamics discussed in the previous section. A central assumption \nof the previous section is that SM parameters (except the auxiliary scale of vacuum energy Λ 0 [ φ ]) should not significantly deviate from their present-day value, such that Standard cosmological history can be taken as the background for and source of the dynamics of ϕ . From Eq. 4, this condition, linearized around the late-time vacuum ϕ = 0, is \nd ζ ( φ ) ≈ d (1) ζ φ ≪ 1 . (68) \nLet us suppose that ϕ ( t i ) = ˙ ϕ ( t i ) = 0, which we will later see is often a consistent assumption. For m ϕ < T 2 g,i /M pl , the effective initial misalignment φ i, eff is equal to the maximum misalignment φ max , and we can use Eq. 65 to estimate (up to a logarithm) \nd (1) ζ φ i, eff ∼ ( d (1) ζ ) 2 ≪ 1 , m ϕ ≲ T 2 g,i M pl . (69) \nOn the other hand, if m φ > T 2 g,i /M pl , then φ max ∼ d (1) g g 2 0 ( T 2 g,i /M pl m ϕ ) 2 , which yields the bound \nd (1) ζ φ i, eff ∼ ( d (1) ζ ) 2 ( T 2 g i m ϕ M pl ) 2 ≪ 1 , m ϕ ≳ T 2 g,i M pl . (70) \nNext we inspect higher order terms in the SM pressure. First, it is necessary to assume \nd (1) ζ φ ≫ d ( n> 1) ζ φ n . (71) \nThis is the simple statement that the interaction potentials must be well described around the late-time minimum by a linear forcing term. This would not be the case if e.g. the minima of the various interaction potentials were to align with the late time minimum ϕ = 0 (i.e. if the leading order interaction were e.g. quadratic). This latter scenario of unique, universal minimum is, for example, proposed in [2], but is not the one studied here, as it leads to suppressed fifth force signatures in the present day. \nAs long as Eq. 71 is satisfied, Eqs. 69 and 70 not only correspond to the requirement that the Standard Model thermal history is only perturbatively altered by the scalar, but also to the requirement that higher order terms in the SM pressure are never dynamically important beyond the leading order. This is because the SM pressure depends on φ precisely through the value of the SM fundamental constants. That is to say, the higher order φ corrections to both the scalar effective potential and the Standard Model pressure are exactly the same. \nIn order to be certain that our calculations are valid, one must also verify that these higher order finiteT terms do not overwhelm the scalar's bare potential. In particular, the Green's function solution Eq. 55 relies on m ϕ and H setting the dynamical timescale for oscillations (both their onset time and frequency). Requiring that the thermally-induced effective mass always be subdominant \nparametrically translates to \n( d (1) ζ ) 2 T 4 M 2 pl ≲ max { m 2 ϕ , H 2 } . (72) \nDividing through by T 4 /M 2 pl however simply recovers Eqs. 69 and 70. We indicate the region where Eqs. 69 and 70 are violated in Fig. 6 in blue and with the label 'nonperturbative,' for which knowledge of the full φ potential and interactions is necessary to predict its relic abundance. \nFinally, we consider the relative size of the different terms in the vacuum potential as defined by the series expansion of M 2 pl Λ 0 [ φ ]. As pointed out already through Eq. 44, the leading φ -dependent term in this expansion defines the vacuum mass m 2 ϕ . Higher order terms will also be be generated. For example, the term ∝ φ 4 generates (through Eq. 4) \nM 2 pl Λ 0 [ φ ] ⊃ 3( d (2) Λ ) 2 -d (4) Λ 4! Λ 0 [0] 4 M 2 pl ϕ 4 ≡ λ ϕ 4 ϕ 4 , (73) \n̸ \ncorresponding to a quartic self-interaction of the ϕ field of dimensionless strength λ ϕ . While we can consistently set all higher order terms in vacuum energy to zero by a judicious choice of d ( n> 2) Λ , setting d (2) Λ = 0 with all d ( n> 2) Λ = 0 leads to a possible benchmark expectation for the size of the quartic self-interaction: \nλ ϕ, benchmark = m 4 ϕ 2Λ 0 [0] M 2 pl . (74)", 'A. Self-interactions': "The preceding discussion has shown that vacuum fluctuations will generically lead to scalar self-interactions, and that consistency of the effective field theory at a given maximum SM temperature places a lower bound on the size of these terms. In this section, we will argue that the extent to which these self-interactions suppress the scalar relic abundance is parametrically weak. In order to make our discussion concrete, we will restrict our attention to the leading terms in the scalar potential \nV ( ϕ ) = 1 2 m 2 ϕ ϕ 2 + 1 3 µ ϕ ϕ 3 + 1 4 λ ϕ ϕ 4 , (75) \nneglecting fifth and higher order terms. \nFirst, we observe that the cubic term is always bounded above as measured about the true vacuum of the potential 9 \nµ 2 ϕ ≤ 9 2 λ ϕ m 2 ϕ . (76) \nWith this limit, one can argue that the cubic is at most of marginal importance relative to the quadratic and quartic terms, except when µ ϕ is close to saturating this bound. To see this, suppose that the cubic is more important than the mass term in the equation of motion: m 2 ϕ < | µ ϕ ϕ | . Using Eq. 76 to eliminate m ϕ , we see that | µ ϕ | < (9 / 2) λ ϕ | ϕ | , which is (up to a factor of 9 / 2) the condition that the quartic dominates over the cubic. While this is not a proof that cubic terms are never important, it shows that they are subdominant for all but a small range of µ ϕ close to the bound Eq. 76, allowing us to reasonably restrict our discussion to the effect of the quartic interaction. \nLet us now focus on the quartic interactions, which we will assume are repulsive so that the potential Eq. 75 is bounded from below. As long as the quartic interactions are dominant over the mass, the scalar amplitude will dilute as ϕ ∝ t -1 / 2 during radiation domination [129]. Since the differential pressure of interactions ∂P int . /∂ φ ∝ T 4 ∝ t -2 redshifts faster than the quartic self-interactions λϕ 3 ∝ t -3 / 2 , the scalar dynamics will be controlled by the plasma up until the time t osc when the self-interactions become important. After this point, we can approximate the scalar as undergoing misalignment with the initial condition that satisfies \nλϕ 3 ( t osc ) ≈ d (1) g g 2 0 √ 4 πG 1 4 t 2 osc . (77) \nUsing the scalar equation of motion without a mass or quartic to solve for the early-time behavior of ϕ , we find that the scalar displacement at this time is approximately \nϕ ( t osc ) ≈ d (1) g M pl log ( · · · ) λ ϕ ( d (1) g ) 2 M 2 pl H 2 RH ≪ 1 , ( d (1) g M pl H 2 RH λ ϕ ) 1 / 3 λ ϕ ( d (1) g ) 2 M 2 pl H 2 RH ≫ 1 . (78) \nThe upper line corresponds to the case where competition between Hubble friction and the differential pressure determines the scalar displacement, whereas the lower line corresponds to the case where the differential pressure competes with the quartic. The relic abundance of a scalar with an initial misalignment ϕ ( t osc ) is approximately \nΩ ϕ ≈ ( m ϕ H eq ) 1 / 2 ( ϕ ( t osc ) M pl ) 2 √ m ϕ m eff ( ϕ ( t osc )) , (79) \nwhere m eff ( ϕ ( t osc )) ≡ √ m 2 ϕ + λ ϕ ϕ 2 ( t osc ). This expression is of the usual form up to the term in the radical, which corrects for the extra dilution due to repulsive selfinteractions. In the small and large λ ϕ limits of Eq. 78, we see that Ω ϕ ∝ ( d (1) g ) 3 / 2 λ -1 / 4 ϕ and ( d (1) g ) 1 / 2 λ -3 / 4 ϕ , resulting in d (1) g ∝ λ 1 / 6 ϕ and d (1) g ∝ λ 3 / 2 ϕ respectively, visible as the kinks in the dotted black lines in Fig. 6. \nOn the other hand, the suppression owing to selfinteractions will not take over unless ϕ explores the nonlinear part of its potential at some point in its history. \nFIG. 6. The available parameter space of a scalar coupled to the fine structure constant, assuming instantaneous reheating when the temperature was T RH ∼ 10 8 GeV, corresponding to a Hubble rate H RH ∼ 10 7 eV. The bold, black line corresponds to the thermal misalignment of the scalar that yields the correct relic abundance, under the assumption that only the scalar's bare mass and linear coupling to the plasma are important. The blue region labeled 'Non-perturbative' indicates where this assumption breaks down, and higher order terms can become important (Eqs. 69 and 70). Similarly, if the scalar has a bare potential with a repulsive quartic interaction, its relic density becomes suppressed if its field range explores parts of the potential where the quartic dominates - we indicate this suppressed relic abundance with the dotted black lines (Sec. V A). Below the dashed yellow line, the scalar mass is not necessarily subject to fine-tuning if the cutoff associated with quantum corrections to the scalar potential is above the maximum temperature of the Universe. The red region labeled 'thermal overclosure' indicates parameter space where thermal ϕ production is expected to overclose the Universe (Eq. 84). The green regions labeled 'Inf. EV' and 'Inf. Var' indicate where the inflationary expectation value and variance of ϕ respectively overwhelm the postinflationary thermal dynamics and our calculations break down (Eq. 86). Finally, the gray regions indicate existing observational bounds on the scalar-QED coupling strength [15, 46-62, 111-114], while the thin red lines represent prospective sensitivities [65-70]. Not visible on this plot are bounds from isocurvature, which are only important at larger H I - see Figs. 2 to 5. \n<!-- image --> \nThis requirement is set by m 2 ϕ ≈ λ ϕ ϕ 2 ( t osc ), which we can interpret as a condition on the quartic coupling. Up to a logarithm, the critical λ ϕ at which self-interactions become important is set by \nλ ϕ ≳ λ crit ≈ 8 πGm 2 ϕ ( d (1) g ) 2 g 4 0 . (80) \nOne may equivalently view Eq. 80 as a condition on d (1) g and m ϕ . Indeed, this is the more appropriate perspective to take when plotting in ( m ϕ , d (1) g ) parameter space, as in Fig. 6 where the critical point is visible as the intersection of the dotted black lines and the solid black line.", 'VI. COSMOLOGICAL CONSTRAINTS': 'In addition to the experimental limits on the various scalar couplings, thermal production, inflationary production, and isocurvature perturbations all place limits the available parameter space that depend on the maximum reheat temperature of the Universe or scale of inflation. \nIn this section, we perform analytical estimates of these bounds, as well as place limits from the perturbative decay of the scalar. In cases where the graphical expression of certain bounds would compromise legibility of Figs. 2 to 5, we illustrate the effect of the bound in Fig. 6.', 'A. Thermal production': 'The hot standard model bath can directly produce ϕ on-shell. For a scalar coupling to some dimensionless parameter g ( ϕ ) = g 0 (1+ d (1) g φ ), we can apply dimensional analysis to estimate the rate at which Standard Model particles produce ϕ \nΓ SM → ϕ ∼ ( d (1) g ) 2 g 2 0 T 3 M 2 pl ∼ ( d (1) g ) 2 g 2 0 H 3 / 2 M 1 / 2 pl ≪ H. (81) \nFor the moderate values of d (1) g we consider, Γ is always slower than the Hubble rate, so ϕ is never produced fast enough to reach thermal equilibrium. On the other hand, the ϕ abundance produced at the moment the Universe \nreaches its maximum temperature can potentially be significant in some parts of parameter space. Assuming the reheat temperature is much larger than the scalar mass T RH ≫ m ϕ , the energy density of the relativistic ϕ satisfy \n˙ ρ ϕ +4 Hρ ϕ ≈ Γ SM → ϕ ρ SM . (82) \nThis equation can be integrated directly, yielding \nρ ϕ ∼ ( d (1) g ) 2 g 2 0 √ H RH M pl ρ SM . (83) \nSince the ϕ were produced at temperature T ∼ √ M pl H RH , they become nonrelativistic when H ∼ m 2 ϕ /M pl , after which point ρ ϕ redshifts as a -3 . The resulting relic abundance as a fraction of the critical density is \nΩ ϕ, th ∼ ( d (1) g ) 2 g 2 0 √ H RH H eq m ϕ M pl , ∼ 10 -9 ( d (1) g ) 2 g 2 0 ( H RH 5 × 10 13 GeV ) 1 / 2 ( m ϕ µ eV ) . (84) \nNote that if m ϕ ≲ KeV then this component will be hot dark matter. We plot this bound in Fig. 6.', 'B. Isocurvature': 'Instantaneous reheating, which we have assumed throughout this manuscript, directly ties the inflationary scale H I to the maximum temperature of the Universe T RH ∼ √ H I M pl , and represents the minimal inflationary Hubble rate corresponding a given reheat temperature. Inflationary fluctuations generate an irreducible spectrum of scalar isocurvature perturbations δϕ ∼ H I , while Planck measurements of the Cosmic Microwave Backrgound (CMB) constrain the fraction of the matter power spectrum in these isocurvature perturbations to be less than few% of the adiabatic perturbations [110]. If ϕ is the dark matter, then δϕ/ϕ ≲ few% × √ 2 × 10 -9 ∼ 10 -6 . Using ϕ ∼ d (1) g g 0 M pl O (log), we find that isocurvature constrains d (1) g g 0 ≳ 10 6 H I /M pl . We indicate isocurvatureconstrained regions of parameter space with dashed lines in Figs. 2 to 5.', 'C. Initial conditions and inflationary production': "Throughout this manuscript, we have assigned vacuum initial conditions to the scalar field at the beginning of radiation domination. However, there is no reason to assume that the scalar field must have started at its vacuum value, and in fact, it generically will not. Inflationary fluctuations will cause the scalar expectation value to take \na random walk during inflation, setting the initial conditions for the scalar field at the beginning of radiation domination. Further, the mean of the random walk is generically not zero, owing to the inflationary fluctuations of the Standard Model particles. \nDuring inflation, the effective temperature of the Standard Model is given by the Hawking temperature T H = H I / 2 π . Assuming that the scalar's thermal potential scales as d (1) g φg 2 0 T 4 , its effective potential during inflation is \nV eff , I ( ϕ ) ∼ 1 2 m 2 ϕ ϕ 2 + d (1) g g 2 0 ϕ √ 2 M pl H 4 I (2 π ) 4 . (85) \nThe homogeneous expectation value of the scalar field is selected from a probability distribution satisfying the Fokker-Planck equation, and after a long period of inflation settles into an equilibrium ∝ exp { -8 π 2 V eff , I ( ϕ ) / 3 H 4 I } [130]. The mean of this distribution is \n⟨ φ ⟩ = -d (1) g g 2 0 2 M 2 pl m 2 ϕ ( H I 2 π ) 4 . (86) \nOn the other hand, the maximum field excursion due to the post-inflationary thermal history of the Universe is \nφ max . ∼ d (1) g g 2 0 × log ( H I m ϕ ) m ϕ ≲ H I , ( H I m ϕ ) 2 m ϕ ≳ H I . (87) \nIf m ϕ ≳ H I , then the post-inflationary field excursion is always much larger than the inflationary expectation value. On the other hand if m ϕ ≲ H I , then the inflationary expectation value dominates when H I ≳ √ M pl m ϕ up to logarithm. We indicate this bound in Fig. 6 in the green region labeled 'Inf. EV.' \nInflationary fluctuations also imply that the variance of the scalar field is \n⟨ φ 2 ⟩ = 3 H 4 I 8 π 2 m 2 ϕ . (88) \nWe indicate where this variance exceeds the (squared) post-inflationary field excursion with the green region labeled 'Inf. Var' in Fig. 6, and as the dotted part of the contours in Figs. 2 to 5.", 'D. Decay': "If the scalar is all of the dark matter, as it is along the contours in Figs. 2 to 5, then its decay rate to Standard Model particles must be slow enough to survive until the present day. In general, however, the decay rate may have to be much faster or slower than the age of the Universe: for example, its decay rate to photons has to be much slower in the ∼ KeV range due to bounds on \nextragalactic X-rays. On the other hand, if the scalar is overproduced relative to the CDM abundance, then it must decay prior to matter radiation equality. In this section, we will estimate the decay rate of the scalar to Standard Model particles, and compute constraints in the particular case where the scalar constitutes all of the dark matter. \na. Photon coupling The operator -(1 / 4) d (1) α EM φF 2 is the most straight forward to treat, since the tree level decay to massless photons is always kinematically accessible (in vacuum). Including the appropriate phase space factors, the decay rate is \nΓ ϕ → γγ = (2 d (1) e ) 2 m 3 ϕ 32 πM 2 pl . (89) \nEven in the case that the decay rate is slower than H 0 , decays to photons are visible and can be constrained by astrophysical observations [112-114], as illustrated in Figs. 2 and 6. \nb. QCD coupling Although the coupling to gluons -(1 / 4) d (1) g 3 φ tr G 2 is superficially similar to the photon coupling, tree-level decay is forbidden since gluons are colored states. For m ϕ ≪ Λ QCD , the leading decay is into photons and takes place at two loops, and we can estimate the decay rate: \nΓ ϕ → γγ ∼ 1 16 π 8 × 3 ( ( 2 3 ) 2 +( 1 3 ) 2 ) α 2 EM 16 π 2 ( d (1) g 3 ) 2 M 2 pl m 3 ϕ , (90) \nwhere the factor of 8 counts the gluons running in the first loop while the other factor multiplying α 2 EM counts the charge-weighted number of quarks. For m ϕ ≫ Λ QCD , it is approximately correct to think of ϕ as decaying directly to gluons, which we estimate using dimensional analysis \nΓ ϕ → gg ∼ 8 16 π ( d (1) g 3 ) 2 M 2 pl m 3 ϕ . (91) \nc. Electron mass coupling A coupling to the electron mass can occur through a number of operators, as illustrated above. Let us first consider a coupling to the electron Yukawa -d (1) y φy e ¯ LHe R , where H is the Higgs doublet, L = ( ν e , e L ) is the left-handed doublet, and e R is the right handed singlet. At low energies, this term corresponds to the electron mass coupling -d (1) y φm e ¯ ee , with m e = y e v/ 2. For m φ < 2 m e , tree level decay is forbidden, and instead decay to two photons is mediated by an electron loop: \nΓ ϕ → γγ = α 2 EM 288 π 3 ( d (1) y e ) 2 M 2 pl m 3 ϕ . (92) \nFor scalar masses above the electron pair production threshold, tree-level decay to electrons dominates: \nΓ ϕ → ee = m 2 e 16 π ( d (1) y e ) 2 M 2 pl m ϕ . (93) \nFinally, for scalars whose mass exceeds that of the Higgs, tree level decays will produce the electroweak symmetric states \nΓ ϕ → HLe R ∼ y 2 e (8 π ) 3 ( d (1) y e ) 2 M 2 pl m 3 ϕ , (94) \nwhere again we have employed dimensional analysis. \nd. Higgs potential coupling When the scalar couples directly to the Higgs potential, its interactions with the other Standard Model particles are precisely those of the Higgs, multiplied by an effective mixing angle (see App. B) \nθ mix . ≡ ( d (1) ν -1 2 d (1) λ ) v √ 2 M pl ( m 2 higgs m 2 higgs -m 2 ϕ ) , (95) \nwhere v is the Higgs VEV and we have assumed m higgs and m ϕ are different enough that θ mix . ≪ 1. In other words, the decay rate of the scalar is precisely that of a Higgs boson with its mass replaced by the scalar's, and multiplied by the mixing angle [131]: \nΓ ϕ → SM = θ 2 mix . Γ h → SM ( m h → m ϕ ) . (96) \nThe resulting decay rate of light scalars to photons is [132, 133] \nΓ ϕ → γγ = | F | 2 ( α EM 4 π ) 2 G F √ 2 θ 2 mix m 3 ϕ 8 π , (97) \n= | F | 2 ( α EM 4 π ) 2 ( d (1) ν -1 2 d (1) λ ) 2 m 3 ϕ 32 πM 2 pl , (98) \nwhere for m ϕ < m e the form factor is F = 7 / 9. If m h ≫ 2 m W , the dominant decay mode of the Higgs is into longitudinally polarized gauge bosons [134]: \nΓ ϕ → WW,ZZ = 3 2 ( d (1) ν -1 2 d (1) λ ) 2 ( m higgs m ϕ ) 4 m 3 ϕ 32 πM 2 pl . (99) \nBecause the mass mixing is suppressed when m ϕ ≫ m higgs , this decay rate is parametrically slow. In the case that ϕ only couples to the Higgs through the mass term in its effective potential, this is a decent approximation for the full decay rate. On the other hand, if the scalar couples to the Higgs through the quartic interaction, then the decay rate scales as \nΓ ϕ → hhh ∼ ( d (1) λ ) 2 λ 2 v 2 (8 π ) 3 M 2 pl m ϕ , (100) \nΓ ϕ → hhhh ∼ ( d (1) λ ) 2 λ 2 (2 π ) 10 2 7 m 3 ϕ M 2 pl , (101) \nwhere we have estimated the four-body phase space factor using Ref. [135].", 'VII. CONCLUSION': "In this manuscript, we have shown that a scalar, which today mediates a fifth force, couples to the early Universe plasma in a manner nearly independent of its low energy interactions. This genericity is a consequence of the tendency for dimensionful scales to transform into dimensionless couplings at higher energies, which in the Standard Model takes place via dimensional transmutation, spontaneous symmetry breaking, and the running of the gauge couplings. Therefore, almost every interaction between a scalar and the Standard Model implies that the scalar couples to a dimension 4 operator at energies above some mass threshold or phase transition. As a result, the scalar's effective potential generically grows as T 4 at high temperatures. This rapid scaling is significant because it leads to scalar dynamics which are highly sensitive to the maximum temperature of the Universe, and which, for high enough temperatures, are very efficient at producing ϕ . Consequently, scalar couplings larger than d (1) ζ ≳ 10 -6 ( m ϕ / eV) -1 / 4 yield an irreducible relic abundance of ϕ constituting any fraction of the total dark matter density, depending on the thermal history of the Universe. Moreover, because couplings in this parameter space lead to the overproduction of dark matter for large enough reheat temperature, the discovery of a fifth force within that space would place an upper limit on the cosmological reheat temperature. \nMany planned experiments will have sensitivity to scalars in this preferred part of parameter space. At smaller scalar masses around 10 -12 eV, the predicted relic abundance is least sensitive to the reheat temperature of the Universe, offering a robust experimental target. Experiments such as [65, 66, 68-70] will be sensitive to this range of couplings and masses. Our work also motivates a greater focus on the eV mass range, where astrophysical constraints and equivalence principle tests are weakest, and which can be populated for reheat temperatures in the GeV to TeV range. Proposals such as [67], which take advantage of eV-scale molecular resonances, have the potential to probe this part of parameter space. \nAlthough our treatment has been fairly general, we have not explicitly treated the case of a scalar coupling only to the light quark masses, since such a treatment necessarily relies on running the QCD gauge coupling at scales where it is nonperturbative. We consider it likely that such a coupling would lead to the same ubiquitous behavior observed for a coupling to any other mass scale in the Standard Model. Nonetheless, any reliable results along these lines seem likely to require the lattice. \nAs displayed, our key figures Figs. 2 to 5 implicitly assume that the SM remains a good EFT of nature at temperatures as high T RH ∼ 10 15 GeV, and one may ask what \nhappens if new physics enters between the electroweak scale and the reheat temperature. A UV completion of SM, excluding ϕ , will contain its own set of fundamental constants from which SM parameters descend. It is likely that the dimensionful Higgs parameter ν will be traded for dimensionless couplings, possibly through symmetry restoration, dimensional transmutation (e.g. in technicolor-like scenarios [136]), or some other mechanism yet to be discovered. So long as the scalar effective field theory remains applicable, the scalar would then couple to dimension 4 operators of the UV completion of the SM, so that the general scaling ∼ T 4 is maintained at high temperatures. Even so, the dimension 4 operators that ϕ couples to constitute dimension 5 operators which must themselves be UV completed, perhaps at some different, higher scale. Whatever the ultimate theory of both ϕ and the SM is, it will contain at most marginal operators, and the T 4 behavior must ultimately be cut off.", 'ACKNOWLEDGMENTS': 'We thank Masha Baryakhtar, Anson Hook, Junwu Huang, Mikko Laine, Ken Van Tilburg, Neal Weiner, Zachary Weiner, and Lawrence Yaffe for helpful discussions and correspondences. We are grateful to Masha Baryakhtar and Zachary Weiner for providing comments on the manuscript. We thank Jedidiah Thompson for collaboration during the early stages of this project. This work was completed in part at the Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Colleges and Universities. D.C. is supported through the Department of Physics and College of Arts and Science at the University of Washington and by the U.S. Department of Energy Office of Science under Award Number DE-SC0024375. Experimental limits and prospects presented in this paper were in part compiled using the GitHub repository associated with Ref. [137].', 'Appendix A: Relativistic degrees of freedom': 'In this appendix, we provide explicit formula for the relativistic degrees of freedom, degrees of freedom in entropy, and charge-square weighted degrees of freedom that we have used in our computations. \ng ∗ = ∑ X =rel. bosons g X ( T X T γ ) 4 + 7 8 ∑ X =rel. fermions g X ( T X T γ ) 4 (A1) \ng ⋆ ( T ) ≈ 2 + 2 × 3 e -( cm W /T ) γ + e -( cm Z /T ) γ + ( 8 × 2 + 7 8 × 3 × 12 -3 ) e -(3 c Λ QCD /T ) γ +2 e -( cm π ± /T ) γ + e -( cm π 0 /T ) γ \n+ 7 8 × 2 [ N eff ( 4 11 ) 4 / 3 + ( 3 -N eff ( 4 11 ) 4 / 3 ) e -( cm e /T ) γ ] + 7 8 × 4 × ( e -( cm e /T ) γ + e -( cm µ /T ) γ + e -( cm τ /T ) γ ) 7 8 × 12 × ( e -( cm c /T ) γ + e -( cm b /T ) γ + e -( cm t /T ) γ ) , (A2) \ng ⋆,S ( T ) ≈ 2 + 2 × 3 e -( cm W /T ) γ + e -( cm Z /T ) γ + ( 8 × 2 + 7 8 × 3 × 12 -3 ) e -(3 c Λ QCD /T ) γ +2 e -( cm π ± /T ) γ + e -( cm π 0 /T ) γ \n+ 7 8 × 2 [ N eff ( 4 11 ) + ( 3 -N eff ( 4 11 )) e -( cm e /T ) γ ] + 7 8 × 4 × ( e -( cm e /T ) γ + e -( cm µ /T ) γ + e -( cm τ /T ) γ ) 7 8 × 12 × ( e -( cm c /T ) γ + e -( cm b /T ) γ + e -( cm t /T ) γ ) , (A3) \ng C ( T ) ≈ 12 4 ( 2 × 1 9 + 4 9 ) e -(3 c Λ QCD /T ) γ + 12 4 × ( 4 9 e -( cm c /T ) γ + 1 9 e -( cm b /T ) γ ) + ( e -( cm e /T ) γ + e -( cm µ /T ) γ + e -( cm τ /T ) γ ) + ( 129 10 csc 2 θ W + 11 2 sec 2 θ W -20 3 ) e -( cT c /T ) γ . (A4) \nFIG. 7. A plot of the relativistic degrees of freedom given in Eq. A2 and Eq. A3 in blue and orange respectively. \n<!-- image --> \nThe constants are c = 1 / √ 6, γ = 3 / 2, N eff = 3 . 046, and T c = √ 2 /λν . \nWe take the number of quarks as a function of temperature to be [138] \nN q ( T ) ≈ 1 log((2 πT/ Λ QCD ) 2 ) ∑ q log (2 πT ) 2 + m 2 q Λ 2 QCD + m 2 q . (A5)', 'Appendix B: The joint scalar-Higgs potential': 'As the Higgs acquires a VEV, it will cause a persistent shift in the vacuum state of ϕ . In fact, the vacuum of the theory is determined as the minimum of the joint Higgs and scalar potential \nV ( H,ϕ ) = -ν 2 ( 1 + 2 d (1) ν ϕ √ 2 M pl ) H † H + λ ( 1 + d (1) λ ϕ √ 2 M pl ) ( H † H ) 2 + 1 2 m 2 ϕ ϕ 2 . (B1) \nIn the broken phase, H = (0 , v + h ) / √ 2, thermal fluctuations are negligible, and the classical expectation value of the fields can be straightforwardly computed by minimizing the joint potential. To leading order in the couplings, \n⟨ φ ⟩ = ( d (1) ν -1 4 d (1) λ ) ν 4 2 λM 2 pl m 2 ϕ , (B2) \nwhile the Higgs VEV v = ν/ √ λ is the same as its Standard Model value to second order in the dilaton couplings. \nThe present-day expectation value of ϕ is zero by convention, so to account for the shift in ground state above and below the electroweak phase transition, we must redefine the scalar field: \nϕ ≡ ˜ ϕ + ⟨ ϕ ⟩ . (B3) \nSince we are working to leading order in d (1) X ( φ ), the effect of this shift is to introduce a linear term in the scalar \npotential at high temperatures: \nV ( H, ˜ ϕ ) = -ν 2 ( 1 + 2 d (1) ν ˜ ϕ √ 2 M pl ) H † H + λ ( 1 + d (1) λ ˜ ϕ √ 2 M pl ) ( H † H ) 2 + 1 2 m 2 ϕ ( ˜ ϕ + ⟨ ϕ ⟩ ) 2 , (B4) \nwhile at low temperatures \nV ( h, ˜ ϕ ) = ( [ d (1) λ -2 d (1) ν ] ν 3 √ λ ˜ φ ) h + 1 2 m 2 h ( 1 + [ 3 2 d (1) λ -d (1) ν ] ˜ φ ) h 2 + µ 0 ( 1 + d (1) λ ˜ φ ) h 3 + 1 4 λ ( 1 + d (1) λ ˜ φ ) h 4 + 1 2 m 2 ϕ ˜ ϕ 2 , (B5) \nwhere m 2 h = 2 ν 2 , µ 0 = λv . At the electroweak phase transition T 2 ∼ 2 ν 2 /λ (up to contributions from gauge fields), the high temperature tadpole m 2 ϕ ⟨ ϕ ⟩ ˜ ϕ is exactly 2 / 3 the size of the contribution from the pressure, and contributes with the same sign. The mass mixing in Eq. B5 can be rotated away, and as a result, we see that ˜ ϕ couples to all the Standard Model particles in precisely the same as h , multiplied by a small mixing angle given in Eq. 95.', 'Appendix C: Scalar dynamics': 'In this section, we provide expressions for quantities relevant to the computation of the scalar relic density.', '1. Initial conditions': 'In our formalism described in Sec. III, the initial conditions of the scalar field are encoded in the coefficients A i and B i , which may be expressed in terms of the ϕ initial conditions \nϕ ( t i ) = M pl ( H eq 2 m ϕ ) 1 / 4 √ 3 π 2 ξ 1 / 4 i ( A i Y 1 / 4 ( ξ i ) -B i J 1 / 4 ( ξ i ) ) , (C1a) \nand \n˙ ϕ ( t i ) m ϕ = M pl ( H eq 2 m ϕ ) 1 / 4 √ 3 π 2 ξ 1 / 4 i ( -A i Y 5 / 4 ( ξ i ) + B i J 5 / 4 ( ξ i ) ) . (C1b)', '2. Overlap integrals': 'Here we compute asymptotic expressions for the overlap integrals relevant for estimating the effective misalignment angles displayed in Eqs. 63, 65 and 67.', 'a. Relativistic massive species': "For a source Eq. 62 that 'turns on' at ξ X,i ∼ T 2 X,i /M pl \n∫ ξ ξ X,i dξ ' ξ ' 1 / 4 Y 1 / 4 ( ξ ' ) ∝ ξ, ξ X,i ≪ ξ ≪ 1 , ξ -1 / 4 , ξ X,i ≪ 1 ≪ ξ, sin ( π 8 + ξ X,i ) ξ X,i 1 / 4 , 1 ≪ ξ X,i ≪ ξ. (C2a) \nand \n∫ ξ ξ X,i dξ ' ξ ' 1 / 4 J 1 / 4 ( ξ ' ) ∝ ξ 3 / 2 , ξ X,i ≪ ξ ≪ 1 , 4 √ 2Γ(3 / 4) √ π , ξ X,i ≪ 1 ≪ ξ, cos ( π 8 + ξ X,i ) ξ X,i 1 / 4 , 1 ≪ ξ X,i ≪ ξ. (C2b)", 'b. 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2024arXiv240907755P | Within the framework of symmetric teleparallel fleft Qright gravity for a connection defined in the noncoincidence gauge we derive the WheelerDeWitt equation of quantum cosmology. Because the gravitational field equation in fleft Qright gravity admits a minisuperspace description the WheelerDeWitt equation is a single inhomogeneous partial differential equations. We assume the powerlaw fleft Qright f0Qmu model and with the application of linear quantum observables we calculate the wavefunction of the universe. Finally we investigate the effects of the quantum correction terms in the semiclassical limit. | 2024-09-01T00:00:00Z | ['2024arXiv240907755P', 'arXiv:2409.07755', '10.48550/arXiv.2409.07755'] | ['General Relativity and Quantum Cosmology', 'High Energy Physics - Phenomenology'] | SemiClassical limit and quantum corrections in noncoincidence powerlaw fQCosmology | 2,024 | 167 | 0.22 | ['EPRINT_HTML', 'EPRINT_PDF'] | 0 | https://arxiv.org/pdf/2409.07755.pdf | {'Semi-Classical limit and quantum corrections in noncoincidence power-law f ( Q ) -Cosmology': "Andronikos Paliathanasis 1, 2, ∗ \n1 Institute of Systems Science, Durban University of Technology, Durban 4000, South Africa 2 Departamento de Matem'aticas, Universidad Cat'olica del Norte, \nAvda. Angamos 0610, Casilla 1280 Antofagasta, Chile \nWithin the framework of symmetric teleparallel f ( Q )-gravity for a connection defined in the non-coincidence gauge we derive the Wheeler-DeWitt equation of quantum cosmology. Because the gravitational field equation in f ( Q )-gravity admits a minisuperspace description the Wheeler-DeWitt equation is a single inhomogeneous partial differential equations. We assume the power-law f ( Q ) = f 0 Q µ model and with the application of linear quantum observables we calculate the wavefunction of the universe. Finally, we investigate the effects of the quantum correction terms in the semi-classical limit.", 'PACS numbers:': 'Keywords: Symmetric teleparallel f ( Q )-gravity; nonmetricity gravity; Wheeler-DeWitt equation; exact solutions;', '1. INTRODUCTION': 'Symmetric teleparallel general relativity [1-6] (STEGR) is a gravitational theory that is equivalent to General Relativity (GR). In STEGR, the geometry of the physical space is described by a metric tensor as in GR, but the autoparallels are defined by a symmetric and flat connection that inherits the symmetries of the metric tensor. This leads to different autoparallels than those of GR, which are constructed by the Levi-Civita connection. \nAssume that ˜ Γ κ µν is a general connection. This can be decomposed [7] as ˜ Γ κ µν = κ µ ν + \n2Γ κ [ µν ] +∆ κ µν , where κ µ ν is the Levi-Civita connection, Γ κ [ µν ] is the torsion tensor, and ∆ κ µν is the symmetric and flat nonmetricity part [8]. Since in STEGR, the connection is flat, it leads to a zero-valued curvature tensor R κ λµν = 0 and it is symmetric, i.e., T κ µν = 0. Thus, only the nonmetricity tensor survives, which leads to the nonmetricity scalar Q . The latter scalar substitutes the Ricci scalar of the Einstein-Hilbert Action Integral, leading to STEGR. As the Ricci scalar R and the nonmetricity scalar Q differ by a boundary term [1], it follows that STEGR is equivalent to GR. \nNowadays, GR is challenged by the analysis of recent cosmological data [9-14]. This has led the scientific community to introduce alternative and modified theories of gravity. Within the symmetry and teleparallel theory, the simplest modification is the f ( Q )-gravity [15, 16], where the Lagrangian function for the gravitational Action Integral is a nonlinear function of the nonmetricity scalar Q . In the linear limit of the function f , the STEGR theory, with or without the cosmological constant term, is recovered. f ( Q )-gravity is the analogue in the framework of STEGR of other modifiend f -theories defined by the Levi-Civita connection or in teleparalellism, see for instance [17-24] and references therein. \nThe dynamical degrees of freedom introduced by the nonlinear f ( Q ) can be attributed to scalar fields [25]. In the scalar field description, f ( Q )-theory is equivalent to a specific case of the scalar nonmetricity theory, where the scalar field is nonminimally coupled to gravity. Therefore, f ( Q )-gravity has properties similar to a Machian theory, even if the theory is not purely Machian. For more details, we refer the reader to [29, 30]. \nf ( Q )-theory suffers from two major problems in the cosmological perturbations of a spatially flat Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) background geometry. In particular, it suffers from strong coupling and the appearance of ghosts [31, 32]. Despite the fact that f ( Q )-theory fails to explain the global evolution of the universe, it provides a unique mechanism leading to the construction of different toy models with interesting applications in gravitational physics. See, for instance, [33-46] and references therein. For other modifications of the STEGR theory, we refer the reader to [47-50]. Moreover, f ( Q )-theory is the simplest mathematical theory which can be applied to understand the effects of the selection of the connection in modified symmetric teleparallel theories. \nThis investigation deals with quantum cosmology within the framwork of f ( Q )-theory and the effects of the quantum correction in the semiclassical limit [52]. Specifically, for \nthe FLRW geometry, we assume a connection defined in the noncoincidence gauge where the field equations admit a minisuperspace description. For this specific connection, the field equations are derived from a point-like Lagrangian with three dependent variables: the scale factor of the FLRW geometry and two scalar fields, which attribute the degrees of freedom provided by the definition of the connection and the nonlinear function f ( Q ). For this two-scalar field cosmological model, we write the Wheeler-DeWitt equation (WDW) of quantum cosmology [53, 54]. We employ the theory of symmetries of differential equations [55] to define quantum observables. These are applied to define similarity transformations and derive the wavefunction of the universe. \nIn the hydrodynamic description of quantum cosmology [56, 57], we make use of the quantum observables to write down conservation laws for the classical field equations and to calculate the classical solutions, because for the power-law f ( Q )-model the classical field equations form an superintegrable Hamiltonian system. The effects of the quantum correction terms related to the nonzero Ricci scalar term of the minisuperspace [53, 54], and of the Bohmian potential term [58-62], are discussed. It follows that the quantum corrections do not affect the general evolution and dynamics of the classical gravitational model. The structure of the paper is as follows. \nIn Section 2 we briefly discuss STEGR and its modification, the f ( Q )-gravity. This gravitational theory has a scalar field description, the theory is equivalent to a specific case of the scalar-nonmetricity gravitational model. Furthermore, the effects of conformal transformations in the theory are investigated, from where we can understand the degrees of freedom introduced by the theory. \nThe case of an isotropic and homogeneous universe in the framework of f ( Q )-gravity is presented in Section 3. We review previous results on the different families of connections in the FLRW background, and explore how the selection of the connection affects the gravitational model and the evolution of the physical space. Moreover, by applying scalar fields, we introduce the minisuperspace description for the field equations. \nFor the connection where the Lagrangian of the field equations is point-like, that is, it describes the motion of classical particles in a curved space under a conservative force term, in Section 4, we derive the WDW equation, which is a single inhomogeneous partial differential equation. For the power-law f ( Q ) model in Section 5, we present the quantum observables for the WDW equation. The latter are used to construct similarity transfor- \nmations aimed at reducing the WDW equation into an ordinary differential equation and providing a closed-form expression for the wavefunction of the universe. In Section 6, we employ the Madelung representation of quantum mechanics to derive the classical and semiclassical limits where quantum correction terms are introduced in the gravitational field equations. For the two models, we explicitly solve the reduced gravitational field equations using the Hamilton-Jacobi theory. It follows that the quantum correction terms do not affect the main dynamics of the classical solution. The two asymptotic limits of the classical cosmological solution describe the self-similar solution related to the power-law model, as well as a cosmological solution with two perfect fluids that do not interact and have constant equation of state parameters. Finally, in Section 7, we draw our conclusions.', '2. SYMMETRIC TELEPARALLEL f ( Q ) -GRAVITY': 'In this Section, we discuss the main properties and definitions of STEGR and introduce the gravitational model under our consideration, which is that of symmetric teleparallel f ( Q )-gravity.', '2.1. STEGR': "In the framework of STEGR, the physical space is described by a four dimensional metric tensor g µν , and the connection Γ κ µν which is symmetric, i.e. Γ κ µν = Γ κ νµ , and flat. Hence, there exists a point transformation x µ → x µ ' , such that all the components of the connection in the new coordinate system are zero, i.e. Γ κ ' µ ' ν ' = 0 [8]. \nBecause of these two fundamental properties for the connection Γ κ µν , the curvature tensor defined as \nR κ λµν (Γ) = ∂ Γ κ λν ∂x µ -∂ Γ κ λµ ∂x ν +Γ σ λν Γ κ µσ -Γ σ λµ Γ κ µσ , (1) \nand the torsion tensor which reads \nT κ µν (Γ) = 1 2 ( Γ κ µν -Γ κ νµ ) , (2) \nare always zero. \nThe fundamental scalar of STEGR is nonmetricity scalar [1] \nQ = Q κµν P κµν , (3) \nin which Q κµν is the nonmetricity scalar Q κµν = ∇ κ g µν , that is, \nQ κµν = ∂g µν ∂x κ -Γ σ κµ g σν -Γ σ κν g µσ , (4) \nand P κ µν is defined as follow \nP κ µν = 1 4 Q κ µν + 1 2 Q κ ( µ ν ) + 1 4 ( Q κ -˜ Q κ ) g µν -1 4 δ κ ( µ Q ν ) (5) \nwhere the vector fields Q µ and ¯ Q µ are given by the expressions \nQ µ = Q ν µν , ˜ Q µ = Q ν µν . (6) \nThe gravitational Action Integral in STEGR is defined [1] \nS = ∫ d 4 x √ -gQ. (7) \nThe nonmetricity scalar Q for the connection Γ κ µν and the Ricci scalar ˆ R ( ˆ Γ ) , for the LeviCivita connection ˆ Γ κ µν of the metric tensor \nˆ Γ κ µν = 1 2 g κλ ( g µκ,ν + g λν,µ -g µν,λ ) (8) \nare related as Q = ˆ R + B , in which B is a boundary term [6] \nB = -1 2 ˚ ∇ λ P λ , (9) \nand P λ = P λ µν ¯ g µν . Consequently, the variation of the Action Integral (7) leads to the field equations of Einstein's GR which follow by the Einstein-Hilbert Action. Finally, operator ˚ ∇ λ in (9) remarks the covariant derivative with respect to the Levi-Civita connection ˆ Γ κ µν (8)", '2.2. f ( Q ) -gravity': "The simplest modification of the STEGR Action Integral (7) is the introduction of the cosmological constant. Nevertheless, inspired by other modifications of GR, the introduction of nonlinear terms in the nonmetricity scalar Q in (7) leads to the introduction of dynamical degrees of freedom which can influence the dynamics and describe the cosmic evolution. \nIn this general concept we assume the gravitational Action Integral to be [15, 16] \nS f ( Q ) = ∫ d 4 x √ -gf ( Q ) , (10) \nwhere f ( Q ) is a smooth differentiable function, and when f '' ( Q ) = 0, the limit of STEGR with or without the cosmological constant term is recovered. With a prime ' we note total derivative with respect to the scalar Q , i.e. f ' ( Q ) = df ( Q ) dQ and f '' ( Q ) = d 2 f ( Q ) dQ 2 . \nThe modified gravitational field equations related to the Action Integral (10) follows from the variation with respect to the metric tensor g µν . \nThe gravitational field equations are [15, 16] \n2 √ -g ∇ λ ( √ -gf ' ( Q ) P λ µν ) -1 2 f ( Q ) g µν + f ' ( Q ) ( P µρσ Q ρσ ν -2 Q ρσµ P ρσ ν ) = 0 . (11) \nequivalently \nf ' ( Q ) G µν -1 2 g µν ( f ( Q ) -f ' ( Q ) Q ) + 2 f '' ( Q ) P λ µν ∇ λ Q = 0 , (12) \nwhere G µν is the equivalent of the Einstein tensor in STEGR, that is, \nG µν = ( P µρσ Q ρσ ν -2 Q ρσµ P ρσ ν ) + 2 √ -g ∇ λ ( √ -gP λ µν ) . (13) \nWhen Q = Q 0 , expression (12) reads \nG µν +Λ eff g µν = 0 , (14) \nwhere Λ eff is an effective cosmological constant term defined as Λ eff = 1 2 f ( Q 0 ) -f ' ( Q 0 ) Q 0 f ' ( Q 0 ) . \nTherefore, for the case where Q = Q 0 and f ( Q 0 ) -f ' ( Q 0 ) Q 0 = 0, the vacuum solutions \nof STEGR are recovered. On the other hand, for the case where Q = Q 0 and f ( Q 0 ) -f ' ( Q 0 ) Q 0 = 0, the effects of a nonzero cosmological constant are present in STEGR. \n/negationslash \nFurthermore, varying the Action Integral (10) with respect to the symmetric and flat connection Γ κ µν yields the equation of motion \n∇ µ ∇ ν ( √ -gf ' ( Q ) P µν σ ) = 0 . (15) \nThe connection is characterized as being defined in the 'coincidence gauge' when equation (15) is identically satisfied. On the other hand, we refer to the connection as being defined in the 'noncoincidence gauge' [15, 16]. \nThe transformation rule for the metric tensor follows that of a tensor field. However, the connection is not a tensor and thus has a different transformation rule. Specifically, the connection is coordinate dependent, though the tensors defined by the connection, such as the curvature tensor, the torsion tensor, and the nonmetricity tensor, are coordinateindependent. \nWhen we consider a specific line element for the metric tensor, this corresponds to the definition of a proper coordinate system. Thus, the components of the connection may not be identically zero in these coordinates, and equation (15) is not trivially satisfied. This means that dynamical degrees of freedom are introduced due to the connection defined in the non-coincidence gauge. These dynamical degrees of freedom associated with the noncoincidence gauge are of geometric origin. As we shall see in the following section, they can be attributed to scalar fields.", '2.3. Equivalency with scalar field theory': "We introduce the Lagrange multiplier λ m , such that the gravitational Action Integral (10) reads \nS f ( Q ) = ∫ d 4 x √ -g ( f ( Q ) + λ m ( Q -ˆ Q )) , (16) \nwhere ˆ Q = ˆ Q ( x κ ) and is the functional expression for the nonmetricity scalar Q . \nVariation with respect to the scalar Q , leads to the equation of motion δS f ( Q ) δQ = 0 , that is, λ m = -f ' ( Q ). \nBy replacing in expression (16) it follows \nS f ( Q ) = ∫ d 4 x √ -g ( f ' ( Q ) ˆ Q +( f ( Q ) -Qf ' ( Q )) ) . (17) \nWe introduce the scalar field φ = f ' ( Q ), and the potential function V ( φ ) = ( Qf ' ( Q ) -f ( Q )), and the latter Action Integral is expressed in the following simpler form \nS f ( Q ) = ∫ d 4 x √ -g ( φ ˆ Q -V ( φ ) ) . (18) \nThis is analogous to O'Hanlon gravity [63] in the framework of symmetric teleparallel theory. Hence, we can say that the nonmetricity f ( Q )-gravity has properties similar to those of a Machian theory [64], although the theory is not purely Machian [29, 30]. \nThe Action Integral (18) is a particular case of the more general theory with a scalar field minimally coupled to gravity, that is, of the scalar-nonmetricity theory defined as \nS STφ = ∫ d 4 x √ -g ( F ( φ ) 2 Q -ω ( φ ) 2 g µν φ ,µ φ ,ν -V ( φ ) ) , (19) \nwhere the gravitational field equations are \nF ( φ ) G µν +2 F ,φ φ ,λ P λ µν + g µν V ( φ ) + ω ( φ ) 2 ( g µν g λκ φ ,λ φ ,κ -φ ,µ φ ,ν ) = 0 . (20) \nThe equation of motion for the scalar field reads \nω ( φ ) √ -g g µν ∂ µ ( √ -g∂ ν φ ) + ω ,φ 2 g λκ φ ,λ φ ,κ + 1 2 F ,φ Q -V ,φ = 0 , (21) \nwhile the equation of motion for the connection is \n∇ µ ∇ ν ( √ -gF ( φ ) P µν σ ) = 0 . (22) \nWe remark that for a linear function F ( φ ) = 2 φ , and ω ( φ ) = 0, the f ( Q )-gravity is recovered, and the field equations (12) become \nφG µν +2 φ ,λ P λ µν + g µν V ( φ ) = 0 . (23) \nMoreover, equation (21) is simultaneously satisfied by the definition of the scalar field potential V ( φ ), while the equation of motion for the connection is simplified to \n∇ µ ∇ ν ( √ -gφP µν σ ) = 0 . (24)", '2.4. Conformal transformation': 'In the previous lines, we learned that f ( Q ) is a partially Machian theory, where, in the scalar field description, the theory is defined in the so-called Jordan frame. In the following lines, we discuss the effects of conformal transformations in f ( Q )-gravity and introduce the equivalent theory in the Einstein frame. \nLet the two conformally related four-dimensional metrics be ¯ g µν and g µν , defined as \n¯ g µν = e 2Ω( x κ ) g µν , ¯ g µν = e -2Ω( x κ ) g µν , (25) \nwhere Ω( x κ ) is a smooth differentiable function. Ω ( x κ ) is known as the conformal factor and defines the transformations. \nThe geometric quantities, which define the f ( Q )-gravity, for the conformally related metrics are related as [30] \n¯ Q λµν = e 2Ω Q λµν +2Ω ,λ ¯ g µν . (26) \nand \n¯ P λ = ¯ P λ µν ¯ g µν = e -2Ω P λ +3Ω ,λ . (27) \nTherefore, the nonmetricity scalars Q and ¯ Q are \n¯ Q = ¯ Q λµν ¯ P λµν = e -2Ω Q + ( 2Ω ,λ P λ +6Ω λ Ω ,λ ) . (28) \nAssume now the Action Integral (18) for the f ( ¯ Q ) -theory in the framework of the space with metric ¯ g µν . Then, the equivalent theory for the metric tensor g µν is given by the Action Integral [28] \n¯ S f ( ¯ Q ) = ∫ d n x √ -g ( Q -1 2 B ln φ + 3 2 φ g µν φ ,µ φ ,ν -V ( ϕ ) φ 2 ) , (29) \nwhere we have assumed that Ω = -1 2 ln φ , and scalar B is the boundary term which relates the Ricci scalar ˆ R ( ˆ Γ ) for the Levi-Civita connection with the nonmetricity scalar, given by expression (9). Indeed, the boundary term is defined as [28] B = -1 2 ˚ ∇ λ P, equivalently, [28] \nB = ( ˜ Q κ -Q κ ) . (30) \nWe observe that the boundary term B plays an important role in the conformal equivalent description of the theory, and it is another scalar field. The definition of the boundary term is directly related to the definition of the connection Γ and the equation of motion (15). \nAt this point, it is important to mention that the conformal equivalency of the gravitational theory in the two frames relates only to the trajectory solutions for the field equations. In general, it is not an equivalence of physical properties. Nevertheless, for the scalar-nonmetricity theory, it has been found that conformal transformations preserve the main eras of the cosmological history and evolution.', '3. ISOTROPIC AND HOMOGENEOUS COSMOLOGY': 'In cosmological scales the physical space is assumed that it is homogeneous and isotropic described by the FLRW line element. The latter in spherical coordinates is expressed as follows \nds 2 = -N ( t ) 2 dt 2 + a ( t ) 2 [ dr 2 1 -kr 2 + r 2 ( dθ 2 +sin 2 θdϕ 2 ) ] , (31) \nin which k is the spatial curvature of the three-dimensional space; N ( t ) is the lapse function, a ( t ) is the scale factor; and H = 1 N ˙ a a , ˙ a = da dt , is the Hubble function. We assume the comoving observer u µ = 1 N δ µ t , thus, the Hubble function is related to the expansion rate θ = u µ ; µ by the expression H = 1 3 θ .', '3.1. Symmetries': 'In the coordinate system ( t, r, θ, ϕ ), the six isometries of the line element (31) are as follows \nK 1 = sin ϕ∂ θ + cos ϕ tan θ ∂ ϕ , K 2 = -cos ϕ∂ θ + sin ϕ tan θ ∂ ϕ , K 3 = ∂ ϕ , \nK 4 = √ 1 -kr 2 ( sin θ cos ϕ∂ r + 1 r ( cos θ cos ϕ∂ θ -sin ϕ sin θ ) ∂ ϕ ) , \nK 5 = √ 1 -kr 2 ( sin θ sin ϕ∂ r + 1 r ( cos θ sin ϕ∂ θ + cos ϕ sin θ ) ∂ ϕ ) , K 6 = √ 1 -kr 2 ( cos θ∂ r -sin θ r ∂ ϕ ) . \nFor k = 0, the latter symmetry vectors form the E 3 ⊗ SO (2) Lie algebra, otherwise for k = 0, the six symmetry vectors are the elements of the SO (4) Lie algebra. \n/negationslash \nThe requirement the connection to inherit the symmetry of the background geometry provides the following set of constraints \nL K I Γ κ µν = 0 , (32) \nwhere L K I is the Lie derivative with respect the vector field K I and I = 1 , 2 , 3 , 4 , 5 , 6. \nThe definition of the Lie derivative depends on the transformation rule. Because the connection has a different transformation rule from that of tensor fields, the Lie derivative differs accordingly. Specifically, in terms of coordinates, the Lie derivative of the connection reads \nL K I Γ κ µν = K I κ , µν +Γ κ µν,r K I r -K I κ ,r Γ r µν + K I s ,µ Γ κ sν + K I s ,ν Γ κ µs . (33) \nBecause the connection is symmetric, expression (33) is simplified as \nL K I Γ κ µν = ∇ ν ∇ µ ( K I κ ) -R κ µνλ K I λ . (34) \nHowever, in symmetric teleparallel theory, the connection is flat, that is, \nR κ µνλ (Γ) = 0 . (35) \nonsequently, the requirement for the connection to inherit the symmetries of the background space is equivalent to the set of differential equations \n∇ ν ∇ µ ( K I κ ) = 0 . (36) \n/negationslash', '3.2. Symmetric and flat connection': 'For the FLRW line element (31) and the requirements (35), (36) it follows that there are found different families of symmetric connections Γ κ µν . One family of connection is defined for k = 0 and three families of connections are defined for k = 0 [65, 66]. \n/negationslash \nThe common nonzero components for the four families of connections are \nΓ r tr = Γ r rt = Γ θ tθ = Γ θ θt = Γ ϕ tϕ = Γ ϕ ϕt = k \nr - \n-γ ( t ) , Γ r rr = kr 1 -kr 2 , Γ r θθ = -r ( 1 -kr 2 ) , Γ r ϕϕ = -r sin 2 θ ( 1 -κr 2 ) , Γ θ rθ = Γ θ θr = Γ ϕ rϕ = Γ ϕ ϕr = 1 , Γ θ ϕϕ = sin θ cos θ , Γ ϕ θϕ = Γ ϕ ϕθ = cot θ. \nFor k = 0, the additional nonzero components for the family Γ k are \nΓ t tt = -k + ˙ γ ( t ) γ ( t ) , Γ t rr = γ ( t ) 1 -kr 2 Γ t θθ = γ ( t ) r 2 , Γ t ϕϕ = γ ( t ) r 2 sin 2 ( θ ) , \nby comparing the latter connections with the notation presented in [65], it follows that γ ( t ) = C 2 ( t ). \nMoreover, for k = 0 , connection Γ A has the additional nonzero components \nΓ t tt = γ ( t ) , \nwhere Γ A is connection Γ ( III ) Q in [65] and γ ( t ) = C 1 ( t ) . \nConnection Γ B is defined for k = 0, with the nonzero components \nΓ t tt = ˙ γ ( t ) γ ( t ) + γ ( t ) , Γ r tr = Γ r rt = Γ θ tθ = Γ θ θt = Γ ϕ tϕ = Γ ϕ ϕt = γ ( t ) , \nand it is connection Γ ( I ) Q of [65] with γ ( t ) = C 3 ( t ) \nFinally, for k = 0, family Γ C has the nonzero components \nΓ t tt = -˙ γ ( t ) γ ( t ) , Γ t rr = γ ( t ) , Γ t θθ = γ ( t ) r 2 , Γ t ϕϕ = γ ( t ) r 2 sin 2 θ, \nwhich is compared with connection Γ ( II ) Q of [65] and γ ( t ) = C 2 ( t ). \nThe connection Γ A is the unique connection defined in the coincidence gauge, and the function γ ( t ) plays no role in the gravitational field equations. Nevertheless, the remaining three families of connections, Γ k , Γ B , and Γ C , are defined in the noncoincidence gauge. In the framework of f ( Q )-gravity, the function γ ( t ) is constrained by the equation of motion (15).', '3.3. Field equations in f ( Q ) -gravity': "In the following lines, we present the field equations for f ( Q )-gravity in an FLRW background geometry. The selection of the connection is crucial in the theory; thus, for each family of connections, a different set of field equations results. \nFor nonzero spatial curvature and connection Γ k we calculate the nonmetricity scalar \nQ ( Γ k ) = -6˙ a 2 N 2 a 2 + 3 γ a 2 ( ˙ a a + ˙ N N ) + 3˙ γ a 2 + k ( 6 a 2 + 3 γN 2 ( ˙ N N + ˙ γ γ -3˙ a a )) . (37) \nThe gravitational field equations in f ( Q )-gravity are \n0 = 3 f ' ( Q ) H 2 + 1 2 ( f ( Q ) -Qf ' ( Q )) -3 γ ˙ Qf '' ( Q ) 2 a 2 +3 k ( f ' ( Q ) a 2 -˙ Qf '' ( Q ) 2 γN 2 ) , (38) \n0 = -2 N ( f ' ( QH ) · -3 H 2 f ' ( Q ) -1 2 ( f ( Q ) -Qf ' ( Q )) + γ ˙ Qf '' ( Q ) 2 a 2 -k ( f ' ( Q ) a 2 + 3 ˙ Qf '' ( Q ) 2 γN 2 ) , (39) \nand the equation of motion for the connection \n0 = ( f '' ˙ Q ) · ( 1 + k a 2 N 2 γ 2 ) + f '' ˙ Q ( ( 1 + 3 k a 2 N 2 γ 2 ) NH + ( 1 -k a 2 N 2 γ 2 ) ˙ N N +2 ˙ γ γ ) . (40) \nWe remark that in the previous field equations, if we set k = 0, we recover the gravitational field equations for the connection Γ C . \nFurthermore, for connection Γ A defined in the coincidence gauge, the nonmetricity scalar \nis \nQ ( Γ A ) = -6 H 2 , (41) \nand the gravitational field equations are \n0 = 3 H 2 f ' ( Q ) + 1 2 ( f ( Q ) -Qf ' ( Q )) , (42a) \n0 = -2 N ( f ' ( Q ) H ) · -3 H 2 f ' ( Q ) -1 2 ( f ( Q ) -Qf ' ( Q )) . (42b) \nFinally, for the family of connection Γ B the nonmetricity scalar reads \nQ = -6 H 2 + 3 γ N ( 3 H -˙ N N 2 ) + 3˙ γ N 2 , (43) \nwhile the gravitational field equations are \n0 = 3 H 2 f ' ( Q ) + 1 2 ( f ( Q ) -Qf ' ( Q )) + 3 γ ˙ Qf '' ( Q ) 2 N 2 (44a) \n0 = -2 N ( f ' ( Q ) H ) · -3 H 2 f ' ( Q ) -1 2 ( f ( Q ) -Qf ' ( Q )) + 3 γ ˙ Qf '' ( Q ) 2 N 2 . (44b) \nBecause Γ B is defined in the noncoincidence gauge, the equation of motion for the connection is as below \n0 = ( f '' ˙ Q ) · + N ˙ Q ( 3 H -˙ N N 2 ) f '' ( Q ) . (45)", '3.4. Minisuperspace description': "A novel property of these cosmological models is that they admit a 'minisuperspace' description. In particular, for each connection, there exists a Lagrangian function whose variation leads to the corresponding field equations. \nFor connections Γ k (and Γ C for k = 0) the corresponding field equations follows from the variation of the (non-canonical) Lagrangian function \nL ( Γ k ) = -3 N aφ ˙ a 2 +3 kNaφ -N 2 V ( φ ) + 3 k 2 N a 3 ˙ Ψ ˙ φ -3 2 aN ˙ Ψ ˙ φ, (46) \nin which φ = f ' ( Q ), V ( φ ) = ( Qf ' ( Q ) -f ( Q )) and ˙ Ψ = 1 γ . We observe that the latter \nLagrangian has kinetic components in the denominator. \nSimilarly, for the connection Γ A the Lagrangian function is \nL ( Γ A ) = -6 N aφ ˙ a 2 -Na 3 V ( φ ) . (47) \nFinally for connection Γ B the corresponding Lagrangian of the field equations is \nL ( Γ B ) = -3 N φa ˙ a 2 -3 2 N a 3 ˙ φ ˙ ψ -N 2 a 3 V ( φ ) , (48) \nwhere now the scalar field ψ is related to the connection as ˙ ψ = γ . Lagrangian function (48) is of the form of a point-like dynamical system. In particular, it describes a constraint dynamical systems where the scale factor and two scalar fields play the role of the particles with interaction U eff = a 3 V ( φ ). \nThe existence of the lapse function in the Lagrangians is necessary in order to reconstruct the constraint equation in each case. \nBetween the above Lagrangian functions, only these who correspond on connections Γ A and Γ B describe dynamical system where expressed in the form L = 1 N K E -NU eff , where K is the kinetic energy, K E = 1 2 G AB ˙ q A ˙ q B and U eff is the effective potential. Thus, for these two dynamical systems we can write the Hamiltonian function in the form \nH ≡ N ( K E + U eff ) = 0 . (49) \nWe make use of this property in order to continue with the quantization of the field equations. The quantization of the field equations which correspond to the connection Γ A has been studied in detailed in . Because of the nature of the Lagrangian, Dirac's method for the quantization of constraint dynamical systems was applied in [67]. Due to the existence of second class constraints the quantization process was different from the usual WDW formalism, where only first class constraints exist. See also the more recent study [68]. \nThe field equations for the connection Γ B described by the point-like Lagrangian function (48) possess only first class constraints, which means that the quantization process is similar to that of the WDW formalism and different from that of the corresponding system for the connection Γ A .", '4. THE WHEELER-DEWITT EQUATION': 'In GR the Wheeler-DeWitt equation follows from the quantization of the constraint equation in the ADM formalism (for discussion we refer the reader to [53, 54, 69]). The WDW equation is a hyperbolic functional differential equation on superspace and represents a family of differential equations at different points. However, when there exist a minisuperspace description, the WDW reduces to a single differential equation. \nIn the minisuperspace description the Hamiltonian constraint is expressed as \nH = N [ 1 2 G AB P A P B + U ( q ) ] = N H ≡ 0 , (50) \nin which G AB is the minisuperspace. \nThe classical field equations are invariant under conformal transformations. This property should also hold for the WDW equation of quantum cosmology. The quantization P A = i /planckover2pi1 ∂ A leads to a differential equation of Klein-Gordon type, which is generally not conformally invariant. Therefore, the Klein-Gordon equation is replaced by the Yamabe equation to ensure conformal invariance. This is achieved by modifying the potential with the term n -2 8( n -1) R , where R is the Ricci scalar of the minisuperspace G AB with dimension n . \nThe WDW equation reads \nH Ψ = [ /planckover2pi1 2 ( 1 2 ∆ -n -2 8( n -1) R ) -U ( q ) ] Φ( q ) = 0 , (51) \nin which ∆ = 1 √ -G ∂ A ( √ -GG AB ∂ B ) is the Laplace operator with respect to the minisuperspace G AB . \nAs we shall see in the following a nonzero Ricci scalar R introduces to the Hamiltonian function of the semiclassical system a potential term which correspond to quantum corrections.', '4.1. Connection Γ B': 'From Lagrangian function (48) we calculate the three-dimensional minisuperspace \nG AB = 6 φa 0 0 0 0 3 2 a 3 0 3 2 a 3 0 , G AB = 1 6 φa 0 0 0 0 2 3 a 3 0 2 3 a 3 0 . (52) \nThus, the corresponding Hamiltonian function is expressed as \nH = 1 12 φa P 2 a + 2 3 a 3 P φ P ψ -a 3 V ( φ ) , (53) \nwith \nP a = 6 φa ˙ a , P φ = 3 2 a 3 ˙ ψ and P ψ = 3 2 a 3 ˙ φ . (54) \nMoreover, the Ricci scalar for the minisuperspace is derived \nR = -3 4 a 3 φ , (55) \nTherefore, the WDW equation (51) for the Hamiltonian function (53), Φ = 0, is expressed \nH as \n1 6 aφ Φ ,aa + 4 3 a 3 Φ ,φψ + 5 12 a 2 φ Φ ,a + 1 3 φa 3 Φ ,ψ + ( 3 32 a 3 φ -2 /planckover2pi1 2 a 3 V ( φ ) ) Φ = 0 . (56)', '4.2. Power-law theory': 'We assume that the function f ( Q ) is a power-law, i.e., f ( Q ) = f 0 Q µ . The power-law function provides a cosmological history that depends on the selection of the connection [70]. However, for all connections, the power-law f ( Q ) function is associated with the existence of self-similar cosmological solutions [71]. For the cosmological model defined by the connection Γ B , the self-similar scaling solution is always unstable, while the unique attractor describes the accelerated de Sitter universe. This physical property remains the same in the presence of matter [74]. \nIn the scalar field description of the f ( Q ) theory, the power-law model f ( Q ) = f 0 Q µ leads to the power-law potential function V ( φ ) = V 0 φ κ , where constants V 0 and κ are related with \nf 0 , µ as \nV 0 = ( µ -1) f -1 µ -1 0 µ -µ µ -1 , κ = µ µ -1 . (57) \nRecall that for values of µ close to 1, the gravitational model describes small deviations from General Relativity. We refer the reader to the interesting discussion on the power-law f ( R ) theory presented in [75]. \nTherefore, the WDW equation (56) for the power-law model is simplified as \n1 6 aφ Φ ,aa + 4 3 a 3 Φ ,φψ + 5 12 a 2 φ Φ ,a + 1 3 a 3 φ Φ ,ψ + ( 3 32 a 3 φ -V 0 /planckover2pi1 2 a 3 φ κ ) Φ = 0 . (58) \nThe solution of the WDW equation (58) provides the wavefunction Φ ( a, φ, ψ ) for this specific cosmological model. \nThe WDW equation is a linear inhomogeneous partial differential equation. In the following lines we employ the method of similarity transformations, also known as Lie symmetry analysis, such that to determine expressions for the wavefunction Φ which satisfy the partial differential equation (58).', '5. SIMILARITY TRANSFORMATIONS': 'Lie symmetry analysis is a robust method for the study of nonlinear differential equations. It provides a systematic way to investigate the algebraic properties and compute solutions for nonlinear and inhomogeneous differential equations [72, 73]. The main characteristic of Lie symmetry analysis is that it allows for the construction of one-parameter point transformations which leave the given differential equation invariant. From these point transformations, it is possible to derive similarity transformations used to facilitate solutions. This procedure is known as the reduction process, and in the case of partial differential equations, the existence of a Lie symmetry is equivalent to the existence of a quantum operator. \nFor a review on the wide applications of symmetry analysis in classical gravitational physics and cosmology, we refer the reader to [55]. In the framework of the WDW equation, the mathematical approach for constructing quantum operators from Lie symmetry analysis is analytically described in [76]. This method has been widely used for constructing the wavefunction of the universe in a broad range of gravitational models [77-84]. \nThe application of the Lie symmetry analysis for the WDW equation (58), i.e. the \nYamabe equation, leads to the following quantum operators \nΞ 1 : ( ∂ ψ -β 1 /planckover2pi1 i ) Φ = 0 , (59) Ξ 2 : ( -( κ +1) 6 a∂ a + φ∂ φ -β 2 /planckover2pi1 i ) Φ = 0 , (60) \nΞ 3 : ( ( κ +1) 6 ln φa∂ a -ln φ φ∂ φ + ( ψ -2 ( κ +1) 3 ln a ) ∂ ψ -( ( K -1) 8 ln φ + β 3 /planckover2pi1 i )) Φ = 0 . (61) \nWith the use of the latter operators we are able to construct the following solution for the wavefunction of the universe, \nΦ( a, φ, ψ ) = a -3 4 + 2 3 i /planckover2pi1 β 1 ( κ +1) φ -κ +1 8 + i /planckover2pi1 λ e -i /planckover2pi1 β 1 ψ ( Φ 1 0 J ¯ λ ( 2 3 /planckover2pi1 i √ 3 V 0 a 3 φ κ +1 2 ) +Φ 2 0 Y ¯ λ ( 2 3 /planckover2pi1 i √ 3 V 0 a 3 φ κ +1 2 )) , (62) \nin which J ¯ λ , Y ¯ λ are the Bessel functions of the first and second kind respectively; and constants λ, ¯ λ are defined as \nλ = 1 9 ( β 1 +9 β 2 + β 1 κ (2 + κ )) , (63) \n¯ λ = i 9 √ β 1 /planckover2pi1 ( 4 β 1 /planckover2pi1 κ ( κ +2) + 9 i ( κ -1) -72 β 2 /planckover2pi1 ) . (64) \nWavefunctions of the form in expression (62) have been explicitly derived before in the literature [83, 84]. Thus, the same analysis for the construction of a Hilbert space can be applied. We conclude the discussion here and shift our focus to the classical limit. \nRecall that for large values of the argument of the Bessel functions, the asymptotic solution is \nΦ( a, φ, ψ ) /similarequal a -3 2 + 2 3 i /planckover2pi1 β 1 ( κ +1) φ -κ +1 4 + i /planckover2pi1 λ e -i /planckover2pi1 β 1 ψ ( ¯ Φ 1 0 cos ( K ) + Φ 2 0 sin ( K ) ) , (65) \nwith K = 2 3 /planckover2pi1 i √ 3 V 0 a 3 φ κ +1 2 -¯ λπ 2 -π 4 ; while in the limit a 3 φ κ +1 2 → 0, the asymptotic behaviour of the wavefunction reads \nΦ( a, φ, ψ ) /similarequal a -3 4 + 2 3 i /planckover2pi1 β 1 ( κ +1)+3 ¯ λ φ -κ +1 8 + i /planckover2pi1 λ -κ +1 2 ¯ λ e -i β 1 /planckover2pi1 ψ . (66) \nIn the following lines, we focus on the analysis of the semiclassical limit of quantum cosmology.', '6. SEMICLASSICAL LIMIT': 'In the Madelung representation [56] of quantum mechanics, that is, in the hydrodynamic approach, we write the wavefunction in the for Φ ( q ) = Ω ( q ) e i /planckover2pi1 S ( q ) , where Ω is the amplitude of the wavefunction. By replacing in the WDW equation (51) and separating the real and imaginary parts we end with the \n1 2 G AB ( ∂S ∂q A )( ∂S ∂q B ) + U ( q ) + /planckover2pi1 2 ( n -2 8( n -1) R1 2Ω ∆(Ω) ) = 0 . (67) \nIn the limit where /planckover2pi1 2 → 0, that is, in the WKB approximation, the Hamilton-Jacobi equation for the classical gravitational field equations is recovered. \nThe new term V Q = -1 2Ω ∆(Ω) is known as the quantum potential in the de Broglie-Bohm representation of quantum mechanics [58, 59] and it depends only on the amplitude of the wavefunction. Furthermore, from the above expression it is clear how the curvature of the minisuperspace contributes in the semiclassical limit. \nThe observables, Q 1 , Q 2 and Q 3 , lead to the conservation laws for the classical system \nI 1 = P ψ , (68) \nI 2 = -( κ +1) 6 aP a + φP φ , (69) \nI 3 = ln φ ( ( κ +1) 6 aP a -φP φ ) + ( ψ -2 ( κ +1) 3 ln a ) P ψ -( K -1) 8 ln φ. (70) \nin which P a = ∂S ∂a , P φ = ∂S ∂φ and P ψ = ∂S ∂ψ .', '6.1. Classical solution': 'The Hamilton-Jacobi equation for the classical system reads \n1 6 aφ ( ∂S ∂a ) 2 + 4 3 a 3 ( ∂S ∂φ )( ∂S ∂ψ ) +2 V 0 a 3 φ κ = 0. \nA 1 ( I 1 , I 2 , κ ) = 2 3 ( κ +1) ( I 1 ( κ +1) 2 -√ ( κ +1) 2 ( I 1 ( ( κ +1) 2 I 1 -18 I 2 )) ) , A 2 ( I 1 , I 2 , κ ) = I 1 ( κ +1) 2 -9 I 2 -√ ( κ +1) 2 ( I 1 ( ( κ +1) 2 I 1 -18 I 2 )) . \nBy using the conservation laws we determine the closed-form solution for the action S ( a, φ, ψ ), that is, \nS ( a, φ, ψ ) = 1 9 ∫ 9 I 2 + √ ( κ +1) 2 ( I 1 ( ( κ +1) 2 I 1 -18 I 2 ) -27 V 0 a 6 φ κ +1 ) -( κ +1) 2 I 1 φ dφ + 2 3 ( κ +1) ∫ √ ( κ +1) 2 ( I 1 ( ( κ +1) 2 I 1 -18 I 2 ) -27 V 0 a 6 φ κ +1 ) -( κ +1) 2 I 1 a da +9 U 0 ( κ +1) ∫ ∫ φ κ √ ( κ +1) 2 ( I 1 ( ( κ +1) 2 I 1 -18 I 2 ) -27 V 0 a 6 φ κ +1 ) dφ da + I 1 ψ. (71) \nThus \nP a = -2 ( I 1 ( κ +1) 2 -√ ( κ +1) 2 ( I 1 ( ( κ +1) 2 I 1 -18 I 2 ) -27 V 0 a 6 φ κ +1 ) ) 3 a ( κ +1) , (72) \nP φ = -I 1 ( κ +1) 2 -9 I 2 -√ ( κ +1) 2 ( I 1 ( ( κ +1) 2 I 1 -18 I 2 ) -27 V 0 a 6 φ κ +1 ) 9 φ , (73) \nP ψ = I 1 , (74) \nBy replacing the momentum terms from (54) we end with a system of three nonlinear firstorder differential equations. \nIn the limit where a 6 φ κ +1 → 0, the field equations are described by the following system (we have assumed the lapse function to be a constant) \nφa 2 ˙ a /similarequal -A 1 ( I 1 , I 2 , κ ) 6 , φa 3 ˙ ψ /similarequal -2 A 2 ( I 1 , I 2 , κ ) 27 , a 3 ˙ φ /similarequal 2 3 I 1 . \nwhere \nWe write the equivalent system \n∫ 1 φ dφ /similarequal 4 I 1 A 1 ∫ 1 a da, ∫ dψ /similarequal 4 A 2 9 A 1 ∫ 1 a da, (75) \nwhere the analytic solution is terms of the scale factor reads \nφ ( a ) /similarequal a 4 I 1 A 1 , ψ ( a ) /similarequal 4 A 2 9 A 1 ln a . (76) \nOn the other hand, when the term a 6 φ κ +1 dominates we end with the following system \n˙ a /similarequal -√ -3 2 V 0 6 ( κ +1) aφ κ -1 2 , (77) \n˙ ψ /similarequal -√ -2 V 0 √ 27 φ κ -1 2 , (78) \n˙ φ /similarequal 2 3 I 1 a -3 . (79) \nThe solution is real only when V 0 < 0. Thus, in this case in a similar way as before we end with the reduced system \n∫ φ κ -1 2 dφ /similarequal -4 √ 2 ( κ +1) 3 | V 0 | I 1 ∫ a -4 da , (80) \n√ ψ ( a ) /similarequal ∫ 4 ( κ +1) 3 φ κ -1 2 1 a da . (81) \nwhich means that the asymptotic solution is \nφ ( a ) /similarequal ( -4 √ 2 ( κ +1) I 1 √ 3 | V 0 | ) 2 κ +1 a -6 κ +1 , ψ ( a ) /similarequal a 3 1 -κ 1+ κ . (82) \nRecall that this solution is valid when a 3 φ κ +1 2 → 0, i.e. ( -4 √ 2( κ +1) I 1 √ 3 | V 0 | ) 2 κ +1 → 0.', '6.1.1. Physical properties of the asymptotic solutions': 'In the following lines we investigate the physical properties of the asymptotic solutions derived before. \nFrom the field equations of connection Γ B , it follows that \nH 2 = 1 6 φ ( V ( φ ) -3 ˙ φ ˙ ψ ) (83) \nHence, for the asymptotic solution at the limit a 6 φ κ +1 → 0, we derive \nH 2 /similarequal H 1 0 ( I 1 , I 2 , κ ) a -6 -8 I 1 I 2 + H 1 0 ( I 1 , I 2 , κ, V 0 ) a 4 I 1 I 2 (1+ κ ) -8 I 1 I 2 . (84) \nThis corresponds to a cosmological solution where the effective fluid in the framework of STEGR corresponds to two perfect fluids with constant equation parameters. \nOn the other hand, in the limit where the term a 6 φ κ +1 dominates the asymptotic behavior for the Hubble function reads \nH 2 /similarequal ¯ H 1 0 ( I 1 , I 2 , κ, V 0 ) a -6+ 12 1+ κ . (85) \nThe latter describes the perfect fluid solution with constant equation of state parameter, which leads to a self-similar spacetime [71]. The equation of state parameter is defined as w eff = 3 -8 1+ κ , from where it follows that acceleration is occurred when -1 < κ < 5 3 .', '6.2. Quantum potential': 'We determine the solution of the field equations in the semiclassical regime where the quantum effects take place and affects the cosmological dynamics. \nFrom the wavefunction (62) we calculate the quantum potential \nV Q ( a, φ, ψ ) = V 0 Q ( I 1 , I 2 , κ ) φa 3 , (86) \nin which V 0 Q ( I 1 , I 2 , κ ) is a constant related to the parameters I 1 , I 2 and κ . \nConsequently, the Hamilton-Jacobi equation (67) becomes \n1 6 aφ ( ∂ ˆ S ∂a ) 2 + 4 3 a 3 ( ∂ ˆ S ∂φ )( ∂ ˆ S ∂ψ ) +2 V 0 a 3 φ κ + 2 ¯ V 0 Q ( I 1 , I 2 , κ ) a 3 φ /planckover2pi1 2 = 0 . \nThe conservation laws presented before are also conservation laws and for the latter Hamil- \ntonian system with the quantum correction. \nThus, the action reads \nψ. \nˆ S ( a, φ, ψ ) = 1 9 ∫ 9 I 2 + √ ( κ +1) 2 ( I 1 ( ( κ +1) 2 I 1 -18 I 2 ) -27 V 0 a 6 φ κ +1 -27 ¯ V 0 Q /planckover2pi1 2 ) -( κ +1) 2 I 1 φ dφ + 2 3 ( κ +1) ∫ √ ( κ +1) 2 ( I 1 ( ( κ +1) 2 I 1 -18 I 2 ) -27 V 0 a 6 φ κ +1 -27 ¯ V 0 Q /planckover2pi1 2 ) -( κ +1) 2 I 1 a da +9 U 0 ( κ +1) ∫ ∫ φ κ √ ( κ +1) 2 ( I 1 ( ( κ +1) 2 I 1 -18 I 2 ) -27 V 0 a 6 φ κ +1 -27 ¯ V 0 Q /planckover2pi1 2 ) dφ da + I 1 (87) \nWe observe that the latter action is of the same function form of (71) resulting the same gravitational field equations as before, with a rescale on the constants I 1 , I 2 and κ . \nConsequently, the behaviour of the semiclassical solution is similar with that derived before for the classical solution without the quantum correction term.', '7. CONCLUSIONS': 'In this study, we reviewed the basic mathematical properties of nonmetricity f ( Q )-gravity. This gravitational model represents the simplest generalization of STEGR by introducing nonlinear terms of the nonmetricity scalar Q into the gravitational Action Integral. f ( Q )-gravity is a particular case of the more general nonmetricity-scalar theory, where a scalar field is nonminimally coupled to gravity. f ( Q )-gravity exhibits Machian properties, even though it is not purely Machian. Due to this characteristic, we discussed the effects of conformal transformations on the gravitational Action Integral and introduced the analogy of the Jordan and Einstein frames in nonmetricity theory. \nWithin the cosmological framework of FLRW geometry, the theory provides four different sets of gravitational field equations. This arises because the connection in STEGR theory is not uniquely defined. The choice of connection leads to the introduction of geometrodynamical degrees of freedom in the field equations, which affects the dynamics of the cosmological parameters. For the coordinate system where the line element for FLRW is expressed in the usual form of (31), there is a family of connections defined in the so-called coincidence gauge. In this case, the connection depends on a gauge function, which does not \naffect the cosmological dynamics. However, for the remaining three families of connections, the geometrodynamical degrees of freedom of the resulting gravitational equations can be attributed to two scalar fields. These two scalar field have a geometric origin related to the dynamical degrees of freedom introduced by the nonlinear function f , and the degrees of freedom introduced by the connection. \nFor the connection where the two scalar fields define a canonical kinetic term, such that the gravitational field equations admit a minisuperspace description, we employed the quantization process to derive the WDW equation of quantum cosmology. We considered the power-law f ( Q ) = f 0 Q µ gravity, thus quantum observables were calculated using the theory of Lie symmetries. In this consideration we focused in a multi-scalar field cosmological model, either if the origin of the scalar field is geometric. \nThe quantum operators constructed by the Lie symmetry analsysis were used to reduce the WDW equation and write a closed-form solution for the wavefunction. We focused on the effects of quantum correction terms in the semiclassical limit. Therefore, we calculated the action by solving the Hamilton-Jacobi equation for the classical system, both with and without quantum correction terms. From the derivation of the Action we were able to write the equivalent reduced classical system. \nWe found that the quantum corrections do not affect the general evolution of the classical solution. However, the quantum potential can rescale the integration constants of the classical solutions. This is an important observation because it states that the initial value problem can be overcome by using the quantum corrections in the semi-classical limit. \nAlthough f ( Q )-cosmology is challenged by strong coupling and the presence of ghosts, this geometric model provides a mathematical framework for a better understanding of the effects of connection choice in gravitational physics. Nonmetricity f ( Q )-theory distinguishes the connection from the metric, in contrast to GR, leading to new physics. 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2024arXiv240904391D | We present a procedure for calculating the heating of and the infrared emission from dust in a homogeneous spherical shell surrounded by a spherically symmetric source of radiation. The results are applicable to newly formed dust either in supernova ejecta or in the circumstellar medium that has been swept up by the expanding shock wave. They can also be applied to the heating and IR emission from dust in clumps or clouds embedded in a homogeneous radiation field. | 2024-09-01T00:00:00Z | ['arXiv:2409.04391', '2024arXiv240904391D', '10.48550/arXiv.2409.04391'] | ['Astrophysics - Astrophysics of Galaxies'] | The External Heating of Dust in a Homogeneous Spherical Shell | 2,024 | 167 | 0.5 | ['EPRINT_HTML', 'EPRINT_PDF'] | 1 | https://arxiv.org/pdf/2409.04391.pdf | {'THE EXTERNAL HEATING OF DUST IN A HOMOGENEOUS SPHERICAL SHELL': 'Eli Dwek 1, 2 and Richard G. Arendt 3, 4, 5 \n1 Emeritus, Observational Cosmology Lab, NASA Goddard Space Flight Center, Mail Code 665, Greenbelt, MD 20771, USA 2 Research Fellow, Center for Astrophysics - Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA 3 Center for Space Sciences and Technology, University of Maryland, Baltimore County, Baltimore, MD 21250, USA 4 Code 665, NASA/GSFC, 8800 Greenbelt Road, Greenbelt, MD 20771, USA 5 Center for Research and Exploration in Space Science and Technology, NASA/GSFC, Greenbelt, MD 20771, USA', 'ABSTRACT': 'We present a procedure for calculating the heating of, and the infrared emission from, dust in a homogeneous spherical shell surrounded by a spherically symmetric source of radiation. The results are applicable to newly formed dust either in supernova ejecta or in the circumstellar medium that has been swept up by the expanding shock wave. They can also be applied to the heating and IR emission from dust in clumps or clouds embedded in a homogeneous radiation field.', '1. INTRODUCTION': 'A recurring astrophysical scenario is that of a spherically symmetric dusty shell that is heated by an external spherically symmetric source of radiation. Examples are dust in the expanding ejecta of a supernova (SN) or dust in the surrounding circumstellar medium (CSM) that is heated by the radiation from an expanding shock wave, or a spherical dusty clump embedded in a homogeneous radiation field (e.g. Sarangi et al. 2018; Sarangi & Slavin 2022). Here we present a procedure for calculating the heating of dust behind a spherically symmetric shock. The analytical equations presented here provide useful insight into the many variables of the problem, and in particular the morphology of the shell for different size cavities and shock proximity. We currently ignore the scattering of the incident radiation in the shell. \nFigure 1a depicts a spherical dusty shell of radius R 2 , containing a spherical cavity of radius R 1 , surrounded by a radiating shock at distance R s from the center, O , of the shell. We consider the intensity of radiation seen by a dust grain located at position A at distance r from O . For evaluating the radiation incident at point A , we adopt a spherical coordinate system centered at A , with the polar axis oriented vertically in the P direction as shown in the figure.', '2. THE INTENSITY OF THE RADIATIVE SHOCK': "We first calculate the unattenuated flux incident on the grain arriving in a cone at angle θ from the polar axis. We assume that the nebula is heated by a shock with a specific luminosity L ν ( λ ), at wavelength λ , where \nL ν ( λ ) = 4 πR 2 s ℓ ϵ ν ( λ, T ) for an optically thin shock =4 πR 2 s π B ν ( λ, T ) for an optically thick shock , (1) \nwhere ℓ = BD and ϵ ν ( λ, T ) [erg s -1 cm -3 Hz -1 ] are, respectively, the radial thickness and specific emissivity of the shock in the optically thin case. The specific luminosity and related quantities are temperature dependent. The specific intensity, or brightness of the shell I ν ( λ ) [erg s -1 cm -2 ster -1 Hz -1 ], is given by \nI ν ( λ ) = ( 1 4 π ) L ν ( λ ) 4 πR 2 s (2) \nThe unattenuated specific flux incident on the grain, F ν [erg s -1 cm -2 Hz -1 ] , is \nF ν ( λ ) = Ω [ I ν ( λ ) / cos α ] \nfor an optically thin shock \[email protected] \n=Ω I ν ( λ ) for an optically thick shock , (3) \nwhere Ω = sin θ dθ dϕ is a unit solid angle of the cone. In the optically thin case the cos α factor takes into account that the path length ℓ ' traversed through the shock is always larger than its radial thickness, and is given by ℓ ' ≈ ℓ/ cos α , where α = ∠ ABO . The angle α is a function of r , θ , and R s and given by \ncos α = [1 -( r sin θ/R s ) 2 ] 1 / 2 . (4) \nIn the optically thick case, the incoming flux does not depend on the angle α , since the cos α reduction in the flux projected into the cone is offset by the factor of 1 / cos α increase in the shock area seen by the dust. \nThe unattenuated specific flux incident on the grain is given by \nF ν ( R s , r, θ, ϕ, λ, τ = 0)= Ω 4 π ϵ ν ( λ ) ℓ [1 -( r sin θ/R s ) 2 ] 1 / 2 for an optically thin shock , = Ω 4 π L ν ( λ ) 4 πR 2 s for an optically thick shock . (5)", '3. EXTINCTION IN THE DUSTY SHELL': "The attenuated flux incident on the dust from the θ direction is \nF ν ( R s , R 1 , R 2 , r, θ, ϕ, λ, τ ) = F ν ( R s , r, θ, ϕ, λ, τ = 0) e -τ ( R 1 ,R 2 ,r,θ,ϕ,a,λ ) , (6) \nThe optical depth, τ , along the path, t , is given by \nτ ( R 1 , R 2 , r, θ, ϕ, a, λ ) = κ ( a, λ ) ρ d t ( R 1 , R 2 , r, θ, ϕ ) , (7) \nwhere κ ( a, λ ) = σ gr ( a, λ ) /m gr is the mass absorption coefficient of the dust at wavelength λ , σ ( a, λ ) = π a 2 Q abs ( a, λ ) is the grain's cross section, Q abs is the absorption coefficient of the dust, m gr = 4 πρ gr a 3 / 3 is the grain's mass, ρ gr is its mass density, a its radius, and ρ d is the mass density of the dust in the shell. \nThe path length t ≡ AE (Figure 1b) may, or may not, traverse the cavity, depending on the direction of the dust viewing angle θ . The critical angle θ c , which defines the boundary between the two cases, is \nθ c = π -arcsin( R 1 /r ) . (8) \nThe path length through the nebula is then given by \nt ( R 1 , R 2 , r, θ, ϕ ) = -r cos θ + ( R 2 2 -h 2 ) 1 / 2 for θ < θ c = -r cos θ + ( R 2 2 -h 2 ) 1 / 2 for θ > θ c (9) -2 ( R 2 1 -h 2 ) 1 / 2 . \nwhere h = r sin θ . \nFigure 1c shows the path length in units of R 2 as a function of angle θ . The thin green lines depict the path length in the absence of a cavity. At r = 0, the grain is at the center of the shell, and with no cavity all path length are independent of θ and equal to R 2 . The presence of a cavity, taken to have a radius R 1 = 0 . 90 R 2 , leaves the path length at all angles below θ c unchanged. However, as shown by the black lines, path lengths are shortened when θ ≥ θ c . The cusps in the figure correspond to the values of θ c for the different values of r . The red and blue lines correspond to the path lengths when r = R 1 and R 2 , respectively.", '4. TEMPERATURE OF THE DUST': "The total power, P ( R 1 , R 2 , r, τ ), absorbed by the grain is given by integrating the incident attenuated specific flux from all directions and frequencies times the grain's cross section \nP ( R 1 , R 2 , r, τ ) = 2 π ∫ ∞ 0 σ gr ( a, λ ) ∫ π 0 F ν ( R s , R 1 , R 2 , r, θ, λ, τ ) sin θ dθ dν . (10) \nThe temperature of a dust grain , T gr , is obtained by equating its heating rate, P , to its luminosity, L gr , \nL gr = 4 m gr ∫ ∞ 0 πB ν ( λ, T gr ( r )) κ ( a, λ ) dν , (11) \nwhere B ν is the Planck function in erg s -1 cm -2 sr -1 Hz -1 . \nFigure 1d depicts the radial profile of the dust temperature with a cavity (dashed lines) and without a cavity (solid lines). Calculations were performed for 0.1 µ m radius silicate grains, heated by an optically thin shock at a distance R s = 5 R 2 , with a luminosity of 10 8 L ⊙ and thickness ℓ = 0 . 1 R s , where R 2 = 1 × 10 16 cm. The temperature profiles are plotted for different radial optical depths at λ ( V ) = 0 . 55 µ m. The radial optical depth at any wavelength λ is given by τ ( λ ) = κ ( λ ) ρ d R 2 . Larger optical depths require an increase in the density of dust grains in the shell. Densities were chosen to give optical depths equal to 1.0, 3.0, 10.0, and 30.0. For τ ( V ) = 1 the resulting density and the dust mass of the filled shell were 4 . 0 × 10 -20 g cm -3 and 8 . 5 × 10 -5 M ⊙ , respectively. The larger dust densities correspond to dust masses of 2 . 5 × 10 -4 , 8 . 5 × 10 -4 , and 2 . 5 × 10 -3 M ⊙ . With no cavity, the dust temperature exhibits the obvious rise towards the surface, where the incident radiation is less attenuated. The same general trend is seen in the presence of a cavity. However the presence of a cavity has the effect of increasing the radiation reaching subshells with radii > R 1 , so that the resulting temperature profile is somewhat higher and slightly rises towards the edge of the cavity. This effect is more pronounced at lower optical depths. The pink horizontal line marked τ ( V ) = 0 depicts the temperature of a grain located at R 2 , that is heated by the shock if it were a point source in the center of the shell.", '5. INFRARED EMISSION FROM THE SHELL': 'The specific infrared (IR) emissivity in the shell is \nϵ ν ( r, a, λ, T gr ( r )) = 4 ρ d πB ν ( λ, T gr ( r )) κ ( a, λ ) , (12) \nand the specific infrared (IR) luminosity of the shell is given by the integral \nL ν ( a, λ ) = 4 π ∫ R 2 R 1 ϵ ν ( r, a, λ, T gr ( r )) r 2 dr . (13) \nFigures 1f and 1g show the internal (intrinsic) specific luminosity for selected subshells { r, r + dr } as a function of wavelength for visual optical depths of τ ( V )=1 (Fig. 1f), and τ ( V )=10 (Fig. 1g). The blue and red curves correspond to the specific luminosities from the subshells without, and with a cavity, respectively. The thick blue line and the thick dashed red line represent the total specific luminosity, summed over all subshells. The cavity in the two figures was characterized by a radius R 1 = 0 . 90 R 2 . \nThe internal specific luminosity in the subshells is characterized by a colder spectrum and lower intensity with increasing optical depth. However, the total spectrum is dominated by the emission from the outer subshells. As a result, at high optical depths, the total spectrum is not greatly affected by the presence of a cavity (Fig. 1g), since the incoming radiation is predominantly absorbed in the outer subshells. At lower optical depths, (Fig. 1f) colder subshells contribute to the total internal spectrum for the cavity free case. The resulting total spectrum is therefore colder compared to that of the shell with a cavity where only the hot dust in the outer subshells contributes to the internal energy.', '6. THE ESCAPE OF THE THERMAL EMISSION FROM THE SHELL': "The observed infrared emission from the dust is the fraction of the total internal energy density that escapes the shell. The escape probability of photons from a spherical homogeneous shell was recently presented by Dwek & Arendt (2024). In that study, the dust emissivity in the shell was assumed to be constant, whereas in the present case it has a radial dependence. The procedure for calculating the escaping radiation is essentially the reverse of that used to calculate the flux of incident radiation that heats the dust. Let ϵ ν ( r, a, λ, T gr ( r )) be the specific energy density at point A , at distance r from the center of the nebula. The specific luminosity escaping from the shell is given by integrating the escaping emission in all directions, and over all subshells \nL ν ( R 1 , R 2 , a, λ ) = ∫ R 2 R 1 r 2 ϵ ν ( r, a, λ, T gr ( r )) [ 2 π ∫ π 0 e -τ ( R 1 ,R 2 ,r,θ,a,λ ) sin θdθ ] dr , (14) \nwhich for τ = 0 reduces to equation (13). \nFigures 1i and 1j represent the total internal and escaping specific luminosity for different optical depths. The solid lines are the internal flux in the shell, and the dotted lines represent the escaping flux. The figures show that the total internal luminosity increases with optical depth, since more of the incident radiation is absorbed by the shell. The figure also shows that the silicate features in the escaping spectrum are suppressed but still detectable at low optical depths, τ ( V ) < 10, and significantly weaker at larger ones. Suppression of these features to create a smooth emerging spectrum requires any emission to be self-absorbed at these IR wavelengths. This requires the dust temperature to be isothermal throughout the subshell that contribute significantly to the emission, so that any absorption or emission features emerging from a subshell are cancelled out by the corresponding emission or absorption in neighboring subshells. The strength of the silicate features will depend on the temperature gradient in the shell and the subshells that are the dominant contributors to the escaping flux. \nFigures 1e, 1h, and 1k are the equivalent of Figs. 1d, 1g, and 1j, but for a shell that is heated by a shock located at a short distance of R s = 1 . 2 R 2 = 1 . 2 × 10 16 cm from the center of the shell. Other shock parameters, its total luminosity and thickness, are identical to the previous case. The figures show that because of the shock's proximity a larger fraction of its luminosity is actually absorbed by the shell. Because the shock's luminosity emerges from a smaller area, its brightness, I ν , is larger, resulting in an increase in dust temperature in all subshells, an effect that is manifested in their spectra, and in that of the escaping radiation.", '7. SUMMARY': "We presented the basic mathematical outline for calculating the internal spectrum from a spherically symmetric shell that is heated by an external spherically symmetric source of radiation. The scenario envisioned here is that of an expanding radiative shock that is heating an interior shell of either ejecta or cooled, swept up, CSM dust. The presence of a cavity affects the emerging radiation, depending on its size. The results are significantly affected by the shock's proximity to the shell, or its luminosity. Our study highlights the interplay between the various global parameters characterizing the shock and the shell. It also shows the many potential degeneracies in fitting any observed spectrum, in which changes in the shock luminosity or distance, in the total dust mass or cavity size, as well as changes in dust composition and size distribution can yield similar results. The results presented here neglect the effect of scattering of the incoming radiation by the dust. Scattering will have an effect on the fraction of the incident UV-optical radiation that penetrates the shell, and on its trajectory and penetration depth. It will have little effect on the escape of the infrared emission from the heated dust. We also neglect the additional heating of the dust by the outgoing IR radiation. Any contribution to the observed emission from collisionally heated dust embedded in the hot dense post-shock gas is assumed to be negligible. \nE.D. thanks the Niels Bohr Institute for their hospitality during the completion of the manuscript, which was supported by a VILLUM FONDEN Young Investigator Grant (project number 25501). Work by R.G.A. was supported by NASA under award number 80GSFC21M0002.", 'REFERENCES': 'Dwek, E., & Arendt, R. G. 2024, Research Notes of the \nAmerican Astronomical Society, 8, 194 \nSarangi, A., Dwek, E., & Arendt, R. G. 2018, ApJ, 859, 66 \nSarangi, A., & Slavin, J. D. 2022, ApJ, 933, 89 \n/uni03B1 \nFigure 1. (a)-(b) Geometrical presentation; (c) Path length through shell; (d, f, g, i, j) model results for shock distance of R s = 5 . 0 × 10 16 cm; (e, h, k) model results for R s = 1 . 2 × 10 16 cm. For sake of clarity, fewer subshells were plotted for the cavity case in (c, f, g, h). Details in text. \n<!-- image -->'} |
2024arXiv240913318A | Coronal heating refers to the physical processes that shape and structure the corona of the Sun and are responsible for its multimillion Kelvin temperatures. These processes are revealed in a number of different observational manifestations and have been studied on theoretical grounds in great detail over the last eight decades. The aim of this Chapter is to give an account of some of those manifestations and to discuss relevant physics that we believe is responsible for them. Coronal heating is closely connected to other magnetohydrodynamic MHD processes occurring in the solar plasma and described in this book such as waves shocks instabilities and magnetic reconnection. | 2024-09-01T00:00:00Z | ['arXiv:2409.13318', '10.48550/arXiv.2409.13318', '2024arXiv240913318A'] | ['Astrophysics - Solar and Stellar Astrophysics'] | Coronal heating | 2,024 | 167 | 0.5 | ['EPRINT_HTML', 'EPRINT_PDF'] | 0 | https://arxiv.org/pdf/2409.13318.pdf | {'Iñigo Arregui a,b and Tom Van Doorsselaere c': '- a Instituto de Astrofísica de Canarias, Vía Láctea s/n, E-38205 La Laguna, Tenerife, Spain,\n- b Departamento de Astrofísica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain,\n- c\n- Centre for mathematical Plasma Astrophysics, Department of Mathematics, KU Leuven, B-3001 \nLeuven, Belgium', 'ABSTRACT': 'Coronal heating refers to the physical processes that shape and structure the corona of the Sun and are responsible for its multi-million Kelvin temperatures. These processes are revealed in a number of different observational manifestations and have been studied on theoretical grounds in great detail over the last eight decades. The aim of this Chapter is to give an account of some of those manifestations and to discuss relevant physics that we believe is responsible for them. Coronal heating is closely connected to other magnetohydrodynamic (MHD) processes occurring in the solar plasma and described in this book such as waves, shocks, instabilities, and magnetic reconnection.', 'KEYWORDS': 'Sun: corona, Sun: magnetic fields, Magnetohydrodynamics (MHD)', '10.1 INTRODUCTION': 'The outermost layer of the solar atmosphere, the corona, is made-up of plasma with a very high electrical conductivity, and is permeated by a highly structured magnetic field displaying a wide range of spatial scales. Plasma and magnetic field interactions are responsible for phenomena like the heating of the solar corona or the acceleration of the solar wind. They are the ultimate source for Sun-Earth interactions. Gathering a satisfactory description of these phenomena still represents a challenge to our understanding of the Sun, and prevent us from mitigating their effects on Earth. \nThe so-called coronal heating problem originated about eight decades ago, when Edlén and Grotrian identified Fe IX and Ca XIV spectral lines in the corona, indicating the presence of fully ionised plasma at 1 million degree Kelvin (Edlén, 1943). Since then, solar physics research has sought to identify the physical processes that might balance the three primary coronal energy loss mechanisms; thermal conduction, radiation, and solar-wind outflow, first quantified by Withbroe and Noyes (1977). \nIn spite of decades of research, the coronal heating problem remains unsolved (see Kuperus, 1969; Withbroe and Noyes, 1977; Kuperus et al., 1981; \nAschwanden, 2005; Klimchuk, 2006; Parnell and De Moortel, 2012; Klimchuk, 2015; Van Doorsselaere et al., 2020b, for reviews than span across different decades). Researchers in the field agree upon the fact that the ultimate reason for the multi-million degree temperatures in the outer solar atmosphere is of magnetic nature and that the energy source lies in the convective motions at the photosphere. The exact physical processes and their relative contribution to the heating of the plasma remain largely unknown or unquantified. \nTwo main types of processes are believed to contribute to the heating: the direct dissipation of magnetic fields by mechanisms such as magnetic reconnection (Sturrock and Uchida, 1981), current cascades (Parker, 1963), viscous turbulence (van Ballegooijen, 1986), or magnetic field braiding (Peter et al., 2004); and the dissipation of magnetic wave energy stored, transported, and transferred to short spatial scales by magnetohydrodynamic (MHD) waves (Alfvén, 1947; Ionson, 1978; Heyvaerts and Priest, 1983). There is ample evidence for the occurrence of both magnetic reconnection (Zweibel and Yamada, 2009) and magnetic wave dynamics (Nakariakov and Verwichte, 2005). They have both been studied extensively, and significant theoretical advancements on their description were made, so a discussion in terms of mutually exclusive mechanisms is beside the point. It is however important to assess the relevance of each mechanism in heating the plasma of the solar atmosphere. \nThe aim of this Chapter is to give an account of some observational manifestations of coronal heating and a discussion on the physical processes that are believed to be most relevant in causing them.', '10.2 CORONAL EMISSION': 'The corona of the Sun is the outermost part of the solar atmosphere. It extends just above the chromosphere towards the interplanetary medium. Light emitted from the white-light corona shows up as a faint and extended halo, with an intensity near the edge of the solar disc that is about a million times fainter than the white-light coming from the disc. The light scattered by the Earth\'s atmosphere is several times brighter that the corona. This makes impossible to observe the white-light corona from Earth unless the disc of the Sun is completely covered. The corona is visible to the naked eye at times of total solar eclipses, when the solar disc is obscured by the Moon (see Figure 10.1). \nThe shape of the corona is generally irregular and varies in time, depending on the stage of the solar magnetic cycle. Near or at sunspot maximum, the corona shows an irregular lobe-form extending around the lunar disc with no preferred solar latitude, although the intensity is reduced near the poles. At sunspot minimum, the corona appears symmetrically elongated about the equator, because active regions are located at low latitudes. It displays long streamers, extending from opposite sides of the Sun and approximately aligned with the solar equator, as well as polar plumes. \nThe origin of coronal white-light emission is the scattering of light emitted \n<!-- image --> \nFIGURE 10.1 Coronal emission and structure On the left, image of the solar corona taken during a total solar eclipse. Red indicates the Fe XI 789.2 nm line, blue the Fe XIII 1074.7 nm line, and green shows the Fe XIV 530.3 nm line. Credit: Habbal/NASA. On the right, X-ray image taken by Yohkoh satellite on 1February 1992 01:44 UT, showing the variable brightness distribution of the corona due to its magnetic structuring, which gives rise to different manifestations of solar activity. Solar X-ray image courtesy of the Yohkoh Legacy data Archive (http://solar.physics.montana.edu/ylegacy) at Montana State University. \n<!-- image --> \nfrom the photosphere. The scattering is due to both free electrons and dust grains. In terms of the different processes that cause coronal emission, we can distinguish the following components: \n- · The K-corona originates from Thomson scattering of photospheric light by free electrons. It displays a featureless continuous emission spectrum like that of the photosphere, but without Fraunhofer lines and is strongly polarised. The absence of Fraunhofer lines is due to the high temperatures and low densities, making electrons travel at very high speeds in the corona. At a temperature of 2 MK, the electrons move with an average speed of 10 000 km s -1 . As a consequence, the Fraunhofer lines of the photospheric spectrum are extremely wide and no longer perceptible as such.\n- · The F-corona arises due to scattering of the Fraunhofer spectrum by the dust particles in the interplanetary atmosphere. It is more an extension of the zodiacal light, rather than a genuine solar phenomenon. This emission shows no polarisation. Because the dust grains move at a much lower speed than the electrons causing the K-corona, the Fraunhofer lines remain visible in the F-coronal emission.\n- · The E-corona arises due to the high temperatures, of the order of a few MK and the low densities around ∼ 10 10 cm -3 in the corona and is the only component produced by the coronal gas itself. Under such extreme conditions, ions produce emission lines. The total integrated brightness of the E-corona is small in comparison to the K- and F-coronae. However, the strong emission relative to the background scattered light, concentrated around specific wavelength, makes possible to observe coronal features through narrow-band \nfilters centred in these emission lines. \nTheexistence of a hot corona can already be deduced from the apparently simple observational manifestations described above. Because there is essentially no dependence on temperature, the intensity is proportional to the line-of-sight integral of the density. The surface brightness of the K-corona depends on the electron number density. Along a given line-of-sigh through the corona, the surface brightness relative to the photospheric brightness can be calculated as the number of electrons times the electron cross-section. Following Phillips (1995), consider a typical coronal streamer of about 1000 km extent. This leads to a density of about 10 14 electrons per cubic metre. The electron density varies with distance from the Sun, but decreases relatively slowly with distance. This is again an indication of high temperature. Using pressure scale arguments and estimating the characteristic length-scale over which density decreases in data for surface brightness variation (Allen, 1973) temperatures as high as 2 000 000 K are obtained Blackwell and Ingham (1967). \nThe coronal spectrum was known already in the 40s of the past century, but its properties were not well understood. The main puzzles were the absence of Fraunhofer lines in the bright continuum observed close to the Sun\'s limb (inner corona), the presence of unidentified bright emission lines, the absence of emission lines of H and metal, the width of coronal emission lines, and the extended size of the corona. It turns out that they are all indications of high kinetic temperature plasma. \nAstrong indication of the hot corona is the presence of so-called \'forbidden" lines in coronal emission and their identification (by Edlén and others) with highly ionized ions, such as the coronal \'green" Fe XIV and \'red" FeX lines at 530.3 and 637.5 nm, respectively. Edlén (1943) identified coronal bright emission lines as being due to elements, such as A, Ca, Fe, Ni, in very high states of ionisation. Very high temperatures and low densities were required to explain the presence of those lines with such ionisation degrees in the coronal spectrum. The intensities of these lines depend on temperature, which controls how much of a particular ion is present relative to all the other ions of an element and also causes the excitation from the ground state to the upper state from which de-excitation leading to the emission is produced. Plasma density is relevant because without low densities electron collisions would produce de-excitation/excitation to another states, preventing the forbidden line to be observed. \nObservations of the corona in visible wave-lengths are limited by the occurrence of eclipses or the use of coronagraphs. In addition, they do not permit us to observe coronal features on the disc because the corona is optically thin at visible wavelengths and optical radiation is not absorbed. The situation is different for radiation in non-visible wavelengths, such as X-rays, extreme ultraviolet (EUV) or radio wavelengths, which are emitted by the corona but not at all by the photosphere. The fact that the primary emission from the corona is in \nEUV and X-ray bands is another clear indicator of the multi-million K coronal temperatures.', '10.3 CORONAL STRUCTURE': 'Solar (and stellar) EUV and X-ray emission requires the existence of a magneticfield-generating dynamo process, which provides the means to connect the convection zone with the atmosphere. Magnetic fields generated within the convective layer feed mechanical energy into the atmosphere and produce plasma heating and structuring. \nAs such, EUV and soft X-ray images of the solar corona show that the brightness distribution is far from uniform (see Figure 10.1). In these wavelengths, coronal features become observable against the solar disc. A highly structured corona becomes apparent, which displays three major types of features: (a) active regions and coronal bright points, which are characterised by strong magnetic fields and closed field configurations; (b) coronal holes, characterised by relatively weaker field strengths and open field configurations; and (c) quiet Sun coronal regions, in which the field is closed, but over larger spatial scales. Each of these features is, in turn, characterised by a different level of X-ray and EUV intensity, clearly indicating the important role of the magnetic field in the energy balance and heating of the corona. The magnetic field structure conditions the geometry of mass and energy flows, guides magnetic waves, and stores magnetic energy that can be released into the surrounding plasma. \nThe structure of the corona in EUV and soft X-ray images is determined by the structure of the magnetic fields that permeate the plasma which, in turn, depends on the cyclic variation of the solar activity (see Figure 10.2). EUV images near solar maximum display a large fraction of the surface covered with active regions located at two well-defined latitude bands above and below the equator. At solar minimum, EUV emission is in general fainter and active regions are scarce. They show up at higher latitudes. At these times, X-ray emission is dominated by smaller magnetic bright points. Open-field regions lead to the so-called coronal holes, which show up at the polar caps during most of the cycle. They have their largest extent during solar minimum and can appear crossing the equator in between the minimum and maximum of the solar cycle. Any plausible coronal heating mechanism, should provide an explanation to these general features and their variation with the solar-activity-cycle.', '10.4 CORONAL DENSITY AND TEMPERATURE DISTRIBUTION': "Gaining knowledge about the physical processes behind the heating of the corona requires the assessment of its physical conditions; most significantly its density, temperature, and their spatial and temporal variation. Early efforts at this respect are reviewed in Allen (1954). \nThe density distribution of the corona can be obtained from the scattering of \nTABLE 10.1 Coronal energy losses [W m -2 ] from Withbroe (1981). \nlight by electrons. Initial studies were able to deduce this density distribution in the form of a sum of powers of radial distance, from the projected distribution of light seen from the Earth, and assuming a given electron distribution in the corona, see, e.g., van de Hulst (1950). The obtained values were mean densities, averaged from observations taken during several eclipses and over volumes that were large enough to include several coronal features. \nEarly coronal temperature estimates were based on the width of coronal emission lines. The absence of hydrogen and metals emission lines indicated that the electron temperature must be above 700 000 K (Goldberg and Menzel, 1948). The observations of Grotrian (1931) led to temperature estimates of about a million degrees. Values of several million K were reported by e.g., Lyot (1937) and Waldmeier (1944). Early methods for estimating coronal temperatures involved the analysis of Doppler broadening of emission lines; line ratios from multiple states of ionisation of ions; the estimate of the density gradient resulting from intensity measurements; the analysis of radio emission at 10-cm wavelength in quiet-corona regions; or the degree of ionisation from ionisation cross-sections and recombination coefficients for the ions concerned (see e.g., Brosius et al., 1994; Noci, 2003, for example). More recent techniques, used either as an alternative or in combination with those, infer the coronal temperature from observed properties of MHD slow-wave modes, using seismology inversion techniques (Marsh et al., 2009; Marsh and Walsh, 2009). \nThe density and temperature of the coronal plasma vary in space and time. Based on the spatial distribution of the temperature, the corona is classically divided into three regions: the lower corona, the region closest to the Sun's surface, has a temperature of around 1 MK and a relatively high density of about 10 8 particles per cubic centimeter. The region just above, the intermediate corona, has a temperature of around 1-2 MK. The density in this region is lower than in the lower corona, around 10 6 particles per cubic centimeter. The upper corona is the region farthest from the Sun's surface and has a temperature of several MK. The density in this region is extremely low, around 10 4 particles per cubic centimeter. \nFigure 1.2 in Golub and Pasachoff (2009) shows the classical schematic representation of the temperature and density variations in the solar atmosphere as a function of height. This is the result of semi-empirical models, based on the computation of solutions to radiation transfer equations for hydrogen, carbon, and other constituents. The solutions are computed on spherically symmetric \nFIGURE 10.2 Coronal emission and the solar activity cycle A collage of images displays coronal emission in extreme ultraviolet light at different stages of the solar cycle (every spring from 1996 to 2016) taken with the Extreme ultraviolet Imaging Telescope (EIT) onboard the Solar and Heliospheric Observatory (SOHO). From Temmer (2021). \n<!-- image --> \nand homogeneous plane-parallel atmosphere models, with physical properties varying with height. They consider optically thick plasmas in non-local thermodynamic equilibrium conditions (see Vernazza et al., 1981, for details). Recent imaging observations of the corona in EUV emission lines demonstrate that the adopted plane-parallel view is extremely simplistic, because of the highly structured nature of the corona they display. It is still of interest to know what the key features are in temperature and density dependence in those simplified atmospheric models. They form the basis for many current numerical computations for the dynamics of the coronal plasma (e.g. Howson and De Moortel, 2022). Fontenla et al. (1993) computed energy balance hydrostatic models of the atmospheric parameters, which are a continuation of the semi-empirical models by Vernazza et al. (1981). Nowadays, a more accurate definition is one that considers the solar corona as any portion of the solar atmosphere with a temperature above 10 5 K (Golub and Pasachoff, 2009). \nThese physical conditions completely determine the energy balance on any given volume of the solar corona. Energy balance estimates for different components of the corona are shown on Table 10.1, with values taken from Table 2.1 in Withbroe (1981) (see also Table 1.1 in Golub and Pasachoff 2009). They consider the amount of energy loss by three mechanisms: thermal conduction, radiation, and plasma outflow. The estimates strongly depend on the particular region of the corona being considered and also on the strength and configuration of the magnetic fields. In coronal holes, the mean temperature and density are lower in comparison, hence conductive and radiative losses are less than in the active region corona. The open magnetic field lines enable large energy losses by plasma outflows, making the total losses higher than in quiet Sun regions. On the other hand, active region energy balance is mainly conditioned by energy lost by thermal conduction and radiation. Coronal energy losses in active regions exceed those in the quiet Sun and coronal hole regions by one order of magnitude, approximately.", '10.5 CORONAL ACTIVE REGION LOOPS': "The temperature and density distribution in the corona are influenced by the magnetic field. The magnetic-field lines in the corona trap plasma and create a highly structured corona, which is far from being homogeneous at different spatial scales. This magnetically dominated corona consist of a variety of structures, such as active regions, coronal loops, X-ray bright points, coronal holes, etc. Coronal loops are the basic building blocks of active regions, areas of the solar corona where the magnetic field is concentrated. They show up as regions with enhanced extreme ultraviolet and X-ray emissions and their temperatures are higher than the surrounding quiet Sun corona. They have a particular morphology and are built by components with different geometrical and physical properties (see Figure 10.3). \nThe central area of an active region is the core, which appears in X-ray \nFIGURE10.3 The structure of coronal active regions EUVimageinthe171Åchannel(1MK) from SDO/AIA (left) and X-ray counterpart from Hinode/XRT (T > 2 MK) showing the different components that conform a typical active region in the corona. From Schmelz and Winebarger (2015). \n<!-- image --> \nwavelengths as a diffuse area where individual loops are difficult to isolate from observations. It is formed by short loops with temperatures in the range of 3 to 5 MK. In the hotter emission lines the loops appear more fuzzy. At the foot-points of these loops, a bright reticulated pattern is observed in EUV emission, the socalled moss. This structure appears to closely follow the photospheric magnetic field distribution. Surrounding the core, we find 1 MK more outlined and longer loops. These are seen to evolve in shorter time-scales in comparison to the core. The periphery of the active region is filled with long and cool fan loops. The background is filled with diffuse emission, which is rather interesting because it is clearly heated, but not connected to strong magnetic field structures. \nPhysical properties discussed in Section 10.4 refer to global conditions, average values that include several coronal features. The analysis of modern highresolution imaging and spectroscopic observations enables us to differentiate the conditions in active regions and their components and offers means to diagnose the properties of plasma heating processes. Current estimates of the temperature and density in the corona are obtained by means of spectroscopic diagnostic techniques. They involve the forward modelling of spectral lines properties, such as their intensity, line-width, and profile. By measuring the intensities of spectral lines over a range of temperatures, the differential emission measure can be determined as a function of the temperature (see Section 10.5.2 below), which informs about the amount of plasma at a given temperature. Some ions have low-lying meta-stable levels whose populations are sensitive to the density in particular density ranges. Taking ratios of the intensities of those selected spectral lines can be used to determine plasma densities, since the ratio of the population of the ground and the meta-stable levels will be density sensitive. Line ratios can also be used to obtain information on the temperature, consid- \n<!-- image --> \nFIGURE 10.4 Equilibrium of active region loops Transition Region and Coronal Explorer (TRACE) observation of an active region with several loops (left) and a simulation of the observed intensity scaled to the hydrostatic thermal scale height of T = 1 MK (right), with a pressure scale height of about 46 Mm. From Aschwanden et al. (2001). Reproduced by permission of the American Astronomical Society (AAS). \n<!-- image --> \nering two spectral lines with differences in excitation energies from a common ground level. The broadening of spectral line profiles gives information on Doppler shift introduced by, e.g., mass flows, waves, or unresolved turbulent velocities. \nObservational analyses of this kind, made with the first imaging observations of coronal loops in EUV wavelengths with TRACE, made apparent a few observational issues. For example, coronal loops are nearly isothermal and have almost constant temperature distributions along their coronal segments, as already noted by Withbroe and Noyes (1977). This is manifested as relatively flat 195 to 171 Å filter ratios along much of their lengths. The pressure scaleheight at 1 MK is about 46 Mm, hence long loops should be gravitationally stratified. Imaging observations, on the contrary, display bright emissions many scale heights above the solar surface (see Figure 10.4). The observed flux is proportional to the emission measure, which has the half pressure scale height of ∼ 23 Mm. Hence, active region loops seem to be over-dense having enhanced intensities in comparison to the properties predicted by simple hydrostatic scaling laws governing the relationship between pressure, temperature, and loop lengths (Rosner et al., 1978; Serio et al., 1981; Aschwanden et al., 2001; Schmelz and Winebarger, 2015). The observationally determined super-hydrostatic scaleheights are incompatible with steady uniform heating models (Winebarger et al., 2003a). Hydrostatic loop models can only explain well the shortest loops (As- \nanden et al., 2001; Winebarger et al., 2003a). On the other hand, models in which heating is localised at the footpoints lead to flatter temperature profiles and larger apex densities, in better accordance with observations (Winebarger et al., 2003a). For instance, Warren et al. (2002) suggested that impulsive heating can produce loops that are overdense relative to the predictions of uniformly heated static loop models. \nAnother striking observed feature is the persistence of loop's bright emission in timescales much longer than the characteristic radiative cooling times in the corona (Lenz et al., 1999a,b). Winebarger et al. (2003b) studied the temporal evolution of five active region loops observed with TRACE. The loops appear first in the 195 Å filter and then in the 171 Å filter. The progression in the appearance of loops from the hotter filters to the cooler filters is consistent with the cooling loops model proposed by Warren et al. (2002). \nWarren et al. (2003) performed detailed hydrodynamic modeling of one of the TRACE loops analyzed by Winebarger et al. (2003b) and compared observed and simulated light curves for the evolution of coronal loops. Some properties of hydrodynamic simulations, such as the cooling and draining times, relate to features of the observed light curves, such as the delay between the appearance of the loop in the different filters. An example is displayed in Figure 10.5, which shows the comparison between observed light curves in 171 and 195 Å with simulation results from monolithic and multi-thread models (Warren et al., 2003). A loop lifetime comparable with the cooling time is compatible with a monolithic (isothermal) loop structure. On the other hand, the observed relatively flat filter ratios and long lifetimes of coronal loops can only be obtained in simulations by considering an ensemble of independent, impulsively heated strands, even if they are 'seen' as a single loop in imaging observations. In general, it is possible to match both the spatial and temporal evolution of the observed loops by assuming that they are actually a collection of threads that are heated sequentially. \nWinebarger and Warren (2004) showed that the analysis of impulsively heated 1 MKactive region loops, as they cool down and show up in different bandpasses, can be used to determine crucial characteristics, such as the magnitude, duration, and location of the energy release. The evolution of the apex density and temperature in such loops depends only on the total energy deposited. This would mean that observations must be made early in the evolution of a loop to be able to determine the heating parameters and to discriminate between different heating scenarios.", '10.5.1 Intensity variations': "A widely used observational diagnostic tool to understand how coronal loops are heated is based on the analysis of the intensity variations of measurements at different EUV or X-ray wavelengths, which is then related to the thermal evolution of the plasma. As first shown by Winebarger and Warren (2005) (see \nFIGURE 10.5 Hydrodynamic evolution of coronal loops Top panel: a comparison between simulated and observed light curves with data from the Transition Region and Coronal Explorer (TRACE) in an active region loop observed on 18 August 1998. In the single loop simulation, the delay between the 195 and 171 Å intensities in the simulated light curve matches the observations, but the loop cools too quickly. In the Middle and Bottom panels, simulated light curves arise from models with a series of threads that are heated sequentially and these models are able to reproduce better some observed characteristics of the observed light curves. From Warren et al. (2003). \n<!-- image --> \nalso Ugarte-Urra et al., 2006; Warren et al., 2007; Ugarte-Urra et al., 2009; Viall and Klimchuk, 2011; Reale, 2014), some coronal loops appear first in X-ray emission (T > 2.5 MK) before showing up as bright structures in EUV imaging observations. This would imply a cooling process in which the peak emission at different wavelengths is reached at consecutive times as the loops' temperature goes down. Improvements in spatial and temporal resolution, as well as the increase in the number of EUV channels, provided by the Solar Dynamics Observatory (SDO: Lemen et al. 2012), have enabled measuring the time-delay between those consecutive peak times, making possible the observational diagnostics of heating/cooling processes in coronal loops and giving information about the frequency of heat deposition. \nThe concept of frequency of heat deposition is based on the so-called Parker nanoflare-hypothesis (Parker, 1988), which postulates that plasma heating in the corona is the result of microscopic and unresolved magnetic reconnection events, uniformly distributed over the full coronal volume. Under this hypothesis, low-frequency heating refers to a scenario in which the repeat time of heating events is much longer that the plasma cooling time (impulsive heating regime). Conversely, high-frequency heating refers to the opposite scenario in which the repeat time is much shorter than the plasma cooling time, which would essentially be akin to a steady heating process. In the intermediate case, plasma cooling and heating repeat times can be of the same order. Much of the observational analyses in recent years have focused on the inference of the impulsive/steady character of the underlying heating mechanism from the analysis of X-ray/EUV imaging and spectroscopic observations. \nA key idea is that when such a small-scale heating event (nanoflare) occurs, the heating phase contributes little to the observed emission and it is the cooling phase which dominates the observed light curves (Bradshaw and Klimchuk, 2011). By means of a time-lag analysis, Viall and Klimchuk (2012) computed the cross-correlation between light curves in six successively cooler SDO/AIA EUV channels (131 Å, 94 Å, 335 Å, 211 Å, 193 Å, and 171 Å) at given locations of an active region. The resulting patterns show intensity peaks in the successively cooler pass-bands being sequentially reached, as the plasma cools through the sequence of EUV wavelengths. Viall and Klimchuk (2015) extended the analysis to the quiet Sun and later Viall and Klimchuk (2017) applied this type of analysis to 15 active regions catalogued by Warren et al. (2012). Their results indicate overwhelmingly positive time lags (cooling plasmas) in all cases, with only a few isolated instances of negative time lags (heating plasma). These observations are interpreted as a manifestation of a impulsive heating scenario in which little emission is produced during the heating phase and observations are indicative of cooling of plasma that has been impulsively heated. Some areas show a time lag of zero which, rather than lack of variability, are interpreted as a manifestation of strong variability consistent with the response of the transition region to heating events (see also Viall and Klimchuk, 2016). \nA number of subsequent studies have applied the same method to simulated \nsynthetic images obtained from models. For instance, Bradshaw and Viall (2016) find that some aspects of the observed light-curves can be reproduced by both high and intermediate-frequency nanoflare models. On the other hand, Lionello et al. (2016) were unable to reproduce the time-lag characteristics from Viall and Klimchuk (2012) in their field-aligned hydrodynamic models. This means that either the time delays may not be representative of the real loop evolution, or that other heating scenarios beyond the impulsive heating and cooling must be considered to explain the observations. Overall, impulsive heating is consistent with EUV loops Ugarte-Urra et al. (2006); Hara et al. (2008), while steady heating would better explain the soft X-ray emission in active region cores (Warren et al., 2010; Winebarger et al., 2011; Tripathi et al., 2011).", '10.5.2 Emission measure distribution': 'Another method to diagnose the plasma physical conditions and the heating characteristics from observations is based on the analysis of the so-called emission measure (EM) distribution, EM ( 𝑇 ) = ∫ 𝑑ℎ 𝑛 2 𝑒 , with 𝑛 e the electron density and the integration is taken alone the line of sight. This enables to determine the temperature distribution of the coronal plasma, with the expectation that this temperature structure can inform about how the corona is heated (Warren et al., 2012). The distributions are derived from the intensities at different spectral lines by computing the EM at the temperature that maximises the contribution function, the so-called peak formation temperature. For each spectral line, it is assumed that the contribution function is constant over a given temperature range around the peak formation temperature and zero otherwise. An example is shown in Figure 10.6, which displays the resulting inverted U curves, the socalled EM loci plots. Each of these curves gives the amount of emission measure that would be needed to produce the observed intensity at each spectral line, if the plasma were isothermal at each temperature. The position of the minima corresponds to the peak formation temperature for each line, since this is the temperature at which the line emits more efficiently and, hence, demands least emission measure (see Del Zanna and Mason, 2018, for a detailed description). \nEmission measure distributions derived from observations can then be compared to those predicted by theoretical models. In these models, the cool portion of the EM(T) distribution, to the left of the peak, is usually described by a powerlaw of the form EM ( 𝑇 ) ∝ 𝑇 𝛼 , with 𝛼 the so-called emission measure slope (Jordan, 1976; Cargill, 1994; Cargill and Klimchuk, 2004). In the nanoflare heating scenario, the value of 𝛼 is an indicator about how often a single strand is reheated and, hence, its observational inference helps to diagnose the heating frequency in nanoflare models. The method has been used to interpret active region core observations in terms of both high- and low-frequency heating (see Table 3 in Bradshaw et al., 2012, and references therein). Overall, the values of the emission measure slope are in the range 2 to 5. The shallower slopes support low-frequency heating and the steeper ones are associated with high-frequency \nFIGURE 10.6 The loci approach to assess the emission measure distribution Emission Measure (EM) loci curves for a selection of lines observed by Hinode/EIS in an active region core (Del Zanna and Mason, 2014). The filled circles represent the values calculated from the Differential Emission Measure (DEM) with Δ log 𝑇 = 0 . 1 K. The triangles, plotted at the temperature 𝑇 max, are the solution using the Jordan and Wilson (1971) approximation. From Del Zanna and Mason (2018). \n<!-- image --> \n(or steady) heating. Low-frequency heating appears to be consistent with observed DEM distributions (Bradshaw et al., 2012; Cargill, 2014). Quasi-steady nanoflare trains are consistent with a large percentage of observed active region cores (Reep et al., 2013; Cargill et al., 2015). \nThe analysis of the emission at different spectral lines with data from space instrumentation, such as the EUV Imaging Spectrometer (EIS) and X-Ray Telescope (XRT) onboard Hinode and the Atmospheric Imaging Assembly (AIA) onboard the Solar Dynamics Observatory (SDO), has demonstrated that wellconstrained temperature measurements can be made over different coronal regions with varying physical conditions. Warren et al. (2012) performed a systematic study of the differential emission distribution in 15 active region cores. The results indicate that the temperature distribution in an active region core is often sharply peaked near 4 MK. When the properties of the emission measure distributions were compared to magnetic properties, such as the total unsigned magnetic flux, the results indicated that high-temperature active region emission is close to equilibrium, while weaker active regions seem to be dominated by 1 MKplasma evolving in the core. \nAnumber of other investigations have analysed the diffuse emission from the cores of active regions by associating the slope to the heating frequency. The overall results are rather inconclusive, with some active region cores showing a predominance of shallow slopes (consistent with low-frequency heating) and some others displaying steeper emission measure slopes (more consistent with high-frequency heating events). The most recent developments aim at applying \nmachine-learning techniques to assess whether or not individual loops or strands are reheated before completely cooling (see e.g., Barnes et al., 2019, 2021, for recent applications).', '10.6 CORONAL DYNAMICS': 'The solar corona is highly dynamic at a variety of time and spatial scales. The continuous emergence, cancellation, and dynamics of photospheric magnetic fields recycle the coronal field in time-scales of just a few hours (Close et al., 2004). Plasma flows, magnetodydrodynamic waves, and magnetic reconnection events are among the fundamental physical processes with relevance to the heating of the corona. Magnetic reconnection is discussed in detail in Chapter 9. It is the process by which magnetic field lines break and reconnect, leading to an energy release that can heat and accelerate charged particles. We focus here on coronal plasma flows and magnetohydrodynamic wave activity.', '10.6.1 Coronal plasma flows': 'Mass flows are among the three primary energy loss mechanisms in the solar atmosphere, together with radiation and thermal conduction (Withbroe and Noyes, 1977). They are present in both magnetically open and closed coronal components. Open magnetic fields in coronal hole regions enable the continuous stream of particles in the form of fast solar wind, with speeds of 700 - 800 km s -1 . The slow solar wind, with speeds of 300 - 500 km s -1 , is believed to originate from streamer belts and the edges of active regions (McComas et al., 2008). Observational signatures of large-scale plasma flows can be obtained from the analysis of white-light data from large-angle spectrometric coronagraphs, such as LASCO onboard SoHO. They allow to study the contribution of different structures to the brightness of the K corona (Wang et al., 2007). At smaller spatial scales, more detailed analyses are possible. They are based on the detection of Doppler velocity displacements, often correlated with measurements of the non-thermal broadening of coronal emission lines. From them, it has been found that persistent flows arise from the edges of active regions, which consist of a steady and quasi-stationary plasma stream (see e.g., Kojima et al., 1999; Sakao et al., 2007; Harra et al., 2008; Hara et al., 2008). Recent observations make use of multi-wavelength, high-resolution instrumentation to track plasma flows across different layers of the solar atmosphere and arising from regions with different magnetic activity level (see Barczynski et al., 2021; Schwanitz et al., 2021, for example). \nBesides their relevance in connection with the solar wind, observations of plasma flows offer an important source of information on the physical characteristics, energy balance, evolution, and heating of active regions structures. The coronal plasma confined in magnetic loops is far from being in a static state. Transient heating and cooling processes continuously fill and drain the \nmaterial along the magnetic field. For example, asymmetric plasma heating in the foot-points of coronal loops creates siphon-type flows that are persistently detected in the form of red/blue-shifts in transition region UV spectral lines and as intensity variations in coronal emission lines. For instance, Winebarger et al. (2001) showed that brightness variations in loops that appear as static structures in imaging EUV observations with TRACE correspond to outflows with velocities between 5 to 20 km s -1 , interpreted as mass flows from the chromosphere into the corona. \nSpectroscopic measurements first obtained with SoHO/SUMER and then improved with Hinode/EIS have been of central importance in detailed studies of persistent loop plasma motions (see Table 8.3 in Aschwanden, 2019, for a list of examples). Measurements of plasma flow speeds in active regions range in between 5 and 100 km s -1 (Harra et al., 2008; Del Zanna, 2008; Doschek et al., 2008; Brooks and Warren, 2012). The measured Doppler flows in active regions show a mixed pattern with redshifts and blueshifts being stronger in cooler and hotter lines, respectively (Del Zanna, 2008). Even when a particular region seems to be dominated by downflows, spatially localised outflows can be found (Doschek et al., 2008). The interpretation given to this is that the outflows might be directed along the long closed loops and/or the open magnetic fields. Obtaining Doppler velocity signatures of such flows is challenging because of a number of reasons. First, projection and integration along the line-of-sight affect their quantification. Also, the interpretation of the measured variations in terms of heating/cooling and/or flows is not straightforward. Finally, signatures of flows and waves are often diffult to distinguish from observations (De Groof et al., 2004; Verwichte et al., 2010; De Moortel et al., 2015). \nIntensity variations are also seen to occur from the top of the loops towards the foot-points. First reported with the use of SoHO/EIT observations by De Groof et al. (2004), they were interpreted as coronal rain by Müller et al. (2005). Nowadays, the occurrence of flowing/falling plasma blobs is widely observed in the corona (Antolin et al., 2010). The phenomenon is believed to be due to the catastrophic cooling of the plasma and is linked to the mechanism of thermal instability. This in turn seems to be a consequence of foot-point-concentrated heating (e.g. Pelouze et al., 2022, , and references therein). Coronal rain clumps fall with typical speeds in the range 30 - 150 km s -1 , that are considerably smaller than the free-fall velocity due to gravity. According to some numerical models, this is caused by the dynamical rearrangement of the coronal pressure, due to the increase in the pressure gradient that opposes gravity as the cool and dense plasma blob falls (Oliver et al., 2014). \nIn the last years, it has been realised that observed phenomena such as coronal rain and, in general, the cool plasma component of the corona, may be of significance to our understanding of the heating of the corona. Observations with instruments onboard the Interface Region Imaging Spectrograph (IRIS) and the Solar Dynamic Observatory (SDO) imaging data indicate that long-period EUVintensity pulsations and periodic coronal rain are widely identified on large \nspatial scales of active regions and over long temporal scales with periods of minutes (Auchère et al., 2014; Şahin et al., 2023). They are believed to originate from multiple evaporation-condensation cycles, produced by the joint action of thermal non-equilibrium and thermal instability. Their presence is believed to pose major observational constraints for coronal heating theories (Antolin and Froment, 2022).', '10.6.2 Coronal waves': 'Early observations of solar coronal structures, using coronal emission lines already pointed out the presence of brightenings in these structures. These brightenings were associated with propagating waves. For example, Vernazza et al. (1975) used coronal EUV lines to search for vertical propagation, such as revealed by time lags between brightenings at different heights, and found evidence for upward propagation of pulses. Antonucci et al. (1984) studied oscillations in the C II, O IV and Mg X UV emission lines observed with Skylab during a loop brightening and found periodic intensity fluctuations with a period of 141 s. These observations of coronal oscillations were restricted to time series analysis without any spatial information. They were based on the measurement of the temporal and spatial variation of spectroscopic properties (such as intensity, line width, and Doppler velocity) of coronal emission lines. The situation changed drastically when high-resolution imaging and spectroscopic observations from instruments onboard e.g., SoHO, TRACE, Hinode, SDO, or Solar Orbiter spacecraft became available. Due to the temperature discrimination and spatial resolution of the EUV and soft X-ray telescopes onboard these satellites, observations of the solar corona have clearly demonstrated the existence of oscillations in the form of standing and propagating waves in solar coronal structures (Nakariakov and Verwichte, 2005; De Moortel, 2005; Aschwanden, 2006; De Moortel and Nakariakov, 2012). In parallel, the development of increasingly sophisticated theoretical and numerical models has enabled to associate them to slow, fast, and Alfvén magnetohydrodynamic (MHD) waves (Roberts, 2000; De Moortel and Nakariakov, 2012). The observed wave dynamics may have some relevance in coronal heating processes (Arregui, 2015; Van Doorsselaere et al., 2020b) as we discuss in the following. \nA defining property of the wave dynamics is their ubiquity. Observations with the Solar Dynamics Observatory (SDO) have demonstrated that Alfvén waves are common in the transition region and corona and that they all seem to share a common origin (McIntosh et al., 2011). Estimates about the energy carried by these disturbances vary, depending on the region of interest. The disturbances seem to be energetic enough to power the quiet corona and coronal hole regions, but not the active region corona. Observations with the highest available resolution instruments, confirm that wave activity is all-pervasive in the EUV corona. Morton and McLaughlin (2013), in a study that combined Hi-C and SDO/AIA data, concluded that this activity was of low energy. Lim \nFIGURE 10.7 Waves in extended regions of the corona Observational signatures of wave propagation in the form of images of the Fe XIII 10747 Å line intensity (left) and Doppler shift in part of the corona observed by CoMP. Adapted from Tomczyk and McIntosh (2009). From Van Doorsselaere et al. (2020b). \n<!-- image --> \net al. (2023) have recently extended the analysis to high-frequency waves (see Section 10.7.2). \nDecaying transverse coronal loop oscillations offered the first imaging evidence for standing waves in the corona associated with the periodic displacement of these structures (Aschwanden et al., 1999; Nakariakov et al., 1999). These large amplitude oscillations have periods of a few minutes and decay rapidly in time. They are caused by energetic events, such as nearby flares or coronal eruptions. Wave damping can be directly observed and measured. Information on periods and damping times has led to numerous developments in coronal seismology (Goossens et al., 2002, 2006; Arregui et al., 2007) and the development of models to explain the damping and possible wave heating (see Sections 10.7.2 and 10.8.2). They are however rather sporadic events (Terradas and Arregui, 2018) and coronal loops are hot irrespective of the presence or absence of lateral displacements. \nTransverse coronal loop oscillations do not always decay in time. Sometimes, they are seen to persist for long periods (Tian et al., 2012), or even have their amplitudes grow in time (Wang et al., 2012). They consist of low-amplitude displacements measured with time-distance analyses of imaging EUV observations and some events display the coexistence of decaying and decayless oscillations in the same area (Nisticò et al., 2013). The ubiquity of decayless oscillations (Anfinogentov et al., 2013, 2015), makes them a relevant component to be considered in the energy balance in the corona. A yet unidentified mechanism must be responsible to counteract the damping of the oscillations by supplying the required energy to the loops. \nObservations made with the Coronal Multichannel Polarimeter (CoMP: Tomczyk et al. 2008) have demonstrated that coronal disturbances are present in extended regions of the corona at all times (Tomczyk et al., 2007; Tomczyk and McIntosh, 2009). Figure 10.7 displays their main features in the form of observations in intensity and Doppler shift measurements. The observed Doppler velocity fluctuations do not produce significant intensity variations and are interpreted as MHD kink waves propagating along the coronal magnetic field. They show signatures of in situ wave damping in the form of a discrepancy in the outward to inward wave power, but energy estimates seem to fall below the required amount to heat the ambient plasma. From the observational side, estimates exist for detected wave energy fluxes (see Table 1 in Van Doorsselaere et al., 2020b, for an overview), yet the main difficulty lies in quantifying the fraction of that energy that gets dissipated. Still, the presence of wave dynamics in extended regions of the corona and their analysis is relevant in the context of both coronal seismology and coronal heating (Verth et al., 2010; Morton et al., 2015, 2016, 2019; Montes-Solís and Arregui, 2020; Morton et al., 2021). Physical processes that may convert wave energy into heating are discussed in Section 10.7.2.', '10.7 CORONAL HEATING MODELS': 'For typical coronal densities of 10 9 cm -3 andatemperature of 10 6 K,theradiative losses of the corona are estimated by 𝑛 2 𝑃 ( 𝑇 ) , where 𝑃 ( 𝑇 ) ∼ 10 22 erg s -1 cm 3 (see Rosner et al., 1978 for a classical citation or Hermans and Keppens, 2021 for a more recent comparison of modern energy loss functions). Converting to SI units, we then have an estimated radiative loss of 𝐿 = 10 -5 W m -3 . This coronal temperature and density lead to an estimate of the internal energy of the plasma as IE = 2 10 -2 J m -3 . From these two estimates, we could deduce an approximate cooling time 𝜏 cool of around \n𝜏 cool = 𝐼𝐸 𝐿 ≈ 2000 s . \nThus, the corona would normally cool down in a matter of half an hour. This firmly points to the existence of a coronal heating mechanism, which keeps it at the observed temperature. Since the corona exhibits the high temperature in all regions, it implies that there must be a coronal heating mechanism operating in all coronal regions and at all times. These heating requirements are listed in Table 10.1. As stated before, the source of the coronal heating is very clearly in the photospheric convective motions. \nCoronal heating models are mostly classified in two groups, namely the DC mechanisms and AC mechanisms. The distinction between these groups of mechanisms is in the time scale of the aforementioned photospheric driver and its comparison to the Alfvén transit time. The Alfvén transit time is the time it takes to cross the particular coronal structure (loop, fibril, plume) with the fastest wave. The fastest wave in coronal structures is usually the Alfvén \nwave (aside from high plasma𝛽 regions, such as prominences or flaring regions, where gas pressure dominates magnetic forces and thermal conduction plays the major role). For a structure of length 𝐿 with an Alfvén speed of 𝑉 A, the Alfvén transit time 𝜏 A is given by \n𝜏 A = 𝐿 𝑉 A . \nHere the Alfvén speed is 𝑉 A = 𝐵 / √ 𝜇𝜌 with magnetic field strength 𝐵 , density 𝜌 and magnetic permeability 𝜇 . \nIf the time scale of the driver 𝜏 is slower than the Alfvén transit time 𝜏 A: \n𝜏 ≫ 𝜏 A , \nit is a quasi-steady driving of the magnetic structure, which is called a DC heating mechanism. When the driver time scale is shorter than the Alfvén transit time \n𝜏 ≪ 𝜏 A , \nmostly waves are launched in the structure, and these are then called AC heating mechanisms.', '10.7.1 DC mechanisms': "The lowest energy state of the coronal magnetic field is in its potential form. It is thus assumed that the magnetic field would return to its potential form if it were not driven from the photospheric convection. The photospheric convection then leads to the creation of a non-potential form of the magnetic field. This has two consequences: (1) the non-potential fields have a slow buildup of currents, and (2) the driving leads to a tangling or braiding of the magnetic field. The latter effect is also known as Parker braiding (Parker, 1972; Pontin and Priest, 2022), and is displayed in Fig. 10.8. With the buildup of currents, the non-potential magnetic energy starts being dissipated by ohmic friction. However, it is thought that this ohmic dissipation of the currents is insufficient to heat the solar corona, because the coronal magnetic Reynolds number 𝑅 m is very high ( 𝑅 m ∼ 10 12 , see Terradas and Arregui, 2018). The ohmic dissipation happens on a dissipative time scale 𝜏 𝜂 , given by (Roberts, 1967) \n𝜏 𝜂 = 𝑅 m 𝜏 A . (10.1) \nThis shows that the ohmic dissipation of the built-up current is extremely slow for typical Reynolds numbers. \nBecause of the slow ohmic dissipation, a lot of non-potential energy can be stored in the coronal magnetic field. The magnetic field configuration becomes increasingly complex. It is thought that that strongly tangled magnetic field eventually reconnects in what is called a nanoflare . These are (as yet unobserved and unobservable) phenomena which are scaled-down versions of flares. The \nFIGURE 10.8 An illustration of Parker's braiding mechanism The footpoint driving by the photospheric motions lead to a non-potential state of the magnetic field with currents (middle panel), and then leads to complicated magnetic structure (right panel). Figure taken from Pontin and Priest (2022). \n<!-- image --> \nnanoflares are caused by a sudden decrease of the Reynolds number: either due to strong converging flows creating a stronger forcing and small lengths scales, or anomalous resistivity attributed to instabilities occurring in the current sheet which effectively lower the Reynolds number several orders of magnitude. \nSince the nanoflares are highly localised phenomena and the thermal conduction is practically only along the magnetic field, the heating by the nanoflares is then spread on single field lines. Thus, it is thought that the scenario of nanoflare heating of the corona leads to the existence of multi-stranded loops, where each individual field line has its own independent thermodynamic evolution (e.g. Aschwanden et al., 2000a; Viall and Klimchuk, 2013). On the other hand, nowadays it is thought that the strand's evolution cannot be entirely independent (Magyar and Van Doorsselaere, 2016), given that they are continuously mixed by the Kelvin-Helmholtz instability operating in transversely oscillation loops (Terradas et al., 2008). Still, despite this criticism, multi-stranded loops models are a popular topic in the coronal heating community (for more detail, see Section 10.8), and the effect of multi-strandedness is both considered in DC heating models (e.g. Reid et al., 2023) and AC heating models (Van Doorsselaere et al., 2008; Luna et al., 2010; Guo et al., 2019a). \nAn important concept in nanoflare heating of the solar corona is the distribution of nanoflares as a function of energy. The reason is that it is postulated that the nanoflares are occurring as scaled versions of regular flares, with a continuous scaling up to some minimum energy. Following Hudson (1991), we can write the histogram 𝑁 ( 𝐸 ) of number of flares as function of energy 𝐸 in a power law form: \n𝑁 ( 𝐸 ) ∼ 𝐸 -𝛼 . (10.2) \nThe total heating 𝐻 due to nanoflares can then be calculated by the integral \n𝐻 = ∫ 𝐸 max 𝐸 min 𝑁 ( 𝐸 ) 𝐸𝑑𝐸, (10.3) \nwhere 𝐸 max is the maximum flare energy and 𝐸 min is the minimum flare energy. The value for the latter (and for the former, for that matter) is currently not know, neither theoretically nor observationally. \nWith the previous power law form, the integral can be evaluated to be \n𝐻 DC ∼ 𝐸 2 -𝛼 2 -𝛼 GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> 𝐸 max 𝐸 min . (10.4) \nIn case 𝛼 < 2, the heating is dominated by the high energy flares, but given their localised nature in active regions (which is in contrast to the corona which is also hot outside of active regions) they are not able to the heat the corona to its observed temperature. Thus, a value of 𝛼 > 2 is desirable if you like to believe in the nanoflare heating of the corona. In the latter case, the heating is dominated by the small-scale nanoflares, which presumably occur everywhere and therefore heat the entire corona (in active regions and quiet Sun alike). \nThe measurement of the power law index of the nanoflare distribution 𝑁 ( 𝐸 ) was a big topic around the turn of the century (Aschwanden et al., 2000b; Parnell and Jupp, 2000). The distribution of nanoflares from different instruments is shown in Fig. 10.9. The currently obtained values for the power law index 𝛼 range between 1.8 - 2.7 (e.g. Purkhart and Veronig, 2022, and references therein) and are thus inconclusive if nanoflares play a major role in the heating of the corona.", '10.7.2 AC mechanisms': "When the driving time scale 𝜏 is shorter than the Alfvén transit time 𝜏 A, the shuffling of the magnetic field happens on fast time scales. Rather than creating tangling, this then results in the generation of (magnetic) waves. In an unstructured corona, these waves are Alfvén and fast waves, but with the structuring imposed by the density contrast in loops or plumes, additionally kink waves are also excited. \nKink waves are transverse motions of an overdense flux tube (Zaitsev and Stepanov, 1975; Wentzel, 1979; Edwin and Roberts, 1983). They are like surface Alfvén waves (transverse waves of an interface), but warped onto a cylindrical (or any other closed) surface. They are driven mostly by magnetic tension (Goossens et al., 2009), have a large parallel vorticity (Goossens et al., 2019), but also have compression and magnetic pressure variations. \nAs for DC heating mechanisms, the waves may contain sufficient energy to heat the solar corona, but the problem is that it is dissipated on long, resistive time scales 𝜏 𝜂 . Thus, it is necessary to have mechanisms to transport the wave \nFIGURE 10.9 Nanoflare energy spectra Observed nanoflare energy distribution from different instruments and the associated power law indices are indicated in the legend of the figure. Figure taken from Aschwanden et al. (2000b). \n<!-- image --> \nFIGURE 10.10 A simulation result of phase mixing The colour scale shows Alfvén waves propagating in a non-uniform plasma, showcasing the effect of phase mixing. Figure taken from McLaughlin et al. (2011). \n<!-- image --> \nenergy from the large scales to the small scales, where the Reynolds number is sufficiently low. Over the course of the years, several mechanisms have been proposed for that. \nThe first such mechanism to consider is phase mixing (Heyvaerts and Priest, 1983). The mechanisms for phase mixing is usually referred to as the phase mixing of Alfvén waves, but the mechanism works equally well for other wave types. The crucial ingredient is the spatial variation of the phase speed of the wave. From a uniform or large-scale driver, the wave fronts on different field lines gradually get out of phase, leading to small length scales. This is shown in Figure 10.10. The wave fronts start from the bottom of the figure as a straight line, but bend due to the Alfvén speed gradient. Then the central part of the figure shows the decrease of wave power, because there the Reynolds number becomes sufficiently low to damp the Alfvén waves. One of the inherent problems of phase mixing is that the wave damping (and the resulting heating) does depend on the transport coefficients for viscosity or resitivity (see Roberts, 2000, for analytical expressions), which are extremely small in the solar corona. \nAnother popular mechanism is called resonant absorption of kink waves (Chen and Hasegawa, 1974). Also for this wave damping mechanism, the presence of an Alfvén speed gradient is crucial. Typically, the mechanism is modelledinacylindrical configuration mimicking a loop, with a smooth variation of the density profile across the loop's edge. In such a system, a transverse wave is found and is called a kink wave . It has a transverse displacement of the whole loop body, in a nearly incompressible way, which has a frequency \n𝜔 k = √︄ 2 𝐵 2 𝜇 ( 𝜌 i + 𝜌 e ) . (10.5) \nFIGURE 10.11 Structure of solar wind wave-power spectral range The formation of a power law due to turbulence, with the energy injection scale, the inertial range and the dissipation scale. Figure taken from Van Doorsselaere et al. (2020b). \n<!-- image --> \nHere 𝜌 i and 𝜌 e are the densities inside and outside the loop respectively (for a derivation, see e.g. Edwin and Roberts, 1983). That kink frequency 𝜔 k is an average of the interior and exterior Alfvén frequency. Thus, at some resonant point, the kink frequency must equal the local Alfvén frequency. Thus, at that resonant point, the loop's global motion is converted from the large scale to small scale, shearing Alfvén waves similar to phase mixing (Soler and Terradas, 2015), allowing to let resistive heating pick up again. This mechanism has the advantage that the damping of the global kink mode happens on a short time scale that does not depend on the resistivity. However, the wave energy in the Alfvén waves at the resonant point only damps at resistive time scales (Terradas and Arregui, 2018). \nA third possible mechanism is the creation of turbulence from non-linear wave interaction. The development of turbulence comes in essence from the (in)famous non-linear term /one.sup in the fluid equations fi 𝑣 · ∇fi 𝑣 . Imagine driving the system with a single frequency 𝜔 . The non-linear term is then acting as a sink at the frequency 𝜔 , but acting as a source at the Fourier component with the double frequency. From that double frequency, one can progressively fill a whole Fourier spectrum with wave power. \nIn the spectral domain, the wave power arranges itself into a power law behaviour. This is readily observed in the solar wind (Bruno and Carbone, \n2013), and it is likely that such a power law also exists in the solar corona (e.g. Tomczyk et al., 2007; Morton et al., 2015). In the solar wind, the spectral range is broken into 3 (or more) regimes, see Figure 10.11. The large-scale regime is called the energy injection scale, where presumably coronal waves are excited at granular or supergranular scales. This regime has a power law index of -1. Then there is the inertial range, where the power law index is -5/3. This can be derived in fluid dynamics from the assumption that the energy cascade rate in this inertial range is independent from the scale, as was first pointed out by Kolmogorov (1962). When the turbulence reaches a small scale, the Reynolds number becomes low and dissipation sets in to damp the fluctuations and heat the corona. Depending on the dissipation mechanism, the power law index there is typically between -2 and -3. \nIn the corona, we believe the turbulence develops from a number of mechanisms. The first one is the interaction of counterpropagating Alfvén waves, as it is also believed to be responsible in the solar wind. During the interaction of the Alfvén wave, they have a short time to non-linearly modify the other wave front, leading to turbulence. A caveat in coronal structures is that the Alfvén waves need to have a variation along their direction of polarisation (Shestov et al., 2022), which necessarily lead to non-zero azimuthal wave numbers 𝑚 in magnetic cylinders. \nAsecond method to create turbulence is through the Kelvin-Helmholtz instability at the shearing boundary of loops oscillating with the kink wave (Terradas et al., 2008). For such standing kink waves, the magnetic shear (i.e. current) is near the footpoints, and does not stabilise the velocity shear. Thus, the classical Kelvin-Helmholtz instability (KHI) takes place because of the rapid change of azimuthal velocity at the loop's boundary, evolving in what is sometimes called Transverse Wave Induced Kelvin-Helmholtz (TWIKH) rolls. The velocity shear is enhanced by resonant absorption (Antolin and Van Doorsselaere, 2019) taking place at the same location. \nThe third mechanism for generating turbulence in the solar corona is called uniturbulence (Magyar et al., 2017), operating in propagating kink waves. In that mechanism, turbulence is not generated from the KHI, because there is a stabilising magnetic field that inhibits the KHI growth. Instead, the kink wave is self-interacting (Magyar et al., 2019) leading to the formation of small scales, kink wave damping and heating (Van Doorsselaere et al., 2020a). \nNowadays, researchers are starting to consider the AC mechanisms also through similar power law descriptions as the DC heating mechanisms (see Equation 10.2). As for the DC heating mechanisms, we may say that the power of AC heating events 𝑃 ( 𝜔 ) follows a power law as a function of the frequency 𝜔 : \n𝑃 ( 𝜔 ) ∼ 𝜔 -𝛿 . (10.6) \nAs for the DC heating mechanisms, we also have to integrate this over the whole \nfrequency domain to find the entire AC heating energy input 𝐻 AC into the corona: \n𝐻 AC ∼ ∫ 𝑃 ( 𝜔 ) 𝑑𝜔 = 𝜔 1 -𝛿 1 -𝛿 GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> 𝜔 𝑚𝑎𝑥 𝜔 𝑚𝑖𝑛 . (10.7) \nIn contrast to the DC heating mechanism, the critical value for the slope is 1. Such considerations are starting to be considered for modern solar telescopes, in casu SolO/EUI, SDO/AIA, (u)CoMP (Morton et al., 2019; Lim et al., 2023). In the latter work, it is tentatively concluded that for small structures, 𝛿 < 0 so that the high frequency waves have the most important contribution in the AC heating.", '10.8.1 Numerical models of DC heating': 'In numerical models of DC heating, there are basically two groups. The first group consider a coronal loop as the hydrodynamic evolution of 1D strands after nanoflare heating. The second group tries to capture the full corona at a large scale, following an ab-initio approach. \nIn the first group of models, the main point that is considered is that nanoflares are very localised heating events. They would suddenly dump their energy on a specific field line, after which the plasma on that field line would evolve hydrodynamically, with the field only providing the geometry but do not restrict the physics. \nAn example of results from such 1D codes are found in Bradshaw and Viall (2016). In that work the loop (bundle) is modelled as 400 independently evolving field lines. On these field lines, a heating function is chosen and parametrised (e.g. continuous heating mimicking waves, or impulsive, stochastic nanoflares). Despite the seemingly simple nature of the model, the thermodynamic evolution is often non-intuitive. The superposition of different strands allows to explain several observational phenomena: for instance in the aforementioned paper, it was shown that these 1D models are compatible with the observed cooling behaviour of coronal loops (Viall and Klimchuk, 2012). \nBecause of their success, such 1D models for coronal loops are also popular in explaining the observed behaviour of flares. Still, critical voices exists saying that the KHI of loops oscillating with kink waves will rapidly mix such individual strands, effectively creating cross-field coupling in the loop (Magyar and Van Doorsselaere, 2016). \nThe second group of models consist of 3D numerical models of a part of the solar corona. They all basically follow the lead of Gudiksen and Nordlund (2005). Such simulations typically take a simulation of the solar convection zone as the bottom boundary condition. Then they let the plasma in the corona evolve self-consistently, driven by the photospheric simulation. It is found that the corona self-organises into loops, whereby it is currently debated what a \ncoronal loop actually is (Bingert and Peter, 2011; Malanushenko et al., 2022). These works seem to point out that the loop is determined as the line-of-sight integration of plasma structures, rather than a magnetic flux tube. \nThe heating in such 3D models is naturally due to the lower Reynolds numbers. In most simulations, the heating is due to numerical resistivity. In the more modern iterations (e.g. using MURAM, Rempel, 2017), a strong weight is given to the compressive viscosity. However, the differentiation between the numerical resistivity or viscosity does not seem to have a large effect on the apparent structure of the resulting corona. \nSuch models are starting to be used to characterise the nature of the nanoflares. For instance, Kanella and Gudiksen (2019) quantify the distribution of the Joule heating events in such a simulation, to understand the spatial distribution and occurrence rates of nanoflares. This is an avenue that could be pursued in the future, providing constraints on the possible coronal heating mechanisms in such 3D simulations.', '10.8.2 Numerical models of AC heating': "Early numerical models considered the heating of coronal loops by resonant absorption in compressible, resistive MHD. The loops were approximated as cylindrical plasma columns and were excited by an external periodic driver (see e.g., Poedts et al., 1989a,b). The driven system reaches a stationary state. Part of the energy from the incident waves is extracted by the coronal loops, gets dissipated, and can heat the plasma. Resonant absorption is very efficient for typical coronal loop parameters as a considerable part of the energy supplied by the external source is dissipated ohmically and converted into heat (Poedts et al., 1990a,b). The process works for wave modes with any azimuthal wavenumber 𝑚 , not only kink modes ( 𝑚 = 1) and can thus play a role in the absorption of p-modes with high azimuthal wave numbers by sunspots (Goossens and Poedts, 1992). As shown by Poedts et al. (1990c), resonant absorption evolves in time with phase mixing towards the resonant point. \nFor phase mixing, the numerical models were previously mostly in 2D (McLaughlin et al., 2011). These heating models were in highly idealised setups, but recently such models started to include the lower atmosphere (Van Damme et al., 2020). Moreover, the transition was made to 3D. In 3D the effects were considered of driving with data-inspired power laws (Pagano and De Moortel, 2019). Moreover, the latest simulations significantly go beyond simplified magnetic field configurations (Howson et al., 2020). The latter considered wave propagation and damping in a 3D configuration induced by magnetic braiding, which offered readily available small scales for enhanced wave dissipation. The conclusion of most of these simulations on phase mixing heating remains the same: it is hard to heat the corona, unless high wave driver amplitudes are considered, in addition to large resistivity (low Reynolds numbers). Thus, it seems that even the small scales introduced by the magnetic braiding do not facilitate \nheating by phase mixing enough to heat the corona. \nAnother group of models considers the heating by torsional Alfvén waves. These find their root as early as Moriyasu et al. (2004); Buchlin and Velli (2007); Antolin et al. (2008). They have 1D field line models for solar coronal loops on which torsional Alfvén waves are driven at the loop's footpoint. The heating is then provided by either the steepening of the Alfvén waves and associated dissipation, or Alfvén wave interaction forming turbulence. A famous example of the latter is van Ballegooijen et al. (2011), who considers the turbulent interaction of a superposition of Alfvén waves in close loop models. Given enough wave driving at the footpoint, the wave heating can create loops with a coronal temperature, even leading to condensations as a consequence of very strong heating. The latest incarnation of this model is Matsumoto (2018), who extended this model to a 3D system. The main issue with these models is that they invariable ignore radial structuring of coronal loops, either by using reduced MHD, or by considering a uniform cross-section before the launching of the waves. Thus, it is rather unclear how such loop models correspond to observations, which are actually density-enhanced with respect to the background coronal plasma. \nIn the last couple of years, there has been a leap into models which do the heating with the KHI and the associated TWIKH rolls (Terradas et al., 2006; Antolin and Van Doorsselaere, 2013). It has been found that coronal loops get fully turbulent (Karampelas and Van Doorsselaere, 2018; Antolin and Van Doorsselaere, 2019). That leads to significant heating, even as much as being able to compensate the loop's radiative losses (Shi et al., 2021). This avenue seems promising, but for now, no observational confirmation exists (aside from the circumstantial Antolin et al., 2015; Pascoe et al., 2020). Still, the predicted damping (Van Doorsselaere et al., 2021) seems to be compatible with the observed dependence of the damping with the amplitude (Nechaeva et al., 2019; Arregui, 2021). On the other hand, the loop heating is only successful for low density loops, but not for higher density loops or driving at a nonresonant frequency (De Moortel and Howson, 2022). Additionally, it does not provide a convincing mechanism to provide mass to the loop from an initially homogeneous corona, and thus does not constitute a self-consistent mechanism for the formation of the loop. \nA final group of wave heating models is by uniturbulence (Magyar et al., 2017). This wave heating mechanism is operating in coronal open regions (Magyar and Nakariakov, 2021), where the coronal plumes may provide the structure for creating the uniturbulence. The damping of the transverse waves is significant (Van Doorsselaere et al., 2020b; Morton et al., 2021), and the resulting turbulent heating must contribute something to the energy budget, on top of the heating by Alfvén wave turbulence from reflected Alfvén waves. This route is interesting to pursue, because it seems from WKB models of the solar wind (van der Holst et al., 2014) that there is insufficient wave driving in coronal holes. As for the KHI heating models, also the observational evidence for uniturbulence is uncertain, although Pant et al. (2019) can self-consistently \nexplain the observed relationship between Doppler shift and spectral line width (McIntosh and De Pontieu, 2012). This also matches well with the observed non-thermal broadening in the corona and solar wind.", '10.9 SUMMARY AND OUTLOOK': "Imaging and spectroscopic observations display a multi-million degree solar corona that cannot be explained by means of thermal energy transport mechanisms. EUV and X-ray emission, primarily produced by highly ionised atoms in the corona, is highly structured by the magnetic field and its overall appearance varies with the solar-activity-cycle. On the large-scale, the corona is the source of the solar wind. The high temperatures in the corona help to accelerate these particles to high speeds, which can now be measured by spacecraft in the solar wind. At the active-region scale, detailed analyses of intensity variations and thermal structure provide constraints to coronal heating theories and modelspecific parameters, such as the frequency of small-scale heating events. The strongly dynamic nature of the corona makes plasma flows, wave dynamics, and magnetic reconnection relevant components in the mass and energy balance of the corona. \nConcerning coronal heating models, it is safe to say that wave heating models made a leap in the last 5 years from 1D models or even cartoon-like models to modern 3D configurations. Despite this step forward, it is still not straightforward to heat coronal loops to the desired temperature. For instance, Shi et al. (2021) show that a coronal loop with density 10 8 cm -3 can be heated to coronal temperatures and sustain it against radiative losses. However, De Moortel and Howson (2022) have shown that this only works for a specific set of parameters (low density, resonant driving). Thus, new ingredients are needed to make AC heating models work. For example, multi-mode driving (Guo et al., 2019b; Afanasyev et al., 2020) and inherent fine structure (Guo et al., 2019a; Howson et al., 2020) will be needed to enhance the heating. \nThe Extreme Ultraviolet Imager onboard ESA's Solar Orbiter (SolO/EUI) is already offering remarkable data on small-scale dynamical phenomena, such as the relaxation of braided coronal magnetic fields through magnetic reconnection (Chitta et al., 2022), quiet-Sun intermittent brightening events (Berghmans et al., 2021), or high-frequency spatially resolved wave dynamics (Petrova et al., 2023; Lim et al., 2023; Zhong et al., 2023). \nThe Marshall Grazing Incidence X-ray spectrometer (MaGIXS) is a slitless imaging spectrograph that enables to obtain spatially resolved soft X-ray (6 - 24 Å) spectra across a wide field of view. The first science results, described in Savage et al. 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2024arXiv240504583D | We present photometric and spectroscopic observations of SN 2023fyq a type Ibn supernova in the nearby galaxy NGC 4388 Dsimeq18Mpc. In addition we trace longstanding precursor emission at the position of SN 2023fyq using data from DLT40 ATLAS ZTF ASASSN Swift and amateur astronomer Koichi Itagaki. Precursor activity is observed up to nearly three years before the supernova explosion with a relatively rapid rise in the final 100 days. The doublepeaked postexplosion light curve reaches a luminosity of sim1043rm ergs1. The strong intermediatewidth He lines observed in the nebular spectrum of SN 2023fyq imply the interaction is still active at late phases. We found that the precursor activity in SN 2023fyq is best explained by the mass transfer in a binary system involving a lowmass He star and a compact companion. An equatorial disk is likely formed in this process sim0.6rm Modot and the interaction of SN ejecta with this disk powers the main peak of the supernova. The early SN light curve reveals the presence of dense extended material sim0.3rm Modot at sim3000rm Rodot ejected weeks before the SN explosion likely due to finalstage core silicon burning or runaway mass transfer resulting from binary orbital shrinking leading to rapid rising precursor emission within sim30 days prior to explosion. The final explosion could be triggered either by the corecollapse of the He star or by the merger of the He star with a compact object. SN 2023fyq along with SN 2018gjx and SN 2015G forms a unique class of Type Ibn SNe which originate in binary systems and are likely to exhibit detectable longlasting preexplosion outbursts with magnitudes ranging from 10 to 13. | 2024-05-01T00:00:00Z | ['2024arXiv240504583D', 'arXiv:2405.04583', '10.48550/arXiv.2405.04583'] | ['Astrophysics - High Energy Astrophysical Phenomena', 'Astrophysics - Solar and Stellar Astrophysics'] | SN2023fyq A Type Ibn Supernova With Longstanding Precursor Activity Due to Binary Interaction | 2,024 | 167 | 0.59 | ['EPRINT_HTML', 'EPRINT_PDF'] | 6 | https://arxiv.org/pdf/2405.04583.pdf | {'SN2023fyq: A Type Ibn Supernova With Long-standing Precursor Activity Due to Binary Interaction': "<!-- image --> \n1 Department of Physics and Astronomy, University of California, 1 Shields Avenue, Davis, CA 95616-5270, USA \n2 TAPIR, Mailcode 350-17, California Institute of Technology, Pasadena, CA 91125, USA \n3 Research Center for the Early Universe (RESCEU), School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan \n4 Steward Observatory, University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721-0065, USA \n5 Gemini Observatory, 670 North A'ohoku Place, Hilo, HI 96720-2700, USA \n6 Department of Physics and Astronomy, Rutgers, the State University of New Jersey, \n136 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA \n7 Department of Physics & Engineering Physics, University of Saskatchewan, 116 Science Place, Saskatoon, SK S7N 5E2, Canada \n8 Department of Physics and Astronomy, Johns Hopkins University, 3400 North Charles Street, Baltimore, MD 21218, USA \n9 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA \n10 W. M. Keck Observatory, 65-1120 M¯amalahoa Highway, Kamuela, HI 96743-8431, USA \n11 IAASARS, National Observatory of Athens, Metaxa & Vas. Pavlou St., 15236, Penteli, Athens, Greece \n12 Las Cumbres Observatory, 6740 Cortona Drive, Suite 102, Goleta, CA 93117-5575, USA \n13 Department of Physics, University of California, Santa Barbara, CA 93106-9530, USA \n14 Steward Observatory, University of Arizona, 933 North Cherry Avenue, Rm. N204, Tucson, AZ 85721-0065, USA \n15 Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138-1516, USA \n16 The NSF AI Institute for Artificial Intelligence and Fundamental Interactions, USA \n17 Itagaki Astronomical Observatory, Yamagata 990-2492, Japan \n18 Department of Astronomy, University of Virginia, Charlottesville, VA 22904, USA \n19 Department of Physics and Astronomy, University of North Carolina, 120 East Cameron Avenue, Chapel Hill, NC 27599, USA", 'ABSTRACT': 'We present photometric and spectroscopic observations of SN 2023fyq, a type Ibn supernova in the nearby galaxy NGC 4388 (D ≃ 18 Mpc). In addition, we trace the three-year-long precursor emission at the position of SN 2023fyq using data from DLT40, ATLAS, ZTF, ASAS-SN, Swift, and amateur astronomer Koichi Itagaki. The double-peaked post-explosion light curve reaches a luminosity of ∼ 10 43 erg s -1 . The strong intermediatewidth He lines observed in the nebular spectrum imply the interaction is still active at late phases. We found that the precursor activity in SN 2023fyq is best explained by the mass transfer in a binary system involving a low-mass He star and a compact companion. An equatorial disk is likely formed in this process ( ∼ 0.6M ⊙ ), and the interaction of SN ejecta with this disk powers the second peak of the supernova. The early SN light curve reveals the presence of dense extended material ( ∼ 0.3M ⊙ ) at ∼ 3000R ⊙ ejected weeks before the SN explosion, likely due to final-stage core silicon burning or runaway mass transfer resulting from binary orbital shrinking, leading to rapid rising precursor emission within ∼ 30 days prior to explosion. The final explosion could be triggered either by the core-collapse of the He star or by the merger of the He star with a compact object. SN 2023fyq, along with SN 2018gjx and SN 2015G, forms a unique class of Type Ibn SNe which originate in binary systems and are likely to exhibit detectable long-lasting pre-explosion outbursts with magnitudes ranging from -10 to -13. \nKeywords: Core-collapse supernovae (304), Circumstellar matter (241), Stellar mass loss (1613)', '1. INTRODUCTION': 'Type Ibn supernovae (SNe) are a subclass of interactionpowered SNe that show narrow helium (He) lines but not hydrogen (H) lines in their spectra (e.g., Smith 2017; Modjaz et al. 2019). Although it has been more than two decades since the discovery of the first Type Ibn SN (SN 1999cp, Matheson et al. 2000), our understanding of Type Ibn progenitors remains limited. The light curves of Type Ibn SNe tend to be short-lived and some of them even resemble the evolution of fast-evolving transients (Ho et al. 2023; Fox & Smith 2019). A general interpretation is that SNe Ibn are Wolf-Rayet/He stars that experience enhanced mass loss right before the SN explosion. The interaction of SN ejecta with the surrounding dense He-rich circumstellar material (CSM) powers some of the SN light curve and ionizes the outer CSM, producing the narrow lines we observe (Pastorello et al. 2007; Hosseinzadeh et al. 2017). \nLight curve modeling of Type Ibn SNe has supported the presence of dense CSM close to the progenitors (Gangopadhyay et al. 2020; Pellegrino et al. 2022; Ben-Ami et al. 2023). Both SNe Ibn and their H-rich counterparts, SNe IIn, have CSM interaction signatures that point to pre-SN mass loss that is much stronger than normal massive-star winds (Smith 2014, 2017). However, the mechanisms driving the enhanced mass loss near the time of explosion remain a subject of active debate. This enhanced mass loss could be attributed to the final-stage stellar activities of massive stars, where the dense CSMcould be produced by eruptive outbursts through pulsational pair instability (Yoshida et al. 2016; Woosley 2017) or wave-driven outbursts excited by late-stage nuclear burning (Quataert & Shiode 2012; Shiode & Quataert 2014; Fuller 2017; Fuller & Ro 2018; Morozova et al. 2020). Alternatively, the dense CSM might be generated through binary interactions (Soker 2013; Smith 2014; Smith & Arnett 2014; Metzger 2022; Wu & Fuller 2022; Dessart et al. 2022; Tsuna et al. 2024a). In this scenario the progenitor does not necessarily have to be a very massive star, as the mass loss would be significantly enhanced by the presence of a binary companion. \nOne way to constrain the progenitor of Type Ibn SNe is by searching for evidence of a massive star or a binary companion in deep images once the SN fades. For example, the low star formation rate at the site of PS1-12sk ruled out a massive star progenitor (Hosseinzadeh et al. 2019). In addition, the absence of evidence for massive star progenitors and the possible detection of binary companions have been reported for \nsome other Type Ibn SNe (Maund et al. 2016; Shivvers et al. 2017). \nAlternatively, a direct way to constrain the mass loss history of SN progenitors is by searching for signs of pre-explosion activity or precursor emission prior to the SN explosion. Precursor emission is commonly observed in Type IIn SNe (e.g., Mauerhan et al. 2013; Smith et al. 2010; Strotjohann et al. 2021; Ofek et al. 2013, 2014; Tartaglia et al. 2016; Pastorello et al. 2013, 2018; Hiramatsu et al. 2024). The bright precursor outbursts in Type IIn SNe may be due to eruptive mass loss from LBV-like progenitors (Gal-Yam et al. 2007; Gal-Yam & Leonard 2009; Smith 2017) or pulsational pair instability outbursts (Smith & McCray 2007; Woosley et al. 2007; Smith 2014). Alternatively, these outbursts could be caused by red supergiants with a compact object companion (Fryer & Woosley 1998; Schrøder et al. 2020; Smith et al. 2024; Tsuna et al. 2024a), or other late-stage binary interaction (Smith & Arnett 2014). To date, precursor emission has been identified in two Type Ibn SNe, SN 2006jc (Pastorello et al. 2007) and SN2019uo (Strotjohann et al. 2021). The precursor outbursts in these events are shorter and fainter compared to those observed in Type IIn SNe, and have been interpreted as resulting from single massive star activities or binary interactions (Pastorello et al. 2007; Foley et al. 2007; Smith et al. 2008; Tsuna et al. 2024a). \nIn this paper we present the optical observations of SN 2023fyq, one of the closest SNe Ibn. The light curves and spectra of this object closely resemble those of Type Ibn SNe. Notably, relatively steady precursor activity is observed up to approximately three years prior to the SN explosion. The detection of precursor emission in SN 2023fyq allows us to investigate the final-stage stellar activity and the nature of its progenitor system. The pre-explosion observations of SN2023fyqare also presented in Brennan et al. (2024), where they identify an asymmetric CSM structure, likely related to unstable stellar activities of the progenitor. \nThe paper is organized as follows: the photometric and spectroscopic observations are described in Section 2. We constrain the reddening and distance of SN 2023fyq in Section 3. We describe the photometric and spectroscopic evolution of SN 2023fyq in Sections 4 and 5. The progenitor scenario and the physical mechanism of precursor activities are discussed in Section 6. We summarize the main results in Section 7.', '2. OBSERVATIONS': "SN 2023fyq was discovered on 2023 April 17 by the Zwicky Transient Facility (ZTF) survey at RA(2000) = 12 h 25 m 45 . s 847, Dec(2000) = + 12 · 39 ' 48 . '' 87 in NGC 4388 \nFigure 1. Composite 𝑔𝑟𝑖 image of SN 2023fyq in NGC 4388 obtained with the Las Cumbres Observatory on 2023 August 11. The position of SN 2023fyq is indicated by white tick markers. \n<!-- image --> \n(De 2023) (see Figure 1). On 2023 June 14 a rapid rebrightening of SN 2023fyq was observed and reported by amateur astronomer Koichi Itagaki (Itagaki 2023). On 2023 June 25 SN 2023fyq was classified as a peculiar Type Ib due the presence of helium lines and the lack of hydrogen lines in the optical spectrum (Valerin et al. 2023). As we will discuss in the paper, a Type Ibn classification is more appropriate for SN 2023fyq because its photometric and spectroscopic evolution match those of Type Ibn SNe. This is consistent with the classification of SN 2023fyq discussed in Brennan et al. (2024). \nIn this section we present the photometric data of SN 2023fyq taken by Las Cumbres Observatory (Brown et al. 2013) via the Global Supernova Project, the Distance Less Than 40 Mpc (DLT40, Tartaglia et al. 2018) survey, ZTF (Bellm et al. 2019; Graham et al. 2019), the Asteroid Terrestrial-Impact Last Alert System (ATLAS, Tonry 2011; Tonry et al. 2018; Smith et al. 2020), the All-Sky Automated Survey for Supernovae (ASAS-SN, Shappee et al. 2014; Kochanek et al. 2017), the Neil Gehrels Swift Observatory (Gehrels et al. 2004), and amateur astronomer Itagaki. We also report the spectroscopic followup of SN 2023fyq taken after the SN explosion. All spectroscopic observations from this paper can be found at https://github.com/yizedong/ SN2023fyq data and will be available on WISeREP (Yaron &Gal-Yam 2012) /one.sup .", '2.1. Photometric Observations': 'For the photometry we adopt a signal-to-noise threshold of 3 for source detections and a signal-to-noise threshold of 5 \nfor computing the upper limit, following the suggestions of Masci (2011). The light curves are shown in Figure 2 and 3.', '2.1.1. Las Cumbres Observatory Observations': 'Our multiband photometric followup campaign with Las Cumbres Observatory was initiated on 2023 July 26. The images were reduced using the PyRAF-based photometric reduction pipeline /l.pc/c.pc/o.pc/g.pc/t.pc/s.pc/n.pc/p.pc/i.pc/p.pc/e.pc (Valenti et al. 2016). Apparent magnitudes were calibrated using the APASS ( 𝑔, 𝑟, 𝑖 ) and Landolt ( 𝑈, 𝐵,𝑉 ) catalogs.', '2.1.2. DLT40 Observations': 'The DLT40 survey is a targeted one-day cadence SN search for very young transients within 40 Mpc (Tartaglia et al. 2018; Yang et al. 2019). \nDLT40 has been monitoring the field of SN 2023fyq since 2014 in the 𝐶𝑙𝑒𝑎𝑟 filer. All of the images have been visually inspected to remove those with bad qualities. A deep template was made with the images taken between 2014 June 20 and 2015 February 01 using Swarp (Bertin et al. 2002). The rest of the images were stacked in windows of 15 days and were then subtracted against the template using HOTPANTS (Becker 2015). We used aperture photometry at the position of SN 2023fyq through a pipeline based on Photutils (Bradley et al. 2022). The photometry was calibrated to the 𝑟 band.', '2.1.3. ZTF Observations': 'ZTF is a time-domain survey using a wide-field camera mounted on the Palomar 48-inch Schmidt telescope (Bellm et al. 2019; Graham et al. 2019). The ZTF public survey searches for transients and variables in the northern sky with a three-day cadence in 𝑔 and 𝑟 filters. \nThe position of SN 2023fyq has been monitored by ZTF since 2018. We obtained the forced photometry from the ZTF Forced Photometry Service (Masci et al. 2023). We removed bad-quality data following the instructions in Masci et al. (2023). For images taken after -300d, the transient was bright enough to be detected in single images, and so the observations were stacked in 1-day time bins. For images taken prior to -300d, the observations were stacked in 15-day time bins to improve the signal to noise ratio (S/N).', '2.1.4. ATLAS Observations': 'The ATLAS survey is an all-sky daily cadence survey (Smith et al. 2020) carried out in two filters, cyan ( 𝑐 ) and orange ( 𝑜 ), roughly equivalent to Pan-STARRS filters 𝑔 + 𝑟 and 𝑟 + 𝑖 , respectively. \nThe position of SN 2023fyq has been monitored by ATLAS since 2015. Forced photometry at the supernova position was obtained from the ATLAS forced photometry server (Shingles et al. 2021). Using the method presented in Young (2022), we stacked the measurements to improve the signal-to-noise \nFigure 2. Photometric limits and detections of SN 2023fyq prior to and after explosion. Detections with S/N > 4 are indicated by large solid symbols, while detections with 3 < S/N ≤ 4 are indicated by hollow symbols. The smaller symbols are nondetection limits with S/N ≤ 3. The precursor activities detected in Type Ibn SN 2006jc ( 𝑅 band) and SN 2019uo ( 𝑟 band) are indicated in the red and green rectangles, respectively. The limits on the precursor activities on Type Ibn SN 2015G are shown with the purple dashed line. All of the bands are in the AB magnitude system. \n<!-- image --> \nratio and obtain deeper upper limits. For images taken after -300d, the observations were stacked in 1-day time bins. For images taken before -300d, the observations were stacked in 15-day time bins.', '2.1.5. ASAS-SN Observations': 'ASAS-SN is an untargeted all-sky survey to a depth of g ∼ 18.5 mag. (Shappee et al. 2014; Kochanek et al. 2017). Weobtained the ASAS-SN reference image subtracted forced photometry from the ASAS-SN sky portal /two.sup .', '2.1.6. Swift Observations': "The position of SN 2023fyq has been observed by the UVOT instrument on the Neil Gehrels Swift Observatory (Gehrels et al. 2004) since 2015. We performed aperture photometry with an apreture size of 3 '' at the position of SN 2023fyq on Swift UVOT images using the High-Energy \nAstrophysics software (HEA-Soft). Background variations in individual images were removed using a 5 '' aperture placed on a blank section of the sky. To remove the underlying galaxy background contamination, we subtracted the flux extracted from Swift UVOT images taken on 2016 November 08. Zero-points were chosen from Breeveld et al. (2011) with time-dependent sensitivity corrections updated in 2020.", "2.1.7. Koichi Itagaki's Observations": "We also incorporated observations taken with Koichi Itagaki's Bitran BN-83MCCD imager mounted on a 0.5m telescope in Okayama Prefecture, Japan. We solved the astrometry of the images using Astrometry.net (Lang et al. 2010). The aperture photometry was performed using a pipeline based on Photutils (Bradley et al. 2022) and was calibrated to r-band magnitudes in the Sloan system (Fukugita et al. 1996).", '2.2. Spectroscopic Observations': "We collected four optical spectra from the FLOYDS spectrograph (Brown et al. 2013) on the 2m Faulkes Telescope \n<!-- image --> \nFigure 3. The light curve evolution of SN 2023fyq. The 𝐶𝑙𝑒𝑎𝑟 filter is calibrated to the 𝑟 band. The hollow symbol indicates the data with 3 < S/N ≤ 4, while the solid symbol indicates the data with S/N > 4. Light curves in the bottom panel have been shifted by the indicated amounts to enhance clarity. All of the bands are in the AB magnitude system. The black dashed line marks the epoch of the first light of the SN ( -11 d), as adopted in the paper. \n<!-- image --> \nSouth in Australia at the Las Cumbres Observatory via the Global Supernova Project. The FLOYDS spectra were reduced following standard procedures using the FLOYDS pipeline (Valenti et al. 2014). We triggered Gemini-North Target of Opportunity (ToO) observations with the Gemini Multi-Object Spectrograph (GMOS; Hook et al. 2004) and the B600 grating on 2023 July 27 and 2023 August 01 through proposal GN-2023A-Q-136. The Gemini spectra were reduced by using the IRAF Gemini package. We triggered further ToO observations with the Andalucia Faint Object Spectrograph and Camera (ALFOSC) on the Nordic Optical Telescope (NOT) at the Spanish 'Roque de los Muchachos' \nObservatory (ORM) on 2023 August 04 through proposal 67112. The NOT ALFOSC spectrum was observed using Grism #4 and a 1. '' 0 slit and was reduced using the PypeIt pipeline (Prochaska et al. 2020; Prochaska et al. 2020). We obtained spectra on 2023 December 12 and 2024 May 1 from the LowResolution Imaging Spectrometer (LRIS; Oke et al. 1995) on the Keck I telescope. The LRIS spectra were reduced in a standard way using the LPipe pipeline (Perley 2019). A low-resolution spectrum was taken on 2024 January 23 with the Goodman High Throughput Spectrograph (GHTS) on the Southern Astrophysical Research Telescope (SOAR; Clemens et al. 2004), and was reduced with the Goodman pipeline (Torres et al. 2017). One spectrum was obtained with the Multi-Object Double Spectrographs (MODS, Pogge et al. 2010) on the twin 8.4 m Large Binocular Telescope (LBT) at Mount Graham International Observatory. The spectrum was reduced using standard techniques, including bias subtraction and flat-fielding using the MODSCCDred package (Pogge 2019) and further reduced with IRAF including cosmic ray rejection, local sky subtraction, and extraction of one-dimensional spectra. A log of the spectroscopic observations is presented in Table A1. We also present an unpublished nebular spectrum of Type Ibn SN 2019kbj taken at 80 d after the peak. The spectrum was taken on 2019 September 23 with the DEep Imaging Multi-Object Spectrograph (DEIMOS, Faber et al. 2003) on the Keck II telescope (Table A1). The DEIMOS spectrum was reduced using the PypeIt pipeline (Prochaska et al. 2020; Prochaska et al. 2020). A detailed analysis of SN 2019kbj has been presented in Ben-Ami et al. (2023).", '3.1. Reddening': 'The empirical correlation between the equivalent width (EW) of the Na I D line and the amount of gas and dust along the line of sight has often been used in extinction estimations (Munari & Zwitter 1997). In order to measure the line-of-sight reddening towards SN 2023fyq, we analyzed the medium-resolution spectrum (R ∼ 1800) taken with Gemini North on 2023 August 1. The measured EW of the host galaxy Na I D 𝜆 5890 (D2) and Na I D 𝜆 5896 (D1) are 0 . 27 ± 0 . 04 ˚ Aand 0 . 15 ± 0 . 04 ˚ A, respectively. The measured EW of the Galactic Na I D2 and Na I D1 are 0 . 23 ± 0 . 02 ˚ A and 0 . 16 ± 0 . 01 ˚ A respectively. Using Eq.9 in Poznanski et al. (2012) and applying the renormalization factor of 0.86 from Schlafly et al. (2010), we found a host extinction of 𝐸 ( 𝐵 -𝑉 ) host = 0 . 037 ± 0 . 09 mag. The Milky Way extinction is measured to be 𝐸 ( 𝐵 -𝑉 ) MW = 0 . 035 ± 0 . 09 mag which is consistent with the Milky Way extinction of 𝐸 ( 𝐵 -𝑉 ) MW = 0.0286 mag from the extinction map by Schlafly & Finkbeiner (2011). We adopt the latter for the Milky Way extinction. \nFigure 4. 𝑟 / 𝑅 Light curve comparison between SN 2023fyq, a sample of Type Ibn SNe, and well-studied normal SESNe. The Vega magnitudes have been converted to the AB magnitude system. The evolution of SN 2023fyq is similar to those of Type Ibn SNe. The SNe used in this plot includes Type IIb SN 1993J (Filippenko et al. 1993), Type Ib SN 2008D (Modjaz et al. 2009), Type Ic SN 2007gr (Hunter et al. 2009)), and Type Ibn SNe: SN 2015U (Tsvetkov et al. 2015; Pastorello et al. 2015a; Hosseinzadeh et al. 2017), iPTF15ul (Hosseinzadeh et al. 2017), iPTF14aki (Hosseinzadeh et al. 2017), iPTF15akq (Hosseinzadeh et al. 2017), SN 2019deh (Pellegrino et al. 2022), SN 2021jpk (Pellegrino et al. 2022), SN 2005la (Pastorello et al. 2008a), SN 2020nxt (Wangq et al. 2024), SN 2018gjx (Prentice et al. 2020), ASASSN-15ed (Pastorello et al. 2015b), SN 2010al (Pastorello et al. 2015c), SN 2015G (Shivvers et al. 2017; Hosseinzadeh et al. 2017), SN 2006jc (Pastorello et al. 2007), SN 2019uo (Gangopadhyay et al. 2020), and SN 2019kbj (Ben-Ami et al. 2023). SN 2018gjx, ASASSN-15ed, SN 2010al, SN 2015G, SN 2006jc, SN 2019uo, and SN 2019kbj will be used for further comparison in the paper, while a broader sample of SNe Ibn are shown in tan. SESNe are shown in grey. \n<!-- image --> \nThroughout the paper, we will adopt a total extinction of 𝐸 ( 𝐵 -𝑉 ) = 0 . 066 ± 0 . 09 mag. \nWe note that Brennan et al. (2024) found a larger host extinction value ( 𝐸 ( 𝐵 -𝑉 ) host = 0 . 4 ± 0 . 1 mag) using the Balmer ratio measured from the host emission lines. The disagreement is probably because this method measures the full column of gas including the background. In this case, there is likely some dust between the SN and the underlying HII region, which is responsible for this greater implied extinction value.', '3.2. Distance': 'The distance of NGC 4388 listed on the NASA/IPAC Extragalactic Database (NED) ranges from 13.6 Mpc to 25.7 Mpc ( 𝜇 = 30.67 - 32.05 mag). We adopt the most recent TullyFisher distance (based on photometry at 3.6 𝜇 m with Spitzer Space Telescope), 18.0 ± 3.7 Mpc ( 𝜇 = 31.28 ± 0.45 mag; Tully et al. 2016).', '4. PHOTOMETRIC EVOLUTION': '<!-- image --> \nFigure 5. The pre- and post-explosion bolometric light curve (upper two panels) and the blackbody temperature and radius evolution (bottom panel) of SN 2023fyq at the precursor phases and the early SN phases. The uncertainties are indicated by the shaded area. \n<!-- image --> \nIn Figure 2 we present the photometric evolution of SN 2023fyq dating back to 2015, illustrating our search for precursor activities. In Figure 3 we take a closer look at the evolution from one year before the SN explosion. All phases mentioned in the paper are with respect to the maximum light in the 𝑟 band, which is measured to be at JD = 2,460,154.3 ± 0.5 after fitting the light curve with a spline function. At ∼ -11 d, a sudden rise of ∼ 1.5 mag within ∼ 17 hrs is clearly observed (see lower panel of Figure 3). As we will discuss below, we attribute this rapid rise to the SN first light. Consequently, we divide the photometric evolution of SN 2023fyq into two phases: the precursor phase ( < -11 d) and the SN phase ( > -11 d).', '4.1. Precursor Detections': 'The precursor is detected from ∼-1000 d to ∼-11 d. There are also single detections at around -2300 d and -1300 d. These detections have 3 < S/N ≤ 4, and are bracketed by nondetections of similar depth. Therefore, they are likely not true detections of precursor emission. As illustrated in Figure 2, the precursor activities remain relatively stable at -10 to -12 mag between ∼ -1300 d and ∼ -100 d. Then, starting from -100 d, the object slowly brightens to ∼-15 mag. Between ∼-2500 and ∼-100 d, the UV observations from Swift only give nondetection limits (See Figure 2). As the precursor gets brighter, at ∼-28 d, a source is detected in the 𝑈𝑉𝑊 1 filter at ∼-13 mag, with similar magnitudes observed in 𝑔 and 𝑜 bands. From -300to -11d, the precursor light curves seem to exhibit multiple bumps, indicative of pre-explosion activities, such as small eruptions, from the progenitor star. As shown in Figure 2, the precursor emission detected in SN 2023fyq appears fainter and longer compared to that observed in Type Ibn SN 2006jc (Pastorello et al. 2007) and SN 2019uo (Strotjohann et al. 2021), even when accounting for uncertainties in the distance measurement of SN 2023fyq. Pre-explosion activities were not detected for Type Ibn SN 2015G down to -13.3 ± 0.5 mag (Shivvers et al. 2017). It should be noted that the precursor searches for SN 2006jc and SN 2019uo only go down to around -13 mag. Therefore, fainter precursor activities like those observed in SN 2023fyq can not be excluded for these events.', '4.2. SN Light Curve': 'The bluer-band ( 𝑈𝑉𝑊 2, 𝑈𝑉𝑀 2, 𝑈𝑉𝑊 1) light curves of SN 2023fyq exhibit a notable bump from -11 d to -4 d, before reaching the second peak and then falling off rapidly. This initial bump in the blue bands is likely attributable to the cooling following shock breakout. For the rest of the bands, the SN light curves show a fast rise and also a fast decline. The peak 𝑟 -band magnitude is measured to be 𝑀 𝑟 = -18 . 5 mag. In Figure 4, we compare the 𝑟 -band light curve of SN 2023fyq with the 𝑟 / 𝑅 -band light curves of a sample of Type Ibn SNe and well-studied normal stripped-envelop SNe (SESNe). At early times SN 2023fyq appears more luminous than the typical SESNe, and the evolution of SN 2023fyq is overall similar to those of Type Ibn SNe. At late times SN2023fyq declines similarly to SN 2018gjx and SN 2015G, but slower than SN 2006jc. The steep decline of SN 2006jc in the optical is likely due to dust formation in the SN ejecta or in the surrounding CSM (e.g., Smith et al. 2008). The slower decline of SN 2023fyq, SN 2018gjx, and SN 2015G at late times could be an indication of less efficient dust formation than in SN 2006jc. However, due to the lack of late-phase observations of Type Ibn SNe, it is not clear if SN 2006jc is really an outlier. SN 2023fyq declines faster than normal SESNe at nebular phases. This may be due to an inefficient trapping of 𝛾 -rays in SN 2023fyq if the light curve tail is \npowered by 56 Ni decay, a power source other than 56 Ni decay, or dust formation in SN 2023fyq.', '4.3. Bolometric Light Curve': 'We constructed the bolometric light curve of SN 2023fyq using data from ZTF, ATLAS, ASAS-SN, Swift, and Itagaki. Since our photometry data come from different sources, the observations may not have been taken simultaneously. To build the spectral energy distribution (SED) in the regions without complete multiband coverage, we reconstruct the multiband light curves using the light curve fitting package presented in Demianenko et al. (2023). This method is able to capture correlations across different observations over time and among various passbands, and compute an approximate light curve within the specified time and wavelength ranges. We have examined different light curve approximation methods presented in Demianenko et al. (2023) and found that the results are not sensitive to the choice of method. We do not extrapolate beyond the observed bands and time frames. The final bolometric light curve is calculated by fitting the SED with a blackbody function using a Markov Chain Monte Carlo (MCMC) routine in the Light Curve Fitting package (Hosseinzadeh & Gomez 2020). For the pre-explosion phase, the temperature cannot be well constrained because there are only three or four bands of data available. Therefore, the blackbody temperatures measured from the pre-explosion spectra of SN 2023fyq in Brennan et al. (2024), after correcting the reddening using the value in our paper, are used as priors for the SED fitting for the precursor phase. This will help constrain the temperature and luminosity evolution during the pre-explosion phase. We present the bolometric light curve of SN 2023fyq, and the corresponding blackbody temperature ( 𝑇 𝐵𝐵 ) and radius ( 𝑅 𝐵𝐵 ), in the precursor phase and the SN phase, in Figure 5. We note that we only focus on the long-term evolution of the bolometric light curve, and small variations in the light curves are not reflected in the final bolometric light curve. \nBefore ∼-100 d, the precursor of SN 2023fyq is in a relatively stable state with a luminosity of ∼ 1 × 10 40 erg s -1 . During that time, 𝑇 𝐵𝐵 and 𝑅 𝐵𝐵 are around 10,000 K and 600 R ⊙ , respectively. After -100 d, SN 2023fyq shows a faster rise and, at ∼-11 d, the luminosity suddenly increases over an order of magnitude (i.e., from ∼ 4 × 10 41 erg s -1 to ∼ 7 × 10 42 erg s -1 ). Later, after a brief decline, the SN reaches its second peak and declines afterwards. The decline of luminosity shortly after ∼-11 d is likely due to the shock cooling after the shock breakout. For 𝑇 𝐵𝐵 , after jumping to ∼ 22,000 K at ∼-11 d, it rapidly declines until entering a brief plateau phase between ∼-5 and 0 d with 𝑇 𝐵𝐵 ≃ 10,000K. After around -40 d, 𝑅 𝐵𝐵 shows a gradual expansion with a velocity of ∼ 700 kms -1 . After -11 d, 𝑅 𝐵𝐵 continuously increase, reflecting an increase of the photospheric radius with \nthe expansion of SN ejecta. The expansion rate of 𝑅 𝐵𝐵 is ∼ 14,000 km s -1 initially, which slows down to ∼ 7000 km s -1 after around -2 d. We note that this change in photospheric velocity could also be attributed to geometric effects. After around 5 d, as will be discussed in the next section, the spectra of SN 2023fyq are dominated by absorption lines from the SN ejecta, so 𝑅 𝐵𝐵 may not accurately reflect the position of the photosphere. We note that this may also influence the accuracy of the bolometric luminosity we obtained.', '5. SPECTROSCOPIC EVOLUTION': 'The spectroscopic evolution of SN 2023fyq is presented in Figure 6. At -1.6 d, the spectrum shows a blue continuum with a prominent He I 𝜆 5876 line. Other He lines, such as He I 𝜆 5015, He I 𝜆 6678, He I 𝜆 7065, and He I 𝜆 7281, are also observed. The He I 𝜆 5876 line shows a rather asymmetric profile (right panel of Figure 6). In the blue wing, the He I 𝜆 5876 line shows a two-component profile, with a narrow absorption feature at ∼-1000 kms -1 and a broad absorption feature at ∼-7000 kms -1 . The velocities reported here come from the absorption minimum. The detection of a two-component He I line profile in SN 2023fyq is consistent with those observed in other Type Ibn SNe (Pastorello et al. 2016), and is likely from different emitting regions. The broad component is from the fast moving ejecta, while the narrow component is likely from the surrounding unshocked He-rich CSM. In the red wing, there is an additional emission component peaking at around 1500 km s -1 . This component is also observed during the pre-explosion phase of SN 2023fyq (Brennan et al. 2024), and could be due to an asymmetric CSM structure formed before the SN explosion. A few days later the object quickly becomes redder, and the Ca II H&K 𝜆𝜆 3934 , 3969 and Ca II 𝜆𝜆 8498, 8542, 8662 lines appear more prominent. No broad hydrogen features are observed in the spectra of SN 2023fyq. However, we can not exclude the presence of narrow hydrogen lines since the spectra are heavily contaminated by the host-galaxy emission. At ∼ 137 d, the spectrum is dominated by strong [O I] 𝜆𝜆 6300, 6364 and [Ca II] 𝜆𝜆 7291, 7323. He lines, such as He I 𝜆 5876 and He I 𝜆 7065 are also strong at this phase. Other lines, including Mg I] 𝜆 4571 and Ca II 𝜆𝜆 8498, 8542, 8662, can be seen in the spectrum. After that, the spectra we have are mainly dominated by the host, while weak [O I] 𝜆𝜆 6300, 6364 lines are still present. \nWe compare the spectra of SN 2023fyq around 0 d and 7 d with other SNe Ibn and normal SESNe at similar phases in Figure 7 and Figure 8. At around 0 d, other SNe Ibn show blue continua plus narrow He I 𝜆 5876 lines in their spectra. The velocities of those narrow He I 𝜆 5876 lines are consistent with that of the narrow component of the He I 𝜆 5876 line in SN 2023fyq. At around 0 d, normal SESNe are redder than SN 2023fyq and other SNe Ibn. This is probably due to the \nFigure 6. Left: The optical spectroscopic evolution of SN 2023fyq. The phase is measured from the 𝑟 -band maximum. The grey bands mark the emission lines from the galaxy. Right: The evolution of the He I 𝜆 5876 line. The pre-maximum spectra marked in grey are from Brennan et al. (2024). The He I 𝜆 5876 line shows a high-velocity component (marked with the blue band) and a low-velocity component (marked with the red band), which may come from the SN ejecta and He-rich CSM, respectively. \n<!-- image --> \npresence of CSM in the SNe Ibn, which is not significant in SESNe. SESNe start to show lines from iron-group elements at this phase, whereas these features are not strong in SN2023fyq or other SNe Ibn at a similar phase. This is likely due to SN 2023fyq having a hotter photosphere at this phase compared to other SESNe. The He lines in Type Ib/c SNe are also much broader than those shown in SN 2023fyq. \nAt around 7 d, SN 2023fyq is very similar to SNe Ibn SN 2018gjx, ASASSN-15ed, SN 2010al, and SN 2015G, which start to show signatures from deeper layers of the ejecta. The He I 𝜆 5876 lines of SN 2018gjx, ASASSN15ed, SN 2010al, and SN 2015G grow broader, with velocities similar to that of the broad component of He I 𝜆 5876 in SN 2023fyq. Interestingly, some similarities between SN 2023fyq and normal SESNe are also observed at around 7 d. To better illustrate this, we flatten the spectrum of SN 2023fyq at ∼ 7 d using SNID following the procedure outlined in Blondin & Tonry (2007) and compare the flattened spectrum with Type Ib and Ic templates at 10 d from \nLiu et al. (2016) in the bottom panel of Figure 8. This comparison clearly indicates that SN 2023fyq exhibits spectral features similar to those of Type Ic SNe, suggesting that its progenitor likely involves a stripped/He star. \nWhen the object enters the nebular phase, the ejecta become optically thin, providing an unique opportunity to study the core of the progenitor star. However, it is challenging to follow up SNe Ibn at nebular phases since they rapidly get fainter. In Figure 9, we compare the nebular spectrum of SN2023fyq at ∼ 136.8d with a few SNe Ibn with late-time observations and normal SESNe at similar phases. The underlying continuum of the background galaxy, obtained from a preexplosion spectrum taken at -504 d as presented in Brennan et al. (2024) when the signal from the host is dominant, is subtracted from the spectrum presented here. SN 2023fyq shows strong intermediate-width He emission lines (full-width halfmaximum (FWHM) velocity of ∼ 4000 km s -1 ), similar to Type Ibn SN 2018gjx and SN 2015G, but the [O I] 𝜆𝜆 6300, 6364 line in SN 2023fyq is significantly stronger than those \nFigure 7. Optical spectral comparison of SN 2023fyq at ∼ 0 d to other Type Ibn SNe and normal SESNe. \n<!-- image --> \nin other objects. Type Ibn SN 2006jc shows only narrow He lines with no signatures of oxygen. SN 2019kbj is overall similar to SN 2006jc but has broader He lines. This is likely because the spectrum of SN 2019kbj presented here is at an earlier phase (80 d). As shown in Pastorello et al. (2008b), the He lines in SN 2006jc became narrower over time. Given the overall similarities between SN 2006jc and SN 2019kbj, we expect the He lines in SN 2019kbj to also become narrower at later phases. SNe Ibn at nebular phases ( ≳ 100 d) seem to fall into two distinct classes, with one still showing only narrow lines and another showing intermediate-width He lines and oxygen lines. Compared to normal SESNe SN 2008D and SN 2007gr, SN 2023fyq shows prominent He emission lines, but otherwise SN 2023fyq is similar to those normal SESNe at the nebular phase. \nOverall, the spectroscopic evolution SN 2023fyq is similar to those of some SNe Ibn. However, the difference between SESNeandSN2023fyqshortlyafterthelightcurvemaximum is less evident. A transition between Type Ibn and Type Ic is clearly observed. Similar behaviors have been reported in several previous studies of other Type Ibn SNe (e.g., Pastorello et al. 2015b; Prentice et al. 2020). If SN 2023fyq is indeed dominated by CSM interaction at peak light, the transition to Type Ic could be due to the CSM-interaction region becoming transparent over time, allowing us to see more signatures from the SN ejecta. It is also possible that the SN ejecta has moved beyond the dense CSM. This suggests that SN 2023fyq is likely exploded from a stripped/He star within He-rich CSM. \n<!-- image --> \nFigure 8. Upper: Optical spectral comparison of SN 2023fyq at ∼ 7 d to other Type Ibn SNe and normal SESNe. Bottom: The optical spectrum taken at ∼ 7 d compared to the mean spectra (the solid lines) and the standard deviations (the shaded regions) of SN Ib and Ic at ∼ 10 d from Liu et al. (2016). SN 2023fyq has several features in common with these normal SESNe, suggesting the progenitor of SN 2023fyq involves a stripped star. \n<!-- image --> \nThe He lines observed at the nebular phase indicate that the interaction with the He-rich CSM is still ongoing. It is natural to link the pre-existing He-rich CSM with the pre-explosion activities of the progenitor system, which likely also produces the precursor emission observed in SN 2023fyq. This topic will be further discussed in Section 6.3.', '6. DISCUSSIONS': 'The detection of sustained precursor emission in SN 2023fyq provides an invaluable opportunity to study the progenitor system of Type Ibn SNe. Below is a summary of the primary observed characteristics of SN 2023fyq: \nFigure 9. Left: Nebular spectral comparison of SN 2023fyq to other Type Ibn SNe with nebular spectra and normal SESNe. The phases are relative to the time of maximum light. A continuum spectrum of the background galaxy is subtracted from the spectrum of SN 2023fyq. At nebular phases, SNe Ibn appear to fall into two distinct classes: one exhibiting only narrow He lines (SN 2019kbj and SN 2006jc), and another displaying intermediate-width He lines and oxygen lines (SN 2023fyq, SN 2015G, and SN 2018gjx). Right: The evolution of the He I 𝜆 5876 line. \n<!-- image --> \n- 1. A long-standing and continuously rising precursor emission starting from years before the SN explosion;\n- 2. The light curve following the explosion exhibits an evolution similar to Type Ibn SNe; the bolometric light curve exhibits two peaks.9\n- 3. The earlyand late-phase spectra both show narrow/intermediate-width He lines. The nebular spectra show prominent [O I] 𝜆𝜆 6300, 6364 emission, suggesting that SN 2023fyq is likely a stripped/He star exploded within He-rich CSM. \nAny progenitor scenario for SN 2023fyq needs to explain the above behaviors. In this section we will discuss the progenitor system and possible powering mechanisms of the precursor and the SN light curve.', '6.1. What Powers The First Peak of The SN Bolometric Light Curve?': 'The light curve of SN 2023fyq reaches its initial peak at around -11 d. The later decrease of luminosity is associated with a prompt decline of 𝑇 𝐵𝐵 and a rapid expansion of 𝑅 𝐵𝐵 . This process is likely the shock cooling phase after the shock breakout. At the beginning of this phase, the expansion of the ejecta is nearly adiabatic, converting the thermal energy into kinetic energy. The rapid decline of the photospheric temperature can produce a decrease in brightness in bluer bands and an increase in brightness in redder bands as the temperature moves through the optical bands, which is consistent with what we see in SN 2023fyq (Figure 3). It is noteworthy that, around the shock breakout, 𝑅 𝐵𝐵 is about 3 × 1000 R ⊙ ( ∼ 2 × 10 14 cm), so the shock breakout likely originates from \nan extended envelope/CSM wind instead of from the stellar surface. A similar conclusion is also drawn by Brennan et al. (2024) based on the pre-explosion spectroscopic and photometric observations of SN 2023fyq. \nWhen 𝑇 𝐵𝐵 drops down to ≃ 10,000 K, it enters a brief plateau phase (Figure 5). Meanwhile, the bolometric light curve reaches the second peak. This 𝑇 𝐵𝐵 plateau phase is likely due to the emergence of another energy source. It is also possible that this 𝑇 𝐵𝐵 plateau phase is partially due to the recombination of He I, and the decrease of 𝑅 𝐵𝐵 expansion rate is due to the recession of the photosphere into the extended envelope. After this process, the outer envelope becomes almost transparent due to the drop of electron scattering opacity. This is consistent with the fact that we start to see more signals, such as Ca lines, from the deeper SN ejecta after 0 d. \nIn conclusion, the first peak of the SN bolometric light curve of SN 2023fyq is likely due to shock breakout in an extended envelope/CSM wind located at ∼ 2000 -3000 R ⊙ .', '6.2. What Powers The Second Peak of The SN Bolometric Light Curve?': 'At 0 d, SN 2023fyq reaches its second peak. It should be noted that all bands (from UV to optical) show peaks at this phase, so this second peak is not an effect of temperature evolution and is instead powered by other formats of energy sources.', '6.2.1. radioactive decay (RAD)?': "We first consider the possibility that the SN light curve around the second peak is powered by the 56 Ni decay. The early light curve evolution of SNe is regulated by the photon diffusion time, which depends on the SN ejecta mass, the ejecta velocity, and the opacity (Arnett 1982). Assuming that the rise time of the light curve is equal to the photon diffusion time and Arnett's law holds for this object, i.e., the peak luminosity is close to the instantaneous decay power at the peak, we can estimate the 56 Ni mass ( 𝑀 𝑁𝑖 ) and the ejecta mass ( 𝑀 𝑒 𝑗 ). We fix the optical opacity 𝜅 𝑜𝑝𝑡 to be 0.1 cm 2 g -1 . Given a peak luminosity of 9 . 5 × 10 42 erg 𝑠 -1 , we get 𝑀 𝑁𝑖 ≃ 0.28 M ⊙ and 𝑀 𝑒 𝑗 ≃ 0 . 54M ⊙ ( 𝑣 𝑝ℎ / 7000kms -1 )( t / 10d ) 2 . \nTherefore, to power the light curve with only 56 Ni decay, around half of the ejecta is composed of 56 Ni. This ratio is much higher than those in typical CCSNe (e.g., Lyman et al. 2016) and similar to those found in Type Ia SNe (e.g., Konyves-T'oth et al. 2020; Graham et al. 2022). If the ejecta is 56 Ni-rich, when the ejecta become optically thin, the optical spectra would be dominated by forbidden lines from Fe and Co. However, as we discussed in Section 5, the nebular spectrum of SN 2023fyq is mainly dominated by He, O and Ca. Therefore, we disfavor the 56 Ni decay as the dominant power source of the early light curve of SN 2023fyq. \nSince the evolution of SN 2023fyq is similar to those of Type Ibn SNe, it is likely that the light curve around the second peak is powered by CSM interaction. It is important to note that, since the spectra after the peak show signals from the SN ejecta but lack prominent narrow He lines, an asymmetric CSM structure must be involved if the second peak is dominated by CSM interaction. \nWe use the model presented in Jiang et al. (2020), which generalizes the self-similar solution to the interaction of stellar ejecta with surrounding CSM originally presented in (Chevalier 1982). In this model, the density of CSM is described by a power law, 𝜌 ∝ 𝑞𝑟 -𝑠 , while the ejecta are divided by an inner region ( 𝜌 𝑒 𝑗 ∝ 𝑟 -𝛿 ) and an outer region ( 𝜌 𝑒 𝑗 ∝ 𝑟 -𝑛 ). We fix the optical opacity ( 𝜅 ) to be 0.1 cm 2 g -1 , 𝑛 = 10, 𝑠 = 0, and 𝛿 = 1 following Pellegrino et al. (2022). The value of 𝜅 ≈ 0 . 1 cm 2 g -1 is motivated by the opacity of singlyionized He at ∼ 10 4 K (e.g., Kleiser & Kasen 2014). We also attempted to fit the data with 𝑠 = 2 (wind-like CSM), but did not achieve a reasonable fit. This result is consistent with the findings reported by Karamehmetoglu et al. (2017), Gangopadhyay et al. (2020), and Ben-Ami et al. (2023). The ejecta velocity (7,000 km s -1 ) is obtained from the velocity of the P-Cygni minimum of the He I lines near peak. The free parameters in our fit are the explosion epoch ( 𝑡 𝑒𝑥𝑝 ), the ejecta mass ( 𝑀 𝑒 𝑗 ), the inner radius of the CSM ( 𝑅 0), the CSM mass ( 𝑀 𝑐𝑠𝑚 ), the density of the CSM at 𝑅 0 ( 𝜌 𝑐𝑠𝑚, 0), and the conversion efficiency of the shock kinetic energy to radiation ( 𝜖 ). \nTo account for the initial shock cooling phase we have incorporated the shock breakout (SBO) model presented by Margalit (2022). This model provides an analytic solution for the shock cooling phase following shock breakout from extended optically thick material, which is suitable for the case of SN 2023fyq. We fix the velocity of the inner envelope at 7,000 km s -1 . Additionally, we introduce two free parameters into our fit: the radius of the extended material ( 𝑅 𝑒 ) and the mass of the extended material ( 𝑀 𝑒 ). \nThe model fit to the observed light curve is performed using an MCMC routine. As illustrated in the upper-left panel of Figure 10, both the initial bump and the subsequent evolution of the light curve are well-fitted by the model. The best-fitting parameters are detailed in Table 1 (CSM+SBO model). It is important to note that the models presented here are likely simplified, so the parameters derived can only be considered as estimations of the order of magnitude. \nShortly after the peak, the spectra of SN 2023fyq exhibit broad absorption lines from the SN ejecta, indicating an optically thin CSM interaction region between the observer and the SN ejecta. However, the model fit indicates that the light curve is still predominantly influenced by the CSM interaction. One possible explanation for this discrepancy is that \nFigure 10. Upper-Left: Fits to the bolometric light curve of SN 2023fyq using a combination of shock breakout and CSM interaction models. Bottom: Fits to the bolometric light curve of SN 2023fyq using a combination of shock breakout, CSM interaction, and 56 Ni decay models. The gap between 30d and 60d in the 56 Ni decay model, indicated by the dashed line, is due to the transition from the photospheric phase to the nebular phase (see Valenti et al. (2008) for more details). The upper-right panel is a zoom-in of the bottom panel to better illustrate the fit close to the SN peak. The initial bump is well-fitted by the shock breakout model. The hollow point is at the precursor phase, so it is not included in the fit. \n<!-- image --> \nour analytical model is oversimplified, leading to an overestimation of the contribution from the CSM interaction. Alternatively, the CSM may not be spherically symmetric. For instance, if the SN were surrounded by a disk/torus-like CSM, strong CSM interaction would mainly occur in the equatorial region. Consequently, an observer looking along the polar direction would observe less obscured signals from the SN ejecta while the majority of the luminosity arises from the CSM interaction. In this case, the narrow lines from the equatorial CSM interaction would not be observed after the interaction region is enveloped by the expanding SN ejecta. The physical picture of this disk-like CSM scenario has been extensively discussed in Smith (2017). \nThe 𝑀 𝑒 𝑗 and 𝑀 𝑐𝑠𝑚 derived for SN 2023fyq are roughly consistent with those found in other studies (e.g., Pellegrino et al. 2022; Ben-Ami et al. 2023). The low ejecta mass implies that the progenitor is likely a low-mass He star. However, this model can only fit the light curve around the peak and cannot explain the light curve flattening at late times (see Figure 5). \nAt later times, the light curve is likely powered by another source of energy.", '6.2.3. RAD+CSM interaction?': 'Since SN 2023fyq is similar to normal SESNe shortly after peak and during nebular phases, it is plausible that a certain amount of 56 Ni is produced during the explosion. Therefore, it is natural to consider 56 Ni decay as an additional energy source. A 56 Ni decay model has been employed to interpret the late-time light curves of many other Type Ibn SNe, often revealing low 56 Ni masses across previous studies (Gangopadhyay et al. 2020; Pellegrino et al. 2022; Ben-Ami et al. 2023). \nWe use the 56 Ni decay model presented in (Arnett 1982; Valenti et al. 2008). The full SN light curve is fitted by a combination of CSM interaction, shock breakout, and 56 Ni decay models. We fix the optical opacity to be 𝜅 = 0.1 cm 2 g -1 and the 𝛾 -ray opacity to be 0.03 cm 2 g -1 . The ejecta velocity is fixed to be 7,000 km s -1 . The best-fit model is shown in the upper-right panel and the bottom panel of Fig- \n, and the best-fit parameters are presented in Table 1 (the CSM+SBO+RAD model). Both the amount of 56 Ni ( ∼ 0.02 M ⊙ ) and the ejecta mass ( ∼ 1 . 2M ⊙ ) are lower than those of SESNe (Lyman et al. 2016). The low ejecta mass implies that the progenitor of SN 2023fyq is less massive than those of normal SESNe right before the SN explosion. One caveat of the model is that we did not consider the CSM interaction at late phases, which may affect the 56 Ni mass we derive here. \nTheparameters we derive here can give some insights about the progenitor of SN 2023yq. The radius of the extended material ( 𝑅 𝑒 ) is around 21 × 10 13 cm ( ∼ 3000 R ⊙ ). This large radius is consistent with the blackbody radius of SN 2023fyq around the shock breakout (Figure 5). This indicates that, at the explosion, the progenitor is surrounded by an extended envelope with a mass of 0.3 M ⊙ at a radius of 𝑅 𝑒 ∼ 3000R ⊙ , consistent with what we discussed in Section 6.1. Considering the width of the narrow line component in the SN spectra (Figure 5) and the narrow lines observed pre-explosion (Brennan et al. 2024), the extended material likely expands with a velocity of ∼ 1000 kms -1 . Such a velocity suggests that the material at around ∼ 3000 R ⊙ was formed within around 20 days before the explosion. \nIn such a scenario, the pre-explosion photophere would be located within the extended material where the optical depth is sufficiently high. For a wind profile 𝜌 ∝ 𝑟 -2 , 𝑅 𝐵𝐵 = 𝜅 / 𝑀 4 𝜋𝜏𝑉 𝑤𝑖𝑛𝑑 is roughly proportional to / 𝑀 / 𝑉 𝑤𝑖𝑛𝑑 , where / 𝑀 is the mass-loss rate, 𝑉 𝑤𝑖𝑛𝑑 is the expansion velocity of the extended material, and 𝜏 is the optical depth at the photosphere. Consequently the expansion of 𝑅 𝐵𝐵 , starting from around -100 d (Figure 5), is likely due to an increase of mass loss. The more pronounced rise between ∼ -40 d and -11 d can be attributed to a more eruptive mass loss immediately preceding the explosion. If the majority of the material characterized by 𝑀 𝑒 is formed during this eruptive phase, the mass loss rate can be estimated to be \n/ 𝑀 ≈ 𝑀 𝑒 𝑉 𝑤𝑖𝑛𝑑 𝑅 𝑒 ≈ 4 . 5 M ⊙ yr -1 Me 0 . 3M ⊙ 3000R ⊙ Re Vwind 1000kms -1 . (1) \nInterestingly, eruptive mass ejections on the order ∼ 0.1-1 M ⊙ are anticipated for low-mass He stars with masses of 2.5-3.2 M ⊙ due to core silicon deflagration or detonation weeks prior to core collapse (Woosley 2019; Ertl et al. 2020). The mass and velocity of the ejected material depend on the amount of silicon that is consumed in the burning process. (Ertl et al. 2020). An ejection mass of ∼ 0.3 M ⊙ with a velocity of ∼ 1000 km s -1 is consistent with the typical values of such events (see figure 14 and table 4 of Woosley 2019). \nThe CSM characterized by 𝑀 𝐶𝑆𝑀 is likely more extended and formed during the earlier phase of the precursor activities. Detailed discussion on this topic are provided in Section 6.3.2. \nIn summary, neither radioactive decay nor CSM interaction alone can be the power source of SN 2023fyq. Approximately \na few weeks before the explosion, about 0.3 𝑀 ⊙ of material is ejected with a velocity of ∼ 1000 km s -1 due to an increase in mass loss from the progenitor. This material expands to a radius of ∼ 3000 R ⊙ at the time of the explosion. After the explosion, the energy deposited by the shock breakout from the extended material produces the initial light curve bump. Around 0 d the light curve is at least partially powered by the interaction between the SN ejecta and the surrounding CSM, with the kinetic energy of the ejecta converted into thermal energy, resulting in a bright peak. After that, as the strength of the CSM interaction decreases over time, the light curve becomes more influenced by radioactive decay, leading to a relatively flat light curve.', '6.3. What Powers The Precursor of SN 2023fyq? 6.3.1. Single Massive Star Activities?': 'SN Precursors have been commonly observed in Type IIn SNe (e.g., Mauerhan et al. 2013; Smith et al. 2010; Ofek et al. 2013, 2014; Tartaglia et al. 2016; Pastorello et al. 2013, 2018; Strotjohann et al. 2021; Hiramatsu et al. 2024), but are rarely found in Type Ibn SNe and Type II SNe. To date, the pre-explosion activities for Type Ibn SNe have only been detected in SN 2006jc (Pastorello et al. 2007) and SN 2019uo (Strotjohann et al. 2021). Searches for precursors in other SNe Ibn yielded only upper limits, ranging from around -15 to -13 mag (e.g., Pastorello et al. 2008a; Shivvers et al. 2017; Wangq et al. 2024). This may be because those SNe Ibn had no precursors or only fainter and shorter ones, and also because most of these events occur at greater distances than SN 2023fyq. Compared to SN 2006jc and SN 2019uo, one unique characteristic of SN 2023fyq is the long-standing precursor emission. Precursor emission observed in SN 2006jc and SN 2019uo is around hundreds of days before the SN explosions with duration of ∼ 10 days. The precursor observed in these events are much shorter and brighter than that in SN 2023fyq (see Figure 2). \nWe first consider the possibility that the precursor of SN 2023fyq is produced by the final-stage stellar activities of a single massive star. In this case, the precursor can be powered by mass ejection driven by wave transport during the late-stage nuclear burning in the core (Quataert & Shiode 2012; Shiode & Quataert 2014; Fuller 2017; Fuller & Ro 2018; Morozova et al. 2020) or pulsational pair instability (Yoshida et al. 2016; Woosley 2017). \nMassive stars with He core masses of 30 - 64 M ⊙ experience pulsational pair instability after carbon burning, producing violent mass ejections before their cores collapse (Woosley 2017). Pulsational pair instability in massive stars have been suggested to be a promising channel of Type Ibn SNe (Yoshida et al. 2016; Woosley 2017; Leung et al. 2019; \nTable 1. Best-fit parameters of the CSM+Shock Breakout model and the CSM+Shock Breakout+RAD model. \nRenzo et al. 2020). The pulsing activities can last for hours to 10,000 years, depending on the He core mass, before the SN explosion (Yoshida et al. 2016; Woosley 2017). In SN 2023fyq, precursor emission is detected for ∼ 3 years before the SN explosion. Therefore, if pulsational pair instability powers the precursor emission of SN 2023fyq, the progenitor would be a He star with a ZAMS mass larger than ∼ 52 M ⊙ (Woosley 2017). However, the outbursts caused by the pulses of these more massive stars are usually energetic and can result in sharply rising light curves, which is inconsistent with the relatively steady precursor emission of SN 2023fyq. Additionally, the low ejecta mass we derived in Section 6.2 does not align with a very massive He star progenitor. Therefore, we disfavor pulsational pair instability as the powering mechanism of precursor emission in SN 2023fyq. \nStrong temperature gradients can form during late-stage nuclear burning in massive stars, which generates convection, exciting internal gravity waves. The gravity waves may carry their energy to the envelope of the star and deposit it there (Quataert & Shiode 2012; Shiode & Quataert 2014; Fuller 2017; Fuller & Ro 2018), which may trigger eruptive mass ejections (Leung & Fuller 2020; Matzner & Ro 2021). The mass ejection itself and the collision between the ejecta generated from multiple outbursts can potentially produce SN precursor emission (Leung & Fuller 2020; Strotjohann et al. 2021; Tsuna et al. 2023). However, it would be difficult to reproduce the time scale of the observed precursor with a single event of dynamical envelope ejection from a stripped star (Tsuna et al. 2024a). This is because the timescale is regulated by radiative diffusion from the precursor ejecta, which is only weeks to months for stripped stars, thus it would work for the precursors of SN 2006jc or SN 2019uo (Tsuna et al. 2024a), but not for SN 2023fyq. In order to produce the precursor emission seen in SN 2023fyq, multiple fine-tuned mass ejections would be needed. Therefore, a more plausible scenario is a continuous mass loss over the timescale of years, with some continuous powering mechanism for the precursor.', '6.3.2. Binary Interaction?': "A low-mass He star in a binary system has been proposed to be a possible progenitor scenario for Type Ibn SNe (Maund et al. 2016; Dessart et al. 2022; Tsuna et al. 2024a), which is supported by the lack of star formation at the site of some \nmembers of the class (Sanders et al. 2013; Hosseinzadeh et al. 2019). In this section we explore the possibility that the progenitor of SN 2023fyq is an exploded stripped star, such as a He star, in a binary system and that the binary mass transfer generated the precursor activities. \nThe stripped SN progenitor in a binary system expands at somepoint in its evolution near core-collapse, filling its Roche lobe and initiating mass transfer onto the companion. Such a scenario is expected for stripped stars with He core masses in the range of 2.5-3 𝑀 ⊙ , which can inflate their envelopes to up to ∼ 100 𝑅 ⊙ at the oxygen/neon burning phase in the final years to decades of their lives (e.g., Wu & Fuller 2022, and references therein). Thus for orbital separations of ∼ (1-few) × 100 𝑅 ⊙ (orbital period of order 100 days for a companion of order ∼ 1 𝑀 ⊙ ), we expect intense mass transfer to initiate during this time period. \nIf the accretor is a compact object, the mass transfer rate is typically orders of magnitude higher than its Eddington rate, / 𝑀 Edd ∼ 2 × 10 -8 M ⊙ yr -1 ( 𝑀 comp / 1 𝑀 ⊙ )( 𝜅 𝑜𝑝𝑡 / 0 . 1 cm 2 g -1 ) -1 (where a radiation efficiency of 10% was assumed), and thus most of the transferred mass actually escapes from the binary system without being accreted onto the compact object. Even if the companion is not a compact object, for large mass transfer rates of ≳ 10 -4 -10 -3 𝑀 ⊙ yr -1 , most of the mass is expected to still escape through the binary's outer Lagrange point (Lu et al. 2023). In either case, this escaped material becomes the CSM that later powers the bright SN. \nIn Section 6.2 we found that the CSM required to power the main SN light curve is around 0 . 6 + 0 . 1 -0 . 1 M ⊙ , which requires a time-averaged mass loss rate of around a few 0 . 1 𝑀 ⊙ yr -1 given that the mass loss is linked to the observed 1000-day precursor. \nFor binary systems exhibiting such high mass loss rates suggested by Wu & Fuller (2022), those with orbital periods ranging from 10 to 100 days are favored. These systems have orbital velocities of ∼ 100 - a few 100 km s -1 . Assuming the velocity of the CSM that escapes the binary system is ∼ 200 kms -1 , the mass loss rate via mass transfer should be at least larger than ∼ 2 × 10 -2 𝑀 ⊙ yr -1 to power the light curve peak (the detailed derivation is shown in Appendix A), which is consistent with what we found in Section 6.2. \nGiven the required / 𝑀 , we can consider two mechanisms to power the precursor emission. The first is a collision of the mass-transfer outflow with external material, which may exist due to a previous mass-transfer episode (e.g., Pejcha et al. 2016; Metzger & Pejcha 2017). While we remain agnostic to the origin of the pre-existing matter, the maximum available power is given by the kinetic luminosity of the outflow as \n𝐿 out ≈ 1 2 / 𝑀𝑣 2 CSM ∼ 1 . 3 × 10 39 erg s -1 × GLYPH<18> / 𝑀 0 . 1 𝑀 ⊙ yr -1 GLYPH<19> GLYPH<16> 𝑣 CSM 200 km s -1 GLYPH<17> 2 . (2) \nThus the precursor may be explained, but only for favorably high CSM velocity as well as high efficiencies for dissipation and radiation conversion close to unity. \nIn the case for a compact object companion, an accretion disk forming around the compact object can be a promising energy source. While most of the transferred mass is removed from the outer L2 point, a small fraction can still accrete onto the companion and form a disk. The disk, if its accretion rate is super-Eddington, can launch a fast radiation-driven wind that can collide with the rest of the mass and dissipate its kinetic energy. \nThe hydrodynamics of the transferred mass has been considered recently in Lu et al. (2023). For a neutron star companion with an orbital separation of 𝑎 ≈ (1-few) × 100 𝑅 ⊙ and mass transfer rate ≫ 10 -3 𝑀 ⊙ yr -1 , most of the mass is indeed lost from the L2 point (their 𝑓 L2 ∼ 1). However the accretion rate can still reach / 𝑀 acc ∼ ( 3-7 ) × 10 -4 𝑀 ⊙ yr -1 (Figure A2 of Lu et al. 2023), which is orders of magnitude larger than the Eddington rate. \nFor a binary mass ratio of 𝑞 = 𝑀 NS / 𝑀 ∗ ≈ 0 . 5, the (Keplerian) circularization radius of the disk is found from the fitting formula in Lu et al. (2023) as \n𝑅 𝑐 ≈ 0 . 10 𝑎 ∼ 7 × 10 11 cm GLYPH<18> 𝑎 100 𝑅 ⊙ GLYPH<19> . (3) \nWe expect a disk wind to be launched roughly where the local luminosity exceeds the Eddington luminosity of the NS, within a disk radius (equation 31 of Lu et al. 2023) \n𝑅 sph ≈ / 𝑀 acc 𝜅 4 𝜋𝑐 ∼ 2 × 10 10 cm GLYPH<18> / 𝑀 acc 5 × 10 -4 𝑀 ⊙ yr -1 GLYPH<19> GLYPH<18> 𝜅 0 . 2 cm 2 g -1 GLYPH<19> , (4) \nwhich is typically less than 𝑅 c for an orbital separation of 𝑎 ∼ 100 𝑅 ⊙ . We have taken the opacity here to be 𝜅 ≈ 0 . 2 cm 2 g -1 as helium is expected to be fully ionized in the interior of the disk. In line with many theoretical works that model superEddington disk winds, we assume a power-law accretion rate / 𝑀 of / 𝑀 ∝ 𝑟 𝑝 ( 𝑅 NS < 𝑟 < 𝑅 sph), where we adopt 𝑅 NS = \n10 km. This means that a fraction of the accreted mass is expelled at each radius, and we assume that the wind velocity is equivalent to the local disk escape velocity. Consequently, the wind kinetic luminosity, integrated over the range of 𝑟 , is estimated as \n𝐿 wind ≈ 𝑝 2 ( 1 -𝑝 ) / 𝑀 acc 𝐺𝑀 NS 𝑅 NS GLYPH<18> 𝑅 NS 𝑅 sph GLYPH<19> 𝑝 ∼ 2 × 10 40 erg s -1 GLYPH<18> / 𝑀 acc 5 × 10 -4 𝑀 ⊙ yr -1 GLYPH<19> 1 / 2 × GLYPH<18> 𝑀 NS 1 . 4 𝑀 ⊙ GLYPH<19> GLYPH<18> 𝜅 0 . 2 cm 2 g -1 GLYPH<19> -1 / 2 (5) \nwhere we have adopted 𝑝 = 0 . 5 in the last equation while a possible range of 0 . 3 ≤ 𝑝 ≤ 0 . 8 is suggested (Yuan & Narayan 2014). We thus find that the disk wind carries the appropriate kinetic luminosity to explain the precursor in the steady-state phase. \nAs the disk wind carries much smaller mass than the rest of the material around the system, its kinetic energy will be efficiently dissipated by their collision. We check that the dissipated energy would be successfully radiated as the precursor. For a wind profile the diffusion timescale in the CSM is \n𝑡 diff ≈ 𝜅 / 𝑀 4 𝜋𝑣 CSM 𝑐 ∼ 8 × 10 4 sec GLYPH<18> / 𝑀 0 . 1 𝑀 ⊙ yr -1 GLYPH<19> GLYPH<18> 𝜅 0 . 1 cm 2 g -1 GLYPH<19> GLYPH<16> 𝑣 CSM 200 km s -1 GLYPH<17> -1 (6) \nand the adiabatic expansion timescale from the dissipation region, whose size is roughly comparable to the orbital separation, is \n𝑡 exp ≈ 𝑎 𝑣 CSM ∼ 3 × 10 5 sec GLYPH<18> 𝑎 100 𝑅 ⊙ GLYPH<19> GLYPH<16> 𝑣 CSM 200 km s -1 GLYPH<17> -1 (7) \nThus we expect that the dissipated energy can be successfully radiated away without adiabatic losses. The radiation will be reprocessed in the CSM, and finally be emitted as optical radiation at 𝑟 ≈ 𝑅 BB. We refer to Tsuna et al. (2024b) for detailed light curve modeling at the precursor phase. \nThe mass loss via the L2 point can form an equatorial disk (e.g., Lu et al. 2023). The interaction of the equatorial disk with the SN ejecta may contribute to the second peak of the SN light curve. In this case, the parameter 𝑀 𝐶𝑆𝑀 mentioned in Section 6.2 roughly characterizes the mass of the equatorial disk. The interaction of SN ejecta with this dense CSM may still continue in the nebular phase, producing the intermediate-width He lines we observe. \nIn this binary scenario, an accretion disk might form inside the ejecta after the SN explosion. The outflow from the disk may dissipate its kinetic energy through collisions with \npreviously ejected matter, potentially becoming an additional power source for the SN light curve. In this case, the amount of CSM derived in Section 6.2.2 and Section 6.2.3 might be overestimated.", '6.3.3. What About The Rise After -100 d In The Pre-explosion Light Curve?': "As we mentioned in Section 4.3, the pre-explosion light curve shows a rapid rise after -100 d, with a more pronounced rise occurring between -40 d and -11 d. This may be associated with eruptive mass loss right before the SN explosion. For the more pronounced rise between -40 d and -11 d, we consider two possibilities: 1) the rise is due to orbital shrinking of the binary, leading to a runaway of mass transfer and resulting in a rapid-rising pre-explosion light curve (i.e., MacLeod et al. 2018). 2) The rise is influenced by the core silicon burning of the He star, which ejects a large amount of material and powers the fast-rising light curve just before the core collapses. \nFor the first case we initially consider the orbital evolution of this binary system over the few-year timescale during which we observe the precursor. The mass loss from the Lagrange point carries away angular momentum as well, which can affect the orbital separation of the binary. This generally leads to shrinking of the orbit, which may have been witnessed as the sharp rise of the light curve as we approach the explosion epoch. From Figure 5 of Lu et al. (2023) we find the orbital shrinking rate for mass ratio 𝑞 = 0 . 5 and 𝑓 L2 = 1 as \n/ 𝑎 𝑎 ≈ (-5 ) / 𝑀 𝑀 ∗ ∼ -( 6 yr ) -1 GLYPH<18> / 𝑀 0 . 1 𝑀 ⊙ yr -1 GLYPH<19> GLYPH<18> 𝑀 ∗ 3 𝑀 ⊙ GLYPH<19> -1 (8) \nwhich means that for a mass loss rate of ∼ 0 . 1 𝑀 ⊙ yr -1 , the orbital separation can significantly shrink in the several years that we observe the precursor. The orbital shrinking of the binary may cause an unstable mass transfer and accretion onto the compact object, resulting in a runaway mass loss. This may explain the rapid rise after around -40 d in the precursor light curve. Given the anticipated significant orbital shrinking within several years for the system under consideration, the shallower rise in the light curve between -100 d and ∼ -40 d is likely also influenced by the orbital shrinking. This may only lead to a gently increase in the accretion rate onto the compact companion, resulting in the rise of the light curve. \nIn this scenario the final SN explosion can be due to the merger of the He star with a compact object (e.g., Chevalier 2012; Soker 2019; Metzger 2022). Such merger-driven explosions have been proposed to explain some long gamma-ray bursts (Fryer & Woosley 1998; Zhang & Fryer 2001; Thone et al. 2011; Fryer et al. 2013), which are usually associated with a subtype of Type Ic SNe that exhibit broad spectral lines. This He-merger scenario can connect the observed rapid increase in the light curve's brightness at the end of the \nprecursor phase with the following SN-like explosion. However, the characteristics of the final explosion post-merger remain poorly understood. For example, the predicted explosion energies are uncertain by many orders of magnitude (Fryer & Woosley 1998; Zhang & Fryer 2001; Schrøder et al. 2020). While the merger-driven explosion might explain the spectral features observed, detailed spectral modeling of these events is still lacking. \nFor the second case, a core-collapse SN explosion is anticipated after significant mass transfer over years from low-mass stripped stars ranging from 2 . 5 to 3 𝑀 ⊙ (Wu & Fuller 2022). Additionally an explosive mass ejection weeks before the explosion due to silicon burning is indeed expected in recent studies for low mass He stars with masses of 2.5 - 3.2 M ⊙ (Woosley 2019). The mass ejected can range from 10 -2 to 1 M ⊙ with velocities from ∼ 100 kms -1 to a few 1000 km s -1 . In Section 6.2 we found that there is likely an eruptive mass loss of ∼ 0.3 M ⊙ a few weeks before the SN explosion with a velocity of ∼ 1000 km s -1 , which is consistent with the silicon burning phase for low-mass He stars. The eruptive mass loss may explain the more pronounced rise of the precursor light curve between ∼ -40 d and -11 d, and the ejected material in turn produces the first SN peak. However, we note that detailed light curve modeling is necessary to confirm this hypothesis. In this case, the shallower rise in the light curve between -100 d and ∼ -40 d is likely still attributed to the orbital shirking of the binary system, like discussed above. \nIn this scenario the final SN explosion results from the core collapse of the He star. This explanation accounts for the observed spectral similarities between SN 2023fyq and SESNe both post-peak and during the nebular phases. \nBoth the merger-driven and core-collapse scenarios can account for certain observed features of SN 2023fyq. In either case, the progenitor system would likely be asymmetric, which aligns with observations of SN 2023fyq. The 56 Ni yields from a merger-driven explosion are likely low (Fryer et al. 2013; Metzger 2022) and, similarly, low 56 Ni production is expected from core-collapse explosions in low-mass helium stars (Woosley 2019). These predictions are consistent with the low 56 Ni mass derived from the late-time light curves of SN 2023fyq. \nAnimportant difference between these two scenarios is that a merger-driven explosion typically results in a single compact object in the remnant, whereas a core-collapse explosion generally leaves behind a compact binary. In the core-collapse scenario, fallback accretion post-explosion could produce observable X-ray emissions approximately 100 to 1000 days after the explosion, which may show time variations tied to the orbital motion of the binary (Kashiyama et al. 2022). This kind of time variations will not be observed in the mergerdriven scenario since there will be only a single compact obejct left. For SN 2023fyq, conducting X-ray follow-up \nyears after the explosion could be helpful in distinguishing between these two scenarios in future studies. \nWe expect X-ray emission when it is transparent to photoionization by oxygen and carbon in the ejecta. Our modeling favors low-mass (a few 𝑀 ⊙ ) helium stars for the progenitor, with carbon-oxygen cores of mass ≈ 1 . 5-2 𝑀 ⊙ . For an explosion ejecta from such progenitors, we infer the mass of carbon/oxygen-rich material to be roughly 𝑀 ej , C / O ∼ 0 . 11 𝑀 ⊙ . The lower limit applies if a neutron star is left behind in the explosion (as in ultra-stripped SNe considered in Kashiyama et al. 2022), and the upper limit is if the bulk of the CO-core is disrupted (e.g. by a merger) and becomes part of the SN ejecta. Adopting the ejecta velocity of 𝑣 ej = 7000 km s -1 and the X-ray photoionization cross section of 𝜎 X ∼ 10 -19 cm 2 ( ℎ𝜈 / keV ) -3 , we expect X-rays with energy ℎ𝜈 to be transparent at \n𝑡 trans ∼ v t 𝜎 X 𝑀 ej , C / O / 14 𝑚 𝑝 4 𝜋𝑣 2 ej ∼ 1 yr GLYPH<18> 𝑀 ej , C / O 0 . 1 𝑀 ⊙ GLYPH<19> 1 / 2 GLYPH<18> ℎ𝜈 5 keV GLYPH<19> -3 / 2 . (9) \nThus follow-up in hard X-rays at years after the explosion is encouraged, although the X-ray luminosity would depend on the uncertain degree of fallback ( ∼ 10 39 - -10 40 erg s -1 at peak, Kashiyama et al. 2022). If the fallback is similar to the ultra-stripped SN models in Kashiyama et al. (2022), we expect the source to be detectable by current X-ray facilities thanks to the proximity of this event. \nIn conclusion the timescale and brightness of the precursor observed in SN 2023fyq before -100 d can be attributed to mass transfer in a binary system. The companion star is likely a compact object, as the energetics of the disk wind launched from super-Eddington accretion onto the compact object can naturally explain the luminosity of the precursor. An equatorial circumbinary disk, formed during the mass transfer, later interacts with the SN ejecta, powering the main SN peak. During the nebular phases the ongoing interaction between the equatorial disk and the SN ejecta produces the intermediate-width He lines observed. The rise of the light curve between -100 d and ∼ -40 d is likely due to orbital shrinking. The more pronounced rise of the light curve starting around -40 d may be linked to 1) an eruptive mass ejection due to final-stage silicon burning, or 2) runaway mass transfer caused by orbital shrinking of the binary system. In the first scenario, the subsequent explosion would result from the core-collapse of the He star. In the second scenario, it would result from the merger of the He star with the compact object. Both scenarios can launch materials into the polar region. The shock breakout from this extended material and the following cooling emission power the first bright SN peak. \n6.4. Connections to Other Transient Phenomena and Implications on The CSM Structure \nIt is noteworthy that the light curve morphology (both the pre- and post-explosion phase) of SN 2023fyq is quite similar to those of luminous red novae (Soker & Tylenda 2003; Tylenda et al. 2011; Mauerhan et al. 2015; Smith et al. 2016; Blagorodnova et al. 2017), which are generally understood to be the product of binary mergers (e.g., Metzger & Pejcha 2017; Soker 2024). The pre-explosion activities in luminous red novae are often associated with binary mass transfer (e.g., Pejcha 2014), and the pre-explosion brightening is due to the increase in the mass-loss rate caused by orbital shrinking. The post-explosion light curves of luminous red novae are double-peaked, in which the first peak is likely from the shock cooling and the second peak is from the interaction between the ejecta and a pre-existing equatorial disc formed during binary mass transfer (Metzger & Pejcha 2017). \nThe scenario for luminous red novae is analogous to what we proposed for SN 2023fyq, and the primary difference is just the explosion energy source. Such an asymmetric CSM structure is consistent with the multi-component profile of the He I 𝜆 5876 line as we discussed in Section 5 and also the asymmetric line profiles observed during the pre-explosion phase of SN 2023fyq (Brennan et al. 2024). Similarities between luminous red novae and interaction-powered SNe have also been reported in previous studies (e.g., Hiramatsu et al. 2024). \nThe SN light curve evolution of SN 2023fyq is similar to those of ultra-stripped SNe (De et al. 2018; Yao et al. 2020). Thefirst bright SN light curve peak in these ultra-stripped SNe is generally understood as a result of shock breakout from the dense CSM ejected weeks before the SN explosion. The second peak of these objects is usually around 10 42 erg s -1 , much fainter than that of SN 2023fyq, and is thought to be powered by 56 Ni decay (De et al. 2018; Yao et al. 2020). It may be that in these objects the CSM is more confined and a more extended ( ∼ 10 15 cm) dense equatorial disk is lacking, resulting in insufficient CSM at these radii to power the second peak through interaction like that observed in SN 2023fyq. \nSNeIbncanshowawidevariety of spectral features at early phases (Hosseinzadeh et al. 2017), which is not surprising if all SNe Ibn experience strong interaction with asymmetric CSM (e.g., Smith et al. 2015; Smith 2017). Only a few SNe Ibn are observed until late phases since they can decline fast. Interestingly, as we show in Figure 9, at late times, these SNe Ibn seem to fall into two distinct classes: Class I that shows broad lines and share many similarities with normal SESNe (SN 2023fyq, SN 2015G, SN 2018gjx) and Class II that is still dominated by narrow emission lines (SN 2006jc, SN 2019kbj). Assuming the progenitors of all these SNe Ibn are He stars, the objects in Class II may be surrounded by more massive CSM and/or have lower explosion energy (Dessart et al. 2022). \nFor the objects in Class I, the intensity of the [O I] 𝜆𝜆 6300, 6364 line can vary significantly among different objects while the other spectral features are quite similar. If the progenitors of all these objects are surrounded by an equatorial disk, the difference in the intensity of the [O I] 𝜆𝜆 6300, 6364 line can be naturally explained by different viewing angles (See Figure 11). If the system is observed from the equatorial direction, the central [O I] 𝜆𝜆 6300, 6364 line forming region can be obscured by the disk. Instead, a polar observer would be able to see the whole nebular emission from the inner ejecta. For both observers, intermediate-width He emission lines from the ongoing interaction of the SN ejecta with the equatorial disk can be seen. \nA disk/torus-like CSM is also invoked in previous studies to explain the spectroscopic evolution of SNe Ibn (Prentice et al. 2020) and SNe IIn (e.g., Smith & Arnett 2014; Smith et al. 2015; Andrews & Smith 2018; Smith & Andrews 2020). Such a disk/torus-like CSM scenario could potentially explain the diversity we see in SNe Ibn in Class I, and is consistent with the precursor model we discussed in Section 6.3.2. This suggests that Class I SNe Ibn may originate from a similar progenitor channel but with variations in viewing angles. \nLong-lasting and relatively stable precursor activities due to binary interaction are commonly seen in luminous red novae (e.g., Tylenda et al. 2011; Mauerhan et al. 2015; Blagorodnova et al. 2017). Given the similarity of the progenitor scenario of luminous red novae and SN 2023fyq, it is possible that precursor activities are not rare in SNe Ibn in Class I. If this is true, the long-lasting and slowly rising pre-explosion emission may serve as a unique early warning for this subclass of Type Ibn SNe. The evolution of the precursor light curves may vary depending on the viewing angle, as the emission could be obscured by the equatorial disk for observers near the equatorial plane. Given that the viewing angle also influences the intensity of the [OI] lines in the nebular spectra, combining the precursor emission with late-time spectroscopy could serve as a unique probe for the progenitor scenario we propose.", '7. SUMMARY': 'The evolution of SN 2023fyq closely resemble that of Type Ibn SNe. The optical spectra post-peak and the nebular spectrum of SN 2023fyq share similarities with those of normal SESNe, implying that the progenitor is a stripped/He star. The SN light curve can be reproduced by a CSM interaction + shock breakout + 56 Ni decay model, implying the presence of dense CSM around the progenitor, a low progenitor mass and a low 56 Ni production. The precursor emission of SN 2023fyq is observed up to around three years before the SN explosion, which is best explained by the mass transfer in a binary system involving a low-mass He star. \nPutting all these together, we summarize a possible timeline for SN 2023fyq: \nFigure 11. A sketch of the possible progenitor system of SN 2023fyq. Upper: around a few years before the explosion, the progenitor (a He star with a mass of ∼ 2.5 - 3 𝑀 ⊙ ) expands at the oxygen/neon burning phase, filling its Roche lobe. This triggers mass transfer onto its companion compact object, resulting in the precursor emission we observe. Around weeks before the explosion, an eruptive mass ejection is triggered through core silicon burning in the low-mass He star or runaway mass transfer due to orbital shrinking, launching dense material to the polar region. The subsequent explosion is likely due to either by core-collapse of the He star or by the merger of the He star with its compact object companion. Bottom: Immediately after the explosion, the shock breaks out from the dense polar material formed weeks before the explosion, producing the first light curve peak. The interaction of SN ejecta with the equatorial disk formed by the pre-explosion binary interaction contributes to the second peak. \n<!-- image --> \n- 1. ∼-1000 d to ∼-100 d (upper panel of Figure 11): A low-mass He star (2.5 - 3 M ⊙ ) expands substantially at the oxygen/neon burning phase, triggering mass transfer to its companion compact object, which produces the precursor emission we observe. The outflow via L2 point produces the He-rich CSM around the progenitor system and forms an equatorial disk ( ∼ 0.6M ⊙ ).\n- 2. ∼-100 d to ∼-11 d: The shrinkage of the orbit leads to an increase in the accretion rate onto the companion compact object, resulting in a rise in the light curve. The more pronounced light curve rise after ∼ -40 d is \nlikely due to either the core silicon burning or the runaway mass transfer caused by orbital shrinking, which triggers an eruptive mass ejection ( ∼ 0.3M ⊙ ) with a velocity of ∼ 1000kms -1 . This launches dense material to the polar region. \n- 3. ∼-11 d (bottom panel of Figure 11): A SN explosion is triggered either by the core-collapse of the He star or by the merger of the He star with a compact object, which sends a shock through the polar material ( ∼ 3000 R ⊙ ). The energy deposited during the shock breakout produces the initial bump of the light curve.\n- 4. ∼-11 d to ∼ 20 d: The SN ejecta collide with the equatorial He-rich CSM ( ∼ 0.6M ⊙ ), converting the kinetic energy of the SN ejecta into thermal energy, contributing to the SN light curve and generating a very blue spectrum with only prominent He lines. With the expansion of the ejecta, the optical depth decreases so that more signals from the SN ejecta are observed.\n- 5. after ∼ 20 d: The strength of the CSM interaction decreases and the SN fades, and radioactive decay likely starts to contribute more to the light curve. Later, the ejecta become more optically thin and the object transitions into the nebular phase. Given our proximity to the polar direction of the system, signals from the inner part of the ejecta are revealed, which closely resemble those of normal SESNe at nebular phases. Additionally, the continuing interaction between the ejecta and the He-rich equatorial CSM produces strong intermediatewidth He emission lines. \nGiven the similarities between SN 2023fyq and other Type Ibn SNe, precursor activities may be common for a certain subclass of Type Ibn SNe. If an equatorial disk is indeed formed during the precursor phase, the precursor emission and the intensity of the [OI] lines at the nebular phases for this class of objects would be dependent on the viewing angle. It is worth noting that this mechanism does not apply to the very brief, singular pre-explosion outburst observed in SN 2006jc and SN 2019uo. For the upcoming LSST survey, a single 30-second visit will achieve a 5 𝜎 depth of approximately 24 mag (Bianco et al. 2022). By stacking images, even deeper limits can be achieved. This enables LSST to effectively constrain the precursors of Type Ibn SNe, such as SN 2023fyq, within 150 Mpc, assuming a typical precursor brightness of -12 mag. A sample of Type Ibn SNe with wellconstrained precursor activities, combined with the late-time spectroscopy, will test the progenitor scenario we propose. We encourage X-ray follow-up on SN 2023fyq in the years following the explosion, as this will help distinguish between a merger-driven explosion and a core-collapse explosion as the mechanism for this event. We also encourage detailed \nspectral and light curve modeling of merger-driven explosions, as well as the silicon burning phase in low-mass He stars just prior to core collapse. By comparing these models with a large sample of observations, we can deepen our understanding of the final stages of stellar evolution.', 'ACKNOWLEDGEMENTS': "Wewouldlike to thank Jim Fuller for the assistance with the manuscript in its early stages. We would like to thank Kyle Davis for sharing the SOAR spectrum from their program. Y.D. would like to thank L.Z. for redesigning and redrawing Figure 11 in the paper. \nResearch by Y.D., S.V., N.M.R, E.H., and D.M. is supported by NSF grant AST-2008108. D.T. is supported by the Sherman Fairchild Postdoctoral Fellowship at the California Institute of Technology. \nTime-domain research by the University of Arizona team and D.J.S. is supported by NSF grants AST-1821987, 1813466, 1908972, 2108032, and 2308181, and by the Heising-Simons Foundation under grant #2020-1864. \nThis work makes use of data from the Las Cumbres Observatory global telescope network. The LCO group is supported by NSF grants AST-1911225 and AST-1911151. \nA.Z.B. acknowledges support from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No. 772086). \nThis publication was made possible through the support of an LSST-DA Catalyst Fellowship to K.A.B, funded through Grant 62192 from the John Templeton Foundation to LSST Discovery Alliance. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of LSST-DA or the John Templeton Foundation. \nBased on observations obtained at the international Gemini Observatory, a program of NSF's NOIRLab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. On behalf of the Gemini Observatory partnership: the National Science Foundation (United States), National Research Council (Canada), Agencia Nacional de Investigaci'on y Desarrollo (Chile), Ministerio de Ciencia, Tecnolog'ıa e Innovaci'on (Argentina), Minist'erio da Ciˆencia, Tecnologia, Inovac¸ ˜oes e Comunicac¸ ˜oes (Brazil), and Korea Astronomy and Space Science Institute (Republic of Korea). \nThis work was enabled by observations made from the Gemini North telescope, located within the Maunakea Science Reserve and adjacent to the summit of Maunakea. We are grateful for the privilege of observing the Universe from a place that is unique in both its astronomical quality and its cultural significance. \nThis work includes observations obtained at the Southern Astrophysical Research (SOAR) telescope, which is a joint project of the Minist'erio da Ciˆencia, Tecnologia e Inovac¸ ˜oes (MCTI/LNA) do Brasil, the US National Science Foundation's NOIRLab, the University of North Carolina at Chapel Hill (UNC), and Michigan State University (MSU). \nSome of the data presented herein were obtained at Keck Observatory, which is a private 501(c)3 non-profit organization operated as a scientific partnership among the California Institute of Technology, the University of California, and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W. M. Keck Foundation. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Maunakea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain. \nThe LBT is an international collaboration among institutions in the United States, Italy and Germany. LBT Corporation Members are: The University of Arizona on behalf of the Arizona Board of Regents; Istituto Nazionale di Astrofisica, Italy; LBT Beteiligungsgesellschaft, Germany, representing the Max-Planck Society, The Leibniz Institute for Astrophysics Potsdam, and Heidelberg University; The Ohio \nState University, and The Research Corporation, on behalf of The University of Notre Dame, University of Minnesota and University of Virginia. \nThis research has made use of the NASA/IPAC Extragalactic Database (NED), which is funded by the National Aeronautics and Space Administration and operated by the California Institute of Technology. \nThis research made use of Photutils, an Astropy package for detection and photometry of astronomical sources (Bradley et al. 2022). \nFacilities: ADS, DLT40 (Prompt5, Prompt-MO), ATLAS, LCOGT (SBIG, Sinistro, FLOYDS), Gemini:North (GMOS), Keck:I (LRIS, DEIMOS), NED, SOAR (Goodman), Swift (UVOT), LBT (MODS) \nSoftware: Astropy (Astropy Collaboration et al. 2013, 2018, 2022), emcee (Foreman-Mackey et al. 2013) HOTPANTS (Becker 2015), Matplotlib (Hunter 2007), NumPy (Harris et al. 2020), PYRAF (Science Software Branch at STScI 2012), Pandas (Wes McKinney 2010), SciPy (Virtanen et al. 2020), SWarp (Bertin et al. 2002), HOTPANTS (Becker 2015), LCOGTSNpipe (Valenti et al. 2016), Light Curve Fitting (Hosseinzadeh & Gomez 2020), LPipe (Perley 2019)", 'A. THE MASS LOSS RATE IN BINARY INTERACTION SCENARIO': 'This appendix calculates the mass loss rate needed for a binary system to explain the observations, as discussed in Section 6.3.2. We begin with estimating the required mass loss rate / 𝑀 of the CSM, which in our scenario is equivalent to the mass transfer rate if the rate is much larger than the Eddington rate and the companion cannot accrete most of the transferred material. The CSM must be optically thick within the observed blackbody radius 𝑅 BB ≈ 600 𝑅 ⊙ at the precursor phase. For a mass loss rate of / 𝑀 , the optical depth at 𝑅 BB is \n𝜏 CSM ( 𝑟 = 𝑅 BB ) ≈ 𝜅 / 𝑀 4 𝜋𝑅 BB 𝑣 CSM ∼ 60 GLYPH<18> / 𝑀 0 . 1 𝑀 ⊙ yr -1 GLYPH<19> GLYPH<18> 𝜅 0 . 1 cm 2 g -1 GLYPH<19> GLYPH<16> 𝑣 CSM 200 km s -1 GLYPH<17> -1 (A1) \nwhere 𝑣 CSM is the velocity of the CSM that escapes the binary system. This is typically the orbital velocity for outflows from mass transfer, which is ∼ 200 km s -1 for the orbital separation of interest (see Section 6.3.2), but the arguments below would not depend much on the adopted value. The value of 𝜅 ≈ 0 . 1 cm 2 g -1 is motivated from that of singly-ionized helium at around 10 4 K (e.g., Kleiser & Kasen 2014). The optical depth then poses a lower limit in / 𝑀 of \n/ 𝑀 ≥ / 𝑀 min ≈ 2 × 10 -3 𝑀 ⊙ yr -1 GLYPH<18> 𝜅 0 . 1 cm 2 g -1 GLYPH<19> -1 GLYPH<16> 𝑣 CSM 200 km s -1 GLYPH<17> (A2) \nTable A1. Spectroscopic observations of SN 2023fyq and SN 2019kbj \nwhich confirms the super-Eddington mass transfer rate /three.sup . As a cross check, we can also roughly infer / 𝑀 from the observed SN. The collision of the SN with the CSM generates a shock that powers the SN light curve. The kinetic energy dissipation rate is \n𝐿 kin = 2 𝜋𝑟 2 GLYPH<18> / 𝑀 4 𝜋𝑟 2 𝑣 CSM GLYPH<19> 𝑣 3 sh ∼ 5 . 5 × 10 43 erg s -1 GLYPH<18> / 𝑀 0 . 1 𝑀 ⊙ yr -1 GLYPH<19> GLYPH<16> 𝑣 CSM 200 km s -1 GLYPH<17> -1 GLYPH<16> 𝑣 sh 7000 km s -1 GLYPH<17> 3 (A3) \nwhere 𝑣 sh is the forward shock velocity. Assuming that the luminosity at the second peak is generated by the interaction with CSM generated in the precursor phase, we infer a mass loss rate of \n/ 𝑀 ∼ 2 × 10 -2 𝑀 ⊙ yr -1 𝜖 -1 GLYPH<18> 𝐿 rad 10 43 erg s -1 GLYPH<19> GLYPH<16> 𝑣 CSM 200 km s -1 GLYPH<17> GLYPH<16> 𝑣 sh 7000 km s -1 GLYPH<17> -3 , (A4) \nwhere 𝜖 = 𝐿 rad / 𝐿 kin ≤ 1 is the radiation conversion efficiency. While this estimate is quite sensitive to the assumed 𝑣 sh, it implies that a similarly high / 𝑀 is also required to explain the SN. The required mass transfer rate of ∼ 0 . 02-0 . 2 𝑀 ⊙ yr -1 for 𝜖 ≈ 0 . 1-1 roughly overlaps with the range obtained from simulations of binaries composed of a low-mass (2 . 5-3 𝑀 ⊙ ) He star and a neutron star, years to decades before the SN (Wu & Fuller 2022, Figure 2).', 'B. SPECTROSCOPIC OBSERVATIONS': 'Table A1 shows a log of the spectroscopic observations of SN 2023fyq and SN 2019kbj.', 'REFERENCES': 'Andrews, J. E., & Smith, N. 2018, MNRAS, 477, 74, \ndoi: 10.1093/mnras/sty584 \nArnett, W. D. 1982, ApJ, 253, 785, doi: 10.1086/159681 Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33, doi: 10.1051/0004-6361/201322068 Astropy Collaboration, Price-Whelan, A. M., Sip"ocz, B. M., et al. 2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f Astropy Collaboration, Price-Whelan, A. M., Lim, P. 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2024arXiv240902003A | PandaX4T and XENONnT have recently reported the first measurement of nuclear recoils induced by the 8B solar neutrino flux through the coherent elastic neutrinonucleus scattering CEnuNS channel. As long anticipated this is an important milestone for dark matter searches as well as for neutrino physics. This measurement means that these detectors have reached exposures such that searches for low mass lesssim 10 GeV dark matter cannot be analyzed using the backgroundfree paradigm going forward. It also opens a new era for these detectors to be used as neutrino observatories. In this paper we assess the sensitivity of these new measurements to new physics in the neutrino sector. We focus on neutrino nonstandard interactions NSI and show that despite the still moderately low statistical significance of the signals these data already provide valuable information. We find that limits on NSI from PandaX4T and XENONnT measurements are comparable to those derived using combined COHERENT CsI and LAr data as well as those including the latest Ge measurement. Furthermore they provide sensitivity to pure tau flavor parameters that are not accessible using stoppedpion or reactor sources. With further improvements of statistical uncertainties as well as larger exposures forthcoming data from these experiments will provide important novel results for CEnuNSrelated physics. | 2024-09-01T00:00:00Z | ['arXiv:2409.02003', '10.48550/arXiv.2409.02003', '2024arXiv240902003A'] | ['High Energy Physics - Phenomenology', 'Astrophysics - Cosmology and Nongalactic Astrophysics', 'High Energy Physics - Experiment', 'Nuclear Experiment', 'Nuclear Theory'] | Implications of first neutrinoinduced nuclear recoil measurements in direct detection experiments | 2,024 | 167 | 0.44 | ['EPRINT_HTML', 'EPRINT_PDF'] | 3 | https://arxiv.org/pdf/2409.02003.pdf | {'Implications of first neutrino-induced nuclear recoil measurements in direct detection experiments': "D. Aristizabal Sierra, 1, ∗ N. Mishra, 2, † and L. Strigari 2, ‡ \n1 Universidad T'ecnica Federico Santa Mar'ıa - Departamento de F'ısica \nCasilla 110-V, Avda. Espa˜na 1680, Valpara'ıso, Chile Department of Physics and Astronomy, Mitchell Institute for Fundamental Physics and Astronomy, Texas A&M University, College Station, Texas 77843, USA \n2 \nPandaX-4T and XENONnT have recently reported the first measurement of nuclear recoils induced by the 8 B solar neutrino flux, through the coherent elastic neutrino-nucleus scattering (CE ν NS) channel. As long anticipated, this is an important milestone for dark matter searches as well as for neutrino physics. This measurement means that these detectors have reached exposures such that searches for low mass, ≲ 10 GeV dark matter cannot be analyzed using the background-free paradigm going forward. It also opens a new era for these detectors to be used as neutrino observatories. In this paper we assess the sensitivity of these new measurements to new physics in the neutrino sector. We focus on neutrino non-standard interactions (NSI) and show that-despite the still moderately low statistical significance of the signals-these data already provide valuable information. We find that limits on NSI from PandaX-4T and XENONnT measurements are comparable to those derived using combined COHERENT CsI and LAr data, as well as those including the latest Ge measurement. Furthermore, they provide sensitivity to pure τ flavor parameters that are not accessible using stopped-pion or reactor sources. With further improvements of statistical uncertainties as well as larger exposures, forthcoming data from these experiments will provide important, novel results for CE ν NS-related physics.", 'I. INTRODUCTION': 'PandaX-4T [1] and XENONnT [2] have recently reported the detection of coherent elastic neutrino-nucleus scattering (CE ν NS) induced by 8 B solar neutrinos. Due to their low energy thresholds and large active volumes these experiments identify the 8 B component of the solar neutrino flux at a significance level of the order of 2 σ . This is the first detection of CE ν NS from an astrophysical source, complementing the recent detections from the stopped-pion source by the COHERENTexperiment [3-5]. Further, this detection probes the CE ν NS cross section at characteristic neutrino energy scales lower than that probed by COHERENT and with a new material target 1 . \nThe detection of solar neutrinos at dark matter (DM) detectors such as PandaX-4T and XENONnT is a milestone in neutrino physics [6-10]. It represents an important step in the continuing development of the solar neutrino program, dating back to over half of a century. From the perspective of solar neutrino physics, it is the second pure neutral current channel detection of the solar neutrino flux, complementing the SNO neutral current detection of the flux using a deuterium target [11]. Its observation was anticipated long time ago to be not only a challenge for DM searches, but also an opportunity for a better understanding of neutrino properties and searches of new physics [12]. \nThe detection of 8 B neutrinos via CE ν NS has important implications more broadly for neutrino physics, astrophysics, and DM. This detection has the potential to provide information on the properties of the solar interior [13]. It also \nhas the potential to probe new physics in the form of nonstandard neutrino interactions (NSI) [14-17], sterile neutrinos [18, 19], neutrino electromagnetic properties [20-23] or new interactions involving light mediators [16, 24]. Detection of solar neutrinos via CE ν NS also is important for interpreting the possible detection of low mass, ≲ 10 GeV, dark matter [25, 26]. A detailed understanding of this signal is of paramount importance for the interpretation of future data. The identification of a possible WIMP signal requires a thorough understanding of neutrino-induced nuclear recoils. \nIn this paper, we examine the sensitivity of the PandaX4T and XENONnT data to NSI. We show that even with this early data these measurements are already capable of providing competitive bounds. In particular, because of neutrino flavor conversion, these measurements are sensitive to all neutrino flavors and so open flavor channels not accessible in CE ν NS experiments relying on π + decay-at-rest or reactor neutrino fluxes. Thus from this point of view these experiments are very unique. \nThe reminder of this paper is organized as follows. In Sec. II we discuss the Standard Model (SM) CE ν NS cross section, define the parameters we use in our calculation and briefly discuss the experimental input employed. In Sec. II we provide a detailed discussion of NSI effects in both propagation and detection. To do so we rely on the two-flavor approximation, which provides rather reliable results up to corrections of ∼ 10% 2 . In Sec. IV, after briefly discussing the main features of both PandaX-4T and XENONnT data, we present the results of our analysis. Finally, in Sec. V we summarize and present our conclusions. In App. A we provide a summary of NSI limits arising from the one-parameter analysis. \nTABLE I. Neutrino flux normalization as recommended in Ref. [35] and inline with the B16(GS98) SSM. For detection only 8 B matters. For propagation we include the whole spectrum.', 'II. CE ν NS CROSS SECTION, 8 B SOLAR NEUTRINO FLUX, EVENT RATES AND EXPERIMENTAL INPUT': 'In the SM, at tree-level the CE ν NS cross section is has no lepton flavor dependence [27], with flavor-dependent corrections appearing at the one-loop level [28-30]. At tree level the scattering cross section reads [27] \nd σ dEr = G 2 F 2 π Q 2 W mN ( 2 -mNEr E 2 ν ) F 2 W ( Er ) . (1) \nFor the nuclear mass, mN , we use the averaged mass number ⟨ A ⟩ = ∑ 9 i = 1 XiAi , where i runs over the nine stable xenon isotopes and Xi refers to i -th isotope natural abundance. QW refers to the weak charge and determines the strength at which the Z gauge boson couples to the nucleus. At tree-level and neglecting q 2 dependent terms ( q referring to the transferred momentum) the weak charge is entirely determined by the vector neutron and proton couplings \nQW = Zg p V + Ng n V , (2) \nwith Z = 54 referring to the nucleus atomic number and N =( ⟨ A ⟩ -Z ) to the number of neutrons. The nucleon couplings are in turn given by the fundamental electroweak neutral current up and down couplings: g p V = 1 / 2 -2sin 2 θ W and g n V = -1 / 2. Because of the value of the weak mixing angle 3 , g n V exceeds g p V by more than a factor 20. Thus, up to small corrections the total cross section scales as N 2 =( A -Z ) 2 . \nEffects due to the finite size of the nucleus are parameterized in terms of the weak-charge form factor, FW , for which different parametrizations can be adopted. However, given the energy scale of solar neutrinos these finite size nuclear effects are small, not exceeding more than a few percent regardless of the parametrization [32, 33]. Although of little \nimpact, our calculation does include the weak-charge form factor. We have adopted the Helm parametrization [34] along with Rn = RC + 0 . 2fm, with RC calculated by averaging the charge radius of each of the nine xenon stables isotopes over their natural abundance. \n8 B electron neutrinos are produced in β + decay processes: 8 B → Be ∗ + e + + ν e . The features of the spectrum as well as its normalization is dictated by the Standard Solar Model (SSM). In our analysis we use the values predicted by the B16(GS98) SSM [36]. The distribution of 8 B neutrino production from the B16(GS98) SSM peaks at around 5 × 10 -2 R ⊙ and ceases to be efficient at 0 . 1 R ⊙ , where the distribution fades away. For the calculation of event rates only the 8 B neutrino flux is required. For the calculation of propagation effects (matter effects), however, we require all possible fluxes. In all cases we adopt neutrino spectra normalization as recommended for reporting results for direct DM searches [35], which are inline with those predicted by the B16(GS98) SSM. The values for those normalization factors along with the kinematic end-point energies for all fluxes are shown in Tab. I. \nCalculation of differential event rate spectra follows from convoluting the CE ν NS differential cross section in Eq. (1) with the 8 B spectral function, namely \ndR dEr = ε NA m Xe mol N 8 B ∫ E max ν E min ν d Φ 8 B dE ν d σ dEr dE ν . (3) \nHere ε refers to exposure measured in tonne-year, NA is the Avogadro number in 1/mol units, m Xe mol = 131 . 3 × 10 -3 kg/mol, N 8 B the 8 B flux normalization from Tab. I, E min ν = √ mNE ν / 2 and E max ν the kinematic end-point of the 8 B spectrum from Tab. I as well. Eq. (3) is valid in the SM, where the CE ν NS differential cross section is flavor universal at tree level. If either through one-loop corrections or new physics the cross section becomes flavor dependent, then the integrand should involve the probability associated with each neutrino flavor (see Sec. III B for a more detailed discussion). The event rate follows from integration of Eq. (3) over recoil energies, with the experimental acceptance A ( Er ) fixed according to the PandaX-4T or XENONnT data sets. Generically it reads \nR = ∫ E max r E min r A ( Er ) dR dEr dEr . (4) \nPandaX-4T perform two types of analyses on their data. First, they perform a combined S1/S2 analysis, in which a neutrino signal event is identified via both prompt scintillation and secondary ionization signals from the nuclear recoil ( paired signal ). The low energy threshold for this analysis is set by the S1 signal, which in terms of nuclear recoil energy is ∼ 1 . 1 keV. The second analysis is an S2 only analysis, in which only the ionization component is used as the signal of an event ( US2 signal ). In this case, the nuclear recoil threshold is lower, ∼ 0 . 3 keV, but the trade-off is an increase in the backgrounds for this sample. \nPandaX-4T present data from two runs: their commissioning run, which they call Run0, and their first science run, which they call Run1. For the paired data set, the exposure is \nTABLE II. PandaX-4T (paired and US2) and XENONnT parameter detector configurations used in the NSI statistical analysis. Values taken from Refs. [1, 2]. \n1.25 tonne-year, and for the US2, the exposure is 1.04 tonneyear. Using a maximum likelihood analysis, PandaX-4T finds a best fitting 8 B event rate from the US2 sample of 75 ± 28 and a paired event rate of 3 . 5 ± 1 . 3. \nThe XENONnT collaboration combined two separate analyses, labelled SR0 and SR1, which when combined amount to an exposure of 3.51 tonne-year. They present acceptances for both an S1 only and an S2 only analysis. For the primary analysis, XENONnT combine the acceptances for S1 and S2 (with a resulting 0.5 keV threshold), and, for this combined exposure, they quote a best fit event rate of 10 . 7 + 3 . 7 -4 . 2 . They point out that this result is in close agreement with: (i) Expectations from the measured solar 8 B neutrino flux from SNO, (ii) the theoretical CE ν NS cross section with xenon nuclei, (iii) calibrated detector response to low-energy nuclear recoils. For the expected event rate, they find 11 . 9 + 4 . 5 -4 . 2 . Calculation of the Z -score-assuming these results to be independent-yields 0 . 2 σ . Thus using either in our statistical procedure produces no sizable deviation in the final results. Tab. II summarizes the detector parameter configurations along with the signals we have employed.', 'III. NEUTRINO NON-STANDARD INTERACTIONS': 'In addition to loop-level corrections, flavor-dependence in the CE ν NS cross section may also be introduced through neutrino NSI [37]. The effective Lagrangian accounting for the new vector interactions can be written as \nL NSI = -√ 2 GF ∑ i = e , µ , τ q = u , d ν i γ µPL ε q i j ν j q γ µ q , (5) \nwhere the ε q i j parameters determine the strength of the effective interaction with respect to the SM strength. Neutrino NSI affect neutrino production, propagation and detection. Since production takes place through charged-current (CC) processes, effects in production are small 4 . Effects on propagation and detection, being due to neutral current, can instead be potentially large. Thus we consider only those two. Propagation effects arise from forward scattering processes which \ninduce matter potentials proportional to the number density of the scatterers. So in addition to the SM matter potential, the new interaction-being of vector type-induces additional matter potentials that affect neutrino propagation and thus neutrino flavor conversion. Detection, instead, becomes affected because of the impact of the new effective interaction on the CE ν NS cross section. All in all, NSI effects on solar neutrinos may be prominent in propagation ⊕ detection. \nNeutrino NSI are constrained by a variety of experimental searches. Here we provide a summary of the main constraints, which does not aim at being complete but rather to provide a general picture of what has been done (for a more detailed account see e.g. Ref. [38]). First of all, global analysis of oscillation data imply tight constraints on the size and flavor structure of matter effects. Thus, those constraints can be translated into limits on NSI parameters [39, 40]. Limits involving global analysis of oscillation data combined with CE ν NS measurements have been also derived [41, 42]. Constraints from CE ν NS data alone, for which only effects on detection apply, have been analyzed using both CsI data releases along with LAr data in Ref. [43], and also the most recent measurement with germanium in Ref. [44]. Further constraints from monojets and missing energy searches at the LHC exist [45, 46]. Involving electrons and at early times, the new interaction can keep neutrinos in thermal contact with electrons and positrons below ∼ 1MeV. Requiring small departures from this value leads to cosmological constraints [47]. In supernovæ, neutrino NSI have as well been considered in e.g. Refs. [48, 49]. \nIn what follows we describe their effects in propagation and in detection. To do so we rely on the two-flavor approximation, well justified up to corrections of the order of 10% because of ∆ m 2 12 / ∆ m 2 13 ≪ 1 and sin 2 θ 13 ≪ 1 [50]. And rather than including the data and constraints discussed above, we focus only on the constraints implied by PandaX-4T and XENONnT.', 'A. Neutrino NSI: Propagation effects': "Electron neutrinos are subject to flavor conversion in the Sun, governed by the vacuum and matter Hamiltonians \ni d dr | ν ⟩ = [ 1 2 E ν U H vac U † + H mat ] | ν ⟩ . (6) \nHere | ν ⟩ T = | ν e , ν µ , ντ ⟩ T refers to the neutrino flavor eigenstate basis, r to the neutrino propagation path, U = U 23 U 13 U 12 ≡ U ( θ 23 ) U ( θ 13 ) U ( θ 12 ) is the 3 × 3 leptonic mixing matrix parametrized in the standard way, H vac = diag ( 0 , ∆ m 2 21 , ∆ m 2 31 ) and in the absence of NSI the matter Hamiltonian is given by H mat = √ 2 GF ne ( r ) diag ( 1 , 0 , 0 ) . Note that because of matter potentials neutrino flavor evolution is more conveniently followed in the flavor basis. \nAs previously pointed out, the presence of neutrino NSI induce new matter potential terms that modify the flavor evolu- \ntion equation, namely \ni d dr | ν ⟩ = [ 1 2 E ν U H vac U † + √ 2 GFne ( r ) ∑ f = e , u , d ε f ] | ν ⟩ , (7) \nwhere the NSI coupling matrices ε f involves the quark relative abundances in addition to the parameters entering in Eq. (17): \nε f = 1 + ε f ee ε f eµ ε f e τ ε f eµ ε f µµ ε f µ τ ε f e τ ε f µ τ ε f ττ . (8) \nExplicitly, ε f i j ( r ) = Yf ( r ) ε f i j ( f = e , u , d ) with Yf ( r ) = nf ( r ) / ne ( r ) . The up- and down-quark relative abundances are written in terms of the neutron relative abundance Yu = 2 + Yn and Y d = 1 + 2 Yn , with the neutron number density calculated from the 4 He and 1 H mass fractions. \nA three-flavor analysis of NSI matter effects demands numerical integration of Eq. (7) for each point in the NSI parameter space. However, an analytical, less CPU expensive and yet precise approach can be adopted in the so-called mass dominance limit ∆ m 2 13 → ∞ [40]. In this approximation, neutrino propagation is properly described in the basis | ˜ ν ⟩ = U T | ν ⟩ ≡ U T 13 U T 23 | ν ⟩ ( propagation basis ). Up to corrections of the order of sin θ 13, the propagating neutrino states are: A mainly electron neutrino state (˜ ν e ), a superposition of muon and tau neutrinos state (˜ ν µ ) and its orthogonal counterpart (˜ ντ ). With these considerations, only ˜ ν e and ˜ ν µ have sizable mixing. Mixing with ˜ ντ for neutrino energies of the order of 10MeV and average SSM quark number densities does not exceed 3 × 10 -2 × ε q i j [15]. With ˜ ντ 'decoupled' from mixing, flavor conversion becomes then a two-flavor problem that can be entirely treated at the analytic level. \nIn two-flavor approximation, the survival probability is given by P ee ( E ν , r ) [40] \nP ee ( E ν , r ) = cos 4 θ 13 P eff ( E ν , r ) + sin 4 θ 13 , (9) \nwhere the r dependence is introduced by the effective probability given by [51] \nP eff ( E ν , r ) = 1 + cos2 θ M ( r ) cos2 θ 12 2 . (10) \nHere θ M ( r ) is the mixing angle in matter and adiabatic propagation has been assumed, thus implying a rather suppressed level-crossing probability ( Pc → 0). With neutrino oscillation data taken from Ref. [50], calculation of the survival probability in Eq. (9) then reduces to the determination of θ M . To do so the following 2 × 2 Hamiltonian has to be diagonalized \nH = 1 4 E ν ( -∆ m 2 21 cos2 θ 12 + A ∆ m 2 21 sin2 θ 12 + B ∆ m 2 21 sin2 θ 12 + B ∆ m 2 21 cos2 θ 12 -A ) . (11) \nIn this expression the A and B terms in the diagonal and offdiagonal entries are given by \nA = 4 √ 2 E ν GFne ( r ) [ cos 2 θ 13 2 -Yq ( r ) ε q D ] , B = 4 √ 2 E ν GFne ( r ) Yq ( r ) ε q N , (12) \nfrom where it can be seen that in the limit ε q i j = 0 and cos θ 13 = 0, A reduces to the SM term and B vanishes. The parameters ε D and ε N result from the rotation from the flavor to the propagation basis and read [40]: \nε q D = -c 2 13 ε q ee + [ c 2 13 -( s 2 23 -s 2 13 c 2 23 )] ε q µµ + ( s 2 23 -c 2 23 s 2 13 ) ε q ττ + s 13 c 13 s 23 ε q eµ + s 13 c 13 c 23 ε q e τ -( 1 + s 2 13 ) c 23 s 23 ε q µ τ , \n2 2 2 (13) ε q N = -s 13 c 23 s 23 ε q µµ + s 13 c 23 s 23 ε q ττ + c 13 c 23 ε q eµ -c 13 s 23 ε q e τ + s 13 ( s 2 23 -c 2 23 ) ε q µ τ , (14) \nwhere ci j ≡ cos θ i j and si j ≡ sin θ i j . The mixing angle in matter thus can be written as \ncos2 θ M ( r ) = ∆ m 2 12 cos2 θ 12 -A √ ( ∆ m 2 12 cos2 θ 12 -A ) 2 + ( ∆ m 2 12 sin2 θ 12 + B ) 2 . (15) \nEqs. (9) and (10) combined with Eqs. (12)-(15) allow the determination of P ee ( E ν , r ) in terms of neutrino oscillation parameters, electron and quark number densities and NSI parameters. The averaged survival probability is then obtained by integrating over r taking into account the distribution of \nneutrino production in the Sun [40]: \n⟨ P ee ( E ν ) ⟩ = ∑αΦα ( E ν ) ∫ 1 0 dr ρ ( r ) P ee ( E ν , r ) ∑αΦα ( E ν ) , (16) \nwhere Φα ( E ν ) refers to the α component of the solar neutrino flux (with α running over all components) and ρα ( r ) to the distribution of neutrino production. For illustration (and only with that aim), in Fig. 1 we show an example of the averaged survival probability as a function of the neutrino energy for the case in which all couplings but ε u ee vanish. As can be seen, the new interaction can either enhance or deplete neutrino flavor conversion depending on its strength and on whether it reinforces or weakens the SM matter potential. With propagation effects already discussed and summarized in Eq. (16) we now \nFIG. 1. Averaged survival probability as a function of neutrino energy for the case in which only ε u ee has a non-vanishing value. This graph aims only at illustrating the impact of neutrino NSI on neutrino propagation in the Sun. The different features are related with the kinematic end-points where certain neutrino fluxes fade away [see Tab. I along with Eq. (16)]. \n<!-- image -->", 'B. Neutrino NSI: Detection effects': "For consistency, the same basis used for neutrino propagation should be used in neutrino detection as well. In doing so the effective Lagrangian in Eq. (17) reads \nL NSI = -√ 2 GF ∑ i = e , µ , τ q = u , d ˜ ν i γ µPL ˜ ε q i j ˜ ν j q γ µ q , (17) \nwhere ˜ ε q = U T ε q U ≡ U T 13 U T 23 ε q U 23 U 13. With the couplings rotated this way the weak-charge in the CE ν NS cross section in Eq. (1) becomes lepton flavor dependent, with the weakcharge in initial-state flavor i given by \nQ 2 ν i = [ + N ( g n V + ˜ ε u ii + 2 ˜ ε d ii ) + Z ( g p V + 2 ˜ ε u ii + ˜ ε d ii )] 2 + ∑ j = i [ N ( ˜ ε u i j + 2 ˜ ε d i j ) + Z ( 2 ˜ ε u i j + ˜ ε d i j )] 2 . (18) \nThe couplings entering in the weak charge can be readily calculated from their definition, with the rotation matrices parametrized for a passive rotation: ˜ ε q i j = ∑ k ,ℓ U ki ε q k ℓ U ℓ j . The effects of the NSI are then clear. By modifying the weakcharge the new interactions can either enhance of deplete the expected reaction rate. Eq. (18) shows that diagonal couplings can produce constructive or destructive interference, while off-diagonal couplings cannot. Note that a proper definition of the flavor basis is, in principle, not possible in the presence of flavor off-diagonal NSI parameters. Strictly speaking then a consistent treatment of such cases requires a density matrix formulation for the calculation of event rates [42]. Arguably, however, differences between the 'standard' approach and the latter are expected to be small provided the off-diagonal parameters are suppressed. That this is the case is somehow expected from data, which do not sizably deviates from the SM expectation. Thus, we adopt the standard procedure regardless of the flavor structure of the parameters considered. \nIn the two-flavor approximation, two neutrino flavors reach the detector: ˜ ν e and ˜ ν µ . Lepton flavor composition of the final state, however, depends on the lepton flavor structure of the interaction. In full generality, the differential event rate is then written as follows \ndR dEr = ∑ k = e , µ , τ ( dR ek dEr + dR µk dEr ) . (19) \nHere the flavored differential event rates are obtained from \n̸ \nEq. (3) by trading QW → Q ν i and by taking into account the survival probability, ⟨ P ee ⟩ , in the first term as well as the oscillation probability to the ˜ ν µ state, 1 -⟨ P ee ⟩ , in the second term. Thus, in the first (second) differential event rate in Eq. (19) couplings ˜ ε q ek ( ˜ ε q µk ) contribute. These couplings are a superposition of the NSI parameters we started with, so in a singleparameter analysis (which we adopt in the first part in Sec. IV) a non-vanishing unrotated NSI parameter can imply the presence of multiple rotated parameters at the cross section level.", 'IV. ANALYSIS AND RESULTS': 'The general problem of assessing the impact of neutrino NSI parameters in neutrino-nucleus event rates involves twelve independent couplings. It is of course a very CPU expensive problem, but not only that. With only a few observables to rely upon, little can be said in the most general case. For practical reasons and as well to make contact with previous analysis, we adopt a single-parameter approach. Towards the end of this section we consider the three lepton flavor diagonal two-parameter cases ( ε u ee , ε d ee ), ( ε u µµ , ε d µµ ) and ( ε u ττ , ε d ττ ); as well to make contact with what has been done previously in the literature (the e and µ cases motivated by previous COHERENT data analysis). It is worth mentioning that because of neutrino flavor mixing multi-ton DM detectors are sensitive \nFIG. 2. Dependence of the ∆χ 2 function on the up-quark NSI parameters for the PandaX-4T [paired and unpaired ionization-only signals (US2)] as well as for XENONnT data sets. Results for the combined analysis are shown as well. The 1 σ and 2 σ confidence level values (horizontal lines) are shown to facilitate reading. \n<!-- image --> \nto τ flavor, which neither reactor nor stopped-pion sources are. From this point of view these measurements are unique. \nWe start with u -quark couplings and proceed by defining a simple χ -square test \nχ 2 = ( R Exp -R SM+NSI σ Exp ) 2 , (20) \nwhere R Exp refers to PandaX-4T and XENONnT event rates central values and (see Tab. II) and R SM+NSI to the SM events rates including as well NSI contributions. Though oversimplified, such χ -square statistic allows to capture the main features of the data sets and their sensitivity to NSI parameters. Results are shown in Fig. 2. First of all, in all cases and with all data sets two minima are found. This result follows from allowing the NSI parameter to have positive and negative values. Because of this range, as we have already pointed out, \nevent rates are symmetric around a small value. Experimental results are thus reproduced in two non-overlapping regions of parameter space. \nOne can see, however, the regions tend to be less pronounced for the XENONnT analysis, regardless of the NSI parameter. Statistical uncertainties are of the order of ∼ 37% in all cases, so they cannot account for this behavior. We thus understand this tendency to be related with measured values and the SM expectation, as we now discuss. We find for the SM predicted values 2.4:46.8:11.3 events for paired:US2:XENONnT. Experimental ranges are on the other hand [2.2,4.8]:[47.0,103.0]:[6.7,14.6] for paired:US2:XENONnT. So, PandaX-4T results tend to prefer values above the SM prediction, while the SM value expected at XENONnT is well within the measured interval. In fact, the expected SM value is 5% away from the midrange, 10.65 \nFIG. 3. Dependence of the ∆χ 2 function on the down-quark NSI parameters for the PandaX-4T [paired and unpaired ionization-only signals (US2)] as well as for XENONnT data sets. Results for the combined analysis are shown as well. The 1 σ and 2 σ confidence level values (horizontal lines) are shown to facilitate reading. \n<!-- image --> \nevents. \nFrom the results one can see that narrower 1 σ level ranges are found for flavor-diagonal parameters. Results for flavor off-diagonal couplings are, instead, wider. This is as well expected. At the cross section level flavor-diagonal contributions add/subtract linearly to the SM contribution, while flavor off-diagonal do quadratically. Since | ε u i j | < 1 . 0, the diagonal components lead to larger deviations than the off-diagonal do for larger values. \nWe provide as well results from a combined analysis, that we have generated by constructing a combined chi-square test χ 2 Combined = χ 2 Paired + χ 2 US2 + χ 2 XENONnT . These results, however, should be interpreted with certain caution. Combining PandaX-4T and XENONnT this way is certainly reliable, but combining paired and US2 data sets might be not because of possible correlations. Very likely the most suitable way \nof combining these data sets is through a covariance matrix. However, such an analysis can only be performed with the full data sets, including backgrounds. it can be noted that the combined analysis is dominated by XENONnT data, with the reason being what we pointed out already: XENONnT measurement is more inline with the SM expectation. \nResults for down-quark couplings are shown in Fig. 3. Differences between these results and those found in the up-quark case are small, a result which is also expected. From a simple inspection of Eqs. (13) and (14) one can see that at the averaged survival probability level they enter in the same functional form. Differences between up and down quarks arise only through their relative abundance, for which in the region of interest (0 . 1 R ⊙ ) Yu differs by no more than 30% from Y d [36]. At the cross section level, the combination of downquark couplings is different from that from the up-quark cou- \nFIG. 4. ∆χ 2 90% CL isocontours in the ε u ee -ε d ee (left graph) and ε u µµ -ε d µµ (middle graph) and ε u ττ -ε d ττ (left graph) planes. Results are shown for the PandaX-4T [paired and unpaired ionization-only signals (US2)] as well as for XENONnT data sets. For comparison results from combined analysis of COHERENT CsI+LAr data [43] are shown as well. Results for the combined analysis have a strong overlapp with those from XENONnT so are not displayed. Note that COHERENT measurements are not sensitive to ε q ττ NSI parameters. \n<!-- image --> \nings [see Eq. (18)]. However, those differences are small and to a certain degree smooth out at the event rate level. \nWe have summarized the 1 σ level ranges following from these two analyzes in Tab. III in App. A. It is worth comparing these results with those derived recently from a combined analysis of COHERENT data [43]. For diagonal couplings these results are rather comparable to those reported in Ref. [43]. More sizable deviations are found for off-diagonal parameters, in particular for ε q eµ and ε q µ τ where the COHERENT combined analysis leads to constraints that exceed those found here by about 20% -50%. Thus, these data sets already provide limits that are comparable with those derived using COHERENT data. Expectations are then that with forthcoming measurements sensitivities to possible new physics in the neutrino sector will improve. Most relevant is the fact that contrary to data coming from stopped-pion sources and/or reactors, measurements from solar neutrino data are sensitive to pure τ flavor parameters. \nFinally, results for the two-parameter analysis are shown in Fig. 4. Overlaid are those derived from COHERENT LAr+CsI combined analysis, in the two cases where they apply. The combined analysis is not displayed because the strong overlapp with the XENONnT data result. It is clear that COHERENT data is moderately more sensitive to NSI effects, but results from PandaX-4T+XENONnT already provide complementary information. We understand this behavior as due to smaller statistical uncertainties in the COHERENT data sets, in particular in the last CsI data set release which largely dominates the fit [43].', 'V. CONCLUSIONS': 'Recent measurements of nuclear recoils induced by the 8 B solar neutrino flux by the PandaX-4T and XENONnT collaborations have opened a new era for both DM searches and neutrino physics. Certainly, for DM searches this implies abandoning the free-background paradigm and adopting new strategies in the quest for DM. For neutrino physics, on the other hand, it provides a new landscape of opportunities that range from precise measurements of the CE ν NS cross section (at energies below those employed in stopped-pion neutrino sources) to searches of new physics that can potentially be hidden in the neutrino sector. This would represent a full program, complementary to all the other CE ν NS related worldwide efforts. \nWith a goal of establishing sensitivity to neutrino physics, in this paper we have studied the sensitivity of the PandaX-4T and XENONnT data sets to neutrino NSI. We have presented a full one-parameter analysis as well as a flavor diagonal twoparameter analysis, the latter with mainly the aim of making contact with previous results derived using COHERENT data. \nIn the one-parameter case, our findings show that with current statistical uncertainties and exposures sensitivities to flavor-diagonal NSI parameters are comparable to those derived using COHERENT data. Sensitivities to flavor offdiagonal parameters are less pronounced, but still competitive with those coming from COHERENT measurements. In the two-parameter case, a comparison with COHERENT recent data analysis demonstrates that with further improvements these experiments have the potential to lead searches for new physics in the neutrino sector through CE ν NS measurements. In particular, and in contrast to reactor or stoppedpion sources, because of neutrino flavor mixing these experiments are sensitive to pure τ flavor observables, providing a', 'Up-type NSI couplings': 'TABLE III. 1 σ CL intervals for ε u i j (upper table) and ε d i j (lower table) derived from PandaX-4T (paired and US2) and XENONnT data sets as well as from a combined analysis of all data. As a function of the NSI parameters, event rates tend to be symmetric around a value close to zero. The non-overlapping intervals in all cases are a result of this behavior.', 'Down-type NSI couplings': 'new channel for this flavor that is difficult to isolate in current solar neutrino data [52]. \nFuture data sets with improved exposures and statistical uncertainties will improve upon the constraints we presented. For example, increasing the exposure by a factor of 5, we checked that sensitivities to ε u ee interactions may improve by about 50%. Given that we are just now working with initial results from Xenon-based DM experiments, it is likely that combined with electron recoil measurements, data from CE ν NS induced by the 8 B solar neutrino flux might lead searches for new physics using this type of technology and perhaps pave the way for unexpected discoveries.', 'Appendix A: Summary of NSI parameters limits': 'In this appendix we collect the 1 σ ranges for up- and downquark NSI parameters. Results are shown in Tab. III. 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2024arXiv240905299L | This paper investigates the secular motion of a massless asteroid within the framework of the doubleaveraged elliptic restricted threebody problem. By employing Poincar variables we analyze the stability properties of asteroid orbits in the presence of planetary perturbations. Our study reveals that periodic orbits identified in the planar configuration maintain stability in the spatial perturbed problem across a wide range of parameter values. These findings supported by numerical simulations contribute to a deeper understanding of asteroid dynamics and have implications for studying exoplanetary systems with highly eccentric host stars. | 2024-09-01T00:00:00Z | ['10.48550/arXiv.2409.05299', 'arXiv:2409.05299', '2024arXiv240905299L'] | ['Astrophysics - Earth and Planetary Astrophysics', 'Mathematics - Dynamical Systems'] | Stability analysis of spatial perturbed elliptic restricted 3body problem with doubleaveraging | 2,024 | 167 | 0.35 | ['EPRINT_HTML', 'EPRINT_PDF'] | 0 | https://arxiv.org/pdf/2409.05299.pdf | {'Stability analysis of spatial perturbed elliptic restricted 3-body problem with double-averaging': 'Yan Luo 2 , Kaicheng Sheng 1 ∗ \n1 \nSchool of Mathematics, Shandong University, Jinan, China 2 Research Centre for Mathematics and Interdisciplinary Sciences, Shandong University, Qingdao, China', 'Abstract': "This paper investigates the secular motion of a massless asteroid within the framework of the double-averaged elliptic restricted three-body problem. By employing Poincar'e variables, the stability properties of asteroid orbits in the presence of spatial perturbations were analysed. The study reveals that periodic orbits identified in the planar configuration maintain stability in the spatial perturbed problem across a wide range of parameter values. These findings, supported by numerical simulations, contribute to a deeper understanding of asteroid dynamics and have implications for studying exoplanetary systems with highly eccentric host stars. \nKeywords : 3-body problem, Double-averaging, Periodic orbit, Linear stability.", '1 Introduction': "The study of the stability of celestial bodies within gravitational systems is a foundational aspect of celestial mechanics. The study of the 3-body problem formed the foundation for understanding the complex dynamic behaviours observed in natural and artificial systems. The restricted 3-body problem plays an important model role in the 3-body problem model. This paper focuses on the double-averaged spatial perturbed elliptic restricted 3body problem (ER3BP) involving a star, a planet, and an asteroid. The study is under the assumption that the mass of the planet is much smaller than the mass of the star. The averaging over fast phases corresponds to the motions of the star-planet system and the asteroid. The classical restricted 3-body problem has been instrumental in developing our understanding of orbital mechanics, particularly in cases where the orbits of the primary bodies are assumed to be circular. However, in reality, most celestial bodies, including stars, planets and asteroids, follow elliptical orbits, which introduces significant complexities to the problem. These complexities necessitate more advanced mathematical techniques and have led to the development of the ER3BP, where the primary bodies follow elliptical orbits. \nDouble-averaging method simplifies the complex gravitational interactions, resulting in a simplified system that retains the essential dynamical features of the original problem while being more tractable for analysis over long timescales. The seminal work of Aksenov [1979] on the double-averaged elliptical restricted 3-body problem laid the foundation for understanding the long-term evolution of orbits in such systems. His work demonstrated how the double-averaging method could be used to reduce the complexity of the problem, making it possible to derive analytical expressions for the evolution of orbital elements over time. \nThe significance of stability in celestial mechanics, particularly in the context of Hamiltonian systems, has been explored by Arnold [1961], who established fundamental results on the stability of equilibrium positions. This work is further elaborated in the comprehensive treatment of classical mechanics by Arnold et al. [2006]. Moreover, the secular evolution of orbits, especially in hierarchical systems, has been investigated by Lidov [1962] and Kozai [1962], whose work on the Lidov-Kozai mechanism has profoundly influenced our understanding of orbital dynamics in N-body systems. The relevance of the Lidov-Kozai mechanism has been further underscored by recent studies, such as those by Katz et al. [2011] and Lithwick and Naoz [2011], who explored its effects in systems with eccentric perturbers. \nThe stability of planar orbits in double-averaged circular problems was established by Neishtadt [1975]. His study indicates that planar orbits also remain stable in the linear approximation of the double-averaged elliptic problem with a sufficiently small eccentricity of the perturber's orbit. The stability of periodic solutions in the restricted 3-body problem, particularly in the presence of elliptical orbits, has also been explored by Leontovich [1962] and Moser [1968], whose work on Hamiltonian systems with two degrees of freedom provided crucial insights into the resonance phenomena that often govern the stability of such systems. Harrington [1968] and Ziglin [1975] analyzed the dynamical evolution of the 3-body problem, while more recent studies, such as those by Naoz et al. [2013] and Sidorenko [2018], have extended these ideas to modern exoplanetary systems. The double-averaged ER3BP specifically benefits from these approaches, offering insights into the stability conditions that govern the long-term behaviour of small bodies in elliptical orbits. \nIn recent years, there has been renewed interest in the study of the double-averaged ER3BP, particularly in the context of exoplanetary systems. The discovery of numerous exoplanets with highly eccentric orbits has prompted researchers to revisit the problem of stability in such systems. Studies by Huang and Lei [2022] and Lei [2022] have focused on the linear stability of the inner case of the double-averaged spatial elliptic restricted 3-body problem, providing new insights into the conditions under which stability can be achieved in these systems. A completed numerical analysis, including all possible situations for the evolution of planar orbits in the double-averaged ER3BP, can be found in the work by Vashkovyak [1982]. \nThe investigation of equilibria within the double-averaged ER3BP, as studied by Palaci'an et al. [2006], and the analysis of apsidal alignment by Neishtadt et al. [2021] analyzed both the linear and nonlinear stability of apsidal alignment in the spatial double-averaged ER3BP, further illustrated the rich dynamical behaviour that emerges in such systems. Linear stability of the equilibria and periodic orbits of the asteroid in spatial double-averaged ER3BP in the inner case with arbitrary inclination is studied in Huang et al. [2024]. \nThis paper aims to provide a comprehensive analysis of the stability of the spatial \nperturbed double-averaged ER3BP, building on the foundational work of Aksenov [1979], Vashkovyak [1981, 1982], Neishtadt et al. [2021] and others. By combining analytical methods with numerical simulations, the long-term stability of the asteroid's orbits in doubleaveraged ER3BP is investigated in linear approximation. The results have important implications for both theoretical and practical aspects of celestial mechanics, providing new insights into the stability of orbits in a wide range of astrophysical systems. Through this work, we hope to contribute to the ongoing exploration of the dynamic and ever-evolving nature of the cosmos. \nA notable characteristic of many exoplanets is their orbits, which often exhibit large eccentricities and inclinations, deviating significantly from the near-circular and coplanar trajectories observed in the solar system [Stephen et al., 2012, Team, 2017]. Our research holds the potential to make substantial contributions to the domain of exoplanetary dynamics [Shevchenko, 2016]. Furthermore, the secular evolution of motions within the non-restricted three-body problem has been extensively studied through the application of averaging techniques [Harrington, 1968, Ziglin, 1975, Michtchenko and Malhotra, 2004], one can extend our work into the non-restricted 3-body problem.", '2 Statement of the problem': "Figure 1: Coordinate frames. \n<!-- image --> \nThe considered spatial perturbed elliptic restricted three-body problem involves a star S , a planet J , and an asteroid A [Brouwer and Clemence, 1961]. Adopting a coordinate system analogous to that in Neishtadt et al. [2021], the star is positioned at the origin O of a Cartesian frame Oxyz . The Oxy plane of this system is defined by the orbital plane of the star-planet system. Consequently, the coordinates of the planet and the asteroid are ( x J , y J , 0) and ( x, y, z ), respectively. A rotating Cartesian frame Ox ' y ' z ' is introduced, where the Ox ' y ' plane aligns with the asteroid's osculating orbital plane. In this frame, the asteroid's coordinates are ( x ' , y ' , 0). The asteroid's orbital motion is characterized by standard osculating elements: semi-major axis ( a ), mean anomaly ( l ), eccentricity ( e ), argument of periapsis ( ω ), inclination ( i ), and longitude of the ascending node (Ω). Refer to Fig. 1 for a visual representation of the system geometry. According to the rotation of the Cartesian \nframe, the transformation between frames Oxyz and Ox ' y ' z ' is \n[ x y z ] T = T Ω ×T i ×T ω × [ x ' y ' 0 ] T , (2.1) \nwhere \nT Ω = cos (Ω) -sin (Ω) 0 sin (Ω) cos (Ω) 0 0 0 1 , T i = 1 0 0 0 cos( i ) -sin( i ) 0 sin( i ) cos( i ) , T ω = cos ( ω ) -sin ( ω ) 0 sin ( ω ) cos ( ω ) 0 0 0 1 . (2.2) \nConsequently, the expressions of x , y and z are \nx = (cos Ω cos ω -cos i sin Ω sin ω ) x ' +( -cos Ω sin ω -cos i sin Ω cos ω ) y ' , y = (sin Ω cos ω +cos i cos Ω sin ω ) x ' +( -sin Ω sin ω +cos i cos Ω cos ω ) y ' , z = (sin i sin ω ) x ' +(sin i cos ω ) y ' . (2.3) \nIt follows from Shevchenko [2016], the planet moves in a prescribed elliptic orbit: \nx J = a J (cos E J -e J ) , y J = a J √ 1 -e J 2 sin E J , l J = E J -e J sin E J . (2.4) \nHere a J , e J , E J , l J are the semi-major axis, the eccentricity, the eccentric anomaly, and the mean anomaly of the planet's orbit. We put a J = 1 for convenience in the following.", '3 Hamiltonian of the system': "We introduce the canonical Delaunay elements ( l, g, h, L, G, H ), where g ( ≡ ω ) and h ( ≡ Ω) are the argument of pericenter and the ascending node of the asteroid. The elements L = √ (1 -µ ) a , G = L √ 1 -e 2 , and H = G cos i correspond to the Keplerian energy, total angular momentum, and z -component of angular momentum, respectively [Brouwer and Clemence, 1961]. To facilitate the dynamical analysis of the asteroid, canonical Poincar'e variables are denoted by \np 1 = L, q 1 = l + g + h, p 2 = √ 2 ( L -G ) cos ( g + h ) , q 2 = -√ 2 ( L -G ) sin ( g + h ) , p 3 = √ 2 ( G -H ) cos h, q 3 = -√ 2 ( G -H ) sin h. (3.1) \nAssuming the planet and star have masses µ and 1 -µ , respectively, such that the total system mass equals unity. µ ≪ 1. The Hamiltonian of the asteroid is given by: \nF = -(1 -µ ) 2 2 L 2 + µU -µ ( x x J + y y J ) , (3.2) \nwhere \nU = -V = -1 √ ( x -x J ) 2 +( y -y J ) 2 + z 2 (3.3) \nis the perturbing gravitational potential which is expressed in terms of the asteroid's coordinates ( x, y, z ). These coordinates can be transformed into Poincar'e variables with equations (2.3) and (3.1). The asteroid's motion in its elliptic orbit is governed by the equations: \nx ' = a (cos E -e ) , y ' = a √ 1 -e 2 sin E, l = E -e sin E, (3.4) \nwhere E denotes the eccentric anomaly [Shevchenko, 2016]. The planet's coordinates, x J and y J , are considered prescribed functions of time. \nThe double-averaged Hamiltonian ¯ F of the asteroid is defined as: \n¯ F = 1 (2 π ) 2 ∫∫ [0 , 2 π ] 2 Fdldl J = -(1 -µ ) 2 2 L 2 + µ ¯ U = -(1 -µ ) 2 2 L 2 -µ ¯ V , (3.5) \nwhere \n¯ U = 1 (2 π ) 2 ∫∫ [0 , 2 π ] 2 Udldl J , (3.6) \nand µ ¯ V = -µ ¯ U is the double-averaged force function of gravity of the planet. \nPerforming the double-averaging over the mean anomalies of the planet and the asteroid, the Hamiltonian becomes independent of q 1 . Consequently, the conjugate momentum p 1 (equivalent to L ) constitutes a first integral of the system. The first term in the doubleaveraged Hamiltonian, ¯ F , thus remains constant. The dynamics of the remaining variables, ( p 2 , q 2 ), ( p 3 , q 3 ), are therefore governed by a two-degree-of-freedom (2-DOF) Hamiltonian system with Hamiltonian µ ¯ U . \nBy introducing the slow time variable τ = µt , the Hamiltonian transforms to ¯ U . Notably, ¯ U depends on the planet's eccentricity, e J , as parameters, and the ratio between the semimajor axis a of the asteroid and a J = 1 of the planet. \nTaking i = 0, the spatial perturbed ER3BP reduces to a planar problem on the invariant plane p 3 = 0, q 3 = 0. This planar system is described by a one-degree-of-freedom (1-DOF) Hamiltonian, ¯ R Θ ( p 2 , q 2 ), where ¯ R Θ ( p 2 , q 2 ) = ¯ U ( p 2 , q 2 , p 3 = 0 , q 3 = 0) is independent of p 3 and q 3 . Following Aksenov [1979] and Neishtadt et al. [2021], the Hamiltonian of the double-averaged planar ER3BP is given by: \n¯ R Θ = 1 (2 π ) 2 ∫∫ [0 , 2 π ] 2 Rdldl J , (3.7) \nwhere \nR Θ = -1 √ ( x -x J ) 2 +( y -y J ) 2 . (3.8) \nNumerical investigations by Vashkovyak [1982] have demonstrated the existence of stationary solutions (equilibria) \nq 2 = 0 , p 2 = p 2 ∗ , (3.9) \nfor some domains in the plane of parameters a , e J in the double-averaged planar ER3BP. Periodic orbits, separatrix and orbits of circulating are included as well. Fixing the parameter a and e J in ¯ R Θ , one can obtain the figure of Θ and e . See Fig. 2. \nFigure 2: Trajectories of Θ, e in double-averaged planar ER3BP. \n<!-- image --> \np 2 ∗ is the root of the equation \n∂ ¯ R ∂p 2 ∣ ∣ ∣ ∣ q 2 =0 = 0 . (3.10) \nThe value of the eccentricity at the equilibrium (3.9) is provided by the equation \ne ∗ = √ 1 -( 1 -p 2 2 ∗ 2 √ a ) 2 . (3.11) \nAs detailed in Vashkovyak [1982], the equilibria defined by equation (3.9) correspond to an apsidal alignment scenario where ω +Ω = 0. (Fig. 3a) The study also identifies periodic orbits in terms of Θ = ω +Ω and the asteroid's eccentricity ( e ) surrounding these equilibria. (Fig. 3b) \n̸ \nTo comprehensively assess the linear stability of asteroid orbits in the spatial perturbed ER3BP, it is essential to consider non-equilibrium cases (i.e., Θ = 0), especially to consider the cases of periodic orbits. The small perturbation case is considered in this paper, which means the inclination i is regarded as a small perturbation. In this case, the formula of x , y and z can be obtained with i = 0 and ω +Ω=Θ: \nx = cos(Θ) x ' -sin(Θ) y ' , y = sin(Θ) x ' +cos(Θ) y ' , z = 0 . (3.12) \nFor small i , proper approximations of formulas (3.1) are obtained: \ncos (Ω) ≈ p 3 √ G · i 2 / 2 , sin (Ω) ≈ -q 3 √ G · i 2 / 2 , cos ( ω ) ≈ p 2 p 3 -q 2 q 3 √ G · i 2 √ p 2 2 + q 2 2 , sin ( ω ) ≈ -p 2 q 3 + p 3 q 2 √ G · i 2 √ p 2 2 + q 2 2 . (3.13) \nFigure 3: Case of equilibria (apsidal alignment) and periodic orbits. \n<!-- image --> \nSubstituting (3.13) into (2.3) and then taking in (3.3), the function U = U Θ with increasing order of p 3 and q 3 can be written as \nU Θ = R Θ + W Θ + O ( p 4 3 + q 4 3 ) , W Θ = 1 2 ( A Θ p 2 3 +2 B Θ , p 3 q 3 + C Θ q 2 3 ) (3.14) \nwhere \nA Θ = yy J G [ ( x -x J ) 2 +( y -y J ) 2 ] 3 / 2 , B Θ = ( xy J + yx J ) 2 G [ ( x -x J ) 2 +( y -y J ) 2 ] 3 / 2 , C Θ = xx J G [ ( x -x J ) 2 +( y -y J ) 2 ] 3 / 2 . (3.15) \nHere all quantities are calculated at i = 0 and Θ = ω +Ω. Calculations are similar to the procedure in Neishtadt et al. [2021]. \nAveraging W Θ over the mean anomaly of the asteroid l and the mean anomaly of the planet l J , the averaged value of W Θ is \n¯ W Θ = 1 2 ( ¯ A Θ p 3 2 +2 ¯ B Θ p 3 q 3 + ¯ C Θ q 3 2 ) , (3.16) \nwhere \n¯ A Θ = 1 4 π 2 ∫∫ [0 , 2 π ] 2 ( A Θ ) d l d l J , ¯ B Θ = 1 4 π 2 ∫∫ [0 , 2 π ] 2 ( B Θ ) d l d l J , ¯ C Θ = 1 4 π 2 ∫∫ [0 , 2 π ] 2 ( C Θ ) d l d l J (3.17) \nare the averaged values of A Θ , B Θ and C Θ respectively. \nThe stability of the periodic orbits of the asteroid is guaranteed in linear approximation if the sequential principal minor ¯ A Θ > 0 and \nD Θ = det [ ¯ A Θ ¯ B Θ ¯ B Θ ¯ C Θ ] > 0 . (3.18) \nThe stability of the periodic orbits of the asteroid will be proved analytically with some conditions and then shown numerically for the general case with the help of Matlab.", '4 Limiting cases': "When the distance between the asteroid and the star is much smaller than between the planet and the star, an expansion over the ratio between a and a J can be performed. This is called the inner case of the double-averaged spatial ER3BP. Huang et al. [2024] considered the double-averaged value of the coefficients in the quadratic form of p 3 and q 3 and averaged W Θ with the same procedure in (3.17). The sequential principal minor \n¯ A Θ , inner ≈ 3 ( 1 + 4 e 2 -5 e 2 cos (Θ) 2 ) 4 G (1 -e 2 J ) 3 2 a 2 , D Θ , inner ≈ 9 (1 + 3 e 2 -4 e 4 ) 16 G 2 (1 -e 2 J ) 3 a 4 , (4.1) \nare calculated to be positive for a is small and e J ↛ 1. The orbits of the asteroid in the double-averaged model are stable in linear approximation for the inner case. \nOn the contrary, when the distance between the asteroid and the star is much larger than the distance between the planet and the star (outer case), an expansion of d = 1 /a can be performed analytically. Taking \nM 1 = x 2 J + y 2 J x 2 + y 2 + z 2 , M 2 = -2 ( xx J + yy J ) x 2 + y 2 + z 2 , (4.2) \nand \nM = M 1 + M 2 , (4.3) \nthe approximate formula of the force function of gravity U = U Θ , outer of the planet up to a 3 is expanded as \nU Θ , outer = -1 √ x 2 + y 2 + z 2 ( 1 -1 2 M + 3 8 ( M 1 M 2 + M 2 2 ) -5 16 M 3 2 ) . (4.4) \nThen, the double-averaged value of the coefficients in the quadratic form of p 3 and q 3 is \n¯ W Θ , outer = 1 2 ( ¯ A Θ , outer p 3 2 +2 ¯ B Θ , outer p 3 q 3 + ¯ C Θ , outer q 3 2 ) . (4.5) \nAveraging over mean anomalies is difficult. By Kepler's equation \nl = E -e sin E, l J = E J -e sin E J , (4.6) \nthe double-averaged value of ¯ W Θ , outer can be derived by \n¯ W Θ , outer = 1 4 π 2 ∫∫ [0 , 2 π ] 2 W Θ , outer d l d l J = 1 4 π 2 ∫∫ [0 , 2 π ] 2 W Θ , outer d l d E d l J d E J d E d E J = 1 4 π 2 ∫∫ [0 , 2 π ] 2 W Θ , outer (1 -e cos E ) (1 -e J cos E J ) d E d E J . (4.7) \n¯ A Θ , outer , ¯ B Θ , outer and ¯ C Θ , outer are derived similarly. Substituting x , y , x J , y J into the doubleaveraged value of the coefficients, the coefficients up to order d 4 are \n¯ A Θ , outer = 3 (1 -e 2 J ) 4 (1 -e 2 ) 3 2 G d 3 -75 e e J cos (Θ) (1 -e 2 J ) 32 (1 -e 2 ) 5 2 G d 4 + O ( d 5 ) , ¯ B Θ , outer = 15 e e J sin (Θ) (17 e 2 J -24) 128(1 -e 2 ) 5 2 G d 4 + O ( d 5 ) , ¯ C Θ , outer = 3 (1 + 4 e 2 J ) 4 (1 -e 2 ) 3 2 G d 3 -15 e e J cos (Θ) (43 e 2 J +34) 64 (1 -e 2 ) 5 2 G d 4 + O ( d 5 ) . (4.8) \nIn the above formulas, as d is small enough, for e ↛ 1, all the coefficients of d 4 in (4.8) are bounded, thus terms of d 4 , as well as higher order terms, are omitted. It is obvious that ¯ A Θ , outer > 0 and ¯ C Θ , outer > 0. The determinant of the quadratic form (4.5) up to order a 7 is \nD Θ , outer = 9 (1 -e 2 J ) (1 + 4 e 2 J ) 16 (1 -e 2 ) 3 G 2 d 6 + 45 e (44 e J +39 e 3 J -83 e 5 J ) cos (Θ) 256 (1 -e 2 ) 4 G 2 d 7 (4.9) \nSimilarly, when d is small, the coefficient of d 7 in (4.9) is bounded as far as e ↛ 1. For small values of d , the sequential principal minor ¯ A Θ , outer > 0 and D Θ , outer > 0, thus (4.5) is a positive definite quadratic form. Consequently, for the outer case, periodic orbits of the double-averaged planar elliptic restricted 3-body problem are stable in the linear approximation with spatial perturbation.", '5 General case': "When the ratio between a and a J = 1 is arbitrary (general case), the orbits of the asteroid and the planet are assumed not in collision, i.e., \na (1 + e ) < 1 -e J for 0 < a < 1 , a (1 -e ) > 1 + e J for a > 1 . (5.1) \nFor periodic orbits around the equilibria in Vashkovyak [1982], the orbits of the asteroid sway up and down periodically. When the sum of the argument of the pericenter and the ascending node of the asteroid is Θ = ω +Ω, denote the distance between the asteroid and the planet by \nd 3 Θ ( l, l J ) = [ ( x -x J ) 2 +( y -y J ) 2 ] 3 / 2 . (5.2) \nThen substituting (3.12) into (3.15), we get \nyy 1 \nΘ Θ \nA Θ = J Gd 3 Θ ( l, l J ) = Gd 3 Θ ( l, l J ) [sin(Θ) x ' y J +cos(Θ) y ' y J ] , B Θ = ( xy J + yx J ) 2 Gd 3 Θ ( l, l J ) = 1 2 Gd 3 Θ ( l, l J ) [ sin(Θ) x ' x J +cos(Θ) y ' x J +cos(Θ) x ' y J -sin(Θ) y ' y J ] , C Θ = xx J Gd 3 ( l, l J ) = 1 Gd 3 ( l, l J ) [cos(Θ) x ' x J -sin(Θ) y ' x J ] . (5.3) \nThe double-averaged value of the quadratic form of p 3 and q 3 is \n¯ W Θ = 1 2 ( ¯ A Θ p 3 2 +2 ¯ B Θ p 3 q 3 + ¯ C Θ q 3 2 ) , (5.4) \nwhere \n¯ A Θ = 1 4 π 2 G ∫∫ [0 , 2 π ] 2 1 d 3 Θ ( l, l J ) [sin(Θ) x ' y J +cos(Θ) y ' y J ] d l d l J , ¯ B Θ = 1 8 π 2 G ∫∫ [0 , 2 π ] 2 1 d 3 Θ ( l, l J ) [ sin(Θ) x ' x J +cos(Θ) y ' x J +cos(Θ) x ' y J -sin(Θ) y ' y J ] d l d l J , ¯ C Θ = 1 4 π 2 G ∫∫ [0 , 2 π ] 2 1 d 3 Θ ( l, l J ) [cos(Θ) x ' x J -sin(Θ) y ' x J ] d l d l J (5.5)", '5.1 General case with small Θ': "In the real cosmos, the Lidov-Kozai effects mostly occur with small changes in eccentricities, thus it is important to consider the small periodic orbits around the equilibrium in Vashkovyak [1982]. In this case, it is reasonable to assume that Θ is small, then \nd 3 Θ ( l, l J ) = [ ( x -x J ) 2 +( y -y J ) 2 ] 3 / 2 = d ' 3 Θ ( l, l J ) + 3 d ' Θ ( l, l J ) ( y ' x J -x ' y J ) Θ + O (Θ 2 ) , (5.6) \nwhere \nd ' Θ ( l, l J ) = √ ( x ' -x J ) 2 +( y ' -y J ) 2 . (5.7) \nThis subsection will prove the positive definiteness of the sequential principal minor of the quadratic form ¯ W Θ with small Θ. The proof is under the assumption that the orbit of the asteroid is inside the planet's orbit. The outside case can be proved similarly. Firstly, let us show \nand \n∫∫ [0 , 2 π ] 2 x ' y J d 3 Θ ( l, l J ) dldl J ∼ O (Θ) , (5.8) \n∫∫ [0 , 2 π ] 2 y ' x J d 3 Θ ( l, l J ) dldl J ∼ O (Θ) . (5.9) \nFigure 4: New orbit and symmetry of orbit. \n<!-- image --> \nThe orbit of the asteroid A concerning ( x ' , y ' ) can be regarded as an orbit of A 0 with x = x ' and y = y ' , in detail, the blue orbit of A 0 can replace the orange orbit of A in Fig. 4a. It can be found that the new orbit (orbit of A 0 ) and the orbit of the planet are symmetrical about the y -axis, See Fig 4b, then \nd ' Θ ( l, l J ) = d ' -Θ (2 π -l, 2 π -l J ) , d ' Θ (2 π -l, l J ) = d ' -Θ ( l, 2 π -l J ) . (5.10) \nAlso, distances with expansions of Θ are \nd 3 Θ ( l, l J ) = d ' 3 Θ ( l, l J ) + 3 d ' Θ ( l, l J ) ( y ' x J -x ' y J ) Θ , d 3 Θ (2 π -l, 2 π -l J ) = d ' 3 Θ ( l, l J ) -3 d ' Θ ( l, l J ) ( y ' x J -x ' y J ) Θ , d 3 Θ ( l, 2 π -l J ) = d ' 3 Θ ( l, 2 π -l J ) + 3 d ' Θ ( l, 2 π -l J ) ( y ' x J + x ' y J ) Θ , d 3 Θ (2 π -l, l J ) = d ' 3 Θ ( l, 2 π -l J ) -3 d ' Θ ( l, 2 π -l J ) ( y ' x J + x ' y J ) Θ . (5.11) \nThen formula (5.8) is proved by \n∫∫ [0 , 2 π ] 2 x ' y J d 3 Θ ( l, l J ) dldl J = ∫∫ [0 ,π ] 2 [ x ' y J d 3 Θ ( l, l J ) -x ' y J d 3 Θ (2 π -l, 2 π -l J ) + x ' y J d 3 Θ (2 π -l, l J ) -x ' y J d 3 Θ ( l, 2 π -l J ) ] dldl J = ∫∫ [0 ,π ] 2 x ' y J [ d 3 Θ (2 π -l, 2 π -l J ) -d 3 Θ ( l, l J ) d 3 Θ ( l, l J ) d 3 Θ (2 π -l, 2 π -l J ) + d 3 Θ ( l, 2 π -l J ) -d 3 Θ (2 π -l, l J ) d 3 Θ (2 π -l, l J ) d 3 Θ ( l, 2 π -l J ) ] dldl J =6Θ ∫∫ [0 ,π ] 2 x ' y J [ -d ' Θ ( l, l J ) ( y ' x J -x ' y J ) d 3 Θ ( l, l J ) d 3 Θ (2 π -l, 2 π -l J ) + d ' Θ ( l, 2 π -l J ) ( y ' x J + x ' y J ) d 3 Θ (2 π -l, l J ) d 3 Θ ( l, 2 π -l J ) ] dldl J ∼ O (Θ) . \nSimilarly, \n∫∫ [0 , 2 π ] 2 y ' x J d 3 Θ ( l, l J ) dldl J =6Θ ∫∫ [0 ,π ] 2 y ' x J [ -d ' Θ ( l, l J ) ( y ' x J -x ' y J ) d 3 Θ ( l, l J ) d 3 Θ (2 π -l, 2 π -l J ) + d ' Θ ( l, 2 π -l J ) ( y ' x J + x ' y J ) d 3 Θ (2 π -l, l J ) d 3 Θ ( l, 2 π -l J ) ] dldl J ∼ O (Θ) , \nwhich proved formula (5.9). Consequently, for small Θ, \n¯ B Θ = Θ 8 π 2 G ∫∫ [0 , 2 π ] 2 x ' x J -y ' y J d 3 Θ ( l, l J ) d l d l J + cos(Θ) 8 π 2 G ∫∫ [0 , 2 π ] 2 y ' x J + x ' y J d 3 Θ ( l, l J ) d l d l J = Θ 8 π 2 G ∫∫ [0 , 2 π ] 2 x ' x J -y ' y J d 3 Θ ( l, l J ) d l d l J + O (Θ) 8 π 2 G ∼ O (Θ) . (5.12) \nThus, ¯ B Θ is of order Θ in the case of small Θ. Numerical checks of ¯ B Θ in Matlab for some values of a , e J are performed with variable Θ and e taken from 0 to 0 . 1 and 0 to 1 respectively. In such a way we verified that ¯ B is small for Θ ∈ [0 , 0 . 1]. \nFigure 5: Value of ¯ B Θ for small Θ. \n<!-- image --> \nBy formula (5.8), the double-averaged value ¯ A Θ is \n¯ A Θ = cos(Θ) 4 π 2 G ∫∫ [0 , 2 π ] 2 y ' y J d 3 Θ ( l, l J ) d l d l J + O (Θ 2 ) 4 π 2 G = 1 2 π 2 G ∫∫ [0 ,π ] 2 [ d 3 Θ ( l, 2 π -l J ) -d 3 Θ ( l, l J ) d 3 Θ ( l, l J ) d 3 Θ ( l, 2 π -l J ) ] y ' y J dldl J . (5.13) \nFor l, l J ∈ (0 , π ), it is obvious that \nd 3 Θ ( l, 2 π -l J ) -d 3 Θ ( l, l J ) d 3 Θ ( l, l J ) d 3 Θ ( l, 2 π -l J ) > 0 . (5.14) \nThus ¯ A Θ is positive. Numerical results of A in Matlab with a = 0 . 3, e J = 0 . 2 and a = 4, e J = 0 . 4 coincide with the analytical conclusion. \nFigure 6: Value of ¯ A Θ for small Θ. \n<!-- image --> \nBy formula (5.9), the double-averaged value ¯ C Θ is \n¯ C Θ = cos(Θ) 4 π 2 G ∫∫ [0 , 2 π ] 2 x ' x J d 3 Θ ( l, l J ) d l d l J + O (Θ 2 ) 4 π 2 G = 1 2 π 2 G ∫∫ [0 ,π ] 2 x ' x J d 3 Θ ( l, l J ) dldl J . (5.15) \nTherefore, proving formula (5.15) is positive is equivalent to proving \n∫∫ [0 ,π ] 2 xx J d 3 Θ ( l, l J ) d l d l J > 0 , (5.16) \n̸ \nwhere x = x ' , i.e. the orbit of asteroid A can be considered as an orbit of A 0 in apsidal alignment case. The case of e J = 0 is the circular problem proved in [Neishtadt, 1975], thus e J = 0 can be assumed. \nFigure 7: Prove of C . \n<!-- image --> \nTaking x > -ae , which means A 0 ( x, y ) is on the right of the orbit of the asteroid, denoted by A 0 ∈ I R , then A ' 0 ( -2 ae -x, y ) is on the left, denoted by A ' 0 ∈ I L . Similarly, \ntaking x > e J , then J ( x J , y J ) ∈ I R J and J ' ( -2 e J -x J , y J ) ∈ I L J . See Fig. 7. The distance A 0 J = d 1 , A ' 0 J ' = d 2 , A 0 J ' = d 3 and A ' 0 J = d 4 . It is shown in Fig. 7a, when d 2 < d 4 , d 2 < d 3 , d 1 < d 4 and d 1 < d 2 , the considered function \nˆ C = xx J d 3 1 + ( -2 ae -x )( -2 e J -x J ) d 3 2 + x ( -2 e J -x J ) d 3 3 + ( -2 ae -x ) x J d 3 4 > xx J d 3 2 + ( -2 ae -x )( -2 e J -x J ) d 3 2 + x ( -2 e J -x J ) d 3 2 + ( -2 ae -x ) x J d 3 2 = 4 aee J d 3 2 > 0 . (5.17) \nIf replacing d 2 < d 4 by d 2 > d 4 , see Fig. 7b, then by rearrangement inequality, \nˆ C ≤ xx J d 3 1 + ( -2 ae -x )( -2 e J -x J ) d 3 4 + x ( -2 e J -x J ) d 3 3 + ( -2 ae -x ) x J d 3 2 > xx J d 3 4 + ( -2 ae -x )( -2 e J -x J ) d 3 4 + x ( -2 e J -x J ) d 3 4 + ( -2 ae -x ) x J d 3 4 = 4 aee J d 3 4 > 0 . (5.18) \nFor other cases, one can prove them by rearrangement inequality directly. Thus, the integral \n∫∫ [0 ,π ] 2 xx J d 3 Θ ( l, l J ) d l d l J = [ ∫∫ I L × I L J + ∫∫ I L × I R J + ∫∫ I R × I L J + ∫∫ I R × I R J ] xx J d 3 Θ ( l, l J ) d l d l J = ∫∫ I R × I R J ˆ C d l d l J > 0 (5.19) \nAs a result, ¯ C Θ is positive. Meanwhile, this is also an analytical proof of variable C in Neishtadt et al. [2021] instead of the numerical work. Numerical calculations of ¯ C Θ were performed in Matlab for some values of a , e J . The variables Θ and e are taken from 0 to 0 . 1 and 0 to 1 respectively with some grids. In such a way we verified that ¯ C is always positive. Fig. 8 are some numerical results with a = 0 . 3, e J = 0 . 2 and a = 4, e J = 0 . 4. \nSince ¯ B Θ ∼ O (Θ) is rather smaller than ¯ A Θ and ¯ C Θ , the determinant D Θ = ¯ A Θ ¯ C Θ -O (Θ 2 ) > 0. Numerical checks are shown in Fig. 9. Thus, ¯ A Θ > 0, D Θ > 0, and ¯ W Θ is a positive definite quadratic form. Hence, small periodic orbits around the equilibria of the double-averaged planar elliptic restricted 3-body problem are stable in the linear approximation as small periodic orbits of the double-averaged spatial perturbed elliptic restricted 3-body problem for small Θ.", '5.2 General case with arbitrary Θ': 'When Θ is not small, expansion of Θ can not be performed. Calculations of the coefficients of the quadratic form ¯ W Θ have been done numerically in Matlab. With the averaging \n<!-- image --> \nFigure 8: Value of ¯ C Θ for small Θ. \n<!-- image --> \n<!-- image --> \nFigure 9: Value of D Θ for small Θ. \n<!-- image --> \nprocedure in (4.7), detailing the expression of double-averaged values of coefficients in the quadratic form: \n¯ A Θ = 1 4 π 2 G ∫∫ [0 , 2 π ] 2 yy J (1 -e cos E ) (1 -e J cos E J ) G [ ( x -x J ) 2 +( y -y J ) 2 ] 3 / 2 d E d E J , ¯ B Θ = 1 8 π 2 G ∫∫ [0 , 2 π ] 2 ( xy J + yx J ) (1 -e cos E ) (1 -e J cos E J ) G [ ( x -x J ) 2 +( y -y J ) 2 ] 3 / 2 d E d E J , ¯ C Θ = 1 4 π 2 G ∫∫ [0 , 2 π ] 2 xx J (1 -e cos E ) (1 -e J cos E J ) G [ ( x -x J ) 2 +( y -y J ) 2 ] 3 / 2 d E d E J . (5.20) \nTaking a J = 1, for chosen values of a and e J , each pair of (Θ , e ) belongs to one certain periodic orbit which is determined by \n¯ R = -1 4 π 2 ∫∫ [0 , 2 π ] 2 (1 -e cos E ) (1 -e J cos E J ) √ ( x -x J ) 2 +( y -y J ) 2 d E d E J . (5.21) \nConsidering values of variables satisfying conditions (5.1), and then plotting the figures of (Θ , e, ¯ A Θ ) and (Θ , e, D Θ ) numerically for some considerable a and e J . The numerical work takes a = 0 . 3 and e J = 0 . 2 in Fig. 10, a = 0 . 2 and e J = 0 . 4 in Fig. 11, a = 4 and e J = 0 . 4 in Fig. 12, a = 10 and e J = 0 . 1 in Fig. 13. All the numerical results demonstrate that the sequential principal minor ¯ A Θ > 0 and D Θ > 0, thus the quadratic form ¯ W Θ is positive defined. Therefore, the numerical calculation gives stability of the periodic orbits of the asteroid in the linear approximation in the double-averaged spatial perturbed elliptic restricted 3-body problem for all values of parameters. \nFigure 10: Value of ¯ A Θ and D Θ for a = 0 . 3 and e J = 0 . 2. \n<!-- image -->', '6 Conclusion': "A complete analysis of secular effects on the motion of a massless asteroid within the framework of the spatial perturbed, double-averaged elliptic restricted three-body problem has been conducted. The stability of the asteroid's orbits was investigated with linearization. \nFigure 11: Value of ¯ A Θ and D Θ for a = 0 . 2 and e J = 0 . 4. \n<!-- image --> \nFigure 12: Value of ¯ A Θ and D Θ for a = 4 and e J = 0 . 4. \n<!-- image --> \nFigure 13: Value of ¯ A Θ and D Θ for a = 10 and e J = 0 . 1. \n<!-- image --> \nNotably, periodic orbits originating from the planar problems are stable within the spatial perturbation across all parameter regions. Numerical simulations corroborate these findings for a variety of periodic orbits. The model's applicability to systems with highly eccentric planets renders it particularly valuable for exoplanet studies.", 'Acknowledgements': 'The authors express their gratitude to Prof. Anatoly Neishtadt for suggestions on the topic and to Prof. Xijun Hu for discussions. Kaicheng Sheng thanks the National Natural Science Foundation of China (NSFC) for the support of this research (Grant: 12371192 & 12271300).'} |
2024arXiv240904458G | Recent studies have demonstrated that a scalar field nonminimally coupled to the electromagnetic field can experience a spininduced tachyonic instability near KerrNewman black holes potentially driving the formation of scalar clouds. In this paper we construct such scalar clouds for both fundamental and excited modes detailing their existence domains and wave functions. Our results indicate that a sufficiently strong coupling between the scalar and electromagnetic fields is essential for sustaining scalar clouds. Within the strong coupling regime black holes that rotate either too slowly or too rapidly are unable to support scalar clouds. Furthermore we observe that scalar cloud wave functions are concentrated near the black holes poles. These findings provide a foundation for future investigations of spininduced scalarized KerrNewman black holes. | 2024-08-01T00:00:00Z | ['2024arXiv240904458G', '10.48550/arXiv.2409.04458', 'arXiv:2409.04458'] | ['General Relativity and Quantum Cosmology'] | Spininduced Scalar Clouds around KerrNewman Black Holes | 2,024 | 168 | 0.16 | ['EPRINT_HTML', 'EPRINT_PDF'] | 0 | https://arxiv.org/pdf/2409.04458.pdf | {'Spin-induced Scalar Clouds around Kerr-Newman Black Holes': "Guangzhou Guo b,a , ∗ Peng Wang a , † Tianshu Wu a , ‡ and Haitang Yang a § \na \nCenter for Theoretical Physics, College of Physics, \nSichuan University, Chengdu, 610064, China and \nb Department of Physics, Southern University of Science and Technology, Shenzhen, 518055, China \nRecent studies have demonstrated that a scalar field non-minimally coupled to the electromagnetic field can experience a spin-induced tachyonic instability near Kerr-Newman black holes, potentially driving the formation of scalar clouds. In this paper, we construct such scalar clouds for both fundamental and excited modes, detailing their existence domains and wave functions. Our results indicate that a sufficiently strong coupling between the scalar and electromagnetic fields is essential for sustaining scalar clouds. Within the strong coupling regime, black holes that rotate either too slowly or too rapidly are unable to support scalar clouds. Furthermore, we observe that scalar cloud wave functions are concentrated near the black hole's poles. These findings provide a foundation for future investigations of spin-induced scalarized Kerr-Newman black holes.", 'I. INTRODUCTION': "The no-hair theorem, which states that stationary black holes are uniquely characterized by their mass, angular momentum and charge [1-5], is a cornerstone of general relativity in the electrovacuum context. Testing this theorem is crucial for advancing our understanding of black hole physics and for constraining the validity of alternative gravitational theories. For example, the black-hole spectroscopy program, which analyzes quasinormal modes extracted from gravitationalwave observations, has emerged as a valuable tool for probing the Kerr nature of astrophysical compact objects [6-8]. \nSince the discovery of the first hairy black hole solution within Einstein-Yang-Mills theory [9-14], numerous counterexamples to the no-hair theorem have appeared [15-17]. In particular, black holes with scalar hair have garnered significant attention due to the potential of scalar fields to model dark energy and dark matter beyond the standard model [18]. A prime example is the presence of ultralight scalar fields outside rotating black holes, which can undergo a superradiant instability [19], leading to the formation of scalar clouds [20-24]. The signatures of these scalar clouds have been used to impose stringent constraints on the scalar field's parameter space, offering valuable insights into dark matter exploration and beyond-the-Standard-Model physics [25-30]. Moreover, \nunder specific conditions, non-linear effects can evolve scalar clouds into stationary hairy black holes [31, 32]. \nAlternatively, non-minimal couplings between a scalar field and curvature invariants have been shown to induce a tachyonic instability in the scalar field [33, 34]. This instability can lead to spontaneous scalarization, endowing black holes with scalar hair only above a certain threshold of spacetime curvature [35-39]. Consequently, spontaneous scalarization allows scalarized black holes to acquire a non-trivial scalar configuration exclusively in regimes of strong gravity, enabling them to evade constraints derived from weak-field gravity tests. Moreover, it has been demonstrated that, within a specific parameter region, the scalar field can exhibit a tachyonic instability near Kerr black holes when the black hole's spin exceeds a certain threshold [40]. Subsequently, this spin-induced tachyonic instability has been shown to be capable of generating hairy (scalarized) black holes at sufficiently high spins [41, 42]. In addition, the dynamics of spontaneous scalarization around Kerr black holes has been examined within the contexts of the Gauss-Bonnet and ChernSimons gravities [43, 44]. For a comprehensive overview of spontaneous scalarization, we refer readers to the review presented in [45]. \nSimilarly, in specific Einstein-Maxwell-scalar (EMS) models featuring a non-minimal coupling between a scalar field and the Maxwell electromagnetic invariant, the coupling, with an appropriate sign, can induce a tachyonic instability in Reissner-Nordstrom (RN) black holes [46]. The evolutions of RN black holes into scalarized RN black holes have been studied, providing valuable insights into spontaneous scalarization. Moreover, for certain parameter regimes, scalarized RN black holes have been found to possess two photon spheres outside the event horizon [47]. This unique feature leads to distinct phenomenology, including black hole images with intricate structures [48-52] and echo signals [53, 54]. Additionally, investigations into superradiant instabilities and non-linear stability of these double photon sphere black holes have been conducted [55, 56]. For a comprehensive analysis of black holes with multiple photon spheres, we refer readers to [57]. \nInterestingly, the tachyonic instability persists even when RN black holes rotate, leading to the formation of scalarized Kerr-Newman (KN) black holes [58]. The existence of these black holes is bounded by bifurcation points, corresponding to scalar clouds supported by KN black holes. However, the presence of scalarized KN black holes is suppressed by the black hole's spin, with a maximum spin threshold beyond which such solutions cease to exist. An analysis of scalar clouds induced by the tachyonic instability around KN black holes has also been conducted, revealing that black holes with sufficiently large spin cannot support scalar clouds [59]. Conversely, if the sign of the coupling constant is reversed, a spin-induced tachyonic instability emerges in KN black holes \nwhen they rotate sufficiently fast [60, 61]. This tachyonic instability can trigger the formation of scalar clouds around KN black holes, marking the onset of spin-induced scalarized KN black holes. The existence domain of spin-induced scalar clouds has been investigated only for a limited range of black hole parameters, particularly in the strong coupling limit. Therefore, a more comprehensive exploration of spin-induced scalar clouds is necessary for a deeper understanding of spontaneous scalarization in KN black holes, providing a foundation for constructing spin-induced scalarized KN black holes. \nThis paper presents a comprehensive investigation of scalar clouds generated by the spin-induced tachyonic instability within the EMS model. The paper is organized as follows. In Sec. II, we introduce the EMS model and its associated scalar clouds, followed by an overview of the computational framework employed to obtain scalar clouds using the spectral method. Sec. IV presents and analyzes our numerical findings. Finally, Section IV summarizes key results and discusses their implications. Throughout this paper, we adopt units where G = c = 4 πϵ 0 = 1.", 'II. SETUP': 'This section commences with a review of the EMS model and the conditions under which it exhibits a spin-induced tachyonic instability. Subsequently, we investigate the scalar clouds generated by this tachyonic instability and present the numerical method for determining their existence domains and wave functions.', 'A. Tachyonic Instability': "A tachyonic instability emerges within the EMS model, where a scalar field Φ is non-minimally coupled to the electromagnetic field A µ through a coupling function f (Φ). Explicitly, the action is given by \nS = 1 16 π ∫ d 4 x √ -g [ R -2 ∂ µ Φ ∂ µ Φ -f (Φ) F µν F µν ] , (1) \nwhere F µν = ∂ µ A ν -∂ ν A µ represents the electromagnetic field strength tensor. Varying the action yields the equation of motion for Φ, \n□ Φ = f ' (Φ) F µν F µν / 4 . (2) \nRemarkably, the inclusion of the scalar-electromagnetic non-minimal coupling term induces the tachyonic instability in the scalar field, leading to spontaneous scalarization in black holes [46, 58]. \nFor spontaneous scalarization to occur, a scalar-free solution with Φ = 0 must exist, from which scalar hair can develop. This condition imposes f ' (0) ≡ df (Φ) /d Φ | Φ=0 = 0, resulting in the series expansion of f (Φ) around Φ = 0, \nf (Φ) = 1 + α Φ 2 + O ( Φ 3 ) , (3) \nwhere α is a dimensionless coupling constant quantifying the strength of the scalar-electromagnetic interaction. Without loss of generality, we set f (0) = 1. \nWithin the EMS model, the rotating scalar-free black hole solution is a KN black hole. Expressed in Boyer-Lindquist coordinates, its metric and vector potential are given by \nds 2 = -△ Σ ( dt -a sin 2 θdφ ) 2 + sin 2 θ Σ [( r 2 + a 2 ) dφ -adt ] 2 + Σ △ dr 2 +Σ dθ 2 , A = Qr dt -a sin 2 θdφ Σ , (4) \nwhere \nΣ = r 2 + a 2 cos 2 θ, △ = r 2 -2 Mr + a 2 + Q 2 . (5) \nHere, Q is the black hole charge, and a represents the ratio of black hole angular momentum J to mass M (i.e., a ≡ J/M ). The event and Cauchy horizons are located at the roots of △ , given by r + = M + √ M 2 -a 2 -Q 2 and r -= M -√ M 2 -a 2 -Q 2 , respectively. For convenience, we introduce the dimensionless reduced black hole charge and spin, defined as \nq ≡ Q/M , χ ≡ a/M . (6) \nTo investigate the stability of the scalar field in the scalar-free black hole background, we adopt the probe limit, neglecting the scalar field's backreaction. Within this approximation, the scalar field obeys the equation of motion \n( □ -µ 2 eff ) Φ = 0 , (7) \nwhere µ 2 eff = αF µν F µν / 2 is the effective mass squared. Self-interactions of the scalar field, which have a minimal impact on the onset of spontaneous scalarization [46, 62, 63], are disregarded. For KN black holes, the effective mass squared becomes \nµ 2 eff = -αq 2 ( ˜ r 4 -6 χ 2 ˜ r 2 cos 2 θ + χ 4 cos 4 θ ) (˜ r 2 + χ 2 cos 2 θ ) 4 M 2 , (8) \nwhere ˜ r ≡ r/M . Given the spatial dependence of µ 2 eff , the tachyonic instability is indicated by \nmin µ 2 eff < 0 . (9) \nMoreover, when the tachyonic instability arises, the minimum value of µ 2 eff becomes increasingly negative for a fixed χ as the magnitude of α or q increases, signifying an amplification of the tachyonic instability with larger | α | or q values. \nThe occurrence of the tachyonic instability has been explored for both α > 0 [58] and α < 0 [60, 61]. In the case of α > 0, the region where µ 2 eff < 0 has been shown to exist outside the event horizon of KN black holes. The spatial extent of this region diminishes with increasing black hole spin, suggesting a potential suppression of the tachyonic instability for rapidly rotating black holes. For α < 0, both spin χ and charge q must be non-zero for the tachyonic instability to arise. Specifically, the parameter space admitting the tachyonic instability is constrained by \nχ ≥ 1 + √ 1 -2 ( 2 -√ 2 ) q 2 2 √ 2 with 0 < q ≤ q cr ≡ √ 2 √ 2 -2. (10) \nThe global minimum value of χ imposes a lower bound, \nχ ≥ χ cr ≡ √ 2 -1 . (11) \nIt is noteworthy that χ 2 cr + q 2 cr = 1, indicating that the KN black hole with q = q cr and χ = χ cr is extremal.", 'B. Scalar Clouds': 'The regular bound-state solutions of Eq. (7) are interpreted as scalar clouds surrounding KN black holes. The tachyonic instability can serve as a driving mechanism for the formation of scalar clouds. Indeed, scalar clouds around KN black holes induced by the tachyonic instability have been recently investigated for the α > 0 case [59]. This paper focuses on scalar clouds in the α < 0 regime. As the formation of such scalar clouds necessitates a tachyonic instability, the bounds on χ and q imposed by Eqs. (10) and (11) constrain their existence domain in the parameter space. \nLeveraging the axial symmetry of KN black holes, we decompose the scalar field Φ into a Fourier series in terms of frequency ω and azimuthal number m , \nΦ( t, r, θ, φ ) = ∫ dω 2 π e -iωt ∑ m e imφ ˜ Φ( ω, r, θ, m ) . (12) \nFor specified ω and m , Eq. (7) reduces to a Partial Differential Equation (PDE) for ˜ Φ( ω, r, θ, m ) with respect to r and θ . As scalar clouds typically serve as seeds for constructing axisymmetric hairy black hole solutions, we focus on stationary, axisymmetric configurations by setting ω = m = 0. For brevity, we denote ˜ Φ(0 , r, θ, 0) by ϕ ( r, θ ) in subsequent discussions. Analogous to hydrogen atoms, wave functions ϕ ( r, θ ) can be characterized by a discrete set of numbers ( n, l ), where the principal quantum number n = 0 , 1 , 2 · · · and the angular momentum quantum number l = 0 , 1 , 2 · · · correspond to the number of nodes of wave functions in the radial and angular directions, respectively. \nTo determine ϕ ( r, θ ), appropriate boundary conditions must be imposed at the event horizon and spatial infinity. Given the regularity of ϕ ( r, θ ) across the event horizon, it can be expanded in a series about r = r + , \nϕ ( r, θ ) = ϕ 0 ( θ ) + ( r -r + ) ϕ 1 ( θ ) + · · · . (13) \nMoreover, the condition of asymptotic flatness necessitates ϕ ( r, θ ) vanishing as r approaches infinity, \nlim r →∞ ϕ ( r, θ ) = 0. (14) \nAdditionally, axial symmetry, coupled with regularity on the symmetry axis, enforces, \n∂ θ ϕ ( r, θ ) = 0, at θ = 0 and π . (15) \nThese boundary conditions uniquely select a discrete set of KN black holes capable of supporting scalar clouds, thereby defining existence surfaces in the ( α, χ, q ) parameter space and existence lines within these surfaces for fixed α in the ( χ, q ) plane. \nAs demonstrated in the α > 0 case, the existence lines of scalar clouds in the ( χ, q ) parameter space delineate boundaries between regions exhibiting excessively strong tachyonic instability for stationary scalar clouds and those with insufficient instability for their formation [58, 59]. For KN black holes constrained by Eqs. (10) and (11), their minimum value of µ 2 eff approaches -∞ as α → -∞ . This observation suggests that, in the limit of α → -∞ , parameter regions defined by (10) and (11) may exhibit excessively strong tachyonic instability for stationary scalar clouds. Conversely, µ 2 eff of black holes outside these constrained regions is always positive, indicating the absence of the tachyonic instability. Consequently, the existence lines of scalar clouds coincide with the boundaries of the constrained parameter regions. Specifically, as α →-∞ , the existence lines \nin the ( χ, q ) parameter space converge to a critical existence line, given by \nχ = 1 + √ 1 -2 ( 2 -√ 2 ) q 2 2 √ 2 for 0 < q ≤ q cr . (16) \nMoreover, as χ increases, the critical existence line extends from ( χ, q ) = ( χ cr , q cr ) to ( χ, q ) = ( 1 / √ 2 , 0 ) .', 'C. Numerical Scheme': 'The wave equation governing ϕ ( r, θ ) in KN black holes is separable, enabling its reduction to ordinary differential equations. However, this study employs a spectral method to directly solve the wave equation for ϕ ( r, θ ), circumventing the need for separability. Consequently, this approach offers a significant advantage for computing scalar clouds around black holes in frameworks beyond general relativity. Spectral methods, a well-established method for solving PDEs [64], approximate the exact solution through a finite linear combination of basis functions. Notably, they exhibit exponential convergence for well-behaved functions, surpassing the linear or polynomial convergence rates achieved by finite difference and finite element methods. Recent investigations have successfully applied spectral methods to the identification of scalar cloud configurations [59], the construction of black hole solutions [65-67] and the calculation of black hole quasinormal modes [68-72]. A comprehensive overview of spectral methods in this context can be found in [65]. \nTo facilitate numerical implementation, we introduce a compact radial coordinate defined as \nx = √ r 2 -r 2 + -r + √ r 2 -r 2 + + r + , (17) \nwhich maps the event horizon and spatial infinity to x = -1 and x = 1, respectively. Under this transformation, the boundary conditions at the event horizon and spatial infinity become \n∂ x ϕ ( x, θ ) = 0 and ϕ (1 , θ ) = 0 , (18) \nrespectively. Without loss of generality, we assume that wave functions ϕ ( x, θ ) possess definite parity with respect to the equatorial plane, thereby permitting the restriction of the analysis to the upper half-domain 0 ≤ θ ≤ π/ 2. For even and odd parities, the boundary condition at θ = π/ 2 is ∂ θ ϕ ( x, θ ) = 0 and ϕ ( x, θ ) = 0, respectively. At θ = 0, we have ∂ θ ϕ ( x, θ ) = 0. \nTo apply the spectral method, the function ϕ ( x, θ ) is decomposed into a spectral expansion as \nϕ ( x, θ ) = N x -1 ∑ i =0 N θ -1 ∑ j =0 α ij T i ( x ) Θ j ( θ ) , (19) \nwhere N x and N θ denote the resolutions in the radial and angular coordinates, respectively, T i ( x ) represents the Chebyshev polynomial, and α ij are the spectral coefficients. The angular basis Θ j ( θ ) is dependent on the parity with respect to θ = π/ 2. Specifically, we adopt \nΘ j ( θ ) = cos (2 jθ ) for even parity cos [(2 j +1) θ ] for odd parity . (20) \nThis choice ensures that ϕ ( x, θ ) automatically satisfies the boundary conditions at θ = 0 and π/ 2. \nTo determine the spectral coefficients α ij , the spectral expansion (19) is substituted into the PDE, followed by discretization at the Gauss-Chebyshev points. This procedure transforms the PDE for ϕ ( x, θ ) into a system of algebraic equations involving α ij . However, Eq. (7) exhibits linear scaling invariance, necessitating an additional constraint to guarantee a non-trivial solution for α ij . This is achieved by setting ϕ ( x, θ ) = 1 at ( x, θ ) = ( -1 , 0). This constraint introduces an extra algebraic equation for α ij through the spectral expansion (19). To balance the number of unknowns and equations, one black hole parameter (e.g., the reduced black hole charge q ) is treated as an additional unknown. The resulting system of algebraic equations for α ij and q is then solved iteratively using the Newton-Raphson method. At each iteration, the linear system of equations is solved using the built-in LinearSolve command in Mathematica. The NewtonRaphson algorithm iterates until successive iterations converge to within a tolerance of 10 -10 . Moreover, while exploring scalar cloud solutions within the ( α, χ, q ) parameter space, the residual of the spectral approximation and the number of nodes are monitored to ensure solution accuracy, maintaining a residual tolerance of 10 -7 . \nIn the Appendix, we perform a convergence test of scalar cloud solutions by plotting the residual error as a function of N x and N θ . The results demonstrate that the error decays exponentially until reaching a round-off plateau below 10 -7 . To balance numerical precision and efficiency, we employ ( N x , N θ ) = (28 , 5) for subsequent numerical computations of ϕ ( x, θ ).', 'III. RESULTS': "In this section, we present numerical results concerning the parameter space of KN black holes that can support scalar clouds for the fundamental and first two excited modes. We also provide representative examples of the corresponding scalar cloud wave functions. \nWe begin by analyzing the fundamental mode of scalar clouds, characterized by nodeless wave functions with ( n, l ) = (0 , 0). The left panel of Fig. 1 displays the existence domain for fundamental clouds within the ( α, χ, q ) parameter space. KN black holes supporting fundamental clouds reside \nFIG. 1. Left Panel: Existence surface of fundamental scalar clouds with ( n, l ) = (0 , 0) in the ( α, χ, q ) parameter space. KN black holes residing on the colored surface admit the scalar clouds. The existence lines for α = -20, -50, -10 2 , -10 3 , -10 4 , -10 5 and -10 6 are shown from right to left. As α increases, this existence surface gradually converges to the critical point B with α = α cr ≃ -13 . 398. Scalar clouds cease to exist when α > α cr . Right Panel: Existence lines for α = -20, -50, -10 2 , -10 3 , -10 4 , -10 5 and -10 6 in the ( χ, q ) space, displayed from top right to bottom left. Both endpoints of the existence lines lie on the extremal KN black hole line (black dashed line), beyond which KN black holes cannot exist (gray region). As α decreases from α cr , the existence line emerges from the critical point B and gradually stretches out. In the limit of α →-∞ , the left segment of the existence line approaches the critical existence line (black dot-dashed line) while the right segment approaches the χ -axis with q = 0. The vertical and horizontal black dotted lines represent χ = χ cr and q = q cr , respectively, with their intersection marking the critical point C . As α →-∞ , the left and right endpoints of existence lines move along the extremal line towards the critical point C and the point at ( χ, q ) = (1 , 0), respectively. \n<!-- image --> \non the colored surface, while existence lines for various fixed α values are also shown. Our findings reveal that as α increases, these existence lines contract and converge towards the critical point B at ( α, χ, q ) ≃ ( -13 . 398 , 0 . 77001 , 0 . 63803), indicated by a black dot. Consequently, there exists a critical value of α , α cr ≃ -13 . 398, beyond which the spin-induced tachyonic instability is insufficient to form scalar clouds. The right panel of Fig. 1 illustrates the same existence lines in the ( χ, q ) plane. The extremal KN black hole line, corresponding to the condition q 2 + χ 2 = 1, is represented by a black dashed line. KN black holes cannot exist in the gray region above this extremal line, imposing an upper limit on the black hole charge q for a given χ . The vertical and horizontal dotted lines correspond to χ = χ cr and q = q cr , respectively. The intersection of these two dotted lines determines the critical point C at ( χ, q ) = ( χ cr , q cr ), marked by a red dot, which lies on the extremal line. Additionally, the black dot-dashed line represents the critical existence line given by Eq. (16). \nFour key characteristics are observed regarding the existence lines: \n- · Termination on Extremal Line: Both endpoints of existence lines lie on the extremal line. We assume that the left and right endpoints of the existence line with a given α locate at ( χ, q ) = ( χ low ( α ) , q up ( α )) and ( χ, q ) = ( χ up ( α ) , q low ( α )), respectively. The existence line decreases as χ increases, implying q low ( α ) < q up ( α ) and χ low ( α ) < χ up ( α ). As χ approaches χ low ( α ) from the right or χ up ( α ) from the left, the charge of the existence line converges to the upper limit set by the extremal line. This implies that, when χ < χ low ( α ) or χ > χ up ( α ), the presence of scalar clouds would necessitate a black hole charge q exceeding its extremal limit. Consequently, scalar clouds cease to exist if χ < χ low ( α ) or χ > χ up ( α ), due to insufficient tachyonic instability.\n- · Shift Toward Smaller q with Decreasing α : For α = α cr , the existence line shrinks to the critical point B with χ low ( α cr ) = χ up ( α cr ) ≃ 0 . 77001 and q low ( α cr ) = q up ( α cr ) ≃ 0 . 63803. As α decreases from α cr , the existence lines shift towards smaller q values with increasing length. This is attributed to the enhancement of the tachyonic instability for more negative α , thereby permitting a lower q to support scalar cloud formation.\n- · Approach Critical Existence Line as α →-∞ : As α goes to -∞ , the segment of the existence line with χ low ( α ) ≤ χ ≤ 1 / √ 2 converges to the critical existence line, consistent with the preceding discussion. Additionally, our results demonstrate that the segment with 1 / √ 2 < χ ≤ χ up ( α ) approaches the q = 0 line as α →-∞ . This observation indicates that, when the coupling constant α is sufficiently strong, a tiny amount of charge can trigger the formation of scalar clouds if χ exceeds 1 / √ 2. Moreover, in this limit, the left endpoint of the existence line approaches the critical point C while the right endpoint approaches the point at ( χ, q ) = (1 , 0).\n- · Serve as Threshold Line : For a given α , the existence line can be considered a threshold line, below which KN black holes exhibit insufficient tachyonic instability to support scalar clouds due to their low q values. Conversely, KN black holes above the existence line possess excessively strong tachyonic instability to sustain stationary scalar clouds. Therefore, nonlinear effects are required to suppress this instability, potentially giving rise to stationary states, such as scalarized KN black holes. \nThe left and right panels of Fig. 2 present density plots of the fundamental cloud existence domain in the ( α, χ ) and ( α, q ) spaces, respectively. Both panels illustrate the absence of scalar \nFIG. 2. Left Panel: Existence domain of fundamental scalar clouds in the ( α, χ ) plane, represented by a density plot where colors indicate the values of q . The domain is confined by the upper and lower boundaries, defined by the right and left endpoints of the existence lines, respectively. These boundaries merge at the critical point B , indicating the upper limit α cr on the coupling constant required to support scalar clouds. The horizontal dashed line represents χ = χ cr , above which the existence domain is located. Right Panel: Existence domain in the ( α, q ) plane, shown as a density plot with colors corresponding to χ values. The domain is bounded by the upper and lower limits, formed by the left and right endpoints of the existence lines, respectively. The horizontal dashed line depicts q = q cr , below which the existence domain is located. \n<!-- image --> \ncloud solutions for α > α cr , where α cr is the α value of the critical point B . In the ( α, χ ) plane, the density plot colors represent the magnitude of q , with grey regions indicating the non-existence of scalar clouds. The existence domain is bounded by the upper limit line, χ up ( α ), and the lower limit line, χ lower ( α ). As α increases towards α cr , the region of existence for scalar clouds contracts, ultimately converging at the critical point B . In the limit of α →-∞ , the upper and lower limits approach 1 and χ cr , respectively. Similarly, the existence domain in the ( α, q ) plane is confined by the upper boundary, q up ( α ), and the lower boundary, q lower ( α ), which merge at the critical point B . In the limit of α →-∞ , the upper and lower boundaries approach q cr and 0, respectively. \nBeyond the fundamental mode, excited modes of scalar clouds can also form around KN black holes, potentially leading to excited states of scalarized black holes. Fig. 3 illustrates existence lines in the ( χ, q ) space for excited scalar clouds with ( n, l ) = (1 , 0) and (0 , 1), which exhibit similarities to the fundamental mode. Specifically, the existence lines of excited clouds lie between the extremal and critical existence lines, with both endpoints resting on the extremal line. As α approaches α cr from below, the existence lines contract and converge to the critical point B , indicating excited clouds cannot form for α > α cr . Moreover, as α becomes more negative, the existence lines shift closer to the critical existence line. To compare the existence lines of fundamental and excited \nFIG. 3. Existence lines in the ( χ, q ) space for excited scalar clouds with ( n, l ) = (1 , 0) ( Left Panel ) and ( n, l ) = (0 , 1) ( Right Panel ). From top right to bottom left, the coupling constant α takes the values of -10 2 , -10 3 , -10 4 and -10 5 , respectively. These existence lines closely resemble those of the fundamental scalar clouds. Representative q and χ values of the existence lines are listed in Tab. I. It is evident that the existence lines for ( n, l ) = (1 , 0) lie above those for ( n, l ) = (0 , 1), suggesting that excited scalar clouds with ( n, l ) = (1 , 0) require a stronger tachyonic instability. \n<!-- image --> \nTABLE I. Black hole charge q and spin χ of representative clouds on the existence lines of fundamental and excited modes for α = -10 2 , -10 3 , -10 4 and -10 5 . For a given α and χ , fundamental clouds with ( n, l ) = (0 , 0) require smaller q than excited clouds, indicating that they are more prone to formation through the tachyonic instability. \nmodes, we provide the q and χ values of representative clouds for various α in Tab. I. It is evident that, for a given α and χ , KN black holes require the smallest q to support fundamental clouds, indicating that a stronger tachyonic instability is needed to form excited clouds. Additionally, the existence line of the ( n, l ) = (0 , 1) excited mode lies just slightly above that of the fundamental mode, suggesting that n = 0 scalar clouds are more easily generated than those with n = 1. \nFinally, we present representative scalar cloud wave functions for ( n, l ) = (0 , 0), (1 , 0) and (0 , 1) in Fig. 4. Selecting three cloud solutions on each existence line with α = -10 3 , all wave \nFIG. 4. Wave function ϕ ( x, θ ) of representative scalar clouds for ( n, l ) = (0 , 0) ( Top Row ), ( n, l ) = (1 , 0) ( Middle Row ) and ( n, l ) = (0 , 1) ( Bottom Row ) with α = -10 3 . For all cases, we set ϕ ( x, θ ) = 1 at ( x, θ ) = ( -1 , 0). Scalar cloud wave functions are concentrated near the black hole's poles, while black hole rotation has a tendency to spread wave functions towards the equatorial plane. \n<!-- image --> \nfunctions exhibit concentrations near the event horizon and the poles. For a fixed r close to the event horizon, the wave functions gradually decrease along the θ direction, reaching a minimum at the equatorial plane. As the black hole's spin increases, the concentration of wave functions tends to spread towards the equatorial plane. It is noteworthy that rapidly rotating black holes with α > 0 display scalar cloud concentrations near the equatorial plane [59]. Beyond these commonalities, ( n, l ) = (1 , 0) scalar clouds feature a radial node, resulting in a valley along the \nθ direction within their wave functions. This valley approaches the event horizon as the spin increases. For ( n, l ) = (0 , 1) scalar clouds, their odd parity with respect to the equatorial plane causes their wave functions to vanish at θ = π/ 2.", 'IV. CONCLUSIONS': "In this paper, we have explored scalar clouds generated by the spin-induced tachyonic instability around KN black holes within the framework of the EMS model, focusing on both the fundamental and excited modes. By employing the spectral method, we have successfully identified the parameter space where such scalar clouds can exist. Our findings reveal that the existence of scalar clouds is contingent upon the interplay between the black hole's charge, spin and the coupling constant α . \nSpecifically, we have determined that for a given α , there exists a distinct existence line in the ( χ, q ) parameter space along which scalar clouds can form. Notably, this existence line intersects the extremal line at both endpoints, implying that the tachyonic instability is insufficient to induce scalar cloud formation for black holes that rotate either too slowly or too rapidly. Additionally, our analysis reveals that the region of the parameter space where scalar clouds exist shrinks as α approaches a critical value, α cr ≃ -13 . 398. This observation suggests that scalar clouds cannot form for α values greater than α cr , a conclusion further supported by the existence domains presented in the ( α, χ ) and ( α, q ) planes. \nPrevious studies [60, 61] have established constraints on the existence domain of scalar clouds, as expressed in Eqs. (11) and (10). These constraints were suggested to be saturated in the strong coupling limit ( α →-∞ ). Our numerical results corroborate these findings. Furthermore, we also showed that for χ > 1 / √ 2, a portion of the existence lines converge towards q = 0 in the strong coupling limit, suggesting that the formation of scalar clouds requires only a minimal amount of charge. \nOur investigation studies the influence of the scalar field mode on scalar cloud formation. While the fundamental mode requires the least charge for formation, excited modes necessitate a stronger tachyonic instability. Additionally, we have observed that scalar cloud wave functions are concentrated near the black hole's poles, differing from the concentration near the equatorial plane in the α > 0 case. As the black hole's spin increases, the concentration of scalar clouds near the poles becomes less pronounced. This wave function behavior is consistent across different modes, although excited scalar clouds exhibit additional features such as radial nodes or odd parity with \nFIG. 5. Logarithmic plot of the residual error as a function of N x ( Top Row ) and N θ ( Bottom Row ) for scalar clouds with ( n, l ) = (0 , 0), (1 , 0) and (0 , 1). In the top row, N θ = 5 is fixed, while in the bottom row, N x = 28 is held constant. All scalar cloud solutions share the same α and a/r 2 + , namely α = -10 3 and a/r 2 + = 0 . 8. Exponential convergence is evident, with a round-off plateau observed. \n<!-- image --> \nrespect to the equatorial plane. \nSince scalar clouds mark the onset of scalarization from scalar-free black holes, the findings presented in this study provide a foundation for future research on non-linear realizations of scalar clouds, namely spin-induced scalarized KN black holes. These explorations may contribute to a deeper understanding of spontaneous scalarization. Additionally, future research could delve into the non-linear dynamics of scalar clouds and their potential implications for black hole stability and related astrophysical phenomena.", 'ACKNOWLEDGMENTS': 'We are grateful to Yiqian Chen for useful discussions and valuable comments. This work is supported in part by NSFC (Grant Nos. 12105191, 12275183, 12275184, 11875196, 12347133 and 12250410250).', 'APPENDIX: CONVERGENCE TEST': "In this appendix, we assess the convergence of our numerical code by calculating fundamental and excited scalar cloud solutions with α = -10 3 and a/r 2 + = 0 . 8 at various resolutions. The top \nrow of Fig. 5 depicts the maximum absolute value of the residual error as a function of the radial resolution N x with N θ = 5. All scalar cloud solutions demonstrate exponential convergence, with a round-off plateau approximately at N x ≥ 30. The bottom row presents the maximum absolute value of the residual error as a function of the angular resolution N θ with N x = 28. While some outliers occur at low θ resolutions, exponential convergence is observed overall, with a convergence plateau reached for N θ ≥ 5. To maintain a residual tolerance of 10 -7 , we adopt ( N x , N θ ) = (28 , 5) in our numerical calculations. \n- [1] Werner Israel. 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2024PhRvD.110f3538I | Perturbative or effective field theory EFTbased fullshape analyses of galaxy clustering data involve nuisance parameters to capture various observational effects such as the galaxydark matter connection galaxy bias. We present an efficient approach to set informative physically motivated priors on these parameters. We extract these priors from simulated galaxy catalogs based on halo occupation distribution HOD models. First we build a joint distribution between EFT galaxy bias and HOD parameters from a set of 10500 HOD mock catalogs. We use the field level EFT technique that allows for cosmic variance cancellation enabling a precision calibration of EFT parameters from computationally inexpensive smallvolume simulations. Second we use neural density estimatorsnormalizing flowsto model the marginal probability density of the EFT parameters which can be used as a prior distribution in full shape analyses. As a first application we use our HODbased priors in a new analysis of galaxy power spectra and bispectra from the BOSS survey in the context of single field primordial nonGaussianity. We find that our priors lead to a reduction of the posterior volume of bias parameters by an order of magnitude. We also find inlineformulammlmath displayinlinemmlmsubsupmmlmifmmlmimmlmrowmmlmiNLmmlmimmlmrowmmlmtextequilmmlmtextmmlmsubsupmmlmommlmommlmn320mmlmnmmlmommlmommlmn300mmlmnmmlmathinlineformula and inlineformulammlmath displayinlinemmlmsubsupmmlmifmmlmimmlmrowmmlmiNLmmlmimmlmrowmmlmtextorthommlmtextmmlmsubsupmmlmommlmommlmn100mmlmnmmlmommlmommlmn130mmlmnmmlmathinlineformula at 68 CL in a combined twotemplate analysis representing a inlineformulammlmath displayinlinemmlmommlmommlmn40mmlmnmmlmommlmommlmathinlineformula improvement in constraints on single field primordial nonGaussianity equivalent to doubling the survey volume. | 2024-09-01T00:00:00Z | ['2024arXiv240213310I', 'arXiv:2402.13310', '2024PhRvD.110f3538I', '10.48550/arXiv.2402.13310', '10.1103/PhysRevD.110.063538'] | ['Cosmology', 'Astrophysics - Cosmology and Nongalactic Astrophysics', 'High Energy Physics - Phenomenology', 'High Energy Physics - Theory'] | Fullshape analysis with simulationbased priors Constraints on single field inflation from BOSS | 2,024 | 168 | 0.48 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML'] | 22 | https://arxiv.org/pdf/2402.13310.pdf | {'Full-shape analysis with simulation-based priors: constraints on single field inflation from BOSS': "Mikhail M. Ivanov, 1, ∗ Carolina Cuesta-Lazaro, 2, 3, 4, † Siddharth \nMishra-Sharma, 5, 1, 6, ‡ Andrej Obuljen, 7, § and Michael W. Toomey 1, ¶ \n1 Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA 2 The NSF AI Institute for Artificial Intelligence and Fundamental Interactions, Cambridge, MA 02139, USA 3 Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA \n4 Center for Astrophysics - Harvard & Smithsonian, \n60 Garden Street, MS-16, Cambridge, MA 02138, USA \n5 The NSF AI Institute for Artificial Intelligence and Fundamental Interactions \n6 Department of Physics, Harvard University, Cambridge, MA 02138, USA \n7 Department of Astrophysics, University of Zurich, \nWinterthurerstrasse 190, 8057 Zurich, Switzerland \nPerturbative, or effective field theory (EFT)-based, full-shape analyses of galaxy clustering data involve 'nuisance parameters' to capture various observational effects such as the galaxy-dark matter connection (galaxy bias). We present an efficient approach to set informative physically motivated priors on these parameters. We extract these priors from simulated galaxy catalogs based on halo occupation distribution (HOD) models. First, we build a joint distribution between EFT galaxy bias and HOD parameters from a set of 10,500 HOD mock catalogs. We use the field level EFT technique that allows for cosmic variance cancellation, enabling a precision calibration of EFT parameters from computationally inexpensive small-volume simulations. Second, we use neural density estimators - normalizing flows - to model the marginal probability density of the EFT parameters, which can be used as a prior distribution in full shape analyses. As a first application, we use our HOD-based priors in a new analysis of galaxy power spectra and bispectra from the BOSS survey in the context of single field primordial non-Gaussianity. We find that our priors lead to a reduction of the posterior volume of bias parameters by an order of magnitude. We also find f equil NL = 320 ± 300 and f ortho NL = 100 ± 130 (at 68% CL) in a combined two-template analysis, representing a ≈ 40% improvement in constraints on single field primordial non-Gaussianity, equivalent to doubling the survey volume.", '1. INTRODUCTION': "The last three decades of precision cosmological observations have yielded significant evidence for new physical phenomena in the form of dark matter, dark energy, and primordial accelerated expansion of the Universe, called cosmic inflation. Future progress in understanding of these phenomena will depend on our ability to extract cosmological information from upcoming large-scale structure surveys, such as DESI [1], Euclid [2], LSST [3], and Roman Space Telescope [4]. \nOn scales larger than ∼ 10 Mpc, large scale structure formation occurs in the mildly non-linear regime. In this regime the data can be described with perturbation theory (its consistent formulation is called effective field theory (EFT) of large-scale structure [5-7]), which provides a systematic framework for modeling galaxy clustering based only on symmetries and scale separation. Currently, this is the only approach to galaxy clustering that offers a sub-percent accuracy for galaxy separations larger than 10 Mpc [8, 9]. Another key advantages of EFT is its efficiency. Namely, EFT model templates can be computed in less than 1 second [10-12]. EFT thus offers an unmatched flexibility in analyzing cosmological models beyond the vanilla ΛCDM model (see e.g. [1214]). All these virtues made EFT a useful tool for the analysis of the galaxy clustering data on large scales, see e.g. [15-18] for a sample of analyses of BOSS data [19]. \nThe main disadvantages of EFT techniques are: (a) the breakdown on small scales (where the statistical power of data can be significant), and (b) the proliferation of free 'nuisance' parameters. These 'nuisance' parameters include, e.g. the classical perturbative bias parameters such as linear, quadratic, and tidal biases b 1 , b 2 , b G 2 etc [20]. These parameters encapsulate effects of smallscale galaxy formation physics and typically are determined from data in actual analyses. However, marginalization over EFT parameters within uninformative priors leads to a significant loss of information in current EFTbased full-shape analyses [21-23]. 1 \nThe loss of information due to free EFT parameters can be avoided by using informative priors. These priors can be extracted from phenomenological or empirical galaxy formation models. In the EFT context, they are UV-complete models available for matching calculations. \nPhenomenological approaches such as the local Lagrangian approximation, co-evolution of galaxies and dark matter, the peak-background split etc. (see [20, 25] and references therein), predict certain relationships between galaxy bias parameters, which have been often used in actual data analyses, see e.g. [26]. However, accurate determination of these parameters from numerical simulations have shown that commonly used phenomenological analytic relations are not very accurate in practice, see e.g. [25, 27-29]. This suggests that their use in data analysis can bias cosmological parameter recovery. An alternative is to match the EFT parameters from empirical galaxy formation models based on hydrodynamical simulations or the halo occupation distribution (HOD) approach [30-33]. \nIn what follows we will focus on the HOD framework. This approach is motivated by the fact that galaxies fundamentally reside inside dark matter halos. Based on that, the HOD framework naturally assumes that key galaxies' properties are derived from those of the halos. Emulators based on HOD models have been successfully applied to data, see e.g. [34-38], which proves that they can reproduce the observed galaxy clustering with sufficient accuracy even though their robustness on small \nscales remains to be determined. \nThere have been notable efforts to determine EFT parameters from hydrodynamical simulations and HOD catalogs [25, 29, 39-41]. In particular, Ref. [25] measured non-linear bias parameters of 3 BOSS-like HOD galaxy samples and 4 halo catalogs using a combination of the power spectrum and bispectrum. Ref. [29] extracted second order galaxy bias parameters from ≈ 30 different samples of galaxies from IllustrisTNG hydrodynamical simulations. The relatively small size of data points in both of these analyses does not allow one to robustly explore the EFT parameter distribution, which is expected to have a complicated correlation structure. In particular, one can estimate that building an accurate distribution for 14 dimensional parameter space of HOD and EFT parameters requires a total of 10 4 samples. 2 Previously, [41] measured Lagrangian bias parameters of a hybrid EFT model for ∼ 8000 galaxy and halo samples. Their results, however, cannot be directly applied to pipelines based on the traditional EFT models. \nIn this paper, we present a new approach for precise determination of priors for EFT-based full shape analyses. The key object of our study is a precise map between EFT parameters and HOD parameters for BOSSlike galaxy catalogs. We build this map from 10,500 mocks, which represents a significant improvement in size over previous EFT measurements. This map can be used in multiple ways. For instance, we consider joint and conditional distributions of EFT and HOD parameters, p ( θ EFT , θ HOD ) and p ( θ EFT | θ HOD ), respectively, which clearly display the response of galaxy bias parameters to variations of HOD parameters. These distributions give us new insights into the physical meaning of galaxy bias parameters and help us connect galaxy bias models on small and large scales. \nFinally, we propose to use the marginal density of the EFT parameters from our samples p ( θ EFT ) as a prior distribution in EFT-based full-shape analyses. Similar ideas have been previously presented in [42] in the context of the Halo Zel'dovich perturbaiton theory model. We demonstrate the power of this approach in a new analysis of the BOSS data in the context of inflation- \nry models with non-local single-field primordial nonGaussianity (PNG) captured by the equilateral and orthogonal templates. This type of non-Gaussianity probes inflaton self-interactions and the propagation speed [4350]. In the past, it has been shown that constraints on single-field PNG from large-scale structure depend significantly on the assumptions about non-linear bias parameters [22, 51-54] (see also [52, 55-60] for recent related work and discussions in the context of multi-field inflation). We find that with our HOD-based priors the constraints improve by ≈ 40%, analogous to a twofold increase of the survey volume. The volume of the posterior distribution of non-linear galaxy bias parameters for each independent chunk of BOSS data shrinks by an order of magnitude. \nOur paper is structured as follows. We describe the technical aspects of how to produce a map of EFT and HOD parameters in Sec. 2. Sec. 3 presents the map and selected conditional distributions. Our PNG constraints on BOSS with HOD-informed priors are presented in Sec. 4. Sec. 5 compares the optimal values of the HOD parameters implied by our EFT full shape analysis to those based on other techniques involving small scale data. Finally, Sec. 6 draws conclusions and discusses directions of further work. Additional plots are presented in Appendix A. Appendix B summarizes details of our normalizing flow training, while Appendix C studies the residual UV-dependence of our results.", '2. THE METHOD': "The main idea of our method is to create a detailed mapping (joint probability density) between EFT and HOD parameters. From the physical point of view, it will be interesting to consider a conditional version of this mapping, p ( θ EFT | θ HOD ), which will describe the dependence of the EFT parameters on the underlying halo model physics. In the context of EFT-based full shape analyses, one is interested in the marginal distribution of the EFT parameters 'informed' by the HOD models, \np ( θ EFT ) = ∫ p ( θ EFT | θ HOD ) p ( θ HOD ) dθ HOD , (1) \nwhere p ( θ HOD ) is the prior distribution of the HOD parameters. Creating p ( θ EFT ) is the main practical goal of our paper. \nTo build a joint EFT/HOD distribution, we measure EFT parameters from a large set of simulations with varying HOD parameters. This poses several challenges. The first one is cosmic variance. A traditional way to determine the EFT parameters is to extract them by fitting a combination of simulated correlation functions, typically the power spectrum and bispectrum, see e.g. [61]. In this approach, the fits have to be performed on large scales where EFT is applicable. But the large scales are also most affected by cosmic variance. As a consequence, a high precision measurement of the EFT parameters requires computationally expensive simulations with large volume. For example, a measurement of the quadratic bias parameters with absolute errorbars σ ≃ 0 . 02 [23, 28] requires a cumulative simulation volume of 566 h -3 Gpc 3 . We solve this problem by employing a field-level EFT [27, 62-75] that allows for cosmic variance cancellation, and thus enables precision calibration of EFT parameters from computationally cheap smallvolume simulations. The field-level EFT also provides a computationally inexpensive way to take into account information beyond the two-point function, which is important in order to break degeneracies between EFT parameters. \nIn what follows we describe in detail the creation of HOD catalogs and field level EFT fits.", '2.1. HOD mocks': 'We build a large set of HOD mocks for BOSS-like galaxies based on the AbacusSummit [76] suite of simulations. For the purposes of this work, we use a set of mocks with an underlying Planck 2018 baseline cosmological model [77]. In principle, the EFT parameters depend on cosmology, so a full analysis of the BOSS data with variations of cosmological parameters will require a distribution that samples cosmological parameters. This will be presented in future work. \nWe produce a suite of 10,500 mock catalogs of galaxies similar to luminous red galaxies of BOSS. Specifically, we generate galaxies whose linear bias is similar to that of BOSS galaxies (around ≈ 2), and number density is less or equal to that of the CMASS sample of BOSS, ¯ n ≈ 3 . 6 · 10 -4 h 3 Mpc -3 . Note that the HOD-to-EFT mapping will have to be calibrated anew for galaxy samples different from that of BOSS, e.g. for emission line \ngalaxies of DESI. \nWe use available N -body simulation boxes with periodic boundary conditions from the small boxes AbacusSummit covariance suite. This suite has a total of 1883 independent realizations of dark matter initial conditions. Each box has a site length 500 h -1 Mpc. Our mocks are produced from snapshots at z = 0 . 5, which matches the typical redshift of the CMASS-type galaxies of BOSS. The mocks are fitted with a resolution of 256 pixels and have been corrected for the cloud-in-cell (CIC) window. Thanks to the small volume of our simulations, we can generate galaxy catalogs and perform EFT fits in a very short time, ∼ 20 and ∼ 40 sec., respectively. \nEach N -body simulation is populated with HOD galaxies [32]. We sample HOD parameters { log M cut , log M 1 , log σ , α , κ , B cen , B sat } from uniform uninformative priors specified in [35]. These are: \nlog M cut ∈ [12 . 4 , 13 . 3] , log M 1 ∈ [13 . 2 , 14 . 4] , log σ ∈ [ -3 . 0 , 0 . 0] , α ∈ [0 . 7 , 1 . 5] , κ ∈ [0 . 0 , 1 . 5] , B cen ∈ [ -0 . 5 , 0 . 5] , B sat ∈ [ -1 . 0 , 1 . 0] . (2) \nIn the HOD model, average numbers of central and satellite galaxies are given by \n⟨ N c ⟩ ( M ) = 1 2 [ 1 + Erf ( log M -log M cut √ 2 σ )] , ⟨ N s ⟩ ( M ) = ⟨ N c ⟩ ( M ) ( M -κM cut M 1 ) α , (3) \nwhere Erf( x ) is the Gauss error function, M cut is the minimum mass of a halo that can host a galaxy, M 1 is the typical (i.e. most probable) halo mass that hosts one satellite galaxy, 3 κM cut is the minimum mass of a halo that can host a satellite, α is the slope of the satellite probability distribution function. \nWe focus on PNG analysis in this work. As mentioned in the Introduction, the only relevant parameters in these case are non-linear bias parameters b 2 , b G 2 which can be extracted from real space mocks. Hence, for our purpose, it is sufficient to consider real space clustering only, and hence we ignore the velocity bias parameters that are relevant in redshift space. \nIn addition, we use B cen , B sat to capture assembly bias for satellites and centrals [78-80]. Specifically, the galaxy \noccupation enhancement is captured by promoting M cut and M 1 to functions of δ R via \nlog M cut → log M cut + B cen ( δ R -0 . 5) , log M 1 → log M 1 + B sat ( δ R -0 . 5) , (4) \nwhere δ R is the (smoothed) dark matter overdensity around halo centers ranked within halo mass bin and normalized to range from 0 to 1, and we use a top-hat filter with radius R = 5 h -1 Mpc for smoothing, following [78]. Note that the main halo itself is excluded when computing δ R , i.e only neighbor halo contributions are taken into account. Positive (negative) values of B sat / cen mean that galaxies of the relevant type preferably form in more (less) dense environments.', '2.2. Field-level EFT': "The standard Eulerian EFT bias model relevant for the one-loop EFT-based full shape analyses is given by [15] \nδ g = b 1 δ + b 2 2 ( δ 2 -σ 2 )+ b G 2 G 2 + b Γ 3 Γ 3 -b ∇ 2 δ ∇ 2 δ + ε , (5) \nwhere and ε is the stochastic density component, δ is the nonlinear dark matter field, ∇ ≡ ∂ i ∂ i , the 'non-local' bias operators are given by [81] \nG 2 = ( ∂ i ∂ j ∇ 2 δ ) 2 -δ 2 , Γ 3 = 4 7 δ G 2 -4 7 ∂ i ∂ j δ ∇ 2 ∂ i ∂ j G 2 ∇ 2 , (6) \nand σ 2 ≡ ⟨ δ 2 ⟩ , enforcing non-renormalization of the background density. The direct use of the above model in field-level EFT, however, leads to a poor match to simulations because the fully non-linear density field δ above has a strong UV sensitivity [27, 65]. This sensitivity also makes it hard to compare field level results with those based on traditional correlation functions because the latter feature renormalized biased parameters as opposed to 'bare' ones in the field-level calculations. In order to compare field-level and correlation function EFT calculations in a more consistent manner, one would want to evaluate the Eulerian bias model (5) with perturbative matter fields, as is done in the Eulerian EFT loop expansion [7, 82]. The perturbative fields, however, miss large contributions from enhanced Zel'dovich displacements [82, 83], which again, leads to a failure of \nFIG. 1. A typical HOD mock galaxy distribution from our set (left), field-level EFT fit to it (center), and the residuals (right). The overdensity field has been smoothed with a R = 4 h -1 Mpc 3D Gaussian filter and the depth of each panel is ≈ 60 h -1 Mpc. \n<!-- image --> \nnaive field-level EFT models to predict the galaxy density field. A way around is to use shifted operators proposed in [65, 71, 84], which have a well controlled smallscale behavior and retain large displacements at the same time. A shifted operator ˜ O is obtained by shifting the Lagrangian operator O ( q ) by the Zel'dovich displacement ψ 1 : \n˜ O ( k ) = ∫ d 3 q O ( q ) e -i k · ( q + ψ 1 ( q )) , (7) \nwhere q are Lagrangian coordinates. Note that in this approach higher order Lagrangian displacements are treated perturbatively, which is appropriate given that they are suppressed just like other operators in perturbation theory [20]. The statistics of shifted operators can be shown to be equivalent to IR-resummed Eulerian EFT [65] (see [85-89] for details of IR resummation). \nIn the context of real-space clustering that we study here, the field level EFT forward model based on shifted operators takes the form \nδ g ∣ ∣ EFT = β 1 ˜ δ 1 + β 2 ( ˜ δ 2 1 ) ⊥ + β G 2 ˜ G ⊥ 2 + β 3 ( ˜ δ 3 1 ) ⊥ , (8) \nwhere ˜ δ 1 is the shifted linear density field δ 1 , β n are scaledependent transfer functions. Importantly, the forward model (8) is built from orthogonalized operators that satisfy \n⟨ ˜ O ⊥ m ˜ O ⊥ n ⟩ = 0 , for m = n, (9) \n̸ \nwhich allows us to remove UV-sensitive two-loop corrections to the transfer functions [27]. \nNote that our model contains an extra cubic operator δ 3 that has a first non-trivial contribution in the bispectrum at the one loop order [23, 90, 91]. We extract the k -dependent transfer functions from each mock and then fit them with the appropriate EFT templates on mildlynonlinear scales. The transfer functions are computed by taking expectation values of the simulated galaxy density field multipled by appropriate shifted operators. Specifically, the transfer function β i of a Fourier space bias operator ˜ O i ( k ) is computed as \nβ i ( k ) = ⟨ δ HOD g ( k ) ˜ O ⊥ i ∗ ( k ) ⟩ ⟨| ˜ O ⊥ i ( k ) | 2 ⟩ , (10) \nwhere δ HOD g is the galaxy density field from simulations. The use of expectation values allows us to connect the transfer functions with renormalized EFT parameters that appear, e.g. in separate universe simulations [92] or in correlation functions. In particular, at the formal one-loop order, we have the following expressions in the k → 0 limit 4 \nβ 1 = b 1 + b ∇ 2 δ k 2 + ( b Γ 3 + b 1 6 + 5 2 b G 2 ) ⟨ ˜ δ 1 ˜ Γ 3 ⟩ ⟨ ˜ δ 1 ˜ δ 1 ⟩ + b 2 2 ⟨ ˜ δ 1 ˜ δ 2 1 ⟩ ⟨ ˜ δ 1 ˜ δ 1 ⟩ + ( b G 2 + 2 b 1 7 ) ⟨ ˜ δ 1 ˜ G 2 ⟩ ⟨ ˜ δ 1 ˜ δ 1 ⟩ -b 1 ⟨ ˜ δ 1 ˜ S 3 ⟩ ⟨ ˜ δ 1 ˜ δ 1 ⟩ , β 2 = b 2 2 , β G 2 = b G 2 + 2 7 b 1 , β 3 = b 3 6 . (11) \nNote that the constant coefficients here are different from those of [65] because we use the Eulerian bias parameters from Eq. (6) (matched at the cubic order), while the parameters used in [65] are more closely related to the Lagrangian bias parameters. The cubic operator ˜ S 3 is the shifted version of \nS 3 = ψ 2 ( q ) · ∇ δ 1 ( q ) , (12) \nwhich is produced by the second order displacement ψ 2 . Note that the presence of this operator is enforced by the equivalence principle. \nIn practice, we use public Hi-Fi mocks 5 to produce the EFT forward model. A typical snapshot generated with the field-level EFT forward model and its residual with the original simulated galaxy field are shown in Fig. 1. \nWe extract the bias parameters from the k → 0 limit of the transfer functions using Eq. (11). In practice we use k max = 0 . 4 h Mpc -1 , for which the one-loop EFT models are reliable [93, 94]. We have checked that a more conservative choice of k max = 0 . 3 h Mpc -1 gives results consistent with those of k max = 0 . 4 h Mpc -1 , but with a somewhat larger scatter in their distribution. This is the reason we adopt k max = 0 . 4 h Mpc -1 as a baseline choice. \nTo account for the scatter in transfer functions on large scales, we adopt error weights for k -bins based on the number of Fourier modes. For b 2 , b G 2 , b 3 , we fit the transfer functions with a polynomial c 0 + c 2 k 2 + c 4 k 4 , and then match c 0 to the constant values of bias parameters in Eq. (11). As far as β 1 is concerned, we calculate the power spectra and cross-spectra of shifted operators in Eq. (11), and use them, along with the best-fit values of b 2 and b G 2 from the previous step, to fit the transfer function at low k , which yields b ∇ 2 δ and b Γ 3 . Plots with typical transfer functions fits can be found in Appendix A. \nThe final ingredient that we need is the distribution of stochasticity parameters, characterizing the power spectrum of the ε field. In practice, we calculate the error power spectrum as \nP err ( k ) = ⟨| δ HOD g ( k ) -δ EFT g ( k ) | 2 ⟩ . (13) \nTheoretical consistency dictates that on large scales it \nshould match the EFT prediction for the stochastic contribution to the galaxy power spectrum [9, 28, 65] \nP err ( k ) = 1 ¯ n ( 1 + α 0 + α 1 ( k k NL ) 2 ) , (14) \nwhere ¯ n = V/N gal is the number density of mock galaxies ( V being the simulation box volume), and we chose k NL = 0 . 45 h Mpc -1 following [28]. Note that the EFT fitting pipelines (e.g. [9, 28]) use parameters P shot and a 0 that are similar to our α 0 and α 1 , respectively. However, as we discuss in detail later, these 'standard' EFT models also absorb additional contributions into the stochastic parameters, which make them somewhat different from our α 0 and α 1 . \nBefore closing this part, let us comment on errors in our measurements of bias parameters. There are two main sources of errors: the residual UV dependence of the transfer functions on large scales and the numerical noise due to the use of a single realization per mock in our fits. The use of orthogonalized shifted operators and k -dependent transfer functions allowed us to remove the bulk of the UV sensitivity of our bias parameters. Within the original field-level method for cosmic variance cancellation, see e.g. [27], the bias parameters stem from a global fit of a constant to many k -bins, which is sensitive to small scale nonlinearity and higher-order loop effects. In order to reduce this sensitivity, one may introduce a relatively aggressive cutoff for the linear fields ([27] used a Gaussian filter with R s = 20 h -1 Mpc). This cutoff generates a mismatch between the field-level and n -point functions results, as the latter are formally extracted in the R s → 0 limit. 6 This is because many EFT codes, e.g. CLASS-PT [10] use dimensional regularization for loop integrals that compute correlation functions, in which all convergent integrals are done without any cutoff, while the divergent pieces are set to zero. In the field level language this means the matching should be done after the orthogonalization (which removes divergent corrections proportional to the mass variance integrals), and without smoothing of the linear fields (i.e. at R s → 0). \nIn this work, however, we follow the method [65] and extract the bias parameters from the lowk limits of the \nFIG. 2. The joint distribution of EFT and HOD parameters extracted from 10,500 HOD mocks for BOSS-like galaxies. Density levels correspond to two-dimensional 1σ and 2σ intervals (i.e. 39.3% and 86.5% of samples). Individual samples are also shown as dots. They are especially pronounced in the tails. \n<!-- image --> \nFIG. 3. Conditional distributions of EFT parameters for different values of log M cut (see legend). Other HOD parameters are kept fixed. Density levels correspond to two-dimensional 1σ and 2σ intervals. \n<!-- image --> \nG \n∇ \ntransfer functions, in which case the UV-sensitive higherorder corrections are absorbed into the k 2 and k 4 polynomials of our fit, leaving the constant part unaffected provided one uses a reasonably high cutoff. As an explicit check, we demonstrate that the lowk transfer functions used in our fits are stable w.r.t. variations of the grid resolution, which provides an effective cutoff for all fields, see Appendix C for more detail. This test shows that our results have converged w.r.t. the R s → 0 limit relevant \nfor the matching to the n -point functions in dimensional regularization. \nAs far as numerical noise is concerned, results of Ref. [65] obtained with a similar box size suggest that the corresponding error is below the level of the scatter induced by variations of HOD models, implying that \n15 \nFIG. 4. Same as Fig. 3, but B sat is varied (see legend) while other parameters are kept fixed. \n<!-- image --> \nthese effects are negligible for parameter constraints. 7", '2.3. Normalizing flow for density estimation and sampling': "The crux of our approach is the ability to effectively model the marginal and HOD-conditional EFT parameter distributions, p ( θ EFT ) and p ( θ EFT | θ HOD ) respectively, given a set of samples { θ EFT , θ HOD } ∼ p ( θ EFT , θ HOD ) as described above. Ideally, we would like to be able to generate new samples from these distributions (i.e., use them as priors), and evaluate the likeli- \nnew set of samples under the modeled distributions. \nGenerative models are a class of statistical models that aim to encode potentially complex target distributions. While high-dimensional distribution modeling is an inherently challenging task, machine learning has brought this into the realm of tractability. Normalizing flows [95], a class of deep generative models, are especially suited for our purposes, as they allow for seamless density estimation as well as sampling, including in the conditional regime. Briefly, normalizing flows model the target density ˆ p ( θ ) as a bijective (invertible) learnable function θ = f φ ( u ) from a simple distribution π ( u ), e.g. a multivariate Gaussian, \nˆ p ( θ ) = π ( u ) ∣ ∣ ∣ ∣ det ( ∂u ∂θ )∣ ∣ ∣ ∣ = π ( f -1 φ ( θ ) ) ∣ ∣ ∣ det J f -1 φ ( θ ) ∣ ∣ ∣ (15) \nwhere ∣ ∣ ∣ det J f -1 φ ( θ ) ∣ ∣ ∣ is the Jacobian determinant of the inverse transformation and is by construction easy to compute. φ are the parameters of the learnable transformation, usually modeled through an appropriate invertible neural network. \nGiven a set of samples { θ } ∼ p ( θ ) (e.g, HOD and EFT parameters), we can maximize Eq. (15) over those samples to train the flow, and thus build an approximation for our target density. The optimal parameters φ ∗ of the transformation are obtained, in practice through stochastic gradient descent or a variant thereof, as \nφ ∗ = arg max φ ⟨ log ˆ p ( θ ) ⟩ θ ∼ p ( θ ) . (16) \nThe target density can then easily be sampled from, by drawing u ∼ π ( u ) from the simple (Gaussian) base density and running the forward transformation f φ ( u ), as well as evaluated for a new sample θ ' . The flows are implemented using the nflows 8 library, with training and evaluation performed using PyTorch [96]. Further details of the normalizing flow implementation, training, and validation are given in Appendix B.", '3. DISTRIBUTIONS OF GALAXY BIAS AND HOD PARAMETERS': 'The joint samples of EFT parameters for given HOD models are displayed in Fig. 2. Numerical values of b ∇ 2 δ \nare given in [ h -1 Mpc] 2 units. We do not show the samples of the HOD parameters as they simply scan over their uniform priors. We see that the EFT parameters are strongly correlated among each other. This is the behavior expected on the basis that the (infinite) entire set of EFT parameters must be produced by only 7 HOD parameters. (In the particle physics context, the situation here is analogous to the textbook matching of chiral perturbation theory to the linear sigma model, where low energy constants are either zero, or obey correlations set by a few parameters of the linear sigma model.) The tightest correlation is between b Γ 3 and b G 2 , which follows the linear law \nb Γ 3 ≈ -3 . 8 b G 2 -0 . 5 . (17) \nNote that this law is steeper than the local Lagrangian model prediction b Γ 3 = -23 12 b G 2 , and the co-evolution model relation b Γ 3 ∼ -1 . 5 b G 2 [25]. The strong correlation between b G 2 and b Γ 3 may be an artifact of our forward model, which absorbs the contribution of the Γ 3 operator into β 1 , thus making our measurement of Γ 3 sensitive to details of the fitting procedure. It will be interesting to compare our results with independent measurements of b Γ 3 from a forward model with the shifted Γ 3 operator. \nAnother important observation is that the higherderivative stochastic counterterm α 1 is very close to zero for most of the samples. A similar pattern was discovered before for the clustering of dark matter halos [65]. It would be interesting to see if this pattern is specific to HOD models, i.e. if other galaxy formation prescriptions can produce larger α 1 matching the natural EFT expectation that this parameter should be proportional to the Lagrangian radius of the host halo. This expectation is supported by the positive correlation between α 1 and b 1 . Note that α 1 is mostly correlated with M cut , M 1 and the satellite environmental assembly bias B sat . These correlations can be understood as follows: non-locality in the galaxy stochasticity is sourced by the stochastic non-locality of the underlying halo (set by M cut ) plus the stochasticity from the satellites, whose spatial distribution increases the non-locality scale, thus generating additional M 1 and B sat dependencies. \nLet us discuss now the correlation between P shot (deviation of white noise amplitude from ¯ n -1 ) and b 1 . Due to halo exclusion effects [97-100], we expect that samples with large enough b 1 should feature sub-Poisson stochastic shot noise, i.e. have negative α 0 . This is in- \ned the case, on average. However, massive halos also tend to have more satellites, which would increase ¯ n , and hence compensate for the sub-Poisson trend produced by the halos. This is also is supported by correlations between α 0 and satellite properties such as B sat . Indeed, the observation that satellite galaxy assembly bias produces super-Poissonian sampling was pointed out before in [101]. This is the reason why we do not see a strong sub-Poisson constant shot noise contribution for galaxies that reside in massive halos. A similar trend was pointed out in a previous work on the field-level calibration of α 0 from HOD [40], with which we find excellent agreement. \nNote that other prominent correlations are b 1 -B cen and b ∇ 2 δ -B sat , which suggest that the measurements of these parameters can be used to diagnose the presence of environmental assembly bias. \nIn order to visualize the response of the bias parameters to variations of the HOD parameters, we study the conditional distribution p ( θ EFT | θ HOD ). It indicates what are the most likely EFT parameters for a given fixed set of HOD parameters. This distribution is extracted by modeling the original samples with normalizing flows as described in Sec. 2.3. \nThe fiducial conditional distribution of EFT parameters that we study is calculated for the set of HOD parameters fixed to the best-fit values of the CMASS BOSS sample [35], \nlog M cut = 12 . 66 , log M 1 = 13 . 66 , α = 1 . 34 , log σ = -0 . 5 , κ = 0 . 03 , B cen = -0 . 43 , B sat = -0 . 22 . (18) \nAs a first example, we plot p ( θ EFT | θ HOD ) for 7 HOD models with different log M cut and all other parameters fixed. The result is shown in Fig. 3. As anticipated from the simple thresholding picture [20], the higher cutoff mass of the halo implies a lower probability of halo formation and hence higher bias of the host halos and galaxies. Note that the higher-derivative bias b ∇ 2 δ is not very sensitive to the cutoff mass, which is at odds with the naive expectation that this parameter should be proportional to the Lagrangian radius of the host halo. It would be interesting to understand the origin of this behavior. This intuition is, however, confirmed by the higher derivative stochastic counterterm α 1 which grows with cutoff mass. \nHaving confirmed that our conditional distribution reflects the basic intuition about the galaxy bias, let us \nconsider now the response of EFT coefficients to a less intuitive parameter, the environment-based assembly (secondary) bias of satellites B sat . The corresponding conditional distributions for 7 samples of galaxies with different B sat (other HOD parameters are fixed) is shown in Fig. 4. The first relevant observation is that local-inmatter-density bias parameters b 1 , b 2 , b 3 display only a weak dependence on B sat . This is consistent with the expectation that these parameters are mostly determined by halo mass. The small residual dependence on B sat can be explained as the tendency to have more/less galaxies for positive/negative B sat , which leads to lower/higher values of local biases. The presence of satellite assembly bias is best reflected by non-local bias parameters b G 2 , b Γ 3 , b ∇ 2 δ and higher-derivative stochastic counterterm α 1 , which is consistent with the fact that assembly bias is an intrinsically non-local property of galaxies. Another interesting observation is that a strong satellite assembly bias leads to super-Poisson shot noise (positive P shot ) regardless of the sign of B sat . \nThe distributions of EFT parameters conditioned to other HOD models have a similar behavior that can be qualitatively understood on the basis of simple physical arguments such as thresholding.', '4. PNG CONSTRAINTS FROM BOSS WITH HOD-INFORMED PRIORS': "In this section, we re-analyze the BOSS data within initial conditions with non-local primordial nonGaussianity. In this analysis, following the standard practice [51, 52, 102], we fix the 'standard' cosmological parameters to the Planck best-fit values, and vary only the EFT and PNG parameters. Single-field PNG is captured by the equilateral and orthogonal templates identical to the ones used in [51]. Note that we use the orthogonal template that has the correct physical scaling in the squeezed limit. \nOur dataset is BOSS DR12 galaxy clustering in redshift space [103]. The BOSS DR12 data is split in four chunks: low-z ( z eff = 0 . 38) and high-z ( z eff = 0 . 61), South and North Galactic Caps. For each chunk, our datavector is { P 0 , P 2 , P 4 , Q 0 , B 0 , B 2 , B 4 , α ∥ , α ⊥ } , where P ℓ , B ℓ are window-free power spectrum and bispectrum multipoles, ℓ = 0 , 2 , 4 [14, 17, 24], Q 0 is the realspace power spectrum proxy [94], while α ∥ , α ⊥ are the \nFIG. 5. Corner plots with 2d and 1d marginalized posterior distribution of the PNG parameters from the full BOSS DR12 dataset and the galaxy bias parameters from BOSS NGC high-z sample. HOD priors on bias parameters are shown in gray. Bias parameters for other samples are shown in Appendix. Density levels correspond to 68% and 95% CL. \n<!-- image --> \nG \npost-reconstructed BAO parameters [104]. We use scale cuts k max P ℓ = 0 . 2 h Mpc -1 , k max B ℓ = 0 . 08 h Mpc -1 , k max Q 0 = 0 . 4 h Mpc -1 and k min = 0 . 01 h Mpc -1 . The covariance matrix is estimated from Multidark Patchy mocks [105, 106]. We assume that EFT parameters in four chunks are independent of each other. Other aspects of our analysis are identical to those of [22, 24]. \nWe first run the usual analysis with the conservative priors on galaxy bias parameters similar to [22]. For ref- \nerence, the priors on the galaxy bias parameter are \nb 1 ∈ [1 , 4] , b 2 ∼ N (0 , 1) , b G 2 ∼ N (0 , 1) , b Γ 3 ∼ N ( 23 42 ( b 1 -1) , 1 ) , (19) \nwhere N ( a, σ ) denotes a normal distribution with mean a and variance σ 2 . \nAs a second step, we replace the conservative priors from the previous analysis with the marginal likelihood p ( θ EFT ), see Eq. (1). We use p ( θ EFT ) to set priors for pa-", 'BOSS PNG with HOD-informed priors': 'TABLE I. Best-fits and 1d marginalized limits for PNG and galaxy bias parameters from BOSS with HOD-informed priors (left panel) and the usual conservative priors (right panel). Upper scripts (1 , 2 , 3 , 4) refer to NGCz3, SGCz3, NGCz1, and SGCz1 samples (z1= 0 . 38, z3= 0 . 61).', 'BOSS PNG with conservative priors': 'rameters θ EFT = { b 1 , b 2 , b G 2 , b Γ 3 } of the BOSS EFT likelihood. The other EFT parameters from our samples, i.e. { b 3 , b ∇ 2 δ , α 0 , α 1 } are not included in the distribution, i.e. are effectively marginalized over, because of the following reasons. \nParameters P shot and a 0 in the BOSS EFT likelihood are quite different from α 0 , α 1 that we have extracted from the HOD mocks. First, P shot in the EFT models currently implemented in CLASS-PT [10] absorbs the constant deterministic contribution from the one-loop autospectrum of δ 2 , see [107] for a recent discussion. This is the first reason why P shot is different from our α shot , which corresponds to a truly stochastic component. Second, our α shot captures departures from the Poissonian shot noise in a periodic box geometry, which is different from the number density of actual galaxies in a survey that is subject to additional weights, e.g. FKP (see e.g. a discussion in [9]). Third, stochastic counterterms \nin the BOSS EFT pipeline additionally absorb contributions from fiber collisions [9, 108, 109]. Given these reasons, we keep using uninformative priors on P shot and a 0 in our analysis. We also do not use a prior on b 3 as this parameter does not appear in our current analysis based on the tree-level bispectrum. Finally, we do not use the prior on b ∇ 2 δ in the current analysis because the contribution of the ∇ 2 δ operator to the galaxy power spectrum in redshift space is degenerate with other redshift space counterterms that we have not studied yet. Their precision measurement with the redshift-space field-level EFT of [71] is left for future work. Since the PNG parameters are primarily degenerate with the quadratic bias parameters b 2 and b G 2 , we do not expect our results to depend significantly on the RSD counterterms and parameters { P shot , a 0 } discussed above. For this reason, we adopt standard conservative priors for them in all analyses presented here. \nAdditionally, we have marginalized over the nonGaussian bias parameter b ζ within conservative priors of [51] in both analyses. In principle, one should extract priors on these parameters from mocks too. However, marginalization over b ζ does not have a noticeable impact on the single-field PNG constraints presented here. We leave a calibration of these parameters from simulations for future work. \nA comment is in order on the redshift-dependence of the EFT parameters. We have calibrated priors at z = 0 . 5, which is somewhat different from z eff ≈ 0 . 4 and 0 . 6 of BOSS chunks. Within the HOD models, the galaxy distribution is determined by local-in-time properties of halos. Therefore, a redshift dependence of bias parameters can be fully compensated by a change of HOD parameters such as M cut , which are varied in our samples. Physically, we expect some residual sensitivity to the past evolution, e.g. to merger and assembly histories, which is captured within EFT by the expansion along the past fluid trajectory [20, 110]. These effects appear only at the two-loop order in the EFT, and hence are irrelevant for our analysis based on the one-loop calculation. All in all, the redshift dependence of the EFT parameters is adequately captured by variations of HOD parameters at the current precision level. \nOur main results are displayed in figure 5 and in table I. A corner plot with EFT parameters for each chunk can be found in Appendix A. The first relevant observation is that the posteriors for non-linear galaxy bias parameters shrink significantly after applying the HOD priors. In particular, the 1d marginalized errorbars on b 2 , b G 2 and b Γ 3 shrink by a factor of few. To quantify the improvement in a more rigorous manner, we introduce a figure of merit (FoM) for bias parameters, defined along the lines of the FoM for the dark energy equation of state. Namely, for each individual BOSS data slice we write \nFoM bias = [det C ( b 1 , b 2 , b G 2 , b Γ 3 )] -1 / 2 , (20) \nwhere C is the covariance matrix of the bias parameters after marginalizing over other parameters in the chain. In the Gaussian case, our FoM is proportional to the inverse volume of the 4-dimensional ellipsoid enclosing the posterior distribution of the bias coefficients. A relevant parameter in our comparison is the FoM ratio between the old and new bias parameter measurements, e.g. for \nthe NGCz3 we have \nFoM bias | HODpriors FoM bias ∣ ∣ cons . priors ∣ ∣ ∣ ∣ ∣ NGCz3 = 60 . 3 , (21) \nwhich can be interpreted as a factor of ≈ 60 reduction in cumulative inverse variance of the posterior bias parameter distribution. The cumulative figure of merit of bias parameters from all four BOSS data chunks is 1 . 2 · 10 7 . \nThe second important observation is that our HODbased measurements of bias parameters are in perfect agreement with the EFT results based on the noninformative priors: the HOD-based posteriors are located almost at the centers of posterior densities from the standard EFT analysis. This is a non-trivial consistency check of both our prior calibration technique and the BOSS EFT pipeline since our measurements of nonlinear bias parameters are dominated by the galaxy bispectrum data 9 , while the HOD-informed priors were extracted from the field-level fits to HOD mocks. One can also note that the constraints on the linear bias parameter b 1 do not improve. This is because the data itself is much more informative than the prior for this parameter, cf. the standard BOSS posterior and the HOD prior in Fig. 5. Indeed, our HOD samples are consistent with a wide range of b 1 , while the actual measurements from BOSS are at the level of few percent. \nThe third important observation is that the 1d marginalized errorbar on f equil NL has narrowed by ≈ 40%, from f equil NL = 806 +490 -510 (conservative priors) to f equil NL = 0 . 323 +0 . 29 -0 . 31 (HOD-informed). The main channel of improvement here is the breaking of degeneracy between f equil NL and b G 2 in the galaxy bispectrum. Since f ortho NL was not significantly correlated with bias parameters to begin with, the improvement for this parameter is only at ≈ 10%, f equil NL = -8 . 2 +130 -150 (conservative priors) vs. f equil NL = 100 +130 -140 (HOD-informed). In line with the previous discussion, let us introduce the FoM for non-Gaussian amplitudes \nFoM PNG = [ det C ( f equil NL , f ortho NL ) ] -1 / 2 , (22) \nwhich calculates the inverse area (up to a factor of 1 /π ) of the error ellipse in the ( f equil NL , f ortho NL ) plane. Then we \nhave \nFoM PNG | HODpriors FoM PNG ∣ ∣ cons . priors = 1 . 76 , (23) \nor a ≈ 40% reduction of the total 2-dimensional posterior variance of PNG parameters. \nFinally, let us mention that the analysis with the conservative priors implies a marginal 2 σ preference for nonzero f equil NL , which cannot be physical as the central value f equil NL = 800 is in strong tension with the CMB measurements [102]. The preference for non-zero f equil NL reduces in our new analysis, which highlights the importance of physically-motivated priors for the full-shape analysis.', '5. COMPARISON WITH SMALL-SCALE MEASUREMENTS': 'An interesting application of the method presented here is the comparison between simulation-based models fitting clustering data on small scales, see e.g. [35-38], and perturbation theory based models constrained by large scales. In particular, we use the conditional model p ( θ EFT | θ HOD ) to convert the HOD posterior chains from [35] into EFT bias parameters. These posteriors were obtained by fitting the two-point correlation function and density split statistics of the CMASS galaxy sample in the scale range 1 < r < 150 h -1 Mpc. The model used for inference is a neural network trained to reproduce the clustering of HOD-based galaxy mocks. \nIn Figure 6, we compare the estimated bias parameters, denoted as HOD fits, with those found in this paper, denoted as EFT fits. Interestingly, although the range of scales used is widely different, both posteriors agree remarkably. Note that some important differences beyond scale ranges are the parameters that are being fitted, since [35] also fits the cosmological parameters, and the statistics used.', '6. DISCUSSION AND OUTLOOK': "We have presented a new framework for deriving informative priors on EFT 'nuisance' parameters from galaxy formation simulations. For concreteness, we focused here on the HOD models, although our approach can be straightforwardly applied to other galaxy formation models. The central object of our method is the \nconditional distribution between parameters of galaxy formation models and EFT parameters. We build this distribution by generating a large sample of EFT parameters extracted from HOD mocks with the field-level technique. In this work, we have used the resulting distribution of the perturbative galaxy bias parameters as a prior in the analysis of the BOSS galaxy clustering data in the context of primordial non-Gaussianity. We have found a significant improvement in the PNG constraints, as well as in the measurement of the galaxy bias parameters. Importantly, the values of bias parameters that we have extracted are fully consistent with previous analyses of BOSS based on conservative priors, as well as with the measurements based on the short-scale density-split statistic. This impressive consistency can be interpreted as a sign of convergence of various independent techniques to model the large-scale structure of the Universe. \nWe note that in this work we have focused on the real space analysis of galaxy clustering that allows us to study perturbative galaxy bias parameters. The informative priors on these parameters are sufficient for our main goal here: to improve limits on single-field primordial non-Gaussianity. A more general analysis, including full redshift-space nuisance parameters and HOD models, is currently in progress. \nA next step in our program will be to carry out a full re-analysis of the BOSS data including cosmological parameters such as σ 8 , Ω m etc. Another important research direction is to extend our calibration of EFT parameters to more general HOD models relevant, e.g. for eBOSS quasars [113-116] and emission line galaxies [9, 117], as well as to other galaxy formation models, such as abundance matching and hydrodynamical simulations. \nNote that the approach presented here can be used for a new type of emulator where small scale data are modeled with simulations, and then large scale clustering is reconstructed from the small simulation box using EFT. Similar ideas have been put forward before (see e.g. [84, 118]). A central tool in this emulator is the conditional distribution between EFT and simulation parameters, identical to the one that we derived here for HOD models. The work on this emulator is currently underway. \nFinally, it would also be interesting to extend our approach to imaging clustering data and intensity mapping, see [84] for recent work in this direction. \nFIG. 6. Comparison of the EFT biased parameters found by small scale simulation-based fits to density split and two-point statistics based on HOD models (HOD fits), and those found through EFT fits to the power spectrum and the bispectrum on large scales (EFT fits). \n<!-- image --> \nCode to reproduce and use the simulation-based priors p ( θ EFT ) and p ( θ EFT | θ HOD ) is available at https:// github.com/smsharma/eft-hod .", 'ACKNOWLEDGMENTS': "We thank Kazuyuki Akitsu, Stephen Chen, Marko Simonovi'c, Fabian Schmidt, and Marcel Schmittfull for useful discussions. The authors also thank the IAIFI Astro-ML Hackathon where the first draft of this work was completed. Monte Carlo Markov Chains with PNG parameters were generated with the Montepython code [119, 120]. This work is supported by the National Science Foundation under Cooperative Agreement PHY-2019786 (The NSF AI Institute for Artificial Intelligence and Fundamental Interactions, http://iaifi. \norg/ ). This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of High Energy Physics of U.S. Department of Energy under grant Contract Number DE-SC0012567. AO acknowledges financial support from the Swiss National Science Foundation (grant no CRSII5 193826). MWT acknowledges financial support from the Simons Foundation (Grant Number 929255).", 'Appendix A: Additional plots': 'Fitting the EFT parameters from the field-level model is visualized in Fig. 7 where we show the data transfer functions and their EFT template fits for a typical mock from our sample. The left panel of Fig. 8 shows the extraction of the stochastic α 0 and α 1 parameters by fitting \nFIG. 7. Transfer functions from our forward model and their EFT template fits for a typical HOD mock. \n<!-- image --> \nFIG. 8. Left panel: Error power spectrum from the forward model and its EFT template fit for a typical HOD mock in our sample. The number density of this mock is ¯ n = 3 . 5 · 10 -4 [ h Mpc -1 ] 3 . Right panel : Power spectra of a typical HOD galaxy field, the deterministic part of the EFT forward model, and the residual between the two (the error power spectrum P err ). \n<!-- image --> \nthe error power spectrum from the data to the EFT template (14). Typical power spectra of the HOD galaxies and the EFT forward model are shown in the right panel of Fig. 8. The difference between the EFT and the HOD density fields by construction is the error field density, whose power spectrum is also shown in Fig. 8. \nIn Fig. 9 we show a corner plot with marginalized posterior distribution for PNG parameters plus linear and quadratic bias parameters for all four BOSS data chunks.', 'Appendix B: Normalizing flow training and validation': 'Normalizing flows are implemented using the nflows library, with training and evaluation performed using PyTorch [96]. Masked Autoregressive Flows [121] are used, with 6 flow transformation, each parameterized using a 2-layer masked autoregressive neural network [122] with GELU activations. 10% of the samples are held out for validation, and the flow is trained with batch size 128 for either 30,000 or 50,000 steps (for the EFT and HOD-conditional EFT distributions, respectively) using the Adam optimizer [123] with learning rate 3 × 10 -4 . \nSince we have a finite number of sampled points for the HOD and EFT parameters, it is imperative to validate that features of our approximated distribution are not artificially sculpted due these points. Figure 10 shows the training (dashed red) and validation (blue) losses over the course of training, with validation being done every 200 steps on a held out set of points. We can see that the loss asymptotes for both the marginal EFT parameter fit (left) and the HOD-conditional fit (right), with no signs of overfitting. \nFinally, as an additional coverage test, we perform a simulation-based calibration analysis [124] of our normalizing flow model. To that end we generate 100 samples from our trained normalizing flow model and compare them to validation samples. In this process, we calculate the rank of each validation sample among the set of 100 posterior samples drawn from the flow, and plot aggregate statistics, shown in Fig. 11. With perfect calibration, these ranks should be uniformly distributed. No significant deviation from uniformity can be observed, validating the calibration and convergence properties of \nour learned posterior.', 'Appendix C: Test of UV-sensitivity': "In this section, we provide a convergence test for our transfer function measurements. This analysis also provides an estimate of the residual UV-dependence of the bias parameters extracted with our field-level technique. \nThe application of the CIC window leads to an implicit smoothing of all fields in our calculations. Note that this smoothing is quite soft due to the particular shape of the CIC window. The cell size of the grid is a proxy of the filtering scale. In our baseline analysis, we use 256 grid points N g , resulting in an effective smoothing scale R s ≃ πk -1 Nyquist = 6 h -1 Mpc. \nIn Fig. 12 we display transfer functions for one of the HOD mocks in our catalog with four choices of the grid size, N g = 64 , 128 , 256 and 512, which corresponds to the smoothing lengths πk -1 Nyquist = 24 , 12 , 6 , 3 h -1 Mpc, respectively. As expected, we see that using a relatively large cutoff R s affects the transfer functions quite significantly. In particular, the R s = 8 h -1 Mpc transfer functions for the non-linear bias parameters take different values in the k → 0 limit. In contrast, the lowk limit of β 1 is unaffected and yields the renormalized linear bias parameter b 1 [65]. \nThe appearance of the scale dependence can be understood from perturbation theory arguments. For example, at the one-loop order β 2 reads, \nβ 2 ( k ) = b 2 2 + ⟨ ˜ δ ⊥ 2 G ⊥ 2 ⟩ ⟨ ˜ δ ⊥ 2 δ ⊥ 2 ⟩ . (C1) \nThe rightmost term above vanishes in the k → 0 limit because the power spectrum of δ 2 is constant on large scales. However, this constant vanishes quickly as one lowers the cutoff, i.e. for R s ≃ 10 h -1 Mpc its value is reduced by a factor of 3 compared to the R s → 0 limit. Hence, the scale dependence of the second term starts playing a more important role as one lowers the cutoff, and ultimately contaminates the lowk limit of β 2 . 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2024MNRAS.534..444A | Coronal mass ejections CMEs are solar eruptions that involve largescale changes to the magnetic topology of an active region. There exists a range of models for CME onset which are based on twisted or sheared magnetic field above a polarity inversion line PIL. We present observational evidence that topological changes at PILs in the photosphere form a key part of CME onset as implied by many models. In particular we study the onset of 30 CMEs and investigate topological changes in the photosphere by calculating the magnetic winding flux using the ARTOP code. By matching the times and locations of winding signatures with CME observations produced by the ALMANAC code we confirm that these signatures are indeed associated with CMEs. Therefore as well as presenting evidence that changes in magnetic topology at the photosphere are a common signature of CME onset our approach also allows for the finding of the source location of a CME within an active region. | 2024-10-01T00:00:00Z | ['arXiv:2409.07261', '10.1093/mnras/stae2110', '10.48550/arXiv.2409.07261', '2024arXiv240907261O', '2024MNRAS.534..444A', '2024MNRAS.tmp.2062A'] | ['Astrophysics - Solar and Stellar Astrophysics'] | Photospheric signatures of CME onset | 2,024 | 168 | 0.51 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF'] | 1 | https://arxiv.org/pdf/2409.07261.pdf | {'O.P.M. Aslam, 1 D. MacTaggart, 1 ★ T. Williams, 2 L. Fletcher 3 , 4 and P. Romano 5': '- 1 School of Mathematics and Statistics, University of Glasgow, Glasgow, G12 8QQ, UK\n- 2 Department of Mathematical Sciences, Durham University, Durham, UK\n- 3 SUPA School of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, UK\n- 4 Rosseland Centre for Solar Physics, University of Oslo, PO Box 1029 Blindern, NO-0315 Oslo, Norway\n- 5 INAF-Osservatorio Astrofisico di Catania, Via Santa Sofia 78, 95123 Catania, Italy \nAccepted 2024 September 09. Received 2024 September 06; in original form 2024 June 12', 'ABSTRACT': 'Coronal mass ejections (CMEs) are solar eruptions that involve large-scale changes to the magnetic topology of an active region. There exists a range of models for CME onset which are based on twisted or sheared magnetic field above a polarity inversion line (PIL). We present observational evidence that topological changes at PILs, in the photosphere, form a key part of CME onset, as implied by many models. In particular, we study the onset of 30 CMEs and investigate topological changes in the photosphere by calculating the magnetic winding flux, using the ARTop code. By matching the times and locations of winding signatures with CME observations produced by the ALMANAC code, we confirm that these signatures are indeed associated with CMEs. Therefore, as well as presenting evidence that changes in magnetic topology at the photosphere are a common signature of CME onset, our approach also allows for the finding of the source location of a CME within an active region. \nKey words: Sun: coronal mass ejections (CMEs) - Sun: magnetic fields - Sun: photosphere', '1 INTRODUCTION': "The energy source of solar eruptions, such as flares and coronal mass ejections (CMEs), lies in the magnetic field (Forbes 2000). As solar active regions emerge and evolve, the magnetic field forms configurations that can become unstable and lead to eruptions, and this scenario is the basis of many models and simulations of CME onset (e.g. Antiochos et al. 1999; Amari et al. 2000; Roussev et al. 2003; Török & Kliem 2007; Aulanier et al. 2010; Ishiguro & Kusano 2017; Jiang et al. 2021). A key element for the possibility of an eruption is that the magnetic field has a suitable magnetic topology . This expression entails two important features. First, that the magnetic field has a suitable connectivity to allow for reconnection to occur efficiently and produce an eruption. Secondly, it describes how inherently twisted the magnetic field is. This latter point is related to magnetic helicity which provides a lower bound for the free magnetic energy required to produce an eruption (see Tziotziou et al. 2014; Liokati et al. 2023, for empirical evidence). \nStudies of magnetic topology have identified particular features that are closely associated with different eruptive events in the solar atmosphere. One such feature, referred to as a bald patch (Titov et al. 1993), is a location where magnetic field lines become tangent to the photosphere. For example, bald patches have been associated with filaments (López Ariste et al. 2006), brightenings (Fletcher et al. 2001), surges (Mandrini et al. 2002) and flares (Lee et al. 2021). The importance of bald patches reflects the role they play in the two elements of magnetic topology outlined above. In terms of connectivity, two distinct bald patches can define separatrices, which are preferential regions for magnetic reconnection (Priest 2014). In \nterms of twist, bald patches are to be expected along polarity inversion lines (Titov & Démoulin 1999). \nBald patches are typically associated with concave up 'U-loops'. However, horizontal magnetic field at the photosphere can be indicative of more general topological behaviour, such as the emergence of a twisted or highly sheared magnetic field. In order to capture topological changes due to (near-)horizontal field at the photosphere, a suitable measure is magnetic winding (Prior & MacTaggart 2020; MacTaggart & Prior 2021; MacTaggart et al. 2021). This quantity is a renormalization of magnetic helicity and provides a direct measure of field line topology. By removing the flux weighting from magnetic helicity, magnetic winding can be used to identify, in particular, topological complexity due to primarily near-horizontal field. Magnetic helicity, by contrast, is dominated by the strongest vertical field due to its flux weighting. For detailed discussions on the properties of magnetic winding, the reader is directed to the citations above. A brief summary of the results required for this work is presented in Appendix A. In magnetograms, it is magnetic winding flux 𝐿 that is calculated and this can indicate where and when there are large topological changes due, in the main, to changing near-horizontal magnetic field at the photosphere. \nThe purpose of this work is two-fold. First, we present evidence that changes in magnetic topology at the photosphere are a common signature of CME onset. We analyze 30 CMEs and identify the topological onset signature at the photosphere by measuring the flux of magnetic winding. Secondly, and in parallel with the first point, we provide an approach for the identification of the location of the the origin of a CME within an active region, based solely on data from observations, i.e. without the explicit need to model the threedimensional magnetic field. \nThe paper is set out as follows. First, the method of our approach \nfor finding signatures of magnetic winding relating to CME onset is outlined in detail. Secondly, we present our results and discuss general features. We then present some example cases in more detail. The paper concludes with a summary and a discussion of the significance of our results.", '2 WINDING SIGNATURE IDENTIFICATION PROCEDURE': 'In this section, we outline the steps involved in the identification of photospheric topological signatures of CME onset, which we will refer to as winding signatures . Examples of outputs from each of the following parts will be presented later, as well as further descriptions of the analysis.', '2.1 Part 1: Time series calculations': 'In this study, we present 30 CME events, making use of events already identified and studied in other works (see the CME list given in Pal et al. (2018) and the flare list given in Liu et al. (2021)). These events have been cross-checked to make sure that each CME occurred at a longitude (relative to the central meridian) within the range (-60 · , + 60 · ) . These limits are typical bounds used in the literature (e.g. Park et al. 2020) and represent the locations where winding (and helicity) calculations become compromised by projection effects. This is because our method of calculating the magnetic winding flux is based on Space-Weather Helioseismic and Magnetic Imager (HMI) Active Region Patches (SHARP) vector magnetograms (Hoeksema et al. 2014). The projection of magnetic field components onto a Cartesian grid is most accurate near the central meridian and the approximation begins to break down at the above limits. \nWith the CME events selected, the first step in identifying the winding signature is to calculate a time series of / 𝐿 ( = d 𝐿 / d 𝑡 ) using the ARTop code (Alielden et al. 2023). This code makes use of SHARP magnetograms, as mentioned above, utilising their full field of view, and the Differential Affine Velocity Estimator for Vector Magnetograms (DAVE4VM) technique (Schuck 2008). For all calculations involving ARTop in this work, all magnetic field of strength lower than 20 Gauss is ignored, an apodizing window of 20 pixels is used for the DAVE4VM calculations and the full resolution of the magnetograms is used for the winding calculations. \nFor each of the 30 events, a time series starting from approximately 20 hours before the recorded coronagraph CME time (details are provided below) until a few hours after this time, is calculated. For events whose active regions are beyond the -60 · longitude limit 20 hours before the recorded coronagraph CME time, the time series are calculated from when the active region has a longitude of -60 · . \nFor each time series, a running mean 𝜇 (based on the previous hour of data) is calculated and an envelope 𝜇 ± 2 𝜎 , with a width of 2 standard deviations in both positive and negative directions, is determined. The ARTop code enables overlaying flare times and strengths on time series for a given active region. Where they occur, flare corresponding to CMEs are also recorded. \nThe peaks in magnetic winding, which we define as spikes of / 𝐿 penetrating the ± 2 𝜎 envelope, near the recorded coronagraph time of the CME, are noted. The largest, in magnitude, of these spikes is taken as the initial marker for the winding signature. Typically, there will be one prominent peak shortly before the coronagraph time of CME and/or its corresponding flare. The analysis we are proposing here is, therefore, not primarily intended to be predictive, but rather a means of providing a deeper analysis of the birth of a given CME in the lower layers of the solar atmosphere. In general, the time series \nof a particular region also may contain other winding spikes that are associated with other topological events that are not directly related to CMEs, such as flares, emergence and motion in penumbrae (see later for more details). A large spike in the magnitude of / 𝐿 corresponds to a large amount of topologically-significant (i.e. twisted and sheared) magnetic field passing through the photosphere within a given time. This, as we will demonstrate, typically occurs in a localized part of the active region on or near a PIL.', '2.2 Part 2: Magnetic winding maps - winding signature time and location': 'At each time, the quantity / 𝐿 is the spatial integral over the map of the field line winding flux / L (see the Appendix for further details). By examining the maps of / L , at the times near the peaks recorded in Part 1, it is possible to determine, by visual inspection, for each CME, the location within active region where the winding signature occurs. The time of the winding signature corresponds to when the structure corresponding to the largest spike first appears in the maps of / L . This is normally co-temporal with the main spike or occurs shortly before ( ∼ 12-24 mins) the main spike. \nProposed mechanisms for CME onset (as in the examples cited previously) are based on sheared or twisted magnetic field above the main PIL of the active region. Thus, time series peaks identified in Part 1 must correspond with signatures at PILs in order to be considered to be related to CME onset (see Moraitis et al. (2024) for a related study on the connection between flares and relative field line helicity at PILs). The strength of the winding signature depends on the particular eruption mechanism that takes place. Later, we will present different pre-eruption magnetic topologies that can lead to strong winding signatures through topological changes at the photosphere.', '2.3 Part 3: Confirmation with ALMANAC': "In order to link the photospheric winding signatures with CME onset, the final stage is to connect their times and locations with the earliest available observations of the CMEs higher in the atmosphere (with respect to the photosphere). Although we base our selection of 30 events on previously recorded CMEs, as detailed below, these use observations in the high atmosphere (from coronagraphs) and, therefore, do not represent the early stage of CME onset. Instead, we need information from heights lower than those available from coronagraph data. \nTo perform this task systematically, we make use of the A utomated Detection of Corona L MA ss Ejecta origi N s for Space Weather A ppli C ations ( ALMANAC ) code (Williams & Morgan 2022). ALMANAC , unlike many widely adopted CME detection methods does not rely upon coronagraph data, but instead utilises data from the Atmospheric Imaging Assembly (Lemen et al. 2012, AIA). The main advantage of ALMANAC is that it does not require geometrical fitting to approximate the CME source location in the low solar corona. Thus, the code does not inherently have large uncertainties due to projection effects caused by fitting a simple 'wire-frame' of a threedimensional object mapped in two dimensions. As such, ALMANAC provides a reliable (low-corona) CME origin that is obtained independently of any winding signatures from the earlier phases of an eruption. \nTo detect potential Earth-directed CMEs, ALMANAC first crops the map size to eliminate off-limb contributions and standardises the intensity across an 8-hour image sequence by thresholding intensities and normalising the data values. It is then smoothed through \nconvolution and subtracted from the normalised data to create a high-bandpass and time-filtered image sequence. Each time step of the time-filtered data is then divided by the median of the absolute values of the unfiltered data to eliminate contribution from 'static' structures such as active regions. The method employs a series of Boolean masks to isolate connected clusters of pixels associated with a potential eruption, and spatio-temporal smoothing of these masks helps avoid the segmentation of regions. The first time step in which a region of sufficient size and duration is identified is used as the CME onset time, whilst the center of mass for the masked pixels at that time provides the central location for the CME. Full details can be found in Williams & Morgan (2022).", '3 SUMMARY OF THE DATA': "We now present results for the 30 selected events. Details are displayed in Table 1. \nWe argue that these data provide strong evidence that the winding signatures at the photosphere are related to CME onset due to the close association of the times and locations of winding signatures to both early observations of CMEs and their associated flares. \nThere is a strong match between the locations of the winding signatures and the CME positions determined by ALMANAC for all events with the exception of the CME in AR11247, for which the difference in both longitude and latitude is greater than 10 · ; and the first event of AR11875, for which the CME was not detected by ALMANAC . We will discuss the event of AR11247 in more detail later. \nDifferences in position are typically only a few degrees, with the largest separation (not including the event of AR11247) being 10 · (for the Carrington longitude of the first event of AR11283). With the exception of the event of AR11247 and the first event of AR11875, the differences in latitude and longitude between the winding measurements from ARTop and the CME locations from ALMANAC are less than the active region dimensions at the photosphere. This is shown in Table 1 by the Δ -quantities, which measure \nΔ 𝑋 = GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> 𝑋 ARTop -𝑋 ALMANAC 𝑋 AR GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> , \nwhere 𝑋 represents longitude or latitude and 𝑋 AR represents the longitudinal or latitudinal extent of the active region. Some deflection is expected for CMEs due to the nearby constraints of surrounding and overlying magnetic field, effects of the rise mechanism (such as twisting, e.g. Vourlidas et al. 2011) and the heliospheric current sheet (Kay et al. 2017). However, as indicated by the Δ -quantities in Table 1, since all these values are less than 1, such deflections are generally still within the photospheric area of the active region (at least for the early onset times presented here). In addition to the physical grounds for deflection, listed above, such a close matching between the locations produced by ARTop and ALMANAC is not necessarily guaranteed since the latter can be influenced by surrounding regions and strong flares. Williams & Morgan (2022) report that for the twenty halo CMEs that they study with ALMANAC , CME locations are within 10 · ± 10 · of X-ray emission values reported by RHESSI for that sample - a typically much larger margin than that which we find based on Δ < 1. \nThe first times listed in Table 1 are those of the winding signatures, the determination of which was outlined in the previous section. These times are constrained by ARTop 's magnetogram cadence, which is 12 minutes (i.e. that of the SHARP magnetograms). The next column of times is that of when the ALMANAC code detects a CME from AIA data (constrained to a 10-minute cadence). The last \ncolumn of times are from existing catalogues of CME observations, in particular those made using the Large Angle and Spectrometric Coronagraph (LASCO) onboard the Solar and Heliospheric Observatory (SOHO) mission (Brueckner et al. 1995) and the Solar TErrestrial RElations Observatory (STEREO) (Kaiser et al. 2008). As mentioned earlier, it is these recorded coronagraph times that were used to select CMEs and help identify the corresponding winding signatures. \nThese times, following the discussion earlier, correspond to observations of different heights in the atmosphere: winding - photosphere, ALMANAC - low atmosphere, coronagraph times - high atmosphere. Therefore, if the winding signatures correspond to CME onset, rather than some later phase of CME evolution, the winding times should be closer to the ALMANAC times than the coronagraph times. This is indeed the case, with all of the winding times preceding the ALAMANC times. Those events for which the winding time is less than 12 minutes (the ARTop cadence) before the ALMANAC time, may be considered to be approximately co-temporal, due to the available time resolution. As mentioned previously, the first CME of AR11875 was not detected with ALMANAC and so we cannot make as clear a connection between the observations at the photosphere and those higher in the atmosphere for this case. \nSince CMEs are normally, though not always, associated with flares, we now compare the winding signature times to the start times of the flares that are associated with CMEs, i.e. those that are closest in time to the recorded coronograph CME times. Table 2 lists the flares associated with each of the 30 events and shows how their start times compare with the winding signature times. \nThe winding signatures precede or are approximately co-temporal with the flare start times for all events except that of AR11810. This event and that of AR11247 are the only ones for which the associated flare is B-class. For C-class and above, there is a consistent picture, providing further evidence that the winding signatures are related to CME onset. It is worth mentioning that since the majority of the winding signatures precede flares, for the events studied, we can exclude the possibility that these signatures are influenced by any instrumental effects related to the inversion of the Fe I line at 617.3 nm due to the strong flare emission.", '4 ANALYSIS OF EXAMPLE REGIONS': 'We now demonstrate the nature of the winding signature for three active regions from Table 1, with different morphologies, connecting it to the global magnetic topology of the active region. For each example region, we display the outputs (time series and maps) from ARTop , from which the winding signature is identified. In order to connect this signature to the global region topology, we approximate the three-dimensional magnetic field at the time of the winding signature by a nonlinear force-free field. We make use of the deep learning method of Jarolim et al. (2023) which produces a solution to the equations \n(∇ × 𝑩 ) × 𝑩 = 0 , (1) \n∇ · 𝑩 = 0 , (2) \nwhere 𝑩 is the magnetic field vector, extrapolating from the SHARP magnetograms of the given region. Clearly, satisfying equation (1) is an approximation (a static representation of a dynamic process) but many studies indicate that force-free extrapolations provide a good representation of the global topology of an active region magnetic field. \nTo test the accuracy of our extrapolations, we consider the metrics \nTable 1. A table of 30 CME events with winding signatures related to CME onset. The columns are, in order, the active region NOAA number, the time of the winding signature, the CME time from ALMANAC , the recorded CME time (LASCO or STEREO), the winding signature coordinates in Carrington longitude and sine latitude (to the nearest degree), the CME location from ALMANAC in Carrington longitude and sine latitude (to the nearest degree) and the relative difference of the winding ( ARTop ) and ALMANAC longitudes and latitudes divided by the longitudinal and latitudinal extensions of the active region respectively (rounded to three decimal places). The Carrington longitudes are normalized to lie within the range [0 · , 360 · ]. \nused by Wheatland et al. (2000) and Jarolim et al. (2023). As a measure of the solenoidality, which indicates how well an extrapolation represents a magnetic field, we calculate the 𝑩 -weighted divergence averaged over the entire domain, and refer to this quantity as 𝜆 , \n𝜆 = GLYPH<28> ∥∇ · 𝑩 ∥ 2 ∥ 𝑩 ∥ 2 GLYPH<29> . \nAs a measure of force-freeness, we first calculate, at every grid point 𝑖 , the sine of the angle between the current density and the magnetic field. Writing 𝐹 𝑖 = ∥(∇ × 𝑩 ) × 𝑩 ∥ 2 , 𝐵 𝑖 = ∥ 𝑩 ∥ 2 and 𝐽 𝑖 = ∥∇ × 𝑩 ∥ 2 for the norms evaluated at grid point 𝑖 , \n𝜎 𝑖 = sin 𝜃 𝑖 = 𝐹 𝑖 𝐵 𝑖 𝐽 𝑖 . \nFrom this, a current-weighted measure of the angle is produced, \n𝜃 𝐽 = arcsin GLYPH<18> " 𝑖 𝐽 𝑖 𝜎 𝑖 Σ 𝑖 𝐽 𝑖 GLYPH<19> . \nThe values of 𝜆 and 𝜃 𝐽 are reported for each extrapolation. We note that the extrapolations presented here are solely for the purpose of giving indications of the magnetic topology within active regions and are not used as the basis of any further topological calculations, as in, for example, Valori et al. (2016) and Thalmann et al. (2020).', '4.1 AR11318': "The first example region we consider, AR11318, has been studied in several other works (Romano et al. 2014; MacTaggart et al. 2021). Compared to other active regions, it is relatively simple - it is a bipolar region in which one flare and one CME occurred. Figure 1 displays the time series of / 𝐿 in the period before the CME. \nOn the time scale of Figure 1, there is a C2.3 flare beginning at 20.32 hours, followed shortly by the first sighting of a CME by LASCO at 21.2 hours. The ALMANAC time for the CME is approximately co-temporal with the C2.3 flare (a 9-minute difference). There is a particularly large spike in / 𝐿 at 19 hours and this, by the method outlined in Section 2, is taken as the winding signature (it is the only spike near the flare and LASCO CME times that penetrates the 2 𝜎 envelope). The location of this signature within AR11318 can be found by considering the maps in Figure 2. \nThere is a rapid change in the field line winding L at one particular location, as seen in Figure 2 (b). Comparing this with the magnetogram in Figure 2 (a), this location corresponds to a small PIL between the two main spots, curving downward at about 97 · Carringtion longitude. Figure 2 (c) displays a zoomed-in window of the the 𝐵 𝑧 map at the location of the winding signature, indicated by contours of / L ( ± 10000 km 2 s -1 ). \nTable 2. The corresponding flares of the 30 CME events. For each event, the GOES flare strength, the winding observation time and the flare start time is shown. \nFigure 1. Time series of the magnetic winding flux rate / 𝐿 before the CME of AR11318. The quantities displayed follow the description in Part 1 of Section 2. The LASCO CME time is at 21.2 hours on this graph. \n<!-- image --> \n<!-- image --> \n(a) \n<!-- image --> \n(b) \nFigure 2. (a) shows a map of 𝐵 𝑧 , (b) a map of / L and (c) a detail of the 𝐵 𝑧 map with contours ( ± 10000 km 2 s -1 ) of / L highlighting the centre of the winding signature at the base of the central curving PIL. All maps correspond to the time of the winding spike identified in Figure 1. \n<!-- image --> \nTo better understand what this feature is, we consider a nonlinear force-free extrapolation, shown in Figure 3. \nThe red field lines, traced from the winding signature location, indicated by 'L', reveal a bald patch which, as described in the Introduction, is a common topological feature related to eruptive phenomena. The winding signature here could refer to the emergence or submergence of the sheared bald patch field lines. Another possible explanation is due to reconnection related to the rising CME flux rope. The extrapolation traces out (shown in blue in Figure 3), a sigmoidal flux rope, which matches well the shape of features observed in AIA 193 Å and 304 Å (Romano et al. 2014).", '4.2 AR11158': "The next example region that we present is much more complex than AR11318. AR11158 is a highly-studied active region due to \nFigure 3. A nonlinear force-free extrapolation of the field lines of AR11318 at the time of the winding signature. The bald patch field lines, shown in red, are traced from the location of the winding signature, indicated by 'L'. Surrounding field lines, shown in blue, trace out a flux rope structure. The metrics for this extrapolation are 𝜆 = 0 . 0018 and 𝜃 𝐽 = 2 . 73 · . \n<!-- image --> \nit possessing both X-class flares and multiple CMEs (e.g. Schrijver et al. 2011; Sun et al. 2012; Tziotziou et al. 2013; Inoue et al. 2015; Kay et al. 2017; Chintzoglou et al. 2019; Moraitis et al. 2021; Lee et al. 2021). Here, we will focus on the second CME from AR11158 listed in Table 1. The time series of / 𝐿 for the period before this CME is shown in Figure 4. Unlike AR11318, there are many flares in this active region. An X2.2-class flare, beginning at 19.73 hours on the time scale of Figure 4, is associated with the CME and is preceded by the largest winding spike at 19.4 hours. The LASCO CME time corresponds to 20.4 hours and the ALMANAC time is approximately co-temporal with the X2.2 flare start time (a 4-minute difference). As before, the location of the winding signature can be seen by considering the maps in Figure 5. \nThere is a clear PIL displayed at the centre of the region in Figure 5 (a) at 35 · Carrington longitude. On the top of the eastern side of the PIL, there is a very strong change in L , corresponding to the winding signature. The global view of this feature is shown in Figure 5 (b) and a zoomed-in section of the 𝐵 𝑧 map overlaid with contours indicating the strongest part of / L (displaying values of ± 60000 km 2 s -1 ) is shown in (c). \nAn approximation of the magnetic topology of the field lines connected to and near the winding signature is shown in Figure 6. At the location of the winding signature, marked in Figure 6 by 'L', there is a strongly-twisted flux rope, whose main axis lies across the central PIL. The core of the flux rope, extending high above the photosphere, is indicated by yellow field lines. The peak of the winding signature corresponds to field lines (shown in red) of the rope that form a highly twisted 'J' structure - a common feature of sigmoidal flux ropes (e.g. Titov & Démoulin 1999; Hood et al. 2012). It is the change of this 'J' structure, in contact with the photosphere, that leads to the winding signature. The rope is surrounded by a magnetic arcade (indicated by blue field lines). This configuration is the starting point for many eruption mechanisms (such as those cited in the Introduction) and matches that found in other extrapolations calculated for this region (e.g. Inoue et al. 2015; Moraitis et al. 2021). \nFigure 4. Time series of the magnetic winding flux rate / 𝐿 before the chosen CME of AR11158. The quantities displayed follow the description in Part 1 of Section 2. The LASCO CME time is at 20.4 hours on this graph. \n<!-- image -->", '4.3 AR11247': "Wenowfocus on the event of AR11247, which is the only example of the cases presented for which there is a large (greater than 10 · ) separation in both longitude and latitude between the winding signature location and the CME location determined by ALMANAC . Although our approach does not seem to identify the coordinates (in space and time) of CME onset in this case, we nevertheless proceed as in previous cases in order to identify what is occurring in this event. As before, we first consider the time series of / 𝐿 , which is shown in Figure 7. \nGiven that the LASCO observation of the CME is at 23.8 hours on this figure, the B4.7 flare is the only possible flare that may be associated with the CME (if, indeed, the CME has an associated flare). However, the ALMANAC observation time is approximately 1 hour before the LASCO time, when there is no obvious winding signature. In order to understand the significance of these results better, we proceed with our previous approach and investigate the winding signature co-temporal with the start of the B4.7 flare, in order to see what connection, if any, it has with the CME. The maps of 𝐵 𝑧 and / L at the time of the winding signature are displayed in Figure 8. \nWefollow our previous approach and select the strongest localized concentration of magnetic field line winding flux, located just above a small PIL south-east of the central negative polarity at 269 · Carrington longitude. As indicated in Table 1, the position of the CME is much further west and south of the winding signature. A possible connection, however, can be found by considering a force-free extrapolation, as shown in Figure 9. \nThe location of the selected winding signature corresponds to a region beneath the null point of a quadrupolar magnetic field. Although this is the classical setup of the breakout model for CMEs (Antiochos et al. 1999), as mentioned previously, the CME does not occur here. Near to this region, the magnetic field has a topology commonto magnetic jets (Shibata et al. 2007; Pariat et al. 2015). The field lines higher in the domain are open, in the sense that they do not connect back down to the lower boundary. This suggests that there \nmay be a connection between the region of the winding signature and magnetic fields outside this domain. Pursuing this line of enquiry, Figure 10 displays the line-of-sight magnetic field superimposed on AIA 171 Å data at the time of the winding signature. \nIn the south-western corner of the map in Figure 10, the dark arc, with a helioprojective longitude of -400', is the rising filament of the CME. Based on the AIA image, the blue field lines from Figure 9 connect to closed loops and not to the CME directly. One possibility, therefore, is that the expansion of the CME acted as a perturbation on the magnetic loops of AR11247, resulting in a localised change in magnetic topology, leading to the winding signature and the weak B-class flare (the ALMANAC time precedes the B4.7 flare time, see Tables 1 and 2). \nThis analysis shows that this particular CME is connected to AR11247butis not ejected from one of its internal PILs (even though it has been previously catalogued as emanating from this region (Pal et al. 2018)). Thus, our approach can help to confirm whether or not a CMEdoes originate from a particular active region. Further, such as in this case, our approach can still be used to analyse the topological complexity of an active region.", "4.4 Regions with 'strong' penumbrae": 'The approach we have outlined in this work can be applied to any SHARP region. For some regions, however, winding signatures at PILs can be masked if there is a particularly dominant sunspot with a developed penumbra. Since a penumbra is a region of magnetoconvection with a near-horizontal magnetic field, magnetic winding is particularly sensitive to field line motions in penumbrae. The result is that the largest spikes in time series of / 𝐿 are dominated by penumbra dynamics and, unless the CME and/or associated flare is very strong, the signatures at the PIL are weaker in comparison and not easily detected as a winding signature. An example of a winding signature dominated by a penumbra, from AR11777, is shown in Figure 11. \nFor active regions such as these, in order to apply the approach \n<!-- image --> \n(a) \n<!-- image --> \n(b) \nFigure 5. (a) shows a map of 𝐵 𝑧 , (b) a map of / L and (c) a detail of the 𝐵 𝑧 map with contours ( ± 60000 km 2 s -1 ) of / L highlighting the centre of the winding signature at the top of the central PIL. All maps correspond to the time of the winding spike identified in Figure 4. \n<!-- image --> \nof this article, the penumbra would have to be masked. This task, however, goes beyond the scope of this work.', '5 SUMMARY AND CONCLUSIONS': "In this work, we present evidence that changes in magnetic topology at the photosphere are a common signature of CME onset. In particular, we show, by means of magnetic winding flux, that strong topological changes in the photosphere, occurring at a polarity inversion line within the active region, typically precede or are approximately co-temporal with early sightings of CMEs. We have found this result by analysing time series and maps of magnetic winding flux in 20-hour periods before the recorded coronograph observations (from LASCO or STEREO) for 30 CME events. From the time series, the \nFigure 6. A nonlinear force-free extrapolation of the field lines of AR11158 at the time of the winding signature. The core of the flux rope is indicated by the yellow field lines. The red field lines, traced from the location of the winding signature, marked with 'L', trace out one of the outer 'J's of the flux rope. Blue field lines trace surrounding field. Note that the region is rotated by almost 180 · with respect to the maps in Figure 5. The metrics for this extrapolation are 𝜆 = 0 . 0025 and 𝜃 𝐽 = 6 . 96 · . \n<!-- image --> \ntimes of winding signatures are identified, and their locations are found by inspection of where the strongest concentrations occur in the field line winding maps. All data related to winding are based on SHARP magnetograms and processed with the ARTop code. In order to provide further evidence that the winding signatures are associated with CME onset, we also searched for the CMEs using the ALMANAC code. This makes use of AIA data and, thus, provides an earlier observation of a CME than LASCO or STEREO. For 28 out of the 30 events studied, there is a good match between the locations of the winding signatures and the CME locations determined by ALMANAC , with differences being less than both the longitudinal and latitudinal extents of the active region at the photosphere. The winding signatures either precede or are approximately co-temporal with the ALMANAC times, which all precede the recorded LASCO or STEREOobservation times. For the remaining two events, data from ALMANAC is unavailable for one, and the other was not directly linked to the region for which it had been catalogued under. \nThestart times of the associated flares of each of the CMEs has also been compared to the times of the winding signatures. For all CMEs associated with flares of class C and above, the winding signatures precede or are approximately co-temporal with the flare start times. \nOur main result is the following. For many active regions, magnetic winding flux can be used to identify, both spatially and temporally, the location of CME onset (and, if present, the associated flare-onset) within an active region. We provide a method to determine these properties that is based on the analysis of HMI and AIA data through the ARTop and ALMANAC codes respectively. Exceptions to this approach include regions for which penumbrae dominate the winding signatures. However, in the many cases for which the approach does work, and as demonstrated, the method allows for the identification of which PIL within the active region is directly related to the dynamics of CME onset. Detection of this location then allows for further investigation of CME onset. \nIn order to define an algorithm for detecting signatures of CME onset, we have given a precise definition of the winding signature in \nFigure 7. Time series of the magnetic winding flux rate / 𝐿 before the CME of AR11247. The quantities displayed follow the description in Part 1 of Section 2. The LASCO CME time is at 23.8 hours on this graph. \n<!-- image --> \nterms of the largest spike in / 𝐿 nearest to the recorded coronograph CMEtime.However, as is clear from the time series presented earlier, there are many other winding spikes and, often, they are associated with flares (but not CMEs). Although we have not focused on them in this work, all such spikes are potentially important in understanding the relationship between magnetic topology and flares, and can be studied in a similar manner to the way presented in this work. \nAs well as identifying the coordinates of CME onset, our approach may also be useful in helping to determine the onset mechanism of a particular CME. The temporal relationship between the winding signature, an associated flare and early CME observations, puts a constraint on which particular mechanism may be at work during CME onset. This feature will be studied in future applications.", 'ACKNOWLEDGEMENTS': 'OPMA, DM and LF acknowledge support from the Leverhulme Trust, grant number RPG-2023-182. DM also acknowledges support from the Science and Technologies Facilities Council (STFC), grant number ST/Y001672/1 and LF acknowledges support from ST/X000990/1. ARTop was partly developed under an Innovation Placement supported by the DiRAC HPC Facility as part of the DiRAC Federation Project. Funding for the DiRAC Federation Project was provided by the UKRI Digital Research Infrastructure programme. ALMANAC was developed under Leverhulme grant RPG2019-361. This research used version 5.1.0 of the SunPy open source software package (The SunPy Community et al. 2020).', 'DATA AVAILABILITY': 'Input files for the ARTop code for the 30 events presented here are available here. \nThe ARTop code is available here and the ALMANAC code is available here.', 'REFERENCES': "Alielden K., MacTaggart D., Ming Q., Prior C., Raphaldini B., 2023, RAS Techniques and Instruments, 2, 398 \nAmari T., Luciani J. F., Mikic Z., Linker J., 2000, ApJ, 529, L49 Antiochos S. K., DeVore C. R., Klimchuk J. A., 1999, ApJ, 510, 485 Aulanier G., Török T., Démoulin P., DeLuca E. E., 2010, ApJ, 708, 314 Brueckner G. E., et al., 1995, Sol. Phys., 162, 357 \nChintzoglou G., Zhang J., Cheung M. C. M., Kazachenko M., 2019, ApJ, 871, 67 \nFletcher L., López Fuentes M. C., Mandrini C. H., Schmieder B., Démoulin P., Mason H. E., Young P. R., Nitta N., 2001, Sol. Phys., 203, 255 Forbes T. G., 2000, J. Geophys. Res., 105, 23153 Hoeksema J. T., et al., 2014, Sol. Phys., 289, 3483 Hood A. W., Archontis V., MacTaggart D., 2012, Sol. Phys., 278, 3 Inoue S., Hayashi K., Magara T., Choe G. S., Park Y. D., 2015, ApJ, 803, 73 Ishiguro N., Kusano K., 2017, ApJ, 843, 101 Jarolim R., Thalmann J. K., Veronig A. M., Podladchikova T., 2023, Nature \nAstronomy, 7, 1171 \nJiang C., et al., 2021, Nature Astronomy, 5, 1126 \nKaiser M. L., Kucera T. A., Davila J. M., St. Cyr O. C., Guhathakurta M., Christian E., 2008, Space Science Review, 136, 5 \nKay C., Gopalswamy N., Xie H., Yashiro S., 2017, Sol. Phys., 292, 78 \nLee J. H., Sun X., Kazachenko M. D., 2021, ApJ, 921, L23 \nLemen J. R., et al., 2012, Sol. Phys., 275, 17 \nLiokati E., Nindos A., Georgoulis M. K., 2023, A&A, 672, A38 \nLiu L., Wang Y., Zhou Z., Cui J., 2021, ApJ, 909, 142 \nLópez Ariste A., Aulanier G., Schmieder B., Sainz Dalda A., 2006, A&A, 456, 725 \nMacTaggart D., Prior C., 2021, Geophysical and Astrophysical Fluid Dynamics, 115, 85 \nMacTaggart D., Prior C., Raphaldini B., Romano P., Guglielmino S. L., 2021, Nature Communications, 12, 6621 \nMandrini C. H., Démoulin P., Schmieder B., Deng Y. Y., Rudawy P., 2002, A&A, 391, 317 \nMoraitis K., Patsourakos S., Nindos A., 2021, A&A, 649, A107 \n- Moraitis K., Patsourakos S., Nindos A., Thalmann J. K., Pariat É., 2024, A&A, 683, A87 \nPal S., Nandy D., Srivastava N., Gopalswamy N., Panda S., 2018, ApJ, 865, \n4 \n<!-- image --> \n(a) \n<!-- image --> \n(b) \nFigure 8. (a) shows a map of 𝐵 𝑧 , (b) a map of / L and (c) a detail of the 𝐵 𝑧 map with contours ( ± 10000 km 2 s -1 ) of / L highlighting several possible locations for the winding signature. The strongest signature corresponds to the region at 269 · . All maps correspond to the time of the winding spike identified in Figure 7. \n<!-- image --> \nPariat E., Dalmasse K., DeVore C. R., Antiochos S. K., Karpen J. T., 2015, A&A, 573, A130 \nPark S.-H., Leka K. D., Kusano K., 2020, ApJ, 904, 6 \nPriest E., 2014, Magnetohydrodynamics of the Sun, doi:10.1017/CBO9781139020732. \nPrior C., MacTaggart D., 2020, Proceedings of the Royal Society of London Series A, 476, 20200483 \nRomano P., Zuccarello F. P., Guglielmino S. L., Zuccarello F., 2014, ApJ, 794, 118 \nRoussev I. I., Forbes T. G., Gombosi T. I., Sokolov I. V., DeZeeuw D. L., Birn J., 2003, ApJ, 588, L45 \nSchrijver C. J., Aulanier G., Title A. M., Pariat E., Delannée C., 2011, ApJ, 738, 167 \nSchuck P. W., 2008, ApJ, 683, 1134 \nShibata K., et al., 2007, Science, 318, 1591 \nSun X., Hoeksema J. T., Liu Y., Wiegelmann T., Hayashi K., Chen Q., Thalmann J., 2012, ApJ, 748, 77 \nThalmann J. K., Sun X., Moraitis K., Gupta M., 2020, A&A, 643, A153 \nThe SunPy Community et al., 2020, The Astrophysical Journal, 890, 68 \nTitov V. S., Démoulin P., 1999, A&A, 351, 707 \nTitov V. S., Priest E. R., Demoulin P., 1993, A&A, 276, 564 \nFigure 9. A nonlinear force-free extrapolation of the field lines of AR11247 at the time of the winding signature. The field lines, shown in red, related to the possible winding signature are traced from the position marked 'L'. Surrounding topological features are revealed by blue field lines. The metrics for this extrapolation are 𝜆 = 0 . 0042 and 𝜃 𝐽 = 18 . 19 · . \n<!-- image --> \nFigure 10. AIA 171 Å with line-of-sight magnetic field 𝐵 𝑧 superimposed of AR11247 at the time of the winding signature from Table 1. \n<!-- image --> \nTörök T., Kliem B., 2007, Astronomische Nachrichten, 328, 743 \nTziotziou K., Georgoulis M. K., Liu Y., 2013, ApJ, 772, 115 \nTziotziou K., Moraitis K., Georgoulis M. K., Archontis V., 2014, A&A, 570, \nL1 \nValori G., et al., 2016, Space Sci. Rev., 201, 147 \nVourlidas A., Colaninno R., Nieves-Chinchilla T., Stenborg G., 2011, ApJ, 733, L23 \nWheatland M. S., Sturrock P. A., Roumeliotis G., 2000, ApJ, 540, 1150 Williams T., Morgan H., 2022, Space Weather, 20, e2022SW003253 \n<!-- image --> \nFigure 11. Maps of (a) 𝐵 𝑧 and (b) / L during the evolution of AR11777. \n<!-- image -->", 'APPENDIX A: MAGNETIC WINDING DEFINITION': 'For detailed descriptions of magnetic winding, we guide the reader to Prior & MacTaggart (2020); MacTaggart & Prior (2021); MacTaggart et al. (2021). Here, we just list the key formulae for completeness. \nThe calculation of magnetic helicity flux from magnetograms has become a standard calculation. For the calculation of magnetic helicity 𝐻 , one formulation of the helicity flux, based on field line winding, is written as \nd 𝐻 d 𝑡 = -1 2 𝜋 ∫ 𝑃 ∫ 𝑃 d d 𝑡 𝜃 ( 𝒙 , 𝒚 ) 𝐵 𝑧 ( 𝒙 ) 𝐵 𝑧 ( 𝒚 ) d 2 𝑥 d 2 𝑦, (A1) \nwhere 𝑃 is the horizontal plane of the photosphere (the magnetogram), 𝒙 and 𝒚 are locations of field lines intersecting 𝑃 and 𝜃 is the pairwise winding angle of 𝒙 -𝒚 . The first part of the integral in equation (A1), involving the rate of change of the winding angle 𝜃 , encodes topological changes in the magnetic field at the photosphere. This part is weighted by the magnetic flux, e.g. at 𝒙 there is a local flux of 𝐵 𝑧 ( 𝒙 ) d 2 𝑥 . \nRenormalizing equation (A1) to remove the flux weighting, produces a direct measure of the field line topology, called magnetic winding, \nd 𝐿 d 𝑡 = -1 2 𝜋 ∫ 𝑃 ∫ 𝑃 d d 𝑡 𝜃 ( 𝒙 , 𝒚 ) 𝜎 𝑧 ( 𝒙 ) 𝜎 𝑧 ( 𝒚 ) d 2 𝑥 d 2 𝑦, (A2) \nwhere 𝜎 𝑧 ( 𝒙 ) is an indicator function taking the sign of 𝐵 𝑧 ( 𝒙 ) , and, therefore, can be -1, 0 or 1. \nEquations (A1) and (A2) represent the total (spatially-integrated) changes in helicity and winding, respectively, over 𝑃 . In this work, equation (A2) is used to create time series. In order to gain spatial information about where strong changes in winding are occurring in an active region, we remove the outer integral from equation (A2) to consider the field line winding rate \nd d 𝑡 L( 𝒙 ) = -𝜎 𝑧 ( 𝒙 ) 2 𝜋 ∫ 𝑃 d d 𝑡 𝜃 ( 𝒙 , 𝒚 ) 𝜎 𝑧 ( 𝒚 ) d 2 𝑦. (A3) \nThis quantity provides the winding rate of the field line at 𝒙 relative to all the other field lines, and it is in this sense that we refer to L as the field line winding. \nThis paper has been typeset from a T E X/L A T E X file prepared by the author.'} |
2024ApJ...974..209G | We carry out idealized threedimensional generalrelativistic magnetohydrodynamic simulations of prograde weakly magnetized and geometrically thick accretion flows where the gas distribution is misaligned from the black hole BH spin axis. We evolve the disk for three BH spins a 0.5 0.75 and 0.9375 and we contrast them with a standard aligned disk simulation with a 0.9375. The tilted disks achieve a warped and twisted steadystate structure with the outer disk misaligning further away from the BH and surpassing the initial 24 misalignment. However closer to the BH there is evidence of partial alignment as the inclination angle decreases with radius in this regime. Standing shocks also emerged in proximity to the BH roughly at 6 gravitational radii. We show that these shocks act to partially align the inner disk with the BH spin. The rate of alignment increases with increasing BH spin magnitude but in all cases is insufficient to fully align the gas before it accretes. Additionally we present a toy model of orbit crowding that can predict the location of the shocks in moderatetofast rotating BHs illustrating a potential physical origin for the behavior seen in simulationswith possible applications in determining the positions of shocks in real misaligned astrophysical systems. | 2024-10-01T00:00:00Z | ['arXiv:2409.09165', '10.3847/1538-4357/ad737d', '10.48550/arXiv.2409.09165', '2024ApJ...974..209G', '2024arXiv240909165G'] | ['Black hole physics', 'Accretion', 'Magnetohydrodynamics', 'General relativity', 'Magnetohydrodynamical simulations', '159', '14', '1964', '641', '1966', 'Astrophysics - High Energy Astrophysical Phenomena'] | Shockinduced Partial Alignment in Geometrically Thick Tilted Accretion Disks Around Black Holes | 2,024 | 168 | 0.46 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF'] | 0 | https://arxiv.org/pdf/2409.09165.pdf | {'Shock-induced partial alignment in geometrically-thick tilted accretion disks around black holes': '<!-- image --> \n1 JILA and Department of Astrophysical and Planetary Sciences, University of Colorado, Boulder, CO 80309, USA', 'ABSTRACT': 'We carry out idealized three-dimensional general-relativistic magnetohydrodynamic (GRMHD) simulations of prograde, weakly magnetized, and geometrically thick accretion flows where the gas distribution is misaligned from the black hole spin axis. We evolve the disk for three black hole spins: a = 0 . 5 , 0 . 75, and 0 . 9375, and we contrast them with a standard aligned disk simulation with a = 0 . 9375. The tilted disks achieve a warped and twisted steady-state structure, with the outer disk misaligning further away from the black hole and surpassing the initial 24 · misalignment. However, closer to the black hole, there is evidence of partial alignment, as the inclination angle decreases with radius in this regime. Standing shocks also emerged in proximity to the black hole, roughly at ∼ 6 gravitational radii. We show that these shocks act to partially align the inner disk with the black hole spin. The rate of alignment increases with increasing black hole spin magnitude, but in all cases is insufficient to fully align the gas before it accretes. Additionally, we present a toy model of orbit crowding that can predict the location of the shocks in moderate-to-fast rotating black holes, illustrating a potential physical origin for the behavior seen in simulations-with possible applications in determining the positions of shocks in real misaligned astrophysical systems. \nKeywords: black hole physics - accretion discs - magnetic field - Magnetohydrodynamics(1964) General relativity(641)', '1. INTRODUCTION': "Hot accretion flows have long been of astrophysical interest. These systems, in contrast to their thin-disk counterparts (Shakura & Sunyaev 1973), exhibit significantly reduced luminosity and operate with accretion rates far below the Eddington limit ( ˙ M ≪ ˙ M Edd ) (Xie & Yuan 2012; Abramowicz & Fragile 2013). This makes them suitable to explain the low/hard state of black hole X-ray binaries (Esin et al. 1997; Dexter et al. 2021) and in low-luminosity Active Galactic Nuclei (AGN) including the supermassive Black Hole in our Galactic Centre, Sagittarius A* (Sgr A*) (Yuan et al. 2002; Narayan & McClintock 2008; Yuan & Narayan 2014). \nTraditionally such systems are assumed to have their angular momentum aligned with the spin of the Black \nCorresponding author: Sajal Gupta \[email protected],[email protected] \nHole (BH). This presumption holds valid for thin, viscous accretion disks with small tilt angles, allowing viscous torques to gradually bring the inner flow into alignment (Bardeen & Petterson 1975; Liska et al. 2019). However, in the case of large tilt, nonlinear behavior such as the tearing of a disk are observed (Nixon & King 2012; Nealon et al. 2015; Liska et al. 2021; Kaaz et al. 2023), and it remains uncertain whether BardeenPetterson alignment can occur under these conditions. Alternatively, if the system has a high accretion rate, the spin of the BH may grow and eventually coincide with the angular momentum of the inflowing material (Volonteri et al. 2005). Furthermore, magneto-spin alignment driven by powerful relativistic jets can also play a crucial role in aligning the strongly magnetized disk with the BH spin (McKinney et al. 2013; Chatterjee et al. 2023). However, the low accretion rates and geometrically thick structure of hot accretion flows such as those observed in Sgr A* and M87 ( ˙ M ∼ 10 -7 -10 -9 M ⊙ /yr for Sgr A ∗ (Dexter & Fragile 2013; White et al. 2020)) \nmake it unlikely that the Bardeen-Petterson effect, cumulative mass accretion, or magneto-spin alignment can induce alignment. Consequently, these systems are more likely to remain misaligned, impacting various physical processes. These include growth rates of supermassive BHs (King et al. 2005; Stella & Vietri 1997), their spin measurements, and observable accretion rates. Additionally, the disk misalignment also affects the structure and size of the BH shadow (Dexter & Fragile 2011; White et al. 2020; Chatterjee et al. 2020) along with the formation and direction of the relativistic jet (King & Nixon 2018; Liska et al. 2019). For instance, recently Cui et al. (2023) observed that the radio jet in M87 precesses with a ∼ 11-year period, attributing this behavior to the misalignment of hot accretion flows. \nPapaloizou & Lin (1995); Nelson & Papaloizou (2000) were among the first to study the evolution of misaligned disks. They developed a linear theory focused on the bending wave regime, and conducted three-dimensional smoothed particle hydrodynamics (SPH) simulations, demonstrating that warps in the low-viscous flows propagate in a wave-like manner rather than diffusing. This causes the disk to remain misaligned even at inner radii, with the disk's plane oscillating as a function of radius (Ivanov & Illarionov 1997; Lubow et al. 2002). Later, Zhuravlev & Ivanov (2011), while developing an analytical model to describe the structure of twisted disk around slowly rotating Kerr BHs, proposed that these oscillations strongly depend on viscosity in the prograde flows and are notably absent for disks with viscosity parameter, α (Shakura & Sunyaev 1973), between 10 -3 and 10 -2 , and effective scale-height, H/r/ √ a ∼ 10 -2 , typically lower than those found in numerical studies of geometrically-thick disks (Fragile et al. 2007; White et al. 2019). Here, a ≡ cJ BH /GM 2 BH , ranging from 0 to 1, is the dimensionless BH spin parameter. Recently, White et al. (2019) applied the linear theory to their GRMHD simulations of tilted accretion flows and observed no such oscillations. Instead, the linear theory predicted an exponential increase in disk tilt with decreasing radius, whereas their study showed that disk modestly warps further away from alignment at large radii before a rapid decline in tilt is noticed, highlighting both the limitations of linear bending wave theory and the tendency of the inner flow to partially align. Throughout our study, we refer to this phenomenon as 'partial alignment', denoting the reduction in inclination angle of the inner disk in its steady-state configuration. Previously, Sorathia et al. (2013a,b); Hawley & Krolik (2018) question the relevance of linear theory as they observed that MHD turbulence in misaligned flows disrupts the radial communication of warps, stressing \nthe importance of strong warps driving non-linear effects such as shocks in the system. \nFragile et al. (2007) carried out simulations of a weakly magnetized hot accretion flow onto a BH of spin a = 0 . 9, and discovered a pair of standing spiral shocks forming close to the BH (Fragile & Blaes 2008). The location of shocks found in their study, along with other GRMHD simulations (White et al. 2019; Generozov et al. 2013), coincides with regions of rapid fall in inclination of the inner flow, raising speculation that the shocks might be responsible for the partial alignment of the inner disk (Mewes et al. 2016; Kawaguchi et al. 2015). Despite this spatial correlation, no conclusive evidence has yet been presented that shocks act to increase the alignment of the inner disk. \nMoreover, the potential mechanism for the formation of shocks in thick, tilted accretion disks is not yet fully understood. Previous studies led by Fragile & Blaes (2008) and Generozov et al. (2013) integrated the fluid trajectories in post-processing and suggested that the convergence of eccentric fluid orbits near the apocenters may be responsible for the shocks formation. However, their analysis were not able to explain the unique geometry of the shocks. It is also unclear how the spatial structure of shocks is affected by the bending and twisting of the disk as well as the spin of the BH. In light of these uncertainties, in this study we attempt to address this speculation of shock-induced partial alignment, and we describe the role of standing shocks in shaping the structure of the disk as well as illustrating a potential mechanism of their origin. \nTo this end we perform GRMHD simulations. Despite the fact that prior 3D SPH simulations have shown a warp in disk shape (Lubow et al. 2002; Nealon et al. 2015; Drewes & Nixon 2021), none have included a magnetic field in the system. In addition, none of them have observed shocks, which appear to be an important non-axisymmetric feature in thick tilted disks. Previous GRMHDsimulations from Fragile and collaborators utilize an artificial viscosity scheme to capture the effects of shocks, whereas the parameter survey led by White et al. (2019) was ran for insufficient time to attain inflow equilibrium across the inner disk. \nWe carry out a series of idealized 3D GRMHD simulations of weakly magnetized, geometrically-thick, tilted accretion disks called standard accretion and normal evolution (SANE) disks, initially making an angle of 24 · with the BH equatorial plane. We evolve the disks for three different BH spins: a = 0.5, 0.75, and 0.9375, allowing us to study the effect of BH spin on the disk structure. To compare the characteristics of misaligned disks with the conventional picture, we simulated an ac- \netion disk with no initial tilt onto a BH of spin a = 0.9375. We self-consistently evolve an entropy equation for the electrons using the Ressler et al. (2015) scheme. The latter enables us to calculate the irreversible heating rate on the fly, enabling us to resolve the spatial structure of the shocks. All the cases are evolved with the same initial conditions, thus making the comparison between different models free of this variable. A brief discussion on the simulation setup and initial conditions are given in section 2. We provide our findings in section 3, and we conclude and discuss our results in section 4.", '2. SIMULATION SETUP': "We performed idealized 3D GRMHD simulations using the publicly available HARMPI code 1 (a parallel, 3D version of HARM (Gammie et al. 2003; Noble et al. 2006)) that solves the GRMHD equations for a SANE disk around a rotating BH defined by the Kerr metric with metric signature ( -,+,+,+). We initialize our simulations with a steady-state hydrodynamic equilibrium gas torus (Fishbone & Moncrief 1976), with inner radius and pressure maximum at r in = 12 r g and r max = 25 r g . Here, r g ≡ GM BH /c 2 denotes one gravitational radii. The torus was seeded with a single loop of poloidal magnetic field, thus facilitating the magneto-rotational instability (MRI) to kickoff. The internal energy and the gas pressure are related by adiabatic index: P gas = (Γ -1) u g , where we chose Γ = 5 / 3. The magnetic field strength is normalized by fixing plasma β -parameter, β =max P gas / max P b = 100, where P b = b µ b µ / 2 is the magnetic pressure. To study the misaligned flows, we rotate the torus with 24 · about the BH equatorial plane, such that the initial angular momentum of the disk lies in the X-Z plane. We evolve the system in modified spherical-polar KerrSchild coordinates ( r, θ, ϕ ), with a grid resolution of 320 × 256 × 160 cells to adequately resolve the MRI turbulence. A higher concentration of cells is implemented in the BH equatorial plane to resolve the gas flow and at the pole to resolve the jet at larger radius. Due to the disk's tilt, orbital parcels traverse regions with varying polar resolution. Specifically, the change in polar resolution reaches up to ∆ θ ∼ 0 . 05 radians over the disk extent. In all the cases we studied, the numerical setup covers the region of (0.88 r H , 10 5 r g ) × (0, π ) × (0,2 π ) in radial, polar and azimuthal directions. Here, r H = r g (1 + √ 1 -a 2 ) is the BH event horizon radius. The outer radial boundary was extended to 10 5 r g using a superexponential radial coordinate. In our simu- \ne boundary conditions (BCs) in the θ -direction. Unlike transmissive polar BCs, which allow fluid to pass through the boundaries with minimal interference (Liska et al. 2018), reflective BCs can induce additional dissipation near the polar axis and potentially enlarge the jet artificially, influencing disk alignment (McKinney et al. 2013). To address these potential issues, we specifically exclude the polar axis region from our averages and remove regions with a magnetization ( σ ≡ b 2 ρ ) greater than one. \nWe utilized Ressler et al. (2015)'s approach and computed the total heating rate at each time step by measuring the difference between the entropy calculated through energy conservation equations and that from entropy conserving equation. Magnetic and/or kinetic energy dissipated at the grid scale due to the turbulence is recaptured as internal energy, which we use to identify shocks. Our procedure is similar to White et al. (2019), with the difference being that we calculated the heating rate at each simulation time-step, which allowed us to incorporate the effects of time-rate change of entropy. \nTo determine the radial dependence of physical quantities, we follow Penna et al. (2010) averaging approach, and define a density-weighted, time-averaged of quantity X as: \n⟨ X ⟩ ρ ( r ) = ∫ ∫ ∫ X ρ ( r, θ, ϕ ) dA θϕ dt ∫ ∫ ∫ ρ ( r, θ, ϕ ) dA θϕ dt (1) \nwhere dA θϕ ≡ √ -gdθdϕ is an area-element in spherical polar Kerr-Schild coordinates, and the integral over dt is a time average in the steady-state. The verticallyintegrated quantities are computed as \n⟨ X ⟩ ρ,θ ( r, ϕ ) = ∫ ∫ X ρ ( r, θ, ϕ ) √ -gdθdt ∫ ∫ ρ ( r, θ, ϕ ) √ -gdθdt (2) \nWe measured the disk thickness using: \nH/r = √ 〈 ( θ ' -θ 0 ) 2 〉 ρ,θ ' ,ϕ ' (3) \nwhere θ 0 ≡ π/ 2 + ⟨ ( θ ' -π/ 2) ⟩ ρ,θ ' ,ϕ ' is the disk midplane location. Here, ( θ ' , ϕ ' ) are the angular coordinates aligned with the disk, which are related to the coordinates ( θ, ϕ ) via Equation B1. Qualitatively, θ ' = π/ 2 represents the local midplane of the disk, and ϕ ' = 0 is set to the local precession angle. The notation ⟨ X ⟩ ρ,θ ' ,ϕ ' denoted a density-weighted, timeaveraged value of quantity X in disk aligned (or tilted) coordinates. At later times, the above equation yields H/r ∼ 0 . 3 -0 . 4 inside r = 20 r g . \nFigure 1. Snapshots from the equilibrium states contrasting the properties of a hot, aligned disk (top row) with tilted accretion flows at BH spin, a ≊ 0 . 94 (bottom row). The color scales are in arbitrary/code units. Bending of the disk is visible in the poloidal slice of the density ( ρ ) and the magnetization ( σ ) (leftmost column), showing the disk midplane (white line), on average, is tilted with respect to equatorial plane of the BH (Z BH = 0). Non-axisymmetric structures in density-weighted, verticallyaveraged snapshots of heating rate ( ˙ q , middle column) signify the standing shocks, which extend well beyond the r ISCO (white dashed circular line) and correlates with low specific angular momentum (( l = -u ϕ /u t , rightmost column) regions. Here r ISCO denotes the innermost stable circular orbit (ISCO) -the smallest possible radius for a stable circular orbit. \n<!-- image -->", '3.1. General Characteristics: Consequences of an initial tilt': 'The focus of this study is to investigate the steadystate properties of tilted flows. We evolve the simulations until a quasi-steady state is achieved, which we determined by monitoring the density-weighted radial profiles of tilt and twist, as well as the time it takes for standing shocks to fully develop. By these criteria, the tilted disks attain the inflow equilibrium up to r ∼ 20 r g within t ∼ 12000 r g /c . Tilt reaches equilibrium around t ∼ 10000 r g /c , while twist profiles develop from the beginning due to frame-dragging precession. We run the \nhigh spin case (a = 0.9375) for t ∼ 24000 r g /c , to be confident about the steady-state whereas the untilted disk is evolved only for t ∼ 10000 r g /c , as we notice negligible changes in relevant physical parameters. \nIn Figure 1, we showcase and compare misaligned accretion disk profiles with the standard aligned case for spin a ≊ 0 . 94. The general trend of the disk bending remains similar across other spins, as demonstrated in Appendix A. The 2D snapshots of different physical quantities for both scenarios are averaged during their respective steady states: from 8000 ≤ t ≤ 10000 r g /c for the aligned case (top row) and from 20000 ≤ t ≤ 22000 r g /c for the misaligned case (bottom row). Since both sim- \n<!-- image --> \nFigure 2. Density-weighted, shell averaged radial profiles of tilt (left) and twist (right) from each simulation (colors). The twist increases as the flow progresses closer to the BH due to frame-dragging precession, which increases with the spin of the BH. The increasing inclination with decreasing radius indicates that the disk warps further away from alignment as it accretes before beginning to align close to the BH (inside ∼ 6 r g ). This rate of partial-alignment also increases as the spin of the BH increases. \n<!-- image --> \nchieved steady states, averaging at different times does not represent distinct evolutionary stages. \nThe left column shows the poloidal slice of the restmass density ( ρ ) and the magnetization ( σ ≡ b 2 ρ ) for aligned and misaligned flows. Comparing these profiles reveals a pronounced disk bending in the tilted flow, indicating persistent misalignment at the steady-state. Perpendicular to the disk midplane (white arrow) is its angular momentum vector ( L disk ), pointing toward the low-density, high-magnetization region called the funnel region. This suggests that jets may emit in the direction of disk angular momentum, and may precess with the disk as remarked by Liska et al. (2018). Near the BH, the tilted flow shows two separate density arms, which are the regions of high compression formed due to standing shocks (Fragile & Blaes 2008; Generozov et al. 2013). \nTo identify standing shocks and their effects, we plot density-weighted, vertically averaged snapshots of the total heating rate ( ≡ ˙ q ), and the specific angular momentum ( l ≡ -u ϕ /u t ) in the central and right columns of Figure 1. In the aligned case (top row), both ⟨ ˙ q ⟩ ρ,θ and ⟨ l ⟩ ρ,θ are axisymmetric as expected. However, in the tilted flow (bottom row), the structure completely changes, and we find significant non-axisymmetric features in both quantities. Furthermore, the deviations from axisymmetry increase with BH spin, as illustrated in Figure 8. \nThe m = 2 azimuthal structure in heating rate ( ˙ q ) plots (cf. figure 1) signifies the pair of standing, spiral shocks. Here the m = 2 mode signifies the Fourier decomposition of the azimuthal structure in the heating rate. These shocks are localized, stronger, and have a more well-defined structure at higher spins. Comparing the heating rate and specific angular momentum in the tilted flow reveals that regions of depleted angular momentum correspond to the standing shocks, suggesting that they may enhance the transport of angular momentum. The geometry of the shocks in the disk-aligned frame is illustrated in Appendix E.', '3.2. Inclination and twist': "We measure the tilt as a function of disk radius by defining inclination as: \nI ( r ) ≡ cos -1 ( J BH · L disk ( r ) | J BH || L disk ( r ) | ) (4) \nwhere J BH = a ˆ z is the angular momentum of BH in natural units. The angular momentum of the disk: L disk is defined as: \n( L disk ) ˆ k = ϵ ijk r ˆ j ( T t ˆ k ) MA (5) \nwhere ϵ ijk is the three-dimensional anti-symmetric Levi-Civeta symbol, and the hats indicate the cartesian coordinates related to spherical coordinates in the conventional way. We call ( L x , L y , L z ) ≡ \nFigure 3. Density-weighted, shell averaged radial profiles of tilt (top row), dimensionless warp ( ˆ ψ ) (middle row) and warp amplitude (bottom row) for tilted simulations (colors). At small radii (shaded areas), the simulations exhibit increased warping ( ˆ ψ ( r ) ≥ 1), indicating the presence of non-linear effects, which coincide with regions of decreasing tilt (top row). \n<!-- image --> \n( L disk (ˆ x ) , L disk (ˆ y ) , L disk (ˆ z ) ). Using the above information, we can write density-weighted, average inclination as: \nI ( r ) = cos -1 ⟨ L z ⟩ ρ √ ⟨ L x ⟩ 2 ρ + ⟨ L y ⟩ 2 ρ + ⟨ L z ⟩ 2 ρ (6) \nand the twist as: \nT ( r ) = tan -1 ( ⟨ L y ⟩ ρ ⟨ L x ⟩ ρ ) (7) \nThe inclination I ( r ) is the measure of how elevated the disk midplane is from the BH equatorial plane, whereas the twist T ( r ) is the measure of the precession of the disk's angular momentum about the spin axis of BH. If the inclination and/or twist is a function of radius, the disk acquires a warped structure and is referred to as a warped disk as mentioned in section 1. \nFigure 2 shows the radial, steady-state profiles of averaged inclination ( I ( r )) [left panel] and twist ( T ( r )) [right panel]. The radial profiles of angles are averaged in their steady-state, particularly, 8000 ≤ t ≤ 10000 for the untilted case, 12000 ≤ t ≤ 14000 for a = 0 . 5, and 0 . 75, and 20000 ≤ t ≤ 22000 for the high spin case ( a ≊ 0 . 94). To get a sense of the accretion flow, it is best to read the plot from right to left. The twist sets up very early in the simulation due to frame-dragging precession and later relaxes to a smaller value due to the transport \nof angular momentum (White et al. 2019). The stronger twist in the high spin case is because frame-dragging precession is proportional to the spin of the BH (Lense & Thirring 1918). Overall, the shape of our twist profile indicates that the disk becomes extremely twisted as one approaches the BH. \nIn the left panel of Figure 2, a tiny tilt of approximately 3 · (grey line) is noticeable in the aligned case. Ideally, we expect a flat line of 0 · . This discrepancy arises due to numerical errors associated with taking the projections of the angular momentum of the disk on the BH equatorial plane, as the out-of-plane angular momentum projections, namely L x and L y , are not conserved. Nevertheless, the associated error is small, suggesting that our routine for determining the inclination is accurate to within about O (3 · ). Our definition also yields the initial tilt and twist far away from the BH, as we expect the tilt and twist to asymptote to an initial value of 24 · and 0 · , which we can see is true for all cases. \nAnalyzing the tilted profiles reveals that inclination increases as one approaches the BH, implying that the disk progressively bends further away from the BH's equatorial plane. Given the simulations' duration, which exceeds the inflow timescale, we conclude that the disk's warping is a stable characteristic, associated with bending waves. However, the predicted high-frequency oscillations from linear theory are virtually absent in our \ntilt profile. This could be due to the small viscosity in our flows, which may suppress the large oscillations and mitigate the radial communication of warps (Zhuravlev & Ivanov 2011). \nThe rapid decline in inclination, especially at very small radii ( r ≤ 6 r g ), causes the disk to partially align with the BH. This descent occurs at a significantly faster rate when the BH has a high spin, around a ≊ 0 . 94, indicating a direct influence of the BH's spin on the inner disk's physical structure. In our observations, we also confirm the presence of shocks in the disk. However, before quantifying their impact on the tilt profile, if any, it is essential to determine their radial extent. \nWe achieve this by identifying the region where the dimensionless warp parameter, represented as ˆ ψ = ψ/ ( H/r ), exceeds 1. Here ψ ( r ) ≡ ∣ ∣ ∣ rd l ( r ) dr ∣ ∣ ∣ signifies the measure of local warp and l ( r ) = L disk ( r ) / | L disk ( r ) | is the unit vector in the direction of disk's angular momentum at radius r. In the presence of small warp ( ˆ ψ > 0, but ≪ 1), strong oscillatory motions are generated in a tilted disk due to warp-induced radial pressure gradients (Ogilvie & Latter 2013). However, when ˆ ψ ( r ) > 1, nonlinear features can emerge. This occurs because, in this regime, local warp surpasses the disk's height, causing these oscillatory motions to couple with the warp. As a result, the radial velocity approaches the sound speed (Sorathia et al. 2013a,b), leading to compression. \nFigure 3 shows that ˆ ψ ( r ) > 1 occurs in the inner flows for spin a = 0 . 75 and 0 . 94 (shaded regions). This aligns neatly with the region of partial-alignment and coincides with the area where shocks form, as observed in the Figure 1. Notably, we do not identify any region with ˆ ψ ( r ) > 1 for the a = 0 . 5 case, as the warping in the lowest spin scenario is less pronounced due to minimal twisting and bending of the disk. In this case where ˆ ψ ( r ) < 1 we do not presume the absence of non-linear effects, but rather that they are much weaker than in the higher spin cases. It is essential to note that the condition ˆ ψ ( r ) > 1 is based on simple physical arguments and should not be interpreted rigidly, i.e. Sorathia et al. (2013b) chose an upper limit of 0.8 instead of 1. \nFigure 4 visualizes the averaged disk plane, represented as a series of rings observed from the perspective of the BH frame. Each ring is characterized by its individual tilt and twist, in accordance with the radial profiles displayed in Figure 2. Notably, the figure distinctly showcases the pronounced warping of the disk, particularly evident in the vicinity of the black hole where the twisting is more intense. \n3.3. Origin of the shocks \nFigure 4. Representation of the averaged disk plane, depicted as tilted and twisted rings for the a ≊ 0 . 94 case. The visualization emphasizes pronounced warping, particularly near the black hole (black sphere). The BH spin axis is vertical in the figure, where the viewpoint camera is located at an elevation of -130 · and an azimuth of -90 · with respect to the BH equatorial plane. \n<!-- image --> \nTo understand the formation of shocks and how they are affected by radial variations of tilt, and twist, we employed two different approaches. First, we draw insights from the work of Ivanov & Illarionov (1997) and developed a toy model following the prescription below: \n- 1. We define a series of centered elliptical orbits with an eccentricity of e = 0 . 1 2 in a disk-aligned frame, with the BH at the center. The diskaligned frame 3 is oriented such that its z-axis ( ≡ Z ' ) is parallel to the disk's angular momentum. The x-axis ( ≡ X ' ) is aligned with the line of nodes, and the y-axis ( ≡ Y ' ) completes the righthanded coordinate system. Initially, these orbits are untwisted, centered and are constrained to the X ' -Y ' plane.\n- 2. We then represent these orbits in the BH reference frame by calculating the angular coordinates ( θ, ϕ ) using Equation B1. This process manifest the geometry of the averaged tilted disk as seen from the BH frame. Figure 4 shows the orientation of such \nFigure 5. Density-weighted, vertically integrated snapshots of the heating rate ( ˙ q ) with centroid positions (black) for all tilted flows, overlaid with locations of maximum crowding of orbits from the toy model (blue) and the predicted shock locations from the basis vector model (green). The close agreement between the centroid locations and predicted shocks location from both methods at BH spins, a = 0 . 75 and ≊ 0 . 94, suggests that the shocks form due to the crowing of elliptical orbits in the flow. The significant discrepancy between the black and blue/green points for radii r > 6 r g is attributed to the large radial variation in the tilt profile compared to the twist. \n<!-- image --> \norbits viewing at a specific angle for the a ≊ 0 . 94 case. \n- 3. To locate the regions of maximum crowding, we identify the points where the distance between neighboring rings is minimized. These points are labeled as the sites of maximum crowding, which we predict to be the locations where shocks form.\n- 4. Finally, we superimpose the predicted sites of maximum crowding on the plot of verticallyintegrated heating rate ( ⟨ ˙ q ⟩ ρ,θ ) and compare them with observed shock-forming regions. \nSecond, based on the work of Ogilvie (1999); Ogilvie & Latter (2013), we defined local basis vectors for our warped disk with a few adjustments: \n- 1. m = rd l ( r ) dr , a unit vector orthogonal to l and pointing where adjacent warped annuli are most separated.\n- 2. ˆ n = l × m | l × m | , a unit vector pointing where adjacent warped annuli are least separated, and thus should point to the location of the shocks.\n- 3. Given the m = 2 nature of shocks, we also considered the vector -ˆ n , as both ˆ n and -ˆ n should point towards the least separated annuli.\n- 4. Finally, we superimposed the x and y components of both vectors, r ˆ n and -r ˆ n on the ( ⟨ ˙ q ⟩ ρ,θ ) plot, \nwhere we multiply by ' r ' to account for the shocks differing locations at different radii. \nThe findings from both our toy model and basis vector model are illustrated in Figure 5, where the black dots represent the centroid of vertically-averaged heating rate, and blue and green dots represent the predicted sites of shock location using the toy model and basis vector model, respectively. Both methods predict the shock locations quite well, especially for higher BH spins ( a ≊ 0 . 94 and 0 . 75). However, mapping the shock structure in the heating rate plot for the low spin case requires a comprehensive fluid trajectory analysis, as done by Generozov et al. (2013), for a meaningful comparison. Because shocks are weaker at lower spins, their m = 2 structure is less distinct. Consequently, using the centroid positions of the vertically-integrated heating rate as a proxy for shock locations introduces errors, particularly at larger radii and for the lowest spin. \nSince both methods are majorly governed by tilt and twist values, and their radial variation, we found that in the very vicinity of the BH, ( r ≤ 6 r g ), twist is the dominant factor in determining the spatial structure of the shocks (see Appendix E) Whereas at larger radii ( r > 6 r g ), the observed disparities between our sites of maximum crowding and the centroid positions are primarily due to the radial dependence of the tilt profile, as at these radii the twist in the disk changes more gradually with radius compared to the tilt. Thus, for radii r > 6 r g , the predicted shock locations are significantly \nFigure 6. Density-weighted, vertically integrated snapshots of change in the rate of inclination for BH spin, a ≊ 0 . 94. The sound agreement between the shock-forming locations depicted by centroid of ( ⟨ ˙ q ⟩ ρ,θ ) (black) and the regions where the tilt is decreasing demonstrates the possible role of shocks in partially aligning the inner disk with the BH. \n<!-- image --> \ninfluenced by the radial variation of the tilt profile. Our observations also indicate that while eccentric orbits are essential in our toy model for identifying regions of local density enhancement, the geometry of these shocks remains largely unaffected as long as the eccentricity remains small ( e < 0 . 2).", '3.4. Shock-induced partial alignment': "To understand the decrease in inclination within the inner flow, we investigate the rate of change of inclination along the path of a fluid parcel. This involves calculating the material derivative of inclination ( I ) using: \nd I dt = ∂ I ∂t + V · ∇I (8) \nwhere V is the coordinate three-velocity, the components of which are given by: V i = u i /u t . \nGiven our focus on the steady-state behavior of tilted flows, the first term on the right-hand side of the equation becomes zero. Expanding the second term yields three additional terms that describe the change in inclination of a fluid parcel subjected to a space-dependent velocity field. To assess the influence of shocks, we compute the vertically-integrated rate of change of inclination, represented as: \n⟨ V · ∇I⟩ ρ,θ = ⟨ V r ⟩ ρ,θ ∂ I ∂r + ⟨ V ϕ ⟩ ρ,θ ∂ I ∂ϕ (9) \nAll the quantities in this equation are density-weighted and vertically averaged within the σ ≤ 1 region to only consider the dense area of the flow and reduce errors \nassociated with high radial velocities in polar regions. In the above equation I is calculated using: \nI ( r, ϕ ) = cos -1 ⟨ L z ⟩ ρ,θ √ ⟨ L x ⟩ 2 ρ,θ + ⟨ L y ⟩ 2 ρ,θ + ⟨ L z ⟩ 2 ρ,θ (10) \nFigure 6 illustrates the evolution of inclination for a fluid parcel moving in the Eulerian frame for the highspin case ( a ≊ 0 . 94). We chose the high spin case because the shock structure is well-defined, and the decline in inclination is much more pronounced compared to lower spins. The plot shows that the region with a negative rate overlaps with the shock-forming region, identified by black dots from the centroid positions of vertically-averaged heating rate. Additionally, the rate exhibits a non-axisymmetric structure similar to the heating rate [cf. Figure 1]. This indicates that the tilt of a fluid parcel decreases as it passes through a shock front, demonstrating the critical role of shocks in reducing the tilt of inner flows. However, fluid parcel also experiences an increase in tilt as it moves out of shock forming regions. This feature, where d I dt is positive, can be attributed to the angular momentum fluxes along with the gravitational torque. \nHowever, it is important to note that the centroid positions do not mark the exact location of the shocks front. Moreover, calculating the rate of change of inclination in the Eulerian frame involves numerical derivatives, which can introduce small errors. Despite these limitations, the regions with decreasing inclination are consistent with the shock-forming regions. The rate of \nFigure 7. Plot comparing the alignment timescale (blue) to the accretion timescale (orange) within the disk. Both timescales are normalized by the local orbital period ( t dyn ). The fact that the gas is accreting much faster than it is aligning indicates the inability of the inner disk to fully align with the spin of the BH. \n<!-- image --> \nchange of inclination for a = 0 . 5 and 0 . 75 are shown in Figure 9. \nFurthermore, we observe that the second term on the right-hand side of equation 9-which represents the flux transport of the change in inclination of a gas parcel through azimuthal motion, is dominant and largely determines the V ·∇I . This causes the sign of d I /dt to flip at the shock position. To ensure the robustness of this result, we performed additional analyses where we defined the rate of change of inclination in three different ways, as detailed in Appendix D. Although these definitions showed minor differences, the overall features of the rate of change of inclination variable remained consistent. \nA natural question arises: if shocks are promoting alignment within the inner disk, why do we not observe complete, mutual alignment between the disk and the black hole, akin to the Bardeen-Petterson effect? Why is partial alignment the prevailing outcome? To address this question, we introduce an alignment timescale: t align ≡ I ( r, t ) / ( -d I ( r, t ) /dt ). The inclusion of a minus sign ensures a positive timescale, considering our expectation that d I ( r, t ) /dt < 0 during the alignment process. \nWe compared this timescale with the inflow timescale: t inflow = r/ ( -⟨ V r ⟩ ρ ). Our hypothesis is that the disk does not completely align because the gas is forced to flow radially inward at a much faster rate than the time it requires to dissipate its out-of-plane angular momentum entirely. Thus, we examine the relationship between the alignment timescale and the inflow/accretion timescale. In the above definitions, ⟨ V r ⟩ ρ represents density-weighted, time averaged radial velocity. The alignment timescale is calculated by taking the ratio of \nradial inclination profile (cf. Figure 2) to the azimuthal average of d I /dt that we calculated using Equation (8). \nFigure 7 illustrates these timescales as a function of radius for the misaligned flow around the BH of spin, a ≊ 0 . 94. The y-axis is normalized by the dynamical timescale ( t dyn = 1 / Ω, where Ω = dϕ dt = ⟨ V ϕ ⟩ ρ ), representing time in periods of flow rotation. The figure depicts that in the inner regions where the tilt is decreasing, t align exceeds t inflow by a factor of ∼ 2 -100. This indicates that gas lacks the required time to attain mutual alignment at a specific radius since it's compelled to migrate inward at a considerably faster pace, preventing it from fully shedding its out-of-plane angular momentum, and hence, leaving the inner disk to partial-alignment only.", '4. CONCLUSIONS & DISCUSSION': "We have presented a series of global, idealized 3D GRMHD simulations of misaligned accretion onto a black hole with different spins, ranging from a = 0 . 5 to 0 . 9375, with an initial inclination of 24 · . Comparing the misaligned simulations to an aligned accretion flow reveals that in all tilted flows, the disks evolve to a highly warped and twisted steady-state structure showing no signs of the Bardeen-Petterson alignment (Bardeen & Petterson 1975). We do, however, discover a potential aligning process for weakly-magnetized, thick accretion disks ( H/r > α ) that brings the inner disk into partial alignment with the BH. The standing shocks in the tilted flows are torquing the inner disk into the BH equatorial plane, likely through the redistribution of angular momentum via the redirection of fluid elements. For instance, while analyzing the evolution of inclination of a comoving fluid [cf. Equations (8) and (9)], we observe that the decrease in inclination at the shock location is far greater due to azimuthal motion than radial motion, suggesting the orbital mixing of fluid elements of different orientations of angular momentum. This importance of azimuthal mixing is also hinted at Liska et al. (2019) and Sorathia et al. (2013a), as Sorathia et al. (2013a) postulated that for alignment to occur azimuthal mixing of angular momentum is required-the second term in equation (9) may quantify such a mechanism. Overall, the presence of spiral shocks results in an even angular momentum distribution (or, less non-axisymmetric) in the azimuthal direction. \nHowever, it is also possible that the partial alignment we observe is a result of the oscillatory nature of bending waves, as suggested by Ivanov & Illarionov (1997) and Zhuravlev et al. (2014). Given that the nonlinear features arise close to the BH, this decrease in inclination \ncould result from oscillatory behavior associated with standing, nonlinear bending waves. \nOur study reveals that the strength of shocks increases with the BH spin, in line with findings of White et al. (2019). This relationship offers an explanation for the absence of shocks observed by Teixeira et al. (2014) for a BH spin of a = 0 . 1. Our findings that shocks partially align the inner flow is consistent with the steeper alignment rate observed in high-spin cases. White et al. (2019) demonstrated that shocks become more pronounced as the initial misalignment of the disk increases for the same BH spin. Combining this finding with our shock-induced partial alignment, it is conceivable that in severely misaligned flows, the inner flow may undergo complete alignment with the spin of the BH. In future research, we intend to develop models to quantify the strength of shocks as a function of varying initial tilt and BH spins, enabling us to better understand the process of alignment between a disk and its black hole. \nWhile our simulations exhibit general characteristics consistent with previous GRMHD studies of hot, tilted flows (Fragile et al. 2007; Fragile & Blaes 2008; Generozov et al. 2013; Teixeira et al. 2014), and to a certain extent in warped thin disks (Kaaz et al. 2023), we did not observe definitive evidence of significant global precession in our system. In the case of high BH spin ( a ≊ 0 . 94), simulated over an extensive period of t ≈ 25 , 000 r g /c , our warped disk exhibited only a modest precession of ≈ 10 · for the radial shell of 15 to 50 r g , suggesting a global precession timescale of ≈ 10 6 r g /c ≈ 0 . 5( M/M ⊙ ) s . This timescale is consistent with the predictions derived from the formulation provided by (Fragile et al. 2007, Equation (43)), where we took inner and outer radius of the evolved disk to be, r i ∼ 15 r g , r 0 ∼ 50 r g as fiducial parameters 4 , and ζ ∼ -0 . 83 (derived from Σ = Σ i ( r/r i ) -ζ ) as estimated in our study. This observation suggests the possibility of the tilted disk precessing as a rigid body under the influence of Lense-Thirring torque (Lense & Thirring 1918). For lower spins, however, the simulation duration was insufficient to capture significant precession, thus limiting our ability to draw meaningful conclusions. \nThe results of our toy model confirm the arguments of Ivanov & Illarionov (1997) and their close agreement with the local basis vector method suggests that nodes of the shocks form predominantly at ± r ˆ n , i.e. at the location where the adjacent warped annuli are least sepa- \nted. Additionally, the simplicity of our model enables us to quantify the effects of tilt and twist. where we discover that the twist predominantly governs the geometry of the shocks, but their location can change if the disk is significantly misaligned. Moreover, while global precession was not prominently observed in our simulations, it is conceivable that, akin to the findings of Fragile & Blaes (2008), spiral shocks might have precessed on a similar timescale if global precession had been present. Considering the pivotal role of twist in determining the locations of maximum crowding, it is plausible that our toy model could predict the location of shocks in scenarios involving global precession of the disk. \nOur model does, however, have limitations. First, it cannot predict the geometry of the shock from the initial misalignment of the torus. Complete knowledge of tilt and twist is needed to predict the spatial structure of shocks. One can avoid performing the simulations by predicting the tilt and twist profile using the linear theory, but as shown by White et al. (2019), the problem is highly non-linear. Secondly, because we use the radial profiles of tilt and twist, we could not predict the vertical structure of the shocks. To determine this we would need to completely track the full parameter variations, such as density, pressure, and velocity across the shock front. Despite having these limitations, our model shows great potential for locating shocks and can be applied to observations of warped disks. While it is true that finding the tilt and especially twist of real warped disks is a highly complex task, one could take the jet inclination as an average tilt and use this as an initial condition in simulations to potentially predict where shocks will develop. \nThe increase in inclination at the outer radii is consistent with the physics of bending waves. However, as shown by White et al. (2019), the straightforward application of the linear theory does not properly yield the observed profile of tilt. The high sensitivity of linear theory to the surface density profiles, Shakura-Sunyaev α viscosity, and the number of assumptions that are violated in GRMHD simulations make the predictions of linear theory inapplicable. An uncharted territory being explored by Zhuravlev & Ivanov (2011) is deriving the governing equations for each component of angular momentum and assessing the contribution of the torque produced due to frame-dragging of the spacetime, pressure forces, Reynolds stress, and Maxwell stress. Approaching the problem in the said manner may also help in determining the source of shocks-induced partial alignment of the inner disk and the dominance of V ϕ ∂ I ∂ϕ in governing d I /dt . \nThe timescale analysis shows why the inner disk never fully aligns. We define an alignment timescale and observed that t align > t inflow , suggesting that the gas does not have enough time to dissipate its out-of-plane angular momentum. While this is a simple, dimensionalitybased estimate, it explains our results quite well and we believe it to be a good order of magnitude check. Also, as one would expect from the 'alignment timescale', our definition is moot in the outer regions of the disk where the flow is significantly more misaligned. \nFinally, standing shocks can accelerate the plasma, and can be the sites of particle acceleration and nonthermal electron distribution in collisionless accretion flows like for Sgr A ∗ (Fragile & Blaes 2008). We suggest that a radiative 3D GRMHD simulation of a misaligned accretion disk rotating around a fast-spinning BH can help us to identify the signatures of standing shocks in spectra. It may also help us to quantify the difference between the spectrum of aligned and misaligned flows. Furthermore, it will allow us to investigate the argu- \nnts of Zhuravlev & Ivanov (2011), who claimed that strong irradiation of the outer parts of the disk by the inner parts could change the spectra of the disk. To effectively probe the effects of shocks, we also propose a parameter survey similar to this, incorporating passive scalars as tracers to investigate the fluid trajectories across the shocks front.", '5. ACKNOWLEDGMENTS': "We thank M. C. Begelman, C. J. White, K. Long, C. Echibur'u -Trujillo, and N. Scepi for stimulating discussions related to this work. We thank the anonymous reviewers for their insightful comments and suggestions, which have significantly improved our research. This work was supported in part by National Science Foundation award AST-2034306, the NASA Astrophysics Theory Program grant 80NSSC20K0527, and by an Alfred P. Sloan Research Fellowship (JD). \nSoftware: Matplotlib (Hunter 2007), NumPy (Harris et al. 2020)", 'A. PROPERTIES OF a = 0 . 5 and 0 . 75 TILTED FLOWS': "The purpose of this appendix is to describe the properties of tilted flows for spins a = 0 . 5 and 0 . 75. In the section 3.1, we describe the warping of the disk and the emergence of non-axisymmetric features in vertically-integrated profiles of ⟨ ˙ q ⟩ ρ, θ and ⟨ l ⟩ ρ, θ for spin a ≊ 0 . 94. These profiles reveal the presence of standing shocks. Similar characteristics are observed in tilted flows with spins a = 0 . 5 and 0 . 75, as depicted in Figure 8. \nThe warping of the disk is evident from poloidal slices of density-magnetization profiles, ρ -σ . As expected, the low-density funnel region, aligned with areas of high magnetization ( σ > 1), stands perpendicular to the disk midplane. Comparing σ profiles of tilted flows with spins a = 0 . 5, 0 . 75, and ≊ 0 . 94 (Figure 1), we note a reduction in the regions with strong magnetization as the BH's spin increases. This phenomenon is attributed to more substantial vertical motions, notably noticeable in the high-spin case, demonstrated in Figure 2, which exhibits pronounced warping. \nThe vertically averaged snapshots of the heating rate ( ˙ q ) reveal the presence of standing shocks. Notably, the degree of non-axisymmetry intensifies as the BH's spin increases, suggesting that the strength of the shocks and their impact on disk partial-alignment and angular momentum transport escalates with higher BH spin.", 'B. TILTED FRAME ANGULAR COORDINATES': "To assess parameters within the tilted frame, which aligns with the disk, we utilize angular coordinates ( θ ' , ϕ ' ) linked to grid coordinates ( θ , ϕ ) through Equation (7) in White et al. (2019). For thoroughness, we present the relationship here: \nθ ' = cos -1 (cos I cos θ +sin I cos T sin θ cos ϕ +sin I sin T sin θ sin ϕ ) , ϕ ' = tan -1 ( -sin T sin θ cos ϕ +cos T sin θ sin ϕ, -sin I cos θ +cos I cos T sin θ cos ϕ +cos I sin T sin θ sin ϕ ) , θ = cos -1 ( -sin I sin θ ' cos ϕ ' +cos I cos θ ' ) , ϕ = tan -1 (sin I sin T cos θ ' +cos I sin T sin θ ' cos ϕ ' +cos T sin θ ' sin ϕ ' , sin I cos T cos θ ' +cos I cos T sin θ ' cos ϕ ' -sin T sin θ ' sin ϕ ' ) (B1) \nFigure 8. Similar to Figure 1, the above plot shows the properties of the tilted flows for BH spin, a = 0 . 5 and 0 . 75. Warping of the disk and standing shocks are apparent from poloidal slices of ρ -σ and density-weighted, vertically averaged ⟨ ˙ q ⟩ ρ,θ and ⟨ l ⟩ ρ,θ profiles. The strength of the shocks is shown to be proportional to the BH spin. However, unlike shocks strength, the strongly magnetized region shrinks with increasing spin due to strong vertical motions that result from enhanced warping. \n<!-- image -->", 'C. RATE OF CHANGE OF INCLINATION FOR a = 0 . 5 and 0 . 75 TILTED FLOWS': 'This section delves into the rate of change of inclination for BH spin values of a = 0 . 5 and 0 . 75. Figure 9 shows that the rate is largely negative in the inner flow, especially for a = 0 . 75. This is consistent with the radial drop in the tilt profile seen in Figure 2. The scattered black points in the plot correspond to centroid positions of vertically-integrated heating rate, which we used as proxy for the determining the location of the shocks front. For the a = 0 . 75 scenario, we observe a sound agreement between the positions of shocks and regions exhibiting negative inclination rates. This suggests that the shocks affect the inclination of disk, particularly compelling the inner flow to align with the spin of the BH. \nHowever, for the a = 0 . 5 scenario, the situation is less straightforward. This is because the shocks are relatively weak, suggesting that their influence is subtler and less discernible through our simple analysis of d I /dt . Additionally, the observed tilt reduction for the low spin is less than 2.5 · , possibly due to errors related to angular momentum projections. \nFigure 10 presents the resulting calculations using the said methods for the high spin case. As evident from the figure, there are noticeable discrepancies in the measured values of d I dt , particularly in the vicinity of the BH. The differences between Methods 2 and 3 are straightforward to interpret: Method 2 considers only the change in the tilt of a fluid parcel as it moves with the mean fluid velocity, whereas Method 3 incorporates both mean and turbulent fluid \n<!-- image --> \nFigure 9. Similar to Figure 6, the above plot shows the change in rate of inclination for the spins of BH a = 0 . 5 (left) and 0 . 75 (right). The good agreement between the black points describing the location of shocks and the regions of negative rate for a = 0 . 75 case elucidates the impact of shocks in aligning the inner disk. However, because the shocks are weak in the low spin case, the decline we observe in the tilt of the inner disk may be due to numerical errors. \n<!-- image -->', 'D. DIFFERENT METHODS TO COMPUTE RATE OF CHANGE OF INCLINATION': 'In this section, we evaluate whether the rate of change of inclination measured in the Eulerian frame at steady-state varies depending on the definition employed. The following table outlines three distinct methods, all based on the same fundamental principle, d I dt = V · ∇I . \nTable 1. Comparison of different methods for calculating d I dt . Method 1 uses vertically-averaged velocities and inclination, Method 2 uses mean fluid velocity and three-dimensional inclination gradients, and Method 3 incorporates both mean and turbulent fluid velocities and three-dimensional inclination gradients. \nFigure 10. The above plot shows the rate of change inclination ( d I dt ) for the BH spin a ≊ 0 . 94, calculated using three different methods. While these methods showed minor differences, the overall features of the plot remained consistent, supporting the reliability of our findings despite the potential numerical errors. \n<!-- image --> \nvelocities. The deviation between Method 1 and other two methods arises primarily due to the different definitions of inclination we chose. Despite these variations, the rates measured by all methods display similar non-axisymmetric structures and regions of decreasing inclination, aligning consistently with the non-axisymmetric structure in the vertically-integrated heating rate. We selected Method 1 for our primary analysis because it defines inclination as the angle between the angular momentum vector of the vertically-averaged disk and the BH spin axis.', 'E. GEOMETRY OF THE SPIRAL SHOCK IN THE DISK-ALIGNED FRAME': "To substantiate the robustness of the m = 2 standing shocks, we quantified the heating rate within the disk-aligned frame. Initially, we transformed the metric tensor to the tilted frame using the relation g µ ' ν ' = ∂x α ∂x µ ' ∂x ζ ∂x ν ' g αζ , where g αζ represents the metric defined in Kerr-Schild coordinates. The Jacobian ( ∂x α ∂x µ ' ) for this transformation is detailed in White et al. (2019). Subsequently, we computed the area element in the new coordinates as √ -g ' , and calculated the angular coordinates θ ' and ϕ ' following the equations outlined in Appendix B. These calculations revealed that at specific radii, the angular coordinates were neither uniformly distributed nor covered the complete polar ([0 , π ]) and azimuthal ([0 , 2 π ]) ranges. \nTo address this, we constructed a uniform angular grid and interpolated the relevant physical quantities onto it. The density-weighted vertical integration of the heating rate was then performed using the equation: \n⟨ ˙ q ⟩ ρ,θ ' ( r, ϕ ' ) = ∫ ∫ ˙ q ( r, θ ' , ϕ ' ) ρ ( r, θ ' , ϕ ' ) √ -g ' dθ ' dt ∫ ∫ ρ ( r, θ ' , ϕ ' ) √ -g ' dθ ' dt (E2) \nFigure 11 illustrates these calculations for the high-spin tilted case and corroborates the persistence of the standing shocks. Notably, the nodes of the shocks predominantly align with the Y Disk = 0 plane, corresponding to a local precession angle of ϕ ' = 0 and π . 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2015ARA&A..53...51S | Modeling galaxy formation in a cosmological context presents one of the greatest challenges in astrophysics today due to the vast range of scales and numerous physical processes involved. Here we review the current status of models that employ two leading techniques to simulate the physics of galaxy formation semianalytic models and numerical hydrodynamic simulations. We focus on a set of observational targets that describe the evolution of the global and structural properties of galaxies from roughly cosmic high noon z 23 to the present. Although minor discrepancies remain overall models show remarkable convergence among different methods and make predictions that are in qualitative agreement with observations. Modelers have converged on a core set of physical processes that are critical for shaping galaxy properties. This core set includes cosmological accretion strong stellardriven winds that are more efficient at low masses black hole feedback that preferentially suppresses star formation at high masses and structural and morphological evolution through merging and environmental processes. However all cosmological models currently adopt phenomenological implementations of many of these core processes which must be tuned to observations. Many details of how these diverse processes interact within a hierarchical structure formation setting remain poorly understood. Emerging multiscale simulations are helping to bridge the gap between stellar and cosmological scales placing models on a firmer more physically grounded footing. Concurrently upcoming telescope facilities will provide new challenges and constraints for models particularly by directly constraining inflows and outflows through observations of gas in and around galaxies. | 2015-08-01T00:00:00Z | ['2015ARA&A..53...51S', '2014arXiv1412.2712S', '10.1146/annurev-astro-082812-140951', '10.48550/arXiv.1412.2712', 'arXiv:1412.2712'] | ['Astrophysics - Astrophysics of Galaxies'] | Physical Models of Galaxy Formation in a Cosmological Framework | 2,015 | 168 | 0.71 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF'] | 1,219 | https://arxiv.org/pdf/1412.2712.pdf | {'Physical Models of Galaxy Formation in a Cosmological Framework': 'Rachel S. Somerville \nDepartment of Physics and Astronomy, Rutgers University [email protected] \nRomeel Dav´e \nDepartment of Physics, University of the Western Cape, Cape Town South African Astronomical Observatories, Cape Town African Institute for Mathematical Sciences, Cape Town [email protected]', 'Key Words': 'galaxy formation, galaxy evolution, numerical simulations, cosmology', 'Abstract': 'Modeling galaxy formation in a cosmological context presents one of the greatest challenges in astrophysics today, due to the vast range of scales and numerous physical processes involved. Here we review the current status of models that employ two leading techniques to simulate the physics of galaxy formation: semi-analytic models and numerical hydrodynamic simulations. We focus on a set of observational targets that describe the evolution of the global and structural properties of galaxies from roughly Cosmic High Noon ( z ∼ 23) to the present. Although minor discrepancies remain, overall, models show remarkable convergence between different methods and make predictions that are in qualitative agreement with observations. Modelers seem to have converged on a core set of physical processes that are critical for shaping galaxy properties. This core set includes cosmological accretion, strong stellar-driven winds that are more efficient at low masses, black hole feedback that preferentially suppresses star formation at high masses, and structural and morphological evolution through merging and environmental processes. However, all cosmological models currently adopt phenomenological implementations of many of these core processes, which must be tuned to observations. Many details of how these diverse processes interact within a hierarchical structure formation setting remain poorly understood. Emerging multi-scale simulations are helping to bridge the gap between stellar and cosmological scales, placing models on a firmer, more physically grounded footing. Concurrently, upcoming telescope facilities will provide new challenges and constraints for models, particularly by directly constraining inflows and outflows through observations of gas in and around galaxies.', '1 INTRODUCTION': "The past decade has seen remarkable progress in measuring the properties of galaxies across the electromagnetic spectrum and over the majority of cosmic history. Wide-field surveys have collected samples of millions of nearby galaxies, spanning roughly six orders of magnitude in galaxy mass and a rich range of galaxy types and environments, from isolated galaxies in voids to rich clusters. Medium-deep surveys have collected samples of tens of thousands of galaxies out to z ∼ 6, and ultra-deep surveys have identified samples of hundreds to thousands of galaxy candidates at z ∼ 6-8, with a few candidates identified (mainly behind lensing clusters) at z ∼ 9-10 (for an overview of recent surveys see Madau & Dickinson 2014). The pan-chromatic wavelength coverage enabled by a suite of ground and space based telescopes has allowed detailed Spectral Energy Distributions (SED) to be constructed for large samples of galaxies, which make it possible to estimate photometric redshifts for galaxies that are too faint to readily obtain spectroscopy, and to estimate intrinsic parameters such as stellar masses and star formation rates (SFR). \nIn addition, high spatial resolution imaging, primarily from the Hubble Space Telescope (HST), and spectroscopy including data from an increasing number of surveys using Integral Field Spectrographs have enabled us to study galaxies' internal structure and kinematics. In particular, the Wide Field Camera 3 on HST has made it possible to study galaxy structure and morphology in the rest-frame optical back to 'Cosmic Noon' - the peak of cosmic star formation (SF) and black hole (BH) accretion activity at z ∼ 2-3 (Madau & Dickinson 2014). We are truly living in a golden age of facilities and databases for studying how galaxies formed and evolved. \nConcurrently over the past decade, advances in numerical methodologies and computing speed have allowed extraordinary progress in our ability to simulate the formation of structure within the paradigm of the Cold Dark Matter (CDM) model (e.g. Klypin et al. 2011, Springel et al. 2005c). A variety of techniques have been developed for computing detailed predictions for the expected observable properties of galaxies based on ab initio (albeit approximate) treatments of the physical processes expected to be important in shaping galaxy formation and evolution. The project of genuinely ab initio computational simulation of galaxy formation is beyond current capabilities, owing to the vast range of spatial scales involved, from the sub-pc scales of individual stars and supernovae, and accretion disks of supermassive BH, to the super-Mpc scales of the 'cosmic web', and to the wide array of poorly understood physical processes. However, by zeroing in on different scales through many different approaches, models are providing fundamental insights into the physical processes that are responsible for molding galaxy properties. Cosmological galaxy formation models have now matured into an essential tool for understanding galaxy evolution, and hence it is timely to review this topic. \nWe focus specifically on state-of-the-art physically-motivated cosmological models of galaxy formation, and ask a series of questions: 1) How well are these models able to predict or reproduce the observed distribution functions of global galaxy properties such as stellar mass, and the evolution of these functions? 2) How well do the models reproduce global scaling relations such as correlations between stellar mass, cold gas fraction, SFR and metallicity? 3) What do these models predict for the demographics of different types of galaxies (e.g. star forming vs. quiescent, or disk-dominated vs. spheroid-dominated)? 4) Are models able to reproduce observed structural scaling relations such as those relating mass with radial size, density, and internal velocity, and the evolution of these relations for different types of galaxies? 5) With regard to all of these questions, what insights have we \ngained into the process of galaxy formation from the successes and failures of our current models? \nThe plan for the rest of this article is as follows. In § 1.1, we give a broad overview of the observational results that we will target in our review. In § 1.2, we review the cosmological background, mainly pointing the reader to other sources. In § 1.3 we give a brief overview of the physical processes that are included in most models of galaxy formation, and in § 1.4 we introduce different tools for modeling galaxy properties. In § 2 we give a more detailed description of the methods used in the models that we will discuss in the remainder of the article, which include numerical hydrodynamic simulations and semi-analytic models. In § 3, we discuss the 'sub-grid' modeling connected with physical processes that are not directly resolved in cosmological simulations, including the formation of stars and supermassive black holes (SMBH), and the impact of 'feedback' from these objects on forming galaxies. In § 4, we discuss the predictions of current models and how they measure up to observations for global properties of galaxies ( § 4.1) and galaxy internal structure and kinematics ( § 4.2). We conclude with a summary and outlook in § 5. A glossary of acronyms is provided at the end of the paper.", '1.1 Observational Targets': 'In this review, we focus on the global and structural properties of the stellar components of galaxies from roughly Cosmic High Noon ( z ∼ 2-3) to the present. We acknowledge that there are many important observations that provide crucial constraints on models that lie beyond this scope. The summary here is quite brief; we will refer the reader to other recent reviews and papers for a more comprehensive overview.', 'Multi-wavelength imaging sur-': "1.1.1 Global Properties: Distribution Functions veys complemented with photometric or spectroscopic redshifts yield estimates of familiar global galaxy properties such as the luminosity and color at various rest-frame wavelengths from the UV to far-IR. In recent years, it has become popular to estimate stellar masses by fitting galaxy SEDs with simple parametric models of galaxy star formation histories combined with stellar population models (Conroy 2013, Walcher et al. 2011). Star formation rates are also estimated using SED modeling, or roughly equivalently using extinctioncorrected rest-UV measures, but more reliably by adding mid- to far-IR photometry and/or nebular emission lines such as H α . We refer to the comoving number density of galaxies as a function of a global property such as luminosity or stellar mass as a distribution function . It has long been known that galaxy distribution functions typically have a characteristic shape often described by a Schechter function (Schechter 1976), which is parameterized by a normalization, a turn-over, and an asymptotic slope to low masses. Examples of luminosity functions, stellar mass functions, and the cold gas (atomic hydrogen) mass function of nearby galaxies from recent large surveys are presented in the review by Blanton & Moustakas (2009). \nTo higher redshifts, galaxy rest-frame optical-NIR luminosity functions and stellar mass functions (SMF) have been measured from medium-deep surveys out to z ∼ 4. At higher redshifts, SMF estimates exist but rely on stellar mass estimates from rest-UV fluxes, which are likely less robust. These measurements have yielded a number of important insights into galaxy assembly: 1) galaxies appear to be continuously building up their mass over cosmic time, in accord with the hierarchical formation picture, and inconsistent with monolithic collapse (Madau & Dickinson 2014). 2) The number density of massive galaxies ( m star > \nM char , where m star is the stellar mass and M char is the characteristic mass in the Schechter function) increases rapidly from z ∼ 4-2, but then stays nearly constant or increases slowly from z ∼ 2-0, indicating that massive galaxies formed and assembled their stars relatively early (Marchesini et al. 2009, Moustakas et al. 2013, Muzzin et al. 2013). 3) The comoving number density of low mass galaxies ( m star < M char ) increases more rapidly than that of more massive galaxies at z < ∼ 1-2, indicating that low mass galaxies formed their stars later and over a longer timescale. This result is sometimes called 'mass assembly downsizing' (Cimatti et al. 2006). \nIt has long been known that the color-luminosity distribution of galaxies is strongly bimodal (e.g. Baldry et al. 2004), with most galaxies falling onto a relatively narrow (in optical colors) 'red sequence' or a broader 'blue cloud'. Spectroscopic indicators of stellar population age as well as UV and IR photometry have confirmed that in the local Universe, the 'red sequence' is largely comprised of 'quiescent' galaxies with predominantly old stellar populations, while the 'blue cloud' represents 'star forming' galaxies with younger stellar populations and significant ongoing star formation (Brinchmann et al. 2004, Kauffmann et al. 2003, Salim et al. 2007, Schiminovich et al. 2007). Because of the strongly bimodal nature of the population, it has become common to draw a line either in the color-luminosity or color-mass plane, or in the specific SFR (sSFR ≡ SFR/m star ) versus m star plane, and to speak of 'red' and 'blue' galaxies or 'star forming' and 'quiescent' galaxies. \nRecent deep surveys have shown that these two populations (star forming and quiescent) can be clearly identified at least up to z ∼ 2, and perhaps up to higher redshifts z ∼ 34 (Brammer et al. 2011, Muzzin et al. 2013). Intriguingly, it appears that the comoving number and mass density of quiescent galaxies has been increasing over time since z ∼ 2, while the number and mass density of star forming galaxies has stayed roughly constant or decreased during this same interval (Bell et al. 2004, 2007, Brammer et al. 2011, Faber et al. 2007, Muzzin et al. 2013). Given that it is the star forming population that is expected to be growing more massive due to the birth of new stars, this result has profound and unexpected implications - it implies that more and more star-forming galaxies must be having their star formation extinguished or 'quenched' as cosmic time progresses. \n1.1.2 Global Properties: Scaling Relations Galaxies show many correlations between their global properties. We refer to such a correlation as a 'scaling relation' when the conditional value of galaxy property y for a fixed value of another property x has a relatively small scatter. Stellar mass is often used as the x variable in galaxy scaling relations. Some well-known examples of global scaling relations with m star are the SFR for star forming galaxies, sometimes known as the 'star forming main sequence' (SFMS; Noeske et al. 2007, Wuyts et al. 2011), the mean fraction of cold gas ( f gas ≡ m gas /m star ) in the interstellar medium (ISM) (Baldry et al. 2008, Peeples & Shankar 2011), and the metallicity of stars or ISM gas (mass-metallicity relation, MZR; Gallazzi et al. 2005, Tremonti et al. 2004, Zahid et al. 2013). Furthermore, some of the tightest known scaling relations in astronomy are those between galaxy properties and the mass of the SMBH they harbor (see Kormendy & Ho 2013, for a comprehensive review). \nDeep multi-wavelength surveys have provided constraints on the evolution of these scaling relations. The normalization of the SFMS has declined by a factor of ∼ 20 since z ∼ 2 (Speagle et al. 2014, and references therein), and a fairly tight sequence appears to be in place up to z ∼ 6 (Salmon et al. 2014, Steinhardt et al. 2014). The MZR seems to have evolved in the sense that galaxies of a given mass had lower gas-phase metallicities at high redshift (Erb et al. 2006, Savaglio et al. 2005, Steidel et al. 2014, Wuyts et al. 2014, Zahid \net al. 2013). There is evidence from measurements of CO (an indirect tracer of molecular hydrogen, H 2 ) in fairly massive high redshift galaxies that the gas fraction of galaxies has decreased significantly over cosmic time since z ∼ 2 (Bothwell et al. 2013b, Genzel et al. 2014, Saintonge et al. 2013, Tacconi et al. 2010, Tacconi et al. 2013). Indirect estimates of cold gas fractions from inverted star formation densities, assuming a fixed relationship between star formation density and cold gas density, also indicate a rapid decrease in cold gas fraction from z ∼ 2 to the present (Erb et al. 2006, Popping et al. 2012, Popping et al. in prep). \nSome scaling relations show clear second-parameter dependences, in the sense that the scatter about a given relation is correlated with some other galaxy property. For instance, the MZR may show a second-parameter dependence on star formation, in the sense that more rapidly star forming galaxies at a given mass have lower metallicities (Lara-L'opez et al. 2010, Mannucci et al. 2010). The cold gas content shows a similar correlation, with high H i -mass galaxies having lower metallicities (Bothwell et al. 2013a, Lara-L'opez et al. 2013). \n1.1.3 Demographics: Correlations with Galaxy Type Since the original discovery of fuzzy 'nebulae' it has been known that galaxies come in different morphological types (Hubble 1926). There are many different methods for quantifying and classifying galaxy morphology, and this subject is reviewed in Conselice (2014); see also the more nearby-Universe focussed discussion in Buta (2013). Although galaxy morphology encompasses many complex facets of galaxy structure including the presence of bulges, thin and thick disks, bars, spiral arms, etc., for the purposes of this article we focus on a single simplified metric: the fraction of a galaxy's light or mass contributed by a flattened, rotationally supported disk , and that contained in an oblate or triaxial, pressure supported bulge or spheroid (often denoted by the bulge-to-disk ratio B/D or bulge-to-total ratio B/T ). The bulge-to-disk ratio is broadly correlated with classical Hubble type (Simien & de Vaucouleurs 1986). We will further simplify much of our discussion by referring to just two classes of galaxies, 'disk-dominated' and 'spheroid dominated' 1 . Unfortunately, there is no standard value for the critical value of B/T used to divide these populations, with values from 0 . 3 < ( B/T ) crit < 0 . 7 used in the literature. As it is difficult to robustly decompose the light of observed galaxies into a spheroid and disk component, other metrics such as the concentration (the ratio of the radius containing 90% of the light to the radius containing 50% of the light) or the 'Sersic index' ( n s ; another measure of the 'slope' of the light profile; e.g. Blanton & Moustakas 2009) are frequently used as rough proxies for morphology. We deliberately avoid using the terms 'early type' and 'late type' as they are sometimes used to refer to galaxy classes divided by morphology and sometimes to those divided according to their stellar populations (star forming vs. quiescent). \nRegardless of how galaxies are classified, there are robust demographic trends for diskdominated vs. spheroid-dominated galaxies. There is a very strong trend between morphology and color or star formation activity, such that disk-dominated galaxies are predominantly blue and star forming, while spheroid-dominated galaxies are largely red and quiescent, with nearly uniformly old stellar populations (e.g. Blanton & Moustakas 2009, Kauffmann et al. 2003, Roberts & Haynes 1994). This trend appears to hold up to z ∼ 2, with the caveat that red optical color becomes a less robust tracer of old stellar populations, \nas many star-forming galaxies at high redshift are reddened by dust. The characteristic Schechter function mass M char is larger for quiescent or spheroid-dominated galaxies, and the slope is much shallower (e.g. Bernardi et al. 2010). Put another way, the fraction of spheroid-dominated galaxies increases strongly with stellar mass and luminosity. Furthermore, as emphasized by Binggeli et al. (1988), different types of galaxies can have luminosity functions that deviate considerably from the Schechter form. \nA number of studies have shown that the probability for a galaxy to be quiescent depends on both its stellar mass and large-scale environment or halo mass (Balogh et al. 2004, Hogg et al. 2004, Peng et al. 2010, Woo et al. 2013). Recent works have shown that the correlation between quiescence and other internal properties such as spheroid fraction, velocity dispersion, and central density is even stronger than that with stellar mass (Bell et al. 2012, Bluck et al. 2014, Cheung et al. 2012, Lang et al. 2014). \n1.1.4 Structural Scaling Relations Both disks and spheroids exhibit correlations between their stellar mass or luminosity, their radial size, and their internal velocity (Bernardi et al. 2010, Courteau et al. 2007, Faber & Jackson 1976, Kormendy 1977, Shen et al. 2003, Tully & Fisher 1977). For disk-dominated galaxies, the radial size is usually characterized by the scale radius r s (the scale radius in the exponential function characterizing the radial light profile; e.g. Mo et al. (2010), Eqn. 2.29 p. 50) and the characteristic velocity is the rotation velocity at the maximum of the rotation curve V rot , which usually occurs at around 2 r s . For spheroid-dominated galaxies, the radial size is characterized by the half light radius or effective radius r e (the radius that contains half of the total luminosity), and the internal velocity is characterized by the (line of sight) velocity dispersion σ . Several of these relationships have names, such as the Tully-Fisher relation for disks ( L -V rot ; Tully & Fisher 1977), and the Faber-Jackson ( L -σ ; Faber & Jackson 1976), and Kormendy ( L -r e ; Kormendy 1977) relations for spheroids. A combination of these three quantities forms a Fundamental Plane ; i.e., galaxies populate a relatively thin plane in L -r -V space, or rescaled versions of these variables (Bender et al. 1992, Burstein et al. 1997, Djorgovski & Davis 1987, Faber et al. 1987). The familiar named bivariate relations are projections of this plane. \nThe slope, scatter, and evolution of these structural scaling relationships for spheroids and disks carry important clues about the formation history and relationship between these objects. For example, 1) the size-mass relationship is considerably steeper for spheroids than for disks at all redshifts (Bernardi et al. 2010, Shen et al. 2003, van der Wel et al. 2014); 2) since z ∼ 2, the size-mass relation for spheroids has evolved much more rapidly than that for disks (Trujillo et al. 2006, van der Wel et al. 2014); 3) the size distribution at fixed mass is narrower for spheroids than for disks (van der Wel et al. 2014) 4) the evolution of the Tully-Fisher and Faber-Jackson relation has been relatively mild (Cappellari et al. 2009, Cenarro & Trujillo 2009, Kassin et al. 2007, Miller et al. 2011, 2012). We note that many high redshift studies present the scaling relations for galaxies divided according to whether they are star forming or quiescent, rather than spheroid or disk dominated, but this seems to make little difference to the qualitative results (van der Wel et al. 2014). \nAnother illustration of the importance of structural-kinematic scaling relations is demonstrated by the distinction between 'classical' bulges and 'pseudo'-bulges (Kormendy 2013, Kormendy & Kennicutt 2004). Classical bulges have centrally concentrated light profiles with Sersic indices n s ∼ 2-3 (where a de Vaucouleur profile has n s = 4), and lie on an extension of the Fundamental Plane for giant ellipticals. Pseudobulges have more extended light profiles that are more similar to disks ( n s ∼ 1) and lie on a different Fundamental \nPlane from classical bulges and giant ellipticals (Kormendy et al. 2009). Furthermore, classical bulges and pseudobulges have different correlations with SMBH mass (Kormendy & Ho 2013). Similarly, the dwarf galaxies that are confusingly termed 'dwarf spheroidals' and 'dwarf ellipticals' obey very different Fundamental Plane relations than do classical bulges and ellipticals of all luminosities (Kormendy & Bender 2012, Kormendy et al. 2009). In fact, dwarf spheroidals and dwarf ellipticals are indistinguishable from dwarf irregulars in their structural parameter correlations. These diverse scaling relations hint at different formation mechanisms for these objects, as reviewed in Kormendy (2013).", '1.2 Cosmological Background': "Our modern theory of cosmology is based on the ansatz that the Universe is homogeneous and isotropic on large scales (the cosmological principle ), and Einstein's theory of General Relativity (GR) that says that the structure of space-time is determined by the mass and energy content of the Universe. Together these allow us to derive equations that describe the evolution of the scale factor (or characteristic size and density) of the Universe in terms of the parameters specifying the mass and energy density. Observations have shown that the Universe started from a much denser, hotter, and nearly homogeneous state and has been expanding for approximately the past thirteen and a half billion years (e.g. Mo et al. 2010, hereafter MvdBW). \nIn this standard picture, quantum fluctuations in the very early Universe were processed during a period of very rapid expansion called inflation to create the small inhomogeneities that are detected via temperature fluctuations in the Cosmic Microwave Background. These tiny fluctuations, viewed at the time when free electrons combined with nuclei to form neutral atoms at a redshift z glyph[similarequal] 1100, have now been studied in exquisite detail with a large number of experiments, including the Wilkinson Microwave Anisotropy Probe and Planck satellites. When combined with other observations such as the distance-redshift relation from Type Ia supernovae, abundances of galaxy clusters, constraints on the present-day expansion rate (Hubble parameter H 0 ) from nearby Cepheid stars, and galaxy clustering (e.g. Baryon Acoustic Oscillations), these measurements yield stringent constraints on the fundamental cosmological parameters (Hinshaw et al. 2013, Planck Collaboration et al. 2013). \nThese combined observations point to a Universe that is geometrically flat and dominated by Dark Matter and Dark Energy, which together account for more than 95% of the energy density of the Universe. The physical nature of both of these mysterious substances is unknown, although there are numerous candidates. In the most popular variant of the standard model, which we will refer to as ΛCDM, the dark matter is 'cold' and collisionless and makes up ∼ 25% of the cosmic mass-energy density, and the dark energy is in the form of a 'cosmological constant' Λ (as expected in the most general form of Einstein's equations of General Relativity), comprising ∼ 70%. The remaining 4% is in baryons (which in this context include leptons), i.e. normal atoms that make up stars, gas, and heavy elements ('metals'). Although these cosmological parameters are still uncertain by up to perhaps ten percent, for the purposes of understanding how galaxies form and evolve, this level of uncertainty is largely irrelevant. \nWith the initial conditions specified, if we neglect 'baryonic' physics, it is relatively straightforward to compute how the density field of the dominant dark matter component evolves as the Universe expands. If we imagine the matter density field as a mountain range, the landscape in the CDM picture is extremely craggy, with many small scale peaks \nsuperimposed on top of the medium and large scale peaks and valleys. As the Universe expands, the background density decreases. When a peak exceeds a critical over-density relative to the background, the region within that peak stops expanding and becomes gravitationally self-bound. Numerical N -body techniques have been used to extensively study and characterize the growth of structure in dissipationless (dark matter only) ΛCDM simulations, as we discuss in § 2.1.1. The gravitationally bound structures that form in these simulations are commonly referred to as dark matter halos , and the abundance, internal structure, shape, clustering, and angular momentum of these halos over cosmic time has been thoroughly quantified (see MvdBW Ch. 6 and 7 and references therein). Based on these dark matter (DM) only simulations, the standard ΛCDM paradigm has been judged to be extremely successful at explaining and reproducing observations on scales larger than a few kpc (e.g. Primack 2005), thereby providing a robust framework upon which to build models of galaxy formation and evolution.", '1.3 Overview of Physical Processes': "In this section we briefly overview the main physical processes that are commonly included in current models of galaxy formation. We discuss these processes and their implementation in more detail in § 2 and § 3. \n- · Gravity - Gravity plays a crucial role in building the 'skeleton' for galaxy formation. The shape and amplitude of the primordial power spectrum of density fluctuations depends on the cosmological parameters and the properties of dark matter. This spectrum, processed by gravity, determines the number of dark matter halos of a given mass that have collapsed at any given time, and how quickly these halos grow over cosmic time via merging and accretion. It also determines how dark matter halos cluster in space. In the standard paradigm, every galaxy is born within one of these dark halos. When halos merge, each containing their own 'central' galaxy, gravity and dynamical friction gradually cause the orbits to decay, until the galaxies merge. Mergers can have important effects on galaxies, including triggering bursts of star formation and accretion onto central supermassive black holes, and transforming galaxy structure and morphology.\n- · Hydrodynamics and Thermal evolution - When an over-dense region composed of gas and dark matter collapses, strong shocks form, increasing the entropy of the gas. The subsequent evolution of the gas is then determined by how efficiently the gas can cool and radiate away its thermal energy. The primary cooling processes relevant for galaxy formation over most of cosmic history are two-body radiative processes. Gas that is hotter than T > ∼ 10 7 K is fully collisionally ionized and cools predominantly via bremsstrahlung (free-free emission). In the temperature range 10 4 < T < 10 7 K, collisionally ionized atoms can decay to their ground state, and electrons can recombine with ions. Below temperatures of 10 4 K, cooling occurs through collisional excitation/de-excitation of heavy elements (metal line cooling) and molecular cooling. \nFollowing collapse and shock-heating, if radiative cooling is inefficient, a pressure-supported quasi-hydrostatic gaseous halo may form. This gas will then gradually cool in what is often referred to as a cooling flow . This is also sometimes referred to as 'hot mode' accretion. Once the gas cools and loses pressure support, it will collapse until it is supported by its angular momentum. If the cooling time of the gas is short compared to the dynamical time, the gas may accrete directly onto the proto-galaxy without ever forming a hot quasi-hydrostatic halo (Birnboim & Dekel 2003, White & Frenk 1991). Cosmological hydrodynamic simulations have shown that this sort of 'cold mode' accretion tends to occur \nwhen gas flows in along relatively cold, dense filaments (Kereˇs et al. 2005). \n- · Star formation - Once gas has collapsed into the central regions of the halo, it may become self-gravitating, i.e. dominated by its own gravity rather than that of the dark matter. As gas cools more rapidly the higher its density, if cooling processes dominate over heating, then a run-away process can ensue whereby Giant Molecular Cloud (GMC) complexes form, and eventually some dense cloud cores within these complexes collapse and reach the extreme densities necessary to ignite nuclear fusion. However, many details of this process remain poorly understood. Moreover, most cosmological simulations are not able to resolve even the scales on which GMC form, much less individual cores. Therefore all existing cosmological simulations implement empirical sub-grid recipes to model star formation.\n- · Black Hole Formation and Growth - The first 'seed' BH may have formed in the early universe either as the remnants of Population III (metal free) stars, via direct collapse of very low angular momentum gas, or via stellar dynamical processes (Volonteri 2010). These seed BH may grow by accreting gas that either has negligible angular momentum, or by forming an accretion disk that drains the gas of angular momentum via viscosity (Netzer 2013). These processes are, again, poorly understood and virtually impossible to model explicitly in cosmological simulations, so are modeled via sub-grid recipes.\n- · Star Formation Feedback - Observations show that less than 10% of the global baryon budget today is in the form of stars. However, in CDM models without some sort of 'feedback' (or suppression of cooling and star formation), we would expect most of the gas to have cooled and formed stars by the present day. Even the pioneers of the earliest models of galaxy formation within a CDM framework recognized this 'overcooling problem', and suggested that energy generated by supernova explosions could heat gas and perhaps blow it out of galaxies, making star formation inefficient (Dekel & Silk 1986, White & Frenk 1991, White & Rees 1978). It is now recognized that there are many processes associated with massive stars and supernovae (e.g. photo-heating, photo-ionization, winds) that could contribute to making star formation inefficient and to driving large-scale winds that reduce the baryon fractions in galaxies (see Hopkins et al. 2012b, for an overview). Once again, most cosmological simulations cannot resolve these physical processes in detail, so nearly all current models implement sub-grid recipes to attempt to capture their effect on galaxy scales.\n- · AGN Feedback - There is strong observational evidence that most or perhaps all spheroid-dominated galaxies (which comprise the majority of all massive galaxies) contain a supermassive black hole (see Kormendy & Ho 2013, for a recent review). A simple calculation indicates that the amount of energy that must have been released in growing these black holes must exceed the binding energy of the host galaxy, suggesting that it could have a very significant effect on galaxy formation (Silk & Rees 1998), however, it is still uncertain how efficiently this energy can couple to the gas in and around galaxies. Observational signatures of feedback associated with Active Galactic Nuclei (AGN) include high-velocity winds, which may be ejecting the cold ISM from galaxies, and hot bubbles apparently generated by giant radio jets, which may be heating the hot halo gas (see Fabian 2012, Heckman & Best 2014, for recent reviews). AGN feedback is also treated with sub-grid recipes in current cosmological simulations.\n- · Stellar populations and chemical evolution - In order to make direct comparisons between models and observations, many modelers convolve their predicted star formation histories with simple stellar population models, which provide the UV-Near IR SED for stellar populations of a single age and metallicity (Conroy 2013), folding in an assumed stellar \nInitial Mass Function (IMF) 2 . Many models now include the important contribution of gas recycling from stellar mass loss self-consistently within simulations (Leitner & Kravtsov 2011). In addition, as stars evolve and go supernova, they produce and distribute heavy elements throughout the gas that surrounds galaxies, evidently polluting the intergalactic medium (IGM) out to fairly large distances from galaxies. Chemical evolution is a critical part of galaxy formation for several reasons: (i) cooling rates at intermediate temperatures are highly enhanced in metal-enriched gas; (ii) the luminosity and color of stellar populations of a given age are sensitive to metallicity; and (iii) heavy elements produce dust, which dims and reddens galaxies in the UV and optical and re-radiates the absorbed energy in the mid-to-far IR. Most cosmological models of galaxy formation now include a treatment of chemical evolution. \n- · Radiative Transfer - The radiation emitted by stars and AGN can have an important impact on galaxy formation. Radiation can directly heat gas, and can also modify cooling rates (especially for metal-enriched gas) by changing the ionization state of the gas. Moreover, the transmission of radiation of different wavelengths through and scattering by dust can greatly impact the measured total luminosity, color, and observationally determined morphological and structural properties of galaxies, especially in the rest-frame UV and optical, which are often all that is available at high redshift. Most current cosmological simulations that are run to low redshift ( z < ∼ 6) do not include radiative transfer self-consistently due to the added computational expense. However, with sufficiently high resolution, radiative transfer through a dusty ISM can be computed in post-processing to estimate the observed pan-chromatic properties of galaxies (e.g. Jonsson et al. 2010) and their line emission (e.g. Narayanan et al. 2008).", '1.4 Overview of Basic Tools': "Theorists have developed a wide range of different tools for modeling galaxy formation and evolution. Here we briefly summarize the most commonly used tools and highlight some significant differences between them. We provide a more detailed description of the methods used in the modeling tools that are the subject of this review in § 2. \nThere is a popular class of what are generally called 'models', including Halo Occupation Distribution (HOD) models (e.g., Berlind & Weinberg 2002, Zheng et al. 2005), Conditional Luminosity Function models (van den Bosch et al. 2007), and sub-halo abundance matching (SHAM) models and related techniques (e.g., Behroozi et al. 2010, Conroy et al. 2006, Moster et al. 2010b, Tasitsiomi et al. 2004). These techniques derive mappings between observable properties of galaxies and predicted properties of dark matter halos, and in general contain no actual modeling of physical processes . Although this family of techniques is extremely useful for gaining insights into the required connection between observable galaxies and dark matter halos, we will not discuss these types of 'models' in detail in this review. \nThe most explicit way to model galaxy formation is using numerical hydrodynamic techniques , in which the equations of gravity, hydrodynamics, and thermodynamics are concurrently solved for particles and/or grid cells representing dark matter, gas, and stars. The advantage of these techniques is that, within the limitations of the adopted numerical resolution, one obtains predictions of the density of each of these three components (as well \nFigure 1: \n<!-- image --> \nVisualization of representative quantities computed by numerical hydrodynamic simulations, from the Illustris project. From left to right, the dark-matter density, gas density, gas temperature, and gas metallicity are shown at different cosmic times (from top to bottom: z = 0, z = 1, z = 2, z = 4). The slice shown has a projected thickness of 21.3 cMpc and shows the whole Illustris simulation box which is 106.5 cMpc on a side. Reproduced from Vogelsberger et al. (2014a). \nas that of heavy elements) over cosmic time. One also obtains predictions for the velocities of the stars and dark matter, and the temperature of the gas. Thus the structure and kinematics of galaxies as well as their global properties and spatial distribution can be studied in great detail (see Fig. 1 and 2 for examples). The main limitation of these techniques is that computational exigencies restrict the dynamic range that can be explicitly simulated. This, combined with our still imperfect understanding of the physics that governs 'smallscale' processes such as star formation, black hole growth, and feedback processes, means that (as already discussed), many important processes must be treated using uncertain and somewhat arbitrary sub-grid recipes. Moreover, computational limitations have historically \nFigure 2: \n<!-- image --> \nA 100 × 100 × 20 cMpc slice through the EAGLE simulation, illustrating the dynamic range that is attainable with state-of-the-art numerical hydrodynamic simulations. The intensity represents the gas density while the color indicates the gas temperatures (blue through green through red from cooler to hotter). The inset shows a region 10 cMpc and 60 ckpc on a side. The zoom in to an individual galaxy with stellar mass 3 × 10 10 M glyph[circledot] shows the optical band stellar light. Reproduced from Schaye et al. (2014). \nmade it difficult to experiment extensively with different sub-grid recipes or to explore the multi-dimensional space of the variables that parameterize these recipes. \nThe other technique that has been widely used to model galaxy formation in a cosmological context is known as 'semi-analytic modeling' (SAM). This method does not explicitly solve fundamental equations for particles or grid cells, but rather adopts a set of simplified flow equations for bulk components (see Baugh 2006, Benson 2010, for reviews). For example, a typical SAM tracks how much gas accretes into halos, how much hot gas cools and turns into stars, how feedback processes remove cold gas from the galaxy or heat the halo gas, how mergers transform disks into spheroids, etc. Fig. 3 shows graphical representations of some of the quantities that can be tracked in a SAM for several example halo 'merger trees'. \nThe computational requirements of these models are enormously reduced compared with \nFigure 3: \n<!-- image --> \nVisualization of representative predictions from a semi-analytic model. Symbol sizes represent the mass of the host dark matter halo; the x-axis is arbitrary. Symbols connected by lines represent halo mergers. Colors represent the mass of different galaxy components (red: hot gas; blue: cold gas; yellow: stars). Several different final host halo masses are shown as indicated on the figure panels. Halos with the same virial mass can have a diversity of merger histories (not shown). Reproduced from Hirschmann et al. (2012a). \nfully numerical simulations. This makes it possible to make predictions for very large volumes, or to simulate galaxies over a larger range of halo mass, and also to extensively explore different sub-grid recipes treating the most uncertain aspects of galaxy formation. Recently, several groups have coupled SAMs with a Bayesian inference approach, and used Markov Chain Monte Carlo techniques to sample the posterior probability distribution of the multi-dimensional space of the model parameters (Henriques et al. 2009, Lu et al. 2011). This is a powerful approach for exploring parameter degeneracies and obtaining more rigorous statistical assessments of the 'goodness of fit' of specific models or model families with observational data (see Bower et al. (2010) for an alternative approach using Bayesian \nEmulator methods). As well, the less explicit nature of SAMs has allowed modelers to bypass some of the numerical issues which for many years caused difficulties in reproducing basic properties of galaxies in numerical simulations. \nThe field has now reached an interesting point where numerical simulations have started to be able to reproduce fundamental observations at a similar level as semi-analytic models. Interestingly, much of this success has been achieved by adopting a similar approach to the one that has long been used by SAMs, namely, 1) parameterizing the physical processes that can't be simulated explicitly, and tuning these parameters to match a subset of observations, 2) experimenting with different sub-grid recipes to achieve the best match to a set of observations. Even the recipes themselves are in many cases very similar to the ones that are commonly implemented in SAMs. Encouragingly, the two techniques have arrived at the same qualitative conclusions about galaxy formation and evolution for all of the topics that we will discuss in this article. For this reason, we structure this article largely in terms of the physical processes and general insights into how they shape galaxy formation, giving examples from both SAMs and numerical simulations.", '2.1 Gravity': "2.1.1 Numerical N -body methods Gravity solvers, or N -body codes, provide the backbone for galaxy formation models, be they SAMs or hydrodynamic simulations. Fundamentally, these codes must determine the force on each mass element from all others by solving Poisson's equation. Numerically solving Poisson's equation to evolve large-scale structure has a long and storied history that has been extensively reviewed elsewhere (e.g. Bagla 2005, Bertschinger 1998, Dehnen & Read 2011), so we greatly limit our discussion here. \nThe basic approach is to subdivide a representative portion of the universe into many particles, compute the forces on these particles from all others, and evolve the system forward in discrete time-steps. In cosmological N -body simulations, the equations are solved within a comoving frame, and the volume is typically evolved with periodic boundaries, under the assumption that there are a space-filling set of identical volumes that approximately represent the larger-scale matter distribution. The expansion rate of the comoving frame is computed using the Friedmann equation (obtained from the Einstein equations within GR, see e.g. MvdBW Ch. 3.2), but the equations actually solved are the familiar Newtonian versions since GR corrections are generally negligible. \nN -body methods are either particle-based, mesh-based, or a hybrid. In galaxy formation, the most popular particle-based approach is the tree code (Barnes & Hut 1986), in which the force from distant groups of particles are approximated via their multipole moments. The particle-mesh (PM) method, in contrast, computes the potential on a grid via a Fourier transform of the density field, and moves particles along potential gradients (Hockney & Eastwood 1988). Both scale with particle number N as O ( N log N ), though PM is considerably faster. Moreover, PM codes intrinsically account for all periodic replicas of the volume, while tree codes must use add-on techniques such as Ewald summation (Hernquist et al. 1991). \nThe advantage of tree codes is that the forces on particles can be accurately represented down to the chosen force softening length glyph[epsilon1] , while PM codes are limited in resolution to their cell size. The ratio of the box length to glyph[epsilon1] defines the dynamic range of the calculation. \nA hybrid Tree-PM approach is thus a popular method to increase dynamic range, in which the small-range forces are more accurately calculated using a tree, while large-range (and periodic) forces are computed via a faster PM method (e.g. Gadget-2 ; Springel 2005). The largest N -body simulations today evolve ∼ 10 12 particles, with a dynamic range exceeding a million. With the advent of new computing technologies such as Graphics Processing Units, there is the potential for even larger computations if such highly-threaded, cache-limited hardware can be effectively utilized; so far this has proved challenging, but progress is being made. \n2.1.2 Dark Matter Halos and Sub-halos A basic ansatz of our current picture of galaxy formation is that galaxies form within dark matter halos. Identifying these objects in N -body simulations is the first step in constructing the merger trees (see below) that form the gravitational backbone for SAMs. On-the-fly halo finding is carried out within some hydro codes as well, in order to use halo properties for sub-grid recipes. \nThe halo mass (or 'virial mass') is usually defined as the mass within a sphere that encloses an average density ∆ vir relative to the background density of the Universe. Similarly, the virial radius is defined as the radius within which the overdensity is equal to this critical value. The actual value of ∆ vir is unfortunately not standardized, and is based on a simple model of the collapse of a uniform spherical overdensity. In an Einstein-de Sitter universe, after collapse and virialization, such a uniform spherical perturbation will have an average density glyph[similarequal] 178 times that of the background (or critical) density (MvdBW Ch. 5.1). Many works use a fixed value of ∆ vir = 200, which is just a rounding up of 178; some apply it relative to the critical density and some relative to the background density. Some works use a redshift and cosmology-dependent value of ∆ vir , as given by the fitting function from Bryan & Norman (1998). These different conventions introduce redshift-dependent differences of as much as a factor of two in halo virial masses, radii, and internal velocities, to which readers must be alert when comparing results from the literature. \nOne of the generic features of the ΛCDM paradigm is that halos have a great deal of 'sub-structure'. This sub-structure arises from objects that collapse and become bound at an earlier time, then get subsumed into a larger virialized structure. A 'sub-halo' is a halo that was once a distinct halo but is now contained within another virialized halo. \nMethods used to identify halos in N -body simulations include 'friends-of-friends' (FOF), Spherical Overdensity (SO), and 6D phase-space based methods. See Knebe et al. (2011) for a comprehensive description and comparison of the results of different halo finders. Different halo finders tend to agree fairly well (within ∼ 10%) for basic halo properties such as mass and peak circular velocity of distinct halos; the cumulative z = 0 halo mass function differs by ± 10% across the 16 halo finders tested in Knebe et al. (2011). However, Klypin et al. (2011) point out that much larger differences between FOF and SO-based finders can arise at high redshift. Identifying substructure is more halo-finder dependent; here 6D phase-space based halo finders such as ROCKSTAR (Behroozi et al. 2013b) were found to perform significantly better. \n2.1.3 Merger Trees In semi-analytic models, the formation of structure through gravitational instability in an expanding Universe is represented via merger trees. A merger tree records the masses of dark matter halos and the times at which these progenitor halos merge together to form a larger halo (see Fig. 3). Merger trees may either be extracted from N-body simulations or constructed using semi-analytic methods. \nThe first proposed methods for constructing merger trees semi-analytically (Cole et al. \n1994, Kauffmann et al. 1993, Somerville & Kolatt 1999) used statistical methods based on the Extended Press-Schechter model (Lacey & Cole 1993). More recent methods apply empirical corrections to achieve better agreement with numerical simulations (e.g. Parkinson et al. 2008). These methods provide an important complement to merger trees extracted from N -body simulations, as they are extremely flexible, and can be used to efficiently explore different cosmologies and power spectra and large dynamic ranges in halo mass. Moreover, N -body based merger trees have their own limitations, as discussed below. \nExtracting merger trees from an N -body simulation appears straightforward on the face of it - one identifies dark matter halos at a series of redshifts or output times, and then identifies which halos at earlier times are 'progenitors' of a given halo identified at some later time. In practice, however, there are complications. 1) Results may be sensitive to the method used for identifying halos, as discussed above. 2) The definition of progenitor is not unique, since the particles from a halo at some time t 1 may end up in different halos at a later time t 2 . 3) A halo may be identified as a sub-halo in one timestep, then move outside of the virial radius of the host again at some later time. 4) Sub-halos are tidally stripped as they orbit within their host halos, and eventually become difficult to identify -most halo finders can no longer robustly identify sub-halos when they drop below 30-40 particles (Knebe et al. 2011). \nFor semi-analytic merger trees and to track sub-structure once the sub-halo can no longer be identified in the N -body simulation, most SAMs include a procedure to estimate the time for a satellite's orbit to decay due to dynamical friction. A variation of the Chandrasekhar formula (MvdBW, § 12.3.1) is generally used for this purpose. Many SAMs use refined versions of this formula based on numerical simulations (e.g. Boylan-Kolchin et al. 2008). Some sub-halos may be tidally destroyed before they reach the center, and their stars added to a diffuse stellar halo. Satellites that survive until they reach the center are assumed to merge with the central galaxy. SAMs then implement a set of recipes for the effect of the merger, which generally include an enhanced 'burst' phase of star formation as well as some sort of morphological transformation (e.g. moving stars from the disk to the spheroid component).", '2.2 Hydrodynamics: Numerical Techniques': "To directly model the visible component of the Universe requires modeling gas physics, i.e. solving the equations of hydrodynamics and evolving them concurrently with the chosen gravity solver. Doing so enormously increases the complexity of the code, resulting in longer calculations with greater intrinsic uncertainties. Most hydro codes are based on solving the Euler equations (e.g. MvdBW p. 366), representing mass, momentum, and energy conservation, typically closed by assuming a non-relativistic ideal gas equation of state. The Euler equations are a form of the Navier-Stokes equations assuming no viscosity or conduction. In most cases, it is necessary to add an artificial viscosity term in order to properly handle convergent flows and shocks. Some experimentation has also been done with adding other physics whereby it is necessary to solve the Navier-Stokes equations directly; see Springel (2010b) for more discussion. \n2.2.1 Lagrangian Methods In galaxy formation, historically the most popular Lagrangian method is Smoothed Particle Hydrodynamics (SPH; see reviews by Monaghan 1992, Springel 2010b). Briefly, in SPH, the particles themselves carry the information about the fluid, which is obtained via a kernel-weighted sum over neighboring particles \ncloser than a smoothing length ( h ): \nX i = Σ j m j ( X j /ρ j ) W ( | r i -r j | , h i , h j ) . (1) \nHere, X i is the quantity to be estimated, m and ρ are the particles' mass and density, and W is the kernel, which is some spherical function of the distance between particles in units of the smoothing length. X i can also be a gradient of a quantity, in which case the gradient propagates through to the kernel which becomes ∇ W . The efficiency and simplicity of evaluating the Euler equations based on these local kernel-smoothed quantities gives SPH many of its key advantages, including natural spatial adaptivity and trivial implementation in three dimensions. \n'Classic' SPH (e.g. Hernquist & Katz 1989, Monaghan 1992) evaluates the density first as a kernel-smoothed average over nearby masses, then the thermal energy to update the pressure, then the hydrodynamic acceleration. A variant of this method is used in the code GASOLINE (Wadsley et al. 2004). A key drawback is that this method does not explicitly conserve energy and entropy in adiabatic flows in the case of variable smoothing lengths. Entropy-conserving (EC-)SPH (Springel 2005) mitigated this flaw by explicitly including variational terms in h as derived from the Lagrangian, and was formulated using entropy as the evolved variable. EC-SPH is employed in the widely-used code Gadget2 (Springel 2005). Subsequently, it was noted that a side-effect of EC-SPH is to create an artificial pressure between cold and hot phases, resulting in a surface tension that causes for example cold clumps moving through a hot medium to be significantly more resistant to disruption than they are in grid-based codes (Agertz et al. 2007). Ironically, classic SPH performs somewhat better in this regard (at the cost of increased particle interpenetration), but all of these versions fail to realistically capture Kelvin-Helmholtz instabilities. \nSeveral promising approaches have been developed recently to mitigate these issues. Read & Hayfield (2012) proposed SPHS, in which they showed that the error in the momentum equation in classic SPH can be reduced by using a different kernel shape with a larger number of neighbors, along with a modification of artificial viscosity to include a higherorder dissipation switch that anticipates shocks. SPHS can yield much better results for a range of surface instability tests, but the required increase in the number of SPH neighbors (442 vs. ∼ 40) slows the calculation and lowers the resolution. \nA different approach was pursued by Saitoh & Makino (2013), following on Ritchie & Thomas (2001). They argued that difficulties in classic or EC-SPH arose from the requirement that the density distribution be differentiable, which is violated at contact discontinuities. They proposed a new formulation that was 'density-independent' (DI-SPH), which used kernel sums to separately obtain the energy density and internal energy, from which the density is inferred. DI-SPH was shown to remove the artificial surface tension and enable improved treatment of surface instabilities (among other tests). Hopkins (2013) reformulated DI-SPH in terms of entropy to incorporate the improved conservation properties of EC-SPH. This pressure-entropy (PE-)SPH provides much improved handling of surface instabilities versus classic SPH, with fewer numbers of neighbors than for SPHS. A modified treatment of artificial viscosity has also been widely implemented, and helps improve the performance of SPH in this regard (e.g. Hu et al. 2014). Thanks to such improvements, current formulations of SPH can now track surface instabilities and associated phenomena to an accuracy that, not long ago, were widely regarded to be challenging for SPH. \n2.2.2 Eulerian Methods A time-honored approach to solving hydrodynamics is to discretize the fluid onto grid cells, and then compute the advection of properties across the \ncell boundaries. This is the basis of Eulerian approaches, which formulate the solution to the Euler equations in the fixed frame, rather than in the fluid frame as in the Lagrangian approach. \nMost current cosmological Eulerian hydro codes employ a high-order Godunov scheme. Here, the Riemann problem is solved across cell faces, which yields a pressure at each cell face, thereby giving the force on the fluid across the cell. The fluid, with all its associated properties, is then advected across the cell face. If the cell is assumed to have uniform properties within it, this is called a (first-order) Godunov solver. Modern codes employ parabolic interpolation, known as the Piecewise Parabolic Method (PPM). Note that while higher order interpolation provides a more accurate solution, it requires using information from neighboring cells which effectively lowers the spatial resolution. \nGiven the dynamic range involved in modeling galaxy formation, a key development was the implementation of Adaptive Mesh Refinement (AMR). Here, cells satisfying some local criteria (typically based on mass) are split into sub-cells, enabling improved resolution in those regions. This effectively achieves some of Lagrangian methods' primary advantage of being naturally adaptive in space and time. Current AMR hydro codes for galaxy formation include Enzo (Bryan et al. 2014), RAMSES (Teyssier 2010), FLASH 3 , and Hydro-Adaptive Refinement Tree (H-ART; Kravtsov et al. 1997). \n2.2.3 Arbitrary Lagrangian-Eulerian Methods Optimally, one would like to unite the advantages of PPM in handling shocks and contact discontinuities with SPH's natural adaptivivity. One approach is to use a deformable mesh, in which the mesh follows the fluid. Such arbitrary Lagrangian-Eulerian codes have historically not played a large role in astrophysics (see e.g. Pen 1998), but this has recently changed with the introduction of Arepo (Springel 2010a). \nArepo uses a Voronoi tesselation to subdivide space around particles. A Voronoi tesselation is a space-filling set of polyhedral cells where the space within a given cell is closer to one particle than any other. The Riemann problem is then solved across the cell faces in order to obtain the force on the particle. The mesh is re-generated as the particles move. In this way, Arepo is able to naturally follow the fluid like a Lagrangian code, while retaining the advantages of Godunov solvers such as excellent handling of contact discontinuities, surface instabilities, and shocks, and the lack of artificial viscosity. \n2.2.4 Advantages and Disadvantages Traditionally, Eulerian methods have enjoyed a superiority in handling strong shocks and surface instabilities, while Lagrangian methods like SPH are more adaptive and provide increased dynamic range for a given CPU expense. However, in recent times the gaps are closing in both directions. Arepo has some important advantages over both Lagrangian and Eulerian methods, particularly EC-SPH (Vogelsberger et al. 2012). \nAn advantage of a particle-based approach such as SPH is that the movement of mass is directly tracked. This makes it more straightforward to follow the mass as it assembles into galaxies, and to track where material ejected from galaxies ends up. It is also straightforward to implement kinetic winds, which as we will discuss below has had substantial success as a sub-grid prescription for galactic outflows. Nonetheless, in mesh codes it is possible to use tracer particles for these purposes. For instance, Arepo has implemented kinetic winds by spawning particles that are decoupled from the hydro mesh, which then later rejoin. \nAMRoffers the key advantange that the mesh can be refined to arbitrarily high resolution, while particle-based methods are limited in resolution by the particle mass. This allows individual systems to be examined in great detail, albeit at great computational cost. For example, Enzo merger simulations by Kim et al. (2009) and H-ART cosmological zoom simulations by Ceverino et al. (2014) both achieved a dynamic range of > ∼ 10 6 , while the most ambitious current SPH simulations can only achieve ∼ 10 5 . \nMore broadly, since all modern codes generally yield similar answers in basic tests relevant to galaxy formation where the answer is approximately known, at this stage it is difficult to identify one code or methodology that is clearly superior to the others. For most properties, differences in sub-grid prescriptions yield much larger variations than differences in hydrodynamical techniques.", '2.3 Thermal evolution': "2.3.1 Cooling and Heating in Numerical Simulations The key difference between baryons and dark matter in galaxy formation is that baryons can dissipate their potential energy via radiative processes. Radiative cooling and photo-ionization heating are thus implemented in essentially all codes, while radiation transport is a growing subfield with specific applications to the epoch of reionization (EoR) and line emission. \nMost simulations today also include cooling from metal line emission, which dominates particularly at 10 5 < ∼ T < ∼ 10 7 Kfor typical warm-hot gas metallicities. Early works employed cooling rates assuming collisional ionization equilibrium (Sutherland & Dopita 1993), but more recent work by Wiersma et al. (2009a) better account for the photo-ionization of metals by the metagalactic radiation field. \nSimulations focusing on the post-EoR universe typically account for photo-ionization heating by assuming all the gas is optically thin and in ionization equilbrium with a spatiallyuniform metagalactic radiation field (e.g. Faucher-Gigu'ere et al. 2009, Haardt & Madau 2012). During the EoR, these assumptions break down, and continuum radiative transfer is necessary in order to properly model the feedback from photo-ionization heating on galaxy growth. Two approaches are used: applying radiative transfer in post-processing to existing density distributions (Iliev et al. 2006), which is useful for evolving large volumes to study the final stages of EoR; and full radiative hydrodynamic codes that evolve the ionizing field together with the baryons, including modeling star formation to self-consistently predict the properties of the sources (Finlator et al. 2011, Iliev et al. 2009, Pawlik & Schaye 2011, Wise & Abel 2011). Given that this review focuses on the post-reionization Universe, we will not discuss this further here. \n2.3.2 Cooling and Cosmological Accretion in SAMs Most semi-analytic models implement some variant of the self-similar cooling flow model originally proposed by White & Frenk (1991) to track the thermal evolution of gas. As the gas enters the halo, it is assumed to be shock-heated to the virial temperature T vir = 35 . 9[ V vir / (km / s)] 2 K, where V vir is the halo virial velocity. One may then calculate the cooling time, which is the time it would take for the gas to radiate away all of its energy: \nt cool = 3 2 µm p kT ρ g ( r )Λ( T, Z h ) . (2) \nHere, µm p is the mean molecular mass, T is the temperature of the gas, ρ g ( r ) is the radial density profile of the gas, Λ( T, Z h ) is the temperature and metallicity dependent cooling \nfunction (e.g. Sutherland & Dopita 1993), and Z h is the metallicity of the hot halo gas. \nThe hot gas is assumed to be distributed with a smooth spherically symmetric density profile. Most models assume that the density profile is described by a singular isothermal sphere ( ρ g ( r ) ∝ r -2 ), although some use different density profiles, such as a Navarro-FrenkWhite (NFW) profile (Navarro et al. 1997) or a cored NFW profile (Cole et al. 2000). \nOne can then solve for the 'cooling radius', within which gas has had time to dissipate all of its thermal energy by cooling. To do this, one must adopt a timescale over which cooling is assumed to have taken place. Common choices for this timescale are the time since the halo has experienced a 'major' (at least 2:1) merger (e.g. Somerville & Primack 1999), or the halo dynamical time t dyn = r vir /V vir (e.g. Springel et al. 2001). It may happen that the model predicts r cool > r vir , indicating that the cooling time is shorter than the dynamical time, corresponding to the 'cold flow' regime described in § 1.3. In this case, most modelers generally assume that gas can flow into the halo on a dynamical time. Although this model is very simple, several studies have shown that the predicted cooling and accretion rates are in surprisingly good agreement with those from numerical hydrodynamic simulations (Benson et al. 2001, Hirschmann et al. 2012a, Monaco et al. 2014, Yoshida et al. 2002).", '2.4 Chemical evolution': "Tracking the enrichment of gas with heavy elements is important for cooling calculations, and for predictions of galactic chemical evolution. Most numerical hydro simulations now include a model for chemical enrichment. Early models tracked only Type II supernova (SN) enrichment, which is closely related to the oxygen abundance. To track other key elements such as carbon and iron, it is necessary to model asymptotic giant branch (AGB) stars whose ejecta dominate the present-day carbon budget, and Type Ia SN that produce the bulk of the iron in stellar-dominated systems. Such delayed feedback sources are now included in most codes, which track a suite of individual elements (Oppenheimer & Dav'e 2008, Wiersma et al. 2009b). The dominant uncertainty typically comes from the metal yield models from SN and stellar evolution, particularly at low metallicities and high masses. Hence at present, absolute abundance predictions should be considered accurate to only a factor of two, but relative trends of metallicity versus other galaxy properties such as stellar mass are likely more robust. \nMost SAMs use a simple instantaneous recycling approximation in which a yield y of heavy elements is produced by stars in each timestep: dM Z = y dm star , where dM Z is the mass of metals produced and dm star is the mass of stars formed. In general these metals are deposited into the cold ISM, although some models deposit some of the metals directly in the hot halo gas. Metals may then be ejected from the cold gas reservoir by winds, and are either deposited in the IGM or in the hot gas halo. Most SAMs treat the yield y as a free parameter rather than taking it from SN yield calculations, and neglect enrichment by Type Ia SNae and AGB stars (so again, the predicted metallicities most closely trace α elements such as oxygen). However a few SAMs in recent years have incorporated more detailed treatments of chemical enrichment, tracking multiple individual elements, and the finite timescales for enrichment and gas recycling from AGB stars, Type Ia, and Type II SNae (Arrigoni et al. 2010, Nagashima et al. 2005, Yates et al. 2013).", '2.5 Initial conditions and zoom simulations': "The generation of standard cosmological initial conditions involves (1) generating a linear matter power spectrum via a transfer function (e.g. Eisenstein & Hu 1999); (2) Gaussianrandom sampling the power spectrum for modes within the simulation volume; and (3) evolving the modes forward in the linear regime via the Zel'dovich approximation; see Bertschinger (1998) for more details. This generates particle positions and velocities sampling the matter field within a specified volume for a specified cosmology, at some specified high redshift that is optimally just before structure within the volume first goes nonlinear (see e.g. MUSIC; Hahn & Abel 2011). \nAn increasingly popular and useful technique is zoom simulations. In zooms, a sub-volume within a cosmologically representative region is evolved at much higher resolution, together with surrounding regions of coarser resolution that provide the tidal field from large-scale structure. After an initial coarse-grained run, a halo or region of interest is selected, and its particles are tracked back to the original initial conditions to define the zoom region . Particles within the zoom region are sampled to finer resolution, including the requisite small-scale power, and the entire volume is run again, typically with hydrodynamics turned on only in the zoom region. In this way, zooms provide an increased dynamic range at a manageable computational cost, albeit only for a single galaxy or halo and its environs. \nSimulations of idealized isolated galaxies, or mergers thereof, provide a valuable testbed to explore detailed physical processes, particularly in the ISM. Initial conditions are typically created in a stable disk configuration (Hernquist 1993a), and then dynamical perturbations grow either from tides induced by a merger or internal stochasticity. Such models can achieve extremely high resolution (by cosmological standards) and can serve to isolate physics of particular interest, hence they remain useful even if they do not fully represent the cosmological baryon cycle.", '3.1 Star Formation and the ISM': "A huge body of observations from UV through near-IR light traces the emission from stars. In order to make contact with these observations, models must attempt to compute how gas in galaxies turns into stars. The ISM is a complex place, with multiple gas phases co-existing at very different densities and temperatures (McKee & Ostriker 1977). Cosmological simulations of more than a single galaxy are still orders of magnitude away from capturing the spatial scales, temperatures, and densities where stars actually form. Moreover, physical processes that are not typically included or captured well in cosmological simulations, such as magnetic fields and turbulence, are thought to play important roles on the scales of dense molecular cloud cores and protostars (McKee & Ostriker 2007). However, advances in our theoretical understanding of star formation as well as better observational characterization of key scaling relations (see Kennicutt & Evans 2012, for a review) have enabled the development of empirical recipes that smooth over much of the small-scale complexity. \nStars are observed to form in the dense, cold, molecular phase of the ISM, and current observations support a (nearly) universal star formation efficiency in molecular gas, where about 1% of the gas is converted into stars per free fall time (Bigiel et al. 2008, 2011, Krumholz et al. 2012, Leroy et al. 2013). Thus the ability to track where molecular gas forms should lead to a more physical approach to modeling star formation. The ISM is observed to become H 2 -dominated at ∼ 1-100 atoms cm -3 . Because gravitational instability is \nthought to be one of the driving forces in the formation of GMC (Dobbs et al. 2014), simply requiring a density threshold for star formation of a few atoms cm -3 may be a good first approximation. However, this also requires high enough resolution ( < ∼ 100 pc) to attain these densities, which is currently achievable only in zoom simulations. \nIn more detail, H 2 formation is catalyzed by dust, and destroyed by Lyman-Werner radiation, so one would expect that H 2 production is thus roughly proportional to metallicity, while destruction depends on the ability to self-shield against interstellar radiation. Some zoom simulations now include a simplified phenomenological treatment of chemical networks and H 2 dust- and self-shielding (Christensen et al. 2012, Gnedin et al. 2009). Fitting functions that attempt to capture the essence of H 2 formation and dissociation and the resulting dependence of H 2 fraction f H 2 on gas density, metallicity, and local UV background have been presented based on these and on idealized (non-cosmological) disk simulations and analytic models (Gnedin & Draine 2014, Gnedin & Kravtsov 2011, Krumholz et al. 2009, McKee & Krumholz 2010). \nAn alternative approach for partioning gas into H i and H 2 is to use the empirical relationship between f H2 and the disk mid-plane pressure, pointed out by Blitz & Rosolowsky (2004, BR). They found that the molecular fraction R mol ≡ Σ H2 / Σ HI was correlated with the disk hydrostatic mid-plane pressure P : R mol = ( P P 0 ) α BR , where P 0 and α BR are free parameters that are obtained from a fit to the observational data. The hydrostatic pressure as a function of radius in the disk can be estimated based on the cold gas surface density, the stellar surface density, and the ratio of the vertical velocity dispersions of the gas and stars (Elmegreen 1989). This approach can be used to estimate f H2 either self-consistently (see below) or in post-processing in numerical simulations or SAMs (Duffy et al. 2012, Obreschkow et al. 2009). \n3.1.1 Numerical Implementation The basic recipe for star formation in many cosmological simulations has not changed markedly from the pioneering work of Katz (1992). Gas that is dense and converging is assigned a SFR based on a Schmidt (1959) law, namely \n˙ ρ ∗ = glyph[epsilon1] ∗ ρ gas t ff ∝ ρ 1 . 5 gas (3) \nwhere the last proportionality arises because the local free-fall time t ff ∝ ρ -0 . 5 . The free parameter glyph[epsilon1] ∗ is typically calibrated to match the amplitude of the observed Kennicutt (1998) relation in simulations of idealized, isolated disks. Long-term SF histories are generally insensitive to glyph[epsilon1] ∗ within reasonable choices (Katz et al. 1996, Schaye et al. 2010), because as discussed later, globally, SF is driven primarily by gas accretion, and over cosmological timescales is not limited by the rate of conversion of gas into stars in the ISM. A somewhat different approach was proposed by Schaye & Dalla Vecchia (2008): they analytically recast the Kennicutt-Schmidt relation as a function of pressure rather than density, assuming a self-gravitating disk. \nStars are generally only allowed to form when the density exceeds some critical value, the choice of which is another free parameter. Springel & Hernquist (2003) incorporated a density threshold based on where the Jeans mass became lower than the particle mass, at which point a sub-grid ISM model is required; this value turned out to be ≈ 0 . 1 cm -3 for typical mass resolutions adopted in cosmological volumes at the time. \nThis simple SF prescription applied to individual disk galaxies was found to quickly collapse gas down to the (artificial) Jeans scale in the simulations, which produced highly clumpy disks that looked nothing like local grand-design spirals. The solution, introduced in \ncosmological runs by Springel & Hernquist (2003) was to artificially overpressurize the ISM, by implementing a sub-grid ISM model based on McKee & Ostriker (1977) that tracked the balance between SN energy input and cooling within a multi-phase ISM. The temperature of the ISM gas (defined as gas above the SF threshold density) is then raised up to as high as 10 6 K. Robertson et al. (2004) extended this model to an arbitrary ISM effective equation of state, and showed that with appropriate overpressurization, this approach can reproduce smooth, stable, gas-rich spirals as observed today. Ironically, as we discuss further in § 4.2.1, it turns out that simulations with no or minimal ISM pressure (Ceverino et al. 2010, 2014) do well at reproducing the clumpy disks that are now known to be common at high redshift ( z ∼ 2; Genzel et al. 2011), though simulations with pressurization can also reproduce these (Genel et al. 2012b). \nIt is clear that real stars do not form at densities of ∼ 0 . 1 atoms cm -3 . Moreover, since the Kennicutt relation is only observed to hold when the ISM is averaged over scales of 0 . 5 -1 kpc, once simulations resolve smaller scales, it becomes dubious to use a SF prescription that is calibrated to match this relation. Thus much recent effort has gone into incorporating more realistic treatments of the ISM into cosmological simulations. Highresolution zoom simulations that simply adopt a higher star formation threshold ( ∼ 5 atoms cm -3 ) and efficient stellar-driven winds (see § 4.2.1 for further discussion) show marked improvement in their ability to produce realistic disks (e.g. Governato et al. 2007, Guedes et al. 2011). Other simulators (e.g. Agertz & Kravtsov 2014, Kuhlen et al. 2012) have incorporated sub-grid recipes to compute the density of molecular hydrogen ρ H 2 and then use that in an equation similar to Eqn. 3 in place of ρ gas - no arbitrary density threshold need then be applied. \nAn exciting development is that cosmological zoom simulations are starting to be able to resolve the Jeans mass/length of gas, corresponding to the scale of molecular cloud complexes, allowing more direct modeling of the multi-phase ISM, (e.g. the FIRE simulations, Hopkins et al. 2013a). Concurrently, ISM simulations including detailed treatments of nonequilibrium chemistry and turbulence are pushing outwards in scale to start 'bridging the gap' with the cosmological runs (e.g. Mac Low & Glover 2012, Walch et al. 2011). Continuing interactions between the galaxy formation and ISM/star formation communities will soon allow us to place our sub-grid recipes on a more secure physical foundation. \n3.1.2 Implementation in Semi-Analytic Models The usual approach to modeling star formation in SAMs is very similar to the approach used in numerical hydro simulations, described above. Gas that has 'cooled' according to the cooling model described in § 2.3.2 loses its pressure support and collapses further, until it is supported by its angular momentum, forming a disk. The initial angular momentum of the halo gas can then be used to estimate the radial size of the disk, as described in § 4.2.4. Some SAMs track the radial structure of the disk in cylindrically symmetric annuli (Avila-Reese et al. 1998, Dutton & van den Bosch 2009, Fu et al. 2010, Kauffmann 1996), while most models assume the disk radial surface density distribution to be an exponential, as is generally the case in observed disk galaxies. \nDifferent SAMs use different but roughly physically equivalent variants of Eqn. 3. Early SAMs typically used an expression of the form \n˙ m ∗ = glyph[epsilon1] ∗ m cold τ ∗ \nwhere ˙ m ∗ is the total star formation rate in the galaxy, m cold is the total cold gas mass in the galaxy, τ ∗ is a characteristic timescale for star formation, and glyph[epsilon1] ∗ is a parameter \nrepresenting the global star formation efficiency. The SF timescale is often assumed to scale with the dynamical time of the dark matter halo, τ ∗ ∝ τ dyn ∝ r H /V H , where r H is the characteristic halo radius and V H is the characteristic halo circular velocity. However, it was quickly realized that a SF law of this form could not reproduce the observed trend of increasing cold gas fractions with decreasing stellar mass in the low redshift universe. Thus, modelers either introduced a SF threshold, such that only the fraction of the cold gas above this threshold was eligible to participate in star formation, or made τ ∗ an explicit function of halo properties, e.g. of V H (Cole et al. 2000), such that the star formation timescale is made longer in lower mass galaxies. \nModels that track disk structure in more detail are able to use empirical laws that are closer to what is actually observed. For example, Somerville et al. (2008) adopted a 'Kennicutt'-like expression, where the star formation rate surface density of the disk is calculated according to Σ SFR = A SF Σ N SF gas for Σ gas > Σ crit (and zero otherwise). The parameters A SF and N SF are taken directly from observations (e.g. Kennicutt 1998), and Σ crit is treated as a free parameter. A similar approach, but with a radius and circular velocity dependent Σ crit based on the Toomre condition for gravitational instability, is adopted in the MPA SAMs (e.g. Croton et al. 2006, Guo et al. 2011, Kauffmann et al. 1999). \nSeveral groups have recently developed SAMs that attempt to track atomic and molecular gas separately (Fu et al. 2010, Lagos et al. 2011b, Popping et al. 2014b, Somerville et al. 2014). Various recipes for H 2 -formation, either employing the metallicity-based fitting functions of Krumholz et al. (2009) and Gnedin & Kravtsov (2011), or alternatively the empirical pressure-based recipe from Blitz & Rosolowsky (2004), have been implemented in these SAMs. Again, the computed density of H 2 may then be used in an empirically calibrated SF law with no need to assume a density threshold - essentially removing all free parameters from the SF recipe (within the observational uncertainties on the slope and normalization of the relationship between Σ SFR and Σ H 2 ). Overall, it appears that the main predictions of SAMs, especially for stellar properties of galaxies, are quite insensitive to the details of the gas partitioning recipe (Berry et al. 2014, Fu et al. 2010, Lagos et al. 2011b, Popping et al. 2014b, Somerville et al. 2014). \nIt is well known that galaxy interactions and mergers can trigger starbursts with enhanced star formation efficiency (SFE), and most SAMs implement a 'burst mode' of star formation in galaxies that have experienced a recent merger. Studies based on hydrodynamic simulations of binary galaxy mergers have shown that the enhancement in the SFE above that in an isolated galaxy is a fairly strong function of the mass ratio of the merger. Many SAMs implement the fitting function introduced by Cox et al. (2008), who parameterized the burst efficiency as e burst = e burst , 0 µ γ , where µ is the merger mass ratio, and e burst is defined as the fraction of the total gas reservoir that is consumed in the burst. \nSubsequent studies have shown that e burst and the burst timescale also depend on the implementation of stellar feedback and the treatment of the ISM (Cox et al. 2008, Robertson et al. 2006b). Hopkins et al. (2009a) found that the burst efficiency depended strongly on the cold gas fraction in the progenitors, with lower burst efficiencies in mergers with higher progenitor gas fractions. However, Moster et al. (2011) did not find a strong correlation with the progenitor cold gas fraction when they including a hot halo in the merger progenitors. Although there have been numerous studies of star formation enhancement in mergers using numerical hydrodynamic simulations of binary mergers (e.g. Cox et al. 2006, 2008, Mihos & Hernquist 1996, Springel 2000), these simulations are not in a cosmological context, and therefore must assume idealized initial conditions. Furthermore, most have not included cosmological accretion or cooling from a hot gas halo. To our knowledge, there has not \nbeen a systematic exploration of the enhancement of star formation activity in mergers using cosmologically-situated hydrodynamic simulations. \nSAMs predict that burst-mode star formation makes a relatively minor contribution to the overall global star-formation rate density at any epoch (e.g. Baugh et al. 2005, Somerville et al. 2008), in agreement with observations (Rodighiero et al. 2011, Schreiber et al. 2014) and cosmological hydro simulations (Kereˇs et al. 2005, Murali et al. 2002). However, mergertriggered bursts may be important for producing certain populations such as ultra-luminous infrared galaxies (ULIRGS) and high-redshift sub-mm detected galaxies (Hayward et al. 2013, Niemi et al. 2012), in agreement with observations that suggest a strong connection between major mergers and starbursts (e.g. Kormendy et al. 2009, Sanders & Mirabel 1996).", '3.2 Black Hole Growth': "The first 'seed' black holes may have been left behind after the explosion of massive stars formed out of primordial gas in the early universe. These 'Pop III' seed BH are expected to have masses of ∼ 100 M glyph[circledot] ; however, such seeds cannot grow into the 10 9 M glyph[circledot] black holes required to power observed quasars at z ∼ 6 -7 if their growth is Eddington-limited. Recently, several mechanisms for creating more massive seed BH (10 4 -10 6 M glyph[circledot] ) have been proposed (see Volonteri 2010, for a review). However, in cosmological simulations, the usual approach is to place seed BH by hand in halos above a critical mass ( M H > ∼ 10 10 -10 11 M glyph[circledot] ). In some cases, seeds of a fixed mass are used, in others, the seed mass is chosen to place the BH on the local M BH -σ relation. The results that we will discuss here are generally insensitive to the details of the seeding procedure. \nOne must then calculate how rapidly these seed BH will accrete gas and grow in mass. The currently predominant model relies on the idea that black hole growth is limited by Bondi accretion of mass within the sphere of influence (Bondi 1952), given by \n˙ M Bondi = α 4 π G 2 M 2 BH ρ ( c 2 s + v 2 ) 3 / 2 , (4) \nwhere M BH is the mass of the BH, c s is the sound speed of the gas, v is the bulk velocity of the BH relative to the gas, ρ is the density of the gas, and α is a boost parameter included because models typically lack the spatial resolution to resolve the Bondi radius (Booth & Schaye 2009, Johansson et al. 2009a). Early models took α to be constant (typically ∼ 100), but some simulators make α a function of density (e.g. Booth & Schaye 2009) and some recent simulations resolve the Bondi radius so can adopt α = 1. Typically, the accretion rate is capped at the Eddington rate. As galaxies merge, their BHs are assumed to merge when they come within some distance of each other, typically a softening length (thereby ignoring GR timescales for BH inspiral). \nThe Bondi accretion model predicts fairly low accretion rates when galaxies are undisturbed, but when strong torques drive gas towards the nucleus as in a major merger, accretion rates can be boosted to levels sufficient to power quasars (Di Matteo et al. 2005, Springel et al. 2005b). This is consistent with the observation that local ULIRGs, which are mostly major mergers, also show strong AGN activity (Sanders & Mirabel 1996). In one paradigm, low accretion rates ( < ∼ 0 . 01 ˙ M Edd , where ˙ M Edd is the Eddington rate) are associated with radiatively inefficient accretion, as in an Advection Dominated Accretion Flow (Blandford & Begelman 1999, Narayan & Yi 1994). In this case, most of the energy is advected into the BH and little emerges as radiation. BH powered at higher accretion \nrates are radiatively efficient and give rise to the population of observed X-ray, UV, and optically luminous quasars and AGN. \nThe assumption of Bondi accretion requires accompanying strong feedback to obtain BHs that follow the M BH -σ relation, as this simple argument demonstrates (Angl'es-Alc'azar et al. 2013a). Consider two BHs of mass M a and M b . If they grow according to the general prescription ˙ M BH = D ( t ) M p BH , then \nd dt ( M a M b ) = D ( t ) M p a M b [ 1 -( M a M b ) 1 -p ] . (5) \nIt is easy to show that the two masses will diverge if p > 1, and they will converge if p < 1. For Bondi accretion p = 2; hence for BHs to converge onto an M BH -σ relation, some strongly self-regulating feedback process must counteract Bondi accretion and make p effectively less than unity. We will discuss possible feedback processes in § 3.3.3, but in general such tuned self-regulation is not so straightforward to arrange, for the usual reason that outward energetic processes tend to escape through paths of least resistence whereas inflows typically arrive through the dense, harder-to-disrupt gas. \nIt is worth emphasizing that the widely used Bondi model implicitly assumes that the accreting gas has negligible angular momentum, which is unlikely to be a good assumption in general. Recently, the problem of dissipating angular momentum to enable BH accretion has received more attention in the cosmological milieu. Hopkins & Quataert (2010, 2011) studied angular momentum transport in disks with non-axisymmetric perturbations both analytically and in simulations, showing that such secular processes can significantly fuel BH growth, as also suggested by Bournaud et al. (2011) and Gabor & Bournaud (2013). Implementing this analytic work into zooms and cosmological runs, Angl'es-Alc'azar et al. (2013a) and Angl'es-Alc'azar et al. (2013b) showed that this torque-limited accretion behaves qualitatively differently than Bondi accretion, since in the Hopkins & Quataert (2011) model, the exponent of BH growth is p = 1 6 . Hence while this model also must incorporate feedback, such feedback does not have to strongly couple to the inflow in order to achieve self-regulation. \nBlack hole accretion in SAMs is of necessity more schematic. In one of the first semianalytic models that incorporated BH growth in the framework of a cosmological model of galaxy formation, Kauffmann & Haehnelt (2000) assumed that all BH growth is triggered by major mergers. Following such an event, they assumed that some fraction of the cold gas was accreted by the BH, with this fraction being a function of the halo circular velocity. A similar recipe is incorporated into the later generations of MPA-SAMs (e.g. Croton et al. 2006, De Lucia & Blaizot 2007, Guo et al. 2011, Henriques et al. 2013). Other SAMs additionally trigger accretion following minor mergers and disk instabilities (Bower et al. 2006, Hirschmann et al. 2012b, Somerville et al. 2008). Some models allow an additional growth channel through a 'Bondi-like' accretion from the hot halo (Fanidakis et al. 2011, Somerville et al. 2008). In the Santa Cruz SAMs (Somerville et al. 2008), black hole growth is parameterized based on the results of hydrodynamic binary merger simulations (Cox et al. 2006, 2008, Robertson et al. 2006b) as characterized by Hopkins et al. (2005b). In this model, rapid black hole accretion is triggered following a major or minor merger. The BH accretes at the Eddington rate until the BH reaches a critical mass, where the energy being radiatied is sufficient to halt further accretion. The accretion rate then declines in a power-law 'blow out' phase until the BH switches off (Hopkins et al. 2005a). \nAll SAMs and numerical cosmological hydrodynamic simulations that explicitly include BHgrowth use the local M BH -σ or M BH -M spheroid relation to calibrate the free parameters \nin the BH accretion recipes. A wide variety of BH growth recipes appear to be able to successfully reproduce this relationship.", '3.3 Feedback Processes': "Feedback can be divided into two general classes, preventive and ejective. Preventive feedback retards star formation by stopping gas from accreting into the ISM, while ejective feedback describes processes that remove the gas from the ISM after it has been accreted. Current wisdom suggests that preventive feedback dominates when the majority of halo gas is near the halo's virial temperature (as in very small dwarfs or massive galaxies), while ejective feedback dominates when most of the halo's gas is well below its virial temperature (as in typical star-forming galaxies). However, individual physical processes can potentially act in both ejective and preventive ways. \n3.3.1 Squelching: Photoionization Suppression Photons above 13.6 eV that ionize hydrogen typically add an ∼ eV-scale amount of latent heat, corresponding to a temperature increase of ∼ 10 4 K. Hence the post-reionization optically-thin IGM has a temperature around this value, which means that gas in halos whose virial temperatures are comparable to 10 4 K will be unable to cool their gas. This temperature corresponds to a halo mass of ∼ 10 8 M glyph[circledot] , implying that photoionization will strongly reduce the baryon content and hence suppress galaxy formation in halos below this mass. This suppression has sometimes been called squelching (Somerville 2002). \nSquelching can have a residual impact on halos much larger than 10 8 M glyph[circledot] , since they are hierarchically assembled in part from squelched halos. Gnedin (2000) showed that the characteristic mass below which halos contain substantially less than their fair share of baryons is well represented by a filtering scale that smooths the baryonic perturbations. Hence one can define a filtering mass , which describes the halo mass that on average contains half the cosmic fraction of baryons. \nThe filtering mass depends on the intricate interplay between photoionization, cooling, and hierarchical growth, which is challenging to model. Early work suggested a roughly constant circular velocity below which baryon accretion is suppressed, of around 30 -50 km/s (e.g. Quinn et al. 1996, Thoul & Weinberg 1996). If extrapolated to today, this would imply halos up to several times 10 10 M glyph[circledot] would be significantly suppressed in baryon content (Gnedin 2000). More recent simulations by Okamoto et al. (2008) found a smaller filtering mass scale, M F ∼ 4 × 10 9 M glyph[circledot] today, but these simulations still assumed ionization equilibrium, did not include metal line cooling, and adopted a uniform meta-galactic ionizing background. Observations of late-type dwarfs with circular velocities < ∼ 42 km/s suggest that their baryon content is much smaller than expected from scaling relations based on larger galaxies (Kormendy & Freeman 2014), thus providing direct constraints on the filtering mass. \nAn additional complication can arise when galactic outflows are included along with squelching, as the two can combine to produce an 'amplification of suppression' that is stronger than the product of the individual effects (Finlator et al. 2012, Pawlik & Schaye 2009). The magnitude of the effect depends on the outflow model implemented, but can be up to a 60% amplification during the EoR. Unfortunately, the high expense of these calculations that include radiative transfer while resolving very small halos prohibits their evolution down to z = 0; hence it is not clear how significant this effect is at later epochs. \nIn semi-analytic models, photoionization squelching is generally implemented by assuming \nthat reionization occurs instantaneously throughout the Universe, at a fixed input redshift. At all later times, the gas that is allowed to accrete into halos is reduced by a factor f coll ( M H , z ). This function is parameterized based on the results of numerical hydrodynamic simulations, and is expressed as a function of the filtering mass (Gnedin 2000, Kravtsov et al. 2004, Okamoto et al. 2008). \n3.3.2 Star Formation Feedback Stars, massive ones in particular, deposit copious amounts of energy and momentum into the ISM during their life and in death. Stellar feedback is invoked to explain two kinds of inefficiencies in galaxies: 1) The efficiency of the conversion of gas into stars within GMC's is puzzlingly low, only about 1% per free fall time (Krumholz et al. 2012); 2) the stellar and baryon fraction within galactic-sized halos is much less than the universal value, ranging from a few to twenty percent (Behroozi et al. 2010, Moster et al. 2010b). The first inefficiency has been ascribed to turbulence generated by stars and SNe within GMCs (e.g. Krumholz et al. 2012, and references therein). For purposes of cosmological simulations, since observations suggest that this efficiency is nearly universal, this can largely be folded into the normalization of the star formation recipe. \nThe second inefficiency is generally ascribed to large-scale galactic outflows powered by massive stars and SNae. Signatures of such outflows, with mass loss rates likely of the same order as the star formation rate or larger, are ubiquitously observed in star forming galaxies (Veilleux et al. 2005). Modeling galactic outflows has therefore become a central challenge for recent simulations. \nEarly work attempted to model stellar feedback via the deposition of thermal energy from SNae in the surrounding gas (e.g. Katz et al. 1996). It was quickly realized that this had almost no effect, because the short cooling times meant that the energy was radiated away very quickly, adding negligible ISM pressure, let alone driving an outflow. Since then, most cosmological models have adopted some sub-grid prescription to enable effective ejective feedback that typically involves either implementing ad hoc 'tricks', such as turning off cooling for some time or super-heating the gas, that attempt to mimic the ISM processes that allow stellar-driven winds to develop in real galaxies, or else implementing outflows via kinetic energy injection. \nA variant on the ISM heating model called 'blast wave' feedback was developed by Stinson et al. (2006) and has been extensively used in Gasoline and RAMSES (Bournaud et al. 2010). Here, after the gas is heated, radiative cooling is shut off for the lifetime of the SN-driven blastwave as predicted by a spherical Sedov solution. This enables the gas to 'feel' the higher pressure and develop a coherent large-scale outflow. While successful in many regards, this model still predicted too much early star formation, so Stinson et al. (2013) added 'early stellar feedback' intended to mimic the energy input from young stellar winds and radiation. \nAnother variant of a purely thermal stellar feedback model was proposed by Dalla Vecchia & Schaye (2012) and implemented in the EAGLE simulations (Schaye et al. 2014). Instead of turning off cooling, the trick for preventing the energy from immediately cooling away involves making the energy deposition stochastic. The mean amount of energy injected per mass of stars formed is set by stellar population models and supernova energetics. The temperature jump of particles receiving a boost is specified (∆ T = 10 7 . 5 K, typically), and a parameter f th determines the probability that a given SPH particle in the vicinity of a star-forming particle will get heated. Hence the gas is heated to much higher temperatures than would be the case if the same amount of energy were continuously added to all of the SPH neighbors, increasing the cooling time and mitigating energy losses. The overall \nefficiency of the feedback can be adjusted by varying f th . Schaye et al. (2014) made f th a function of the local gas metallicity and density, as they found that this most successfully reproduced the observed SMF and galaxy sizes. \nApopular approach introduced by Navarro & White (1993) and implemented into Gadget2 by Springel & Hernquist (2003) is to simulate outflows by giving gas 'kicks', rather than trying to overpressurize ISM gas by adding thermal energy. Such kinetic outflows are less directly tied to the physics generating outflows, but enable greater control over outflow parameters in order to both mimic observed outflows more closely and assess the impact of varying the outflow parameters. In such models, hydrodynamics is sometimes shut off ('decoupled') for some period of time to mimic the collective power of supernovae blowing a chimney through the ISM; it is unclear whether this provides a more physical description of outflow propagation through the ISM, but it generally does result in better resolution convergence (Dalla Vecchia & Schaye 2008). These models are parameterized by a mass loading factor η ≡ ˙ M out / ˙ M ∗ and a wind velocity v wind , which together determine how many particles to kick and how hard to kick them. Springel & Hernquist (2003) assumed a constant mass loading factor and constant wind velocity, and showed that this yielded a cosmic star formation history in much better agreement than a model without outflows. Oppenheimer & Dav'e (2006) showed that adopting scalings motivated by analytic 'momentum-driven' wind models (Murray et al. 2005) produced better agreement with many galaxy and IGM properties including the galaxy mass-metallicity relation, the enrichment history of the IGM, and the galaxy stellar mass function (see also Dav'e et al. 2013, 2011b, Finlator & Dav'e 2008, Oppenheimer & Dav'e 2008). For momentum-driven winds, the mass loading factor scales as η ∝ σ -1 , and the wind velocity scales as v wind ∝ σ , where σ is the velocity dispersion of the galaxy. The Illustris simulations (Vogelsberger et al. 2014a) also employ kinetic winds by creating and launching decoupled wind particles, and rejoining them back into the gas mesh after recoupling. They adopt scalings expected for 'energy-driven' winds, namely η ∝ σ -2 . \nMost semi-analytic models parameterize star formation feedback in a similar manner, based on the approach introduced in White & Frenk (1991) and Kauffmann et al. (1993). In each timestep, the SAM computes the rate at which cold gas is ejected from the disk by a wind: \n˙ m ej = glyph[epsilon1] w ( V 0 V c ) α w ˙ m ∗ \nwhere ˙ m ∗ is the star formation rate in the galaxy, V c is the circular velocity of the galaxy, V 0 is an arbitrary normalization parameter, and glyph[epsilon1] w and α w are treated as tunable free parameters. For α w = 1 or α w = 2, this is equivalent to the 'momentum driven' or 'energy driven' wind scalings discussed above. One must then decide what happens to the ejected gas, and here different modelers diverge more widely. Some fraction of the ejected gas may escape the dark matter halo, and may be tracked in an 'ejected' reservoir from which it is allowed to accrete into the halo again over a longer timescale. Otherwise, the ejected gas is added to the halo hot gas reservoir. SAMs generally implement some sort of model, of varying complexity, to arrange that the fraction of ejected gas that escapes the halo is larger at lower halo V H , and assymptotes to unity above V H glyph[similarequal] 120-150 km/s (or a halo mass of a few × 10 12 M glyph[circledot] ). \n3.3.3 AGN Feedback Observational phenomena associated with accreting black holes include electromagnetic radiation, relativistic jets, and less-collimated non-relativistic outflows (Krolik 1999). There are several different physical mechanisms whereby the large \namounts of energy and momentum produced by AGN can couple with the gas in and around galaxies, possibly regulating the growth of the black hole itself, and potentially suppressing cooling and star formation on galactic scales. At the most basic level, AGN can heat gas up (thermal feedback), drive winds that eject gas (kinetic feedback), and ionize or photo-dissociate gas (radiative feedback). The main heating mechanisms are Compton, photo-ionization, and photo-electric heating. Radiation may also drive winds via pressure on spectral lines, free electrons, or dust. These winds may originate in the torus or accretion structure near the black hole, the broad line region (BLR), larger nuclear scales ( ∼ kpc), or all of the above. Winds arising on 'small' (BLR/accretion disk) scales may drive galaxy-scale winds by shocking and sweeping up ISM gas - or they may simply vent out of the galaxy without ejecting much mass. In addition, highly relativistic giant radio jets may heat the intra-cluster medium through bubbles, weak shocks, and sound waves (Fabian 2012, McNamara & Nulsen 2007). \nFocussing first on the processes associated with the radiatively efficient ('radiative mode', sometimes called 'quasar mode' or 'bright mode') of BH growth, one of the major dynamical questions is whether AGN-driven winds are primarily 'energy driven' or 'momentum driven'. As in the case of stellar driven winds, the question is how quickly and efficiently is the thermal energy generated when the wind shocks the surrounding gas radiated away. Momentum cannot, of course, be radiated away, and so if most of the thermal energy is quickly dissipated, we term the wind 'momentum driven'. If radiative losses are negligible, we term it 'energy driven'. Clearly real winds may often be somewhere in between. The significance of this distinction is that the momentum flux of swept-up material in an energy-conserving outflow is 'boosted' due to work done by the hot shocked gas (an effect familiar from the Sedov-Taylor phase in supernova remnants). \nIt has been argued that in the dense cold gas that must surround rapidly accreting black holes, cooling times are short and winds must be predominantly momentum-driven (Debuhr et al. 2011, King 2005, Ostriker et al. 2010). However, observations of AGN-driven outflows suggest 'boost' factors of ˙ p/ ˙ p rad ∼ 2-30 (e.g. Moe et al. 2009, Sturm et al. 2011), with an average probably around 10, where ˙ p rad = L AGN /c is the radiative momentum flux output by the AGN. Faucher-Gigu'ere & Quataert (2012) argued recently based on analytic calculations that AGN-driven outflows are likely to be largely energy-conserving in many situations relevant to observed systems, particularly for 'fast' ( v w ∼ 10 , 000-30,000 km/s) winds. \nOne of the earliest three dimensional simulations of AGN feedback in galaxies was presented in Springel et al. (2005b) and Di Matteo et al. (2005). Here, the BH accretion rate was modelled using the Bondi approach outlined above, and the resulting bolometric luminosity was assumed to be proportional to the BH accretion rate. A fixed fraction of the bolometric luminosity was deposited into the neighboring gas particles as thermal energy. These simulations did not use cosmological initial conditions, but considered binary mergers of idealized disk galaxies without hot gas halos. This work showed that deposition of about 5% of the bolometric luminosity was able to drive strong outflows that eventually halted further accretion onto the BH and also removed nearly all cold gas from the galaxy, resulting in quenching of star formation (Springel et al. 2005a). Furthermore, the models produced self-regulated BH growth, leading to a tight M BH -σ relationship in agreement with the observed one. A similar approach has been used in a large number of subsequent studies. Although these studies, taken at face value, suggest that purely energy driven winds can regulate BH growth and drive large-scale outflows, it is likely that radiative losses were artificially suppressed due to the highly pressurized ISM model adopted in these \nsimulations. \nMoreover, these simulations neglected the expected momentum deposition. Several recent works have implemented momentum-driven winds in hydrodynamic simulations via radiation pressure on dust (Debuhr et al. 2011, 2010) and via BLR winds (Choi et al. 2014a, 2012) and found that these winds can play a significant role in modulating the growth of the black hole and the galaxy. The star formation remains quenched over a much longer timescale in the simulations that include momentum-feedback, because the density of hot gas near the center of the halos is significantly reduced (Choi et al. 2014b). \nThe other important class of feedback processes is connected with highly collimated jets of relativistic particles ('jet mode' or 'radio mode'; see the recent reviews by Fabian 2012 and Heckman & Best 2014). The kinetic energy in these jets can exceed the total bolometric luminosity of the AGN by several orders of magnitude. While jets may be observed at many wavelengths, there is a class of sources detected at radio wavelengths that do not exhibit the classical signatures of 'radiatively efficient' AGN - no UV, X-ray, or IR excess, and no highly ionized emission lines. Optically, these objects resemble normal massive early type galaxies. They are associated with radiatively inefficient accretion, and with extremely low accretion rates onto the central BH. The radio jets are observed to correspond, in many cases, with 'bubbles' visible in X-ray images, regions filled with hot plasma presumably heated by shocks from the jet's interaction with the ICM. Studies of the bubble energetics have shown that there is easily enough energy deposited in the ICM to offset cooling; in fact, in groups and low-mass clusters the energy probably exceeds the requirements for balancing cooling by up to an order of magnitude. Radio galaxies are common in massive early type galaxies in groups and clusters, and bubbles and/or radio sources are seen in 95% of 'cool core' clusters (clusters with short central cooling times). \nOnce again, the energetics are such that one expects this 'jet mode' feedback to have a significant impact on galaxy formation, but many details of the physics remain unclear. The main puzzle is how such highly columnated bi-polar jets can nearly isotropically heat a large volume of intragroup or cluster gas (Vernaleo & Reynolds 2006). The bubbles provide an important clue - these bubbles rise buoyantly in the hot atmosphere, reaching fairly large radii. Heating may occur via turbulent mixing of bubbles with the ICM (Scannapieco & Bruggen 2008), viscous dissipation of weak shocks (Ruszkowski et al. 2004), or cosmic ray heating (Sharma et al. 2009). Although some recent simulations that attempt to explicitly model jet heating in 3D have claimed greater success at averting the cooling flow problem (Gaspari et al. 2011, Li & Bryan 2014), all of these simulations neglect a cosmological formation history, with merging and accretion, as well as star formation and stellar feedback. A detailed physical understanding of how the jets couple to the surrounding hot gas and how effective they are in regulating cooling flows over long timescales remains lacking (see also Babul et al. 2013, Cielo et al. 2014). \nSijacki et al. (2007) were the first to attempt to include both the 'radiative' and 'jet' modes of AGN feedback in numerical cosmological simulations, albeit in a simplified way, necessitated by the relatively coarse numerical resolution. Above a critical black hole accretion rate ( ∼ 0 . 01 times the Eddington rate), the AGN was assumed to be radiatively efficient and a fraction of the AGN bolometric luminosity was deposited in the gas as thermal energy. Below the critical accretion rate, the AGN is assumed to be radiatively inefficient and to produce jets which inflate bubbles - however they do not directly simulate the jet. Instead they insert bubbles by hand, with energy and radius scaled to the black hole mass as motivated by analytic models for radio cocoon expansion. \nSimilar approaches have now been implemented in a few sets of cosmological simulations. \nThe Illustris simulations, using the Arepo moving mesh code, also use the Bondi accretion model and a similar feedback scheme to that of Sijacki et al. (2007). In addition, the Illustris simulations include a simplified treatment of photo-ionization and photo-heating due to the AGN radiation field. A somewhat different approach is taken in the EAGLE (Schaye et al. 2014) and OWLS (Schaye et al. 2010) simulations - they adopt a variant of the 'stochastic thermal feedback' model used for star formation feedback, described in § 3.3.2, in which an average energy injection rate is given by E BH ∝ ˙ m acc c 2 , where ˙ m acc is the accretion rate onto the BH and c is the speed of light. The injected energy is stored by each BH until it can stochastically heat some minimum number of particles with a temperature increase ∆ T . The value of ∆ T may depend on resolution, and is higher than for the stellar feedback model ∆ T ∼ 10 8 . 5 -10 9 K (Schaye et al. 2014). Other simulations do not explicitly follow black hole growth and associated feedback, but include heuristic 'quenching' mechanisms based on surrogate galaxy or halo properties (Gabor & Dav'e 2012, Gabor et al. 2011). \nA large number of groups have also implemented 'radiative mode' AGN-driven winds and 'jet mode' AGN heating in semi-analytic models. Although the details differ from model to model, there are a number of common elements that are widely adopted: 1) A distinction is made between black hole fueling via cold gas (which is typically assumed to be driven into the nucleus by mergers and/or disk instabilities; see § 3.2), and hot gas which is generally assumed to accrete via a cooling flow. 2) BH accretion fueled by the merger/disk instability driven mode is associated with radiatively efficient accretion at a significant fraction of the Eddington rate; accretion fueled by hot gas is assumed to lead to very sub-Eddington, radiatively inefficient accretion associated with the 'jet mode'. 3) The 'jet mode' is assumed to be activated only in the presence of a quasi-hydrostatic hot halo, i.e. when the halo is predominantly accreting via the 'hot mode' discussed earlier. 4) The 'jet mode' is able to extract a certain fraction of the BH mass in the form of energy, which is used to offset cooling, or is assumed to be able to establish heating-cooling balance when the BH mass exceeds a critical value. \nFor example, in the Croton et al. (2006) model, the 'jet mode' accretion rate is modeled as: \n˙ m BH , R = κ AGN ( m BH 10 8 M glyph[circledot] )( f hot 0 . 1 )( V vir 200 km / s ) 3 \nwhere f hot is the fraction of the total halo mass in the form of hot gas, m BH is the mass of the black hole, and V vir is the virial velocity of the halo. The cooling rate computed as described in § 2.3.2 is offset by a heating term, such that the effective cooling rate is: \n˙ m cool , eff = ˙ m cool -L AGN 1 2 V 2 vir \nwhere L AGN = glyph[epsilon1] rad ˙ m BH c 2 with glyph[epsilon1] rad = 0 . 1 the conversion of accreted rest mass into energy. Other SAMs use similar scalings, some with more attempted explicit connection with the invoked physical processes and/or with observations (e.g. Monaco et al. 2007, Somerville et al. 2008), but these appear to produce similar results at z = 0, and even for the redshift evolution of massive galaxies (Fontanot et al. 2009). \nIn addition to 'jet mode' feedback, some SAMs implement AGN-driven winds. Somerville et al. (2008) adopted momentum driven wind scalings associated with 'radiative mode' AGN activity: \ndM out dt = glyph[epsilon1] wind glyph[epsilon1] rad c V esc ˙ m acc \nwhere glyph[epsilon1] wind represents the effective coupling efficiency, V esc is the escape velocity of the galaxy, and ˙ m acc is the BH accretion rate in the radiatively efficient mode. See also Fontanot et al. (2006) for an alternate implementation of 'radiative mode' wind feedback in SAMs. \nExamples of SAMs that do not follow BH growth explicitly, but instead implement more heuristic halo-based quenching include Cattaneo et al. (2006) and Lu et al. (2011). In these models, cooling is simply switched off when the halo mass exceeds a critical value, which may depend on redshift.", '4 RESULTS FROM CURRENT MODELS: INSIGHTS AND PUZZLES': "We return for a moment to Fig. 1 and 2 to illustrate some general insights into the process of galaxy formation and evolution in the ΛCDM framework that have arisen from numerical simulations. Starting with the left column of Fig. 1, we see that structure formation in the dark matter component proceeds via the formation of sheets or giant walls, which form filaments where they intersect. Dark matter halos form at the intersection of filaments, which funnel dark matter and gas into halos like tributaries flowing into a lake. Comparing the first and second columns of Fig. 1, one can see that there is a very strong correspondence between the dark matter and gas density fields on large scales. This illustrates that gas flows on large scales are dominated by gravity. Moving to the third column of Fig. 1, we can see that the gas surrounding massive halos is hot, and larger regions become heated as time progresses. This heating is in part due to shock heating as halos collapse, but in these simulations is in large part due to star formation and AGN feedback. Finally, examining the rightmost column of Fig. 1, we see that metals are dispersed to quite large distances from galaxies, and polluted regions again fill a larger comoving volume over time. Fig. 2 shows how filaments of relatively cold gas can sometimes penetrate some distance into hot halos - these supply the 'cold mode' accretion discussed earlier (sometimes called 'stream fed' accretion). The inset in Fig. 2 emphasizes how small galaxies are compared with the structures seen in the 'cosmic web'.", '4.1 Global Properties': "4.1.1 Stellar Mass Assembly Over Cosmic Time A fundamental observational target for modelers is reproducing the statistical distributions of global properties for galaxy populations at different cosmic epochs, such as luminosity functions (LF), stellar mass functions, and cold gas mass functions. It has been realized for some time that the observed local LF or SMF is not 'naturally' reproduced by galaxy formation models based within the ΛCDM paradigm: CDM models generically predict that the slope of the mass function of dark matter halos has a slope of α H ∼ -2, while the slope of the observed galaxy SMF locally is much shallower ( α g glyph[similarequal] -1 . 3). A number of authors suggested that supernova feedback could flatten out the low-mass slope by suppressing star formation in low-mass halos (Dekel & Silk 1986, Larson 1974, White & Frenk 1991). Furthermore, although ΛCDM predicts an exponential cut-off or 'knee' in the halo mass function, with a similar functional form to that of the observed SMF, the halo mass function turn-over is at much larger masses. Although the cooling times in these massive, group and cluster-sized halos are predicted to be somewhat longer than in low-mass halos (Blumenthal et al. 1984, Rees & Ostriker 1977), this turns out to be insufficient to explain the very inefficient star formation required to reconcile the abundance of massive galaxies with that of dark matter halos. \nAfter decades of effort, theoretical models of galaxy formation are now fairly successful \nFigure 4: \n<!-- image --> \nGalaxy stellar mass function at redshifts z ∼ 0-4. In the z = 0 . 1, z = 1, and z = 2 panels, black square symbols show a double-Schechter fit to a compilation of observational estimates. Observations included in the fit are: z = 0 . 1 - Baldry et al. (2008), Moustakas et al. (2013); z = 1 and z = 2 panels - Tomczak et al. (2014), Muzzin et al. (2013). The fits shown at z = 1 and z = 2 are interpolated to these redshifts from adjacent redshift bins in the original published results. The formal quoted 1 σ errors on the estimates shown in these three panels are comparable to the symbol size, and are not shown for clarity (the actual uncertainties are much larger, but are difficult to estimate accurately). In the z = 0 . 1 panel, the estimates of Bernardi et al. (2013) are also shown (open gray circles). In the z = 4 panel we show estimates from Duncan et al. (2014, squares), Caputi et al. (2011, crosses), Marchesini et al. (2010, circles, for z = 3-4), and Muzzin et al. (2013, pentagons, z = 3-4). Solid colored lines show predictions from semi-analytic models: SAGE (Croton et al. in prep, dark blue), Y. Lu SAM (Lu et al. 2013, magenta), GALFORM (Gonzalez-Perez et al. 2014, green), the Santa Cruz SAM (Porter et al. 2014, purple), and the MPA Millennium SAM (Henriques et al. 2013). The dotted light blue line shows the Henriques et al. (2013) SAM with observational errors convolved (see text). Colored dashed lines show predictions from numerical hydrodynamic simulations: EAGLE simulations (Schaye et al. 2014, dark red), ezw simulations of Dav'e and collaborators (Dav'e et al. 2013, bright red) and the Illustris simulations (Vogelsberger et al. 2014b, orange). \nat reproducing the SMF of galaxies at z ∼ 0 by invoking a plausible, if still in most cases schematic, set of physical processes. Fig. 4 shows a compilation of predictions of recent numerical hydrodynamic simulations and semi-analytic models for the SMF from z = 4 to z ∼ 0. These models are all taken directly from the original publications and no attempt has been made to calibrate them to the same set of observations or to correct for the \nslight differences in cosmology 4 . This success has been obtained by 'tuning' not only free parameters but also the recipes associated with the sub-grid physics (star formation, stellar feedback, AGN feedback). Predictions of the build-up of stellar mass over cosmic time, with these recipes and parameters held fixed, present a more stringent test of the models. \nIn a broad brush sense the model predictions are generally encouraging. A very general prediction of ΛCDM-based models is that galaxies built up their stellar mass gradually over time, which is supported by observations. All models predict efficient early star formation ( z > ∼ 4) in low mass halos, and steep stellar mass and rest-UV luminosity functions at these early epochs, in agreement with observations. Models including AGN feedback or heuristic quenching predict that massive galaxies formed earlier and more rapidly than lower mass galaxies, again in qualitative agreement with observations. Most models even demonstrate very good quantitative agreement, within the errors on stellar mass estimates, between predicted and observed SMF and LF for massive galaxies ( m star > M char ). Note that in Fig. 4, most of the theoretical predictions for the stellar masses have not been convolved with the expected uncertainties that are inherent in the observational estimates. Including these in a simplified manner brings the model predictions into better apparent agreement with the observations on the massive end (e.g. Henriques et al. 2013, Lu et al. 2013), as shown here for the MPA SAM as an illustration. For a more detailed study of this issue see Mitchell et al. (2013). \nAs can be seen as well in Fig. 4, models currently have greater difficulties reproducing the abundances and assembly histories of low-mass galaxies at intermediate redshifts. Fontanot et al. (2009) demonstrated that three independently developed SAMs overproduce galaxies with m star < ∼ 10 10 M glyph[circledot] by a factor of ∼ 2-3 over the redshift range 4 < ∼ z < ∼ 0 . 5. Weinmann et al. (2012) showed that a qualitatively similar problem exists for SAMs and for hydrodynamic simulations. This problem appears to persist even in the latest state-of-the-art cosmological hydrodynamic simulations, as seen in Fig. 4, and already pointed out in the case of Illustris by Torrey et al. (2014). As discussed in Fontanot et al. (2009), several different sets of observations suggest that massive galaxies form early and rapidly, while low-mass galaxies form later and with a more extended timescale - the phenomenon that is often referred to as 'downsizing' or 'staged' galaxy formation (Noeske et al. 2007). The overproduction of low-mass galaxies is a symptom of the failure of current models to reproduce this mass-dependence in the star formation histories of galaxies. Low-mass dark matter halos actually have earlier formation times than high-mass halos - the opposite of the trend seen in observations (Conroy & Wechsler 2009). In current simulations, the star formation histories closely trace the DM mass accretion histories, thus similarly failing to reproduce the observed trend. \nIt seems clear that the sub-grid recipes controlling star formation and/or stellar feedback need to be modified in order to solve this problem. Henriques et al. (2013) found that making the stellar feedback stronger and modifying the timescale for the re-accretion of ejected gas led to significant improvement in the MPA-SAM for the predicted abundances of low-mass galaxies as well as other observed properties at z < ∼ 3. White et al. (2014) investigated several classes of empirical solutions to this problem, including modifying the efficiency of stellar driven galaxy outflows, modifying the timescale for gas to turn into stars, and modifying the timescale for gas to be accreted (or re-accreted) into galaxies. They concluded that solutions that modified the outflow efficiencies and accretion timescales were the most promising. Moreover, Torrey et al. (2014) experimented with changing the \ncoupling strength and velocity of the stellar driven winds, and found that this can change the normalization of the SMF at the low-mass end, but cannot change the evolutionary shape, which is what is required to solve this problem. \nA convenient way to assess the success of a cosmological simulation in reproducing the galaxy SMF or LF is via empirical constraints on the relationship between stellar mass (or luminosity) and halo mass, as derived by 'galaxy-halo mapping' techniques such as SHAM and HOD, and other methods such as galaxy-galaxy lensing, clustering, satellite kinematics, and X-ray observations (Behroozi et al. 2010, 2013a, Moster et al. 2013, 2010b, and references therein). Different methods and groups are generally all in broad agreement that star formation feedback plays a crucial role in shaping this relationship for halos with M H < ∼ 10 12 M glyph[circledot] , with photo-ionization squelching perhaps also playing a significant role below halo masses of about a few 10 10 M glyph[circledot] (this mass scale remains uncertain; see § 3.3.1). At larger halo masses, M H > ∼ 10 12 M glyph[circledot] , there is a general consensus that AGN feedback probably plays an important role, although other processes (such as gravitational heating) may contribute as well (Birnboim & Dekel 2011, Johansson et al. 2009b, Khochfar & Ostriker 2008). In order to reproduce the slope of the stellar mass-halo mass ( m star -M halo ) relation at low masses, most models adopt stellar feedback recipes that either assume or result in mass loading factors that increase fairly strongly with decreasing halo mass or circular velocity, similar to the energy- or momentum-driven wind scalings discussed in § 3.3.2. \nThere is a broad though not universal consensus that AGN feedback implemented purely via deposition of thermal energy associated with the radiatively efficient mode of BH growth (as in e.g. Di Matteo et al. 2005) does not by itself suppress cooling and star formation in massive halos enough (or on long enough timescales) to satisfy observational constraints. Although thermal energy deposition can temporarily slow or halt cooling, after several Gyr, the gas starts to re-cool and form stars (Choi et al. 2014b, Gabor & Dav'e 2012). An exception is the stochastic thermal feedback model implemented in EAGLE, which reproduces the observed stellar fractions very well, though there is still tension between the predicted temperatures of the hot gas in group- and cluster-sized halos and X-ray observations (Schaye et al. 2014). Another solution on the high-mass end is nearly constant injection of energy via 'jet mode' feedback, although as discussed in § 3.3.3, implementations of this process in cosmological simulations remain schematic. Inclusion of the momentum deposition associated with the radiatively efficient mode also appears to be able to suppress cooling for longer (Choi et al. 2014b). \n4.1.2 Global Scaling Relations: Gas, Star formation and Metals Galaxies are comprised of stars, gas, metals, black holes, and dark matter. The scaling relations between these properties as a function of mass and redshift provide crucial constraints on galaxy growth, and are in principle among the most direct ways to constrain baryon cycling processes. \nThe basic origin of many scaling relations can be understood in a simple framework based on mass balance in the ISM (alternately called an 'equilibrium', 'bathtub', or 'gas regulator' model): \n˙ M inflow = ˙ M ∗ + ˙ M outflow + ˙ M gas , (6) \nwhere the terms are the baryonic mass inflow rate, SFR, mass outflow rate, and rate of change of the mass in the gas reservoir. When averaged over cosmological timescales, ˙ M gas is expected to be small compared to the other terms (Dav'e et al. 2012, Dekel & Mandelker 2014, Finlator & Dav'e 2008), though its evolution can have important effects (Lilly \nFigure 5: \n<!-- image --> \nThe average star formation rate in bins of stellar mass, for redshift bins from z = 0-4. Grey and black symbols show observational estimates: z = 0 . 1 - Salim et al. (2007, open circles); z = 1 and z = 2 Whitaker et al. (2014, pentagons, interpolated in redshift from the published results); z = 4- Steinhardt et al. (2014, crosses); Salmon et al. (2014, circles); all panels - fit to data compilation from Speagle et al. (2014, squares). Colored lines show predictions from semi-analytic models and numerical hydrodynamic simulations; key is the same as in Fig. 4. Note that the observational estimates shown are for star forming galaxies; different methods have been used to isolate the 'star forming sequence' from 'quiescent' galaxies. Some of the modelers have applied a cut to select star forming galaxies, but some have not. \net al. 2013). Inflow into halos is driven primarily by gravitational accretion from the IGM (Dekel et al. 2009, Kereˇs et al. 2005). The rate at which dark matter halos grow, the halo mass accretion rate ( ˙ M halo ), is well-characterized in ΛCDM, and roughly given by ˙ M halo ∝ M halo (1 + z ) 2 . 5 (Dekel et al. 2009, Faucher-Gigu'ere et al. 2011). However, preventive feedback within galaxy halos can retard gas accretion into the ISM, and outflows can remove fuel for star formation even after it enters the ISM, so ˙ M inflow may not trace ˙ M halo . \nWe can rewrite equation 6 as \nsSFR = ζ (1 + z ) 2 . 5 ( m star /M halo ) × (1 + η ) , (7) \nwhere ζ is the fraction of material entering the virial radius that makes it into the ISM, and η ≡ ˙ M outflow / ˙ m star is the outflow mass loading factor. The dependence of sSFR on m star and z therefore reflects the evolving combination of accretion and feedback. \nFigure 5 shows a comparison of SFR vs. m star for the SAMs and simulations shown in Figure 4, along with a compilation of recent observational determinations as described in the figure caption. All models generally reproduce the near-unity slope, at all redshifts. Most models match the amplitude at z ∼ 0, although the turnover at high masses due to quenching can vary significantly (and can be sensitive to the definition of 'star forming' galaxies), and models tend to predict a steeper trend at low masses. By z ∼ 1 -2, it is clear that most models fall below the observations, a long-standing discrepancy first highlighted in Daddi et al. (2007). The redshift dependence of the sSFR is generically difficult to match in models because it differs strongly in the intermediate redshift regime (4 < ∼ z < ∼ 0 . 5) from the dependence predicted by ˙ M halo (Dav'e 2008, Sparre et al. 2014). By z = 4, some models are able to match the data, though others continue to fall substantially short. The normalization of the predicted SFR vs. m star relation depends on resolution and the calibration of the sub-grid parameters - e.g. Schaye et al. (2014) show (their Fig. 11) that a higher resolution simulation in the EAGLE suite, re-calibrated to the SMF, predicts a higher SFR at m star < ∼ 10 10 , in better agreement with the observations. However, the redshift dependence of the sSFR is roughly unchanged (Furlong et al. 2014). \n<!-- image -->", 'Figure 6:': "The average metallicity of cold gas in bins of stellar mass, for redshift bins from z = 0-4. Grey and black symbols show observational estimates: z = 0 . 1 - Peeples et al. (2014, filled circles); Andrews & Martini (2013, stars). In all panels, the filled squares show the compilation of Zahid et al. (2013). Colored lines show predictions from semi-analytic models and numerical hydrodynamic simulations; key is the same as in Fig. 4. \nFigure 7: \n<!-- image --> \nThe average cold gas fraction in bins of stellar mass, for redshift bins from z = 0-4. Grey and black lines and symbols show observational estimates: z = 0 . 1 - binned results from the compilation of Peeples et al. (2014, filled squares). In all panels, the open squares show the predictions of the 'equilibrium model' presented in Saintonge et al. (2013), which are in good agreement with their data compilation and with the estimates of Genzel et al. (2014, shown as solid triangles), extending up to z ∼ 3. The solid gray lines and light gray shaded areas show the empirical total cold gas fraction (( M HI + M H 2 ) /m star ) estimates from Popping et al. (2014a). Dotted gray lines show the molecular fraction estimates ( M H2 /m star ) from Popping et al. (2014a). Note that the z = 0 observational estimates shown are for H i +H 2 , while the z > 0 estimates are based on CO and most closely trace H 2 . Colored lines show predictions for the total cold gas fraction from semi-analytic models and numerical hydrodynamic simulations; color key is the same as in Fig. 4. \nThese trends can be generally understood in the mass balance framework. From abundance matching, the m star -M halo relation is constrained to evolve mildly with redshift (Behroozi et al. 2013a, Moster et al. 2013). If ζ and η also evolve slowly, then sSFR should evolve as ∼ (1 + z ) 2 . 5 . This is indeed roughly the evolution observed out to z ∼ 2 (e.g. Lilly et al. 2013, Speagle et al. 2014, Whitaker et al. 2014). However, to higher redshift the evolution slows, suggesting that either the m star -M halo relation or η is higher, or that ζ is lower. The assumption of ˙ M gas ≈ 0 may be faulty at very early epochs if the inflowing gas cannot be processed in the ISM fast enough, which observations suggest may be the case at z > ∼ 4 (Papovich et al. 2011). It had been suggested that the efficiency of converting ISM gas into stars is reduced owing to the lower metallicity at early epochs which results in less efficient formation of molecular gas (Krumholz & Dekel 2012), but hydrodynamic simula- \ns incorporating H 2 formation modeling suggest that this effect is not large enough to solve the problem at the observed mass scales (Christensen et al. 2012, Somerville et al. 2014). Moreover, the EAGLE simulations also include a metallicity-dependent density threshold for star formation as proposed in Schaye (2004), representing the same physical effect, but still suffer the same problem. Hence although it is encouraging that most models are now able to predict the sSFR evolution to within a factor of 2-3 and predict roughly the right qualitative trend, such discrepancies, if real, could be pointing to a need to revise our basic understanding of the physical processes regulating star formation at these epochs. \nThe mass dependence of the sSFR also poses interesting challenges to models. In detail, the halo mass accretion rate has a super-linear dependence on M halo , which would naively translate into a positive slope for sSFR( m star ). Observations indicate a sub-unity slope, becoming shallower with time (see Figure 5). Part of this may be due to the fact that for M halo > ∼ 10 11 M glyph[circledot] , an increasing amount of halo inflow is gravitationally shocked into hydrostatic equilibrium (Birnboim & Dekel 2003, Gabor & Dav'e 2012, Kereˇs et al. 2005). Simulations show this is sufficient to explain the mildly negative slope in moderate-sized halos (Dav'e et al. 2011b, Faucher-Gigu'ere et al. 2011), but is insufficient to explain the rapid increase in quenched galaxies at high masses, which requires additional feedback, likely associated with AGN. \nMoreover, models with outflows tuned to reproduce the observed SMF (hence m star -M halo ) predict a flat or positive slope for sSFR( m star ) at M halo < ∼ 10 11 M glyph[circledot] , while observations show a negative slope. The stochasticity of star formation in dwarf galaxies (Tolstoy et al. 2009) may result in a duty cycle whereby observed samples preferentially select dwarfs that are in a high sSFR state, but observations show that essentially all isolated dwarfs in the nearby universe are star-forming (Geha et al. 2012). This is an aspect of the 'dwarf galaxy conundrum' highlighted in Weinmann et al. (2012) and White et al. (2014) and discussed above, which remains a puzzle: in current models that are normalized to fit the present-day SMF, dwarf galaxies not only form their stars too early (resulting in the low-mass excess at intermediate redshift seen in Fig. 4), they also have sSFR that are well below the observed values (see also Torrey et al. 2014). At higher redshifts, the mass dependence is in good agreement with existing observations, but deeper near-IR data is needed to test if a similar discrepancy occurs in mass-selected samples of dwarfs at z > ∼ 1. \nAnother key scaling relation is the stellar mass-gas phase metallicity relation (MZR), which can also be understood from equation 6. Given a metal yield y per unit star formation, the metallicity will be the yield times the SFR, divided by the amount of accreting gas, i.e. \nZ = y ˙ M ∗ ˙ M inflow ≈ y 1 + η , (8) \nwhere this approximation is valid in the limit of no recycled (i.e. previously enriched) accretion into the ISM (Finlator & Dav'e 2008). Wind recycling is generally more rapid in more massive galaxies (Oppenheimer et al. 2010), which will tend to make the MZR steeper. Also, outflows that are significantly more enriched than the ISM can result in a lower metallicity. Hence the MZR primarily reflects galactic outflows, modulated by wind recycling and the 'metal loading factor' (Peeples & Shankar 2011). \nFigure 6 shows the predicted MZR in our suite of SAMs and simulations, compared with observations. We emphasize that, due to uncertainties in the theoretical yields of at least a factor of ∼ 2, and differences of ∼ 30 percent in the adopted value of solar metallicity in different simulations, the absolute normalizations of the predicted MZR should not be given as much weight as the trends with mass and redshift. We also show a recent compilation of \nobservational estimates. Gas-phase abundance measures are sensitive to calibration (Kewley & Ellison 2008), but it is usually the case that relative abundances are more consistent among various indices. Hence the slope of the observed MZR is more robustly known than the amplitude, though the amplitude should still be accurate to within a factor of 2 -3. We show MZR determinations at z ∼ 0 . 1, z ∼ 1, and z ∼ 2 converted to the same calibration, from Zahid et al. (2013). We also show the local MZR from Peeples et al. (2014), which uses the average of all of the calibrations presented in Kewley & Ellison (2008), and the local 'direct method' MZR from Andrews & Martini (2013). \nAt z = 0, most models produce roughly the correct metallicity for galaxies with stellar masses of a few × 10 10 M glyph[circledot] , but predicted MZRs are typically steeper than the observed relations to low masses and have a less pronounced turnover to high masses (EAGLE, which produces a very shallow MZR, is a notable exception). To higher redshifts, models generally predict slow evolution, about a factor of two at a given stellar mass from z = 2 → 0, which is roughly consistent with available observations. \nTo explore the origin of the slope discrepancy, note that equation 8 shows that (in the absence of recycling and metal-enriched outflows), when η glyph[greatermuch] 1 as is generally the case at low masses in these models, the observed MZR Z ∝ m 0 . 3 ∗ (Tremonti et al. 2004) implies η ∝ m -0 . 3 ∗ . When such a scaling (which is similar to the momentum driven wind scaling) is implemented into hydrodynamic simulations, this produces good agreement with the observed MZR (Dav'e et al. 2011a, Finlator & Dav'e 2008), but the predicted SMF is somewhat too steep at the faint end (Dav'e et al. 2011b). Ameliorating this by incorporating a steeper mass dependence of η results in an MZR that is too steep (Dav'e et al. 2013). Accounting for wind recycling does not help this problem- Oppenheimer et al. (2010) highlighted the importance of wind recycling in shaping the SMF at intermediate masses, but in general wind recycling is more important at higher masses, which further steepens the MZR. In general, current simulations have difficulty simultaneously matching the low-mass ends of the SMF and the MZR, suggesting that enrichment in low-mass galaxies is not fully understood. This problem was also discussed in the context of the Illustris simulations by Torrey et al. (2014), who speculated that this tension may suggest that preventative, rather than ejective, feedback is dominant in low-mass galaxies. \nCold gas scaling relations provide information on the fuel for star formation. CO measurements are currently the best tracer of molecular gas content, although there remain significant uncertainties in the conversion factor from CO to H 2 ( X CO ; Bolatto et al. 2013), particularly to higher redshifts. Observations show that low-mass galaxies are more gasrich, with f gas ∝ m -0 . 57 ∗ (Peeples & Shankar 2011). Direct estimates of the H 2 fraction of galaxies to high redshift from CO and dust-based methods (corrected for selection effects) indicate a rise in m H 2 / ( m H 2 + m star ) back in cosmic time to z ∼ 2, then a plateau or possibly a slight decline (Geach et al. 2011, Genzel et al. 2014, Saintonge et al. 2013, Scoville et al. 2014, Tacconi et al. 2013). Empirical estimates of H 2 and total gas fraction based on extended SHAM modeling (Popping et al. 2014a) indicate a similar behavior. \nFigure 7 shows a comparison of the cold gas fractions ( ≡ m cold /m star ) in models, where we defined cold gas in the numerical simulations as that having a hydrogen number density n H > 0 . 13 cm -3 . At z = 0, models generally reproduce the steeply rising gas fractions to low masses, though some have gas fractions significantly below the observed ones at the lowest masses. Gas fractions tend to be fairly sensitive to the prescription used to turn that gas into stars, which varies significantly between models. This generic trend of gas fraction with galaxy mass in the models arises from two physical effects that make the global SFE lower in low-mass galaxies: stronger stellar feedback (Brooks et al. 2007), \nand less efficient formation of H 2 (Christensen et al. 2012, Popping et al. 2014b). Models generally predict rising gas fractions at a given mass to earlier epochs, in broad agreement with observational and SHAM-based estimates out to z ∼ 2, though gas fractions from the a priori models tend to be lower than the empirical SHAM predictions at z ∼ 2-1; perhaps this is another manifestation of the 'dwarf galaxy conundrum' discussed above. Models that track H 2 formation generally predict that galaxies become increasingly H 2 -dominated at higher masses and at high redshift (Fu et al. 2010, Lagos et al. 2011a, Popping et al. 2014b). \nAtomic hydrogen (H i ) can be detected in emission in nearby galaxies, and in distant galaxies via absorption. Since H i represents a transient phase of accretion from the ioinized IGM to the molecular ISM, it is necessary to include both self-shielding and molecular gas formation physics in order to model it, neither of which are straightforward at typical cosmological, or even zoom, resolutions. Nonetheless, SAMs and simulations can broadly reproduce H i mass functions and scaling relations (Dav'e et al. 2013, Duffy et al. 2012, Lagos et al. 2011a, Obreschkow et al. 2009, Popping et al. 2009, 2014b). H i may be a particularly good tracer of environmental processes including satellite quenching (Lagos et al. 2014, Rafieferantsoa et al. 2014), because it is usually arises in the more loosely-bound outskirts. In addition, SAMs and numerical simulations are being used to study the nature of H i seen in absorption (Lyman-limit and Damped Lymanα systems), and its connection with galaxies identified in emission (Berry et al. 2014, Bird et al. 2014, Rahmati et al. 2013, Rahmati & Schaye 2014). These studies provide important complementary constraints on disk formation and feedback processes. \nSo far, we have only considered mean scaling relations, which can be understood in terms of the average accretion rate into the ISM. In the accretion-driven scenario, galaxies fluctuate around the scaling relations, and the timescale to return to the mean is comparable to that required to double the mass of the galaxy. Hence the scatter of the scaling relations reflects the frequency and efficacy of 'perturbing' events. In particular, mergers can drive significant departures from the mean scalings. For example, galaxies that lie significantly above the SF main sequence are observed to have concentrated, spheroid-like (high Sersic) light profiles (Wuyts et al. 2011), as expected if they are driven by major mergers. Reproducing the scatter in the observed scaling relations over cosmic time is a stringent challenge that models are only beginning to tackle (e.g. Sparre et al. 2014). \nFor the mass-metallicity relation, the scatter is seen to be well-correlated with SFR, in the sense that galaxies at a given mass with low metallicity have high SFR (Lara-L'opez et al. 2010, Mannucci et al. 2010) and high H i content (Bothwell et al. 2013a, Lara-L'opez et al. 2013). This is a natural outcome of the accretion rate fluctuation scenario, since a galaxy that undergoes an uptick in accretion will increase its SFR and gas content, owing to a larger gas supply, and lower its metallicity since the accreted gas (or infalling galaxy) will tend to have lower metallicity (Dav'e et al. 2011a). This so-called 'fundamental metallicity relation' has two aspects, namely this second-parameter trend, and the claim by Mannucci et al. (2010) that it is invariant with redshift from z ∼ 0 -2 . 5. However, calibration uncertainties in metallicity measures owing to evolving ISM conditions (Kewley et al. 2013) make the redshift independence difficult to robustly confirm, and even the existence of this second-parameter trend with SFR is not as clear at higher redshifts (Sanders et al. 2014, Steidel et al. 2014). \n4.1.3 Demographics of Star-Forming and Quiescent Galaxies The existence of quiescent galaxies, that almost entirely ceased forming stars many billions of years ago, is an \nadditional indication of the need for some sort of 'quenching' mechanism - processes that prevent gas from cooling and/or forming stars. Peng et al. (2010) coined the terms 'mass' and 'environmental' quenching. In view of the strong correlations between quiescence and other galaxy internal properties (see § 1.1), we prefer the terms 'internal' and 'environmental' quenching. Some of the discussion here will mirror that in § 4.1.1, however, the requirements for producing the correct internal and environmental statistical correlations for quiescent galaxies are more stringent than simply reproducing the stellar mass function - models that reproduce the latter are not guaranteed to reproduce the former. \nThe massive galaxies that are predominantly early type and quiescent in the observed universe are expected to reside within massive dark matter halos ( > ∼ 10 12 M glyph[circledot] ). These halos are expected theoretically, and known observationally through X-ray observations, to be filled with hot gas at virial temperatures of a few × 10 6 -10 8 K that is gravitationally shockheated on infall, and enriched to about a third of solar. Assuming hydrostatic equilibrium, this gas should be cooling fairly rapidly, at rates of hundreds to thousands of solar masses per year. The absence of the signatures of gas cooling below about one-third of the virial temperature in clusters, along with the absence of large amounts of cold gas or young stars, constitutes the classical 'cooling flow' problem (McNamara & Nulsen 2007, and references therein). This problem has its counterpart in theoretical models, in that it has proven difficult to find plausible physical mechanisms that can suppress cooling and keep galaxies in massive halos as quiescent as they are observed to be. \nSimulations without any sort of 'quenching' mechanism (such as AGN feedback) produce inverted color-magnitude relations (more massive and luminous galaxies are more likely to be blue and star forming) without any hint of bimodality (Gabor et al. 2011, Somerville et al. 2008). The first generation of SAMs that included AGN feedback were able to qualitatively reproduce the observed bimodality of the color and sSFR distribution and the fraction of quiescent galaxies as a function of stellar mass (Bower et al. 2006, Croton et al. 2006, Kimm et al. 2009, Somerville et al. 2008); certainly, including AGN feedback greatly improved the results relative to the old models. In these models, the mechanism that was primarily or entirely responsible for quenching was the 'jet mode' type of feedback described in § 3.3.3, in which star formation dies out because the hot gas halo is continually heated so the supply of new cold gas is cut off. More heuristic models, in which cooling is simply shut off when the dark matter halo exceeded a certain critical mass, performed nearly as well as models that explicitly implemented 'jet mode' AGN feedback (Cattaneo et al. 2006, Kimm et al. 2009). Some recent SAMs reproduce the observed tighter correlation of the quiescent fraction with B/T than with stellar mass, while others do not, suggesting that this could provide constraints on quenching mechanisms (Lang et al. 2014). \nSpringel et al. (2005a) showed that including thermal AGN feedback in hydrodynamic simulations of isolated binary mergers was able to drive powerful winds that evacuated most of the cold gas from the galaxy, leading to strong quenching of star formation. These results motivated semi-empirical models positing that quenching associated with mergers, rapid black hole growth, and 'radiative mode' AGN feedback could explain the growth of the quiescent early type population (Hopkins et al. 2008a,b). However, subsequent work with semi-analytic models and cosmologically-based hydro simulations suggested that thermal feedback associated with the 'radiative mode' of BH accretion, when implemented using algorithms similar to those of Springel et al. (2005b), is not able to produce longlived quiescent galaxies, since it fails to prevent subsequent accretion which reactivates star formation within a Gyr or two (see e.g. Choi et al. 2014b, Gabor et al. 2011). \nFollowing the approach presented in Sijacki et al. (2007), the Illustris simulations ex- \nplicitly included both local thermal heating associated with black hole accretion above a critical rate (representing 'radiative mode'), and more distributed heating associated with low BH accretion rates (representing 'jet mode'). They produced a bimodal color magnitude diagram, with a red galaxy fraction as a function of stellar mass and environment in good agreement with observations by Peng et al. (2010) and others (Vogelsberger et al. 2014a). The observed red sequence colors have proven difficult to reproduce quantitatively in all types of models (Gabor & Dav'e 2012, Guo et al. 2011, Vogelsberger et al. 2014a), but significant uncertainties remain in this regime in the stellar population models that are used to predict such colors from models. In contrast, the 'stochastic thermal' AGN feedback model as implemented in EAGLE does not explicitly have two distinct modes, and can still reproduce quenched galaxy observations at a similar level (Schaye et al. 2014). Cosmological zoom-in simulations including fast momentum-driven AGN winds also appear to be able to quench and maintain quiescence over long timescales without any explicit 'jet mode' type feedback (Choi et al. 2014b, Choi et al. in prep). Conversely, Gabor & Dav'e (2014) suggested that the presence of a hot halo kept hot by AGN feedback is sufficient to quench a galaxy, without the need for additional radiative mode feedback, showing that this reproduces both internal and environmental quenching as observed. Hence there remains much debate over the relative importance of these two AGN feedback modes, whether one or both are required, and even whether they are distinct. \nReproducing the observed patterns of 'environmental quenching' has provided another challenge to models. Peng et al. (2012) showed that when SDSS galaxies were identified as 'satellites' or 'centrals' using a group catalog, the fraction of quiescent centrals depended only on stellar mass, while the fraction of quiescent satellites depended on both mass and environment. Certainly there are many candidate processes that could preferentially quench satellites, such as harrassment, tidal stripping, or ram pressure stripping. In many SAMs, galaxies are not allowed to accrete any new gas from the hot halo or the IGM once they become satellites (sometimes called 'strangulation'). This is known to produce far too high a fraction of quiescent satellites (Font et al. 2008, Kimm et al. 2009, Weinmann et al. 2006b). Instead, satellite quenching seems to take a surprisingly long time, perhaps many Gyr (Wetzel et al. 2012). Hydrodynamic simulations indeed show that infalling satellites remain star-forming for at least a Gyr (Simha et al. 2009), as it takes time for the hot gas and dark matter from the halo in which the satellite galaxy was born to be stripped away. Including this delayed stripping of the hot gas halo, without including any other environmental effects (e.g. tidal or ram pressure stripping of the cold gas in satellites) improves satellite statistics in SAMs (Font et al. 2008, Weinmann et al. 2010), though some tension with observations remains (Hirschmann et al. 2014). A particularly curious observational result is 'galaxy conformity', in which halos with red central galaxies preferentially have red satellites (Weinmann et al. 2006a). This effect extends even beyond the virial radius to surrounding centrals, and it is not reproduced at the observed level in SAMs (Kauffmann et al. 2013). It can be reproduced in 'age abundance matching' models, an extension of abundance matching that uses halo formation times to assign SFR or colors (Hearin et al. 2014), but the physics that drives conformity remains unclear. Satellite and environmental quenching has not yet been extensively investigated in self-consistent cosmological simulations, but there is clearly much to be learned by doing so and this is an area where much progress can be made in the near future.", '4.2 Internal Structure and Kinematics': "Figure 8: \n<!-- image --> \nTop: Gas surface density and simulated observations of a spiral galaxy formed in a high-resolution zoom-in simulation. Top left: H i gas surface density (gray) and H 2 surface density (red). Top middle and right: optical images of stellar emission, showing the galaxy in face-on and edge-on orientations. Middle left: rotation curve for the same initial conditions, but with different sub-grid physics for treating cooling (primordial only vs. metal-line) and the ISM (H 2 -based cooling and SF). Middle right: distribution of scaled specific angular momentum for the dark matter and baryons in these same simulations. Bottom: star formation histories for the same three models shown in the middle panels. Reproduced from Christensen et al. (2014). \n4.2.1 Formation of Galactic Disks The longstanding conventional paradigm to explain the origin of galactic disks posits that gas accreting from the halo conserves its specific angular momentum j , thereby settling into a disk (Fall & Efstathiou 1980, Mo et al. 1998). While modern cosmological simulations support this basic paradigm, they suggest that the full story is much more complicated. \nThe average specific angular momentum of galactic disks is indeed comparable to that expected from conserving the j from the halo (Dalcanton et al. 1997, Dutton & van den Bosch 2012). However, the distribution of j within disks predicted from simple infall is strongly inconsistent with observations, in the sense that observed galaxies have a strong \ndeficit of lowj gas, a mild deficit of highj gas, and a large excess of intermediatej gas (Bullock et al. 2001, van den Bosch et al. 2001). This suggests that some process removes lowj gas and deposits it at intermediatej . \nEarly numerical hydrodynamic simulations of disk galaxy formation suffered an even more severe 'angular momentum catastrophe', as they produced disks with much lower average j than the halo, indicating that a large amount of j was being lost during the formation process. For many years, simulations were only able to produce very compact disks with large spheroids, and were unable to produce spirals even as late-type as the Milky Way (Navarro & Steinmetz 2000, Sommer-Larsen et al. 1999, Steinmetz 1999). Moreover, these galaxies exhibited centrally-peaked rotation curves in disagreement with observed flat rotation curves, did not lie on the observed Tully-Fisher relation, and formed far too large a fraction of their available baryons into stars. It was gradually realized that the origin of this angular momentum catastrophe lay in too-efficient star formation and gas consumption in small objects at high redshift. These then assembled into low redshift galaxies via relatively gas-poor mergers, which are very efficient at dissipating angular momentum and building spheroids (Maller & Dekel 2002). \nImplementing more efficient star formation feedback has proven to be the key to solving all of these problems (e.g. Governato et al. 2007, Guedes et al. 2011). Stellar-driven winds preferentially remove low angular momentum gas from the centers of galaxies, and deposit it in the disk outskirts after re-torquing in the halo (Brook et al. 2012, Ubler et al. 2014). Feedback also makes star formation less efficient, keeping galaxies gas rich, which makes disks more resiliant to mergers (Governato et al. 2009, Robertson et al. 2006a). Finally, the baryonic mass of small infalling satellites, particularly at early epochs, is greatly reduced, thereby mitigating early spheroid growth via merging. \nFig. 8 shows a state-of-the-art high resolution zoom-in simulation of a disk galaxy using the GASOLINE code (Christensen et al. 2012). One can see that it is now possible to form very late-type and even bulgeless galaxies (Christensen et al. 2014). The same simulations predict a z = 0 m star -M halo relation in agreement with observational constraints (Munshi et al. 2013). In dwarf galaxies, the same 'blast wave' feedback model can impulsively heat the dark matter, removing the central cusp generically predicted in dark matter simulations, and thereby producing rotation curves in better agreement with observations (Governato et al. 2010, Oh et al. 2011, Pontzen & Governato 2012). As a bonus, Brooks & Zolotov (2014) have shown that destroying the central cusps in dwarf galaxies leads to enhanced tidal stripping of satellites. The combined effects of energetic stellar feedback and enhanced stripping produces satellites with internal kinematics that agree with observed dwarf spheroidal galaxies in the Local Group, plausibly resolving the 'too big to fail' problem pointed out by Boylan-Kolchin et al. (2011). While these successes may be specific to a particular sub-grid model for feedback that may or may not be fully accurate, nonetheless it suggests it is possible to form disk galaxies with realistic properties within a ΛCDM universe provided that the sub-grid treatment of stellar feedback and the ISM possess certain key features. First, stellar feedback must be effective at keeping galaxy-wide star formation efficiencies low, and stellar winds must preferentially remove low-angular momentum material. Second, star formation should occur only in very dense, highly clustered environments like those that are expected to form GMCs, not smoothly distributed over the whole disk, which helps to make stellar feedback more efficient because the star formation is highly clustered as in real galaxies. \nRecent observations of disks during Cosmic Noon have presented new challenges for models. Disks at z ∼ 2 are observed to be substantially puffier, having rotation velocity V rot \ndivided by gas dispersion σ of ∼ 3 as opposed to ∼ 10 for today's disks (Forster Schreiber et al. 2009). Many z ∼ 2 disks also have large, bright clumps that comprise a substantial portion of the disk star formation (Guo et al. 2012), though significantly less of the stellar mass (Wuyts et al. 2012), and are generating outflows (Genzel et al. 2011). While most of these objects are sufficiently massive by z ∼ 2 to likely evolve into ellipticals today, lower mass objects that will evolve into today's disks generally have even higher V rot /σ . Understanding the origin of these properties and the evolution of the population to z = 0 has become a major cottage industry. \nAs discussed earlier, ISM pressurization was found to be necessary to stabilize disks against fragmentation (Robertson et al. 2004) and form spiral galaxies like those observed today. With the discovery of clumpy disks at z ∼ 2 (Elmegreen & Elmegreen 2005, Forster Schreiber et al. 2009), the abundant clumps that formed in simulations without ISM pressurization began to be touted as a 'feature' (Bournaud et al. 2007, Ceverino et al. 2010, Dekel et al. 2009). However, such models produced overly high stellar fractions, typically > ∼ 50% whereas abundance matching constraints suggest a value of ∼ 10 -20% (Behroozi et al. 2010, Moster et al. 2010b, Wake et al. 2011). After implementing more efficient stellar feedback due to radiation pressure from young stars, Ceverino et al. (2014) reduced the stellar fractions by a factor of 2-3, but are they are still somewhat high (Moody et al. 2014). Models with significant ISM pressurization, either imposed (Genel et al. 2012) or selfconsistently generated (Faucher-Gigu'ere et al. 2013), matched stellar fraction constraints and still produced massive clumps, but these were less prominent and quicker to disrupt. \nThe clump formation can be understood analytically in the context of the Toomre (1964) Q ≡ c s Ω /πG Σ, where c s is the sound speed, Ω is the angular speed, and Σ is the local surface density. If Q < 1 gravitational collapse can overcome shearing disruption, and the region is unstable. Clumps are observed to be regions with Q glyph[lessmuch] 1 (Genzel et al. 2011), and simulations indicate that clumps are unstable regions self-regulated by gravity (Ceverino et al. 2010). The characteristic mass scale for instability is m clump < ∼ 10 9 M glyph[circledot] , in good agreement with observations suggesting clumps up to these masses (Genzel et al. 2011). If this is the basic origin of clumps, then they are expected to become less prominent in disks at later epochs, reducing in mass as disks settle (Dekel et al. 2009). Simulations show that disks do indeed settle towards z = 0, in accord with observations (Kassin et al. 2014). If cosmological accretion drives turbulence in the ISM, then the settling may be due to the decreasing accretion rate at lower redshifts (Genel et al. 2012a), though Hopkins et al. (2013b) argue that accretion does not drive the turbulence in galactic disks. It remains to be demonstrated that a single sub-grid ISM model can simultaneously reproduce the ∼ 10 9 M glyph[circledot] clumps in turbulent highz disks along with thin disks with < ∼ 10 6 M glyph[circledot] clumps today like the Milky Way. Nonetheless there is at least a plausible description for the evolution of clumps in disks across cosmic time. \n4.2.2 Formation of Spheroid-dominated Galaxies Since the seminal work of Toomre (1977), it has been recognized that nearly equal mass ('major') mergers can efficiently remove angular momentum from stellar disks, producing dispersion-dominated spheroids (Barnes 1988, 1992, Hernquist 1992, 1993b, Mihos & Hernquist 1996, Toomre 1977). Unequal mass ('minor') mergers down to mass ratios of ∼ 1:10 can thicken disks and build up the spheroid component of galaxies (Moster et al. 2010a, Walker et al. 1996). Mergers are expected to be ubiquitous in the hierarchical CDM paradigm. Thus the most basic picture of the origin of the two dominant classes of galaxy morphologies is that smooth accretion of gas produces disks, and mergers destroy disks and build spheroid-dominated \ngalaxies. A merger-driven formation mechanism for spheroid-dominated galaxies has been implemented in most SAMs since the earliest such models (Baugh et al. 1996, Kauffmann 1996, Kauffmann et al. 1993, Somerville & Primack 1999), motivated by the studies based on binary mergers simulated with numerical hydrodynamics. These early works and others over the past decade (e.g. De Lucia et al. 2006, Parry et al. 2009) have shown that this picture can qualitatively reproduce many of the observed correlations pertaining to galaxy morphology, namely, spheroid dominated galaxies are predicted to be more massive, more common in massive halos, redder, and to have older stellar populations. In empirical support of this picture, it has been shown that the observed rate of mergers derived from pair counts and visually identified interacting galaxies is in plausible statistical agreement with the build-up of the quiescent, spheroid-dominated population (Hopkins et al. 2008a, Robaina et al. 2010). \nMore recent work has led to a refinement of the merger picture. Numerical hydrodynamic simulations showed that gas-rich mergers do not drive efficient angular momentum loss, and so lead to re-formation of disk-dominated galaxies (Hopkins et al. 2009a, Robertson et al. 2006a, 2004). In addition, following the formation of a spheroid via a merger, newly accreted gas can re-form a disk. Thus, a picture has developed in which morphological transformation and morphological demographics are intimately linked with feedback and quenching . It is known that the massive early-type galaxies in the local universe formed most of their stars at least 8-10 Gyr ago, around z ∼ 2-4 (Thomas et al. 2005, Trager et al. 2000). We also know that the average massive star-forming galaxies at z ∼ 2 are quite gas rich (Genzel et al. 2014, Tacconi et al. 2013). Thus, in order to produce a spheroid-dominated population at z = 0, some process had to consume or remove much of the gas from their progenitors before they merged, and prevent significant amounts of new gas from cooling. This appears to point qualitatively towards a combination of 'ejective' and 'preventative' feedback, perhaps linked with two different modes of AGN feedback. \nIt has been suggested that spheroids may also form and grow in situ due to internal gravitational instabilities. There are two different kinds of internal processes that may grow bulges, which are frequently grouped together under the term 'disk instabilities', but which are physically quite distinct and are thought to produce fundamentally different kinds of bulges. We have already discussed the formation of giant clumps in Toomre-unstable disks (see § 4.2.1), sometimes called 'violent disk instabilities' (Dekel et al. 2009). If these clumps survive and migrate to the galaxy center, they may form a classical bulge (Bournaud et al. 2011, Dekel et al. 2009, Elmegreen et al. 2008). However, there remains some debate about the importance of clumps in feeding spheroid growth. Simulations implementing kinetic feedback that were able to match m star -M halo constraints suggested that clumps mostly disrupt before reaching the center (Genel et al. 2012b). Hopkins et al. (2012a) also found that in simulations of isolated disks (not cosmological) with a suite of physically motivated stellar feedback physics, even large clumps mostly blow themselves apart while in the disk, thereby only modestly contributing to spheroid growth. However, some recent simulations suggest that clumps can survive substantially longer than a disk dynamical time and grow a spheroid (Bournaud et al. 2014, Mandelker et al. 2014). Comparing stellar and SFR maps, Wuyts et al. (2012) showed that clump lifetimes are ∼ 100 -200 Myr, which would suggest disruption unless inward migration can occur on a single dynamical time or less, but radial age gradients of clumps suggest somewhat longer lifetimes (Genzel et al. 2011). Additionally, as the giant clumps orbit within the disk, even if they disrupt before reaching the center, they may drive inflows of gas into the galaxy nucleus, via the same sort of physics as merger-induced nuclear inflows (Bournaud et al. 2011). \nThe other process that is referred to as a 'disk instability' is not really a (global) instability at all. It involves the secular transfer of mass into a compact, dynamically hot component via the formation of a bar (Combes et al. 1990, Hohl 1971, Ostriker & Peebles 1973, Toomre 1964). The topic of galactic bars is largely outside of the scope of this review, but a few points are worth briefly noting. First, viewed side-on, bars may be identified as 'boxy bulges' (our Galaxy is a familiar example), but if viewed face-on these structures would not be identified as bulges (Combes et al. 1990). It is generally impossible to robustly distinguish bars from bulges in distant galaxies. Second, secular processes can redistribute angular momentum and mass within the disk, building a pseudobulge (Kormendy 2013, Kormendy & Kennicutt 2004). In constrast to the violent disk instabilities described above, the disk essentially remains in dynamical equilibrium during this secular evolution . The fundamental differences between classical bulges and pseudobulges are briefly summarized in § 4.2, and a much more complete discussion is given in Kormendy & Kennicutt (2004). The stronger correlation between black hole mass and classical bulge mass recently emphasized by Kormendy & Ho (2013) is presumably evidence that the processes that build classical bulges (mergers and violent disk instabilities) are most closely connected with black hole fueling. \n4.2.3 Demographics of Spheroid- and Disk-Dominated Galaxies Explaining the demographics of galaxies of different morphologies is another challenge for theory. Detailed quantitative statistical comparisons between the predictions of cosmological simulations and observations of galaxy morphological demographics are difficult, because up until now, most observational studies of galaxy morphology have used classifiers that are not straightforward for models to predict. Semi-analytic models predict the fraction of stellar mass or light in a spheroid component ( B/T ), while most observational studies use visual morphological classification or statistics such as Sersic index or concentration. A few observational studies have carried out decompositions into spheroid and disk contributions (Bluck et al. 2014, Gadotti & Kauffmann 2009, Simard et al. 2011), but there are large uncertainties in these decompositions as well (see e.g. Benson et al. 2007, Tasca & White 2011). There is a large dispersion in observational estimates of galaxy morphological demographics derived from different surveys and classification methods. \nA number of studies have compared the predictions of SAMs with observational estimates of luminosity or stellar mass functions divided by galaxy morphology, or with the fraction of disk- or spheroid-dominated galaxies as a function of stellar mass (e.g. Benson et al. 2007, Guo et al. 2011, Parry et al. 2009, Porter et al. 2014). These studies all found fairly good agreement between the predictions of these different SAMs and the observations, but interestingly the dominant mechanism that drives spheroid growth is different in different models, as we discuss further below. \nAlthough the details of the prescriptions differ, all semi-analytic models that attempt to track galaxy morphology assume that mergers destroy disks and build spheroids. However SAM-based studies have found, to varying degrees, that non-merger related mechanisms for spheroid growth may be needed. The most commonly invoked alternative to mergers is a 'disk instability' mode as described above. This is assumed to occur when the mass in the disk exceeds a critical value that depends on the angular momentum of the disk material. The implementation of this process varies widely between models, leading to significantly differing conclusions about its importance. The GALFORM SAMs assume that when a disk becomes unstable, all of the stars and gas in the disk are moved to a spheroid component. They find that these disk instabilities are the dominant channel for spheroid growth except \nat the highest stellar masses (Parry et al. 2009). Other SAMs (De Lucia & Blaizot 2007, Guo et al. 2011, Porter et al. 2014) make a more moderate assumption, that just enough stars or stars and gas are moved from the disk to the spheroid to return the system to stability. These models find that disk instabilities appear to be needed to reproduce the observed numbers of spheroid dominated galaxies at intermediate masses (Porter et al. 2014), but are sub-dominant in driving spheroid growth at all masses. An important and apparently robust prediction is that models with a 'disk instability' driven channel for spheroid growth appear to form massive spheroids earlier than models in which spheroids form only via mergers (De Lucia et al. 2011, Porter et al. 2014). These predictions can now be confronted with observations from the new generation of medium-deep surveys with HST (Brennan et al. in prep). \nThe flip side of producing enough spheroid dominated galaxies is the challenge of producing galaxies that are close to pure disks, which are perhaps surprisingly frequently observed in the real Universe (Fisher & Drory 2011, Kormendy et al. 2010). If mergers destroy disks and build spheroids, and nearly all halos of all masses have experienced mergers during the course of their formation history, as predicted by ΛCDM, is it possible to reconcile the existence of these objects with the ΛCDM picture? Hopkins et al. (2009) showed that including the suppression of disk destruction in mergers with high gas fraction progenitors alleviates this problem, bringing predictions into agreement with observations in a semianalytic model - the majority of mergers occur at high redshift when galaxy gas fractions are expected to have been fairly high. Furthermore, Moster et al. (2010a, 2012) showed that accounting for the presence of both cold gas in the disk and hot gas in the halo decreases disk heating due to minor mergers by a factor of 2-3 relative to previous calculations that included dissipationless components (stars and dark matter) only. However, Porter et al. (2014) showed that adding a disk instability driven channel for spheroid formation, tuned to reproduce the abundances of spheroid-dominated galaxies, may leave behind too few objects with extremely low B/T < ∼ 0 . 2. Detailed studies with larger samples of galaxies simulated at high resolution in a full cosmological context are required to determine whether this is truly a fundamental problem for ΛCDM, but it remains a serious concern. \nExtensive detailed predictions on morphological demographics from numerical cosmological simulations have not yet appeared in the literature. Such studies should be possible with the new generation of simulations, and detailed analysis of these simulations should help shed light on the physical mechanisms that are responsible for shaping galaxy morphology. \n4.2.4 Structural Scaling Relations The existence of structural scaling relations for galaxies, the relationship between the structure of disks and spheroids at a given mass scale, and the evolution of these relations over cosmic time, encode crucial information about galaxy formation and provide stringent constraints for models. \nWhat physics determines the internal structure of galaxies? The most basic picture is that dark matter and diffuse gas acquire angular momentum through tidal torques and mergers (Peebles 1969, Vitvitska et al. 2002), leading to dark matter and gaseous halos with a broad log-normal distribution of spin parameters . The dimensionless spin parameter is usually defined as \nλ ≡ J | E | 1 / 2 GM 5 / 2 \nwhere M , J , and E are the mass, angular momentum, and total energy of the system, respectively (MvdBW, p. 502). If we assume, perhaps na¨ıvely, that the halo gas conserves its angular momentum as it cools and collapses to form a disk, and that the post-collapse \ndisk surface density profile has an exponential form, then the disk scale radius will be given by \nr s = 1 √ 2 λr vir F -1 R F -1 / 2 E \nwhere r vir is the virial radius of the halo, and F R and F E are functions that account for the initial density profile of the dark matter halo and the contraction of the inner halo due to the increased gravitational force after the gas falls in (Mo et al. 1998). The rotation velocity can then be calculated by adding the contribution of the exponential disk and the contracted halo in quadrature. \nIn spite of its simplicity, this model does remarkably well at reproducing the size-mass relation for disk galaxies and its evolution since z ∼ 2 (Dutton et al. 2011, Firmani & AvilaReese 2009, Somerville et al. 2008). Recently, high-resolution numerical hydro simulations have also been shown to be quite successful at reproducing the size-mass relation for galactic disks and its evolution (Aumer et al. 2013, Brooks et al. 2011). As discussed above, hydro simulations have only recently been able to successfully reproduce disk sizes, and including strong star formation driven outflows that preferentially remove the low angular momentum material appears to be a crucial component of this success. Recent simulations suggest that accretion by 'cold streams' may bring most of the gas into galaxies, with an average specific angular momentum that is a factor of ∼ 2-3 higher than that of the dark matter halo (Stewart et al. 2013). About a factor of 2-3 of this angular momentum is then lost via torques within the mis-aligned disk and via outflows (Danovich et al. 2014). The success of the simple model for predicting disk sizes may therefore be simply a happy accident. \nMuch of the convincing evidence for the importance of mergers in producing spheroiddominated galaxies comes from the success of merger simulations in reproducing structural properties of classical bulges and elliptical galaxies. For example, early work (Barnes 1992, Hernquist 1992) showed that mergers transform rotationally supported disks with exponential light profiles into slowly rotating remnants with luminosity profiles that are welldescribed by an r 1 / 4 form over a large radial range. More recently, it has been shown that remnants of binary disk mergers lie on the observed fundamental plane (Hopkins et al. 2009b, Robertson et al. 2006b). \nA striking recent observation is that, at fixed stellar mass, spheroid-dominated galaxies at z ∼ 2 have much smaller sizes and central densities higher by orders of magnitude compared to today's (e.g. Barro et al. 2013, Trujillo et al. 2006, van der Wel et al. 2014, van Dokkum et al. 2014, 2008). For dissipationless (dry) mergers on parabolic orbits (hence with small orbital energy), energy conservation and the virial theorem can be used to show that, given a progenitor mass ratio η ≤ 1 and ratio of squares of their velocity dispersions of glyph[epsilon1] ≤ 1, the ratio of final to initial radius is given by \nr f r i = (1 + η ) 2 1 + glyph[epsilon1]η (9) \n(Naab et al. 2009). For a 1:1 merger, η = glyph[epsilon1] = 1, hence r f /r i = 2, which leads to a modest surface density reduction of a factor of four. One can show that, for a given total mass increase, the size is increased much more by a series of minor mergers than by a single major one. Numerical simulations confirm such a size increase in dissipationless mergers, which generally move galaxies along the mass-size relation (Boylan-Kolchin et al. 2005). This can reproduce the observed size increase and central density reduction (Naab et al. 2009, Oser et al. 2012) since z ∼ 2 for cosmologically-plausible merger histories (Gabor & Dav'e 2012), via minor mergers depositing material predominantly in the outskirts. \nIf disks (star forming galaxies) are continuously being transformed into spheroids (quiescent galaxies), as the demographic observations indicate (see § 1.1), how then can we understand the very different slopes and evolution of the size-mass relationship for disks and spheroids? Several recent works have pointed out that accounting for the effects of dissipation in gas-rich mergers, can lead to important changes in the scaling relations (Covington et al. 2008, Covington et al. 2011, Hopkins et al. 2010, Hopkins et al. 2009b, Porter et al. 2014, Shankar et al. 2010, Shankar et al. 2013). In the presence of gas, energy is dissipated, which can lead to merger remnants that are smaller and denser than their progenitors. Porter et al. (2014) implemented a recipe for computing spheroid sizes and velocity dispersions based on a simple analytic model including the effects of gas dissipation, tuned to binary merger simulations, self-consistently within the Santa Cruz SAM. They showed that without any tuning the model predicts rapid size evolution of spheroid-dominated galaxies since z ∼ 2, along with the weaker evolution in the Faber-Jackson relation, in very good quantitative agreement with the observed structural relations. \nIn this picture, dissipation plays a major role in explaining the different slope, scatter, and evolution of the size-mass relation for spheroid dominated (quiescent) galaxies relative to disks. Lower mass spheroids have lower mass progenitors, which have higher gas fractions at all redshifts. More gas means more dissipation and smaller remnants, thus a steeper size-mass relation. Progenitors at higher redshifts have higher gas fractions than those at lower redshift, so the size-mass relation for spheroids 'tilts away' from that for disks more, contributing to more rapid size evolution especially for the lower-mass spheroids. The decrease in scatter occurs because disks with higher angular momentum have larger radii and lower gas densities, resulting in less efficient star formation. These large radius disks therefore end up with higher gas fractions, and experience more dissipation when they merge, producing smaller remnants. Similarly, the observed tilt of the Fundamental Plane can be explained by the expected trends in galaxy gas content with mass and redshift, and the physics of gas dissipation in mergers (Covington et al. 2011, Hopkins et al. 2009b, Porter et al. 2014).", '5 Summary and Outlook': "Galaxy formation models set within the hierarchical CDM paradigm have made remarkable progress over the past decade. In this review, we have focused on the methods and phenomenology of models that attempt to track astrophysical processes and predict galaxy properties within a cosmological framework. We identified a set of key observations that current models strive to reproduce, and which describe the assembly of galaxies from Cosmic Noon ( z ∼ 2-3) to the present. These observations include distribution functions of global properties such as stellar mass functions and global scaling relations such as those between stellar mass and SFR, gas fraction, and ISM metallicity. In addition, observations are now starting to provide measurements of galaxy demographics, how the break-down of the galaxy population in terms of star-forming and quiescent, and disk and spheroid dominated objects, has evolved over this time period. The observed relationships between global and structural properties (such as light profile shape, size or internal density, and kinematics) and their evolution provide even stronger constraints on models. We described how well current state-of-the-art galaxy formation models are able to reproduce these observations, and what we have learned from their successes and failures about the physics of galaxy formation. \nAlthough many discrepancies with observations remain, overall we would give today's \nsuite of galaxy formation models a passing grade. Summarizing the scorecard we have discussed in detail in this article: \n- · Qualitatively, hierarchical models correctly predict the build-up of stellar mass over cosmic time, with massive galaxies forming earlier and more rapidly than low-mass galaxies. Quantitatively, most models agree with observed galaxy number densities from z ∼ 4-0 at least at the factor of 2-3 level. However, models tend to predict that galaxies have nearly self-similar star formation histories, while observations imply a stronger mass dependence for these histories (sometimes known as 'downsizing'). This is part of a set of linked discrepancies connected with low-mass galaxies that we termed the 'dwarf galaxy conundrum', which remains an open puzzle for models.\n- · Models predict qualitatively the right slope and evolution of mean global scaling relations between stellar mass and SFR, gas fraction, or gas phase metallcity. These linked correlations can be understood at the most basic level via a very simple flow model describing an approximate equilibrium between galactic inflows and outflows. Quantitatively, compared with our current observational estimates, models tend to predict a SFMS and MZR that are too steep, and possibly gas fractions that are too low at intermediate redshift, in galaxies with low stellar masses. These are additional symptoms of the dwarf galaxy conundrum mentioned above. Also, models have difficulty reproducing the observed redshift dependence of the sSFR at any mass, indicating that real galaxies deviate from the simple equilibrium model.\n- · Models can qualitatively explain the existence of two basic morphological types, disks and spheroids, via two different assembly modes. Disks are formed via smooth accretion of diffuse gas, which largely conserves its angular momentum, while spheroids are formed via gas-poor mergers that efficiently transfer angular momentum. Recently, numerical simulations demonstrated the ability to form pure disks in at least some cases - a major achievement as previous generations of simulations were only able to form spheroid-dominated galaxies. The strong feedback in such models also results in rotation curves and Local Group satellite demographics in better agreement with observations, which had previously been identified as a fundamental challenge to cold and collisionless dark matter. However, it is still unclear how well models match observed morphological demographics and their evolution in detail. There is still much debate about how efficiently mergers can build spheroids, how this depends on the parameters of the merger and the gas fraction of the progenitors, and the role of other processes such as secular evolution and violent disk instabilities.\n- · Models predict the correct qualitative trends between stellar population demographics (the fraction of SF and quiescent galaxies) and internal properties such as stellar mass: galaxies with higher stellar masses, higher spheroid fractions, and higher central densities have a higher probability of being quiescent. Some models correctly reproduce the dependence of quiescent fraction on environmental parameters such as large scale density as well, but the physics specific to the quenching of satellite galaxies remains imperfectly understood. Quantitatively, models still have difficulty reproducing observed color or sSFR distributions in detail. Models have not yet extensively confronted the emerging measurements of stellar population demographics at high redshift.\n- · Many models are able to at least qualitatively reproduce the observed sizes and internal velocities of observed galaxies, and scaling relations such as the Kormendy \n(size-mass), Tully-Fisher (mass-velocity), and Fundamental Plane relations. Reproducing observed disk sizes in numerical simulations has been a multi-decade struggle, and the solution has emerged through a combination of greatly increased resolution, more physical treatment of the ISM, and the effective implementation of stellar winds that preferentially remove low-angular momentum gas. Correctly reproducing the structural scaling relations and their evolution for both disks and spheroids , as well as the correct overall evolution of the number densities of these two populations, remains an open challenge for models. \nAlthough there remain a wide range of models, and a healthy diversity of computational methods, virtually all models implement a qualitatively similar set of core physical processes. While it is possible that all models are being led down the garden path due to their reliance on phenomenology, the concordance among models using different methods is encouraging, and strongly suggests that we are making fundamental progress in at least identifying the main physical players involved. Some of the core processes identified include the prevalance of cold smooth accretion in building disks and fueling star-forming galaxies, the ubiquity and efficiency of star formation-driven outflows, the importance of black hole-related feedback in quenching star formation in massive galaxies, merger-driven morphological evolution that depends on the gas content of progenitors, and various physical processes that uniquely impact satellite galaxies once they fall into a larger halo containing hot gas. In addition, the convergence towards a similar qualitative view of the types of processes that are needed in different circumstances, based on more empirical considerations (e.g., preventative vs. ejective feedback, internal vs. environmental quenching, etc.) is also encouraging. \nMany of these processes connect stellar scales to cosmological scales, making ab initio modeling nearly impossible, and forcing models to rely on phenomenological prescriptions to describe sub-grid physics, which must be calibrated in some way by observations. It is clear that many model results are sensitive to the details of these sub-grid recipes and their implementation, leading to a valid concern that these models may have little genuine predictive power (Haas et al. 2013a,b). There are perhaps two ways to combat this concern. First, although the sub-grid recipes and their parameters are tuned to match a subset of observations, the current suite of available observations is diverse and rich enough that by confronting models with as wide as possible a set of complementary constraints, and by exploring different sub-grid recipes and implementations, one can isolate the approach that satisfies the broadest set of constraints. Second, by studying 'small scale' simulations (for example, of the ISM and the formation of individual stars, or regions near SMBH), one may hope to place the sub-grid recipes used in our cosmological simulations on a physically grounded foundation. Zoom techniques are now enabling simulations that are starting to bridge the gap between the scales of individual stars and SMBH and galactic scales. Although it will not be feasible to simulate cosmological volumes with these techniques in the near future, they will allow us to learn much about the interface between the 'micro'scales of stars and BH and the 'macro' scales of galaxies. \nIn addition, there are physical processes that may be important in regulating galaxy formation, but which are not commonly included in current 'mainstream' models. These include turbulence, magnetic fields, cosmic rays, and self-consistent radiative transfer. It is important to carry out experiments to determine the importance of these processes in shaping the observable properties of galaxies, and there has been significant recent progress on this front as well (e.g. Hanasz et al. 2013, Kotarba et al. 2011, Mendygral et al. 2012, Pfrommer 2013, Scannapieco & Bruggen 2010, Wise & Abel 2011). \nIdeally, we would obtain direct observational confirmation (or refutation) of the set of core processes that models currently invoke. However, in many cases this is challenging. Smooth gas accretion (i.e. in small enough lumps that adiabatically add to the fuel supply without disrupting galactic structure) is expected to be very diffuse and in a phase that is difficult ( T ∼ 10 4 K) to nearly impossible ( T ∼ 10 5 K) to detect. The key parameter characterizing outflow efficiency in models is the mass loss rate, but since outflows are highly multi-phase it is difficult to account for all the mass (Veilleux et al. 2005). We observe the signatures of black hole activity in the form of AGN and jets associated with massive galaxies, but it is difficult to observationally constrain how efficiently this energy couples to surrounding gas to enact quenching. We can observe signposts and signatures of mergers in the form of close pairs and morphologically disturbed galaxies, but their rate is difficult to quantify precisely and their effect is difficult to directly constrain observationally. We can measure the statistics of galaxies in different environments, but it has been difficult with existing samples to disentangle the correlations between environment and internal properties, and to locate the environments at high redshift that are the progenitors of typical groups and clusters in the local Universe. \nHowever, there are several important observational developments taking place now, or on the horizon, that will challenge and help to refine our models of galaxy formation. First, a new generation of sub-mm and radio interferometers (including ALMA, NOEMA, JVLA, Apertif, ASKAP, MeerKAT, and the SKA) will literally revolutionize our ability to characterize the cold gas in the ISM of galaxies out to high redshifts (Carilli & Walter 2013). Second, high-resolution spectroscopy in the rest-frame UV is now able to probe the diffuse gas and metals in the circumgalactic medium of galaxies for galaxy-targetted sightlines spanning a diverse range of galaxy types, from nearby galaxies to z ∼ 2-3 (e.g. Peeples et al. 2014, Prochaska et al. 2013, Rudie et al. 2012, Tumlinson et al. 2013). This provides constraints on the gas and metals that have been ejected by the winds invoked by our models, which probably comprise a much larger fraction of the halo baryon budget than the stars and cold ISM within galaxies. Third, Integral Field Unit spectrographs on ground based telescopes and on JWST will allow us to better characterize stellar and AGN driven winds and to study spatially resolved stellar population parameters and kinematics for large samples of nearby and high-redshift galaxies. Finally, high-resolution, wide-field multi-wavelength imaging such as will be possible with WFIRST will enable us to study galaxy internal properties and demographics over a much larger range of environments, allowing us to better disentangle internal and environmental forces and accumulating better statistics for rare events such as mergers and luminous AGN. \nWe thus live in interesting times where modelers are now offering some specific and nontrivial challenges to observers to go out and confirm, or rule out, key physical processes. Just because a given mechanism is not observed does not mean it is not occuring; one must carefully assess whether that mechanism is expected to be observable. A general trend is that models make the most direct predictions about gas-related processes, particularly inflows and outflows in the baryon cycle, with the growth of stellar and black hole components being almost a side-effect. Hence, in principle, observations that trace gas processes directly offer the greatest potential for new advances and constraints. Modelers and observers must work together to identify key tests that can be conducted with present and upcoming facilities in order to constrain the core physical processes. The emerging interplay between galaxy formation models and state-of-the-art telescopes is the hallmark of a healthy and vibrant area of research. \nThe way forward for galaxy formation models is fairly clear, but immensely challenging. \nAs a blueprint, consider the Lymanα forest: several decades ago, studying the interplay of gas dynamics with cosmological structure formation led to a revolution in our understanding that eventually resulted in the Ly α forest becoming a pillar of precision cosmology. Our goal should be to equivalently turn galaxy formation into a precision field, where parameterized recipes are tied to the physics of small scale processes in such a way that the parameters no longer need to be empirically tuned, but are constrained by our physical understanding of those processes (e.g. stellar evolution models, or BH accretion disk models). Numerical simulations on different scales (zooms and cosmological volumes) and semi-analytic models have crucial and complementary roles to play in this process, helping to better understand the physics in detail as well as to synthesize and parameterize it within a ΛCDM context. It is almost surely the case that the physical processes included in models so far will not be sufficient to fully describe galaxy evolution, and there will be many twists and surprises forthcoming. Hence there is much work to be done, but it appears that cosmological models of galaxy formation are on a secure foundation for the exciting journey ahead.", 'Acknowledgements': 'It would take many more pages to thank all the colleagues who have provided valuable insights and participated in discussions that have shaped this work, and we apologize for the inevitable choices we had to make to review this vast topic while conforming to page limits. But we would particularly like to thank Andrew Benson, Richard Bower, Rob Crain, Darren Croton, Michelle Furlong, Violeta Gonzalez-Perez, Bruno Henriques, Yu Lu, Joop Schaye, Paul Torrey, Mark Vogelsberger, and their collaborators for providing the data from their models and simulations and for constructive comments on this article. We also thank Avishai Dekel, Thorsten Naab, and Gergo Popping for comments. We especially thank our Scientific Editor, John Kormendy, for his thorough reading of the paper, and for comments and suggestions that improved the article. rss gratefully acknowledges the generous support of the Downsbrough family. RD acknowledges support from the South African Research Chairs Initiative and the South African National Research Foundation. This work was supported in part by NASA grant NNX12AH86G.', 'Glossary of Acronyms': 'AGB: asymptotic giant branch \nAGN: active galactic nucleus \nALMA: Atacama Large Millimeter/submillimeter Array \nAMR: adaptive mesh refinement \nASKAP: Australian Square Kilometre Array Pathfinder \nBH: black hole \nBLR: broad line region \nB/T : \nbulge to total ratio \nCDM: cold dark Matter \nckpc: comoving kilaparsec \ncMpc: comoving megaparsec \nEC-SPH: entropy-conserving SPH \nEoR: epoch of reionization \nDI-SPH: density-independent SPH \nFOF: friends of friends \nGMC: giant molecular cloud \nGR: General Relativity \nH i : neutral hydrogen \nHOD: halo occupation distribution \nHST: Hubble Space Telescope \nIGM: intergalactic medium \nIMF: initial mass function \nIR: infrared \nISM: interstellar medium \nJVLA: Jansky Very Large Array \nJWST: James Webb Space Telescope \nLF: luminosity functions \nMeerKAT: http://www.ska.ac.za/meerkat/index.php \nMZR: mass-metallicity relation \nNFW: Navarro-Frenk-White \nNOEMA:NOrthern Extended Millimeter Array; http://iram-institute.org/EN/noema-project.php \nPE-SPH: pressure-entropy SPH \nPM: particle-mesh \nPPM: Piecewise Parabolic Method \nSAM: semi-analytic model \nSDSS: Sloan Digital Sky Survey \nSED: spectral energy eistribution \nSF: star formation \nSFE: star formation efficiency \nSFMS: star forming main sequence \nSFR: star formation rate \nSKA: Square Kilometer Array \nSMBH: supermassive black hole \nSMF: stellar mass function \nSN: supernova \nsSFR: specific star formation rate \nSHAM: sub-halo abundance matching \nSO: spherical overdensity \nSPH: smoothed particle hydrodynamics \nULIRG: ultra-luminous infrared galaxies \nUV: ultraviolet \nWFIRST: Wide-Field Infrared Survey Telescope \nΛCDM: cold dark matter with a cosmological constant (Λ)', 'Literature Cited': "- 1. 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2024arXiv240911669Z | We study the spontaneous scalarization of Bardeen black holes whose tachyonic instability triggers the formation of scalarized charged black holes SCBHs. In this case we find infinite n012cdots branches of SCBHs with magnetic charge g. The n 0 branch of SCBHs can be found for the coupling parameter alpha geq alphan0g with both quadratic 1alpha varphi2 and exponential ealpha varphi2 couplings where alphan0g represents the threshold of tachyonic instability for the Bardeen black holes. Furthermore it is shown that the n 0 branch for both couplings is stable against radial perturbations. This stability shows that this branch can be used for further observational implications. | 2024-09-01T00:00:00Z | ['2024arXiv240911669Z', 'arXiv:2409.11669', '10.48550/arXiv.2409.11669'] | ['General Relativity and Quantum Cosmology'] | Spontaneous scalarization of Bardeen black holes | 2,024 | 168 | 0.17 | ['EPRINT_HTML', 'EPRINT_PDF'] | 0 | https://arxiv.org/pdf/2409.11669.pdf | {'No Header': '1', 'Spontaneous scalarization of Bardeen black holes': 'Lina Zhang 1 , 2 ∗ , Qiyuan Pan 1 , 2 † , Yun Soo Myung 3 ‡ , De-Cheng Zou 4 § \nKey Laboratory of Low Dimensional Quantum Structures and Quantum Control of Ministry of Education, \n2 \nSynergetic Innovation Center for Quantum Effects and Applications, and Department of Physics, Hunan Normal University, Changsha, Hunan 410081, China Institute of Interdisciplinary Studies, Hunan Normal University, Changsha, Hunan 410081, China 3 Institute of Basic Sciences and Department of Computer Simulation, Inje University, Gimhae 50834, Republic of Korea 4 College of Physics and Communication Electronics, Jiangxi Normal University, Nanchang 330022, China \n(Dated: September 19, 2024) \nWe study the spontaneous scalarization of Bardeen black holes, whose tachyonic instability triggers the formation of scalarized charged black holes (SCBHs). In this case, we find infinite ( n = 0 , 1 , 2 , · · · ) branches of SCBHs with magnetic charge g . The n = 0 branch of SCBHs can be found for the coupling parameter α ≥ α n =0 ( g ) with both quadratic (1αϕ 2 ) and exponential ( e -αϕ 2 ) couplings, where α n =0 ( g ) represents the threshold of tachyonic instability for the Bardeen black holes. Furthermore, it is shown that the n = 0 branch for both couplings is stable against radial perturbations. This stability shows that this branch can be used for further observational implications.', 'I. INTRODUCTION': 'Spontaneous scalarization is a dynamic process that imparts scalar hair to black holes (and other compact objects) without changing the predictions in the weak field limit [1-5]. This phenomenon is a strong gravity phase transition caused by tachyonic instability resulting from the nonminimal coupling between scalar fields and spacetime curvature or matter. Black hole spontaneous scalarization has been extensively studied [3-11], including cases involving rotation [12, 13] and spin-induced scalarization [14-18]. These black holes are found to be entropically favorable compared to bald (general relativity) solutions and their n = 0 branches are stable [19-21]. The nonlinear dynamics of scalarized black holes in scalar-Gauss-Bonnet(sGB) gravity, including mergers and stellar core collapse, have been examined [22-26]. Additionally, spontaneous scalarization has been explored \nin other alternative theories of gravity [27-32]. These includes the Einstein-Maxwell-scalar theory with exponential and quadratic scalar couplings [33, 34]. \nIn general relativity, singularity theorems [35] suggest that singularities are inevitable inside black holes. It is worth noting that these are considered nonphysical and may be avoided in an alternative theories of gravity. In this context, Bardeen [36] proposed the first regular black hole solution, which is spherically symmetric and free of singularities. The physical source of Bardeen black holes was initially unclear. By the end of the last century, nonlinear electromagnetic sources were proposed to explain the matter content [37, 38], suggesting that regular black holes could be obtained due to nonlinear electric charge or magnetic monopoles. Other similar solutions were also found when using nonlinear electrodynamics [39-42]. We note that regular black holes are of great interest for understanding fundamental issues in physics, including singularities and nonlinear electrodynamics [43, 44]. In this work, hence, we would like to study the spontaneous scalarization of Bardeen black holes by introducing two scalar field couplings. \nThe work is organized as follows. In Sec. II, we introduce the Einstein-nonlinear electrodynamics theory coupled with scalar field. Sec. III is devoted to discuss the tachyonic instability of the Bardeen black holes. In Sec. IV, we consider two scalar field coupling forms to derive the n = 0 branch of SCBHs numerically. We wish to analyze the stability of n = 0 branch of SCBHs in Sec. V. Finally, we close the work with discussions and conclusions in Sec. VI.', 'II. THE THEORETICAL FRAMEWORK': 'We consider Einstein-nonlinear electrodynamics theory with scalar coupling function described by the following action functional \nI = 1 16 π ∫ d 4 x √ -g [ R2( ∇ ϕ ) 2 -4 ˜ f ( ϕ ) L ( F ) ] , (1) \nwhere R is the scalar curvature, ϕ is the scalar field and a coupling function ˜ f ( ϕ ) depending on ˜ f ( ϕ ). Further, L ( F ) is a nonlinear function of F = F 2 = F µν F µν with F µν = ∂ µ A ν -∂ ν A µ defined by \nL ( F ) = 3 2 sg 2 ( √ 2 g 2 F 2 / 2 1 + √ 2 g 2 F 2 / 2 ) 5 2 , (2) \nwhere the parameter s is given by s = | g | 2 M , g and M are free parameters associated with the magnetic charge and mass, respectively. \nVarying the action with respect to g µν , ϕ , and A µ gives three field equations \nG µν ≡ 2 T µν = 2 ˜ f ( ϕ ) [ 4 ∂ L ( F ) ∂ F F µη F η ν -g µν L ( F ) ] +2 ∂ µ ϕ∂ ν ϕ -( ∇ ϕ ) 2 g µν , (3) \n∇ 2 ϕ = ∂ ˜ f ( ϕ ) ∂ϕ L ( F ) , (4) \n∇ µ [ 4 ˜ f ( ϕ ) ∂ L ( F ) ∂ F F λµ ] = 0 . (5) \nTaking into account ϕ = 0, the Bardeen black hole solution is obtained by solving Eqs.(3)(5) [37, 38] \nds 2 Bardeen = -f ( r ) dt 2 + dr 2 f ( r ) + r 2 dθ 2 + r 2 sin 2 θ dφ 2 , (6) \nwith the metric function \nf ( r ) = 1 -2 Mr 2 ( r 2 + g 2 ) 3 2 . (7) \nHere g and M are the magnetic charge and mass of Bardeen black hole, respectively. In this case, the magnetic field strength is expressed as \nF µν = 2 δ θ [ µ δ φ ν ] g sin θ, (8) \nwhere we have F θφ = g sin θ ( A φ = -g cos θ ) and F 2 = 2 g 2 r 4 . In this case, computing the energymomentum tensor T ν µ = diag[ -ρ, p r , p t , p t ], there is the violation of strong energy condition ( ρ + p r +2 p t < 0) at the center, implying the regular black hole [37, 38].', 'III. INSTABILITY FOR BARDEEN BLACK HOLES': 'We briefly mention the tachyonic instability of Bardeen black hole as it serves as the starting point for spontaneous scalarization. In this paper, we choose two coupling forms: ˜ f ( ϕ ) = 1 -αϕ 2 , representing a quadratic coupling with parameter α and ˜ f ( ϕ ) = e -αϕ 2 , denoting an exponential coupling. Based on the Klein-Gordon equation (4), the linearized equation for a perturbed scalar δϕ is expressed as \n¯ ∇ 2 δϕ +2 α L ( F ) δϕ = 0 , (9) \nwhich determines the tachyonic instability of Bardeen black hole. The last term in (9) represents an effective mass term, leading to the instability of Bardeen black hole which is contingent on the coupling parameter α . Considering M = 0 . 5 and g = 0 . 2 as a typical nonextremal Bardeen black hole, one can yield an outer horizon r = r + = 0 . 935 from f ( r ) = 0 in Eq. (7), for example. \nFIG. 1: (a) The α -dependent potential V ( r, α, g = 0 . 2) as a function of r ∈ [ r + , 3 . 0] and α ∈ [0 . 01 , 30] for the outer horizon radius r + = 0 . 935( M = 0 . 5 , g = 0 . 2). The shaded region along the α -axis represents the negative region of the potential. (b) Plots of potentials V ( r, α, g = 0 . 2) with three different values α = { 10 , α th = 12 . 712 , 20 } from top to bottom near the V -axis. \n<!-- image --> \n(b) \nNow, we use the separation of variables for the spherically symmetric Bardeen background (6) given by \nϕ ( t, r, θ, φ ) = u ( r ) r e -iωt Y lm ( θ, φ ) . (10) \nChoosing a tortoise coordinate r ∗ , defined by r ∗ = ∫ dr f ( r ) , we obtain the radial part of the scalar equation as \nd 2 u dr 2 ∗ + [ ω 2 -V ( r ) ] u ( r ) = 0 . (11) \nHere the scalar potential V ( r ) is expressed as \nV ( r ) = f ( r ) [ l ( l +1) r 2 + 2 M [ r 2 -g 2 (2 + 3 α ) ] ( g 2 + r 2 ) 5 / 2 ] . (12) \nThe s ( l = 0)-mode is permissible for scalar perturbations and can therefore be used to assess the instability of Bardeen black hole. From now on, we will focus on the l = 0 mode. From the potential (12), the sufficient condition for stability requires that the potential be positive definite outside the event horizon, expressed as V ( r ) ≥ 0 [45]. However, deriving the instability condition from potential (12) is challenging, so we observe the negative region near the horizon as a signal of instability. We show the negative region of potential (12) as a function of r and α in Fig. 1(a). Fig. 1(b) indicates that the width and depth of the negative region in V ( r, α ) increase with α . If \n- \n- \n- \n- \n- \nV \n( \n0.04 \n0.02 \n0.00 \n0.02 \n0.04 \n0.06 \n0.08 \n0.10 \nr \n) \n1.0 \n1.5 \n2.0 \n2.5 \n3.0 \nr \nthe potential V ( r ) is negative in the near-horizon, it is conjectured that this may lead to a growing perturbation in the spectrum, indicating tachyonic instability of a Bardeen black hole. However, this is not always true. \nFIG. 2: Three curves of Ω in e Ω t as a function of α are used to determine the thresholds of instability [ α th ( g )] around a Bardeen black hole. We find α th ( g ) = 17 . 338(0 . 175) , 12 . 712(0 . 200) , 7 . 251(0 . 250) when three curves cross α -axis. \n<!-- image --> \nA key factor in determining the stability of a black hole is whether the scalar perturbation decays over time. The linearized scalar equation (11) around a Bardeen black hole permits an unstable (growing) mode such as e Ω t for scalar perturbations, signaling instability in the black hole. Notably, this instability often leads to the emergence of scalarized black holes. Therefore, we solve equation (11) numerically after substituting ω = -i Ω, by imposing boundary conditions of a purely ingoing wave at the near-horizon and a purely outgoing wave at infinity. From Fig. 2, we read off the threshold of instability [ α th ( g )]. Thus, the instability bound can be determined numerically by \nα ≥ α th ( g ) , (13) \nwith α th ( g ) = 17 . 338(0 . 175) , 12 . 712(0 . 200) , 7 . 251(0 . 250). On the other hand, stable Bardeen black holes exist for α < α th ( g ). For g = 0 . 2, Fig. 1(b) shows that α < α th = 12 . 712 corresponds to stable Bardeen black holes, while α ≥ α th corresponds to unstable Bardeen black holes. \nTo check the instability bound (13), we need to precisely determine α th ( g ), as it influences the formation of scalarized black holes. This can be verified by solving for a static scalar solution [scalar cloud: ϕ ( r )] to the linearized equation (11) with u ( r ) = rϕ ( r ) and ω = 0 in the Bardeen background. For l = 0, M = 0 . 5, and g = 0 . 2, requiring an asymptotically normalizable solution yield a discrete set for α n ( g ), where n = 0 , 1 , 2 , · · · denotes the number of zero crossings of ϕ ( r ) \n(or order number). See Fig. 3 for static scalar solutions ϕ ( z ) with z = r/ 2 M , M = 0 . 5, and g = 0 . 2. The n = 0 scalar mode represents the fundamental branch of scalarized black holes, while the n = 1 , 2 scalar modes indicate other branches. Actually, infinite ( n = 0 , 1 , 2 , · · · ) branches of SCBHs appear from infinite scalar modes. This is a key result for spontaneous scalarization. We note that { α 0 , α 1 , α 2 } correspond to the first three bifurcation points for emerging the n = 0 , 1 , 2 branches. As is shown in Fig. 2, we confirm that for given g = 0 . 2, \nα th ( g ) = α n =0 ( g ) , (14) \nwhich means that the instability threshold for Bardeen black holes means a formation of the largest n = 0 branch of SCBHs. \nFIG. 3: Plot of radial profiles ϕ ( z ) = u ( z ) /z as a function of z = r/ 2 M for M = 0 . 5 and g = 0 . 2, showing the first three static perturbed scalar solutions. The number n of zero nodes describes the n = 0 , 1 , 2 SCBHs. \n<!-- image -->', 'IV. SCALARIZED CHARGED BLACK HOLES': "All scalarized charged black holes will be generated from the onset of scalarization ϕ n ( r ) in the unstable region of Bardeen black hole [ α ( g ) ≥ α th ( g )]. In order to find scalarized charged black holes numerically, one proposes the metric ansatz and fields \nds 2 SBH = -N ( r ) e -2 δ ( r ) dt 2 + dr 2 N ( r ) + r 2 ( dθ 2 +sin 2 θdφ 2 ) , ϕ = ϕ ( r ) = 0 , A φ = A φ ( r ) . (15) \n/negationslash \nin which N ( r ) = 1 -2 m ( r ) /r , and δ ( r ) is the function of r . \nSubstituting the metric ansatz and fields (15) into Maxwell equation (5), we can obtain a vector potential solution A φ = -g cos θ , namely the magnetic field solution of F θφ = g sin θ and F 2 = 2 g 2 r 4 like Bardeen black hole solution. This implies that we do not need to have an approximate solution for A φ . \nWe mention again that n = 0 branch of SCBHs appears for α ( g ) ≥ α th ( g ). In particular, we consider two coupling forms: ˜ f ( ϕ ) = 1 -αϕ 2 and ˜ f ( ϕ ) = e -αϕ 2 . Using these forms, we construct the n = 0 branch of SCBHs numerically for M = 0 . 5 and g = 0 . 2. Similarly, we may construct other branches of SCBHs. \nNow, we introduce the scalar ϕ ( r ). Plugging the metric ansatz and fields (15) into Eqs. (3)-(4) results in three equations for { δ ( r ) , m ( r ) , ϕ ( r ) } as \nδ ' ( r ) + rϕ ' 2 ( r ) = 0 , (16) \n6 g 2 Mr 2 ˜ f ( ϕ ) ( g 2 + r 2 ) 5 / 2 + r ( r -2 m ) ϕ ' 2 ( r ) -2 m ' ( r ) = 0 , (17) \nr ( r -2 m ) ϕ '' ( r ) -{ m [2 -2 rδ ' ( r )] + r [2 m ' ( r ) + rδ ' ( r ) -2] } ϕ ' ( r ) -3 g 2 Mr 2 ˜ f ' ( ϕ ) ( g 2 + r 2 ) 5 / 2 = 0 , (18) \nwhere the prime ( ' ) indicates differentiation with respect to the argument. An approximate solution in the near-horizon is \nm ( r ) = r + 2 + m 1 ( r -r + ) + · · · , (19) \nδ ( r ) = δ 0 + δ 1 ( r -r + ) + · · · , (20) \nϕ ( r ) = ϕ 0 + ϕ 1 ( r -r + ) + · · · , (21) \nwhere three coefficients are determined by \nm 1 = 3 g 2 Mr 2 + ˜ f ( ϕ 0 ) ( g 2 + r 2 + ) 5 / 2 , δ 1 = -r + ϕ 2 1 , \nϕ 1 = 3 g 2 Mr + [ ( g 2 + r 2 + ) 5 / 2 +6 g 2 Mr 2 + ˜ f ( ϕ 0 ) ] ˜ f ' ( ϕ 0 ) ( g 2 + r 2 + ) 5 -36 g 4 M 2 r 4 + ˜ f ( ϕ 0 ) 2 . (22) \nThe near-horizon solution involves two parameters, ϕ 0 = ϕ ( r + , α ) and δ 0 = δ ( r + , α ), which are determined by matching (19)-(21) with the asymptotic solution in the far-region \nm ( r ) = M -Q 2 s 2 r + · · · , ϕ ( r ) = Q s r + · · · , δ ( r ) = Q 2 s 2 r 2 + · · · , (23) \nwhich incorporates the Arnowitt-Deser-Misner mass M and the scalar charge Q s . \nConsequently, for quadratic coupling, we obtain the n = 0 branch of SCBH solution shown in Fig. 4(a) for α = 13 . 048 at g = 0 . 2. The metric function N ( r ) has a slightly different horizon at ln r = -0 . 0683 compared to the Bardeen horizon at ln r = -0 . 0671, but it nearly coincides with the Bardeen metric function f ( r ) as ln r increases. Also, δ ( r ) decreases as ln r increases, while δ Bardeen ( r ) remains zero because e -2 δ ( r ) = 1 for the Bardeen case. Similarly, it is shown that \nscalar hair ϕ ( r ) decreases as ln r increases. Similarly, for exponential coupling, we obtain a SCBH solution for n = 0 branch [see Fig. 4(b)]. \n<!-- image --> \nFIG. 4: Plots of a SCBH solution with g = 0 . 2, and M = 0 . 5 for α = 13 . 048 (Quadratic coupling) and α = 13 . 667 (Exponential coupling) in the n = 0 branch of α ≥ 12 . 712. It shows metric functions δ ( r ), N ( r ), and f ( r ) for the Bardeen black hole, and scalar hair ϕ ( r ). We note that metric function N ( r ) has a horizon at ln r = -0 . 0683 while f ( r ) for Bardeen black hole takes a horizon at ln r = -0 . 0671. \n<!-- image -->", 'V. STABILITY OF SCALARIZED BLACK HOLES': "Now, we are in a position to analyze the stability of n = 0 branch of SCBHs. For this purpose, we choose three magnetic charges: g = 0 . 175, 0 . 200, and 0 . 250 with corresponding bifurcation points given by α n =0 = { 17 . 338 , 12 . 712 , 7 . 251 } , respectively. We consider two coupling forms: ˜ f ( ϕ ) = 1 -αϕ 2 and ˜ f ( ϕ ) = e -αϕ 2 . \nFirstly, we introduce radial (spherically symmetric) perturbations around the SCBHs as \nds 2 RP = -N ( r ) e -2 δ ( r ) [1 + /epsilon1H 0 ( t, r )] dt 2 + dr 2 N ( r )[1 + /epsilon1H 1 ( t, r )] + r 2 ( dθ 2 +sin 2 θdϕ 2 ) , ϕ ( t, r ) = ϕ ( r ) + /epsilon1 δϕ ( t, r ) r , (24) \n/negationslash \nwhere ϕ ( r ), N ( r ), and δ ( r ) represent the background SCBH solution, and H 0 ( t, r ), H 1 ( t, r ), and δϕ ( t, r ) represent the perturbations about it. We do not need to introduce a perturbation for gauge field A φ . Here, /epsilon1 ( /epsilon1 /lessmuch 1) is a control parameter for the perturbations. From now on, we focus on analyzing the l = 0 (s-mode) propagation,neglecting all higher angular momentum modes ( l = 0). In this case, all perturbed fields except the scalar field δϕ may be considered redundant. \nConsidering the separation of variables \nδϕ ( t, r ) = ϕ 1 ( r ) e Ω t , (25) \nwe derive the Schrodinger-type equation for scalar perturbations as \nd 2 ϕ 1 ( r ) dr 2 ∗ -[ Ω 2 + V SCBH ( r ) ] ϕ 1 ( r ) = 0 , (26) \nwhere r ∗ is the tortoise coordinate defined by dr ∗ dr = e δ ( r ) N ( r ) , and its potential reads as \nV SCBH ( r ) = e -2 δ ( r ) N ( r ) r 2 ( g 2 + r 2 ) 5 / 2 [ ( g 2 + r 2 ) 5 / 2 -6 g 2 Mr 2 ˜ f ( ϕ ) -( g 2 + r 2 ) 5 / 2 N ( r ) + 12 g 2 Mr 3 ˜ f ' ( ϕ ) ϕ ' ( r ) -2 r 2 ( g 2 + r 2 ) 5 / 2 ϕ ' ( r ) 2 +12 g 2 Mr 4 ˜ f ( ϕ ) ϕ ' ( r ) 2 +3 g 2 Mr 2 ˜ f '' ( ϕ ) ] (27) \nFor quadratic coupling, as suggested by Fig. 5(a), the potentials around the n = 0 branch show small negative regions in the near-horizon, which may indicate instability. However, a small negative region in the potential V SCBH with α = 12 . 713 (or g = 0 . 2) does not necessarily imply instability and may instead indicate stability. The linearized scalar equation (26) around the n = 0 branch may support either a stable (decaying) mode with Ω < 0 or an unstable (growing) mode with Ω > 0. \nTo fix it, we have to solve Eq. (26) numerically with vanishing ϕ 1 ( r ) at the horizon and infinity. From Fig. 6(a), we find that the n = 0 black hole is stable against the l = 0 scalar mode. Additionally, we show that the stability (or instability) of n = 0 black holes is independent of the magnetic charge g . \nFIG. 5: Three scalar potentials V SCBH for l = 0 scalar mode around the n = 0 branch. Even though they contain small negative regions in the near-horizon, these turn out to be stable black holes. \n<!-- image --> \n(a)Quadratic coupling: ˜ f ( ϕ ) = 1 -αϕ 2 \n(b)Exponential coupling: ˜ f ( ϕ ) = e -αϕ 2 \n<!-- image --> \nFor exponential coupling, we also obtain the potential V SCBH for n = 0 branch (see Fig. 5(b)), which is very similar to the potentials shown in Fig. 5(a). The n = 0 branch exhibits a large positive region outside the horizon, suggesting stability. \nTo determine the stability or instability of scalarized black holes, we need to solve the exponential version of equation (26) numerically. This is done by imposing the boundary condition \nthat the redefined scalar field ˜ ϕ 1 ( r ) has an outgoing wave at infinity and an ingoing wave at the horizon. From Fig. 6(b), we find that the n = 0 black hole is stable against the l = 0 scalar mode because its Ω is negative. This indicates that introducing the exponential coupling does not affect the stability of scalarized Bardeen black holes. \n(a)Quadratic coupling: ˜ f ( ϕ ) = 1 -αϕ 2 \n<!-- image --> \nFIG. 6: The negative Ω is given as a function of α for the l = 0 scalar mode around the n = 0 branch, showing stability. Here we consider three different cases of g = 0 . 175, 0 . 200, and 0 . 250. Three dotted curves start from α n =0 = 17 . 338, 12 . 712, and 7 . 251. Three red lines denote the unstable Bardeen black holes [see Fig. 2]. \n<!-- image -->", 'VI. DISCUSSIONS': 'In this work, we investigated the spontaneous scalarization of Bardeen black holes. The computational process is as follows: detecting tachyonic instability of Bardeen black holes → predicting scalarized Bardeen black holes (bifurcation points) → obtaining the n = 0 branch of SCBHs with both quadratic and exponential couplings → performing the (in)stability analysis of this branch. \n/negationslash \nFirstly, we note that the Bardeen black hole is unstable for α > α n =0 ( g ) [see Figs. 6(a) and 6(b)], while it is stable for α < α n =0 ( g ). Here, α n =0 ( g ) denotes the threshold of instability for the Bardeen black hole and indicates the boundary between Bardeen and n = 0 branch. Consequently, the n = 0 branch can be found for any α ≥ α n =0 ( g ) with both quadratic and exponential couplings. We also find that the bifurcation point α n =0 ( g ) increases as g decreases. Therefore, the tachyonic instability becomes harder to realize for smaller magnetic charges. We expect to have infinite ( n = 0 , 1 , 2 , · · · ) branches of SCBHs because all SCBHs are found by spontaneous scalarization. All other branches ( n = 0) seem to be unstable against radial perturbations as suggested by Refs. [33, 34]. \nFinally, we have shown that the n = 0 branch of SCBHs, obtained with both quadratic and \nexponential couplings, are stable against radial perturbations. Since the n = 0 branch of SCBHs is stable, it is considered as an end point of the Bardeen black hole. Hence, observational implications of this branch are possible to occur [46].', 'Acknowledgements': "Q. Y. Pan is supported by National Natural Science Foundation of China (Grant Nos. 12275079 and 12035005). D. C. Zou is supported by National Natural Science Foundation of China (Grant No. 12365009) and Natural Science Foundation of Jiangxi Province (No. 20232BAB201039). \n- [1] T. Damour and G. Esposito-Far'ese, Phys. Rev. Lett. 70 , 2220 (1993).\n- [2] T. Damour and G. Esposito-Far'ese, Phys. Rev. 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2024ApJ...975..260Z | Polarization of electromagnetic waves carries a large amount of information about their astrophysical emitters and the media they passed through and hence is crucial in various aspects of astronomy. Here we demonstrate an important but longoverlooked depolarization mechanism in astrophysics when the polarization vector of light travels along a nonplanar curve it experiences an additional rotation in particular for radio waves. The process leads to depolarization which we call geometric depolarization GDP. We give a concise theoretical analysis of the GDP effect on the transport of radio waves in a randomly inhomogeneous plasma under the geometrical optics approximation. In the case of isotropic scattering in the coronal plasma we show that the GDP of the angle of arrival of the linearly polarized radio waves propagating through the turbulent plasma cannot be ignored. The GDP effect of linearly polarized radio waves can be generalized to astrophysical phenomena such as fast radio bursts and stellar radio bursts etc. Our findings may have a profound impact on the analysis of astrophysical depolarization phenomena. | 2024-11-01T00:00:00Z | ['2024arXiv240912365Z', '10.3847/1538-4357/ad7d0a', 'arXiv:2409.12365', '2024ApJ...975..260Z', '10.48550/arXiv.2409.12365'] | ['Plasma astrophysics', 'Radio astronomy', 'Radio bursts', 'Radio transient sources', '1261', '1338', '1339', '2008', 'Astrophysics - High Energy Astrophysical Phenomena'] | The Twisting of Radio Waves in a Randomly Inhomogeneous Plasma | 2,024 | 169 | 0.47 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF'] | 0 | https://arxiv.org/pdf/2409.12365.pdf | {'The Twisting of Radio Waves in a Randomly Inhomogeneous Plasma': '<!-- image --> \n1 School of Electrical and Electronic Engineering, Anhui Science and Technology University, Huangshan Avenue 1501, Bengbu 233030, China 2 School of Astronomy and Space Science, Nanjing University, Xianlin Road 163, Nanjing 210023, China \n3 Key laboratory of Modern Astronomy and Astrophysics, Nanjing University, Ministry of Education, Nanjing 210023, China', 'ABSTRACT': "Polarization of electromagnetic waves carries a large amount of information about their astrophysical emitters and the media they passed through, and hence is crucial in various aspects of astronomy. Here we demonstrate an important but long-overlooked depolarization mechanism in astrophysics: when the polarization vector of light travels along a non-planar curve, it experiences an additional rotation, in particular for radio waves. The process leads to depolarization, which we call 'geometric' depolarization (GDP). We give a concise theoretical analysis of the GDP effect on the transport of radio waves in a randomly inhomogeneous plasma under the geometrical optics approximation. In the case of isotropic scattering in the coronal plasma, we show that the GDP of the angle-of-arrival of the linearly polarized radio waves propagating through the turbulent plasma cannot be ignored. The GDP effect of linearly polarized radio waves can be generalized to astrophysical phenomena, such as fast radio bursts and stellar radio bursts, etc. Our findings may have a profound impact on the analysis of astrophysical depolarization phenomena. \nKeywords: Plasma astrophysics (1261); Radio astronomy (1338); Radio bursts (1339); Radio transient sources (2008)", '1. INTRODUCTION': "According to Fermat's principle, light travels in straight lines when it propagates through an isotropic and homogeneous medium, the polarization state of it will remain constant over the whole path (Landau & Lifshitz 1960). Furthermore, while the medium is nonuniform, the polarization state might produce a nontrivial rotation to the polarization plane (also called Rytov rotation (Rytov 1938), this is a classical example of the well-known geometric phase, or Berry phase (Berry 1984), see Anandan 1992 and references therein) while the trajectory of the ray is not a planar curve, in other words, the torsion of the space curve is non-zero. Therefore, the rotation of the plane of polarization might give appreciable cumulative effects when the light travels a relatively long distance. \nCorresponding author: Ze-Lin Zhang \nHalf a century ago, under the geometrical optics approximation, Rytov rotation has already been discussed in the context of depolarization of linearly polarized light as it propagates through the Earth's atmospheric turbulence (Kravtsov 1970). Together with the method calculating the depolarization of the linearly polarized light based on diffraction theory (Strohbehn & Clifford 1967; Tatarskii 1967), both methods have demonstrated that the depolarization effects-'geometric' depolarization (GDP) and 'diffractional' depolarization (DDP)can be neglected for optical and microwave systems in the Earth's atmosphere (Wheelon 2006). However, the results mentioned above may not hold outside the Earth's atmosphere for other wavelengths. For instance, environments like the solar radio flares via plasma emission mechanics (Pick & Vilmer 2008) and even in the burst of radio waves from magnetosphere of pulsars (Beloborodov 2022; Bacchini & Philippov 2024). \nKnowing the polarization behavior of radio emission is the key to understanding the radiation mechanisms of corresponding astrophysical processes and gaining insight into their turbulent plasma environments (Lyu- \ntikov 2022; Lower et al. 2024). Now let us consider the propagation of linearly polarized radio waves in nonuniform turbulent plasmas. Due to the inhomogeneity of the medium, the propagation directions of radio waves vary from place to place and the Rytov polarization plane rotation occurs accordingly. If the medium is anisotropic, it can be inferred that this rotation will also contribute to the Faraday rotation angle as well (Ferri'ere et al. 2021) and it will further refresh our understanding of the strength and geometry of astrophysical magnetic fields (Broderick & Blandford 2010). However, tracking the path of light in complex astrophysical environments is not an easy task. Thanks to the techniques of the numerical three-dimensional ray-tracing that are continually evolving (MacDonald & Marscher 2018; Kontar et al. 2019), we can simulate the paths of the rays in turbulent plasmas and analyze the depolarization processes via their torsion information. \nThere are various instabilities present in astrophysical plasmas. The analysis of instabilities is fundamental to the study of coherent radiation. The energy transfer in coherent radiation within the plasma can be achieved through the resonant wave-particle interaction (Melrose 2017). In the case of weak magnetic field, non-thermal particles can excite Langmuir wave via beam-plasma instability in the plasma (the electromagnetic radiations are generated in anisotropic turbulent plasmas) (Bacchini & Philippov 2024). The energy from Langmuir turbulence is partially converted into the energy of transverse radio waves and it primarily occurs near the fundamental frequency and the second harmonic of the plasma oscillation frequency. In the study of solar atmospheric observations, it has been found that many radio burst processes exhibit characteristics of coherent radiation, including that the frequency of radiation often appears near the plasma fundamental frequency or its second harmonic (Kontar et al. 2017). \nIn this paper, we consider the case that linearly polarized radio wave propagation from solar radio emission to Earth-based observers. Kansabanik (2023) reported a robust imaging-based evidence for linearly polarized emission in metre-wavelength solar radio bursts. Due to the fact that the propagation path of light in a randomly inhomogeneous plasma is wavelength-dependent (Tribble 1991; Zhang et al. 2016), and considering that the validity of the geometrical optics approximation is also wavelength-dependent (Landau & Lifshitz 1960; Budden 1985), we assume that the radio waves generated in the outer solar corona during flares are created through the mechanisms of plasma emission for simplicity, then the radiation is generated close to the plasma frequency or its harmonic (Pick & Vilmer 2008). Thus, our results \nbecome independent of the wavelength in the frequency range of 0.1-500 MHz (Kontar et al. 2019). We applied the three-dimensional stochastic description of ray and the techniques of differential geometry for the calculations (Chernov 1967; Kontar et al. 2019). In the case of isotropic coronal scattering, the GDP of the angleof-arrival of the radio waves propagating through the turbulent plasma is of the order of 1 -10 3 rad/AU by selecting appropriate parameters. Our results can be generalized to other astrophysical radio burst phenomena that exhibit coherent radiation characteristics (including plasma emission processes), such as radio emission from pulsar magnetospheres triggered by plasma instabilities.", '2. THEORETICAL MODEL': 'When investigating the propagation of radio waves in a turbulent medium, the solution based on geometrical optics approximation leads to the main constraint: λ ≪ l i , where λ is the wavelength, l i is the inner scale of turbulent plasma. It means the wavelength is much smaller than the space scale of the inhomogeneity of the background field. Hence, the stochastic description of rays in three-dimensional space and the definition of the ray diffusion coefficient can be analyzed via differential geometry methods from a pedagogical viewpoint.', '2.1. Three-dimensional Ray Statistics Model': 'Chernov (1967) assumed a classic Gaussian spatial correlation function for the refractive index of a randomly inhomogeneous medium. The geometric description (known as the tantrix sphere, referring to Berger & Prior 2006 and Thorne & Blandford 2017) of this model is presented in Figure 1. \nLet ξ be the angle between two tangents at different moments to the ray, k 1 and k 2 , separated by a distance ∆ s . If ∆ s equals a few times the correlation distance l c , then ⟨ ξ ⟩ = 0 and \n⟨ ξ 2 ⟩ = 4 D ∆ s, (1) \nwhere ⟨·⟩ denotes the value of ensemble average and D is defined as the ray diffusion coefficient. As shown in Figure 1, for tiny values of ξ , ∆ θ and ∆ ϕ , the incremental changes of the polar angle θ will be given by \n∆ θ ≃-ξ cos ψ (2) \nand the azimuthal angle ϕ will be given by \n∆ ϕ ≃ ξ sin ψ/ sin θ, (3) \nwhere ψ is the angle between the vector k 2 -k 1 and the plane spanned by k 2 and the z axis. The angles ξ , \nFigure 1. (a) The unit tangent vectors ( k 1 and k 2 ) of the ray at different moments in unit k -space (the tantrix sphere of the ray, or the sphere of tangent directionsk ). (b) The ray in R 3 space and its projections onto the three mutually perpendicular coordinate planes ( x -O -y , y -O -z and z -O -x ) are drawn by orange curves. ∆ s is the arc-length between k 1 and k 2 . The green arrows represent the unit wave vectors of light at different moments. The polar angle θ and azimuthal angle ϕ in the two subplots correspond to each other. \n<!-- image --> \nψ , θ , and ϕ are statistically independent of one another and of any angle in a different increment along the ray. The angle ψ with uniform probability-density distribution satisfying ⟨ cos 2 ψ ⟩ = 1 / 2, together with Equations (1) and (2) give \n⟨ ∆ θ 2 ⟩ = 2 D ∆ s. (4) \nThe mean-square value of polar angle θ (known as the quivering angle) for the electromagnetic wave traveled a distance s through the inhomogeneous medium will be given by \n⟨ θ 2 ⟩ = 2 D s. (5)', '2.2. The Ray Diffusion Coefficient': 'In the case of isotropic Gaussian spectrum of density fluctuations, the angular scattering rate per unit distance (also called the ray diffusion coefficient) is given by (Chernov 1967; Arzner & Magun 1999; Kontar et al. 2019) \nD ≡ D θθ = 1 2 d ⟨ θ 2 ⟩ d s = √ π 4 ϵ 2 √ 2 l c ω 4 pe ( ω 2 -ω 2 pe ) 2 , (6) \nwhere ϵ 2 = ⟨ δn 2 ⟩ /n 2 is the variance of relative density fluctuation and n = ⟨ n ⟩ is the ensemble average of plasma density, l c is the correlation length. The ray diffusion coefficient D θθ derived from the power-law spectrum of isotropic density fluctuations can be expressed in the same form by replacing l c with an equivalent scale length l eq given as (Zhang et al. 2021) \nl c ≡ l eq = π -3 / 2 l 2 / 3 o l 1 / 3 i , (7) \nwhere l i and l o are inner and outer scales delineating the inertial range of the turbulence separately.', '2.3. The Differential Geometry of Polarized Waves': "As shown in Figure 2, we define a polarization trihedral ( O -k -e -h , PT for short), where k , e and h are the wave, electric-field and magnetic-field vectors of the ray at any infinite tiny increment d s . The right-handed orthonormal frame O -k -e -h is called the Darboux frame (Stoker 1989), which rotates an angle ∆ α relative to the Frenet-Serret frame ( O -k -n -b , where n and b are the normal and binormal vectors of the ray). Using theories of differential geometry (Chru'sci'nski & Jamioglyph[suppress]lkowski 2004), we have \nk = d r d s , n = 1 κ d 2 r d s 2 , b = 1 κ d r d s × d 2 r d s 2 , (8) \nwhere κ is the curvature of the ray r ( s ). Since the light waves are transverse, the vectors e and h are always co-planar with vectors n and b . Then we have \ne = (cos ∆ α ) n +(sin∆ α ) b , h = -(sin ∆ α ) n +(cos∆ α ) b \n(9) \nand the angle ∆ α is determined by the torsion τ of the spacial ray as ∆ α = -τ ∆ s , and it is also known as the geometric phase mentioned above (or topological phase, see Vinitski ˘ i et al. 1990), as presented in Figure 2. To complete the geometrical optics prescription of the field, a construction rule should be added: the PT does not rotate about the ray, which means \nΩ p · k = 0 , (10) \nFigure 2. Polarization trihedral (represented by green, blue, and gray arrows, separately) transports along the ray (represented by orange curve) in R 3 space with non-zero torsion. The angle ∆ α between the electric field vectors e 1 and e 2 at different moments is equal to the geometric phase. \n<!-- image --> \nwhere Ω p is the angular velocity vector of the PT (Lewis 1966). To examine Ω p , let us identify arc-length s with time t , and introduce the Frenet-Serret equations \nd k d t = κ n , d n d t = -κ k + τ b , d b d t = -τ n . (11) \nIf Ω = ω 1 u 1 + ω 2 u 2 + ω 3 u 3 is the angular velocity of the moving trihedral O -u 1 -u 2 -u 3 (a rigid body), then from the properties of Ω : d u / d t = Ω × u and Equations (9) and (11), we obtain u 1 = k , u 2 = e , u 3 = h , ω 1 = 0, ω 2 = κ sin ∆ α , ω 3 = κ cos ∆ α , then the angular velocity of PT is given by \nΩ p = ( κ sin ∆ α ) e +( κ cos ∆ α ) h = κ b (12) \nwhich implies Equation (10). Compare Equation (12) with Equation (8), we have \nΩ p = d r d s × d 2 r d s 2 . (13) \nReferring to Saleh (1967), the change of the polarization angle ∆ α can be defined as the cumulative rotation \nof the polarization vector around z axis and then \n∆ α = Ω p · e z ∆ s, (14) \nwhere e z is the unit vector of the z axis. Now we set \nd r d s = (sin θ cos ϕ, sin θ sin ϕ, cos θ ) . (15) \nApplying Equations (3), (8), (12), (14-15) and noting that ∆ ϕ ≃ d ϕ/ d s ∆ s , then we have \n∆ α ≃ sin 2 θ ∆ ϕ = ξ sin θ sin ψ, (16) \nwe noticed that ⟨ ξ ⟩ = 0 implies ⟨ ∆ α ⟩ = 0. From the Equations (1) and (4) and the fact that ξ , θ and ψ are statistically independent, then gives \n⟨ (∆ α ) 2 ⟩ = 4 D 2 s ∆ s, (17) \nwe noticed that ⟨ (∆ α ) 2 ⟩ depends on the position along the ray, s . Equation (10) indicates the polarization vector does not rotate around the ray itself, then we can conclude that ∆ α should be related to θ , see Equation (5). Summing the uncorrelated ∆ α of different increments along the ray of length L . The total change of the polarization angle as ⟨ α ⟩ = 0 and ⟨ α 2 ⟩ = 2 D 2 L 2 . Compare with Equation (5), we have √ ⟨ α 2 ⟩ ∼ ⟨ θ 2 ⟩ . \nSubstituting for the ray diffusion coefficient D from Equation (6), we can estimate the root-mean-square (r.m.s) value of the polarization angle as \n√ ⟨ α 2 ⟩ ≡ √ 2 D θθ L = √ π 4 ϵ 2 ω 4 pe ( ω 2 -ω 2 pe ) 2 L l c . (18) \nThis is the key formula in this paper, and the following calculations are based on this formula to estimate the impact of GDP on the angle-of-arrival. It was found to increase linearly with the length of the path L .", '3.1. GDP in a Randomly Inhomogeneous Plasma': "Despite recent observations indicating that the scattering of radio waves in coronal plasma exhibits anisotropy (Kontar et al. 2017), for simplicity, we continue to assume the isotropic scattering and it does not affect the discussion of the GDP effect. The main reasons for the above consideration are three: firstly, under the geometrical optics approximation, an isotropic and smoothly inhomogeneous medium is equivalent to an anisotropic medium (Bliokh & Bliokh 2004), it is sufficient to make the torsion of the ray path non-zero; secondly, in the case of anisotropic scattering, the impact of the background magnetic field on polarization should be considered, such as the Faraday depolarization process, while this will not impact the GDP process, it will \nFigure 3. Upper panels: (a) The r.m.s. values of GDP √ ⟨ α 2 ⟩ against ϵ and diffractive length scale l c . (b) The r.m.s. values of GDP √ ⟨ α 2 ⟩ against ϵ and the heliocentric distance R ⊙ which is from the Sun to the Earth. Lower Panels: (c) The r.m.s. values of quantity ⟨ e ' 2 ⟩ against the heliocentric distance R ⊙ and the minimum wave-number k m from the Sun to the Earth. (d) The r.m.s. values of quantity ⟨ e '' 2 ⟩ against the heliocentric distance R ⊙ and the minimum wave-number k m from the Sun to the Earth. \n<!-- image --> \ncomplicate the issue; thirdly, plane waves propagated in an anisotropic medium are completely linearly polarized in certain polarization planes (Landau & Lifshitz 1960), and it is consistent with the polarization state we considered here. Since the refractive index of the unmagnetized turbulent (isotropic and inhomogeneous) plasma n ref = (1 -ω 2 pe /ω 2 ) 1 / 2 is significantly deviate from unity for the angular frequency of radio waves ω close to the local plasma frequency ω pe in the turbulent plasma of the solar atmosphere, the impact of the den- \ninhomogeneity along the wave path is significant in the transmission of solar radio bursts generated by plasma processes. \nAssuming the radio point source with an isotropic distribution of directions k of the ray and with a frequency ω = 2 ω pe (second harmonic emission). We adopt the square root of the variance of relative density fluctuation ϵ varies from 0.05 to 0.1 (Krupar et al. 2018). The distance from source to observer L = 1 AU ≃ 1 . 496 × 10 13 cm. For radio waves with frequencies from 3 MHz to \n300 MHz, the Fresnel scale r F = √ λL/ 2 π varies from 1 . 54 × 10 7 cm to 1 . 54 × 10 8 cm, this is consistent with the description in Narayan 1992. In the strong scattering environment appropriate for electromagnetic waves near the plasma frequency, the waves quickly become isotropic. In the case of strong scattering close to the sun, r F should be much larger than the scale of the 'diffractive' length l c (the correlation scale in Equation (6)), so we set it within the range of 10 5 cm to 10 6 cm. According to Equation (18), the r.m.s. values of GDP are close to 10 3 rad/AU which is shown in Figure 3 (a). The above calculations are performed in the case of the isotropic Gaussian spectrum of density fluctuations, the plasma density fluctuations can be considered as effectively static since the group velocity of density fluctuations is much less than the speed of light. \nIn addition, in situ observations indicate an inverse power-law spectrum of density fluctuations ∝ q -( p +2) with the exponent p ≈ 5 / 3 as observed and q is the wave-vector of electron density fluctuations (Alexandrova et al. 2013). The inner scale of the solar wind turbulence l i = ( r/R ⊙ ) × 10 5 cm can be viewed as a dissipation scale (the electron gyro-radius) (Alexandrova et al. 2013; Verscharen et al. 2019), where the solar radius R ⊙ = 6 . 955 × 10 10 cm, and the heliocentric distance r varies from R ⊙ to 215 R ⊙ (photon propagates from the Sun to the Earth). We choose the outer scale l o = 0 . 25 R ⊙ ( r/R ⊙ ) 0 . 82 (Zhang et al. 2021). According to Equation (7), l c varies from 5 . 6 × 10 7 cm to 6 . 3 × 10 9 cm where the geometrical optics approximation still holds for the radio waves with wavelengths from 10 2 cm to 10 4 cm. According to Equation (18), the maximum value of r.m.s. of is close to 2 rad/AU which is shown in Figure 3 (b). \nThe three-dimensional Monte Carlo ray-tracing simulations by Chen et al. (2020) indirectly support our theoretical results, suggesting that space curve (the path of the ray) with torsion will cause linearly polarized radio waves to accumulate a geometric phase (the angle-of-arrival related to the GDP process) as it propagates through the isotropic and randomly inhomogeneous plasma. What needs to be noted is that if the polarization surface returns to its initial direction upon reaching the observer, different gauges only bring differences of integer multiples of 2 π , which has no corresponding observable effects. \n3.2. GDP vs. DDP \nTo compare the effects of GDP and DDP on the propagation of radio waves in an inhomogeneous and isotropic plasma, we adopt the density profile n ( r ) [cm -3 ] of the plasma used for the solar scattering simulations by Chen \nFigure 4. Top side: the variation of the normalized quantity C 2 n ( r ) /n 2 ( r ) with the heliocentric distance R ⊙ which varies from the Sun to the Earth. Bottom side: the spatial evolution trend of squared fractional density perturbation amplitude ⟨ δn 2 i ⟩ /n 2 as it changes with the heliocentric distance R ⊙ , where l i ( r ) = 2 . 5 × 10 4 ( r/R ⊙ -1) 1 . 3 cm, see Equation (21) and Kontar et al. (2023). The data presented in the figure is derived from Coles & Harmon 1989; Marsch & Tu 1990; Spangler 2002; Kellogg & Horbury 2005. The deviation between data and simulation is essentially controlled within an order of magnitude. \n<!-- image --> \net al. (2020) and Kontar et al. (2019, 2023) is \nn ( r ) = 4 . 8 × 10 9 ( R ⊙ r ) 14 +3 × 10 8 ( R ⊙ r ) 6 +1 . 4 × 10 6 ( R ⊙ r ) 2 . 3 , (19) \nwhere the solar radius R ⊙ = 6 . 955 × 10 10 cm, and the heliocentric distance r varies from 2 R ⊙ to 215 R ⊙ . The amplitude of the Kolmogorov (1941) density turbulence spectrum varies with distance r from the Sun as \nC 2 n ( r ) ≃ 3 . 5 × 10 3 ( r R ⊙ -1 ) -4 . 7 cm -20 / 3 , (20) \nit is the normalization coefficient of the power spectrum and is directly related to the density variance, then from Equations (19) and (20), we can calculate the normalized quantity C 2 n ( r ) /n 2 ( r ) [cm -2 / 3 ], as shown in the top side of Figure 4 and it close to a constant for distances \n> 10 R ⊙ . The profiles of the dissipation length scale l i allow us to estimate the amplitude of the density fluctuations at the dominant inner scale. The squared fractional density perturbation amplitude at l i is \n⟨ δn 2 i ⟩ n 2 = 4 πl 2 / 3 i C 2 n ( r ) n 2 ( r ) . (21) \nIt is close to the values of ϵ 2 which are shown in the upper panels of Figure 3 when the ray travels more than 10 R ⊙ away from the Sun, as shown in the bottom side of Figure 4. The low rotation measure in metrewavelength solar radio bursts indicates that the linear polarized emission has encountered much lower electron densities, implying that it originates at a much higher position within the solar corona (Dey et al. 2024). \nAgain, assuming L = 1 AU, the wavelength λ = 300 cm, wave-number k = 2 π/λ . Following Kravtsov (1970)'s train of thought, we estimate the quantities ⟨ e ' 2 ⟩ (where e is the electric-field polarization vector as mentioned in Section 3.1, its mean-square value is proportional to the square of the angle-of-arrival variance in Equation (18)) and ⟨ e '' 2 ⟩ (as the average intensity of the orthogonal component of the field divided by that of the incident plane wave, see Wheelon 2006 for more details) with Kolmogorov's spectral density \nΦ N ( κ ) = AC 2 N κ -11 / 3 exp ( -κ 2 /k 2 m ) , (22) \nwhere the refractive index structure function C 2 N ≃ C 2 n /n 2 , the constant A ≈ 0 . 033, the minimum wavenumber k m = 2 π/L , and the above form is valid for κ ≫ 2 π/L (Spangler 2002; Kontar et al. 2023). After some calculations, we once again obtain the wavelengthindependent estimation of GDP (model-independent: not limited to plasma emission mechanism) as \n⟨ e ' 2 ⟩ ≡ 2 π 4 L 2 [∫ ∞ 0 κ 3 Φ N ( κ )d κ ] 2 = N 1 C 4 N L 2 k -2 / 3 m (23) \nand DDP that is proportional to the distance L as \n⟨ e '' 2 ⟩ ≡ π 2 L 2 k 2 ∫ ∞ 0 κ 5 Φ N ( κ )d κ = N 2 C 2 N L k 2 k -7 / 3 m , (24) \nwhere the coefficients N 1 = π 4 A 2 Γ 2 (1 / 6) / 2 ≈ 1 . 643 and N 2 = π 2 A Γ(7 / 6) / 4 ≈ 0 . 076. The results are depicted in the lower panels of Figure 3. We introduce the relative depolarization ratio η to measure the impact of GDP and DDP on radio waves which is given by \nη ≡ √ ⟨ e ' 2 ⟩ √ ⟨ e '' 2 ⟩ . (25) \nIn our case, the ratio η ≃ 10 4 -10 5 ≫ 1, which means that the impact of GDP on polarized radio waves is not \nonly non-negligible but also significantly greater than the impact of DDP. For simplicity, we can perform a quick calculation to confirm the results we reached in the upper panels of the Figure 3, let C 2 N = 10 -11 cm -2 / 3 (strong scattering case), k m = 10 -5 cm -1 , and λ = 300 cm, using Equations (23) and (24), then we have ⟨ e ' 2 ⟩ ≈ 4 . 13 and η ≈ 17538 ≫ 1.", '4.1. Discussion': "Our aforementioned calculations are not only applicable to the propagation of linearly polarized radio waves in the turbulent plasma of the solar wind but also to more general cases, such as the radio emission from the magnetosphere of pulsars, which is used to explain the phenomenon of fast radio bursts. According to Rytov (1938), the Rytov rotation happens, as the radio wave propagates along a non-planar curve. The GDP effect examined in isotropic and smoothly inhomogeneous scattering of linearly polarized radio waves results from the superposition of randomly oriented polarizations of the waves traveling along different random paths. \nHowever, the situation is different for circular polarization (Gorodnichev et al. 2014). Circularly polarized light can be considered as a superposition of two linearly cross-polarized radio waves shifted in phase by π/ 2. In the case of isotropic coronal scattering, both components of the polarized waves will undergo Rytov rotation, but the phase shift between them remains unchanged. Therefore, a circularly polarized radio wave traveling along any random trajectory is unaffected by the Rytov rotation. \nIt should be noted that the focus of polarimetric solar radio studies has been solely on circular polarization for decades (Kansabanik et al. 2022). Grognard & McLean (1973) concluded that it is hard to detect any linear polarization in solar radio emissions at metre-wavelength (30-300 MHz) due to the large differential Faraday rotation experienced by the emission while traveling through the corona will suffer depolarization. Chapter 6 of Kansabanik (2023) discussed the reasons in detail. However, Dey et al. (2022) found that some type III bursts show the presence of linearly polarized emission. Recently, Dey et al. (2024) presented the first robust imagingbased evidence for linearly polarized emission in solar radio bursts at metre-wavelength. Therefore, our analysis still remains valid in solar radio bursts. In general, there are countless discussions on the depolarization of electromagnetic waves by turbulent media. However, we have discussed the GDP phenomenon induced by the propagation of linearly polarized radio waves in astrophysical turbulent plasma from a new perspective. The \nFigure 5. Cartoon illustration of depolarization effects. (a) Ideal case: the propagation of radio waves in an anisotropic plasma with uniform distribution along the line-of-sight. The dynamical phase φ DP caused by Faraday rotation effect. (b) Actual case: total phase of depolarization in a turbulent magnetized plasma equals to the dynamical phase φ DP caused by Faraday rotation effect plus the geometric phase φ GP (blue arrows represent the nontrivial rotation to the polarization plane) caused by the twisting path with non-zero torsion (the diagram has, to some extent, magnified the degree of distortion of the path of the ray.). \n<!-- image --> \nfurther refinement of this method still requires development from the following aspects: \n- · Ray tracing techniques (Kontar et al. 2019; Chen et al. 2020). Numerical ray tracing simulations of radio waves propagating through plasma resulted in the polarization position angles displaying wavelength dependencies (Lower et al. 2024). The impact of wavelength dependence on Rytov rotation should be considered in other plasma environments. Detailed numerical simulation can provide credible information on the curvature and torsion of the ray paths, then the geometric phase can be calculated.\n- · Faraday conversion and rotation effects. As we mentioned before, Rytov rotation is an example of the geometric phase. According to Bliokh & Bliokh (2004), the geometrical optics approximation in an isotropic and smoothly inhomogeneous medium is anisotropic. Budden & Smith (1976) has shown that the additional 'phase memory' (one type of geometric phase) is important for radio waves in an isotropic ionosphere. From Berry (1986) and Berry (1990), we can conclude that if the plasma is anisotropic, with an axis of birefringence, and an axis of gyrotropy that is locally fixed by the direction of the background magnetic field, then the slow rotation of these axes will produce a geometric phase. In the radio frequency band, the Faraday rotation effect will cause any linearly polarized wave to lose its initial polarization orientation characteristics. Liu & Qin (2012) \ndiscussed the impact of the geometric phase on Faraday rotation and concluded that the magnitudes of geometric and dynamical Faraday rotation angles are of the same order using typical parameters of the Tokamak plasma. It might be used for astrophysical plasma diagnostics, as shown in Figure 5. Therefore, a reconsideration of the pathdependence depolarization in the Faraday rotation measure could refresh our understanding of the strength and geometry of astrophysical magnetic fields. The interplay between works in the community of optics and those in the community of astrophysics, such as Macquart & Melrose 2000 and Lyutikov 2022 may inspire more new ideas for both. \n- · Combining with more observations. The GDP effect deserves the attention of radio astronomers. With the rapid development of radio observation technology, such as the Square Kilometre Array (SKA), the Five-hundred-meter Aperture Spherical Telescope (FAST), and the Very Large Array (VLA) et al., we are eager to gain a deeper understanding of the radio sources (pulsar magnetospheres, blazar jets, gamma-ray bursts afterglows, fast radio bursts (FRBs) and other radio transient phenomena) via polarization information of radio waves (Altunin 1981; Urata et al. 2019; Luo et al. 2020; Kansabanik et al. 2022; Kansabanik 2023; Kumar et al. 2023; Zhang 2023; Lower et al. 2024). Recently, Bastian et al. (2022) reported detections of linear polarization in stellar radio bursts. Feng \net al. (2022) reported the detection of highly linearly polarized radio emissions from the fast radio burst sources, such as FRB 20121102A, FRB 20180916B, and FRB 20201124A show frequencydependent behaviors. Hence, the consideration of the frequency dependence of the GDP effect for the propagation of radio waves, as well as the discussion on the robustness of the geometric phase against ambient perturbations, are worth further research. Additionally, it is worth mentioning that non-stationary gravitational lensing and plasma lensing effects which could bend the path may also affect the polarization vector of electromagnetic waves (Dyer & Shaver 1992; Crisnejo & Gallo 2018; Er et al. 2023).", '4.2. Conclusions': "To sum up, we demonstrate an important but long-neglected decoherence mechanism in astrophysics. We propose a toy model which including the threedimensional stochastic description of rays and the techniques of differential geometry to estimate the impact of GDP on the angle-of-arrival of polarized radio waves to the Earth's observer. Based on the geometrical optics approximation, we assume that the radio waves generated in the outer solar corona during flares are created through the mechanisms of harmonic emission of plasma for simplicity. Thus, our results become independent of \nthe wavelength. Then we applied the three-dimensional stochastic description of ray and the techniques of differential geometry for analysis. In the case of isotropic coronal scattering, the angle-of-arrival of the linearly polarized radio waves duo to GDP can not be neglected (in the isotropic case). This is the first time the concept and related methods of geometric phase have been introduced into the field of radio astronomy to study polarization issues. Our results can be generalized to other astrophysical radio burst phenomena that exhibit coherent radiation characteristics, such as radio emission from pulsar magnetospheres triggered by plasma instabilities (in the anisotropic case). Our results are expected to profoundly influence the analysis of astrophysical depolarization, and it is essential for understanding astrophysical processes through a substantial amount of polarization information from electromagnetic radiation. \nZe-Lin Zhang would like to thank Xiang-Yu Wang, Jian Liu, and Pei-Jin Zhang for useful discussions. This work is supported by the Natural Science Foundation of Education Department of Anhui Province under Grant No. 2024AH050330 and the Talents Introduction Project of Anhui Science and Technology University under Grant No. DQYJ202202 and the National Natural Science Foundation of China under the grant No. 12393852.", 'REFERENCES': "Alexandrova, O., Chen, C. H. K., Sorriso-Valvo, L., Horbury, T. S., & Bale, S. D. 2013, SSRv, 178, 101, \n- -. 1990, Proceedings of the Royal Society of London Series A, 431, 531, doi: 10.1098/rspa.1990.0149 \ndoi: 10.1007/s11214-013-0004-8 Altunin, V. 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2024Natur.626..975A | The identification of sources driving cosmic reionization a major phase transition from neutral hydrogen to ionized plasma around 600800 Myr after the Big BangSUP13SUP has been a matter of debateSUP4SUP. Some models suggest that high ionizing emissivity and escape fractions fSUBescSUB from quasars support their role in driving cosmic reionizationSUP56SUP. Others propose that the high fSUBescSUB values from bright galaxies generate sufficient ionizing radiation to drive this processSUP7SUP. Finally a few studies suggest that the number density of faint galaxies when combined with a stellarmassdependent model of ionizing efficiency and fSUBescSUB can effectively dominate cosmic reionizationSUP89SUP. However so far comprehensive spectroscopic studies of lowmass galaxies have not been done because of their extreme faintness. Here we report an analysis of eight ultrafaint galaxies in a very small field during the epoch of reionization with absolute magnitudes between MSUBUVSUB 17 mag and 15 mag down to 0.005LSUPSUP refs. SUP1011SUP. We find that faint galaxies during the first thousand million years of the Universe produce ionizing photons with logSUBionSUB Hz ergSUP1SUP 25.80 0.14 a factor of 4 higher than commonly assumed valuesSUP12SUP. If this field is representative of the largescale distribution of faint galaxies the rate of ionizing photons exceeds that needed for reionization even for escape fractions of the order of 5. | 2024-02-01T00:00:00Z | ['2024Natur.626..975A', 'arXiv:2308.08540', '2023arXiv230808540A', '10.48550/arXiv.2308.08540', '10.1038/s41586-024-07043-6'] | ['Astrophysics - Astrophysics of Galaxies'] | Most of the photons that reionized the Universe came from dwarf galaxies | 2,024 | 169 | 0.67 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML'] | 101 | https://arxiv.org/pdf/2308.08540.pdf | {'Most of the photons that reionized the Universe came from dwarf galaxies': "Hakim Atek 1* , Ivo Labbé 2 , Lukas J. Furtak 3 , Iryna Chemerynska 1 , Seiji Fujimoto 4 , David J. Setton 5 , Tim B. Miller 6 , Pascal Oesch 7,8 , Rachel Bezanson 5 , Sedona H. Price 5 , Pratika Dayal 9 , Adi Zitrin 2 , Vasily Kokorev 9 , John R. Weaver 10 , Gabriel Brammer 8 , Pieter van Dokkum, Christina C. Williams 12,13 , Sam E. Cutler 10 , Robert Feldmann 15 , Yoshinobu Fudamoto 16,17 , Jenny E. Greene 14 , Joel Leja 18,19,20 , Michael V. Maseda 21 , Adam Muzzin, Richard Pan 23 , Casey Papovich 24,25 , Erica J. Nelson 26 , Themiya Nanayakkara 2 , Daniel P. Stark 27 , Mauro Stefanon 28 , Katherine A. Suess 29,30 , Bingjie Wang 18,19,20 and Katherine E. Whitaker 8,10 \nInstitut d'Astrophysique de Paris, CNRS, Sorbonne Université, 98bis \n1 Boulevard Arago, 75014, Paris, France. \n- 2 Centre for Astrophysics and Supercomputing, Swinburne University of Technology, Melbourne, VIC 3122, Australia.\n- 3 Physics Department, Ben-Gurion University of the Negev, P.O. Box 653, Be'er-Sheva 84105, Israel. \n4 Department of Astronomy, The University of Texas at Austin, Austin, TX 78712, USA. \n5 Department of Physics and Astronomy and PITT PACC, University of Pittsburgh, Pittsburgh, PA 15260, USA. \n6 \nCenter for Interdisciplinary Exploration and Research in Astrophysics \n(CIERA) and Department of Physics & Astronomy, Northwestern \nUniversity, IL 60201, USA. \n- 7 Department of Astronomy, University of Geneva, Chemin Pegasi 51, 1290 Versoix, Switzerland. \n8 \nCosmic Dawn Center (DAWN), Niels Bohr Institute, University of \nCopenhagen, Jagtvej 128, København N, DK-2200, Denmark. \n- 9 Kapteyn Astronomical Institute, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands. \n29 Department of Astronomy and Astrophysics, University of California, Santa Cruz, 1156 High Street, Santa Cruz, CA 95064 USA. 30 Kavli Institute for Particle Astrophysics and Cosmology and Department of Physics, Stanford University, Stanford, CA 94305, USA. \nThe identification of sources driving cosmic reionization, a major phase transition from neutral Hydrogen to ionized plasma around 600-800 Myr after the Big Bang, 1, 2, 3 has been a matter of intense debate. 4 Some models suggest that high ionizing emissivity and escape fractions ( 𝑓 esc ) from quasars support their role in driving cosmic reionization. 8, 9 Others propose that the high 𝑓 esc values from bright galaxies generates sufficient ionizing radiation to drive this process. 5 Finally, a few studies suggest that the number density of faint galaxies, when combined with a stellar-mass-dependent model of ionizing efficiency and 𝑓 esc can effectively dominate cosmic reionization. 6, 7 However, so far, low-mass galaxies have eluded comprehensive spectroscopic studies owing to their extreme faintness. Here we report an analysis of eight ultra-faint galaxies (in a very small field) during the epoch of reionization with absolute magnitudes between 𝑀 UV ∼ -17 to -15 mag (down to 0.005 𝐿 ★ 10,11 ). We find that faint galaxies during the Universe's first billion years produce ionizing photons with log( 𝜉 ion / Hz erg -1 ) = 25 . 80 ± 0 . 14 , a factor of 4 higher than commonly assumed values. 22 If this field is representative of the large scale distribution of faint galaxies, the rate of ionizing photons exceeds that needed for reionization, even for escape fractions of order five per cent. \nWe combine ultra-deep JWST imaging data with ancillary Hubble Space Telescope ( HST ) imaging of the gravitational lensing cluster Abell 2744 (A2744 hereafter) in order to photometrically select extremely faint galaxy candidates in the epoch of reionization. A crucial component of the present study is the use of strong gravitational lensing to amplify the intrinsically-faint flux of distant sources. An accurate estimate of the magnification factor is required to retrieve the intrinsic luminosity of sources. This step relies on a good knowledge of the total mass distribution in the galaxy cluster. Here we use the most recent lensing model ( v1.1 ) published for the UNCOVER survey. The magnification factors for our galaxy sample range from 𝜇 ∼ 2 to 𝜇 ∼ 27. The values are reported in Table 1, together with 1 𝜎 uncertainties. The second part of the UNCOVERprogram consists of ultra-deep follow-up spectroscopy with the NIRSpec instrument. We used the Multi-Shutter Assembly to obtain multi-object spectroscopy in 7 pointings, totaling an exposure time ranging from 2.7 to 17.4 hours. Figure 1 shows the position of these sources in the A2744 field with the associated regions of high magnification and the configuration of the NIRSpec slits. Simultaneous spectral fits to the continuum and the emission lines provide estimates of the spectroscopic", 'Dwarf galaxies reionized the Universe': 'Fig. 6 : Stellar population simultaneous fitting to the NIRSpec spectra and NIRCam photometry . Panel a: Two representative sources (IDs 18924 and 16155) are shown. The best-fit Bagpipes model (red curve) is plotted over the observed NIRSpec spectrum (black curve), together with the error spectrum (gray curve). The NIRCam photometric measurements are represented with black points with their associated 1-sigma uncertainties. Panel b: Posterior distribution function for the main physical properties of ID 16155. When relevant, the parameters are corrected for magnification. Panel c: Same as panel b, for source ID 18924. \n<!-- image -->', 'References': '- 37 Oke, J. B. & Gunn, J. E. Secondary standard stars for absolute spectrophotometry. Astrophys. J. 266 , 713-717 (1983).\n- 38 Bezanson, R. et al. The JWST UNCOVER Treasury survey: Ultradeep NIRSpec and NIRCam ObserVations before the Epoch of Reionization. arXiv e-prints arXiv:2212.04026 (2022).\n- 39 Weaver, J. R. et al. The UNCOVER Survey: A first-look HST+JWST catalog of 50,000 galaxies near Abell 2744 and beyond. arXiv e-prints arXiv:2301.02671 (2023).\n- 40 Rieke, M. J. et al. Performance of NIRCam on JWST in Flight. PASP 135 , 028001 (2023).\n- 41 Brammer, G. Grizli: Grism redshift and line analysis software. Astrophysics Source Code Library, record ascl:1905.001 (2019).\n- 42 Lotz, J. M. et al. The Frontier Fields: Survey Design and Initial Results. Astrophys. J. 837 , 97 (2017).\n- 43 Steinhardt, C. L. et al. The BUFFALO HST Survey. Astrophys. J. Suppl. Ser. 247 , 64 (2020).\n- 44 Jakobsen, P. et al. The Near-Infrared Spectrograph (NIRSpec) on the James Webb Space Telescope. I. Overview of the instrument and its capabilities. Astron. Astrophys. 661 , A80 (2022).\n- 45 Ferruit, P. et al. The Near-Infrared Spectrograph (NIRSpec) on the James Webb Space Telescope. II. Multi-object spectroscopy (MOS). Astron. Astrophys. 661 ,', 'Methods': 'Throughout the paper, we use AB magnitudes 37 and a standard cosmology with H 0 = 70 km s -1 Mpc -1 , Ω Λ = 0 . 7, and Ω 𝑚 = 0 . 3.', 'Observations and sample selection': 'The UNCOVER dataset consists of both imaging and spectroscopic observations of the lensing cluster A2744. The imaging observations and data reduction are described in detail in the survey and catalog papers. 38,39 Here, we briefly summarize the imaging and photometric products used in the present paper. HST imaging consists of 7 broadband filters (F435W, F606W, F814W, F105W, F125W, F140W, F160W). The NIRCam 40 images include short-wavelength (SW) broadband filters (F115W, F150W, F200W), long-wavelength (LW) broadbands (F277W, F356W, F444W), and one medium-band filter (F410M). Data were processed, and drizzled into 0.04 arcsec pix -1 mosaics using the Grism redshift and line analysis software for space-based spectroscopy ( G rizli; v1.6.0.dev99). 41 In terms of ancillary data, the HFF program 42 has obtained deep optical and NIR observations of the core area of A2744 with the Advanced Camera for Surveys (435W, F606W, F814W), and Wide-Field Camera Three (F105W, F125W, F140W, F160W). A wider area around the cluster has also been covered by the BUFFALO program 43 in almost identical broadband filters (without F435W and F140W). All HST observations were drizzled to the same pixel scale and the same orientation as the NIRCam mosaics. \nThe second part of the UNCOVER program consists of ultra-deep follow-up spectroscopy with the NIRSpec instrument. 44 Data were obtained between July 31st and August 2nd 2023. Observations use the Prism mode and the Multi-Shutter Assembly 45 of NIRSpec to observe more than 650 targets. In order to optimize background subtraction, each target has been observed with a 3-slitlet nodding strategy. Observations were split into 7 pointings, with important overlap at the center, providing total on-target exposure times ranging from ∼ 2 . 7 ℎ to ∼ 17 . 4 ℎ . The spectral resolution is wavelength-dependent and varies between 𝑅 ∼ 30 to 𝑅 ∼ 300 over the full wavelength range 𝜆 ∼ 0 . 6 -5 . 3 𝜇𝑚 . Data were reduced using the JWST/NIRSpec analysis software msaexp version 0.6.10. The processing is based on level 2 MAST products, using the CRDS context file jwst\\_1100.pmap . The software performs basic reduction steps, including flat-field, bias, 1/f noise and snowballs correction, wavelength and photometric calibrations of individual exposure frames. 46 The extraction of 1D spectra from individual exposures is operated on inverse-weighted stack of 2D spectrum in the dispersion direction, following an optimal extraction procedure. 47 Then the software fits a Gaussian profile along the cross-dispersion direction to define the 1D extraction aperture. Finally, we compute the final deep 1D spectrum by inversevariance stacking the individual spectra. In order to account for slit loss effects, we apply a wavelength- (broadband-) dependent correction factor to re-scale the 1D spectrum to the observed NIRCam aperture photometry. We show an example of the imaging and spectroscopic data in Figure 5. A clear Lyman-break at rest-frame wavelength 𝜆 rest = 1216 Å is observed, together with multiple strong emission lines, including H 𝛼 +[N/i.pc/i.pc], [O/i.pc/i.pc/i.pc] 𝜆𝜆 4960 , 5008, H 𝛽 , H 𝛾 , and [O/i.pc/i.pc] 𝜆 3727 \nThe selection of our sample combines several criteria to constrain the photometric redshifts of the sources. First, we applied a color-color selection, based on a fluxdropout in the HST optical filters caused by rest-frame Lyman -𝛼 absorption by residual intergalactic Hydrogen gas. This selection consolidates most of the sources identified in the Hubble Frontier Fields (HFF) data 26,48 at 6 < 𝑧 < 9. Second, we performed spectral energy distribution (SED) fitting with the Eazy 49 software to estimate photometric redshifts, assuming a flat luminosity prior and the corr\\_sfhz library of stellar population templates. The allowed redshift range was set to 0 . 01 < 𝑧 < 20. The sources have been selected to have best-fit photometric solutions lying 6 < 𝑧 < 9 at the heart of the epoch of reionization. The final selection was then performed according to the intrinsic luminosity, combining high magnifications ( 𝜇 ≳ 2) and faint observed luminosities in F150W, resulting in intrinsic absolute UV magnitudes of order 𝑀 UV ≳ -17.', 'Spectral fitting': 'In order to determine the spectroscopic redshift of the sources, we fit spectral templates using msaexp , which based on the SED fitting software Eazy . 50 The code combine a set of templates to fit simultaneously the continuum, including the Lyman break caused by the IGM absorption, and the emission lines. For our analysis, we adopt the corr\\_sfhz\\_13 template library which include redshift-dependent SFHs, which are known to perform better than the default fsps\\_full library in recovering the true redshift. 39 Wesearch for the best-fit solution over the redshift interval 0 < 𝑧 < 15. The spectroscopic redshifts are reported in Table 2. Once the best-fit redshift is found, we refit the spectra, fixing the redshift to 𝑧 spec , with a set of spline functions to measure the continuum and Gaussians in order to measure the emission line fluxes. Examples of the best-fit model plotted over the observed spectrum are presented in Figure 1.', 'Strong lensing': "We use version v1.1 of the UNCOVER lensing model, 12 which is publicly available in the latest UNCOVER data release DR-1. The model is based on the parametric approach by Zitrin et al. 51 which has been re-written to be fully analytic, i.e. not limited by a grid-resolution. 52,12 The UNCOVER lens model of A2744 was constructed on a wealth of ground- and space-based data, including deep HST and JWST imaging, and Multi Unit Spectroscopic Explorer 53 (MUSE) spectroscopic redshifts of both cluster members and multiple images. 54,55,17,56 It comprises 421 cluster member galaxies identified in the ∼ 45 arcmin 2 UNCOVER field-of-view and five smooth cluster-scale dark matter (DM) halos. The model is constrained with 141 multiple images belonging to 48 sources and achieves an image reproduction RMS of Δ RMS = 0 . 51 '' in the lens plane. Thanks to the massive cluster substructures identified with UNCOVER, 12 the critical area of the cluster is 1 . 5 × larger than inferred from HFF data and the total source plane area with 𝜇 > 4 of ∼ 4 arcmin 2 for a source at redshift 𝑧 s = 6. The model uncertainties on the amplification values are derived from a Markov Chain Monte Carlo (MCMC) procedure within the modeling code Zitrin-analytic . 57,58 The temperatures of the MCMC are chosen to reflect typical", '16 Dwarf galaxies reionized the Universe': 'Fig. 5 : UNCOVER JWST data for galaxy 16155 at 𝑧 spec = 6 . 88. The top panels show image cutouts in seven different filters at increasing wavelength including ancillary HST/ACS data in F814W, and UNCOVER JWST imaging in F115W, F150W, F200W, F277W, F356W, and F444W bands (left to right). The central panel shows the UNCOVER NIRSpec data, with the 2D spectrum on top of the 1D optimally extracted spectrum (black with gray 1𝜎 uncertainty ranges). The red lines show the best-fit msaexp template spectrum. The observed-frame wavelengths of key emission lines are indicated as vertical dashed lines. The bottom panels show a zoomed in version of three different parts of the spectrum around the Ly 𝛼 break (left), around the [O/i.pc/i.pc/i.pc]+H 𝛽 emission lines (middle) and the H 𝛼 line (right). \n<!-- image --> \nsystematics inherent to parametric lens modeling techniques. 58 In order to better estimate systematics uncertainties inherent to models, we used an independent mass model for A2744 17 for comparison. The magnification factors are in good agreement within 1𝜎 uncertainties, except for two objects that have a difference at the level of 1.5 and 2𝜎 . Moreover, the statistical uncertainties derived from each model are of \nthe same order. We have incorporated these systematic uncertainties in the quoted errors on the amplification factors.', 'Bagpipes': 'We infer global physical properties from SED-fitting using the Bagpipes software package 59,60 ). Before fitting, all models are convolved with the NIRSpec/Prism instrumental resolution curve provided by the Space Telescope Science Institute ( jwst\\_nirspec\\_Prism\\_disp.fits ), assuming that the flight performance is 1.3 times better than stated, and is consistent with an earlier work where a factor 1/0.7 is introduced for modeling 𝑧 > 10 galaxies. 61 Additionally, we fit with a wavelengthindependent velocity smoothing (0 < log ( 𝑣 smooth ) < 3 . 3) as a nuisance parameter. We adapt the following model grid: 62 stellar population models, the MILES spectral library, 63,64 CLOUDY nebular emission models, 65 and 66 dust model (with 0 < 𝐴 𝑣 < 5 and 0 . 3 < 𝑛 < 2 . 5 as free parameters). The stellar and gas phase metallicity are tied to the same value, and also included as a free parameter in the range -2 < log ( Z / Z ⊙ ) < 0 . 3. The ionization parameter is also left free in the range of -3 . 5 < log ( U ) < -1 . 0. We parameterize the star formation history as a delayed𝜏 model (SFR ∝ -𝑡 / 𝜏 , which can flexibly produce rising or falling star formation histories for this range of 𝜏 at the redshift of the sample), with the age ( -3 < log(age) < 0 . 48) and 𝜏 (0.01 < 𝜏 <5) as free parameters. This parameterization has been shown to reliably recover star formation rates and the mass formed in recent star formation, but is potentially susceptible to under-estimating stellar masses due to outshining by the youngest stellar population. 67 Redshift is restricted to vary in a narrow range around the best-fitting spectroscopic redshifts ( ± 0.1). We fit for a polynomial calibration vector of order 2 after applying a wavelength-independent calibration to scale the normalization of the spectrum to the photometry. The Bagpipes white noise model is used to allow for underestimated errors up to a factor of 10. A signal-to-noise ceiling of 20 is imposed on both our photometry and spectroscopy to account for systematic issues with the flux calibration. Sampling is performed via PyMultinest , 68,69 with the default Bagpipes convergence criteria. The most important physical properties derived from this procedure are presented in Table 2, and example fits and posteriors are presented in Figure 6. \nIn addition to the star-formation rate (SFR) derived from SED-fitting, we also compute the SFR based on H 𝛼 or H 𝛽 recombination line. The H 𝛼 indicator traces massive short-lived stars on a timescale of a few Myr, whereas the UV emission indicates an SFRaveraged on a longer timescale up to a few 100 Myr. We report high values for the SFR(H 𝛼 )/SFR(UV) ratio in the range [5-60], indicating recent bursts of star-formation in these young systems. This is in line with their specific star-formation rates (sSFR). Given their low stellar masses, these sources have log(sSFR(H 𝛼 ) / yr -1 )=[ -7 . 4 , -6 . 2], which means that they can double their stellar mass within 2 to 20 Myr.', 'BEAGLE': 'Werun an additional spectral fit with the BayEsian Analysis of GaLaxy sEds 70 tool ( BEAGLE ) on the magnification-corrected spectra. BEAGLE uses the latest version of the Bruzual & Charlot stellar population synthesis models 71 and nebular emission templates computed with CLOUDY . 72,73 We then assume a Chabrier 74 initial stellar mass function (IMF), an SMC dust attenuation law, 75 the latest Inoue et al. analytic \nIGM attenuation models, 32 and a delayed exponential SFH as for our Bagpipes fit. All other parameters are left free to vary with uniform or log-uniform priors: stellar mass log ( 𝑀 / M ⊙ ) ∈ [ 4 , 10 ] , current (10 Myr) SFR log ( 𝜓 / M ⊙ yr -1 ) ∈ [-2 , 4 ] , maximum stellar age log ( 𝑡 age / yr ) ∈ [ 6 , 𝑡 universe ] , star-formation e-folding time log ( 𝜏 / yr ) ∈ [ 5 . 5 , 9 . 5 ] , stellar metallicity log ( 𝑍 / Z ⊙ ) ∈ [-2 . 2 , -0 . 3 ] , effective V -band dust attenuation optical depth ˆ 𝜏 𝑉 ∈ [ 0 , 3 ] , effective galaxy-wide ionization parameter log 𝑈 ∈ [-4 , -1 ] , gas-phase metallicity log ( 𝑍 gas / Z ⊙ ) ∈ [-2 . 2 , -0 . 3 ] , and dust-to-metal mass ratio 𝜉 d ∈ [ 0 . 1 , 0 . 5 ] . The posterior distribution of the physical properties that we derive with BEAGLE agree well with the Bagpipes results presented in Table 2.', 'Contribution of galaxies to reionization': 'Using the present spectroscopically-confirmed sample of ultra-faint galaxies, we have the opportunity to put constraints on the UV luminosity function. We first describe the selection procedure of our sample and associated biases. The original sample has been selected in HFF studies, 26,48 based on HST observation. The selection of the original photometric sample is based on Lyman break criteria, which identify the dropout due to continuum absorption by the neutral IGM blueward of Ly 𝛼 . The photometric redshift through SED fitting were only measured to refine the redshift solution. In addition, three sources were selected from the UNCOVER imaging data based on their photometric redshifts. Regarding these three sources, strong emission lines tend to help put stronger constraints on the photometric redshift estimates, resulting in narrower best-fit solutions, which could favor strong-line emitters in the sample selection. 76 For the selection of this spectroscopic sample, we primarily focus on faint intrinsic magnitudes, typically 𝑀 UV ≳ -17 mag, as can be seen in Figure 2. Their apparent magnitude ranges from 𝑚 𝐹 150 𝑊 = 27 . 4 to 29.7 AB mag. While there is an intentional bias to select intrinsically faint galaxies in this study, this is less the case regarding observed magnitudes. Finally, during the Multi-Shutter Assembly design, we assigned equal weights to all galaxies. Therefore, the only bias introduced here is the optimization of the number of sources that are included in one mask configuration. Therefore, galaxies that did not make it to the final sample were simply excluded for mask optimization reasons. \nFirst, we compute an initial UV LF based on the present sample binned in four magnitude bins and a survey volume, which depends on the original selection of the source. For the five HFF sources, we use the source plane effective volume 26 as a function of the magnitude bin. We rebin the original HFF sample to match the new magnitude bins. A scaling factor is then applied to the LF points to match the HFF completeness-corrected counts. Finally, we apply a correction factor based on the success rate of the spectroscopic confirmation. For the three sources outside the HFF area, we recompute the source plane effective survey volume using the new lensing model and assuming a similar completeness function across the field. We perform the same exercise of rescaling the number counts in each magnitude bin. The final UV LF is calculated by combining all galaxies with their associated corrected effective volume. Regarding uncertainties, we use the HFF volume uncertainties for the HFF sources, and the updated volume uncertainties for the new sample, respectively. We \nFig. 7 : Spectroscopic constraints on the UV luminosity function . The UV luminosity function as determined from our spectroscopic sample is represented by orange points. Also shown, the photometric determination from the HFF data, 26 together with the best-fit Schechter function (blue curve) and a modified Schechter with a potential turnover (teal curve). The shaded region of each curve represent the 1 -𝜎 uncertainties \n<!-- image --> \nnote that the HFF uncertainties include systematic effects derived from a comparison between four independent models. 26 For the three sources outside the HFF coverage, the comparison to another independent model 17 shows that the uncertainties are negligible for such small amplification factors. We also include Poisson errors and cosmic variance in the final LF results. For the survey volume probed by our program, we estimated cosmic variance to be around 𝜎 CV ∼ 30%. 77 Overall, In Figure 7, we show that our measurements (orange points) are in good agreement with the faint-end of the photometric UV LF derived from HFF observations 26 (gray points). \nIn our efforts to determine whether galaxies can reionize the Universe, we proceed to calculate another crucial parameter: the production efficiency of ionizing radiation 𝜉 ion . This quantity is defined as the ratio between the LyC photon production rate in the units of s -1 , and the observed non-ionizing UV luminosity density 𝐿 UV estimated at 1500 Å in units of erg s -1 Hz -1 : \n𝜉 ion = 𝑁 ( 𝐻 0 ) 𝐿 𝑈𝑉 [ 𝑒𝑟𝑔 -1 𝐻𝑧 ] , (1) \nwhere 𝑁 ( 𝐻 0 ) can be estimated from the H 𝛼 Balmer line 78 assuming a case B recombination theory: 79 \n𝐿 ( 𝐻𝛼 ) [ 𝑒𝑟𝑔 𝑠 -1 ] = 1 . 36 × ( 1 -𝑓 𝑒𝑠𝑐 ) 10 -12 𝑁 ( 𝐻 0 ) [ 𝑠 -1 ] (2) \nwhere 𝐿 ( 𝐻𝛼 ) is in units of erg s -1 , and 𝑓 esc is the escape fraction of Lyman continuum radiation. In this calculation, we assume that 𝑓 esc =0, meaning that all Lyc photons are reprocessed into the Balmer lines. The derived 𝜉 ion value can be considered as a lower limit, since higher 𝑓 esc will lead to a higher 𝜉 ion . The H 𝛼 emission line is not detected in source ID 18924. In this case, we use the H 𝛽 luminosity and a case B conversion factor. Our measurements are reported in Figure 3, together with literature results. 80,81,82,83,84,85 These uncertainties due to potential field-to-field variations can also affect the ionizing properties of galaxies. We incorporated cosmic variance errors ( 𝜎 UV ∼ 30%) to the 𝜉 ion values for both 𝑀 UV > -16 . 5 mag and 𝑀 UV < -16 . 5 mag (cf. Figure 3). \nSpectroscopic measurements in brighter galaxies have also reported 𝜉 ion values higher than canonical values. 88 Several JWST studies have measured 𝜉 ion in faint galaxies at the epoch of reionization. 86,23 However, their results are based on emission line fluxes inferred from broadband excess, rather than direct spectroscopic measurements. In particular, JWST medium-band photometric data have been used to infer 𝜉 ion for galaxies over the redshift range 3 < 𝑧 < 7. 87 These measurements offer an opportunity to explore a larger population of galaxies through wide-area imaging. They report ionizing efficiencies in the range log( 𝜉 ion / Hz erg -1 ) =25 . 31 -25 . 39, where galaxies with strong Ly 𝛼 emission tend to have the highest values. These values are smaller than our average measurements for the faintest galaxies. However, the vast majority of their sample is at significantly lower redshifts. Their redshift distribution has two peaks at 𝑧 = 3 and 𝑧 = 5, and the median redshift of their sample is 𝑧 = 4 . 02, which is well below the epoch of reionization. Among their sample of 370 galaxies, only ∼ 25 galaxies lie within the epoch of reionization. Therefore, their average 𝜉 ion value is not representative of the epoch of reionization. Furthermore, this difference is smaller if we take into account the dynamical range of UV magnitudes explored in their study. Although they find a weak dependency of 𝜉 ion with 𝑀 UV , their sample consists of galaxies with UV magnitudes ranging from -23 to -15.5 mag, compared to our subsample of 𝑀 UV > -16 . 5 mag. While imaging-based measurements can be complementary to spectroscopic studies, they have larger uncertainties (in the range 𝜎 = 0.43 - 0.64), due to the way emission line fluxes are inferred, and are model-dependent, since the continuum is derived from SED-fitting. \nFinally, we derive constraint on the LyC escape fraction using indirect indicators calibrated in a large sample of nearby LyC emitting galaxies. Since the LyC emission is impossible to measure at the epoch of reionization, large efforts have been devoted in the last two decades to determine the escape fraction in 𝑧 < 4 galaxies, and more importantly establish indirect methods, which can be transferable to reionization sources. This was precisely the motivation of the recent Low-redshift Lyman continuum survey (LzLCS). 89 Among the different physical properties of LyC leakers, the observed UV continuum slope 𝛽 has been identified as a promising proxy of 𝑓 esc . 28 Here, we use the UV slopes derived from the best-fit models of Bagpipes and \nthe LzLCS relation to infer 𝑓 esc . The derived values for the present sample range from 4.5% to 15.6%. Despite large uncertainties (around 50%), only two sources have 𝑓 esc values that can reach below 4% at 1-sigma. \nNow with all three properties in hand, we can assess the contribution of faint galaxies to cosmic reionization. We compute the ionizing photon emissivity of galaxies, which is the product of the total UV luminosity density 𝜌 UV , derived from integrating the UV LF, and the production efficiency 𝜉 ion . The result will depend on the faint integration limit, which is set the faintest bin of the spectroscopic LF at 𝑀 UV =-15 mag. By multiplying this quantity by the escape fraction 𝑓 esc , we obtain the total ionizing photon rate density that is available to ionize the IGM. Down to 𝑀 UV =-15 mag, modest 𝑓 esc values around 5% are sufficient to maintain reionization. \nWe measure the gas-phase oxygen abundance using the strong optical lines diagnostic. Specifically, we use the R3=log([O/i.pc/i.pc/i.pc] 𝜆 5007 /H 𝛽 ) and adopt the most recent empirical calibrations at high-redshift. 90,91,92 All of our sources show detections of [O/i.pc/i.pc/i.pc] 𝜆 5007 and H 𝛽 . Furthermore, we also separate our sample into two bins according to the EW(H 𝛽 ) at EW(H 𝛽 )=100Å to account for ionization parameter variations. 90 Also, for a given value of R3, the calibration defines two metallicity solutions. Although these sources have likely low metallicities, we use the O32=log([O/i.pc/i.pc/i.pc] 𝜆𝜆 4949,5007/[O/i.pc/i.pc] 𝜆𝜆 3727,3729) ratio to distinguish between the two branches. For most of the sources, the [O/i.pc/i.pc] is not detected, which provides a lowerlimit on O32, which is found to vary in the range O32=[0.7-1.4]. Using the O32 metallicity indicator, the resulting values are all compatible with the low-metallicity branch solution. The metallicity measurements are reported in Table 2. \nIn addition to the physical properties that may ease the escape of ionizing photons from these galaxies, metallicity can also inform us on the ionizing properties of this population. We measured the gas-phase metallicity using the R3 = [O/i.pc/i.pc/i.pc]/H 𝛽 line ratio based on the most recent calibrations. 93 We find extremely low metallicities, ranging from 12+log(O/H)= 6.70 to 7.46, which corresponds to 1% to 6% of the solar metallicity. Such low metallicities are often suggestive of strong ionizing radiation from massive stars. 94 At the same time, these distant low-mass galaxies, considered as the building-blocks of present-day galaxies, are expected to be metal-poor, owing to their supposedly pristine gas conditions. We note that our estimate relies on the calibration of strong lines diagnostics, which are prone to significant uncertainties at 𝑧 > 6. On average, these estimates lie with 1 to 2 𝜎 intervals from each other.', 'Size Measurements': "To further characterize these sources, we measure their sizes by fitting their morphology in the NIRCam F150W filter with a Sérsic profile. 95 Measurements of the half-light radii for all eight galaxies are performed using the pysersic 96 package. For each source, we assume a single Sersic profile and mask all nearby sources in the photometric catalog. 39 The priors for the half-light radius, Sersic index, axis ratio and position angle are all uniform and varied from 0 . 5 -10 pixels, 0 . 65 -4, 0 . 1 -1 and 0 -2 𝜋 respectively. Priors central position and flux are represented as Gaussian distributions with the location and width based on the photometric catalog. 39 A flat sky background is fit simultaneously. The Posterior distribution is explored using the \nNo-U Turn (NUTS) 97 sampler implemented in numpyro 98 with 2 chains for 1,000 warm-up and sampling steps each. \nSeveral of our objects are significantly distorted by the gravitational lensing which means that we need to take the shear into account when deriving the half-light radius. In order to do that, we use our lensing model to derive the tangential and radial magnifications, defined as 𝜇 = 𝜇 t 𝜇 r , from the deflection field at each source's position and redshift. Since our objects are sheared along the tangential direction, we use the tangential component of the magnification to correct the half-light radii. \nThe derived effective radii, corrected for magnification and taking the shear into account, vary between 𝑟 eff = 30 to 300 pc. Overall, these constraints show that these sources are small, in broad agreement with an extrapolation to lower masses of the size-mass relation derived at similar redshifts, 99 albeit with significant scatter. Such small sizes and high sSFRs also supports the scenario of stochastic star formation histories in these systems or dust ejection, 100 owing to their small dynamical time and a low gravitational potential.", '24 Dwarf galaxies reionized the Universe': 'A81 (2022). \n- 46 Heintz, K. E. et al. Extreme damped Lyman𝛼 absorption in young star-forming galaxies at 𝑧 = 9 -11. arXiv e-prints arXiv:2306.00647 (2023).\n- 47 Horne, K. An optimal extraction algorithm for CCD spectroscopy. 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E., Topping, M. W., Reddy, N. A. & Brammer, G. B. Direct T\\_e-based Metallicities of z=2-9 Galaxies with JWST/NIRSpec: Empirical Metallicity Calibrations Applicable from Reionization to Cosmic Noon. arXiv e-prints arXiv:2303.08149 (2023).\n- 94 Stanway, E. R. & Eldridge, J. J. Initial mass function variations cannot explain the ionizing spectrum of low metallicity starbursts. Astron. Astrophys. 621 , A105 (2019).\n- 95 Sérsic, J. L. Influence of the atmospheric and instrumental dispersion on the brightness distribution in a galaxy. Boletin de la Asociacion Argentina de Astronomia La Plata Argentina 6 , 41-43 (1963).\n- 96 Pasha, I. & Miller, T. B. pysersic: A Python package for determining galaxy structural properties via Bayesian inference, accelerated with jax. arXiv e-prints arXiv:2306.05454 (2023).\n- 97 Hoffman, M. D., Gelman, A. et al. The no-u-turn sampler: adaptively setting path lengths in hamiltonian monte carlo. J. Mach. Learn. 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Computing in Science and Engineering 9 , 90-95 (2007). \n- 104 Brammer, G. msaexp: NIRSpec analyis tools. Zenodo (2022).\n- 105 Harris, C. R. et al. Array programming with NumPy. Nature 585 , 357-362 (2020).\n- 106 Hoffman, M. D. & Gelman, A. The No-U-Turn Sampler: Adaptively Setting Path Lengths in Hamiltonian Monte Carlo. arXiv e-prints arXiv:1111.4246 (2011). \n107 Phan, D., Pradhan, N. & Jankowiak, M. Composable Effects for Flexible and Accelerated Probabilistic Programming in NumPyro. arXiv e-prints arXiv:1912.11554 (2019). \n108 Virtanen, P. et al. SciPy 1.0: fundamental algorithms for scientific computing in Python. Nature Methods 17 , 261-272 (2020). \nData Availability. The NIRCam and HST imaging data are available on the UNCOVER webpage: https://jwst-uncover.github.io/. The NIRSpec spectroscopic data are publicly available through the Mikulski Archive for Space Telescopes ( MAST ; https://archive.stsci.edu/), under program ID 2561. The UNCOVER Lensing products are available at https://jwst-uncover.github.io/DR1. html#LensingMaps. \nCode Availability. Astropy, 101,102 Bagpipes, 59,60 BEAGLE, 70 EAzY, 50 Matplotlib, 103 msaexp v0.6.10, 104 NumPy, 105 NUTS, 106,107 PyMultinest, 68,69 pysersic, 96 SciPy, 108 GrizLi https://github.com/gbrammer/grizli \nAcknowledgments. H.A. and IC acknowledge support from CNES, focused on the JWST mission, and the Programme National Cosmology and Galaxies (PNCG) of CNRS/INSU with INP and IN2P3, co-funded by CEA and CNES. H.A thanks the Cosmic Dawn Center (DAWN) for their support. DAWN is funded by the Danish National Research Foundation under grant No. 140. IL acknowledges support by the Australian Research Council through Future Fellowship FT220100798. P.D. acknowledges support from the NWO grant 016.VIDI.189.162 (\'ODIN") and from the European Commission\'s and University of Groningen\'s CO-FUND Rosalind Franklin program. A.Z. acknowledges support by Grant No. 2020750 from the United StatesIsrael Binational Science Foundation (BSF) and Grant No. 2109066 from the United \nStates National Science Foundation (NSF), and by the Ministry of Science & Technology, Israel. The work of C.C.W. is supported by NOIRLab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. \nAuthor Contributions. H.A. led the analysis and article writing. L.J.F. and A.Z. constructed the lens model and extracted lensing related quantities. S.F produced figures. I.L. and R.B. are the PIs of the UNCOVER program. R.B. and I.L. designed the observations and reduced the spectra. J.W. and B.W. produced the catalogs used for target selection. P.D. provided simulations to interpret the observational results obtained. V.K. produced line measurements. I.C. estimated survey volumes. D.J.S. ran SED fitting analysis. T.B.M. measured the galaxy sizes. All authors contributed to the manuscript and aided the analysis and interpretation. \nAuthor Information. Correspondence and requests for materials should be addressed to [email protected]. Reprints and permissions information is available at www.nature.com/reprints. The authors declare no competing interests.'} |
2024arXiv240909458Z | Bouncing cosmologies while offering a compelling alternative to inflationary models face challenges from the growth of vector perturbations during the contracting phase. While linear vector instabilities can be avoided with specific initial conditions or the absence of vector degrees of freedom we demonstrate the significant role of secondary vector perturbations generated by nonlinear interactions with scalar fluctuations. Our analysis reveals that in a broad class of singlefield matter bounce scenarios these secondary vector perturbations inevitably get unacceptably large amplitudes provided the curvature fluctuations are consistent with cosmic microwave background observations. This finding underscores the crucial importance of scalarinduced vector perturbations in bouncing cosmology and highlights the need for further investigation into their potential impact on the viability of these models. | 2024-09-01T00:00:00Z | ['2024arXiv240909458Z', '10.48550/arXiv.2409.09458', 'arXiv:2409.09458'] | ['General Relativity and Quantum Cosmology', 'Astrophysics - Cosmology and Nongalactic Astrophysics'] | Constraining matter bounce scenario from scalarinduced vector perturbations | 2,024 | 169 | 0 | ['EPRINT_HTML', 'EPRINT_PDF'] | 0 | https://arxiv.org/pdf/2409.09458.pdf | {'Mian Zhu a Chao Chen b,c, 1': '- a Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, 30348 Krakow, Poland\n- b Department of Physics, School of Science, Jiangsu University of Science and Technology, Zhenjiang, 212003, China\n- c Jockey Club Institute for Advanced Study, The Hong Kong University of Science and Technology, Hong Kong, China \nE-mail: [email protected], [email protected] \nAbstract. Bouncing cosmologies, while offering a compelling alternative to inflationary models, face challenges from the growth of vector perturbations during the contracting phase. While linear vector instabilities can be avoided with specific initial conditions or the absence of vector degrees of freedom, we demonstrate the significant role of secondary vector perturbations generated by non-linear interactions with scalar fluctuations. Our analysis reveals that in a broad class of single-field matter bounce scenarios, these secondary vector perturbations inevitably get unacceptably large amplitudes, provided the curvature fluctuations are consistent with cosmic microwave background observations. This finding underscores the crucial importance of scalar-induced vector perturbations in bouncing cosmology and highlights the need for further investigation into their potential impact on the viability of these models.', '1 Introduction': 'Inflation [1], the standard paradigm of the early-universe cosmology, provides a natural way to explain the formation of large-scale structures (LSS) and the observation of cosmic microwave background (CMB). Nonetheless, inflationary cosmology may suffer from the initial singularity problem [2-5] and the trans-Planckian problem [6-8]. These challenges motivate us to explore alternative early universe scenarios such as the non-singular bouncing cosmology [9-12], where a contraction phase takes place in prior to the expansion phase. While bouncing cosmology offers an intriguing alternative for the early universe, it faces significant challenges. Conceptual issues [13, 14] and its compatibility with CMB observations [15, 16] remain critical concerns. There are also extensive debate surrounding specific problems of bouncing cosmologies [17-36] and proposed solutions [37-50], a comprehensive review of these challenges is available in the reference [51]. \nIn this paper, we highlight another challenge for bouncing cosmology, the overproduction of vector perturbations, a problem overlooked in the community. Early studies [52, 53] demonstrated that linear vector perturbations scale as S i ( k ) ∝ a -2 , leading to its growth that can break down the perturbation theory. Resolving this issue typically requires specific model constructions or finely-tuned initial conditions for vector perturbations. For instance, a single-field bouncing scenario lacks vector degrees of freedom, preventing primordial vector fluctuations from vacuum fluctuations. \nHowever, secondary vector perturbations inevitably arise from non-linear interactions with primordial curvature fluctuations ζ . Those fluctuations cannot be arbitrarily fine-tuned, as the power spectrum of curvature fluctuation P ζ is determined by CMB observations. In Ref. [54], scalar-induced vector perturbations (SIVP) are investigated in specific collapsing universes with theoretical considerations. For the first time, we in this paper connect the power spectrum P ζ to CMB observations, establishing a lower bound for the energy density of SIVP. Specifically, we work in matter bounce scenario [55], a simple-yet-significant bouncing scenario where nearly scale-invariant curvature fluctuation is generated in a matterdominated contraction phase (i.e., the effective equation-of-state parameter is zero). Our results demonstrate that the energy density of SIVP becomes comparable to the background \nenergy density at the end of the matter contraction phase, provided the contraction is driven by a k-essence scalar field. This significant back-reaction poses a serious challenge to the viability of the matter bounce scenario. \nThis paper is organized as follows: Section 2 introduces the theoretical framework. Section 3 presents the calculation of the energy density ratio of SIVP to the background. We conclude in Section 4. Technical details are provided in the appendices. Throughout the paper, we set the Planck mass M p = 1 . A dot denotes derivative with respect to cosmic time t , and a prime denotes differentiation with respect to τ , unless otherwise specified.', '2 Theoretical setup': "We work in a spatially-flat FLRW universe \nd s 2 = -d t 2 + a ( t ) 2 d x i d x i = a ( τ ) 2 ( -d τ 2 +d x i d x i ) , (2.1) \nwhere d τ = d t/a is the conformal time. In matter bounce, the scalar factor scales as a ∝ τ 2 , and can be parameterized as \na ( τ ) = ( τ/τ 0 ) 2 , τ < τ 0 < 0 , (2.2) \nwhere τ 0 labels the end of the contraction phase. It will also be useful to define a comoving Hubble parameter H ≡ a ' /a = 2 /τ < 0 . The background energy density is given by the Friedmann's equation, ρ bg ( τ ) = 3 H 2 = 12 τ 4 0 /τ 6 . In the framework of k-essence theory [56], \nS = ∫ d 4 x √ -g [ R 2 + K ( ϕ, X ) ] , X ≡ -1 2 ∂ µ ϕ∂ µ ϕ , (2.3) \nthe quadratic action for curvature fluctuation ζ is [57-59] \nS (2) ζ = ∫ d τ d 3 x z 2 s 2 [ ζ ' 2 -c 2 s ( ∂ i ζ ) 2 ] , z 2 s = 3 a 2 c 2 s , (2.4) \nwhere we used the fact that the effective slow-roll parameter ϵ ≡ -˙ H/H 2 = 3 / 2 in the matter contraction phase, and we regarded the sound speed for curvature perturbations, c s ≡ K ,X / ( K ,X +2 XK ,XX ) , as a constant for simplicity. Working in the Fourier space with a canonical mode function v k = z s ζ k , the dynamical equation for curvature perturbations becomes \nv '' k + ( c 2 s k 2 -2 τ 2 ) v k = 0 . (2.5) \nImposing the vacuum initial condition, we get the expression for curvature fluctuations as \nζ k ( τ ) ≡ v k ( τ ) z s = e -ikc s τ c s √ 6 c s k ( 1 -i c s kτ ) ( τ 0 τ ) 2 . (2.6) \nIn contrast to the vanilla slow-roll inflation case, the curvature perturbations grow on superhorizon scales | kτ | ≪ 1 in the matter contraction phase (see e.g., Ref. [60]). Hence, one needs to evaluate the curvature power spectrum at the end of the contraction phase: \n⟨ ζ ⃗ k ζ ⃗ p ⟩ ( τ = τ 0 ) = (2 π ) 3 δ ( ⃗ k + ⃗ p ) | ζ ⃗ k | 2 = (2 π ) 3 δ ( ⃗ k + ⃗ p ) c s 6 k ( 1 + 1 k 2 c 2 s τ 2 0 ) . (2.7) \nFrom the definition of the scalar power spectrum, \n⟨ ζ ⃗ k ζ ⃗ p ⟩ = (2 π ) 3 δ ( ⃗ k + ⃗ p ) 2 π 2 k 3 P ζ ( k ) , (2.8) \nwe derive, \nP ζ ( k, τ 0 ) = k 2 c s 12 π 2 ( 1 + 1 k 2 c 2 s τ 2 0 ) ≃ 1 12 π 2 c s τ 2 0 , (2.9) \nwhich is scale-invariant. \nIn the FLRW universe, the most general perturbed metric, including only vector perturbation, is given by [61] \nd s 2 = a 2 ( τ ) [ -d τ 2 -2 G i d τ d x i +( δ ij + F ij )d x i d x j ] , (2.10) \nwhere F ij satisfies F ij = ∂ i F j + ∂ j F i and ∂ i F i = 0 , and G i is divergent free, ∂ i G i = 0 . Since there is no vector degree of freedom in our setup, F i and G i should be regarded as secondorder fluctuations induced by the first-order perturbations. We work in the Newtonian gauge where F i = 0 , and the metric involving scalar and vector perturbations can be written as \nd s 2 = a 2 ( τ ) [ -e 2Φ d τ 2 -2 G i d τ d x i + e -2Φ δ ij d x i d x j ] , (2.11) \nwhere the scalar perturbation Φ is related to the curvature fluctuation via [62, 63] \nζ = Φ + H H 2 -H ' (Φ ' + H Φ) . (2.12) \nThe vector power spectrum, defined as \n⟨ G λ ( ⃗ k ) G s ( ⃗ p ) ⟩ ≡ (2 π ) 3 δ ( ⃗ k + ⃗ p ) δ λs 2 π 2 k 3 P G ( τ, ⃗ k ) . (2.13) \nAs discussed above, G i can be sourced by Φ (or equivalently ζ via Eq. (2.12)) through their nonlinear coupling, its power spectrum is computed as \nP G ( τ, k ) = ∫ ∞ 1 √ 2 d t ∫ 1 √ 2 -1 √ 2 d s (1 -2 s 2 )(2 t 2 -1)(2 st +1) 2 4( t 2 -s 2 ) 2 ×P ζ ( k √ 2 ( t -s ) , τ ) P ζ ( k √ 2 ( t + s ) , τ ) |I ( t, s, z ) | 2 , (2.14) \nwhere z ≡ kτ . We summarize the computational details in App. A.", '3 Energy density of SIVP': "The energy density of SIVP is given by [64] \nρ V ( τ, ⃗x ) = 1 4 a 2 ∂ i G j ( τ, ⃗x ) ∂ i G j ( τ, ⃗x ) . (3.1) \nFrom Eq. (3.1), the energy density of SIVP is related to P G as \nρ V ( τ ) = 1 2 a 2 ∫ d kk P G ( τ, ⃗ k ) . (3.2) \nThe back-reaction is represented by the ratio of the energy density of SIVP against the background one at τ = τ 0 : \nδ V ≡ ρ V ( τ 0 ) ρ bg ( τ 0 ) = 1 24 ∫ ( kτ 0 ) P G ( k, τ 0 )d( kτ 0 ) . (3.3) \nThe P ζ on super-horizon scales is associated to CMB observations: \nP ζ ( ⃗ k, τ 0 ) ≃ 1 12 π 2 c s τ 2 0 = A s , (3.4) \nwhere A s = 2 . 1 × 10 -9 from Planck collaboration [65]. We denote the scales of fluctuations that 'cross the horizon' at the beginning/end of contraction phase to be k min and k max : \nk min = c -1 s |H ( τ ini ) | = -2( c s τ ini ) -1 , k max = c -1 s |H ( τ 0 ) | = -2( c s τ 0 ) -1 , (3.5) \nwhere τ ini labels the initial time of matter contraction phase. The modes with k min < k < k max becomes super-horizon during the matter contraction phase, and we adopt a minimal curvature power spectrum for super-horizon perturbations \nP ζ = { A s , k min < k < k max , 0 , otherwise . (3.6) \nIt is possible that modes with k < k min or k > k max become super-horizon before or after the matter contraction phase, hence also give a positive contribution to SIVP. Adopting the ansatz in Eq. (3.6) captures the dominant contribution to SIVP from modes entering the horizon during the contraction phase, providing a lower bound sufficient to analyze the instability. \nThe scale of scale-invariant curvature fluctuation indicated by CMB observations ranges from k C /a today ≃ 10 -4 Mpc -1 to k L /a today ≃ 1 Mpc -1 . To match CMB data, modes with k = k C must be well within the horizon ( k C ≫ k min ), while k L must be super-horizon at τ = τ 0 ( k L ≤ k max ). Introducing a dimensionless scaling factor \ng ≡ τ ini τ 0 = k max k min , (3.7) \nand g must be larger than 10 4 . \nFigure 1 . Left: The SIVP power spectra (2.14) at the end of matter contraction τ 0 as functions of c s with g = 10 4 ; Right: The SIVP power spectra (2.14) as functions of g with c s = 1 . \n<!-- image --> \nThe resulting vector power spectrum is determined by the dimensionless parameters c s and g , as τ 0 is fixed by c s through Eq. (3.4). We numerically evaluate P G and present the result in Fig. 1. In App. B, we demonstrate that P G scales as A 2 s c -2 s g 4 log g . Consequently, \nδ V ∝ A 2 s c 4 s g 4 (log g + O (1)) , (3.8) \nfrom our numerical results. We further confirm this scaling through analytical estimations presented in App. C. As a comparison, the energy density of linear curvature perturbations scales as g 2 assuming a scale-invariant P ζ . As a result, the back-reaction problem associated with SIVP can be more severe than that of curvature fluctuations due to its g 4 log g scaling. \nThe value of δ V is presented in Table 1, revealing that δ V is less than unity only when c s ≃ 1 and g ≃ 10 4 . In all other cases, δ V > 1 , indicating either a significant back-reaction on the background evolution or a breakdown of perturbation theory. Considering the consistency relation in the context of k-essence theory during matter contraction, we have r = 24 c s [33], where r is the tensor-to-scalar ratio constrained by r 0 . 002 < 0 . 044 [66]. This constraint implies c s < 0 . 02 , inevitably leading to an excessively large δ V . Even if we artificially set c s = 1 , the parameter g is significantly larger than 10 4 in reality because k min ≫ k C . Combining these arguments, we conclude that cosmological models where nearly scale-invariant curvature fluctuations on CMB and LSS scales originate from a matter contraction phase governed by a minimally coupled k-essence field are constrained by the overproduction of SIVP, rendering such models invalid. \nTable 1 . The values of δ V in Eq. (3.3) for various values of c s and g .", '4 Conclusion': 'Vector fluctuations play a crucial role in bouncing cosmologies, particularly during the contraction phase. This study, for the first time, combines the concept of secondary vector fluctuations induced by scalar fluctuations with observational constraints on curvature perturbations, revealing an overproduction of these modes in a matter contraction phase governed by a k-essence scalar field. This finding highlights the importance of vector modes in bouncing cosmologies and motivates further investigation into their impact on various bouncing scenarios, including matter bounces with more complex actions [47], Ekpyrotic scenarios [37], and scenarios where the bouncing phase significantly influences the evolution of curvature fluctuations [67]. \nOur findings motivate further exploration of perturbation theory within the context of bouncing cosmology. Additionally, our results could be revisited by replacing the cut-off of \nthe curvature power spectrum with regularized primordial fluctuations. This approach could, in principle, yield more accurate results. However, this area currently lacks sufficient research. Additionally, the growth of anisotropic shear during the contraction phase, whose energy density scales as ρ ∝ a -6 , warrants investigation into scalar-induced shear, which could provide additional theoretical constraints on bouncing models. The secondary vector fluctuations in bouncing cosmology in the presence of vector field(s) deserves future investigation due to its potential connection with topics such as primordial magnetogenesis [68].', 'Acknowledgement': 'M.Z. thanks Yi Wang for discussions which inspired this work, and Xian Gao, Chunshan Lin and George Zahariade for their critical comments. M.Z. also thanks the correspondence with Yi-Fu Cai and Sabino Matarrese for the discussions on several technical problems. We are grateful to Atsuhisa Ota for his contributions during the initial stage of this work. M.Z. was supported by grant No. UMO 2021/42/E/ST9/00260 from the National Science Centre, Poland. C.C. thanks the support from the Jockey Club Institute for Advanced Study at The Hong Kong University of Science and Technology. C.C. is supported by NSFC (Grants No. 12433002).', 'A Scalar-induced vector perturbations': "This appendix derives the second-order vector fluctuations by solving the second-order Einstein equations, following a similar approach used in the study of scalar-induced gravitational waves (SIGW) (see, for example, the review [69]). Throughout this section, superscripts will be used to denote the order of perturbation. For instance, ρ (2) represents the perturbed energy density at second order. The second-order vector G (2) i is determined by the ij components of the Einstein equations, \n̸ \nG (2) j i = T (2) j i , i = j . (A.1) \nAlthough it is possible to derive the secondary vector fluctuations using momentum constraints (as there are no vector degrees of freedom in our specific scenario), the method of computing with Einstein equations will prove useful in future studies involving vector fields. Examples include primordial magnetogenesis [68] and baryon asymmetry [70]. For detailed explanations of both methods and their equivalence, see Refs. [71, 72]. \nThe computation of geometric quantities based on the metric perturbations in Eq. (2.11) is straightforward. We present some useful expressions below (utilizing the identity H ' = -H 2 / 2 , applicable in the matter bounce scenario, and the notation ∂ 2 ≡ ∂ i ∂ i ), \nR = -2 e 2Φ a 2 [ ( ∂ Φ) 2 -2 ∂ 2 Φ ] , G (0)0 0 = -3 H 2 a 2 , (A.2) \nG (2) j i = 1 2 a 4 d d τ [ a 2 ( ∂ i G j + ∂ j G i )] -1 a 2 [ 2 ∂ i Φ ∂ j Φ + δ j i (4ΦΦ '' +5Φ ' 2 -( ∂ l Φ) 2 +12 H ΦΦ ' ) ] , (A.3) \nwhere R is the intrinsic curvature defined for a constant-time hypersurfece associated with the metric (2.11). \nFor the matter sector, we obtain \nT ν µ = ( ρ + P ) u µ u ν + Pδ ν µ +Σ ν µ , (A.4) \n̸ \nwhere ρ and P are the energy density and the pressure, respectively; u µ is the four-velocity of the observer; Σ µν is the anisotropic stress subject to the conditions Σ 00 = Σ 0 i = 0 , Σ ij = Σ ji and δ ij Σ ij = 0 . For simplicity, we will set the anisotropic stress to be zero, and leave the study of Σ ij = 0 case in the future work. From (A.4) we get \nT (2) j i = ( ρ (0) + P (0) ) u (1) i u (1) j + P (2) δ j i . (A.5) \nThe four-velocity by definition is normalized according to u µ u ν g µν = -1 . Along with the definition u µ = g µν u ν , one arrive \nu (0)0 = ( -g (0) 00 ) -1 / 2 , u (1)0 = 1 2 ( -g (0) 00 ) -3 / 2 g (1) 00 , (A.6) \nand accordingly \nu (1) i = -2 a 3 H 2 ( ∂ i Φ ' + H ∂ i Φ) , u (1) i = -2 3 a H 2 ( ∂ i Φ ' + H ∂ i Φ) . (A.7) \nThe rest quantities are to be determined by the perturbed Einstein equations G ( n ) µν = T ( n ) µν . For instance, \nρ (0) = -T (0)0 0 = -G (0)0 0 = 3 H 2 a 2 , (A.8) \nP (2) = 1 3 T (2) i i -1 3 ( ρ (0) + P (0) ) u (1) i u (1) i = 1 3 G (2) i i -1 3 ( ρ (0) + P (0) ) u (1) i u (1) i . (A.9) \nWe organize some useful expressions as below, \nρ (0) = 3 H 2 a 2 , P (0) = 0 , (A.10) \nP (2) = -1 9 a 2 H 2 [ 4( ∂ i Φ ' ) 2 + H 2 ( ∂ i Φ) 2 +8 H ∂ i Φ ∂ i Φ ' +45 H 2 Φ ' 2 ] . (A.11) \nFrom Eqs. (A.5), (A.3), (A.10), and (A.11), we obtain the following equation, \n∂ i G j ' +2 H ∂ i G j -4 ∂ i Φ ' ∂ j Φ ' +8 H ∂ i Φ ' ∂ j Φ+10 H 2 ∂ i Φ ∂ j Φ 3 H 2 +( i ←→ j ) = 0 . (A.12) \nNow we are about to derive the equation for SIVP. The curvature fluctuation is related to the Φ through the relation (2.12). While directly converting Φ to ζ using Eq. (2.12) is challenging, we can leverage the fact that curvature fluctuations grow on super-horizon scales. By adopting the ansatz in Eq. (3.6), we focus on the contributions from modes that are super-horizon at the end of the contraction phase. This allows us to utilize the simplified relationship Φ = 3 2 ζ , derived by combining Eqs. (2.12) and (2.6) in the super-horizon regime. We will use this relationship to convert Φ to the curvature fluctuation ζ in the following calculations. \nIn order to extract the transverse vector modes, we define a projection vector [73], \nV kl i ≡ 1 ∇ 2 ∂ l T k i ≡ 1 ∇ 2 ∂ l ( δ k i -∂ k ∂ i ∇ 2 ) , (A.13) \nwhich is able to project a term S kl into a transverse vector G i such that \nG i = V kl i S kl , (A.14) \nand it is helpful to list the following relationships, \nV kl i δ kl = 0 , V kl i ∂ k ∂ l Φ = 0 , V kl i ∂ k G l = 0 , V kl i ∂ l G k = G i . (A.15) \nWe then get the equation of G i as \nG ' i +2 H G i = V kl i S kl , (A.16) \nwhere the source term is \nS kl ≡ 15 ∂ k ζ∂ l ζ + 6( ∂ k ζ ' ∂ l ζ + ∂ k ζ∂ l ζ ' ) H + 6 ∂ k ζ ' ∂ l ζ ' H 2 . (A.17) \nAs a final step, we move to the Fourier space. We choose a pair of polarization vector { e ( ˆ k ) , ¯ e ( ˆ k ) } , which are orthogonal to each other and ⃗ k , satisfying: \ne λ i ( ˆ k ) e σ,i ( ˆ k ) = δ λσ , e λ i ( ˆ k ) k i = 0 , ∑ λ e λ,i ( ˆ k ) e λ,j ( ˆ k ) = δ ij -k i k j k 2 . (A.18) \nThe vector perturbation becomes \nG i ( τ, ⃗x ) = ∑ λ ∫ d 3 ⃗ k (2 π ) 3 e i ⃗ k · ⃗x G λ ( τ, ⃗ k ) e λ i ( ˆ k ) , (A.19) \nand we have \nV ab i S ab ( τ, ⃗x ) = ∫ d 3 ⃗ k (2 π ) 3 ik b k 2 ( δ a i -k a k i k 2 ) e i ⃗ k · ⃗x S ab ( τ, ⃗ k ) , = ∑ λ ∫ d 3 ⃗ k (2 π ) 3 ik b k 2 e λ i ( ˆ k ) e λ,a ( ˆ k ) e i ⃗ k · ⃗x S ab ( τ, ⃗ k ) , (A.20) \nwhere S ab ( τ, ⃗ k ) is the Fourier transform of S ab ( τ, ⃗x ) . Hence the equation for vector mode becomes \nG λ ' ( τ, ⃗ k ) + 2 H G λ ( τ, ⃗ k ) = S λ ( τ, ⃗ k ) , (A.21) \nwhere \nS λ ( τ, ⃗ k ) = ik m k 2 e λ,n ( ˆ k ) S nm ( τ, ⃗ k ) = -∫ d 3 ⃗ p (2 π ) 3 ik m k 2 e λ,n ( ˆ k ) p n p m f ( ⃗ p, ⃗ k, τ ) ζ ⃗ p ( τ 0 ) ζ ⃗ k -⃗ p ( τ 0 ) , (A.22) \nwith the evolution kernel, \nf ( ⃗ p, ⃗ k, τ ) = 15 T ( pτ ) T ( | ⃗ k -⃗ p | τ ) + 6 H 2 T ' ( pτ ) T ' ( | ⃗ k -⃗ p | τ ) + 6 H [ T ' ( pτ ) T ( | ⃗ k -⃗ p | τ ) + T ( pτ ) T ' ( | ⃗ k -⃗ p | τ )] . (A.23) \nNote that the prime in (A.23) denotes derivative with respect to the argument instead of τ . Here, T is the transfer function defined as \nζ ⃗ k ( τ ) = T ( kτ ) ζ ⃗ k ( τ 0 ) , τ < τ 0 < 0 . (A.24) \nWith the help of Eq. (2.6), we have \nT ( kτ ) = τ 3 0 τ 3 e ic s k ( τ 0 -τ ) i -c s kτ i -c s kτ 0 . (A.25) \nNow we are about to evaluate the two-point correlation function of G λ , defined as \n⟨ G λ ( ⃗ k ) G s ( ⃗ p ) ⟩ ≡ (2 π ) 3 δ ( ⃗ k + ⃗ p ) δ λs 2 π 2 k 3 P G ( τ, ⃗ k ) . (A.26) \nThe general solution of G i is given by \nG λ ( τ, ⃗ k ) = 1 a ( τ ) 2 ∫ τ a (˜ τ ) 2 S λ (˜ τ, ⃗ k )d˜ τ . (A.27) \nApplying the bouncing background (2.2) and specifying the integration range, we have \nG λ ( τ, ⃗ k ) = ∫ τ τ ini d˜ τ ( ˜ τ τ ) 4 S λ (˜ τ, ⃗ k ) , (A.28) \nwhere τ ini is the conformal time at the beginning of the contraction phase. The two-point correlation function of G λ becomes \n⟨ G λ ( τ, ⃗ k ) G s ( τ, ⃗ k ' ) ⟩ = ∫ τ τ ini d˜ τ 1 ∫ τ τ ini d ˜ τ 2 ( ˜ τ 1 τ ) 4 ( ˜ τ 2 τ ) 4 ⟨ S λ (˜ τ 1 , ⃗ k ) S s (˜ τ 2 , ⃗ k ' ) ⟩ = -1 k 2 k ' 2 ∫ τ τ ini d˜ τ 1 ∫ τ τ ini d˜ τ 2 ( ˜ τ 1 τ ) 4 ( ˜ τ 2 τ ) 4 ∫ d 3 ⃗ p (2 π ) 3 ⟨ ζ ⃗ p ζ ⃗ k -⃗ p ζ ⃗ q ζ ⃗ k ' -⃗ q ⟩ × k m p m p n k ' i q i q j e λ,n ( ˆ k ) e s,j ( ˆ k ' ) f ∗ ( ⃗ p, ⃗ k, ˜ τ 1 ) f ( ⃗q, ⃗ k ' , ˜ τ 2 ) . (A.29) \nAssuming a Gaussian distribution of curvature fluctuation and with the help of Eq. (2.8), the contraction of the four-point correlator is decomposed as \n⟨ ζ ⃗ p ζ ⃗ k -⃗ p ζ ⃗ q ζ ⃗ k ' -⃗ q ⟩ = ⟨ ζ ⃗ p ζ ⃗ q ⟩⟨ ζ ⃗ k -⃗ p ζ ⃗ k ' -⃗ q ⟩ + ⟨ ζ ⃗ p ζ ⃗ k ' -⃗ q ⟩⟨ ζ ⃗ k -⃗ p ζ ⃗ q ⟩ = (2 π ) 6 2 π 2 p 3 2 π 2 | ⃗ k -⃗ p | 3 δ ( ⃗ k + ⃗ k ' ) [ δ ( ⃗ p + ⃗q ) + δ ( ⃗q + ⃗ k -⃗ p ) ] P ζ ( ⃗ p ) P ζ ( ⃗ k -⃗ p ) . (A.30) \nAdopting the coordinates of two polarization vectors and ⃗ k as \ne ( ˆ k ) = (1 , 0 , 0) , ¯ e ( ˆ k ) = (0 , 1 , 0) , ⃗ k = (0 , 0 , k ) , ⃗ p = p (sin θ cos ψ, sin θ sin ψ, cos θ ) , (A.31) one can simplify the expressions as \n⟨ G λ ( τ, ⃗ k ) G s ( τ, ⃗ k ' ) ⟩ = 8 π 5 k 3 δ λs δ ( ⃗ k + ⃗ k ' ) ∫ ∞ 0 d y ∫ 1+ y | 1 -y | d x y 2 x 2 [ 1 -( 1 + y 2 -x 2 2 y ) 2 ] × ( 1 + y 2 -x 2 2 y ) 2 P ζ ( ky ) P ζ ( kx ) ∣ ∣ ∣ ∣ ∫ z z ini d˜ z ˜ z 4 z 4 f ( x, y, ˜ z ) ∣ ∣ ∣ ∣ 2 , (A.32) \nwith the introduction of auxiliary variables \nx ≡ | ⃗ k -⃗ p | k , y ≡ p k ; z ≡ kτ < 0 , z 0 ≡ kτ 0 < 0 . (A.33) \nFollowing the convention in the study of induced gravitational waves, we further define \ns = y -x √ 2 , t = y + x √ 2 , (A.34) \nand the total vector power spectrum is calculated as \nP G ( τ, ⃗ k ) = ∫ ∞ 1 √ 2 d t ∫ 1 √ 2 -1 √ 2 d s (1 -2 s 2 )(2 t 2 -1)(2 st +1) 2 4( t 2 -s 2 ) 2 ×P ζ ( k √ 2 ( t -s ) , τ ) P ζ ( k √ 2 ( t + s ) , τ ) |I ( s, t, z ) | 2 , (A.35) \nwhere \nI ( s, t, z ) = ∫ z z ini d˜ z ˜ z 4 z 4 f ( s, t, ˜ z ) , (A.36) \nis the time integral, z ini corresponds to the value of z at far past, i.e., z ini = kτ ini . Explicitly, we have \nf ( s, t, ˜ z ) ≡ 3 z 6 0 e -ic s ( √ 2 t ˜ z -2 z 0 ) ( t 2 -s 2 ) 3 ˜ z 6 ( c s z 0 -i ) 2 [ 2 √ 2 ic s t ˜ z (14 -c 2 s ˜ z 2 ( t 2 -s 2 )) +28 -2 c 2 s (9 t 2 -5 s 2 )˜ z 2 + c 4 s ˜ z 4 ( t 2 -s 2 ) 2 ] . (A.37) \nThe time integral (A.36) can be integrated out analytically 1 , \nF (˜ z ) ≡ ∫ d˜ z ˜ z 4 z 4 f ( s, t, ˜ z ) = 3 z 6 0 e -i ( √ 2 c s t ˜ z -2 z 0 ) √ 2˜ zt 3 z 4 ( t 2 -s 2 ) 3 (1 + ic s z 0 ) 2 [ -i ˜ z 3 c 3 s t 2 ( t 2 -s 2 ) 2 + ic s ˜ z ( s 4 -14 s 2 t 2 +21 t 4 ) -√ 2 c 2 s t ˜ z 2 ( s 4 -4 s 2 t 2 +3 t 4 ) + 28 √ 2 t 3 ] . (A.38) \nNotice that the specific time scale τ 0 appears as we set a reference time scale τ 0 in the definition of transfer function (A.24). Introducing g ≡ z ini /z 0 to label the duration of contraction phase and an auxiliary variable u ≡ c s z 0 , we have \n|I ( z 0 ) | 2 = 9 u 2 (1 + u 2 ) -2 2 c 2 s t 6 ( t 2 -s 2 ) 6 [ I 1 + I 2 cos ( √ 2(1 -g ) tu ) + I 3 sin ( √ 2(1 -g ) tu )] , (A.39) \nwhere \nI 1 = ( g 4 +1 ) ( t 2 -s 2 ) 4 t 4 u 6 +8 ( g 2 +1 ) ( 2 s 2 -3 t 2 ) ( s 2 -t 2 ) 2 t 4 u 4 +2 ( s 8 -28 s 6 t 2 +126 s 4 t 4 -140 s 2 t 6 +105 t 8 ) u 2 +1568 ( 1 + g -2 ) t 6 , (A.40) \nI 2 = 2 g 2 t 2 u 4 ( s 2 -t 2 ) 2 [ s 4 -t 2 u 2 ( s 2 -t 2 ) 2 -14 s 2 t 2 +21 t 4 ] -4 gt 2 u 2 ( s 4 -4 s 2 t 2 +3 t 4 ) [ u 2 ( s 4 -4 s 2 t 2 +3 t 4 ) -28 t 2 ] +2 u 2 ( s 4 -14 s 2 t 2 +21 t 4 ) [ -s 4 + t 2 u 2 ( t 2 -s 2 ) 2 +14 s 2 t 2 -21 t 4 ] +112 g -1 t 4 [ u 2 ( s 4 -4 s 2 t 2 +3 t 4 ) -28 t 2 ] , (A.41) \nI 3 = 2 √ 2 g 2 t 3 u 3 ( s 2 -t 2 ) 2 [ u 2 ( s 4 -4 s 2 t 2 +3 t 4 ) -28 t 2 ] -2 √ 2 gtu 3 ( s 4 -4 s 2 t 2 +3 t 4 ) [ -s 4 + t 2 u 2 ( t 2 -s 2 ) 2 +14 s 2 t 2 -21 t 4 ] -2 √ 2 tu ( s 4 -14 s 2 t 2 +21 t 4 ) [ u 2 ( s 4 -4 s 2 t 2 +3 t 4 ) -28 t 2 ] +56 √ 2 g -1 tu [ t 4 u 2 ( s 2 -t 2 ) 2 -t 2 ( s 4 -14 s 2 t 2 +21 t 4 ) ] . (A.42)", 'B Vector Power Spectrum from Numerical Evaluation': "We're interested in the energy density of the induced vector perturbation, \nρ V ( ⃗x, τ ) = 1 4 a 2 ∂ i G j ( ⃗x, τ ) ∂ i G j ( ⃗x, τ ) , (B.1) \nwhich is related to the vector power spectrum as \nρ V ( τ ) = 2 × 1 4 a 2 ∫ k P G ( τ, ⃗ k )d k = 1 2 a 2 ∫ k P G ( τ, ⃗ k )d k . (B.2) \nThe factor of 2 arises from the two polarizations of vector perturbations. The vector energy density, ρ V , is directly determined by the vector power spectrum, P G . Therefore, we will focus on evaluating the vector power spectrum numerically in the following. \nFirst, Fig. 2 demonstrates that the oscillatory terms in Eq. (A.39), namely the I 2 and I 3 terms, contribute significantly on larger scales. Conversely, on scales k ≃ k max , the value of Eq. (A.39) is primarily determined by the non-oscillatory term I 1 . In both cases, the power spectrum rapidly diminishes on scales k ≥ k max due to the cutoff of the curvature power spectrum in Equation (3.4). On scales k < k max , P G exhibits a strong blue tilt, which can be parameterized as \nP G ( τ 0 , k ) = A G ( k k max ) n G , k < k max , (B.3) \nfrom which we obtain the ratio of ρ V versus the background energy density value ρ bg , \nδ V ≡ ρ V ( τ 0 ) ρ bg ( τ 0 ) = 1 24 ∫ ( kτ 0 ) P G ( k, τ 0 )d( kτ 0 ) , (B.4) \nFigure 2 . Comparison of the vector power spectrum P G calculated using Eq. (A.39) (including oscillatory terms) with P G evaluated using only the non-oscillatory term I 1 . The parameters used are c s = 1 and g = 10 4 . \n<!-- image --> \nas \nδ V ( τ 0 ) = 1 24 ∫ ( kτ 0 ) P G ( k, τ 0 )d( kτ 0 ) < 1 6 c 2 s ∫ 1 0 A G ( k k max ) n G +1 d ( k k max ) = A G 6 c 2 s ( n G +2) . (B.5) \nAs Fig. 2 illustrates, the non-oscillatory terms primarily determine the amplitude A G of the vector power spectrum, while the oscillatory terms are responsible for its spectral index n G . However, neglecting the oscillatory terms leads to a difference in the calculated n G by a factor of order unity. For example, with c s = 1 and g = 10 4 , we find n G = 8 . 2 when excluding oscillatory terms and n G = 6 . 0 when using the full expression in Equation (A.39). This discrepancy can be understood as follows: When k ≃ k max , we have u ≃ -2 and g ≫ 1 , making I 1 the dominant term since I 1 = O ( g 4 ) while I 2 and I 3 are of order O ( g 2 ) . Conversely, on larger scales where k ≃ k min , we have u ≃ -2 g -1 , resulting in I 1 = O ( g 0 ) and I 2 , I 3 = O ( g -1 ) . In this case, the oscillatory terms become significant. For simplicity, we will proceed with the numerical calculations by evaluating only the contributions from the non-oscillatory term. As we will demonstrate later, the resulting δ V is significantly larger than unity. Therefore, any potential loss of a factor of order unity in this numerical approximation will not significantly impact our conclusions. \nFigure 1 displays the resulting vector power spectrum P G ( τ 0 , k ) for various values of c s and g . The figure reveals that the amplitude A G is proportional to c -2 s g 4 . More precisely, as we will demonstrate below, A G exhibits the following behavior for g ≫ 1 : \nA G ≃ A 2 s c 2 s g 4 ( C 1 log g + C 2 ) + O ( g 3 ) . (B.6) \nThe value of A G is shown in Fig. 3. A direct fit of the numerical result gives C 1 = 15 . 45 and C 2 = -14 . 20 . App. C provides a semi-analytical explanation for Eq. (B.6). \nNow we conclude the vector power spectrum for different model parameters in Table 2. The log g dependence in Eq. (B.6) is implicitly reflected in the running of the spectral indices. \nFigure 3 . The value of A G as a function of g with c s = 1 fixed. \n<!-- image --> \nTable 2 . Vector power spectra (B.3) and the values of δ V for various values of c s and g . \nAs argued in Sec. 3, g ≫ 10 4 , making the log g term in Eq. (B.6) dominant. Therefore, we arrive at the scaling, \nA G ∝ A 2 s c 2 s g 4 log g , δ V ∝ A 2 s c 4 s ( n G +1) g 4 log g . (B.7) \nAs a comparison, the energy density of linear curvature perturbations is estimated as [64] \nρ ζ ≡ -1 2 R→ ρ ζ ∝ ∫ k max k min dln k k 2 a 2 P ζ , (B.8) \nusing Eq. (A.2). Notice that ρ ζ scales as g 2 assuming a scale-invariant curvature power spectrum P ζ . Thus, the energy density of SIVP exhibits a faster growth rate with respect to g , leading to a more severe back-reaction problem during the contraction phase.", 'C Analysis on the SIVP Power Spectrum': 'In this appendix, we conduct a semi-analytical study on the vector power spectrum. Using the identity \n| a 2 | -2 | ab | + | b 2 | ≤ | a -b | 2 ≤ | a 2 | +2 | ab | + | b 2 | , (C.1) \nwe get \nJ 1 -J 2 + J 3 ≤ |F ( z 0 ) -F ( gz 0 ) | 2 ≤ J 1 + J 2 + J 3 , (C.2) \nFigure 4 . The integration range of momentum integrals. We take g = 4 here for illustrative purposes. \n<!-- image --> \nwith \nJ 1 ≡ |F ( z 0 ) 2 | , J 2 ≡ 2 |F ( z 0 ) F ( gz 0 ) | , J 3 ≡ |F ( gz 0 ) 2 | . (C.3) \nOur primary interest lies in determining the value of A G , which corresponds to the value of P G ( k, τ 0 ) at k = k max . For the remainder of this discussion, we will fix k to be equal to k max . The integration range relevant to the power spectrum is defined by the following identity: \nk min ≤ k 1 , k 2 ≤ k max ; | k 1 -k 2 | ≤ k max , k 1 + k 2 ≥ k max , (C.4) \nwhich we depict in Fig. 4. In terms of variable s and t , it becomes \n∫ √ 2 g +1 √ 2 g d t ∫ √ 2 -t t -√ 2 d s + ∫ g +1 √ 2 g 1 √ 2 d t ∫ t -√ 2 g √ 2 g -t d s . (C.5) \nOne can straightforwardly calculate the integrations. However, the resulting formulae is tediously long and we will use a simpler strategy. We observe that contributions to the integrals mainly comes from the boundary near the points t = 1 √ 2 , s = ± 1 √ 2 , where we can expand the integrand as a series of ( t -1 √ 2 ) , which greatly simplify the calculations. Further, we will expand the results around g →∞ , which gives \n∫ d s d t (1 -2 s 2 )(2 t 2 -1)(2 st +1) 2 4( t 2 -s 2 ) 2 J 1 ≃ 27 g 4 z 2 0 (49 + c 2 s z 2 0 ) 56(1 + c 2 s z 2 0 ) , (C.6) \n∫ d s d t (1 -2 s 2 )(2 t 2 -1)(2 st +1) 2 4( t 2 -s 2 ) 2 J 3 ≃ 3 g 4 c 2 s z 4 0 (45 -84 c 2 s z 2 0 +140 c 4 s z 4 0 ln g ) 280(1 + c 2 s z 2 0 ) 2 , (C.7) \nup to O ( g 3 ) contributions. We are interested in the value of P G at k = k max , that is z 0 c s = -2 and \n∫ d s d t (1 -2 s 2 )(2 t 2 -1)(2 st +1) 2 4( t 2 -s 2 ) 2 P 2 ζ J 1 ( k = k max ) ≃ 1431 g 4 A 2 s 350 c 2 s , (C.8) \n∫ d s d t (1 -2 s 2 )(2 t 2 -1)(2 st +1) 2 4( t 2 -s 2 ) 2 P 2 ζ J 3 ( k = k max ) ≃ 6 g 4 A 2 s 875 c 2 s (2240 ln g -291) . (C.9) \nThe momentum integral for J 2 is a bit tricky, since it contains absolute values. For our purpose, we notice that near the pole, the real part of F ( z 0 ) is always no less than its imaginary part. Thus we take |F ( z 0 ) | ≤ √ 2 ℜ ( F ( z 0 )) . On the other hand, we have F ( gz 0 ) to be governed by its g 2 term in the g →∞ limit. Thus we arrive at \n∫ d s d t (1 -2 s 2 )(2 t 2 -1)(2 st +1) 2 4( t 2 -s 2 ) 2 P 2 ζ J 2 ( k = k max ) ≃ 4032 g 4 A 2 s 125 c 2 s . (C.10) \nCombining all pieces, we get \n15 . 36 log g -30 . 16 ≤ A G c 2 s A 2 s g 4 ≤ 15 . 36 log g +34 . 35 . 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2024arXiv240900540K | The galaxy M87 is one of the prime targets for high resolution radio imaging pursuing the ringlike shadow of its supermassive black hole the innermost regions of accretion flow and the formation of the relativistic jet. However it remains challenging to observe both jointly. Only recently global mmVLBI array GMVAALMA observations at 86 GHz in 2018 were able to reconstruct the M87 black hole shadow and the extended jet emission simultaneously. In order to analyze the ring and jet of M87 conventional CLEAN algorithms were mainly employed alongside the RML method SMILI in the previous work. To test the robustness of the reconstructed structures of M87 GMVAALMA observations at 86GHz we estimate the ring diameter width and the extended jet emission with the possible central spine by two different novel imaging algorithms resolve and DoGHiT. Overall reconstructions are consistent with the results reported in the previous paper. The ring structure of the M87 is resolved at higher resolution and the posterior distribution of M87 ring features is explored. The resolve images show that the ring diameter is 60.9 2.2 muas and width is 16.0 0.9 muas. The ring diameter is 61.0 muas and width is 20.6 muas by DoGHiT. The ring diameter is therefore in agreement with the estimation 6448 muas by SMILI and the geometrical modeling. Two bright spots in the ring are reconstructed by four independent imaging methods the substructure in the ring is therefore most likely originated from the data. A consistent limbbrightened jet structure is reconstructed by resolve and DoGHiT albeit with a less pronounced central spine. Modern datadriven imaging methods confirm the ring and jet structure in M87 complementing traditional VLBI methods with novel perspectives on the significance of recovered features. They confirm the result of the previous report. | 2024-08-01T00:00:00Z | ['10.48550/arXiv.2409.00540', 'arXiv:2409.00540', '2024arXiv240900540K'] | ['Astrophysics - High Energy Astrophysical Phenomena', 'Astrophysics - Instrumentation and Methods for Astrophysics'] | Imaging the black hole shadow and extended jet of M87 | 2,024 | 169 | 0.6 | ['EPRINT_HTML', 'EPRINT_PDF'] | 1 | https://arxiv.org/pdf/2409.00540.pdf | {'Imaging the black hole shadow and extended jet of M87': "Jong-Seo Kim 1, ⋆ , Hendrik Müller 1 , 2 , Aleksei S. Nikonov 1 , Ru-Sen Lu 1 , 3 , 4 , Jakob Knollmüller 5 , Torsten A. Enßlin 6 , 7 , Maciek Wielgus 1 , and Andrei P. Lobanov 1 , 8 \n- 1 Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany\n- 2 Jansky Fellow of National Radio Astronomy Observatory, 1011 Lopezville Rd, Socorro, NM 87801, USA\n- 3 Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai, People's Republic of China\n- 4 Key Laboratory of Radio Astronomy, Chinese Academy of Sciences, Nanjing, People's Republic of China\n- 5 Radboud University, Heyendaalseweg 135, 6525 AJ, Nijmegen, The Netherlands\n- 6 Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 85748 Garching, Germany\n- 7 Ludwig-Maximilians-Universität, Geschwister-Scholl-Platz 1, 80539 Munich, Germany\n- 8 Institut für Experimentalphysik, Universität Hamburg, Luruper Chaussee 149, 22761, Hamburg, Germany \nReceived August 29, 2024", 'ABSTRACT': 'Context. The galaxy M87 is one of the prime targets for high resolution radio imaging pursuing the ring-like "shadow" of its supermassive black hole, the innermost regions of accretion flow, and the formation of the relativistic jet. However, despite the physical process of ring-like structure and jet are closely interconnected, and this connection may unravel the physics behind the jet-launching mechanism, it remains challenging to observe both jointly. Only recently, global mm-VLBI array (GMVA) + ALMA observations at 86 GHz in 2018 were able to reconstruct the M87 black hole shadow and the extended jet emission simultaneously. In order to analyze the ring and jet of M87, conventional CLEAN algorithms were mainly employed alongside the regularized maximum likelihood method SMILI in the previous work. \nAims. To test the robustness of the reconstructed structures of M87 GMVA + ALMA observations at 86GHz, we estimate the ring diameter, width, and the extended jet emission with the possible central spine by two di ff erent novel imaging algorithms: resolve and DoG-HiT . \nMethods. We performed Bayesian self-calibration and imaging with uncertainty estimation by resolve . We reconstructed the image using closure amplitudes and phases only by DoG-HiT . \nResults. Overall reconstructions are consistent with the results reported in the previous paper. The ring structure of the M87 is resolved at higher resolution and the posterior distribution of M87 ring features is explored. The resolve images show that the ring diameter is 60 . 9 ± 2 . 2 µ as and width is 16 . 0 ± 0 . 9 µ as. The ring diameter is 61 . 0 µ as and width is 20 . 6 µ as by DoG-HiT . The ring diameter is therefore in agreement with the estimation (64 + 4 -8 µ as) by SMILI and the geometrical modeling. Two bright spots in the ring are reconstructed by four independent imaging methods, the substructure in the ring is therefore most likely originated from the data. A consistent limb-brightened jet structure is reconstructed by resolve and DoG-HiT , albeit with a less pronounced central spine. \nConclusions. Modern data-driven imaging methods confirm the ring and jet structure in M87, complementing traditional VLBI methods with novel perspectives on the significance of recovered features. They confirm the result of the previous report. \nKey words. techniques: interferometric - techniques: image processing - techniques: high angular resolution - methods: statistical galaxies: active - galaxies: jets - galaxies: individual (M87)', '1. Introduction': 'Continued improvements of sensitivity and uv-coverage of very long baseline interferometry (VLBI) allow to image and analyze radio emission from the close vicinity of supermassive black holes (SMBH) and inside the formation zone of relativistic jet in active galactic nuclei (AGNs). The recent Event Horizon Telescope (EHT) observations have captured the black hole "shadow" in M87* and SgrA* (Event Horizon Telescope Collaboration et al. 2019a, 2022), and the Global mm-VLBI Array (GMVA) and Very Long Baseline Array (VLBA) observations have provided high fidelity images of the extended jet emission in radio-loud AGNs (Walker et al. 2018; Kim et al. 2018; Okino et al. 2022). The nearby galaxy M87 with a SMBH mass of 6 . 5 × 10 9 M ⊙ (Event Horizon Telescope Collaboration et al. 2019b) and redshift z = 0 . 004283 (Cappellari et al. 2011) pro- \nes the unique opportunity to study the black hole accretion disk and jet launching mechanism simultaneously due to the large angular scales of 0.08 pc or 260 R g (where R g = GM / c 2 ) per milliarcsecond. So far, the black hole shadow and relativistic jet of M87 at radio frequencies are observed independently due to instrumental limitations and characteristics of the source in di ff erent frequency regimes. \nOnly lately, observations of M87 with the GMVA and the phased Atacama Large Millimeter / Submillimeter Array (ALMA) in 2018 was able to detect the black hole shadow and extended jet emission altogether. Lu et al. (2023) employed a CLEAN (Högbom 1974; Clark 1980) self-calibration and imaging method in order to reconstruct the ring and jet structures in M87 with a large field of view. Additionally, the ring in M87 was reconstructed with a smaller field of view using a regularized maximum likelihood (RML) based imaging software called SMILI (Akiyama et al. 2017). The global Millimeter VLBI Array (GMVA) observations at 86 GHz play a significant role to \nresolve the core and extended jet emission of M87. Prior to EHT M87 observation in 2017 (Event Horizon Telescope Collaboration et al. 2019a), GMVA M87 observation in 2014 and 2015 (Kim et al. 2018) provided the core and edge-brightened jet of M87 image down to 13 R g . In 2018, GMVA M87 observation was conducted jointly with the phased ALMA. The joint observation with ALMA facilitates the detection of the ring with extended jet emission since ALMA provides the longest northsouth baselines with improved sensitivity. These findings, for the first time, connected the central ring-like phenomenon which is interpreted as the black hole shadow with the innermost jet. This work is accompanied with recent groundbreaking observations of the jet in M87 on a variety of scales, ranging from the (full polarimetric results on) horizon scales (Event Horizon Telescope Collaboration et al. 2019a, 2021, 2023, 2024), to the strongly edge-brightened innermost jet (Walker et al. 2018; Kim et al. 2018, 2023), as well as to the triple-peaked, helical structure and dynamics in the large scale jet (Asada et al. 2016; Hada 2017; Nikonov et al. 2023; Cui et al. 2023). It is imperative to connect the compact scale structures of M87* to the large scale structures to understand AGN jet formation. The observations presented by Lu et al. (2023) play a significant role in connected horizon scales to jet scales. \nHowever, the data reduction process of GMVA + ALMA is challenging due to the Fourier coverage sparsity of uv-coverage, tropospheric phase corruption at mm wavelength, low signal-tonoise ratio (SNR), and inhomogeneous antenna statistics due to the di ff erent sensitivity of antennas and characteristics of polarization receivers between ALMA (linear) and the rest of the antennas (circular). This di ffi culty jeopardizes the interpretation of some of the features in the reconstruction, namely the substructure in the ring (two brighter spots) and the jet (inner ridge line). An independent assessment of the robustness of these features is needed, to facilitate the much awaited scientific interpretation of these features. \nThe latest advancement of forward modeling imaging algorithms enables us to generate more robust results from sparse mm-VLBI data sets. As an example, Bayesian imaging is a probabilistic approach reconstructing the posterior distribution using the Bayes\' theorem. Hence, Bayesian imaging is able to estimate uncertainty of parameters, such as image features and instrumental gains, however the image reconstruction is comparatively computationally demanding. RML methods reconstruct an image by minimizing the data fidelity term with regularizers. The forward modeling approaches fit the model to the data directly in visibility domain and the model in Bayesian imaging and RML methods tend to be flexible and can be modified easily. We are able to encode knowledge about the source and measurement setup explicitly in the prior distribution and regularizers. As a result, both of the imaging approaches outperform traditional inverse modeling CLEAN algorithm and we can reconstruct reproducible images with improved resolution in less supervised fashion. A detailed comparison between CLEAN and forward modeling approaches can be found in Arras et al. (2021); Müller et al. (2024). In this work, we reconstruct images by two imaging algorithms: we perform Bayesian self-calibration and imaging jointly by the Bayesian imaging software resolve (Junklewitz et al. 2016; Arras et al. 2022; Roth et al. 2023; Kim et al. 2024) and reconstruct an image with closure amplitudes and closure phases only (Chael et al. 2018) using RML-based DoG-HiT software (Müller & Lobanov 2022, 2023a,b). Those two independent imaging methods are utilized in order to estimate the robustness of the M87 ring and extended jet emission from the GMVA + ALMA observation in 2018. They quantify the robust- \nss of the recovered features from two alternative, supplementary perspectives, by self-calibration and imaging in a probabilistic point of view and by imaging only with closure quantities without potential self-calibration bias. \nThis manuscript is structured as follows. In section 2, we explain resolve and DoG-HiT image reconstruction methods. In section 3, we show two image reconstruction results and then analyze the robustness of the M87 ring structure and jet emission. In section 4, we summarize our results.', '2.1. Bayesian self-calibration and imaging by resolve': "resolve 1 is an open-source Bayesian imaging software for radio interferometric data (Junklewitz et al. 2016; Arras et al. 2022; Roth et al. 2023; Kim et al. 2024). In resolve , samples of potential images and antenna-based gain solutions which are consistent with the data are reconstructed by Bayes' theorem in variational inference sense (Blei et al. 2016; Knollmüller & Enßlin 2019; Frank et al. 2021). In this paper, we used Metric Gaussian Variational Inference method (Knollmüller & Enßlin 2019, MGVI) to estimate the posterior distribution of the sky brightness distribution and antenna-based gains. The MGVI method enables us to perform high-dimensional Bayesian inference with a ff ordable computational resources. The probabilistic approach can be advantageous for the M87 GMVA + ALMA observation due to the sparse uv-coverage, large and heterogeneous data uncertainties. For details of the M87 GMVA + ALMA data, we refer to the section 1 and 2 of the supplementary information in Lu et al. (2023). \nWe reconstructed the resolve image in Figure 1 with a spatial domain of 2048 × 1024 pixels and a field of view of 4 mas × 2 mas from a-priori calibrated (without self-calibration) data. The Bayesian self-calibration is performed simultaneously with the imaging. The number of posterior samples are 100 and the χ 2 n value of the final result is 1 . 1. The wall-clock time for the resolve reconstruction is 5 . 5 hours on a single node of MPIfR cluster with 25 MPI (Message Passing Interface) tasks. \nSince GMVA + ALMAarray is highly inhomogeneous, it is a reasonable assumption that each antenna gain has di ff erent temporal correlation structure. In resolve , we utilized Gaussian process prior with non-parametric correlation kernel in NIFTy software 2 for the sky brightness distribution (image) and gain prior model, the spatial correlation between image pixels and temporal correlation between gains can therefore be inferred from the data without manual steering of gain solution interval constraints (see Figure A.2). The amplitude gain prior is assumed to be correlated and di ff erent correlation kernels are inferred per antenna. Gain amplitude for right-hand circular polarization (RCP) and left-hand circular polarization (LCP) are assumed to have the same correlation structure to stabilize the selfcalibration and image reconstruction. The phase gain is assumed to be uncorrelated since the phase coherence time in GMVA observation is comparable with data averaging time (10 seconds) and it can be even shorter under poor weather conditions. \nFurthermore, the posterior distribution of desired parameters, such as each pixel in the image and antenna-based gain solutions can be estimated in resolve . In other words, the reliability of the reconstructed parameters and the image features, such as the ring structure and extended jet emission, can be quantified \nby estimated uncertainties from posterior samples. As an example, if one antennna is problematic, then it would result in high uncertainty in its gain solution. As a result, in Bayesian imaging, the high uncertainty of these data points can be self-consistently taken into account in the image reconstruction. More details about Bayesian self-calibration and imaging method for VLBI data and validation with synthetic data can be found in Kim et al. (2024).", '2.2. Imaging with closure quantities by DoG-HiT': 'DoG-HiT is a Regularized Maximum Likelihood (RML) imaging algorithm that models the image by multiscalar wavelet basis functions (Müller & Lobanov 2022). The basis functions are fitted to the uv-coverage, o ff ering a neat separation between covered and non-covered Fourier coe ffi cients, i.e. gaps in the uvcoverage (for more details on the wavelets, we refer to Müller & Lobanov 2023a). The image is recovered by a sparsity promoting forward-backward splitting framework which e ff ectively calculates the multiresolution support, i.e. the set of all statistically significant wavelet scales to represent the image. It has been demonstrated that the multiresolution support is a beneficial prior information that allows for the reconstruction even for sparse and weakly constrained settings (Müller & Lobanov 2022, 2023b). \nIn this work, we aim to validate the results presented in Lu et al. (2023) by a self-calibration independent technique, i.e. by estimating the robustness of the estimate against the gaincalibration. Therefore, we perform closure-only imaging as pioneered by Chael et al. (2018). We represent the DoG-HiT image (see Figure 1) in a square field of view of 4096 µ as by 512 × 512 pixels. The reconstruction was run on a standard notebook for roughly thirty minutes. The χ 2 n to the closure phases was 1 . 36 and to the closure amplitudes 1 . 1. For the reconstruction with DoG-HiT , we first select a set of wavelet basis functions, called a dictionary, fitted to the uv-coverage of the observation. Then we run DoG-HiT with all large scale wavelets that were significant to fit the extended, di ff use jet emission. Then we use this image as an initial guess and add all small-scale wavelets relevant to represent the central ring, and minimize the χ 2 -metric to the closure phases and closure amplitudes. \nIn this framework we directly fit to the self-calibration independent closure amplitudes and closure phases. For a given number of antennas more closure quantities can be constructed than visibilities. However, not all the measurements are independent since they can be represented as linear combinations of other closure triangles / quadrilaterals (Twiss et al. 1960; Blackburn et al. 2020; Thyagarajan et al. 2022). Since the number of statistically independent closure phases / amplitudes is therefore smaller than the number of independent visibilities, e ff ectively leading to a number of degeneracies such as the lost information of the total flux density and the absolute source position, we have to account for the larger freedom in the models. There are two opposite strategies to achieve this. We could either try to explore the multimodality / degeneracies inherent to closure quantities, e.g. by recently proposed multiobjective optimization schemes (Müller et al. 2023; Mus et al. 2024, 2023), or we utilize a more constraining prior information that resolves the degeneracies. For the purpose of the latter approach, the multiresolution support proved successful and has been implemented in DoG-HiT .', '3. Results': 'The M87 GMVA + ALMA images at 86GHz by resolve , DoG-HiT , and CLEAN (Lu et al. 2023) are shown in Figure 1. resolve and DoG-HiT image fits files and results are archived in zenodo 3 .', '3.1. Estimation of ring-like features in M87': 'Azoom-in into the central compact emission region is presented in Fig. 2. In the visibility domain, the presence of the visibility null amplitude at around 2.3 G λ and the phase jump around the null amplitude (see Fig. S2 in Lu et al. 2023) are analogous to EHT M87 observations in 2017, 2018 (Event Horizon Telescope Collaboration et al. 2019a, 2024), which is the strong evidence of the M87 ring structure. Figure 5 (middle panel) shows the visibility domain representation of resolve and DoG-HiT images, namely the posterior mean model visibility amplitude by resolve and amplitude of Fourier transformed DoG-HiT image. The visibility null amplitudes in resolve and DoG-HiT are located at around 2.3 G λ , which is consistent with the results in Lu et al. (2023). The visibility amplitude null is shorter than M87 EHT observation in 2017, 2018 (at around 3.4 G λ ). It implies that the M87 ring at 3 mm is larger than the ring at 1 mm since the baseline location of the visibility null amplitude scales inversely with the ring diameter (Lu et al. 2023). The M87 ring diameter at 86 GHz (64 + 4 -8 µ as) is estimated in Lu et al. (2023) by geometric modeling and SMILI images. Furthermore, they found that the thick ring is preferred over a thin ring by imaging analysis and model fitting. \nIn this manuscript, we estimate the M87 ring features, such as diameter and width, by using two independent imaging methods resolve and DoG-HiT . To validate the robustness of the ring structure, posterior distribution of the M87 ring diameter, width, and ellipticity is estimated by VIDA software (Tiede et al. 2022) from 100 resolve posterior sample images. VIDA is an image feature extraction tool treating each image as a probability distribution and comparing the image to the geometrical model image (template) by utilizing Kullback-Leiber (KL) or Bhattacharyya (Bh) divergences as the objective function. From the corresponding template, the ring features of each posterior samples image are estimated. In this analysis, a flexible ring model ( CosineRingwFloor ) is used. \nFigure 3 shows that the M87 ring diameter by resolve is 60 . 9 ± 2 . 2 µ as and 61 . 0 µ as by DoG-HiT . The estimated ring diameter by resolve and DoG-HiT is within errors of the estimation (64 + 4 -8 µ as) in (Lu et al. 2023). The width is 16 . 0 ± 0 . 9 µ as by resolve and 20 . 6 µ as by DoG-HiT . The discrepancy of ring width from resolve and DoG-HiT results from the sparse uvcoverage beyond the visibility null amplitude. We note that the ring diameter and width show anti-correlation, which is due to the finite resolution of the telescope array. The e ff ective radius of the ring decreases with a larger ring width (see the Appendix Gof Event Horizon Telescope Collaboration et al. 2019c), which explains the dependence between the estimated diameter and width. This anti-correlation is also shown by SMILI reconstructions (see Figure S14 in Lu et al. 2023). The ellipticity of the ring, τ , is defined as τ = 1 -b / a , where a is the semi-major axis lengths and b is the semi-minor axis lengths of elliptical ring. The τ is 0 . 06 ± 0 . 04 µ as by resolve and 0 . 04 by DoG-HiT , which means there is no significant ellipticity of the M87 ring. \n2 \nRelative RA [mas] \nFig. 1. GMVA + ALMA M87 image reconstructions at 86 GHz. Each image presents results obtained by a di ff erent algorithm. The top posterior mean image was reconstructed using resolve Bayesian self-calibration and imaging method. The middle image was obtained using DoG-HiT closure amplitudes and closure phases only imaging. The bottom image was obtained using CLEAN self-calibration and imaging in Lu et al. (2023). The CLEAN image is convolved with an elliptical beam, which is represented as an ellipse with sizes 79 × 37 µ as, P.A. = -63 · in the bottom left corner. \n<!-- image --> \nDetailed description of the VIDA.jl software can be found in Tiede et al. (2022). \nGMVA + ALMA M87 observation (2018, April 14) was conducted a week before EHT M87 observation (2018, April 21 and 25), which is shorter than the expected timescale for decorrelation of the emission pattern (Georgiev et al. 2022). The flux density of the compact region 200 µ as × 200 µ as) at 1 mm is 0 . 5 ± 0 . 1 \nJy by DIFMAP , eht-imaging , and SMILI softwares (see Table 2 in Event Horizon Telescope Collaboration et al. (2024)). We note that the compact region flux density constraints from EHT 2018 data was challenging due to the lack of short baseline coverage (Event Horizon Telescope Collaboration et al. 2024). The total flux density at 3 mm is 0 . 57 ± 0 . 03 Jy on mas scales, and the flux density of the compact region at 3 mm is 0 . 33 ± 0 . 02 Jy by \n0 \n2 \nFig. 2. The ring of the M87 image reconstruction at 86 GHz from the Figure 1. The color shows intensity in Jy mas -2 according to the linear color bar located at the top of the figure. Each image presents results obtained by a di ff erent algorithm, whose names are indicated in the lower right corners. The top left image is the resolve posterior mean image using the Bayesian self-calibration and imaging method. The top right image is the DoG-HiT reconstruction using closures only imaging. The bottom left image represents the CLEAN image with the over-resolved 37 µ as circular beam in Lu et al. (2023). The bottom right shows the SMILI image in Lu et al. (2023). All images were processed by a Gaussian interpolation. \n<!-- image --> \n- \n- \n- \n- \nRelative RA [ µ as] \nresolve . DoG-HiT images estimate 0.43 Jy in the compact field of view. As a result, the spectral index α of the M87 compact region is slightly positive α ∼ 0 . 4. This implies a mixed optical depth in the core, under a caveat that the emitting region at 3 mm is larger than at 1 mm, following the ring diameter analysis. The observed ratio of flux densities is reasonably consistent with predictions of the numerical models, that typically indicate inhomogeneous optical depth in the compact region.', '3.2. Extended jet emission of M87': 'Our reconstructions of the limb-brightened M87 jet structure broadly agree with the one described in Lu et al. (2023): we see an edge-brightened jet anchored to the vicinity of the ring-like feature. One peculiar feature in the image reported by Lu et al. (2023) is the presence of bright spine along the jet axis. This central ridge-line may be related to the triple-helix structure in \nthe jet of M87 that has been observed at larger scales (Nikonov et al. 2023). \nHowever, it is questionable whether this feature of the image represents a real structure on-sky, or appears as a consequence of imaging artifacts. Particularly, it has been recently demonstrated that CLEAN deconvolution errors are prone to produce inner ridge lines for edge-brightened jet configurations (Pashchenko et al. 2023). In fact, the central spine is less prominent in the DoG-HiT and resolve reconstructions. \nDoG-HiT and resolve allow for the quantification of the robustness of the central spine from two independent perspectives: by calibration-independent imaging with a minimal human bias in DoG-HiT , and by uncertainty estimation in resolve . For DoG-HiT , the small scale structure and the large scale (diffuse) structure are represented by di ff erent wavelets, ultimately expressed by the multiresolution support. This allows us to estimate the robustness of the central ridge by a jackknife test, i.e. \nFig. 3. The posterior distribution of the M87 ring diameter, thickness, and ellipticity estimated from 100 resolve posterior sample images. The red marks correspond to the estimation of the ring diameter and ring thickness obtained by the DoG-HiT reconstruction (Diameter : 61.0 µ as, Thickness : 20.6 µ as, Ellipticity : 0.04). \n<!-- image --> \nwe cut the di ff use emission at the location of the central spine and recalculate the fit statistics to the (calibration-independent) closure quantities. To this end, we applied the following strategy: We flagged long baselines (to focus the analysis on the di ff use emission), then we fitted the closure phases and closure amplitudes with DoG-HiT only varying coe ffi cients in the multiresolution support, and calculate the updated fit statistics. Next, we mask out the di ff use, central spine from the multiresolution support, and refit the closure quantities. Finally, we compare the scoring with a central spine and without. \nThis strategy resembles a standard strategy in VLBI, often applied in the discussion of the existence of a counter-jet. However, we note some key advantages of the strategy applied by us, compared to CLEAN . First, DoG-HiT directly fits closure quantities, hence the conclusions that we can draw are less dependent on the phase and amplitude self-calibration. Second, we perform the jackknife test on a multiscalar domain, allowing us to divide the emission more clearly into small and large scale structures. Finally, DoG-HiT directly fits a model to the data that is physically reasonable, opposed to CLEAN (e.g. see the discussion in Müller & Lobanov 2023a). Hence, we compare the fit quality of the approximated on-sky representation rather than an unphysical list of CLEAN components (which would question the interpretability of the χ 2 -statistics). \nWe obtain χ 2 cph = 1 . 104, χ 2 cla = 1 . 494 when fitting the data with a central spine, and χ 2 cph = 1 . 061, χ 2 cla = 1 . 592 when fitting without a central spine. Neither mode is strongly favored. From this study, we cannot report the conclusive detection of a central spine in the image. \nAn alternative perspective on the robustness of image features is o ff ered by the built-in uncertainty quantification in resolve . In Figure 4, the transverse flux intensity profiles of the M87 jet emission are depicted. The mean and standard deviation \nof the transverse jet profile can be obtained from 100 posterior sample images by resolve . The intensity profile of the jet at 0 . 25 , 0 . 5 , 0 . 75 mas shows that the edge-brightened structure (two peaks) is prominent, however a significant central spine structure is not seen. The standard deviation of the pixel fluxes at the central spine is not particularly higher than the limb-brightened feature. Therefore, the noticeable central spine of the M87 jet is not detected in resolve image. The central spine in the images obtained by CLEAN in Lu et al. (2023) may be a consequence of CLEAN artifacts resulting from CLEAN windows and sparsity promoting CLEAN sky model. Further observations with additional short baseline antennas are required to conclude the detection of the central spine. The edge-brightened morphology that we recover is both consistent with the earlier observations at 86 GHz (Kim et al. 2018) and well-motivated theoretically (e.g., Yang et al. 2024). The counter jet is not detected consistently in the three images, it is therefore not discussed further in this work.', '3.3. Substructure of the M87 ring': "A feature that appears consistent across all four di ff erent imaging algorithms SMILI , CLEAN , resolve , and DoG-HiT are the two bright blobs in the ring. While images are recovered with various resolutions, the recovered ring emission always shows the double structure within the ring towards the top and the bottom, at consistent positions across methods. Note that the limbbrightened structure that are recovered by resolve close to the ring seem to connect to the ring exactly at the positions where the brighter blobs within the ring occur consistently for all four reconstructions. Hence, it may be natural to interpret the double pattern in the ring as a physical phenomenon. In this subsection we present some discussion on how real these features may be. \nFirst, it is noticeable, that these brighter regions in the ring form a double structure that is point-symmetric to the center of the ring. That supports the interpretation as an imaging artifact, especially due to the sparse coverage at long baselines. In fact, Lu et al. (2023) have tested SMILI and CLEAN reconstructions on synthetic data and found that similar structures are artificially introduced by the imaging procedure, compare section 4 and particularly figure S8 in Lu et al. (2023). \nMultiple well-understood artifacts may cause such a double structure. Here, we discuss the three most natural scenarios. First, it could be caused by specific choices of the regularization assumption inherent to the respective imaging algorithm. Second, the structure could be introduced artificially by residual gain e ff ects. Finally, the structure may be described by a residual sidelobe structure, hence essentially a consequence of the sparsity of the uv-coverage. In what follows, we will discuss each of these concerns individually. \nThe double structure appears for all four imaging methods. Four imaging methods that were utilized here, approach the image reconstruction from four vastly di ff erent perspectives: CLEAN recovers the structure in an inverse modelling framework essentially processing a sparsity promoting regularization approach (Lannes et al. 1997), SMILI approaches the image by a weighted sum of multiple handcrafted data and regularization terms as a RMLalgorithm (Akiyama et al. 2017), DoG-HiT processes multiscale functions in the context of compressive sensing (Müller & Lobanov 2022), and resolve estimates the posterior distribution of image and gains from the prior model encoding source and instrument information and likelihood (i.e. the data) (Kim et al. 2024). We note that the prior in Bayesian imaging can be interpreted as regularizers in RML methods, and vice versa (Kim et al. 2024). Multiscale functions in DoG-HiT and Gaussian pro- \n5 \nFig. 4. M87 jet transverse profiles and intensity map obtained by resolve . Intensity in the map is represented by false-color according to the colorbar used in Figure 1 in logarithmic scale. The image is rotated 18 · clockwise. Intensity plots at the bottom of the figure show flux density profiles of the jet at 0.25, 0.5 and 0.75 mas from the phase center. Each vertical line corresponds to a location where profiles were extracted. \n<!-- image --> \ncess prior in resolve are flexible and does not ask for the double structure in the ring as a prior knowledge. The fact that all four independent methods using a variety of regularization / prior information lead to a similar structure challenges the interpretation of the double structure as an artifact from the assumptions and prior information applied by the imaging procedure. \nThe structure may be a consequence of unsolved gain residuals. In fact, it is a possible issue of the alternating self-calibration and cleaning procedure to produce 'phantom' structures point symmetric to the origin that has been reported in practice for a long time. We note however, that the double feature also appears in DoG-HiT reconstructions which are independent against gain corruptions (closure-only imaging), making a potential cause by the calibration of the phases less likely, supported by resolve which solves for the self-calibration with imaging simultaneously. \nFinally, could the double structure be introduced by the sparsity of the uv-coverage? That is a possibility that never can be eradicated completely, simply since the observation fundamentally misses relevant visibilities in the gaps of the uv-coverage. Consequently, there are missing information. In fact, the Fourier domain representation of a double source is a fringe (compare Fig. 5), which is exactly the kind of artifacts not fully cleaned residuals may introduce. This issue has been identified by synthetic data tests by Lu et al. (2023). Moreover, the occurrence of the double structure may be related to the lack of long westeast baselines. Nevertheless, we argue that the recovered double structure is represented by the measured visibilities. We show in Fig. 5 the central ring image (top panels) and the amplitudes of the full Fourier transform of the reconstructions (middle panels) for DoG-HiT (left panels) and resolve (right panels). A ring feature is identified in the Fourier domain by a first zero that is clearly covered by observations (compare, e.g., to the model \nfitting discussions in Lu et al. 2023). Moreover, the visibility domain representation of DoG-HiT and resolve show the ellipticity of the null visibility amplitude points in Figure 5 (middle panel). We note that the location of the null visibility points in uv-domain is elliptical, with a elongation in the direction of the jet. This may result in bright spots in the image aligned perpendicularly to the jet, related to the stretch of the image domain ring. The data from multiple antennas (EF, KP, OV, PV, YS) at the longest west-east baselines implies that the ellipticity of the null visibility points is originated from the data. \nTo analyze the uv-coverage sparsity corresponding to the M87 ring substructure, we subtract a uniform ring feature from the images (defined by 60% of the respective emission peak) to extract the double feature on top of the ring. The resulting double patterns are depicted in the bottom panels of Figure 5, and their respective amplitudes in the Fourier domain in the last row. The observed uv-points are overplotted with red crosses. Observations span the main fringe and the first side-lobe uniformly, the fringes are not produced exclusively in the gaps of the uvcoverage. These findings, as well as the striking similarity between multiple imaging approaches, constitute a convincing argument for the physical nature of bright spots along the ring. A definitive answer however may not be given with the quality of the existing data set, particularly in the presence of the synthetic data tests performed by Lu et al. (2023), and follow-up observations are needed. \nIf the emission forming the ring image is dominated by the accretion disk, elongation in the direction perpendicular to the ring may be a simple geometric e ff ect for the inclined observer. While the 230 GHz ring image is strongly asymmetric with respect to the jet axis (Event Horizon Telescope Collaboration et al. 2024), our 86 GHz reconstructions exhibit high degree of symmetry. In numerical general relativistic magnetohydrodynamic (GRMHD) simulations of accretion, consistent behavior appears for retrograde accretion (negative black hole spins). In that case 230 GHz image asymmetry is driven primarily by the spin e ff ects (Event Horizon Telescope Collaboration et al. 2019d), dominant very near the event horizon. In the 86 GHz image, formed by a more extended emission, these e ff ects are balanced by the Doppler boost enhancing brightness on the opposite side of the black hole, resulting in a ring structure elongated perpendicularly to the jet axis, with bright spots on both sides of the central brightness depression.", '4. Conclusions': "Lu et al. (2023) presents observations of the core and jet in M87 observed with the GMVA + ALMA at 86GHz. The image contains a ring-like feature that looks similar to the one reported by the EHT (Event Horizon Telescope Collaboration et al. 2019a, 2024), but with approximately 1.5 times larger ring diameter resulting from the null visibility points and phase jump at 2.3 G λ . For the first time, this ring feature is connected to a innermost jet in the same image, possibly providing constraints on the launching mechanism of the jet. Furthermore, the recovered image contains several fainter features that may be of great importance for the scientific interpretation, especially in the context of recent works on the large scale jet structure in M87 (Kim et al. 2018; Cui et al. 2023; Nikonov et al. 2023; Kim et al. 2023). \nTo validate the results reported in Lu et al. (2023), we apply two more imaging algorithms specially designed to study the robustness of recovered features: by Bayesian self-calibration and imaging with resolve , and by closure-only imaging with DoG-HiT . The distinctive features of resolve and Dog-HiT , \nnamely the probabilistic approach and the multiscale waveletbased deconvolution algorithm, allow us to quantify the robustness of recovered M87 ring and extended jet emission. We obtained the posterior distribution of ring diameter, width, and ellipticity by analysis of the resolve posterior sample images. We confirm the M87 ring-like structure at 86 GHz with the diameter of 60 . 9 ± 2 . 2 µ as, the thickness of 16 . 0 ± 0 . 9 µ as, and the ellipticity is 0 . 06 ± 0 . 04 by resolve and the diameter of 61 . 0 µ as, the thickness of 20 . 6 µ as, and the ellipticity is 0 . 04 by DoG-HiT . The estimated ring diameter is consistent with the estimate (64 + 4 -8 µ as) in Lu et al. (2023). \nFurthermore, image reconstructions by resolve and DoG-HiT show that the ring is embedded in a strongly edgebrightened large scale jet structure agreeing with the findings reported in Lu et al. (2023). Among the upper and lower arm of the edge-brightened jet, the CLEAN reconstruction presented in Lu et al. (2023) features a third, central spine that may be interpreted together with a triple-helix structure at larger scales (Nikonov et al. 2023). However, the central spine structure by resolve and DoG-HiT reconstructions is less prominant compared to CLEAN reconstruction. Our analysis shows that this central spine is neither necessary to fit the data, nor supported by the uncertainty quantification by resolve in image domain. The validation by two independent imaging methods implies that the central spine is fainter than previous report. \nFinally, all utilized imaging algorithms coincide on the same substructure in the ring consisting of two bright spots to the North and the South, co-located with inner anchor points of the edge-brightened jet to the horizon scale. Stemming from a coincidence of this phenomenon across a variety of imaging algorithms, and its representation in the Fourier domain, we argue that is more likely that this substructure of the ring results from the data, and not an imaging or self-calibration artifact, although artifact originated from sparse uv coverage is not ruled out. The potential physical origin of these structures is unclear. If real, they might be transient emission structures during the observational period, however, their alignment perpendicular to the jet suggests otherwise. They also might be permanent structures potentially linked to the disk-jet transition, as their location within the disk seems to coincide with the position the jet edges point to. Finally, they may be a result of an interplay of Doppler boost and black hole spin e ff ects, as seen in numerical simulations of retrograde accretion. \nAcknowledgements. We thank Jae-Young Kim for helpful suggestions and feedback on drafts of the manuscript, Jack Livingston and Thomas Krichbaum for comments and informative discussions. J. K. and A. N. received financial support for this research from the International Max Planck Research School (IMPRS) for Astronomy and Astrophysics at the Universities of Bonn and Cologne. This work was supported by the M2FINDERS project funded by the European Research Council (ERC) under the European Union's Horizon 2020 Research and Innovation Programme (Grant Agreement No. 101018682). This research has made use of data obtained with the Global Millimeter VLBI Array (GMVA), which consists of telescopes operated by the MPIfR, IRAM, Onsala, Metsahovi, Yebes, the Korean VLBI Network, the Greenland Telescope, the Green Bank Observatory and the Very Long Baseline Array (VLBA). The VLBA and the GBT are facilities of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. The data were correlated at the correlator of the MPIfR in Bonn, Germany.", 'References': 'Akiyama, K., Kuramochi, K., Ikeda, S., et al. 2017, ApJ, 838, 1 Arras, P., Bester, H. L., Perley, R. A., et al. 2021, A&A, 646, A84 Arras, P., Frank, P., Haim, P., et al. 2022, Nature Astronomy, 6, 259 Asada, K., Nakamura, M., & Pu, H.-Y. 2016, ApJ, 833, 56 Blackburn, L., Pesce, D. W., Johnson, M. D., et al. 2020, ApJ, 894, 31 \nKim et. al.: Imaging the ring and extended jet of M87 \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nJy \n/ \nmas \n2 \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nJy \n/ \nmas \n<!-- image --> \nFig. 5. Reconstructions of the central feature by DoG-HiT (left panels), and resolve (right panels). Top panels: reconstructions of the compact emission. Top-medium panels: The central ring feature. Medium panels: The recovered amplitudes of the central feature, with the uv-coverage overplotted (red points). Medium-bottom panels: The central ring feature, cutted at 60% of the respective peak brightness, showcasing the double blob pattern on top of the ring. Bottom panels: The amplitudes and the uv-coverage of the double pattern alone. \n<!-- image --> \n0.8 \nFig. A.1. resolve image pixel-wise relative uncertainty, which is the sky brightness posterior standard deviation normalized by the posterior mean by the resolve reconstruction from the top panel of Figure 1. \n<!-- image -->', 'Appendix A: Uncertainty estimation by resolve': 'Figure A.1 shows the pixel-wise relative uncertainty of the resolve image in Figure 1. The relative uncertainty is defined as the sky brightness posterior standard deviation normalized by the posterior mean from 100 posterior sample images. Lower relative uncertainty values of the M87 ring emission and limb-brightened are estimated. However, the relative uncertainty values in the central spine are higher compared to the limbbrightened jet emission. Figure A.2 depicts the posterior amplitude gains by resolve . ALMA (AA) amplitude gain solutions show smooth behavior with lower posterior standard deviations compared to other arrays due to the high sensitivity of ALMA array. The gain amplitudes with high uncertainties result from the absence of the data (BR: < 4h, EF: > 6h, GB: < 4h, KP: < 3h, LA: < 3h, PT: < 3h, and YS: RCP gains). For instance, the RCP gain amplitude for YS antenna has uniformly distributed high uncertainties since only single polarization mode (LCP) was observed for YS antenna. \nFig. A.2. resolve posterior amplitude gains. The gain as a function of time is illustrated as a thin line with a semi-transparent standard deviation. The left and right columns of the figure show gains from the right (RCP) and left (LCP) circular polarizations correspondingly. Each row represents an individual antenna, whose abbreviated name is indicated in the bottom left corner of each LCP plot. \n<!-- image -->'} |
2024OJAp....7E..58E | We report discovery and spectroscopic followup of 21 astrometric binaries containing solartype stars and dark companions with masses near 1.4 M. The simplest interpretation is that the companions are dormant neutron stars NSs though ultramassive white dwarfs WDs and tight WDWD binaries cannot be fully excluded. We selected targets from Gaia DR3 astrometric binary solutions in which the luminous star is on the main sequence and the dynamicallyimplied mass of the unseen companion is a more than 1.25M and b too high to be any nondegenerate star or close binary. We obtained multiepoch radial velocities RVs over a period of 670 days spanning a majority of the orbits dynamic range in RV. The RVs broadly validate the astrometric solutions and significantly tighten constraints on companion masses. Several systems have companion masses that are unambiguously above the Chandrasekhar limit while the rest have masses between 1.25 and 1.4 M. The orbits are significantly more eccentric at fixed period than those of typical WD MS binaries perhaps due to natal kicks. Metalpoor stars are overrepresented in the sample 3 out of 21 objects 14 have FeH 1.5 and are on halo orbits compared to 0.5 of the parent Gaia binary sample. The metalpoor stars are all strongly enhanced in lithium. The formation history of these objects is puzzling it is unclear both how the binaries escaped a merger or dramatic orbital shrinkage when the NS progenitors were red supergiants and how they remained bound when the NSs formed. Gaia has now discovered 3 black holes BHs in astrometric binaries with masses above 9 M and 21 NSs with masses near 1.4M. The lack of intermediatemass objects in this sample is striking supporting the existence of a BHNS mass bimodality over 4 orders of magnitude in orbital period. | 2024-07-01T00:00:00Z | ['arXiv:2405.00089', '10.48550/arXiv.2405.00089', '2024arXiv240500089E', '10.33232/001c.121261', '2024OJAp....7E..58E'] | ['Astrophysics - Solar and Stellar Astrophysics', 'Astrophysics - Astrophysics of Galaxies'] | A population of neutron star candidates in wide orbits from Gaia astrometry | 2,024 | 169 | 0.66 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML'] | 22 | https://arxiv.org/pdf/2405.00089.pdf | {'A POPULATION OF NEUTRON STAR CANDIDATES IN WIDE ORBITS FROM GAIA ASTROMETRY': "Kareem El-Badry 1 , 2 , Hans-Walter Rix 2 , David W. Latham 3 , Sahar Shahaf 4 , Tsevi Mazeh 5 , Allyson Bieryla 3 , Lars A. Buchhave 6 , Ren'e Andrae 2 , Natsuko Yamaguchi 1 , Howard Isaacson 7 , 8 , Andrew W. Howard 1 , Alessandro Savino 7 , and Ilya V. Ilyin 9 \n1 \nDepartment of Astronomy, California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125, USA 2 Max-Planck Institute for Astronomy, Konigstuhl 17, D-69117 Heidelberg, Germany 3 Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA 4 Department of Particle Physics and Astrophysics, Weizmann Institute of Science, Rehovot 7610001, Israel 5 School of Physics and Astronomy, Tel Aviv University, Tel Aviv, 6997801, Israel 6 DTU Space, National Space Institute, Technical University of Denmark, Elektrovej 328, DK-2800 Kgs. Lyngby, Denmark 7 Department of Astronomy, University of California Berkeley, Berkeley, CA 94720, USA 8 Centre for Astrophysics, University of Southern Queensland, Toowoomba, QLD, Australia and 9 Leibniz-Institut fur Astrophysik Potsdam (AIP), An der Sternwarte 16, D-14482 Potsdam, Germany \nVersion July 15, 2024", 'ABSTRACT': "We report discovery and spectroscopic follow-up of 21 astrometric binaries containing solar-type stars and dark companions with masses near 1 . 4 M ⊙ . The simplest interpretation is that the companions are dormant neutron stars (NSs), though ultramassive white dwarfs (WDs) and tight WD+WD binaries cannot be fully excluded. We selected targets from Gaia DR3 astrometric binary solutions in which the luminous star is on the main sequence and the dynamically-implied mass of the unseen companion is (a) more than 1 . 25 M ⊙ and (b) too high to be any non-degenerate star or close binary. We obtained multi-epoch radial velocities (RVs) over a period of 700 days, spanning a majority of the orbits' dynamic range in RV. The RVs broadly validate the astrometric solutions and significantly tighten constraints on companion masses. Several systems have companion masses that are unambiguously above the Chandrasekhar limit, while the rest have masses between 1.25 and 1.4 M ⊙ . The orbits are significantly more eccentric at fixed period than those of typical WD + MS binaries, perhaps due to natal kicks. Metal-poor stars are overrepresented in the sample: three out of 21 objects (14%) have [Fe / H] ∼ -1 . 5 and are on halo orbits, compared to ∼ 0 . 5% of the parent Gaia binary sample. The metal-poor stars are all strongly enhanced in lithium. The formation history of these objects is puzzling: it is unclear both how the binaries escaped a merger or dramatic orbital shrinkage when the NS progenitors were red supergiants, and how they remained bound when the NSs formed. Gaia has now discovered 3 black holes (BHs) in astrometric binaries with masses above 9 M ⊙ , and 21 NSs with masses near 1 . 4 M ⊙ . The lack of intermediate-mass objects in this sample is striking and significant, supporting the existence of a BH/NS mass bimodality over four orders of magnitude in orbital period. \nSubject headings: stars: neutron - binaries: spectroscopic - stars: evolution", '1. INTRODUCTION': "Most stars with initial masses ≳ 8 M ⊙ leave behind neutron stars (NSs) when they die. Several thousand NSs are known in the Milky Way, a large majority of which are radio pulsars. Most ( > 99%; Lorimer 2008) young pulsars are isolated. Yet, a large majority of the massive stars from which NSs form are in binaries, triples, and higher-order multiples (e.g. Sana et al. 2012; Moe & Di Stefano 2017). The apparent mismatch in multiplicity properties of NSs and their progenitors hints that most massive binaries are destroyed during or prior to the formation of a NS. This destruction can come as a result of binary interaction leading to a stellar merger, or due to the binary becoming unbound during a supernova (SN), which can impart a kick of order 250 km s -1 on the newborn NS (e.g. Hobbs et al. 2005; Faucher-Gigu'ere & Kaspi 2006). \nAll known companions to young pulsars are massive OB stars (Kaspi et al. 1996; Bassa et al. 2011; Shannon \nCorresponding author: [email protected] \net al. 2014; Lyne et al. 2015; Andersen et al. 2023; van der Wateren et al. 2023). The same is true for all detached NS + main sequence (MS) binaries detected in X-rays (e.g. Reig 2011). This likely reflects the fact that binaries containing a massive star are more likely to survive mass transfer and a SN when the companion is also a massive star. \nHowever, NSs with low-mass stellar companions do exist, and in fact make up the majority of all known binary NSs. The known systems are all currently accreting from a binary companion ('low-mass X-ray binaries'; LMXBs) or recycled, meaning that past accretion from a companion spun up the pulsar and buried its magnetic field, slowing subsequent spin-down (e.g. Bhattacharya & van den Heuvel 1991). LMXBs and recycled pulsars are over-represented in observed samples because they can have long lifetimes (up to and exceeding the age of the Universe) compared to normal young pulsars, which are only detectable for ∼ 10 7 yrs. The companions to most recycled pulsars are white dwarfs (WDs), low-mass stars, and brown dwarfs that have transferred mass to the \nNS via stable Roche lobe overflow. Models predict that the initial masses of typical companions were at most (1 -2) M ⊙ (Podsiadlowski et al. 2002). \nAlthough they have not yet been unambiguously detected, there is little doubt that non-interacting binaries containing a low-mass MS star and a NS exist: such systems are the progenitors of LMXBs and millisecond pulsars. A few candidates for such objects have been identified, including (a) young pulsars with roughly solarmass companions and eccentric orbits (PSR B1820-11 and PSR J1954+2529 Phinney & Verbunt 1991; Parent et al. 2022), and (b) MS stars with unseen companions that may be NSs (e.g. Mazeh et al. 2022; Yuan et al. 2022; Zheng et al. 2022; Yi et al. 2022; Escorza et al. 2023; Lin et al. 2023; Zhao et al. 2023). The nature of these candidates as NS + MS binaries is quite uncertain. Among the objects in group (a), there is no doubt of the presence of a NS, but the companions - which have not been detected electromagnetically - may be massive WDs. Among those in group (b), the NS has not been detected, and the minimum dynamically implied mass is well below the Chandrasekhar limit. Many of the candidates in group (b) are likely to host massive WDs rather than NSs. \nBy precisely monitoring the astrometric light-centroid 'wobble' of nearly two billion stars, the Gaia mission opens a new window on the Galactic binary population (see El-Badry 2024a, for a recent review). The mission's 3rd data release ('DR3') in June 2022 included orbital solutions for about 1 . 7 × 10 5 astrometric binaries, including 3 × 10 4 joint astrometric + radial velocity (RV) solutions (Gaia Collaboration et al. 2023b). Although DR3 employed stringent quality cuts (Halbwachs et al. 2023) and represents only a small fraction of the binary sample that will be accessible in future data releases, the DR3 binary sample was already more than an order of magnitude larger than all samples of binary orbits in the previous literature. Gaia's particular sensitivity to longperiod orbits ( P orb ∼ (1 -3) years in DR3) - and the fact that astrometric data provides constraints on binary inclinations that are not accessible with RVs alone - has already enabled the discovery of unexpected binary populations, including three stellar-mass black holes (BHs) in au-scale orbits (El-Badry et al. 2023a,b; Gaia Collaboration et al. 2024) and a population of WD + MS binaries with similar orbital separations (Shahaf et al. 2023b,a; Yamaguchi et al. 2024). The population of wide NS + MS binaries studied in this paper is closely related to these two populations. \nHere we present results from a follow-up program of Gaia astrometric binaries suspected to contain NSs. One of our candidates, Gaia NS1 (J1432-1021), was already studied in detail by El-Badry et al. (2024). This object has the highest inferred dark companion mass of any of the objects in our sample, and it is the only object for which the minimum companion mass from RVs alone is well above the Chandrasekhar mass, independent of astrometric constraints on the inclination. We suspect that most of the other binaries in the sample also host NSs, but because the unseen companions have masses near the maximum WD mass, other possibilities cannot be ruled out definitively. \nThe remainder of this paper is organized as follows. \nSection 2 describes selection of our initial sample from Gaia DR3, and Section 3 summarizes our spectroscopic follow-up and measurement of metallicities. We infer parameters of the luminous stars by fitting their spectral energy distributions in Section 4. In Section 5, we carry out joint fits of the astrometry and our follow-up RVs. In Section 6, we discuss the nature of the dark companions, the binaries' possible formation histories, their Galactic orbits and possible abundance anomalies, and the BH/NS mass distribution. We summarize our findings in Section 7. We discuss spurious astrometric solutions in Appendix A and limits on possible optical contamination from WD companions in Appendix B. Tables of RVs and orbital parameters are provided in Appendices C and D.", '2. SAMPLE SELECTION': "We selected targets from Gaia DR3 following the general approach outlined by Shahaf et al. (2019). In brief, the astrometric 'triage' algorithm seeks to identify sources whose astrometric orbits are so large - given their orbital period - that they cannot be explained by any luminous star companion, or by a companion that is a close binary containing two luminous stars. Shahaf et al. (2023b) applied this algorithm to astrometric binaries published in Gaia DR3, producing a catalog of 177 candidates in which the astrometric solution and assumed luminous star mass implies the secondary must be a WD, NS, or BH. These classifications are, however, contingent on the validity of the Gaia astrometric solutions, which are in some cases spurious. \nCalculating the mass of a star's unseen companion from its astrometric solution requires an estimate of the mass of the star. In constructing their candidate sample, Shahaf et al. (2023b) used the IsocLum mass estimates calculated by Gaia Collaboration et al. (2023b). These estimates, which are available through the Gaia archive in the gaiadr3.binary masses catalog, were inferred by comparison of the extinction-corrected colors and absolute magnitudes to a grid of PARSEC isochrones, with a prior that the metallicity is close to solar. These masses can thus be overestimated if the metallicities are subsolar, or underestimated if they are supersolar. Their validity is also contingent on the validity of the extinction estimates and on the assumption that a single star contributes to the observed photometry. We improve the mass estimates in Section 4. \nShahaf et al. (2023b) noted that their compact object binary candidate sample appeared to fall within two populations in the mass-eccentricty plane: one with a mean mass close to 0 . 6 M ⊙ and eccentricities below 0.2, and another with an apparent mean mass of ∼ 1 . 3 M ⊙ and a broad eccentricity distribution. It would be natural, they noted, to identify these two populations with WDs and NSs. Our follow-up has shown that the division is probably not so simple: some systems in the high-mass, high-eccentricity sample unambiguously host white dwarfs, as revealed by strong UV excess (e.g. Ganguly et al. 2023). Other objects have high eccentricities but companion masses below 1 M ⊙ , which are implausibly low for a NS. Despite these caveats, our follow-up has shown that most of the low-eccentricity, lower-mass compact object candidates identified by Shahaf et al. (2023b) are WDs, and at least some of the higher-eccentricity, \nhigher-mass companions are NSs. We return to this discussion in Section 6.2. \nWe initiated spectroscopic follow-up for a majority of the NS candidates identified by Shahaf et al. (2023b) in June 2022. We also carried out RV follow-up observations of some astrometrically-selected NS candidates not included in the Shahaf et al. (2023b) sample because their orbital periods slightly exceed 1000 days. Several of our candidates were also listed in other samples of compact object candidates from Gaia DR3, including the sample curated by Andrews et al. (2022). The observations presented here were obtained before June 2024, but our program is ongoing.", '2.1. Rejection of spurious astrometric solutions': 'For some candidates, our RV follow-up soon showed the Gaia astrometric solution to be spurious or to have significantly underestimated uncertainties. Examples are shown in Appendix A. About a quarter of candidates with good astrometric quality flags turned out to be spurious. 1 While the fraction of all astrometric solutions that are spurious is small, RV follow-up over a significant fraction of an orbit is critical for vetting astrometric solutions of unusual objects: our follow-up has demonstrated that incorrect solutions do exist, even among solutions with favorable goodness of fit and other Gaia quality flags.', '2.2. Completeness of the sample': 'Among the NS candidates identified by Shahaf et al. (2023b) and Andrews et al. (2022), our sample includes all systems that (a) are brighter than G = 15, (b) have best-fit companion masses M 2 > 1 . 25 M ⊙ from joint fitting of astrometry and RVs, (c) were not found to have spurious solutions or significantly underestimated astrometric uncertainties through RV follow-up, and (d) were observed over at least half an orbit. Properties of these sources are summarized in Table 1. The Shahaf et al. (2023b) and Andrews et al. (2022) samples mainly contain sources near the main sequence with inferred luminous star masses M ⋆ ≲ 1 . 3 M ⊙ . 2 While NS companions to evolved stars and more massive MS stars are likely to exist, these in most cases cannot be distinguished from luminous stars or tight luminous-star binaries based on astrometry alone. \nTable 3 in Appendix A provides a summary of our RV follow-up and our current assessment of the viability of all candidates from the Shahaf et al. (2023b) and Andrews et al. (2022) samples. We defer a full description of our follow-up program - including RVs of suspected WDs and all objects that turned out to have spurious astrometric solutions - to future work. We suspect that our sample contains most of the NS-hosting binaries with M 2 ≳ 1 . 25 M ⊙ and astrometric solutions published in DR3. A handful of likely good candidates are not included because our RV follow-up has not yet \ncovered enough of the orbits to confirm the astrometric solution with high confidence, and another handful were excluded because they are too faint ( G > 15) to be amenable for RV follow-up with the instruments at our disposal. Our sample does not contain any low-mass NSs with M < 1 . 25 M ⊙ , a mass limit below which some NSs likely do exist (Ferdman et al. 2014; Martinez et al. 2015). As we discuss throughout the paper, it becomes increasingly challenging to distinguish between NSs and massive WDs at lower masses.', '2.3. Summary of the sample': "Basic properties of our NS candidates are listed in Tables 1 and 4. Figure 1 compares our candidates to the full sample of astrometric binaries (solution types Orbital and AstroSpectroSB1 ) published in DR3. Three candidates have AstroSpectroSB1 solutions: J0152-2049, J2145+2837, and J1150-2203; the rest have Orbital solutions. The upper left panel of Figure 1 shows the sources on the extinction-corrected color-magnitude diagram. We estimate the extinction for sources in the north ( δ > -30 deg) using the 3D dust map from Green et al. (2019); we use the map from Lallement et al. (2022) for sources farther south. All our targets are solar-type stars on the main sequence, with absolute magnitudes and colors suggesting luminous star masses of (0 . 7 -1 . 3) M ⊙ . Several candidates are near the blue edge of the main sequence. A potential concern is that this could be due to blue excess from a hot WD companion. However, our analysis of the sources' full spectral energy distributions - in particular, the lack of UV excess - speaks against this possibility (Appendix B). \nThe upper right panel of Figure 1 shows orbital periods and distances. Most of the binaries in our sample are within 1 kpc of the Sun and have periods between 100 and 1000 days. There are no binaries with orbital periods close to 1 year owing to the degeneracy between such orbits and parallactic motion. The period distribution peaks near 600 days, likely because short-period orbits are smaller and can only be resolved at close distances, while significantly longer orbits would not have been well-sampled during the ∼ 1000-day observing window for DR3 solutions. The period distribution of the NS candidate sample is fairly similar to that of all astrometric binaries. \nThe lower left panel of Figure 1 shows the sources' distribution on the sky in Galactic coordinates, with the Galactic center at the center. Both our candidates and the full astrometric binary sample are distributed all across the sky. Some evidence of the Galactic disk is evident in the distribution of all binaries, but the distribution is heavily affected by the Gaia scanning law, and most of the NS candidates are at high latitude. \nFinally, the lower right panel of Figure 1 shows the masses of the luminous stars in our sample and the minimum mass ratio inferred from their astrometric mass ratio function (AMRF; Shahaf et al. 2019). The luminous star masses plotted here are taken from the gaiadr3.binary masses table following Shahaf et al. (2023b); more accurate masses for our candidates are measured in Section 4. The astrometric mass ratio functions are also calculated based on Gaia data alone following Shahaf et al. (2023b), without accounting for our follow-up RVs. By virtue of our selection, the objects in \nFig. 1.Black points show all binaries with astrometric orbital solutions published in Gaia DR3. Red points show the 21 objects presented in this work, which have astrometrically-inferred companion masses M 2 > 1 . 25 M ⊙ . Upper left : dereddened color-magnitude diagram. Most candidates are solar-type stars near the main-sequence. Upper right : orbital period and distance. Candidates have orbital periods of ∼ (100 -1000) d and are within ∼ 1 kpc of the Sun. Lower left : Galactic coordinates, with the Galactic center in the middle of the plot. Imprints of the Gaia scanning law are visible; most NS candidates are at high latitude. Bottom right : luminous star mass and minimum mass ratio, M 2 /M ⋆ . The NS candidates have some of the highest estimated mass ratios in the astrometric binary sample. \n<!-- image --> \n- \n<!-- image --> \nq \n<!-- image --> \n/circledot \nour sample have among the largest minimum mass ratios of binaries with solutions published in DR3. \nThat the unseen companions to objects in our sample are massive can be appreciated intuitively from Figure 2, which compares their periods and the physical size of their photocenter orbits to other binaries with astrometric solutions in DR3, including Gaia BH1 and BH2. According to Kepler's 3rd law, more massive and darker companions are found above and to the left of the population of luminous binaries and triples in this parameter space (see El-Badry et al. 2023a). NS companions are expected to be found between the populations of luminous binaries and BH companions, and this is indeed where our candidates are clustered. Close inspection of Figure 2 will reveal that a few additional binaries fall above the candidates studied here, suggestive of higher masses. These are primarily sources where our RV follow-up showed the astrometric solution to be spurious, and red giants, for which a more massive MS companion could not be ruled out.", '3. SPECTROSCOPIC FOLLOW-UP': 'We measured multi-epoch RVs for all targets, primarily using high-resolution spectrographs on two 2m-class telescopes. Our targets are bright, with most having apparent magnitudes G = 13 -14. We additionally obtained single-epoch higher-SNR spectra for most targets with 8-10m class telescopes. We describe all the spectroscopic observations below.', '3.1. FEROS': 'We obtained 129 spectra with the Fiberfed Extended Range Optical Spectrograph (FEROS; Kaufer et al. 1999) on the 2.2m ESO/MPG telescope at La Silla Observatory (programs P109.A-9001, P110.A-9014, P111.A-9003, P112.A-6010, and P113.26XB). Some observations used 2 × 2 binning to reduce readout noise at the expense of spectral resolution; the remainder used 1 × 1 binning. The resulting spectra have resolution R ≈ 40 , 000 (2 × 2 binning) and R ≈ 50 , 000 (1 × 1 binning). Exposure times ranged from 1200 to 3600 seconds, depending on source brightness. We reduced the data using the CERES pipeline (Brahm et al. 2017), which performs bias-subtraction, flat fielding, wavelength cali- \nTABLE 1 \nBasic properties of the sample. M ⋆ is the inferred mass of the luminous star from fitting the SED and spectroscopic metallicity. P orb and M 2 are the orbital period and mass of the unseen companion. These quantities, as well as the eccentricity and parallax, ϖ , are from joint fits of the Gaia astrometry and our RV follow-up. G is the apparent magnitude measured by Gaia , and N RVs is the number of follow-up RVs we have measured. The source J1432-1021 was already studied by El-Badry et al. (2024) under the name Gaia NS1. \nFig. 2.Comparison of NS candidates presented here (red stars) to the rest of the Gaia DR3 astrometric binary sample. At fixed period, dark and massive companions produce larger photocenter orbits than normal stellar companions. For typical solar-type primaries, NS companions produce photocenter orbits that are smaller than those of BH binaries, but larger than those of luminous binaries and triples. \n<!-- image --> \nbration, and optimal extraction. The pipeline measures and corrects for small shifts in the wavelength solution during the course a night via simultaneous observations of a ThAr lamp with a second fiber.', '3.2. TRES': 'We obtained 155 spectra using the Tillinghast Reflector Echelle Spectrograph (TRES; F"ur\'esz 2008) mounted on the 1.5 m Tillinghast Reflector telescope at the Fred Lawrence Whipple Observatory (FLWO) on Mount Hopkins, Arizona. TRES is a fibrefed echelle spectrograph with a wavelength range of 390-910 nm and a resolving power of R ∼ 44 , 000. The spectra were extracted as', '3.3. MIKE': "We obtained 7 spectra with the Magellan Inamori Kyocera Echelle (MIKE) spectrograph on the Magellan Clay telescope at Las Campanas Observatory (Bernstein et al. 2003). We used the 0.5' slit and exposure times ranging from 600 to 1200 seconds, yielding spectral resolution R ∼ 40 , 000 on the blue side and R ∼ 55 , 000 on the red side. The typical SNR at 5800 ˚ A was ∼ 30 and the total wavelength coverage was ∼ 3330 -9680 ˚ A. We reduced the spectra with the MIKE Pipeline within CarPy (Kelson et al. 2000; Kelson 2003) and subsequently flux-calibrated them using observations of a standard star. We merged the orders into a single spectrum, weighting by inverse variance in the overlap regions.", '3.4. HIRES': 'We obtained 6 spectra using the High Resolution Echelle Spectrometer (HIRES; Vogt et al. 1994) on the 10m Keck I telescope on Maunakea. The data were obtained and reduced using the standard California Planet Survey setup (CPS; Howard et al. 2010), including use of the C2 decker (0.86 arcseconds × 14 arcseconds), which yields spectra with R ≈ 55 , 000 and wavelength coverage over most of 3700-8000 ˚ A. We used 600 second exposures, yielding a typical SNR of 40 per pixel at 6000 ˚ A. The CPS reduction includes sky-subtraction using the long C2 decker. We merged spectra from individual orders using the same procedure as with the MIKE data.', '3.5. PEPSI': 'We obtained 10 spectra using the Potsdam Echelle Polarimetric and Spectroscopic Instrument (PEPSI; Strassmeier et al. 2015) spectrograph on the Large Binocular Telescope in binocular mode. We used the 300 µ m fiber, \nwhich has a diameter of 2.3 arcsec on sky, and the CD2 and CD5 cross-dispersers on the blue and red side, respectively. Exposure times ranged from 300 to 1200s. The spectra were reduced and orders were merged as described in Strassmeier et al. (2018); they cover the wavelength ranges of 4222-4792 ˚ A and 6236-7433 ˚ A with spectral resolution R ≈ 50 , 000.', '3.6. ESI': 'We obtained 1 spectrum with the Echellette Spectrograph and Imager (ESI; Sheinis et al. 2002) on the 10m Keck II telescope on Maunakea. We used a 300 second exposure with the 0.3 arcsec slit, yielding a resolution R ≈ 12000 and SNR ≈ 50, with useful wavelength coverage of 3900-10000 ˚ A. We reduced the data using the MAuna Kea Echelle Extraction (MAKEE) pipeline, which performs bias-subtraction, flat fielding, wavelength calibration, and sky subtraction, and we refined the wavelength solution using telluric absorption lines.', '3.7. Other spectroscopy': 'Early in our follow-up program, we measured RVs for several candidates using spectra from lower-resolution spectrographs, including Keck/DEIMOS (Faber et al. 2003), Magellan/MagE (Marshall et al. 2008), and Palomar P200/DBSP (Oke & Gunn 1982). These RVs were used to rule out some candidates with spurious solutions but were not included in our final analysis due to their larger uncertainties ( ≳ 3 kms -1 ). A few objects in our sample also have archival spectra from the LAMOST survey (Cui et al. 2012). However, these typically have RV uncertainties of at least a few km s -1 - about 100 times larger than our observations - so we opted not to include them in our analysis.', '3.8. RVs and offsets': 'We measure RVs for each echelle order by crosscorrelating a synthetic spectral template with the normalized spectrum. We report the median across orders as the measured RV for each epoch and calculate the uncertainty as the standard deviation across orders divided by the square root of the number of orders. We use a Kurucz spectral template from the BOSZ library (Bohlin et al. 2017) with surface gravity log( g/ cms -2 ) = 4 . 0 and effective temperature and metallicity matched to each target (Section 3.9). We use non-rotating templates for all targets except J0230+5950, which is rotating with v sin i ≈ 15 kms -1 . \nFor the FEROS spectra, we use 15 orders covering wavelengths between 450 and 670 nm when calculating RVs. For the TRES spectra, we use 31 orders covering 420-670 nm. The median uncertainty of the FEROS RVs is ≈ 0 . 06 km s -1 , while the median uncertainty of the TRES RVs is ≈ 0 . 05 km s -1 . \nAs described by El-Badry et al. (2024), we fit for a single global RV offset between TRES and FEROS. We found an offset of 0 . 16 kms -1 , in the sense that RV FEROS = RV TRES -0 . 16 kms -1 . This offset has already been applied to our reported RVs. Because we generally only obtained one spectrum per target for MIKE, HIRES, PEPSI, and ESI, we did not fit for offsets for these instruments, but instead adopted a conservative \n0.5 km s -1 minimum uncertainty for all the RVs measured with these instruments. All the RVs are listed in Appendix D and provided in machine-readable form as supplementary material.', '3.9. Metallicities': 'The radius and temperature of a MS star of a given mass depend on its metallicity, so metallicity estimates are important for reliable mass estimates of the luminous stars, and in turn, mass estimates of their companions. We collate metallicity measurements from several sources:', '3.9.1. BACCHUS': 'For the MIKE, HIRES, and FEROS spectra, we measured metallicities and atmospheric parameters using the Brussels Automatic Code for Characterizing High accUracy Spectra (BACCHUS; Masseron et al. 2016; Hayes et al. 2022). The code performs 1D LTE spectral synthesis to determine stellar parameters from Fe excitation/ionization balance; i.e., the requirement that lines with different excitation potentials all imply the same Fe abundance. The reported metallicity [Fe/H] is the mean Fe abundance calculated over lines in the VALD atomic linelist with a wavelength coverage of 4200 to 6700 ˚ A. Here we assume that the detailed abundance pattern traces solar values; detailed abundances of these stars will be investigated in future work. The errors reported by BACCHUS represent the scatter in the implied abundances between the different lines and methods of abundance calculations but do not take into account other systematic uncertainties.', '3.9.2. SPC': 'We fit the TRES spectra using the Stellar Parameter Classification (SPC) tool (Buchhave et al. 2012), which cross-correlates the normalized TRES spectra with a grid of synthetic spectra in the wavelength range of 5050 to 5360 ˚ A, centered on the Mg I b triplet. SPC infers the metallicity [M/H], effective temperature, T eff , and surface gravity log g by fitting the peaks of the crosscorrelation functions with a three-dimensional polynomial in stellar parameters. Given systematic uncertainties in the synthetic stellar spectra, error floors on the derived [M/H] and T eff values are ∼ 0.08 dex and ∼ 50 K, respectively (Buchhave et al. 2012; Furlan et al. 2018).', '3.9.3. Gaia XP metallicities': 'All of the stars in our sample have metallicity estimates (along with T eff and log g ) calculated as described by Andrae et al. (2023) using Gaia XP low-resolution spectra. For bright stars within the temperature range of our sample, the expected precision of these metallicities is ∼ 0 . 1 dex. However, the metallicities published by Andrae et al. (2023) are expected to be less reliable for astrometric binaries - even those with dark companions - because they are calculated based on the gaiadr3.gaia source parallax, which comes from a 5-parameter astrometric solution that neglects orbital motion. These parallaxes can be significantly biased for astrometric binaries with large photocenter wobbles. We therefore recalculate metallicities for our targets using the parallaxes in the gaiadr3.nss two body orbit catalog, otherwise using \nFig. 3.Metallicities measured from high-resolution spectra (yaxis), and from low-resolution Gaia XP spectra (x-axis). The XP metallicities are inferred with a modified version of the Andrae et al. (2023) model that uses the parallax from the astrometric binary solution rather than the one from the 5-parameter solution reported in the gaia source catalog. The two independent metallicity measurements are in good agreement. \n<!-- image --> \nthe same empirically trained machine-learning models as Andrae et al. (2023).', '3.9.4. Metallicity comparisons': 'For each source, we report both the Gaia XP metallicity and a metallicity measured from a high-resolution spectrum in Table 2. Several sources were observed with more than one high-resolution spectrograph. In these cases, we adopt the metallicity from the highest-SNR spectrum. The results are reported in Table 2 and shown in Figure 3. The agreement between the Gaia XP and high-resolution metallicities is good, implying that the XP metallicities are robust for this sample and that highresolution spectra will not be essential for bulk metallicity measurements in future follow-up efforts, though they are required for RV measurements. \nWe also investigated the consistency of metallicities measured from high-resolution spectra by different pipelines. For sources with more than one high resolution spectrum, we found a median absolute deviation of 0.08 dex between measurements, which is comparable to the reported uncertainties.', '4. SPECTRAL ENERGY DISTRIBUTIONS AND LUMINOUS STAR MASSES': "We inferred the masses and evolutionary states of the luminous stars by fitting their broadband spectral energy distributions (SEDs) with single-star models. In the optical, we use the synthetic photometry in the SDSS ugriz bands constructed from Gaia BP/RP spectra and empirically calibrated as described by Gaia Collaboration et al. (2023a). We supplement this with near-infrared photometry from the 2MASS (Skrutskie et al. 2006) and WISE (Wright et al. 2010) surveys. To search for possible light contributions from hot WD companions, we also \nretrieved UV photometry from GALEX (Martin et al. 2005), but we did not include it in our SED fits. \nThe SEDs of all sources are shown in Figure 4. While optical and near-infrared photometry is available for all sources, several sources are outside the published footprints of GALEX in one or both of its UV bands. In cases where a source was within the footprint of a GALEX observation but was was not detected, we plot the 3 σ upper limit. Sources with no upper limits or detection shown in the NUV or FUV were not observed in that band. \nWe fit the SEDs using MINESweeper (Cargile et al. 2020), a code designed for joint modeling of stellar photometry and spectra. We only use the photometric modeling capabilities of the code but place a prior on the present-day metallicity from spectroscopy. The free parameters of the fit are the initial mass and metallicity of each star, its age (as parameterized by the 'equivalent evolutionary point'), the parallax, and the foreground extinction. For each call to the likelihood function, the mass, metallicity, and equivalent evolutionary point are converted to a radius and effective temperature using MIST isochrones (Choi et al. 2016), and are then used to predict a model SED that is compared to the data. We place priors on the parallax from the Gaia 12-parameter binary solution and set an age upper limit of 13 Gyr. We also use a metallicity prior from the high-resolution spectra (Table 2), and an extinction prior from 3D dust maps. We again use the Green et al. (2019) map for sources with declination δ > -30 deg, and the Lallement et al. (2022) map for sources farther south. \nThe SEDs and best-fit models of all candidates are shown in Figure 4, and best-fit parameters for each target are listed in the upper right corner of each panel. Most of the luminous stars have masses near 1 M ⊙ . This is also true for the full astrometric binary catalog published in Gaia DR3 and mostly reflects the distance and flux limits of the astrometric binary sample. Several of the MS stars are somewhat evolved (i.e., their radii are up to 70% larger than stars of the same mass at the zero-age main sequence). \nWe removed one initial target, Gaia DR3 ID 747174436620510976, from the sample because it was not possible to obtain a good single-star fit to the SED without changing the Gaia parallax. The star has near solarmetallicity star and an SED that suggests T eff ≈ 5450 K and R ≈ 0 . 60 R ⊙ . However, this radius is too small for any main sequence star of the observed effective temperature. A possible explanation is that the Gaia parallax is underestimated and the true radius is ≈ 0 . 9 R ⊙ , as expected for a main-sequence star. This would also cast doubt on the other parameters of the astrometric solution, and our follow-up RVs are in modest tension with the predictions of the astrometric solution (Appendix A), so we removed the source from the sample. \nThe best-fit parameters from SED fitting are listed in Table 2. We set a minimum uncertainty of 0 . 03 M ⊙ on M ⋆ to account for systematic uncertainties in the stellar models. This uncertainty propagates to the uncertainties on M 2 calculated in the next section. \nOur mass constraints are based on single-star evolutionary models. This is reasonable because the expected lifetimes of the companions' progenitors are much shorter than the lifetimes of the luminous stars. This means than any mass transfer would have occurred before the \nFig. 4.Spectral energy distributions and best-fit single-star models. The parameters of the best-fit model are listed in each panel. UV photometry is plotted where available, but not included in the fit. All targets' SEDs are reasonably well-fit by a single-star model. Parameters from SED fitting are listed in Table 2. \n<!-- image --> \nTABLE 2 \nParameters of the luminous stars. We list both metallicities measured from high-resolution spectra and those measured from low-resolution Gaia XP spectra following Andrae et al. (2023). The high-resolution metallicities are used as a prior when fitting the broadband SEDs. The metallicities are compared in Figure 3, and the SED fits are shown in Figure 4. \nluminous stars experienced any significant evolution. As discussed by El-Badry et al. (2024), there is also no plausible evolutionary scenario in which the luminous stars have significantly lower masses than implied by singlestar models.", '5. JOINT FITTING OF ASTROMETRY AND RVS': "Our joint modeling of the RVs and Gaia astrometry assumes a Keplerian 2-body orbit. The luminous star has mass M ⋆ , while the companion has mass M 2 . The orbit is specified by its period, P orb , eccentricity, e , periastron time, T p , and three angles describing the orientation: the inclination, i , longitude of the ascending node, Ω, and argument of periastron, ω . The center-of-mass motion of the binary is described by a systemic velocity, γ , and five astrometric parameters: the right ascension α and declination δ , the parallax ϖ , and the proper motions µ ∗ α and µ δ . \nThe semimajor axis a is set by Kepler's 3rd law: \na = ( P 2 orb G ( M ⋆ + M 2 ) 4 π 2 ) 1 / 3 , (1) \nwhile the semimajor axis of the luminous star's orbit is \na 1 = a ( q 1 + q ) , (2) \nwhere q = M 2 /M ⋆ is the mass ratio. The RV semiamplitude of the star's orbit is \nK ⋆ = 2 πa 1 P orb √ 1 -e 2 sin i. (3) \nFinally, the angular photocenter semimajor axis is given by \n˚ a 0 = a d ( q 1 + q -ϵ 1 + ϵ ) . (4) \nHere d is the distance to the binary and ϵ is the flux ratio in the G -band. We define ϵ = F G, 2 /F G,⋆ , where F G, 2 and F G,⋆ represent the G -band flux of the companion and the luminous star, respectively. For a dark companion ( ϵ = 0), the photocenter simply traces the primary (i.e., the star whose RVs are being measured). A nonzero value of ϵ increases the primary semimajor axis, a 1 , that corresponds to a given photocenter semimajor axis. \nThe Gaia astrometric solutions are expressed as constraints on 4 or 6 Thiele-Innes parameters, for Orbital and AstroSpectroSB1 solutions, respectively: \nA =˚ a 0 (cos ω cos Ω -sin ω sin Ω cos i ) (5) \nB =˚ a 0 (cos ω sin Ω + sin ω cos Ω cos i ) (6) \nF = -˚ a 0 (sin ω cos Ω + cos ω sin Ω cos i ) (7) \nG = -˚ a 0 (sin ω sin Ω -cos ω cos Ω cos i ) (8) \nC = a 1 sin ω sin i (9) \nH = a 1 cos ω sin i. (10) \nIn the convention used here and in the Gaia archive, the astrometric parameters A,B,F , and G have angular units (mas), while the spectroscopic parameters C and H have physical units (au). In the Gaia data processing, C and H are constrained by the measured RVs, while \nA,B,F , and G are constrained by astrometry (see Pourbaix et al. 2022). Our standard fit has 14 free parameters: P orb , e , M ⋆ , M 2 , i , Ω, ω , ϖ , α , δ , µ ∗ α , µ δ , T p , and γ . In the fiducial modeling, we assume a dark companion ( ϵ = 0). We also experimented with a 15-parameter fit in which ϵ is left free (Section 5.4). \nFor each call to the likelihood function, we construct the predicted vector of Gaia -constrained parameters, θ Gaia . For objects with Orbital solutions, this is given by \nθ Gaia = [ α, δ, ϖ, µ ∗ α , µ δ , A, B, F, G, e, P orb , T p ] , (11) \nwhile for those with AstroSpectroSB1 solutions, \nθ Gaia = [ α, δ, ϖ, µ ∗ α , µ δ , A, B, F, G, C, H, γ, e, P orb , T p ] . (12) \nWe then calculate a likelihood term that quantifies the difference between these quantities and the Gaia constraints: \nln L Gaia = -1 2 ( θ Gaia -µ Gaia ) ⊺ Σ -1 Gaia ( θ Gaia -µ Gaia ) . (13) \nHere µ Gaia and Σ Gaia represent the vector of best-fit parameters constrained by Gaia and their covariance matrix, which we construct from the corr vec parameter reported in the Gaia archive. \nWe additionally predict the RVs of the luminous star at the array of times t i at which we obtained spectra. This requires calculating K ⋆ from Equation 3 and iteratively solving Kepler's equation (e.g. Murray & Dermott 1999). When then calculate a RV term in the likelihood, \nln L RVs = -1 2 ∑ i (RV pred ( t i ) -RV i ) 2 σ 2 RV ,i , (14) \nwhere RV i and RV pred ( t i ) are the measured and predicted RVs, with their uncertainties σ RV , i . The full likelihood is then given by \nln L = ln L Gaia +ln L RVs . (15) \nTo assess the relative importance of the Gaia data and our follow-up RVs in constraining the orbits, we also carry out a fit in which we omit the ln L RVs term from Equation 15. This essentially samples from the Gaia covariance matrix, but it incorporates our constraints on M ⋆ , transforms from the Thiele-Innes coefficients to a more physically interpretable set of parameters, and allows a direct constraint on M 2 . \nWe use flat priors on all parameters except M ⋆ , for which we use a normal distribution informed by our SED fit (Section 4). We sample from the posterior using emcee (Foreman-Mackey et al. 2013) with 64 walkers, taking 3000 steps per walker after a burn-in period of 3000 steps. We choose the initialization point for the sampler by transforming the best-fit Gaia parameters to Campbell elements using nsstools (Halbwachs et al. 2023).", '5.1. Inclination sign degeneracy': "Astrometric orbits describe the plane-of-the-sky motion of a binary's photocenter. With astrometry alone, the data are always equally consistent with an orbit that has inclination + i and one that has inclination -i . Mathematically, this is evident from the fact that A , B , F , and \nG depend only on cos i , an even function. RVs break this degeneracy. However, because orbit-fitting posteriors are usually multi-modal, our MCMC sampling often fails to converge if initialized far from the best-fit solution. For this reason, we separately initialize two sets of MCMC chains near the positive and negative astrometry-only solutions, and retain only the chain that reaches a higher maximum likelihood.", '5.2. Results': 'The results of all our joint fits are reported in Table 4 in Appendix C. Constraints on a few parameters are also listed in Table 1. Figures 5 and 6 compare our measured RVs to predictions of the Gaia -only solution (cyan) and the Gaia +RVs solution (black). We plot predicted RV curves for 50 random samples from the posterior in order to show the uncertainty in these predictions. For the Gaia -only fits, we fix the center-of-mass RV (which is not constrained by astrometry) to the maximum-likelihood value from the joint fit. For all the targets shown here, the RVs are consistent with predictions of the Gaia -only solution, meaning that at least some samples predict RV curves that overlap with the observed RVs. \nNot surprisingly, the predicted RV curves from the joint RVs+astrometry fits are much more tightly constrained than those from the Gaia -only fits. The residuals in Figure 6 show that for most targets, the RVs are generally consistent with the joint solution within (1 -2) σ . Some systems have χ 2 /N RVs < 1 . 0, suggesting that their RV uncertainties are overestimated. This could occur if a small number of outlier orders dominate the standard deviation in the RV uncertainty calculation. A few candidates have χ 2 /N RVs = 1 . 0 -2 . 0, implying modestly underestimated RV uncertainties.', '5.2.1. Consistency of the astrometric and RV constraints': "Figure 7 compares our constraints on several fitting parameters, as well as parameters that are transformations of them, from the Gaia -only and Gaia +RVs fits. For most systems, the uncertainties in all parameters are much smaller in the Gaia +RVs fits than in the Gaia -only fits. This is true even for parameters such as the inclination that are not constrained directly by RVs, because the Gaia -only constraints include significant covariances between parameters. \nThe two sets of parameters are generally consistent within (1 -2) σ , suggesting that the astrometric solutions and their uncertainties are generally reliable. There is little evidence of systematic biases in most of the Gaia -only parameters, though the inferred M 2 values are on average lower in the joint fits than in the Gaia -only fits. As we discuss in Section 5.3, this is likely a consequence of how the candidates were selected. \nEl-Badry et al. (2024) found that the astrometry-only and astrometry+RVs constraints for Gaia NS1 (J14321021) were inconsistent at the 3 σ level, suggesting that the astrometric uncertainties for that source were underestimated. Similarly, Chakrabarti et al. (2023) and Nagarajan et al. (2023) found evidence for underestimated uncertainties in the astrometric orbit of Gaia BH1. Figure 7 implies that most of our candidates have more robust astrometric uncertainties: indeed, Gaia NS1 has the most discrepant solution among all the objects in the \nsample. This could arouse suspicion that the companion's mass is simply overestimated. However, El-Badry et al. (2024) showed that RVs alone require the companion to be the most massive in the sample, even if the astrometric orbit constraints are disregarded. \nConsidering the 4 Thiele-Innes parameters A , B , F , and G , only 22 out of 84 parameters across 21 objects differ by more than 1 σ , and only 5 differ by more than 2 σ . The Gaia -only and Gaia +RVs parallaxes are consistent within 1 σ for 17 candidates, and within 2 σ for 20 candidates.", '5.3. Mass functions': 'Figure 8 compares several limits on the unseen companion masses between the Gaia -only and Gaia +RVs fits. The left panel shows the RV mass function, \nf ( M 2 ) = P orb K 3 ⋆ 2 πG ( 1 -e 2 ) 3 / 2 (16) \n= M 2 ( M 2 M 2 + M ⋆ ) 2 sin 3 i (17) \nwhich represent a lower limit on the companion mass that corresponds to the limiting case where the orbit is edgeon and the luminous star is massless. K ⋆ is the RV semiamplitude as measured from the joint fit (Equation 3). \nThe middle panel shows the astrometric mass function, \nf ( M 2 ) ast = ( ˚ a 0 ϖ ) 3 ( P orb yr ) -2 . (18) \nFor a dark companion, \nf ( M 2 ) ast = M 2 ( M 2 M 2 + M ⋆ ) 2 . (19) \nThat is, f ( M 2 ) ast represents a lower limit on the companion mass that incorporates the known inclination from astrometry but treats the luminous star as massless. Finally, the right panel shows the minimum companion mass implied by the RV mass function and our assumed luminous star mass, for an edge-on orbit. We obtain this by setting i = 90 deg and solving Equation 17 for M 2 , setting M ⋆ to the value inferred from the SED (Table 2). We propagate uncertainties on all parameters by calculating these mass functions directly from the MCMC samples. \nAlthough the Gaia -only and Gaia +RVs mass functions are generally consistent, the Gaia +RVs constraints on average favor lower mass functions. This means that, on average, the observed RV semi-amplitudes are somewhat lower than predicted by the astrometric solutions. Since our targets are selected from the tails of the companion mass function distribution (e.g. Figure 1), it is expected that more objects in the sample will scatter toward higher masses than toward lower masses. That is, binaries with noise scattering their best-fit masses toward lower values would not have entered the sample in the first place.', '5.4. Flux ratio constraints': 'Our modeling thus far has assumed that the companion is dark. In this case, the semimajor axis of the photocenter orbit, ˚ a 0 , will be identical to the semimajor axis of \n1 \n- \nFig. 5.Comparison of our measured RVs (red) to predictions of the Gaia astrometric solutions (cyan). Individual lines show random draws from the posterior. For AstroSpectroSB1 solutions, the systemic RV is set to the value measured by Gaia ; for Orbital solutions, it is set to the best-fit value from our joint astrometry+RV fits (Figure 6). Systems with long periods ( P orb ≳ 700 days) have significant uncertainties in their Gaia -only periods and phases at the time of our observations. The RVs are broadly consistent with the Gaia predictions. \n<!-- image --> \n1 \n- \nFig. 6.Joint fits of our RVs (red) and the Gaia constraints. Lower sub-panels show residuals compared to the best-fit solution. Inclusion of RVs in the fit yields much tighter constraints than those from Gaia alone (Figure 5). The RV residuals are generally consistent with 0 within their (1 -2) σ uncertainties. \n<!-- image --> \nFig. 7.Orbital and astrometric parameters inferred from joint fits of the Gaia solution and our RVs (vertical axis) and the Gaia solution alone (horizontal axis). The photocenter semi-major axis, ˚ a 0 , and the Thiele-Innes coefficients, A TI , B TI , F TI , and G TI , are derived quantities, while the other parameters are free parameters in our fits. Uncertainties on most parameters are much smaller in the joint fits, but the two sets of constraints are generally consistent at the (1 -2) σ level. Inclinations are mirrored about 90 deg (i.e., 100 deg is shown as 80 deg) for visualization only. \n<!-- image --> \nGaia \nonly \nthe star whose RVs are being measured, ˚ a 1 . To explore the possibility of a luminous companion, we tried repeating the fit while adding the flux ratio ϵ (Equation 4) as \na free parameter. \nWe first tried fitting with no bounds on ϵ . In this case, we found negative best-fit values of ϵ for a ma- \nV \nR \n+ \na \ni \na \nG \nFig. 8.Comparison of mass functions for the unseen companions inferred from the Gaia solution alone (horizontal axis) and joint fitting of that solution and our RVs (vertical axis). Left and middle panels show the spectroscopic and astrometric mass functions. Right panel shows the minimum companion mass implied by the RV mass function if the orbit were edge-on . The mass functions inferred from Gaia +RVs are in most cases consistent with those from Gaia alone, but they are systematically lower. Because our candidates were selected on the basis of high astrometric mass functions, and massive companions are intrinsically rare, their companion masses are overestimated on average. That is, systems with underestimated masses did not enter the sample. \n<!-- image --> \njority of the sources, with a median and ± 1 σ range of ϵ = -0 . 01 ± 0 . 03 across the full sample. Most sources have ϵ constrained with a 1 σ uncertainty of σ ϵ ≲ 0 . 02. Negative values of ϵ are unphysical, but mathematically allowed (e.g. Equation 4). A preference for negative ϵ simply reflects observed RV variability with a lower amplitude than predicted by the best-fit astrometric solution with ϵ = 0; i.e., it is a consequence of the same selection effect that causes the mass functions in Figure 8 to be overestimated. \nNext, we imposed a constraint that ϵ must be positive. In this case, the posterior constraints on ϵ pile up against 0 for most sources. The median upper limit on ϵ is ϵ < 0 . 012 (1 σ ), or ϵ < 0 . 032 (2 σ ). That is, the combination of RVs and astrometry strongly disfavor luminous companions, because luminous companions would imply a larger a 1 for fixed ˚ a 0 , and this would imply a larger RV variability amplitude than is observed.', '5.5. Astrometric phase coverage': "One possible concern is that the Gaia observations could have resulted in poor orbital phase coverage. This can occur, for example, for orbital periods that are close to a year, half a year, or several periods related to the Gaia scanning law (El-Badry et al. 2023a; Holl et al. 2023). While actual epoch-level astrometric data are not published in DR3, it is possible to predict when a source should have been observed using the Gaia observation scheduling tool (GOST) 3 . Given a source's coordinates, GOST returns a list of observation times when the scanning law predicts that the source will transit across the Gaia focal plane. \nFigure 9 shows sky-projected photocenter orbits for all candidates. Individual gray lines show predictions for 50 random draws from the posterior of the joint fit. Red points show interpolations of the scan times predicted by GOST onto the orbit corresponding to the marginal- \nized posterior median. Overall, most of the sources have good astrometric phase coverage, with Gaia observations sampling most of the predicted orbital ellipse. \nIt is not guaranteed that Gaia will actually obtain data on each source at the predicted times, as GOST does not account for gaps between CCDs and issues such as micrometeoroids impacts that cause temporary gaps in the datastream. In each panel of Figure 9, we list both the number of visibility periods predicted by GOST and the number actually used in calculating the astrometric solution ( visibility periods used ; a visibility period is a group of observations separated from other observations by at least 4 days). The number of visibility periods used ranges from 80 -100% of the number predicted by GOST, suggesting that the scan times shown in Figure 9 are a reasonable but not perfect approximation of those used in producing the astrometric solutions. The median number of visibility periods used is 21, significantly higher than the minimum of 12 that is required to constrain a 12-parameter astrometric binary solution.", '5.6. Astrometric solution quality diagnostics': "As an additional check on the reliability of our candidates' astrometric solutions, Figure 10 compares several quality diagnostics of these candidates to the larger sample of all astrometric binaries from Gaia DR3. Cyan points show the Gaia astrometric solutions alone, while red points show joint Gaia +RVfits. Hollow points show three sources we found to have spurious solutions after RV follow-up (Appendix A). \nFirst considering the Gaia -only solutions, the NS candidates have goodness of fit values and parallax uncertainties that are broadly representative of the larger astrometric binary sample. They have higher-significance photocenter ellipse measurements (i.e., larger ˚ a 0 /σ ˚ a 0 ) than most other astrometric binaries, reflecting the fact that NS companions produce large photocenter orbits at fixed period (Figure 2). The sources with spurious solutions have similar goodness of fit values to those with \n2 \nFig. 9.Sky-projected orbits of the luminous stars. The angular orbits have been multiplied by distance, so that the displayed orbital sizes correspond to relative physical sizes. Red points mark Gaia observations predicted by GOST , showing that all of the targets in our sample are expected to have good astrometric phase coverage. Note that we do not have access to the actual measured ∆RA and ∆Dec values; only to the predicted scan times. Red text in each panel indicates the number of visibility periods that are predicted by GOST , as well as the number actually used in calculating the astrometric solution. \n<!-- image --> \ngood solutions, though their parallax uncertainties are on average larger at fixed apparent magnitude. \nThe uncertainties from the joint Gaia +RV fits are often significantly smaller than those from Gaia alone. \nEven the parallax, which is not directly constrained by RVs, has smaller uncertainties in the joint Gaia + RVs fit, because the Gaia solutions include significant covariances between parallax and other orbital parameters that are directly constrained by RVs. We emphasize, however, that the uncertainties shown in Figure 10 are purely statistical. The parallax zeropoint and uncertainties of the astrometric binary solutions have yet to be investigated in detail. Nevertheless, we conclude from Figure 10 and the good agreement between measured and predicted RVs that the astrometric solutions of the candidates in our sample are robust. We also conclude that spurious solutions cannot always be easily identified on the basis of their astrometric uncertainties and quality flags.", '6.1. Nature of the unseen companions': "The orbits of the binaries in our sample are wellcharacterized, and the companion masses are measured with typical uncertainties of a few percent. However, we have not detected light from the companions, leaving their astrophysical nature uncertain. Our joint fitting of astrometry and RVs places upper limits of a few percent on the G -band flux ratio, ϵ = F G, 2 /F G,⋆ (Section 5.4). As we show below, this rules out all plausible non-degenerate companions. \nWe begin by considering the astrometric mass function (Equation 19). For a luminous companion with flux ratio ϵ and mass ratio q = M 2 /M ⋆ , this quantity can be expressed as (e.g. Shahaf et al. 2019): \nf ( M 2 ) ast = M ⋆ | q -ϵ | 3 (1 + ϵ ) 3 (1 + q ) 2 . (20) \nGiven an observationally constrained M ⋆ and f ( M 2 ) ast , we can solve Equation 20 for q , and thus for M 2 , for any possible ϵ . In Figure 11, the black line shows this constraint for a representative source in our sample, J2244-2236, which has M ⋆ = 1 . 00 ± 0 . 03 M ⊙ and f ( M 2 ) ast = 0 . 503 ± 0 . 007 M ⊙ . The dynamically-implied mass of the companion increases with its assumed luminosity. Physically, this reflects the fact that a brighter companion dilutes the light from the primary more, such that a higher companion mass is required to explain the same photocenter orbit. \nWe can then consider possible companion types: \n- 1. A single MS star : The dotted cyan line in Figure 11 shows the expected flux contribution versus mass relation for a MS companion. We take this from a 5 Gyr-old, solar-metallicity MIST isochrone. Because the mass supplied by such a companion is much less than the astrometric constraint for all flux ratios, a single MS companion is ruled out.\n- 2. An inner binary containing two MS stars : The yellow dashed line in Figure 11 shows the mass and flux expected if the companion is a binary containing two MS stars, so the full system is a triple. We assume the two stars have equal mass, since this yields the highest mass-to-light ratio. We use the same isochrone as for a single star but assume twice the total mass and light. This still falls short of \nthe astrometric mass constraint, so the companion cannot be a close MS+MS binary. \n- 3. An inner triple containing three MS stars : The red dot-dashed line in Figure 11 shows the case where the companion is a triple containing three MS stars (i.e., the full system is a quadruple). We assume all three inner components have the same mass. Even this falls below the astrometric constraint for all flux ratios, so the companion cannot be an inner triple. In addition, a quadruple system with four stars in such a tight orbit would be unlikely to be stable over long timescales.\n- 4. An inner binary containing a WD and a MS star : The blue dashed line in Figure 11 corresponds to an inner binary containing a 1 M ⊙ WD and a MS star. We assume that the WD is sufficiently cold that it does not contribute any light in the optical. Such an inner binary could reproduce the observed astrometric mass function if the MS star has a mass of ≈ 0 . 6 M ⊙ , in which case it would contribute ≈ 4% of the total light. However, as discussed in Section 5.4, the RVs rule out a flux ratio above 2% for this source, because a 1 . 6 M ⊙ companion would produce larger-amplitude RV variability. Only a tight binary containing a WD with M ≳ 1 . 1 M ⊙ and a low-mass MS star could match the data.\n- 5. An inner binary containing two WDs : All of the objects in our sample could in principle be explained by a tight inner binary containing two cold and relatively massive WDs. The total mass of the inner binaries would need to be near the Chandrasekhar mass. Chandrasekhar-mass close WD+WD binaries are relatively rare: despite decades of dedicated searches, none have been conclusively identified. Empirical limits suggest a space density comparable to the space density of neutron stars (e.g. Badenes & Maoz 2012). As discussed by El-Badry et al. (2024), there are significant evolutionary challenges to forming a massive WD+WD binary within the orbits of the observed luminous stars: such triples are likely to become dynamically unstable during their evolution, particularly given that the initial orbits of the luminous stars would have to have been significantly tighter than observed today when mass loss from the inner binary is accounted for. Nevertheless, tight WD+WD binaries cannot be ruled out based on the observed data alone.\n- 6. A single ultramassive WD : Several of our candidates have best-fit masses above the maximum WD mass mass, which is near 1 . 37 M ⊙ (Althaus et al. 2022). However, about half have masses below this limit, and given the astrometric uncertainties, a majority of our candidates are consistent at the fewσ level with having M 2 ≲ 1 . 37 M ⊙ . If the companions are single WDs, all of them would be among the highest-mass WDs known (e.g. Cognard et al. 2017; Caiazzo et al. 2021; Miller et al. 2023; Yamaguchi et al. 2024). \nGiven that our candidates are selected from the upper tail of the inferred companion mass distri- \n10 \nFig. 10.Comparison of our NS candidates (cyan/red) to all sources with astrometric orbital solutions published in DR3 (black). Hollow symbols show three initial candidates we found to have spurious solutions (Appendix A). Upper left: astrometric goodness of fit . Large values indicate a poor formal fit. There is a discontinuity at G = 13 owing to a quirk of the Gaia data processing: brighter sources typically have larger goodness of fit values because the uncertainties in their individual-epoch astrometric data are underestimated. Lower left: parallax uncertainty. Cyan points show the uncertainty reported in Gaia DR3, while red points show the uncertainty from our joint RV + astrometry fit. Right panels: parallax over error (top) and photocenter semimajor axis over error (bottom) as a function of orbital period. The diagonal 'cliff' is a result of quality cuts imposed on the published solutions. The NS candidates have typical goodness of fit values and astrometric uncertainties for their apparent magnitudes and periods, indicative of unproblematic astrometric solutions. The sources with spurious solutions have slightly larger parallax uncertainties than average but otherwise do not stand out from the sources with reliable solutions. \n<!-- image --> \nbution, the possibility that companion masses are overestimated systematically should be taken seriously. However, at least one object in the sample - Gaia NS1 (J1432-1021) - has a companion mass high enough that it cannot be a single WD: this object has M 2 = 1 . 90 ± 0 . 03, about 17 σ above the maximum WD mass. \nAs we discuss below, almost all the binaries in our sample have higher eccentricities than typical WD+MS binaries at similar periods. A simple interpretation is that the companions are NSs and the eccentricities are a result of natal kicks, but the data also suggest that systems containing massive WDs have higher typical eccentricities than those containing ∼ 0 . 6 M ⊙ WDs, so we cannot completely rule out a scenario in which many of the unseen companions are ultramassive WDs. \n- 7. A single neutron star : The inferred masses of the unseen companions in our sample are typical of neutron stars in pulsar binaries ( Ozel & Freire 2016). We consider this the simplest and most plausible scenario.", '6.2. Period-eccentricity relation': 'Figure 12 (left panel) shows the period-eccentricity relation for our candidates. We compare them to the larger sample of ∼ 3000 WD+MS binary candidates selected from Gaia astrometry by Shahaf et al. (2023a) on the basis of their AMRF. Our candidates - which differ from the Shahaf et al. (2023a) sample only in that they have inferred companion masses above 1 . 25 M ⊙ - have significantly higher eccentricities than those systems at fixed period. The low eccentricities of the WD+MS binaries are likely a result of tidal circularization when the progenitors of the WDs were giants. If the unseen companions in our sample are NSs, their higher eccentricities could could be naturally understood as a result of natal kicks. \nAlthough the eccentricities of the WD+MS binary candidates in Figure 12 are much lower on average than those of ordinary MS+MS binaries at the same periods, the large majority of them are not consistent with zero, and are much higher than expected for binaries that have gone through long periods of stable mass transfer (Phinney 1992; Lorimer 2008). This may be a consequence of these systems having first initiated mass transfer on the asymptotic giant branch rather than on the first giant \nFig. 11.Constraints on the mass of the unseen companion in J2244-2236, a typical system, as a function of the G -band flux ratio. Solid black line shows the constraint from the astrometric mass function (Equations 20), assuming M ⋆ = 1 . 00 M ⊙ ; a completely dark companion would imply M 2 ≈ 1 . 44 M ⊙ . If the companion contributes some light, its astrometrically-implied mass increases. Dotted cyan line shows the expected flux ratio and mass for a single MS companion. Because this is always below the black line, no single MS companion can explain the orbit. The same is true for an equal-mass inner binary (yellow dashed) or an equal-mass inner triple (red dot-dashed). An inner binary containing a 1 M ⊙ WD and a ≈ 0 . 6 M ⊙ MS star (blue dashed) could explain the astrometric mass function with the WD contributing 4% of the light in the G -band, but this violates the flux ratio constraint from RVs (see text). A tight WD+WD binary (not shown, because it could have F 2 /F tot ≈ 0), could explain the orbit, but it is unclear how such a system could form. \n<!-- image --> \nbranch and/or having gone through common envelope evolution rather than stable mass transfer. Similar eccentricities are observed in other populations of WD+MS binaries at these periods, such as blue stragglers and barium stars (Mathieu & Geller 2009; Jorissen et al. 2019; Escorza et al. 2019). \nThe right panel of Figure 12 shows eccentricities and dark companion masses. A majority of the objects from the Shahaf et al. (2023a) sample with M 2 > 1 . 25 M ⊙ are included in our sample: those that are not are either faint ( G > 15 . 0) or were excluded because our follow-up showed them to have spurious astrometric solutions or lower dark companion masses. It is evident that while most WD+MS candidates with M 2 ≲ 0 . 8 M ⊙ have loweccentricity orbits ( e < 0 . 2), systems with M 2 ≳ 1 . 0 M ⊙ tend to be more eccentric. It is tempting to simply attribute this dichotomy to WD vs. NS companions (e.g. Shahaf et al. 2023b). However, the eccentric population seems to extend to lower masses than expected for NSs. We have also identified a handful of systems in the course of our follow-up that have M 2 > 1 M ⊙ , e > 0 . 5, and clear UV excess, which points to a WD companion rather unambiguously. These eccentricities could be a result of faster orbital inspiral and/or more asymmetric mass loss in the super-AGB progenitors of mas- \nzard et al. 2010; El-Badry & Rix 2018), or eccentricity-pumping due to massive circumbinary disks formed through the mass transfer process (Dermine et al. 2013). Kozai-Lidov oscillations in systems containing a wide tertiary are another possibility.', '6.3. Galactic orbits': "To explore the Galactic stellar populations our NS candidates are members of and imprints of possible natal kicks, we show their locations in the Toomre diagram in Figure 13. For each binary, we use the center-of-mass velocity from the joint fit and the parallax and proper motion from the Gaia binary solution to calculate the current 3D motion of the binary's center of mass in Galactocentric cylindrical coordinates. We assume that the Sun is 20 pc above the Galactic midplane and 8.12 kpc from the Galactic center and has a 3D velocity vector ( V R, ⊙ , V ϕ, ⊙ , V Z, ⊙ ) = ( -12 . 9 , 245 . 6 , 7 . 78) kms -1 (Drimmel & Poggio 2018). We perform the same calculation for the other 33,467 binaries with AstroSpectroSB1 solutions. We do not consider Orbital solutions because their center-of-mass RVs are not known. \nThe results are shown in Figure 13. We compare the NS candidates in our sample to all binaries with AstroSpectroSB1 solutions (left panels) and to those with d = 0 . 4 -1 . 0 kpc (right panels; these have a similar distance distribution to the NS candidates). Dashed lines centered on V ϕ = 225kms -1 show approximate boundaries of the Galactic thin disk (total velocity < 70 kms -1 ), thick disk (70 -180 kms -1 ), and halo ( > 180 kms -1 ) (e.g. Bensby et al. 2014). Three objects in our sample - J1739+4502, J1432-1021, and J0152-2049 are on halo orbits, with total velocities of 370, 350, and 290 kms -1 with respect to the local standard of rest. These objects all have metallicities [Fe / H] < -1 . 2, implying that their high velocities are mainly the result of membership to an old stellar population, not natal kicks. On the other hand, the 6 systems in the 'thick disk' region of the Toomre diagram have metallicities close to solar. This suggests that natal kicks may have played a role in kinematically heating their orbits. The median and middle 68% range of midplane distances, | z | , are respectively 0.37 kpc and (0 . 11 -0 . 55) kpc. This makes the objects in our sample kinematically cold compared to NS LMXBs, which typically have | z | ≳ 1 kpc (van Paradijs & White 1995; Jonker & Nelemans 2004). This is not surprising since only NSs formed with weak kicks will remain bound in the wide orbits to which astrometry is sensitive. \nMetal-poor stars on halo orbits seem to be significantly overrepresented in the NS candidate sample. In particular, 3 of the 21 objects in the sample are unambiguously on halo orbits. In comparison, 0 of the 319 WD + MS binary candidates selected by Shahaf et al. (2023b) with AstroSpectroSB1 solutions have halo-like orbits! Among 11420 AstroSpectroSB1 orbits with d = 0 . 4 -1 . 0 kpc, 61 (i.e., 0.5%) have halo-like orbits, 1636 (14.3%) have thick disk-like orbits, and 9723 (85.1%) have thin disk-like orbits. In contrast, in the NS sample, 3/21 (14%) have halo-like orbits, 6/21 (29%) have thick disklike orbits, and 12/21 (57%) have thin disk-like orbits. Since the halo orbits of the NS candidates cannot be the result of natal kicks, the high fraction of halo or- \nFigure 14 compares the spectra of these three stars to those of their closest spectral doppelgangers. The latter are identified using the wavelength range 5650 -5850 ˚ A, which does not contain any lithium lines. The three NS candidates have Li I λ 6708 lines that are clearly significantly stronger than those of the comparison stars shown in Figure 14, and indeed, stronger than those of any stars observed by GALAH with similar atmospheric parameters. \n<!-- image --> \n/circledot \nFig. 12.Left: Period-eccentricity diagram for NS+MS candidates from this work (black) compared to WD+MS binary candidates selected by Shahaf et al. (2023b) from Gaia DR3 on the basis of their astrometrically-constrained mass ratios. For NS+MS candidates, uncertainties are smaller than symbols. Most of the WD candidates have low eccentricities, probably resulting from (incomplete) tidal circularization when the WD progenitor was a red giant. Our candidates have significantly higher eccentricities than the WD candidates at all periods, which may be the result of kicks during NS formation. Right: mass-eccentricity diagram. Our candidates are all selected to have M 2 > 1 . 25 M ⊙ . Solid and dashed lines show median and middle 68% across both red and black points. Binaries with M 2 = (1 . 0 -1 . 2) M ⊙ , most of which likely contain high-mass WDs, also have higher eccentricities than systems containing lower-mass WDs. \nbits suggest that NSs formed from low-metallicity stars are more likely to survive in Gaia -detectable binaries. As discussed by El-Badry (2024b), BH companions also seem to be overrepresented in the Gaia sample at low metallicity.", '6.4. Lithium enhancement': "To search for spectroscopic anomalies that could result from mass transfer from the unseen companions' progenitors, we compared their spectra to spectra of similar stars observed by the GALAH survey (Buder et al. 2021). We followed the same procedure described by El-Badry et al. (2024): in brief, we compared the rest-frame, resolution matched, and continuum-normalized spectra of our candidates pixel-by-pixel to all high-SNR spectra in GALAH DR3 and visually inspected the closest matches. Most of our candidates have unremarkable spectra and abundances, and we defer a full abundance analysis to future work. However, we highlight one feature that is striking: all three of the metal-poor halo stars in the sample are strongly enhanced in lithium. \nThe sources J1432-1021, J1739+4502, and J0152-2049 have Li I λ 6708 lines with equivalent widths of 114, 197, and 141 m ˚ A. Comparing these values to Kurucz model spectra calculated with ATLAS/SYNTHE (Kurucz 1979, 1993) and applying NLTE corrections from Wang et al. (2021), we find Li abundances of A(Li) = 2 . 90 ± 0 . 08, A(Li) = 3 . 53 ± 0 . 09, and A(Li) = 3 . 11 ± 0 . 08. These abundances represent more than a factor of 100 enhance- \nin surface lithium abundance compared to normal stars of similar temperature and metallicity (Figure 15). \nThe origin of the excess lithium is not yet understood. Possibilities include pollution by products of hot bottom burning in super-AGB stars, pollution by supernova ejecta, and spallation by high-energy particles (see ElBadry et al. 2024, for further discussion). Because metalpoor stars have thin convective envelopes, only their surface layers are expected to have been polluted. This is likely the reason we only find strong lithium enhancement in the metal-poor stars in our sample: the thickness of the convective envelope is more than 50 × greater at solar metallicity than at [Fe / H] = -1 . 5, meaning that any accreted material will be diluted 50 times more at solar metallicity, and abundance anomalies will be much more subtle. Assuming lithium is mixed uniformly into the outer 0.2% of the stars by mass, (1 -3) × 10 -11 M ⊙ of lithium must have been accreted in order to explain the observed enhancement. \nOur inferred A(Li) are higher than the predicted yields of super-AGB stars with M ≲ 8 M ⊙ , but compatible with predictions for 8 M ⊙ models (see Ventura & D'Antona 2010, who note that these yields depend sensitively on the assumed winds). This could support a scenario where the companions are NSs formed through electron-capture SNe, or ultramassive WDs. However, this scenario is difficult to reconcile with the fact that one of the binaries in question has M 2 = 1 . 90 ± 0 . 03 M ⊙ , whereas electron capture SNe are expected to produce low-mass NSs.", '6.5. Constraints on kicks and mass loss': "Models predict that dynamical channels are less efficient for forming wide NS binaries than for BH binaries (Tanikawa et al. 2024). Therefore, we assume the systems in our sample formed from primordial binaries and use their current orbits to place limits on their pre-SN masses and natal kicks. \nFig. 13.Toomre diagram of NS candidate binaries compared to all binaries with AstroSpectroSB1 solutions published in Gaia DR3. In the left panels, black points show the full DR3 catalog. In the right panels, they show only binaries with distances 0.4-1 kpc, similar to the NS candidates. Points are colored by metallicity. Dashed lines show total velocities of 70 and 180 km s -1 with respect to the local standard of rest, approximately separating the thin disk, thick disk, and halo. The three candidates with halo orbits (labeled) all have low metallicities. Halo and thick-disk orbits are significantly over-represented in the NS candidate sample. Natal kicks may be responsible for kinematically heating the orbits of these objects somewhat, but our results imply that NS candidates are over-represented in low-metallicity populations. \n<!-- image --> \nWe consider the evolutionary state of the binary that likely immediately preceded the current state: a circular orbit containing a stripped helium star of mass M Hestar and a 1 M ⊙ MS companion. Given that the orbits in our sample are too tight to accommodate red supergiants at their maximum radii of ∼ 1000 R ⊙ , the binaries would likely have gone through a common envelope event prior to reaching this stage. Survival of such a common envelope event and a wide final orbit is in fact nontrivial (e.g. \nKotko et al. 2024), but here we consider only the effects of the SN. \nIf there is no natal kick, the binary will be unbound if its total mass decreases by more than a factor of 2 during the SN (e.g. Blaauw 1961). The expected postSN eccentricity in this case is \ne = ∆ M M NS + M ⋆ , (21) \nFig. 14.Evidence for lithium enhancement in the three halo stars. Red lines show the spectra of the three NS candidates. Black lines show spectra of three stars observed by the GALAH survey with similar physical parameters to each target and spectra that closely match those of the target in most lines. All three of our targets have unusually strong Li I λ 6708 lines, indicative of lithium enhancement. \n<!-- image --> \nFig. 15.Lithium abundances of the three metal-poor halo stars in our sample (Figure 14), compared to metal-poor stars observed by the GALAH survey. All three stars are strongly enhanced in lithium, with A(Li) higher than any of the ∼ 10 3 stars with similar stellar parameters observed by GALAH. \n<!-- image --> \nWhere M NS is the mass of the NS, ∆ M = M Hestar -M NS is the mass lost during the explosion, and M ⋆ is the mass of the luminous star. For typical targets in our sample, e ≈ 0 . 4 and M ⋆ = 1 M ⊙ . This would imply that only of order 1 M ⊙ was lost during the SN if natal kicks were weak. \nKicks complicate this analysis: it is possible for a wellaimed natal kick to keep a binary bound - and even maintain a low eccentricity orbit - with arbitrarily large mass loss during the SN. This, however, requires finetuning, and most orbits will simply be unbound if kicks are strong and mass loss is significant. Inferring the full distribution of kick velocities from observations of binaries like those in our sample is difficult, since NSs born with strong kicks will not make it into the sample in the first place. We can, however, still constrain the kicks and mass loss that likely occurred in the binaries that did survive. \nWe model the combined effects of natal kicks and instantaneous mass loss using Monte Carlo simulations. \nFig. 16.Survival probability (top) and median predicted eccentricity (bottom) for wide helium star + MS binaries when the He star explodes and leaves behind a NS. We assume the initial orbit is circular and consider initial helium star masses of 3, 4, and 5 M ⊙ , and initial periods of 100, 300, and 1000 days. In all cases, we model the companion as a 1 M ⊙ MS star and adopt a final NS mass of 1 . 4 M ⊙ . The NS is born with a velocity v kick in a random direction. For a 3 M ⊙ He star, orbits are most likely to remain bound if the kick is slow. For M Hestar > 3 . 8 M ⊙ , the binary will be unbound by mass loss unless a fortuitously aligned kick allows the NS to catch the escaping companion. \n<!-- image --> \nFollowing Brandt & Podsiadlowski (1995), we predict post-SN orbits for a range of v kick , M Hestar , and preSN orbital period. For each choice of these values, we generate N = 10 6 kick directions distributed uniformly on a sphere and calculate the post-SN orbit via energy and angular momentum conservation. In all cases, we assume a 1 M ⊙ companion, a 1 . 4 M ⊙ NS, and an initially circular orbit. \nThe results are shown in Figure 16. The top panels show the fraction of all orbits that remain bound, and the bottom panels show the median eccentricity of those that do. The probability of remaining bound is highest for low M Hestar and low v kick . However, a significant fraction of orbits remain bound even for M Hestar of 5 or 10 M ⊙ when paired with a suitable kick velocity. Higher values of M Hestar require stronger kicks to remain bound: for a given M Hestar , binaries are most likely to survive when v kick is about half the pre-SN orbital velocity of the MS star. The median predicted post-SN eccentricity of surviving binaries is typically 0 . 4 -0 . 8. These values are slightly higher than the median eccentricity of 0.4 within our sample, but given that Gaia is less sensitive to higheccentricity orbits, the observed eccentricity distribution appears consistent with originating mainly from kicks. \nLight curve modeling of stripped-envelope SNe suggests typical ejecta masses of (1 -3) M ⊙ (e.g. Lyman et al. 2016), which would correspond to He star masses of ∼ (3 -5) M ⊙ for the NSs in our sample. For kicks with v kick ≲ 50 kms -1 , about 30% of binaries forming from such He stars at periods typical of our sample would be expected to survive. Such kick velocities are quite low compared to typical values inferred from observations of young pulsars (e.g. Hobbs et al. 2005; Faucher-Gigu'ere \n& Kaspi 2006), but within the range required for NSs to be retained within globular clusters (Ivanova et al. 2008). Although only a minority of NSs formed in binaries with the periods and companions typical of our sample are likely to survive, the relatively low space density of wide NS+MS binaries suggested by our sample compared to the predicted space density of all dormant NSs 4 allows for a scenario in which only the lowv kick tail of the NS population survives in binaries. \nCuriously, the most massive NS in the sample, Gaia NS1 (J1432-1021), has the lowest-eccentricity orbit, requiring the lowest kicks. This appears inconsistent with SNe simulations, which predict high-mass NSs to receive the strongest kicks (Muller et al. 2019; Burrows et al. 2023). It is possible that Gaia NS1 formed through a different channel than the other objects in the sample, but larger sample is required to investigate this possibility quantitatively.", '6.6. Future evolution to symbiotic X-ray binaries': "When the MS stars in our sample leave the main sequence, they will begin transferring mass to the dark companions: first by winds, and eventually by Roche lobe overflow. Rodriguez et al. (2024) have calculated \n4 An accurate estimate of the space density of wide NS+MS binaries is presently difficult to calculate due to uncertainties in the Gaia selection function. However, the nearest NS candidate in our sample is at d ≈ 250 pc. The can be compared to d ≈ 120 pc, the distance to the nearest known young NS (Walter et al. 2010), and d ≈ 20 pc, a predicted distance to the nearest NS based on the Galactic model of Sweeney et al. (2022). These distances suggest that there are ∼ 10 3 dormant NSs for every one in a binary like the objects in our sample. \nmodels representative of this evolution. Their calculations suggest that the binaries will be X-ray bright for at least ∼ 5 Myr during Roche lobe overflow, and most likely for a few tens of Myr due to wind accretion at earlier times. Since the X-ray bright phase is predicted to be 100-1000 times shorter-lived than the current X-ray faint phase, one expects X-ray bright systems to be rare, but they are discoverable to large distances. \nAt least a few neutron stars with solar-mass red giant companions are known to exist in symbiotic X-ray binaries (e.g. Hinkle et al. 2006, 2019). One system that is likely representative of our candidates' future evolution is GX 1+4 (Davidsen et al. 1977; Hinkle et al. 2006), which contains a ≲ 1 M ⊙ giant in a 1161-day orbit at a distance of ≈ 4 kpc. Another similar system is the symbiotic Xray binary IGR J16194-2810 (Masetti et al. 2007), which contains a ∼ 1 M ⊙ red giant at a distance of ≈ 2 . 1 kpc in a 193-day orbit with a NS companion (Nagarajan et al. 2024; Hinkle et al. 2024). Given their X-ray properties, these systems undoubtedly contain NSs. This provides some support for the interpretation that the dark objects in our binaries are also NSs, despite the uncertainties associated with forming NS+MS binaries in such wide orbits.", '6.7. Is there a BH/NS mass gap?': 'Figure 17 shows the mass distribution of dark companions identified thus far from Gaia astrometry: the 21 NS candidates presented here, and three BHs studied previously (El-Badry et al. 2023a,b; Nagarajan et al. 2023; Gaia Collaboration et al. 2024). All the NS candidates have best-fit masses between 1.25 and 2 . 0 M ⊙ , while the BHs have masses above 9 M ⊙ . We have not detected any dark companions with intermediate masses, and our follow-up has been complete for sources published in DR3 with astrometric solutions implying M 2 ≳ 2 M ⊙ . The fact that we detected 21 dark companions with M 2 < 2 M ⊙ - which produce smaller astrometric wobbles than would low-mass BHs - strongly suggests that our search would have detected 3 -5 M ⊙ BHs if they existed in our search sample. \nFigure 18 compares the masses and orbital periods of BH and NS binaries identified with Gaia astrometry to BHs and NSs found with other methods. Yellow points show NSs in radio pulsars binaries ( Ozel & Freire 2016; Fonseca et al. 2021). The hollow yellow symbol shows the companion to PSR J0514-4002E, whose nature is uncertain (Barr et al. 2024). Red points show highand low-mass BH X-ray binaries (Remillard & McClintock 2006; Corral-Santana et al. 2016; Miller-Jones et al. 2021). Cyan lower limits show spectroscopic BH binary candidates (Giesers et al. 2019; Shenar et al. 2022; Mahy et al. 2022). The BH and NS samples are both far from complete. Nevertheless, Figure 18 shows rather unambiguously that the BH/NS mass distribution is bimodal over at least 4 orders of magnitude in orbital period. \nOur results thus support the presence of mass gap between NSs and BHs, similar to the gap reported for X-ray binaries (Bailyn et al. 1998; Kreidberg et al. 2012) and gravitational wave sources (Abbott et al. 2023). The existence of a mass bimodality does not depend much on the nature and exact mass of objects proposed to be in the mass gap (e.g. Casares et al. 2022; Barr et al. 2024), \nas such objects are rare, at least in binaries. Of course, it is possible that the mass distribution of BHs in binaries is different from the mass distribution of all BHs, since (for example) low-mass BHs could experience stronger kicks and more frequently be unbound (Burrows et al. 2023).', '7. CONCLUSIONS': "The first set of orbital solutions from the Gaia mission led to the identification of more than 50 candidate neutron star (NS) + main sequence (MS) binaries in auscale orbits. We are carrying out a spectroscopic followup program that yields radial velocities (RVs) and stellar parameters for these binaries. We have now covered a majority of an orbital period for most of these candidates with RVs. This allows us to tighten constraints on orbits through joint fitting of RVs and astrometry to and root out spurious astrometric orbits when the predictions of the Gaia solution do not match RVs (Appendix A). Our main results are as follows: \n- 1. Summary of the sample : We have constructed a sample of 21 candidate NS + MS binaries containing solar-type MS stars and dark companions with best-fit masses of 1 . 25 -1 . 90 M ⊙ (Figure 1). Most systems have periods of 400 to 1000 days and distances of 0.4 to 1 kpc. We select candidates as those with photocenter wobbles too large to be explained by a normal luminous companion or an unresolved inner binary containing two luminous stars. At fixed period and luminous star mass M ⋆ , NS companions produce photocenter orbits that are larger than WDs and non-degenerate companions can produce, but smaller than those produced by BH companions (Figure 2).\n- 2. Quality of the Gaia orbital solutions : RVs for all the objects in our final sample are in good agreement with predictions of the Gaia orbital solutions (Figure 5), and joint fitting of RVs and astrometry leads to well-constrained orbits (Figure 6). For most objects, constraints from the Gaia -only and Gaia +RVs solutions are consistent within 1 σ (Figure 7). However, some initial candidates were removed from the sample after RV follow-up revealed tensions with the Gaia solution (Figure 19 and Table 3). Most but not all of these objects have large goodness of fit , indicative of a problematic astrometric solution. On average, companion masses constrained by joint RV+astrometry fits are lower than the pure-astrometry estimates (Figure 8); this is an expected consequence of selecting candidates from the upper tail of the companion mass distribution. The objects all have unproblematic Gaia quality flags and astrometric uncertainties typical of their apparent magnitudes (Figure 10). Most of the orbits are predicted to have been sampled uniformly in phase by Gaia (Figure 9).\n- 3. Eccentricities : Most of the binaries in our sample have fairly eccentric orbits, with a median eccentricity of 0.4. Their eccentricities are significantly higher than those of typical WD companions at similar periods (Figure 12), and are consistent \n<!-- image --> \n/circledot \n<!-- image --> \n/circledot \nFig. 17.Mass distribution of dark companions revealed by follow-up of Gaia astrometric binaries. The 21 sources with masses M dark = 1 . 25 -1 . 9 M ⊙ are presumed NSs introduced in this paper. The two objects with masses near 9 M ⊙ are presumed BHs (El-Badry et al. 2023a,b). A third BH has significantly higher mass, M dark ≈ 33 M ⊙ (Gaia Collaboration et al. 2024). No dark companions with mass between 2 and 8 M ⊙ have been detected, even though the effective search volume for such sources is larger than for the NSs.Fig. 18.Masses and orbital period of Galactic BHs and NSs in binaries with well-constrained masses. The candidates presented here are shown in magenta and are compared to NSs in pulsar binaries (yellow), BHs in astrometric binaries (black), BHs in X-ray binaries (red), and BHs in spectroscopic binaries (cyan). A mass bimodality exists over at least 4 orders of magnitude in orbital period. \n<!-- image --> \nwith eccentricities expected due to kicks during NS formation (Figure 16). However, high-mass WDs ( M ≳ 1 M ⊙ ) also have higher eccentricities than ∼ 0 . 6 M ⊙ WDs, so the eccentricities of our candidates do not guarantee that they are NSs. \n- 4. Metallicities and chemical abundances : We measured metallicities for the MS stars from highresolution spectra (Figure 3). A majority of them have metallicities near solar ( -0 . 5 < [Fe / H] < 0 . 5), but low metallicity halo stars are over-represented. Three out of 21 targets (14%) have [Fe / H] ∼ -1 . 5 \nand space velocities of ∼ 300 kms -1 (Figure 13). Only 0.5% of all binaries with Gaia astrometric solutions at comparable distance are on halo orbits. This suggests that low-metallicity massive stars are more likely to form NSs in wide orbits with lowmass companions. \nAll three low-metallicity stars are strongly enhanced in lithium (Figure 14 and 15). Lowmetallicity stars have much thinner convective envelopes than solar-metallicities stars of the same mass and evolutionary state, so accreted material \nwill be diluted less and abundance anomalies will be more detectable at low metallicity. The origin of the lithium is unclear: one possibility is accretion of Li-rich winds from super-AGB stars (Cameron & Fowler 1971; Ventura & D'Antona 2010). However, at least one the Li-enhanced objects has a companion with M 2 = 1 . 90 ± 0 . 03 M ⊙ , which is considerably higher than the expected mass of high-mass WDs or NSs formed from electron capture SNe. \n- 5. Nature of the companions : We have not detected radiation from the companions, but constraints on their masses and orbits can narrow down the possibilities. Joint fitting of astrometry and RVs places tight upper limits on the flux ratio, with a median 2 σ upper limit of ϵ < 0 . 03. This rules out all plausible nondegenerate companions (Figure 11). Several different possibilities remain, including single NSs, single massive WDs, and tight WD+WD, WD+NS, or WD+MS binaries. For most of the companions to be single massive WDs, their masses would have to be overestimated at the fewσ level. Scenarios involving inner binaries are difficult to explain with evolutionary models - triple systems tend to become unstable during their evolution when the outer orbit is as tight as our candidates - but should not be dismissed entirely for want of imagination. It is quite possible that the sample contains more than one kind of dark companion.\n- 6. BH/NS mass distribution : The Gaia mission has now enabled astrometric discovery of three BHs, 21 candidate NSs, and thousands of WDs in astrometric binaries. The mass distribution of BHs and NS candidates is shown in Figure 17. There is a conspicuous gap in the mass distribution between 2 and 8 M ⊙ . This may be in part a result of small number statistics, but the distribution does seem to disfavor a population of lower-mass BHs that significantly outnumber higher-mass BHs. Since the astrometric search volume is larger for higher-mass objects, our sample suggests that low-mass BHs are significantly less common in astrometric binaries than are NSs. When combined with samples of BHs and NSs discovered with other methods, our sample reveals a clear mass bimodality that extends over 4 orders of magnitude in orbital period (Figure 18). \nIt will likely remain difficult to conclusively establish the nature of the companions for some time. Radio detection of the companions as pulsars could prove that they are NSs, and efforts are underway to search for radio pulsations from most of our candidates. However, radio detection of any individual candidate seems a priori unlikely: young NSs are only detectable as radio pulsars for ≲ 10 Myr - only ∼ 0 . 1% of the expected lifetime of these binaries - and in the simplest evolutionary scenarios for their formation, there is no reason to expect the NSs to be recycled. Given the wide separations of the binaries and weak winds of the main-sequence companions, an X-ray detection due to accretion is also not expected (e.g. Rodriguez et al. 2024). WD companions are not expected to be detectable at any wavelength unless they \nare fortuitously young. Models involving inner binaries can be tested with high-precision RV observations (Nagarajan et al. 2023), but only if the inner binary has a period longer than a few days. \nWe briefly discuss two possible avenues for determining the nature of the companions in the absence of a direct detection. First, higher-precision astrometric orbits from future Gaia data releases will tighten mass constraints. The companion mass uncertainties are in most cases limited by the astrometric inclination uncertainties, and more precise measurements could more firmly rule out the possibility that the companions are ultramassive WDs. Future Gaia releases will also include epoch astrometry for all these sources, which will allow for further checks of the consistency between the astrometry and RV data. \nSecond, ongoing RV follow-up of astrometric binaries containing WDs will provide a more complete mapping of the period - eccentricity relation for post-interaction binaries containing WDs. This relation is particularly uncertain for massive WDs in wide orbits, making it difficult to distinguish between WDs and NSs on the basis of their eccentricities. Features in the relation near the WD-NS transition mass may make it possible to statistically differentiate between WD and NS scenarios, even if the nature of companions in individual systems remain uncertain.", 'ACKNOWLEDGMENTS': "We thank the referee for a constructive report, and Josh Simon, Casey Lam, Kyle Kremer, Jim Fuller, and Thomas Tauris for useful discussions. We are grateful to Yuri Beletsky, Sam Kim, Angela Hempel, R'egis Lachaume, Gil Esquerdo, Perry Berlind, and Mike Calkins for observing help. This research was supported by NSF grant AST-2307232. HWR acknowledges support from the European Research Council for the ERC Advanced Grant [101054731]. This research benefited from discussions in the ZTF Theory Network, funded in part by the Gordon and Betty Moore Foundation through Grant GBMF5076, and from collaboration at the 'Renaissance of Stellar Black-Hole Detections in The Local Group' workshop hosted at the Lorentz Center in June, 2023. \nThis research made use of pystrometry, an open source Python package for astrometry timeseries analysis (Sahlmann 2019). This work made use of Astropy, 5 a community-developed core Python package and an ecosystem of tools and resources for astronomy (Astropy Collaboration et al. 2022). \nThis work has made use of data from the European Space Agency (ESA) mission Gaia ( https://www.cosmos.esa.int/gaia ), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/ dpac/consortium ). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. \nThis paper includes data gathered with the 6.5 meter Magellan Telescopes located at Las Campanas Observatory, Chile. Some of the data presented herein were ob- \ntained at Keck Observatory, which is a private 501(c)3 non-profit organization operated as a scientific partnership among the California Institute of Technology, the University of California, and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W. M. \nKeck Foundation. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Maunakea has always had within the Native Hawaiian community. 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G., et al., 2015, Astronomische Nachrichten, 336, 324', 'REJECTION OF SPURIOUS SOLUTIONS AND SUMMARY OF ALL FOLLOW-UP': "For some candidates, our RV follow-up showed the Gaia astrometric solution to be spurious or to have significantly underestimated uncertainties. Measured and predicted RVs for three such sources are shown in Figure 19. The source shown in the top panel, with DR3 source id 6593763230249162112, was characterized as a high-quality NS candidate by both Andrews et al. (2022) and Shahaf et al. (2023b). The Gaia orbital solution has goodness of fit = 1.67, indicative of an unproblematic astrometric solution. Its DR3 astrometric solution is based on 20 visibility periods, which is typical for solutions published in DR3. There does not appear to be any reason to mistrust the solution based on the quality flags published in DR3. Yet, comparison of the observed and predicted RVs leaves little doubt that the astrometric solution is seriously in error. \nThe same is true for source 3869650535947137920, which is shown in the middle panel of Figure 19 and is included in the sample from Shahaf et al. (2023b). The source's goodness of fit is 5.46, which is quite normal for a source with G < 13 (Figure 10). The first several months of our follow-up showed the RVs to be evolving in a manner consistent with the astrometric solution's predictions. However, the RVs began to diverge from those predictions in the second season of our follow-up. Although the qualitatively similar shape of the predicted and observed RV curves suggests the astrometric solution is not completely spurious, the observed degree of disagreement suggests that the astrometric uncertainties are significantly underestimated. \nFinally, source 747174436620510976, shown in the bottom panel of Figure 19, is an example of a case where the shape of the observed RV curve is consistent with the astrometric predictions, but there is a fewσ offset in phase. Our analysis of the source's SED also suggested that the parallax is overestimated, leading us to \n- Strassmeier K. G., Ilyin I., Steffen M., 2018, A&A, 612, A44 Sweeney D., Tuthill P., Sharma S., Hirai R., 2022, MNRAS, 516,\n- 4971\n- Tanikawa A., Wang L., Fujii M. S., 2024, arXiv e-prints, p. arXiv:2404.01731\n- Tremblay P. E., Bergeron P., 2009, ApJ, 696, 1755\n- Ventura P., D'Antona F., 2010, MNRAS, 402, L72\n- Vogt S. S., et al., 1994, in Crawford D. L., Craine E. 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The Gaia solution has goodness of fit = -0 . 96, indicative of a good solution, and was included in both the Shahaf et al. (2023b) and Andrews et al. (2022) candidate lists. \nThese examples highlight the importance of RV followup: while the sources shown in Figure 19 clearly are binaries, and least two of the three likely have orbital parameters not too far from those inferred from astrometry alone, their astrometric uncertainties are unlikely to be reliable. Since most NSs have masses near the Chandrasekhar limit, small problems with the astrometric solution can seriously change our conclusions about the nature of a candidate. Epoch astrometry from Gaia DR4 may allow us to obtain more useful astrometric constraints from sources like those shown in Figure 19. \nIn Table 3, we provide a summary of our follow-up of all NS candidates from Shahaf et al. (2023b) and Andrews et al. (2022). For the Shahaf et al. (2023b) sample, we list candidates for which they inferred M 2 > 1 . 25 M ⊙ . The Andrews et al. (2022) sample only includes targets for which they infer M 2 > 1 . 4 M ⊙ . A majority of the candidates ruled out by RV follow-up have significantly higher goodness of fit than typical sources of the same apparent magnitude (Figure 10). Several candidates with incomplete RV follow-up, particularly those with G < 15 that are inaccessible with FEROS and TRES, remain to be studied.", 'BLUE EXCESS DUE TO WD COMPANIONS': 'As can be seen in Figure 1, several of our candidates are at the blue edge of the main sequence. Compared to stars near the middle of the main sequence, this amounts to a blue color excess of 0.05 to 0.1 mag. Most of our candidates have also been observed by GALEX , with NUV detections that rule out significant ( ≳ 0 . 2 mag) UV excess. Here we investigate whether the optical blue excess could be due to a WD companion. \nWe first consider a 1 M ⊙ MSstar in a binary with a single 1 . 3 M ⊙ WD companion. We construct the combined spectral energy distribution of the pair, modeling the WD with a Koester DA model with log( g/ cms -2 ) = 9 . 0 \nTABLE 3 \nAll the NS candidates from Andrews et al. (2022, A22) and Shahaf et al. (2023b, S23). We list the orbital period according to the Gaia astrometric solution, the apparent magnitude and goodness of fit , and the status of our follow-up. Candidates for which the status is colored in green are modeled in detail in this work. Those for which status is colored in red are disfavored by RV follow-up. Blue text indicates that detailed follow-up is presented elsewhere, and black text indicates that insufficient RVs have been obtained for a verdict to be reached. \nFig. 19.Examples of NS+MS candidates we rejected after RV follow-up. In the top panel, the RVs are grossly inconsistent with the predictions of the Gaia solution, suggesting that the solution is spurious, or at least has a large accumulated phase error not captured in the Gaia uncertainties. Middle panel shows an example in which the RVs evolve in a manner somewhat consistent with the Gaia solution prediction, but do not match it quantitatively. This indicates that the uncertainties of the astrometric parameters are likely underestimated, casting doubt on the companion mass constraint. Bottom panel shows a marginal case: the RVs qualitatively track the astrometric solutions predictions, but a phase offset suggests that the period uncertainty is underestimated. We exclude all these objects - even though they may indeed host NSs or massive WDs - retaining only sources with good agreement between astrometry and RVs. \n<!-- image --> \n(Tremblay & Bergeron 2009; Koester 2010) and the MS star with the same models used in Section 4. We then calculate the combined total magnitude of both objects using pyphot . The solid lines in Figure 20 show the resulting color excess as a function of WD effective temperature. We define the color excess in each band as the color of the combined MS + WD pair minus the color of the MS star alone. \nBecause a 1 . 3 M ⊙ WDis very small, the predicted blue excess in the optical is negligible except at very high temperatures. On the other hand, effective temperatures above ≈ 25 , 000 K would lead to a detectable FUV source (here we assume a distance of 500 pc), which represents a strong UV excess easily distinguishable from a single MS star. A weaker, but still significant, NUV excess is also predicted. On the basis of these calculations, we conclude that - given their high dynamical masses - there is no plausible way for single WD companions to significantly change the optical colors of our candidates without \nsimultaneously causing a large UV excess, which is not observed. \nNext, we consider the possibility of a close WD+WD binary of total mass 1 . 3 M ⊙ . In this case, a broader range of combined SEDs are possible, depending on the mass ratio and cooling ages of both WDs. For simplicity, we consider the case where both WDs have the same mass and effective temperature. In this case, the blue excess is significantly stronger at fixed T eff , reflecting the fact that two 0 . 65 M ⊙ WDs have ≈ 18 times the total surface area of a single 1 . 3 M ⊙ WD. \nWith two 0 . 65 M ⊙ WDs, a ≈ -0 . 04 mag blue excess is possible in the optical if both WDs have T eff ∼ 50 , 000 K. However, this would be accompanied by an excess of several magnitudes in both the NUV and FUV bands, and is thus ruled out. The same is true for a 1 . 0 + 0 . 3 M ⊙ WD binary. In this case, the flux is dominated by the 0 . 3 M ⊙ WD, which could cause a significant optical blue excess for T eff ≳ 30 , 000 K. Here too, the excess would be accompanied by a much larger excess in the UV.', 'ALL FITS': 'Table 4 lists constraints from joint fitting of RVs and Gaia astrometry for all sources.', 'GaiaDR3 ID6481502062263141504': 'Fig. 20.Predicted color excess due to a WD companion. In all cases, we assume a 1 M ⊙ solar-type primary with a 1 . 3 M ⊙ companion, representative of the objects in our sample. Solid blue lines show the case where the companion is a single WD, while dashed black line shows an equal-mass WD+WD binary and dotted cyan lines show a (1 . 0 + 0 . 3) M ⊙ WD binary. Top panel shows the optical blue excess in the Gaia G BP -G RP bands, while the middle and bottom panels show the UV excess in the GALEX NUV and FUV bands. Lighter lines in the bottom panel show cases where the system would be fainter than 20.0 mag in the FUV band at a distance of 500 pc and would likely not be detected. WD+WD binaries lead to much more blue and UV excess than single WD companions of the same mass. Even WD companions that are quite hot produce negligible blue excess in the optical. \n<!-- image -->', 'GaiaDR3 ID5530442371304582912': 'Orbital period \nP orb [days] \n497 . 6 ± 0 . 4 \nSemi-major axis \na [au] \n1 . 593 ± 0 . 021 \nPhotocenter semi-major axis \na 0 [mas] \n0 . 965 ± 0 . 014 0 . 168 ± 0 . 004 \nEccentricity \ne \nInclination \ni [deg] \n129 . 2 ± 1 . 1 \nPeriastron time \nT \np \n[JD-2457389] \n23 ± 4 \nAscending node angle \nΩ[deg] ω [deg] \n170 . 8 ± 1 . 8 \nArgument of periastron \n331 . 9 ± 1 . 9 \nLuminous star mass \nM ⋆ [ M ⊙ ] \n0 . 90 ± 0 . 05 \nNeutron star mass \nM 2 [ M ⊙ ] \n1 . 28 ± 0 . 04 \nRV semi-amplitude \nK ⋆ [kms - 1 ] \n16 . 03 ± 0 . 09 \nRV mass function \nf ( M 2 ) RVs [M ⊙ ] - 1 \n0 . 204 ± 0 . 004 \nCenter-of-mass RV Parallax \nγ \n[kms \n] \n41 . 10 ± 0 . 08 \nϖ \n[mas] \n1 . 035 ± 0 . 014 \nGoodness of fit \nχ \n2 \nRVs \n/N \nRVs \n0 . 2', 'GaiaDR3 ID4922744974687373440': 'Orbital period \nP orb [days] \n561 . 83 ± 0 . 29 \nSemi-major axis \na [au] \n1 . 717 ± 0 . 016 \nPhotocenter semi-major axis \na 0 [mas] \n2 . 344 ± 0 . 026 \nEccentricity \ne \n0 \n. \n795 \n± \n0 \n. \n005 \nInclination \ni [deg] T p [JD-2457389] \n85 . 9 ± 0 . 8 \nPeriastron time \n- 36 . 7 ± 1 . 2 \nAscending node angle \nΩ[deg] \n259 . 7 ± 0 . 7 \nArgument of periastron \nω \n[deg] \n308 . 4 ± 0 . 5 \nLuminous star mass \nM ⋆ [ M ⊙ ] \n0 . 802 ± 0 . 031 \nNeutron star mass \nM 2 [ M ⊙ ] \n1 . 34 ± 0 . 04 \nRV semi-amplitude \nK \n⋆ f ( M 2 ) RVs [M ⊙ ] - 1 \n[kms \n- \n1 \n] \n34 . 2 ± 0 . 8 \nRV mass function \n0 . 519 ± 0 . 020 \nCenter-of-mass RV Parallax \nγ \n[kms \n] \n52 . 58 ± 0 . 07 \nϖ [mas] 2 \n2 . 183 ± 0 . 016 \nGoodness of fit \nχ \nRVs \n/N \nRVs \n0 . 3', 'GaiaDR3 ID1434445448240677376': 'Orbital period \nP orb [days] \n570 . 94 ± 0 . 31 \nSemi-major axis \na \n[au] \n1 \n. \n835 \n± \n0 \n. \n018 \nPhotocenter semi-major axis \na 0 [mas] \n1 . 437 ± 0 . 010 \nEccentricity \ne i [deg] \n0 . 3093 ± 0 . 0010 \nInclination \n55 . 1 ± 0 . 5 \nPeriastron time \nT p [JD-2457389] \n- 84 . 6 ± 1 . 5 \nAscending node angle \nΩ[deg] \n8 . 1 ± 1 . 1 \nArgument of periastron \nω [deg] \n110 . 34 ± 0 . 15 \nLuminous star mass \nM ⋆ [ M ⊙ ] \n1 . 16 ± 0 . 05 \nNeutron star mass \nM 2 [ M ⊙ ] \n1 . 362 ± 0 . 030 \nRV semi-amplitude \nK ⋆ [kms - 1 ] \n16 . 266 ± 0 . 014 \nRV mass function \nf ( M 2 ) RVs [M ⊙ ] γ [kms - 1 ] \n0 . 2189 ± 0 . 0007 \nCenter-of-mass RV Parallax \n- \n18 . 047 ± 0 . 012 \nϖ [mas] \n1 . 452 ± 0 . 010 \nGoodness of fit \nχ 2 RVs /N RVs \n2 . 3', 'J1150-2203;': 'GaiaDR3 ID3494029910469026432', 'GaiaDR3 ID1694708646628402048': 'Orbital period \nP orb [days] \n632 . 65 ± 0 . 21 \nSemi-major axis \na [au] \n1 . 868 ± 0 . 023 \nPhotocenter semi-major axis \na 0 [mas] \n1 . 961 ± 0 . 010 \nEccentricity \ne i [deg] \n0 . 2668 ± 0 . 0010 \nInclination \n105 . 7 ± 0 . 4 \nPeriastron time \nAscending node angle \nArgument of periastron \nLuminous star mass \nNeutron star mass \nRV semi-amplitude \nRV mass function \nCenter-of-mass RV Parallax \nGoodness of fit \nJ0217-7541;', 'GaiaDR3 ID4637171465304969216': 'Orbital period \nP \norb a [au] \n[days] \n636 \n. \n1 \n± \n0 \n. \n7 \nSemi-major axis \n1 \n. \n936 \n± \n0 \n. \n016 \nPhotocenter semi-major axis \na 0 [mas] \n1 . 348 ± 0 . 012 \nEccentricity \ne \n0 . 3228 ± 0 . 0033 \nInclination \ni \n[deg] \n132 . 3 ± 0 . 7 \nPeriastron time \nT p [JD-2457389] \n18 . 4 ± 2 . 6 \nAscending node angle \nΩ[deg] \n95 . 1 ± 1 . 6 \nArgument of periastron \nω \n[deg] \n29 . 4 ± 0 . 5 \nLuminous star mass \nM \n⋆ ⊙ M 2 [ M ⊙ ] \n[ \nM \n] \n0 . 996 ± 0 . 033 \nNeutron star mass \n1 \n. \n396 \n± \n0 \n. \n033 \nRV semi-amplitude \nK \n⋆ \n[kms \n- \n1 \n] \n15 . 09 ± 0 . 04 \nRV mass function \nf \n( \nM \n2 \n) \nRVs \n[M \n⊙ \n] \n0 . 1919 ± 0 . 0013 \nCenter-of-mass RV \nγ \n[kms \n- \n1 \n] \n59 \n. \n641 \n± \n0 \n. \n027 \nParallax \nϖ [mas] \n1 . 193 ± 0 . 012 \nGoodness of fit \nχ \n2 \n/N \nRVs \n0 . 8', 'GaiaDR3 ID5580526947012630912': 'Orbital period \nP orb [days] \n654 . 6 ± 0 . 6 \nSemi-major axis \na \n[au] \n2 \n. \n132 \n± \n0 \n. \n028 \nPhotocenter semi-major axis \na 0 [mas] \n1 . 357 ± 0 . 027 0 . 721 ± 0 . 013 \nEccentricity \ne i [deg] \nInclination \n171 \n. \n47 \n± \n0 \n. \n34 \nPeriastron time \nT p [JD-2457389] \n52 . 1 ± 1 . 5 \nAscending node angle \nΩ[deg] \n362 . 1 ± 1 . 6 \nArgument of periastron \nω \n[deg] \n331 \n. \n4 \n± \n1 \n. \n0 \nLuminous star mass \nM \n⋆ \n[ \nM \n⊙ \n] \n1 . 32 ± 0 . 06 \nNeutron star mass \nM \n2 \n[ \nM \n⊙ \n] \n1 . 70 ± 0 . 07 \nRV semi-amplitude \nK \n⋆ \n[kms \n- \n1 \n] \n4 . 25 ± 0 . 24 \nRV mass function \nf \n( \nM \n2 \n) \nRVs \n[M \n⊙ \n] \n0 . 00175 ± 0 . 00023 \nCenter-of-mass RV \nγ \n[kms \n- \n1 \n] \n- 10 . 49 ± 0 . 05 \nParallax \nϖ \n[mas] \n1 \n. \n130 \n± \n0 \n. \n011 \nGoodness of fit \nχ \n2 \n/N \nRVs \n0 . 2', 'J1739+4502;': 'GaiaDR3 ID1350295047363872512 \nOrbital period \nP orb [days] \n657 \n. \n4 \n± \n0 \n. \n6 \nSemi-major axis \na [au] \n1 . 914 ± 0 . 018 \nPhotocenter semi-major axis \na 0 [mas] \n1 . 376 ± 0 . 016 \nEccentricity \ne \n0 . 6777 ± 0 . 0018 \nInclination \ni [deg] T p [JD-2457389] \n144 . 7 ± 0 . 5 \nPeriastron time \n179 \n. \n7 \n± \n2 \n. \n3 \nAscending node angle \nΩ[deg] \n70 . 5 ± 1 . 5 \nArgument of periastron \nω [deg] \n334 . 36 ± 0 . 22 \nLuminous star mass \nM ⋆ [ M ⊙ ] \n0 \n. \n781 \n± \n0 \n. \n030 \nNeutron star mass \nM \n2 \n[ \nM \n⊙ \n] \n1 . 38 ± 0 . 04 \nRV semi-amplitude \nK \n⋆ \n[kms \n- \n1 \n] \n15 \n. \n909 \n± \n0 \n. \n034 \nRV mass function \nf \n( \nM \n2 \n) \nRVs \n[M \n⊙ \n] \n0 . 1091 ± 0 . 0007 \nCenter-of-mass RV \nγ \n[kms \n- \n1 \n] \n- \n264 \n. \n202 \n± \n0 \n. \n026 \nParallax \nϖ \n[mas] \n1 \n. \n126 \n± \n0 \n. \n013 \nGoodness of fit \nχ \n2 \nRVs \n/N \nRVs \n1 \n. \n3', 'GaiaDR3 ID2426116249713980416': 'Orbital period \nP orb [days] \n719 . 8 ± 0 . 9 \nSemi-major axis \na [au] \n2 . 075 ± 0 . 021 \nPhotocenter semi-major axis \na 0 [mas] \n2 . 040 ± 0 . 014 \nEccentricity \ne \n0 \n. \n3993 \n± \n0 \n. \n0021 \nInclination \ni [deg] \n145 . 1 ± 0 . 4 \nPeriastron time \nT p [JD-2457389] \n- \n190 \n. \n8 \n± \n3 \n. \n4 \nAscending node angle \nΩ[deg] \n73 . 8 ± 1 . 6 \nArgument of periastron \nω \n[deg] \n172 . 41 ± 0 . 33 \nLuminous star mass \nM \n⋆ \n[ \nM \n⊙ \n] \n0 . 94 ± 0 . 04 \nNeutron star mass \nM 2 [ M ⊙ ] \n1 . 362 ± 0 . 034 \nRV semi-amplitude \nK \n⋆ \n[kms \n- \n1 \n] \n11 . 57 ± 0 . 07 \nRV mass function \nf \n( \nM \n2 \n) \nRVs \n[M \n⊙ \n] \n0 . 0891 ± 0 . 0016 \nCenter-of-mass RV Parallax \nγ \n[kms \n- \n1 \n] \n22 . 22 ± 0 . 04 \nϖ \n[mas] \n1 \n. \n661 \n± \n0 \n. \n019 \nGoodness of fit \nχ \n2 \nRVs \n/N \nRVs \n1 \n. \n4', 'GaiaDR3 ID6328149636482597888': 'Orbital period \nP orb [days] \n730 . 9 ± 0 . 5 \nSemi-major axis \na \n[au] \n2 \n. \n208 \n± \n0 \n. \n016 \nPhotocenter semi-major axis \na \n0 \n[mas] \n2 . 132 ± 0 . 016 \nEccentricity \ne \n0 . 1203 ± 0 . 0022 \nRVs \nRVs \nT p [JD-2457389] \n- 257 . 1 ± 1 . 1 \nΩ[deg] \n132 \n. \n5 \n± \n0 \n. \n6 \nω \n[deg] \n55 \n. \n36 \n± \n0 \n. \n14 \nM \n⋆ \n[ \nM \n⊙ \n] \n0 . 91 ± 0 . 05 \nM 2 [ M ⊙ ] \n1 \n. \n258 \n± \n0 \n. \n032 \nK \n⋆ \n[kms \n- \n1 \n] \n18 . 584 ± 0 . 018 \nf \n( \nM \n2 \n) \nRVs \n[M \n⊙ \n] \n0 \n. \n3767 \n± \n0 \n. \n0010 \nγ \n[kms \n- \n1 \n] \n- 69 . 617 ± 0 . 012 \nϖ \n[mas] \n1 \n. \n812 \n± \n0 \n. \n010 \nχ \n2 \n/N \nRVs \n2 . 3 \nRVs \nInclination \ni \n[deg] \n68 . 8 ± 0 . 5 \nPeriastron time \nT \np \n[JD-2457389] \n186 \n. \n4 \n± \n2 \n. \n3 \nAscending node angle \nΩ[deg] \n82 . 7 ± 0 . 8 \nArgument of periastron \nω \n[deg] \n259 . 4 ± 0 . 7 \nLuminous star mass \nM ⋆ [ M ⊙ ] \n0 . 790 ± 0 . 030 \nNeutron star mass \nM 2 [ M ⊙ ] \n1 . 898 ± 0 . 030 \nRV semi-amplitude \nK \n⋆ f ( M 2 ) RVs [M ⊙ ] γ [kms - 1 ] \n[kms \n- \n1 \n] \n21 . 79 ± 0 . 04 \nRV mass function \n0 . 766 ± 0 . 004 \nCenter-of-mass RV \n133 \n. \n457 \n± \n0 \n. \n033 \nParallax \nϖ [mas] 2 \n1 . 367 ± 0 . 011 \nGoodness of fit \nχ \n/N \nRVs \n0 . 7 \nJ1048+6547; \nGaiaDR3 ID1058875159778407808 \nOrbital period \nP orb [days] \n827 ± 5 \nSemi-major axis \na \n[au] \n2 . 344 ± 0 . 031 \nPhotocenter semi-major axis \na 0 [mas] \n1 . 301 ± 0 . 017 \nEccentricity \ne \n0 \n. \n357 \n± \n0 \n. \n009 \nInclination \ni [deg] T p [JD-2457389] \n107 . 8 ± 1 . 0 \nPeriastron time \n- \n47 ± 6 \nAscending node angle \nΩ[deg] \n159 \n. \n9 \n± \n1 \n. \n4 \nArgument of periastron \nω \n[deg] \n288 . 5 ± 2 . 4 \nLuminous star mass \nM \n⋆ \n[ \nM \n⊙ \n] \n0 \n. \n99 \n± \n0 \n. \n05 \nNeutron star mass \nM 2 [ M ⊙ ] \n1 . 52 ± 0 . 07 \nRV semi-amplitude \nK \n⋆ \n[kms \n- \n1 \n] \n19 . 0 ± 0 . 5 \nRV mass function \nf \n( \nM \n2 \n) \nRVs \n[M \n⊙ \n] \n0 . 479 ± 0 . 030 \nCenter-of-mass RV \nγ \n[kms \n- \n1 \n] \n56 . 8 ± 1 . 2 \nParallax \nϖ \n[mas] \n0 \n. \n916 \n± \n0 \n. \n016 \nGoodness of fit \nχ \n2 \n/N \nRVs \n1 . 4', 'GaiaDR3 ID1801110822095134848': 'Orbital period \nP orb [days] \n889 \n. \n5 \n± \n0 \n. \n7 \nSemi-major axis \na \n[au] \n2 \n. \n405 \n± \n0 \n. \n029 \nPhotocenter semi-major axis \na 0 [mas] \n5 . 923 ± 0 . 026 \nEccentricity \ne \n0 . 5840 ± 0 . 0035 \nInclination \ni \n[deg] \n125 . 29 ± 0 . 24 \nPeriastron time \nT p [JD-2457389] \n- 392 . 4 ± 0 . 9 \nAscending node angle \nΩ[deg] \n167 \n. \n7 \n± \n0 \n. \n4 \nArgument of periastron \nω \n[deg] \n243 \n. \n03 \n± \n0 \n. \n33 \nLuminous star mass \nM \n⋆ \n[ \nM \n⊙ \n] \n0 . 95 ± 0 . 05 \nNeutron star mass \nM \n2 \n[ \nM \n⊙ \n] \n1 \n. \n396 \n± \n0 \n. \n035 \nRV semi-amplitude \nK \n⋆ \n[kms \n- \n1 \n] \n17 . 60 ± 0 . 12 \nRV mass function \nf \n( \nM \n2 \n) \nRVs \n[M \n⊙ \n] \n0 \n. \n269 \n± \n0 \n. \n005 \nCenter-of-mass RV \nγ \n[kms \n- \n1 \n] \n- 43 . 19 ± 0 . 11 \nParallax \nϖ \n[mas] \n4 \n. \n137 \n± \n0 \n. \n016 \nGoodness of fit \nχ \n2 \nRVs \n/N \nRVs \n1 . 7', 'J0230+5950;': 'GaiaDR3 ID465093354131112960 \nOrbital period \nP orb [days] \n1029 \n± \n5 \nSemi-major axis \na \n[au] \n2 . 713 ± 0 . 020 \nPhotocenter semi-major axis \na \n0 \n[mas] \n3 . 82 ± 0 . 06 \nRVs \nRVs \nTable 5 lists all the RVs used in our analysis. \nTABLE 4 Orbit fitting results for all candidates.', 'RADIAL VELOCITIES': 'J2102+3703 \n2460285.6430 \n16 \n. \n12 \n± \n0 \n. \n04 \nJ2102+3703 \n2460307.5776 \n12 \n. \n99 \n± \n0 \n. \n04 \nJ2102+3703 \n2460464.8749 \n20 \n. \n87 \n± \n0 \n. \n04 \nJ0742-4749 \n2459905.8042 \n58 \n. \n32 \n± \n0 \n. \n06 \nJ0742-4749 \n2460038.5929 \n43 \n. \n73 \n± \n0 \n. \n50 \nJ0742-4749 \n2460091.5001 \n35 \n. \n10 \n± \n0 \n. \n17 \nJ0742-4749 \n2460228.8210 \n28 \n. \n03 \n± \n0 \n. \n05 \nJ0742-4749 \n2460286.7148 \n33 \n. \n71 \n± \n0 \n. \n07 \nJ0742-4749 \n2460299.7489 \n35 \n. \n99 \n± \n0 \n. \n05 \nJ0742-4749 \n2460333.6623 \n43 \n. \n42 \n± \n0 \n. \n09 \nJ0742-4749 \n2460395.5552 \n57 \n. \n40 \n± \n0 \n. \n05 \nJ0152-2049 \n2459815.8462 \n60 \n. \n86 \n± \n0 \n. \n11 \nJ0152-2049 \n2459817.8988 \n60 \n. \n71 \n± \n0 \n. \n11 \nJ0152-2049 \n2459853.8745 \n57 \n. \n73 \n± \n0 \n. \n08 \nJ0152-2049 \n2459878.8281 \n55 \n. \n80 \n± \n0 \n. \n08 \nJ0152-2049 \n2459903.6764 \n54 \n. \n02 \n± \n0 \n. \n08 \nJ0152-2049 \n2459921.6314 \n52 \n. \n97 \n± \n0 \n. \n08 \nJ0152-2049 \n2460111.9030 \n37 \n. \n40 \n± \n0 \n. \n06 \nJ0152-2049 \n2460186.7817 \n25 \n. \n29 \n± \n0 \n. \n05 \nJ0152-2049 \n2460222.8075 \n21 \n. \n91 \n± \n0 \n. \n07 \nJ0152-2049 \n2460263.7682 \n64 \n. \n93 \n± \n0 \n. \n06 \nJ0152-2049 \n2460284.6708 \n66 \n. \n08 \n± \n0 \n. \n10 \nJ0152-2049 \n2460284.6789 \n66 \n. \n16 \n± \n0 \n. \n07 \nJ0152-2049 \n2460315.5915 \n64 \n. \n05 \n± \n0 \n. \n19 \nJ0152-2049 \n2460340.5730 \n61 \n. \n76 \n± \n0 \n. \n05 \nJ0152-2049 \n2460345.6194 \n61 \n. \n16 \n± \n0 \n. \n08 \nJ0003-5604 \n2459813.8841 \n52 \n. \n24 \n± \n0 \n. \n08 \nJ0003-5604 \n2459829.6990 \n51 \n. \n25 \n± \n0 \n. \n06 \nJ0003-5604 \n2459832.7276 \n50 \n. \n93 \n± \n0 \n. \n16 \nJ0003-5604 \n2459903.6453 \n47 \n. \n00 \n± \n0 \n. \n07 \nJ0003-5604 \n2459923.5775 \n45 \n. \n90 \n± \n0 \n. \n06 \nJ0003-5604 \n2460098.9090 \n35 \n. \n87 \n± \n0 \n. \n12 \nJ0003-5604 \n2460113.8234 \n35 \n. \n37 \n± \n0 \n. \n11 \nJ0003-5604 \n2460139.7994 \n38 \n. \n17 \n± \n0 \n. \n05 \nJ0003-5604 \n2460185.6890 \n87 \n. \n74 \n± \n0 \n. \n07 \nJ0003-5604 \n2460221.7244 \n71 \n. \n15 \n± \n0 \n. \n06 \nJ0003-5604 \n2460284.6157 \n60 \n. \n13 \n± \n0 \n. \n05 \nJ0003-5604 \n2460337.5476 \n55 \n. \n07 \n± \n0 \n. \n07 \nJ1733+5808 \n2460031.8949 \n- \n3 \n. \n46 \n± \n0 \n. \n04 \nJ1733+5808 \n2460049.9539 \n- \n3 \n. \n67 \n± \n0 \n. \n04 \nJ1733+5808 \n2460071.8604 \n- \n4 \n. \n51 \n± \n0 \n. \n04 \nJ1733+5808 \n2460092.7797 \n- \n6 \n. \n89 \n± \n0 \n. \n03 \nJ1733+5808 \n2460109.9133 \n- \n10 \n. \n09 \n± \n0 \n. \n04 \nJ1733+5808 \n2460123.8108 \n- \n13 \n. \n63 \n± \n0 \n. \n05 \nJ1733+5808 \n2460157.7645 \n- \n24 \n. \n95 \n± \n0 \n. \n04 \nJ1733+5808 \n2460201.6402 \n- \n35 \n. \n21 \n± \n0 \n. \n03 \nJ1733+5808 \n2460217.6235 \n- \n36 \n. \n03 \n± \n0 \n. \n04 \nJ1733+5808 \n2460235.6037 \n- \n35 \n. \n66 \n± \n0 \n. \n02 \nJ1733+5808 \n2460353.0203 \n- \n23 \n. \n53 \n± \n0 \n. \n03 \nJ1733+5808 \n2460388.0004 \n- \n19 \n. \n74 \n± \n0 \n. \n03 \nJ1733+5808 \n2460418.9187 \n- \n16 \n. \n62 \n± \n0 \n. \n03 \nJ1150-2203 \n2459953.9957 \n36 \n. \n57 \n± \n0 \n. \n04 \nJ1150-2203 \n2459979.9657 \n35 \n. \n69 \n± \n0 \n. \n03 \nJ1150-2203 \n2460002.7907 \n34 \n. \n62 \n± \n0 \n. \n03 \nJ1150-2203 \n2460008.8457 \n34 \n. \n23 \n± \n0 \n. \n04 \nJ1150-2203 \n2460038.6835 \n32 \n. \n17 \n± \n0 \n. \n50 \nJ1150-2203 \n2460055.7516 \n30 \n. \n12 \n± \n0 \n. \n02 \nJ1150-2203 \n2460072.6913 \n27 \n. \n65 \n± \n0 \n. \n02 \nJ1150-2203 \n2460089.6143 \n24 \n. \n40 \n± \n0 \n. \n03 \nJ1150-2203 \n2460103.5881 \n20 \n. \n80 \n± \n0 \n. \n03 \nJ1150-2203 \n2460103.6773 \n20 \n. \n94 \n± \n0 \n. \n03 \nJ1150-2203 \n2460110.5077 \n18 \n. \n65 \n± \n0 \n. \n04 \nJ1150-2203 \n2460286.8120 \n32 \n. \n44 \n± \n0 \n. \n03 \nJ1150-2203 \n2460308.0275 \n33 \n. \n92 \n± \n0 \n. \n03 \nJ1150-2203 \n2460333.7586 \n35 \n. \n47 \n± \n0 \n. \n03 \nJ1150-2203 \n2460344.9676 \n35 \n. \n96 \n± \n0 \n. \n03 \nJ1150-2203 \n2460364.9310 \n36 \n. \n63 \n± \n0 \n. \n03 \nJ1150-2203 \n2460386.8766 \n37 \n. \n22 \n± \n0 \n. \n06 \nJ1150-2203 \n2460397.6330 \n37 \n. \n48 \n± \n0 \n. \n03 \nJ1150-2203 \n2460407.8571 \n37 \n. \n59 \n± \n0 \n. \n03 \nJ1150-2203 \n2460445.6831 \n38 \n. \n11 \n± \n0 \n. \n03 \nJ1449+6919 \n2459980.9622 \n- \n77 \n. \n13 \n± \n0 \n. \n05 \nJ1449+6919 \n2460016.9387 \n- \n73 \n. \n74 \n± \n0 \n. \n05 \nJ1449+6919 \n2460040.9170 \n- \n71 \n. \n06 \n± \n0 \n. \n03 \nJ1449+6919 \n2460058.9230 \n- \n68 \n. \n98 \n± \n0 \n. \n05 \nJ1449+6919 \n2460066.8727 \n- \n67 \n. \n94 \n± \n0 \n. \n03 \nJ1449+6919 \n2460090.8304 \n- \n64 \n. \n91 \n± \n0 \n. \n04 \nJ1449+6919 \n2460110.7739 \n- \n62 \n. \n17 \n± \n0 \n. \n04 \nJ1449+6919 \n2460123.7389 \n- \n60 \n. \n43 \n± \n0 \n. \n08 \nJ1449+6919 \n2460131.6774 \n- \n59 \n. \n29 \n± \n0 \n. \n05 \nJ1449+6919 \n2460161.6674 \n- \n55 \n. \n10 \n± \n0 \n. \n04 \nJ1449+6919 \n2460203.6256 \n- \n50 \n. \n11 \n± \n0 \n. \n03 \nJ1449+6919 \n2460306.0214 \n- \n59 \n. \n32 \n± \n0 \n. \n04 \nJ1449+6919 \n2460338.9694 \n- \n69 \n. \n48 \n± \n0 \n. \n05 \nJ1449+6919 \n2460359.9881 \n- \n75 \n. \n04 \n± \n0 \n. \n03 \nJ1449+6919 \n2460388.9268 \n- \n80 \n. \n58 \n± \n0 \n. \n04 \nJ1449+6919 \n2460407.9069 \n- \n82 \n. \n97 \n± \n0 \n. \n02 \nJ1449+6919 \n2460430.8414 \n- \n84 \n. \n55 \n± \n0 \n. \n03 \nJ1449+6919 \n2460442.8494 \n- \n85 \n. \n05 \n± \n0 \n. \n03 \nJ1449+6919 \n2460473.7228 \n- \n85 \n. \n21 \n± \n0 \n. \n05 \nJ0217-7541 \n2459815.8817 \n65 \n. \n41 \n± \n0 \n. \n11 \nJ0217-7541 \n2459905.6561 \n77 \n. \n98 \n± \n0 \n. \n04 \nJ0217-7541 \n2459984.5744 \n69 \n. \n87 \n± \n0 \n. \n06 \nJ0217-7541 \n2460013.5032 \n62 \n. \n79 \n± \n0 \n. \n08 \nJ0217-7541 \n2460112.9135 \n50 \n. \n19 \n± \n0 \n. \n11 \nJ0217-7541 \n2460185.7575 \n48 \n. \n89 \n± \n0 \n. \n10 \nJ0217-7541 \n2460222.8307 \n49 \n. \n26 \n± \n0 \n. \n05 \nJ0217-7541 \n2460285.6091 \n51 \n. \n48 \n± \n0 \n. \n03 \nJ0217-7541 \n2460299.7070 \n52 \n. \n08 \n± \n0 \n. \n03 \nJ0217-7541 \n2460338.5986 \n54 \n. \n36 \n± \n0 \n. \n04 \nJ0639-3655 \n2459903.7571 \n- \n12 \n. \n07 \n± \n0 \n. \n05 \nJ0639-3655 \n2459919.7583 \n- \n12 \n. \n04 \n± \n0 \n. \n04 \nJ0639-3655 \n2460038.5444 \n- \n8 \n. \n82 \n± \n0 \n. \n50 \nJ0639-3655 \n2460076.5173 \n- \n4 \n. \n55 \n± \n0 \n. \n13 \nJ0639-3655 \n2460089.4660 \n- \n5 \n. \n97 \n± \n0 \n. \n08 \nJ0639-3655 \n2460185.8288 \n- \n9 \n. \n71 \n± \n0 \n. \n32 \nTRES TRES TRES FEROS MIKE FEROS FEROS FEROS FEROS FEROS FEROS FEROS FEROS TRES TRES FEROS FEROS FEROS FEROS FEROS TRES TRES FEROS TRES FEROS TRES FEROS FEROS FEROS FEROS FEROS FEROS FEROS FEROS FEROS FEROS FEROS FEROS TRES TRES TRES TRES TRES TRES TRES TRES TRES TRES TRES TRES TRES TRES TRES FEROS TRES MIKE TRES FEROS FEROS TRES TRES FEROS FEROS TRES FEROS TRES TRES TRES FEROS TRES TRES TRES TRES TRES TRES TRES TRES TRES TRES TRES TRES TRES TRES TRES TRES TRES TRES TRES TRES TRES FEROS FEROS FEROS FEROS FEROS FEROS FEROS FEROS FEROS FEROS FEROS FEROS MIKE FEROS FEROS FEROS \n<!-- image --> \nTABLE 5 Radial velocities for all targets. A machine-readable version of the table is included as supplemental material. \n<!-- image --> \nThis paper was built using the Open Journal of Astrophysics L A T E X template. The OJA is a journal which provides fast and easy peer review for new papers in the astro-ph section of the arXiv, making the reviewing process simpler for authors and referees alike. Learn more at http://astro.theoj.org .'} |
2024arXiv240901616J | The radial gradient of gasphase metallicity is a powerful probe of the chemical and structural evolution of starforming galaxies closely tied to disk formation and gas kinematics in the early universe. We present spatially resolved chemical and dynamical properties for a sample of 26 galaxies at 0.5 lesssim z lesssim 1.7 from the MSA3D survey. These innovative observations provide 3D spectroscopy of galaxies at a spatial resolution approaching JWSTs diffraction limit and a high spectral resolution of Rsimeq2700. The metallicity gradients measured in our galaxy sample range from 0.05 to 0.02 dexkpc1. Most galaxies exhibit negative or flat radial gradients indicating lower metallicity in the outskirts or uniform metallicity throughout the entire galaxy. We confirm a tight relationship between stellar mass and metallicity gradient at zsim1 with small intrinsic scatter of 0.02 dexkpc1. Our results indicate that metallicity gradients become increasingly negative as stellar mass increases likely because the more massive galaxies tend to be more disky. This relationship is consistent with the predictions from cosmological hydrodynamic zoomin simulations with strong stellar feedback. This work presents the effort to harness the multiplexing capability of JWST NIRSpecMSA in slitstepping mode to map the chemical and kinematic profiles of highredshift galaxies in large samples and at high spatial and spectral resolution. | 2024-09-01T00:00:00Z | ['arXiv:2409.01616', '2024arXiv240901616J', '10.48550/arXiv.2409.01616'] | ['Astrophysics - Astrophysics of Galaxies'] | MSA3D Metallicity Gradients in Galaxies at zsim1 with JWSTNIRSpec Slitstepping Spectroscopy | 2,024 | 169 | 0.6 | ['EPRINT_HTML', 'EPRINT_PDF'] | 3 | https://arxiv.org/pdf/2409.01616.pdf | {'MSA-3D : Metallicity Gradients in Galaxies at z ∼ 1 with JWST/NIRSpec Slit-stepping Spectroscopy': "Mengting Ju, 1 Xin Wang, 1, 2, 3 Tucker Jones, 4 Ivana Bariˇsi'c, 4 Themiya Nanayakkara, 5 Kevin Bundy, 6 Claude-Andr'e Faucher-Gigu'ere, 7 Shuai Feng, 8, 9 Karl Glazebrook, 5 Alaina Henry, 10 Matthew A. Malkan, 11 Danail Obreschkow, 12, 13 Namrata Roy, 14 Ryan L. Sanders, 15, 16 Xunda Sun, 1 Tommaso Treu, 11 and Qianqiao Zhou 1 \n1 \nSchool of Astronomy and Space Science, University of Chinese Academy of Sciences (UCAS), Beijing 100049, China 2 National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, China 3 Institute for Frontiers in Astronomy and Astrophysics, Beijing Normal University, Beijing 102206, China 4 Department of Physics and Astronomy, University of California, Davis, 1 Shields Avenue, Davis, CA 95616, USA 5 Centre for Astrophysics and Supercomputing, Swinburne University of Technology, Hawthorn, VIC 3122, Australia 6 UCO/Lick Observatory, University of California, Santa Cruz, 1156 High Street, Santa Cruz, CA 95064, USA 7 Department of Physics & Astronomy and CIERA, Northwestern University, 1800 Sherman Ave, Evanston, IL 60201, USA 8 College of Physics, Hebei Normal University, 20 South Erhuan Road, Shijiazhuang, 050024, China 9 Hebei Key Laboratory of Photophysics Research and Application, 050024 Shijiazhuang, China 10 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA 11 Department of Physics and Astronomy, University of California, Los Angeles, CA 90095-1547, USA 12 International Centre for Radio Astronomy Research (ICRAR), M468, University of Western Australia, Perth, WA 6009, Australia 13 Australian Research Council, ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), Australia 14 Center for Astrophysical Sciences, Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD, 21218 15 Department of Physics and Astronomy, University of Kentucky, 505 Rose Street, Lexington, KY 40506, USA 16 Department of Physics and Astronomy, University of California, Davis, One Shields Ave, Davis, CA 95616, USA", 'ABSTRACT': "The radial gradient of gas-phase metallicity is a powerful probe of the chemical and structural evolution of star-forming galaxies, closely tied to disk formation and gas kinematics in the early universe. We present spatially resolved chemical and dynamical properties for a sample of 25 galaxies at 0 . 5 ≲ z ≲ 1 . 7 from the MSA-3D survey. These innovative observations provide 3D spectroscopy of galaxies at a spatial resolution approaching JWST's diffraction limit and a high spectral resolution of R ≃ 2700. The metallicity gradients measured in our galaxy sample range from -0.03 to 0.02 dex kpc -1 . Most galaxies exhibit negative or flat radial gradients, indicating lower metallicity in the outskirts or uniform metallicity throughout the entire galaxy. We confirm a tight relationship between stellar mass and metallicity gradient at z ∼ 1 with small intrinsic scatter of 0.02 dex kpc -1 . Our results indicate that metallicity gradients become increasingly negative as stellar mass increases, likely because the more massive galaxies tend to be more 'disky'. This relationship is consistent with the predictions from cosmological hydrodynamic zoom-in simulations with strong stellar feedback. This work presents the effort to harness the multiplexing capability of JWST NIRSpec/MSA in slit-stepping mode to map the chemical and kinematic profiles of high-redshift galaxies in large samples and at high spatial and spectral resolution. \nKeywords: galaxies: High-redshift galaxies - galaxies: star formation - galaxies: abundances galaxies: kinematics and dynamics", '1. INTRODUCTION': "Corresponding author: Mengting Ju, Xin Wang \[email protected] \[email protected] \nGas-phase metallicity is a crucial parameter for studying gas inflows and outflows. Processes such as star formation, gas accretion, and galaxy mergers can lead to local enrichment and dilution of the interstellar medium (e.g, Thielemann et al. 2017; Maiolino & Mannucci \n2019). Metallicity gradients are commonly used to describe the distribution of oxygen abundance in the interstellar medium. The radial gradient of metallicity is a powerful probe of the chemical and structural evolution of star-forming galaxies, closely tied to their disk formation and gas kinematics. Studying these gradients, particularly in galaxies at high redshift, offers valuable information on the population evolution of galaxies (Tremonti et al. 2004; Mannucci et al. 2010; Cresci et al. 2010; Queyrel et al. 2012; Swinbank et al. 2012; Jones et al. 2010a, 2013; Troncoso et al. 2014; Ho et al. 2015; Wang et al. 2019, 2020, 2022a; Ju et al. 2022; Wang & Lilly 2023; Wang et al. 2024a; He et al. 2024; Cheng et al. 2024; Venturi et al. 2024). \nIn the local Universe, galaxies typically show a decrease in metallicity from their center to their outskirts (e.g., Searle 1971; S'anchez et al. 2014; Belfiore et al. 2017; Kreckel et al. 2019; S'anchez-Menguiano et al. 2020). This pattern is referred to as a 'negative metallicity gradient' (Zaritsky et al. 1994; van Zee et al. 1998). More recent large Integral Field Unit (IFU) surveys (e.g., CALIFA, S'anchez et al. 2012; MaNGA, Bundy et al. 2015; SAMI, Bryant et al. 2015) have helped establish demographic trends. The metallicity gradients vary with galaxy morphology and stellar mass: massive disk galaxies tend to have steeper gradients, while low-mass galaxies and analogs of high-redshift systems exhibit flatter gradients (Belfiore et al. 2017; Carton et al. 2018; S'anchez-Menguiano et al. 2018; Mingozzi et al. 2020). However, while trends are observed in units of dex kpc -1 , S'anchez et al. (2014) suggest that local galaxies may have a common gradient when normalized to the effective radius ( ∼ -0 . 1 dex R -1 e , up to 2 R e ) in galaxies with stellar mass M ∗ > 10 9 . 5 M ⊙ . \nNegative metallicity gradients can be explained by the inside-out growth of galaxies, where star formation initially occurs in the central regions, enriching the gas near the center first. This inside-out star formation process influences the distribution of gas-phase metallicity across galaxies (Samland et al. 1997; Dav'e et al. 2011; Hemler et al. 2021). In this model, metals can be redistributed by various physical processes, including stellar feedback and rotation, leading to flat gradients which can be observed in isolated high-redshift galaxies (Yuan et al. 2011; Jones et al. 2013). However, inside-out growth can not predict the positive gradients observed in some high-redshift galaxies. Indeed, there are many works suggesting that stellar feedback (Gibson et al. 2013; Ma et al. 2017; Sharda et al. 2021), mergers, and interactions (Rich et al. 2012) contribute to flattening metallicity gradients. For example, feedbackdriven winds expel metal-rich material from galaxies, \nwhich can then re-accrete in external regions, homogenizing the distribution of metals (Pilkington et al. 2012; Angl'es-Alc'azar et al. 2014; Muratov et al. 2017; Pandya et al. 2021). Detailed descriptions of these processes are found in Venturi et al. (2024). \nWhile negative metallicity gradient slopes are well measured at z ∼ 0, it is unclear whether they persist in the early Universe. Simulations such as IllustrisTNG (Hemler et al. 2021) have shown that at z < 2, more massive galaxies typically have flatter or even positive (also known as 'inverted') metallicity gradients. In contrast, the Feedback in Realistic Environments (FIRE) simulations predict that more massive galaxies have steeper negative gradients (Ma et al. 2017; Bellardini et al. 2021, 2022; Sun et al. 2024). Observations of high-redshift galaxies have found that a larger fraction of these galaxies exhibit flat or inverted metallicity gradients compared to local galaxies (Yuan et al. 2011; Jones et al. 2013; Wuyts et al. 2016; Leethochawalit et al. 2016; Wang et al. 2017, 2019, 2022b; Simons et al. 2021; Cheng et al. 2024). \nNotably, there is a large dispersion in metallicity gradient measurements at high redshifts across previous work, which may be due to large measurement errors and/or heterogeneous methods employed in the analysis. Importantly, kiloparsec-scale angular resolution is crucial to avoid biases in the spatially resolved analysis (Yuan et al. 2013), but was previously only possible with space-based slitless spectroscopy (with challenges due to low spectral resolution, e.g. Wang et al. 2019, 2020, 2022b) or adaptive optics (AO) assisted IFU surveys (with low observing efficiency, e.g. Jones et al. 2013; Leethochawalit et al. 2016), and in both cases with a limited set of emission lines. It is challenging to perform spatially resolved spectroscopy of high-redshift galaxies, primarily due to seeing limitations. For example, the MaNGA survey, based on a ground-based telescope, has a site seeing of approximately 1 . '' 5, corresponding to about 12 kpc at z ∼ 1 (Law et al. 2015). This resolution exceeds the typical size of the kind of high-redshift galaxies that are thought to be Milky Way progenitors. Even excellent seeing of 0 . '' 5 corresponds to about 4 kpc at z ∼ 1, still insufficient to resolve the effective radii of typical high-z galaxies. \nAlthough IFS combined with AO (e.g., Keck/OSIRIS, VLT/SINFONI and ERIS, Gemini/NIFS) can achieve adequate spatial resolution, this is primarily effective at wavelengths λ > 1 . 5 µ m, while instruments such as MUSE with Narrow Field Mode and laser tomography adaptive optics (Bacon et al. 2010) can deliver good resolution reaching red optical wavelengths. AO observations have modest Strehl ratios in typical extragalactic \nfields and especially at shorter wavelengths, and are subject to beam smearing effects in all cases (e.g., Burkert et al. 2016). AO data have proven sufficient to analyze metallicity gradients with strong optical emission lines at z ≳ 1 . 5 (e.g., Yuan et al. 2011; Jones et al. 2013), but it is hard to measure metallicities near z = 1 using optical emission lines such as H β , [O III ] λ 5007, H α . While JWST offers the necessary spatial and spectroscopic resolution at the relevant wavelengths, its IFS modes are limited to observations of single objects. This makes it expensive to study metallicity gradients for large samples of galaxies. \nIn this work, we present measurements of metallicity gradients for 25 galaxies at z = 0 . 5 -1 . 7 using JWST/NIRSpec micro-shutter assembly (MSA) observations in a novel slit-stepping mode from the MSA3D project. This new observing mode allows multiplexed IFS measurements, providing a sufficient gain in efficiency to constrain metallicity gradients for a large high-redshift galaxy sample. By observing 43 galaxies at z ∼ 1 simultaneously, the MSA-3D project significantly reduces the required integration time compared to an equivalent survey with the NIRSpec IFU. In JWST Cycle 1, few projects have obtained as extensive 3D spectroscopy on galaxies at z ∼ 1 as MSA-3D . Our sample is arguably the best to date at z ∼ 1 in many ways. We have (1) high angular resolution, approaching the diffraction limit of a 6.5-meter telescope; (2) exquisite S/N of nebular emission lines detected across multiple scale radii of highz galaxies; (3) far better spectral resolution than grism surveys; (4) coverage of a larger number of diagnostic emission lines as compared to most ground-based AO-assisted spectroscopy; (5) a decently large and uniform sample. For further details and description of the observations, we refer the readers to the MSA-3D project overview paper (Bariˇsi'c et al. 2024). \nThe paper is organized as follows. In Section 2, we introduce the observation strategies of the slit-stepping mode, our targets, and emission lines fitting. In Section 3, we analyze the metallicity maps and the metallicity gradients of our targets. We discuss redshift and mass-dependent evolution in the metallicity gradient slopes in Section 4, and summarize the results in Section 5. We adopt the standard concordance cosmological model of H 0 = 69 . 32 km s -1 Mpc -1 , Ω M = 0 . 2865. Throughout the paper, we abbreviate the forbidden lines with [O III ] λλ 4959 , 5007 = [O III ], [N II ] λλ 6548 , 6584 = [N II ], [S II ] λλ 6717 , 6731 = [S II ], if presented without wavelength values.", '2. DATA': "2.1. Sample and observations \nThe MSA-3D project obtained its first observations from March 29 to March 30, 2023 (JWST Cycle 1, GO2136, PI: Jones). Using JWST/NIRSpec, we observed 43 star-forming galaxies at z = 0 . 5 -1 . 7 in the Extended Groth Strip (EGS) field using a slit-stepping mode to obtain multiplexed integral field spectroscopy (IFS). The field contains extensive photometric and spectral survey data. We selected galaxies with reliable redshifts from the CANDELS survey (Koekemoer et al. 2011) and the 3D-HST survey catalogue (Skelton et al. 2014; Momcheva et al. 2016), ensuring that emission lines such as H α , [N II ], [O III ], and H β are observable using the G140H/F100LP grating/filter. Properties of the 43 targeted galaxies are listed in Table 1. Most of the targets are located on the 'star-forming main sequence,' which is characterized by a specific star formation rate of approximately 10 -9 yr -1 at z ∼ 1 (e.g., Whitaker et al. 2014). The NIRSpec grating and filter, G140H/F100LP, covered a wavelength range of 0 . 97 µm ≤ λ ≤ 1 . 82 µm with a spectral resolution of R ∼ 2700 to map strong nebular emission lines. We employed a consistent MSA mask setup across 63 distinct pointings, moving one slit width (0 . '' 2) along the dispersion axis in each of the 9 steps, and one barshadow width (0 . '' 075) along the crossdispersion axis in each of the 7 steps to avoid the influence of the bar between all shutters. The total exposure time was 20.4 hours and the effective integration time at each spatial position was 117 minutes. Detailed observing strategies and data processing methods are discussed in the MSA-3D project overview paper (Bariˇsi'c et al. 2024). The FoV in the datacubes spans 1 . '' 8 × (2 . '' 0 -3 . '' 0), depending on the number of slitlets (3-5) used to observe each galaxy. The resolution element in a typical reconstructed data cube is about 0 . '' 20 × 0 . '' 08 ( ∼ 1 . 63 × 0 . 65 kpc at z = 1). In this paper, we analyze data cubes that were interpolated onto a new grid with a spatial sampling of 0 . '' 08 × 0 . '' 08. \nWe present an example of the observed spectrum from the central spaxel of galaxy ID8365 ( z = 1 . 68) in Figure 1. The stellar mass of this galaxy is log( M ∗ /M ⊙ ) = 9 . 79 and its SFR is 9.76 M ⊙ yr -1 . These values were obtained from the UVCANDELS catalog, and are reported in Table 1. The properties were estimated using Dense Basis SED modeling based on HST observations ranging from F275W to F160W filters (Sun et al. 2023; Wang et al. 2024b; Mehta et al. 2024). In panel (a), we present a three-color image obtained from JWST/NIRCam imaging using the F115W, F277W, and F444W filters. For this galaxy, we selected five slitlets for observation using the slit-stepping method (indicated by white rectangles). The red box represents the FoV of the resulting pseudo-IFU datacube, which is 1 . '' 8 × 3 . '' 0. \nFigure 1. An example of the observed spectrum from the spaxel in the galaxy center and Gaussian models of the observed emission lines. Panel (a): The color-composite image of galaxy ID8365 is made from the JWST/NIRCam imaging (blue: F115W, green: F277W, red: F444W). The red box (1 . '' 8 × 3 . '' 0) represents the entire field of view of our slit-stepping 3D spectroscopy and the white boxes mark the relative sizes of the 5 open microshutter slitlets. The observing strategy of the MSA-3D project acquires spectroscopy of our targets in 63 pointings in total, moving slitlets in 9 steps of 0 . '' 2 in the dispersion direction and 7 dithers of 0 . '' 075 in the cross-dispersion direction. For further details, refer to Bariˇsi'c et al. (2024). Panel (b) shows the full observed spectrum from the spaxel in the galaxy center indicated as a green box in panel (e), with the orange curves being the Gaussian-fitted emission line models. We detect pronounced nebular emission features in the galaxy center. We also zoom-in on the line fitting results in panels (c) and (d). The grey vertical dashed lines represent the locations of H β , [O III ], H α , [N II ], and [S II ]. Panel (e): the flux map of the H α emission line, where the ellipse represents the resolution element. \n<!-- image --> \nThe chip gap occurs near 1.4 µ m (corresponding to a rest wavelength of ∼ 5223 ˚ A for this target). The spectrum exhibits very distinct emission lines, particularly strong ones like the [O II ] λλ 3727 , 3729, H β , [O III ], H α , [N II ], [S II ] emission line. The MSA-3D data provide excellent S/N of these emission lines in individual spaxels.", '2.2. Emission line fitting': "In this work, our primary focus is on the metallicity gradients, which are derived from the ratios of the emission line fluxes. Our sources are star-forming galaxies, with strong emission lines (as shown in Figure 1). We fit Gaussian profiles to various emission lines to determine their line flux, velocity, and velocity dispersion. Specifically, a three-parameter Gaussian curve was fitted to the following emission lines for each spaxel: H β , [O III ], H α , [N II ], [S II ]. In many cases, not all of the lines are observed, depending on the source redshift and wavelength \ncoverage. Table 1 lists the emission lines observed for each galaxy. \nWe typically do not detect Balmer absorption or other stellar features in the spectra of individual spaxels. Therefore, we fit the spectra using a straight line to model the continuum, plus Gaussian profiles for the emission lines. Based on spectral energy distribution modeling of our targets, we find that stellar Balmer absorption has little effect on the results (e.g., Zahid et al. 2011), causing the metallicity to be overestimated by ≲ 0 . 05 dex with negligible effect on the gradient slopes. Each spectrum was fitted in two regions with rest-frame wavelengths near 4820-5100 ˚ A (including H β and [O III ]) and 6520-6800 ˚ A (including H α , [N II ], and [S II ]). We fit the emission lines plus continuum simultaneously. All lines are fit with the same redshift (i.e., velocity) and width (i.e., dispersion), using the values measured for H α . To calculate intrinsic line widths, we correct for the instrument resolution by subtracting it in quadrature from the best-fit value. The resolution curve for \nTable 1. Measured properties for the parent sample of 43 galaxies on which we perform slit-stepping 3D spectroscopy in the MSA-3D project. \nNotes: a The stellar population properties presented here are obtained from the UVCANDELS catalog. \nthe G140H grating shows that the average instrumental FWHM across the emission line wavelength range is approximately 5.2 ˚ A, corresponding to an instrumental dispersion of roughly 50 km s -1 . \nIn Figure 1, we show an example of Gaussian emission line fits with orange lines in panels (c) and (d). The emission lines are distinct in the spaxel of galaxy ID8365, yielding a high S/N for the line flux measurements. For instance, the flux of H α is (133 . 83 ± 2 . 28) × 10 -20 erg / s / cm 2 and the equivalent width (EW) is (44 . 40 ± 1 . 57) ˚ A in the rest frame. The uncertainties in the flux and EWs are calculated based on the residu- \netween the best-fit linear and Gaussian models and the observed spectra. This statistical uncertainty represents the emission S/N, and does not include additional sources of uncertainty such as underlying stellar absorption. The flux map of the H α emission line for galaxy ID8365 is shown in panel (e). The ellipse indicates the resolution element (0 . '' 20 × 0 . '' 08). \nThe sky coverage of the IFS data cubes is larger than the effective radii of our targets (Figure 1), and in general, there are many spaxels with no detectable emission. Therefore, it is necessary to select the reliable spaxels associated with each galaxy for analysis. We select the \nspaxels from the target sources using the properties of H α , requiring the S/N of the H α flux to be greater than 10 and the S/N of the H α EW to be greater than 5. The EW S/N > 5 requirement is typically not a limiting factor, as we exclude only 10% of spaxels which otherwise meet the emission line flux requirement. In addition, we exclude unreliable spaxels by visually inspecting the 3-color images and spectra. Consequently, we obtain spaxels from the target galaxies that are suitable for analysis. The resulting maps of all galaxies analyzed in this paper, including their H α flux, observed velocity, and observed velocity dispersion, are shown in Figure A1 in Appendix A.", '3.1. Measuring chemical abundances': "Most metallicity diagnostics are calibrated on H II regions. Therefore, before obtaining gas-phase metallicity maps, we need to verify that the emission lines originate from star-forming regions. Isolating individual H II regions is highly challenging in the highz universe, due to limited resolution (S'anchez et al. 2014). In this work, we exclude regions not classified as star-forming in the BPT diagram (Baldwin et al. 1981; Kewley et al. 2001; Kauffmann et al. 2003) using the [N II ]/H α versus [O III ]/H β ratios. However, because of detector gaps and limited wavelength ranges, not all four emission lines are detected in each galaxy (Table 1). Cid Fernandes et al. (2011) demonstrated a tight correlation between [N II ]/H α and EW(H α ), indicating that pure starforming regions have EW(H α ) > 3 ˚ A and log([N II ]/H α ) < -0 . 4. For galaxies where H II regions cannot be identified using the BPT diagram, we instead use the WHAN diagram of EW(H α ) and log([N II ]/H α ). Figure A1 in Appendix A presents the BPT or WHAN diagrams and their maps color-coded according to the positions of individual regions for galaxies in our sample. We compared the results of WHAN and BPT diagrams for the 10 galaxies where both diagnostics are available. In both diagrams, spaxels classified as non-star-forming are largely in the outer regions of low S/N and constitute a small fraction of the total spaxels (Figure A1). Additionally, the majority of spaxels classified as starforming by the WHAN diagram were also classified as star-forming in the BPT diagram, indicating strong consistency between these diagnostic methods. This analysis indicates that WHAN diagrams can be effectively utilized for our sample to provide consistent results in cases where the BPT diagram is not available. \nVarious strong-line methods use different indicators to measure gas-phase metallicity. Examples include \nO3N2 = log [O III ] λ 5007 / H β [N II ] λ 6584 / H α (Pettini & Pagel 2004; \nMarino et al. 2013), N2 = log [N II ] λ 6584 H α (StorchiBergmann et al. 1994; Denicol'o et al. 2002; Marino et al. 2013), and the N2S2H α diagnostic (using H α , [N II ], and [S II ]; Dopita et al. 2016; Cameron et al. 2019). O3N2 and N2 are among the most widely used indicators for calculating gas-phase metallicity. These indicators are minimally affected by dust extinction since they use pairs of emission lines which are close in wavelength. Given the emission line coverage of our targets (Table 1), the O3N2 indicator can be used for 20 galaxies, the N2 indicator for 38 galaxies, and N2S2H α for 33 galaxies. While N2S2H α has some advantages over N2, it requires better S/N such that there are fewer spaxels with reliable measurements. Thus, for this work, we use the N2 indicator to give the largest homogeneous sample. While we use both WHAN and BPT diagrams to select star-forming spaxels, we have verified that these give consistent metallicity gradients for the 10 galaxies where both diagnostics are available. It is important to use the same calibration for all galaxies in our analysis in order to avoid systematic differences (e.g., Kewley & Ellison 2008; L'opez-S'anchez et al. 2012; Peimbert et al. 2012). The N2 values for our targets range from -2.5 to -0.3, and we use the fit given by Pettini & Pagel (2004) which is valid across this range: \n12 + log(O / H) = 9 . 37 + 2 . 03 × N2 + 1 . 26 × (N2) 2 +0 . 32 × (N2) 3 (1) \nwith a systematic 1σ uncertainty of 0.18 dex. We thus combine this scatter in quadrature when deriving metallicities using this strong line calibration. \nWe also visually inspect the resulting metallicity maps of these galaxies. Spaxels with uncertainty of > 0 . 25 dex in metallicity were discarded before deriving the metallicity gradient. Discarding these spaxels has no significant effect on the results. To ensure sufficient spatial coverage to measure a reliable gas-phase metallicity gradient, we required galaxies to have a minimum of 20 spaxels located in the star-forming regions of the BPT or WHAN diagram. This requirement results in 25 galaxies suitable for metallicity gradient measurements (out of 38 galaxies with suitable wavelength coverage). Of these, 10 galaxies relied on the BPT diagram to determine their star-forming spaxels, while the remaining galaxies utilized the WHAN diagram. The metallicity maps for each galaxy are shown in Figure A1 in Appendix A.", '3.2. Deriving abundance gradients': "In order to measure the de-projected distance (R/kpc) of each spaxel from the center, it is essential to deter- \nFigure 2. The distribution of metallicity gradient slopes, in units of dex kpc -1 . The black error bar shows the median 1σ uncertainty, highlighting the very small measurement uncertainties afforded by the high-quality MSA-3D data set. The metallicity gradients of our 25 galaxies are predominantly negative or flat, with slopes typically more positive than those of local spiral galaxies. \n<!-- image --> \nmine the galaxy center, position angle, and axis ratio (b/a). These structural parameters were measured using multi-wavelength HST photometry from the CANDELS survey (Koekemoer et al. 2011). We used the position angle and b/a values obtained from the CANDELS catalog (Stefanon et al. 2017). The centers of the galaxies are determined as the peaks of the continuum maps from the data cubes, using rest wavelengths around 5500 ± 25 ˚ A whenever possible. If this is not feasible, wavelengths around 6500 ± 25 ˚ A are used. Both regions represent optical continuum which is free of strong emission lines. The metallicity gradient (i.e., O/H as a function of de-projected radius) for each galaxy is shown in Figure A1 in Appendix A. We also adopt the effective radius R e as reported in Stefanon et al. (2017) to examine metallicity as a function of normalized distance R / R e for each galaxy. Figure A1 includes both physical (R/kpc) and normalized (R / R e ) radii, on the lower and upper axes respectively. We perform a linear fit to the gradients from 0.5 kpc to the edge of the field of view. This fit is shown by the red dashed line, with the slope of the line indicated in the subplot. The central region is not used in order to mitigate beam smearing effects. We additionally fit a linear model within the range 0.5 - 2.5 R e , following a similar approach to that used in Belfiore et al. (2017) for purposes of comparison. The distribution of gradients in physical (at > 0 . 5 kpc) units is shown in Figure 2. The gradient slopes measured in our sample range from -0.03 to 0.02 dex kpc -1 , with a median uncertainty of 0.01 dex kpc -1 . We note that the errors of our gradient measurements are primarily \ndominated by the scatter of 0.18 dex from the metallicity inference based on the Pettini & Pagel (2004) strong line calibration. Most galaxies exhibit negative or flat radial gradients, indicating that metallicity is lower or similar in the outskirts compared to the centers. However, the slopes are typically less negative than local galaxies, suggesting that flatter or even inverted metallicity gradients are more common at z ∼ 1 compared to z ∼ 0 (see Section 4). \nThe slopes of gradients can be misleading when radial profiles are not linear. Therefore, it is more accurate to examine the complete profiles (e.g., Oyarz'un et al. 2019). In the subplots of Figure A1, we also show the metallicity profiles represented by the average values in radial bins. In observing the profiles and slopes of these galaxies, we identified some interesting phenomena. The best-fit linear slope alone is insufficient to fully characterize the metallicity gradient in some galaxies. For instance, although the slopes of galaxies ID4391 and ID13416 are similar, their profiles differ significantly. These distinct profiles may indicate different galaxy formation processes, with ID13416 likely undergoing a more quiescent state than ID4391. In this paper, we attempt to quantify these profiles by calculating the root-mean-square error (RMSE), which is the square root of the mean of the differences between the observed metallicity in spaxels and predicted values provided by the linear models. The RMSE scatter in our sample ranges from 0.04 to 0.11 dex kpc -1 . The majority display relatively low scatter, such that the radial metallicity profile is well approximated with a linear fit. Only 4 out of 25 galaxies have metallicity gradients associated with large RMSE scatter ( > 0 . 1 dex kpc -1 ).", '4.1. Metallicity gradients versus redshift and stellar mass': "According to the inside-out growth scenario, galaxies quickly form a compact core, resulting in very steep metallicity gradients in high-redshift galaxies. As galaxies evolve and their sizes increase, these gradients tend to flatten. This is similar to the predictions obtained with 'weak' feedback in the MUGS simulation (Gibson et al. 2013). In the same study, the MAGICC simulation with 'enhanced' feedback was presented. In this simulation, the stronger feedback distributes energy and recycles the interstellar medium (ISM) over larger scales, resulting in a 'flat' and nearly time-invariant metallicity gradient. This demonstrates how the redshift evolution of gradient slopes is sensitive to feedback processes and galactic outflows. \n<!-- image --> \nFigure 3. Correlations among the metallicity gradient, redshift, and stellar mass. Left: Metallicity gradients as a function of redshift. The red points with 1σ error bars correspond to our results. The orange and blue lines correspond to the MUGS ('weak' feedback) and MAGICC ('enhanced' feedback) simulations from Gibson et al. (2013), respectively. The green line represents the predicted evolution of metallicity gradients from the TNG50 simulations (Hemler et al. 2021), the magenta line shows predictions from the FIRE-2 zoom-in simulations (Sun et al. 2024) and the yellow line corresponds to the results from EAGLE simulations (Tissera et al. 2022). The grayscale shows a density histogram derived from previous observational results in the literature at z = 0 . 5 -3 (Swinbank et al. 2012; Queyrel et al. 2012; Jones et al. 2013; Leethochawalit et al. 2016; Wuyts et al. 2016; Molina et al. 2017; Carton et al. 2018; Forster Schreiber et al. 2018; Patr'ıcio et al. 2019; Wang et al. 2017, 2019, 2020, 2022b; Curti et al. 2020; Simons et al. 2021; Li et al. 2022; Venturi et al. 2024). The gradients in our sample are flat and nearly redshift-invariant, showing higher values than those from the MUGS, MAGICC, and TNG50 simulations, but similar to the results of the FIRE-2 zoom-in simulations. Right: Metallicity gradients as a function of stellar mass. The red points represent our results, and the red line represents the best linear fit. The blue and green lines correspond to local galaxies (from Belfiore et al. 2017) and star-forming galaxies at cosmic noon (from Cheng et al. 2024), respectively. The magenta line is the mass-metallicity gradient relationship predicted by the FIRE-2 simulations. The grayscale density histogram in this panel is derived from results in the literature in the same range z = 0 . 5 -1 . 7 as our sample (Swinbank et al. 2012; Queyrel et al. 2012; Leethochawalit et al. 2016; Wuyts et al. 2016; Molina et al. 2017; Carton et al. 2018; Forster Schreiber et al. 2018; Patr'ıcio et al. 2019; Wang et al. 2017, 2019, 2020, 2022b; Curti et al. 2020; Simons et al. 2021). Our work reveals a mass dependence of metallicity gradients at ∼ 2σ significance, showing that more massive galaxies exhibit more negative gradients, consistent with the FIRE-2 simulation results. \n<!-- image --> \nThe left panel of Figure 3 shows metallicity gradient slopes measured from our sample (red points) as a function of redshift, alongside other observational and theoretical results. The orange and blue lines represent the MUGS and MAGICC simulations, respectively. These curves are for a single example galaxy. The green line corresponds to the results from the TNG50 simulation (Hemler et al. 2021), the yellow line illustrates EAGLE simulations (Tissera et al. 2022), and the magenta line shows the Feedback in Realistic Environments (FIRE2) simulations (Sun et al. 2024). All these lines are for population averages. The FIRE-2 curve corresponds to a set of cosmological zoom-in simulations and shows the average metallicity gradient for eight galaxies with stellar masses ranging from log M ∗ / M ⊙ ≈ 9 - 10.5 at z = 1. The grayscale is a density histogram of measurements from the observational literature at z = 0 . 5 -3 (Swinbank et al. 2012; Queyrel et al. 2012; Jones et al. 2013; Leethochawalit et al. 2016; Wuyts et al. 2016; Molina et al. 2017; Carton et al. 2018; Forster Schreiber et al. 2018; Patr'ıcio et al. 2019; Wang et al. 2017, 2019, 2020, 2022b; Curti et al. 2020; Simons et al. 2021; Li et al. 2022; Venturi et al. 2024). The metallicity gradients \nmeasured for our sample are relatively flat and nearly time-invariant over the observed redshift range, aligning with the results of other studies. They are slightly higher (more positive) than those predicted by the MUGS, MAGICC, and TNG50 simulations, but similar to the results of the FIRE-2 and EAGLE simulations. This suggests that feedback from star formation can significantly impact the metallicity gradient, particularly in galaxies experiencing intense star formation. \nTracing the growth of a galaxy through observations is nearly impossible. However, galaxies grow in mass as they evolve. Therefore, we can observe the changes in the metallicity gradient from small to high-mass galaxies as representing different stages in their growth (Hemler et al. 2021). In the right panel of Figure 3, we show the relationship between metallicity gradient slope and stellar mass (with masses listed in Table 1). We find a significant correlation: more massive galaxies have more negative gradient slopes. We now quantify the mass dependence. Considering that our sample consists of only 25 galaxies, we used bootstrap resampling to account for sample variance effects. Using 1000 iterations, we derived a linear model, resulting in the following equa- \ntion: \ngradient(dex kpc -1 ) = -0 . 0140[ ± 0 . 0062] × [log(M ∗ / M ⊙ ) -10] -0 . 0097[ ± 0 . 0029] . (2) \nThe Pearson correlation coefficient is -0 . 36 and the Pvalue is 0 . 07, indicating a fairly strong anti-correlation. We calculated an RMSE of 0.01 dex kpc -1 for the slopes of these galaxies concerning this correlation. Our study therefore reveals a strong negative correlation between the metallicity gradient and galaxy mass, statistically significant at ∼ 2σ . More massive galaxies tend to be more extended, providing a greater number of usable spaxels for calculating the metallicity gradient. On average, 107 spaxels per galaxy are employed in this study, with more massive galaxies typically having more usable spaxels. However, at fixed stellar mass, there is no clear correlation between the number of spaxels and the metallicity gradient slope. This supports the reliability of the relationship between stellar mass and the metallicity gradient (Equation 2). We consider the derived mass-metallicity gradient relation to be robust. \nThe root mean square (RMS) of the slopes for these 25 galaxies is 0.02 dex kpc -1 , which is significantly smaller than the dispersion of the other observations in Figure 3 (grayscale). The overall RMS of these literature measurements is approximately 0.06 dex kpc -1 , and we calculated the RMS of the slopes provided by various individual studies, resulting in a mean value of 0.05 dex kpc -1 . Therefore, the dispersion of our sample is much smaller than that of the other observations in Figure 3 (grayscale), thanks to the high data quality enabled by the MSA-3D slit-stepping observing strategy. \nThe relationship between metallicity gradients and mass for local galaxies is illustrated with a blue line in Figure 3 (right panel). For low-mass galaxies, the gradient appears to be flat or slightly inverted. As the mass increases, the gradient steepens, and for massive galaxies above log M ∗ / M ⊙ > 10 . 5 it becomes flatter again but remains negative (Belfiore et al. 2017). The flattening in massive galaxies may result from the metallicity reaching equilibrium, while the flattening in low-mass galaxies may be due to strong feedback, gas mixing, and wind recycling. At z ∼ 1, our sample shows that more massive galaxies exhibit more negative gradients, similar to the FIRE-2 simulation results. The flat gradients of the low-mass galaxies are similar to those found in local galaxies and can be attributed to strong stellar feedback. However, we do not see a flattening of slopes in the most massive galaxies in our z ∼ 1 sample, and their steep slopes are not explained by the inside-out growth scenario. This difference at high masses com- \nz ∼ 0 may be due to higher gas fractions and star formation rates at z ∼ 1, such that gas metallicities have not reached equilibrium and are more susceptible to processes such as radial gas flows. \nMetallicity gradients are valuable indicators of the spatial distribution of metallicity within galaxies, but they do not capture all relevant characteristics (e.g., galaxy ID6199, ID9424 and ID11225). In galaxy ID9424, two distinct regions with varying metallicity are visible on the metallicity map, yet this is not reflected in the radial distribution alone. Azimuthal variations in metallicity, alongside radial gradients, may also provide critical insights. Bellardini et al. (2021, 2022) explored such azimuthal variations through FIRE simulations, showing that at early times (high redshift), local star formation leads to metal inhomogeneities in the azimuthal direction, and that the radial gradient may not fully describe the metallicity distribution. In galaxy ID9424, several strong H α clumps are located far from the galactic center and exhibit significantly lower metallicity. This may be attributed to gas accretion, which introduces azimuthal scatter without substantial radial variation. As galaxies evolve, they transition from being azimuthally dispersed to radially dominated systems, forming well-settled disks characterized by more pronounced radial distributions and greater azimuthal uniformity. The non-radial variations in these MSA-3D metallicity maps can be used to infer ISM enrichment and mixing scales at z ∼ 1, and compare them with the ∼ 1 kpc scales found for z = 0 (Metha et al. 2021; Li et al. 2023). We leave such an analysis for future work.", '4.2. The connection between galaxy chemo-structural evolution and disk formation: two case studies': "The evolution of metallicity gradients is thought to be closely related to the formation of well-established disks and thus the emergence of the Hubble sequence. Gas-phase gradients require limited radial mixing from processes such as mergers/interactions, inflows, and feedback-driven outflows, in addition to near-circular orbits (e.g., Ma et al. 2017; Bellardini et al. 2021, 2022; Sun et al. 2024). These conditions are also associated with disk settling. In this section, we investigate the connection between metallicity gradients and galaxy disk dynamics with two case studies in the MSA-3D sample. In addition, Galaxy ID8512 is presented in Bariˇsi'c et al. (2024), showing a well-settled rotating disk together with a steep metallicity gradient. Galaxy ID9960 has the steepest gas-phase metallicity gradient in our sample (Table 1), and exhibits a massive and regularly rotating disk. We also highlight the ability to measure its metallicity gradient despite the presence of an ac- \nve galactic nucleus (AGN). In contrast, Galaxy ID4391 has a turbulent disk and a complex spatial metallicity distribution, possibly influenced by interaction with a nearby companion. These examples demonstrate our ability to probe the physical drivers of metallicity gradients. We plan to carry out a comprehensive study using our MSA-3D dataset in future work.", '4.2.1. Galaxy ID9960': "Among the 25 galaxies examined in this work, galaxy ID9960 at z = 1 . 51 exhibits a very steep negative metallicity gradient, surpassing even that of local galaxies (see Figs. 3 and A1). It has a stellar mass of log(M * /M ⊙ ) = 11.20, the most massive galaxy within our sample. In panel (a) of Figure 4, we present the spectrum from the central spaxel of this galaxy. Unfortunately, the H β and [O III ] lines fall in the chip gap, such that we cannot determine the star formation and AGN regions through the BPT diagram. We instead use the WHAN diagnostic diagram. As shown in Figure A1, the central region of this galaxy is classified as a sAGN (i.e., Seyfert), while the outer regions are classified as star forming. We conclude that there is a complex excitation structure comprising both star formation and a central AGN, which can be separated thanks to the angular resolution and minimal beam smearing of these data. We reiterate that the derived metallicity gradient slope is based exclusively on the spaxels classified as star forming, where the adopted strong line calibrations are applicable. \nTo further examine the nature of the nuclear activity, we plot the spectrum of the central spaxel zoomed in around H α , [N II ], and [S II ] in panel (b) of Figure 4. The spectrum exhibits narrow line emission, with a velocity dispersion of about 160 km s -1 for H α at the galaxy center. We do not observe a strong broad ( σ ≳ 1000 km s -1 ) emission component, as is seen in Type 1 AGNs. The spectral features thus indicate that this galaxy hosts a Type 2 AGN. Furthermore, panels (c) to (e) of Figure 4 present images obtained from the HST/ACS F606W (rest-frame wavelength ∼ 2400 ˚ A), JWST/NIRCam F115W 1 (rest-frame ∼ 4600 ˚ A), and JWST/NIRCam F356W (rest-frame ∼ 1.41 µ m) filters (Koekemoer et al. 2011; Wang et al. 2023). While the central region remains indistinct in the rest-frame UV image (F606W), a faint but discernible disk structure is apparent. The lack of a UV-bright nucleus is also indicative of a Type 2 AGN. \nWe now consider the gas kinematics of galaxy ID9960, to examine whether the strong metallicity gradient is associated with a rotation-dominated star forming disk. \nWhile we see evidence of AGN emission and possible AGN-driven outflows (Figure 4), these signatures are confined to the central region. The extended emission is dominated by star formation. As shown in Figure A1, ID9960's H α velocity map clearly exhibits a mature rotating disk. It satisfies all standard disk classification criteria (e.g., Schreiber et al. 2018) including a 'spider diagram' pattern in the velocity field. The rotation curve extracted along the major axis is shown in panel (f) of Figure 4. To correct the observed (line-of-sight) velocity map for inclination angle i , we use the axis ratio b/a assuming a thin disk: cos( i ) = b/a . The axis ratio of 0.48 from the CANDELS photometric catalog leads to i = 61 · . We model the rotation following the method described in Ju et al. (2022), adopting an arctangent rotation curve (e.g., Jones et al. 2010b): \nV ( R ) = v 0 + 2 π v c arctan( R/R t ) . (3) \nHere R t is the scale radius, v c is the asymptotic rotation velocity and v 0 is an overall systemic velocity. We use the nested sampling code nautilus (Lange 2023) to find the best-fit rotating disk model, presented in panel (f) of Figure 4. Black dots represent the median observed velocity within the distance bins, while the red line corresponds to the best-fit model. The model matches the data to within a RMSE of 33 km s -1 , comparable to the measurement scatter. The v c = 250 +6 -5 km s -1 is similar to the Milky Way ( ∼ 230 km s -1 ; e.g., Bland-Hawthorn &Gerhard 2016). The estimated dynamical mass within the half-light radius (6 kpc) is approximately 10 11 M ⊙ , compatible with the stellar mass and a modest fraction of gas and dark matter. \nThe ratio of the rotation velocity of a gas disk to its velocity dispersion, v c /σ , is a key indicator of dynamical support. Typically, the calculation uses v c as the maximum rotational velocity and σ 0 as the intrinsic velocity dispersion from the outer regions of galaxies (where rotation curves are flat; e.g., Wisnioski et al. 2015). In this work, we calculate the ratio of v c to the velocity dispersion σ 0 = 48 . 9 ± 1 . 6 km s -1 (median and sample standard error) of the spaxels located at radius > 3 kpc. The resulting ratio is v c /σ 0 = 5 . 13 ± 0 . 20, which is typical of massive z ∼ 1 galaxies. According to the classifications of Girard et al. (2020) and Kassin et al. (2012), for example, the v/σ ratio is used to classify galaxies as 'rotationally supported' when v/σ > 1. Regular or well-settled rotation is identified by v/σ > 3, while 1 < v/σ < 3 indicates irregular or disturbed rotation. Hence, the v/σ of this galaxy is high enough to qualify as regularly rotating. \nIn summary, galaxy ID9960 exhibits a steep metallicity gradient and a well-settled star-forming disk. This \nThe steep metallicity gradient and the clear rotation signatures of galaxy ID9960 supports the strong connection between chemo-structural evolution and disk formation. \n<!-- image --> \nFigure 4. Detailed spectroscopic and kinematic properties of galaxy ID9960 - the steepest gradient source in our sample. Panel (a): The spectrum from the spaxel in the galaxy center, with dashed lines highlighting the emission lines of H β , [O III ], H α , [N II ], and [S II ]. H β and [O III ] unfortunately both fall within the chip gap. Panel (b): A zoomed-in spectrum of panel (a). The vertical dashed lines mark the wavelengths of Na I D λλ 5890 , 5896, H α , [N II ], and [S II ]. The high flux ratio of [N II ]/H α and the strong blueshifted absorption of Na I D indicate the presence of AGN-driven outflows (e.g., Dopita et al. 1998; Rupke et al. 2005; Davies et al. 2024). Panels (c) to (e): image stamps of galaxy ID9960 in the filters of HST/ACS F606W, JWST/NIRCam F115W, and JWST/NIRCam F356W. The red box marks the entire field of view of our slit-stepping 3D spectroscopy. Panel (f): The one-dimensional velocity profile (black and grey dots) with the best-fitting model shown in red. The green dots represent the intrinsic velocity dispersion, corrected for the instrument line spread function as described in the text. We derive v c /σ 0 = 5 . 13 ± 0 . 20, which together with the two-dimensional velocity map (Figure A1) indicates that galaxy ID9960 has a regularly rotating disk. \nsupports a physical picture in which strong rotational support is necessary for sustaining a radial abundance gradient. This galaxy represents a compelling case at z ≈ 1 . 5 for co-evolution of chemical profiles and gas kinematics, signifying the emergence of a modern Hubble sequence.", '4.2.2. Galaxy ID4391': "In contrast to galaxy ID9960 introduced above, galaxy ID4391 shows a more complex chemical profile and more turbulent gas kinematics, possibly affected by interaction with a neighboring galaxy. First of all, its metallicity log(O / H) does not follow a linear trend with radius (Figure 5), such that the derived metallicity gradient depends on the radial range used. Within the range 0.5 - 3.5 kpc ( ∼ 2 . 5 R e ), the data are well described with a gradient of best-fit slope -0 . 074 ± 0 . 012 dex kpc -1 . However, the metallicity increases at R ≳ 5 kpc. Fitting the range 0.5 - 10 kpc, the gradient 'flattens' to 0 . 0013 ± 0 . 0117 dex kpc -1 . \nThe quality of our MSA-3D IFS data enables a detailed characterization of the complex metallicity spatial distribution. As shown in the 2D metallicity map \nin panel (b) of Figure 5, the increasing O/H at large radius is largely driven by a tail-like structure extending to the lower left, with metallicity comparable to that found in the galaxy's center. We verify that this structure is robust by additionally measuring metallicity using the O3N2 ratio (as calibrated by Marino et al. 2013). Panels (c) and (d) of Figure 5 show the same general trend with both the N2 (red) and O3N2 (blue) diagnostics. This tail structure, spanning a galactocentric radius R ≃ 2 . 5 -5 R e , is connected to a neighboring galaxy visible in the image cutout. While this source falls outside our IFS coverage (see panel (a) of Figure 5) and is not spectroscopically confirmed, its photometric redshift z ∼ 1 is similar to ID4391, with a mass ratio of approximately 0.2 (Yung et al. 2022). We speculate that the high-metallicity tail may be related to a merger or gas flows associated with this neighboring galaxy. \nTo explore the dynamical evidence of possible mergerinduced turbulence, we examine the velocity and velocity dispersion maps of galaxy ID4391 shown in Figure A1. The velocity dispersion is clearly amplified along the direction to the possible merging companion, which \nThe unusual metallicity profile might be attributed to galaxy mergers or gas inflows, both of which are also capable of significantly disturbing gas motions. \n<!-- image --> \nFigure 5. Kinematic properties of galaxy ID4391, with complex chemical profile verified by different metallicity indicators. Panel (a): the NIRCam/F115W image stamp of galaxy ID4391. Panel (b): the metallicity map obtained using the O3N2 method. Panels (c) and (d): metallicity gradients with galactocentric radii in units of kpc and R e units determined by the N2 method (red dots) and the O3N2 method (blue dots). The points and error bars show the metallicity profiles using averages in radial bins. The black line in the two panels represents the slope for local galaxies (Belfiore et al. 2017). Both methods display a distinct gradient from the galaxy center to the outskirts, with a turning point located near 2.5 R e . Panel (e): The onedimensional velocity profile (black and grey dots) with the best-fitting model is shown in red. Panel (f): The one-dimensional intrinsic velocity dispersion is presented on the same spatial axis as the velocity profile. The gray dots represent the intrinsic velocity dispersion, while the green error bars indicate the median and standard deviation of the gray dots within radial bins. The v c /σ is 1 . 91 ± 0 . 14, indicating irregular rotational motion and suggesting that this galaxy likely has a thick disk. \nroughly corresponds to the kinematic major axis. Using the same method as for ID9960 above, we constructed a rotating disk model for this galaxy. The velocity and dispersion profiles are shown in panels (e) and (f) of Figure 5, along with the best-fit disk model. We obtain v c = 82 +4 -3 km s -1 with an inclination angle i = 67 +3 -4 degrees. The median intrinsic velocity dispersion σ 0 of the spaxels located beyond 3 kpc (where the rotation curve is approximately flat) is 42 . 7 ± 1 . 8 km s -1 . The v c /σ 0 ratio is 1 . 91 ± 0 . 14, indicating irregular rotational motion according to classification criteria, consistent with a thick gas disk, likely perturbed by gravitational interactions from the companion galaxy.", '5. SUMMARY': "In this paper, we analyze 3D spectroscopy of 43 starforming galaxies at redshifts z = 0 . 5-1.7 obtained using a novel multiplexed slit-stepping approach via the MSA3D survey (Bariˇsi'c et al. 2024). The observations cover multiple strong rest-frame optical emission lines (e.g., H β , [O III ], H α , [N II ], and [S II ]) with spectral resolution R ∼ 2700. This allows us to spatially map the emission line flux, EWs, velocity field, and velocity dispersion. From the line flux ratios, we investigate the gas-phase \nmetallicity distributions and analyze trends in the radial metallicity gradients. \nWe focus on the subset of 25 galaxies in our sample with homogeneous metallicity gradients measured using the N2 method. The gradient slopes range from -0.03 to 0.02 dex kpc -1 , with most galaxies showing negative or flat gradients. The gradients show a clear negative correlation with stellar mass, statistically significant at the ∼ 2 σ level, with more massive galaxies having steeper gradients. The scatter of the gradient slope measurements is very small, with an RMS of 0.02 dex kpc -1 . Compared with simulation results, we find that the gradients are similar to predictions from the FIRE-2 simulations, which incorporate feedback in cosmological galaxy formation. Relatively flat gradients in low-mass galaxies can be attributed to strong, time-dependent stellar feedback. The steeper gradients in more massive galaxies in our sample are not explained by a simple inside-out growth scenario. The steep gradients may arise from high gas fractions and star formation rates at z ∼ 1 compared to z ∼ 0, which prevent the metallicity from reaching equilibrium and make it more susceptible to effects such as radial gas flows. \nWe highlight two exemplary galaxies within our sample, illustrating the ability of these data to perform both sample-wide analyses and detailed case studies. Galaxy ID9960 has the most negative metallicity gradient and highest stellar mass within the sample, with log(M * /M ⊙ ) = 11.20. It contains a rotation-dominated star forming disk with v c /σ = 5 . 13 ± 0 . 20. The chemical profiles and gas kinematics of this massive z ∼ 1 . 5 disk galaxy likely signify the end stage of the co-evolution of chemical profiles and disk structure, where the steep abundance gradient is maintained by strong rotational support. In contrast, galaxy ID4391 shows a complex spatial metallicity distribution which is not adequately described by a simple radial gradient. The gradient is steeply negative within 2.5 effective radii, while beyond this range it becomes positive. This galaxy features a thick rotating disk, with increased velocity dispersion in the direction of a possible interacting companion. Its gradient might be affected by gas accretion induced by merging. These galaxies may offer key insight into the tight connection between galaxy chemo-structural evolution and disk assembly, as the modern Hubble sequence emerges. \nThis paper presents demographic trends of metallicity gradients in the z ∼ 1 galaxy population. This work takes advantage of MSA multiplexing to secure IFS datasets for a sample of 43 galaxies simultaneously, from which we obtain 25 robust metallicity gradient measurements. We find a tight relationship between stellar mass and metallicity gradient at z ∼ 1, which provides a crucial benchmark for advancing our understanding of galaxies' chemical evolution. Notably, the high-quality data provided by MSA-3D were obtained with only ∼ 30 hours of JWST time. An equivalent survey with NIRSpec's IFU would require an order of magnitude more time. This work thus demonstrates the remarkable efficiency gain of our slit-stepping strategy (see also Bariˇsi'c et al. 2024). To date, three other JWST programs have adopted a similar slit-stepping approach (i.e., GO-3426 (PI: Jones), GO-2123 (PI: Kassin) and GO-4291 (PI: Kassin)). We envision further programs to expand the \navailable sample in terms of galaxy demographics and redshift, enabling a more comprehensive study of the correlation between the chemical evolution of galaxies and the properties of their disks across their most formative epochs.", 'ACKNOWLEDGEMENTS': "We thank the anonymous referee for the constructive comments, which significantly improved the manuscript. This work is supported by the National Natural Science Foundation of China (grant 12373009), the CAS Project for Young Scientists in Basic Research Grant No. YSBR-062, the Fundamental Research Funds for the Central Universities, and the science research grant from the China Manned Space Project. XW acknowledges the support by the Xiaomi Young Talents Program, and the work carried out, in part, at the Swinburne University of Technology, sponsored by the ACAMAR visiting fellowship. This work is based on observations made with the NASA/ESA/CSA James Webb Space Telescope. The data were obtained from the Mikulski Archive for Space Telescopes at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-03127 for JWST. These observations are associated with program JWST-GO-2136. We acknowledge financial support from NASA through grant JWST-GO-2136. TJ acknowledges support from a Chancellor's Fellowship and a Dean's Faculty Fellowship, and from NASA through grant 80NSSC23K1132. CAFG was supported by NSF through grants AST2108230 and AST-2307327; by NASA through grant 21ATP21-0036; and by STScI through grant JWST-AR03252.001-A. \nFacilities: JWST (NIRSpec MSA) \nSoftware: nautilus (Lange 2023) \nThe JWST data presented in this article were obtained from the Mikulski Archive for Space Telescopes (MAST) at the Space Telescope Science Institute. The specific observations analyzed can be accessed via doi:10.17909/s8wp-5w10.", 'REFERENCES': "Angl´es-Alc´azar, D., Dav´e, R., ¨ \nOzel, F., & Oppenheimer, B. D. 2014, ApJ, 782, 84, doi: 10.1088/0004-637X/782/2/84 \nBacon, R., Accardo, M., Adjali, L., et al. 2010, in Society of \nPhoto-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 7735, Ground-based and Airborne Instrumentation for Astronomy III, ed. I. S. McLean, S. K. Ramsay, & H. Takami, 773508, doi: 10.1117/12.856027 \nFigure A1. A comprehensive spatially-resolved view of our galaxy sample with metallicity gradient measurements. For each galaxy we show 8 subplots on two rows. The first row displays, from left to right: the 3-color image, H α emission flux map, observed H α velocity map, and intrinsic H α velocity dispersion map (corrected for instrument resolution). The second row shows the gas-phase metallicity map using the N2 method, the metallicity gradient with the best linear fit represented by the red dashed line, a map color-coded according to the positions of individual spaxels in the BPT or WHAN diagram, and the BPT or WHAN diagram used to classify the ionization properties in each galaxy. The red boxes in the three-color images indicate the FoV (1 . '' 8 × (2 . '' 0 -3 . '' 0)) for each of our sources. The ellipses on the maps represent the size of the resolution element of our MSA-3D integral field spectroscopy. \n<!-- image --> \nFigure A1. (continued) For subplots other than the metallicity maps and the metallicity gradients, we show spaxels with S / N of Flux(H α ) > 10 and S / N of EW(H α ) > 5. For the metallicity maps and gradients, we plot only the subset of those spaxels classified as star-forming with statistical uncertainty of < 0 . 25 dex in metallicity. For the metallicity gradient subplots, the bottom and top abscissas are the galactocentric radius in units of kpc and effective radius R e , respectively. In these subplots we report the best-fit metallicity gradient slope for each galaxy based on individual spaxels, along with the scatter (i.e., the RMSE between metallicity of individual spaxels and the linear fit). The red points and error bars show the metallicity profiles using averages in radial bins. \n<!-- image --> \nFigure A1. continued \n<!-- image --> \nFigure A1. continued \n<!-- image --> \nFigure A1. continued \n<!-- image --> \nFigure A1. continued \n<!-- image --> \nFigure A1. continued \n<!-- image --> \nFigure A1. continued \n<!-- image --> \nFigure A1. continued \n<!-- image --> \nBaldwin, J. A., Phillips, M. 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2024mpsp.book..203S | This Chapter outlines the basic properties of waves in solar partially ionized plasmas. It provides a summary of the main sets of equations from the singlefluid formalism to the multifluid one giving examples for purely hydrogen and for hydrogenhelium plasmas. It then discusses the solutions for waves under the singlefluid frame the influence of the ambipolar diffusion diamagnetic effect and the Hall effect on the propagation dissipation and mode conversion of the magnetohydrodynamic waves. The Chapter continues by outlining the wave solutions in the multifluid formalism the influence of the elastic interparticle collisions into the propagation damping and dissipation of different magnetohydrodynamic modes. Both parts discuss linear and nonlinear wave solutions and the effects of the gravitational stratification of the solar atmosphere. | 2024-05-01T00:00:00Z | ['2024mpsp.book..203S', '2024arXiv240911528K', '10.1016/B978-0-32-395664-2.00011-6', 'arXiv:2409.11528', '10.48550/arXiv.2409.11528'] | ['Astrophysics - Solar and Stellar Astrophysics'] | MHD waves in the partially ionized plasma from single to multifluid approach | 2,024 | 169 | 0.42 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML'] | 0 | https://arxiv.org/pdf/2409.11528.pdf | {'MHD waves in the partially ionized plasma: from single to multi-fluid approach': 'Elena Khomenko 𝑎,𝑏 and David Martínez-Gómez 𝑎,𝑏 \n(a) Instituto de Astrofísica de Canarias, La Laguna, Tenerife, Spain; \n(b) Dpto de Astrofísica, Universidad de La Laguna, La Laguna, Tenerife, Spain', 'ABSTRACT': 'This Chapter outlines the basic properties of waves in solar partially ionized plasmas. It provides a summary of the main sets of equations, from the single-fluid formalism, to the multi-fluid one, giving examples for purely hydrogen, and for hydrogen-helium plasmas. It then discusses the solutions for waves under the single-fluid frame: the influence of the ambipolar diffusion, diamagnetic effect, and the Hall effect on the propagation, dissipation, and mode conversion of the magnetohydrodynamic waves. The Chapter continues by outlining the wave solutions in the multi-fluid formalism: the influence of the elastic inter-particle collisions into the propagation, damping and dissipation of different magnetohydrodynamic modes. Both parts discuss linear and non-linear wave solutions, and the effects of the gravitational stratification of the solar atmosphere.', 'KEYWORDS': 'Waves, Energy transfer, Sun, Atmosphere', '6.1.1 Single-fluid multi-species plasma': 'Partially ionized (PI) plasmas are a common type of plasmas in the cosmic and in the laboratory environments. PI plasmas are usually rather cold and/or rarefied media with temperatures that do not allow hydrogen to ionize. In the Sun, partially ionized plasma fills the whole volume of the photosphere and the chromosphere, and may even be present at some locations in the (much hotter) transition regions and solar corona. Propagation of waves and shocks in such plasmas gives rise to a series of new effects, such as damping, dissipation, dispersion, or appearance of the new modes with mixed properties. The solar plasma is composed by particles of different atomic elements in different ionization stages. The mathematical description of such partially ionized plasma depends on the degree of the collision coupling of different neutral and charged components. \nWhere collision coupling is strong enough, a single fluid approach can be used. In that case, the conservation equations take the following form. \n𝜕𝜌 𝜕𝑡 + ∇ · ( 𝜌 𝑽 ) = 0 , (6.1) \n𝜕 ( 𝜌 𝑽 ) 𝜕𝑡 + ∇ · ( 𝜌 𝑽𝑽 + 𝑃 I ) = 𝑱 × 𝑩 + 𝜌 𝒈 , (6.2) \n𝜕 𝜕𝑡 GLYPH<18> 𝑒 + 1 2 𝜌𝑉 2 GLYPH<19> + ∇ · GLYPH<18> 𝑽 ( 𝑒 + 1 2 𝜌𝑉 2 ) + 𝑃 𝑽 + 𝒒 + 𝑭 𝑅 GLYPH<19> = 𝑱 · 𝑬 + 𝜌 𝑽 · 𝒈 . (6.3) \nThe macroscopic velocity, 𝑽 , pressure, 𝑃 , internal energy, 𝑒 , and heat flux, 𝒒 , are defined through the summation over all species composing the plasma, see Khomenko et al. (2014): \n𝑽 = " 2 𝑁 + 1 𝛼 = 1 ( 𝜌 𝛼 𝑽 𝛼 ) 𝜌 , (6.4) 𝑃 = 2 𝑁 + 1 ∑︁ 𝛼 = 1 GLYPH<18> 𝑃 𝛼 + 1 3 𝜌 𝛼 𝑤 2 𝛼 GLYPH<19> , 𝑒 = 3 2 𝑃 + 2 𝑁 ∑︁ 𝛼 = 1 𝜒 𝛼 , 𝒒 = 2 𝑁 + 1 ∑︁ 𝛼 = 1 GLYPH<18> 𝒒 𝛼 + 5 2 𝑃 𝛼 𝒘 𝛼 + 1 2 𝜌 𝛼 𝑤 2 𝛼 𝒘 𝛼 GLYPH<19> , \nwhere the quantities with sub-index 𝛼 refer to individual plasma components ( 𝑁 ions, 𝑁 neutrals of different chemical elements, and electrons), 𝜌 is the total plasma density, 𝒘 𝛼 is the drift velocity of each specie taken with respect to 𝑽 , 𝒘 𝛼 = 𝑽 𝛼 -𝑽 , and 𝜒 𝛼 is the total potential energy of ionization level of a species 𝛼 (the ionization energy does not apply to electrons for obvious reasons). \nFor completeness, equations above include radiative energy flux, 𝑭 𝑅 . In the solar atmosphere, energy exchange by radiation plays a major role. A complex treatment for non-local equilibrium radiative transfer is especially relevant for the chromosphere, where the multi-fluid effects are also more pronounced. As for the heat conduction (also viscosity, not included in the equations above), under strong plasma magnetization, it acts differently in the directions parallel and perpendicular to the magnetic field. With weakening the collisions in the chromosphere, the classical approximations (as those from standard plasma physics books by, e.g., Spitzer, 1962; Braginskii, 1965; Bittencourt, 1986; Balescu, 1988) may become invalid, and more detailed treatment becomes necessary, see Hunana et al. (2022). In the following we will not discuss the effects of conductivity, viscosity, and radiation because they are analogous to those for waves in fully ionized plasmas. \nThe system 6.1-6.3 has the same form as an MHD system. The energy conservation equation is written for the sum of the internal and kinetic energies and its right hand side contains the Joule heating term, 𝑱 · 𝑬 , where 𝑱 is electric current, \n𝑱 = (∇ × 𝑩 )/ 𝜇 0 , (6.5) \nand 𝑬 is electric field. The form of the latter depends on the presence of extra species in a plasma. Therefore, both the energy conservation equation, and the induction equation will have additional terms due to neutrals, as described in the next section.', "6.1.1.1 Generalized Ohm's law for single-fluid description": "The expression for the electric field required for the single-fluid system 6.1-6.3 is obtained from the generalized Ohm's law (GOL). Derivation of the GOL can be found in many plasma physics books, for example in Braginskii (1965); Krall and Trivelpiece (1973); Bittencourt (1986). For the specific case of the Sun, derivation of the multi-species GOL is provided in, e.g., Khomenko et al. (2014) and Ballester et al. (2018). In the single-fluid description, the electric field is computed in the system of reference of the center of mass velocity of the whole plasma, including its neutral components, 𝑽 , and the GOL takes the following form, \n[ 𝑬 + 𝑽 × 𝑩 ] = 𝜂𝜇 0 𝑱 + 𝜂 𝐻 𝜇 0 [ 𝑱 × 𝑩 ] | 𝐵 | -𝜂 𝐻 𝜇 0 ∇ 𝑃 𝑒 | 𝐵 | -𝜂 𝐴 𝜇 0 [( 𝑱 × 𝑩 ) × 𝑩 ] | 𝐵 | 2 + 𝜉 𝑛 𝛼 𝑛 [ 𝑮 × 𝑩 ] , (6.6) \nThis expression contains at the right hand side the Ohmic, Hall, Biermann battery, ambipolar, and diamagnetic terms. In the expression for the battery term, 𝑃 𝑒 is the electron pressure. The 𝑮 vector is a combination of partial pressure gradients and is given by, \n𝑮 = 𝜉 𝑛 ∇( 𝑃 𝑒 + 𝑃 𝑖 ) + ( 1 -𝜉 𝑛 )∇ 𝑃 𝑛 , (6.7) \nwhere 𝑃 𝑖 and 𝑃 𝑛 is the pressure of ions and neutrals, respectively. The coefficients of all the terms in Eq. 6.6 are provided as follows, in the units of [ 𝑚 2 𝑠 -1 ] , \n𝜂 = 𝛼 𝑒 𝜇 0 𝑒 2 𝑛 2 𝑒 ; 𝜂 𝐴 = 𝜉 2 𝑛 | 𝐵 | 2 𝜇 0 𝛼 𝑛 ; 𝜂 𝐻 = | 𝐵 | 𝜇 0 𝑒𝑛 𝑒 , (6.8) \nwith 𝜉 𝑛 = 𝜌 𝑛 / 𝜌 being the neutral fraction, 𝜌 𝑛 neutral mass density, 𝑛 𝑒 electron numberdensity, 𝑒 electron charge. Coefficients 𝛼 𝑛 and 𝛼 𝑒 are neutral and electron collision parameters, that take into account collisions between all species in the plasma, \n𝛼 𝑛 = 𝑁 ∑︁ 𝛽 = 1 𝜌 𝑒 𝜈 𝑒𝑛 𝛽 + 𝑁 ∑︁ 𝛼 = 1 𝑁 ∑︁ 𝛽 = 1 𝜌 𝑖 𝛼 𝜈 𝑖 𝛼 𝑛 𝛽 ; 𝛼 𝑒 = 𝑁 ∑︁ 𝛼 = 1 𝜌 𝑒 𝜈 𝑒𝑖 𝛼 + 𝑁 ∑︁ 𝛽 = 1 𝜌 𝑒 𝜈 𝑒𝑛 𝛽 . (6.9) \nwhere sub-index 𝑛 𝛽 indicate neutrals of the type 𝛽 , 𝑖 𝛼 indicate ions of the type 𝛼 . The expressions for the collision frequencies between ions and neutrals ( 𝜈 𝑖 𝛼 𝑛 𝛽 ) and electrons with neutrals ( 𝜈 𝑒𝑛 𝛽 ), can be taken from Spitzer (1962), and the expressions for collisions between electrons and ions ( 𝜈 𝑒𝑖 𝛼 ) can be taken from Braginskii (1965): \n𝜈 𝑖 𝛼 𝑛 𝛽 = 𝑛 𝑛 𝛽 √︄ 8 𝑘 𝐵 𝑇 𝜋𝑚 𝑖 𝛼 𝑛 𝛽 𝜎 𝑖𝑛 ; 𝜈 𝑒𝑛 𝛽 = 𝑛 𝑛 𝛽 √︄ 8 𝑘 𝐵 𝑇 𝜋𝑚 𝑒𝑛 𝛽 𝜎 𝑒𝑛 ; 𝜈 𝑒𝑖 𝛼 = 𝑛 𝑒 𝑒 4 ln Λ 3 𝜖 2 0 𝑚 2 𝑒 GLYPH<18> 𝑚 𝑒 2 𝜋𝑘 𝐵 𝑇 GLYPH<19> 3 / 2 , (6.10) \nwhere 𝑚 𝑖 𝛼 𝑛 𝛽 = 𝑚 𝑖 𝛼 𝑚 𝑛 𝛽 /( 𝑚 𝑖 𝛼 + 𝑚 𝑛 𝛽 ) , 𝑚 𝑒𝑛 𝛽 = 𝑚 𝑒 𝑚 𝑛 𝛽 /( 𝑚 𝑒 + 𝑚 𝑛 𝛽 ) are the reduced masses, 𝜎 𝑖𝑛 and 𝜎 𝑒𝑛 are collision cross sections (see, e.g., Huba, 2013), Λ is the Coulomb logarithm. \nThe generalized induction equation is obtained by using GOL, Eq. 6.6, together with the Faraday's law, \n𝜕 𝑩 𝜕𝑡 = ∇ × GLYPH<20> 𝑽 × 𝑩 -𝜂𝜇 0 𝑱 -𝜂 𝐴 𝜇 0 𝑱 ⊥ + ∇ 𝑃 𝑒 𝑒𝑛 𝑒 -𝜂 𝐻 𝜇 0 [ 𝑱 × 𝑩 ] | 𝐵 | -𝜉 𝑛 𝛼 𝑛 [ 𝑮 × 𝑩 ] GLYPH<21> , (6.11) \nwhere we have defined the current perpendicular to magnetic field as, 𝑱 ⊥ = -( 𝑱 × 𝑩 ) × 𝑩 /| 𝐵 | 2 . The relative importance of each of the non-ideal terms in the generalized induction equation, Eq. 6.11, can be evaluated by comparing the values of these coefficients in a solar atmospheric model. \nFigure 6.1 shows the coefficients 𝜂 𝐴 , 𝜂 𝐻 , and 𝜂 as a function of horizontal distance and height in a numerically computed solar magneto-convection model with average magnetic field of 50 G (Khomenko et al., 2018). The height range covers from slightly below the photosphere to the middle chromosphere. The ion fraction (bottom panel) at these layers drops as low as 𝜉 𝑖 = 10 -4 -10 -5 (very weakly ionized), while recovering values around 1 (fully ionized) below the photosphere and in the upper chromosphere. As a consequence, both ambipolar, 𝜂 𝐴 , and Hall, 𝜂 𝐻 coefficients take large values, exceeding by several orders of magnitude the Ohmic coefficient, 𝜂 , from the middle photosphere up. The Hall coefficient dominates over the ambipolar one in the photosphere, at heights between 0 and 0.5 Mm, while the ambipolar coefficient dominates over all three at heights above 1 Mm. One can also observe strong horizontal variations in the values of the coefficients caused by the variations in the atmospheric parameters, temperature, density, and magnetic field. At locations with strong magnetic flux tubes, as those seen around horizontal coordinates 3 and 4.5 Mm, both 𝜂 𝐴 , and 𝜂 𝐻 have high values through the whole photospheric layers. Therefore, it can be expected that both ambipolar and Hall effects would significantly affect wave behavior in magnetized photosphere and chromosphere. \nOnce the generalized induction equation is obtained, it can be used to derive the total energy conservation equation, including non-ideal terms, \n5 \nFIGURE6.1 Three top panels: coefficients of the generalized induction equation defined in Eq. 6.8, ambipolar, 𝜂 𝐴 , Hall, 𝜂 𝐻 , and Ohmic 𝜂 . Bottom panel gives the ion fraction, 𝜉 𝑖 = 𝜌 𝑖 / 𝜌 . All the quantities are shown along a horizontal direction and height in a 2D cut of a 3D magneto-convection model, computed with the Mancha3D code (Khomenko et al., 2018). \n<!-- image --> \n𝜕𝑒 tot 𝜕𝑡 + ∇ · h 𝑽 GLYPH<16> 𝑒 tot + 𝑃 + | 𝑩 | 2 2 𝜇 0 GLYPH<17> -𝑩 ( 𝑽 · 𝐵 ) 𝜇 0 + 𝒒 + 𝑭 𝑅 i (6.12) +∇ · h ( 𝜂 𝐴 + 𝜂 ) [ 𝑱 ⊥ × 𝑩 ] -∇ 𝑃 𝑒 × 𝑩 𝑒𝑛 𝑒 + 𝜂 𝑝 𝑮 ⊥ i = 𝜌 𝑽 · 𝒈 , \nwith 𝑒 tot defined as, \n𝑒 tot = 1 2 𝜌 𝑽 2 + 𝑒 + 𝑩 2 2 𝜇 0 , (6.13) \n𝑮 ⊥ = -𝑮 × 𝑩 × 𝑩 | 𝐵 | 2 ; 𝜂 𝑝 = 𝜉 𝑛 | 𝐵 | 2 𝜇 0 𝛼 𝑛 . (6.14) \nand \nIt can be seen that the Hall effect does not affect the energy of the system, while both Ohm and ambipolar effects provide a dissipation term, proportional to the sum of both coefficients, 𝜂 and 𝜂 𝐴 . Since 𝜂 𝐴 exceeds 𝜂 by several orders of magnitude in the chromosphere, it can be expected that the ambipolar effect will be the main source of dissipation for chromospheric waves. The battery effect also has its corresponding counterpart in the energy equation, though in practice, its influence into the energy is rather small for the typical parameters of solar partially ionized plasma. The last term in the square brackets, the 𝑮 ⊥ -term, is frequently (and unjustifiably) omitted. \nAlternatively, energy conservation can be cast for the internal energy only, \n𝜕𝑒 𝜕𝑡 +∇· ( 𝑽 𝑒 + 𝒒 + 𝑭 𝑅 ) + 𝑃 ∇· 𝑽 = 𝜂𝜇 0 𝐽 2 + 𝜂 𝐴 𝜇 0 𝐽 ⊥ 2 -𝑱 · ∇ 𝑃 𝑒 𝑒𝑛 𝑒 + 𝑱 · 𝜉 𝑛 𝛼 𝑛 [ 𝑮 × 𝑩 ] . (6.15) \nThe right hand side of this equation contains Joule heating terms, related to the Ohmic and ambipolar diffusion, as well as the battery and 𝐺 -term. All these non-ideal terms directly affect the internal energy of the system. The Joule heating terms are always positive and lead to the internal energy increase. The ambipolar term is expected to be the leading one in the solar chromosphere. The Ohmic term is proportional to the square of total currents, 𝐽 2 , while the ambipolar term only affects the currents perpendicular to the magnetic field, 𝐽 ⊥ . This makes the fundamental difference between both effects. \nEquations 6.1, 6.2, 6.11 and either of Eq. 6.12 or 6.15 provide a closed set of equations of single-fluid MHD that can be used for the wave analysis. The advantage of the single-fluid MHD is that one does specifically address the question of plasma composition except when computing the ambipolar, Hall and battery coefficients. Therefore, large scale simulations can be easily performed.", '6.1.2 Two-fluid multi-species plasma': "Once the collision coupling weakens with height in the solar atmosphere, all individual species behave in a slightly different way, moving with different velocities. Strictly speaking, this requires considering particles of different chemical elements and ionization stages as separate fluids, see e.g., Khomenko et al. (2014). However, such a description may result unpractical since many equations need to be solved. A good approximation is to assume that the difference in behavior between neutrals and charges is larger than between the neutrals/charges of different kinds themselves, since the latter feel the presence of the magnetic field and the former do not. This assumption allows to decrease the number of equations for different species to just two, for an average neutral particle and an average charged particle. This brings to the following system of equations, \n𝜕𝜌 n 𝜕𝑡 + ∇ · ( 𝜌 n 𝑽 n ) = 𝑆 𝑛 , (6.16) \n𝜕𝜌 c 𝜕𝑡 + ∇ · ( 𝜌 c 𝑽 c ) = -𝑆 𝑛 , (6.17) \n𝜕 ( 𝜌 n 𝑽 n ) 𝜕𝑡 + ∇ · ( 𝜌 n 𝑽 n 𝑽 n + 𝑃 n I ) = 𝜌 n 𝒈 + 𝑹 n , (6.18) \n𝜕 ( 𝜌 c 𝑽 c ) 𝜕𝑡 + ∇ · ( 𝜌 c 𝑽 c 𝑽 c + 𝑃 c I ) = 𝑱 × 𝑩 + 𝜌 c 𝒈 -𝑹 n , (6.19) \n𝜕 𝜕𝑡 GLYPH<18> 𝑒 n + 1 2 𝜌 n 𝑽 2 n GLYPH<19> +∇· GLYPH<18> 𝑽 n ( 𝑒 n + 1 2 𝜌 n 𝑽 2 n ) + 𝑃 n · 𝑽 n + 𝒒 n + 𝑭 𝑛 𝑅 GLYPH<19> = -𝜌 n 𝑽 n · 𝒈 + 𝑀 n , (6.20) \n𝜕 𝜕𝑡 GLYPH<18> 𝑒 c + 1 2 𝜌 c 𝑽 2 c GLYPH<19> +∇· GLYPH<18> 𝑽 c ( 𝑒 c + 1 2 𝜌 c 𝑽 2 c ) + 𝑃 c · 𝑽 c + 𝒒 c + 𝑭 𝑐 𝑅 GLYPH<19> = -𝜌 c 𝑽 c · 𝒈 + 𝑱 · 𝑬 -𝑀 n . (6.21) \nIn these equations, the sub index 'n' is for neutrals and 'c' is for charges, the rest of the notations is the same as in the single-fluid case. The definitions of the partial pressures ( 𝑃 𝑛 and 𝑃 𝑐 ), the heat flux vectors ( 𝒒 𝑛 and 𝒒 𝑐 ), the internal energies ( 𝑒 𝑛 and 𝑒 𝑐 ), and the radiative energy fluxes ( 𝑭 𝑛 𝑅 and 𝑭 𝑐 𝑅 ) all involve summation over the species present in the plasma, similar to Eq. 6.4, but separately for the charges (ions and electrons) and neutrals (Khomenko et al., 2014). \nThe two-fluid description defined above requires a stronger coupling between charged particles than between charged and neutral particles (Zaqarashvili et al., 2011). An example calculation of collision frequencies between the charged and neutral particles is given in Fig. 6.2 for a 2D cut of a numerically computed model of solar magneto-convection, same as in Fig. 6.1. It can be observed that collisions between charged particles (electrons and ions, top panel) are orders of magnitude more frequent in the chromosphere, above ∼ 0.5 Mm, compared to the collisions between ions and neutrals (bottom panel). Therefore, neutrals can be considered much weakly coupled to the rest of the plasma. This justifies grouping the fluids into two, charged and neutral, components. Under some circumstances, the coupling between different neutral components becomes weak enough to justify splitting the neutrals into more fluids. This approach has been applied in the case of hydrogen-helium plasma in Zaqarashvili et al. (2011), and will be discussed below. \nThe two-fluid system of equations presents several additional terms at the right hand side, 𝑆 𝑛 , 𝑹 𝑛 and 𝑀 𝑛 . Those terms describe elastic and inelastic collisions between the fluid and couple each pair of equations. While the equations Eq. 6.16-6.21 are generic, the collision terms are not. At this point one needs to specify the type of colliding particles, plasma composition, and the type of collisions.", '6.1.3 Purely hydrogen plasma': "One of the widely used cases is the one for purely hydrogen plasmas (Leake et al., 2012, 2014; Popescu Braileanu et al., 2019a). \nFIGURE 6.2 Collision frequencies between electrons and ions (top) and ions and neutrals (bottom) defined in Eq. 6.10 shown along a horizontal direction and height in a 2D cut of a 3D magnetoconvection model, computed with the Mancha3D code (Khomenko et al., 2018). \n<!-- image --> \nFor a purely hydrogen plasma, the internal energies and partial pressures of the neutral and charged fluid are linked as follows, \n𝑒 𝑛 = 𝑃 𝑛 /( 𝛾 -1 ) ; 𝑒 𝑐 = 𝑃 𝑐 /( 𝛾 -1 ) + 𝑛 𝑒 𝜙 ion , (6.22) \nwhere 𝜙 ion is hydrogen ionization energy. \nThe collision terms for this case can be specified in a relatively simple form: \n𝑆 n = 𝜌 c Γ rec -𝜌 n Γ ion , (6.23) \n𝑹 n = 𝜌 c 𝑽 c Γ rec -𝜌 n 𝑽 n Γ ion + 𝐾 col 𝜌 c 𝜌 n ( 𝑽 c -𝑽 n ) , (6.24) \n𝑀 n = GLYPH<18> 1 2 Γ rec 𝜌 c 𝑽 2 c -1 2 𝜌 n 𝑽 2 n Γ ion GLYPH<19> + 1 𝛾 -1 𝑘 B 𝑚 n GLYPH<16> 𝜌 c 𝑇 c Γ rec -𝜌 n 𝑇 n Γ ion GLYPH<17> + 1 2 GLYPH<16> 𝑽 2 c -𝑽 2 n GLYPH<17> 𝐾 col 𝜌 c 𝜌 n + 1 𝛾 -1 𝑘 B 𝑚 n ( 𝑇 c -𝑇 n ) 𝐾 col 𝜌 c 𝜌 n . (6.25) \nThe collision term in the continuity equations, 𝑆 𝑛 , takes into account inelastic collisions between the particles, i.e., those where the particle identity is modified. For the purely hydrogen plasma those are ionization and recombination processes, with the corresponding rates, Γ ion and Γ rec . The simplified treatment given by Eqs. 6.16 and 6.17 only considers the hydrogen atom transferring between two, neutral and ionized states and does not explicitly include the excitation states. Both ionization and recombination can be collisional or radiative. For the complete treatment of hydrogen ionization balance, one can refer to the work by Leenaarts et al. (2007). Otherwise, a simplified treatment is \nfrequently used, as in Meier (2011), Leake et al. (2012), Popescu Braileanu et al. (2019a) or Murtas et al. (2022). In those later works the excitation is by electron impact and recombination is spontaneous, these approximations have limited validity for the range of parameters found for partially ionized chromospheric and photospheric plasmas, and are mostly valid for coronal conditions. \nThe collision term in the momentum equation, 𝑹 𝑛 , Eq. 6.24, includes the momentum exchange during ionization/recombination processes (terms proportional to Γ ), and elastic collisions (term proportional to 𝐾 col ). The latter can be linked to the 𝛼 𝑛 collision parameter given by Eqs. 6.9, but, more generically, it can also include the charge exchange contribution (Meier and Shumlak, 2012; Popescu Braileanu et al., 2019a), \n𝐾 col = 𝛼 𝑛 𝜌 𝑐 𝜌 𝑛 + 𝐾 cx col . (6.26) \nAdding up the charge exchange collisions together with the rest of elastic collisions in a single collision parameter 𝐾 col is possible only for hydrogen plasma. The mathematical treatment of the charge exchange reaction is more complex once other chemical species are present. \nThe collision term in the energy equation, 𝑀 𝑛 , Eq. 6.25, contains four groups of contributions: kinetic energy exchange in ionization/recombination, thermal energy exchange in ionization/recombination, kinetic energy exchange in elastic collisions and thermal exchange in elastic collisions. \nSince Eqs. 6.16-6.21 are written in conservation form, and the energy equation is written in terms of the sum of internal and kinetic energies, the collision terms are completely symmetric and sum to zero if the equations for neutrals and charges are added up. The role of collisions for heating the plasma can be better understood if one considers the conservation equations of internal energy only, \n𝜕𝑒 n 𝜕𝑡 + ∇ · GLYPH<0> 𝑽 n 𝑒 n + 𝒒 𝑛 + 𝑭 𝑛 𝑅 GLYPH<1> + 𝑃 n ∇ · 𝑽 n = 𝑄 n , (6.27) \n𝜕𝑒 c 𝜕𝑡 + ∇ · GLYPH<0> 𝑽 c 𝑒 c + 𝒒 𝑐 + 𝑭 𝑐 𝑅 GLYPH<1> + 𝑃 c ∇ · 𝑽 c = 𝑱 · 𝑬 + 𝑄 c . (6.28) \nThe collision terms, 𝑄 , in the internal energy equation have a different the form compared to those refined by Eqs. 6.25. \n𝑄 n = 1 2 Γ rec 𝜌 c ( 𝑽 c -𝑽 n ) 2 + 1 𝛾 -1 𝑘 B 𝑚 n GLYPH<16> 𝜌 c 𝑇 c Γ rec -𝜌 n 𝑇 n Γ ion GLYPH<17> + 1 2 ( 𝑽 c -𝑽 n ) 2 𝐾 col 𝜌 c 𝜌 n + 1 𝛾 -1 𝑘 B 𝑚 n ( 𝑇 c -𝑇 n ) 𝐾 col 𝜌 c 𝜌 n , (6.29) \n𝑄 c = 1 2 Γ ion 𝜌 n ( 𝑽 c -𝑽 n ) 2 -1 𝛾 -1 𝑘 B 𝑚 n GLYPH<16> 𝜌 c 𝑇 c Γ rec -𝜌 n 𝑇 n Γ ion GLYPH<17> + 1 2 ( 𝑽 c -𝑽 n ) 2 𝐾 col 𝜌 c 𝜌 n -1 𝛾 -1 𝑘 B 𝑚 n ( 𝑇 c -𝑇 n ) 𝐾 col 𝜌 c 𝜌 n . (6.30) \nWhile some of the contributions sum to zero (those corresponding to thermal energy exchange, second and forth group of terms), those corresponding to the kinetic energy exchange are always positive in both neutral and charges energy equations. According to these equations, if there is a velocity difference between charges and neutrals, it will produce frictional heating of the medium, contributing to the internal energy increase. \nClosing the two-fluid system of equations requires the generalized Ohm's law for the electron field, and generalized induction equation for the evolution of the magnetic field. The GOL for the two-fluid case is expressed in the frame of reference of the charged fluid ( 𝑽 𝑐 ≈ 𝑽 𝑖 ), and has the following form, \n𝑬 + 𝑽 c × 𝑩 = 𝜂𝜇 0 𝑱 + 𝜂 H 𝜇 0 𝑱 × 𝑩 | 𝐵 | -𝜂 H 𝜇 0 ∇ 𝑃 e | 𝐵 | -𝜂 D ( 𝑽 c -𝑽 n ) , (6.31) \nwith the coefficients \n𝜂 = 𝜌 𝑒 ( 𝜈 𝑒𝑖 + 𝜈 𝑒𝑛 ) 𝜇 0 𝑒 2 𝑛 2 𝑒 ; 𝜂 𝐻 = | 𝐵 | 𝜇 0 𝑒𝑛 𝑒 ; 𝜂 𝐷 = 𝜌 𝑒 ( 𝜈 𝑒𝑛 -𝜈 𝑖𝑛 ) 𝑒𝑛 𝑒 . (6.32) \nNotice that 𝜂 𝐷 has different units (m/(t q)) from 𝜂 and 𝜂 𝐻 (l 2 /t). This GOL has similar contributions as in the single-fluid case, Eq. 6.6, i.e. Ohmic, Hall, and battery terms. However, due to the change of the system of reference, the ambipolar term is not present. Instead, the last term contributing to the electric field is the one proportional to the charges-neutral velocity difference. The coefficient multiplying this term, 𝜂 𝐷 is proportional to the electron mass and it is generally very small in the solar context (Martínez-Gómez et al., 2021). This last term should not be confused with the ambipolar term.", '6.1.4 Hydrogen-helium plasmas': 'The consideration of hydrogen-only plasma is a good first approximation for the study of the solar atmosphere dynamics, since hydrogen is by far the most abundant element in the Sun. Nevertheless, a more accurate description can be achieved by including the second most abundant element, i.e., helium, whose abundance is about 10% of that of hydrogen (in terms of particles number). Since a helium particle is around 4 times more massive than a hydrogen particle, even such a low abundance of helium can have an important influence on the propagation of waves, and on the overall dynamics of the solar plasma in the multi-fluid description. \nFor the range of temperatures of the solar atmosphere, helium can be found in its three possible states of ionization: neutral, singly ionized, and doubly ionized. Together with the two ionization states of hydrogen, and electrons, the plasma is then composed by six different kinds of particles. This circumstance opens several options for a multi-fluid approach, depending on the degree of coupling assumed for each pair of species. Below we discuss two alternative descriptions \nof hydrogen-helium plasmas most frequently used for waves studies in the Sun: the three-fluid and the five-fluid models.', '6.1.4.1 Three-fluid model': 'The three-fluid model assumes that the temperature of the plasma is not high enough to allow for the presence of doubly-ionized helium (HeIII). It additionally considers that protons (p), singly-ionized helium (HeII) and electrons (e) are strongly coupled, so they are treated as a single fluid. The other two fluids of the model are the neutral hydrogen (H) and neutral helium (He), respectively. Thus, the continuity, momentum, and energy conservation equations are given by (Zaqarashvili et al., 2011), \n𝜕𝜌 s 𝜕𝑡 + ∇ · ( 𝜌 s 𝑽 s ) = 0 , 𝑠 ∈ { c , H , He } , (6.33) \n𝜕 ( 𝜌 c 𝑽 c ) 𝜕𝑡 + ∇ · ( 𝜌 c 𝑽 c 𝑽 c + 𝑃 c I ) = 𝑱 × 𝑩 + 𝜌 c 𝒈 + 𝑹 c , (6.34) 𝜕 ( 𝜌 H 𝑽 H ) 𝜕𝑡 + ∇ · ( 𝜌 H 𝑽 H 𝑽 H + 𝑃 H I ) = 𝜌 H 𝒈 + 𝑹 H , 𝜕 ( 𝜌 He 𝑽 He ) 𝜕𝑡 + ∇ · ( 𝜌 He 𝑽 He 𝑽 He + 𝑃 He I ) = 𝜌 He 𝒈 + 𝑹 He , \n𝜕𝑒 c 𝜕𝑡 + ∇ · ( 𝑽 c 𝑒 c ) + 𝑃 c ∇ · 𝑽 c = 𝑱 · 𝑬 + 𝑄 c , (6.35) 𝜕𝑒 H 𝜕𝑡 + ∇ · ( 𝑽 H 𝑒 H ) + 𝑃 H ∇ · 𝑽 H = 𝑄 H , 𝜕𝑒 He 𝜕𝑡 + ∇ · ( 𝑽 He 𝑒 He ) + 𝑃 He ∇ · 𝑽 He = 𝑄 He . \nIn these equations, the density of charges is given by 𝜌 c = 𝜌 p + 𝜌 HeII + 𝜌 e ≈ 𝜌 p + 𝜌 HeII , and the center of mass velocity of the charged component is given by \n𝑽 c = 𝜌 p 𝑽 p + 𝜌 HeII 𝑽 HeII + 𝜌 e 𝑽 e 𝜌 p + 𝜌 HeII + 𝜌 e ≈ 𝜌 p 𝑽 p + 𝜌 HeII 𝑽 HeII 𝜌 c . (6.36) \nThe internal energies 𝑒 s are defined without taking into account the ionization energy, i.e. simply by 𝑒 s = 𝑃 s /( 𝛾 -1 ) . Notice that, for simplicity, Eqs. 6.35 do not include thermal and radiative energy fluxes, compared to Eqs. 6.20-6.21. The collision terms in the momentum Eq. 6.34 are given by, \n𝑹 c = -GLYPH<0> 𝛼 pH + 𝛼 pHe + 𝛼 HeIIH + 𝛼 HeIIHe GLYPH<1> 𝑽 c + 𝜉 p ( 𝛼 HeIIHe + 𝛼 HeIIH ) 𝒘 -𝜉 HeII GLYPH<0> 𝛼 pH + 𝛼 pHe GLYPH<1> 𝒘 + GLYPH<0> 𝛼 pH + 𝛼 HeIIH GLYPH<1> 𝑽 H + GLYPH<0> 𝛼 pHe + 𝛼 HeIIHe GLYPH<1> 𝑽 He , \n𝑹 H = -GLYPH<0> 𝛼 pH + 𝛼 HeIIH + 𝛼 HeH GLYPH<1> 𝑽 H + 𝛼 HeH 𝑽 He + GLYPH<0> 𝛼 pH + 𝛼 HeIIH GLYPH<1> 𝑽 c , \n(6.37) (6.38) \n𝑹 He = -GLYPH<0> 𝛼 pHe + 𝛼 HeIIHe + 𝛼 HeH GLYPH<1> 𝑽 He + 𝛼 HeH 𝑽 H + GLYPH<0> 𝛼 pHe + 𝛼 HeIIHe GLYPH<1> 𝑽 c . (6.39) \nHere, the mass fractions of protons and singly ionized Helium are defined as, 𝜉 p = 𝜌 p / 𝜌 c , 𝜉 HeII = 𝜌 HeII / 𝜌 c , with respect to the total mass density of charges. The drift velocity of plasma is given by 𝒘 = 𝑽 p -𝑽 HeII . In practice, the terms proportional to 𝒘 are dropped out arguing their smallness, since there is no consistent way for computing them in the frame of this model (Zaqarashvili et al., 2011). \nThe quantities 𝛼 st ≡ 𝜌 s 𝜈 st = 𝜌 t 𝜈 ts represent the friction coefficients of collisions between species \'s\' and \'t\'. These parameters are similar to those defined in Eq. 6.9, but neglecting collisions with electrons due to their much lower mass. \nThe collision frequencies between different species, 𝜈 st , are modified compared to Eqs. 6.10, taking into account the possibility of different temperatures of the species. For collisions between one neutral and one ionized species, \n𝜈 st = 𝑛 𝑡 𝑚 t 𝑚 t + 𝑚 s GLYPH<18> 8 𝑘 𝐵 𝑇 s 𝜋𝑚 s + 8 𝑘 𝐵 𝑇 t 𝜋𝑚 t GLYPH<19> 1 / 2 𝜎 st . (6.40) \nThe choice of the cross-sections, 𝜎 st , plays an important role when it comes to quantitative conclusions regarding wave damping in the solar chromosphere (Soler et al., 2015a). For collisions between two ions, \n𝜈 st = 𝑛 t 𝑍 2 s 𝑍 2 t 𝑒 4 ln Λ 3 𝜖 2 0 𝑚 s 𝑚 st GLYPH<18> 2 𝜋𝑘 𝐵 𝑇 s 𝑚 s + 2 𝜋𝑘 𝐵 𝑇 t 𝑚 t GLYPH<19> -3 / 2 , (6.41) \nwhere 𝑍 s , t is the charge number of the species \'s" or \'t\', and the Coulomb\'s logarithm, ln Λ , is generalized for the case of plasma with several temperatures (Spitzer, 1962; Vranjes and Krstic, 2013). The charge exchange collisions are neglected in this model. \nThe terms 𝑄 c , 𝑄 H and 𝑄 HeII represent the energy exchange due to elastic collisions between each pair of species. \n𝑄 s = ∑︁ 𝑡 ≠ 𝑠 𝑄 𝑠𝑡 ; 𝑠, 𝑡 ∈ { c , H , He } , (6.42) \nwhere \n𝑄 st = 1 2 ( 𝑽 t -𝑽 s ) 2 𝐾 st coll 𝜌 s 𝜌 t + 𝐴 st 1 𝛾 -1 𝑘 B 𝑚 s + 𝑚 t ( 𝑇 t -𝑇 s ) 𝐾 st coll 𝜌 s 𝜌 t . (6.43) \nHere 𝐴 st = 4 for collisions between electrons and neutral particles and 𝐴 st = 3 for collisions between the remaining species (Draine, 1986). The relation between 𝐾 coll and 𝛼 st is, 𝐾 st coll 𝜌 s 𝜌 t = 𝛼 st . Note that Eq. 6.43 is valid in the limit of small drift velocities between the species (see the corrections made by Schunk, 1977). Ionization/recombination has been neglected in this model, so the corresponding terms in the continuity Eqs. 6.33 are null, compared to Eqs. 6.16 and 6.17. \nFinally, the generalized Ohm\'s law for the three-fluid Hydrogen-Helium model is given by, \n𝑬 + 𝑽 c × 𝑩 = 𝛼 ep + 𝛼 eHeII 𝑒 2 𝑛 2 e 𝑱 -∇ 𝑃 e 𝑒𝑛 e + 𝑱 × 𝑩 𝑒𝑛 e , (6.44) \nwhere one recognizes Ohmic, battery and Hall terms. The smaller terms proportional to the drift velocities are removed.', '6.1.4.2 Five-fluid model': 'The five-fluid model treats each ionisation state of hydrogen and helium as a different fluids, which interact with each other by means of elastic collisions. No strong coupling is assumed between each pair of ionization states (MartínezGómez et al., 2017). Therefore, each state 𝑠 ∈ { p , H , He , HeII , HeIII } has a corresponding full set of mass, momemtum and pressure evolution equations, which, following the works by Schunk (1977) and Draine (1986), are given by: \n𝜕𝑛 s 𝜕𝑡 + ∇ · ( 𝑛 s 𝑽 s ) = 0 , (6.45) \n𝜕 ( 𝜌 s 𝑽 s ) 𝜕𝑡 + ∇ · ( 𝜌 s 𝑽 s 𝑽 s + 𝑃 s I ) = 𝑞 s 𝑛 s ( 𝑬 + 𝑽 s × 𝑩 ) + 𝜌 s 𝒈 + ∑︁ t ≠ s 𝑹 st , (6.46) \n𝜕𝑒 s 𝜕𝑡 + ∇ · ( 𝑽 s 𝑒 s ) + 𝑃 s ∇ · 𝑽 s = ∑︁ t ≠ s 𝑄 st , (6.47) \nwhere 𝑚 s is the mass, 𝑛 s the number density, and 𝑞 s = 𝑍 s 𝑒 is the electric charge (with 𝑍 s being the signed charge number). \nIn this approach, electrons are not strongly coupled to any of the other charged species but their dynamics is not described by a full set of temporal evolution equations. The number density of electrons is computed from the assumption of quasi-neutrality of the plasma, " s 𝑍 s 𝑛 s ≈ 0, so that 𝑛 e ≈ " 𝑀 i 𝑍 i 𝑛 i , where 𝑀 is the number of ionized species. Then, due to their low mass in comparison with that of the rest of species, the inertia of electrons is neglected and their momentum equation is used to compute the following expression for the electric field: \n𝑬 + 𝑽 e × 𝑩 = -∇ 𝑃 e 𝑒𝑛 e + 𝑚 e 𝑒 𝒈 + 1 𝑒𝑛 e ∑︁ j ≠ e 𝑹 ej . (6.48) \nFurthermore, the current density is given by \n𝑱 = ∑︁ s 𝑞 s 𝑛 s 𝑽 s = 𝑒𝑛 e ( 𝑽 𝑖 -𝑽 e ) , (6.49) \nwhere \n𝑽 𝑖 ≡ " 𝑀 j 𝑍 j 𝑛 j 𝑽 j 𝑛 e (6.50) \nis the velocity of ions. Then, the velocity of electrons can be expressed as 𝑽 e = 𝑽 𝑖 -𝑱 /( 𝑒𝑛 e ) , and the electric field can be rewritten in the system of reference of the whole plasma as, \n𝑬 + 𝑽 𝑖 × 𝑩 = 𝑱 × 𝑩 𝑒𝑛 e -∇ 𝑃 e 𝑒𝑛 e + 𝑚 e 𝑒 𝒈 + 1 𝑒𝑛 e ∑︁ j ≠ e 𝑹 ej . (6.51) \nHence, the only remaining evolution equation for electrons is the energy one, Eq. 6.47. \nThe terms R st and 𝑄 st in Eqs. 6.46 and 6.47 represent the momentum and the energy transfer due to elastic collisions between two species \'s\' and \'t\'. The R st term is given by \n𝑹 st = 𝜌 s 𝜌 t 𝐾 st coll ( 𝑽 t -𝑽 s ) (6.52) \nand the 𝑄 st is the one given by Eq. 6.43. The ionization/recombination is again neglected.', '6.2 WAVES IN STRONGLY COLLISION-COUPLED PLASMAS': 'In the photosphere and low chromosphere of the Sun, the collision frequency between plasma species is much higher than the typical frequencies of the observed waves, see Fig. 6.2. This justifies the use of a single-fluid approximation. According to Sect. 6.1.1.1, the presence of neutrals leads to the modified GOL, where the ambipolar term is expected to be the lead, followed by the modified Hall term. The ambipolar diffusion and Hall effect have been studied extensively in the context of solar atmospheric waves, both using the analytical theory, idealized, and realistic numerical simulations (see the review by Ballester et al., 2018). The section below gradually builds the theory of wave propagation in partially ionized plasma under a single-fluid approximation, from linear wave theory to shocks and the effects of gravitational stratification.', '6.2.1 Basic equations of linear theory': "In order to study linear waves, one has to consider Eqs. 6.1-6.3 (or Eq. 6.15), together with the induction Eq. 6.11, and split the variables into the background component (variables with subindex '0') and a small perturbation (variables with subindex '1'), \n𝑽 = 𝑽 1 ( 𝒓 , 𝑡 ) ; 𝑩 = 𝑩 0 ( 𝒓 ) + 𝑩 1 ( 𝒓 , 𝑡 ) ; 𝑱 = 𝑱 1 ( 𝒓 , 𝑡 ) = ∇ × 𝑩 1 / 𝜇 0 ; (6.53) 𝑃 = 𝑃 0 ( 𝒓 ) + 𝑃 1 ( 𝒓 , 𝑡 ) ; 𝜌 = 𝜌 0 ( 𝒓 ) + 𝜌 1 ( 𝒓 , 𝑡 ) . \nHere we assumed that background magnetic field is current-free, i.e. 𝑱 0 = ∇× 𝑩 0 / 𝜇 0 = 0. This condition has repercussion for the energy conservation and the induction equation. We assume no velocity is present in the equilibrium. \nWe will consider an ideal gas with no ionization/recombination, so that 𝑒 = 𝑃 /( 𝛾 -1 ) . In the following we will also neglect the battery term due to its smallness. The linearized set of equations becomes, \n𝜕𝜌 1 𝜕𝑡 + 𝜌 0 ∇ · 𝑽 = 0 , (6.54) \n𝜌 0 𝜕 𝑽 𝜕𝑡 + ∇ 𝑃 1 = [ 𝑱 1 × 𝑩 0 ] + 𝜌 1 𝒈 , (6.55) \n𝜕𝑃 1 𝜕𝑡 + 𝛾𝑃 0 ∇· 𝑽 +( 𝑽 · ∇) 𝑃 0 = -( 𝛾 -1 ) h ∇·( 𝒒 + 𝑭 𝑹 ) -𝑱 1 · 𝜉 𝑛 𝛼 𝑛 [ 𝑮 0 × 𝑩 0 ] i , (6.56) \n𝜕 𝑩 1 𝜕𝑡 = ∇ × GLYPH<26> [ 𝑽 × 𝑩 0 ] + 𝜂𝜇 0 𝑱 1 -𝜂 𝐻 𝜇 0 [ 𝑱 1 × 𝑩 0 ] | 𝐵 0 | + (6.57) 𝜂 𝐴 𝜇 0 [[ 𝑱 1 × 𝑩 0 ] × 𝑩 0 ] | 𝐵 0 | 2 -𝜉 𝑛 𝛼 𝑛 [ 𝑮 1 × 𝑩 0 ] GLYPH<27> . \nThe equation of internal energy evolution has been rewritten in terms of pressure. Expressions for radiation and heat conduction will be specified below, when considering particular applications. The Ohmic and ambipolar heating terms in the energy equation are of the second order and therefore they have been neglected in the linear approximation. \nThese equations are generic and do not include specific assumptions about the equilibrium, except that it is current-free. If the ionization fraction can be assumed smoothly varying in space, the partial pressure 𝑮 0 , 1 term can be simplified to \n𝜉 𝑛 𝛼 𝑛 𝑮 0 , 1 = Ξ ∇ 𝑃 0 , 1 ; Ξ = 𝜉 2 𝑛 𝜉 𝑖 ( 1 + 𝜉 𝑖 ) 𝛼 𝑛 (6.58) \nwith 𝜉 𝑖 = 1 -𝜉 𝑛 being the ionization fraction. This way, unknown partial pressures of ions and neutrals, are eliminated in favor of the total plasma pressure. \nSeveral particular analytical solutions of the linearized system of equations have been developed, e.g., for the cases of a homogeneous unbounded plasma, and, in some particular cases, solutions for gravitationally stratified isothermal atmosphere. Below we consider some of the most common solutions.", '6.2.2 Waves in homogeneous plasmas': 'In MHD description, a homogeneous unbounded plasma supports three types of waves, fast and slow magnetoacoustic waves and the Alfvén wave. The singlefluid, quasi MHD description does not change this picture, and three distinct wave modes are still present. However, these modes acquire different properties, thanks to the additional damping and dispersion mechanisms present in the partially ionized plasmas. \nIn the absence of gravity, one can choose the reference system such that the homogeneous magnetic field is directed along the 𝑥 axis, 𝑩 0 = 𝐵 0 ˆ 𝑥 . All equilibrium parameters are constant in space, i.e. 𝑃 0 , 𝜌 0 , 𝑇 0 = const . Following Forteza et al. (2008), we assume the presence of the heat conduction with \n𝒒 = -𝜅 · ∇ 𝑇 , and heat-loss function with ∇ · 𝑭 𝑹 = 𝜌𝐿 ( 𝜌, 𝑇 ) . For the moment, we will neglect the Hall effect. Then, the linearized equations take the following form, \n𝜕𝜌 1 𝜕𝑡 + 𝜌 0 ∇ · 𝑽 = 0 , (6.59) \n𝜌 0 𝜕 𝑽 𝜕𝑡 + ∇ 𝑃 1 = 1 𝜇 0 [[∇ × 𝑩 1 ] × 𝑩 0 ] , (6.60) \n𝜕𝑃 1 𝜕𝑡 + 𝛾𝑃 0 ∇ · 𝑽 = ( 𝛾 -1 ) h ∇ · ( 𝜅 · ∇ 𝑇 1 ) -𝜌𝐿 ( 𝜌, 𝑇 ) i (6.61) \n𝜕 𝑩 1 𝜕𝑡 = ∇ × {[ 𝑽 × 𝑩 0 ] + 𝜂 [∇ × 𝑩 1 ]+ (6.62) 𝜂 𝐴 [[[∇ × 𝑩 1 ] × 𝑩 0 ] × 𝑩 0 ] | 𝐵 0 | 2 -Ξ [∇ 𝑃 1 × 𝑩 0 ] GLYPH<27> . \nIn a homogeneous plasma the coefficients 𝜂, 𝜂 𝐴 , Ξ , 𝜅 are constant. Therefore, one can look for the solution for perturbations in the form of plane waves. Without loss of generality, one can assume the wave vector 𝑘 lying in the 𝑥 -𝑧 plane, while all the vector quantities having 3 spatial dimensions. This way, all the perturbed quantities are proportional to, 𝑓 ∼ exp (-𝑖𝜔𝑡 + 𝑖𝑘 𝑥 𝑥 + 𝑖𝑘 𝑧 𝑧 ) , and it is trivial to obtain a linear system of the scalar equations for perturbations from Eq. 6.59-6.62. \nPartially ionized plasmas in the Sun are often studied in the context or solar prominences. In the work by Carbonell et al. (2004), the heat loss function, 𝐿 ( 𝜌, 𝑇 ) has been specified for prominence conditions as a difference between the optically thin radiative losses, and some generic heating function, as, \n𝐿 ( 𝜌, 𝑇 ) = 𝜒 ∗ 𝜌𝑇 𝛼 -ℎ𝜌 𝑎 𝑇 𝑏 . (6.63) \nThe coefficients 𝜒 ∗ and 𝛼 for the thin radiative losses depend on temperature (Hildner, 1974; Rosner et al., 1978; Milne et al., 1979), while the exponents 𝑎 and 𝑏 of the heating function depend on the heating mechanisms. Without the loss of generality, the linearized version of the heat-loss term becomes (Carbonell et al., 2004), \n𝜌𝐿 ( 𝜌, 𝑇 ) lin = 𝜌 1 𝐿 + GLYPH<18> 𝜕𝐿 𝜕𝜌 GLYPH<19> 𝑇 𝜌 1 𝜌 0 + GLYPH<18> 𝜕𝐿 𝜕𝑇 GLYPH<19> 𝜌 𝑇 1 𝜌 0 = 𝜌 1 𝐿 + 𝐿 𝑇 𝜌 1 𝜌 0 + 𝐿 𝜌 𝑇 1 𝜌 0 . (6.64) \nThe heat conduction in partially ionized plasmas is anisotropic, therefore the heat conduction tensor can be decomposed into the parallel and perpendicular components with respect to the magnetic field, \n𝜅 = 𝜅 | | ˆ 𝒃 ˆ 𝒃 + 𝜅 ⊥ ( I -ˆ 𝒃 ˆ 𝒃 ) . (6.65) \nAssuming that the dominant component of the heat conduction is the electron heat conduction along magnetic field lines, and that the neutral heat conduction is isotropic, Forteza et al. (2008) reduced Eq.6.65 to \n𝜅 = 𝜅 | | ,𝑒 ˆ 𝒃 ˆ 𝒃 + 𝜅 𝑛 I . (6.66) \nUnder these approximations, and for the chosen geometry, the dispersion relations for the Alfvén waves and for magneto-acoustic waves decouple from each other (Forteza et al., 2008). The one for the Alfvén waves has the following form, \n𝜔 2 -𝑘 2 𝑥 Γ ( 𝜃 ) 2 = 0 , (6.67) \nand the one for magneto-acoustic waves is, \n𝑎 5 𝜔 5 + 𝑎 4 𝜔 4 + 𝑎 3 𝜔 3 + 𝑎 2 𝜔 2 + 𝑎 1 𝜔 + 𝑎 0 = 0 . (6.68) \nThe coefficients of these dispersion relations are given by the following expressions, \nΓ ( 𝜃 ) = 𝑐 2 𝐴 -𝑖𝜔 ( 𝜂 𝐶 + 𝜂 tan 2 𝜃 ) ; (6.69) \n𝑎 0 = -𝑖𝑘 2 𝑘 2 𝑥 𝑐 2 𝐴 𝜌 0 ( 𝐴𝑇 0 -𝐻𝜌 0 ) ; (6.70) \n𝑎 1 = -𝑘 2 GLYPH<20> 𝑐 2 𝑆 𝑐 2 𝐴 𝑘 2 𝑥 -( 𝐴𝑇 0 -𝐻𝜌 0 ) Ψ 𝜌 0 GLYPH<21> ; (6.71) \n𝑎 2 = 𝑖𝑘 2 " ( 𝐴𝑇 0 -𝐻𝜌 0 ) 𝜌 0 + 𝐴𝑇 0 𝑐 2 𝐴 𝑃 0 + 𝑐 2 𝑆 Ψ # ; (6.72) \n𝑎 3 = 𝑘 2 GLYPH<20> ( 𝑐 2 𝑆 + 𝑐 2 𝐴 ) + 𝐴𝑇 0 𝜂 𝐶 𝑃 0 GLYPH<21> ; (6.73) \n𝑎 4 = -𝑖 GLYPH<18> 𝐴𝑇 0 𝑃 0 + 𝑘 2 𝜂 𝐶 GLYPH<19> ; (6.74) \n𝑎 5 = -1 , (6.75) \nwhere we have introduced Cowling conductivity, 𝜂 𝐶 = 𝜂 + 𝜂 𝐴 (Cowling, 1945). The coefficients 𝐴 , 𝐻 and Ψ are defined as, \n𝐴 = ( 𝛾 -1 )( 𝜅 | | ,𝑒 𝑘 2 𝑥 + 𝜅 𝑛 𝑘 2 + 𝜌 0 𝐿 𝑇 ) ; (6.76) \n𝐻 = ( 𝛾 -1 )( 𝐿 + 𝜌 0 𝐿 𝜌 ) ; (6.77) \nΨ = 𝑘 2 𝜂 𝐶 -𝑘 2 𝑧 𝑐 2 𝐴 𝜌 0 Ξ . (6.78) \nIn the equation above, 𝑘 2 = 𝑘 2 𝑥 + 𝑘 2 𝑧 is the total wave vector, 𝑐 𝑆 = √︁ 𝛾𝑃 0 / 𝜌 0 is the sound speed and 𝑐 𝐴 = 𝐵 0 / √ 𝜇 0 𝜌 0 is the Alfvén speed taking the total plasma density. \nBelow we discuss the solutions for the magneto-acoustic and Alfvén waves separately, starting by the former.', '6.2.2.1 Magneto-acoustic waves': 'Equation 6.68 can be simplified by introducing a modified, complex, sound speed, \nΛ 2 = 𝑇 0 𝜌 0 𝐴 -𝐻 -𝑖𝜔𝑐 2 𝑆 𝑇 0 𝜌 0 𝐴 -𝐻 -𝑖𝜔 . (6.79) \nWith this definition, the dispersion relation for magneto-acoustic waves shortens to, \nGLYPH<16> 𝜔 2 -𝑘 2 Λ 2 GLYPH<17> GLYPH<16> 𝑖𝜔𝑘 2 𝜂 𝐶 + 𝜔 2 GLYPH<17> -𝑘 2 𝑐 2 𝐴 ( 𝜔 2 -Λ 2 𝑘 2 𝑥 ) + 𝑖 Λ 2 𝑐 2 𝐴 𝑘 2 𝑘 2 𝑧 Ξ 𝜌 0 𝜔 = 0 . (6.80) \nIn the absence of heat conduction, heating and radiation losses, the modified sound speed Λ is equal to the common sound speed 𝑐 𝑆 , and the dispersion relation, Eq.6.80, reduces to the one reported in Forteza et al. (2007). Furthermore, it can be verified that, by taking the neutral fraction to zero, and neglecting Ohmic diffusion, this dispersion relation simplifies to the known case for magneto-acoustic waves in homogeneous unbounded plasmas (Priest, 2014). \n𝜔 4 -𝜔 2 𝑘 2 ( 𝑐 2 𝑆 + 𝑐 2 𝐴 ) + 𝑘 2 𝑘 2 𝑥 𝑐 2 𝐴 𝑐 2 𝑆 = 0 . (6.81) \nThe adiabatic dispersion relation for magneto-acoustic waves, i.e., Eq. 6.80 with Λ = 𝑐 𝑆 , is 4th order in 𝜔 and supports 4 wave modes. However, compared to the ideal case, Eq. 6.81, the former equation is not bi-quadratic, and has imaginary terms. It means that waves traveling in the opposite directions will, generically, have different properties, and will be damped or amplified in time/space, depending on the conditions of the plasma. \nIn Forteza et al. (2007), adiabatic Eq. 6.80 was solved assuming temporal damping, that is, 𝜔 = 𝜔 𝑅 + 𝑖𝜔 𝐼 , with 𝜔 𝐼 ≪ 𝜔 𝑅 (weak damping approximation). Under this approximation, the real part of the wave frequency remains the same as for magneto-acoustic waves in a fully ionized plasma, i.e. one given by Eq. 6.81. The approximate expression for the imaginary part of the frequency becomes, \n𝜔 𝐼 ≈ -𝑘 2 𝑘 2 𝑧 𝑐 2 𝐴 𝑐 2 𝑆 Ξ 𝜌 0 + 𝜂 𝐶 𝑘 2 ( 𝜔 2 𝑅 -𝑘 2 𝑐 2 𝑆 ) 4 𝜔 2 𝑅 -2 𝑘 2 ( 𝑐 2 𝑆 + 𝑐 2 𝐴 ) . (6.82) \nIn the limit of low plasma 𝛽 , with 𝑐 𝑆 ≪ 𝑐 𝐴 , and 𝜔 fast , slow 𝑅 ≈ { 𝑘𝑐 𝐴 , 𝑘 𝑥 𝑐 𝑆 } for the fast and slow wave, respectively, the damping simplifies to \n2 𝜔 fast 𝐼 ≈ -𝜂 𝐶 𝑘 2 -𝜂 𝐴 𝑘 2 𝑧 𝑐 2 𝑆 𝑐 2 𝐴 ≈ -𝜂 𝐶 𝑘 2 , (6.83) \nfor the fast wave, and \n2 𝜔 slow 𝐼 ≈ -𝑘 2 𝑧 𝑐 2 𝑆 𝑐 2 𝐴 GLYPH<18> 𝜂 + 𝜂 𝐴 1 1 + 𝜉 𝑖 GLYPH<19> , (6.84) \nFigure 6.4 illustrates the damping over period times for the fast (top) and \n<!-- image --> \nFIGURE 6.3 Damping time over wave period for the fast magneto-acoustic wave (left) and for the slow magneto-acoustic wave (right) as a function of ionization fraction, defined as ˜ 𝜇 = 1 /( 1 + 𝜉 𝑖 ) . The data are computed according to expressions Eq. 6.83 and 6.84 (triangles) and from the numerical solution of the adiabatic dispersion relation for magneto-acoustic waves (Eq. 6.80). The computations are done for conditions typical for solar prominences: 𝐵 0 = 10 𝐺 , 𝑇 0 = 8 kK, 𝜌 0 = 5 × 10 -11 kg m -3 ( 𝑐 𝐴 = 126 km s -1 , 𝑐 𝑆 = 10 . 5 km s -1 ), 𝑘 𝑥 𝑥 0 = 𝜋 / 2, 𝑘 𝑧 𝑥 0 = 0 . 1, 𝑥 0 = 3 Mm. Figure from Forteza et al. (2007). \n<!-- image --> \nfor the slow wave. Notice that the diamagnetic effect (the term proportional to Ξ ) only affects the slow wave in this approximation and that, without taking it into account, the damping for the slow wave would reduce to 2 𝜔 slow 𝐼 ≈ -𝑘 2 𝑧 ( 𝑐 2 𝑆 / 𝑐 2 𝐴 ) 𝜂 𝐶 . Similar expressions for the damping times were also derived in Soler et al. (2009). \nIt can be observed from the expressions for the 𝜔 fast 𝐼 and 𝜔 slow 𝐼 coefficients, that the damping is proportional to the Cowling diffusivity 𝜂 𝐶 , i.e. the sum of the Ohmic and ambipolar diffusivities, and that both cause similar effects. Since 𝑐 𝑆 ≪ 𝑐 𝐴 , the damping is much stronger for the fast wave than for the slow wave. \nForteza et al. (2007) computed the damping times over the wave period for the typical solar prominence conditions in the adiabatic case. Their calculations are illustrated in Fig. 6.3 for the fast wave (left) and for the slow wave (right). The results are presented as a function of ionization fraction, where the value of ˜ 𝜇 =0.5 means fully ionized plasma and ˜ 𝜇 = 1 means fully neutral gas. It can be observed that, indeed, the fast mode damping due to neutrals is several orders of magnitude larger than that of the slow mode. Nevertheless, even for the fast mode, significant damping, 𝜏 𝐷 / 𝑃 ≈ 1, is only achieved for very low ionization fractions. Most contribution into this damping comes from the ambipolar diffusion, since for prominence parameters, 𝜂 𝐴 values significantly exceed the values of the Ohmic diffusion coefficient, 𝜂 . \nWhen non-adiabatic effects are present, the dispersion relation, Eq.6.80 is of the 5th order. In addition to the fast and slow magneto-acoustic modes, there is a thermal mode, as described by Carbonell et al. (2004) Since the properties of the thermal mode in partially ionized plasma are similar to those of the fully ionized case, here we do not discuss them further. \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFIGURE 6.4 Damping time over wave period for the fast magneto-acoustic wave (top) and for the slow magneto-acoustic wave (bottom) as a function of wave number 𝑘 in a non-adiabatic partially ionized plasma, calculated by numerically solving the dispersion relation, Eq.6.68, adopted from Forteza et al. (2008). The following parameters were used for the calculation: ionization fraction ˜ 𝜇 = 1 /( 1 + 𝜉 𝑖 ) = 0 . 8 (left panels) and ˜ 𝜇 = 0 . 99 (right panels), 𝐵 0 = 10 𝐺 , 𝜃 = 45 𝑜 , 𝑇 0 = 8 kK, 𝜌 0 = 5 × 10 -11 kg m -3 and the parameters of the heating-cooling function from Hildner (1974). Different lines correspond do different combinations of damping mechanisms. Solid line: ion-neutral collisions and thermal mechanisms; dotted line: only ion-neutral collisions; dashed line: only thermal mechanisms; dashed-dotted line: only heating-cooling and electron thermal conduction. \n<!-- image --> \nslow (bottom) non-adiabatic magneto-acoustic waves as a function of the wave number 𝑘 , as computed by numerically solving Eq.6.80 by Forteza et al. (2008). The shaded area in this figure marks the range of the wave numbers typically detected in solar prominences. The influence of different damping mechanisms is shown by different lines styles. The damping of the fast wave is dominated by radiative cooling at small 𝑘 , and by ion-neutral collisions at large 𝑘 for both neutral fractions considered (upper panels). Variation of the ionization fraction produces the change of the wavelength of the dominant damping mechanisms. For prominence conditions, ion-neutral collisions become the dominant damping mechanism for large neutral fraction, while radiation plays significant role at lower neutral fractions (shaded area at the upper panel). \nFor the slow wave and large neutral fraction ˜ 𝜇 = 0 . 99 (bottom right panel) the ratio of the damping time to period shows three minima (maximum attenuation) \nassociated to the typical scales of the different damping mechanisms. That at smallest 𝑘 is due to radiative cooling effects, the one at intermediate 𝑘 is due to ion-neutral collisions, and the one at largest 𝑘 is due to neutral thermal conduction. For ˜ 𝜇 = 0 . 8, the latter two peaks merge. For the prominence conditions, radiation plays the most important role in the damping of the slow wave, regardless the neutral fraction. \nThere is another important effect that can be observed in Fig. 6.4. Namely, that the fast waves disappear after a certain wavelength. This effect is related to the presence of the cut-off wavelength for the fast mode in partially ionized plasma. It can be illustrated by considering propagation parallel to the magnetic field, i.e., 𝑘 = 𝑘 𝑥 . In this case the dispersion relation Eq. 6.80 reduces to, \n( 𝜔 2 -𝑘 2 𝑥 Λ 2 )( 𝑖𝜔𝑘 2 𝑥 𝜂 𝐶 + 𝜔 2 -𝑘 2 𝑥 𝑐 2 𝐴 ) = 0 . (6.85) \nThe fast waves decouples from the slow and thermal waves, and its dispersion relation (second bracket in the last expression) can be solved to obtain the complex temporal frequency, \n𝜔 = -𝑖𝑘 2 𝑥 𝜂 𝐶 2 ± 𝑘 𝑥 2 √︃ 4 𝑐 2 𝐴 -𝑘 2 𝑥 𝜂 2 𝐶 . (6.86) \nTherefore, in order to have non-zero real part of the wave frequency, 𝜔 𝑅 ≠ 0, one has to fulfill the condition, \n𝑘 𝑥 < 2 𝑐 𝐴 𝜂 𝐶 = 𝑘 𝑐 , (6.87) \nwhere 𝑘 𝑐 is the cutoff wavelength. For wave numbers greater that the critical wavelength, the wave is completely damped and is not propagating in time domain. This effect will be discussed in more details when considering magnetoacoustic waves in a two-fluid approximation and it will be shown that the cutoff wavelength disappears.', '6.2.2.2 Alfvén waves': 'According to the dispersion relation Eq.6.67, the Alfvén waves are not affected by the thermal effects and are only affected by the ion-neutral collisions. By solving the dispersion relation, the real and imaginary parts of the wave frequency can be obtained, \n𝜔 = 𝜔 𝑅 + 𝑖𝜔 𝐼 = -𝑖𝑘 2 𝑥 ( 𝜂 𝐶 + 𝜂 tan 2 𝜃 ) 2 ± 𝑘 𝑥 2 √︃ 4 𝑐 2 𝐴 -𝑘 2 𝑥 ( 𝜂 𝐶 + 𝜂 tan 2 𝜃 ) 2 . (6.88) \nThe behavior of the Alfvén wave is similar to that of the fast wave in the adiabatic case, see Fig.6.5. In the non-adiabatic case, the damping of the fast wave is modified by the thermal effects at lower frequencies, compared to the \nFIGURE 6.5 Damping time over wave period for the Alfvén wave as a function of wave number 𝑘 in a non-adiabatic partially ionized plasma, calculated by numerically solving the dispersion relation, Eq.6.67, adopted from Forteza et al. (2008). The atmospheric parameters used for the calsulation are the same as in Fig.6.4. The ionization fraction ˜ 𝜇 = 0 . 5 (dotted line), ˜ 𝜇 = 0 . 6 (dashed line), ˜ 𝜇 = 0 . 8 (solid line), ˜ 𝜇 = 0 . 99 (dashed-dotted line). \n<!-- image --> \nAlfvén wave. By comparing the curves for different ionization fraction in Fig.6.5, one can observe that the ratio of the damping time to period decreases when going to higher neutral fractions, similar to the fast wave. For the typical prominence conditions, the Alfvén waves are only weakly damped (shaded area in Fig.6.5). \nOne can also observe the presence of a cutoff wave number after which the Alfvén wave does not propagate. This cutoff number can be computed from the dispersion relation, Eq. 6.88 and the following result is obtained \n𝑘 𝑥 < 2 𝑐 𝐴 cos 𝜃 ( 𝜂 𝐶 + 𝜂 tan 2 𝜃 ) = 𝑘 𝐴 𝑐 . (6.89) \nThis cutoff number depends on the ionization fraction through 𝜂 𝐶 , and on the propagation angle 𝜃 with respect to the magnetic field. Usually 𝑘 𝐴 𝑐 > 𝑘 𝑐 , and both become equal for the parallel propagation, then the fast and Alfvén waves become degenerate and cannot be distinguished. By considering the fully ionized plasmas, 𝜂 𝐶 = 𝜂 , the value of 𝑘 𝐴 𝑐 for the resistive plasmas can be recovered (Ferraro and Plumpton, 1961). \nSimilarly as for the fast wave, the sharp cutoff wave number for the Alfvén waves disappears in the two-fluid description, as discussed below.', '6.2.2.3 Aflvén waves and the Hall effect': "The Hall effect is a non-ideal effect that does not necessarily require the presence of neutrals. In a fully ionized plasma this effect is due to the different gyrofrequencies of ions, Ω ci , and electrons, Ω ce , i.e., mainly the difference in ion and electron mass. For a range of ion-electron collision frequencies, the electrons \nare magnetized ( Ω ce > 𝜈 ie ) while the ions are not ( Ω ci < 𝜈 ie ). The Hall effect in fully ionized plasmas becomes important at frequencies of the order of the ion-cyclotron frequency. \nThe Hall effect in partially ionized plasmas can be enhanced by collisions between ions and neutrals, which can work to demagnetize the ions further. In order to describe the action of the Hall effect, let us consider a linearized set of equations, neglecting the diamagnetic effect, radiation and thermal conduction, and considering a homogeneous isothermal plasma. We use linear continuity and momentum Eqs.6.59 and 6.60, together with the induction equation containing the Ohmic, ambipolar and Hall terms (Pandey and Wardle, 2008), \n𝜕 𝑩 1 𝜕𝑡 = ∇ × GLYPH<26> [ 𝑽 × 𝑩 0 ] + 𝜂 [∇ × 𝑩 1 ] -𝜂 𝐻 [∇ × 𝑩 1 ] × 𝑩 0 | 𝐵 0 | + (6.90) 𝜂 𝐴 [[[∇ × 𝑩 1 ] × 𝑩 0 ] × 𝑩 0 ] | 𝐵 0 | 2 GLYPH<27> . \nThe system is closed by an isothermal energy equation 𝑃 0 = 𝑐 2 𝑆 𝜌 0 . \nBy comparing in order of magnitude the Hall and convective terms, one can see that the Hall effect in partially ionized plasma becomes important at frequencies of the order of the Hall frequency (Forteza et al., 2008; Cally and Khomenko, 2015), \n𝜔 𝐻 = 𝜉 𝑖 Ω ci . (6.91) \nThe Hall frequency can be several orders of magnitude lower than the ioncyclotron frequency at locations with very cold plasma and low ionization fractions near the solar temperature minimum. Following the discussion in Section 6.1.1.1, the Hall effect is expected to be the dominant non-ideal effect in the middle-upper photosphere of the Sun. \nPandey and Wardle (2008) and Zaqarashvili et al. (2012) derived the dispersion relation for magneto-acoustic waves and Alfvén waves, including the Hall effect, in the form, \nn GLYPH<2> 𝜔 2 - ( 𝑐 2 𝐴 -𝑖𝜔𝜂 𝐴 ) 𝑘 2 cos 2 𝜃 + 𝑖𝜔𝑘 2 𝜂 GLYPH<3> + 𝑘 2 sin 2 𝜃 h 𝜔 ˜ 𝜔 𝑐 2 𝐴 + 𝑖𝜔𝜂 𝐴 io × (6.92) GLYPH<2> 𝜔 2 - ( 𝑐 2 𝐴 -𝑖𝜔𝜂 𝐴 ) 𝑘 2 cos 2 𝜃 + 𝑖𝜔𝜂𝑘 2 GLYPH<3> -𝜂 2 𝐻 𝑘 4 𝜔 2 cos 2 𝜃 = 0 , \nwhere ˜ 𝜔 2 = 𝜔 2 -𝑘𝑐 2 𝑆 . It can be verified that, in the absence of the Hall effect, one recovers the dispersion relations for the fast and Alfvén waves previously discussed in the sections above (6.2.2.1, 6.2.2.2), see Forteza et al. (2007, 2008). \nFor propagation parallel to magnetic field, 𝜃 = 0, the dispersion relation for the Alfvén waves from Wardle (1999) is recovered, \n𝜔 2 + 𝑖𝜔𝜂 𝐶 𝑘 2 -𝑘 2 𝑐 2 𝐴 = ± 𝜂 𝐻 𝜔𝑘 2 . (6.93) \nIn the low frequency limit, for waves with frequencies much lower than the Alfvén frequency, 𝜔 ≪ 𝑘𝑐 𝐴 , this dispersion relation has the following solution \nfor the real and imaginary parts of the wave frequency (Wardle, 1999; Forteza et al., 2008; Zaqarashvili et al., 2012) \n𝜔 𝑅 = ± 𝑐 2 𝐴 𝜂 𝐻 𝜂 2 𝐶 + 𝜂 2 𝐻 = ± 𝜔 𝐻 𝜂 2 𝐻 𝜂 2 𝐶 + 𝜂 2 𝐻 , (6.94) \n𝜔 𝐼 = -𝑐 2 𝐴 𝜂 𝐶 𝜂 2 𝐶 + 𝜂 2 𝐻 = -𝜔 𝐻 𝜂 𝐻 𝜂 𝐶 𝜂 2 𝐶 + 𝜂 2 𝐻 . (6.95) \nIt can be observed that, in the presence of the Hall current, the real part of the wave frequency is small, but always non-zero. It means that the sharp cutoff wavelength, present in partially ionized plasmas due to ion-neutral collisions, disappears in the presence of the Hall effect. An explanation for that is the following. When the Hall current is included, electrons can have a different dynamics to those of the ions. Ions may not be able to follow the magnetic field fluctuations due to the effect of ion-neutral collisions, but it is easier for electrons to remain coupled to the magnetic field, i.e., to stay magnetized. Therefore, ionneutral collisions cannot completely suppress the fluid oscillations because of the distinct behavior of electrons when Hall's current and/or electron inertia are included (see the discussion in Pandey and Wardle, 2008).", '6.2.3 Effects of gravitational stratification': 'Solar atmosphere is a strongly gravitationally stratified medium. The effects of neutrals, considered before, apply for homogeneous plasmas and are valid only locally. There are only few attempts to consider fully analytical solutions in partially ionized gravitationally stratified medium. Even without taking into account partial ionization, the height dependence of the coefficients in the wave equation allows for analytical solution only in some limited cases. For example, the case of an isothermal gravitationally stratified atmosphere with arbitrary inclined constant magnetic field has an exact solution in terms of Meier function or hypergeometric functions (Zhugzhda and Dzhalilov, 1984; Cally, 2001). \nAnother approach to attack the problem of the wave propagation in a gravitationally stratified atmosphere is by means of a local dispersion relation. This technique has been widely used to study acoustic-gravity waves, as well as adiabatic MHD waves. In this case, the atmosphere is approximated by layers with locally homogeneous properties. This approach gives approximately precise results as far as the wavelength of the perturbation is much smaller than density/pressure scale height (0-order WKB approximation). Nevertheless, gradients caused by gravitational stratification introduce new effects and the local dispersion relation approach may not always describe the variations of the waveforms with height. \nThe presence of gradients and strong vertical stratification allows for the process of wave-mode transformation. The classical mode transformation happens when the fast magneto-acoustic mode travelling through the solar interior \n(a 𝑝 -mode, which is essentially acoustic below the photosphere) emerges at the solar surface and encounters the layer, located somewhere in the photosphere or chromosphere, where acoustic and Alfvén speeds are similar. At this layer, part of the energy of the fast magneto-acoustic mode is transformed into a slow magneto-acoustic mode (also essentially acoustic) propagating along the magnetic field, and a fast magneto-acoustic mode (now essentially magnetic) that will eventually refract and reflect back to the surface due to the gradients of the Alfvén speed (Cally, 2006; Khomenko and Collados, 2006). At the upper turning point of the fast (magnetic) mode, an Alfvén mode can be generated through the secondary mode transformation (Cally and Goossens, 2008). This transformation can be considered as a geometry-induced mode transformation since it results from the stratification of the solar atmosphere and its efficiency depends on the orientation of the magnetic field. Mode transformation is expected to play an important role for waves propagating in solar active regions (Khomenko and Collados, 2015). Partial ionization affects both, propagation of different wave modes through the stratified atmosphere, and the mode transformation process. We discuss these effects in the sections below.', '6.2.3.1 Local dispersion relation': 'In this section, a local dispersion relation for waves in a gravitationally stratified atmosphere is derived, taking into account Ohm, ambipolar and Hall effects. It considers a particular case, which includes the majority of the modes and effects discussed above, but now adding a stratification. The diamagnetic effect, heat conduction and radiation are neglected. \nA Cartesian reference system is chosen such that spatially constant equilibrium magnetic field vector lies in the 𝑥 -𝑧 plane, \n𝑩 = { 𝐵 𝑥 0 , 0 , 𝐵 𝑧 0 } . \nIn this system, a perturbation propagates in the vertical 𝑧 direction, and the perturbed vector quantities have the following components, \n𝑽 = { 𝑉 𝑥 ( 𝑧 ) , 𝑉 𝑦 ( 𝑧 ) , 𝑉 𝑧 ( 𝑧 )} , 𝑩 1 = { 𝐵 𝑥 1 ( 𝑧 ) , 𝐵 𝑦 1 ( 𝑧 ) , 0 } . \nSince the horizontal magnetic field components are functions of the vertical coordinate only, in order to fulfill the divergence-free condition, the vertical component must be constant with 𝑧 , therefore, 𝐵 𝑧 1 = 0. \nIn the spirit of the local dispersion relation we assume that all the background quantities are locally constant with height withing a narrow layer. One then can apply the equations layer by layer, by varying the values of the background quantities. By imposing all the first-order quantities varying as 𝑓 ∼ exp (-𝑖𝜔𝑡 + 𝑖𝑘 𝑧 𝑧 ) , the system Eq. 6.54-6.57 becomes (Kazeminezhad and Goodman, 2006), \n-𝑖𝜔𝜌 1 + 𝑖𝑘 𝑧 𝑉 𝑧 = 0 , (6.96) \n-𝑖𝜔𝜌 0 𝑉 𝑥 -𝑖𝑘 𝑧 𝐵 𝑥 1 𝐵 𝑧 0 𝜇 = 0 , (6.97) \n0 \n-𝑖𝜔𝜌 0 𝑉 𝑦 -𝑖𝑘 𝑧 𝐵 𝑦 1 𝐵 𝑧 0 𝜇 0 = 0 , (6.98) \n-𝑖𝜔𝜌 0 𝑉 𝑥 + 𝑖𝑘 𝑧 𝑃 1 + 𝜌 1 𝑔 + 𝑖𝑘 𝑧 𝐵 𝑥 1 𝐵 𝑥 0 𝜇 0 = 0 , (6.99) \n-𝑖𝜔𝐵 𝑥 1 -𝑖𝑘 𝑧 𝑉 𝑥 𝐵 𝑧 0 + 𝑖𝑘 𝑧 𝑉 𝑧 𝐵 𝑥 0 + 𝜂 𝐶 𝑘 2 𝑧 𝐵 𝑥 1 + 𝜂 𝐻 cos 𝜃𝑘 2 𝑧 𝐵 𝑦 1 = 0 , (6.100) \n-𝑖𝜔𝐵 𝑦 1 -𝑖𝑘 𝑧 𝑉 𝑦 𝐵 𝑧 0 + ( 𝜂 + 𝜂 𝐴 cos 2 𝜃 ) 𝑘 2 𝑧 𝐵 𝑦 1 -𝜂 𝐻 cos 𝜃𝑘 2 𝑧 𝐵 𝑥 1 = 0 , (6.101) \n𝑃 1 = 𝑐 2 𝑆 𝜌 1 , (6.102) \nwhere 𝑐 2 𝑆 = 𝛾𝑃 0 / 𝜌 0 is the background sound speed and 𝜃 is magnetic field inclination angle with respect to the vertical 𝑧 direction. After combining these equations a single dispersion relation can be obtained, \nGLYPH<16> 𝜔 2 -𝑘 2 𝑧 𝑐 2 𝐴𝑧 + 𝑖𝜂 𝐶 𝑘 2 𝑧 𝜔 GLYPH<17> GLYPH<16> 𝜔 2 -𝑘 2 𝑧 𝑐 2 𝐴𝑧 + 𝑖 ( 𝜂 + 𝜂 𝐴 cos 2 𝜃 ) 𝑘 2 𝑧 𝜔 GLYPH<17> GLYPH<16> 𝜔 2 -𝑘 2 𝑧 𝑐 2 𝑆 + 𝑖𝑔𝑘 𝑧 GLYPH<17> -(6.103) \n-GLYPH<16> 𝑘 2 𝑧 𝜔𝜂 𝐻 cos 𝜃 GLYPH<17> 2 GLYPH<16> 𝜔 2 -𝑘 2 𝑧 𝑐 2 𝑆 + 𝑖𝑔𝑘 𝑧 GLYPH<17> -𝑘 2 𝑧 𝜔 2 𝑐 2 𝐴𝑥 GLYPH<16> 𝜔 2 -𝑘 2 𝑧 𝑐 2 𝐴𝑧 + 𝑖 ( 𝜂 + 𝜂 𝐴 cos 2 𝜃 ) 𝑘 2 𝑧 𝜔 GLYPH<17> = 0 . \nThis equation is 6th order in 𝜔 and allows for three wave modes with properties resembling fast, slow and Alfvén modes. In the case of the absence of the magnetic field, it reduces to, \nGLYPH<16> 𝜔 2 + 𝑖𝜂𝑘 2 𝑧 𝜔 GLYPH<17> 2 GLYPH<16> 𝜔 2 -𝑘 2 𝑧 𝑐 2 𝑆 + 𝑖𝑔𝑘 𝑧 GLYPH<17> = 0 . (6.104) \nThe first bracket contains an entropy mode ( 𝜔 = 0) and an evanescent perturbation in 𝐵 𝑥 1 and 𝐵 𝑦 1 with 𝜔 = -𝑖𝑘 2 𝑧 𝜂 . The second bracket is a usual acoustic-gravity mode propagating vertically in a stratified atmosphere. The propagation of this mode is subject to the acoustic cut-off with the value defined by temperature ( 𝜔 𝑐 = 𝑔 / 2 𝑐 𝑆 ). Under approximations of the local dispersion relation, the propagation of this mode is unaffected by any of the non-ideal effects. Notice that, this conclusion goes in line with the discussion from Section 6.2.2.1. According to Section 6.2.2.1, slow magneto-acoustic waves propagating along the magnetic field are unaffected by non-ideal effects to the first order (this can be seen by setting Ξ = 0 and 𝑘 = 𝑘 𝑧 in the adiabatic Eq. 6.80). \nAnother particular case is obtained by setting 𝐵 𝑥 0 = 0. This way we consider waves propagating vertically along the vertical magnetic field. The dispersion \nrelation Eq. 6.103 reduces to, \nGLYPH<26> GLYPH<16> 𝜔 2 -𝑘 2 𝑧 𝑐 2 𝐴𝑧 + 𝑖𝜂 𝐶 𝑘 2 𝑧 𝜔 GLYPH<17> 2 -GLYPH<16> 𝑘 2 𝑧 𝜔𝜂 𝐻 GLYPH<17> 2 GLYPH<27> GLYPH<16> 𝜔 2 -𝑘 2 𝑧 𝑐 2 𝑆 + 𝑖𝑔𝑘 𝑧 GLYPH<17> = 0 . (6.105) \nYet again we obtain longitudinally propagating acoustic-gravity mode in the second bracket. The first bracket corresponds to the Alfvén wave (or fast wave, which is indistinguishable in this limit), affected by the Ohmic and ambipolar dissipation and Hall effect, with the dispersion relation, \n𝜔 2 -𝑘 2 𝑧 𝑐 2 𝐴𝑧 + 𝑘 2 𝑧 𝜔 ( 𝑖𝜂 𝐶 ± 𝜂 𝐻 ) = 0 . (6.106) \nIt can be seen that, under the approximations done to derive the local dispersion relation, the propagation of this wave is not affected by gravity, and we recover the case described in Section 6.2.2.3. \nFinally, by setting 𝐵 𝑧 0 = 0 we recover the case of waves propagating transverse to the magnetic field in the direction of stratification, \n( 𝜔 2 + 𝑖𝜂𝑘 2 𝑧 𝜔 ) GLYPH<2> ( 𝜔 2 + 𝑖𝜂 𝐶 𝑘 2 𝑧 𝜔 )( 𝜔 2 -𝑘 2 𝑧 𝑐 2 𝑆 + 𝑖𝑔𝑘 𝑧 ) -𝑘 2 𝑧 𝜔 2 𝑐 2 𝐴𝑥 GLYPH<3> = 0 . (6.107) \nNotice that the Hall term, due to its particular shape, does not affect waves in this case. Yet again, the fist bracket in the dispersion relation corresponds to a fully damped perturbation. The second bracket describes modes with mixed properties, and the analytical solution is impractical. It represents the generalization of the fast magneto-acoustic mode affected by resistive dissipation and gravity. In the absence of resistive effects, it reduces to the one of the fast mode in a gravitationally stratified atmosphere, with properties analogous to the acoustic-gravity modes, but with the propagation speed, and the gravitational cut-off, dependent on the magnetic field, \n𝜔 2 -𝑘 2 𝑧 ( 𝑐 2 𝑆 + 𝑐 2 𝐴𝑥 ) + 𝑖𝑘 𝑧 𝑔 = 0 . (6.108) \nThe dispersion equation for acoustic gravity or fast modes can be solved for 𝑘 𝑧 recovering its real and imaginary parts, 𝑘 𝑧 = 𝑘 zR + 𝑖𝑘 zI . The imaginary part gives the wave amplitude growth with height, 𝑘 zI = 𝑔 / √︃ 4 ( 𝑐 2 𝑆 + 𝑐 2 𝐴𝑥 ) . Wave amplitude growth in a stratified atmosphere is a usual property derived from the conservation of the kinetic energy. Since 𝜂 𝐶 introduces damping, it can be expected that the effects of the gravitational stratification and damping will be competing with each other. Depending on the height, magnetic field strength, and wave frequency, the amplitude growth can be overcome by the damping.', '6.2.3.2 Propagation of Alfvén waves in a stratified atmosphere': 'Local dispersion relation is usually not a good approximation for fast and Alfvén waves because their wavelength in the solar atmosphere strongly increases with height due to the Alfvén speed growth. Consider an example of torsional Alfvén \nwaves propagating along a an expanding magnetic flux tube. Using cylindrical coordinate system, the following coupled linearized equations for 𝑉 𝜃 and 𝐵 1 𝜃 components can be obtained (Zaqarashvili et al., 2013). \n𝜕 𝜕𝑡 ( 𝑟𝑉 𝜃 ) = 𝑩 0 · ∇ 𝜇 0 𝜌 0 ( 𝑟𝐵 1 𝜃 ) , (6.109) \n𝜕𝐵 1 𝜃 𝜕𝑡 = 𝑟 ( 𝑩 0 · ∇) " 𝑉 𝜃 𝑟 + 𝜂 𝐶 𝐵 2 0 𝑩 0 · ∇ 𝑟 2 ( 𝑟𝐵 1 𝜃 ) # . (6.110) \nFollowing the original work by Zaqarashvili et al. (2013), the Cowling diffusion coefficient is defined taking into account collisions between electrons, neutral and ionized Hydrogen and Helium. By defining 𝑈 𝜃 = 𝑉 𝜃 / 𝑟 these two equations can be combined into a single wave equation, \n𝜕 2 𝑈 𝜃 𝜕𝑡 2 = 𝐵 0 𝑠 𝜇 0 𝜌 0 𝑟 2 𝜕 𝜕𝑠 " 𝑟 2 𝐵 0 𝑠 𝜕 𝜕𝑠 𝑈 𝜃 + 𝜇 0 𝜌 0 𝜂 𝐶 𝐵 2 0 𝑠 𝜕𝑈 𝜃 𝜕𝑡 !# , (6.111) \nwith the derivatives taken along the magnetic field direction 𝑠 , and being 𝐵 0 𝑠 the magnetic field along 𝑠 . This equation can be simplified at locations close to the flux tube axis, so that 𝐵 0 𝑠 𝑟 2 ≈ const: \n𝜕 2 𝑈 𝜃 𝜕𝑡 2 = 𝑐 2 𝐴 𝜕 2 𝜕𝑠 2 " 1 + 𝜂 𝐶 𝑐 2 𝐴 𝜕 𝜕𝑡 ! 𝑈 𝜃 # . (6.112) \nEquation 6.112 can be Fourier-analyzed in the temporal domain, but not in the spatial domain, since the coefficients depend on the stratification. Nevertheless, based on approximate calculations in the FAL93-F model (Fontenla et al., 1993), the ratio 𝜂 𝐶 / 𝑐 2 𝐴 does not vary much with height in the chromosphere, and can be assumed approximately constant (Zaqarashvili et al., 2013). Additionally, if a thin flux tube approximation can be assumed valid, then the density fall off with height is compensated by the magnetic field dependence, making the Alfvén speed constant. Under these conditions, Eq. 6.112 reduces to a homogeneous Alfvén equation in a form similar to Eq. 6.67, and the conclusions about the damping and cutoff wave number hold valid (Forteza et al., 2008), \n𝜔 2 + 𝑖𝜂 𝐶 𝑘 2 𝑠 𝜔 -𝑘 2 𝑠 𝑐 2 𝐴 = 0; 𝜔 = ± 𝑘 𝑠 𝑐 𝐴 v t 1 -𝜂 2 𝐶 𝑘 2 𝑠 4 𝑐 2 𝐴 -𝑖 𝜂 𝐶 𝑘 2 𝑠 2 . (6.113) \nGiven the atmospheric parameters from FAL93-F, Zaqarashvili et al. (2013) evaluated the typical values of the damping times over the wave period, 𝜏 𝑑 / 𝑇 0 , to be between ≈ 1 -70 for wave periods of 1 -60 sec. When collisions with Helium are taken into account, the damping is about 20-30% stronger. \nAt the upper part of magnetic flux tubes the field expands and the thin flux tube approximation is not valid anymore. In this situation one has to consider \nFIGURE 6.6 Height dependence of the velocity of an upward propagating torsional Alfvén wave in the atmosphere with parameters from FAL93-F model atmosphere. Green lines: fully ionized plasma; blue lines: partially ionized hydrogen plasma; red lines: partially ionized hydrogen-helium plasma. Left panel: period of 3 s, right panel: period of 10 s. Figure from Zaqarashvili et al. (2013) \n<!-- image --> \nexponentially changing Alfvén speed in Eq. 6.112. In such a case, assuming 𝑐 𝐴 = 𝑐 𝐴 0 exp (-𝑠 / 2 𝐻 ) , the Fourier-transformed Eq. 6.112 reduces to, \n𝜕 2 𝑈 𝜃 𝜕𝑠 2 + exp GLYPH<16> -𝑠 𝐻 GLYPH<17> 𝑘 2 𝑠 0 𝑈 𝜃 = 0; 𝑘 2 𝑠 0 = 𝜔 2 𝑐 2 𝐴 0 -𝑖𝜔𝜂 𝐶 0 . (6.114) \nThe values with sub-index \'0\' are taken at a reference height. The solution of this equation is a standard Bessel or Hankel function. Figure 6.6 illustrates the solution for 𝑈 𝜃 for short-period waves. It can be observed that wave amplitudes increase with height due to the effects of stratification. However, the wave forms in partially ionized plasma are visibly damped at higher heights, compared to the case of the fully ionized solution (green lines). The damping is more significant when the Helium atoms are taken into account. The shorter period waves experience a stronger dependence on the ion-neutral collisions, as expected from analytical solutions for homogeneous plasma. It can be also observed that, due to the effects of stratification, the effective wavelength of perturbations becomes very large, so that these waves turn out to be almost evanescent. This MHD effect is not changed by partial ionization, however, the amplitudes of these long-wavelength waves are visibly reduced due to ion-neutral collisions. Overall, the work by Zaqarashvili et al. (2013) allows to quantify that shortperiod (<5 s) torsional Alfvén waves damp quickly in conditions close to those in chromospheric network, owing to ion-neutral collision, while this damping is not so important for waves with longer periods.', '6.2.3.3 Propagation of magneto-acoustic waves in a stratified atmosphere': 'In Section 6.2.2.1, it was shown how ambipolar diffusion affects fast and slow magneto-acoustic waves, concluding that fast waves can be significantly damped. Slow waves are also susceptible to ambipolar damping, but to a lower degree. \nFIGURE 6.7 Snapshots of the simulation of magneto-acoustic wave pulse excited at the top of the photosphere and propagating through the chromosphere. Panels from left to right show different moments of time. Upper panels: pure MHD case, bottom panels: ambipolar diffusion is included. The ambipolar diffusion coefficient changes almost linearly on log 10 scale from 10 3 m 2 s -1 at 0.5 Mmto 10 11 m 2 s -1 at 2.1 Mm. Black lines are magnetic field lines. Magnetic field has a strength of 17.4 G and is inclined by 45 𝑜 to the vertical gravity direction. The wave period is of 5 sec. Plasma 𝛽 changes from 𝛽 > 1 to 𝛽 < 1 at about 1.25 Mm. Figure from Popescu Braileanu and Keppens (2021). \n<!-- image --> \nThese conclusions were obtained for conditions close to those in solar prominences. When it comes to the propagation of fast and slow magneto-acoustic waves in the gravitationally stratified solar chromosphere, analytical solutions become impractical and numerical approach allows to get better insights. \nPopescu Braileanu and Keppens (2021) considered a fully numerical solution for the propagation of magneto-acoustic waves launched from the upper photosphere upwards and propagating through the stratified chromosphere permeated by a constant inclined magnetic field, and with thermodynamic conditions given, approximately, by solar 1D model VALC (Vernazza et al., 1981). An illustrative example from these numerical experiments is shown in Figure 6.7. The upper panel shows how a Gaussian pulse at the bottom boundary generates a fast magneto-acoustic wave package. On its way to the chromosphere this wave package suffers a refraction and reflection due to the gradients of the Alfvén speed, a typical behavior seen in many previous works (e.g., Bogdan et al., 2003; Khomenko and Collados, 2006). The plasma 𝛽 = 1 layer is located around 1.25 Mm in their experiment, so mode transformation is taking place (Cally, 2006). In this particular case, the fast wave propagating through 𝛽 = 1 layer does not generate significant slow wave due to the mode transformation, because of the relatively large field inclination and high wave frequency (Cally, 2006). It is interesting to compare the ideal MHD case to the case when the ambipolar diffusion was included in the model (bottom panel). In the latter case, it is \nclear that the fast wave is "eaten out" by the ambipolar diffusion and is unable to complete the refraction trajectory. According to Popescu Braileanu and Keppens (2021), in most cases considered in their work, the high frequency fast waves are significantly damped before they are reflected. This damping is stronger if the waves propagate across the magnetic field. The damping increases with wave frequency, and magnetic field strength, due to the increase of 𝜂 𝐴 . The results of these numerical experiments are in qualitative agreement with the analytical wave theory in homogeneous plasmas (Forteza et al., 2007, 2008), and they allow to quantify the effect of ambipolar diffusion on waves for the solar case.', '6.2.3.4 Hall-induced mode transformation': "Thegeometrical fast-to-Alfvén mode transformation is intrinsically a 3D process. This transformation cannot happen when the wave vector lies in a plane defined by the magnetic field vector and the direction of the stratification (gravity or strong pressure gradient). So necessarily, the wave propagation should happen with an angle to the plane. In partially ionized plasmas, the Hall effect is able to assist the process of the mode transformation by naturally adding a 3rd dimension to the problem. It happens as a result of the Hall effect producing perturbations of the current in the direction perpendicular to the plane defined by the magnetic field and the gravity. \nTo illustrate this effect, following Cally and Khomenko (2015), consider a single-fluid system of equations of mass, momentum and induction with only the Hall effect, and cold plasma ( 𝑃 ≪ 𝑃 mag , 𝑐 𝑆 ≈ 0) approximation (linearized Eqs. 6.59, 6.60 and induction equation, Eq. 6.90 with only the Hall term). The velocity is substituted by the Lagrangian displacement, 𝑽 = 𝜕 𝝃 / 𝜕𝑡 . The plasma is gravitationally stratified along the 𝑥 direction, and an inclined magnetic field is contained in the 𝑥 -𝑧 plane, while the wave vector is directed outside of this plane. Using the Fourier transform of the variables for the coordinates where no stratification exist, 𝜉 ( 𝑥, 𝑦, 𝑧, 𝑡 ) = 𝜉 ( 𝑥 ) exp [ 𝑖 ( 𝑘 𝑦 𝑦 + 𝑘 𝑧 𝑧 -𝜔𝑡 )] , the following coupled system of equations for the displacement perturbation are obtained: \n𝜕 2 ∥ + 𝜕 2 ⊥ + 𝜔 2 𝑐 2 𝐴 ! 𝜉 ⊥ = -𝑖𝑘 𝑦 𝜕 ⊥ 𝜉 𝑦 + 𝑖 h 𝑖 𝑘 𝑦 𝜕 ⊥ ( 𝜖 𝜉 ⊥ ) - ( 𝜕 2 ∥ + 𝜕 2 ⊥ )( 𝜖 𝜉 𝑦 ) i , 𝜕 2 ∥ + 𝜔 2 𝑐 2 𝐴 -𝑘 2 𝑦 ! 𝜉 𝑦 = -𝑖𝑘 𝑦 𝜕 ⊥ 𝜉 ⊥ + 𝑖 h ( 𝜕 2 ∥ -𝑘 2 𝑦 )( 𝜖 𝜉 ⊥ ) -𝑖 𝑘 𝑦 𝜕 ⊥ ( 𝜖 𝜉 𝑦 ) i . \nThis system describes the propagation of coupled fast and Alfvén waves. The equations use a change of the reference system linked to the wave propagation directions, ( 𝒆 ⊥ , 𝒆 𝑦 , 𝒆 ∥ ) . The (∥ , ⊥) plane is the same as ( 𝑥, 𝑧 ) plane, the ∥ and ⊥ directions are the one along the magnetic field and perpendicular to it. \nIn these equations 𝜖 is the Hall parameter, defined as a ratio between the wave frequency and the Hall frequency, \n𝜖 = 𝜔 /( Ω ci 𝜉 𝑖 ) = 𝜔 / 𝜔 𝐻 . (6.115) \n<!-- image --> \nFIGURE 6.8 Left panel: Hall parameter, 𝜖 , as a function of height for different wave frequencies in the solar atmosphere. The vertical line corresponds to the photospheric level, 𝑧 = 0 km, negative heights are below solar surface. Right panel: Velocity 𝑣 perp = 𝑣 𝑦 of the Alfvén wave generated after the Hall-induced mode transformation as a function of magnetic field inclination angle. The magnetic field in this experiment was inclined by 10 degrees with respect to the stratification direction. Figure from González-Morales et al. (2019). \n<!-- image --> \nIf the Hall effect is absent ( 𝜖 = 0), the equations are only coupled through nonzero 𝑘 𝑦 , i.e. for wave propagation outside of the 𝑥 -𝑧 plane. However, if the Hall effect is present, the equations for the fast and Alfvén waves are coupled even if 𝑘 𝑦 = 0. As discussed in Sect. 6.2.2.3, a small ionization fraction around the temperature minimum allows for the significant decrease in the frequency of waves for which the Hall effect becomes important. It can be expected that typical thermodynamic conditions in the Sun's atmosphere create the so-called Hall window, with the extension and amplitude as the one shown at the left panel of Figure 6.8, taken from González-Morales et al. (2019). These values of 𝜖 were computed using the parameters from the Model-S of Christensen-Dalsgaard et al. (1996) (below the photosphere) with the chromospheric model VAL-C by Vernazza et al. (1981). Inside the Hall window, Hall coupling produces a continuous oscillation between the fast-mode and Alfvén-mode states (Cally and Khomenko, 2015). \nAccording to the theoretical mode conversion model in a cold plasma by Cally and Khomenko (2015), the Hall coupling preferentially occurs where the wave vector is nearly parallel to the guide field. This is illustrated in Figure 6.9. There, the Hall-induced mode conversion coefficient is shown as a function of the magnetic field inclination, for several parameters of the Hall window, and the values of 𝜖 . The conversion efficiency has a broad maximum for the vertical fields. It is also sensitive to the location and extension of the Hall window, and its amplitude decreases with decreasing the value of 𝜖 . \nNumerical calculations by González-Morales et al. (2019), performed for parameters applicable to the solar atmosphere are in a general agreement with the cold plasma model of the Hall-induced conversion. The right panel of Figure 6.8 shows the amplitudes of the Alfvén waves generated after Hallinduced transformation at chromospheric heights, as a function of inclination \n<!-- image --> \nFIGURE 6.9 Conversion coefficient from fast to Alfvén wave for the Hall-induced conversion model by Cally and Khomenko (2015), as a function of magnetic field inclination angle and 𝑘 𝑦 = 0. Left and right panels are different by the location and thickness of the Hall window, indicated in the figure. Solid, dashed and dotted curves are for progressively smaller value of non-dimensional Hall coefficient 𝜖 parameter, indicated in the figure. \n<!-- image --> \nangle between the magnetic field and the wave propagation direction (in this experiment, the field was inclined by 10 degrees to the vertical). The amplitude of the Alfvén waves exponentially increases with frequency, and is a sensitive function of the inclination angle. The maximum amplitudes are reached for waves with frequencies of 1Hz, and make up to ∼ 10% of the amplitude of the fast waves entering the Hall window. \nIn a stratified solar atmosphere, the Hall-induced conversion is a two-step process (González-Morales et al., 2019; Raboonik and Cally, 2019). Firstly, a classic geometrical mode transformation at the 𝑐 𝑆 = 𝑐 𝐴 layer has to happen, producing fast and slow magneto-acoustic waves from essentially acoustic solar 𝑝 modes. After that, the newly produced fast (magnetic) wave is able to convert to the Alfvén wave though the Hall effect. For this two-stage process to be effective, particular conditions must be fulfilled, namely, the 𝑐 𝑆 = 𝑐 𝐴 layer must be located below the Hall window layer (see the left panel of Fig. 6.8), this can be fulfilled for intermediate field strengths of the order of hG. \nRaboonik and Cally (2019) demonstrated how the classic and the Hallinduced mode transformation work together in a stratified solar atmosphere to produce both up-going and down-going Alfvén waves. According to their results, a down-going slow wave, produced after the incident fast wave reflection/conversion, can couple to the down-going Alfvén wave through the Hall effect. This coupling will be strongest for the horizontal wave numbers oriented opposite to the field inclination, and magnetic field strength of the order of 100 G in order to place the Hall window at the optimum location. Alternatively, an up-going slow wave injected from below can couple to the up-going Alfvén wave. Unlike the Hall-mediated fast-Alfvén coupling, the slow-Alfvén coupling occurs lower at the atmosphere and for much lower wave frequencies (those at with typical acoustic-gravity waves are evanescent). \nFIGURE 6.10 Vertical energy fluxes computed for three different prescriptions of ambipolar diffusion (columns): 𝜂 𝐴 = 0 (left), Gaussian ambipolar diffusion region with 𝜖 𝐴 = 𝜖 𝐴 0 exp (-( 𝑥 -𝑥 0 ) 2 / 2 𝜎 2 ) , 𝜖 𝐴 0 = 𝜎 = 1 (middle); 𝜖 𝐴 = 0 . 05 between 𝑥 = -2 and 𝑥 = 5 (right). Top row: 𝜃 = 30 𝑜 , 𝜙 = 0 𝑜 (no Alfvén conversion). Bottom row: 𝜃 = 30 𝑜 , 𝜙 = 30 𝑜 (conversion to mostly up-going Alfvén waves). Blue/red curves are for fast/Alfvén wave flux, solid/dashed lines are for upward/downward flux. Black curve is the total flux. Figure from Cally and Khomenko (2018). \n<!-- image -->", '6.2.3.5 Influence of ambipolar diffusion on the mode transformation': 'Ambipolar diffusion in the solar atmosphere is expected to be the largest in the middle-upper chromosphere. Right in the same regions, plasma 𝛽 is also expected to be around unity, and therefore geometrical mode transformation is expected to take place. Here we follow a theoretical study by Cally and Khomenko (2018) of how ambipolar diffusion affects the process of the mode transformation from fast to Alfvén waves, using a cold plasma approximation. \nConsider a uniform background magnetic field, inclined with respect to the gravity 𝑥 -direction by an angle of 𝜃 , and contained in the 𝑥 -𝑧 plane. The wave vector lies outsize of the plane formed by the gravity and the magnetic field, forming an azimuth angle 𝜙 . It is again convenient to consider three characteristic directions, ( 𝒆 ⊥ , 𝒆 𝑦 , 𝒆 ∥ ) . The non-ideal induction equation contains only the ambipolar term. In this case, the wave equation for the displacement 𝝃 takes the following form, \n𝜕 2 𝑡 𝝃 = 𝑐 2 𝐴 ( 𝜕 2 | | - ∇ ⊥ ∇·) 𝝃 + 𝜂 𝐴 𝑐 2 𝐴 𝜕 𝑡 𝝃 ! , (6.116) \nwhere the derivative ∇ ⊥ is taken in the direction perpendicular to the magnetic field. It can be seen that there is no structural change in the Eq. 6.116, compared to the standard wave equation without 𝜂 𝐴 , i.e. there are no additional derivatives due to the ambipolar effect, that would lead to a different physics (Cally and Khomenko, 2018). In this regard, the ambipolar diffusion differs greatly from \nthe Hall effect considered by Cally and Khomenko (2015), which produced its own mode conversion mechanism. \nAssumingavertically stratified atmosphere such that 𝜔𝐻 2 / 𝑐 2 𝐴 = exp (-𝑥 / 𝐻 ) = 𝑠 , and introducing a dimensionless ambipolar diffusion parameter 𝜖 𝐴 ( 𝑠 ) = 𝜔𝜂 𝐴 / 𝑐 2 𝐴 , which is a function of 𝑠 only, Eq. 6.116 can be rewritten as, \n(( 1 -𝑖𝜖 𝐴 ) -1 𝜕 2 𝑡 -𝑐 2 𝐴 𝜕 2 | | ) 𝑿 = 𝑐 2 𝐴 ∇ ⊥ ∇ · 𝑿 , (6.117) \nwhere 𝑿 = ( 1 -𝑖𝜖 𝐴 ) 𝝃 . Dropping the compression term at the right hand side this equation reduces to the ambipolar-damped Alfvén wave equation discussed in Section 6.2.2.2. \nEquation 6.117 can be Fourier analyzed in the 𝑦 and 𝑧 directions perpendicular to the stratification direction, and then it splits into two equations for the ⊥ and 𝑦 components of 𝑿 , \n𝜕 2 𝑥 + 𝜔 2 ( 1 -𝑖𝜖 ) 𝑐 2 𝐴 -𝑘 2 𝑧 𝑋 ⊥ = -𝑖𝑘 𝑦 ( sin 𝜃𝜕 𝑥 -𝑖𝑘 𝑧 cos 𝜃 ) 𝑋 𝑦 , (6.118) \n( cos 𝜃𝜕 𝑥 + 𝑖𝑘 𝑧 sin 𝜃 ) 2 + 𝜔 2 ( 1 -𝑖𝜖 ) 𝑐 2 𝐴 -𝑘 2 𝑦 𝑋 𝑦 = -𝑖𝑘 𝑦 ( sin 𝜃𝜕 𝑥 -𝑖𝑘 𝑧 cos 𝜃 ) 𝑋 ⊥ \n! ! . \nThe magnetic Poynting flux can be obtained from the following expression, \n𝑭 mag = 1 𝜇 0 𝑅𝑒 [ 𝑬 ∗ 1 × 𝑩 1 ] = 𝜔𝐵 0 𝜇 0 𝐼𝑚 [(∇ · 𝑿 ) 𝑿 ∗ + ( 𝑿 ∗ · 𝜕 | | 𝑿 ) 𝒆 | | ] , (6.119) \nwith 𝑬 1 = 𝑩 × 𝜕 𝑡 ( 𝝃 + 𝜂 𝐴 𝑽 / 𝑐 2 𝐴 ) being the 1st order perturbation of the electric current, and ∗ denoting a complex conjugation. The computation of magnetic flux allows to explore the influence of the ambipolar effect on the mode conversion. The main conclusions of these calculations are summarized in Figure 6.10. Overall, the conversion picture stays the same in the presence of ambipolar diffusion. In the case of no Alfvén conversion ( 𝜃 = 30 𝑜 , 𝜙 = 0 𝑜 , top row), the net wave flux is zero for 𝜂 𝐴 = 0 case since the fast wave is perfectly reflected. When 𝜂 𝐴 is distinct from zero, it can be observed the presence of an asymmetry between the up-going and down-going waves, the ambipolar diffusion reduces the non-converting fast mode flux by about 50%. In the second case ( 𝜃 = 30 𝑜 , 𝜙 = 30 𝑜 , bottom row), when Alfvén conversion is present, about 25% of the energy is going to the upward Alfvén wave when 𝜂 𝐴 = 0. This amount is significantly reduced in the cases of 𝜂 𝐴 ≠ 0 (middle and right panels). As it can be seen from the height dependence of the fluxes, the upward propagating Alfvén waves, produced after the transformation, are almost immune to further ambipolar dissipation. This would reduce the impact for heating by waves produced by this mechanism. \nFigure 6.10 from Cally and Khomenko (2018) is computed for non-dimensional 𝜖 𝐴 coefficients. For the values of 𝜂 𝐴 or 𝜖 𝐴 computed as in Section 6.1.1.1 for \nthe parameters of the solar atmosphere, the effect of ambipolar diffusion on the mode conversion would be too weak to measurably affect waves with frequencies typically observed in the Sun (around 3-5 mHz). In order to produce a measurable effect, one has to invoke a concept of turbulent ambipolar diffusion.', '6.2.4 Non-linear perturbations and plasma heating': "Waves are one of the best candidates to bring energy to heat the upper solar atmosphere. In particular, heating theories based on Alfvén waves are very promising, because Alfvén waves do not shock at lower layers, and due to their incompressibility they are not affected by damping through viscosity, conductivity or radiation (Section 6.2.2.2). Efficient dissipation of these waves can be achieved through the ambipolar diffusion mechanism. The linear wave analysis, as the one considered above, does not allow to compute the dissipation of the wave energy from the equations, since the Joule dissipation term in the singlefluid approach is a second order term. Therefore, the linear analysis excludes important aspect of the wave heating. Many works suggest that Joule dissipation of currents by ambipolar mechanism has a potential to provide sufficient energy to maintain the temperature of the solar chromosphere (see, e.g., Goodman, 1996, 2004; Judge, 2008; Krasnoselskikh et al., 2010; Khomenko and Collados, 2012; Martínez-Sykora et al., 2012). MHD waves, such as fast or Alfvén waves, constantly produce currents across magnetic field lines, and these currents are susceptible to non-linear resistive dissipation, enhanced through ion-neutral interaction (de Pontieu and Haerendel, 1998; Goodman, 2000, 2011a; Kazeminezhad and Goodman, 2006; Goodman and Kazeminezhad, 2010; Shelyag et al., 2016; Khomenko et al., 2018). \nIn order to understand the role of neutrals in the wave plasma heating, the internal energy equation, Eq. 6.15, can be expressed separating different heating terms on the right hand side, \n𝜕𝑒 𝜕𝑡 + ∇ · ( 𝑽 𝑒 ) = 𝑄 𝐽 + 𝑄 𝑐 + 𝑄 visc -𝑄 R - ∇ · q . (6.120) \nwhere 𝑄 𝐽 = 𝜂𝐽 2 + 𝜂 𝐴 𝐽 ⊥ 2 is resistive Joule heating term, 𝑄 𝑐 = -𝑃 ∇· 𝑽 is shock compressional heating term, 𝑄 visc is viscous heating, 𝑄 R is radiative cooling, and the last term is total thermal energy flux. If the initial configuration of the magnetic field is current-free, then 𝑄 𝐽 0 = 0, otherwise, 𝑄 𝐽 0 ≠ 0. The latter value can be used as a reference value to estimate the importance of the resistive wave heating. It is also instructive to compare resistive heating to the other contributions. \nGoodman and Kazeminezhad (2010) have studied propagation of shock wave perturbation developed from an initially sinusoidal fast-mode wave driver in an atmosphere with solar-like stratification (model FAL). In their 1.5D experiment the magnetic field is horizontal, 𝐵 𝑥 ( 𝑧 ) , and it is stratified in the vertical direction due to gravity. The background magnetic field is not current-free. The magnitude \nof the current, and the associated resistive heating produced in the stationary atmosphere is used as a reference value for comparison with the model where waves are driven and develop shocks. The results of the simulation show that, at shock fronts the current density, 𝐽 1 , can exceed by 2-3 orders of magnitude the background current density, 𝐽 0 . The resistive heating rate, 𝑄 𝐽 1 , increases by 4-6 orders of magnitude in shocks relative to the stationary value 𝑄 𝐽 0 . The ratio between the ambipolar and total resistive heating rates in the experiment by Goodman and Kazeminezhad (2010) shows that almost all the resistive heating above 500 km is due to ambipolar diffusion. The compressional heating in shocks can exceed by 2-3 order of magnitude the background heating 𝑄 𝐽 0 . The values of the compressional shock heating are significantly above the resistive one in their experiment at all heights. The height-integrated heating rates differ almost a factor of 300. Therefore, resistive heating by itself stays significantly below the shock compresisonal heating. Similar conclusions were also reached in Arber et al. (2016), who studied 1.5D propagation of a spectrum of Alfvén waves non-linearly coupled to the slow waves, producing shocks. Nevertheless, this conclusion does not mean that the resistive effects are not important. The resistivity defines the width of the shock fronts over which the compression happens. This way it sets the total heating rates through the atmosphere. \nIn the solar chromosphere, the ambipolar diffusion dominates by orders of magnitude the Ohmic diffusion, therefore it can be considered the dominant resistive mechanism. The non-linear action of the ambipolar diffusion on the propagation of the magnetic Poynting flux can be determined by writing the total energy conservation equation as, \n𝜕𝑒 tot 𝜕𝑡 + ∇ · 𝑺 = 0 . (6.121) \nTherefore, according to the divergence theorem, the time variation of the total energy in the closed volume is zero if no energy flux enters/exists through the boundary. The total energy flux, including heat flux and radiative energy flux is given by \n𝑺 = GLYPH<18> 𝜌𝑉 2 2 + 𝑃 + 𝑒 GLYPH<19> 𝑽 + 𝑺 EM + 𝒒 + 𝑭 𝑅 . (6.122) \nThis expression includes the electromagnetic Poynting flux \n𝑺 EM = 𝑬 × 𝑩 𝜇 0 . \nTaking the expression for the electric field from the generalized Ohm's law, Eq. 6.6, (without the battery and diamagnetic terms), \n𝑺 EM = -[ 𝑽 × 𝑩 ] × 𝑩 𝜇 0 + 𝜂 𝑱 × 𝑩 + 𝜂 𝐴 𝑱 ⊥ × 𝑩 = 𝑺 ideal EM + 𝑺 non -ideal EM . (6.123) \ncan be split into the ideal and resistive contributions. This expression shows that, since the total energy must be conserved, if a given amount of the ideal Poynting \n<!-- image --> \nFIGURE 6.11 Left: 3D rendering of the magnetic field lines in a flux tube covering from the photosphere to the chromosphere. Grey surfaces are the locations of plasma 𝛽 = 1. Yellow contours show the locations of the strong currents generated by propagating torsional Alfvén wave with a frequency of 25 mHz. Red contours show the locations of the temperature enhancement of 500 K relative to the ideal case without ambipolar diffusion. Right: Height dependence of the Poynting flux absorption, where 0 means no absorption, and 1 means total absorption. Figure from Shelyag et al. (2016). \n<!-- image --> \nflux is generated at the boundary of a volume (for example by means of a wave perturbation), after propagating though a volume where ambipolar (and Ohmic, to a lower extent) diffusion are acting, the amount of 𝑺 ideal EM will be decreased. The difference between 𝑺 ideal EM at the entrance and exit of the domain remains in the volume and is converted into the thermal energy. \nThe absorption of the Poynting flux has been verified by means of simulations of torsional Alfvén waves propagating along magnetic flux tube from the photosphere to the chromosphere (Shelyag et al., 2016), see Figure 6.11. These simulations show how the locations with enhanced perpendicular current density spatially coincide with locations of the temperature enhancements (left panel). The absorption coefficient of the Poynting flux (right panel), defined as the ration between the Poynting flux in simulations with/without ambipolar diffusion, is a function of height, reaching maximum of about 80% absorption in the chromosphere, where the ambipolar diffusion is the largest. Therefore, heating has been achieved by dissipation of torsional Alfvén waves. In a more complex situations, realistic numerical simulations also demonstrate the ability of the ambipolar diffusion to dissipate into heat incompressible magnetic waves (e.g., Alfvén waves), see Khomenko et al. (2018); González-Morales et al. (2020); Khomenko et al. (2021).", '6.3 WAVES IN TWO-FLUID HYDROGEN PLASMAS': 'Previous sections have shown the importance of the presence of neutral species for the evolution of plasmas in the solar atmosphere. The interaction between \nthe ionized and the neutral species plays a relevant role in processes such as the propagation and the mode transformation of MHD waves, or heating of solar plasma. These conclusions have been obtained through the application of single-fluid models, which assume that there is a strong coupling between all the components of the plasma. Therefore, they are strictly applicable to scenarios where the dynamics of the ionized and the neutral particles have very small differences. \nThe strong-coupling assumption is perfectly valid for the photosphere, the lower regions of the chromosphere, or the cores of solar prominences, where the large densities lead to very high collisional frequencies. But it becomes less accurate, for instance, at the upper layers of the chromosphere or at the transition regions between solar prominences and the corona. There the plasma is more rarefied, and the charged-neutral collision frequencies might not be so much higher than the frequencies of the waves that propagate through those environments. As it was shown by the works of, e.g., Khomenko et al. (2016), Wiehr et al. (2019, 2021) or Zapiór et al. (2022), it is possible to measure drift velocities between the ionized and neutral species of the order of hundreds of meters per second at the edges or at rapidly evolving regions of solar prominences. These results point out the necessity of more general fluid models that allow for a larger ion-neutral decoupling than the single-fluid approach. \nIn the present section, we consider two-fluid hydrogen plasmas, with the charged and neutral components interacting by means of elastic collisions. Using the two-fluid model we reconsider some of the topics studied in the previous section and underline the differences with respect to the single-fluid model. We start by considering the linear regime in order to study the propagation of MHD waves in both homogeneous and stratified partially ionized plasmas. We then discuss the nonlinear regime in the context of plasma heating caused by collisions between different species, and the effects of the charged-neutral interaction on the formation and propagation of shocks in the solar chromosphere.', '6.3.1 Basic equations of linear theory': 'Following the same procedure as in Section 6.2.1, let us assume that each variable 𝒇 is the sum of an equilibrium value, 𝒇 0 , and a small-amplitude perturbation, 𝒇 1 . In addition, let us consider a static background, so there is no equilibrium flows, that is, 𝑽 𝛼, 0 = 0 . Then, applying these assumptions to Eqs. 6.16-6.21, the following set of two-fluid linear equations is obtained: \n𝜕𝜌 n , 1 𝜕𝑡 + 𝜌 n , 0 ∇ · 𝑽 n , 1 + 𝑽 n , 1 · ∇ 𝜌 n , 0 = 0 , (6.124) \n𝜕𝜌 c , 1 𝜕𝑡 + 𝜌 c , 0 ∇ · 𝑽 c , 1 + 𝑽 c , 1 · ∇ 𝜌 c , 0 = 0 , (6.125) \n𝜌 n , 0 𝜕 𝑽 n , 1 𝜕𝑡 = -∇ 𝑃 n , 1 + 𝜌 n , 1 𝒈 + 𝛼 cn GLYPH<0> 𝑽 c , 1 -𝑽 n , 1 GLYPH<1> , (6.126) \n𝜌 c , 0 𝜕 𝑽 c , 1 𝜕𝑡 = -∇ 𝑃 c , 1 -∇ ( 𝑩 0 · 𝑩 1 ) 𝜇 0 + ( 𝑩 0 · ∇) 𝑩 1 𝜇 0 + 𝜌 c , 1 𝒈 -𝛼 cn GLYPH<0> 𝑽 c , 1 -𝑽 n , 1 GLYPH<1> , (6.127) \n𝜕𝑃 n , 1 𝜕𝑡 + 𝛾𝑃 n , 0 ∇ · 𝑽 n , 1 + 𝑽 n , 1 · ∇ 𝑃 n , 0 = 0 , (6.128) \n𝜕𝑃 c , 1 𝜕𝑡 + 𝛾𝑃 c , 0 ∇ · 𝑽 c , 1 + 𝑽 c , 1 · ∇ 𝑃 c , 0 = 0 , (6.129) \n𝜕 𝑩 1 𝜕𝑡 = ( 𝑩 0 · ∇) 𝑽 c , 1 -𝑩 0 ∇ · 𝑽 c , 1 -1 𝜇 0 𝑒𝑛 e ∇ × [(∇ × 𝑩 1 ) × 𝑩 0 ] , (6.130) \nwhere the evolution equations for the energies have been rewritten in terms of the pressure of each fluid (Eq. 6.22), and \n𝛼 cn = 𝜌 c 𝜈 cn = 𝜌 n 𝜈 nc = 𝐾 col 𝜌 c 𝜌 n , (6.131) \nis known as the friction coefficient. The friction coefficient generally depends on the complete densities of the two fluids, that is, the sum of the background and the perturbation values. However, for the linear approximation we assume that it only depends on the background densities, 𝜌 c , 0 and 𝜌 n , 0 . \nThe equations above do not include any assumptions on the properties of the equilibrium state apart from being static. Therefore, they can be applied to homogeneous and inhomogeneous backgrounds, as in the cases of unbounded uniform plasmas or of gravitationally stratified atmospheres, respectively. Both scenarios are explored in the following sections.', '6.3.2 Waves in homogeneous plasmas': 'It has been shown in Section 6.2.2 that MHD waves in homogeneous unbounded plasmas in a single-fluid description can be classified into two main categories: Alfvén waves and magnetoacoustic waves. A multi-fluid description increases the number of waves that are allowed in the system. In the two-fluid model analyzed in this section, the additional waves are related to acoustic modes of the neutral species. Due to a large number of available modes, their study can be cumbersome. It is helpful to make use of the general properties of the modes, which allows to simplify the mathematical description. As it has been shown in Chapter 5 , Alfvén waves are incompressible and propagate vorticity perturbations, while magnetoacoustic waves are compressible and they do not propagate vorticity. Therefore, by choosing an appropriate set of variables (vorticity for Alfvén waves and compressibility for magnetoacoustic waves), the equations that describe the dynamics of both kinds of waves decouple and they can be studied separately.', '6.3.2.1 Alfvén waves': 'This section follows Soler et al. (2013b) to derive the properties of Alfvén waves in the two-fluid homogeneous plasmas. The equilibrium state is given by \na uniform and unbounded partially ionized plasma, unaffected by gravity, and embedded in a uniform and straight magnetic field oriented along the 𝑧 -direction, 𝑩 = 𝐵 𝑧 ˆ 𝑧 . The Hall term is neglected in this section. \nWe perform a Fourier analysis by assuming that the spatial dependence of the perturbations is given by exp GLYPH<0> 𝑖𝑘 𝑥 𝑥 + 𝑖𝑘 𝑦 𝑦 + 𝑖𝑘 𝑧 GLYPH<1> , where 𝑘 𝑥 , 𝑘 𝑦 , and 𝑘 𝑧 are the components of the wavenumber in the 𝑥 -, 𝑦 -, and 𝑧 - directions, respectively. Then, we define the 𝑧 -component of the vorticity of the neutral and charged fluids as \nΓ n = (∇ × 𝑽 n ) · ˆ 𝑧 = 𝑖𝑘 𝑥 𝑉 n , y -𝑖𝑘 𝑦 𝑉 n , x , (6.132) \nΓ c = (∇ × 𝑽 c ) · ˆ 𝑧 = 𝑖𝑘 𝑥 𝑉 c , y -𝑖𝑘 𝑦 𝑉 c , x . (6.133) \nBy applying the rotational operator ( ∇× ) over Eqs. 6.126, 6.127, and 6.130, and combining the resulting expressions, we obtain the following evolution equations for Γ n and Γ c : \n𝜌 n 𝜕 Γ n 𝜕𝑡 + 𝛼 cn Γ n = 𝛼 cn Γ c , (6.134) \n𝜌 c 𝜕 2 Γ c 𝜕𝑡 2 + 𝛼 cn 𝜕 Γ c 𝜕𝑡 + 𝜌 c 𝑘 2 𝑧 𝑐 2 A Γ c = 𝛼 cn 𝜕 Γ n 𝜕𝑡 , (6.135) \nwhere 𝑐 A = 𝐵 𝑧 / √ 𝜇 0 𝜌 c is the Alfvén speed defined through the density of the charged component. \nThen, assuming that the temporal dependence of the perturbations is proportional to exp (-𝑖𝜔𝑡 ) and combining the previous equations, the dispersion relation for Alfvén waves is obtained: \n𝜔 3 + 𝑖 𝛼 cn 𝜌𝜉 c 𝜉 n 𝜔 2 -𝑐 2 A 𝑘 2 𝑧 𝜔 -𝑖 𝛼 cn 𝜌𝜉 n 𝑐 2 A 𝑘 2 𝑧 = 0 , (6.136) \nwhere 𝜌 = 𝜌 c + 𝜌 n , 𝜉 c = 𝜌 c / 𝜌 , and 𝜉 n = 𝜌 n / 𝜌 . The dispersion relation can also be written in terms of the ionization fraction, 𝜒 = 𝜌 n / 𝜌 c , and the neutral-charged collision frequency, 𝜈 nc = 𝛼 cn / 𝜌 n : \n𝜔 3 + 𝑖 ( 1 + 𝜒 ) 𝜈 nc 𝜔 2 -𝑘 2 𝑧 𝑐 2 A 𝜔 -𝑖𝜈 nc 𝑘 2 𝑧 𝑐 2 A = 0 . (6.137) \nTo study the properties of standing Alfvén waves, we solve Eq. 6.137 for a real wavenumber 𝑘 𝑧 and allow for a complex temporal frequency, 𝜔 = 𝜔 R + 𝑖𝜔 I . Since Eq. 6.137 is a cubic equation in 𝜔 , it has three solutions. Exact analytic solutions are too complex to provide any useful information on the physics of these waves. However, some information can be extracted by performing the change of variable 𝜔 = 𝑖𝑠 , which leads to the following expression \n𝑠 3 + ( 1 + 𝜒 ) 𝜈 nc 𝑠 2 + 𝑘 2 𝑧 𝑐 2 A 𝑠 + 𝜈 nc 𝑘 2 𝑧 𝑐 2 A = 0 , (6.138) \nand then computing its polynomial discriminant. Eq. 6.138 is a cubic equation in which all the coefficients are real. Thus, its discriminant is given by: \nΛ = -𝑘 2 𝑧 𝑐 2 A h 4 ( 1 + 𝜒 ) 3 𝜈 4 nc -GLYPH<16> 𝜒 2 + 20 𝜒 -8 GLYPH<17> 𝜈 2 nc 𝑘 2 𝑧 𝑐 2 A + 4 𝑘 4 𝑧 𝑐 4 A i . (6.139) \nEquation 6.138 has one real root and two complex conjugate roots when Λ < 0, a multiple real root when Λ = 0, and three distinct real roots when Λ > 0. The complex roots of Eq. 6.138 correspond to damped oscillatory solutions of Eq. 6.137, while the real solutions of Eq. 6.138 correspond to evanescent solutions of Eq. 6.137, that is, solutions with its real part of the frequency equal to zero. \nIn the absence of collisions ( 𝜈 nc = 0) the discriminant becomes Λ = -4 𝑘 6 𝑧 𝑐 6 A < 0, which means that Eq. 6.138 has one real root and two complex conjugate roots \n𝑠 = ± 𝑖𝑘 𝑧 𝑐 A , 𝑠 = 0 , (6.140) \nwhich correspond to the frequencies \n𝜔 = ± 𝑘 𝑧 𝑐 A , 𝜔 = 0 . (6.141) \nThe two non-zero solutions correspond to the solutions for Alfvén waves in a fully ionized plasma. \nIn a general case with 𝜈 nc ≠ 0, it is possible to find the wavenumbers that satisfy Λ = 0. At those wavenumbers, denoted by 𝑘 -𝑧 and 𝑘 + 𝑧 , the nature of the solutions changes. The values of 𝑘 -𝑧 and 𝑘 + 𝑧 are given by, \n𝑘 ± 𝑧 = 𝜈 nc 𝑐 A " 𝜒 2 + 20 𝜒 -8 8 ( 1 + 𝜒 ) 3 ± 𝜒 1 / 2 ( 𝜒 -8 ) 3 / 2 8 ( 1 + 𝜒 ) 3 # -1 / 2 . (6.142) \nSince 𝑘 𝑧 was assumed real, it can be verified from Eq. 6.142 that there is a minimum value of the ionization fraction that allows for Λ = 0. This minimum value is 𝜒 = 8, which corresponds to the critical values of the wavenumbers 𝑘 + 𝑧 = 𝑘 -𝑧 = 3 √ 3 𝜈 nc / 𝑐 A . For values larger than the minimum ionization fraction, the relation 𝑘 + 𝑧 < 𝑘 -𝑧 is fulfilled. Outside the interval GLYPH<0> 𝑘 + 𝑧 , 𝑘 -𝑧 GLYPH<1> , the discriminant is negative, which means that two of the solutions of the dispersion relation correspond to damped Alfvén waves, while the remaining solution is evanescent. The interval GLYPH<0> 𝑘 + 𝑧 , 𝑘 -𝑧 GLYPH<1> is known as the cut-off region (Kulsrud and Pearce, 1969; Soler et al., 2013b), since for wavenumbers inside this interval Λ > 0, all three roots of Eq. 6.138 are real, and the solutions of Eq. 6.137 are purely imaginary. Therefore, there are no propagating waves in the cut-off region. \nWhen 𝑘 𝑧 > 𝑘 -𝑧 , there is a weak coupling between the charged and neutral fluids and, thus, disturbances in the magnetic field affect only the charged fluid, as if the plasma were fully ionized. Conversely, when 𝑘 𝑧 < 𝑘 + 𝑧 the collisional interaction is strong enough to couple both fluids, so they behave as a single fluid. In the intermediate situation, when 𝑘 𝑧 ∈ GLYPH<0> 𝑘 + 𝑧 , 𝑘 -𝑧 GLYPH<1> , collisions between the charged and the neutral particles efficiently dissipate perturbations in the magnetic field before enough inertia is transferred to the neutral fluid. The result is that oscillations are suppressed inside that interval of wavenumbers. \nAnother path to obtain information on the properties of Alfvén waves in partially ionized plasmas without fully solving the dispersion relation is to consider \nFIGURE 6.12 Normalized wave frequency, 𝜔 𝑅 / 𝑘 𝑧 𝑐 A (top panels), and normalized damping rate, 𝜔 𝐼 / 𝑘 𝑧 𝑐 A (bottom panels), as functions of the normalized collision frequency, 𝜈 nc / 𝑘 𝑧 𝑐 A for the Alfvén waves in partially ionized plasma. Solid and dashed lines represent numerical results of the oscillatory and evanescent modes, respectively. Symbols correspond to analytical approximations. Left panels show the case with 𝜒 = 2 (intermediate plasma ionization), while right panels show the case with 𝜒 = 20 (weak plasma ionization). Shaded areas on the right panels mark the position of the cut-off region. Figure from Soler et al. (2013b). \n<!-- image --> \ncertain approximations. For that, one inserts the expression 𝜔 = 𝜔 R + 𝑖𝜔 I into Eq. 6.137 and assumes that the damping rate is small compared to the wave frequency, | 𝜔 I | ≪ | 𝜔 R | . This leads to the following approximate expressions for 𝜔 R and 𝜔 I for two propagating modes, \n𝜔 𝑅 ≈ ± 𝑘 𝑧 𝑐 A v t 𝑘 2 𝑧 𝑐 2 A + ( 1 + 𝜒 ) 𝜈 2 nc 𝑘 2 𝑧 𝑐 2 A + ( 1 + 𝜒 ) 2 𝜈 2 nc , (6.143) \n𝜔 𝐼 ≈ -𝜒𝜈 nc 2 GLYPH<2> 𝑘 2 𝑧 𝑐 2 A + ( 1 + 𝜒 ) 2 𝜈 2 nc GLYPH<3> 𝑘 2 𝑧 𝑐 2 A . (6.144) \nThe approximate solution for the remaining purely imaginary mode is 𝜔 = 𝑖𝜖 , with \n𝜖 ≈ -𝜈 nc 𝑘 2 𝑧 𝑐 2 A + ( 1 + 𝜒 ) 2 𝜈 2 nc 𝑘 2 𝑧 𝑐 2 A + ( 1 + 𝜒 ) 𝜈 2 nc . (6.145) \nNow, approximate results for some interesting limits can be obtained. For \ninstance, in the weak coupling regime, that is, when 𝜈 nc ≪ 𝑘 𝑧 𝑐 A , we get \n𝜔 𝑅 ≈ ± 𝑘 𝑧 𝑐 A , 𝜔 𝐼 ≈ -𝜒𝜈 nc 2 , 𝜖 ≈ -𝜈 nc , (6.146) \nwhich shows that in this limit the damping of Alfvén waves is independent from the wavenumber of the perturbation. On the other hand, in the strong coupling limit ( 𝜈 nc ≫ 𝑘 𝑧 𝑐 A ), the approximate results are: \n𝜔 𝑅 ≈ ± 𝑘 𝑧 𝑐 A √︁ 1 + 𝜒 , 𝜔 𝐼 ≈ -𝜒 2 ( 1 + 𝜒 ) 2 𝑘 2 𝑧 𝑐 2 A 𝜈 nc , 𝜖 ≈ - ( 1 + 𝜒 ) 𝜈 nc . (6.147) \nIn this regime, 𝜔 R is inversely proportional to the factor √︁ 1 + 𝜒 , which implies that the presence of neutral species in the plasma reduces the frequency of the Alfvén waves in comparison to the fully ionized case. In addition, the damping is proportional to 𝑘 2 𝑧 and, thus, the damping is more efficient for larger wavenumbers (or short wavelengths). \nThe assumption made to derive the previous approximations is not valid for waves with 𝑘 𝑧 ∈ GLYPH<0> 𝑘 + 𝑧 , 𝑘 -𝑧 GLYPH<1> , because in the cut-off region 𝜔 R = 0. The goodness of the approximate expressions can be checked in Fig. 6.12, which shows a comparison between the exact solutions (black lines) and the approximations (red symbols). Figure 6.12 also shows that the cut-off region is only present for large values of 𝜒 (large neutral fraction). Another important conclusion that can be extracted from the bottom panels of Fig. 6.12 is that the damping of the propagating modes due to the charged-neutral collisions is more efficient around the value 𝜈 nc /( 𝑘 𝑧 𝑐 A ) ≈ 1, that is, when the collision and the oscillation frequencies are of the same order of magnitude.', '6.3.2.2 Magneto-acoustic waves': "The same equilibrium conditions as in the previous section are assumed for magneto-acoustic waves. The spatial and temporal dependence of the perturbations are also the same. However, in this case one has to consider compressibility perturbations instead of the vorticity ones (Soler et al., 2013a). The compressibility of the neutral and the charged fluids, respectively, is defined as \nΔ n = ∇ · 𝑽 n = 𝑖𝑘 𝑥 𝑉 n , x + 𝑖𝑘 𝑦 𝑉 n , y + 𝑖𝑘 𝑧 𝑉 n , z , (6.148) \nΔ c = ∇ · 𝑽 c = 𝑖𝑘 𝑥 𝑉 c , x + 𝑖𝑘 𝑦 𝑉 c , y + 𝑖𝑘 𝑧 𝑉 c , z . (6.149) \nThen, one computes the divergence of Eqs. 6.126, 6.127, and 6.130, and combine the resulting expressions with Eqs. 6.124, 6.125, 6.128, and 6.129 to obtain the two following coupled equations for ∇ n and ∇ c : \nGLYPH<16> 𝜔 2 -𝑘 2 𝑐 2 S , n GLYPH<17> Δ n = -𝑖𝜈 nc ( Δ n -Δ c ) , (6.150) \n45 \nGLYPH<16> 𝜔 4 -𝜔 2 𝑘 2 GLYPH<16> 𝑐 2 S , c + 𝑐 2 A GLYPH<17> + 𝑘 2 𝑘 2 𝑧 𝑐 2 A 𝑐 2 S , c GLYPH<17> Δ c = -𝑖𝜈 cn 𝜔 3 ( Δ c -Δ n ) + 𝑖𝜈 cn 𝜔 + 𝑖 ( 𝜈 nc + 𝜈 cn ) 𝑘 2 𝑘 2 𝑧 𝑐 2 A GLYPH<16> 𝑐 2 S , c Δ c -𝑐 2 S , n Δ n GLYPH<17> , (6.151) \nwhere 𝑘 2 = 𝑘 2 𝑥 + 𝑘 2 𝑦 + 𝑘 2 𝑧 , and 𝑐 S , n and 𝑐 S , c are the sound speed of the neutral and the charged fluid, respectively, given by \n𝑐 S , c = √︄ 𝛾𝑃 c , 0 𝜌 c , 0 and 𝑐 S , n = √︄ 𝛾𝑃 n , 0 𝜌 n , 0 . (6.152) \nThe combination of Eqs. 6.150 and 6.151 yields the dispersion relation for magneto-acoustic waves in a partially ionized plasma (Ballester et al., 2018), \nh GLYPH<16> 𝜔 4 + 𝑖𝜈 cn 𝜔 3 -𝑘 2 GLYPH<16> 𝑐 2 A + 𝑐 2 S , c GLYPH<17> 𝜔 2 GLYPH<17> GLYPH<0> 𝜔 + 𝑖 ( 𝜈 cn + 𝜈 nc ) GLYPH<1> + 𝑘 2 𝑘 2 𝑧 𝑐 2 A 𝑐 2 S , c ( 𝜔 + 𝑖𝜈 nc ) i GLYPH<16> 𝜔 2 -𝑘 2 𝑐 2 S , n + 𝑖𝜈 nc 𝜔 GLYPH<17> + 𝜈 cn 𝜈 nc 𝜔 GLYPH<2> 𝜔 3 GLYPH<0> 𝜔 + 𝑖 ( 𝜈 cn + 𝜈 nc ) GLYPH<1> -𝑘 2 𝑘 2 𝑧 𝑐 2 A 𝑐 2 S , n GLYPH<3> = 0 , (6.153) \nwhich is a seventh order equation of 𝜔 , so it has seven different solutions. Due to its complexity, it must be solved numerically. Nevertheless, the study of limiting cases can provide useful information about the nature of the solutions. \nBy neglecting the collisional interaction, 𝜈 nc = 𝜈 cn = 0, the dispersion relation becomes \n𝜔 GLYPH<16> 𝜔 4 -𝑘 2 GLYPH<16> 𝑐 2 A + 𝑐 2 S , c GLYPH<17> 𝜔 2 + 𝑘 2 𝑘 2 𝑧 𝑐 2 A 𝑐 2 S , c GLYPH<17> GLYPH<16> 𝜔 2 -𝑘 2 𝑐 2 S , n GLYPH<17> = 0 . (6.154) \nThis expression shows that four of the seven modes are magneto-acoustic modes related to the charged fluid (forward and backward propagating slow and fast modes), two modes correspond to acoustic waves of the neutral fluid (forward and backward propagating acoustic modes). The remaining solution, with 𝜔 = 0, is the entropy mode (Goedbloed and Poedts, 2004). \nIn the opposite limit, when 𝜈 cn →∞ and 𝜈 nc →∞ (strong coupling between the fluids), the dispersion relation simplifies to, \n𝜔 3 ' › › 𝜔 4 -𝜔 2 𝑘 2 𝑐 2 A + 𝑐 2 S , c + 𝜒𝑐 2 S , n 1 + 𝜒 + 𝑘 4 𝑐 2 A GLYPH<16> 𝑐 2 S , c + 𝜒𝑐 2 S , n GLYPH<17> ( 1 + 𝜒 ) 2 cos 2 𝜃 ' fi fi = 0 , (6.155) \n« \n‹ \nwhere 𝜃 is the angle between the wavevector, 𝒌 , and the equilibrium magnetic field, 𝑩 . There are three entropy modes with 𝜔 = 0. The frequencies of the other four modes are given by \n𝜔 2 = 𝑘 2 𝑐 2 A + 𝑐 2 S , c + 𝜒𝑐 2 S , n 2 ( 1 + 𝜒 ) ± 𝑘 2 𝑐 2 A + 𝑐 2 S , c + 𝜒𝑐 2 S , n 2 ( 1 + 𝜒 ) 1 -4 𝑐 2 A GLYPH<16> 𝑐 2 S , c + 𝜒𝑐 2 S , n GLYPH<17> cos 2 𝜃 GLYPH<16> 𝑐 2 A + 𝑐 2 S , c + 𝜒𝑐 2 S , n GLYPH<17> 2 1 / 2 , (6.156) \nFIGURE 6.13 Real part (left) and absolute value of the imaginary part (right) of frequency of magneto-acoustic waves, as a function of the averaged collision frequency, ˜ 𝜈 ≡ 2 𝛼 cn /( 𝜌 c + 𝜌 n ) , for oblique propagation at an angle 𝜃 = 𝜋 / 4, with 𝛽 c ≡ 𝑐 2 S , c / 𝑐 2 A = 0 . 04. From top to bottom, panels represent the cases with 𝜒 = 0 . 2 (strong ionization), 𝜒 = 2 (intermediate ionization), and 𝜒 = 20 (weak ionization). All frequencies are normalized by 𝑘𝑐 i , with 𝑐 i ≡ 𝑐 S , c , respectively. Figure from Soler et al. (2013a). \n<!-- image --> \nwhere the signs '+/-' correspond to the forward and backward modified fast/slow waves. Here, the term 'modified' is used to underline that these modes have modified properties compared to the fully ionized case. \nIt is interesting to note that in the uncoupled case there are three different kinds of propagating modes (the slow and fast waves of the charged fluid plus the neutral acoustic mode), but in the strongly coupled case there are only two kinds of propagating modes (the modified versions of the slow and fast magnetoacoustic modes). The reason is that, in this limit, the neutral acoustic \nmodehastransformed into an evanescent mode, with 𝜔 = 0. Another remarkable feature of the strong coupling limit is that the wave frequencies are real, meaning the absence of collisional damping, as in the completely uncoupled case. \nIn an intermediate collisional coupling regime, the different modes have mixed properties, which strongly depend on the physical conditions of the two fluids (magnetic field strength, ionization degree) and the direction of propagation with respect to the background magnetic field. Therefore, for an arbitrary value of the collision frequency, 𝜈 nc , there is no simple analytic solution of the dispersion relation and no general trend can be described for each mode. The solutions must be explored by numerically solving Eq. 6.153. \nTo illustrate the numerical solutions that can be obtained from Eq. 6.153 for a particular set of parameters, Fig. 6.13 shows the results for magneto-acoustic waves obliquely propagating in a strongly magnetized plasma (that is, with a small value of plasma 𝛽 ). The top panels of Fig. 6.13 show how, for the case of 𝜒 = 0 . 2 (strong ionization), the slow and fast modes transform into the modified slow and fast modes as the averaged collision frequency, ˜ 𝜈 ≡ 2 𝛼 cn /( 𝜌 c + 𝜌 n ) , increases or how the acoustic mode related to the neutral fluid turns into an entropy mode. However, for an intermediate ionization degree of 𝜒 = 2, the neutral acoustic mode transforms into the modified slow wave as ˜ 𝜈 increases, and the slow magneto-acoustic wave is the one that becomes an entropy mode. Finally, the bottom panels of Fig. 6.13 reveal the presence of cut-off regions of the slow and fast magneto-acoustic modes for the case of weak ionization, 𝜒 = 20. Similar to Section 6.3.2.1, waves within the cut-off regions have 𝜔 R = 0 and they do not propagate.", '6.3.3 Waves in stratified partially-ionized plasmas': "The results from the previous section are applicable to scenarios where the equilibrium state of the plasma is homogeneous. Nevertheless, for the short-period waves (with wavelengths smaller than the pressure scale height), propagation through the stratified atmosphere can be approximately studied by locally solving Eqs. 6.137 and 6.153 for the parameters of the plasma corresponding to every height. For instance, this method has been used by Soler et al. (2013a) to study the propagation of magnetoacoustic waves in the solar chromosphere. They found that fast waves with wavelengths of 1 -10 km are strongly damped due to charged-neutral collisions at heights of 1500 -2000 km above the photosphere, while the damping of slow waves is more important at heights of 1000 -1500 km. \nNevertheless, it has to be kept in mind that local application of the homogeneous dispersion relation is only an approximation. It misses two important effects related to the stratification of the background atmosphere, i.e. the existence of the gravitational cut-off frequency, and the increase of the wave amplitude with height caused by the exponential density fall-off (Mihalas and Mihalas, 1984). Fortunately, these two effects can still be addressed by the linear theory. \nBelow we consider a particular case of how the charge-neutral interaction affects the propagation of fast magneto-acoustic waves in the solar chromosphere, taking also into account the influence of the gravitational stratification, following the work by Popescu Braileanu et al. (2019b). \nThe background atmosphere is gravitationally stratified in the 𝑧 direction, 𝒈 = -𝑔 ˆ 𝒛 . The plasma temperature, and a purely horizontal magnetic field, 𝐵 0 , 𝑥 both vary with height. Applying these physical conditions to Eqs. 6.124-6.130 we get that the linear evolution of fast magneto-acoustic waves is described by the following set of equations: \n𝜕𝜌 n , 1 𝜕𝑡 + 𝜌 n , 0 𝜕𝑉 n , z 𝜕𝑧 + 𝑉 n , z 𝑑𝜌 n , 0 𝑑𝑧 = 0 , (6.157) \n𝜕𝜌 c , 1 𝜕𝑡 + 𝜌 c , 0 𝜕𝑉 c , z 𝜕𝑧 + 𝑉 c , z 𝑑𝜌 c , 0 𝑑𝑧 = 0 , (6.158) \n𝜌 n , 0 𝜕𝑉 n , z 𝜕𝑡 = -𝜌 n , 1 𝑔 -𝜕𝑃 n , 1 𝜕𝑧 + 𝐾 col 𝜌 n , 0 𝜌 c , 0 GLYPH<0> 𝑉 c , z -𝑉 n , z GLYPH<1> , (6.159) \n𝜌 c , 0 𝜕𝑉 c , z 𝜕𝑡 = -𝜌 c , 1 𝑔 -𝜕𝑃 c , 1 𝜕𝑧 -1 𝜇 0 GLYPH<18> 𝜕𝐵 1 , 𝑥 𝜕𝑧 𝐵 0 , 𝑥 + 𝑑𝐵 0 , 𝑥 𝑑𝑧 𝐵 1 , 𝑥 GLYPH<19> + 𝐾 col 𝜌 n , 0 𝜌 c , 0 GLYPH<0> 𝑉 n , z -𝑉 c , z GLYPH<1> , \n(6.160) \n𝜕𝑃 n , 1 𝜕𝑡 = 𝑐 2 S , n 𝜕𝜌 n , 1 𝜕𝑡 + 𝑐 2 S , n 𝑉 n , z 𝑑𝜌 n , 0 𝑑𝑧 -𝑉 n , z 𝑑𝑃 n , 0 𝑑𝑧 , (6.161) \n𝜕𝑃 c , 1 𝜕𝑡 = 𝑐 2 S , c 𝜕𝜌 c , 1 𝜕𝑡 + 𝑐 2 S , c 𝑉 c , z 𝑑𝜌 c , 0 𝑑𝑧 -𝑉 c , z 𝑑𝑃 c , 0 𝑑𝑧 , (6.162) \n𝜕𝐵 1 , 𝑥 𝜕𝑡 = -𝐵 0 , 𝑥 𝜕𝑉 c , z 𝜕𝑧 -𝑉 c , z 𝑑𝐵 0 , 𝑥 𝑑𝑧 . (6.163) \nNotice that the background densities and pressures, 𝜌 n , 0 , 𝜌 c , 0 , 𝑃 n , 0 , and 𝑃 c , 0 , are all functions of height. Consequently, the Alfvén and sound speeds, 𝑐 A , 𝑐 S , n , and 𝑐 S , c , and the collision coefficient 𝐾 col also depend on height. \nThe height variation of the background pressures of the neutral and the charged fluids are given by the conditions of (magneto-)hydrostatic equilibrium, \n𝑑𝑃 n , 0 𝑑𝑧 = -𝑚 H 𝑔 𝑘 B 𝑇 𝑃 n , 0 , (6.164) \n𝑑 GLYPH<0> 𝑃 c , 0 + 𝑃 m , 0 GLYPH<1> 𝑑𝑧 = -𝑚 H 𝑔 2 𝑘 B 𝑇 𝑃 c , 0 , (6.165) \nwhere 𝑃 m , 0 ≡ 𝐵 2 0 , 𝑥 /( 2 𝜇 0 ) is the magnetic pressure. Ideal equation of state has been used. The solution of Eq. 6.164 is \n𝑃 n , 0 = 𝑃 n , 0 ( 𝑧 0 ) exp GLYPH<18> -𝑚 H 𝑔 𝑘 B ∫ 𝑧 𝑧 0 1 𝑇 ( 𝑧 ' ) 𝑑𝑧 ' GLYPH<19> . (6.166) \nEquation 6.165 can be solved by assuming that the pressure of the charged fluid and the magnetic pressure have the same exponential dependence on height, \n𝑃 c , 0 = 𝑃 c , 0 exp (-2 𝐹 ( 𝑧 )) , (6.167) \n𝑃 m , 0 = 𝐵 0 , 𝑥 ( 𝑧 ) 2 2 𝜇 0 = 𝐵 0 , 𝑥 ( 𝑧 0 ) 2 2 𝜇 0 exp (-2 𝐹 ( 𝑧 )) + 𝐶, (6.168) \nwith 𝐶 being an integration constant. By imposing 𝐹 ( 𝑧 ) to satisfy 𝐹 ( 𝑧 ) ≥ 0 and 𝐹 ( 𝑧 0 ) = 0, one obtains: \n𝐹 ( 𝑧 ) = 𝑚 H 𝑔 4 𝑘 B 𝑃 c , 0 ( 𝑧 0 ) 𝑃 c , 0 ( 𝑧 0 ) + 𝐵 0 , 𝑥 ( 𝑧 0 ) 2 /( 2 𝜇 0 ) ∫ 𝑧 𝑧 0 1 𝑇 ( 𝑧 ' ) 𝑑𝑧 ' . (6.169) \nGiven the equilibrium conditions, linearized Eqs. 6.157-6.163 can be combined into two coupled differential equations for the vertical velocity component of each fluid: \n𝜕 2 𝑉 n , z 𝜕𝑡 2 = 𝑎 n ( 𝑧 ) 𝜕 2 𝑉 n , z 𝜕𝑧 2 + 𝑏 n ( 𝑧 ) 𝜕𝑉 n , z 𝜕𝑧 + 𝐾 col 𝜌 c , 0 GLYPH<18> 𝜕𝑉 c , z 𝜕𝑡 -𝜕𝑉 n , z 𝜕𝑡 GLYPH<19> , (6.170) \n𝜕 2 𝑉 c , z 𝜕𝑡 2 = 𝑎 c ( 𝑧 ) 𝜕 2 𝑉 c , z 𝜕𝑧 2 + 𝑏 c ( 𝑧 ) 𝜕𝑉 c , z 𝜕𝑧 + 𝐾 col 𝜌 n , 0 GLYPH<18> 𝜕𝑉 n , z 𝜕𝑡 -𝜕𝑉 c , z 𝜕𝑡 GLYPH<19> , (6.171) \nwhere \n𝑎 c ( 𝑧 ) = 𝑐 2 S , c + 𝑐 2 A , 𝑎 n ( 𝑧 ) = 𝑐 2 S , , (6.172) \n𝑏 c ( 𝑧 ) = 1 𝜌 c , 0 𝑑 GLYPH<0> 𝜌 c , 0 𝑎 c GLYPH<1> 𝑑𝑧 and 𝑏 n ( 𝑧 ) = 1 𝜌 n , 0 𝑑 GLYPH<0> 𝜌 n , 0 GLYPH<1> 𝑑𝑧 . (6.173) \nCombining both velocity equations and assuming the perturbations in the form { 𝑉 c , z ( 𝑧, 𝑡 ) , 𝑉 n , z ( 𝑧, 𝑡 )} = { ˜ 𝑉 c , z ( 𝑧 ) , ˜ 𝑉 n , z ( 𝑧 )} exp (-𝑖𝜔𝑡 ) , the following expression is obtained: \n𝑎 c 𝑎 n 𝜕 4 ˜ 𝑉 c , z 𝜕𝑧 4 + ( 𝑎 n 𝑏 c + 𝑎 c 𝑏 n ) 𝜕 3 ˜ 𝑉 c , z 𝜕𝑧 3 + GLYPH<2> 𝑏 c 𝑏 n + 𝜔 2 ( 𝑎 c + 𝑎 n ) + 𝑖𝐾 col 𝜔 GLYPH<0> 𝑎 c 𝜌 c , 0 + 𝑎 n 𝜌 n , 0 GLYPH<1> GLYPH<3> 𝜕 2 ˜ 𝑉 c , z 𝜕𝑧 2 + 𝜔 GLYPH<2> 𝜔 ( 𝑏 c + 𝑏 n ) + 𝑖𝐾 col GLYPH<0> 𝑏 c 𝜌 c , 0 + 𝑏 n 𝜌 n , 0 GLYPH<1> GLYPH<3> 𝜕 ˜ 𝑉 c , z 𝜕𝑧 + ˜ 𝑉 c , z 𝜔 3 GLYPH<2> 𝜔 + 𝑖𝐾 col GLYPH<0> 𝜌 c , 0 + 𝜌 n , 0 GLYPH<1> GLYPH<3> = 0 , (6.174) \nwhich is a fourth-order differential equation with non-uniform coefficients. An approximate solution for this equation can be obtained by taking into account that the waves considered in this analysis are short-period waves. Therefore, a WKB approximation can be applied to this equation by assuming that the dependence of the velocity perturbations on height is \n{ ˜ 𝑉 n , z , ˜ 𝑉 c , z } = { 𝑈 n ( 𝑧 ) , 𝑈 c ( 𝑧 )} · exp [ 𝑖𝜙 ( 𝑧 )] , (6.175) \nwith \n𝜙 ( 𝑧 ) = -∫ 𝑧 0 𝑘 ( 𝑧 ' ) 𝑑𝑧 ' . (6.176) \nIn addition, the velocity amplitude and the wavenumber gradients are assumed small and of the same order, so \n𝜕𝜙 𝜕𝑧 = -𝑘 = -𝑘 0 -𝜖 𝑘 1 ( 𝑧 ) and 𝑈 c , n ( 𝑧 ) = 𝑈 c , n , 0 + 𝜖𝑈 c , n , 1 ( 𝑧 ) , (6.177) \nwhere 𝜖 is a small parameter. The second (and higher-order) derivatives of 𝑘 1 ( 𝑧 ) and 𝑈 c , n , 1 are set to zero. \nAfter taking into account Eq. 6.177 to compute the derivatives of ˜ 𝑉 c , z , the following approximate dispersion relation is obtained from Eq. 6.174: \nGLYPH<16> 𝜔 2 -𝑘 2 𝑎 c -𝑖𝑘𝑏 c GLYPH<17> GLYPH<16> 𝜔 2 -𝑘 2 𝑎 n -𝑖𝑘𝑏 n GLYPH<17> -𝑖𝜔𝐾 col 𝜌 0 GLYPH<16> -𝜔 2 + 𝑘 2 𝑎 + 𝑖𝑘𝑏 GLYPH<17> = 0 , (6.178) \nwhere 𝑏 = GLYPH<0> 𝜌 n , 0 𝑏 n + 𝜌 c , 0 𝑏 c GLYPH<1> / 𝜌 0 , 𝑎 = GLYPH<0> 𝜌 n , 0 𝑎 n + 𝜌 c , 0 𝑎 c GLYPH<1> / 𝜌 0 , and 𝜌 0 = 𝜌 c , 0 + 𝜌 n , 0 . Note that in the limit 𝐾 col 𝜌 0 / 𝜔 ≫ 1 (high collision limit) this dispersion relation reduces to the single-fluid expression, -𝜔 2 + 𝑘 2 𝑎 + 𝑖𝑘𝑏 = 0, while in the opposite limit, 𝐾 col 𝜌 0 / 𝜔 ≫ 1, the separate dispersion relations of charges and neutrals are recovered. Furthermore, if the effect of stratification is neglected, that is, if the coefficients 𝑏 c , 𝑏 n , and 𝑏 are set equal to zero, while 𝑎 c , 𝑎 n and 𝑎 are now constant, Eq. 6.178 becomes the dispersion relation for fast magneto-acoustic waves propagating in a uniform atmosphere: \nGLYPH<16> 𝜔 2 -𝑘 2 𝑎 c GLYPH<17> GLYPH<16> 𝜔 2 -𝑘 2 𝑎 n GLYPH<17> + 𝑖𝜔𝐾 col 𝜌 0 GLYPH<16> 𝜔 2 -𝑘 2 𝑎 GLYPH<17> = 0 , (6.179) \nwhich is equivalent to Eq. 6.153 if we set 𝑘 𝑧 = 0 in that equation. Note that in Section 6.3.2.2 the background magnetic field was oriented along the vertical direction while in the present analysis it is purely horizontal. \nThe exact numerical solution of Eq. 6.178 for the case of waves with periods of 5 s propagating in the solar atmosphere is presented in Fig. 6.14. There are four different solutions, corresponding to a fourth order polynomial in 𝑘 . Two of the solutions propagate upwards, with 𝑘 R > 0, and two downwards, with 𝑘 R < 0. One of the solutions propagating in each direction (black and red curves) has very large real and imaginary wavenumbers, meaning that it has a very small propagation speed and a very strong collisional damping. The combined effect of stratification and collisional interaction can be more clearly seen in the remaining modes (yellow and green curves). For instance, it can be seen that the upward propagating mode (green curve) has a positive 𝑘 I up to a height around 1 . 4 Mm, which means that the amplitude of this wave increases as it moves from dense to rarefied regions. However, this wave is also affected by the collisional damping and at a certain height this effect is able to balance the amplitude growth due to stratification. Finally, at higher layers the damping due to collisions dominates and the imaginary part of the wavenumber becomes negative, so the amplitude \nFIGURE 6.14 Solutions of the dispersion relation Eq. 6.178 for waves with a period of 5 s propagating in the solar chromosphere. Upper and lower panels represent the real and the imaginary parts of the wavenumber as functions of height. Figure from Popescu Braileanu et al. (2019b). \n<!-- image --> \nof the perturbation decreases. For the wave propagating downwards ( 𝑘 R < 0, yellow curve), a positive 𝑘 I corresponds to a decrease in the amplitude. In this case, the gravitational stratification also produces a positive imaginary part of the wavenumber because the wave propagates from low density to high density layers. \nAs a next approximation, the derivatives of ˜ 𝑉 c , z and ˜ 𝑉 n , z , computed before, can be inserted into the individual velocity equations, Eqs. 6.170, and 6.171, and after separating the 0th-order terms from the 1st-order terms, the following expressions for the velocity amplitudes can be obtained: \n𝑈 c , n ( 𝑧 ) = 𝑈 c , n , 0 exp GLYPH<18> ∫ 𝑧 0 𝑑𝑘 𝑑𝑧 𝑖𝑎 c , n 𝑏 c , n -2 𝑖𝑘𝑎 c , n 𝑑𝑧 ' GLYPH<19> , (6.180) \nwhich provide the full profile of the velocity perturbations as functions of height. This way, by using Eq. 6.175 and the expression { 𝑉 c , z ( 𝑧, 𝑡 ) , 𝑉 n , z ( 𝑧, 𝑡 )} = { ˜ 𝑉 c , z ( 𝑧 ) , ˜ 𝑉 n , z ( 𝑧 )} exp (-𝑖𝜔𝑡 ) , the full temporal and spatial evolution of the waves can be reconstructed. \nFigure 6.15 shows a comparison between the solutions computed from the \nFIGURE 6.15 Comparison between results from the full numerical solution of the system of linear Eqs. 6.124-6.130 (black lines) and approximate analytical solutions (red dotted lines) for fast magneto-acoustic wave propagating in the solar chromosphere. Panels show snapshots of the velocity of charges, 𝑉 c , z , as a function of height, 𝑧 . Panel from left to right, from top to bottom corresponds to wave periods of 1, 5, 7.5, and 20 seconds, respectively. Figure from Popescu Braileanu et al. (2019b). \n<!-- image --> \ndispersion relation following the procedure described above, and the results of numerically solving the full set of linear equations given by Eqs. 6.124-6.130. It can be seen that the two solutions are in good agreement, except that the approximate solution is not sufficiently precise in the upper layers. Each panel of Fig. 6.15 corresponds to a different period of the wave and it clearly shows how the amplitude of the velocity perturbations increases as the waves propagate upwards until it reaches the height where the collision damping dominates over the growth due to stratification and then the amplitude starts to decrease. Waves with shorter periods are more affected by the dissipation caused by the chargedneutral collisions.", '6.3.4 Plasma heating': 'The linear theory of partially ionized plasmas shows that the charged-neutral collision interaction is a damping mechanism. As magnetohydrodynamic waves propagate, their amplitude reduces due to the charge-neutral collisions and, consequently, the kinetic and magnetic energies carried by those waves also decrease. However, the linear theory does not provide answers on what happens with the energy that is removed. Is it lost? Where does it go? It is transformed', 'into another kind of energy?': 'To provide an answer to these questions, it is necessary to resort to the full non-linear equations presented in Section 6.1.2. According to Eqs. 6.276.30, there is always an increase of the internal energy of each component of the plasma when colliding charged and neutral fluids have different velocities. Therefore, the energy of magnetohydrodynamic waves is transformed into the internal energy, leading to plasma temperature increase, in a process known as frictional plasma heating. \nThe process of frictional heating is frequently invoked in the context of models of chromospheric heating by magnetohydrodynamic waves (Srivastava et al., 2021). For example, it has been considered in the models of dissipation and heating by Alfvén waves by, e.g., Goodman (2011b); Tu and Song (2013); Soler et al. (2015b); Martínez-Gómez et al. (2018). This section follows the approach by Song and Vasyli¯unas (2011) to study the process of plasma heating by Alfvén waves. \nTo study the energy of the whole system, evolution equations of the internal energy of neutrals and charges, Eqs. 6.27 and 6.28, are added up. Here we do not consider processes of thermal conduction, viscosity, ionization and recombination. Thus, the following expression for the total internal energy of the two-fluid plasma is obtained: \n𝜕𝑒 𝜕𝑡 + ∇ · ( 𝑽 n 𝑒 n + 𝑽 c 𝑒 c ) + 𝑃 n ∇ · 𝑽 n + 𝑃 c ∇ · 𝑽 c = 𝑄, (6.181) \nwhere 𝑒 = 𝑒 n + 𝑒 c and 𝑄 is given by \n𝑄 = 𝑱 · ( 𝑬 + 𝑽 c × 𝑩 ) + 𝜈 cn 𝜌 c | 𝑽 c -𝑽 n | 2 . (6.182) \nThe latter term represents the frictional heating of plasma (Vasyli¯unas and Song, 2005). It contains the contribution from Ohmic heating, which is caused by collisions of ions and neutral particles with electrons, and from the frictional heating due to collisions between the charged and neutral species. \nThis form of Eq. 6.182 does not allow yet to link the plasma heating to the parameters of waves. In order to get better insights, Vasyli¯unas and Song (2005) performed several manipulations to the set of two-fluid non-linear equations. By neglecting gravity and pressure gradients, the momentum equations for neutrals and charges, Eq. 6.18 and 6.19, can be approximated as \n𝜌 n 𝜕 𝑽 n 𝜕𝑡 = 𝜌 n 𝜈 nc ( 𝑽 c -𝑽 n ) , (6.183) \n𝜌 c 𝜕 𝑽 c 𝜕𝑡 = 𝑱 × 𝑩 -𝜌 c 𝜈 cn ( 𝑽 c -𝑽 n ) . (6.184) \nAssuming the velocity perturbations to be proportional to exp (-𝑖𝜔𝑡 ) , Eq. 6.183 gives the difference of velocities between the two fluids, \n𝑽 c -𝑽 n = -𝑖𝜔 𝜈 nc 1 1 -𝑖𝜔 / 𝜈 nc 𝑽 c . (6.185) \nFrom Ohm\'s law, Eq. 6.31, the Ohmic heating can be expressed as \n𝑱 ∗ · ( 𝑬 + 𝑽 c × 𝑩 ) = 𝑚 e 𝜈 e | 𝑱 | 2 𝑛 e 𝑒 2 , (6.186) \nwhere the symbol \'*\' denotes the complex conjugate and 𝜈 e = 𝜈 ec + 𝜈 en . Combining Eqs. 6.184 and 6.185, the Lorentz force term becomes, \n𝑱 × 𝑩 = 𝜌 n 𝜈 nc ( 𝑽 c -𝑽 n ) GLYPH<20> 1 + 1 𝜒 -𝑖𝜔 𝜈 nc 1 𝜒 GLYPH<21> . (6.187) \nThis equation is used to compute the product ( 𝑱 × 𝑩 ) · ( 𝑱 × 𝑩 ) ∗ = | 𝑱 | 2 | 𝑩 | 2 (where it has been taken into account that 𝑱 · 𝑩 = 0), and obtain the expression: \n| 𝑱 | 2 = 𝑱 · 𝑱 ∗ = GLYPH<16> 𝜌 n 𝜈 nc 𝐵 GLYPH<17> 2 " GLYPH<18> 1 + 1 𝜒 GLYPH<19> 2 + GLYPH<18> 𝜔 𝜒𝜈 nc GLYPH<19> 2 # | 𝑽 c -𝑽 n | 2 . (6.188) \nBy inserting Eq. 6.188 into Eq. 6.186, the Ohmic heating becomes, \n𝑱 ∗ · ( 𝑬 + 𝑽 c × 𝑩 ) = 𝜈 e 𝜈 cn Ω ce Ω ci " GLYPH<18> 1 + 1 𝜒 GLYPH<19> 2 + GLYPH<18> 𝜔 𝜒𝜈 nc GLYPH<19> 2 # 𝜌 n 𝜈 nc | 𝑽 c -𝑽 n | 2 . (6.189) \nFinally, summing the contributions of the Ohmic and the frictional heating and taking into account that | 𝑽 c -𝑽 n | 2 = ( 𝑽 c -𝑽 n ) · ( 𝑽 c -𝑽 n ) ∗ , the time-averaged total heating rate is given by \n⟨ 𝑄 ⟩ = ( 1 + 𝜈 e 𝜈 cn Ω ce Ω ci " GLYPH<18> 1 + 1 𝜒 GLYPH<19> 2 + GLYPH<18> 𝜔 𝜒𝜈 nc GLYPH<19> 2 #) 𝜔 2 𝜈 nc GLYPH<0> 1 + 𝜔 2 / 𝜈 2 nc GLYPH<1> 𝜌 n ⟨ 𝑽 2 c ⟩ . (6.190) \nIn this expression, Ω ce and Ω ci are the electron and ion cyclotron frequencies. By defining 𝜅 ≡ 𝜈 e 𝜈 /( Ω ce Ω ci ) , with 𝜈 ≡ 𝜈 cn + 𝜈 nc , the heating rate can also be expressed as, \n⟨ 𝑄 ⟩ = 𝜒𝜌𝜔 2 ⟨ 𝑽 2 c ⟩ 𝜈 GLYPH<2> 1 + ( 1 + 𝜒 ) 2 𝜔 2 / 𝜈 2 GLYPH<3> GLYPH<20> 1 + 𝜅 GLYPH<18> 𝜒 + 1 𝜒 GLYPH<19> GLYPH<18> 1 + 𝜔 2 𝜈 2 GLYPH<19> GLYPH<21> . (6.191) \nIt is interesting to check the behavior of the total heating rate for certain ranges of the frequency of the perturbation. For instance, if the oscillation frequency is much lower than the collision frequency, 𝜔 ≪ 𝜈 /( 𝜒 + 1 ) = 𝜈 nc , Eq. 6.191 simplifies to \n⟨ 𝑄 ⟩ = 𝜌𝜔 2 ⟨ 𝑽 2 c ⟩ 𝜈 [ 𝜒 + 𝜅 ( 𝜒 + 1 )] , (6.192) \nFIGURE 6.16 Heating rate divided by mean square velocity fluctuation, ⟨ 𝑄 ⟩/⟨ 𝑽 2 𝑐 ⟩ (in our notation), as a function of wave frequency, 𝜔 , for different heights above the solar surface, and for different ionization degrees. Green, red, and blue lines correspond to the ionization degrees 𝜒 = 1 . 97 × 10 -4 , 9 . 5 × 10 3 and 1 . 49, and heights ℎ = 50 km, 1032 km, and 2024 km, respectively. Figure from Song and Vasyli¯unas (2011). \n<!-- image --> \nshowing that in the limit of strong collision coupling the heating rate has a quadratic dependence on the frequency 𝜔 . In the intermediate frequency range, 𝜈 nc ≪ 𝜔 ≪ 𝜈 , we have \n⟨ 𝑄 ⟩ = 𝜌𝜈 ⟨ 𝑽 2 c ⟩ 1 + 𝜅 𝜒 , (6.193) \nand the heating rate is independent from 𝜔 . Finally, in the high-frequency limit, 𝜔 ≫ 𝜈 , the heating rate becomes, \n⟨ 𝑄 ⟩ = 𝜌𝜔 2 ⟨ 𝑽 2 c ⟩ 𝜈 𝜅 𝜒 + 1 , (6.194) \nand the heating rate recovers its quadratic dependence on 𝜔 . \nThis three different regimes of the heating rate dependence on frequency are clearly visible in Fig. 6.16, where the complete Eq. 6.191 is used. This figure gives the results for three different heights in the solar atmosphere. According to these results, most of the collisional plasma heating occurs at the lower altitudes of the solar chromosphere, with higher-frequency waves producing the largest amounts of heating. Song and Vasyli¯unas (2011) applied Eq. 6.191 to a 1-dimensional model of the solar chromosphere and computed the total frequency-integrated heating rate as 𝑄 tot ( 𝑧 ) = ∫ ∞ 0 ⟨ 𝑄 ( 𝑧, 𝜔 )⟩ 𝑑𝜔 , as a function of 𝑧 . They found values of 𝑄 tot of the order of 10 -1 erg cm -3 s -1 at the lower chromosphere and of the order of 10 -2 erg cm -3 s -1 at the middle and upper chromosphere. These values are consistent with the empirical estimates of the energy input required to balance the radiative losses (Withbroe and Noyes, 1977; Vernazza et al., 1981). Song and Vasyli¯unas (2011) found that, for a given wave energy flux coming from the photosphere, the heating is stronger in regions with \nFIGURE 6.17 Results from numerical simulations of propagation of fast magneto-acoustic waves in the solar chromosphere in the linear regime (black lines) and non-linear regime (red dotted lines). Each panel represents the vertical 𝑧 -component of the velocity of the charged fluid at a fixed time as a function of height above the solar surface for a different wave period ( 𝑃 = 1 , 5 , 7 . 5 , and 20 s). Figure from Popescu Braileanu et al. (2019b). \n<!-- image --> \na weaker magnetic field. The reason is that in those regions the electron cyclotron motion becomes less important in comparison with the effect of collisions and, thus, the Joule heating is enhanced.', '6.3.5 Multi-fluid shocks': 'This chapter further explores the non-linear plasma dynamics under the twofluid approximation and studies the formation and evolution of shocks using the mathematical framework described in Section 6.1.2.', '6.3.5.1 Propagation and dissipation of shocks': 'In order to illustrate the influence of non-linearities, we reconsider the model of fast magneto-acoustic waves propagation in a gravitationally stratified solar chromosphere permeated by a horizontal magnetic field, already discussed in Section 6.3.3, but now for waves with larger amplitudes. \nFigure 6.17 shows the results from numerical simulations performed by Popescu Braileanu et al. (2019b) for different periods of the driver that triggers the propagation of fast magneto-acoustic waves at the bottom of the chromosphere. It includes a comparison between the linear regime, where the set of \nEqs. 6.124-6.130 is solved (same data were already presented in Fig. 6.15), and the non-linear regime, where the full version of the two-fluid equations is considered, Eqs. 6.16-6.21. In both linear and non-linear cases, one can observe a combination of two different effects: on the one hand, the growth of the amplitude due to the gravitational stratification and, on the other hand, the decrease of the amplitude at higher layers caused by the charged-neutral collisional damping. However, the non-linear regime (represented by the red dotted lines) has another feature that is not present in the linear one: the waves do not conserve their sinusoidal shapes but their fronts steepen as they propagate upwards. Due to the large amplitude of the perturbations, the phase speed of waves at the different parts of the wavefront is noticeably modified: crests propagate faster than valleys, and a saw-tooth profile is formed, characteristic for a shock wave. It can be seen that the shock formation process strongly varies with the period of the waves. The saw-tooth profiles appear at lower heights for shorter-period waves, while the longer-period ones remain in the linear regime up to higher layers of the atmosphere. \nThe formation of shocks clearly affects the attenuation of the waves. It enhances the effect of collisional damping, leading to a smaller amplitude of the waves compared to the linear regime, and modifying the height at which the waves reach their largest amplitudes. The reason for this enhancement is that the spatial scale of the shock fronts becomes closer to the mean charge-neutral collision free-path, which is the scale where the largest decoupling occurs and, thus, the damping due to collisions is more efficient, as discussed in Sections 6.3.2.1 and 6.3.2.2. As the waves propagate further upwards, the collisional damping becomes strong enough for the waves to (almost) recover their linear sinusoidal profiles due to the smallness of their amplitudes. This effect can be clearly appreciated at the top right and bottom panels of Fig. 6.17, where saw-tooth profiles appear at the intermediate layers of the atmosphere but are not present at the higher layers. \nSimilar results are obtained for the case of slow magneto-acoustic waves, propagating aligned to the magnetic field, for which the magnetic field only serves as a wave guide (Zhang et al., 2021). Figure 6.18, obtained from twofluid numerical simulations byZhang et al. (2021), shows that the development of shock fronts in acoustic waves is faster for shorter-period waves and that longer-period waves reach much larger amplitudes in the upper layers of the chromosphere. This figure also demonstrates that the dissipation of the wave energy causes an increment of the temperature of the plasma. The rise of temperature starts at lower heights for waves with shorter periods, as indicated by the vertical lines. \nUpto this point, we have not discussed the influence of ionization and recombination on the propagation of waves, although the corresponding terms have been included into the general equations of the two-fluid model, Eqs. 6.23-6.25. Only a few works have addressed this issue under a multi-fluid framework, using a rather simplified model (Maneva et al., 2017; Zhang et al., 2021). The results \n<!-- image --> \nFIGURE 6.18 Snapshots of the temperature fluctuations of charges from the two-fluid simulations of acoustic waves propagation in the solar chromosphere. Left panel: simulations including ionization/recombination and elastic collisions; right panel: only elastic collisions are included. Vertical dotted lines indicate approximate heights at which kinetic energy decay starts. Figure from Zhang et al. (2021). \n<!-- image --> \nof the simulations by Zhang et al. (2021), including ionization and recombination are illustrated at the left panel of Fig. 6.18, and can be compared to the same simulation but without taking these effects into account (right panel). It can be observed that a larger increase of the plasma temperature is obtained when ionization and recombination are not included in the model. According to Zhang et al. (2021), the inclusion of ionization and recombination enhances the decoupling between ions and neutrals and the collisional heating, as a consequence of the modifications of the equilibrium atmosphere conditions. This might seem contradictory in view of the smaller temperature increment achieved in the model with ionization/recombination. However, it has an easy explanation. The process of removing an electron from a neutral particle (ionization) requires a considerable amount of energy. Therefore, a significant part of the collisional heating is used into ionizing the neutral fluid, so there is less energy available to increase the temperature of the plasma.', '6.3.5.2 Formation of shock sub-structure': 'To conclude this section, we briefly discuss another interesting consequence that the charged-neutral interaction has on propagating shocks: the formation of internal sub-structure where each fluid has a very distinct behavior (Snow and Hillier, 2020). This effect cannot be appreciated in the results shown in Figs. 6.17 and 6.18 because it occurs at very small scales and it requires a very good spatial resolution to be properly captured. To study how the charged-neutral interaction modifies the \'internal" structure of the shocks, Snow and Hillier (2019) numerically solved the system of Eqs. 6.16-6.21, without taking into account the processes of ionization or recombination, thermal conduction, or viscosity, and using the ideal version of the induction equation. They considered \nFIGURE 6.19 Snapshots of numerical simulations of stationary shock sub-structure in fully and partially ionized plasmas with 𝛽 = 1 . 0, 𝐵 𝑥 = 0 . 1, and 𝜉 n = 0 . 9. Black lines represent the singlefluid MHD case, red and blue lines represent the neutral and charged components, respectively, of the two-fluid case. Panel a): 𝑥 - component of the velocity. Panel b): 𝑦 -component of magnetic field. Panel c): " 𝑦 -component of the velocity. Panel d): pressure; green line shows the total pressure, 𝑃 n + 𝑃 c . The two-fluid solution is plotted after 2500 collisional times. Adapted from Snow and Hillier (2019). \n<!-- image --> \nthe following initial conditions (given in non-dimensional units) for their highresolution numerical simulations \n𝐵 𝑥 = 0 . 1 , (6.195) \n𝐵 𝑦 = GLYPH<26> -1 . 0 if x > 0 1 . 0 if x < 0 , (6.196) \n𝜌 n = 𝜉 n 𝜌, (6.197) \n𝜌 c = 𝜉 c 𝜌 = ( 1 -𝜉 n ) 𝜌, (6.198) \n𝑃 n = 𝜉 n 𝜉 n + 2 𝜉 c 𝑃 = 𝜉 n 𝜉 n + 2 𝜉 c 𝛽 𝐵 2 0 2 , (6.199) \n𝑃 c = 2 𝜉 c 𝜉 n + 2 𝜉 c 𝑃 = 2 𝜉 c 𝜉 n + 2 𝜉 c 𝛽 𝐵 2 0 2 , (6.200) \nThen, they compared the results for a partially ionized plasma with those for a fully ionized scenario. Figure 6.19 shows the results for the case with 𝛽 = 1 . 0, 𝐵 𝑥 = 0 . 1, and 𝜉 n = 0 . 9. It can be clearly seen that the shock transition for \nthe partially ionized case (red and blue lines) has a larger complexity than the fully ionized one (black lines). The transitions between the pre- and post-shock values of the different variables in the fully ionized case are sharper while for the partially ionized case they occur along larger distances. For instance, the slowmodeshockfront has a discontinuous jump in the MHD simulation but it presents a finite width with more complex sub-structure in the two-fluid simulation. In addition, the collision coupling leads to overshooting (sudden increase) in the neutral velocity and pressure, and to a reversal of the magnetic field across the shock front. The magnetic field reversal is more prominent for larger values of the equilibrium neutral pressure. Although it is not shown in Fig. 6.19, an overshooting is also present in the neutral density, which is accompanied by a decrease in the total Alfvén speed (the Alfvén speed computed taking into account the total density of the plasma). \nThese results are a clear example of how the neutral and the charged fluids have different dynamics at very small spatial scales and how single-fluid models are not able to properly describe the full behavior of the plasma. The use of multi-fluid framework is fundamental for analyzing the small-scale dynamics of partially ionized plasmas.', '6.4 WAVES IN HYDROGEN-HELIUM PLASMAS': 'This section makes a further step in the multi-fluid modeling of solar waves and describes the effects of including Helium, additionally to Hydrogen, into the model. According to Section 6.1.4, using the multi-fluid perspective, the inclusion of Helium can be done in various ways depending on the number of helium ionization states that are considered as separate fluids. Due to the complexity of the hydrogen-helium multi-fluid models, the present section focuses only on the study of Alfvén waves: subsection 6.4.1 focuses on the influence of neutral hydrogen and helium on the damping of Alfvén waves in the solar chromosphere, while subsection 6.4.2 analyses the effect of collisions on resonances and cut-off regions related to the presence of multiple ionized species in the plasma.', '6.4.1 Three-fluid model': "Following Zaqarashvili et al. (2011), we consider a plasma composed by three different fluids: charges, neutral hydrogen and neutral helium, and apply linearization to the set of Eqs. 6.33-6.44. Taking into account that Alfvén waves are incompressible, the following condition must be fulfilled for all fluids, \n∇ · 𝑽 s = 0 for 𝑠 ∈ { H , He , c } . (6.201) \nThis condition simplifies the set of equations as there is no need to include the continuity and pressure equations into consideration. \nLet us assume that the equilibrium state is static and homogeneous, the background magnetic field is oriented along the 𝑧 -direction, 𝑩 0 = ( 0 , 0 , 𝐵 0 ,𝑧 ) 𝑡 \nand the waves propagate in 𝑥 -𝑧 -plane, so that 𝜕 / 𝜕𝑦 = 0, and they are polarized along the 𝑦 -direction. The resulting set of linear equations for Alfvén waves in a hydrogen-helium plasma is given by \n𝜕𝑉 c , y 𝜕𝑡 = 𝐵 0 ,𝑧 𝜇 0 𝜌 c , 0 𝜕𝐵 1 , 𝑦 𝜕𝑧 -𝛼 H + 𝛼 He 𝜌 c , 0 𝑉 c , y + 𝛼 H 𝜌 c , 0 𝑉 H , y + 𝛼 He 𝜌 c , 0 𝑉 He , y , (6.202) \n𝜕𝑉 H , y 𝜕𝑡 = 𝛼 H 𝜌 H , 0 𝑉 c , y -𝛼 H + 𝛼 HeH 𝜌 H , 0 𝑉 H , y + 𝛼 HeH 𝜌 H , 0 𝑉 He , y , (6.203) \n𝜕𝑉 He , y 𝜕𝑡 = 𝛼 He 𝜌 He , 0 𝑉 c , y -𝛼 He + 𝛼 HeH 𝜌 He , 0 𝑉 He , y + 𝛼 HeH 𝜌 He , 0 𝑉 H , y , (6.204) \n𝜕𝐵 1 , 𝑦 𝜕𝑡 = 𝐵 0 ,𝑧 𝜕𝑉 c , y 𝜕𝑡 , (6.205) \nwhere 𝑉 c , y , 𝑉 H , y , and 𝑉 He , y are the velocity perturbations of the charged fluid, the neutral hydrogen and neutral helium, respectively, 𝐵 1 , 𝑦 is the perturbation of magnetic field, and 𝜌 c , 0 , 𝜌 H , 0 , and 𝜌 He , 0 are the equilibrium values of densities. In addition, the following definitions have been used: \n𝛼 H = 𝛼 pH + 𝛼 HeIIH , 𝛼 He = 𝛼 pHe + 𝛼 HeIIHe . (6.206) \nAccording to the original work by Zaqarashvili et al. (2011), collisions between neutral hydrogen and neutral helium in solar prominence and spicule conditions do not have a strong effect on the propagation of Alfvén waves, since 𝛼 HeH / 𝛼 pHe ∼ 𝑛 H / 𝑛 p < 1. Therefore, the terms proportional to 𝛼 HeH can be dropped from Eqs. 6.202-6.205. \nA dispersion relation can be obtained by performing a Fourier and normal mode analysis, assuming the perturbations proportional to exp [ 𝑖 ( 𝑘 𝑧 𝑧 -𝜔𝑡 )] . Then, the combination of Eqs. 6.202-6.205 produces the following expression: \n𝜔 4 + 𝑖 GLYPH<18> 𝛼 H 𝜌 H + 𝛼 He 𝜌 He + 𝛼 H + 𝛼 He 𝜌 c GLYPH<19> 𝜔 3 -GLYPH<18> 𝑘 2 𝑧 𝑐 2 A + 𝛼 H 𝛼 He 𝜌 𝜌 H 𝜌 He 𝜌 c GLYPH<19> 𝜔 2 -𝑖𝑘 2 𝑧 𝑐 2 A GLYPH<18> 𝛼 H 𝜌 H + 𝛼 He 𝜌 He GLYPH<19> 𝜔 + 𝛼 H 𝛼 He 𝜌 H 𝜌 He 𝑘 2 𝑧 𝑐 2 A = 0 , (6.207) \nwhere 𝜌 = 𝜌 H + 𝜌 He + 𝜌 c is the total density of the plasma (note that, for simplicity, the subscript '0' has been dropped from the symbols for the equilibrium densities). This dispersion relation is a fourth order polynomial in the frequency 𝜔 , which means that it has four different roots. Two of these roots correspond to Alfvén waves damped by the collisions between the three different species of the plasma. In addition there are two purely imaginary solutions, which correspond to evanescent modes associated with the neutral hydrogen and neutral helium. \nTo study the effect of helium of the propagation of Alfvén waves, it is illustrative to compare the predictions from Eq. 6.207 with those obtained in the absence of helium. The dispersion relation describing the properties of Alfvén \nFIGURE 6.20 Normalized damping rate (imaginary part of the frequency) of Alfvén waves, 𝜛 𝐼 = 𝜔 𝐼 /( 𝑘 𝑧 𝑐 A ) as a function of the Alfvén frequency, normalized by the hydrogen collision frequency, 𝑎 = 𝑘 𝑧 𝑐 A / 𝜈 H , for four different sets of plasma temperatures and number densities. Blue crosses correspond to the damping rates due to ion collision with neutral hydrogen only, while red asterisks include the effect of collisions with neutral helium particles. Red (blue) solid lines represent the damping rates derived in the single-fluid approach with (without) neutral helium. Figure from Zaqarashvili et al. (2011). \n<!-- image --> \nwaves in a hydrogen-only two-fluid plasma can be obtained from Eq. 6.207 by setting 𝛼 He = 0, \n𝜔 3 + 𝑖 𝛼 H 𝜌 H GLYPH<18> 1 + 𝜌 H 𝜌 c GLYPH<19> 𝜔 2 -𝑘 2 𝑧 𝑐 2 A -𝑖 𝛼 H 𝜌 H 𝑘 2 𝑧 𝑐 2 A = 0 , (6.208) \nwhich is equivalent to Eqs. 6.136 or 6.137. The comparison of the damping rates of Alfvén waves computed from Eqs. 6.207 and 6.208 is presented in Fig. 6.20. Each panel corresponds to a different set of plasma conditions (temperature and number densities), which represents a certain height in the solar chromosphere. The main conclusion that can be extracted from Fig. 6.20 is that, in general, the presence of neutral helium in the chromospheric plasma enhance the damping of Alfvén waves caused by elastic collisions. This effect is larger for frequencies closer to charge-neutral collision frequencies.", '6.4.2 Five-fluid model': 'Below, the five-fluid model detailed in Section 6.1.4.2 is used to investigate the properties of Alfvén waves. In this model, all possible ionization states of hydrogen and helium are taken into account and treated as separate fluids. We follow the procedure described by Martínez-Gómez et al. (2017). \nAs a first step, the expression for the electric field, Eq. 6.51, is introduced into the momentum equation, Eq. 6.46, and in the Faraday\'s law to get the induction equation. Again, since we are only interested in Alfvén waves, we consider the condition of incompressibility, ∇ · 𝑽 s = 0 for 𝑠 ∈ { e , p , H , He , HeII , HeIII } . Therefore, the terms related to the pressure gradients in the momentum equations are neglected, and there is no need to consider the continuity and pressure equations, Eqs. 6.45 and 6.47. \nAfter assuming a uniform and static background, the linear equation for the temporal evolution of the velocity perturbation of each ionization state 𝑠 ∈ { p , H , He , HeII , HeIII } becomes, \n𝜕 𝑽 s 𝜕𝑡 = 𝑍 s 𝑒 𝑚 s GLYPH<20> ( 𝑽 s -𝑽 𝑖 ) × 𝑩 0 + (∇ × 𝑩 1 ) × 𝑩 0 𝑒𝑛 e 𝜇 0 + 𝜂𝜇 0 𝑱 1 GLYPH<21> + 𝑍 s 𝑛 e 𝑚 s ∑︁ 𝑡 ≠ 𝑒 𝛼 et ( 𝑽 t -𝑽 𝑖 ) + ∑︁ 𝑡 ≠ 𝑠 𝜈 st ( 𝑽 t -𝑽 s ) , (6.209) \nwhere the presence of gravity has been neglected, 𝒈 = 0. The weighted mean velocity of ions, 𝑽 𝑖 , and the resistivity, 𝜂 , are given by \n𝑽 𝑖 = 𝑍 p 𝑛 p 𝑽 p + 𝑍 HeII 𝑛 HeII 𝑽 HeII + 𝑍 HeIII 𝑛 HeIII 𝑽 HeIII 𝑛 e , (6.210) \n𝜂 = 𝛼 ep + 𝛼 eH + 𝛼 eHe + 𝛼 eHeII + 𝛼 eHeIII 𝑒 2 𝑛 2 e 𝜇 0 . (6.211) \nThe linear induction equation is \n𝜕 𝑩 1 𝜕𝑡 = ∇ × " 𝑽 𝑖 × 𝑩 0 -(∇ × 𝑩 1 ) × 𝑩 0 𝑒𝑛 e 𝜇 0 -𝜂𝜇 0 𝑱 1 -1 𝑒𝑛 e ∑︁ 𝑡 ≠ 𝑒 𝛼 et ( 𝑽 t -𝑽 𝑖 ) # . (6.212) \nThe last term in Eq. 6.209 contains a dependence on the velocity of electrons, 𝑽 e but this model does not include an evolution equation for 𝑽 e . To remove that explicit dependence on 𝑽 e , one can rewrite this collision term as \n∑︁ 𝑡 ≠ 𝑠 𝜈 st ( 𝑽 t -𝑽 s ) = ∑︁ 𝑡 ≠ 𝑠,𝑒 𝜈 𝑠𝑡 ( 𝑽 s -𝑽 s ) + 𝜈 se ( 𝑽 e -𝑽 s ) , (6.213) \nand then apply Eq. 6.49 so that 𝑽 e = 𝑽 𝑖 -𝑱 1 /( 𝑒𝑛 e ) . Finally, applying Ampère\'s law one gets the following expression \n∑︁ 𝑡 ≠ 𝑠 𝜈 st ( 𝑽 s -𝑽 t ) = ∑︁ 𝑡 ≠ 𝑠,𝑒 𝜈 st ( 𝑽 t -𝑽 s ) + 𝜈 se ( 𝑽 𝑖 -𝑽 s ) -𝜈 se ∇ × 𝑩 1 𝑒𝑛 e 𝜇 0 , (6.214) \nwhich shows that, when resistivity is considered, even neutral species are affected by the magnetic field due to their coupling with electrons. \nIn order to derive the dispersion relation for Alfvén waves in this multi-fluid approach, one assumes that the background magnetic field is oriented along the 𝑥 -direction, that is 𝑩 0 = { 𝐵 𝑥 , 0 , 0 } , and that the waves propagate along the same direction, so 𝒌 = { 𝑘 𝑥 , 0 , 0 } . After performing normal mode analysis and Fourier analysis in space, with perturbations proportional to exp (-𝑖𝜔𝑡 + 𝑖 𝒌 · 𝒓 ) , one arrives to a set of twelve equations where the 𝑦 -and 𝑧 -components are coupled, unlike the case of two-fluid ideal Alfvén waves. The consideration of additional equations for ions causes Alfvén waves to be circularly polarized, instead of linearly polarized. \nTo study the properties of circularly polarized waves it is useful to define the following variables: \n𝑉 s , ± = 𝑉 s , y ± 𝑖𝑉 s , z , 𝐵 1 , ± = 𝐵 1 , 𝑦 ± 𝑖𝐵 1 ,𝑧 , (6.215) \nwhere the \'+/-\' sign corresponds to the left/right-hand polarization ( 𝐿 / 𝑅 modes). Through this step, a system of twelve coupled equations is transformed into two independent systems of six equations, which are considerably easier to manage, and provide a clearer view on the properties of the waves. The resulting equations for the velocity and magnetic field perturbations are, \n𝜔𝑉 s , ± = Ω s GLYPH<2> ± GLYPH<0> 𝑉 s , ± -𝑉 𝑖, ± GLYPH<1> -𝑘 𝑥 𝜂 𝐻 𝐵 1 , ± GLYPH<3> ∓ 𝑖𝜂 𝑍 s 𝑒𝑘 𝑥 𝑚 s 𝐵 1 , ± + 𝑖 𝑍 s 𝑛 e 𝑚 s ∑︁ 𝑡 ≠ 𝑒 𝛼 et GLYPH<0> 𝑉 t , ± -𝑽 𝑖, ± GLYPH<1> + 𝑖 ∑︁ 𝑡 ≠ 𝑠,𝑒 𝜈 st GLYPH<0> 𝑉 t , ± -𝑉 s , ± GLYPH<1> + 𝑖𝜈 se GLYPH<0> 𝑉 𝑖, ± -𝑉 s , ± GLYPH<1> ± 𝑖𝜈 se 𝑘 𝑥 𝜂 𝐻 𝐵 1 , ± , (6.216) \n𝜔𝐵 1 , ± = -𝑘 𝑥 𝐵 𝑥 𝑉 ± -GLYPH<2> 𝑖𝜂𝑘 2 𝑥 ± 𝑘 2 𝑥 𝐵 𝑥 𝜂 𝐻 GLYPH<3> 𝐵 1 , ± ± 𝑖 𝑘 𝑥 𝑒𝑛 e ∑︁ 𝑡 ≠ 𝑒 𝛼 et GLYPH<0> 𝑉 t , ± -𝑉 ± GLYPH<1> , (6.217) \nwhere Ω s = 𝑍 s 𝑒𝐵 𝑥 / 𝑚 s is the cyclotron frequency of species 𝑠 . \nThe dispersion relation for each polarization can be obtained by combining Eqs. 6.216 and 6.217. However, due to the large number of equations, it is easier to express the systems in a matrix form, and solve the matrix equations, \n𝐴 ± · 𝒖 ± = 0 , (6.218) \nFIGURE 6.21 Real part (top) and imaginary part (bottom) of wavenumber for Alfvén waves excited by a periodic driver as functions of the driver frequency, 𝜔 , normalized by the proton cyclotron frequency, Ω ci (in our notation). Left panels: solutions neglecting charged-neutral collisions (black lines), full solution (red lines). Right panels: solutions neglecting resistivity, i.e. collisions with electrons (black lines), full solution (red lines). The diagonal black lines is Alfvén frequency, 𝜔 A = 𝑘 𝑥 𝑐 𝐴 . The dotted vertical lines show the positions of the frequencies of cyclotron resonances, Ω p > Ω HeIII > Ω HeII . The labels 𝐿 and 𝑅 refer to left-handed and right-handed polarization. Figure from Martínez-Gómez et al. (2017). \n<!-- image --> \nwhere 𝒖 ± = { 𝑉 p , ± , 𝑉 HeII , ± , 𝑉 HeIII , ± , 𝑉 H , ± , 𝑉 He , ± , 𝐵 1 , ± } are the vectors of unknowns and 𝐴 ± are the matrices of coefficients, which contain the dependence on the frequency, 𝜔 , the wavenumber, 𝑘 𝑥 , and the parameters of the plasma, see the original work by Martínez-Gómez et al. (2017). The solution of Eq. 6.218 is given by the characteristic equation of each matrix: \nD ± ( 𝜔, 𝑘 𝑥 ) ≡ det 𝐴 ± = 0 . (6.219) \nThe resulting dispersion relations are too complex to be shown here and exact analytical solutions are not practical. The exact solutions, discussed below, have been obtained numerically. Nevertheless, some information can yet be extracted by taking into account the general structure of the dispersion relations and the results from previous sections. \nThe dispersion relation for each polarization state is a convoluted polynomial of sixth order in 𝜔 ± , but only of the second order in 𝑘 𝑥, ± . Therefore, in contrast with the case of Alfvén waves in a single-fluid plasma, there is a clear asymmetry in the number of solutions depending on whether the wave driver is \nFIGURE 6.22 Snapshots from simulations of Alfvén waves generated by a periodic driver in a region with physical conditions of higher chromosphere. Panels a-d show the results for wave frequencies, 𝜔 = { 10 -4 , 10 -3 , 10 -2 , 10 -1 } Ω p . Red solid, blue dashed, and green dashed-dotted lines represent the 𝑦 -component of the velocity of protons, neutral hydrogen, and neutral helium. Red diamonds and the black crosses, represent the velocity for the singly and doubly ionized helium. The vertical dotted and dashed-dotted lines show the position of points moving at the Alfvén speed, and the modified Alfvén speed. The curved black dotted lines represent the collision damping predicted by the dispersion relation, Eq. 6.219. Figure from Martínez-Gómez et al. (2017). \n<!-- image --> \nimpulsive or periodic. When the dispersion relation is solved as a function of a real wavenumber, 𝑘 𝑥 (impulsive driver), Eq. 6.219 has six different complex solutions for each polarization, 𝜔 ± = 𝜔 R , ± + 𝑖𝜔 I , ± . For low wavenumbers, two of the modes correspond to the forward and backward Alfvén waves. Other two modes are related to the presence of the two neutral species (H and He) and, although 𝜔 R , ± is not exactly zero, they have very low frequencies. The remaining solutions are high-frequency modes associated with the cyclotron motions of the \nions (see Martínez-Gómez et al., 2016, 2017). For high wavenumbers, one of the Alfvén waves turns into a whistler mode (with a frequency that keeps increasing with the wavenumber) while the other one becomes an ion-cyclotron wave, with a frequency that tends to the cyclotron frequency of singly-ionized helium, Ω HeII ) , (Cramer, 2001). The nature of the neutral-related modes hardly changes, while the frequency of the two remaining solutions tends to the cyclotron frequency of the remaining ions, Ω HeIII and Ω p . Thus, in this high-wavenumber range, there is one ion-cyclotron mode for each ionized fluid. All the modes are damped due to collisions between the different species. The effect is more pronounced for the high-frequency modes, such as the whistler and ion-cylotron waves. \nWhen a periodic driver is considered, and Eq. 6.219 is solved as a function of a real frequency, 𝜔 , each polarization has only two possible complex solutions, 𝑘 𝑥, ± = 𝑘 R , ± + 𝑖𝑘 I , ± . Figure 6.21 shows an example of this scenario, where the solutions have been numerically obtained for plasma parameters corresponding to an upper chromospheric region. In a multi-ion plasma with no collisions between its components, the left-hand polarized or ion-cyclotron modes have resonances when the frequency of the driver is equal to any of the cyclotron frequencies (Cramer, 2001). It means that the wavenumber tends to infinity at those frequencies, leading to a zero phase speed, so the wave does not propagate. Thus, the effect of the driver is to increase the amplitude of the fluctuations of the ion fluid associated with the given cyclotron frequency at the location where the driver is applied. Furthermore, beyond each resonance, there is a cut-off region where the waves become evanescent and then a small region where the propagation is again allowed. As it can be seen at the top panels of Fig. 6.21, when collisions are taken into account, the wavenumber of the ion-cyclotron modes (denoted by the tag 𝐿 ) remains finite at the cyclotron frequencies, i.e., at 𝜔 = Ω HeII , 𝜔 = Ω HeIII and 𝜔 = Ω p . In addition, there is no evidence of the presence of cut-off regions where 𝑘 R = 0. Therefore, it can be concluded that, one of the important effects of elastic collisions is to remove the cyclotron resonances and cut-off regions, i.e., waves are allowed to propagate for any driver frequency. However, the ion-cyclotron modes are still strongly damped by the collisional interaction, as shown in the bottom panels of Fig. 6.21. \nAs a final remark for this section, it is interesting to briefly abandon the analysis of the dispersion relations and come back to the full set of equations presented in Section 6.1.4.2. Figure 6.22 shows the results from several numerical simulations of Alfvén waves generated by a periodic driver. Each panel corresponds to a different frequency of the driver, 𝜔 , and the different lines and symbols represent the perturbation of velocity of each of the five fluids of the plasma. It is clear how all the fluids are strongly coupled for a low driver frequency and how their behavior start to decouple as the frequency of the driver is increased. Figure 6.22 also shows that the amplitude of the waves decreases due to elastic collisions as they propagate, in good agreement with the predictions obtained from Eq. 6.219.', '6.5 SUMMARY AND OUTLOOK': 'This chapter focuses on describing the properties of waves in partially ionized plasmas with a different degree of collisional coupling between the plasma and neutral components. It describes the change of the physics of waves, compared to those in a classical ideal MHD case. It is shown in what way both linear and non-linear wave solutions are affected by plasma partial ionization, and how the gravitational stratification adds an additional flavor for waves propagating in a stratified solar atmosphere. The content of this chapter can be summarized in the following way: \n- · The complexity of the mathematical description of waves in partially ionized plasmas depends on the degree of collisional coupling. In general terms, the photosphere and low chromosphere, and in the dense cores of solar prominences, for waves with typical solar frequencies, a single-fluid approximation can be applied. In the middle-upper chromosphere and in the transition layer between prominences and corona, low collisional frequencies made multifluid approach to be necessary.\n- · In the single-fluid approximation, the number of MHD modes in the system remains unchanged compared to ideal MHD and usual fast, slow and Alfvén modes can still be distinguished, but with modified properties. The main partial ionization effects that influence waves in this approximation are the ambipolar diffusion, modified Hall effect, and diamagnetic effect.\n- · Fast and Alfvén waves can be significantly damped by ambipolar diffusion, while the slow mode is only weakly affected by this mechanism. Nevertheless, mechanisms such as radiative cooling and thermal conduction can be of the same importance for the damping of the fast and slow modes, depending on their wavenumber. The Alfvén mode is unaffected by the latter.\n- · In certain conditions, cutoff wavenumbers appear for the fast and Alfvén waves due to ambipolar diffusion, a similar effect to that of the Ohmic diffusion. However, once the Hall effect is taken into account, the hard cutoff disappears due to the distinct behavior of ions and electrons in this case.\n- · Modeling of fast and Alfvén waves in a gravitationally stratified atmosphere in a single fluid approximation revealed that the ambipolar diffusion produces significant damping of waves with periods of 1-10 s in the upper chromosphere. The damping is stronger for the shorter period waves.\n- · Mode conversion in a stratified atmosphere is affected by the ambipolar and the Hall effects. The ambipolar diffusion reduces the amount of the energy going into the converted Alfvén wave since the fast wave energy flux in the conversion region is reduced. The Hall effect enables a new mechanism of mode transformation by coupling the fast and Alfvén waves through the so-called Hall window, typically located in the photosphere.\n- · Resistive heating in shocks due to ambipolar diffusion can be an efficient chromospheric heating mechanism. Nevertheless, 1D computations show so far that compressional heating in shocks overcomes the efficiency of the \nresistive heating. Resistivity plays an important role by defining the width of the shock front over which the compression happens. \n- · In the multi-fluid case, additional wave modes exist, compared to the usual MHD modes. These modes are related to the acoustic modes of the neutral fluid(s), and to the whistler modes and ion-cyclotron modes of the ionized fluid(s).\n- · In the multi-fluid approximation, all the modes can be damped due to ionneutral collisions, but to a different degree. In general, it holds that fast and Alfvén waves are damped more. Similarly, in the two-fluid approximation, both Alfvén and magneto-acoustic modes show the presence of cut-off regions, where the waves cannot propagate, affected by inter-particle collisions.\n- · Fast and slow magneto-acoustic waves propagating in a gravitationally stratifiedatmosphere are affected by two competing effects: wave amplitude growth due to the stratification, and wave amplitude damping due to ion-neutral collisions in the two-fluid approximation. These competing effects extend the range of wave periods affected by ion-neutral effects up to 20-30 s. Since the wave with larger periods are able to propagate relatively undamped to the higher chromosphere, they suffer more significant ion-neutral decoupling, and consequently, produce more significant frictional heating.\n- · Frictional heating produces regimes where the heating rates scale either as a square of the wave frequency or linear with the wave frequency. 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2024JCAP...10..004A | Galaxy clusters are expected to be both dark matter DM reservoirs and storage rooms for the cosmicray protons CRp that accumulate along the clusters formation history. Accordingly they are excellent targets to search for signals of DM annihilation and decay at ray energies and are predicted to be sources of largescale ray emission due to hadronic interactions in the intracluster medium ICM. In this paper we estimate the sensitivity of the Cherenkov Telescope Array CTA to detect diffuse ray emission from the Perseus galaxy cluster. We first perform a detailed spatial and spectral modelling of the expected signal for both the DM and the CRp components. For each case we compute the expected CTA sensitivity accounting for the CTA instrument response functions. The CTA observing strategy of the Perseus cluster is also discussed. In the absence of a diffuse signal nondetection CTA should constrain the CRp to thermal energy ratio X SUB500SUB within the characteristic radius R SUB500SUB down to about X SUB500SUB lt 3 10SUP3SUP for a spatial CRp distribution that follows the thermal gas and a CRp spectral index SUBCRpSUB 2.3. Under the optimistic assumption of a pure hadronic origin of the Perseus radio minihalo and depending on the assumed magnetic field profile CTA should measure SUBCRpSUB down to about SUBCRpSUB 0.1 and the CRp spatial distribution with 10 precision respectively. Regarding DM CTA should improve the current groundbased ray DM limits from clusters observations on the velocityaveraged annihilation crosssection by a factor of up to 5 depending on the modelling of DM halo substructure. In the case of decay of DM particles CTA will explore a new region of the parameter space reaching models with SUBSUB gt 10SUP27SUP s for DM masses above 1 TeV. These constraints will provide unprecedented sensitivity to the physics of both CRp acceleration and transport at cluster scale and to TeV DM particle models especially in the decay scenario. | 2024-10-01T00:00:00Z | ['10.1088/1475-7516/2024/10/004', '2023arXiv230903712C', 'arXiv:2309.03712', '2023arXiv230903712T', '10.48550/arXiv.2309.03712', '2024JCAP...10..004A'] | ['cosmic ray experiments', 'dark matter experiments', 'galaxy clusters', 'gamma ray experiments', 'Astrophysics - High Energy Astrophysical Phenomena', 'Astrophysics - Cosmology and Nongalactic Astrophysics', 'High Energy Physics - Phenomenology'] | Prospects for ray observations of the Perseus galaxy cluster with the Cherenkov Telescope Array | 2,024 | 169 | 0.61 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML'] | 6 | https://arxiv.org/pdf/2309.03712.pdf | {'Prospects for γ -ray observations of the Perseus galaxy cluster with the Cherenkov Telescope Array': "- K. Abe 1 , S. Abe 2 , F. Acero 3 , 4 , A. Acharyya 5 , R. Adam 6 , 7 , ∗ ,\n- A. Aguasca-Cabot 8 , I. Agudo 9 , A. Aguirre-Santaella 10 ,\n- J. Alfaro 11 , R. Alfaro 12 , N. Alvarez-Crespo 13 , R. Alves Batista 10 ,\n- J.-P. Amans 14 , E. Amato 15 , E. O. Angüner 16 , L. A. Antonelli 17 ,\n- C. Aramo 18 , M. Araya 19 , C. Arcaro 20 , L. Arrabito 21 , K. Asano 2 ,\n- Y. Ascasíbar 10 , J. Aschersleben 22 , H. Ashkar 7 ,\n- L. Augusto Stuani 23 , D. Baack 24 , M. Backes 25 , 26 , A. Baktash 27 ,\n- C. Balazs 28 , M. Balbo 29 , O. Ballester 30 , A. Baquero Larriva 13 , 31 ,\n- V. Barbosa Martins 32 , U. Barres de Almeida 33 , 34 , J. A. Barrio 13 ,\n- P. I. Batista 32 , I. Batkovic 35 , R. Batzofin 36 , J. Baxter 2 ,\n- J. Becerra González 37 , G. Beck 38 , J. Becker Tjus 39 ,\n- W. Benbow 40 , J. Bernete Medrano 41 , K. Bernlöhr 42 , A. Berti 43 ,\n- B. Bertucci 44 , V. Beshley 45 , P. Bhattacharjee 46 ,\n- S. Bhattacharyya 47 , B. Bi 48 , N. Biederbeck 24 , A. Biland 49 ,\n- E. Bissaldi 50 , 51 , J. Biteau 52 , 53 , O. Blanch 30 , J. 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Hamal 54 , W. Hanlon 40 , S. Hara 118 , V. M. Harvey 58 ,\n- T. Hassan 41 , L. Heckmann 43 , M. Heller 64 ,\n- S. Hernández Cadena 12 , ∗ , O. Hervet 96 , J. Hie 119 , N. Hiroshima 2 ,\n- B. Hnatyk 100 , R. Hnatyk 100 , J. Hoang 96 , D. Hoffmann 75 ,\n- W. Hofmann 42 , J. Holder 120 , D. Horan 7 , P. Horvath 121 ,\n- D. Hrupec 122 , M. Hütten 2 , ∗ , M. Iarlori 123 , T. Inada 2 ,\n- F. Incardona 83 , S. Inoue 102 , F. Iocco 90 , 18 , M. Iori 103 ,\n- M. Jamrozy 124 , P. Janecek 54 , F. Jankowsky 125 , C. Jarnot 119 ,\n- P. Jean 119 , I. Jiménez Martínez 41 , W. Jin 5 , C. Juramy-Gilles 55 ,\n- J. Jurysek 54 , M. Kagaya 2 , D. Kantzas 95 , V. Karas 126 ,\n- H. Katagiri 127 , J. Kataoka 128 , S. Kaufmann 61 , D. Kerszberg 30 ,\n- B. Khélifi 78 , R. Kissmann 129 , T. Kleiner 32 , G. Kluge 130 ,\n- W. Kluźniak 131 , J. Knödlseder 119 , Y. Kobayashi 2 , K. Kohri 132 ,\n- N. Komin 38 , P. Kornecki 14 , K. Kosack 3 , G. Kowal 99 , H. Kubo 2 ,\n- J. Kushida 1 , A. La Barbera 71 , N. La Palombara 73 , M. Láinez 13 ,\n- A. Lamastra 17 , J. Lapington 133 , P. Laporte 14 , S. Lazarević 104 ,\n- F. Leitgeb 32 , M. Lemoine-Goumard 93 , J.-P. Lenain 55 ,\n- F. Leone 134 , G. Leto 83 , F. Leuschner 48 , E. Lindfors 135 ,\n- M. Linhoff 24 , I. Liodakis 135 , S. Lombardi 17 , F. Longo 136 ,\n- R. López-Coto 9 , M. López-Moya 13 , A. López-Oramas 37 ,\n- S. Loporchio 50 , 51 , P. L. Luque-Escamilla 137 , O. Macias 138 ,\n- J. Mackey 60 , P. Majumdar 139 , D. Malyshev 92 , D. Mandat 54 ,\n- M. Manganaro 140 , G. Manicò 110 , 134 , M. Mariotti 35 , S. Markoff 138 ,\n- I. Márquez 9 , P. Marquez 30 , G. Marsella 141 , 110 , G. A. Martínez 41 ,\n- M. Martínez 30 , O. Martinez 142 , C. Marty 119 , A. Mas-Aguilar 13 ,\n- M. Mastropietro 17 , G. Maurin 46 , D. Mazin 2 , 43 , D. Melkumyan 32 ,\n- A. J. T. S. Mello 82 , 143 , J.-L. Meunier 55 , D. M.-A. Meyer 36 ,\n- M. Meyer 27 , D. Miceli 20 , M. Michailidis 48 , J. Michałowski 144 ,\n- T. Miener 13 , J. M. Miranda 142 , A. Mitchell 92 , M. Mizote 145 ,\n- T. Mizuno 146 , R. Moderski 131 , M. Molero 37 , C. Molfese 88 ,\n- E. Molina 37 , T. Montaruli 64 , D. Morcuende 13 , 9 , K. Morik 24 ,\n- G. Morlino 15 , A. Morselli 108 , E. Moulin 57 , V. Moya Zamanillo 13 ,\n- K. Munari 83 , T. Murach 32 , A. Muraczewski 131 , H. Muraishi 147 ,\n- S. Nagataki 102 , T. Nakamori 148 , R. Nemmen 34 , 149 , N. Neyroud 46 ,\n- L. Nickel 24 , J. Niemiec 144 , D. Nieto 13 , M. Nievas Rosillo 37 ,\n- M. Nikołajuk 150 , K. Nishijima 1 , K. Noda 2 , D. Nosek 151 ,\n- V. Novotny 151 , S. Nozaki 43 , P. O'Brien 133 , M. Ohishi 2 ,\n- Y. Ohtani 2 , A. Okumura 152 , 153 , J.-F. Olive 119 , B. Olmi 154 , 15 ,\n- R. A. Ong 155 , M. Orienti 62 , R. Orito 156 , M. Orlandini 59 ,\n- E. Orlando 136 , M. Ostrowski 124 , I. Oya 67 , A. Pagliaro 71 ,\n- M. Palatiello 63 , G. Panebianco 59 , D. Paneque 43 ,\n- F. R. Pantaleo 51 , 50 , R. Paoletti 157 , J. M. Paredes 8 ,\n- N. Parmiggiani 59 , S. R. Patel 52 , B. Patricelli 17 , 158 , D. Pavlović 140 ,\n- M. Pech 54 , M. Pecimotika 140 , 159 , U. Pensec 55 , 14 , M. Peresano 81 , 80 , J. Pérez-Romero 10 , 47 , ∗ , G. Peron 78 , M. Persic 160 , 161 ,\n- P.-O. Petrucci 162 , O. Petruk 45 , G. Piano 105 , E. Pierre 55 ,\n- E. Pietropaolo 123 , F. Pintore 71 , G. Pirola 43 , S. Pita 78 , C. Plard 46 ,\n- F. Podobnik 157 , M. Pohl 36 , 32 , M. Polo 41 , E. Pons 46 , G. Ponti 163 ,\n- E. Prandini 35 , J. Prast 46 , G. Principe 136 , C. Priyadarshi 30 ,\n- N. Produit 29 , E. Pueschel 32 , G. Pühlhofer 48 , M. L. Pumo 134 , 110 ,\n- M. Punch 78 , F. Queiroz 164 , 165 , A. Quirrenbach 125 , S. Rainò 77 ,\n- R. Rando 35 , S. Razzaque 166 , 115 , S. Recchia 81 , M. Regeard 78 ,\n- P. Reichherzer 84 , 39 , A. Reimer 129 , O. Reimer 129 \n, \n- A. Reisenegger 11 , 167 , W. Rhode 24 , D. Ribeiro 168 , M. Ribó 8 ,\n- T. Richtler 169 , J. Rico 30 , F. Rieger 42 , C. Righi 163 , L. Riitano 86 ,\n- V. Rizi 123 , E. Roache 40 , G. Rodriguez Fernandez 108 ,\n- J. J. Rodríguez-Vázquez 41 , P. Romano 163 , G. Romeo 83 ,\n- J. Rosado 13 , A. Rosales de Leon 55 , G. Rowell 58 , B. Rudak 131 ,\n- C. B. Rulten 61 , F. Russo 59 , I. Sadeh 32 , L. Saha 40 , T. Saito 2 ,\n- H. Salzmann 48 , D. Sanchez 46 , M. Sánchez-Conde 10 , ∗ ,\n- P. Sangiorgi 71 , H. Sano 2 , M. Santander 5 , A. Santangelo 48 ,\n- R. Santos-Lima 34 , A. Sanuy 8 , T. Šarić 170 , A. Sarkar 32 ,\n- S. Sarkar 84 , K. Satalecka 135 , F. G. Saturni 17 , V. Savchenko 171 ,\n- A. Scherer 11 , P. Schipani 76 , B. Schleicher 91 , 49 , J. L. Schubert 24 ,\n- F. Schussler 57 , U. Schwanke 172 , G. Schwefer 42 ,\n- M. Seglar Arroyo 30 , S. Seiji 1 , D. Semikoz 78 ,\n- O. Sergijenko 100 , 173 , 174 , M. Servillat 14 , V. Sguera 59 ,\n- R. Y. Shang 155 , P. Sharma 52 , H. Siejkowski 175 , A. Sinha 13 ,\n- C. Siqueira 23 , V. Sliusar 29 , A. Slowikowska 176 , H. Sol 14 ,\n- A. Specovius 92 , S. T. Spencer 92 , 84 , D. Spiga 163 , A. Stamerra 17 , 177 , \n47 \nS. Stanič \n, T. Starecki \n178 \n, R. Starling \n133 \n, Ł. Stawarz \n124 \n, \n- C. Steppa 36 , T. Stolarczyk 3 , J. Strišković 122 , Y. Suda 106 ,\n- T. Suomijärvi 52 , H. Tajima 152 , D. Tak 32 , M. Takahashi 152 ,\n- R. Takeishi 2 , S. J. Tanaka 179 , T. Tavernier 54 , L. A. Tejedor 13 ,\n- K. Terauchi 180 , R. Terrier 78 , M. Teshima 43 , W. W. Tian 2 ,\n- L. Tibaldo 119 , O. Tibolla 61 , F. Torradeflot 181 , 41 , D. F. Torres 182 ,\n- E. Torresi 59 , G. Tosti 163 , 44 , L. Tosti 44 , N. Tothill 104 ,\n- F. Toussenel 55 , V. Touzard 119 , A. Tramacere 29 , P. Travnicek 54 ,\n- G. Tripodo 141 , 110 , S. Truzzi 157 , A. Tsiahina 119 , A. Tutone 71 ,\n- M. Vacula 121 , 54 , B. Vallage 57 , P. Vallania 80 , 183 , C. van Eldik 92 ,\n- J. van Scherpenberg 43 , J. Vandenbroucke 86 , V. Vassiliev 155 ,\n- M. Vázquez Acosta 37 , M. Vecchi 22 , S. Ventura 157 ,\n- S. Vercellone 163 , G. Verna 157 , A. Viana 23 , N. Viaux 184 ,\n- A. Vigliano 63 , C. F. Vigorito 80 , 81 , V. Vitale 108 , V. Vodeb 47 ,\n- V. Voisin 55 , S. Vorobiov 47 , G. Voutsinas 64 , I. Vovk 2 ,\n- T. Vuillaume 46 , S. J. Wagner 125 , R. Walter 29 ,\n- M. Wechakama 69 , 70 , R. White 42 , A. Wierzcholska 144 , M. Will 43 ,\n- D. A. Williams 96 , F. Wohlleben 42 , A. Wolter 163 ,\n- T. Yamamoto 145 , R. Yamazaki 179 , T. Yoshida 127 , T. Yoshikoshi 2 ,\n- M. Zacharias 125 , 26 , G. Zaharijas 47 , D. Zavrtanik 47 ,\n- M. Zavrtanik 47 , A. A. Zdziarski 131 , A. Zech 14 , V. I. Zhdanov 100 ,\n- M. Živec 47 , J. Zuriaga-Puig 10 , and P. De la Torre Luque 185\n- 1 Department of Physics, Tokai University, 4-1-1, Kita-Kaname, Hiratsuka, Kanagawa 2591292, Japan\n- 2 Institute for Cosmic Ray Research, University of Tokyo, 5-1-5, Kashiwa-no-ha, Kashiwa, Chiba 277-8582, Japan\n- 3 Université Paris-Saclay, Université Paris Cité, CEA, CNRS, AIM, F-91191 Gif-sur-Yvette Cedex, France\n- 4 FSLAC IRL 2009, CNRS/IAC, La Laguna, Tenerife, Spain\n- 5 University of Alabama, Tuscaloosa, Department of Physics and Astronomy, Gallalee Hall, Box 870324 Tuscaloosa, AL 35487-0324, USA\n- 6 Université Côte d'Azur, Observatoire de la Côte d'Azur, CNRS, Laboratoire Lagrange, France\n- 7 Laboratoire Leprince-Ringuet, CNRS/IN2P3, École polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France\n- 8 Departament de Física Quàntica i Astrofísica, Institut de Ciències del Cosmos, Universitat de Barcelona, IEEC-UB, Martí i Franquès, 1, 08028, Barcelona, Spain\n- 9 Instituto de Astrofísica de Andalucía-CSIC, Glorieta de la Astronomía s/n, 18008, Granada, Spain\n- 10 Instituto de Física Teórica UAM/CSIC and Departamento de Física Teórica, Universidad Autónoma de Madrid, c/ Nicolás Cabrera 13-15, Campus de Cantoblanco UAM, 28049 Madrid, Spain\n- 11 Pontificia Universidad Católica de Chile, Av. Libertador Bernardo O'Higgins 340, Santiago, Chile\n- 12 Universidad Nacional Autónoma de México, Delegación Coyoacán, 04510 Ciudad de México, Mexico\n- 13 IPARCOS-UCM, Instituto de Física de Partículas y del Cosmos, and EMFTEL Department, Universidad Complutense de Madrid, E-28040 Madrid, Spain\n- 14 LUTH, GEPI and LERMA, Observatoire de Paris, Université PSL, Université Paris Cité, CNRS, 5 place Jules Janssen, 92190, Meudon, France\n- 15 INAF - Osservatorio Astrofisico di Arcetri, Largo E. Fermi, 5 - 50125 Firenze, Italy\n- 16 TÜBİTAK Research Institute for Fundamental Sciences, 41470 Gebze, Kocaeli, Turkey\n- 17 INAF - Osservatorio Astronomico di Roma, Via di Frascati 33, 00040, Monteporzio Catone, Italy\n- 18 INFN Sezione di Napoli, Via Cintia, ed. G, 80126 Napoli, Italy\n- 19 CCTVal, Universidad Técnica Federico Santa María, Avenida España 1680, Valparaíso, Chile\n- 20 INFN Sezione di Padova, Via Marzolo 8, 35131 Padova, Italy\n- 21 Laboratoire Univers et Particules de Montpellier, Université de Montpellier, CNRS/IN2P3, CC 72, Place Eugène Bataillon, F-34095 Montpellier Cedex 5, France\n- 22 Kapteyn Astronomical Institute, University of Groningen, Landleven 12, 9747 AD, Groningen, The Netherlands \n- 48 Institut für Astronomie und Astrophysik, Universität Tübingen, Sand 1, 72076 Tübingen, Germany\n- 49 ETH Zürich, Institute for Particle Physics and Astrophysics, Otto-Stern-Weg 5, 8093 Zürich, Switzerland\n- 50 Politecnico di Bari, via Orabona 4, 70124 Bari, Italy\n- 51 INFN Sezione di Bari, via Orabona 4, 70126 Bari, Italy\n- 52 Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France\n- 53 Institut universitaire de France (IUF)\n- 54 FZU - Institute of Physics of the Czech Academy of Sciences, Na Slovance 1999/2, 182 21 Praha 8, Czech Republic\n- 55 Sorbonne Université, CNRS/IN2P3, Laboratoire de Physique Nucléaire et de Hautes Energies, LPNHE, 4 place Jussieu, 75005 Paris, France\n- 56 University of Zagreb, Faculty of electrical engineering and computing, Unska 3, 10000 Zagreb, Croatia\n- 57 IRFU, CEA, Université Paris-Saclay, Bât 141, 91191 Gif-sur-Yvette, France\n- 58 School of Physics, Chemistry and Earth Sciences, University of Adelaide, Adelaide SA 5005, Australia\n- 59 INAF - Osservatorio di Astrofisica e Scienza dello spazio di Bologna, Via Piero Gobetti 93/3, 40129 Bologna, Italy\n- 60 Dublin Institute for Advanced Studies, 31 Fitzwilliam Place, Dublin 2, Ireland\n- 61 Centre for Advanced Instrumentation, Department of Physics, Durham University, South Road, Durham, DH1 3LE, United Kingdom\n- 62 INAF - Istituto di Radioastronomia, Via Gobetti 101, 40129 Bologna, Italy\n- 63 INFN Sezione di Trieste and Università degli Studi di Udine, Via delle Scienze 208, 33100 Udine, Italy\n- 64 University of Geneva - Département de physique nucléaire et corpusculaire, 24 rue du Général-Dufour, 1211 Genève 4, Switzerland\n- 65 Armagh Observatory and Planetarium, College Hill, Armagh BT61 9DG, United Kingdom\n- 66 School of Physics, University of New South Wales, Sydney NSW 2052, Australia\n- 67 Cherenkov Telescope Array Observatory, Saupfercheckweg 1, 69117 Heidelberg, Germany 68 Unitat de Física de les Radiacions, Departament de Física, and CERES-IEEC, Universitat\n- Autònoma de Barcelona, Edifici C3, Campus UAB, 08193 Bellaterra, Spain\n- 69 Department of Physics, Faculty of Science, Kasetsart University, 50 Ngam Wong Wan Rd., Lat Yao, Chatuchak, Bangkok, 10900, Thailand\n- 70 National Astronomical Research Institute of Thailand, 191 Huay Kaew Rd., Suthep, Muang, Chiang Mai, 50200, Thailand\n- 71 INAF - Istituto di Astrofisica Spaziale e Fisica Cosmica di Palermo, Via U. La Malfa 153, 90146 Palermo, Italy\n- 72 Universidade Cruzeiro do Sul, Núcleo de Astrofísica Teórica (NAT/UCS), Rua Galvão Bueno 8687, Bloco B, sala 16, Libertade 01506-000 - São Paulo, Brazil\n- 73 INAF - Istituto di Astrofisica Spaziale e Fisica Cosmica di Milano, Via A. Corti 12, 20133 Milano, Italy\n- 74 INFN Sezione di Pisa, Edificio C - Polo Fibonacci, Largo Bruno Pontecorvo 3, 56127 Pisa \n- 102 RIKEN, Institute of Physical and Chemical Research, 2-1 Hirosawa, Wako, Saitama, 351-0198, Japan\n- 103 INFN Sezione di Roma La Sapienza, P.le Aldo Moro, 2 - 00185 Roma, Italy\n- 104 Western Sydney University, Locked Bag 1797, Penrith, NSW 2751, Australia\n- 105 INAF - Istituto di Astrofisica e Planetologia Spaziali (IAPS), Via del Fosso del Cavaliere 100, 00133 Roma, Italy\n- 106 Physics Program, Graduate School of Advanced Science and Engineering, Hiroshima University, 739-8526 Hiroshima, Japan\n- 107 Department of Physics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan\n- 108 INFN Sezione di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Rome, Italy 109 Alikhanyan National Science Laboratory, Yerevan Physics Institute, 2 Alikhanyan Brothers St., 0036, Yerevan, Armenia\n- 110 INFN Sezione di Catania, Via S. Sofia 64, 95123 Catania, Italy\n- 111 Université Paris Cité, CNRS, CEA, Astroparticule et Cosmologie, F-75013 Paris, France 112 Universidad Andres Bello, República 252, Santiago, Chile\n- 113 Núcleo de Astrofísica e Cosmologia (Cosmo-ufes) & Departamento de Física, Universidade Federal do Espírito Santo (UFES), Av. Fernando Ferrari, 514. 29065-910. Vitória-ES, Brazil\n- 114 Astrophysics Research Center of the Open University (ARCO), The Open University of Israel, P.O. Box 808, Ra'anana 4353701, Israel\n- 115 Department of Physics, The George Washington University, Washington, DC 20052, USA\n- 116 Université Paris Cité, Université Paris-Saclay, CEA, CNRS, AIM, F-91191 Gif-surYvette, France\n- 117 King's College London, Strand, London, WC2R 2LS, United Kingdom\n- 118 Learning and Education Development Center, Yamanashi-Gakuin University, Kofu, Yamanashi 400-8575, Japan\n- 119 IRAP, Université de Toulouse, CNRS, CNES, UPS, 9 avenue Colonel Roche, 31028 Toulouse, Cedex 4, France\n- 120 Department of Physics and Astronomy and the Bartol Research Institute, University of Delaware, Newark, DE 19716, USA\n- 121 Palacký University Olomouc, Faculty of Science, Joint Laboratory of Optics of Palacký University and Institute of Physics of the Czech Academy of Sciences, 17. listopadu 1192/12, 779 00 Olomouc, Czech Republic\n- 122 Josip Juraj Strossmayer University of Osijek, Trg Ljudevita Gaja 6, 31000 Osijek, Croatia\n- 123 Dipartimento di Scienze Fisiche e Chimiche, Università degli Studi dell'Aquila and GSGC-LNGS-INFN, Via Vetoio 1, L'Aquila, 67100, Italy\n- 124 Astronomical Observatory, Jagiellonian University, ul. Orla 171, 30-244 Cracow, Poland 125 Landessternwarte, Zentrum für Astronomie der Universität Heidelberg, Königstuhl 12, 69117 Heidelberg, Germany\n- 126 Astronomical Institute of the Czech Academy of Sciences, Bocni II 1401 - 14100 Prague, Czech Republic\n- 127 Faculty of Science, Ibaraki University, Mito, Ibaraki, 310-8512, Japan\n- 128 Faculty of Science and Engineering, Waseda University, Shinjuku, Tokyo 169-8555, Japan \n153 Kobayashi-Maskawa Institute for the Origin of Particles and the Universe, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8602, Japan \n154 INAF - Osservatorio Astronomico di Palermo 'G.S. Vaiana', Piazza del Parlamento 1, 90134 Palermo, Italy \n155 Department of Physics and Astronomy, University of California, Los Angeles, CA 90095, USA \n- 156 Graduate School of Technology, Industrial and Social Sciences, Tokushima University, Tokushima 770-8506, Japan\n- 157 INFN and Università degli Studi di Siena, Dipartimento di Scienze Fisiche, della Terra e dell'Ambiente (DSFTA), Sezione di Fisica, Via Roma 56, 53100 Siena, Italy\n- 158 University of Pisa, Largo B. Pontecorvo 3, 56127 Pisa, Italy\n- 159 Rudjer Boskovic Institute, Bijenicka 54, 10 000 Zagreb, Croatia\n- 160 INAF - Osservatorio Astronomico di Padova, Vicolo dell'Osservatorio 5, 35122 Padova, Italy \n161 INAF - Osservatorio Astronomico di Padova and INFN Sezione di Trieste, gr. coll. Udine, Via delle Scienze 208 I-33100 Udine, Italy \n162 Univ. Grenoble Alpes, CNRS, IPAG, 414 rue de la Piscine, Domaine Universitaire, 38041 Grenoble Cedex 9, France \n163 INAF - Osservatorio Astronomico di Brera, Via Brera 28, 20121 Milano, Italy \n164 International Institute of Physics, Universidade Federal do Rio Grande do Norte, 59078970, Natal, RN, Brasil \n165 Departamento de Física, Universidade Federal do Rio Grande do Norte, 59078-970, Natal, RN, Brasil \n166 Centre for Astro-Particle Physics (CAPP) and Department of Physics, University of Johannesburg, PO Box 524, Auckland Park 2006, South Africa \n- 167 Departamento de Física, Facultad de Ciencias Básicas, Universidad Metropolitana de Ciencias de la Educación, Avenida José Pedro Alessandri 774, Ñuñoa, Santiago, Chile \n168 Department of Physics, Columbia University, 538 West 120th Street, New York, NY 10027, USA \n169 Departamento de Astronomía, Universidad de Concepción, Barrio Universitario S/N, Concepción, Chile \n- 170 University of Split - FESB, R. Boskovica 32, 21 000 Split, Croatia \n171 EPFL Laboratoire d'astrophysique, Observatoire de Sauverny, CH-1290 Versoix, Switzerland \n172 Department of Physics, Humboldt University Berlin, Newtonstr. 15, 12489 Berlin, Germany \n173 Main Astronomical Observatory of the National Academy of Sciences of Ukraine, Zabolotnoho str., 27, 03143, Kyiv, Ukraine \n- 174 Space Technology Centre, AGH University of Science and Technology, Aleja Mickiewicza, 30, 30-059, Kraków, Poland\n- 175 Academic Computer Centre CYFRONET AGH, ul. Nawojki 11, 30-950, Kraków, Poland 176 Institute of Astronomy, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University in Toruń, ul. Grudziądzka 5, 87-100 Toruń, Poland\n- 177 Cherenkov Telescope Array Observatory gGmbH, Via Gobetti, Bologna, Italy\n- 178 Warsaw University of Technology, Faculty of Electronics and Information Technology, Institute of Electronic Systems, Nowowiejska 15/19, 00-665 Warsaw, Poland\n- 179 Department of Physical Sciences, Aoyama Gakuin University, Fuchinobe, Sagamihara, Kanagawa, 252-5258, Japan\n- 180 Division of Physics and Astronomy, Graduate School of Science, Kyoto University, Sakyoku, Kyoto, 606-8502, Japan\n- 181 Port d'Informació Científica, Edifici D, Carrer de l'Albareda, 08193 Bellaterrra (Cerdanyola del Vallès), Spain\n- 182 Institute of Space Sciences (ICE, CSIC), and Institut d'Estudis Espacials de Catalunya (IEEC), and Institució Catalana de Recerca I Estudis Avançats (ICREA), Campus UAB, Carrer de Can Magrans, s/n 08193 Cerdanyola del Vallés, Spain\n- 183 INAF - Osservatorio Astrofisico di Torino, Strada Osservatorio 20, 10025 Pino Torinese (TO), Italy\n- 184 Departamento de Física, Universidad Técnica Federico Santa María, Avenida España, 1680 Valparaíso, Chile\n- 185 Stockholm University and the Oskar Klein Centre for Cosmoparticle Physics, Stockholm, Sweden\n- ∗\n- Corresponding authors (alphabetical order): \nRémi Adam ([email protected]), \nSergio Hernández-Cadena ([email protected]), \nMoritz Hütten ([email protected]), \nJudit Pérez-Romero ([email protected]), \nMiguel A. Sánchez-Conde ([email protected]) \nAbstract. Galaxy clusters are expected to be both dark matter (DM) reservoirs and storage rooms for the cosmic-ray protons (CRp) that accumulate along the cluster's formation history. Accordingly, they are excellent targets to search for signals of DM annihilation and decay at γ -ray energies and are predicted to be sources of large-scale γ -ray emission due to hadronic interactions in the intracluster medium (ICM). In this paper, we estimate the sensitivity of the Cherenkov Telescope Array (CTA) to detect diffuse γ -ray emission from the Perseus galaxy cluster. We first perform a detailed spatial and spectral modelling of the expected signal for both the DM and the CRp components. For each case, we compute the expected CTA sensitivity accounting for the CTA instrument response functions. The CTA observing strategy of the Perseus cluster is also discussed. In the absence of a diffuse signal (non-detection), CTA should constrain the CRp to thermal energy ratio X 500 within the characteristic radius R 500 down to about X 500 < 3 × 10 -3 , for a spatial CRp distribution that follows the thermal gas and a CRp spectral index α CRp = 2 . 3 . Under the optimistic assumption of a pure hadronic origin of the Perseus radio mini-halo and depending on the assumed magnetic field profile, CTA should measure α CRp down to about ∆ α CRp ≃ 0 . 1 and the CRp spatial distribution with 10% precision, respectively. Regarding DM, CTA should improve the current ground-based γ -ray DM limits from clusters observations on the velocityaveraged annihilation cross-section by a factor of up to ∼ 5 , depending on the modelling of DM halo substructure. In the case of decay of DM particles, CTA will explore a new region of the parameter space, reaching models with τ χ > 10 27 s for DM masses above 1 TeV. These constraints will provide unprecedented sensitivity to the physics of both CRp acceleration and transport at cluster scale and to TeV DM particle models, especially in the decay scenario.", '1.1 Diffuse γ -ray emission from cosmic-rays and dark matter in galaxy clusters': "Clusters of galaxies are the largest virialized structures in the Universe, with masses up to about 10 15 M ⊙ . They are dominated by dark matter (DM; ∼ 85 % in mass) and permeated by the intracluster medium (ICM; ∼ 10 -15 % in mass), whose physical properties are shaped by the hierarchical growth of structures through the merging of subclusters and the smooth accretion of surrounding matter [1, 2]. While the ICM is mostly thermal, these energetic merging events do not only dissipate the kinetic energy into heat via shock waves and turbulence, but may also accelerate cosmic-rays (CR) in the ambient magnetic field [3]. Galaxies in galaxy clusters account only up to a few percent of the total mass, yet they can also directly inject CR via active galactic nuclei (AGN) feedback or star formation activity [4-6]. \nDirect evidence for the presence of CR electrons (CRe) and magnetic fields have been found in a growing number of galaxy clusters, thanks to the radio observations of diffuse synchrotron emission [7-9]. These sources are classified as radio halos (that roughly follow the thermal ICM) and radio relics (elongated and peripheral [10]). Radio halos are further classified as giant radio halos ( ∼ Mpc size), associated with cluster mergers [11], suggesting that they are powered by the energy dissipated during these events, and mini-halos in more relaxed clusters, which extend on 100 -300 kpc scales and are generally confined in the \ncore of cool-core clusters [12, 13]. Recent observations have made the phenomenology more complex, showing that in a number of cases mini-halos are surrounded by larger scale (usually very steep spectrum) emission, similar to giant radio halos [14-16]. Furthermore, it has been discovered that radio emission may extend on scales larger than those of giant radio halos, in the form of radio bridges [17, 18] and mega-halos [19]. These emerging evidences from observations suggest a more complex picture of non-thermal phenomena in galaxy clusters that may require a revision of current classification schemes. Galaxy clusters are also expected to act as storehouses for the CR protons (CRp) and heavier nuclei [20] due to their long lifetimes, once they are injected in the ICM via several mechanisms (cosmic shocks and galaxies via supernovae explosions, starbursts or AGN) [3, 21, 22]. These CRp should interact hadronically in the ICM to produce γ -ray emission [23-27]. Such interactions imply the production of high-energy secondary electrons, which will eventually contribute to the clusterscale radio emission. According to the current theoretical picture for radio halos and relics, the emitting electrons are (re)accelerated by turbulence and shocks, respectively [28-33]. The large volumes that are probed by radio halos in which the ICM is tenuous disfavour an important contribution from secondary electrons in these sources [3], although secondaries can contribute to the population of the seed particles to reaccelerate [32, 34, 35]. On the other hand, pure hadronic models may still explain the smaller mini-halos in dense cores [36, 37], although a number of evidences suggests that gas motions may play a major role [38, 39]. Cosmological numerical simulations of CR in clusters have obtained quantitative predictions for the expected γ -ray emission [40-42], which has proved useful when searching for clusters in γ -ray observations [43]. Nevertheless, many uncertainties related to acceleration and transport physics affect the expected γ -ray signal [3, 22, 44] and cosmological simulations including full turbulent reacceleration physics have not been obtained yet. In this context the study of γ -ray emission from galaxy clusters plays a central role. \nIn the past, galaxy clusters have provided strong gravitational evidence in favour of the existence of DM [45-47]. Thus, being DM-dominated, these objects also represent natural astrophysical targets for current DM search efforts. Since the underlying nature of DM (and thus its potential signatures) is still unknown [48], galaxy clusters have been used to probe the properties of the DM particle with a variety of techniques (e.g., [49-52]). One of the most promising ones is the search for DM-induced γ -ray signals ([53-55], for reviews), expected from the annihilation or decay of the so-called Weakly Interacting Massive Particles (WIMPs), one of the most studied DM particle candidate. WIMPs (e.g., [56-59]) would be produced thermally in the Universe via the 'freeze-out' mechanism and would have masses O (0 . 1 -100) TeV 1 . They can arise from several theoretical frameworks, ranging from minimal extensions of the Standard Model [56, 62, 63] to extra-dimensions [64, 65] and others [6669]. The expected γ -ray flux from their annihilation or decay in astrophysical objects mainly depends on the target DM density (squared, for annihilation) and its distance to Earth. Thus, for DM decay, local galaxy clusters can yield the highest expected fluxes compared to other possible targets, as they are the most massive structures in the Universe. As for DM annihilation, clusters can provide fluxes comparable to the ones from dwarf spheroidal galaxies (dSphs, [70]), as long as the DM interactions expected in their substructures are taken into account [71-73]. These substructures, usually referred to as subhalos, are a natural result of the Λ CDM hierarchical growth of structures [74-76], and their abundance is expected to be comparatively significant in clusters, as they are the largest exponents of structure \nformation at present. Despite the optimal characteristics of galaxy clusters to be used for γ -ray DM searches, the main drawback with respect to other targets is the predicted γ -ray emission from more conventional astrophysical processes. Indeed, the expected γ -rays from hadronic interactions of the CRp in the ICM can act as a complex background to search for a DM-induced signal using standard analysis techniques. \nThe search for diffuse γ -ray emission from galaxy clusters has been going on for over two decades, both using space-based observations in the GeV band [43, 77-85] and ground-based observations at energies > 100 GeV [86-90]. Yet, such signal remained elusive 2 . Nonetheless, these non-detections allowed to constrain the CRp to thermal energy ratio down to few percent [94], challenging the understanding of diffusive shock acceleration in the ICM when combined with radio data [95-97], although the large modelling uncertainty affects the predictions [21]. The stringent γ -ray limits were also used to disfavour hadronic models for nearby radio halos [98, 99] and to test models based on the reacceleration of secondary particles [92, 99]. \nThe non-detection of γ -ray emission from clusters is in agreement with the lack of DMinduced γ -ray signals from other promising astrophysical targets, especially dSphs [100-102]. DM searches in clusters mainly targeted very massive and local objects [70, 103, 104], since the expected flux is proportional to the mass of the objects and decreases with the distance squared. Yet, the lack of a DM signal in clusters has allowed to provide also strong constraints for annihilating WIMPs from a combined analysis of various clusters [105-112] or single cluster observations [83, 113, 114]. These DM annihilation limits are nevertheless not at the level of discarding thermal WIMP models 3 . The situation comparatively improves for WIMP DM decay. Indeed, lower limits on the WIMP lifetime derived from the observation of clusters [106, 115, 116] are among the most constraining ones at present. Some authors have also studied the possibility that some past hints of detection of γ -rays in clusters were due to WIMP DM [107, 117, 118], however these works were inconclusive due to the faintness of such signals, indeed never confirmed. \nThe Cherenkov Telescope Array 4 (CTA, [119]) will be the next generation ground-based γ -ray observatory. It will be amongst the most sensitive γ -ray telescope from 20 GeV to 300 TeV. CTA will be based at two sites: La Palma, in the North, and Paranal in the South, allowing us to observe sources in a large fraction of the sky. CTA will provide a major improvement, up to one order of magnitude in sensitivity and up to a factor 2 in angular resolution, with respect to previous instruments 5 . The study of CR physics and DM are among the main drivers of CTA science [120]. In particular, CTA will allow us to search for diffuse γ -ray emission from galaxy clusters. One of the proposed CTA key science projects consists in the observation of the Perseus galaxy cluster, which is among the most promising targets for such observations. These observations should complement DM searches in the Galactic center [121], dwarf galaxies, the Large Magellanic Cloud [122] or the search of axionlike particles, and should be used to probe fundamental physics [123].", '1.2 The Perseus cluster as a promising γ -ray target': 'The Perseus cluster (Abell 426) is the brightest cluster in the X-ray sky [124]. It is one of the most massive nearby clusters and presents the typical properties of a relaxed cool-core cluster with a dense core. Nonetheless, two main cold fronts have been identified and interpreted as the result from the sloshing due to minor mergers [125, 126]. The Perseus cluster hosts a radio mini-halo [127-135], and X-ray cavities associated with the radio lobes of the central AGN, NGC 1275 (3C84), indicate that the feedback is important in the cluster center [136, 137]. AGN activity may also be responsible for weak shocks and turbulence [138, 139], which could reaccelerate particles, in addition to direct CR injection from the AGN. The contribution to the radio mini-halo from hadronic interactions, direct CR injection from AGN and the role of turbulence remains unknown. \nTwo AGN from the Perseus cluster are known γ -ray emitters: NGC 1275 [140] and IC 310 [141]. Both sources are variable in time. NGC 1275 is the central galaxy of the cluster and IC 310, located about 0.6 deg southwest from the X-ray peak, is consistent with a narrow-angle tail radio galaxy infalling into the cluster [134]. A few other remarkable radio galaxies are present in the cluster volume: NGC 1265, NGC 1272, CR 15 [134], that were not detected at γ -ray energies 6 . While these sources are expected to contaminate the CTA data when searching for diffuse emission, they are also contributing to inject CR into the ICM, which could eventually contribute to large-scale γ -ray emission. \nThe Perseus cluster has been recognised as one of the best targets for searches of CRinduced γ -ray emission [42, 87]. This is because the mini-halo traces a dense region where hadronic collisions and the production of secondaries is maximised. While its central γ -ray bright galaxy, NGC 1275, prohibits reliable constraints on the diffuse ICM component from Fermi -LAT, the better angular resolution of CTA and the larger accessible energy range probed is expected to allow us separating the different sources of a possible γ -ray emission. In fact, the expected mild angular extent of the cluster (about 1 deg) due to its proximity (about 75 Mpc) implies that CTA is expected to resolve the diffuse emission if bright enough, but also that the angular extension of the signal is smaller than the field of view diameter by a factor of 5 to 10, so that systematic effects associated with the background modelling are limited. Along this line, also due to its large mass and proximity, Perseus stands out as one of the best clusters to search for DM-induced γ -ray emission. Indeed, the expected annihilation/decay flux is comparable to the one from other promising local galaxy clusters [142], such as Coma, Fornax, Ophiuchus, Hydra or Centaurus - see [70, 104] 7 as well as other traditional DM targets such as dwarf satellite galaxies or nearby galaxies. \nA previous search for diffuse TeV γ -ray emission towards the Perseus cluster was performed using the MAGIC telescopes [87, 88, 90]. Not having detected the signal, they reported upper limits on the CR to thermal pressure ratio assuming different models. For instance, using a spectral photon index slope of 2.2 and a relatively compact profile for the CRp, they obtained an upper limit of ∼ 1 -2 % on this ratio. This provided the best limit on the CR content of a cluster obtained from ground-based γ -ray observations so far, at a similar level to that obtained with Fermi -LAT [43] 8 . Fermi -LAT data from Perseus have been recently \nused to obtain constraints on the velocity-averaged DM annihilation cross-section as well [111]. The obtained limits are more than two orders of magnitude above from the reference value of the thermal relic cross-section. Additionally, MAGIC observations were used to set constraints on the DM decay lifetime [116], these being among the strongest constraints for DM masses in the TeV, range up to date reaching the value of 2 × 10 26 s. \nFinally, the location of the Perseus cluster in the sky allows for low zenith-angle observations from the CTA Northern Array, guaranteeing the best sensitivity of the array over its whole energy range. For all these reasons, the Perseus cluster was selected as the prime target for diffuse γ -ray emission searches from galaxy clusters with CTA, as one of the Key Science Projects (KSP) [120]. We refer to this previous work for further details about this choice. \nThe paper is organized as follows. Section 2 and Section 3 present the cluster modelling of the cluster in the context of CR and DM induced γ -ray emission, respectively. The observation setup and the background sky modelling are discussed in Section 4. Section 5 and Section 6 provide the results on the CTA sensitivity to CR and DM physics, respectively. We conclude in Section 7. A few appendices complement the paper. \nThroughout this paper, we assume a flat Λ CDM cosmology with H 0 = 70 km s -1 Mpc -1 , Ω M = 0 . 3 , and Ω Λ = 0 . 7 . The coordinates of the Perseus cluster center are taken as the ones of its central galaxy NGC 1275, R.A., Dec = 49.9507, 41.5117 deg 9 and its redshift is z = 0 . 017284 [145], corresponding to a luminosity distance of d L ≃ 75 Mpc. Given our reference cosmological model, 1 deg in the sky corresponds to a 1.265 Mpc distance at the redshift of Perseus. We adopt the characteristic angular radius θ 500 = 59 . 7 ± 0 . 4 arcmin 10 obtained by [146] using the fit of the Planck universal pressure profile 11 . This corresponds to a physical radius R 500 = (1 . 26 ± 0 . 01 ) Mpc and to a mass M 500 = (5 . 77 ± 0 . 12) × 10 14 M ⊙ . In the paper, International System of Units is used unless specified.', '2 Modelling the γ -ray emission associated with cosmic-rays': "The prediction of the diffuse γ -ray emission induced by CR in the ICM requires the detailed modelling of the physical components at play, their interactions, and the underlying radiative processes. In this section, we describe such a model based on the MINOT software [147] 12 . MINOT is dedicated to compute the observable properties of the ICM (radio synchrotron, thermal Sunyaev-Zel'dovich effect, X-ray thermal bremsstrahlung, inverse-Compton, γ -rays from hadronic interactions, and neutrino emission) given the user-defined physical state of the cluster. The code describes galaxy clusters as spherically symmetric objects. Here, we discuss essentially the calibration of the input model using external data assuming different scenarios, which we feed to MINOT . This includes the thermal gas density and pressure (Section 2.1.1), the magnetic field strength (Section 2.1.2), and the CR spatial and spectral distributions (Section 2.1.3). This allows us to perform predictions for the γ -ray signal and estimate model uncertainties using the method implemented in MINOT .", '2.1 Modelling the intracluster medium components': 'We model the Perseus cluster assuming spherical symmetry, considering only radial profiles to describe its ICM components. This assumption is expected to be fairly accurate given the fact that the Perseus cluster is overall a relaxed system. \nWhile clusters are diffuse objects with no clear definitions of their extension, we expect density and pressure discontinuities near the virial radius [148, 149]. We consider a maximum radial extent of the cluster and define this truncation radius as R tr = 3 R 500 = 3 . 78 Mpc ≃ 2 R 200 using the measurement from [150] as a reference. The exact choice of this value does not significantly affect our results, but this allows us to perform numerical integrations over a well defined region.', '2.1.1 Thermal gas': 'The modelling of the thermal gas is necessary to compute the CR induced γ -ray emission for two reasons: 1) the nuclei (protons, helium and heavier elements) are involved in hadronic interactions that lead to the emission of γ -rays and secondary electrons, which require modelling of the thermal gas density; 2) the thermal pressure, or thermal energy, is used for the relative normalization of the CR energy. \nThe thermal electron density is modeled as a double β -model [151], following [152]: \nn e ( r ) = n 0 , 1 ( 1 + ( r r c, 1 ) 2 ) -3 β 1 / 2 + n 0 , 2 ( 1 + ( r r c, 2 ) 2 ) -3 β 2 / 2 . (2.1) \nThe core parameters are taken from the XMM-Newton measurement [152, 153]. While [152, 153] used the Einstein telescope results from [154] for the peripheral outskirts region parameters, we use instead the more recent Suzaku measurement [146]. We have ( n 0 , 1 , r c, 1 , β 1 , n 0 , 2 , r c, 2 , β 2 ) = ( 4 . 6 × 10 -2 cm -3 , 57 kpc , 1 . 2 , 3 . 6 × 10 -3 cm -3 , 278 kpc , 0 . 71 ) , when accounting for the different cosmological models. \nWe rely on X-ray spectroscopic measurements to describe the gas temperature, as \nk B T ( r ) = 7 × ( 1 + ( r r t, 1 ) 3 )( 2 . 3 + ( r r t, 1 ) 3 ) -1 ( 1 + ( r r t, 2 ) 1 . 7 ) -1 keV , (2.2) \nwith k B the Boltzmann constant. The first terms allow us to describe the core temperature as provided by [152], where r t, 1 = 73 . 8 kpc, given our cosmological model. We introduce the last term in order to account for the temperature drop in the outskirt as measured by [146], where we set r t, 2 = 1600 kpc. \nGiven the thermal electron density and temperature, we compute the electron thermal pressure as \nP e ( r ) = n e ( r ) k B T ( r ) . (2.3) \nAdditionally, we assume an ICM helium mass fraction of 0.2735 and we model the metal abundances as constant, with Z Z ⊙ = 0 . 3 [155], using solar abundances from [156]. The electron number density and thermal pressure, together with the abundances are used to compute the full thermodynamic properties of the thermal component following [147]. \nIn Figure 1, we present the thermal electron number density, the temperature and the electron pressure radial profiles. We compare our model to other parameterizations available in the literature. In the core, all density profile models agree since they all rely on XMMNewton data [152]. In the outskirts, the agreement is good up to R 500 . Beyond, ROSAT [157] \nand Suzaku [146] agree well but the Einstein telescope model [154] leads to a larger electron number density. The temperature profile is typical of that of a cool-core cluster. Our model matches the [152] model in the core and the [146] model in the outskirt according to their domain of validity. All the pressure profile models agree well in the outskirt except for the model derived from the extrapolation of the [152] temperature in the outskirt, which overpredicts the pressure by a factor of a few. Similarly, in the core, the [146] models do not agree with the more direct measurement based on [152] where they extrapolate the profile with an isothermal component. The observed differences are due to the different scales probed by the respective instruments and the extrapolation of the profiles. \nGiven the definition of our thermal model and the data used to constrain it, we are able to accurately describe the cluster from its core (10 kpc) to the outskirts (2 Mpc). The uncertainties associated with the thermal model are expected to be negligible compared to the uncertainties associated with the non-thermal component. In appendix A, we also present a validation of our thermal model using the Planck Compton parameter map, showing that our model accurately describes the pressure profile of the cluster. We note that because the temperature model based on [152] is valid only up to about 200 kpc, its use leads to an overestimation of the thermal energy by a factor of a few, depending on the details of the line-of-sight integration of the model, when extrapolated beyond its validity region. This is what is done in [87, 88, 90] up to the virial radius ( ∼ R 200 ), which should affect their constraints on the CR by a similar amount.', '2.1.2 Magnetic field strength': 'The modelling of the magnetic field strength is necessary when considering jointly the γ -ray emission and the radio synchrotron emission. However, the magnetic field strength and the structure in galaxy clusters remain poorly known to date [158]. We thus consider several approaches in order to model the magnetic field distribution in the Perseus cluster, which will allow us to quantify the associated systematic effect. \nOur first approach relies on the scaling of the magnetic field to the thermal gas density. In this case, the magnetic field strength is given by \n⟨ B ⟩ ( r ) = ⟨ B ref ⟩ ( n e ( r ) n e , ref ) η B , (2.4) \nwhere ⟨ B ref ⟩ and n e , ref are magnetic field and density normalization parameters, respectively. We first compute the magnetic field strength using the rotation measure estimate from [159], which gives ⟨ B ⟩ (10 kpc) ∼ 25 µ G, and set η B = 2 / 3 assuming magnetic field flux conservation. \nThe Coma cluster is one of the only clusters for which the magnetic field strength profile was measured [160]. As a second approach, we thus assume that the Coma and Perseus clusters have the same magnetic field strength to gas density ratio. In this case, we use the values η B = [0 . 4 , 0 . 5 , 2 / 3 , 0 . 9] , corresponding to ⟨ B ref ⟩ = [3 . 9 , 4 . 7 , 5 . 0 , 5 . 4] µ G with n e , ref = 3 . 42 × 10 -3 cm -3 , allowing us to account for the uncertainties in the measurement [160]. \nThe structure of the Kelvin-Helmholtz instability visible in Perseus was used to infer the thermal to magnetic pressure ratio, β pl = P gas /P B ∼ 200 , for the overall cluster volume prior to sloshing [125]. In a last approach, we use this measurement to infer the magnetic \nFigure 1 . Perseus cluster thermal gas and magnetic field models. Top left panel: thermal electron number density profile. The model by [152] combines XMM-Newton observations in the core and Einstein observations in the outskirt [154]. The red curve is similar, but we have replaced the outskirt model by that obtained in [157] with ROSAT. Top right panel: gas temperature profile. The model by [152] is constrained by the data up to about 200 kpc. The model by [146] is constrained by the data down to about 200 kpc. Bottom left panel: thermal electron pressure profile. In the case of [146], both the combination of gas temperature and density, and the Planck universal pressure profile fit to the their data are shown. Bottom right panel: Magnetic field strength models according to scaling from literature measurements. The truncation radius is visible at 3 R 500 . \n<!-- image --> \nfield strength profile, as \n⟨ B ⟩ ( r ) = ( 2 µ 0 β pl P gas ( r ) ) 1 / 2 , (2.5) \nwhere µ 0 is vacuum permeability. \nIn Figure 1 (bottom right panel), we present our models of the magnetic field strength profiles of the Perseus cluster. All models provide a similar order of magnitude, but the differences reflect the difficulty in having an accurate description. The scatter between models nearly reaches an order of magnitude in the core and in the outskirt, with the best agreement between all models around 200 -600 kpc. We note that the model with η B = 0 . 9 and ⟨ B 0 ⟩ = 5 . 4 µ G implies a core magnetic field that reaches about 50 µ G, corresponding to an energy density that is one third of the thermal pressure ( β pl ∼ 3 ) and thus very high compared to physical expectations. Assuming the pure hadronic model, a higher magnetic field will lead \nto a lower γ -ray flux for a fixed radio emission. Thus, the highest magnetic field model gives a conservative estimate of the expected γ -ray emission. In the following, as a baseline, we use the model based on [159] with η B = 2 / 3 that corresponds to an intermediate estimate.', '2.1.3 Cosmic-ray model': 'The γ -ray emission induced by hadronic interactions is directly related to the spatial and spectral distribution of CRp in the ICM. They are modeled according to a radial profile and a canonical power-law in momentum space, \ndN CRp dpdV ( E,r ) = A CRp p -α CRp n e ( r ) η CRp . (2.6) \nThe CR radial profile assumes a scaling with respect to the gas density so that only one parameter, η CRp , is necessary to describe the spatial distribution. Physically, this parameter allows us to account for the CR dynamics and the competition between advection and streaming, which remains poorly known. We also consider the case where the CRp profile is scaled with respect to the thermal pressure profile, and thus related to the thermal energy. The value of the power-law slope α CRp is related to the acceleration of CR such as the associated Mach number distribution [161]. The normalization A CRp is computed given the value of the CRp to thermal energy ratio, X CRp ( R ) = U CRp ( R ) U th ( R ) , by integrating Equation 2.6 accordingly. The thermal energy is computed by integrating the pressure profile. The CR energy is computed by integrating the CR distribution over the volume and between E p, min = 1 . 22 GeV, i.e., the energy threshold of proton-proton interactions, and E p, max = 10 PeV, above which some CRp could escape the cluster [42], although the exact value does not affect our results. In the following, we refer to X CRp ( R 500 ) as X 500 and use this reference for normalization. The value of this parameter has been predicted using numerical simulations, being X 500 ∼ 1 % [42]. \nThe CRe are modeled accounting for two contributions: the primary electrons that are independent from the CRp, and the secondary electrons that are produced from hadronic interactions assuming stationarity. The primary electrons, whenever considered, are modeled similarly to the CRp (Equation 2.6), but with different spectral model to account for energy losses (e.g., power-law or exponential cutoff power-law). We will show in Section 2.3 that in practice, primary CRe are irrelevant for our purpose. Secondary CRe are computed as detailed in [147].', '2.2 Non-thermal radiative processes': 'Given the ICM model, we compute the different observables associated with the γ -ray emission from hadronic interactions, the radio synchrotron, and inverse-Compton emission. The calculations are done using the MINOT software [147].', '2.2.1 Hadronic interactions and γ -ray emission': 'The hadronic production rate of γ -rays is computed by integrating the collision rate of protonproton interactions multiplied by the energy distribution of γ -rays produced per collision, over the energy of the CR. The computation is based on the parameterization from [162], and its implementation follows the work by [163]. In this paper, we use the Pythia8 proton-proton interaction model and include corrections for proton-nuclei collision [147]. \nOnce the rest frame production rate of γ -rays is computed, the radial profile and energy spectrum of γ -rays, as would be observed from Earth, are obtained by line of sight integration \nand eventually integrating over the energy or the solid angle, respectively. We also account for the Universe opacity as a function of photon energy using the extra-galactic background light (EBL) model from [164], but this choice does not affect our results given the fact that the corresponding cutoff is at sufficiently high energy, of about 30 TeV [26, 27]. \nIn proton-proton interactions, in addition to γ -rays, secondary electrons are obtained following the same scheme. In this case, however, the prescription from [165] is used instead of that from [162]. The spatial and spectral distribution of secondary CRe is then obtained from the particle injection rate by applying energy losses assuming a steady state scenario. \nModelling uncertainties associated with the particle interaction rates are expected to be accurate to about 30% at CTA energies, for the same parent CRp population (see Figure 7 from [147]). This is estimated by comparing the output from the different parametrizations implemented in MINOT (from [162]: Pythia8 , Geant4 , QGSJET , SIBYLL , and [165]). See also [166] for another recent determination of the γ -ray production cross-section.', '2.2.2 Inverse-Compton γ -ray emission': 'Leptonic γ -ray emission arises via the inverse-Compton scattering of CRe (secondaries, but also primary CRe whenever considered) onto cosmic microwave background (CMB) photons. The inverse-Compton MINOT calculations are based on the analytical approximation given by [167]. We will see later that, in practice, inverse-Compton emission should be negligible (from both primary or secondary electrons) for CTA observations and it will be ignored in the following sections.', '2.2.3 Synchrotron radio emission': 'Synchrotron emission should be modeled when considering radio data in order to calibrate our model or check that it does not imply excess radio signal compared to observations. The MINOT software computes synchrotron emission following [168], for which uncertainties are negligible for this work. This assumes that the orientation of the magnetic field is randomized as it is expected for radio halos.', '2.3 Calibration of the model parameters': 'In this section, we discuss the different methodologies employed to set our model parameters.', '2.3.1 Baseline model': 'As a baseline, we consider the results obtained from numerical simulations and cosmic-ray transport description (e.g., [42, 44]; see also [43] for an application with Fermi -LAT) in order to set the value of the free parameters of the model: η CRp , α CRp and X 500 (see Eq. 2.6). According to these works, the radial distribution of CR is expected to roughly scale with the thermal gas density ( η CRp = 1 ) when advection by the turbulent gas dominates, but may flatten if diffusion and streaming become significant. The spectral slope is expected to be α CRp ∼ 2 . 3 and a normalization, X 500 , of a few percent is usually expected. However, large uncertainties are associated with these values, such as the transport of CR for which the streaming velocity is poorly known [44], or the details of acceleration mechanism [21]. Our baseline parameter set is defined as ( X 500 , η CRp , α CRp ) = ( 10 -2 , 1 . 0 , 2 . 3 ) , but we explore a large range of values, as given in Table 1. \nIn Figure 2, we show the hadronic γ -ray observables associated with the models that we consider and how they vary as a function of the parameters. We also provide the associated radial profile of the ratio between integrated CRp energy and thermal energy, X CRp ( < R ) . \nTable 1 . Summary of the parameter values and their explored range, and the γ -ray flux at CTA energies for the hadronic and inverse-Compton emission (given as: reference value, [min, max]). The flux F 500 is computed within θ 500 by cylindrical integration for energies above 150 GeV and given in units of 10 -14 cm -2 s -1 . In the case of pure hadronic and pure leptonic models, the central value corresponds to the best-fit model and the interval corresponds to the 68% confidence level. The changes in α CRp for the different magnetic field models is only due to degeneracies in the parameter space and correlations with η B . It is not a physical effect and the difference remains well below statistical uncertainties. \nAll the models are calibrated to have the same normalization X 500 = 10 -2 . In the CTA energy range, the γ -ray flux decreases when increasing the slope α CRp and the profile gets more compact when increasing the scaling η CRp . We note that for a fixed normalization at R 500 , the flux computed within R 500 also increases with η CRp because of the increased protonproton collision rate due to the spatial overlap of the CR and the target gas. We note that the inverse-Compton emission arising from secondary CRe in the baseline model is always subdominant compared to the hadronic emission.', '2.3.2 Pure hadronic scenario': 'It is also possible to use radio data of the Perseus mini-halo to calibrate our model parameters in the case of a pure hadronic scenario. To do so, we extract the Very Large Array (VLA) surface brightness profile at 1380 MHz from [129] and use the spectrum measured at 327, 609, and 1395 MHz from [169], which were taken from [131]. We use an aperture of 15 arcmin, as it corresponds to the mini-halo extent given in this work. The profile is affected by the presence of NGC 1275 in the core. In addition the VLA observation may suffer from the absence of adequately short baselines that are necessary to properly detect the mini-halo emission at large scales. For this reason, we exclude the data points below 23 kpc, and those above 80 kpc in the fitting. We also assume a 10% uncertainty on the measured profile [170]. \nWe fit the radio data using a Markov Chain Monte Carlo (MCMC) technique with the emcee package [171]. We use a Gaussian likelihood function and fit simultaneously the spectrum and the profile. In addition to our three physical parameters, we consider a nuisance parameter to account for the inter-calibration uncertainty between the spectrum and the profile, which were extracted from different instruments, and marginalise over this value using a Gaussian prior centered on unity and with a standard deviation of 10%. Flat priors are used for the other parameters: η CRp ∈ [0 , 5] , α CRp ∈ [2 , 4] and X 500 ∈ [0 , 0 . 2] . These limits are defined according to realistic physical expectations of the parameter values. The upper limit of 0.2 on X 500 corresponds to the expected hydrostatic mass bias of galaxy clusters, itself dominated by turbulent motions plus magnetic fields, so that X 500 > 0 . 2 would be unrealistic (see [172] for a review about the mass of galaxy clusters). For a given model allowed by the radio data, we compute the γ -ray observables associated with hadronic interactions and \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 2 . Observable properties of the hadronic γ -ray emission for different values of the model parameters. The inverse-Compton emission is also reported as a thin line with similar color and style on the top panels. Top left panel : γ -ray emission spectrum in the case of hadronic processes (within R 500 ). Top right panel : γ -ray emission profile, as a function of the projected radius, in the case of hadronic processes (computed within [0 . 15 , 50] TeV). Bottom left panel : fraction of the enclosed γ -ray flux, as a function of the projected radius (computed within [0 . 15 , 50] TeV). Bottom right panel : enclosed CRp to thermal energy ratio, as a function of physical radius. Note that the two bottom panels do not depend on α CRp . The magnetic field model used when computing the secondary CR electrons that induce the inverse-Compton signal is the one from [159]. \n<!-- image --> \ninverse-Compton emission. We reproduce the constraint for some of the considered magnetic field models discussed in Section 2.1.2. The magnetic field model based on [159] with η B is used as a reference and we quantify the implications of this choice in Appendix B. \nIn Figure 3, we show the pure hadronic best-fit model, its 68% enclosed confidence interval, 100 model realizations randomly sampled from the MCMC chains, and the considered data points. The pure hadronic model provides a good description of the data that we have considered. Given the reference magnetic field model, it favors parameter values of ( X 500 , η CRp , α CRp ) ∼ ( 5 × 10 -2 , 0 . 8 , 2 . 5 ) , which is compatible with the MAGIC upper limit [90], but we stress that uncertainties are large and that the parameters are degenerate. Accordingly, the γ -ray observables are affected by large uncertainties, especially on the CR spectral slope, which leads to large uncertainties for the expected flux in the CTA energy range. We also stress that systematic effects may affect the radio data that we used. Moreover, this approach relied on a magnetic field model, which limits our prediction. Accounting for different parameterization of the magnetic field, we have bracketed the associated systematic uncertainty. Overall, our estimate is expected to be accurate within a factor of about \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 3 . Constraints on the radio and γ -ray observables in the case of the pure hadronic model, assuming the magnetic field model based on [159] and η B = 2 / 3 . The best-fit models are shown as solid blue lines (pink, in the case of inverse-Compton emission). The dashed lines provide the enclosed 68% confidence region and 100 model realizations are shown as a light lines. We also report the baseline model for the γ -ray prediction. Top left panel : radio spectrum within a 15 arcmin aperture diameter. The data were taken from [169]. Top right panel : radio profile at 1380 MHz. The data were taken from [129]. Following [170], we use 10% uncertainties and reject the points below 23 kpc (contamination from NGC 1275) and above 80 kpc (possible issues with large scale flux recovery). The grey areas correspond to the discarded data. Bottom left panel : γ -ray spectrum prediction, computed by cylindrical integration within θ 500 . Bottom right panel : γ -ray projected profile prediction, computed in the range 150 GeV - 50 TeV. \n<!-- image --> \ntwo in this scenario (see Appendix B).', '2.3.3 Pure leptonic contribution': "Although it is not physically motivated, as it would require a continuous in situ injection of CRe without protons, we consider the pure leptonic scenario as an exercise to address the inverse-Compton contribution. We use the radio data including CRe only and compute the γ -ray emission due to inverse-Compton. Since high-energy CRe suffer from major energy losses [173], the only way to observe inverse-Compton emission at CTA energies would be a scenario in which fresh CRe are injected in the ICM, as for the hadronic model. We consider that all the radio emission arises from a continuous particle injection with a power-law energy spectrum, and we apply energy losses given the magnetic field strength, the CMB at the \ncluster's redshift and the thermal gas. Given these assumptions, the γ -ray emission at CTA energies is similar to the inverse-Compton generated in the pure hadronic model, and thus much smaller than typical expectations from hadronic interactions in numerical simulations or in the pure hadronic model. In practice, continuous injection is not guaranteed and the amount of inverse-Compton emission may thus drop dramatically in the CTA energy range. This is why our computation only provides an upper limit to the inverse-Compton signal.", '2.3.4 Turbulent reacceleration model': 'The most favored scenario for the acceleration of relativistic electrons generating large-scale emission in the giant radio halos in galaxy clusters is based on turbulent reacceleration [29, 30, 32, 35, 174-177]. Turbulent reacceleration has been proposed also for mini-halos as an alternative to pure hadronic models [39, 169]. Turbulence may reaccelerate both primary (CRp and CRe) and secondary particles if they are present in the ICM. In these models, for a given radio luminosity, the level of γ -rays depends on the ratio between primary and secondary electrons that are reaccelerated. Even in the case where only primary CRp and their secondary products are considered, the γ -ray emission that is generated in these models is smaller than the one in pure hadronic models [92, 99]. As a consequence, if turbulence plays an important role in the acceleration process in the mini-halo volume, the γ -ray spectrum calculated in Section 2.3.1 provides an optimistic view of the expected signal. Modeling the expected γ -ray signal in the CTA energy band from the Perseus cluster in the case that the radio mini-halo originates from turbulent reacceleration is beyond the aim of this paper. Yet as a reference we just mention that modellings in the case of the Coma cluster, which hosts a giant radio halo, predict a γ -ray signal 4-6 times smaller than that in the pure hadronic scenario [32, 92]. Note that rescaling the CRp normalization of our pure hadronic model ( X 500 = 4 . 5 × 10 -2 ) by the same amount would give a value that is in line with that of our baseline model ( X 500 = 10 -2 ), so that our baseline model is in qualitative agreement with what one can expect from turbulence reacceleration. \nIn the next sections, we will use the models discussed in Section 2.3 to address the CTA sensitivity to the CR physics of the Perseus cluster in the context of the proposed KSP observations.', '3 Modelling the γ -ray emission associated with dark matter': "In this section, we build a DM profile for the Perseus cluster following the most up-to-date results both from observations and numerical cosmological simulations. \nAssuming DM is completely composed of WIMPs [57, 178], we can compute the expected DM-induced γ -ray emission from any astrophysical source as \nd Φ γ dE (∆Ω , l.o.s, E ) = dϕ γ dE ( E ) × J (∆Ω , l.o.s ) . (3.1) \nFor a given energy E , line of sight l.o.s , and solid angle ∆Ω subtended by the region of interest, d Φ γ dE is the DM-induced γ -ray flux, dϕ γ dE contains the spectral information about the expected emission, and J , referred as the 'astrophysical factor', encloses the details of the spatial morphology of the putative signal. In order to perform this DM flux computation, we will assume that WIMPs annihilate into Standard Model (SM) particles in the halo of Perseus (in the following, the annihilation scenario) or that they decay into SM particles \n(in the following, the decay scenario). According to these two scenarios, the expression in Equation 3.1 becomes these two, respectively: \nd Φ ann γ dE (∆Ω , l.o.s, E ) = < σv > 8 πm 2 χ dN γ dE 0 ∣ ∣ ∣ ∣ E 0 =(1+ z ) E × J ann (∆Ω , l.o.s ); d Φ dec γ dE (∆Ω , l.o.s, E ) = 1 4 πm χ τ χ dN γ dE 0 ∣ ∣ ∣ ∣ ∣ E 0 =(1+ z ) E × J dec (∆Ω , l.o.s ) , (3.2) \nwhere dN γ dE is the number of photons per unit energy we expect from a given annihilation channel and DM particle mass m χ , < σv > is the velocity-averaged annihilation cross-section in the annihilation scenario and τ χ is the mean lifetime of the DM particle in the decay scenario. In deriving these expressions, we have assumed Majorana WIMPs (for Dirac WIMPs it would be necessary to multiply by a factor 1/2 [178]. To obtain the corresponding fluxes, we use for dN γ dE the results from [179], including electro-weak corrections. For the thermallyaveraged cross-section, we expect a value around 3 × 10 -26 cm 3 s -1 [180, 181] to produce the observed DM relic abundance [182], while the main bound for the decay lifetime to account for the observed DM density is the current age of the Universe ∼ 10 17 s, despite the efforts of trying to obtain tighter constraints from just gravitational probes [183]. We recall that for decaying DM only half of the energy budget stored in m χ is available per SM particle produced in two-body decays. We can also define the spatial morphology of the signal, the so-called J-factor, as: \nJ ann (∆Ω , l.o.s ) = ∫ ∆Ω d Ω ∫ l.o.s ρ 2 DM ( r, l ) dl ; J dec (∆Ω , l.o.s ) = ∫ ∆Ω d Ω ∫ l.o.s ρ DM ( r, l ) dl, (3.3) \nwhere ∆Ω = 2 π (1 -cos α int ) , being α int the integration angle and ρ DM ( r ) the DM density profile. \nFrom Equation 3.3 the main dependencies in both scenarios can be deduced [72]: \nJ ann ∝ M 200 c 3 200 d 2 L ; J dec ∝ M 200 d 2 L , (3.4) \nwhich are the mass M 200 13 , the luminosity distance to the Earth d L and the concentration c 200 14 for the annihilation case. From these dependencies, one can clearly see why local galaxy clusters are the best targets to consider to test decaying DM [184], since they are the most massive objects in the Universe. For the case of annihilation it may not seem so straightforward. Yet, in the last years some studies where performed [70] comparing the suitability of galaxy clusters and dwarf spheroidal galaxies (dSphs) for γ -ray DM searches, concluding that galaxy clusters can also provide competitive results. One key element for this outcome is to consider the distribution of smaller halos that galaxy clusters should host according to the Λ CDM structure formation theory [76], usually called subhalos. The role of these substructures come into play through the c 200 dependency, since we expect subhalos of \na given mass to be much more concentrated than main halos of the same mass. In contrast, in the decay scenario (see Equation 3.4) the subhalos do not provide a sizable contribution since their masses are really low compared to the distance. Thus, we will not consider them for this scenario. \nThe morphology of the expected DM γ -ray emission is strictly determined by the DM density profile ρ DM and in case of annihilation, particularly by the DM substructure. In the next sections, we present the models we build for the DM distribution in the main halo of Perseus as well as a model for the population of subhalos.", '3.1 Perseus main halo': 'The main DM halo of Perseus can be described through a Navarro-Frenk-White (NFW) density profile [185, 186]: \nρ NFW ( r ) = ρ 0 ( r r s )( 1 + r r s ) 2 , (3.5) \nwhere r s is the scale radius and ρ 0 the normalization of the DM density. Introduced as the result of DM-only N-body simulations, the NFW model falls in the family of "cuspy"-like profiles. \nTo obtain the parameters of the NFW profile, we need to consider a concentration-mass ( c -M ) relation. Consistently, this relation should comprehend the mass scales involved in cluster physics. Taking this into account, we use the parametrization developed in [72] for main halos. This concentration-mass relation was found to have an associated 1 σ scatter of 0.14 dex. We build the DM density profile starting from the measured mass of Perseus. We consider the halo to fulfill the spherical collapse model for overdensities ∆ = 200 times the critical density of the Universe ρ crit . This allows us to calculate M 200 shown in Table 2 from the measured X-ray mass, M 500 , provided in [146]. \nWe can also obtain the radius R 200 that contains an enclosed mass M 200 , as \nR 200 = ( M 200 4 3 π ∆ ρ crit ) 1 / 3 , (3.6) \nwhose corresponding projected angle is θ 200 = arctan ( R 200 d L ) . Now we can easily compute the scale radius in Equation 3.5 as \nr s ≡ R 200 c 200 . (3.7) \nFinally, the normalization of the density profile ρ 0 is obtained by imposing the recovery of M 200 after the integration over the cluster volume of the profile ( M 200 = ∫ R 200 0 ∫ 4 π 0 ρ NFW ( r ) r 2 drd Ω ) and isolating ρ 0 : \nρ 0 = 2 ∆ 200 ρ crit c 200 3 f ( c 200 ) , (3.8) \nwhere f ( c 200 ) = 2 c 2 200 ( ln (1 + c 200 ) -c 200 1+ c 200 ) . Table 2 lists the corresponding parameters of the obtained NFW DM density profile following the above description. \nIt is interesting to compare the value we obtained for c 200 and reported in Table 2 with that obtained from observational data. Indeed, from X-ray observations an observational value of c 200 = 5 . 0 ± 0 . 5 was inferred [187], which well matches our adopted c 200 . \nTable 2 . NFW density profile parameters for the Perseus galaxy cluster; see Equations (3.5-3.8) for the definition of each parameter and accompanying text for details on their derivation.', '3.2 The role of substructures': "According to the Λ CDM hierarchical bottom-up structure formation scenario [188, 189], galaxy clusters should host a significant amount of substructure or subhalos. Due to stripping processes, these subhalos are expected to have higher concentrations than main field halos of the same mass (e.g., [73]). Because of these high concentrations, we expect them to have a considerable impact in the annihilation fluxes (Equation 3.4). This enhancement effect in J ann -factor, usually called boost factor B , was already estimated in [70] for a selected list of local clusters. These authors found that halo substructure in galaxy clusters could provide boost factors of the order B ∼ 40 , in general agreement with current, more refined boost calculations from N-body simulations [73, 190]. In order to account for the contribution of these substructures to the J ann -factor, we factorize the computation in the following way: \nJ ann (∆Ω , l.o.s ) = ∫ ∆Ω 0 d Ω ∫ l.o.s ( ρ main ( r ) + N sub ∑ i ρ sub ( r ) ) 2 dl = = J ann main + < J ann sub > + < J ann cross -prod >, (3.9) \nwhere N sub is the total number of subhalos, ρ main is the DM profile of Perseus' main halo obtained in Section 3.1 and ρ sub the DM profile for each subhalo. Due to the complex dynamics that substructures undergo (e.g., tidal stripping, dynamical friction, interaction with baryons, etc. - see [189, 191-193]), the precise survival probability of the smallest subhalos is not yet known [194-197]. In any case, when it comes to the subhalo density profile ρ sub , deviations from NFW are expected to be pronounced mainly in the outermost subhalo regions, where mass losses are severe. Therefore we keep adopting the NFW profile also to model the subhalo inner structure. We remark that due to their small extension, we do not expect individual subhalos to be spatially resolved by CTA. Therefore, we decided to average their contribution in < J ann sub > . We then model the population of subhalos as: \nd 3 N dV dMdc = N sub dP V dV ( R ) dP M dM ( M ) dP c dc ( M,c ) , (3.10) \nwhere P i with i = V, M, c represents the probability distribution in the volume of the main halo V , of the subhalo masses M and the subhalo concentrations c . The probability distributions in Equation 3.10 have been modeled based on results of DM-only cosmological simulations, namely from Via Lactea-II (VL-II, [198, 199]), Aquarius [200, 201] and the work of [73]. We describe each of these distributions in the following: \n- · Subhalo distribution within the main halo dP V dV ( r ) : As we assume that Perseus main halo is spherically symmetric, the distribution within the volume collapses to the distribution in radius, so in the following we refer to it simply as the Subhalo Radial Distribution \n(SRD). Based on N-body simulations, we choose to follow the antibiased relation [199] 15 , defined as: \ndP V dV ( r ) = ρ main ( r ) r/r a 1 + r r a , (3.11) \nwhere r a works as a scale radius and it is defined by the fraction of the total mass that will be in form of subhalos f sub . For the distribution of subhalos, we also consider the number of substructure levels to N lvl = 2 (subhalos inside subhalos) 16 . \n- · Subhalo mass distribution dP M dM ( M ) : Also known as the subhalo mass function (SHMF), is usually modelled using a power-law (e.g., [204]): \ndP M dM ∝ M -α . (3.12) \nValues range between α = 1 . 9 [200, 201] and α = 2 . 0 [198]. This slope is key to evaluate the contribution of substructure to the J ann -factor [73]. For higher values of α , i.e., a more numerous population of small subhalos, we obtain proportionally higher boosts. As a way to account for this uncertainty, we will define different benchmark models covering different physical scenarios for the abundance of subhalos. A value of α = 1 . 9 , more in line with latest simulations results [189, 205], will lead to conservative values for the J ann -factor, while α = 2 . 0 will yield an upper bound to subhalo boost values. The mass budget in both cases is a fraction of the total mass, f sub . Assuming a value for the minimun subhalo mass of M min = 10 -6 M ⊙ and that the maximum subhalo mass in terms of the host is M % max = 0 . 01 17 , we need different values of f sub to conserve the total mass. Integrating the SHMF with the different values of α for the selected M min and M % max , we obtain a value of f sub = 0 . 182 for α = 1 . 9 , and f sub = 0 . 319 for α = 2 . 0 . \n- · Subhalo concentrations dP c dc ( c, M ) : In the same way we assumed a concentration-mass relation for the main halo, we select a ( c -M ) relation to describe subhalo DM profiles. We adopt the state-of-the-art ( c -M ) subhalo model by [73], which includes a radial dependence of the concentration within the main halo: \nc 200 ( M 200 , x sub ) = c 0 [ 1 + 3 ∑ i =0 [ a i log 10 ( M 200 10 8 h -1 M ⊙ )] i ] × [1 + b log 10 ( x sub )] , (3.13) \nwhere c 0 = 19 . 9 , a i = [ -0 . 195 , 0 . 089 , 0 . 089] , b = -0 . 54 and x sub refers to subhalo distance with respect to the center of the host halo. The importance of including a ( c -M ) relation specifically derived for subhalos resides, as already stated, in the fact that subhalo concentrations are known to be higher than that of field halos of the same mass [73]. Thus, their contribution to the J ann -factor is expected to be critical. \nNote that, in deriving this relation, [73] assume NFW profiles for subhalos, keeping our modelling consistent. Every concentration-mass relation is known to exhibit an intrinsic scatter [73, 206], yet it is highly computationally expensive to take it into account for each subhalo in the field halo. Then, for the sake of this study we decide to neglect it, as its impact on the J ann -factor will lie within the spread we have by considering different values for α in the SHMF. \nAfter the detailed discussion on the parametrization followed for the subhalo population modelling, we establish three benchmark models for the computation of the J-factors. Each of them represents an expected different level of contribution of the subhalo population to the annihilation flux: \n- · MIN: considers the Perseus main halo and neglects the existence of substructures.\n- · MED: the SRD follows the antibiased relation. We adopt α = 1 . 9 for the slope of the SHMF (Equation 3.12), using a coherent subhalo mass fraction of f sub = 0 . 182 .\n- · MAX: similar to the MED model but we choose α = 2 . 0 for the slope of the SHMF (Equation 3.12), with a coherent adjustment of the subhalo mass fraction to f sub = 0 . 319 . \nThese models and their values are summarized in Table 3. With the definition of the above benchmark models, we aim to bracket a wide range of possible substructure scenarios. This will translate into a bracketing for the possible values of the J ann -factor, being the MED model our best guess and for which we will produce the main results, and the MIN and MAX models as realistic lower and upper limits to the contribution levels of substructures. For decaying DM, we will use a realistic value for J dec assuming the MIN model, since subhalos do not provide a sizeable contribution to it (see Equation 3.4). \nTable 3 . Summary of the defined benchmark DM models for the annihilation interaction scenario. From left to right, ρ sub is the main DM profile for both the main halo and the subhalos, ( c -M ) main is the concentration-mass relation used for Perseus main halo, SRD is the Subhalo Radial Distribution, ( c -M ) sub is the concentration-mass relation for the subhalos, α is the slope of the SHMF (Equation 3.12) and f sub is the fraction of the total mass bound in substructures, see text for more details. Regarding the models, 'SC+14' refers to the concentration model of [72], antibiased to the SRD from [199] from Equation 3.10, 'M+17' to the ( c -M ) subhalo model of [73] from Equation 3.13.", '3.3 Dark matter annihilation and decay fluxes': 'Once defined the DM models we use for Perseus cluster and for its expected subhalo population, we are interested in computing their γ -ray induced fluxes, the J-factors. We use the publicly available CLUMPY code [202, 207, 208] to compute them. The obtained integrated Jfactors for the cluster and for each benchmark model are summarized in Table 4 and shown in the left panels of Figure 4. The right panels in this same figure show the differential J-factors as a function of the angle from the center of Perseus. \nTable 4 . Perseus integrated J-factors in the case of the three benchmark annihilation scenarios (MIN, MED, MAX - see Table 3) and the J dec -factor for the decay scenario. In all cases, we integrate the DM signal up to R 200 ( α int = θ 200 ). See text for details. \nFigure 4 . Integrated J-factors (Equation 3.3) versus the integration angle α int ( left panels ) and differential J-factors versus the radial angle θ ( right panels ) for the annihilation scenario ( top panels ) and decay scenario ( bottom panels ) for the three substructure benchmark models correspondingly. In all the panels θ 200 and the projected angle of the scale radius, θ s , are shown as a measure of the containment of the emission and the extension of the cluster. \n<!-- image --> \nFrom the J ann -factor values given in Table 4 for each benchmark model and from Figure 4 upper panels, we can better understand the impact of including subhalos in our calculations. Indeed, substructures do not only boost the expected annihilation signal but also modify its spatial morphology, subhalos being particularly important in the outskirts of the cluster. The latter is the main reason to also take into account the expected extension of the cluster DM emission in the analysis itself, as it will be described later on. Although the precise \nproperties of the subhalo population within the cluster is the main origin of uncertainty in the computation of the J ann -factors, we also wanted to check the impact of the uncertainty in the M 200 value and the one from the scatter in the concentration-mass relation. The mass uncertainty from the study of [146] is too small to have an impact itself, thus we compute two different values of M 200 assuming extreme values for the hydrostatic mass bias ( b HSE = 0 , 0 . 3 ) and propagate this to the J ann -factor computation. This results into σ J ann = 0 . 002 -0 . 005 dex correspondingly, which is two orders of magnitude smaller than the variation introduced from the substructure benchmark models ( σ J ann ∼ 1 dex). If we also include the 0.14 dex from the scatter of the concentration-mass relation for each of the above extreme mass values, we obtain σ J ann = 0 . 2 dex, a J ann -factor uncertainty similar to what is typically obtained for dSphs [100]. Note though that this σ J ann is still well within the uncertainty related to the different considered models for the substructure population. Therefore, from now on we only consider as a theoretical uncertainty the spread among the three substructure benchmark models. As for decay, we compute the uncertainty of J dec from only the use of different mass values and from the scatter in the concentration-mass relation as well, since the substructures do not play a role for this case. We found σ J dec ⪅ 10 -3 dex for all cases, thus we decided to disregard this in the following. \nTo quantify the enhancement in the total annihilation flux due to the presence of subhalos, we define a subhalo boost factor as: \nB = J X ann /J MIN ann -1 , (3.14) \nwhere X can be either MED or MAX ( B = 0 means no substructure contribution in our definition 18 ). For Perseus and our MED and MAX models, we obtain boost factors of B = 9 . 2 and B = 59 . 3 , respectively. \nWe can compare our obtained J-factors for the annihilation scenario with the results in [70] 19 . For the MIN scenario (no substructure inclusion), these authors obtain log 10 ( J ann ) = 17 . 35 (units of GeV 2 cm -5 ), very similar to our results (Table 4). For the case where they include substructure (which, according to their modelling, would be roughly similar to our MED case), the J ann -factors increase up to log 10 ( J ann ) = 18 . 23 , also compatible with our MED results (Table 4). The boost factor that they obtain between their two models is B = 6 . 6 (following our definition in Equation 3.14), a value just around one point lower than our results for the MED model (Table 4). This slight mismatch is expected due to some differences in the subhalo modelling in each case. In order to perform a comparison with more recent results, we can examine the boost values we should expect from using the ( c -M ) relation of [73]. In this case, for a Perseus-like galaxy cluster halo, they obtain B ∼ 9 for α = 1 . 9 and B ∼ 72 for α = 2 . 0 . The agreement with our MED scenario is absolute, while for MAX we get a relative discrepancy of ∼ 20% 20 . \nA similar comparison can be performed for the decay scenario as well. Given their large masses and relatively close distances, galaxy clusters are the perfect laboratories for probing DM decay and, accordingly, there have been several previous studies in this matter (see refs. in Section 1). One of the latest works on DM decay in the TeV range focused on Perseus and \nconsisted of an observational campaign performed by the MAGIC collaboration [116]. In this work, authors used the main halo model introduced in [70] to compute the expected decay flux, obtaining a value of log 10 ( J dec ) = 19 . 18 (units of GeV cm -2 ), compatible within 5 % with our value from Table 4. \nAs a last step, we create 2D spatial templates of the expected DM annihilation/decay emissions using CLUMPY . Four maps are created, three for annihilation (each one corresponding to one of our benchmark models in Table 3), and one more for decay. These maps are all shown in Figure 5. As seen already in Figure 4, the presence of substructures mainly impacts the morphology of the DM signal in the outer regions of the cluster. \nFigure 5 . Two-dimensional spatial templates of the expected DM emission in Perseus, quantified in terms of the differential J-factors on the z axis, for the three considered MIN, MED, MAX annihilation scenarios ( top panels and left bottom panel ) and for decay ( bottom right panel ). See Tables 3 and 4 for details on each DM model. \n<!-- image -->', '4 Observation setup': 'Before considering the CTA data analysis, we discuss the configuration of the observations. In addition to the instrumental background due to CR induced air showers, we present the modelling of the known sources located in the Perseus cluster region which may affect the analysis. We also investigate how the observing strategy will impact the CTA sensitivity to a putative cluster signal. \nTable 5 . CTA observation configuration summary.Notes. † This IRF corresponds to the full configuration with 15 MST and 4 LST. More recent IRFs are now also available, corresponding to the CTA initial configuration including 9 MST and 4 LST (prod5 [209]). The corresponding on axis sensitivity is reduced by a factor of about 30% at high energies ( E ≳ 500 GeV), but the off-axis sensitivity is slightly improved.', '4.1 CTA pointing configuration': 'The CTA observing setup will consist in a set of pointings, with a radial offset with respect to the Perseus cluster center, θ pointing . Such offset is necessary to analyse the data using classical ON-OFF techniques used in Imaging Air Cherenkov Telescopes (IACTs), in whose case the background is estimated using a mirror region equivalent to the region of interest assuming the azimuthal symmetry of the background with respect to the pointing center [210]. For template-based analysis, in which the full region of interest is modeled using dedicated templates that describe the different sky components and are fit to the data, such offset could be set to zero. Various position angles of the pointing offset can also be considered. This would allow us to cover a larger field around the cluster and thus to increase the chance of serendipitous discovery of new sources. The CTA field of view depends on the energy, given the field of view of the respective telescope class dominating each energy regime: about 4.3 deg diameter for the large-sized telescope (LST) and 7.5 deg diameter for the medium-sized telescope (MST). See [119] and [120]. \nA total of 300 hours are proposed for observations of the Perseus cluster in the course of the CTA key science program and this is what we assume as our ON-source time for the remaining. Nonetheless, we consider a dead-time fraction of 5% so that the effective observing time is 285 hours. The observing time will be split into the different pointings. The pointing directions are split between 8 positions, equally spaced from 0 to 315 deg around the cluster reference center, with steps of 45 deg. Nevertheless, we stress that our results do not significantly depend on the number of position directions that we consider. \nBecause the Perseus cluster field is only observable from the CTA north site (La Palma), we consider the instrument response function (IRF, which connects the sky γ -ray signal to the actual measurement) associated with long observations [211] from the north site 21 . We neglect any variation of the zenith angle of the target source during the observations and assume that the zenith angle is fixed to 20 deg for all pointings. For comparison, the zenith angle ranges from 12 to 36 deg for the MAGIC observations presented in [88], also performed from La Palma and within which range there is no significant difference. The IRFs are also used to model the instrumental background. Note that IRFs describing the initial CTA array configuration are now available: 9 MST and 4 LST instead of the 15 MST and 4 LST used for the present work. They would imply a reduced sensitivity by about 30% at high energies, compared to the IRFs used here, although the off-axis sensitivity improved and could make \nup for some of the deficit in telescope number. \nWe provide a summary of the CTA observation setup in Table 5. A schematic view of the CTA observations configuration is provided in Figure 6. We include NGC 1275 and IC 310 (see discussions in Section 4.2). The X-ray peak (reference center) is aligned with NGC 1275 and IC 310 is offset toward the southwest by about 0.6 deg but remains within θ 500 . For a given pointing, the OFF regions are spread over a circle around the pointing for which the radius corresponds to the pointing offset from the source. This is illustrated by the yellow (OFF) and orange (ON) filled regions for which we have used a radius of 0.5 deg. Also shown are the locations of those sources in the Fermi -LAT 4FGL catalog [212, 213] within five degrees from the cluster center.', '4.2 Background sky': 'In addition to the diffuse emission from the Perseus cluster, it is necessary to model other sources that are located around the target cluster. At very high energies, two γ -ray point sources have been detected by the MAGIC telescopes, namely NGC 1275 and IC 310. \nNGC 1275, the brightest galaxy in the Perseus cluster, is a variable source presenting flare activities up to fifty times its mean flux at E > 100 GeV, with observed day-by-day variability [214]. It is not possible to predict what will be the exact state of the source at the time of CTA observations. Yet, we expect that a large majority of the data will be obtained when the source is in a quiescent state and periods with high intensity can be removed from analysis. We therefore assume that the spectrum of NGC 1275 follows the quiescent energy spectrum obtained from MAGIC observations as presented in [90]. This spectrum is well described, as a function of energy E , by a simple power-law given by \ndN dEdSdt = 2 . 1 × 10 -11 ( E 200 GeV ) -3 . 6 cm -2 s -1 TeV -1 . (4.1) \nIC 310, a member galaxy of the Perseus cluster, is also a variable source presenting high-amplitude and short duration flares [215]. While its spectral shape was observed to not significantly change, its amplitude does by a factor of up to ∼ 7 . Similarly to NGC 1275, its spectrum is modeled as a power-law, \ndN dEdSdt = 0 . 741 × 10 -12 ( E 1 TeV ) -1 . 81 cm -2 s -1 TeV -1 , (4.2) \ncorresponding to the quiescent state of the source [215]. This source state corresponds to the large majority of the reported observations. We note that, because we apply EBL absorption a posteriori when performing simulations, we use the intrinsic MAGIC best-fit model. The equatorial coordinates of IC 310 are set to (R.A., Dec.) = (49.179,+41.325) deg. \nIn addition to NGC 1275 and IC 310, several sources have been detected at lower energies with Fermi -LAT around the Perseus cluster. In appendix C we list these sources and estimate their fluxes in the CTA energy range. Given their location, none of them should affect the analysis. \nFinally, we consider the contribution from the Galactic diffuse emission in the region around the Perseus cluster using the work by [216]. Within θ 500 , the emission is expected to be smooth and present a soft gradient. Compared to other contaminant sources, the instrumental background, and the cluster diffuse emission (e.g., baseline CR model), the Galactic γ -ray foreground should be largely subdominant over all the considered energy range. We refer to Appendix D for more details. Because of the uncertainties in the foreground model, its spatial structure and the fact that it is subdominant, it is not considered in the following. \nFigure 6 . Schematic view of the Perseus cluster region and the assumed observation setup. An illustration of the ON-source and OFF-source regions is shown for one pointing, assuming a 0.5 deg aperture radius. IC 310 and NGC 1275 are observed at very high energy and are shown as cyan stars. 4FGL sources within a 5 deg radius around the cluster center are also depicted as blue crosses. For comparison, the point spread function (PSF) of CTA at 1 TeV is shown in the bottom left corner. \n<!-- image -->', '4.3 Towards an optimal observing strategy': 'In this Section, we explore how the observation setup parameters described in Section 4.1, together with the background sky discussed in Section 4.2, will affect the expected detection significance. To do so, and as it will be the case in Section 5, we use the ctools software 22 [217]. We focus on the CR component since the observation setup selection will be driven by the CR case. We have also checked the influence of different setups for the DM case \n(Appendix I) and the chosen one is appropriate for both science cases. Our background model includes both the instrumental background and the two AGNs NGC 1275 and IC 310. We consider the case of ON-OFF analysis, and the case of template fitting. \nON-OFF analysis We first consider the case of a classical ON-OFF analysis. A pointing offset, θ pointing , is necessary to define the OFF regions that will be used to monitor the background (see Figure 6). In addition to the pointing offset, the aperture radius of the ON region will affect the detection significance depending on the shape of the signal. For instance, a small aperture will be favored for compact sources because it will lead to a lower instrumental background while keeping most of the signal photon counts. By contrast, a larger aperture will be preferred for more extended sources. The size of the ON region is also limited by the pointing offset, for which we aim at testing the impact. Here, we request at least 3 independent OFF regions so that the radius of the ON region, θ ON , is constrained to θ ON ≤ √ 2 θ pointing . Such condition will also affect the maximum significance for a given pointing offset. We also consider the possibility of masking the known point sources. We use an energy-dependent aperture radius proportional to the point spread function (PSF) 68% containment angle θ PS ( E ) = N PSF × PSF( E ) , where N PSF can also be varied and for which the optimal value may depend on the diffuse cluster model. \nIn order to optimise the observation setup, we compute the expected significance according to [218]: \nσ = √ 2 × ( d × ln ( d m ) + m -d ) . (4.3) \nThe quantity d corresponds to the expected photon counts (between 150 GeV and 50 TeV) of the diffuse cluster signal plus background in the ON region, after accounting for the pointsource mask. The quantity m corresponds to the expected photon counts associated with the background only for the same region. The background includes both the instrumental background and the point sources, assuming that their spectral energy distribution (SED) is known (see Section 4.2) and kept fix. The cluster and point-source models are convolved with the IRFs. We compute σ for a set of pointing offset values, for each of which we vary the values of the ON region aperture θ ON and the size of the point-source mask θ PS . The counts are summed over all the energy bins. \nTemplate fitting Beside the classical ON-OFF analysis technique, we consider the case of template fitting, which might be more appropriate for diffuse sources assuming that the background can be properly modeled [121], as it will be further explored in Section 5. To do so we perform simulations of the observations by varying the pointing offset, and use the maximum likelihood technique to fit the data and recover the test statistic value. We fit for the normalization of the cluster model, for the instrumental background normalization and spectral tilt, and for the normalization and spectral index of the two point sources. We use the square root of the recovered test statistic ( √ TS ) as an estimate of the significance, with \nTS = 2 × ln L ( H 1 ) L ( H 0 ) , (4.4) \nwhere L ( H 1 ) is the Poisson likelihood associated with the assumed emission model (i.e., including the cluster) and L ( H 0 ) corresponds to the Poisson likelihood of the null-hypothesis (i.e., without including the cluster) [219]. We perform this test for the different diffuse cluster models. We perform several simulations for each case to obtain a measurement of the uncertainty on the recovered significance. \nResults and discussions In Figure 7, we show the expected significance of the ON-OFF analysis as a function of θ PS and N PSF in the case of θ pointing = 1 . 5 deg. Note that the pointsource mask aperture cannot be larger than the size of the ON region because NGC 1275 sits at the center of the cluster. Although the best significance always corresponds to no pointsource mask ( θ PS = 0 ), because the point source was accounted for in the background, we can observe that the recovered significance as a function of the mask and the ON region aperture highly depends on the cluster signal. This implies that the precision in the modeling of NGC 12175 will be more critical as the cluster diffuse emission is more compact (higher value of η CRp ). For instance, when fixing the ON region aperture to its optimal value, masking the point source with N PSF = 1 will reduce the significance by about 1% for η CRp = 0 and about 50% for η CRp = 1 . 5 . In practice, the optimization of the mask aperture will also depend on how well the central point source, NGC 1275, can be modeled. For instance, understanding the tails of the PSF could be a major issue when searching for diffuse emission, but we leave these investigations for future work, when the real CTA data will be available. We also observe that the best ON region aperture increases with the extent of the cluster (parameter η CRp ), going from about 0.4 deg to 1 deg for the considered cases and using 1.5 deg as the pointing offset. We also compute the maximal value of the significance as a function of the pointing offset, as shown in Figure 8, left panel. As expected, the shape of the curve depends on the diffuse cluster model and we can observe maximal values between about 0.2 deg and 1.5 deg for all the cases. \nThe results of the template fitting analysis are presented in Figure 8, right panel, where we can observe the normalized significance as a function of the pointing offset for all four cluster models considered (fixing X 500 = 10 -2 , α CRp = 2 . 3 , and varying the spatial parameter η CRp = (0 , 0 . 5 , 1 , 1 . 5) ). The significance smoothly decreases with the pointing offset, depending on the combination of the effective area and the instrumental background. Unlike for the ON-OFF analysis, no significant significance decrease is observed for small pointing offsets. We note that the background associated with NGC 1275, relative to the cluster signal, does not depend on the offset and acts as an extra background. For this reason, the sensitivity as a function of the radius is flattened by the presence of the central galaxy. We do not observe any significant impact of the cluster diffuse model in this case. The significance remains relatively constant up to a radius of about 1 deg and vanishes beyond. \nAccording to the results presented above and in order to allow for ON-OFF analyses while maximizing the significance of the template analyses, the pointing offset will be set to 1.0 deg in the following. This would enable to choose between either an ON-OFF or a template-based analysis after the actual observations have been made without a significant lose of sensitivity in each case.', '5 CTA sensitivity to CR induced γ -ray emission from Perseus': 'In this section, we use the cluster models defined in Section 2 together with the observation setup and background sky defined in Section 4 in order to predict the CTA sensitivity to CR induced γ -ray emission from the Perseus cluster. This is done via the use of the publicly available package KESACCO (Keen Event Simulation and Analysis for CTA Cluster Observations 23 ). KESACCO is a python package that is based on the ctools software [217], and dedicated to perform the simulation and analysis of galaxy clusters observed with CTA. In Appendix F, we validate our results using the gammapy software [220]. We first discuss the data preparation. \n=1.5 \n=1.0 \nFigure 7 . Normalized signal-to-noise ratio as a function of the radius of the ON region and the radius used to define the point-source mask (in units of N PSF ), for different cluster models defined via the η CRp parameter. Black lines correspond to 90%, 99% and 99.9% of the peak significance. The pointing offset was set to 1.5 deg in the present case. \n<!-- image --> \nON-OFF \nFigure 8 . Normalized significance as a function of the pointing offset in the case of the ON-OFF analysis (left) and the template analysis (right). \n<!-- image --> \nTemplate \nWe then quantify the degeneracy between the different signal and background components. Finally, we perform the analysis of simulated data using different methods and under different assumption for the signal to address the sensitivity of CTA to the ICM induced γ -ray emission.', '5.1 Data preparation': 'The analysis and results presented in this section are based on event files simulated according to the setup defined above. The events are binned in energy and according to their sky coordinates. We consider 30 bins in energy and 0.02 deg sky pixels to make sure that the PSF and spectrum sampling is sufficient, unless otherwise specified. The region of interest is centered on the Perseus cluster and is 3 deg wide in diameter. Although we consider eight different pointing positions, we use a stacked analysis in which all data from multiple observations are stacked into a single counts cube for each sky and energy bin. Unless otherwise stated, we focus on 3D template analysis as it is expected to be more appropriate in the case of such extended sources and given the presence of the two AGN. Indeed, this method will allow us to constrain the shape of the diffuse Perseus cluster γ -ray emission in a straightforward way. \nThe CTA sensitivity is expected to be best around a few TeV, but covers energies from 20 GeV to 300 TeV. Because of the EBL attenuation, the differential flux of all sources which we account for in this paper (all located at the same redshift) is expected to drop drastically around 30 TeV (see Figure 2, top left panel). Therefore, we consider a maximum energy of 50 TeV for the analysis. We also conservatively increase the low energy threshold to 150 GeV. This choice is driven by the fact that we aim at constraining the spectrum and the shape of the ICM induced emission in the presence of the steep spectrum source, NGC 1275, which strongly dominates at low energy where the PSF is also larger. Such a threshold will allow us to avoid any significant bias that could arise due to mismodeling, but its value could be adapted when the true data will be available. This choice will not apply to the DM search in Section 6.', '5.2 Sky components and degeneracy': 'We first investigate how the components of the sky model compare to each other and how degenerate they are. We focus on the region enclosed within 1 deg from the cluster center( ∼ θ 500 ). This region contains the two point sources and a large fraction of the cluster flux. In Figure 9 (left panel), we present the stacked sky map including both the astrophysical and instrumental background, computed over the full energy range. The location of the different pointing centers is shown, falling very close to the radius θ 500 . The location of the two point sources are indicated as green circles and they are both clearly visible in the map. The cluster diffuse emission (in the case of our baseline model here) is blended with NGC 1275 and cannot be clearly distinguished at this stage. Figure 9 (right panel) shows how the different model components compare to each other. The instrumental background is the dominant component, by about an order of magnitude, depending on the energy. NGC 1275 dominates at low energy and IC 310 dominates at high energy. The cluster emission is subdominant in this case (baseline cluster model) given the area that is considered. Note that the signal from IC 310 can be easily masked given its location. \nSecondly, we perform the joint likelihood fit of all the model free parameters: normalization and spectral index of the instrumental background, normalization and spectral index of both NGC 1275 and IC 310, and amplitude of the cluster model. We extract the correlation matrix between these components as reported in Figure 10 for the baseline cluster model. The parameters associated with IC 310 are not significantly degenerate with any other model components given its sky location. On the other hand, small degeneracies are observed between the cluster amplitude and the instrumental background (correlation ∼ 0 . 10 ), as well as the parameters associated with NGC 1275. We note that the degeneracy depends on the cluster model parameters. For instance, a more extended cluster presents less degeneracy \nFigure 9 . Left panel: combined counts per pixel map, over the full energy range (150 GeV - 50 TeV). The bottom left grey circle gives the PSF at 1 TeV, and also accounts for an extra 0.1 deg smoothing used for visual purpose. The grey crosses give the pointing coordinates, the green circles show the position of the point sources, the grey dashed lines show the radius θ 500 and the location of the cluster center is indicated by the cyan cross. Right panel: counts per bin measured as a function of energy within one degree from the cluster center. Contributions from the instrumental background, the point sources and the cluster diffuse emission are shown. This is in the case of the baseline model. The normalized residual is also shown at the bottom of the figure. \n<!-- image --> \nwith NGC 1275 but is more degenerate with the instrumental background. We also note that increasing the size of the region of interest leads to a slightly lower instrumental background degeneracy because more leverage is available. The degeneracy also depends on the energy range used to select the data. \nAccording to the degeneracy between the model components and the amplitude of these components within the central region of the field of view, we expect that the instrumental background will lead to most of the uncertainties associated with the cluster model constraints that can be extracted. The presence of NGC 1275 is also significantly affecting the properties that can be extracted for the Perseus cluster.', '5.3 Probing the parameter space with CTA': 'We first estimate the parameter space constraint that can be obtained with CTA. We perform event simulations without including the cluster signal, but still account for it afterwards. \nWe use our ctools framework to fit for the cluster together with the other sky model components and extract the upper limit on the cluster flux normalization. We also extract the corresponding upper limit on the overall cluster γ -ray flux. This is reproduced for 50 simulations in order to compute the mean upper limit and its standard deviation (implying an uncertainty on the mean and standard deviation of the upper limit of less than 5%, see appendix E), and for the different models that we test: spectral indices α CRp = [ 2 . 0 , 2 . 2 , 2 . 4 , 2 . 6 , 2 . 8 , 3 . 0 ] and spatial scalings η CRp = [ 1 . 5 , 1 . 0 , 0 . 5 , 0 . 0 ] . A higher spatial scaling indicates a more compact cluster, as can be seen by referring to Section 2 and Figure 2. \nFigures 11 and 12 show the expected parameter space constraints in case of nondetection, and the flux upper limit in the range [0 . 15 -50] TeV for 285 hours of observations, respectively. In both figures, the exclusion limit at 95% confidence interval is shown, with the shaded areas representing the standard deviation. Figure 11 shows the upper limit in γ -ray flux as a function of the spectral index α CRp , for different spatial scalings. A flux upper \nFigure 10 . Correlation matrix between the different free parameters included in our model. The cluster model corresponds to the baseline model. \n<!-- image --> \nlimit down to 10 -13 cm -2 s -1 is expected with CTA. Figure 12 shows X CRp ( R 500 ) , the CR to thermal energy density ratio, as a function of spectral index. In the case of non-detection we are able to put a constraint down to X CRp ( R 500 ) ∼ 10 -4 for a very compact CRp distribution. The most recent constraints put forth on diffuse γ -ray emission from Perseus come from the MAGIC telescope observations and analysis by [90]. As mentioned in Section 2.3.3, the γ -ray and thermal emission modelling used by MAGIC have overestimated the thermal energy, meaning that the MAGIC constraints in the left panel are actually too optimistic. Either way, it is evident that CTA will be able to put a constraint that is about an order of magnitude deeper than MAGIC. \nBoth figures show that a more compact cluster would give better constraints. From the previous sections we know that in the CTA energy range, a lower spectral index gives more diffuse γ -ray emission (see Figure 2), hence the non-detection of diffuse γ -ray emission puts a better constraint for lower spectral index numbers. \nWe also show that CTA will be able to exclude the pure hadronic best-fit model, assuming [159] magnetic field model with η B = 2 / 3 . Accounting for statistical uncertainties and assuming the most pessimistic magnetic field model (about 30% change, see Section B), CTA should still be able to exclude significantly most of the parameters allowed in the pure hadronic scenario, although a small region of the parameter space with high α CRp and low η CRp may still be viable in the case of non-detection, despite the higher value of X 500 that would be implied. \nFigure 11 . Exclusion limit at 95% confidence level for the parameter X CRp , as a function of α CRp and for different values of η CRp , assuming non-detection of the signal. The shaded area represent the 68% confidence interval. The limits obtained by MAGIC [90] are reported for their isobaric model (roughly corresponding to our η CRp = 1 model) and their extended model (roughly corresponding to our η CRp = 0 . 5 model). We also report the baseline model and the best-fit parameters in the case of the pure hadronic scenario (where η CRp = 0 . 78 , and given the reference magnetic field model based on [159]) in black. To highlight the statistical uncertainties in the pure hadronic model, a set of 500 samples randomly extracted from the MCMC chains is shown, color-coded by the value of η CRp , together with the 68 and 95% confidence intervals reported as grey lines. \n<!-- image -->', '5.4 Measuring the cosmic-ray properties in the case of detection': 'In this section, we consider the baseline and the pure hadronic models, and produce the corresponding event simulations. We then investigate the sensitivity of CTA to constrain the model parameters. We consider two different scenarios. In the first one (Section 5.4.1), we focus only on the recovered spectral properties by constraining the background model independently from the ICM and assuming that the cluster spatial shape is known. Then, in Section 5.4.2, we consider simultaneously both the spectral and spatial properties together with all background components. The two methods are completely independent from each other. In the following, we analyse and discuss representative simulations of the data corresponding to our two models.', '5.4.1 Spectral constraints only': 'We first focus on constraining the spectral energy distribution only. We start by using a likelihood fit to constrain the Perseus cluster normalization and all the other free sky parameters (see Figure 10). To obtain the detection significance, we apply the Test Statistics defined \nFigure 12 . Same as Figure 11 for the flux upper limit for energies in the range [0 . 15 , 50] TeV, with the same color code. \n<!-- image --> \nin Equation 4.4. As a reference, we obtain a test statistic value of √ TS ∼ 6 . 5 in the case of the baseline model, and √ TS ∼ 11 for the pure hadronic model. We then extract the spectrum of the diffuse cluster component thanks to the dedicated csspec function (from ctools ) using the true input cluster spatial template to do so, which we keep fixed. The instrumental background is left free but the point sources are kept fixed to their maximum likelihood values. This procedure allows us to extract the Perseus cluster flux normalization in each independent energy bin. Additionally, the csspec function is used to recover the full likelihood scan for the normalization in each bin. \nIn order to measure the posterior likelihood in the parameter space, we employ an MCMC approach using the emcee package [171], following the method used by [92]. The sampling is performed according to the log likelihood function defined as \nln L ( ⃗ θ | D ) = ∑ i interp ( ln L i ( F scan , F i ( ⃗ θ )) ) . (5.1) \nThe parameters ⃗ θ ≡ ( X 500 , α CRp ) correspond to the normalization and the CRp spectral index, respectively, and D to the data points. In each energy bin i , L i is the likelihood scan extracted with csspec as a function of the flux F scan . This quantity is interpolated at the location of any model to be tested against the data, F i ( ⃗ θ ) , and summed over all bins. We use flat priors ( X 500 > 0 , 2 < α CRp < 5 ) on the parameters and check a posteriori that the limits do not affect our results. \nOnce the MCMC has converged, we remove the burn-in phase and the chains provide us also with the posterior probability distribution function. We also recompute 100 SED models using parameters randomly sampled from the chains and measure the median and 68% confidence limit envelop on the recovered spectrum. The best-fit model is computed using the parameters that correspond to the maximum likelihood point in the chains. \nt \nd \nS \nd \nE \nd \nFigure 13 . Spectral constraints. The left panels correspond to the baseline model while the right panels correspond to the pure hadronic best-fit model (see Table 1). Top panels : recovered spectrum (red points) and their error bars (according to the likelihood scan curvature), best-fit model (black), 68% confidence level (blue region) and a set of 100 realizations (light blue). The residual normalized by the error bars, χ , is shown at the bottom. The dashed lines correspond to χ = ± 2 . Bottom panels : constraints on the parameter space. Contours provide the 68% and 95% confidence region and the black star represents the injected model. The blue shaded region on the one dimensional histograms provides the 68% confidence region after marginalization. \n<!-- image --> \nIn Figure 13, we present the recovered spectrum and the constraints in the parameter space obtained in the case of the baseline and pure hadronic best-fit models. In both cases the best-fit describes the data well, as can be observed in the residual. The constraints in the parameter space show that both input models are recovered with the 68% confidence interval. The pure hadronic model allows us to better constrain the model given the higher signal-tonoise ratio. We note that the normalization and the spectral index of the CRp distributions are highly degenerate. This is particularly true for CTA because no leverage is available near the spectral bump around 1 GeV. Because of this degeneracy, an amplitude up to four times the input value is still allowed in the case of the baseline model. This reduces to about twice the input value in the case of the pure hadronic model, for which the signal is stronger. The error on the spectral slope is about 0.1 and 0.07 for the baseline and pure hadronic models, respectively. While the details are beyond the scope of this paper, we note that combining \nlower energy data (e.g., Fermi -LAT) to CTA, even in the case of non-detection, could help break the parameter degeneracy.', '5.4.2 Joint spectral-imaging constraints': 'The analysis described in Section 5.4.1 is standard, but intrinsically limited by the fact that the cluster profile is a priori unknown. This might lead to significant systematic effects when recovering the spectral energy distribution if it is improperly modeled. In addition, the uncertainties in the background (diffuse and point sources) are expected to affect the constraints on the cluster diffuse emission, as discussed in Section 5.2. Although the error bars on the extracted cluster spectrum are expected to account for the correlation between the different components, the respective parameters are not co-varied when constraining the model to reduce the computing time. To mitigate these possible biases, we consider the jointfit of the full sky model simultaneously, as described below. This allows us to sample the full parameter space. While the analysis described in Section 5.4.1 is relatively quick, the one described hereafter is significantly more demanding in term of computing time, and the two are therefore complementary. \nWe compute the IRF convolved model counts cube, in spatial and energy bin i , as \nM i ( ⃗ θ ) = C i ( X 500 , α CRp , η CRp ) + ∑ j ∈ [1 , 2] PS ( j ) i ( A ( j ) PS , α ( j ) PS ) + B i ( A bkg , α bkg ) . (5.2) \nThe component C i ( X 500 , α CRp , η CRp ) corresponds to the cluster model, which depends on the parameters X 500 , α CRp , and η CRp . The components PS (1 , 2) i ( A (1 , 2) PS , α (1 , 2) PS ) represent the pointsource contribution, NGC 1275 and IC 310, respectively. The component B i ( A bkg , α bkg ) is the diffuse instrumental background, for which we assume a fixed shape with a floating spectral index and normalization. The parameter space includes nine parameters, \n⃗ θ ≡ ( X 500 , α CRp , η CRp , A (1 , 2) PS , α (1 , 2) PS , A bkg , α bkg ) (5.3) \nAs done in Section 5.4.1, we use an MCMC method to sample the parameter space. The likelihood function is defined according to Poisson statistics as \nln L ( ⃗ θ | D ) = ∑ i ˜ M i ( ⃗ θ ) -d i ln( ˜ M i ( ⃗ θ )) , (5.4) \nwith d i the data counts cube. Since the convolution of a model with the IRF is computationally expensive, it is not possible to compute M i ( ⃗ θ ) for all the MCMC test parameters. Therefore, we compute the models associated to all the components on predefined grids and ˜ M i ( ⃗ θ ) corresponds to the model M i ( ⃗ θ ) after interpolation at the exact location of the requested parameters. Each model component amplitude only accounts for a linear scaling and does not need to be included into the grid. The cluster model grid is thus bi-dimensional, defined as a function of the non-trivial parameters α CRp and η CRp (typically 10 × 10 points allows us to recover any model with percent level precision). For the point sources and the instrumental background, the slopes α PS , bkg are the only relevant parameters and the grids are therefore one-dimensional. The range over which the model grids are computed correspond to flat prior and we check a posteriori that it does not affect our result. We note that an extra prior is used for the cluster normalization, as X 500 ∈ [0 , 20 × X (input model) 500 ] to avoid sampling nonphysical models in the case the signal-to-noise ratio is too low to provide a reliable constraint on this parameter. In order to highlight the impact of the point sources on the cluster \nparameters constraint, we also perform the fit using a prior on the point-source parameters. The latter is Gaussian, centered on the input parameter values and with a standard deviation corresponding to 5% of the parameter values. \nTable 6 . Recovered parameters of the joint spectral-imaging analysis for the baseline model, baseline model plus prior on NGC 1275, and the pure hadronic best-fit model cases. \n- \n- \nFigure 14 presents the posterior distribution in the full parameter space for the baseline cluster model (the case of the pure hadronic model is available in Appendix G).Table 6 also reports the recovered parameters and their uncertainties in the different cases. The cluster normalisation, the point sources and the instrumental background parameters are shown relative to their input values. We can see that for this simulated data, the input model is recovered within 95% confidence interval for all the parameters (and 68% for most of them). While the observed posterior distribution is non-trivial for most of the parameters, we have tested that in the limit of high S/N, the parameter space is well described by a multivariate Gaussian function. As discussed in the case of the spectral only analysis (Figure 13), strong degeneracy is observed between the cluster normalization and the CRp slope. On the other hand, the cluster profile parameter is not strongly degenerate with the other cluster parameters, but is anti-correlated with the parameters associated with NGC 1275. Indeed, as the cluster profile gets more compact, it resembles more to the point source so that increasing η CRp leads to a smaller point-source normalization. As the amplitude and slope of NGC 1275 are positively correlated, the same trend is observed between η CRp and α (1) PS . The cluster normalization is also anti-correlated with NGC 1275 parameters for similar reasons. No significant correlation is observed between the cluster model and IC 310, as expected given its sky coordinates. The instrumental background normalization and slope is slightly correlated with the cluster parameters, in particular with the profile parameter, since lowering the value of η CRp leads to a flatter signal that slightly mimics the instrumental background. The correlation is thus positive between η CRp and A bkg . As expected, the amplitude and spectral slope of the instrumental background and point sources are correlated. Compared to the spectral constraints only, the error bars on the parameter α CRp increase from 0.10 to 0.17, highlighting the importance of constraining simultaneously all the components. The error on the cluster profile parameter, η CRp , is about 0.26. The normalization is constrained to X 500 ≲ 0 . 1 (at 68% limit), which is significantly larger than in the case of spectral constraints only. \nWhen using the prior on point source, we observe a strong reduction of the volume allowed by the cluster parameters (especially avoiding too high values of the normalization), in agreement with the degeneracy between components. On the other hand, the instrumental background parameters are nearly unchanged. The posterior constraints on NGC 1275 parameters is strongly improved, as can be expected due to the prior, but the constraints on IC 310 is nearly unchanged because the CTA data already provide a constraint better than \n1 \nFigure 14 . Posterior constraint on the full parameter space in the case of the baseline model. Contours provide the 68% and 95% confidence interval. The black star shows the input model parameters. The constraints are reported in the standard case (blue) and in the case where we assume a 5% prior on the flux and spectral indices of point sources (orange). The CRp normalisation is given relative to the input one, X CRp, 0 = 10 -2 . \n<!-- image --> \n5% on the corresponding parameters. \nWe note that the steep spectrum of NGC 1275 makes it very sensitive to any uncertainty in IRF convolution of the model, which in turns may affect the cluster reconstruction when all parameters are allowed to vary, due to degeneracy. This may cause notable systematic effects if improperly accounted for in the case of a low significance, but we leave its detailed investigation for future work when the true data will be taken. \nIn Figure 15, we show the input data, model and best-fit residual (including the cluster or not) after summing over all the energy bins. We can see that the best-fit model describes \nFigure 15 . Comparison between the data and the best-fit model, for the full energy range. Units are in counts per pixel. Top left panel: data (baseline model only). Top right panel: best-fit model (baseline model only). Bottom and middle left panel: residual between the data and the background components (including point sources) for the baseline and pure hadronic models, respectively. Bottom and middle right panel: total residual between the data and the model for the baseline and the pure hadronic models, respectively. Contours provide the signal-to-noise ratio with 2 σ spacing. The maps where smoothed with a Gaussian kernel with 0.15 deg for visualization purpose. \n<!-- image --> \nthe data well. The signal is dominated by the instrumental background on large scales, and by the point sources on small scales. Once these components are subtracted, we can observe \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 16 . Left panel: cluster diffuse emission counts recovered within an aperture radius of 1 deg, after instrumental background and point-source subtraction. Right panel: cluster diffuse emission profile computed over all the energy bins, after instrumental background and point-source subtraction. The top panel is in the case of the baseline model and the bottom panel for the best-fit pure hadronic model. \n<!-- image --> \nthe faint signal associated with the cluster emission. \nIn Figure 16, we show the recovered cluster diffuse emission spectrum (left) and profile (right) after subtracting the point sources and the instrumental background. The spectrum is computed within 1 deg radius, which nearly corresponds to θ 500 and the profile is computed by summing over the energy bins. The spectrum is overall well constrained over all the CTA energy range for this specific model, allowing us to recover the spectral distribution of the CRp. The cluster emission is detected on the profile up to about 0.2 deg radius (given the binning of 0.05 deg per radial bin), which allows us to constrain the shape of the signal well.', '5.5 Discussions': 'The CTA observations are expected to provide unprecedented constraints on the diffuse γ -ray emission from the Perseus cluster. In this section we review its sensitivity to cluster scale CR physics and discuss the assumptions of the present analysis and caveats. \nToward a multi-wavelength view of the Perseus cluster from radio to γ -rays We first highlight how CTA observations would compare to data at other wavelengths, in particular in terms of the scales that are probed, in Figure 17. We consider the most optimistic case that is the pure hadronic model (see Figure 15 for the baseline model). We compare a realization of the CTA (background subtracted) image to radio data (VLA, 1.4 GHz), X-ray \n<!-- image --> \n<!-- image --> \nFigure 17 . Comparison between the CTA simulated map (background subtracted, pure hadronic best-fit model case, with [159] plus η B = 2 / 3 magnetic field) and multi-wavelength data. Left panel: CTA surface brightness together with signal-to-noise ratio (with 2 σ steps). The map was smoothed with a 0.15 deg Gaussian for visual purposes and the bottom left grey circle accounts for the effective map resolution. The top left subplot shows a zoom comparison between CTA and the radio mini-halo as imaged by the VLA at 1.4 GHz (extracted from the NASA/IPAC extragalactic database). VLA contours are logarithmically spaced. Top right panel: ROSAT [221] X-ray image extracted from the ROSAT All Sky Survey. Contours are logarithmically spaced. Bottom right panel: Compton parameter from [222]. The central region appears negative due to the strong contamination from NGC 1275, and another radio source is visible on the West. Contours are linearly spaced. \n<!-- image --> \n(ROSAT All Sky Survey), and tSZ ( Planck ). The CTA signal, while slightly resolved, is expected to be relatively compact and is only detected up to about θ 500 / 4 . The mini-halo is also very compact (at least at 1.4 GHz), slightly extended when compared to the CTA image angular resolution. In contrast, the X-ray and tSZ images that trace the thermal gas density squared and the pressure, respectively. We note that the tSZ signal is contaminated by two radio galaxies that appear negative on the map, including NGC 1275 at the center. The γ -ray signal morphology is expected to compare well with X-ray data assuming that the CR follow the thermal gas well. In contrast, if the CR follow a distribution comparable to the temperature it would resemble more the tSZ image ( ∝ P e = n e k B T ). \nThe energy budget of the Perseus cluster The cluster ICM is dominated by its thermal component, but non-thermal contributions (turbulence, magnetic fields, cosmic-rays) should \nbe non-negligible and are only poorly known to date. The amplitude of the CTA detection (or upper limit) will provide a direct measurement (or a further constraint) of the energy stored in the CR component, and thus the CR pressure. Noteworthy, this will also provide a measurement of one of the contributions to the hydrostatic mass bias, which is one of the key ingredients for cluster cosmology (see [172] for a review). Nonetheless, even in the case of high significance detection, the CTA constraints on this parameter would have large uncertainties due to the degeneracy with other parameters (see Figure 13 and 14), unless strong priors are available for the shape and the spectrum of CRp via other wavelengths. This could be the case thanks to high quality radio observations, assuming specific scenario (e.g., pure hadronic model). The use of lower energy data, such as Fermi -LAT, could also help breaking the degeneracy between the parameters of the cluster components, but also provide a useful prior on the spectrum of NGC 1275. The measurement of the CR to thermal energy will be easier in the case of a harder CRp spectrum and a more compact profile. \nCosmic-ray acceleration mechanism and transport If the cluster is detected, the CTA data will allow us constraining the spectral index of the CRp that induce the γ -ray emission. The CRp spectrum (and amplitude) reflects the acceleration mechanism at play and gives estimates of the CR acceleration efficiency. Such measurement will therefore have important implications to discriminate between the various models that remain poorly constrained to date [3, 21]. The CTA sensitivity to diffuse emission will, however, loosen as α CRp increases. For instance, assuming that the CR spatial distribution follows that of the thermal gas, it will not be possible to detect the cluster for a CR slope larger than 2.6 if the CR to thermal energy ratio is lower than about 5% (Figure 11 and 12). Assuming that the CR pressure support within R 500 is less than 20%, as can be reasonably expected (e.g., [172]), CTA will be limited to probe α CRp < [2 . 6 , 2 . 9] depending on the spatial distribution ( η CRp ∈ [0 , 1 . 5] ). \nThanks to the resolved observations provided by CTA, the data will also be sensitive to the spatial profile of the γ -ray emission, which is related to the spatial distribution of the CRp in the cluster. In turn, the cluster CR profile will give precious information about the production sites and transport processes (advection and turbulent motion imply a more compact profile, while diffusion and streaming flatten it). In this paper, we have chosen to model the CR profile using a scaling with respect to the thermal gas density, which allows us to test the effect of transport via a single parameter, η CRp . We stress that this choice is not unique and the present work could be extended to test any model. However, given the current limited knowledge about the CR production and transport in the ICM, simple models might already be sufficient for such investigations. The precision on the CRp spatial distribution will depend on CR to thermal energy ratio, the spectral distribution of CR, and the shape of the CR profile itself since a more compact distribution will lead to a larger signal-to-noise ratio. For hard spectra, CTA should be able to detect the diffuse Perseus emission even in the case of flat CR distributions, provided that the CR to thermal energy ratio is sufficient ( X 500 ≳ 10 -2 ). \nProbing the time integrated AGN and starburst activity in Perseus In this paper, we have considered that the cluster ICM was filled with CRp following a given spectral and spatial distribution, regardless of their detailed origin. While they may arise from shocks and turbulence, it is also possible that accumulated injection from compact sources play an important role [223]. Therefore, the CTA observations will be sensitive to the time integrated AGN cosmic-ray outburst and starburst activity. Since the number of AGNs and starbursts in the Perseus cluster can be estimated, this will provide a unique way to estimate the average \nenergy injection via CRp from such sources in galaxy clusters. The level of γ -ray emission will depend on how CR from AGN and starbursts are processed, so that acceleration, transport and injection are closely linked to each other. \nConstraints on the magnetic field strength in the pure hadronic model The cluster non-thermal diffuse emission that we aim at probing with CTA is physically connected with the diffuse synchrotron emission observed at radio wavelengths, the latter being produced by the entangled combination of CRe and magnetic field (see Section 2.2). Therefore, it is in principle possible to constrain the magnetic field strength using the joint analysis of radio and γ -ray data, assuming a given physical scenario such as the pure hadronic model (as done for the Coma cluster [98, 99]). In the present analysis, we have assumed a fixed magnetic field model ([159] plus η B = 2 / 3 as a reference, with < B (10kpc) > ∼ 25 µ G) when considering the pure hadronic scenario. For instance, neglecting uncertainties arising from the radio data, our reference magnetic field model implies a test statistic TS = 125 ( ≡ 11 σ detection) for CTA in the pure hadronic scenario. Lowering the γ -ray emission, so that the cluster is not detected, would imply lowering the amount of secondary CRe by the same amount. The magnetic field can then be adjusted so that the radio synchrotron emission remains the same. \nIn Appendix B, we show how the different models of magnetic field considered in this paper affect our model parameters and translate into γ -ray emission (Table 1). For example, the model giving the lowest magnetic field profile, by a factor of about 2.5 within 100 kpc from the center compared to our reference, implies a normalization that is 60% larger with nearly unchanged CR slope and spatial distribution, and therefore an increased γ -ray emission by about 60% as well. This shows that in turn, constraints can be obtained on the magnetic field strength when fixing the radio and γ -ray observables. Given the current uncertainties in the radio spatial and spectral distribution of the Perseus cluster, we only provide a qualitative discussion. However, we note that current and future radio observations 24 should provide an unprecedented measurement of the diffuse radio emission from the Perseus cluster, which could then be combined with CTA observations to measure the magnetic field. This could be further combined with independent measurements of the magnetic field (e.g., Faraday rotation measure) to test CR physics and magnetic field in the ICM and particle acceleration mechanisms. \nAnalysis choices and CTA configuration The analysis presented in this paper rely on several choices that are expected to affect our sensitivity results. We discuss the most relevant ones hereafter. \nMost of the analysis described in this paper has been performed using a template fitting analysis. Indeed, it was necessary in order to not only test for the detection of the cluster, but also model and constrain the spectral and spatial distribution of the emission. Because it uses more information than classical ON-OFF technique, the template fitting analysis is expected to provide more accurate constraints. On the other hand, it might be more sensitive to systematic effects due to the mismodelling of the data. \nWe decided to select events in the energy range between 150 GeV and 50 TeV. While the exact value of the upper bound does not affect the results, because no signal is expected above ∼ 30 TeV, reducing the minimal energy is expected to increase statistics and improve the CTA constraints. On one hand, we use a conservative value that is relatively large compared to what can be expected with CTA. On the other hand, low energy data may be more difficult \nto model and in addition to the instrumental background, NGC 1275 strongly dominates at low energies. Therefore, it is likely that the improvement in sensitivity will not be major when pushing to lower energy thresholds. \nWe conservatively selected events within a 3 × 3 deg 2 region around the cluster. While no significant cluster diffusion emission is expected beyond this region, it is likely that the corresponding data will help constraining the instrumental background, provided that it can be properly modeled. This might, in turn, help breaking the cluster - instrumental background degeneracy and slightly improve the CTA constraints. \nA new version of the CTA IRFs has been released since the analysis presented here was completed. They correspond to the initial array configuration, made of 4 LST plus 9 MST (the so-called CTAO Alpha configuration, prod5-IRFs) instead of the one with 4 LST plus 15 MST (prod3-IRFS) used here. The reduction in sensitivity is mostly effective at high energy ( E ≳ 500 GeV), by a factor of up to 30%, but improved off-axis sensitivity. Therefore we would expect the results presented in this paper to be affected by such sensitivity reduction. Assuming the initial array configuration would imply a reduction of the upper limits (Figure 11 and 12) and a widening of the uncertainties on the cluster recovered parameters (Figures 13 and 14) by up to 30%, although the improved off-axis sensitivity should mitigate this effect. Cluster models with harder spectra (small α CRp ) should be the most affected because a larger part of their flux arises from the high energy part of their spectrum.', '6 CTA sensitivity to DM induced γ -ray emission from Perseus': "In this section we investigate the sensitivity of CTA to the DM emission model presented in Section 3 for the cluster. The results shown in the following have been obtained using the gammapy (v.0.18.2) open-source code [220] 25 . \nPerseus represents a complex environment where multiple γ -ray sources coexist. Indeed, with the current generations of IACTs, these astrophysical sources have posed large difficulties to the search for a putative DM signal in the area, as it is hard to avoid 'contamination' from the astrophysical sources and to disentangle between these and DM, both spectrally and spatially. In contrast, due to its expected improvement in sensitivity, angular resolution and larger FoV, CTA will allow for a much better discrimination by means of analysis techniques that were traditionally discarded for this kind of telescopes and used only for all-sky γ -ray facilities such as Fermi -LAT. In particular, in our DM search we perform a templatebased analysis, which allows to take into account the rest of expected γ -ray emissions as backgrounds, and is in line with the analysis previously performed in Section 5 for CRs and also with recent studies in the Galactic center region by the CTA Consortium [121]. This type of analysis, although computationally expensive, is expected to provide not only the highest DM sensitivity but also to yield the most realistic one 26 . Yet, mainly for the sake of comparison with previous IACT works and in order to provide a first-order sensitivity study in our work as well, we also carry out the so-called ON-OFF analysis, i.e. the most standard analysis strategy adopted by some current IACTs. Full details of such analysis are left for Appendix I with the corresponding comparison with template-based results. \nTo perform the simulation of the DM-induced γ -ray emission from Perseus, we adopt the observation setup described in Section 4, i.e. a pointing with one degree offset from the cluster center. Similarly to the CR analysis, we stack the data from the different pointings in a single 3D cube, containing the proposed 300h observation time. We adopt ten energy bins, starting from 50 GeV up to 100 TeV. This allows us to exploit the whole range of CTA's sensitivity as well as to overlap with the DM parameter space previously explored by the Fermi -LAT (at the lowest considered energies) and by the H.E.S.S and MAGIC telescopes (in the TeV regime). We use a spatial binning of 0.02 deg per pixel and our field of view diameter is set to 5 deg. \n0.05 - 0.12 TeV \n0.12 - 0.30 TeV \nFigure 18 . Left panel: Simulation of an observation of the Perseus region with CTA. The color informs on the expected counts. This example is for the MED annihilation model, with a DM particle of m χ = 10 TeV, b ¯ b channel and < σv > = 3 × 10 -26 cm 2 s -1 . We adopted the 'Baseline' model for the CR-induced emission (Section 2.3.1) and included the two AGNs in the region via the descriptions in Equations 4.1 and 4.2. We only show the first four energy bins of our analysis, each of them corresponding to one panel. The white dashed circle goes over the different proposed pointings (all of them at 1 deg from the cluster's center). Right panel: Comparison of the spectra of the different simulated γ -ray components integrated up to R 200 . We also include the spectrum of a m χ = 10 TeV DM particle decaying via the b ¯ b channel. We recall that all these emissions do not happen through the entire CTA FoV. \n<!-- image --> \nTo model the DM distribution in the cluster, we use the 2D templates described in Section 3.3 for each of the three benchmark models in the case of annihilation and for the one of decay. For annihilation, we simulate a putative γ -ray signal assuming a branching ratio of 100% to the b ¯ b annihilation channel, a DM mass m χ = 10 TeV 27 and a velocity-averaged cross-section matching the thermal value of < σv > = 3 × 10 -26 cm 3 s -1 . For decay, we adopt τ χ = 10 27 s as the DM particle lifetime. The expected γ -ray contribution from the existing AGNs in the area is accounted for via the models described in Section 4.2, while the CRinduced γ -ray emission is described using the spectral and spatial template corresponding to the 'Baseline' benchmark model (Section 2.3.1). We remark that those γ -rays originated from both the AGNs and CRs are considered as backgrounds in our DM search, together \nwith the CTA instrumental background modelled by the IRFs. In total, we create four sets of simulations, one per each DM template (MIN, MED, MAX and DEC - see Table 3), and always adopting the same combination of backgrounds. An example of a simulation can be seen in Figure 18 for the MED annihilation model. We also compare in the same figure the spectra of the different γ -ray sources in Perseus to the ones for annihilation and decay. This comparison shows that in the annihilation scenario the γ -ray flux is expected to be orders of magnitude below the astrophysical backgrounds for realistic cross-section values. Yet, in the case of decay, we can have comparable fluxes (adopting τ χ = 10 27 s, which is indeed very close to current constraints, [224]). \nFinally, in order to obtain statistically meaningful results, we create 100 different simulations with the same observation setup, and use the corresponding mean for the data analysis itself. A study of the stability of our simulation results with respect to the number of simulations is included in Appendix J, which shows that a reasonable compromise in terms of convergence and computation time is indeed achieved after a few tens of simulations in almost all cases. Thus, in total, we produce 400 simulations, 100 per each benchmark DM model (MIN, MED, MAX and DEC), which we now proceed to analyze.", '6.1 Template-fitting analysis and DM sensitivity': "To search for DM emission, we first define for convenience the normalization of the DMinduced γ -ray flux A χ , as < σv > = A χ × < σv > thermal , for the three annihilation models, and τ χ = τ ref /A χ (Equation 3.2), where τ ref = 10 27 s, for the decay scenario. We fit this normalization for each simulation using the iminuit [225] backend of gammapy to the theoretical emission of different DM particle candidates. To cover the available mass range for WIMPs, we perform the fit to fourteen DM masses across the defined energy range and for two representative annihilation/decay channels, i.e. b ¯ b and τ + τ -28 . \nTogether with the DM normalization, we apply a joint-likelihood fitting including the relevant parameters of the considered astrophysical backgrounds. For example, since the CR-related flux is considered as a background for this analysis, we only allow for an overall normalization on the CR-induced γ -ray flux for the 'Baseline' model. Then, the complete flux model to fit the simulations is defined, in spatial and energy bin i , as: \nM i ( ⃗ θ ) = DM i ( A χ ) + CR i ( A CR ) + ∑ j ∈ [1 , 2] PS ( j ) i ( A ( j ) PS , α ( j ) PS ) + B i ( A bkg , α bkg ) . (6.1) \nIn total, we fit eight different parameters: \n⃗ θ ≡ ( A χ , A CR , A (1 , 2) PS , α (1 , 2) PS , A bkg , α bkg ) , (6.2) \nwhere all the parameters have been previously defined (see Section 5.4.2 and Equation 5.2). The likelihood function is the one corresponding to Poissonian data (Equation 5.4). We also adopt a flat prior on A χ , to allow only for zero or positive values 29 . This joint-likelihood fit will enable to check for intrinsic correlations and to account for them when estimating the size of the uncertainties. To obtain the detection significance of the DM-induced γ -ray emission, we apply the Test Statistics as defined in Equation (4.4). We define a detection when TS = 25 , which roughly corresponds to 5 σ . \n<!-- image --> \nFigure 19 . Sensitivity of CTA to a DM annihilation signal from the Perseus cluster. Curves represent the 95% C.L. upper limits on the velocity-averaged cross-section versus the DM mass for the MED annihilation model. The green (yellow) band shows the 1 σ ( 2 σ ) scatter of the projected limits. The black dashed line is the thermal relic cross-section ( < σv > thermal = 3 × 10 -26 cm 3 s -1 ). Left panel: Cross-section upper limits for the b ¯ b channel ( right panel for τ + τ -channel) in comparison with the most recent results on DM-annihilation searches in galaxy clusters using Fermi -LAT ([111] with orange dotted lines; [106] with blue dot-dashed lines) and H.E.S.S ([114], with purple dashed lines). All of our results are shown for the template fitting analysis, unless stated otherwise. \n<!-- image --> \nIn neither of the MIN, MED, MAX annihilation or decay scenarios, and for none of the channels considered, a statistically significant detection is predicted. We proceed to compute the 95% confidence level (C.L.) upper limits (lower limits for decay) of the DM normalization parameter A χ using the likelihood profile method [219, 226]. For the DM normalization, we assume a one-sided distribution (we recall the flat prior A χ ≥ 0 ), thus the 95% C.L. corresponds to a change of ∆ TS = 2 . 71 , with respect to the best fit.", '6.2 Projected sensitivity to annihilating DM': "Assuming the templates and spectra for DM annihilation (Section 3), we compute the 95% C.L. upper limits for A χ for all considered cases, convert this value to < σv > (the standard parameter to be constrained for annihilating DM) and average the results for 100 simulations. \nIn Figure 19, we show the CTA projected sensitivity as 95% C.L. upper limits on the velocity-averaged cross-section versus the DM mass, for the MED annihilation model and the two considered representative annihilation channels. As it can be seen, the most constraining limits are obtained for the τ + τ -channel, although also the b ¯ b channel yields similar results for DM masses above ∼ 10 TeV. In all cases, our exclusion limits are more than ∼ O (10 2 ) above the thermal relic cross-section value. Yet, we note that they will be the most constraining ones (in the TeV energy range) considering galaxy clusters as DM targets. This can also be seen in both panels of Figure 19, where we show a comparison to recent works. The latest cluster DM limits obtained by currently operating IACTs come from the observation of the Fornax cluster (14.5 hours) by the H.E.S.S Collaboration [114], and are around one order of magnitude weaker than our CTA predictions at a few TeV (i.e. at the peak of H.E.S.S. sensitivity). We also compare our CTA predictions with limits from Fermi -LAT in the subTeV WIMP mass range. In [111], authors analyze 12 years of Fermi -LAT data for a sample of five clusters and, in the absence of a signal, set constraints only for the b ¯ b channel. Perseus is the cluster yielding their weakest limits while Fornax gives the most constraining ones. The \nCTA sensitivity predictions for Perseus are always better than the obtained in [111] in the whole range but below 100 GeV, where both results are still comparable. But even for Fornax, CTA's improvement in sensitivity will allow us to set the tightest DM constraints for DM masses above 1 TeV. In the case of the τ + τ -annihilation channel, we can compare our results with those in [106], where authors combine 3 years of Fermi -LAT data from 8 galaxy clusters to derive the corresponding DM limits. As a consequence of using a considerable reduced amount of data compared to [111], their limits are considerably weaker in comparison and cannot compete to those from CTA. In conclusion, CTA DM limits from Perseus will be the most constraining ones from these class of targets in the TeV energy range. \nBy far, the largest uncertainty on the obtained cross-section upper limits is the modelling and contribution of cluster DM subhalos to the total J ann -factor. This was the reason to build three different annihilation benchmark models in the first place (MIN, MED and MAX - Section 3.2), each of them representative of very diverse levels of substructure and their contribution to the annihilation flux. To properly quantify the impact of this uncertainty in our limits, in addition to the 95% C.L mean upper limit for the MED case presented above, we also compute limits for the MIN and MAX benchmark models. These results are shown in Figure 20 for the τ + τ -annihilation channel, and reveal that our limits can be modified substantially depending on the considered subhalo scenario. In particular, the boost factor associated to our MED model ( B MED = 9 . 2 ) translates into an improvement of a factor ∼ 7 with respect to the MIN benchmark model. In the case of the MAX model ( B MAX = 59 . 3 ), the limits improve up to O (10) times the MED model constraints. 30 Previous studies have also accounted for the subhalo contribution in their limits. In [111], for instance, authors show that their boosted DM model improves their limits only by a factor ∼ 1 . 5 , indeed a natural consequence of the low contribution of their substructure model to the annihilation flux. In earlier work by [106], authors gained up to ∼ O (10 2 ) of improvement in their constraints due to subhalos, however they assumed boosts B = 500 -1200 , which from recent studies can be considered as overly optimistic (e.g., [190] and references therein). As for the constraints from the H.E.S.S Collaboration [114], their boosted MED ( B = 10 ) and MAX ( B = 100 ) models improved their limits by a factor ∼ 7 and ∼ 75 , respectively, in reasonable agreement with our results (we note though that, even with these improvements in their limits, our CTA projected constraints for the MED model are more restrictive). All in all, our results in Figure 20 illustrate the key role that halo substructure can have to discover/rule-out WIMP DM. \nOther uncertainties affect the results in this section as well, namely the one coming from the scatter in the concentration-mass relation and the uncertainty in the estimation of Perseus mass. Yet, the level of these uncertainties ( σ J ann = 0 . 2 dex; see Section 3.3) is well below the one induced by the modelling of the subhalo population ( σ B MAX = 2 dex). As a result, we note that the variation of upper limits shown in Figure 20 among our three different subhalo models encompasses by far these other, second-order uncertainties. Finally, we remark that these results are for the use of the 'Baseline' CR model, since different models for the CR densities may also impact our results (see Appendix H). \nFigure 20 . 95% C.L. upper limits of the velocity-averaged cross-section versus the DM mass for the MIN (dot-dashed line), MED (solid) and MAX (dashed) (see Table 4) subhalo models and the τ + τ -annihilation channel. The green (yellow) band represents the 1 σ ( 2 σ ) scatter of the projected limits. The black dashed line represents the thermal relic cross-section ( < σv > thermal = 3 × 10 -26 cm 3 s -1 ). All of our results are shown for the template fitting analysis, unless stated otherwise. \n<!-- image -->", '6.3 Projected sensitivity to decaying DM': 'Similarly to the annihilating DM case, we compute 95% C.L. upper limits for A χ , convert this value to τ χ (the standard parameter to be constrained for decaying DM) and average the results of 100 simulations. We note that as A χ and τ χ are inversely proportional, the upper limits on A χ translating into lower limits for τ χ . \nWe show, in Figure 21, the projected sensitivity of CTA as the 95% lower limits on the DM particle lifetime versus the DM mass for our DEC model. As for the annihilation constraints, the most constraining limits are for the τ + τ -channel, reaching similar values than the b ¯ b channel for masses above ∼ 10 TeV. We also compare our CTA projections to the most up-to-date constraints on DM decay in clusters. The most recent work comes from the MAGIC Collaboration [116], where authors analyze 202h of data targeting the Perseus cluster. Their limits yield the tightest constraints for masses above ∼ 1 TeV for the τ + τ -channel, yet CTA will improve these limits by more than two orders of magnitude at lower masses and up to a factor 7 for masses higher than 5 TeV. This large improvement in the low mass range is probably twofold: i) the improvement of up to one order of magnitude in the sensitivity of CTA with respect to MAGIC 31 in the lower energy range; ii) the adoption of a mask of 0.1 deg in the MAGIC analysis to avoid contamination from NGC 1275, thus loosing a considerable amount of data. Another comparable work from existing IACTs comes from [115], where authors analyze 14.5h of H.E.S.S. observations of the Fornax cluster. Though covering a smaller range of DM masses, their DM decay limits, both for the b ¯ b and τ + τ -channels, are of the same order of magnitude than the MAGIC limits for Perseus [116]. Lastly, we also compare in Figure 21 our CTA predictions to the constraints obtained using 3 years \n] \ns \n[ \n] \ns \n[ \n10 \n10 \n10 \n10 \n10 \n10 \n27 \n26 \n25 \n24 \n23 \n22 \nbb \n0.1 \n1 \n10 \n100 \nFigure 22 . 95% C.L. mean lower limits of the lifetime of the DM particle versus the DM mass (black lines) for b ¯ b ( left panel ) and τ + τ -( right panel ) decay channels, in comparison with the limits obtained by [224] (corresponding to NFW 5 σ free source fits, in dashed red lines). The green (yellow) band shows the 1 σ ( 2 σ ) scatter of the projected limits. All of our results are shown for the template fitting analysis, unless stated otherwise. \n<!-- image --> \nm \n[TeV] \nFigure 21 . Sensitivity of CTA to a DM decay signal from the Perseus cluster, at 95% C.L., in terms of the mean lower limits of the lifetime of the DM particle versus the DM mass. The green (yellow) band shows the 1 σ ( 2 σ ) scatter of the projected limits. Left panel: Mean lifetime lower limits for the b ¯ b channel ( right panel for τ + τ -channel) in comparison with the most recent results on DM decay in galaxy clusters using MAGIC data (olive long-dashed lines; [116]), Fermi -LAT data (blue dot-dashed lines; [106]) and H.E.S.S data (purple dashed lines; [115]). All of our results are shown for the template fitting analysis, unless stated otherwise. \nof Fermi -LAT data from 8 galaxy clusters [106]. These limits prevail up to DM masses of a few TeV, where CTA starts to yield first comparable and then, quickly, stronger constraints for both decay channels. From all these comparisons, we conclude that CTA will not only be able to test an unexplored region in the DM decay parameter space, but should yield the most stringent constraints from γ -ray DM decay searches above 1 TeV 32 . \n<!-- image --> \n+ \n0.1 \n1 \n10 \n100 \nThis work \nFermi \n-LAT ( \nAckermann \n+12), MW Halo \nm \n[TeV] \nFinally, we compare in Figure 22 our CTA projections to the most constraining DM decay results from γ -ray observations so far [224]. In the latter work, the Fermi -LAT Collaboration \nThis work \nMAGIC ( \nAcciari \n+18), Perseus \nHuang \n+11, Clusters combined \nCirelli \n+12, Fornax \n] \ns \n[ \n10 \n10 \n10 \n27 \n26 \n25 \nanalyzes 1 year of LAT data from the Milky Way halo. The CTA exclusion curves, as already seen in previous comparisons with Fermi -LAT limits, become more constraining above DM masses around a few TeV (few hundreds of GeV for the case of the τ + τ -channel) 33 . From these comparisons, we conclude that the null-detection of a DM decay signal in Perseus with CTA would still provide unprecedented constraints on DM decay models at the TeV mass scale.', '7 Summary and conclusions': "In this work, we have analyzed the CTA sensitivity to detect diffuse γ -ray emission from the Perseus galaxy cluster. We have considered the possible contribution from CR and from DM annihilation and decay. We have assumed 300 hours of observations as proposed by the CTA consortium as a key science project [120]. \nWe built a CR-induced γ -ray emission model using the MINOT software [147] (Section 2). Our model relies on a description of the thermal gas pressure and density, the magnetic field strength, and a parameterization of the spatial and spectral distribution of the CRp. We calibrated the model components according to available data in the literature (Figure 1). The thermal part is expected to be accurate from the core to the outskirt and was kept fixed. The magnetic field was modeled by considering available measurements and different assumptions and scaling to estimate the associated systematic uncertainty. The CRp parameters were calibrated according to several scenarios including a baseline model and the pure hadronic model (Figure 1 and Table 1). The description of the background sky was made according to the literature, noting that at least two point sources will affect the CTA observations (Section 4.2). The observation setup was investigated focusing on the offset between the cluster center and the pointing (Figure 8). Finally, the CTA sensitivity to diffuse γ -ray emission was investigated (Section 5). We considered both the case of non-detection in which we computed the expected exclusion limit that CTA will obtain (Figure 11), and the case of specific models for which a detection is expected. In the case of detection, we investigated the constraints that CTA will be able to provide regarding the spatial and spectral properties of the γ -ray diffuse emission (Figures 13 and 14). \nOur main findings with regard to γ -ray emission are as follows. \n- · The pure hadronic model, as calibrated using existing radio data, implies about 5% of CRp energy relative to the thermal energy within R 500 , a CRp spectral index of ∼ 2 . 3 and a CRp profile slightly shallower than the thermal gas density, assuming a magnetic field strength model based on existing data, although we note that parameters are strongly degenerate. Even when fixing the magnetic field strength, the uncertainty on the γ -ray flux corresponds to a factor of about two. This large uncertainty is mainly due to the limited spectral coverage of the available radio data that we used. The systematic uncertainty associated with the magnetic field implies an additional factor of a few in the uncertainty, depending on the radius.\n- · According to our modelling and the CTA IRF, we found that a pointing offset of about 1 deg was an optimal choice. This is obtained by considering both ON-OFF analysis and template-fitting techniques and for both CR and DM related analysis. \n- · In the case of non detection, we find that CTA should improve the current limits on the CRp content of the Perseus cluster by about an order of magnitude. Assuming a standard scenario ( α CRp = 2 . 3 and η CRp = 1 ), CTA should be able to constrain the CRp to thermal energy ratio within R 500 , X 500 , down to about a 3 × 10 -3 .\n- · Assuming the pure hadronic model, CTA should allow us to detect the ICM induced diffuse emission with a high significance. The spectral index of the CRp should be constrained to an uncertainty of about ± 0 . 1 and the spatial distribution down to about 10% precision. \nCTA observations of Perseus will allow us to address fundamental questions related to the underlying mechanism that accelerates particles in the ICM (such as shocks, turbulent reacceleration, the direct injection of CR from AGN), the direct injection of CR from AGN. CTA will also allow us to test the physics associated with CR transport in the ICM. According to the results presented in this paper, CTA will provide unprecedented constraints on the physics associated with particle acceleration in galaxy clusters. \nIn our work, we also investigated the potential of CTA to search for DM in Perseus. To model its DM content, we assume the DM to be entirely composed of WIMPs. We consider two different DM scenarios, where the WIMP annihilates or decays into SM particles. We build the DM density profile of the cluster accounting for the smooth DM distribution in the main halo plus the abundant DM in the form of substructures, or subhalos, that these massive objects are expected to host. The main halo is modelled starting from X-ray data, that allows for an estimate of the cluster's mass. For that mass, we then build a tailored NFW profile for Perseus following results from numerical cosmological simulations at these scales. The contribution of the subhalo population to the expected DM flux is the main uncertainty in our DM annihilation model (subhalos do not play a role for decay), and simply reflects the existing ongoing debates in the literature on the precise properties of these objects. To bracket this uncertainty, we define three benchmark models (Table 3), each of them representative of the different contribution that subhalos may have in the computation of the DM annihilation flux: the MIN model, where substructures are completely neglected; the MED model, our best guess according to the most recent results from numerical simulations; and the MAX model, defined to provide an upper limit to the role that substructures could play for the DM flux. From these, we compute the corresponding J ann factors for the MIN, MED and MAX model and J dec for the MIN case (summarized in Table 4) and produce 2D spatial templates of the expected emissions (Figure 5). The effect of substructures in the J ann -factor is quantified in terms of the so-called subhalo boost factor . More precisely, we find B MED ∼ 9 and B MAX ∼ 59 for the MED and MAX models, respectively. As expected, substructure affects mainly the morphology and strength of the DM signal in the outskirts of the cluster (Figure 4). \nAdopting these DM templates and a particular DM spectrum (Figure 18), we then create simulations of CTA observations of Perseus, that take also into account the γ -ray emission from the CR (baseline model), the two AGNs in the area (NGC 1275 and IC 310) and the instrumental background. All of these components act as background in our DM search analysis. In a next step, we fit these observations to b ¯ b and τ + τ -annihilation/decay channels and several DM masses using a template-fitting approach. In the absence of a DM signal in all the considered DM scenarios, we find the following results: \n- · DM annihilation: We obtain 95% C.L. upper limits on the annihilation cross-section for different DM masses (Figure 19). The best results are for the τ + τ -annihilation \nchannel, reaching values of < σv > ∼ 5 × 10 -24 cm 3 s -1 for DM masses up to ∼ 1 TeV. For masses above few TeV, our limits become less constraining as the sensitivity of the CTA weakens, reaching values up to 10 -21 cm 3 s -1 . These limits are between 2-4 orders of magnitude above the value of the thermal-relic cross-section. Different prescriptions for the subhalo population in Perseus weaken (MIN) or strengthen (MAX) our results by a factor ∼ O (10) (Figure 20). \n- · DM decay: We obtain 95% C.L. lower limits on the DM particle lifetime for different DMmasses (Figure 21). The best results are again for the τ + τ -decay channel, reaching values of τ χ ∼ 10 27 s for DM masses in the range of 10-30 TeV. \nTo put these DM results into the more general context: for WIMP annihilation, our prospects show that CTA will be able to provide the best constraints from γ -ray DM searches in galaxy clusters above 1 TeV (Figure 19). Note, though, that CTA is not expected to reach the thermal relic cross-section value from these observations 34 . On the other hand, the sensitivity of CTA to DM decay in Perseus will be particularly significant. Indeed, comparing our predictions with other existing constraints, CTA will test an unexplored region of the DM decay parameter space for TeV WIMPs. This will allow CTA to set unprecedented constraints on the DM particle lifetime at these masses (Figures 21 and 22). All these DM results not only demonstrate the superb capabilities of CTA to search for DM and to test the preferred DM models, especially for heavy WIMPs, but also the excellent potential of galaxy clusters as excellent targets for γ -ray DM searches.", 'Acknowledgments': "This work was conducted in the context of the CTA Consortium (mainly the CTA DMEP and CR Working Groups and the CTA Galaxy Clusters Task Force). We gratefully acknowledge financial support from the following agencies and organizations: \nState Committee of Science of Armenia, Armenia; The Australian Research Council, Astronomy Australia Ltd, The University of Adelaide, Australian National University, Monash University, The University of New South Wales, The University of Sydney, Western Sydney University, Australia; Federal Ministry of Education, Science and Research, and Innsbruck University, Austria; Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ), Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP), Fundação de Apoio à Ciência, Tecnologia e Inovação do Paraná - Fundação Araucária, Ministry of Science, Technology, Innovations and Communications (MCTIC), Brasil; Ministry of Education and Science, National RI Roadmap Project DO1-153/28.08.2018, Bulgaria; The Natural Sciences and Engineering Research Council of Canada and the Canadian Space Agency, Canada; CONICYT-Chile grants CATA AFB 170002, ANID PIA/APOYO AFB 180002, ACT 1406, FONDECYT-Chile grants, 1161463, 1170171, 1190886, 1171421, 1170345, 1201582, Gemini-ANID 32180007, Chile, W.M. gratefully acknowledges support by the ANID BASAL projects ACE210002 and FB210003, and FONDECYT 11190853; Croatian Science Foundation, Rudjer Boskovic Institute, University of Osijek, University of Rijeka, University of Split, Faculty of Electrical Engineering, \nMechanical Engineering and Naval Architecture, University of Zagreb, Faculty of Electrical Engineering and Computing, Croatia; Ministry of Education, Youth and Sports, MEYS LM2015046, LM2018105, LTT17006, EU/MEYS CZ.02.1.01/0.0/0.0/16\\_013/0001403, CZ.02. \n1.01/0.0/0.0/18\\_046/0016007 and CZ.02.1.01/0.0/0.0/16\\_019/0000754, Czech Republic; Academy of Finland (grant nr.317636 and 320045), Finland; Ministry of Higher Education and Research, CNRS-INSU and CNRS-IN2P3, CEA-Irfu, ANR, Regional Council Ile de France, Labex ENIGMASS, OCEVU, OSUG2020 and P2IO, France; Max Planck Society, BMBF, DESY, Helmholtz Association, TU Dortmunt University grant DFG SFB 1491, Germany; Department of Atomic Energy, Department of Science and Technology, India; Istituto Nazionale di Astrofisica (INAF), Istituto Nazionale di Fisica Nucleare (INFN), MIUR, Istituto Nazionale di Astrofisica (INAF-OABRERA) Grant Fondazione Cariplo/Regione Lombardia ID 2014-1980/RST\\_ERC, Italy; ICRR, University of Tokyo, JSPS, MEXT, Japan; Netherlands Research School for Astronomy (NOVA), Netherlands Organization for Scientific Research (NWO), Netherlands; University of Oslo, Norway; Ministry of Science and Higher Education, DIR/WK/2017/12, the National Centre for Research and Development and the National Science Centre, UMO-2016/22/M/ST9/00583, Poland; Slovenian Research Agency, grants P1-0031, P1-0385, I0-0033, J1-9146, J1-1700, N1-0111, and the Young Researcher program, Slovenia; South African Department of Science and Technology and National Research Foundation through the South African Gamma-Ray Astronomy Programme, South Africa; The Spanish groups acknowledge the Spanish Ministry of Science and Innovation and the Spanish Research State Agency (AEI) through the government budget lines PGE2021/28.06.000X.411.01, PGE2022/28.06.000X.411.01 and PGE2022/28.06.000X.711.04, and grants PGC2018-095161-B-I00, PGC2018-095512-B-I00, PID2019-104114RB-C31, PID 2019-107847RB-C44, PID2019-104114RB-C32, PID2019-105510GB-C31, PID2019-104114RBC33, PID2019-107847RB-C41, PID2019-107847RB-C43, PID2019-107847RB-C42, PID2019107988GB-C22; the 'Centro de Excelencia Severo Ochoa' program through grants no. SEV2017-0709, CEX2019-000920-S, CEX2020-001007-S; the 'Unidad de Excelencia María de Maeztu' program through grants no. CEX2019-000918-M, CEX2020-001058-M; the 'Ramón y Cajal' program through grant RYC-2017-22665; the 'Juan de la Cierva-Incorporación' program through grants no. IJC2018-037195-I, IJC2019-040315-I. They also acknowledge the La Caixa Banking Foundation, grant no. LCF/BQ/PI21/11830030; the 'Programa Operativo' FEDER 2014-2020, Consejería de Economía y Conocimiento de la Junta de Andalucía (Ref. 1257737), PAIDI 2020 (Ref. P18-FR-1580) and Universidad de Jaén; 'Programa Operativo de Crecimiento Inteligente' FEDER 2014-2020 (Ref. ESFRI-2017-IAC-12), Ministerio de Ciencia e Innovación, 15% co-financed by Consejería de Economía, Industria, Comercio y Conocimiento del Gobierno de Canarias; the 'CERCA' program of the Generalitat de Catalunya; and the European Union's 'Horizon 2020' GA:824064 and NextGenerationEU. Swedish Research Council, Royal Physiographic Society of Lund, Royal Swedish Academy of Sciences, The Swedish National Infrastructure for Computing (SNIC) at Lunarc (Lund), Sweden; State Secretariat for Education, Research and Innovation (SERI) and Swiss National Science Foundation (SNSF), Switzerland; Durham University, Leverhulme Trust, Liverpool University, University of Leicester, University of Oxford, Royal Society, Science and Technology Facilities Council, UK; U.S. National Science Foundation, U.S. Department of Energy, Argonne National Laboratory, Barnard College, University of California, University of Chicago, Columbia University, Georgia Institute of Technology, Institute for Nuclear and Particle Astrophysics (INPAC-MRPI program), Iowa State University, the Smithsonian Institution, V.V.D. is funded by NSF grant AST-1911061, Washington University McDonnell \nCenter for the Space Sciences, The University of Wisconsin and the Wisconsin Alumni Research Foundation, USA. \nThe research leading to these results has received funding from the European Union's Seventh Framework Programme (FP7/2007-2013) under grant agreements No 262053 and No 317446. This project is receiving funding from the European Union's Horizon 2020 research and innovation programs under agreement No 676134. This research has made use of the CTA instrument response functions provided by the CTA Consortium and Observatory, see http://www.cta-observatory.org/science/ctao-performance/ for more details. \nJPR work was supported by grant SEV-2016-0597-17-2 funded by MCIN/AEI/10.13039/ 501100011033 and 'ESF Investing in your future'. MASC was also supported by the Atracción de Talento contracts no. 2016-T1/TIC-1542 and 2020-5A/TIC-19725 granted by the Comunidad de Madrid in Spain. The work of JPR and MASC was additionally supported by the grants PGC2018-095161-B-I00 and CEX2020-001007-S, both funded by MCIN/AEI/10.13039/ 501100011033 and by 'ERDF A way of making Europe'. MH acknowledges funding from the Max Planck Society, the University of Tokyo, and the ICRR Inter-University Research Program in the Fiscal Years 2021 and 2022. PTL is supported by the Swedish Research Council under contract 2019-05135. SHC work is supported by UNAM-PAPIIT IG101323. \nThis research made use of gammapy , 35 a community-developed core Python package for TeV γ -ray astronomy [220]. This research made use of Astropy, a community-developed core Python package for Astronomy [230], in addition to NumPy [231], SciPy [232], Healpy [233] and Ipython [234]. Figures were generated using Matplotlib [235]. \nWe acknowledge support from the CNRS/IN2P3 Computing Center (Lyon - France) for providing computing and data-processing resources needed for this work. We thank the support of the Hydra HPC cluster in Instituto de Física Teórica (IFT UAM-CSIC) for the computing time and resources.", 'A Validation of the thermal model using Planck data': "The thermal Sunyaev-Zel'dovich (tSZ) effect provides a direct measurement of the electron thermal pressure in galaxy clusters. In order to validate our thermal gas pressure model calibration, which is based on the indirect inference of the pressure from X-ray observations, we compare the tSZ prediction of our model to Planck data [222] obtained with MILCA [236]. \nWe extract a MILCA sky patch centered on the Perseus cluster and compute its profile. Uncertainties are computed according to the standard deviation of the map in regions free from emission. We project our tSZ model on the same sky patch, convolve the map to the 10 arcmin Planck beam and compare it to the data in Figures 23 (map) and 24 (profile). Radio sources (including NGC 1275) are masked within 30 arcmin. \nOur model, as calibrated using X-ray information, provides an excellent match to the Planck Compton parameter map and profile, and thus to the cluster thermal pressure and thermal energy distribution. The model is validated up to ≳ 2 R 500 . In the core the bright radio source associated with NGC 1275 prohibits the validation below 30 arcmin, but this is where high quality X-ray data are available. We verify that extrapolation of the model by [152] leads to an overestimation of the tSZ signal by a factor of a few. \nFigure 23 . Comparison between the Compton parameter map measured by Planck and that inferred from our thermal model. Left panel : raw data. Middle : model. Right panel : residual. The dashed circle indicates θ 500 . Radio sources appear as negative point sources on the Compton parameter map. The two white circles are masks for NGC 1275 and another radio source. \n<!-- image --> \nFigure 24 . Comparison between the Compton parameter profile measured by Planck and that inferred from our thermal model. We also show the result of the extrapolation of the model by [152]. \n<!-- image -->", 'B Impact of the magnetic field on the model calibration': 'In Figure 25, we show how the choice of the magnetic field strength model affects the observable in terms of radio synchrotron emission and γ -ray emission in the case of the pure hadronic scenario. The model parameters were fixed to { X CRp , 500 , α CRp , η CRp } = { 0 . 075 , 2 . 5 , 0 . 75 } . Since the radio data are limited to the cluster core, the observable profile is essentially sensitive to the core magnetic field. While the hadronic emission does not depend on the magnetic field, the inverse-Compton signal is affected by the magnetic field via the synchrotron losses that modify the spectral and spatial distribution of the secondary CRe. However, matching the CRp model normalization so that a given magnetic field model would fit the radio data would also imply changing the normalisation of the hadronic and inverse-Compton model, accordingly. \nWe also note that in the CTA energy range, the radio data implies that the inverseCompton signal is too small to be detected, and much smaller than the expected hadronic signal. The predicted level of emission would be similar in a pure leptonic model assuming continuous injection (as it is the case in the hadronic model, via secondary particle production). However, in the case of discontinuous particle injection, energy losses are expected to lead to a cutoff in the CR spectrum, which will drastically reduce the amount of inverseCompton emission at CTA energies. The level of emission seen in Figure 25 can thus be considered as an upper limit for the inverse-Compton emission, and it is therefore neglected in the present paper. \nIn Figure 26, instead we fit for the CRp parameters { X CRp , 500 , α CRp , and η CRp } by matching the radio synchrotron emission to the data in the pure hadronic scenario (as in Section 2.3.2). We show the constraints on the recovered pure hadronic parameter space for three extreme magnetic field models. As we can see, changing the magnetic field model essentially affects the normalization, and only minor variations are observed on the other parameters. This also mainly leads to a change in the normalization of the γ -ray observables. \n2 \nFigure 25 . Same as Figure 3, but showing the dependence of the pure hadronic model observables on the magnetic field model (see Section 2.1.2 and Figure 1). The color code is the same for all panels. Top left panel : radio synchrotron spectrum. Top right panel : radio synchrotron profile. Bottom left panel : γ -ray spectrum. The hadronic component, which does not depend on the magnetic field, is shown in black and the inverse-Compton in color, for the different magnetic field models. The shaded pink area corresponds to the CTA energy range. Bottom right panel : γ -ray profile. The dashed lines correspond to the CTA PSF and θ 500 , respectively. \n<!-- image --> \nFigure 26 . Constraints on the parameter space of the pure hadronic model, for three extreme magnetic field models. See Section 2.1.2 and Figure 1. \n<!-- image -->', 'C Fermi -LAT sources around the Perseus cluster': 'No sources other than NGC 1275 and IC 310 has been detected within the CTA field of view in the direction of Perseus at very high γ -ray energies yet. Thus, we do not consider any other object in the modelling of the sky. Yet, the improved CTA sensitivity and increased field of view may allow us to detect other γ -ray emitters. To highlight such opportunities, we use the Fermi -LAT catalog, 4FGL-DR2 [212, 213], and select all sources detected within 5 deg from the Perseus reference center. These sources are listed in Table 7 and their locations are reported in Figure 6. NGC 1275 and IC 310 are included in the 4FGL source list. The spectra of all of these sources, at Fermi -LAT energies, are best described by power-laws, except for the pulsar PSR J0340+4130 and NGC 1275. For reference, and comparison to detected VHE sources NGC 1275 and IC 310, we extrapolate their fluxes up to the CTA energy range at 100 GeV. Most of the obtained values correspond to ≲ 10 -11 cm -2 TeV -1 s -1 , which is smaller but comparable to the extrapolation of the fluxes of NGC 1275 and IC 310 at the same energy (about 25 and 5 × 10 -11 cm -2 TeV -1 s -1 ) according to equations 4.1 and 4.2. However, we stress that a cutoff in the spectra is possible, especially for distant sources affected by EBL absorption, and these numbers provide optimistic fluxes. Additionally, some of the sources may be variable so that these values only provide rough estimates. Even in the case of a detection, none of these sources is expected to affect the results presented in this paper because of the large angular separation from the cluster center. \nTable 7 . 4FGL-DR2 sources located within 5 deg from the Perseus cluster center. \n× \nNotes. ⋆ Detected at very high γ -ray energies ; † Taken from the NASA/IPAC extragalactic database ; PL: power-law ; LP: LogParabola ; PLSEC : PLSuperExpCutoff ; RDG: radio galaxies; BCU: blazar candidates of uncertain type ; FSRQ: flat-spectrum radio quasars type of blazar; PSR: pulsar.', 'D Galactic foreground estimate': 'The Galactic foreground is estimated using the models developed by [216, 237]. We consider the min and max models as two different estimates. In Figure 27, we present the spatial distribution of the diffuse foreground signal at 1 TeV, and the spectrum integrated in different apertures around the cluster center. We can observe that the signal presents a soft gradient over the considered region, plus a few clumps. The latter are not correlated with the cluster and none of them is located within θ 500 . Even in the case of a large aperture of about 1 deg ( ≃ θ 500 ), the integrated diffuse foreground is expected to be subdominant compared to the cluster CR induced signal within the same region (baseline CR model). When reducing the aperture, the Galactic foreground becomes completely negligible compared to the cluster emission since the latter is much more compact. \n<!-- image --> \nFigure 27 . Left panel: Surface brightness image of the Galactic foreground contribution at 1 TeV ( min model). Right panel: spectral energy distribution of the Galactic foreground estimates and comparison to the baseline cluster model and the contribution from NGC 1275. Three different aperture radius are used, as indicated in the legend. The min model is given in red and the max model is given in pink. See [216, 237] for more details about the modelling. \n<!-- image -->', 'E Convergence of the CR parameter constraints versus the number of simulations': 'In Section 5.3, we derived exclusion limits on the CR normalization as a function of the CR spectral and spatial distribution parameters. This was done by averaging the upper limits obtained over a given number of simulations in order to reduce the Poissonian noise. \nFigures 28 shows the evolution of the constraint on the CR normalisation upper limit (mean and standard deviation), as a function of the number of simulations performed. We can see that averaging the results over 50 simulations is enough to obtain uncertainties on the mean and the standard deviation of the exclusion limit lower than 10%, which is negligible compared to other sources of uncertainties discussed in the paper. Therefore, we use a set of 50 simulations in the analysis. \n<!-- image --> \nFigure 28 . Evolution of the upper limit on the parameters X 500 ≡ X CRp ( R 500 ) (95% confidence level), as a function of the number of simulations performed and for the different model parameters ( α CRp , η CRp ) presented in Figure 11. Left : normalized mean upper limit. Right : normalized mean standard deviation. Note that the results obtained for the different models are correlated because the same underlying Poissonnian realization is used for the different models. \n<!-- image -->', 'F Validation of the cosmic-ray analysis with gammapy': 'In this appendix, we validate our baseline cosmic-ray analysis, which relies on ctools [217], using the alternative software gammapy [220]. First, we compute the number count prediction given the sky model discussed in Section 4 using the gammapy framework and a similar setup as the one used for ctools , but we use only one fixed pointing direction here. We verify that for this setup the two count cubes agree within a few percent, so that the residual difference is negligible compared to other sources of uncertainties. We then reproduce the results presented in Section 5.3 and Figure 11 with gammapy . In this case, the 95% upper limits are computed using the iminuit software [225]. To reduce the computing time, we free only the normalization of each sky component in the fit and compute how the upper limit changes when considering the full parameter space (normalization plus spectral parameters) from test cases. The corresponding correction, of about 50% depending on the model, are applied to the results. As shown in Figure 29, gammapy and ctools give consistent results on the constraints obtained in the cosmic-ray parameter space within error bars. The residual differences are smaller than other sources of uncertainties presented in the paper (e.g., the cluster modeling) and are likely due to the fact that only the normalization parameters are fitted, or the slight differences in the setup. We conclude that using either ctools or gammapy for the cosmic-ray analysis will not affect the results presented in the paper. \nFigure 29 . Comparison of the exclusion limit on the parameter X CRp obtained with ctools and gammapy . The four panels give the results for the different values of η CRp . The error bars represent the standard deviation of the upper limits and the data points give the means. A set of 50 simulations is used both for ctools and gammapy . \n<!-- image -->', 'G Cosmic-ray parameter constraints in the pure hadronic model': 'In this appendix, we report the constraints in the parameter space obtained as discussed in Section 5.4.2, in Figure 30. In the case of this data realization, the input parameters are recovered within 68% confidence, or right at the limit, for all parameters. \nFigure 30 . Example of posterior constraint on the full parameter space, as in Figure 14, but in the case of the pure hadronic model. Contours provide the 68% and 95% confidence interval. The black star shows the input model parameters. \n<!-- image -->', 'H Interplay between the DM components and the astrophysical γ -ray sources': 'In Sections 6.2 and 6.3, we have discussed the results of the template-fitting analysis in terms of the DM parameters. Now we will investigate the results obtained for the rest of free parameters (Equation 6.2) and scrutinize the correlations between them. \nAppendix K (Table K) shows the recovered mean values for the parameters regarding the astrophysical backgrounds (CRs, NGC 1275, IC 310, instrumental). In all of the cases we recover, within 1 σ , the input values that were used to produce the simulations in the first place, independently of the DM channel or mass that we fit. These results reinforce the idea that the DM component seems to be, at most, mildly correlated with the parameters describing the astrophysical sources in the area. \nFigure 31 . Correlation matrix obtained for a fit to the MED annihilation model, a DM mass of m χ = 10 TeV and the τ + τ -channel. The color scale ranges from -1 for parameters completely degenerated and anticorrelated (in red), to 1 for those degenerated and correlated (in blue). See text for discussion on the found correlations. \n<!-- image --> \nTo properly quantify the correlation between the different parameters we compute the correlation matrix. For this, we use an MCMC method based on the emcee python package [171]. We follow the same methodology explained in Section 5.4.2 for the astrophysical parameters ( A CR , A PS1 , α PS1 , A PS2 , α PS2 , A bkg , α bkg ). We simplify our analysis only including the CR normalization due large amount of computation time that it is needed. For the DM normalization ( A χ ), we use a flat prior ( A χ ≥ 0 ) to avoid sampling non-physical values. Since this approach is computationally very expensive, we only perform the fit for a representative DMmass of m χ = 10 TeV, the τ + τ -channel and the MED annihilation model, and use a total of 30 simulated observations. The obtained correlation matrix is shown in Figure 31. Indeed, the DM normalization is only mildly anticorrelated with the CR normalization, showing little \ncorrelation with the parameters describing NGC 1275. This implies that the results obtained for the limits on DM cross-section and DM lifetime should be robust. We caution that this correlation matrix may be dependent on the considered DM scenario (annihilation/decay), channel or m χ , that were not explored in their totality given the constraints on the computation time. In Figure 31, we can also appreciate that, with respect to the correlation matrix shown in Figure 10, which did not include DM fluxes, the correlation of the CR normalization with the normalization of NGC 1275 is strengthened. Also, NGC 1275 parameters change their correlation sign. In the case of a detection of γ -rays from Perseus, these correlations should be investigated in depth and properly taken into account in the analysis.', 'I CTA sensitivity to DM under the ON-OFF observational setup': "The ON-OFF method is one of the most standard observational and analysis approaches adopted by the existing IACTs to obtain DM constraints (e.g., [114, 116, 120]). This choice is mainly motivated by their limited FoV and angular resolution, which makes it very difficult a (optimal) use of spatial information in large regions of the sky, as needed for a templatefitting analysis. In fact, a large number of IACT DM analyses for clusters even consider the DM-induced γ -ray fluxes as a point-like source in the target center containing the integrated DM flux of the whole object, this way neglecting its extension as a first approximation to the DM search. \nFor the sake of comparison, in this appendix we first compute the CTA sensitivity to DM in Perseus assuming the DM-induced emission as a point-like source. We will only include the instrumental background and will neglect the rest of γ -ray sources in the area. This methodology, although over-simplistic, will provide a first-order evaluation of the CTA sensitivity to DM in the object and is expected to yield the most stringent (unrealizable) constraints. Yet, as described in Section 4.2, the Perseus cluster hosts a very bright AGN, NGC 1275, in its center, which in a real observation may be difficult to neglect. Thus, in a second step, and in order to perform a more refined, realistic ON-OFF analysis, we use the DM templates to account for the spatial extension of the DM emission and, as adopted in [116], we place a circular mask of 0.1 deg. radius in the center of Perseus to avoid AGN contamination. In the observational setup described in Section 4.1, IC 310 is neither in the ON nor in the OFF regions, thus it is not included in this analysis either. Since in this section we focus on the sensitivity to the DM emission, we neglect the CR component in the following, which indeed has already been properly discussed and considered for obtaining the main DM results in this paper, i.e. those derived via the template-fitting analysis in Section 6. In total, we create 4 simulations for the ON-OFF analysis, two assuming the MED annihilation and DEC decay scenarios, both integrated to look like a point-like source in the center of Perseus; and two more simulations adopting the MED and DEC DM templates, this time including a central mask. \nWe follow the likelihood maximization method and the likelihood ratio test ( TS ), as done in the template-fitting analysis, to search for a signal. The likelihood for the ON-OFF method corresponds to a product of Poissonian likelihoods for the ON and OFF regions, described for each energy bin (i-th) and ON-OFF regions (j-th) as: \nL ( A χ | D ) = ∏ ij ( N S ij + κ ij N B ij ) N ON ij N ON ij ! e -( N S ij + κ ij N B ij ) × ( N B ij ) N OFF ij N OFF ij ! e -N B ij , (I.1) \nwhere N S ij is the number of expected signal events in the ON region, N B ij is the number of expected background events, κ is the normalization factor to account for potentially different background acceptance in the ON and OFF regions, N ON ij is the number of observed photons in the ON region and N OFF ij the same but for the OFF regions 36 . In this case, we only fit one parameter, the DM normalization A χ , resulting in profiling a one-dimensional likelihood, with the corresponding flat prior of positive normalization values. To perform this analysis, we use the gammapy software package for γ -ray data analysis, especially the function FluxPointsEstimator , with the iminuit backend. \nNo hint of detection neither for annihilation nor for decay is obtained in our simulations, thus we proceed to obtain the corresponding upper (lower for decay) DM limits. For this analysis, we use a number of Poissonian realizations as suggested by our studies in Appendix J. In particular, we average 100 realizations in order to obtain statistically meaningful and stable results. We also check several variations of this final configuration (referred as 'Case 1' in the following) to test the impact of the details of the observational strategy in the DM limits. In Table 8 we show the different setups here evaluated and Figure 32 the corresponding upper limits in each case. For the latter, we adopt the MED annihilation case and use the 2D spatial template for the DM emission, yet we neglect the mask for computational reasons in cases 2, 3, and 4. \nTable 8 . Different observational setups tested in the ON-OFF analysis of this section. θ pointing is the pointing offset with respect to the center of Perseus, where the ON region is centered; θ ON is the aperture radius of the ON region (which for all the considered setups coincide with the aperture radius of each of the OFF regions), N OFF is the number of OFF regions considered, and κ is the normalization parameter introduced in Equation I.1. Our default configuration is 'Case 1' (highlighted in bold in the table), i.e. the same introduced in Section 4.1. \nFigure 32 shows that all the tested cases lie within the 1 σ scatter band of the 'Case 1' configuration. Indeed, the discrepancy that is observed at the lowest considered energies is due to the inclusion of the mask for the 'Case 1' configuration while no mask is present for the rest of cases. We conclude that the configuration selected in Section 4.1 is also optimal for DM searches in Perseus based on the ON-OFF method. \nWe show in Figure 33 the 95% C.L. limits for the canonical 'Case 1' ON-OFF setup, for the different assumptions on the spatial extension of the DM-induced emission (i.e., point-like and extended), for both the MED annihilation and DEC decay models. At a first glance, we can see that the best limits are reached for the point-like source assumption in all of the cases, as expected. This is even more pronounced at the lowest DM masses considered, since in the case of 'Extended+mask' the mask removes a comparatively larger fraction of photons, this way weakening the limits. The comparison with other IACT results on DM searches in galaxy clusters show that CTA prospects are up to more than O (10) constraining, depending on the \nFigure 32 . 95% C.L. mean upper limits to the annihilation cross-section under the different ON-OFF configurations shown in Table 8. In all cases we adopt the MED model and the τ + τ -channel, and consider the spatial extension of the DM-induced emission in Perseus. The green band represents the 1 σ scatter of the projected limits around 'Case 1'. The mean is computed over 50 Poisson realizations for computational reasons. Note that no mask is included for cases 2, 3 and 4, which explains some of the observed differences among curves. See text for discussion. \n<!-- image -->", 'DM mass range and annihilation/decay channel 37 .': "Finally, we also explore the impact of the different analysis pipeline on our limits. In the right panel of Figure 34, together with our template-fitting analysis results for annihilation (solid line, MED model, see Section 6.2), we show the results obtained for the canonical 'Case 1' ON-OFF setup ('ON-OFF - Extended+mask', dot-dashed line). We can appreciate the loss in sensitivity in the lower mass range, mainly due to the central mask. In the high mass range (above ∼ 1 TeV), the limits become a factor up to ∼ 4 times more constraining than the template-fitting ones, thanks to an over-simplistic modelling of the rest of astrophysical sources. In the same right panel of Figure 34, we also include the annihilation results for the point-like source assumption. We can notice that this approach ('ON-OFF - PS'; dashed line) is the most optimistic scenario, as expected, improving the limits of the template-fitting analysis by a factor ∼ 2 -2 . 5 , yet being within the 2 σ scatter of the template-fitting results in all the explored mass range. Although useful and relevant to understand the absolute sensitivity reach for CTA in an idealistic scenario, we recall that these 'ON-OFF - PS' limits do not describe a realistic science case and correspond to an overly simplistic setup. An important conclusion from these comparisons among different analyses is that, despite the very different methods and assumptions on the modelling of γ -ray sources in the area, we obtain upper limits that lie within the 2 σ scatter of the template-fitting results. This demonstrates not only the robustness of the found results but also points towards a low correlation of the DMparameters with respect to those corresponding to the rest of γ -ray sources in the cluster, including CRs (further investigated in Appendix H). This effect is also quantified for the decay \nFigure 33 . Left panels (right panels): 95% C.L. mean upper (lower) limits for the annihilation MED model (DEC decay model) for τ + τ -( top panels ) and b ¯ b ( bottom panels ) for the ON-OFF 'Case 1' configuration (see Table 8). The solid black line shows the results considering the spatial extension of the DM emission plus a mask of 0.1 deg in the center of Perseus, while the dot-dashed black line corresponds to the results for the over-simplistic point-like DM source assumption; see text for details. The green band represents the 1 σ scatter of the projected limits. We also show for comparison the results from the MAGIC observations of Perseus (olive long-dashed lines; [116]) and the results from the H.E.S.S observations of Fornax (purple dashed lines; [114, 115]). \n<!-- image --> \nscenario in the left panel of Figure 34. We show our canonical results for the template-fitting analysis (solid line, see Section 6.3), together with the limits resulting from the 'Case 1' ON-OFF setup (ON-OFF - Extended+mask, dot-dashed line) and the simplistic ON-OFF analysis for which we assume the DM emission to be a point-like source (ON-OFF - PS, dashed line). As expected, the most simplistic approach (ON-OFF - PS) is the one providing the most optimistic constraints, in this case being a factor ∼ 2 -2 . 5 better than the templatefitting results, yet lying within the 2 σ scatter of the latter. We remark that the aim of this overly simplistic approach is simply to understand the maximum sensitivity reach of CTA in an idealistic, unrealizable scenario. As for the results of the 'ON-OFF - Extended+mask' method, the effect of the mask is clearly visible in the lower mass range as a worsening with respect to the canonical limits, while there is a light improvement for masses above a few TeV, still being withing the 1 σ scatter band. Thus, the use of the different analysis methods produce mean lower limits within the 2 σ scatter of our canonical, template-fitting results, in agreement with that found for annihilation as well. \n10 \n10 \n] \n1 \ns \n3 \nm \n10 \nc \n[ \n> \nv \n10 \n< \n10 \n10 \n20 \n21 \n22 \n23 \n24 \n25 \n< \nv \n> \nthermal \n0.1 \n1 \n10 \n100 \nFigure 34 . Limits for the τ + τ -channel for three different analysis methods for annihilation ( left panel ) and decay ( right panel ). The solid line refers to the template-fitting approach (MED model in case of annihilation, see Sections 6.2 and 6.3); the dot-dashed line is for the ON-OFF 'Case 1' configuration, and the dotted line indicates our most simplistic analysis, i.e. ON-OFF assuming a point source for all the DM in the cluster. The green (yellow) band shows the 1 σ ( 2 σ ) scatter of the projected limits using the template-fitting approach. \n<!-- image --> \nm \n[TeV]", 'J Convergence of the DM fits versus the number of simulations': 'The simulated observations that we use as CTA data (for more details see Section 6) produce the corresponding photons and events assuming a Poisson distribution. To obtain stable and representative prospects, we need to average the results over a certain number of different outcomes of the Poisson randomization process. To guarantee a number of realizations high enough to have trustful results, at the same time representing a good compromise in terms of computational time, we analyze the evolution of the 1 σ and 2 σ scatter bands corresponding to the DM upper limits (for the MED annihilation scenario) with respect to the number of simulations that are considered in the computation of the mean upper limits. This is shown in Figure 35 and Figure 36 for the template-fitting and ON-OFF analysis methods, respectively (note though that in the latter case we only study the 1 σ band). \nIn Figure 35 we can see that the 1 σ bands converge around 80 realizations to a value of ∼ 0.1 dex, mostly independently of the DM mass or annihilation channel. The 2 σ bands seem to converge instead after around one hundred realizations, to a value between 0.2-0.4 dex depending on the mass (no clear correlation is observed). With these results at hand, we decide to average our DM limits over 100 Poissonian realizations for the template-fitting analysis. \nIn Figure 36 the 1 σ band converges much more quickly, around 50 realizations for any channel or DM mass. In contrast to the case of the template fitting, we appreciate a clear correlation of the convergence value with DM mass: larger masses lead to higher 1 σ values. This is surely related to the likelihood we used for the ON-OFF method (Equation I.1), which is known to result in biased estimates in case of observations with very low counts, as it is the case of large WIMP masses. Indeed, our simulations including a DM annihilation flux plus the CTA instrumental background have an extremely low number of photons, resulting in a higher error of the estimated upper limit. We decide to present the mean DM limits from the ON-OFF results after averaging over 100 Poissonian realizations. \n+ \nTemplate fitting \nON-OFF - PS \nON-OFF - Extended+mask \nFigure 35 . Convergence of the 1 σ ( top panels ) and 2 σ ( bottom panels ) bands in the case of the template-fitting analysis technique, for b ¯ b ( left panels ) and τ + τ -( right panels ) channels and for different values of the fitted DM mass. \n<!-- image --> \nFigure 36 . Convergence of the 1 σ band for the case of the ON-OFF analysis technique, for b ¯ b ( left panel ) and τ + τ -( right panel ) channels, and for different values of the fitted DM mass. \n<!-- image -->', 'K Recovered astrophysical parameters from the DM template-fitting': 'In Table K we show the best-fit values (averaged over 100 simulations) obtained for all the parameters involved in the modelling of the γ -ray sources considered in the analysis (see Equation 6.2). The shown results are for the case of including a DM template corresponding to the MED annihilation model. The recovered best-fit values are all compatible with the input value used to create the simulations within the 1 σ error. Also, we tested that these values do not seem to be correlated with either the considered DM mass or annihilation channel of the fit, indeed recovering the same value within 10% of the error in all cases. \n1 \nσ \nstatistical error. \nTable 9 . Recovered values, with their corresponding errors, of the parameters describing the astrophysical sources in Perseus (see Equation 6.2). In all cases, we adopt the MED annihilation model for the DM component considered in the template-fitting. The error corresponds to the symmetrical', 'L CTA sensitivity to DM in Perseus with ctools': 'In this appendix, we describe the analysis performed with ctools software to search for γ -ray DM emission in the simulated CTA observations (see Section 4.1) of the Perseus cluster. The main goal is to perform a comparison with the results already obtained with the gammapy software, presented in Section 6, and to also quantify the compatibility among them.', 'L.1 Data preparation': "We simulate three different sets of observations of the Perseus cluster using the ctools public code [217]. ctools is a software to simulate and analyze data for γ -ray observatories. It is based on the gammalib C-library. To generate the observations, we use the ctobssim tool to simulate γ -ray events in ten energy bins starting from 30 GeV to 100 TeV, same as used for the gammapy analysis (check Section 6). The total duration of the observations is 300 h, obtained after stacking 300 individual observations of 1 h duration each (Section 4.1). \nThe first two data sets correspond to classical ON-OFF analyses, used in current IACTs. The first ON-OFF setup, as it is explained in Section I for gammapy , we assume that the DMinduced γ -ray emission is described by a point source (PS). We also neglect the contributions of the other γ -ray sources in Perseus, and only consider the instrumental background contribution that is modeled via the IRFs. Also following Section I, the second ON-OFF data set uses a more realistic modeling of the Perseus cluster by including the emission of NGC 1275. We use the DM templates to model the spatial morphology of the DM signal (see Section 3.3). Additionally, we place a circular mask of 0.1 deg radius, as in [116], to block out the bright emission from NGC 1275 (Equation 4.1). The instrumental background is modeled by the \nIRFs. In the end, we also create four sets of simulations, as done with gammapy , for the ON-OFF observations: two for the MED model annihilation scenario, i.e., one assuming the point-like source ('PS') approximation and another one for the extended source plus the mask ('ES+Mask'); and two for DEC decay scenario, 'PS' and 'ES+Mask' setups. \nThe last set of simulations refers to only one circular ON region of 3 deg radius, considering the contribution of all the γ -ray sources in the Perseus cluster. NGC 1275 and IC 310 emissions are described by Equations 4.1 and 4.2, respectively. The CR-induced γ -ray emission is described using the 'Baseline' model (Section 2.3.1). We consider a total of four scenarios for the DM-induced γ -ray emission, i.e. (MIN, MED, MAX) for DM annihilation, and (DEC) for DM decay. \nFinally, for every set of observations we simulate a total of 100 realizations, to consider the statistical background fluctuations and compute mean parameter values, and as 1 σ and 2 σ bands.", 'L.2 DM analysis pipeline with ctools': 'We follow the same analysis strategy described in Section 6.1 for gammapy . In the case of ctools , the DM analysis pipeline is available in the ctadmtool public code 38 . ctadmtool integrates three different steps in the calculation of exclusion limits. In the first step, the γ -ray flux induced by annihilation/decay of DM is estimated, given the parameters of the DM candidate and the spatial emission template. We use PPPC4DMID [238] to interpolate to the desired values of DM mass. ctadmtool computes this γ -ray flux to the number of mass points that the user wants to explore. The second step, we use the ctlike tool to estimate the parameters (Equation 6.2) that fit the observation, get the correlation matrix and TS (Equation 4.4) for every component, and compute the TS profiles as a function of the DM normalization. The TS profile is computed by letting free the parameters of all other components in the cluster. In the absence of a signal, the final step is to estimate the upper limits (95% C.L., ∆ TS = 2 . 71 with respect to the best fit) to the flux and convert to exclusion limits of the DM parameters. \nWe assume that the DM normalization, A χ , can only take physical values ( A χ ≥ 0 ), and that a signal detection occurs when TS ≥ 25 . We adopt 10 logarithmically-spaced values of the DM mass in the range from 50 GeV to 100 TeV, and assume two representative DM channels, b ¯ b and τ + τ -.', 'L.3 CTA sensitivity to DM under the ON-OFF observational setup with ctools': "We do not find a DM signal neither in the annihilation nor in the decay scenarios for the different sets of ON-OFF observations. Thus, we proceed to compute the 95% C.L. ULs to the DM-induced γ -ray flux and, from there, calculate exclusion limits of annihilation crosssection and decay lifetime as a function of the DM mass. We show in Figure 37 the 95% C.L. exclusion limits for the 'PS' and 'ES+Mask' ON-OFF observational setups, and for the four sets of observations (Section L.1). We observe that the best limits are obtained for the (unrealistic) 'PS' case, in agreement with the results obtained with gammapy (Section I). Placing a mask on NGC 1275 ('ES+Mask') weakens the limits up to O (10) for DM mass below 1 TeV (10 TeV) for annihilation/decay channels to τ + τ -( b ¯ b ). \nFigure 37 . Left panels (right panels): 95% C.L. mean upper (lower) limits for the annihilation MED model (DEC decay model) for τ + τ -( top panels ) and b ¯ b ( bottom panels ), for the ON-OFF configurations. The solid black line shows the results considering the spatial extension of the DM emission plus a mask of 0.1 deg in the center of Perseus ('ES+Mask'), while the dot-dashed black line corresponds to the results for the point-like ('PS') DM source assumption. The green (yellow) band represents the 1 σ ( 2 σ ) scatter of the projected limits. We also show for comparison the results from the MAGIC observations of Perseus (purple dashed lines; [116]) and from H.E.S.S observations of Fornax (purple dashed lines for annihilation [114]; olive dashed lines for decay [115]). \n<!-- image -->", 'L.4 CTA sensitivity to DM based on template fitting with ctools': "In this case, as done with gammapy in Section 6.1, we first check with ctadmtool if NGC 1275 can potentially contribute to the DM- and CR-induced γ -ray emission components in the Perseus cluster, while ideally it should not. To estimate the effect of this 'contamination' we extract the TS for every component as a function of the DM mass, for 100 realizations of the observations, and compute the mean value of the TS for the (MIN, MED, MAX) annihilation and (DEC) decay scenarios. \nFigure 38 shows the mean value of the TS for the MED DM-induced γ -ray emission, CR-Baseline and NGC 1275 as a function of the DM mass used in the fit. For clarity, we only show the results for MED DM scenario, but same results are obtained for the rest of cases. We observe that for DM masses below ∼ 1 TeV the TS of NGC 1275 has a decrement of almost a factor 2 with respect to that obtained above ∼ 10 TeV. This decrement in NGC 1275 TS \nFigure 38 . Mean TS values associated to three emission components in the Perseus cluster. The purple, red and olive lines correspond, respectively, to the MED DM emission model with annihilation to channel τ + τ -, the CR-Baseline model, and NGC 1275. The black dashed line is the detection threshold ( TS ≥ 25 ). We observe that for masses below 1 TeV, the TS for NGC 1275 decreases to the half of its value for DM masses above ∼ 10 TeV. This change seems to be associated with a detection of the DM-induced γ -ray emission that it is not observed neither for the ON-OFF setups nor in the gammapy -based analyses. See text for more details. \n<!-- image --> \nis possibly associated with the apparent positive detection of DM-induced γ -ray signal, not observed neither in the previous ON-OFF analysis with ctadmtool nor in the full gammapy analysis. Moreover, we notice that the CR component has also significant variations in the TS that are possibly correlated with the decrement in NGC 1275 TS , starting for DM masses below 10 TeV. These variations though do not change the fact that we always have a detected CR-induced γ -ray signal (Baseline model). We do not show the TS of IC 310 because it is constant for all the DM masses considered in the fit. \nThe behaviour and our interpretation of these TS curves as a function of the DM mass is supported by the correlation matrix. Figure 39 shows the average correlation matrix obtained for 100 realizations of the 300 h observations considering all the γ -ray emission components in the cluster. We select four different DM masses according to the different behaviours of the TS in Figure 38. For DM masses below 1 TeV, we see that the DM normalization is anti-correlated to the free parameters of CRs and NGC 1275. We can also observe a mildly anti-correlation with the instrumental background parameters. It is interesting that the anti-correlation between DM and the instrumental background disappears as the DM mass increases. We can also observe that the sign of correlation between DM and NGC 1275 parameters changes around DM masses of ∼ TeV, and then the value of the correlation decreases as the DM mass increases. This is different to the correlation matrix obtained with gammapy (Figure 31), where there is not obvious correlation between NGC 1275 model-parameters and DM Normalization. We also find a strong anti-correlation for DM masses above ∼ 1 TeV be- \nFigure 39 . Correlation matrices of the free parameters of the emission models in the Perseus cluster (Equation 6.2), for four different DM masses. Shown are the mean values of the correlation matrix obtained for 100 realization of observations assuming the MED annihilation scenario. \n<!-- image --> \nCR and NGC 1275, also differing with respect to the behaviour obtained with gammapy . IC 310 model parameters do not show any correlation with other parameters, which supports the fact that its TS is constant in the whole range of considered DM masses, as mentioned before. \nFrom these results, we conclude that the template fitting analysis with the setup described in this section is not sufficient when using ctadmtool , since the strong NGC 1275 emission makes the analysis particularly tricky, and will possible lead to a fictitious detection of a DM-induced γ -ray signal. \nIn light of the previous results, in the following we propose and describe a specific analysis strategy to avoid the leaking of NGC 1275 emission into the other emission templates when using ctools . This alternative strategy will enable us to properly compute projected limits to velocity-averaged cross-section and lifetime of DM particles. The strategy is based on placing a mask in the center of the Perseus cluster so as to block out a significant fraction of the emission from NGC 1275. The main drawback of this approach is that, as the CRs and DM emission models also peak in this central region, we will be less sensitive to a putative \nemission from both contributions. \nTable 10 . Angular sizes (radii) of the mask applied to the simulation in the center of the Perseus cluster. The size is set to 2 times the value of the angular resolution of the CTA North Array at the energy corresponding to the lower extreme of each energy interval [120]. \nWe select five different sizes for the mask, each of them applied to a particular energy range between 30 GeV and 100 TeV. More precisely, the angular size of the mask is set to 2 times the value of the CTA angular resolution at the energy corresponding to the lower extreme of each energy interval. Table 10 provides the corresponding mask sizes and energy ranges. With this definition of the mask, we guarantee that we remove ∼ 95% of the photons coming from NGC 1275. More importantly, with this analysis configuration we do not detect a DM signal in the simulation data and confirm that the TS values are not stable across all considered energies. In the same way, we do not detect a signal associated with CR-induced γ -ray emission, either. \nIn the absence of a clear DM signal, we compute 95% C.L. exclusion limits to the annihilation cross-section (decay lifetime) versus the DM mass. Figure 40 shows the exclusion limits for the MED annihilation scenario for both b ¯ b and τ + τ -channels, while Figure 41 shows the results obtained for the DEC decay scenario. In both figures, we follow the same convention to show the results as in Sections 6.2 and 6.3, and compare to recent results from other experiments as well. Note that results from both the template-fitting and ON-OFF extended analyses, both for annihilation and decay, are in good agreement for DM masses above 1 TeV. This is expected, as the masks adopted in both types of analyses at these energies is comparable.", 'L.5 Comparison with gammapy': "Finally, in this section we show the comparison between the results obtained via the two analysis pipelines used in this paper to search for a DM-induced γ -ray signal in the Perseus cluster. The results obtained with ctools are shown in Sections L.3 and L.4, while those with gammapy are presented in Sections 6.2, 6.3, and I. Figures 42 and 43 show the comparison for annihilating and decaying DM for the different observational setups considered in this work: template fitting, ON-OFF (ES+Mask), and ON-OFF (PS). In the following, we only focus the discussion in the comparison for the MED annihilation scenario. For the DEC scenario, as well as MIN and MAX annihilation models, conclusions are similar. \nIn the case of the template-fitting analysis, we observe that the results obtained with ctools are less restrictive in comparison to the limits obtained with gammapy for DM masses below 1 TeV (10 TeV) for τ + τ -( b ¯ b ) annihilation channel. This difference is a consequence of the restrictive mask placed on NGC 1275 in the alternative setup used for ctools , which significantly decreases the CTA sensitivity at lower energies (i.e., DM masses). The larger difference for b ¯ b is due to the fact that in this case the annihilation spectrum has its maximum \nFigure 40 . Sensitivity of CTA to a DM annihilation signal from the Perseus cluster, obtained via a template-fitting analysis with ctools and adopting a mask with those properties in Table 10; see text for details. Curves represent the 95% C.L. upper limits on the velocity-averaged cross-section for the MED annihilation model. The green (yellow) band shows the 1 σ ( 2 σ ) scatter of the projected limits. The black dashed line is the thermal relic cross-section ( < σv > thermal = 3 × 10 -26 cm 3 s -1 ). Top left panel: Upper limits for the two considered annihilation channels, b ¯ b channel (dashed) and τ + τ -(solid). Top right panel: Limits for the τ + τ -channel for three different analysis methods. The solid line refers to the template-fitting placing a mask in the center of the cluster; the dot-dashed line is for the ON-OFF analysis assuming a point-like source (PS) for the DM-induced γ -ray emission, and the dotted line refers to the ON-OFF analysis when we consider a DM spatially-extended emission and place a mask in the center of the Perseus cluster (Section L.1). Bottom panels: Cross-section upper limits for the b ¯ b ( left panel ) and τ + τ -( right panel ) channels in comparison with the most recent results on DM-annihilation searches in galaxy clusters using Fermi -LAT ([111], dotted, and [106], purple dot-dashed lines) and H.E.S.S ([114]; blue dashed lines). \n<!-- image --> \nlocated at ∼ 1 / 20 of the considered DM mass, thus even still at high masses of around few TeV, the major contribution to the photon spectra comes from lower energies, where the CTA sensitivity is lower. Above 1 TeV or 10 TeV, depending on the annihilation channel, the results between ctools and gammapy are in good agreement, indeed being within the 2 σ statistical fluctuations. \nFor the ON-OFF (ES+Mask) analysis, we observe that the projected limits obtained with ctools and gammapy are in good agreement for DM masses above 1 TeV, always within the 1 σ bands. Similarly to the previous case, we also observe a decrease in the sensitivity with ctools for DM masses below 1 TeV, being the magnitude of this decrement comparable to the decrease of sensitivity observed in the template-fitting method. We find the same effect \nFigure 41 . Sensitivity of CTA to a DM decay signal from the Perseus cluster, at 95% C.L., in terms of the mean lower limits of the lifetime of the DM particle versus DM mass. The green (yellow) band shows the 1 σ ( 2 σ ) scatter of the projected limits. Top left panel: Mean lifetime lower limits for the two considered decay channels, b ¯ b (dashed line) and τ + τ -(solid). Top right panel: Mean lifetime lower limits for the τ + τ -channel for three different analysis methods: template-fitting with a mask on NGC 1275 (solid line), ON-OFF analysis assuming a point-like source (PS) for the DM-induced γ -ray emission (dot-dashed), and ON-OFF analysis considering a spatially-extended DM emission with a mask in the center of the Perseus cluster (dotted line). See Section L.1 for details. Bottom panels: Mean lifetime lower limits for the τ + τ -( left panel ) and b ¯ b ( right panel ) channels in comparison with the most recent results on DM decay in galaxy clusters using MAGIC data (olive dashed lines; [116]), Fermi -LAT data (black dot-dashed lines; [106]) and H.E.S.S data (purple dashed lines; [115]). \n<!-- image --> \nfor b ¯ b , in concordance with the results obtained for the template fitting as well. As discussed in previous subsections of this appendix, we believe that the size of the mask used for this analysis, of 0 . 1 deg radius, is not sufficient to entirely cover the emission from NGC 1275, which impacts our sensitivity at the lowest energies, where the angular resolution is worse. We did not increase the the size of the mask placed on NGC 1275, as our goal is to compare the performance of the ctools and gammapy DMpipelines under the same observation setups. \nFinally, for the ON-OFF (PS) analysis, we observe that the results from ctools and gammapy are in excellent agreement, within 1 σ of statistical fluctuations, and independent of the assumed DM annihilation channel. \nAs an overall conclusion from these exercises of comparison between ctools and gammapy , we find a good agreement between both analysis pipelines at high DM masses, either in basic or more complex analysis setups and scenarios. However, for lower DM mass, typically below \nFigure 42 . Sensitivity of CTA to a DM annihilation signal from the Perseus cluster, at 95% C.L., in terms of the mean upper limits of the velocity-averaged cross-section of the DM particle versus the DM mass, obtained via the two CTA analysis softwares used in this work, ctools (solid line) and gammapy (dotted line). The green (yellow) band shows the 1 σ ( 2 σ ) scatter of the projected limits obtained with ctools , and the gray bands (when visible) represent the 1 σ and 2 σ bands for gammapy . The black dashed line is the thermal relic cross-section ( < σv > thermal = 3 × 10 -26 cm 3 s -1 ). Top left panel: Mean cross-section upper limits for the τ + τ -annihilation channel obtained with ctools (solid) and gammapy (dotted), in both cases adopting the template-fitting analysis approach (Sections 6.1 and L.4). Top right panel: Mean cross-section upper limits for the b ¯ b channel for ctools (solid) and gammapy (dotted) for the template fitting (Sections 6.1 and L.4). Bottom panels: Mean crosssection upper limits for the τ + τ -channel for the MED annihilation scenario, and ON-OFF 'ES + Mask' ( left panel ) and ON-OFF 'PS' analyses ( right panel ), as well as their comparison with the gammapy results (Section I). \n<!-- image --> \n1 TeV, ctools is less sensitive to disentangling the emission from multiple components. For the Perseus cluster, the main reason is that NGC 1275 overshines or eclipses the contribution of the DM- and CR-induced γ -ray emissions, inducing a signal leaking in these templates that causes unstable, untrustable results. Indeed, when the emission of NGC 1275 is artificially decreased by placing a mask on it, conveniently chosen according to CTA's angular resolution properties, we are capable of recovering the sensitive obtained with gammapy .", 'References': "[1] G.M. Voit, Tracing cosmic evolution with clusters of galaxies , Reviews of Modern Physics 77 (2005) 207 [ astro-ph/0410173 ]. \nFigure 43 . Sensitivity of CTA to a DM decay signal from the Perseus cluster, at 95% C.L., in terms of the mean lower limits of the lifetime of the DM particle versus the DM mass, obtained via the two CTA analysis softwares used in this work, ctools (solid line) and gammapy (dotted line). The green (yellow) band shows the 1 σ ( 2 σ ) scatter of the projected limits obtained with ctools ,and the gray bands (when visible) represent the 1 σ and 2 σ bands for gammapy . Top left panel: Mean lifetime lower limits for the τ + τ -decay channel obtained with ctools (solid) and gammapy (dotted), in both cases adopting the template-fitting analysis approach (Sections 6.1 and L.4). Top right panel: Mean lifetime lower limits for the b ¯ b channel for ctools (solid) and gammapy (dotted) for the template fitting (Sections 6.1 and L.4). 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2024arXiv240909496K | Positronium spectrum and lifetimes are known with a high precision. The situation is different for positronium moving across a magnetic field. The total momentum does not commute with the Hamiltonian and is replaced by conserved pseudomomentum. The internal dynamics is not separated from the motion of the system as a whole. The Coulomb potential well is distorted and a wide outer potential well is created. We analytically determine the energy spectrum for a broad range of the magnetic field and pseudomomentum values. We locate the region of these parameters for which the ground state resides in the outer well. The results may play a role in the suppression of pulsars radio emission polar cap problem. | 2024-09-01T00:00:00Z | ['10.48550/arXiv.2409.09496', '2024arXiv240909496K', 'arXiv:2409.09496'] | ['High Energy Physics - Phenomenology', 'Astrophysics - High Energy Astrophysical Phenomena'] | Metamorphosis of Positronium Moving Across a Magnetic Field | 2,024 | 170 | 0.26 | ['EPRINT_HTML', 'EPRINT_PDF'] | 0 | https://arxiv.org/pdf/2409.09496.pdf | {'Metamorphosis of Positronium Moving Across a Magnetic Field': 'B. O. Kerbikov ∗ \nLebedev Physical Institute, Moscow 119991, Russia', 'A. A. Simovonian †': 'Moscow Institute of Physics and Technology, Dolgoprudny 141700, Moscow Region, Russia \nPositronium spectrum and lifetimes are known with a high precision. The situation is different for positronium moving across a magnetic field. The total momentum does not commute with the Hamiltonian and is replaced by the conserved pseudomomentum. The internal dynamics is not separated from the motion of the system as a whole. The Coulomb potential well is distorted and a wide outer potential well is created. We analytically determine the energy spectrum for a broad range of the magnetic field and pseudomomentum values. We locate the region of these parameters for which the ground state resides in the outer well. The results may play a role in the suppression of pulsars radio emission (polar cap problem).', 'I. INTRODUCTION': "There are several reasons for placing focus on theoretical and experimental studies of positronium (Ps). It is an ideal system for testing the accuracy of the QED calculations with unprecedented precision [1-3]. Ps may serve as a testing ground for the search of possible effects beyond the Standard Model [1, 4-7]. Currently a vital interest in Ps stems from its possible role in suppressing the one-photon e + e -pair creation in the neutron star (NS) magnetospheres [8-11]. Moving across pulsar magnetic field (MF) which may be of order 10 12 G Ps undergoes spectacular and deep transformation. This phenomenon is the subject of the present study. \nFor the system with no net electrical charge, like the hydrogen atom or Ps, moving in MF the center-of-mass (COM) momentum is not a conserved quantity. The complete separation of the internal dynamics from the COM motion turns to be impossible [12-17]. The spectrum and the wave function of the system parametrically depend on the pseudomomentum eigenvalues. Interaction of the COM motion and the relative motion leads to the transformation of the spectrum and the wave function. The Coulomb potential well (CW) is distorted and above a certain value of the pseudomomentum an additional outer, or «magnetic», potential well (MW) is formed. For hydrogen the decentered states were predicted long ago in [18] and since then studied by a number of authors [19-23]. \nTo the best of the authors knowledge for Ps the pseudomomentum K was first introduced in [24] and in a more elaborate form in [25]. In these works calculations were performed for K = 0 only. In [26] the formation of Ps in NS magnetosphere predicted in [8] was reconsidered in the pseudomomentum formalism. In [10] the photonPs conversion in pulsar magnetosphere was investigated \nwith pseudomomentum implicitly introduced. It is important to note that in neither of the above publications the formation of the MW was discussed in spite of the fact that for the hydrogen atom this phenomenon was known since 1976 [18]. The delocalized states in Ps were first discussed in [27, 28]. The authors performed a thorough study of the spectrum in the pseudomomentum formalism. Calculations in [27, 28] were performed using the numerical adaptive finite elements method. By contrast, we rely on the analytical methods different for various intervals of MF strength and K values. Explicit formulas of the present work allow to analyse the dependence of the spectrum on B and K values and to investigate the asymptotic regimes. Some overlap of the present research with [27, 28] is unavoidable. \nIt is necessary to remind some basic Ps and MF properties and characteristics. Throughout this work we follow the conditions ℏ = c = 1 , α = e 2 = 1 / 137 . The ground state binding energy of Ps is E B = -me 4 / 4 ≃ -6 . 8 eV = -R ∞ / 2 , where m ≡ m e = 0 . 51 MeV. Ps Bohr radius is a B ( Ps ) = 2 /me 2 = 2 a 0 ≃ 1 . 06 · 10 -8 cm, where a 0 ≃ 1 /me 2 = 0 . 529 · 10 -8 cm is the hydrogen Bohr radius. In the above system of units 1 MeV 2 = 1 . 45 · 10 13 G. MF Landau radius is ℓ = 1 / √ e B, the cyclotron frequency is ω c = e B /m , B a = m 2 e 3 = 2 . 35 · 10 9 G is the atomic unit of MF strength, ℓ ( B a ) = a 0 . The atomic field strength B ' a for Ps defined as l = 1 / √ e B ' a = a B ( Ps ) is equal to B ' a = B a / 4 . Our main focus will be on MF in the range γ = B / B a > 1 . \nThe work is organized as follows. In Sec.II we formulate the Hamiltonian of two particles with opposite electric charges moving in MF. The pseudomomentum K is introduced and it is shown that the internal wave function and the energy spectrum depend on the value of K. In Sec.III the critical value K c for the formulation of the MW is derived. The energy spectrum equation in the adiabatic approximation and in the «shifted» representation is derived in Sec.IV. In Sec.V the configuration with two separated potential wells is investigated. In Sec.VI the situation when the potential wells overlap is consid- \ned. In Sec.VII we consider some implications of the Ps wave function transformation. Sec.VIII contains the summary of the work.", 'II. TWO COUPLED PARTICLES WITH OPPOSITE CHARGES MOVING ACROSS MAGNETIC FIELD': "The physics of bound quantum system moving in MF is intricate. Due to lack of translational invariance the total kinetic (mechanical) momentum ˆ P kin = ∑ N i =1 (ˆ p i -e i A i ) is not a conserved quantity [12-23]. For a neutral system like Ps it is possible to construct a quantity K called pseudomomentum which commute with the Hamiltonian [13-17]. The Hamiltonian of two particles with opposite electric charges e 1 = e > 0 , e 2 = -e and equal masses m 1 = m 2 = m in a constant MF can be written as \nˆ H = 1 2 m [ ˆ p 1 -e A ( r 1 ) ] 2 + 1 2 m [ ˆ p 2 + e A ( r 2 ) ] 2 + V ( η ) , (1) \nwhere η = r 1 -r 2 , V ( η ) = -e 2 / | η | for Ps, spindependent and σ B terms are temporarily omitted. MF is assumed to be homogeneous and directed parallel to the z-axis B = (0 , 0 , B ) . We use the symmetric gauge A = 1 / 2 B × r = 1 / 2 B ( -y, x, 0) . Next we introduce the center-of-mass coordinate R = 1 / 2 ( r 1 + r 2 ) and momenta operators \nˆ π = -i ∂ ∂ η , ˆ P = ˆ p 1 + ˆ p 2 = -i ∂ ∂ R . (2) \nThe Hamilton operator takes the form \nˆ H = 1 4 m ( -i ∂ ∂ R -e 2 B × η ) 2 + + 1 m ( π -e 2 B × R ) 2 + V ( η ) . (3) \nThe pseudomomentum operator commuting with ˆ H reads \nˆ K = 2 ∑ i =1 ( p i + 1 2 e i B × r i ) = -i ∂ ∂ R + e 2 B × η . (4) \nˆ K is the integral of motion and eigenfunctions Ψ( R , η ) of ˆ H are eigenfunctions of ˆ K \nˆ K Ψ( R , η ) = K Ψ( R , η ) , (5) \nwhere K is the eigenvalue of ˆ K . Let us represent Ψ( R , η ) as Ψ( R , η ) = exp( i ν R ) φ K ( η ) with yet unknown ν . It can be found from (4) and (5) that \nˆ K Ψ( R , η ) = ( ν + e 2 B × η ) exp( i ν R ) φ K ( η ) = = K exp( i ν R ) φ K ( η ) , (6) \nhence K = ν + ( e/ 2) B × η , and we get Ψ( R , η ) in the form \nΨ( R , η ) = exp [ i ( K -e 2 B × η ) R ] φ K ( η ) . (7) \nThe action of ˆ H given by (3) on the factorized wave function (7) yields the eigenvalue equation for φ K ( η ) \n[ ˆ π 2 m + K 2 4 m -e 2 m [ K × B ] · η + e 2 4 m [ B × η ] 2 + + V ( η ) ] φ K ( η ) = E · φ K ( η ) . (8) \nWe see that unlike the free-field case the internal wave function φ K ( η ) as well as the eigenvalue E depend on K . Collective and internal motion are connected through motional Stark term e/ 2 m [ K × B ] · η . The electric field induced by this term is directed perpendicular to B . It is instructive to rewrite ˆ H in terms of pseudomomentum \nˆ H = 1 4 m ( ˆ K -e B × η ) 2 --1 m ( π -e 2 B × R ) 2 + V ( η ) . (9) \nFrom the Hamilton's equations of motion or directly from (9) one finds the relation between the center-of-mass velocity V = ˙ R and K \nV = 1 2 m ( K -e B × η ) . (10) \nFor a given MF the velocity V of Ps is determined by pseudomomentum and the electric dipole moment. The expectation value of V is obtained upon averaging (10) over Ps wave function.", 'III. THE OUTER POTENTIAL WELL FORMATION': 'The motion of Ps across MF results in a transformation of the potential shape. For a fixed value of MF a second potential well starts to be formed at a certain critical value K c of the pseudomomentum. This phenomenon has been studied for the hydrogen atom [18-23], Ps [27, 28] and quarkonium [29]. \nFrom (8) and (9) we may write the internal motion effective potential U eff ( η ) as \nU eff ( η ) = K 2 4 m + e B 2 m K x y -e B 2 m K y x + + e 2 B 2 4 m ( x 2 + y 2 ) -e 2 √ x 2 + y 2 + z 2 , (11) \nwhere η = ( x, y, z ) . The first term is spatially independent and plays the role of an additive constant to the \nbinding energies. The evolution of the potential shape as a function of K is governed mainly by the interplay of the Stark x, y terms and the diamagnetic forth term in (11). The potential possesses azimuthal symmetry and one can set either K x or K y equal to zero. We choose K y = 0 , then K = ( K , 0 , 0) . The projection of (11) onto x = z = 0 plane is \nU eff ( y ) = K 2 4 m + e B 2 m K y + e 2 B 2 4 m y 2 -e 2 | y | . (12) \nThe condition for the potential minimum at y < 0 ∂ U eff /∂y = 0 yields the equation \ny 3 + K e B y 2 -2 m B 2 = 0 . (13) \nFor the outer magnetic well (MW) and a saddle point (SP) to exist for K > 0 , y < 0 this equation must have three real roots. It implies the following condition on K and B \nK 3 > K 3 c = 27 2 ( me 2 )( e B ) = 27 2 ( e B a 0 ) . (14) \nThe three roots are \ny 1 = K ℓ 2 3 [ 2 cos α 3 -1 ] , (15) \ny 2 , 3 = -K ℓ 2 3 [ 1 + cos α 3 ± √ 3 sin α 3 ] , (16) \nwhere \ncos α = 27 a 0 ℓ 2 K 3 -1 = 2 K 3 c K 3 -1 . (17) \nTherefore y 2 < y 3 < 0 < y 1 . It means that y 1 is an unphysical solution, y 2 ≡ y m corresponds to the MW minimum, y 3 ≡ y s - to the saddle point. Note that at fixed value of MF and increasing K ≫ K c (16) yields \ny m →-K e B = -K l 2 , y s →-√ 2 em BK = -l √ 2 a 0 K , y m y s → 3 √ 3 2 ( K K c ) 3 / 2 . (18) \nThis shows that with increasing K the MW tends to separate from the CW for any given value of MF. In the same limit K ≫ K c the MW minimum approaches zero energy E = 2 m from below U eff ( y m ) →-e 2 ( e B ) / K. \nThe effective potential (12) along the y direction is shown in Fig.1 for B = 6 · 10 7 G and three different values of K. The solid black and red horizontal lines correspond to E = 2 m and I = e B /m . The quantity I is the ionization potential. In absence of the Coulomb interaction it is equal to the sum of the individual electron and positron Landau ground state energies. The motion in a plane perpendicular to the direction of MF is at large distances dominated by the contribution of the diamagnetic \nFIG. 1. The effective potential (12) in eV as a function of y in a 0 = 1 /m e e 2 = 0 . 529 · 10 -8 cm for MF B = 6 · 10 7 G, K c = 2 . 61 keV and three different values of K. The solid black horizontal line corresponds to E = 2 m , the red horizontal line to the ionization threshold I = e B /m . \n<!-- image --> \nterm. This term provides confinement and ionization is possible only along MF direction. The presence of the cubic term in (13) makes the evolution with K of the effective potential plotted in Fig.1 reminiscent of the first order phase transition.', 'IV. THE ENERGY SPECTRUM EQUATION': "The spectral problem (8) for the Hamiltonian \nˆ H = K 2 4 m -1 m ∆ η -e 2 m [ K × B ] · η + e 2 4 m [ B × η ] 2 -e 2 | η | (19) \ndoes not admit an exact solution. For the exciton and the hydrogen atom moving in MF some analytical approximations has been developed [13, 16, 18-20]. Numerical calculations for Ps have been performed in [27, 28]. We wish to solve the problem relying on analytical methods as far as possible. Following [13, 20, 22] it is appropriate to use the «shifted» representation η ' = η -η c , where η c is the difference between the coordinates of the gyromotion guiding centers of e + and e - \nη c = K 1 × B e B 2 -K 2 × B -e B 2 = = -[ B × K ] e B 2 = -2 m e B 2 [ B × V ] , (20) \nwhere V is given by (10). Note that with B directed along the z -axis η cz = 0 . Henceforth we shall use the following notations: η ' = ( η ' x , η ' y , z ) ≡ ( ρ ' , z ) , η c = ( η cx , η cy ) ≡ ρ 0 and the coordinate z is not affected by the shift. Inserting η ' = η -η c into (19) one obtains the \n«shifted» Hamiltonian \nˆ H ' = K 2 z 4 m -1 m ∂ 2 ∂z 2 -1 m ( ∂ 2 ∂η ' 2 x + ∂ 2 ∂η ' 2 y ) + + e 2 4 m B 2 ρ ' 2 -e 2 √ ( ρ ' + ρ 0 ) 2 + z 2 . (21) \nThe Stark term has been eliminated in this representation. The pseudomomentum K enters into (21) through ρ 0 and only then via an explicit term K 2 z / 4 m . In strong MF when γ = B / B a ≫ 1 , a 0 ≫ ℓ the Schrödinger equation with the Hamiltonian ˆ H ' is best solved by the wave function expansion over the complete set of Landau orbitals [30] in the ρ ' plane. Retaining only lowest Landau level (LLL) we arrive at the adiabatic approximation [31, 32] \nφ K ( ρ ' , z ) = R ( ρ ' ) f ( z ) , (22) \nwhere R ( ρ ' ) is the LLL wave function \nR ( ρ ' ) = 1 √ 2 πℓ exp ( -ρ ' 2 4 ℓ 2 ) , (23) \nand f ( z ) is a longitudinal part. Adiabatic approximation is increasingly accurate with γ increasing. Substituting (22)-(23) into the Schrödinger equation ˆ H ' φ K = E φ K , acting by ∂ 2 /∂ ρ ' 2 on R ( ρ ' ) , multiplying by R ( ρ ' ) and integrating over d ρ ' , we obtain \nd 2 f dz 2 + m [ E -e B m + U ( z ) ] f ( z ) = 0 , (24) \nU ( z ) = e 2 2 πℓ 2 ∫∫ R 2 d ρ ' exp( -ρ ' 2 / 2 ℓ 2 ) √ ( ρ ' + ρ 0 ) 2 + z 2 . (25) \nThe term K 2 z / 4 m in (21) does not influence the spectrum and is omitted. \nTransition from the Schrödinger equation H ' φ K = E φ K with H ' given by (21) to (24) may be considered as averaging over the fast MF variables. \nAs already noted, I = e B /m in (24) is zero-point energy of the LLL, or the ionization threshold. The energy E in (24) is given by \nE = I + E B , (26) \nwhere E B is the binding energy. The sates with ( E -I ) < 0 are bound states corresponding to the closed channels. States with ( E -I ) > 0 lie above the ionization threshold and form a series of autoionizing resonances. With this definition (24) takes the form \n( -1 m d 2 dz 2 -U ( z ) ) f ( z ) = E B f ( z ) . (27)", 'V. THE SEPARATED POTENTIAL WELLS': 'According to (14) the outer potential well is formed when K 3 > K 3 c = (27 / 2) me 3 B = (27 / 2)(1 /a 0 ℓ 2 ) . Under \nthis condition different configurations of the two potential wells and a barrier between them are possible. The shape of the resulting configuration dictates the most appropriate methods to solve eq.(24). The problem has three parameters with the dimension of length \na B ( Ps ) = 2 a 0 = 2 me 2 , ℓ = 1 √ e B , | ρ 0 | = | y 0 | = K e B . (28) \nHere | y 0 | is the absolute value of the projection of ρ 0 = -[ B × K ] /e B 2 onto the x = z = 0 plane and K = ( K , 0 , 0) in this projection. Important to note that according to (18) y 0 = -K /e B coincides with the position of the MW minimum y m in the limit K ≫ K c . Therefore one can interpret | y 0 | as the distance from the bottom of the MW to the Coulomb center. The two wells, the Coulomb and the magnetic ones, are actually separated if | y 0 | ≫ a B ( Ps ) . The use of the adiabatic approximation (22) implies that a B ( Ps ) ≫ ℓ . We come to the conclusion that the conditions for the double-well regime in the adiabatic approximation are the following \nK > K c , | y 0 | ≫ a B ( Ps ) ≫ ℓ. (29) \nFor the hydrogen atom the above situation was investigated in [18]. \nNext we return to eqs.(24)-(25). Inequalities (29) permit to take the square root out of the integral (25) at ρ = 0 , i.e., at y = 0 in the projection under consideration. Then (27) takes the form \nd 2 f dz 2 + m ( E -e B m + e 2 √ y 2 0 + z 2 ) f ( z ) = 0 . (30) \nThe characteristic distance for the dynamics of f ( z ) is z ∼ ℓ while according to (29) | y 0 | ≫ ℓ . Therefore the potential energy in (30) can be expanded in a series in powers of z 2 . As a result (30) reduces to the linear oscillator equation, see Fig.2 \nd 2 f dz 2 + m ( E -e B m + e 2 | y 0 | -e 2 | y 0 | 3 z 2 2 ) f ( z ) = 0 . (31) \nThe energy levels are given by \nE n = e B m -e 2 | y 0 | + ( n + 1 2 ) √ 2 e 2 m | y 0 | 3 . (32) \nIt is instructive to express E n in atomic energy units a.u. = 2 R ∞ = 27 . 2 eV. One has \nE n -e B /m a.u. = -a B ( Ps ) 2 | y 0 | + + 1 2 ( a B ( Ps ) | y 0 | ) 3 / 2 ( n + 1 2 ) . (33) \nWe remind that according to (29) | y 0 | ≫ a B ( Ps ) . Eq.(33) describes the spectrum in a wide, shallow parabolic potential. Imposing the condition ( E n -e B /m ) < 0 we \nFIG. 2. The illustrative plot of the oscillator approximation. The solid line is the projection of U eff on the x = z = 0 plane. The dashed line is the parabolic approximation. \n<!-- image --> \nTABLE I. The energies of the lowest Landau states I = e B /m , | y 0 | = K /e B and the ground states binding energies | E B | for the MF in the range (10 10 -10 14 ) G and K > K c . \nconclude that (32)-(33) are valid up to \nn ≲ √ me 2 K 2( e B ) = √ | y 0 | a B ( Ps ) . (34) \nFormally, beyond this restriction there is a condensation of infinite number of levels close to the ionization threshold. The transition from bound to autoionization states is far from being thoroughly studied [19, 33, 34]. \nThe results of the ground states binding energies \nE B = E n =0 -I = -e 2 | y 0 | ( 1 -1 2 √ a B ( Ps ) | y 0 | ) (35) \nare presented in Table I. The characteristic bound states energies are of the order of a few eV.', 'VI. THE OVERLAPPING POTENTIAL WELLS': "As already explained, the configuration of the potential depends upon the values of the four basic parameters: K , a B ( Ps ) , ℓ, | ρ 0 | = | y 0 | and on their dimensionless ratios. Under the condition (29) the outer well is formed, the adiabatic approximation is applicable and the two wells are isolated. Now we wish to consider the situation when the outer well is formed, the adiabatic approximation is \nd 2 dξ 2 f ( ξ ) + ( -1 4 + ν | ξ | ) f ( ξ ) = 0 , \n(40) \nwith ν related to the binding energy according to \nE B = -1 ma 2 B ν 2 . (41) \nNote that ν = 1 corresponds to E B = E 0 ≃ -6 . 8 eV the ground state energy of Ps. Eq. (40) is the Whittaker's confluent hypergeometric equation [41]. The solution with positive ν that decreases exponentially at infinity is f ( ξ ) = const · W ν, 1 / 2 ( ξ ) . To match with the interior solution it suffices to keep the first order in z terms. For 0 < ν ≪ 1 one can write [37] \nW ν, 1 / 2 ( 2 z νa B ) = -1 + 2 z a B ( ln 2 z a B + 1 2 ν + C 1 ) , (42) \nvalid but the two wells overlap. This corresponds to the following conditions \nK > K c , ℓ ≪ a B ( Ps ) , | y 0 | ≪ a B ( Ps ) . (36) \nNote that we are not comparing | y 0 | and ℓ since their ratio depends on the value of K. In this section we partly rely on methods developed in [13, 35-40]. \nOur task is to solve eq.(24) under the conditions (36). Firstly, we note that in the adiabatic approximation the size of Ps in the transverse plane is determined by the wave function (23) and is of the order of ρ ∼ ℓ ≪ a B ( Ps ) . Therefore the density distribution of | φ K ( ρ , z ) | 2 takes the elongated shape in z -direction. This gives birth to the idea to replace the potential (25) by a MF independent one-dimensional potential. At | z | → ∞ (25) has the following asymptotic form \nU ( z ) → e 2 | z | + O ( | y 0 | 2 | z | 3 ) . (37) \nFor (24) with the account of (26) this yields \n( -1 m d 2 dz 2 -e 2 | z | ) f ( z ) = E B f ( z ) . (38) \nThis is a one-dimensional Schrödinger equation with a Coulomb potential studied in [35-40]. As a pure mathematical problem, eq. (38) does not have a complete set of solutions [38]. A way to overcome this problem is to remove the singularity at the origin by the use of a cutoff at z = z 0 [35, 36]. We follow an alternative approach of [37, 39]. Eq. (38) is solved inwards from | z | = ∞ and at z = z 0 the logarithmic derivative is matched with the same quantity of the approximate inner solution. This procedure lead to an equation for the energy spectrum. \nIt is convenient to change the variable z to ξ = 2 z/νa B [35] with ν and ξ being dimensionless. Whereupon (38) reduces to \nd 2 dξ 2 f ( ξ ) + [ 1 4 ( mν 2 a 2 B ) E B + ν | ξ | ] f ( ξ ) = 0 , (39) \nor \nwhere C is the Euler constant ( C = 0 . 5772 ... ). The logarithmic derivative reads \nη ( ν, z ) ext = ∂ ∂z ln W ν, 1 / 2 ( 2 z νa B ) = = -1 νa B -2 a B ( ln 2 z νa B + C ) . (43) \nNext we consider the inner solution. The potential given by (25) is an even function of z and hence the solutions may be either even or odd. We focus on even solutions leaving the odd ones for another publication. Following [37] we present the potential (25) in the following form \nU ( z ) = e 2 2 πℓ 2 ∫∫ d ρ exp( -ρ 2 / 2 ℓ 2 ) √ ( ρ + ρ 0 ) 2 + z 2 = = e 2 ℓ ϑ ( | z | ℓ ) , (44) \nwhere ϑ ( ξ ) is analytic for | ξ | < ∞ . Introducing the variable t = z/ℓ we rewrite eq. (27) in the following form \nd 2 dt 2 f ( t ) + ( 2 ℓ a B ϑ ( t ) -ℓ 2 a 2 B ν 2 ) f ( t ) = 0 , (45) \nwhere the expression (41) for E B has been used. Unlike the two and three dimensions for a one-dimensional Coulomb potential U ( | z | ) = -e 2 / | z | there is no continuity of the wave function and its derivative at z = 0 . Therefore in zero order we take ℓ = 0 (the infinite MF). In this case there are two solutions of (45) \nf (0) ( t ) = const , f (0) ( t ) = const · t. (46) \nThe first one corresponds to even solution. To first order in ℓ we have the equation \nd 2 dt 2 f ( t ) + 2 ℓ a B ϑ ( t ) f (0) ( t ) = 0 . (47) \nThe solution reads \nf ( t ) = const ( 1 -2 ℓ a B t ∫ 0 dt 1 t 1 ∫ 0 dt 2 ϑ ( t 2 ) ) . (48) \nFor f (0) ( t ) = const, i.e., for even states, the solution is finite and non-vanishing at z = 0 . Its logarithmic derivative is \nη ( z ) int = -2 a B z/ℓ ∫ 0 ϑ ( t ) dt, (49) \nwhere according to (44) \nϑ ( t ) = 1 2 πℓ ∫∫ d ρ exp( -ρ 2 / 2 ℓ 2 ) √ ( ρ + ρ 0 ) 2 + z 2 . (50) \nThis is the point where the dependence on pseudomomentum K comes into play through ρ 0 = -[ B × K ] /e B 2 , or y 0 = -K /e B, K = ( K , 0 , 0) , see (28) above. Insertion of (50) into (49) yields \nη ( z ) int = -2 a B ∫∫ d ρ exp( -ρ 2 / 2 ℓ 2 ) 2 πℓ 2 z/ℓ ∫ 0 dt √ u 2 + t 2 , (51) \nwhere u 2 = ( ρ + ρ 0 ) 2 /ℓ 2 . Integrating over dt on obtains \nη ( z ) int = -2 a B ∫∫ d ρ exp( -ρ 2 / 2 ℓ 2 ) 2 πℓ 2 [ ln ( 1 + + √ 1 + ( ρ + ρ 0 ) 2 z 2 ) +ln z | ρ + ρ 0 | ] = J 1 + J 2 , (52) \nwith J 1 and J 2 corresponding to the contribution of the first and second terms inside the square brackets. In the strong field limit ℓ → 0 and for z/ℓ → ∞ the integration of the first term is straightforward and gives J 1 ≃ -(2 /a B ) ln 2 . Evaluation of J 2 is more complicated and is presented in the Appendix A. Summing the two contributions we obtain \nη ( z ) int = -2 a B ( ln z ℓ + 1 2 ln 2 + 1 2 ln C + + 1 2 x ∫ 0 dt exp( -t ) ln x t ∣ ∣ ∣ ∣ x = ℓ 2 K 2 2 ) . (53) \nEquating the logarithmic derivatives (43) and (53) we obtain the equation for ν of the Ps ground state \n1 ν +ln 1 ν 2 = ln a 2 B 2 ℓ 2 -C --x ∫ 0 dt exp( -t ) ln x t ∣ ∣ ∣ ∣ x = ℓ 2 K 2 2 = = ln γ +ln2 -C Λ ( ℓ 2 K 2 2 ) , (54) \nwhere γ = B / B a , B a = m 2 e 3 = 2 . 35 · 10 9 G is the atomic MF, Λ( x ) is the integral entering into (54). The derivation of (54) is based on the expansion (42) valid for ν ≪ 1 which corresponds to the ground state. The excited and odd states will be the subject of the forthcoming investigation. At x → 0 the function Λ( x ) behaves as Λ( x ) ≃ x + O ( x 2 ) . For ℓ K ≪ 1 the term Λ in (54) gives the following contribution to the ground state binding energy \nE B -E B 0 = | E B 0 | - | E B | = 1 ma 2 B ν 0 ( ℓ 2 K 2 2 ) , (55) \nwhere E B 0 and ν 0 correspond to (54) with Λ = 0 . At x →∞ one has Λ( x ) ≃ ln x + C + O ( e -x ) . We remind that equation (54) for ν has been derived under the condition \nFIG. 3. Dependence of the binding energy for the ground state (41) on the pseudo-momentum K. The blue dashed line corresponds to the limiting case of small K (B2), the red line corresponds to the limiting case of large K (B3). B = 10 10 G. \n<!-- image --> \n(36). Therefore the value of K can not be arbitrary large, namely K ℓ 2 ≪ a B , or K ≪ γ/a B ( Ps ) , γ = B / B a . For e.g. B = 10 12 G the condition on K is K ≪ 3 MeV. \nFor the neutral two-body system at rest in MF the eigenvalue equation (54) has been derived and treated by several authors most probably starting from [32]. In a refined form with a detailed discussion it was presented in [37] and later in [38]. The dependence on pseudomomentum in the form of the integral Λ in (54) was given in [13] with a reference to [37]. It is interesting to see the relationship between (54) and the well-known result from Landau and Lifshitz [30] for the hydrogen ground state energy in MF. This estimate is logarithmically accurate and reads \nE B = -me 4 2 ln 2 a 2 B ( H ) ℓ 2 . (56) \nTo compare (56) with (54) we first recast (54) for the hydrogen. It amounts to the replacement of a B ( Ps ) by a B ( H ) and m in (41) by 2 m since the reduced mass of H is twice of Ps. Then we bring down (54) to the minimal form 1 /ν = ln[ a 2 B ( H ) / 2 ℓ 2 ] and obtain \nE B = -me 4 2 ln 2 a 2 B ( H ) 2 ℓ 2 , (57) \nwhich coincides with (56) with logarithmic accuracy. For Ps the same approximation yields for B = 10 10 G \nE B = -me 4 2 ln 2 a 2 B ( Ps ) 2 ℓ 2 ≃ -30 eV . (58) \nThe numerical solution of (54) for this value of B and K = 0 gives E B ≃ -9 eV. \nThe dependence of the Ps ground state binding energy on K is shown in Fig.3. The difference between the results of the Landau-type formula (58) and what gives the numerical solution of (54) is, as we see, considerable. The same is true for hydrogen where at B = B cr = m 2 /e = \n4 . 414 · 10 13 G (56) leads to | E B 0 | = 1320 eV while the exact calculation gives | E B 0 | = 448 eV [38]. \nAs has been said in the Introduction, the main focus of our interest to the problem of Ps moving across a MF lies in the magnetospheres of NSs [8-11]. Therefore it is necessary to present the result for the Ps moving in MF ground state energy obtained by solving the BetheSalpeter equation [10, 42]. Using the notation of the present work the result of [10, 42] reads \nE B = -me 4 4 ln 2 a 2 B ( Ps ) 4 ℓ 2 (1 + ℓ 2 P 2 x ) . (59) \nThe gauge used by the authors is A x = -B y, A 0 = A y = A z = 0 , B x = B y = 0 , B = B z . The gauge of the present work is A = 1 / 2[ B × r ] = 1 / 2 · B ( -y, x, 0) . The quantity P x is the transverse centre-of-mass momentum P x = p p x + p e x ( p p x refers to e + and p e x to e -) which is a constant of motion in the gauge of [10]. It is analogous to the pseudomomentum K x ≡ K of the present work through we are not immediately aware of the exact relation between the two. To find a bridge between (54) and (59) we take (54) in the following truncated form \n1 ν ≃ ln a 2 B ( Ps ) 2 ℓ 2 -Λ ( ℓ 2 K 2 2 ) . (60) \nAt ℓ 2 K 2 / 2 ≪ 1 one has Λ( x ) ≃ x ≃ ln(1 + x ) and (60) yields \nE B = -me 4 4 ln 2 a 2 B ( Ps ) 2 ℓ 2 (1 + ℓ 2 K 2 / 2) , (61) \nwhich closely resembles (59). In the opposite limit of large ℓ 2 K 2 / 2 ≫ 1 one has Λ( ℓ 2 K 2 / 2) ≃ ln( ℓ 2 K 2 / 2) . This leads to \nE B = -me 4 4 ln 2 a 2 B ( Ps ) ℓ 4 K 2 . (62) \nThis corresponds to the limit ℓ 2 P 2 x ≫ 1 in (59). We may draw a conclusion that our basic equation (54) and the result (59) from Bethe-Salpeter equation are neither identical nor contradictory to each other.", 'VII. IMPLICATIONS': "We have shown that the Ps wave function and the energy spectrum experience deep transformation as it moves through a MF. This carries far-reaching physical implications. Here we point out some new physical effects and postpone the detailed discussion for later. The first phenomenon to consider is the emergence of longlived Ps [27, 28]. The Ps decay rate is proportional to the square of the wave function | ψ (0) | 2 evaluated at contact. For the MF strength B ≫ ∆ / 4 µ 0 = 3 . 63 T Ps has a maximum symmetry spin state with spin-down e -and spin-up e + [7, 30, 43-45]. Here ∆ = 8 . 4 · 10 -4 eV is the \nhyperfine splitting, µ 0 = 5 . 79 · 10 -5 eV / T is the Bohr magneton. Therefore it is sufficient to consider only the coordinate part of the Ps wave function. \nAccordingly to (22)-(23) the square of the Landau ground state wave function at η = r 1 -r 2 = 0 , i.e., at ρ ' = -ρ 0 , z = 0 reads \n| φ K (0) | 2 = 1 2 πℓ 2 exp ( -ρ 2 0 2 ℓ 2 ) f 2 (0) = = 1 2 πℓ 2 exp ( -K 2 ℓ 2 2 ) f 2 (0) . (63) \nFor the separated potential wells f ( z ) satisfies the oscillator equation (31). The ground state solution is \nf ( z ) = ( mω π ) 1 / 4 exp ( -mω 2 z ) , (64) \nwhere \nω 2 = 2 e 2 m | y 0 | 3 = 2 e 2 m K 3 ℓ 6 . (65) \nTherefore \n| φ K (0) | 2 = 1 2 πℓ 2 √ mω π exp ( -K 2 ℓ 2 2 ) . (66) \nThis has to be compared with | φ (0) | 2 = 1 /πa 3 B ( Ps ) for Ps at rest without MF. From (66) it follows that the probability density at the origin exponentially drops for K 2 ℓ 2 / 2 ≫ 1 . Recall that the outer potential well is formed when K > K c = (27 / 2 a 0 ℓ 2 ) 1 / 3 . For B = 10 12 G one has K c ≃ 70 keV. When K ≫ K c the square of the wave function at contact dramatically drops and Ps is actually stable against annihilation. As a side remark we mention that the parameter K 2 ℓ 2 / 2 in (66) enters also in the eigenvalue equation (54). \nAnother important property of the Ps decentered configuration is that it has a giant electric dipole moment [13, 27, 28, 42, 43]. From the physical considerations it is clear that the average distance between particles if given by the difference (20) between the coordinates of the gyro-motion guiding center of e + and e - \n⟨ η ⟩ = K 1 × B e B 2 -K 2 × B -e B 2 = -B × K e B 2 . (67) \nRepeating the arguments that led to (12), consider the gauge K y = 0 , K = ( K , 0 , 0) . Then ⟨ η ⟩ y = y 0 = -K /e B and \nd = ey 0 = -K B . (68) \nAs noted before, and as it is clear from Fig.2, y 0 coincides with the position of the MW minimum at K ≫ K c . According to (29) in the configuration with the decentered MW | y 0 | ≫ a B ( Ps ) . It means that the dipole momentum can exceed by many orders of magnitude the \nvalue corresponding to the atomic unit of length. The Ps with a giant dipole moment plays an important role in the NS magnetosphere at the polar gap [42]. The electric field exerts a torque on the Ps dipole and causes it to rotate with angular velocity proportional to d. According to [42] this rotation prevents the increase of the distance between e + and e -and thereby prevents the ionization of Ps inside the polar gap. \nLet us briefly touch upon a long standing contradictory problem of Ps one-photon annihilation in strong MF [10, 24, 25, 44-46]. The starting point of the discussion was Carr and Sutherland work [44]. The authors claimed that in MF the one-photon annihilation from the ground state of Ps is possible and calculated the corresponding rate. In [24] Wunner and Herold disputed this claim since in their view the state considered as a ground one in [44] was in fact an excited level. The analysis of [24] was based on the pseudomomentum formalism. The authors of [24] investigated the one-photon annihilation only from the K = 0 state. This state was self-evidently considered as a ground state of Ps. The conclusion of [24] was that onephoton annihilation from the Ps ground state is strictly forbidden. \nWe find it appropriate to shift the emphasis from the problem of one-photon annihilation from the K = 0 state to annihilation of Ps in the real physical conditions. We mean Ps moving across NS magnetosphere, or in AGN jets [43], or Ps in a strong laboratory MF. The relation between the centre-of-mass velocity V , pseudomomentum K , MF and the internal coordinate η is given by (10). This relation does not imply the physical reasons for the extra edge of the K = 0 state. As explained earlier, with the increase of K above the critical value, K c the Coulomb states are pushed above the ionization threshold and all of the bound states reside in the outer MW. The intersection of the photon dispersion line and the series of the Ps decentered states deserves a close attention.", 'VIII. SUMMARY AND FUTURE PROSPECTS': 'Wehave investigated the transformation of the Ps wave function and energy spectrum taking place when Ps is moving through a MF. The main difference from the unmoving case is that the internal dynamics can not be separated from the centre-of-mass motion. The spectrum and the wave function parametrically depend on the generalized momentum operator K (4) also called pseudomomentum. When K exceeds the critical value (14) the outer «magnetic» potential well is formed in addition to the distorted Coulomb one. Two principal configurations are possible: \n- (i) The separated Coulomb and outer potential wells with a potential barrier in between.\n- (ii) The overlapping wells with bilocalized wave function. \nBoth cases correspond to K > K c and to MF strong enough for adiabatic approximation to be legitimate \nγ = B B a = ω c 2 R y ≫ 1 . (69) \nThe first configuration corresponds to the relation (29) between the three main parameters | y 0 | , a B ( Ps ) and ℓ . The second is realized under the condition (36). With K continuously increasing above K c the former Coulomb states are pushed above the ionization threshold and the spectrum resides in the decentered outer well. The character of this evolution depends on the value of MF. The states in the outer well have small annihilation rate and giant dipole moment. \nSome interesting problems of Ps moving across MF remain for future studies. Among them is a bunch of questions relate to the possible transformation of photon into Ps in the pulsar magnetosphere [8-11, 26, 43, 45]. As explained earlier it does not suffice to consider the onephoton annihilation from the K = 0 state. For this and for several other problems it is necessary to construct a kind of a wave packet made of different pseudomomentum values. \nAnother utmost interesting subject is magnetically stimulated Ps center-of-mass chaotic diffusion motion [47-50]. Roughly speaking, the center-of-mass undergoes a transition from regular to Brownian motion if the internal motion changes from regular to chaotic. Worth mentioning that diffusion motion may be important for high precision experiments with antihydrogen atoms [49, 50].', 'IX. ACKNOWLEDGMENTS': 'The authors would like to thank M.A. Andreichikov for useful remarks and discussion.', 'Appendix A: Calculation of the integral J 2': "Let's transform the original expression for J 2 to the following form \nJ 2 = ∫∫ R 2 d 2 ρ exp ( -ρ 2 2 ℓ 2 ) ln z | ρ + ρ 0 | = = 2 πℓ 2 ln z ℓ --πℓ 2 ln 2 + πℓ 2 C --ℓ 2 ∞ ∫ 0 τdτ exp( -τ 2 ) J ( τ ) , (A1) \n- [1] G. S. Adkins, D. B. Cassidy, and J. Perez-Rios, Phys. Rept. 975 , 1 (2022).\n- [2] S. G. Karshenboim, Phys. Rept. 422 , 1 (2005).\n- [3] S. G. Karshenboim, Int. J. Mod. Phys. A 19 , 3979 (2004). \nwhere C is Euler's constant, numerically equal to C ≃ 0 . 57721 . By J ( τ ) we mean the following integral \nJ ( τ ) = 2 π ∫ 0 dφ ln ( 1 + 2 ρ 0 √ 2 τℓ cos φ + ρ 2 0 2 τ 2 ℓ 2 ) , (A2) \nwhich can be calculated by isolating the perfect square in the expression under the logarithm \nJ ( τ ) = 2 π ∫ 0 dφ ln ( 1 + 2 ρ 0 √ 2 τℓ cos 2 φ + ρ 2 0 2 τ 2 ℓ 2 ) = = 4 π ln | 1 -a | +1+ a 2 = = { 0 , when a < 1 4 π ln a, when a ≥ 1 , (A3) \nwhere the notation a = ρ 0 / √ 2 τℓ was introduced. Thus, integration over τ in J 2 terminates at the value τ = ρ 0 / √ 2 ℓ , that is \nJ 2 = 2 πℓ 2 ln z ℓ -πℓ 2 ln 2 + πℓ 2 γ --ℓ 2 ρ 2 0 / 2 ℓ 2 ∫ 0 dt exp( -t ) ln ρ 2 0 2 tℓ 2 . (A4)", 'Appendix B: Asymptotic behavior of function Λ': "Function Λ( x ) has the form \nΛ( x ) = x ∫ 0 dye -y ln x y . 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2020LRSP...17....3L | Nearly all energy generated by fusion in the solar core is ultimately radiated away into space in the solar atmosphere while the remaining energy is carried away in the form of neutrinos. The exchange of energy between the solar gas and the radiation field is thus an essential ingredient of atmospheric modeling. The equations describing these interactions are known but their solution is so computationally expensive that they can only be solved in approximate form in multidimensional radiationMHD modeling. In this review I discuss the most commonly used approximations for energy exchange between gas and radiation in the photosphere chromosphere and corona. | 2020-12-01T00:00:00Z | ['2020LRSP...17....3L', '10.1007/s41116-020-0024-x', '2020arXiv200203623L', 'arXiv:2002.03623', '10.48550/arXiv.2002.03623'] | ['The Sun', 'Magnetohydrodynamics', 'Radiative transfer', 'Astrophysics - Solar and Stellar Astrophysics'] | Radiation hydrodynamics in simulations of the solar atmosphere | 2,020 | 170 | 0.43 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML'] | 23 | https://arxiv.org/pdf/2002.03623.pdf | {'No Header': 'Living Reviews in Solar Physics manuscript No. \n(will be inserted by the editor)', 'Radiation hydrodynamics in simulations of the solar atmosphere': 'Jorrit Leenaarts \nReceived: date / Accepted: date \nAbstract Nearly all energy generated by fusion in the solar core is ultimately radiated away into space in the solar atmosphere, while the remaining energy is carried away in the form of neutrinos. The exchange of energy between the solar gas and the radiation field is thus an essential ingredient of atmospheric modeling. The equations describing these interactions are known, but their solution is so computationally expensive that they can only be solved in approximate form in multi-dimensional radiation-MHD modeling. In this review, I discuss the most commonly used approximations for energy exchange between gas and radiation in the photosphere, chromosphere, and corona. \nKeywords The Sun · Magnetohydrodynamics · Radiative transfer', 'Contents': 'Institute for Solar Physics, Department of Astronomy, Stockholm University \nAlbaNova University Centre, SE-106 91 Stockholm, Sweden \nE-mail: [email protected]', '1 Introduction': "The interaction of matter and radiation is an indispensable ingredient of solar and stellar atmospheric modeling. Around 98% of the energy generated in the solar core is transported outwards first by radiative diffusion, then convection, and ultimately escapes into space in the photosphere of the Sun where the overlying solar material becomes transparent. The remainder of the energy escapes the Sun in the form of neutrinos (Turck-Chi'eze and Couvidat 2011). The layers above the photosphere (chromosphere and corona) are hotter than radiative equilibrium models predict. Therefore deposition of non-thermal energy and conversion into heat must occur in these layers. The radiative energy losses (ignoring the ∼ 10 -6 fraction of energy carried away by the solar wind, Le Chat et al. 2012) from the chromosphere and corona must balance this energy deposition in a time and space-averaged sense. Radiation is thus an essential ingredient in setting the structure of the outer solar atmosphere and the response of the atmosphere to non-thermal energy deposition. \nModeling the interaction of radiation and matter in the solar atmosphere is in general a difficult problem. The specific intensity, the fundamental quantity used to characterise the radiation field, depends on seven parameters: three space dimensions, two angles describing direction, frequency, and time. In addition, the radiative transfer problem is non-linear and non-local: local changes to the intensity through absorption and emission depend on the intensity itself, and radiation emitted at one place in the atmosphere can influence other locations. \nWriting general equations that describe the interaction of radiation and matter is not so difficult. Because of limitations in computing time these equations have so far been solved rather completely in one-dimensional geometry only. The Radyn code (Carlsson and Stein 1992, 2002), is perhaps the most well-known example, but other codes are used too (e.g., Flarix, see Kaˇsparov'a et al. 2009). \nHowever, solving these general equations in 2D and 3D is still beyond current computational capabilities. This is problematic because modeling convection requires at least 2D geometry, and fully modeling the rich physics caused by the interaction of the magnetic field with matter requires 3D geometry. \nSince the pioneering simulations of Nordlund (1982), a plethora of codes that aim to model solar and/or stellar atmospheres in 3D have been developed. The ones that I am aware of are: Stagger (e.g., Stein and Nordlund \n1998, but many different versions of this code exist), MURaM (Vogler 2004; Rempel 2017), Bifrost (Gudiksen et al. 2011), COBOLD (Freytag et al. 2012), Stellarbox (Wray et al. 2015), RAMENS (Iijima and Yokoyama 2015; Iijima 2016), MANCHA3D (Khomenko et al. 2017, 2018), ANTARES (Leitner et al. 2017), and RADMHD (Abbett and Fisher 2012). The latter is the only code that does not use the radiation treatment developed by Nordlund (1982), and instead uses a simpler but much faster method. \nRadiation-hydrodynamics in the solar atmosphere is a vast topic. In this review I limit myself to describing only the most commonly used approximations for radiative transfer in the photosphere, chromosphere, and corona in these multi-dimensional radiation-MHD codes. I do not discuss results obtained with any of these codes. An excellent review of solar magnetoconvection as studied using radiation-MHD simulations is given in Stein (2012). The field of radiation-MHD modeling of the combined photosphere, chromosphere, transition region, and/or corona has developed tremendously during the last 15 years. To my knowledge no recent review covering this development exists. Example starting points for studying applications are Carlsson et al. (2016); Mart'ınez-Sykora et al. (2017); Khomenko et al. (2018), and Cheung et al. (2019). \nIn the convection zone the radiation diffusion approximation holds to a high degree of precision and the energy exchange between radiation and the solar gas is close to zero. In the photosphere the gas loses large amounts of energy in the form of radiation which then largely escapes into space. The LTE assumption for the source function and opacity is still not too bad and even the grey approximation is still reasonably accurate. I devote a large fraction of this review discussing approximations for the photosphere and low chromosphere in Sect. 3. \nThe situation in the chromosphere and transition region is more complex. The chromosphere is optically thin for optical continuum radiation and most radiative energy exchange takes place in a few strong spectral lines. Radiation scattering is important so that non-LTE effects must be taken into account, the ionisation balance of hydrogen and helium is out of equilibrium so that assuming LTE or statistical equilibrium to compute opacities or source functions is in general no longer accurate. Modeling energy exchange taking into account these complexities is discussed in Sections 5 and 6. \nIn the corona the physics of radiative energy losses and gains becomes again somewhat less complex. Most coronal structures are optically thin for all frequencies except in the radio regime. The radio regime lies in the far tail of the Planck function and does not contribute significantly to the radiative losses. For all other frequencies it can be assumed that photon absorption does not take place and that all photons emitted by the gas escape from the corona. It is discussed in Sect. 4.", '2 Fundamentals': 'Excellent books that review the fundamentals of radiation hydrodynamics and modeling of stellar atmospheres are Mihalas and Mihalas (1984) and Hubeny and Mihalas (2014). The first book focusses on fundamental theory, while the second one discusses modeling of 1D static and moving atmospheres. Below I briefly touch upon some general aspects that are relevant for the methods discussed in Sect. 3 - 7.', '2.1 The MHD equations including radiation': "The dominant paradigm for multidimensional modeling of the solar atmosphere has been the magnetohydrodynamics (MHD) approximation. The MHD equations for the density, momentum and internal energy including radiation terms can be written as \n∂ρ ∂t = -∇ · ( ρ v ) , (1) \n∂ p ∂t = -∇ · ( v ⊗ p -τ ) -∇ P + J × B + ρ g -∇ P rad , (2) \n∂e ∂t = -∇ · ( e v ) -P ∇ · v + Q + Q rad , (3) \nwith ρ the mass density, v the velocity vector, p the momentum density, τ the stress tensor, P the gas pressure, J the current density, B the magnetic field vector, g the acceleration due to gravity, P rad the radiation pressure tensor, e the internal energy, Q rad the heating or cooling owing to radiation, while Q expresses energy exchange by any other processes such as dissipation of currents, heat conduction, and viscosity. \nThe assumptions under which MHD is valid tend to be fulfilled in the photosphere and convection zone as well as the corona, but break down in the chromosphere and transition region where the frequencies of collisions between particles become smaller than the cyclotron frequencies of ions and electrons. (e.g., Khomenko et al. 2014). \nSeveral radiation-MHD codes have been extended to include some effects beyond MHD, through including the ambipolar diffusion term, the Hall term, and/or Biermann's battery term, to the induction equation (Mart'ınez-Sykora et al. 2012; Cheung and Cameron 2012; Khomenko et al. 2017, 2018) These inclusions retain the single fluid description. This greatly simplifies the treatment of the radiative term Q rad , because energy lost or gained by the gas modifies the internal energy of the gas as a single entity, instead of modifying the energy of the electrons and different species of atoms, ions, and molecules separately. \nEfforts are underway to move beyond single-fluid radiation-MHD to a multi-fluid description, treating neutrals, ions, and/or electrons as separate fluids each with their own temperatures and velocities. Radiative transitions \nTable 1 Comparison of energy densities and radiative flux \ncan modify the internal energy and change the ionisation state of atoms, and in a multi-fluid description these must be computed in detail (e.g., Khomenko et al. 2014). This review does not discuss radiation-hydrodynamics in multifluid models.", '2.2 Energy density of radiation and matter': 'It is useful to compare the energy density and flux of the radiation field to the energy density of the gas in the solar atmosphere. To get an estimate we assume that the radiation field is isotropic and given by the Planck function at T rad = 5777 K. Then the energy density is \nE rad = 4 σ c T 4 rad = 0 . 84 J m -3 , (4) \nwith σ the Stefan-Boltzmann constant. Assuming that the solar surface is a radiating blackbody at the same temperature gives the radiative flux at the surface as \nF rad = σT 4 rad = 6 . 3 × 10 7 Wm -2 , (5) \nTable 1 compares the radiative energy density to the internal energy density e of the solar gas for typical photospheric, chromospheric and coronal values assuming the solar gas has an ideal gas equation of state ( e = nk B T ). The radiative energy density in the chromosphere and corona is corrected for the non-isotropy of the radiation above the photosphere. The energy density of the radiation is much lower than the energy density of the gas in the photosphere and chromospere, but in the corona they are about equal. The corona is however optically thin for most photospheric radiation and only little absorption or scattering of radiative energy by the gas occurs.', '2.3 Radiation pressure and force': 'The force exerted by the radiation pressure tensor is typically ignored under normal solar conditions because it is small compared to other forces. To illustrate this one can compare the radiation pressure of isotropic black body radiation to the gas pressure in the photosphere. This pressure is \n4 σ 3 c T 4 = 0 . 27 Pa (6) \nat a photospheric temperature of T = 5700 K, while the gas pressure in the photosphere is roughly 10 4 Pa. Similarly, one can compute a rough estimate of the upward acceleration of photospheric material by radiation: \na rad ≈ κF c = 3 × 10 -3 ms -2 , (7) \nwith κ the Rosseland opacity per mass unit and F the frequency-integrated radiative flux, both taken in the photosphere. The radiative acceleration is much smaller than the downward-directed solar surface gravity acceleration g = 274 ms -2 .', '2.4 Energy exchange between radiation and matter': 'In absence of an absorbing medium, the monochromatic radiative flux divergence ∇· F ν is zero. If a medium (in the solar case a gas or plasma) is present, then the rate per volume with which the material gains energy from the radiation field is given by \nQ rad = -∫ ∞ 0 ∇· F ν d ν = -∇· F , (8) \nwith F the total radiative flux. The total radiative flux is an integral over frequency, and as far as the internal energy of the gas is concerned, the exact distribution of the flux over frequency is not important. Typically, radiationMHDsimulations of the solar atmosphere aim to reproduce the correct detailed behaviour of the gas only. The computation of the radiation field only has to reproduce the correct heating and cooling, but does not need to reproduce the correct spectral energy distribution. This allows for large simplifications in treating the radiation without sacrificing too much accuracy in the value of the total flux divergence.', '2.5 Explicit expression of the radiative flux divergence': 'The monochromatic flux is defined in terms of the intensity I ν as \nF ν = ∮ n I ν d Ω, (9) \nwith n the unit vector pointing in the direction of Ω . Substitution of this equation into the integral over all directions of Eq. (14) yields an expression of the total radiative flux divergence in terms of emissivity or source function S ν = j ν / ( κ ν ρ ), intensity and opacity: \n∇· F = ∫ ∞ 0 ∮ ( j ν -κ ν ρI ν ) d Ω d ν (10) = ∫ ∞ 0 ∮ κ ν ρ ( S ν -I ν ) d Ω d ν, \nwith κ ν the extinction coefficient per unit mass. If one assumes that both the emissivity and extinction coefficient do not depend on direction then Eqs. (10) reduce to \nwhere \n∇· F = ∫ ∞ 0 4 πκ ν ρ ( S ν -J ν ) d ν, (11) \nJ ν = 1 4 π ∮ I ν d Ω (12) \nis the angle-averaged intensity. \nThis assumption is reasonable for bound-free transitions and other continuous processes. For bound-bound transitions it is not valid when flow velocities are of the same order or larger than the thermal speed √ 2 kT/m + v 2 turb , with m the mass of the atom or ion that is involved, and v turb the microturbulent velocity. Nevertheless, the assumption is almost always taken to be valid because it allows large simplifications.', '2.6 Light travel time and hydrodynamical timescales': "Typical bulk flow velocities in the solar atmosphere range from 1 km s -1 in convective upflows to > 300 kms -1 in chromospheric evaporation following solar flares (Graham and Cauzzi 2015). Alfv'en velocities up to 2 . 2 × 10 3 kms -1 have been measured in the solar corona by studying properties of coronal loop oscillations (e.g., Aschwanden et al. 1999; Pascoe et al. 2018). These values are much lower than the speed of light. \nThe typical hydrodynamic timescale in the photosphere is ∼ 5 minutes, it is ∼ 1 minute in the chromosphere and a few minutes in the corona. The light crossing time is of the order of a second even for the largest coronal structures. \nBecause of the low flow speeds and long hydrodynamic timescales compared to light crossing times, it is typically assumed that light travel times can be ignored in solar radiation hydrodynamics. That means that the computation of the radiative flux divergence at a given time t only depends on the state of the atmosphere at time t . The time-dependent transfer equation for the intensity in direction n then simplifies from \n1 c ∂I ν ∂t + n · ∇ I ν = j ν -α ν I ν (13) \nto \nn · ∇ I ν = j ν -α ν I ν , (14) \nwith j ν the emissivity and α ν = κ ν ρ the extinction coefficient per volume. In other words, it is assumed that the radiation field adapts instantaneously to changes in the state of the solar gas.", '2.7 Diffusion approximation': 'At very large optical depth the following approximation holds for J ν : \nJ ν ≈ S ν + 1 3 d 2 S ν d τ 2 ν , (15) \nwhere the second derivative should be taken along the direction of the gradient of S ν 1 . For the flux the following approximation holds: \nF ν ≈ -4 π 3 κ ν ρ ∇ S ν . (16) \nAt depths where the optical depth at all frequencies is much larger than unity, one can derive the diffusion approximation for the total radiative flux from from Eq. (16) if one also assumes that the source function is equal to the Planck function B ν : \nF ≈ -16 3 σT 3 κ R ρ ∇ T, (17) \nwith the Rosseland opacity defined as \n1 κ R = (∫ ∞ 0 1 κ ν d B ν d T d ν )/(∫ ∞ 0 d B ν d T d ν ) . (18) \nMethods to compute the total flux divergence that rely on the solution of the radiative transfer equation must converge to the diffusion approximation at large depth. Methods that require a numerical approximation of the source function must use one of at least second order in order to recover Eq. (15).', '3 Radiative transfer in the photosphere: multi-group radiative transfer': 'Multi-group radiative transfer was first introduced in Nordlund (1982). It is a method that approximates the frequency integral in Eq. (11) by a sum over a limited number N so called radiation groups (or radiation bins): \n∫ ∞ 0 4 πκρ ( S ν -J ν ) d ν ≈ N ∑ i =1 4 πκ i ρ ( S i -J i ) , (19) \nwith κ i , S i , and J i the opacity, source function and radiation field in each group, as defined below. \nThe rationale of the method is the realisation that the Λ -operator that produces the angle-averaged radiation field from the source function depends on the opacity only, and is a linear operator. While it is customary to write Λ ν [ . . . ] because the opacity changes with frequency, a better notation would \nbe Λ κ [ . . . ]. For two frequencies ν 1 and ν 2 with identical opacities everywhere in the atmosphere κ 1 = κ 2 = κ one can thus write \nJ ν 1 + J ν 2 = Λ ν 1 [ S ν 1 ] + Λ ν 2 [ S ν 2 ] = Λ κ [ S ν 1 + S ν 2 ] (20) \nIn order to derive the approximation in Eq. (19) one first approximates the frequency integral with a sum over discrete frequencies ν j with summation weights w j . These frequencies are then grouped into N bins, each labeled i , that have similar opacities. Finally, Eq. (20) is used to define a bin-integrated source function: \n∫ ∞ 0 κ ( S ν -J ν ) d ν glyph[similarequal] ∑ j w j κ j ( S j -J j ) (21) = N ∑ i =1 ∑ j ( i ) w j κ j ( S j -J j ) = N ∑ i =1 ∑ j ( i ) w j κ j ( S j -Λ j [ S j ]) (22) ≈ N ∑ i =1 κ i ∑ j ( i ) w j S j -Λ i [ ∑ j ( i ) w j S j ] ≡ N ∑ i =1 κ i ( S i -Λ i [ S i ]) (23) ≡ N ∑ i =1 κ i ( S i -J i ) . (24) \nHere j ( i ) is the set of all frequencies with similar opacities that are placed in bin i . The bin-integrated source function is constructed from the frequencydependent source function using \nS i = ∑ j ( i ) w j S j (25) \nWhat is left is now is to define how to group frequency points in the various bins, how to define the representative opacity for each bin κ i , and how to define the frequency-dependent source function S j . The opacity and source function are in general functions of at least frequency, temperature and density. Additional dependencies are on the velocity field in the atmosphere, the ray direction, possible location-dependent abundances, and in the case of non-LTE radiative transfer, on the radiation field. \nSection 3.1 discusses how to sort frequencies into opacity groups. In Sect. 3.2 it is assumed that the source function is given by the Planck function. This assumption is relaxed to allow for coherent scattering in Sect. 3.3.', '3.1 Sorting frequencies into groups': "Grouping frequencies and defining a bin opacity are necessarily somewhat crude. The density and thus opacities drop roughly exponentially with height, and opacities at a given point in the atmosphere vary over many orders of \nFig. 1 Illustration of the principle of τ -sorting. The horizontal axis is frequency, while the vertical axis shows the Rosseland optical depth in the 1D reference atmosphere. The solid line is the Rosseland optical depth where the monochromatic optical depth unity is reached. The frequencies are divided into four bins using the three border values τ 1 R -τ 3 R . Adapted with permission from Ludwig (1992). \n<!-- image --> \nmagnitude with frequency. The aim of the multi-group opacities is to approximate the full radiative transfer with sensitivity spanning from the low-opacity continua that form in the photosphere to strong lines that form in the mid and ideally even the upper chromosphere. \nThe de-facto standard method for grouping opacities is called τ -sorting: First a 1D reference atmosphere is chosen, for which the opacities and vertical optical depth scales at all frequency points j are computed. The atmosphere is then divided into various height bins. Frequencies at which optical depth unity is reached in height bin i are assigned to radiation bin i . This height sorting can in principle be done directly on a geometrical height scale, but most radiation-MHD codes follow Nordlund (1982). They set the borders between the height bins in terms of the Rosseland optical depth scale τ R : Let the borders be defined at a number of Rosseland optical depths τ k R . A frequency ν j then belongs to bin i if \nτ k -1 R < τ R ( τ ν = 1) ≤ τ k R . (26) \nA common choice is to put the border values τ k R equidistantly spaced in log τ R . The method is illustrated in Fig. 1. \nex \nj \n( \ni \n, \nl \n) runs over those \nB \ne \nff \nect of the sorting pro- \nstrated in Figs. 2 and 3. \ny from the ATLAS9 stel- \nT \n= \n4470 K and log \np \n= \noptical depth at 500 nm \n). After applying the tau- \nns the spectrum is a step \nvalues (Fig. 3). \npproach the exact opac- \nsmall ODF intervals, the \ned to converge to the ex- \nthe detailed reference \ng process) in the limit \n. This is due to the fact \nbetween bin-levels \nτ \nl \nref \nwill comprise frequen- \nwith di \nff \nerent height \n. It should be noted that \n∆ \nν \ni \ndenotes the average \ni+1 \nFig. 2. \nMonochromatic opacity as a function of wavelength for \nT \n= \nτ \nref \nFig. 1. \nSchematic illustration of the \nτ \n-sorting procedure using ODFs. \nWithin each ODF interval \n∆ \nν \nquency bins \nΩ \nl \n, \ndepending on the height where \nτ \nal.: Non-grey RT in numerical simulations of the solar photosphere \n∆ν \nν \ning procedure using ODFs. \ns \n∆ \nν \nwhere \nτ \nij \nare sorted to fre- \nij \n= \n1 is reached \ne major requirement for \n. For greater depths, the \ne inappropriate, but this \nn transfer becomes in- \nay. \nrealized in a convenient \nep \nj \nwithin ODF inter- \ngives \nτ \nij \nas a function of \nFig. 1). The ODF steps \ngrated quantities are ob- \nsteps in a given bin, viz. \n(16) \n(17) \nwithin one bin are similar, which is the major requirement for \nval \nBl \nand \nare used to model the line opacities. \nFig. 2. Monochromatic opacity as a function of wavelength for T = 4470 K and log p = 4 . 12, corresponding to log τ 500 = -2 in the solar atmosphere. ATLAS9 (Kurucz 1993) Opacity Distribution Functions are used to model the line opacities. κ P , l = 1 Bl ∑ i ∆ ν i B ∆ ν i ∑ j ( i , l ) w j ( i , l ) κ i j ( i , l ) , (17) where, for a given bin index l , the index j ( i , l ) runs over those steps in ∆ ν i ,whichareelementsof Ω l . B ∆ ν i denotes the average of B over the ODF interval ∆ ν i . The e ff ect of the sorting procedure on the opacity spectrum is illustrated in Figs. 2 and 3. Figure 2 shows the ODF-based opacity from the ATLAS9 stellar atmosphere code (Kurucz 1993) for T = 4470 K and log p = Fig. 3. Opacity as a function of wavelength for a τ -sorted multigroup model with five opacity bins, for the same thermodynamical conditions as in Fig. 2. using ODFs as basis for sorting frequencies introduces an additional error since the rearrangement of spectral lines inherent in the ODF concept a ff ects the way frequencies are classified. However, since the ODF solution approximates the exact soluFig. 2 Illustration of the concept of group mean opacity. Left-hand panel: Monochromatic extinction coefficients based on Opacity Distribution Functions for T = 4470 K and P = 1 . 3 × 10 4 Pa, corresponding roughly to the upper photosphere. Right-hand panel: The opacity that is 'assigned' to each wavelength using a 5-group scheme. The large continous variation in the left-hand panel is replaced by only five discrete opacities. Note that the high monochromatic opacities in the UV are replaced by a much lower group opacity. Adapted with permission from Vogler et al. (2004), copyright by ESO. \n<!-- image --> \n4 \n. \n12 (corresponding to a continuum optical depth at 500 nm \nlog \nτ \n500 \n= \n2 in the solar atmosphere). After applying the tau- \nsorting procedure with five opacity bins the spectrum is a step \n- \ntion well, the error incurred this way is small compared to the \nconsequence of the binning procedure (see Sect. 3.5). \nThe choice of a reference atmosphere is another possible \nFig. 3. Opacity as a function of wavelength for a τ -sorted multigroup model with five opacity bins, for the same thermodynamical conditions as in Fig. 2. function which can assume five discrete values (Fig. 3). In contrast to the ODFs, which approach the exact opacity spectrum in the limit of infinitely small ODF intervals, the multigroup solution can not be expected to converge to the exact solution (i.e. the solution based on the detailed reference spectrum which was used in the sorting process) in the limit of a very large number of opacity bins. This is due to the fact that, even for infinitely small intervals between bin-levels τ l ref (or κ l in the case of κ -sorting), each bin will comprise frequencies from di ff erent parts of the spectrum with di ff erent height profiles of the corresponding opacities. It should be noted that source of error connected with the τ -sorting scheme. It is plausible that the opacity binning method with τ -sorting shows good results in calculations of static 1D model atmospheres if the 'exact' solution (i.e., the 1D atmosphere resulting from the ODF approach) is chosen as the reference atmosphere. In multidimensional time-dependent simulations, however, the physical parameters may deviate considerably from a 1D reference stratification. As a consequence, the assignment of frequencies to bins might lead to an inappropriate representation of the opacities. In order to test the applicability of the τ -sorting procedure in non-planeparallel cases, we performed test Opacities are generally well-approximated by their LTE values in the photosphere. LTE opacities in a static atmosphere are functions of temperature, density and elemental composition only, but even then it requires a large effort to accurately compute them. For historical reasons this is commonly done using an intermediate step called Opacity Distribution Functions (ODFs). ODFs were developed originally to speed up computations used in modeling of 1D LTE radiative equilibrium stellar atmospheres (e.g., Gustafsson et al. 1975; Kurucz 1979). \nusing ODFs as basis for sorting frequencies introduces an additional error since the rearrangement of spectral lines inherent in the ODF concept a ff ects the way frequencies are classified. However, since the ODF solution approximates the exact solution well, the error incurred this way is small compared to the The method used to compute an appropriate mean opacity in a bin i from the opacities κ j depend on whether one assumes an LTE or non-LTE source function and are described in Sects. 3.2 and 3.3. An illustrative solution assuming an LTE source function is given in Fig 2. \nconsequence of the binning procedure (see Sect. 3.5). \nThe choice of a reference atmosphere is another possible \nsource of error connected with the \nτ \n-sorting scheme. It is plau-", 'sible that the opacity binning method with τ -sorting shows good results in calculations of static 1D model atmospheres if 3.2 Multi group radiative transfer with LTE source function': "the 'exact' solution (i.e., the 1D atmosphere resulting from the \nODF approach) is chosen as the reference atmosphere. In multidimensional time-dependent simulations, however, the physical parameters may deviate considerably from a 1D reference stratification. As a consequence, the assignment of frequencies to bins might lead to an inappropriate representation of the opacities. In order to test the applicability of the τ -sorting procedure in non-planeparallel cases, we performed test The source function in the solar atmosphere is in general not equal to the Planck function because at some height in the atmosphere densities become too low to set up Saha-Boltzmann populations through collisions. However, detailed non-LTE computations in 1D models show that in the deep photosphere (roughly defined here as -100 km < z < 200 km, with the z = 0 defined as the location where τ 500 nm = 1), the source function is almost exactly equal to the Planck function (see Fig. 36 of Vernazza et al. 1981), and the opacities are close to their LTE values. At larger heights this is no longer the case for lines and continua of neutral atoms with a low ionisation potential because they tend to be over-ionized by the radiation from below. Nevertheless, one can expect that assuming LTE for both the source function and the opacity is a good approximation for computing the flux divergence in the photosphere. \ni \n, the ODF steps \n∆ \nν \nij \nare sorted to fre- \nij \n= \n1 is reached \nThe source function in group i is then given by \nS i = ∑ j ( i ) w j B j . (27) \nThis expression only depends on temperature and can thus easily be precomputed and stored in a 1D lookup table. \nOnce the opacities κ j are grouped into the N bins, one still needs to compute an appropriate bin opacity κ i . Choosing the numerical equivalent of the Rosseland opacity in each bin ensures that the diffusion approximation is recovered deep in the atmosphere: \nThis choice of bin opacity is however not appropriate at low optical depths, where photons are mainly in the free streaming regime. Following approximations valid at small optical depth put forward in Mihalas (1970), and further developed in Ludwig (1992), it turns out that the Planck-mean opacity κ B is a good choice for the outermost layers of the atmosphere: \n1 κ R i = ∑ j ( i ) w j 1 κ j d B j d T / ∑ j ( i ) w j d B j d T . (28) \nκ B i = ∑ j ( i ) w j κ j B j ∑ j ( i ) w j B j . (29) \nSomewhere in the atmosphere one should make a transition from one opacity definition to the other. This can be achieved through defining the group opacity as \nκ i = Wκ B i +(1 -W ) κ R i , (30) \nW = e -τ i /τ 0 . (31) \nwhere τ 0 is an adjustable parameter of order unity and τ i the vertical optical depth. Computing τ i in the MHD simulation is somewhat computationally expensive and instead it is estimated, for example through using the relation between mass density and optical depth in the 1D reference atmosphere used for the sorting of frequencies into bins (Ludwig 1992; Vogler et al. 2004). \nAnother option is to base the transition on the local mean free path: \nW = e -l i / ( κ i ρ ) , (32) \nwhere l i is a typical length scale over which the bin-integrated source function changes (Skartlien 2000). The advantage of this method is that it does not depend on the properties of a 1D reference model. \nFor a fixed elemental composition, κ i depends only on a combination of any two thermodynamic parameters (for example e and ρ ). Like the groupintegrated source function, they are commonly precomputed and put in a 2D lookup table. \nExtensive discussions of multi-group radiative transfer assuming an LTE source function can be found in the PhD-thesis of Ludwig (1992, in German) and in Vogler et al. (2004). \n3.3 Multi group radiative transfer with non-LTE source function \nThe assumption of LTE for the source function is accurate in the photosphere, but breaks down in the chromosphere and transition region. Here the energy exchange is dominated by the resonance lines of H i , Ca ii , Mg ii , and He ii (see for example Fig. 49 of Vernazza et al. 1981). These lines can have photon destruction probabilities \nglyph[epsilon1] = C ul A ul + C ul + B ν B ul < 10 -4 , (33) \nwith C ul the downward collisional rate coefficient and A ul and B ul Einstein coefficients for spontaneous and induced deexcitation. Scattering should therefore be taken into account. \nScattering has a strong damping effect on the amplitude of ∇· F ν /ρ . In LTE it is given by \n∇· F LTE ν ρ = 4 πκ ν ( B ν -J ν ) . (34) \nIf one assumes the presence of a coherently scattering line in a two-level atom at frequency ν , then the source function becomes \nS ν = (1 -glyph[epsilon1] ) j ν + glyph[epsilon1]B ν . (35) \nIn that case the flux divergence per mass unit is \n∇· F NLTE ν ρ = 4 πglyph[epsilon1]κ ν ( B ν -J ν ) , (36) \nso that for a given difference between B ν and J ν , the amplitude of the radiative energy exchange in non-LTE can be orders of magnitude smaller than in LTE. Skartlien (2000) extended the method presented in Sect. 3.2 to include coherent two-level scattering, as an approximation to the much more complicated full non-LTE radiative transfer problem. He assumed that the monochromatic extinction coefficient can still be computed in LTE. This assumption is rather accurate for the resonance lines of H i , Ca ii , Mg ii , whose lower levels are the ground state of a dominant ionisation state. In addition he assumed that the scattering coefficient in a spectral line is given using the approximation by van Regemorter (1962), so that it is independent of the actual line, and only depends on frequency, temperature and electron density. Starting from the monochromatic two level source function S ν = (1 -glyph[epsilon1] ν ) J ν + glyph[epsilon1] ν B ν and following a reasoning similar to the one given in Sect. 3.2, he arrives at an expression for the group source function: \nS i = glyph[epsilon1] i J i + t i . (37) \nHere glyph[epsilon1] i represents a group-averaged scattering probability, and t i the groupintegrated thermal production of photons. He also derives an expression for the group extinction coefficient κ i . Similar to the LTE case, glyph[epsilon1] i , t i , and κ i have different expressions for the diffusion regime and in the free streaming \nFig. 3 Illustration of short and long characteristics used for solving the transfer equation in 2D decomposed domains. Grey dots indicate grid points where the intensity should be computed, with grey lines connecting those grid points. Blue lines indicate subdomain boundaries, where it is assumed that the horizontal boundaries are periodic. Red arrows indicate two examples of long characteristics, while the orange arrows indicate short characteristics. Note that the long characteristics cross multiple subdomain boundaries and wrap around the periodic horizontal boundary Each subdomain contains a piece of the long characteristic of variable length. \n<!-- image --> \nregime in the outer atmosphere. An important difference is that the streaming quantities now depend on the monochromatic mean intensity J ν in a 1D reference atmosphere. This means that the quantities must be recalculated for each different target of simulations (e.g., sunspots or quiet Sun). Simulations containing a variety of structures might suffer from inaccuracies because a 1D atmosphere cannot be representative of all atmospheric structures. \nAnother difference compared to the method employing an LTE source function is that the computation of Q rad now requires solution of Eq. (37) together with the transfer equation because J i = Λ [ S i ]. This is typically done through accelerated Λ -iteration. Because of the multidimensional geometry, a local approximate Λ -operator (which is equivalent to Jacobi-iteration, see Olson et al. 1986) is efficient and easily coded, such as in the Oslo Stagger Code (Skartlien 2000). Gauss-Seidel iteration (Trujillo Bueno and Fabiani Bendicho 1995) offers superior convergence speed and has been implemented in the Bifrost code (Hayek et al. 2010; Gudiksen et al. 2011).", '3.4 Solving the transfer equation': "Most modern radiation-MHD codes are parallelized to make use of large supercomputers with distributed memory. Typically, the simulated domain is represented on a 3D Cartesian grid. The domain is split into smaller subdomains and each CPU handles the required computations on its own subdomain, communicating information to other subdomains as needed. This architecture scales well for the MHD equations: they are local and require communication with neighbouring subdomains only. \nRadiation is however intrinsically non-local. An emitted photon can traverse many subdomains before undergoing another interaction with the solar gas, so communication between subdomains is not necessarily local. This problem is shared between radiation-MHD codes that aim to compute a reasonable approximation of the flux divergence, and non-LTE radiative transfer codes such as Multi3d (Leenaarts and Carlsson 2009) and PORTA ( ˇ Stˇep'an and Trujillo Bueno 2013), wich aim to accurately compute the emergent spectrum from a given model atmosphere. Consequently, there is a large amount of literature addressing efficient solutions of the transfer equation in multidimensional geometries and/or decomposed domains (e.g., Kunasz and Auer 1988; Auer 2003; Vogler et al. 2005; Heinemann et al. 2006; Hayek et al. 2010; Ibgui et al. 2013; ˇ Stˇep'an and Trujillo Bueno 2013). I will only briefly touch upon some aspects. \nSolving for the flux F or for the angle-averaged radiation field J requires computation of the intensity for a number of different directions at all points on the numerical grid. In practice there are two methods that are used in radiation-MHD codes: short characteristics (SCs) and long characteristics (LCs). Both are illustrated in Fig. 3. \nShort characteristics solve the transfer equation along short line segments (orange in Fig. 3), starting at a grid cell boundary and ending at a grid point where the intensity is desired. Along the line one typically computes a numerical approximation of the formal solution: \nI ( τ ) = I ( τ = 0) e -τ + ∫ τ 0 S ( t ) e t -τ d t, (38) \nwhere the optical thickness scale has its zero point at the start of the SC and τ is the optical thickness along the entire SC. Intensities I ( τ ) are computed at the grid points (grey circles in Fig. 3). Computation of I ( τ = 0), which is needed at the start of the orange arrow, thus requires interpolation from the grid points. SC methods are therefore somewhat diffusive and coherent beams of photons disperse. High-order interpolation schemes can alleviate, but not eliminate, this diffusion. In practice this diffusion is typically not a problem, given that the photosphere is emitting photons everywhere and both the source function and the resulting radiation field are rather smooth. \nThe transfer equation is solved along all SCs in a sequential order, starting from a known boundary condition (the diffusion approximation at the bottom of the atmosphere for rays going up, and typically zero for SCs going down from the top of the atmosphere). The method was introduced for 2D cartesian \ngeometry by Kunasz and Auer (1988). An in-depth description of the method in 3D Cartesian geometry is given in Ibgui et al. (2013). The ordered fashion in which SCs must be computed leads to complications with spatial domain decomposition. An example method of how to achieve reasonable parallelism despite this ordering can be found in ˇ Stˇep'an and Trujillo Bueno (2013). Short characteristics can be easily computed along any angle. Typical ray quadratures (i.e., the set of angles chosen to numerically compute Eq. (9) or (12)) that are in use are the angle sets from Carlson (1963), or equidistant in azimuth and using a Gaussian quadrature in inclination. \nBruls et al. (1999) present a method to compute SCs on unstructured grids. It is not currently in use in the common radiation-MHD codes that use fixed Cartesian grids, but might be of great use for codes that use unstructured or adaptive grids. \nThe long characteristic method traces rays from the lower boundary of the domain to the upper boundary (red line segments in Fig. 3). An LC does generally intersect only a few grid points. Interpolation of the source function and the opacity from the grid points to the LC and interpolation of the intensity along the LC back to the grid points is thus necessary. The transfer equation along an LC can be solved using the formal solution (Eq. (38)), or by solving the second-order form of the transfer equation (Feautrier 1964). \nLong characteristics allow photons to travel in a straight line and are thus not diffusive. Efficient parallel algorithms exist for solving along LCs in decomposed domains (Heinemann et al. 2006). However, this parallel algorithm is only easily implemented when LCs cross cell boundaries exactly through grid points, which limits the application to grids with fixed spacing and a maximum of 26 directions: both directions along three axes, six face diagonals, and 4 space diagonals in a cubic grid (e.g., Popovas et al. 2019). Arbitrary ray quadratures can be implemented at the expense of code simplicity. \nBoth the SC and LC method can handle exactly horizontal rays, but such rays require implicit solution methods in case of periodic boundary conditions in the horizontal direction, which is the default for solar atmosphere radiation-MHD simulations. This adds additional coding complications in the usual case that parallelism is implemented through spatial domain composition. Horizonal ray directions are therefore usually avoided. \n3.5 Computation of the heating rate from the intensity and source function \nAnalytically, the two expressions for the heating rate in a bin Q i = -∇· F i = 4 πκ i ρ i ( J i -S i ) are equivalent. In case of actual numerical computation this is no longer the case. \nIn the deep atmosphere the diffusion approximation holds. Eq. (15) shows that while S i and J i increase with depth because of the increase in temperature, their difference becomes smaller, and at some point roundoff errors become noticeable. These errors are then amplified by the exponential increase of the density with depth. Using the source function and radiation field to \ncompute Q i is thus unstable in the deep layers. Instead, computing the flux from the intensity using the numerical equivalent of Eq. (9) and then taking the divergence is stable in the deep layers. Because the radiative flux is small compared to the energy density of the gas, an error in computation of its divergence will not lead to large errors in the internal energy. \nIn the upper layers the situation is reversed: the radiative flux is large compared to the energy density of the gas (see Table 1 and Eq. (4)), and a small error in computation of the flux divergence from the intensity will lead to a large error in Q i and e . Using 4 πκ i ρ i ( J i -S i ) is stable however, because of the generally large split between S i and J i . \nFollowing the suggestion by Bruls et al. (1999), most 3D codes that use short characteristics compute Q i using the flux divergence at large optical depth, and switch to using the source function and radiation field around an optical depth of 0.1. \nAn alternative to the above switching scheme is to rewrite the transfer equation in terms of the quantity K I = S -I (dropping bin indices i here for brevity). The quantity K I is proportional to the cooling rate in a specific ray direction. The transfer equation then transforms into \nd K I d τ = d S d τ -K I , (39) \nwhich in its integral form is given by \nK I ( τ ) = K I ( τ 0 )e τ -τ 0 + ∫ τ τ 0 e τ ' -τ d S d τ ' d τ ' . (40) \nThe total heating rate is then computed from \nQ = ∮ K I d Ω (41) \nThis equation does not suffer from the numerical precision problems caused by the cancellation of the subtraction S and I . Eq. (40) has the same form as the formal solution of the normal transfer equation and can be solved efficiently using a variety of methods. If the source function is known (such as when assuming LTE) then solving straight for K is possible without having to solve for I first. Heinemann et al. (2006) describe an elegant method for solving for K in decomposed domains using long characteristics and a direct solution of the transfer equation. In case of a non-LTE source function, such as in Sect. 3.3, then computing S requires solving the transfer equation to obtain I and J . Computing Q from K I afterwards then offers little benefit over computing Q straight from ∇· F or S -J . \nNordlund (1982) implemented a similar method as Heinemann et al. (2006), but based on the second-order form of the transfer equation: \nd 2 P d τ 2 = P -S. (42) \nFig. 4 Summary table of the three major binary choices in multi-group radiative transfer from computing radiative losses and gains in the photosphere. \n<!-- image --> \nwhere \nP = 1 2 ( I ( Ω ) + I ( -Ω )) , (43) \nthe average of an ingoing and an outgoing ray (e.g., Hubeny and Mihalas 2014, p. 387). Defining K P = P -S one arrives at \nd 2 K P d τ 2 = K P -d 2 S d τ 2 . (44) \nThis equation has the same form as the second order transfer equation and can be solved efficiently along a long characteristic using the method of Feautrier (1964). \n3.6 Summary and examples of photospheric radiative transfer \nFigure 4 shows a summary table of the three major binary choices in multigroup radiative transfer from computing radiative losses and gains in the photosphere: LTE or non-LTE source function, short characteristics or long characteristics, and solving for K or solving for I in order to compute the flux divergence. Each of the resulting 8 options is numbered. \nThe simulation by Nordlund (1982) is of Type 5. MURaM (Vogler 2004; Rempel 2017), RAMENS (Iijima and Yokoyama 2015; Iijima 2016), and MANCHA3D (Khomenko et al. 2017, 2018) are Type 2 codes. Stellarbox (Wray et al. 2015) is Type 6. COBOLD (Freytag et al. 2012) has both Type 2 and Type 6 options, but for stellar atmosphere simulations only Type 2 is supported. \nFig. 5 Example of the vertically emergent intensity computed with a 4-group non-LTE scheme. The flux divergence along the horizontal white line is shown in Fig. 6. The model was computed with the Bifrost code (Gudiksen et al. 2011). \n<!-- image --> \nThe Oslo version of the Stagger code 2 (Type 8) and the Bifrost code (Gudiksen et al. 2011) (Type 4) are as of October 2019 the only codes using a non-LTE source function. \nFigures 5 and 6 demonstrate the result of a 4-group computation in a radiation-MHD simulation. The details of this particular simulation can be found in Carlsson et al. (2016). \nFigure 5 shows the vertically emergent intensity in each radiation group. Bins 1 and 2 grouped areas of the spectrum with generally low opacities. The corresponding images are indeed reminiscent of observed optical continuum \nFig. 6 Atmospheric structure and flux divergence per mas unit Q i /ρ = -∇ · F i /ρ = 4 πκ i ( J i -S i ) in a vertical slice along the white line in Fig. 5. The top row shows the temperature and density, the two bottom rows show the flux divergence per mass unit in the four opacity groups. Brown is cooling, blue is heating, and the brightness scale for the flux divergence panels is clipped at 20% of the maximum of the absolute value to enhance contrast. The maximum and minimum values are indicated in each panel. The black curves indicate the τ i = 1 height in each bin. Note that the maxima of the heating and cooling per unit mass do not coincide with the τ = 1 height. The flux divergence in this simulation is artificially set to zero at the height where the entire atmosphere above has a maximum optical thickness of 10 -5 . The total radiative losses above this height are negligible, but the local losses and gains per unit mass are not. \n<!-- image --> \nimages of the photosphere. Bins 3 and 4 contain opacities of stronger lines, and resemble images taken in the wings of the Ca ii H&K lines (e.g., Rutten et al. 2004). The images are dominated by reversed granulation; the small bright structures are caused by magnetic field concentrations. \nFigure 6 shows the heating rate per mass in each radiation bin. The prominent funnel shape in the mass density panel is caused by the presence of a strong magnetic field concentration that fans out with height. All bins show strong cooling at the top of the granules (Red color just below z = 0 . 0 Mm), and bins 2 - 4 show heating just above the granules. There is strong cooling per mass unit in the chromosphere above z = 1 Mm. While the optical thickness of the chromosphere above this height is small because of the low density, the heating rate per mass is independent of the mass density, and depends on the value of κ i and the size of ( S i -J i ) only. \nSKARTLIEN \nF IG . 3.È Upper panels: Horizontally averaged radiative heating per volume unit shown in a bilogarithmic plot. The units are arbitrary. Thick lines are the exact group solutions, while the thin lines are the approximate group solutions. The vertical line segments in the uppermost panel show the heights where the average optical depth per group is unity, as measured along the vertical line. L ower panel: Horizontally averaged amplitudes of radiative heating (average absolute value) relative to the amplitude of the exact total heating. Thick lines are the exact amplitude ratios, while the thin lines are ratios from the approximate group solutions. Fig. 7 Average radiation heating and cooling per volume ( Q ) as a function of height in a 3D radiation-hydrodynamics simulations using a 4-group non-LTE radiative transfer scheme. Upper panels: Horizontally averaged radiative heating per volume unit shown in a bilogarithmic plot. The units are arbitrary. Thick lines are the exact group solutions, while the thin lines are the approximate group solutions. The vertical line segments in the uppermost panel show the heights where the average optical depth per group is unity, as measured along the vertical line. Lower panel: Horizontally averaged amplitudes of radiative heating (average absolute value) relative to the amplitude of the exact total heating. Thick curves are the exact amplitude ratios, while the thin curves are ratios from the approximate group solutions. Adapted with permission from Skartlien (2000), copyright by AAS. \n<!-- image --> \napproximate solution. The horizontal averages ( \nthin lines \n) \ncoincide very well with the horizontal averages of the exact solutions ( thick lines ). The upper panels show cooling below 0.0 Mm in all groups, and heating in groups 2, 3, and 4 immediately above 0.0 Mm. The lower panel shows the amplitude of the approximate and exact Ñux divergence (horizontal average of the absoFigure 7 demonstrates the accuracy of approximating the spectrum by only a few frequency groups in the non-LTE scheme of Skartlien (2000) on the horizontally averaged values of Q i . It does however not test the assumptions of coherent scattering, LTE equation of state and LTE opacity. The absolute error in Q = ΣQ i is of the order of a few percent, while the error in individual \nlute value) relative to the amplitude of the \ntotal exact \nÑux \ndivergence, \ni.e., \nand \n* \no \nT \n/ \nS \no \n/ \no \nT \n/ \nS \no \n/ \n(brackets denote horizontal average). We see that the exact \nS \no \n/ \nrelative amplitudes ( \nthick lines \n) coincides very well with the \napproximate values ( \nthin lines \n). \nA sample of the spatial structure is displayed in Figure 4, \nwhere we have shown the exact heating per mass unit in all \ngroups as gray-scale images in a vertical slice. Black con- \ntours mark locations of zero heating, and lighter shades of \ngray means positive heating. Gas in layers immediately \nabove the cooling layer is heated in all groups in expanding \nÑow above granules. Granules are seen as curved horizontal \nstructures. As up-Ñowing gas expands and cools, the tem- \nperature falls below the radiation temperature, and the gas \nis radiatively heated. Note also heating of the cool region \ni \ntot \no \nT \nS \no \n/ \ni \ntot \no \nT \nIG \nF \n. \n4.ÈExact group radiative heating per mass \nthrough the simulation. Full drawn black contours \nheating. Heating is found at lighter shades of gray, and \nshades. Dash-dotted black contours show the zero level \ngroup heating and coincide well with the zero level of \nWhite curves are the horizontal averages of the radiative \nunit (normalized to \nÐ \nt the plotting window). Black \nzero level for these curves, and positive values are to the \nL ower panel: \nTemperature in the same vertical slice. \nthe granular layer at the height 0.0 Mm, and heating \nall groups, and also heating of the cool region \nchromosphere. \nbelow 3000 K in the chromosphere as \nfrom below is converted to thermal energy via \nThe dash-dotted black contours show the \nthe approximate heating and coincide well \ngroups i can be as large as 50% (group 3 at z = 0 . 3 Mm). Note that Q is dominated by group 1. The absolute value of Q is decreasing with height because of its dependence on the mass density. \nIn Fig. 8 the difference between assuming an LTE or non-LTE source function is demonstrated. The expression for the flux divergence in the multigroup scheme is the same as for the monochromatic case: \n∇· F i ρ = 4 πglyph[epsilon1] i κ i ( S i -J i ) , (45) \nwhere glyph[epsilon1] i = 1 in LTE, and glyph[epsilon1] i ≤ 1 in non-LTE. In the latter case glyph[epsilon1] i can be as low as 10 -4 . A strict comparison between the scattering and non-scattering cases is not possible, because of the different definition of the group-mean opacities, source functions and thermal emission terms. Nevertheless, the main result from this figure is that the LTE scheme vastly overestimates the radiative cooling in the chromosphere, because glyph[epsilon1] i = 1 in LTE. \nThe amplitude of the cooling and heating in groups 2 - 4 between z = 0 Mm and z = 0 . 3 Mm is however much larger in non-LTE than in LTE. Skartlien (2000) speculates that this is caused by the smoothing effect that scattering has on J i , but a thorough investigation of this effect has never been done.", '4 Radiative losses in the transition region and corona': "The corona is optically thin at all wavelengths except for the radio regime. Its radiative losses are dominated by a myriad of EUV lines from highly ionised stages of many different elements (e.g., Curdt et al. 2004; Woods et al. 2012). \nIn the transition region, which I loosely define here as that part of the atmosphere where hydrogen is ionised but the temperature is below 100 kK, the emission is dominated by lines of lower ionisation stages (typically twice to four times ionised). For most lines, and for most regions on the sun, the TR is optically thin. In solar flares this is not necessarily the case: Kerr et al. (2019) showed that the TR can have appreciable optical thickness in the Si iv 140 nm lines. \nNon-equilibrium ionisation effects play a role in the transition region (Olluri et al. 2013; Golding et al. 2017) and corona (Hansteen 1993; Bradshaw et al. 2004; Dzifˇc'akov'a et al. 2016). Figure 1 of Hansteen (1993) shows an increase in radiative losses at a given density and temperature in the transition region by a factor two in a 1D hydrodynamic situation when non-equilibrium ionisation is used instead of instantaneous ionisation equilibrium. \nA fully general treatment of TR and coronal radiative losses in a radiationMHD simulation would thus involve solution of the full 3D non-equilibriumnon-LTE radiative transfer problem for most ionisation stages for a wide range of elements. This is currently impossible for 3D simulations because of limits on computation speed. \nNeglecting radiative non-local transfer effects through assuming optically thin radiative transfer (i.e., assuming J ν = 0 in the rate equations) alleviates \nJ \nand source \nbut the scaling \nfrom the new \nprevious LTE \ncaused mainly by \nfrom di†ering \nshow the \nHorizontal aver- \ncorrespond to \nequal to \nthe scaling by \nall group \nline scat- \nsource func- \nsnapshot that \nthe same set \nresults. \nVol. 536 \nF IG . 7.ÈHorizontal averages of radiative heating per mass unit. The units are arbitrary, and the scaling is the same for all Ð gures. The amplitude of the Ñux divergence for the previous LTE method (old) is much larger above 0.3 Mm than for the current method. This e†ect is caused mainly by the larger di†erences between the source function and the mean intensity. Around 0.0 Mm, the new method produces larger amplitudes in the sense that hot regions are cooled more and cooler regions are heated more. L ower panel: Sum of all groups. The main di†erence above 1.0 Mm comes from the contribution in group 4. Fig. 8 Horizontal averages of radiative heating per mass unit ( Q/ρ for a 4-group scheme assuming LTE and non-LTE. The units are arbitrary, and the scaling is the same for all figures. The amplitude of the flux divergence for the LTE method (labeled old ) is much larger above 0.3 Mm than for the non-LTE method (labeleld n ew). This effect is caused mainly by the larger differences between the source function and the mean intensity. Around 0.0 Mm, the new method produces larger amplitudes in the sense that hot regions are cooled more and cooler regions are heated more. Lower panel: Sum of all groups. The main difference above 1.0 Mm comes from the contribution in group 4. Adapted with permission from Skartlien (2000), copyright by AAS. \n<!-- image --> \ntering would imply that \nThe values of the mean \nJ \n* \n\\ \nS \n*. \nintensities are also di†ering, mainly because of di†erences in \ni \ni \nthe source function, and partially because of the di†erent \nheights for which the optical depths are unity. \nDiamonds and triangles show the heights for where the \nhorizontally averaged optical depth \nis unity, for previous \nand current methods, respectively. The mean intensities in \nq \ni \noptically thin regions are approximately \nS \nJ \n( \nq \ni \n* \nT \n^ \n1 \nS \nS \n* \nT \nThe group mean opacities that determines the \n2 \ni \n\\ \n1). \noptical depths are seen in the bottom panel of Figure 6. The \ni \nnew opacities are higher than the old opacities above 0.3 \nalready much of the problems without sacrificing much accuracy in most circumstances. If one furthermore excludes hydrogen and helium, the justification being that these elements are fully ionised at high temperatures and do not contribute to line cooling, then changes in ionisation of the elements do not influence the pressure and temperature. \nThe problem reduces then to solving the rate equations \nglyph[negationslash] \nglyph[negationslash] \n∂n i ∂t + ∇· ( n i v ) = n l ∑ j,j = i n j P ji -n i n l ∑ j,j = i P ij , (46) \nwhere i sums over all energy levels and ionisation stages of each element under consideration. The rate coefficients P ij contain collisional (de-)excitation and collisional ionisation recombination terms and spontaneous radiative deexcitation and recombination terms, but no radiative terms that involve absorption of an existing photon. Ignoring absorption therefore only allows for cooling. The radiative loss rate per volume in a bound-bound transition between a lower level i and upper level j is then given by \nQ ij = hν ij A ji n j , (47) \nand a similar expression can be written for bound-free transitions. The total cooling rate per volume can then be computed by summing the contributions of all transitions of all elements and including electron-ion free-free radiation. A more detailed description of this method in a 1D radiation-hydrodynamics code is given in Hansteen (1993). To my knowledge this method has not been implemented in 3D codes. \nThe option of computing non-equilibrium ionisation was added to the Bifrost code by Olluri et al. (2013), but they did not implement the resulting radiative cooling. \nInstead, the default method to compute radiative losses in the corona is to assume statistical equilibrium (i.e., the left-hand-side of Eq. (54) is assumed to be zero. Together with the assumption of no photon absorption these two assumptions together are often called the 'coronal approximation'. The cooling a a spectral line of element X in ionisation stage m with upper level j and lower level i can then be written as: \nQ ij = hν ij A ji n e n j,m n m n m n X n X n H n e n H , (48) \n≡ G ( T, n e ) n e n H . (49) \nHere n j,m /n m is the fraction of all ions in ionisation stage m in level j , n m /n X is the fraction of all atoms of species X in ionisation stage m , and n X /n H is the abundance of element X relative to hydrogen. The function G ( T, n e ) is only weakly dependent on the electron density: the upper level population is dominated by collisional excitation from the ground state, so that n e n m ∼ A ji n j,m , and the rate coefficients setting up the ionisation balance are almost linear in the electron density. A residual electron density dependence remains \nE. Landi & M. Landini: Radiative losses of optically thin coronal plasmas \nFig. 5. Percentual differences between total emissivity curves calculated assuming different values of the electron density. Full line : 10 8 vs. 10 10 cm -3 ; Dashed line : 10 8 vs. 10 12 cm -3 ; Dash-dotted line : 10 8 Fig. 6. Fig. 9 Sensitivity of the coronal loss function Λ ( T, n e ) to the electron density. Full line: 10 14 vs. 10 16 m -3 ; dashed line: 10 14 vs. 10 18 m -3 ; dash-dotted line: 10 14 vs. 10 20 m -3 . Adapted with permission from Landi and Landini (1999), copyright by ESO. \n<!-- image --> \nvs. 10 \n14 \ncm \n- \n3 \n. \nequilibrium, while for lower densities collisional de-excitation in G ( T, n e ) through collisional deexcitation and density-dependent dielectronic recombination (Summers 1972, 1974). \nPercentual difference between the total emissi \ntained with the old and new version of the Arcetri \nadopted electron density is 10 \n10 \ncm \n- \n3 \n. \nAs big improvements have been done in the \nsion of the Code versus the older version \n& Monsignori Fossi 1990, we have performed a \nbetween the present results and those obtained \nversion of the Arcetri Code. The adopted element \nare from Allen 1973. There are three main dif \nthe two versions of the Arcetri Code: \n(a) \nthe old \nall \nline intensities using the \nCoronal \n(b) \nthe collision rates were calculated \n(c) \nradiative data came from different \nbecomes negligible compared to radiative decay and the Coronal ModelApproximation (yielding density insensitive line Contribution Functions ) may be adopted. Fig. 5 displays the percentual difference Perc.Diff ( i ) = η ( T,N e = 10 8 ) -η ( T,N i e ) η ( T,N e = 10 8 ) (6) (with N i e = 10 10 , 10 12 and 10 14 cm -3 ) between total emissivity calculated imation , A total coronal radiative loss function Λ ( T, n e ) can thus be constructed through summing up Q ij for all levels and ionisation stages of all relevant elements and adding the contribution from continuum processes. The electron density dependence of Λ ( T, n e ) is weak: Landi and Landini (1999) performed a sensitivity study for electron densities in the range 10 14 - 10 20 m -3 , and found a difference in the value of Λ of at most 20% (see Fig 9). It is therefore reasonably accurate to pre-compute Λ at a fixed typical coronal electron density (say n e = 10 16 m -3 ). \nthan in the present version of the Code. \nThus, the present comparison allows to check \nof different assumptions in level population \nresulting total plasma emissivity. \nFig. 6 displays the percentual difference \ncurves calculated at different densities as a function of electron temperature. As expected, the greatest differences are found with the curves at 10 12 --10 14 cm -3 , which are very similar, because density-dependence affects line emissivity mostly between 10 8 and 10 10 cm -3 . Differences are always smaller than 25% and show a marked temperature dependence, being hightors and A larger uncertainty can be introduced by inaccuracies of the elemental abundances. The coronal loss function is dominated by losses from C, Si, O, and Fe, and the abundances of these elements have a large influence. The abundance of C and O is debated after 3D non-LTE computations (Asplund et al. 2004, 2005) gave a different result than calculations using 1D models Grevesse and Sauval (1998). \nest at transition region and coronal temperatures and decreasing down to zero at the edges of the selected temperature range. The maximum at coronal temperatures is given by the presence of a host of strong density dependent lines formed in quiet corona, mainly from Fe, Mg and Si ions. The high temperature tail is dominated by strong, density insensitive lines and freefree continuum; the low temperaure tail is dominated by density insensitive transition region and chromospheric lines and for this Perc.diff Another complication is that the most accurate abundances are derived from photospheric lines, but coronal abundances are generally different from abundances in the photosphere. Feldman et al. (1992) compared coronal abundances for 15 elements to photospheric abundances, and Fig. 10 show the difference in the loss function when using photospheric or coronal abundances. Unfortunately it appears that no systematic re-investigation of coronal abundances have been made since 1992. Furthermore, the coronal abundances are not constant, but depend on the coronal structure (see for example Sec 2.1 \nreason there are small differences between computations carried \nout assuming different density values. \n3.2. Effect of different datasets and approximations \nin level population computation \nLevel populations are strongly sensitive to any change or prob- \nlem in the atomic parameters, collision strengths and transition \nprobabilities as well as in the approximation adopted for their \ncalculations, and this affects line radiation. It is therefore im- \nportant to check the effects of different transition probabilities \ndatasets on the resulting total emissivity curve. \nη \n- \nη \nold \nbetween the two versions of the Code as a function \ntemperature. It is possible to see that rather high dif \nto 60%) are found at transition region temperatures, \ndiscrepancies occur at coronal temperatures. In the \ntion of Fig. 6 the older version of the Arcetri \ntotal emissivity than the more recent version at \ntemperatures. This is due to the presence of few \ntransitions from O \niv \n, O \nv \n, C \niv \nwhose emissi \ndifferent values in the two versions of the Code; \nis due both to the use of different datasets and to \napproximations used in level population \nan overestimation of line emissivity for these \nold version of the Code. The negative section of \nis due to the much larger number of lines \nversion. \n= \nη \nold \nnew \n926 \nK. P. Dere et al.: CHIANTI - an atomic database for emission lines. IX. \nFig. 5. Radiative loss rate for coronal abundances ( upper curve ) and photospheric abundances ( lower curve ). 1 . 6 Fig. 10 Radiative loss functions Λ computed assuming n e = 10 15 m -3 as function of temperature. Upper curve: coronal abundances from Feldman et al. (1992). Lower curve: photospheric abundances from Grevesse and Sauval (1998). Adapted with permission from Dere et al. (2009), copyright by ESO. \n<!-- image --> \n∫ \nemission measure ( \nn \ne \nN \nHd \nV \n) has been calcu \nization equilibria discussed in Sect. 4. The lo \nprocesses of bremsstrahlung, radiative \nation and two-photon radiation. An electron d \nwas used and two sets of elemental abundan \nabundances of Feldman et al. (1992)andthe \ndances of Grevesse & Sauval (1998). The val \nloss rate are shown in Fig. 5 and in Table 1. \nThe most recent calculation of radiati \nthose of Landi & Landini (1999); Colgan et \n& Landini (1999) used the Arcetri spectr \nCHIANTI 2 database. Their Fig. 1 shows cal \nsity of 10 \n10 \ncm \n- \n3 \nfor the coronal abundances \net al. 1992)andtheionizationequilibriaofAr \n(1985) and Arnaud & Raymond (1992). A c \nthat the calculations of Landi & Landini ( \nto our current values. The major di \nff \nerences \nwhere the current loss rate is a factor of \n× \n10 \n4 \nK where the current loss rate is a \nthan the values of Landi & Landini (1999). \nhigher temperature is primarily due to the f \ndid not include transitions for the hydrogen-li \n46%lower in the case of oxygen, with the changes largely resulting from the application of 3D model atmospheres to the interions. The di to di ff of Cranmer and Winebarger 2019, and references therein). Abundances might thus well be the dominant source of uncertainty in the coronal loss function. \nff \nerences at the lower \nerences in the way the Arcetri code and \nlate the radiative losses due to the continuum. \nThere are significant di \nff \nerences between \nColgan et al. (2008) and those of both Landi \nand our current values. Detailed comparisons \n(2008) are not possible as none of the atomi \nauthors' radiative loss calculation were publi \npretation of photospheric absorption lines instead of 1D models. Since solar photospheric abundances are a vital reference point for many fields in astronomy the new abundances have caused a re-assessment of many previous works. The most dramatic has been in the study of the solar interior, where models had preThe CHIANTI atomic database 3 (Dere et al. 1997, 2019) provides an extensive compilation of critically assessed atomic data. In addition to the data it delivers software packages in both the IDL and Python languages for using this data and easy generation of loss functions using a variety of abundances and other input options. \nnote that for most ions radiative losses are \nsmall number of strong fine structure tran \nconfigurations, and so modeling these trans \nviously been in excellent agreement with the precise helioseisComputation of Q in the corona is then straightforward: \nmic measurements of parameters such as the sound speed profile. Oxygen is the dominant contributor to opacity in the region Q = -Λ ( T ) n e n H , (50) \natomic data is crucial to deriving accurate \nmates. Colgan et al. (2008)treatedentireconfi \nlevels in their calculations with no fine struc \nno attempt was made by the authors to asse \ntheir approach for the dominant low-lying con \nCHIANTI fine structure levels are used for \njust below the convection zone, and the revised abundance significantly a ff ected the solar structure in this region, breaking the excellent agreement with measurements. The crucial role of oxygen in the solar interior models has led to a number of di ff erent measurements and re-analyses of oxygen lines and photospheric models (e.g., Centeno & Socaswhere Λ ( T ) is computed using a 1D lookup table. The hydrogen density is easily computed from the mass density, and the electron density can be approximated accurately assuming full ionisation of both H and He at high temperatures. In the low temperature regime, where H i , He i , and He ii exist in significant amounts, one can employ precomputed tables of the electron density assuming LTE or coronal equilibrium. \nmost accurate atomic data for transitions be \nare chosen following assessments of the best a \nliterature. We are thus confident in the accurac \nradiative losses for the ion models used in the \n24. New IDL procedures \nThe parameters that are necessary to calculat \nsections and rate coe \nffi \ncients, r \nand ionization equilibria are now included \ndatabase. Accordingly, new Interactive Data \nfunctions \nioniz\\_cross \n, \nioniz\\_rate \n, \nmake\\_ioneq\\_all \nare also provided. \nioniz\\_cross \nreturns the ionization cross s \nfied ion and electron energy in eV. \nioniz\\_rate \nreturns the ionization rate coe \nfied ion and temperature in K. \nrecomb\\_rate \nreturns the recombination r \nspecified ion and temperature in K \nmake\\_ioneq\\_all \ncreates a new ionization e \nNavarro 2008; Meléndez & Asplund 2008; Ca ff au et al. 2008, and references therein) leading to oxygen abundances ranging between the value given in Grevesse & Sauval (1998) and that given in Grevesse et al. (2007). This makes it di ffi cult for a project such as CHIANTI which seeks to provide a single set of reference abundances for users. Our solution is to retain the In order to avoid contribution from the convection zone, photosphere, and chromosphere, one needs to multiply Q with a cutoff function that drives Q to zero in the lower atmosphere. Bifrost employs a soft cutoff function: exp( -P/P 0 ), with P the gas pressure and P 0 a typical pressure at the top of the chromosphere (Gudiksen et al. 2011). MURaM uses a hard cutoff at T=20,000 K (Rempel 2017). \nthe CHIANTI software, and to provide the Grevesse et al. (2007) \nabundances as an option that can be selected by the user. \nAlthough the change in the oxygen abundance is the most \nsignificant in the Grevesse et al. (2007) tabulation, it is also to \nbe noted that the neon and argon abundances have changed by \na similar amount. This is on account of the fact that neither of \nthese elements can be measured in the solar photosphere, and \nso they have usually been measured relative to oxygen by other \nmeans (spectroscopy of the solar upper atmosphere, solar ener- \ngetic particle measurements). \nCHIANTI users are encouraged to bear in mind the ongo- \ning debate over solar photospheric abundances and to assess the \nimpact on their analyses. \n23. Radiative loss rates \nFig. 11 Radiative losses in the transition region and corona in a radiation-MHD simulation computed with the MURaM code. Top: temperature; middle: mass density; bottom: -Q , i.e., a positive value means radiative cooling. Also note the strong corrugation of the chromosphere-TR boundary. Figure computed from a simulation by Danilovic et al. (2019). \n<!-- image --> \nThe radiative losses in the TR and corona tend to have sharp peak in the transition region owing to the quadratic density-dependence. Rempel (2017) noticed that the relatively large grid spacing of the simulations compared to the thickness of the TR can lead to inaccuracies in the computation of Q . He proposed a scheme using subgrid interpolation to improve the accuracy. \nA demonstration of Q and its relation to temperature and density is shown in Figure 11 for a simulation carried out with the MURaM code. The cooling is largest (note the logarithmic scale) in the transition region, just at the border between the TR and the chromosphere, where it typically is in the range of 0.1-1.0 W m -3 , and it quickly drops down to coronal values around 10 -4 - 10 -5 W m -3 . Zooming in reveals that the strongly-emitting layer is often only a few pixels wide. \n> \n00 \nLO \nLO \ni-D \na \nCO \nCTi \n704 \nVERNAZZA, AVRETT, AND LOESER \nVol. 45 \nFig. 49. - Net radiative cooling rates for Ca n, Mg n, H , H, and other constituents calculated from model C at 50.4 nm, all calculated from model C. In all the calculations we have carried out for various atoms and ions, namely, H, H ', neutral and ionized He, C, Si, Mg, Ca, and O, and neutral Fe, Al, Na, and B, these are the In Table 29 we list the integrated net cooling rates fïïdh for the Ca n and Mg n lines, La, and H', i.e., the area under each $ as a function oî h in Figure 49. Negative values of 0 in the temperature-minimum reFig. 12 Net radiative cooling in the 1D semi-empirical VAL3C model atmosphere. The cooling between z = 700 km and z = 2120 km in this model is dominated by five lines from Ca ii and two lines from Mg ii . At larger heights H i Ly α alone is the dominant radiative cooling agent. Adapted with permission TODO:ASK PERMISSION from Vernazza et al. (1981), copyright by AAS. \n<!-- image --> \nonly line and continuum rates that appear to exceed \n- 4 \n10 \nergs cm \n-3 \ns ' \n1 \nin absolute value. \nThe calculated net radiative cooling rates for the Ca n \ngion are assumed to be zero in each integral. Athay \n(1976, Table IX-1) estimates that the contributions to \ndh have the following order of importance: Balmer \ncontinuum, Ha, Ca n IR triplet, Mg n, Ca n K and H, \nFe i lines, and Na i D. Athay states, however, that these", 'and Mg n resonance Unes are quite sensitive to the convergence and internal consistency of our numerical 5 Radiative transfer in the chromosphere': "solution. Some of our preliminary solutions gave maxi- \nmum rates for these lines 2 to 3 times larger than those \nestimates are not based on detailed computations of the \nnet energy loss or gain in the bound-free continua. The \nshown in Figure 49. order we determine is Usted in Table 29. ' American Astronomical Society ¥ Provided by the NASA Astrophysics Data System The assumptions underpinning the methods for photospheric radiative transfer as presented in Sect. 3 are no longer valid in the chromosphere: the chromosphere has a low opacity except in a few strong spectral lines, in ultraviolet continua below 160 nm, and in (sub-)mm continua above 160 µ m. The lines that dominate the radiative energy exchange are the H&K and infrared triplet of Ca ii , the h&k lines of Mg ii , and H i Ly α (see Fig. 12). The opacity in these lines is severely underestimated in the construction of a bin-averaged opacity. The source function is no longer described by the Planck function, and the assumptions of coherent scattering or complete redistribution are very inaccurate for these strong lines, which should instead be modelled using partially coherent scattering (PRD). The ionisation balance of hydrogen and helium is out of equilbrium. \nProper inclusion of the radiative cooling in the chromosphere (even when ignoring non-equilibrium effects) thus involves solving the 3D non-LTE radiative statistical-equilbrium transfer problem including PRD. This is in principle possible using dedicated radiative transfer codes, but it is computationally expensive. A single solution to the problem for a single atom costs a CPU time of ∼ 10 s per grid cell (Sukhorukov and Leenaarts 2017), which is very large compared to the ∼ 5 µ s CPU time per grid cell per timestep for radiationMHD simulations (e.g., Gudiksen et al. 2011). Simplifications that speed up the computation are thus required. \nCarlsson and Leenaarts (2012) developed techniques to do so by describing the net effect of all the radiative transfer as a combination of (1) an optically thin radiative loss function which represents the local energy loss through radiation per atom in the right ionisation stage per electron, (2) the probability that this energy escapes the atmosphere, and (3) the fraction of atoms in the ionisation stage under consideration. These three factors must all be determined empirically because there are no obvious general physics-based approximations. \nThe method approximates the radiative loss per volume owing to species X in ionisation state m as \nQ X m = -L X m E X m ( τ ) n X m n X A X n H ρ n e ρ. (51) \nHere, L X m is the optically thin radiative loss function per electron and per particle of element X in ionisation stage m , E X m ( τ ) is the photon escape probability as function of some depth parameter τ , n Xm n X is the fraction of element X in ionisation stage m , and A X the abundance. \nThe quantities L X m , E X m , and n X m /n X are determined from a 1D radiationhydrodynamic simulation including non-equilibrium ionisation computed with the Radyn code (Carlsson and Stein 1992, 2002) for H i . For Ca ii and Mg ii they were determined from a 2D radiation-MHD simulation with Bifrost that provided the atmospheric structure and subsequent statistical equilibrium radiative transfer calculations including PRD using Multi3d (Leenaarts and Carlsson 2009). \nThe loss function L X m can be computed for each grid cell in the simulation by summing the net downward radiative rates multiplied with the energy difference of the transition, summed over all relevant transitions. The joint probability density function (JPDF) of L X m and gas temperature for Ca ii is shown in the top panel of Fig. 13. Radiative transfer effects make that L X m is no longer a unique function of T , but the red curve indicates an approximate fit. The ionisation degree can be computed from the atomic level populations in the calibrating simulations (see the bottom panel of Fig. 13). Again they are no longer clean functions of temperature, but the spread is minor and a sensible fit as function of temperature can be made. Finally, the empirical escape probability E X m is computed from the total radiative losses in the simulation, the radiative loss function and the ionisation degree. The middle panel of Fig. 13 shows E Ca ii with the vertical column mass as depth parameter. Again there is considerable spread in the JPDF. \nuchmorelikely \nP states than to \ntransition \nhe ground state \nmber of radia- \nstate through a \n30 A&A539, A39 (2012) \nhe original col- \n(4) still holds. \nlevels give a \nmes are \nsmaller \nstate. In con- \nnot caused by \nof the values \nlowest temper- \n(not visible in \ndip downwards. \nsses in lines and \ns lead to emit- \nhere as function \ncurves give the \nhere increased \norption of pho- \nexcitation pro- \nto Eq. (4) \nsible routes in \nthese processes \nr equations and \nempirical es- \nling as obtained \nnd the optically \neally we would \nction of optical \ne total cooling \nl depth scale to \nA&A539, A39 (2012) \nlate the escape \nationale behind \nnce lines is pro- \nion stage under \ncase for both \non of the atmo- \nother advantage \nhydrogen particle \nsolid), Lyman- \nβ \nontinuum (blue) \nto calculate \nan MHD simu- \nLyman lines \ns than hydrogen \nregion where \ned for calculat- \ngion stays at a \nlation \n2 \n. \nWe ex- \nsimulation and \noid most points \ne, the hydrogen \nnction. Instead, \nescape probability \nperatures above \nonal to the neu- \ned fit (red). Only \nincluded. \nlose to the total \n14 \ner spread, both \ncolumn density \n4 \n10 \n- \ncm \noptical depth at \ncape probabil- \n× \n)) start to break \nthe Lyman tran- \nline center and the \ng \n- \n1 \nFig. 4. \n<!-- image --> \nFig. 7. \nFig. 5. \nMg \ndepth at Lyman- \nLyman- \nand Mg \nNote that the adopted relations are not the averages of the \nto contribute and H- \nα \neventually dominates below 7 kK. The \nhydrogen optically thin radiation dominates over contributions from other elements below 32 kK. Figures 4, 5 show the probability density function of the total radiative losses per singly ionized calcium and magnesium atom, respectively, as function of temperature for the Bifrost simulation. This cooling was computed in detail us2 points shown in Figs. 6 -8 but the ratio between the average of the actual cooling and the average of the optically thin cooling such that points with large cooling get larger weight. The adopted fit may therefore look like a poor representation of the PDFs of Figs. 6-8. If these figures had only included the points with the largest cooling, the correspondence would have been Fig. 10. Probability density function of the fraction of Ca atoms in the form of Ca as function of temperature in the Bifrost test atmosphere between heights of 0.5 Mm and 2.0 Mm. Curves show the adopted fit to the PDF (red), the coronal approximation (green) and the coronal approximation with a two-level atom (blue). Fig. 13 Chromospheric radiative losses in Ca ii using empirically calibrated recipes. Top: JPDF of radiative loss function and temperature in the chromosphere of a radiation-MHD model. The blue curve is the coronal approximation, the red curve the adopted fit. Ca ii actually heats (meaning a negative loss function, not shown here because of the logarithmic axis) at low temperatures, and that is why the red curve appears a bad fit. Middle: escape probability, with the adopted fit in red. Bottom: fraction of all Ca as Ca ii . Red is the adopted fit, blue the ionisation balance under coronal equilibrium conditions and green the balance assuming LTE. Adapted with permission from Carlsson and Leenaarts (2012). \nis high to rather \nattering proba- \nwards in the at- \nhave an escape \nith low escape \nway on the \nradiation field. \nn mass density, \nas the adopted \nto the radiative \nhydrogen atom \nlculate the aver- \ner space for the \ncooling accord- \nnormalization \nmore obvious. \ning the radiative transfer code Multi3d (Leenaarts & Carlsson \n2009)assumingstatisticalequilibriumandplane-parallelgeom- \netry with each column in the snapshot treated as an independent \n1D atmosphere. \n4.3. Ionizationfraction \nThe probability density function is close to the total col- \nlisional excitation curve for both Ca \nIn Sects. 4.1-4.2 we derived expressions for the radiative losses \nper atom in the relevant ionization stage per electron. In order \nand Mg \n. \nIn the case \nto convert these losses to actual losses per volume, the frac- \nof \nCa \nCa \n3d \n2 \nthis \nmay be a bit surprising at first glance. The \nD doublet is metastable and we do not include tran- \nsitions from it to the ground state. This implies Eq. (2) does not \ntion of atoms in the ionization stage under study (H \n, Ca \nand \nFig. 6. \nbeen excluded. \nCa \nFig. 14. \nCooling as function of column mass from hydrogen transitions \nat \nt \n= \n3170 s in the RADYN simulation. Total cooling from the detailed \nFig. 16. As Fig. 15 but for the low-mid chromosphere. detailed solution with the approximate one and not as pictures that at each location either reduces to Q chr( z ) = 2 πχ ( z ) ∫ 1 -1 I ( z ,µ Computing this chromospheric radiative loss function in a radiation-MHD simulation is fast and simple; it only involves computing the values of the fitted quantities from 1D lookup tables. The electron density can for example be determined from a 2D lookup table computed assuming LTE. This is not particularly accurate but simple. A more realistic electron density can be obtained by computing the non-equilibrium ionisation balance of hydrogen and/or helium (see Sect. 6). \n<!-- image --> \nemissivity associated with this loss is \nη \n= \n- \nQ \ncor \n4 \nπ \n· \nBecause the coronal radiative losses are integrated o \n31 quency one has to choose a single representative opacity \npute the absorption. A decent choice is to use the \nionization edge of the ground state of neutral helium ( \nby \nα \n). The opacity \nχ \nis then given by \nχ \n= \nα \nN \nwhere \nN \nHeI is the number density of neutral helium, \nN \nnumber density of helium, \nA \nHe is the abundance of he \native to hydrogen, \nN \nH is the total number density of h \nparticles and \nquantity \nN \nHeI \nN \nH \nρ \n/ \nN \nHe can be pre-computed from a \nputation as is done for H, Ca and Mg in Sect. 4.3. \nThe most accurate method for computing the abso \ncoronal radiation in the chromosphere is to compute the \nation field resulting from the coronal emissivity and the \ntion using the radiative transfer equation: \n- \nχ \nI \nthe intensity and \ns \nthe geometrical path length \nray. The solution can be computed using standard long \ncharacteristics techniques in decomposed domains ( \net al. 2006; Hayek et al. 2010). Once the intensity is kn \nFig. 15. Average cooling as function of height in the Bifrost test atmosphere. Total cooling (black), H (turquoise), Mg (blue) and Ca (red). According to the detailed calculations (solid) and according to the recipes (dashed). d I d s = η with I Fig. 14 Test of the validity of tabuled chromospheric radiative losses. The curves show the horizontally averaged radiative cooling (i.e., -Q/ρ as a function of height in a 2D Bifrost radiation-MHD simulation. The colored lines show the results computed from a 2D nonLTE radiative transfer computation (solid) and the approximate recipe (dashed) are shown. For hydrogen there is no detailed radiative transfer computation because the recipe was computed from a 1D simulation with the Radyn code. The black solid land dashed lines show the total cooling from all three elements combined. Adapted with permission from Carlsson and Leenaarts (2012). \nheating rate can be directly computed as \nχ \nQ chr = with Ω Figure 14 displays a comparison of this simple recipe with a detailed calculation. The recipe does a surprisingly good job in reproducing the average cooling given the simplicity of the method. However, in individual grid cells large errors in Q might occur, caused by the spread of values around the red curves in Fig. 13. \n∫ \nΩ \n- \nthe solid angle. \nThe method described above is accurate, but \nfaster method can be obtained by treating each colum \nMHD simulation as a plane-parallel atmosphere and a \nη \nor \nχ \nis zero. For each \nquantities then become only a function of height \nz \nand \nµ \n= \n4 \nπχ \nJ \n)d \n1D \n, \nthe cosine of the angle with respect to the ver \nIf we furthermore assume the emitting region is lo \ntop of the absorbing region, we can ignore rays that p \nward from the interior of the star in the integral of \nThe intensity that impinges on the top of the absorbin \nfor inward pointing rays is simply the integral of the e \nof the mean chromospheric energy balance. 7. Heating from incident radiation field Part of the optically thin radiative losses from the corona escapes outwards, but an equal amount of energy is emitted towards the with µ A39, page 8 of 10 Carlsson and Leenaarts (2012) also describe techniques to model the absorption by the chromosphere of radiation emitted in the corona. Half of this radiation is emitted downwards and the majority will be absorbed in the chromosphere. The corona emits mainly in the far UV regime, and the dominant source of extinction at these wavelengths are the H i , He i , and He ii continua. The coronal loss function is a frequency-integrated quantity (see Sect. 4), so in order to model the extinction one has to estimate a representative opacity. Carlsson and Leenaarts (2012) propose to use the opacity at the edge of the He i continuum at 50 nm. The contribution to the flux divergence is then given by \nQ abs = 4 πκ HeI ρJ cor , (52) \nI \nd \nΩ \nη \n= \n4 \nπχ \nJ \n3D \n- \nη, \nis a constant dependent on the abundan \n= \nαρ \nN \nH \nρ \nA \nN \nHeI \nN \nHe \n, \nα \nHeI \nHe \nwith κ HeI the representative opacity. The angle-averaged radiation field J cor is computed from the transfer equation: \nd I d s = η -κ HeI ρI, (53) \nwith the emissivity given by η = -Q cor / 4 π . The quantity Q cor is the coronal loss function as defined in Eq. (50). In other words: it is assumed that only the corona contributes to the production of photons, the chromosphere can only absorb these photons, and scattering is ignored. \nBecause most radiation-MHD codes already include a 3D radiative transfer module used for the photospheric radiation, one can relatively easily implement the absorption of coronal radiation at a very modest increase in computation time. Simpler 1D recipes that ignore the spreading of radiation in the horizontal directions are also possible. They offer little else besides a marginal increase in computation speed, and can lead to spuriously large heating when coronal radiation emitted by a small localized source is forced to be absorbed in a single vertical column. \nThe population of neutral helium in the ground state needed to compute κ HeI can be computed assuming LTE populations without a too large error. Computing it taking into account non-equilibrium ionisation is more accurate but much slower (see Sect. 6). \nFigure 15 gives an example of how the various approximations for radiative cooling act in different atmospheric regimes. The photospheric cooling rate per mass is largest in the photosphere, but has a small contribution up into the chromosphere. Coronal losses are relevant throughout the entire corona, but are largest in the transition region. The chromospheric losses are largest just below the transition region owing to Ly α cooling (see also Fig. 12). Modest cooling owing to Ca ii and Mg ii lines and heating from the absorption of coronal radiation happens somewhat deeper in the chromosphere. The lower-right panel, which combines photospheric, chromospheric, and coronal losses, clearly shows that the largest radiative losses per mass unit occur in the transition region.", '6 The equation of state and non-equilibrium ionisation': "Non-equilibrium radiative transfer and non-equilibrium ionisation play a role in the solar atmosphere wherever magneto-hydrodynamic timescales are short compared to timescales on which an atomic system reacts to changes. The combined continuity and rate equation for the level population n i in level i of an atomic species with N energy levels is given by: \nglyph[negationslash] \nglyph[negationslash] \n∂n i ∂t + ∇· ( n i v ) = N ∑ j =1 ,j = i n j P ji -n i N ∑ j,j = i P ij , (54) \nwhere P ij is the rate coefficient for transitions from level i to level j . If one ignores the advection term and assumes that all the P ij are constant in time \nFig. 15 Example of chromospheric radiative losses and gains in comparison to the photospheric and coronal losses and gains in a 3D simulation computed with the Bifrost code. Top row: temperature and density in a vertical slice through the simulation. The panel labeled Q photo shows the radiative losses per mass unit computed with the method described in Sect. 3; Q chromo displays the chromospheric losses and absorption of coronal radiation per mass unit using the methods from Carlsson and Leenaarts (2012); Q corona shows the losses per mass unit computed as described in Sect. 4. The panel labeled Q total shows the sum of the previous three panels, i.e., all radiative losses and gains in the simulation. Brown color indicates cooling, blue color represents heating. \n<!-- image --> \nthen the equations for all levels together form a set of coupled first-order differential equations \n∂ n ∂t = P n , (55) \nwith n the vector of level populations and P the rate coefficient matrix. This system has the solution: \nn ( t ) = N ∑ i =1 c i a i e λ i t , (56) \nwhere a i are the eigenvectors of the rate matrix with corresponding eigenvalues λ i , and c i are constants that depend on the initial condition. One of the eigenvalues is zero, and the corresponding eigenvector represents the equilibrium solution. All other eigenvalues are negative and have an absolute value smaller than one. If the system starts away from the equilibrium solution, then \nilibrium is \nþ \nR \n21NRB \nÞ \n: \n34 CARLSSON & STEIN \nð \n14 \nÞ \nis small compared to \ns individually, then the \nradiative rates will make \non timescale given by \ntransitions can either \n(if NRB \n> \n0) the relax- \non timescale (at each \nnumerical simulation. \ny: The value of the hydro- \na given time step and \non timescale calcula- \nnsistent with this state of \nsolving the equations of \nthe initial population \nhisatmospherewasthen \nrature by 1% throughout. \nthis perturbed state were \nistical equilibrium giv- \n1 \nÞ \n. \nThe time evolution \n/C28 \nrelax \ncal simulation, from the \nVol. 572 \n<!-- image --> \norders of magnitude too short. Eigenvalues calculated from a rate matrix \nof protons at a given the full rate equations, utionwascastinatwoon timescale was calfit to ¼/C0 t : ð 15 Þ Fig. 6. -Relaxation timescale as function of column mass. The numerically determined relaxation timescale is given as the thick solid line. Timescales determined from eigenvalues of the rate matrix are also shown for several cases: full rate matrix ( dotted line ), collisions only ( dot-dashed line ), Lyman transitions in detailed balance ( thin solid line ), and Ly /C11 treated with aconstantnetradiativebracket( dashed line ). The electron density is given as the thick dashed curve. The ionization/recombination timescale becomes very long in the chromosphere. Eigenvalues calculated from a rate matrix with all the radiative and collisional rates give a timescale several Fig. 16 Relaxation timescale for hydrogen ionisation as function of column mass in a 1D radiation hydrodynamics simulation computed with the RADYN code. The numerically determined relaxation timescale is given as the thick solid line. Timescales determined from eigenvalues of the rate matrix are also shown for several cases: full rate matrix (dotted line), collisions only (dot-dashed line), Lyman transitions in detailed balance (thin solid line), and Ly α treated with a constant net radiative bracket (dashed line). The electron density is given as the thick dashed curve. Adapted with permission from Carlsson and Stein (2002), copyright by AAS. \nwith all Lyman radiative transitions in detailed balance except for Ly \n/C11 \n, \nionization/recomwhich is included using its net radiative bracket, give a timescale that closely matches the numerical result. it will evolve towards equilibrium on a characteristic timescale given by \nis shown in Figure 6 \ncreases outward from the \nin the midchromosphere \nτ = 1 min( | λ i | ) , λ i = 0 . (57) \nglyph[negationslash] \nhebaseofthetransition of the rate matrix, Pij transition rate per atom the processes controlcan be represented The numerically determined relaxation timescale is compared with the timescale from the eigenvalue calculation, which is the inverse of the smallest (in absolute value) nonzero eigenvalue, in Figure 6. We have calculated the eigenvalues using several di ff erent assumptions A detailed discussion of the time dependence of atomic level populations is given in Judge (2005). One of the results from that paper is that the slowest time scale tend to be associated with processes that have small net rate coefficients. The smallest rate coefficients are often associated with ionisation and recombination processes (as well as transitions involving metastable levels). \n/C21 \ni t \n: \nlevel 6 is the continuum. \nponding to the \ni \nth eigen- \non the initial conditions. \neigenvector is the equili- \nare all negative, since the \nequilibrium value. \n; \nð \n16 about the rate matrix. Note, first of all, that when all the processes are included in the rate matrix the timescale obtained from the eigenvalues ( dotted line ) is orders of An illustrative solution for a two-level atom was derived in Carlsson and Stein (2002). The equilibration timescale is given by \nÞ \nð 17 Þ same general pattern as the numerical timescale but is generally slightly larger. When radiative rates are added, but with all the Lyman transitions assumed to be in detailed balance, the timescale from the eigenvalues of the rate matrix ( thin solid line )reproducethenumerical with C 12 and C 21 the upward and downward collisional rate coefficient, R 12 and R 21 the radiative rate coefficients and the term between square brackets the net radiative bracket. The collisional coefficients depend linearly (or quadratically in case of collisional recombination) on the electron density and \nmagnitude smaller than found in the numerical solution ( thick solid line ). The timescale obtained from the rate matrix with only collisional rates ( dot-dashed line )hasthe τ = 1 /( C 21 + C 12 + R 21 [ 1 -n 1 n 2 R 12 R 21 ]) . (58) \ntimescale up to the peak in the midchromosphere. This \nindicates that in this region the large Lyman radiative \nrates are in fact changing with time so as to maintain \nnearly detailed balance as the populations change. The \ngeneral behavior of the relaxation time is thus controlled \nby the collisional processes. The timescale increases from \nthe photosphere to the midchromosphere, where the elec- \ntron density has a minimum approximately as the inverse \nof the collisional recombination rate ( \nn \n/C0 \n2 \ne \n). \nFrom the \nmidchromosphere to the transition region the timescale \n/ \ntions, \nexponentially on temperature. For hydrogen one thus expects a maximum in the chromosphere where the temperature and density are relatively low and strong lines are close to detailed balance so that the net radiative bracket is small. \nCarlsson and Stein (2002) did a detailed calculation of this timescale for hydrogen and found that the timescale can be as long as 10 5 s in the chromosphere (see Fig. 16), and that this is the timescale on which ionisation equilibrium is established. \nA similar calculation for helium was done for helium by Golding et al. (2014), finding timescales of the order 10 2 s to 10 3 s in the chromosphere and transition region, again associated with the ionisation equilibrium \nThese timescale are larger than the hydrodynamical timescales in the chromosphere and transition region. This has severe consequences for the treatment of radiation hydrodynamics. Because hydrogen and helium are majority species they do not only contribute to the radiative flux divergence, but their ionisation state also influences the temperature, pressure and electron density. The assumption that the equation of state (EOS) can be computed assuming LTE is no longer accurate. \nThe relaxation timescale itself depends on the radiation field, so that a proper treatment of the EOS now requires solution of Eq. (54) together with an equation for charge conservation \nn e = ∑ i,j,k ( j -1) n ijk , (59) \nand energy conservation \ne = 3 2 k B T ∑ i,j,k n ijk + n e + ∑ i,j,k n ijk E ijk . (60) \nHere n ijk are the atomic level populations of species i in ionisation state j and excitation state k and E ijk is the sum of the dissociation, ionization and excitation energy of a particle in state ijk . The rate coefficients P ij contain the radiation field so that the transport equation (Eq. (14)) must be solved too. As stated before, this is too computationally expensive to be of use in 3D simulations. \nSollum (1999) developed approximations to the chromospheric radiation field in hydrogen transitions based on detailed 1D calculations with the Radyn code. He found that the angle-averaged and profile-function-averaged radiation field in a given transition can be approximated by a constant above a certain height in the chromosphere, and by the local Planck function at larger depth. In between these two limits he specifies a smooth transition. His recipes allow for an extreme simplification because the solution of the coupled set of Eqs. (54), (59), and (60) can then be solved without having to solve the transfer equation. A limitation of his method is that the Lymanα line was set in detailed radiative balance. This limitation makes the solution less accurate in the very upper chromosphere and transition region. \n630 \nJ. Leenaarts et al.: Non-equilibrium hydrogen ionization in 2D simulations of the solar atmosphere \nFig. 3. Time slices of the gas temperature ( first row ), the ionization degree of hydrogen ( second row ), number density of hydrogen in the n = 2level ( third row ), and the n = 2columndensity( fourth row )inamagneticelement( left-hand column ) and in the internetwork ( right-hand column ). The upper-left magnetic element panel shows dynamic fibrils pushing the corona upward with 3 min periodicity. The upper-right internetwork panel shows rather unstructured shocks and a slowly varying height of the transition region. The snapshot used in Figs. 1 and 2 is indicated by a black dotted line. of about 2500 K. These low values result from the reverse process: over-ionization compared with LTE. More energy remains stored as ionization energy, leaving less kinetic energy for the radiative transfer uses the group-mean opacity, scattering probability and Planck function based on the corresponding LTE (or coronal equilibrium) temperature and electron density. The raFig. 17 Temperature and hydrogen ionisation degree n H i / ( n H i + n H ii ) as function of time in two columns of a 2D radiation-MHD simulation that includes non-equilibrium ionisation of hydrogen. Left-hand panel: a column in a magnetic element. Right-hand panel: a column with weak magnetic field. The left-hand panel shows regular shock waves with 3-minute period, while the right-hand panel shows a more irregular temperature structure. In both cases the ionisation degree does not follow the temperature structure. Instead it is rather constant. Adapted with permission from Leenaarts et al. (2007), copyright by ESO. \n<!-- image --> \ngas particles. \ndiative cooling in the chromosphere and transition region, where \ndeviations from equilibrium are largest, is thus inconsistent with \nFig. 3. Time slices of the gas temperature ( first row ), the ionization degree of hydrogen ( second row ), number density of hydrogen in the n = 2level ( third row ), and the n = 2columndensity( fourth row )inamagneticelement( left-hand column ) and in the internetwork ( right-hand column ). The upper-left magnetic element panel shows dynamic fibrils pushing the corona upward with 3 min periodicity. The upper-right internetwork panel shows rather unstructured shocks and a slowly varying height of the transition region. The snapshot used in Figs. 1 and 2 is indicated by a black dotted line. 4. Discussion and conclusions 4.1. Limitationsofthesimulation Our implementation of non-equilibrium hydrogen ionization has various limitations. First, the assumption that all Lyman transitions are in detailed balance is justified up to the transition region (Sollum 1999). However, the transition region is optically thin in most Lyman features, requiring detailed radiative transfer modeling to evaluate their influence on the hydrogen populations. Second, the multi-group radiative transfer within the simthe non-equilibrium temperature and electron density as computed in the simulation. Third, the cool parts of the simulation chromosphere often reach the limiting temperature of 2400 K allowed in the simulation. It is not clear how low the actual chromospheric minima may reach because radiative heating in the hydrogen continua and other strong spectral features is not taken into account in the simulation, only their radiative cooling. 4.2. Discussion His method was implemented in the Oslo Stagger Code by Leenaarts et al. (2007), and used to perform a 2D radiation-MHD simulation of the solar atmosphere with a non-equilibrium EOS. Helium was still treated assuming LTE populations. The temperature and hydrogen ionisation degree in this simulation is demonstrated in Fig. 17. As a consequence of the long equilibration timescale, the ionisation degree in the chromosphere does not follow the temperature. The ionisation degree is instead rather constant in time for a given Lagrangian fluid element. \nOur implementation of non-equilibrium hydrogen ionization has various limitations. First, the assumption that all Lyman transitions are in detailed balance is justified up to the transition region (Sollum 1999). However, the transition region is optically thin in most Lyman features, requiring detailed radiative transfer modeling to evaluate their influence on the hydrogen populations. Second, the multi-group radiative transfer within the simulation, which sets the radiative cooling and heating, employs LTE ionization. For given internal energy and mass density, the lation. It is not clear how low the actual chromospheric minima may reach because radiative heating in the hydrogen continua and other strong spectral features is not taken into account in the simulation, only their radiative cooling. 4.2. Discussion From the analysis of our simulation we obtain the following picture. The internetwork chromosphere is irregularly pervaded These authors also implemented a semblance of radiative transfer in the Ly α line. They assigned a single-bin source function based on the net radiative rate and representative opacity to the line, and used those to solve the transfer equation ignoring scattering. The resulting radiation field is then used to add an upward radiative rate coefficient in Eq. (54). In this way the recipes by Sollum (1999) were extended to the very upper chromosphere, but in a rather crude fashion. \nof about 2500 K. These low values result from the reverse process: over-ionization compared with LTE. More energy remains stored as ionization energy, leaving less kinetic energy for the gas particles. 4. Discussion and conclusions 4.1. Limitationsofthesimulation radiative transfer uses the group-mean opacity, scattering probability and Planck function based on the corresponding LTE (or coronal equilibrium) temperature and electron density. The radiative cooling in the chromosphere and transition region, where deviations from equilibrium are largest, is thus inconsistent with the non-equilibrium temperature and electron density as computed in the simulation. Third, the cool parts of the simulation chromosphere often reach the limiting temperature of 2400 K allowed in the simuulation, which sets the radiative cooling and heating, employs LTE ionization. For given internal energy and mass density, the From the analysis of our simulation we obtain the following picture. The internetwork chromosphere is irregularly pervaded Golding et al. (2016) developed approximations for the radiation field in helium transitions. The physics of helium ionisation is somewhat more complicated than for hydrogen because ionisation in the chromosphere is mainly driven by UV radiation produced in the corona. The resulting recipes take this into account, but require the solution of the transfer equation in seven radiation bins in order to approximate the radiation field in the continua of helium as well as the He ii 30.4 nm line. \nThe increased realism of an EOS that includes non-equilibrium ionisation comes at a rather steep price in computational efficiency: A simulation with a \nnon-equilibrium EOS is around three (hydrogen only) to five times (hydrogen and helium) slower than a similar simulation with an LTE EOS. \nNon-equilibrium ionisation has strong consequences on the structure of the chromosphere and transition region. Most importantly, ionisation can no longer function as efficiently as an energy buffer when the internal energy density of a gas parcel changes. \nIn LTE energy must be expended to ionise hydrogen and helium if the internal energy density is increased before the temperature can rise. Likewise, a decrease in internal energy leads to an instantaneous transfer of ionisation energy to thermal energy, slowing down the temperature decrease. This leads to characteristic bands of 'preferred temperatures' in joint probability density functions of radiation-MHD simulations assuming LTE (see Fig. 18). The temperature of these bands are associated with the temperatures where H i , He i , and He ii ionise according to the Saha-Boltzmann equations. \nIf the ionisation balance is computed in non-equilibrium, and the ionisation/recombination timescale is long, then an increase in internal energy leads directly to a temperature increase. Temperature decreases are also stronger than in LTE because ionisation energy cannot be released quickly enough to counteract cooling. The bands of preferred temperature disappear, and the gas in the chromosphere behaves somewhat like and ideal gas. A clear example of this effect is the increased the amplitude of the temperature jump in acoustic shocks (see Carlsson and Stein 2002; Leenaarts et al. 2007). \nLow-temperature areas in the chromosphere have a higher electron density than predicted by the Saha-Boltzmann equations, and the reverse is true for high-temperature areas. This has en effect on the formation of chromospheric lines because the source function couples to the local temperature mainly through collisions with electrons. Non-equilibrium ionisation is also expected to have an effect on the efficiency of heating through ambipolar diffusion (e.g., Mart'ınez-Sykora et al. 2017; Khomenko et al. 2018).", '7 Other developments': "7.1 Fast approxmiate radiative transfer in the photosphere \nAbbett and Fisher (2012) propose a method for quick evaluation of the radiative losses in the photosphere as an alternative to the methods described in Sect. 3. They assume that each column in a simulation can be treated as an independent plane-parallel atmosphere. The angle-averaged radiation field at vertical optical depth τ ν in a column is then given by: \nJ ν ( τ ν ) = 1 2 ∫ ∞ 0 S ν ( τ ' ν ) E 1 ( | τ ν -τ ' ν | ) d τ ' ν . (61) \nThe Astrophysical Journal, \n817:125 \n( \n13pp \n) \n, 2016 February 1 \nGolding, Leenaarts, & Carlsson \nFigure 2 \n<!-- image --> \nFigure 8. \nJoint probability density function of height and logarithmic temperature. The \n/uniFB01 \ngure includes data from the time interval 1000 \n- \n3000 s. The three horizontal \nplateaus ( at 6, 10, and 22 kK ) in the LTE simulation indicate preferred temperatures when using the LTE equation of state. These temperatures are associated with the LTE ionization of H I, He I, and He II. The plateaus vanish when we introduce non-equilibrium hydrogen and helium ionization. Fig. 18 Joint probability density functions of height and temperature in four radiationMHD simulations of the solar atmosphere. The simulations differ in their treatment of the equation of state. LTE: LTE equation of state; HION: hydrogen in non-equilibrium, helium in LTE, hydrogen Lyman transitions in detailed balance. LYA-HION: as HION but with radiative transfer in Lyman transitions; HELIUM: both hydrogen and helium in nonequilibrium. The three horizontal plateaus (at 6, 10, and 22 kK) in the LTE simulation indicate preferred temperatures when using the LTE equation of state. These temperatures are associated with the LTE ionisation of H I, He I, and He II. The plateaus vanish when we introduce non-equilibrium hydrogen and helium ionisation. Adapted with permission from Golding et al. (2016). 14 \nFigure 9. \nDifferential \n1000 \n- \n3000 s. The HELIUM run DEM does not have a bulge associated with \nthe preferred temperature at \nT \n/uniF0A0 \nvalue than the other three runs in the temperature range 11 \nFigure 1 Left: Temperature at the RADMHD model photosphere. Right: Magnetic-field lines threading the low atmosphere over a small subdomain (the box in the left frame indicates the approximate size of the corresponding subdomain). The gray slice indicates the average height of the visible surface. The domain spans 24 × 24 × 12 Mm 3 at a resolution of 512 × 512 × 256. Fig. 19 Temperature at a constant height in the photosphere of a 3D simulation computed with the RADMHD code. The panel spans a size of 24 × 24 Mm 2 and has a grid spacing of 21.3 km. The convection pattern strongly resembles the result of simulations that model radiation with higher fidelity, despite the strong simplifications in the computation of the radiative losses. Adapted with permission from Abbett and Fisher (2012), copyright by Springer. \n<!-- image --> \nTemperature in the \nRADMHD \nlow chromosphere showing a reverse granulation pattern. Lighter \n(darker) colors indicate hotter (cooler) temperatures. In the models, this occurs because the radiative cooling \ndiminishes with height, and the \np \nv \nwork of converging and diverging flows above the intergranular lanes \nbegins to dominate. The horizontal slice spans 21 \n× \n12 Mm \n2 \nat a resolution of 448 \n× \n256. \n∇ \n· \nOccurrence of helium ion fractions as a function of temperature for \nHe II in the top panel, and He III in the bottom panel. Each column has been \nnormalized to increase readability. Median values and CHIANTI values are \nW.P. Abbett, G.H. Fisher \nThe first exponential integral E 1 ( x ) peaks sharply at x = 0, so that this expression can be approximated as \nJ ν ( τ ν ) ≈ 1 2 S ν ( τ ν ) ∫ ∞ 0 E 1 ( | τ ν -τ ' ν | ) d τ ' ν (62) \n≈ S ν ( τ ν ) ( 1 -E 2 ( τ ν ) 2 ) (63) \nThis result is inserted into the equation for the radiative cooling (Eq. (11)) to yield \nIf one assumes an LTE source function, then the frequency integral can be approximated using a reasoning similar to the one given in Sect.3.2, and together with some additional assumptions one arrives at the final result: \nQ ≈ -2 πρ ∫ ∞ 0 κ ν S ν E 2 ( τ ν ) d ν. (64) \nQ ≈ -2 κ B ρσT 4 E 2 ( τ B ) , (65) \nwhere κ B is the Planck-averaged opacity (see Sect. 3.2), τ B the optical depth computed from this opacity, and σ is the Stefan-Boltzmann constant. With this scheme, computation of the radiative heating only requires a simple 2D table lookup to get κ B , and a column-by-column integration over depth to compute τ B . \nFigure 19 shows the resulting temperature structure in the photosphere in a simulation where the radiative losses are computed in this simplified fashion. The method is simple and extremely fast and is well suited for problems that do not require very accurate radiative losses, but nevertheless want the radiative losses to drive reasonably realistic-looking convection.", '7.2 Escape probability method': 'An interesting new method that speeds up non-equilibrium radiative transfer was proposed by Judge (2017). The major time consuming task in radiative transfer is the evaluation of the formal solution for all required frequencies and angles in order to construct the angle-averaged and frequency-averaged radiation field \n¯ J = 1 4 π ∫ ∞ 0 ∫ Ω I ν d Ω d ν, (66) \nwhich appears in the radiative rate coefficients of Eq. (54). \nJudge (2017) suggests to compute ¯ J from the equation \nd d τ ( S -¯ J ) = q 1 / 2 d d τ ( q 1 / 2 S ) , (67) \n(see Frisch and Frisch 1975; Hummer and Rybicki 1982), where τ is a vertical optical depth parameter and the function q an escape probability function that only depends on the vertical optical depth. This equation is approximate only: \nthe main approximations are that the source function varies only slowly over an optical path length and that horizontal structure in the atmosphere can be ignored. \nThe big advantage of using Eq. (67) is that it replaces the repeated evaluations of the transfer equation in order to compute ¯ J in a transition with the solution of a single integral along vertical columns in the atmosphere. Solving the statistical equilibrium non-LTE radiative transfer problem problem in a MURaM test atmosphere is ∼ 100 times faster than the full method. \nJudge (2017) shows that this method can also be used to solve non-equilibrium problems (i.e, it can be used to simultaneously solve Eqs. (54), (59), and (60)). Radiation-MHD simulations using this method to compute chromospheric radiative energy exchange or the equation state have so far not been reported on. It would be very interesting to see whether the method is fast enough to be used in practice and how it compares to the methods described in Sect. 6.', '8 Conclusions and outlook': "In this review I presented the most commonly used methods to approximate the transfer of energy between solar plasma and the radiation field in radiationMHD simulations. The theory and methods for computing radiative energy exchange in the photosphere are perhaps the most well-developed and wellstudied. They are also the most accurate: Pereira et al. (2013) compared a 3D hydrodynamics model of the upper solar convection zone that employed accurate abundances and opacity calculations, together with multi-group LTE radiative transfer with 11 groups. The resulting model reproduces the observed continuum center-to-limb intensity variation, the absolute flux spectrum, the wings of hydrogen lines and the distribution of continuum intensities caused by granulation to a high degree of precision. It seems therefore safe to say that we can model photospheric radiative energy exchange sufficiently accurate for almost all purposes. \nRadiative losses in the transition region and corona are much less accurate when using the standard method of an optically loss curve computed from statistical equilibrium, no absorption of radiation and a fixed set of abundances as described in Sect. 4. Inaccuracies caused by ignoring non-equilibrium ionisation effects appear to be largest in the transition region and during solar flares, and might well lead to a factor two error in the value of the loss function Λ ( T ). The choice of abundance has a smaller effect in the transition region, but can lead to differences up to a factor three at higher temperatures in the corona. A systematic test of these effects in 3D radiation-MHD simulations has so far not been done. \nRadiative transfer and non-equilibrium ionisation in the chromosphere is probably the least studied and the methods described in Sections 5 and 6 are likely the most inaccurate of all method used in the photosphere-chromospherecorona system. A long sequence of approximations is needed to order to make the problem computationally feasible. Testing the accuracy of all approxima- \nFig. 20 Brightness temperature in the nominal line core of H α in a simulated active region. The radiation-MHD simulation was performed with the MURaM code by Danilovic et al. (2019). The simulation used a single-bin LTE treatment of chromospheric radiative transfer. Nevertheless, the simulation resembles observations rather well. The 3D non-LTE radiative transfer calculation used to obtain the H α intensity was done with the Multi3d code. \n<!-- image --> \ntions combined can only be done in 1D geometry (using RADYN or similar codes), and has so far not been done. A particular worry is the crude way in which the radiative transfer in the Lyα line is implemented in the nonequilibrium ionisation method, \nInterestingly, the way how chromospheric radiative transfer is implemented, and the accuracy of the method used seems however to have only a minor effect on the overall structure of the simulated chromospheres. A MURaM simulation of active regions that only contains single-bin (gray) LTE radiative transfer and a coronal loss function produce chromospheres whose structure resembles the real chromosphere (see for example Bjørgen et al. (2019) and Fig. 20). \nI argue that this is a consequence of the physics of the chromosphere in the MHD approximation. Thermal conduction in the chromosphere is not efficient, and the only way a Lagrangian fluid element can lose energy is through radiation. In a chromosphere that is in a statistically stable state, the dissipation of non-thermal energy into heating and the radiative cooling must be in balance in a volume and time-integrated sense. An increase in non-thermal energy deposition must be accompanied by an increase in radiative losses. In other words: radiative losses are a reaction to heating, and the chromosphere will adapt its thermodynamic state until heating and cooling are in balance again. \nHere is the catch: the radiative losses are very sensitive to temperature, but the mechanisms that dissipate non-thermal energy are not. The radiative losses assuming LTE scale as T 4 , while the radiative losses in the coronal equilibrium approximation scale exponentially with temperature under chromospheric conditions (see Fig. 10). The non-LTE chromospheric loss function will lie between these two extremes most of the time. \nIn the MHD approximation there are only two mechanisms that irreversibly convert non-thermal energy into heat: viscosity and electrical resistance. The first one scales as T 1 / 2 (assuming a dilute gas of rigid elastic spheres), while the second scales as ln T/T -3 / 2 (assuming Spitzer resistivity). In practice the viscosity and resistance are replaced by numerical terms that are independent of temperature in almost all numerical simulations. \nWhen the description of the chromospheric radiative losses or the equation of state is not correct, then the simulation will change the temperature compared to the correct solution until balance between energy gains and losses is achieved again. The steep temperature dependence of the radiative losses makes the temperature change modest (say 1000 K-2000 K if our methods are not too bad), but the effect on the non-thermal energy dissipation rate is small. The result is a chromosphere with the wrong temperature, but almost correct dissipation, density and velocity structure. \nIt thus seems that one can study certain aspects of chromospheric physics without a too complicated treatment of the radiative losses and equation of state 4 . This does however not mean that the work described in Sections 5 and 6 can be ignored. The only proper way to validate models is to compute the various diagnostics (line profiles and continua) and compare to observations. The emergent intensities in the diagnostics sensitively depend on the temperature and electron density. Comparison of models with observations (such as in Leenaarts et al. 2013; Bjørgen et al. 2018) show that current models are not getting the intensities right, and that they need to be refined. The treatment of chromospheric radiative energy exchange is very likely one of the aspects in need of further improvement. \nThe reverse comparison is also true: inferred model atmospheres generated through non-LTE inversions of observations (e.g., de la Cruz Rodr'ıguez and van Noort 2017; de la Cruz Rodr'ıguez et al. 2019) can only be used to constrain physical models of the chromosphere if we are sure that the treatment of radiation in the models is done sufficiently accurately. \nSo what can be improved? I propose the following non-exhaustive list: \n- -It would be interesting to test, and if necessary improve, the accuracy of Skartlien's multi-bin radiative transfer with scattering in the mid and upper chromosphere. If successful, this would increase the accuracy compared to the radiative loss tables of Carlsson and Leenaarts (2012), which ignore 3D effects.\n- -The tables for Ca ii and Mg ii of Carlsson and Leenaarts (2012) should be updated. They were calibrated on a single 2D radiation-MHD simulation with a weak magnetic field using 1.5D radiative transfer. A much larger variety of models is available now, and 3D non-LTE radiative including PRD is now possible (Sukhorukov and Leenaarts 2017).\n- -The accuracy of the non-equilibrium ionisation methods should be more critically assessed. Comparison against the full physics is only possible in 1D but this will already give more insights in the possible deficiencies of the model, and in particular of the treatment of Ly α .\n- -The escape probability method for approximating the chromospheric radiation field discussed in Sect. 7.2 should be tested and developed further.\n- -It should be investigated whether non-equilibrium ionisation of elements that are important for radiative losses in the TR and corona can be implemented in a sufficiently computationally efficient fashion. The ionisation balance of helium can be modeled with only one energy level per ionisation stage and parametrized ionisation/recombination rates that include the effects of the excited levels. 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2024MNRAS.534..621D | We have numerically demonstrated that simulated cool star coronae naturally form condensations. If the star rotates slowly with a corotation radius greater than the Alfvn radius i.e. inlineformulatexmath idTM0001 notationLaTeXRmathrmK gt RmathrmAtexmathinlineformula these condensations will form below the corotation radius inlineformulatexmath idTM0002 notationLaTeXRmathrmKtexmathinlineformula and simply fall back to the stellar surface as coronal rain. If however the star is more rapidly rotating inlineformulatexmath idTM0003 notationLaTeXRmathrmK lt RmathrmAtexmathinlineformula not only rain will form but also slingshot prominences. In this case condensations collect into a large mass reservoir around the corotation radius from which periodic centrifugal ejections occur. In this case some 51 per cent of the coronal mass is cold gas either in rain or prominences. We find that 21 per cent of the mass lost by our simulated fast rotating star is cold gas. Studies of stellar massloss from the hot wind do not consider this component of the wind and therefore systematically underestimate massloss rates of these stars. Centrifugal ejections happen periodically between every 7.517.5 h with masses clustering around inlineformulatexmath idTM0004 notationLaTeX1016texmathinlineformula g These results agree well with observational statistics. Contrasting the fast and slow rotating magnetospheres we find that there are two distinct types of solutions highlying and lowlying loops. Lowlying loops only produce coronal rain whereas highlying loops produce both rain and slingshots. | 2024-10-01T00:00:00Z | ['10.48550/arXiv.2409.07297', 'arXiv:2409.07297', '2024arXiv240907297D', '10.1093/mnras/stae2131', '2024MNRAS.534..621D', '2024MNRAS.tmp.2087D'] | ['Astrophysics - Solar and Stellar Astrophysics'] | Simulating stellar coronal rain and slingshot prominences | 2,024 | 170 | 0.49 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF'] | 0 | https://arxiv.org/pdf/2409.07297.pdf | {'Simulating stellar coronal rain and slingshot prominences': 'S. Daley-Yates 1 ⋆ , Moira M. Jardine 1 \n1 School of Physics and Astronomy, University of St Andrews, North Haugh, St Andrews, Fife, Scotland KY16 YSS, UK', 'ABSTRACT': "We have numerically demonstrated that simulated cool star coronae naturally form condensations. If the star rotates slowly, with a co-rotation radius greater than the Alfv'en radius (i.e. R K > R A ), these condensations will form below the co-rotation radius R K and simply fall back to the stellar surface as coronal rain. If, however, the star is more rapidly rotating, ( R K < R A ), not only rain will form but also 'slingshot prominences'. In this case, condensations collect into a large mass reservoir around the co-rotation radius, from which periodic centrifugal ejections occur. In this case, some 51% of the coronal mass is cold gas, either in rain or prominences. We find that 21% of the mass lost by our simulated fast rotating star is cold gas. Studies of stellar mass-loss from the hot wind do not consider this component of the wind and therefore systematically underestimate mass-loss rates of these stars. Centrifugal ejections happen periodically, between every 7.5 - 17.5 hours with masses clustering around 10 16 g, These results agree well with observational statistics. Contrasting the fast and slow rotating magnetospheres, we find that there are two distinct types of solutions, high lying and low lying loops. Low lying loops only produce coronal rain whereas high lying loops produce both rain and slingshots. \nKey words: Sun: filaments, prominences - stars: coronae - stars: magnetic field - stars: activity", '1 INTRODUCTION': "Much of our understanding of the structure and dynamics of stellar coronae has come from X-ray or UV observations of the hot (1-10 × 10 6 K) gas trapped in the star's corona by the confining effect of the magnetic field. The hot plasma that escapes to form the stellar wind is not dense enough to be observed directly and is mainly studied through its impact on the rotational evolution of the star. The mechanisms responsible for heating the coronal gas to these temperatures are still a matter of debate, many decades after the existence of stellar coronae was discovered. Studies of both this background heating and also the intermittent powerful heating in stellar flares have however been re-invigorated by the realisation of their impact on the evolution of exoplanetary atmospheres (Segura et al. 2005; Airapetian et al. 2020). \nObservations of the Sun, however, show that coronal gas can also exist in a cool ( 10 4 K) phase. These are temperatures typical of the Sun's chromosphere, but they are found within the solar corona both in large, quasi-stable prominences (masses typically 10 15 g) and also in smaller, dynamic clumps of 'coronal rain' (Antolin & Froment 2022; S¸ahin et al. 2023). This rain is often found to be falling at speeds close to, but typically less than, the free-fall speed, with values in the range 100 -150 km s -1 (Antolin & Froment 2022), or ≃ 40 km s -1 (S¸ahin et al. 2023). Sun-as-a-star measurements give similar values. Namekata et al. (2022b) found 95 km s -1 while Otsu et al. (2022) reported velocities up to 200 km s -1 . \n- ⋆ E-mail: [email protected] \n© \nThis is consistent with the view that these clumps are condensations that are falling along magnetic field lines. They are often found after a flare has ablated chromospheric material which rises into the corona, raising the local density to the point where it cools and then falls back towards the surface. This cycle of heating and cooling regulates the total mass of the corona - indeed above some active regions the cool gas can comprise some 50% of the active region volume (S¸ahin & Antolin 2022). \nStudies of this cool gas phase in the coronae of other stars originally focused on the very large, relatively stable 'slingshot' prominences (Collier Cameron & Robinson 1989a,b). These cool clouds are detected as transient absorption features that move from blue to red through the H α line profile. They often recur at the same rotation phase, suggesting that they are co-rotating with the star. The time taken for the absorption feature to move through the line profile gives the distance of the absorbing material from the rotation axis directly, showing that they typically form close to or beyond the Keplerian co-rotation radius, where the outward centrifugal force begins to dominate over the inward gravitational force. Since their original discovery, they have been detected in a range of stars, from those still in the T Tauri phase (Skelly et al. 2008, 2009) to those whose disks have dissipated, but which are still rotating rapidly (Collier Cameron & Woods 1992; Hall & Ramsey 1992; Byrne et al. 1996; Eibe 1998; Barnes et al. 2000, 2001; Petit et al. 2005; Dunstone et al. 2006a; Dunstone 2008; Leitzinger et al. 2016; Cang et al. 2020, 2021; Zaire et al. 2021). They have even been observed on binary stars such as the K supergiant 32 Cyg (Schroeder 1983). \nThe prominence masses derived from these observations are typically (2 -6 × 10 17 g). This is several orders of magnitude greater than large solar prominences, but consistent with predictions of theoretical models that use stellar surface magnetic maps as inputs (Villarreal D'Angelo et al. 2018; Waugh & Jardine 2019; Villarreal D'Angelo et al. 2019; Waugh et al. 2021). When these prominences become unstable, they will be ejected (if they are beyond the co-rotation radius) and so may remove enough angular momentum from the star to contribute significantly to the wind torques (Faller & Jardine 2022; Waugh & Jardine 2022). This suggests that prominence ejection may provide a significant fraction of the extra torque required for stellar wind models to explain the observed rotational evolution (Gallet & Bouvier 2013, 2015; See et al. 2018; Evensberget & Vidotto 2024). \nThe large distance from the stellar surface at which these 'slingshot' prominences are supported means that they are expected to form and be ejected in a cyclic process (Jardine et al. 2020). This is because the sonic point of the upflow that forms the prominence is typically below the prominence formation site (close to the co-rotation radius). As a result, the growth in mass of the prominence can not be checked by information propagating back down to the surface, leading to runaway growth and eventual centrifugal ejection. The continued upflow ensures that another prominence forms and this limit-cycle behaviour continues. The observationally-derived masses and lifetimes of prominences therefore provide crucial information on the mass-loss rates of these stars. This is particularly important for very active stars where other methods have so far been unable to provide measurements of wind mass-loss rates (Wood et al. 2021). \nIn addition to these large and quasi-stable structures, cool coronal plasma has also been detected in smaller-scale, more dynamic structures. Transient, red-shifted absorption features were detected in the earliest studies (Byrne 1987; Houdebine et al. 1993; Eibe et al. 1999). These were interpreted as 'failed prominences' - material that had condensed, but failed to find a stable location in which to accumulate and so simply drained back to the surface. More recent surveys have now shown that these transient features can be seen both shifted to the red and also to the blue (Fuhrmeister et al. 2018; Vida et al. 2019; Namekata et al. 2022a), with estimated masses in the range 10 13 -10 14 kg (Vida et al. 2019). \nTypically, these features are moving at speeds below the escape speed, although some have been seen at greater speeds that indicate that they may be undergoing ejection from the star. These fast-moving features have in many cases been observed to occur at the same time as extremely powerful 'superflares' detected with Lamost (Kanodia et al. 2022; Wu et al. 2022) or TESS (Namekata et al. 2021; Inoue et al. 2023; Namizaki et al. 2023; Namekata et al. 2024). The possibility that these ejections are associated with the stellar equivalent of solar coronal mass ejections makes their study even more important. Solar coronal mass ejections are strong sources of energetic particles that can ionise the upper layers of planetary atmospheres. Their frequency and power on younger solar analogues, or exoplanet hosts, may be an important aspect of exoplanetary evolution. \nOriginally, theoretical studies of this 'coronal rain' were confined to the solar case (Antiochos et al. 1999; Karpen et al. 2006; Antolin et al. 2010; Froment et al. 2018; Li et al. 2022) with many focused on the role of the heating mechanism Xia \net al. (2012); Fang et al. (2013); Zhou et al. (2021); Ruan et al. (2021). Recently, we published a study in which the concept of coronal rain is extended to the stellar case (Daley-Yates et al. 2023). By simulating a moderately-rotating solar-like star, we demonstrated the formation and subsequent draining of large-scale coronal condensations (Daley-Yates et al. 2023). These condensations were triggered by enhanced footpoint heating. The resultant coronal rain had line-of-sight velocities in the range 50 km s -1 (blue shifted) to 250 km s -1 (red shifted), typical of those inferred from stellar H α line asymmetries. Since this star was only rotating moderately quickly, its co-rotation radius lay beyond its corona. As a result, although these condensations were formed at all heights within the corona, condensations could not be formed beyond the co-rotation radius, and hence could not escape the star. \nIn the present paper we extend this pilot study to a more rapidly-rotating star to examine the outward ejection of such condensations; for the first time numerically demonstrating the slingshot mechanism. We also include an updated version of the slower rotation case described above so that we can contrast the two rotation states. Therefore we can see how faster rotators exhibit the slingshot mechanism and slower ones do not, within the same numerical prescription.", '2 MODELLING': "For the basic equations solved in our simulations including the form of the magnetohydrodynamic (MHD) equations, please see our previous paper (Daley-Yates et al. 2023) and the paper for the simulation code we use, MPI-AMRVAC (Keppens et al. 2021). \nIn the following sections we will introduce the key concepts of the cooling instability, a phenomenological heating prescription and the magnetosphere classification needed to understand the results presented in Section 4. We also describe how the magnetic field depends on rotation and therefore how the heating rate and magnetic field can be specified by the star's rotation rate.", '2.1 Stellar multi-phase gas': 'The magnitude of radiative energy loss in a plasma depends on its temperature and composition, represented by a loss function Λ( T ). Between chromospheric and coronal temperatures (10 4 -10 6 K), Λ( T ) is non-linear, with cooler temperatures radiating away energy more efficiently. This leads to a runaway effect where coronal gas can suddenly cool, condensing to either long-lived prominences or short-lived coronal rain. It is the hot coronal plasma co-existing with the cooler condensations that we call a multi-phase gas. For a comprehensive study of the cooling function, the cooling instability and its role in numerical simulations see Hermans & Keppens (2021).', '2.2 Magnetosphere classification': "Here we introduce a classification scheme where a star's magnetosphere is described as either dynamical (DM) or centrifugal (CM) based on whether the co-rotation radius ( R K ) is inside the Alfv'en radius ( R A ) or vice versa. This scheme is based on MHD simulations of massive stars by ud-Doula & \nOwocki (2002); ud-Doula et al. (2006); ud-Doula et al. (2008), see also the work of Petit et al. (2013). Indeed, the concept of centrifugal outbreak events that may be responsible for Xray flares in massive stars (ud-Doula et al. 2006) may be very similar to slingshot prominence ejection in cool stars. This notion was extended to cool stars by Villarreal D'Angelo et al. (2017) who postulated that both cool stars and massive stars trap material in their magnetospheres by centrifugal support in the same manner, despite having completely different underlying wind driving physics. Cool star winds are believed to be thermally driven (Parker 1958) while massive star winds are believed to be radiatively driven (Castor et al. 1975). \nIf R K > R A then the star is a relatively slow rotator. All the magnetic loops have summits below the co-rotation radius, so that at all points along their length the effective gravity points downwards towards the stellar surface. Plasma that condenses in one of these loops will simply fall under the action of this effective gravity, back towards the stellar surface. If R K < R A then the star is a relatively fast rotator. The summits of the tallest loops may be beyond the co-rotation radius. In this case, plasma that condenses there will tend to fall outwards but may be supported against centrifugal ejection by the tension of the magnetic field. Thus regions exist between R K and R A where there is mechanical stability and where plasma can accumulate and form a stable prominence. See the work of Waugh & Jardine (2019); Waugh et al. (2021); Waugh & Jardine (2022) for a comprehensive analytic study of prominence stability and how it impacts cool star mass-loss. \nTo explore these concepts, we have simulated two stars: one with a centrifugal magnetosphere ( P ∗ = 0 . 38 days) and one with a dynamical magnetosphere ( P ∗ = 3 . 8 days) 1 . We interpret our results in Section 4 through this dynamical vs centrifugal magnetosphere classification scheme.", '2.3 Stellar wind heating model': 'The following sections detail the specifics of our heating model and how we parameterise it with observational statistics of cool star magnetic fields. We show how the strength of the field, and therefore the heating, depends on rotation rate.', '2.3.1 Scaling laws': "We can measure the magnetic field strengths of stars using the Zeeman-Doppler Imaging (ZDI) and Zeeman Broadening (ZB) techniques (please see Donati & Landstreet (2009) and references therein for details). We note that ZDI is sensitive to a star's large-scale magnetic field while ZB shows the smallscale magnetic field. \nIn all the following equations we include the saturation of the magnetic field at periods below P sat = 1 . 6 days in accordence with the data from Vidotto et al. (2014) and Reiners \n1 We chose the value of P ∗ = 0 . 38 days as it is the rotation rate of the K3 dwarf star BO Microscopii (BO Mic, HD 197890), AKA Speedy Mic , a quintessential rapid rotating cool star with R K < R A (giving a centrifugal magnetosphere). The choice of P ∗ = 3 . 8 days is simply an order of magnitude lower so that R K > R A (giving a dynamical magnetosphere). \nFigure 1. Scaling laws for magnetic field strength as a function of rotation period for cool stars. These are normalised such that the stellar radius and mass have solar values for all periods. These relations are from Reiners et al. (2022) in the case of the smallscale field and Vidotto et al. (2014) in the case of the large-scale field. The purple dots indicate the field values for P ∗ = 0 . 38 days and the green dots a value of P ∗ = 3 . 8 days, corresponding to the values used in our two simulations. \n<!-- image --> \net al. (2022). By small-scale fields we mean magnetically active regions on the surface and for these we use the scaling relations of Reiners et al. (2022): \nB s = 8570 G P -1 . 25 ( P > P sat ) (1) \nB s = 5300 G P -0 . 16 ( P < P sat ) . (2) \nThese are simplified versions of equations 2 and 3 in Reiners et al. (2022), we have removed the dependence on M ∗ , L ∗ and R ∗ 2 so that we assume that the magnetic field strengths depend only on rotational period. \nFor large scale fields, which here we mean the dipole component, we use the scaling relations of Holzwarth & Jardine (2007), \nB w = B ⊙ ( P P ⊙ ) -1 . 32 ( P > P sat ) (3) \nB w = B ⊙ ( P sat P ⊙ ) -1 . 32 ( P < P sat ) . (4) \nUnlike Holzwarth & Jardine (2007) we use the exponent of Vidotto et al. (2014) and account for magnetic field saturation. B ⊙ is the solar polar dipole field strength ∼ 10 G at solar maximum (Vidotto et al. 2014). These scaling relations and the values used in the simulations are plotted in Fig. 1.", '2.3.2 heating model': 'The coronae and winds of our simulated stars are driven by a phenomenological heating model similar to that of Lionello \n2 This is because we have simulated stars with only solar values for M ∗ , L ∗ and R ∗ and they appear in Reiners et al. (2022) in solar units, therefore M ∗ = 1, L ∗ = 1 and R ∗ = 1. \nTable 1. Stellar simulation parameters for the two magnetosphere types, centrifugal (CM) and dynamic (DM). \net al. (2009) and Downs et al. (2010). The following equation was derived by Abbett (2007) and describes an empirical relation between the unsigned magnetic flux and energy deposited in the corona (see Fig 1. of Pevtsov et al. (2003) for this relation). The heating rate is given by \nQ = cϕ α ψ ζ ∫ ψdV . (5) \nWhere c = 0 . 8940 and α = 1 . 1488 are fit parameters from Bercik et al. (2005). ϕ is the unsigned magnetic flux, given by \nϕ = ∮ r = R ∗ | B r | ds (6) \nand ψ is the local heating weighting function which we simply take as the magnitude of the magnetic field as \nψ = | B | exp ( -r -R ∗ λ ) . (7) \nWe also include an exponential envelope function limiting the heating to the lower corona, in the same way as Downs et al. (2010). Because the volume integral of the weighting function is done over the entire simulation, this envelope function ensures the radial extent of the numerical grid does not impact the heating rate. \nThe length scale of the envelope function has different values for the small-scale and large-scale heating. For the smallscale heating it is limited to 40 Mm, which is approximately the height of active regions. \nWhen performing test simulations we found that for fixed size, large-scale heating envelopes, the wind from the star ( r > 10 R ∗ ) would not maintain its temperature and would drop to chromospheric values. To address this, we use a scaling law based on rotation rate, that allows the heating to extend higher into the corona for faster rotating stars. Since faster rotating stars have stronger magnetic fields and therefore larger closed magnetospheres, this results in heating at greater altitudes. This scaling law is \nλ w = λ 0 ( P P ⊙ ) -2 ( P > P sat ) (8) \nλ w = λ 0 ( P sat P ⊙ ) -0 . 4 ( P < P sat ) . (9) \nThe exponents were determined experimentally, from trialand-error simulations. \nThe heating equations were applied to both the small- and large-scale fields with the resulting total heating being simply \nQ total = Q s + Q w , where the subscripts s and w refer to the heating rates derived from the small-scale (surface) and large-scale (wind) magnetic field strengths from equations 1 and 4 respectively. \nAll the variables used for modelling the heating and defining the physical parameters of our simulated stars are summarized in Table. 1.', '3 NUMERICAL MODELLING': 'As the simulations we present here are based on those in our previous study, we provide only a brief description of the simulation setup and highlight what we have changed for the purpose of the new simulations. For more details please see Daley-Yates et al. (2023). \nWe solve the MHD equations with optically thin radiative losses and thermal conduction using the parallel, block based, adaptive mesh code MPI-AMRVAC (Xia et al. 2018; Keppens et al. 2021). Our computational grid extends between r ∈ { 1 , 50 } R ∗ and θ ∈ { 0 , π } radians. We used a resolution of 1024 cells in the radial direction and 640 cells in the poloidal direction. This differs from our previous study, in which we achieved an effective resolution of approximately twice this value via the use of additional refinement levels. In our present study we chose to use a static grid as we found in testing that the numerical overheads of adaptive mesh refinement outweighed the gains in resolution. This is because we needed high resolution both far from the star to resolve the slingshot prominences, as well as close to the star to capture the condensation behaviour there (in our previous study only the latter condensations were present). \nIn the next section we will present and discuss the results of our two simulations conducted with the framework described above.', '4 RESULTS AND DISCUSSION': 'Here we present the results of our simulations, broken down into dynamical and centrifugal magnetospheres and describe the major phases seen in both. We will also show in the case of the centrifugal magnetosphere how the gas is partitioned into hot and cold phases, how the maximum temperature responds to centrifugal breakout, the impact this has on the mass-loss rate and finally the statistics of the simulated slingshots.', '4.1 Magnetosphere types': 'Here we will lay out the differences between the dynamical and centrifugal magnetospheres and demonstrate that it is the CM that generates slingshot prominences while the DM only generates coronal rain. As the results for the CM are more extensive, we will start with the DM results.', '4.1.1 Dynamical magnetosphere': 'Fig. 2 illustrates three distinct stages of coronal rain formation. Initially, evaporation from the chromosphere increases the density in the closed magnetosphere. Once the density in the closed-field region is high enough, cooling by radiation leads to thermal runaway and gas cools to chromospheric temperatures (10 4 K). We see sympathetic cooling across field lines and cold gas extends from the apexes of the loops down to the chromosphere before finally the closed magnetosphere returns to only hot gas. These stages are summarised as: \n- (i) The chromospheric evaporation stage.\n- (ii) The cooling stage.\n- (iii) The condensation and evacuation stage. \nOnce established, these stages repeat in a cyclic manner for the rest of the simulation. This process is identical to solar coronal rain (Antolin 2020). While the formation of stellar coronal rain is observable and an indicator of magnetic activity, it does not impact the properties of the extended wind, for example mass-loss rate, and therefore plays no role in the long term evolution of the star.', '4.1.2 Centrifugal magnetosphere': "The formation phases of the slingshot prominence process are shown in Fig. 3. First, as in the DM case, evaporation from the chromosphere increases the density in the closed magnetosphere. Second, the thermal instability leads to gas cooling to chromospheric temperatures within the closed magnetosphere, both above and below R K . \nThird, cold gas below R K drains to the surface forming coronal rain, while cold gas beyond R K undergoes centrifugal breakout, elongating to a thin sheet as it leaves the closed magnetosphere. This thin sheet remains intact as it passes through the magnetic Y-null point at the apex of the helmet streamer. This is important as it demonstrates that the timescale for breakout of a slingshot (and therefore the time between subsequent breakouts) is not simply the timescale of the tearing instability but rather the time taken for the cold gas pressure to exceed the magnetic tension. This is a function of the rate of evaporation of gas form the chromosphere and therefore the feeding rate of the stable region above R K . \nFourth, the current sheet, unstable to the tearing instability, divides into a series of plasmoids. Finally, the plasmoids coalesce into larger structures as they pass beyond R A . \nIn Fig. 3, the dotted white line indicates the Kepler corotation radius and the solid cyan line shows the Alfv'enic Mach surface. Crucially, R K is inside R A , signifying a CM, in contrast to Fig. 2 where R K is either at or outside R A . The five stages of the CM are therefore: \n- (i) The chromospheric evaporation stage.\n- (ii) The cooling stage. \n(iii) Condensation and draining below , and centrifugal breakout above R K . \n- (iv) The tearing instability produces plasmoids.\n- (v) The plasmoids coalescence beyond R A . \nA major departure in the CM case from the DM case is that there is no stage where the magnetosphere is wholly evacuated of cold gas. Low lying loops, well below R K , do exhibit this, but between R k and R A cold gas is constantly forming and draining. See Fig. 4 for not only the temperature but also the density profiles including a close up of both. We have concentrated on the temperature and density as these quantities completely define the presence of condensations. Please see Appendix A for plots of the velocity components. We will see in the following section that this cold gas component forms a significant fraction of the total magnetosphere mass.", '4.2 Time series': 'Below we analyse the time dependent quantities that show how the multi-phase gas evolves over the simulation (which last for a total of 393 hr). This maximum time was chosen to strike a balance between simulating a sufficient number of slingshots and the computational resources used. We concluded that 393 hr allowed us to draw firm conclusions and to perform a limited statistical comparison to observations. Fig. 8 shows our time series results.', '4.2.1 Mass balance': 'The top plot of Fig. 5 shows a series of the cold gas mass fraction in both simulation volumes. Here, as with the rest of the analysis, we define the boundary between hot and cold gas as 80000 K. For the CM, once a quasi-steady-state has been reached, the cold gas component is 51% of the total gas in the simulation (excluding the chromosphere) and remains at approximately this value for the rest of the simulation. For the DM, the cold gas component is 2%, a much smaller fraction of the magnetosphere. It is also intermittent, going to zero during the DM stage (i) as described in Section 4.1.1.', '4.2.2 Maximum temperature as an observational proxy': "In the middle plot of Fig. 5 we show the time series of the maximum temperature ( T max ) in the simulation. This is determined by finding T max on the entire numerical grid at each time step. The location of T max is always in the region of the reconnected field, upstream of the recently ejected prominence. Each successive breakout is accompanied by a spike ( T max ≲ 10 8 K) which is up to an order of magnitude above the background coronal temperature ( ∼ 2 × 10 7 K). This behaviour is also seen in the DM simulation (though with a smaller T max ), which has no centrifugal breakout, so this cannot be attributed to the slingshots. In both the CM and DM case, between reconnection events, the magnetosphere has to relax back to equilibrium. This is evidenced in Fig. 2 where the field lines are stretched out over the frames shown. \nAt ∼ 30 hr after the condensations have formed (either into coronal rain in the DM case or slingshots in the CM case), there is a pinching of the helmet streamer and reconnection of the magnetic field. This accompanies the spike \nFigure 2. Time series of the 2D temperature profile for the dynamical magnetosphere case. There are three distinct phases of prominence formation: First, the thermal instability leads to gas cooling to chromospheric temperatures within the closed magnetosphere. Second, cold gas below R K drains to the surface. Third, once all the cold gas has drained to the surface, there is only hot gas left in the magnetosphere. The dotted white line indicates the Kepler co-rotation radius and the cyan solid line shows the Alfv'enic Mach surface. A movie version of this figure is available online. \n<!-- image --> \n∗ \nFigure 3. Time series of the 2D temperature profile for the centrifugal magnetosphere case. There are five distinct phases of the slingshot prominence process. From top to bottom: Evaporation raises the density of the mangetosphere. Second, the thermal instability leads to gas cooling to chromospheric temperatures within the closed magnetosphere, not just at R K . Third, cold gas below R K drains to the surface and gas beyond R K undergoes centrifugal breakout, elongating to a current sheet. Forth, the current sheet, unstable to the tearing instability, divides into a series of plasmoids. Finally, the plasmoids coalesce into larger structures as they pass beyond R A . The dotted white line indicates the Kepler co-rotation radius and the cyan solid line shows the Alfv'enic Mach surface. A movie version of this figure is available online. \n<!-- image --> \n∗ \nFigure 4. Example simulation snapshot showing density and temperature profiles at a time coinciding with the ejection of prominence material. The upper two panels show the large-scale overview with the ejected prominence at ∼ 30 R ∗ . The bottom two panels show a zoomed-in portion of the inner magnetosphere. This region shows prominence material in three different stages: gas that is falling back to the stellar surface, that is suspended at co-rotation and that is undergoing centrifugal ejection. \n<!-- image --> \n∗ \n∗ \nin T max , which may have a signature in observations. Stellar flares are observed in UV, x-ray and radio wavelengths (Benz & Gudel 2010) and are characterised by a sudden rise in emission and a decay to background levels. \nIf we contrast the DM to the CM, we see a lower background temperature, less frequent spikes and lower T max spikes. \nThe amplitude of our heating model in our simulations \nis proportional to the magnetic field strength. The CM has B 0 ∼ 800 G and the DM B 0 ∼ 200 G dipole moments. Therefore the CM is necessarily hotter as there is more energy being deposited into the simulation than in the DM case. This explains why the baseline T max is lager for the CM. The spikes are due to reconnection, with a larger field strength in the CM case, there is more magnetic energy released by reconnection than the DM case, leading to a lager T max spike for the CM. \n) \n( \nFigure 5. Top: time series of the cold gas mass fraction in the both simulation volumes. Here, as with the rest of the analysis, we define the boundary between hot and cold gas as 80000 K. Once a quasi-steady-state has been reached ( t > 300 hr), the cold gas component of the CM has reached a time average of 51% (horizontal blue dashed line) and for the DM has reached 2% (horizontal orange dashed line) of the total gas in the simulations (excluding the chromosphere). Middle: The maximum temperature in the two simulations. Both exhibit rapid temperature spikes followed by a decay back to coronal temperatures. The origin of these spikes is the thermal energy released through magnetic reconnection that accompanies the tearing of the current sheet illustrated in Fig. 3. The difference in occurrence rates of these spikes is due to the rotational periods of the two simulated stars. Bottom: mass-loss rates for the two magnetosphere types and in the case of the centrifugal magnetosphere, the division between hot and cold gas. In steady state ( t > 300 hr), we find that mass-loss rate in cold gas is 21% that of the total. We also find that the dynamical magnetosphere is losing mass an order of magnitude more slowly than the centrifugal case. \n<!-- image --> \nThe entire helmet streamer region is subsonic, therefore shock heating is not responsible for the high temperature, contrary to the massive star case where the entire wind is supersonic almost from the surface of the star. This is consistent with the observational result that flare activity is correlated with rotation rate for young stellar objects. The faster a star spins, the more it flares (Gunther et al. 2020; Vida et al. 2024). \nCalculating synthetic observations from our simulation result would give us a direct comparison to observations, this will be the subject of a dedicated future study.", '4.2.3 Mass-loss rates': "The mass-loss rates are measured by integrating the radial component of the momentum over a surface at r = 25 R ∗ . The results are shown in the bottom plot of Fig. 5 for both the CM and DM. We show both the cold ( ˙ M c ) and hot ( ˙ M h ) components in the CM, but only the hot component for the \nDM case, as no cold gas escapes the the star. Another difference between the two magnetosphere types is the magnitude of ˙ M h . Our DM has mass-loss rates that are always lower by approximately an order of magnitude. This is consistent with the difference in heating rates used in the two simulations (see Table 1). The faster rotator has more heating, giving greater chromospheric evaporation, leading to a denser magnetosphere and wind, resulting in the higher mass-loss rate. \nTurning to ˙ M c , for times t > 200 hr, the simulation starts to generate slingshots. The mass flux associated with each slingshot is plotted as the blue spikes overlaying the background ˙ M h . By integrating the area under these curves we find that the accumulated mass-loss in cold gas is 21% of the total mass-loss, a significant fraction. \nTo better understand the mass-loss associated with an individual slingshot, we plot a single breakout event in Fig. 6. During the event, ˙ M c is 35 times larger than ˙ M h . If we re- \nFigure 6. Mass-loss rate for a single slingshot event for both the background hot gas (orange line) and cold gas (blue line) from the CM. At its peak, the cold gas mass-loss is 35 times greater than the background hot gas mass-loss. \n<!-- image --> \nll that the mass-loss calculation is carried out over a surface containing the star at r = 25 R ∗ and for the hot gas includes the fast and slow wind at all latitudes, this means that for the period of a slingshot, the star's mass-loss is dominated by cold gas.", '4.2.4 Low- and high-lying loop solutions': 'Cold gas in the magnetosphere from either the DM or CM is highly dynamic. Simple formation and draining occurs cyclically for the DM but in the CM case, cold gas drains, escapes outwards or is suspended all at the same time. To better understand the nature, intermittent or otherwise, of each of these states we plot the radial mass distribution of both simulations in Fig. 7 in the manner of ud-Doula et al. (2008) and Daley-Yates et al. (2019). \nThe DM shows condensation and draining in the form of coronal rain. The CM shows strikingly different behaviour. After an initial coronal rain event, a reservoir forms above R K and remains for the length of the simulation. Coronal rain still occurs in a similar manner to the DM. This can be seen below the reservoir of stable material. There is also draining from the bottom of the reservoir to the stellar surface, similar to coronal rain. Above the reservoir cold gas breaks out of the closed magnetosphere, indicating slingshot events. These form upward trending tracks from the reservoir to the edge of the simulation domain 3 . The gradient of these tracks indicate the velocity of the slingshots as they leave the magnetosphere. We do not attempt to quantify the velocities from Fig. 7 as we measure this directly from simulation outputs and present it in the next section. We note however that the gradient of the tracks decreases beyond > 30 R ∗ , indicating that the slingshots begin to slow down as they leave the magnetosphere. \nFig. 7 illustrates the two types of loops that form in the \n3 The apparent dotted nature of the tracks in Fig. 7 is due to the cadence of the simulation outputs which is every 1000 s (with inflate cadence the tracks would be continuous) \nmagnetosphere of a young, fast rotating star. Low-lying loops support the formation of coronal rain. We have seen and reported the low-lying loop solutions in our previous paper Daley-Yates et al. (2023). Indeed low-lying loop solutions are seen on the Sun as Solar coronal loops that exhibit Solar coronal rain. The second, high-lying loops, support the formation of slingshots and are not seen on the Sun. The division between these two solutions depends on the position of the corotation radius. R K must coincide with closed magnetic loops capable of forming condensations. Fig. 7 shows the position of R K for both magnetospheres. The presence of high-lying loop solutions does not preclude the existence of low-lying loops. In the CM we see both families of solutions. These two loop solutions were predicted by the analytic work of Jardine & Collier Cameron (1991); Waugh & Jardine (2019); Waugh et al. (2021); Waugh & Jardine (2022). We report here numerical conformation of this idea.', '4.3 Slingshot statistics': 'Here we investigate the statistical properties of the CM simulation. In total there are 18 slingshots recorded. The time between slingshots varies from 0.9 - 17.9 hrs (approximately 0 . 075 -1 . 5 stellar rotations) with an average of 7.7 hrs. Breakout times around 2.5 hr are due to multiple blobs of gas being ejected as part of the same breakout event, and are in fact one slingshot. We do not count these as separate events since the blobs can and do coalesce at larger radii. For the 7.5 17.5 hrs range it is perhaps useful to compare to the observations of the K3 dwarf star Speedy Mic (BO Mic, HD 197890) by Dunstone et al. (2006a) who measured stable prominences lasting for more than 13 stellar rotations. They found prominence structures at an average height of 2 . 85 ± 0 . 54 R ∗ , a very similar height to our stable reservoir situated at > 2 . 21 R ∗ in our CM simulation (see Fig. 7). We argue that what Dunstone et al. (2006a) observed was not cold gas making up slingshot prominence, but the stable reservoir which supplies the cold gas to the slingshots. The times between our slingshots is also much shorter than the 13 rotations that they report. This is consistent with the idea that this is in fact the timescale for the reservoir to evolve and not the time taken for individual prominences to breakout as slingshots. \nOur slingshots have velocities between 556 - 1315 km/s with an average of 813 km/s. The velocities do not agree with Vida et al. (2019) whose values for the blue shifted velocities are between 0 - 800 km/s. In our case the slingshots are traveling along the line of sight, so all samples are the fastest possible. For the observations, the lines of sight velocities are components and not the total velocities, making our velocity distribution an upper limit on the observations. This is what we see when we compare them to Fig. 7 of Vida et al. (2019). We have one outlier in the velocity beyond the results of Vida et al. (2019) with a value of 1315 km/s. \nInoue et al. (2023) reported observing the very active RS CVn-type binary V1355 Orionis releasing a superflare of 7 × 10 35 ergs along with a feature showing an H α excess travelling between 760 - 1690 km/s with a mass between 9 . 5 × 10 18 g - 1 . 4 × 10 21 g. While this velocity range includes our outlier, the mass range far exceeds any of the slingshots we measure in our simulation which are between 1 . 61 × 10 15 - 1 . 68 × 10 17 g with an average of 3 . 95 × 10 16 g. The K-type subgiant component of V1355 Orionis is a larger star than those we have \n) \n( \n) \n( \nFigure 7. Time series of the radial mass distribution for only the cold gas component. Both plots show the equatorial co-rotational radius ( R K ) as the dashed line. Top: dynamical magnetosphere showing the formation of cold gas and its subsequent fall back to the stellar surface, coronal rain. Bottom: centrifugal magnetosphere showing an altogether more nuanced picture. The coronal rain seen in the dynamical magnetosphere is still present, but here it is only a subset of the overall behaviour. On top of this is a reservoir of cold gas which supplies gas for the periodic breakout of slingshot. These slingshots can be seen as upward moving tracks in the colour map above this reservoir. \n<!-- image --> \nsimulated and therefore has different values for R K and R A and may support larger prominences. It has a co-rotation radius of 1 . 8 R ∗ above its surface, making it comparable in this respect to our simulated CM star. However for the slingshot mechanism to be responsible for the superflare on V1355 Orionis, its magnetic field would have to be sufficient to give it a centrifugal magnetosphere. Unfortunately we do not have any information on the magnetic field strength or structure of V1355 Orionis, so our speculation ends here and can only state that the velocity range of our simulated slingshots does coincide with the velocity associated with this superflare. \nOur mass range of 1 . 61 × 10 15 - 1 . 68 × 10 17 g for our slingshots agrees with the statistics of Vida et al. (2019) remarkably well. To underline this, we plot the normal fit to their data that they report over our statistical sample. Both quantities are normalised for ease of comparison. There is a tight agreement between observation and our results. Dunstone et al. (2006b) reported masses for the largest prominences of Speedy Mic to be 0 . 5 × 10 17 - 2 . 3 × 10 17 g, which agrees well with our results. Both the statistical sample of Vida et al. (2019) and this single stellar observation agree much better with our simulated mass range of 1 . 61 × 10 15 - 1 . 68 × 10 17 g \nthan the superflare observation of V1355 Orionis, despite the velocities being so different. We stress our results are not directly comparable to that of Vida et al. (2019), as each data point in their observations is a snapshot of the whole magnetosphere of an individual star. Our statistics are made up of only the slingshots as measured when they pass through r = 25 R ∗ , and we ignore any other gas in the simulation, cold or otherwise. As such we regard our results as a possible subset, embedded in the complete results presented by Vida et al. (2019). We also only look at their blue-shifted results as this is relevant to our slingshots.', '5 CONCLUSIONS': "We have performed numerical simulations of stellar coronae that demonstrate the formation of massive clumps of cool gas condensing out of the hot corona. The onset of condensation depends on the heating process. An increase in coronal heating evaporates chromospheric material into the corona, raising the density sufficiently high for the onset of thermal collapse. The subsequent dynamics of these clumps, however, depend on the star's rotation rate. \nFigure 8. Statistical distribution of the 18 separate slingshot prominences seen in our CM simulation. From top to bottom: time between breakouts, line of sight velocities of each breakout and finally the masses of the breakouts. This bottom plot also a fit to the observational results reported by Vida et al. (2019). In order to make the mass results comparable to the observational statistics, we have normalised the counts. \n<!-- image --> \n<!-- image --> \n<!-- image --> \n( \n) \nWe find two regimes that can be classified by the ratio of the co-rotation radius R K to the Alfv'en radius R A . This centrifugal-dynamical magnetosphere framework was first developed in the massive stars community by ud-Doula et al. (2008) and Petit et al. (2013) and expanded to cool stars by Villarreal D'Angelo et al. (2017). The more slowly-rotating \nof our two stars is in the dynamical regime (DR: R K < R A ). In this case, condensations form below the co-rotation radius and fall back towards the stellar surface. We can identify this pattern of downflows as the stellar equivalent of solar coronal rain. This behaviour is also seen in the more rapidlyrotating of our two stars, which is in the centrifugal regime (CR: R K > R A ). In this case, condensations not only form below the co-rotation radius but also at and above it. At the co-rotation radius they grow to form a quasi-stable mass reservoir, from which there are regular centrifugal breakout events as the mass grows beyond the point of magnetic confinement and some of it is ejected from the star. We identify the large stable reservoir with stellar 'slingshot prominences' and the ejecta with the fast-moving absorption features with which they are associated. \nIn the slowly-rotating dynamical magnetosphere case, the hot wind carries away mass at a rate 3 . 6 × 10 -14 M ⊙ / yr which is a little greater than the present-day Sun. There is no massloss from the condensations. In the faster-rotating centrifugal magnetosphere case, the hot gas removes mass more rapidly, at a rate of 9 . 4 × 10 -14 M ⊙ / yr. In addition, however, the cold gas ejected from the large reservoir at co-rotation radius removes mass at a rate of 2 . 7 × 10 -14 M ⊙ / yr. This comprises 21% of the total mass lost. Indeed, some 51% of the coronal mass is in cold gas. This is potentially a significant contribution that is unaccounted for in models of the hot wind. \nThe distribution of clump masses clusters around 10 16 g and the line of sight velocities range between 600 -1000 km/s, with a single outlier > 1200 km/s. These results agree well with the observational statistics of Vida et al. (2019) for clump masses, but as discussed in Section 4.3, our simulated velocities agree with the upper limit of the observations. \nBoth regimes display well-defined periodicities. In the slowly-rotating dynamical magnetosphere case, there are periodic down-flows where the cold gas drains completely from the corona. The interval between these events is ∼ 75 hr. In the faster-rotating centrifugal case, the region below the co-rotation radius also shows regular draining events, both from clumps that form below the co-rotation radius, and also from the large mass reservoir at the co-rotation radius itself. The part of the mass reservoir that lies about the co-rotation radius also loses mass periodically. These centrifugal breakout events have periods from 7.5 - 17.5 hr (there is a shorter period of 2.5 hr due to fragmentation into sub-clumps). \nBoth our simulation have been limited to 2D, in future studies we intend to move to 3D. This increase in dimensionality will allow us to quantify such things as: the number of prominence structures a star can support, weather they form a disk or discrete structures and allow us investigate observational signatures with a strong geometric component, such as H α tracks. \nContrasting the two magnetosphere types that we have simulated shows that there are two distinct types of solutions, high lying and low lying loops. Both are seen in the centrifugal magnetosphere but only the low lying loops are seen in the dynamic magnetosphere. Low lying loops only produce the stellar equivalent of solar coronal rain whereas high lying loops produce not only rain but also what has been dubbed 'slingshot prominences'. The modern day Sun only exhibits the low lying loops, but in the past when it was rotating faster it would have had both low and high loop types. This means that the early solar system planets would have \nevolved under the influence of slingshots. As a result we propose the slingshot mechanism as a new type of space weather for young stellar systems.", 'DATA AVAILABILITY': 'A CSV file of the data presented in Fig. 8 is provided online.', 'ACKNOWLEDGEMENTS': "The authors thank the reviewer for their helpful comments and suggestions; which improved the quality and content of the publication. SD-Y and MJ acknowledge support from STFC consolidated grant number ST/R000824/1. This work was performed using the DiRAC Data Intensive service at Leicester, operated by the University of Leicester IT Services, which forms part of the STFC DiRAC HPC Facility (www.dirac.ac.uk). The equipment was funded by BEIS capital funding via STFC capital grants ST/K000373/1 and ST/R002363/1 and STFC DiRAC Operations grant ST/R001014/1. DiRAC is part of the National e-Infrastructure. This research was supported by the International Space Science Institute (ISSI) in Bern, through ISSI International Team project 545 ('Observe Local Think Global: What Solar Observations can Teach us about Multiphase Plasmas across Physical Scales'). For the purpose of open access, the authors have applied a Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript version arising.", 'REFERENCES': "A42 \nSee V., et al., 2018, MNRAS, 474, 536 \nSegura A., Kasting J. F., Meadows V., Cohen M., Scalo J., Crisp \nD., Butler R. A. H., Tinetti G., 2005, Astrobiology, 5, 706 Skelly M. B., Unruh Y. C., Collier Cameron A., Barnes J. R., Donati J. F., Lawson W. A., Carter B. D., 2008, MNRAS, 385, 708 \nSkelly M. B., Unruh Y. C., Barnes J. R., Lawson W. A., Donati J. F., Collier Cameron A., 2009, MNRAS, 399, 1829 \nVida K., Leitzinger M., Kriskovics L., Seli B., Odert P., Kov'acs O. 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A., 2023, arXiv e-prints, p. arXiv:2305.08775 ud-Doula A., Owocki S. P., 2002, ApJ, 576, 413 ud-Doula A., Townsend R. H. D., Owocki S. P., 2006, ApJ, 640, L191 ud-Doula A., Owocki S. P., Townsend R. H. D., 2008, MNRAS, 385, 97-108", 'APPENDIX A: ANGULAR COMPONENTS': 'Figure A1 shows the velocity components. \nFigure A1. Radial (top), poloidal (middle) and azimuthal (bottom) velocity in the observers reference frame, at 305 hr from the start of the simulation. \n<!-- image --> \n) \n( \n) \n( \n∗ \n<!-- image --> \n) \n( \n∗ \n<!-- image --> \n∗'} |
2024Natur.630..836W | Interactions between exoplanetary atmospheres and internal properties have long been proposed to be drivers of the inflation mechanisms of gaseous planets and apparent atmospheric chemical disequilibrium conditionsSUP1SUP. However transmission spectra of exoplanets have been limited in their ability to observationally confirm these theories owing to the limited wavelength coverage of the Hubble Space Telescope HST and inferences of single molecules mostly HSUB2SUBO ref. SUP2SUP. In this work we present the panchromatic transmission spectrum of the approximately 750 K lowdensity Neptunesized exoplanet WASP107b using a combination of HST Wide Field Camera 3 WFC3 and JWST NearInfrared Camera NIRCam and MidInfrared Instrument MIRI. From this spectrum we detect spectroscopic features resulting from HSUB2SUBO 21 CHSUB4SUB 5 CO 7 COSUB2SUB 29 SOSUB2SUB 9 and NHSUB3SUB 6. The presence of these molecules enables constraints on the atmospheric metal enrichment MH is 1018 solarSUP3SUP vertical mixing strength logSUB10SUBKSUBzzSUB 8.49.0 cmSUP2SUP sSUP1SUP and internal temperature gt345 K. The high internal temperature is suggestive of tidally driven inflationSUP4SUP acting on a Neptunelike internal structure which can naturally explain the large radius and low density of the planet. These findings suggest that eccentricitydriven tidal heating is a critical process governing atmospheric chemistry and interiorstructure inferences for most of the cool lt1000 K superEarthtoSaturnmass exoplanet population. | 2024-06-01T00:00:00Z | ['2024Natur.630..836W', 'arXiv:2405.11018', '2024arXiv240511018W', '10.1038/s41586-024-07514-w', '10.48550/arXiv.2405.11018'] | ['Astrophysics - Earth and Planetary Astrophysics'] | A high internal heat flux and large core in a warm Neptune exoplanet | 2,024 | 170 | 0.66 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML'] | 27 | https://arxiv.org/pdf/2405.11018.pdf | {'A High Internal Heat Flux and Large Core in a Warm Neptune Exoplanet': "Luis Welbanks 1* , Taylor J. Bell 2,3 , Thomas G. Beatty 4 , Michael R. Line 1 , Kazumasa Ohno 5,6 , Jonathan J. Fortney 5 , Everett Schlawin 7 , Thomas P. Greene 3 , Emily Rauscher 8 , Peter McGill 9 , Matthew Murphy 7 , Vivien Parmentier 10 , Yao Tang 5 , Isaac Edelman 2 , Sagnick Mukherjee 5 , Lindsey S. Wiser 1 , Pierre-Olivier Lagage 11 , Achr'ene Dyrek 11 and Kenneth E. Arnold 4 \n1 School of Earth and Space Exploration, Arizona State University, Tempe, AZ, USA. 2 Bay Area Environmental Research Institute, NASA's Ames Research Center, Moffett Field, CA, USA. \n- 3 Space Science and Astrobiology Division, NASA's Ames Research Center, Moffett Field, CA, USA. \n4 Department of Astronomy, University of Wisconsin-Madison, Madison, WI, USA. 5 Department of Astronomy and Astrophysics, University of California Santa Cruz, Santa Cruz, CA, USA. \n6 Division of Science, National Astronomical Observatory of Japan, Tokyo, Japan. 7 Steward Observatory, University of Arizona, Tucson, AZ, USA. \n8 Department of Astronomy, University of Michigan, Ann Arbor, MI, USA. 9 Space Science Institute, Lawrence Livermore National Laboratory, Livermore, CA, USA. 10 Laboratoire Lagrange, Observatoire de la Cˆote d'Azur, Universit'e Cˆote d'Azur, Nice, France. 11 Universit'e Paris-Saclay, Universit'e Paris Cit'e, CEA, CNRS, AIM, F-91191 Gif-sur-Yvette, \n- France. \n*Corresponding author(s). E-mail(s): [email protected] \nInteractions between exoplanetary atmospheres and internal properties have long been hypothesized to be drivers of the inflation mechanisms of gaseous planets and apparent atmospheric chemical disequilibrium conditions 1 . However, transmission spectra of exoplanets has been limited in its ability to observational confirm these theories due to the limited wavelength coverage of HST and inferences of single molecules, mostly H 2 O(ref. 2 ). In this work, we present the panchromatic transmission spectrum of the approximately 750 K, low-density, Neptune-sized exoplanet WASP-107b using a combination of HST WFC3, JWST NIRCam and MIRI. From this spectrum, we detect spectroscopic features due to H 2 O (21 σ ), CH 4 (5 σ ), CO (7 σ ), CO 2 (29 σ ), SO 2 (9 σ ), and NH 3 (6 σ ). The presence of these molecules enable constraints on the atmospheric metal enrichment (M/H is 10-18 × Solar 3 ), vertical mixing strength (log 10 K zz =8.4-9.0 cm 2 s -1 ), and internal temperature ( > 345 K). The high internal temperature is suggestive of tidally-driven inflation 4 acting upon a Neptune-like internal structure, which can naturally explain the planet's large radius and low density. These findings suggest that eccentricity driven tidal heating is a critical process governing atmospheric chemistry and interior structure inferences for a majority of the cool ( < 1,000K) super-Earth-to-Saturn mass exoplanet population. \nThe mass of WASP-107b is similar to Neptune (1.78 M N ), but its extreme low density is suggestive of a Hydrogen/Helium (H/He) envelope-to-core-mass ratio ( > 85%; ref. 5 ) more like Jupiter/Saturn (around 90%) than like Neptune/Uranus (5-15 % ; ref. 6 ). This high of an envelope mass fraction for such a low mass planet presents challenges to the standard coreaccretion paradigm of planet formation 7 -it is unclear how such a low mass planetary core (inferred to be < 5 M ⊕ ; ref. 5 ) could accrete such a massive gaseous envelope, but then stop short of fully growing into a 'Jupiter' 5 . An alternative hypothesis 4 is that tidal heating could inflate the planetary envelope, reducing the need for such a envelope-to-core-mass ratio. \nIn addition to mass and radius constraints, measurements of the atmospheric abundances of molecules containing carbon (C), oxygen (O), nitrogen (N), and sulphur (S) can be used to jointly constrain both scenarios. The abundances of molecules in the atmosphere will be set by the intrinsic elemental inventory and the chemical processes primarily driven to disequilibrium induced by transport and photochemistry 8, 9 . The former constrains the partitioning of material between the envelope and core, while the latter is sensitive to the amount of tidal heating, altering the deep atmosphere temperature and consequently, the molecular abundances at their quenched values 1 . One of the key challenges in interpreting exoplanet atmosphere compositions is disentangling the intrinsic elemental inventory from different chemical processes, given the presence (or lack-there-of) of molecular spectral features. \nInitial reconnaissance spectroscopy with the HST Wide Field Camera-3 (WFC3, 0.8-1.6 µ m) presented 2, 10 a muted water vapor absorption feature (1.4 µ m) and a surprising lack of methane (CH 4 ) absorption (1.6 µ m) - expected to be in abundance under solar elemental abundances and chemical equilibrium at temperatures below approximately 1,000 K (refs. 9, 11 ). The presence of water but lack of methane could indicate either a low carbon-to-oxygen ratio (C/O) envelope or else be due to the quenching of methane at deeper, hotter layers 1, 2 . The constraints on the water abundance were not precise enough to determine the bulk envelope metal enrichment. \nThe broad wavelength coverage (0.6-28 µ m) offered by JWST provides access to multiple molecular bands of the major C, O, N, and S species, \ncritical to enabling the precise atmospheric abundance constraints necessary for breaking the aforementioned degeneracies. As part of the MANATEE NIRCam+MIRI GTO program (JWST-GTO1185; ref. 12 ), we collected two new transit observations of WASP-107b using JWST NIRCam's F322W2 (2.4-4.0 µ m) and F444W (3.9-5.0 µ m) filters and grism 13 on 14 January 2023 and 4 July 2023, respectively. We analyzed the JWST/NIRCam observations with three separate data analysis pipelines ( Eureka! 14 , Pegasus (Beatty et al., in prep.), and tshirt 15 ) which all agree well within error (Fig. 1 and Extended Data Fig. 2); we ultimately chose to adopt Eureka! 's analysis for our fiducial NIRCam spectrum when performing atmospheric modelling. In addition, we incorporate recently published JWST/MIRI LRS observations 16 as well as the previously published HST/WFC3 G102 and G141 observations 2, 10 . To ensure a uniform set of orbital and limb-darkening parameters between the many different instruments, we re-analyzed the HST/WFC3 observations using Pegasus and the JWST/MIRI LRS observations using Eureka! . We find that these re-analyzed WFC3 and MIRI/LRS spectra are consistent with the literature spectra aside from a constant offset caused by the improved orbital solution. All together, these data give us a panchromatic dataset with observations spanning 0.8-12.2 µ m, with continuous coverage between 2.45 µ mand 12.2 µ m. \nAmong the chemical species expected in exoplanet atmospheres 9, 17 (see Methods), the final NIRCam spectrum shown in Figure 2 shows prominent absorption features due to H 2 O (2.5-3.2 µ m, detected at 21 σ ), CO 2 (2.66-2.86 µ m and 4.24.5 µ m, detected at 29 σ ), CO (4.5-4.9 µ m, detected at 7 σ ), and SO 2 (3.94-4.1 µ m, detected at 9 σ ), with weaker features due to CH 4 (3.2-3.5 µ m, detected at 5 σ ) and NH 3 (2.9-3.1 µ m, detected at 6 σ ), with reported detections from the 1-dimensional radiativeconvective-photochemical equilibrium models (1DRCPE) described below. These features are complemented by two additional H 2 O bands in HST/WFC3 (around 1.13 µ m and 1.4 µ m) and another in MIRI (between 5-7 µ m), along with another strong SO 2 feature (7.1-7.7 µ m) and a weak NH 3 feature (10.311 µ m) in MIRI, shown in Figure 3. The panchromatic spectrum exhibits a slight downward slope from blueto-red, indicative of aerosol scattering 18, 19, 20 , and a strong concavity across MIRI, previously attributed to silicate cloud particulate resonance features 16 . The \nFig. 1 | Spectroscopic and broadband NIRCam lightcurves of the transit of WASP-107b. The raw spectroscopic F322W2 and F444W transit lightcurves are shown in panels a and b after spectral binning (0.015 µ m bins) but without any temporal binning. Masked values (for example, cosmic rays) have been colored black. Even from these raw data, it is possible to visually identify the increased transit depth ≲ 3 µ m primarily caused by H 2 O, from 3.9-4.1 µ m caused by SO 2 , and from 4.3-4.5 µ m caused by CO 2 . The F322W2 broadband (2.45-3.95 µ m) and F444W broadband (3.89-4.97 µ m) transit lightcurves are shown panels c and d with gray points without error bars. Black points with 1 σ error bars show temporally binned data with a cadence of 5 minutes; note that the error bars are typically smaller than the point size. BJD TDB is the date in the Barycentric Julian Date in the Barycentric Dynamical Time system. \n<!-- image --> \nFig. 2 | Independent reductions of the transmission spectrum of WASP-107b. a, The NIRCam F322W2 and F444W transmission spectra as reduced by the Eureka! , Pegasus , and tshirt pipelines are shown in different colors at a constant wavelength binning of ∆ λ = 0 . 015 µ m and with 1 σ error bars. All three analyses show clear agreement on the overall shape of the spectrum, with features from H 2 O ( ∼ 2.5-3.5 µ m), CH 4 ( ∼ 3.2-3.8 µ m), SO 2 ( ∼ 3.9-4.1 µ m), and CO 2 ( ∼ 4.2-4.6 µ m) all clearly visible by-eye in each of the reductions. b, The absorption cross sections of the six detected species, many of which are visually identifiable in panel a . \n<!-- image --> \npresence of both CO 2 and SO 2 features is indicative of envelope metal enrichment above solar and photochemistry 21 , while the relatively weak CH 4 feature is suggestive of CH 4 depletion 2, 10, 16 . \nTo rigorously infer the atmospheric composition and internal temperature from the observed transmission spectrum, we employ Bayesian inference with the transmission spectra generated from a suite of 1-dimensional radiative-convective-photochemical equilibrium models (1D-RCPE) 11 (see Methods). This self-consistent method properly captures the degeneracies between the intrinsic atmospheric composition and chemical processes. The final result from this process are samples from a posterior-probability distribution that represents the constraints on T irr , T int , [M/H], C/O, and cloud properties, and an estimate of the Bayesian model evidence, which we use to draw conclusions about WASP-107b's atmosphere. Figure 3 shows that this model setup adequately explains the observed panchromatic spectrum, capturing nearly all of the salient features. From this process we find a metallicity of 10 -18 × solar 3 (at 68 % confidence, median of 12 . 3 × solar) and a sub-solar carbon-tooxygen ratio (C/O = 0 . 33 +0 . 06 -0 . 05 ). Furthermore, the \nFig. 3 | Interpretation of WASP-107b's transmission spectrum. The observed transmission spectrum of the planet with 1 σ error bars is compared to the best-fit one-dimensional radiative-convective-photochemical equilibrium models (1D-RCPE model, χ 2 /N data =1.4) shown at a spectral resolution of R=300. The colored shaded regions show the contributions from individual gases to the best fit model. The gray dashed line shows the clear atmosphere component of the model, that is, the gas contributions without the presence of clouds or hazes. Panel a shows the broad wavelength spectrum (0.8-12 µ m) of the planet. Panel b shows the fit residuals with the corresponding 2 σ data error envelope. Panel c shows the NIRCam observations, while d shows HST WFC3 spectra and e shows JWST MIRI observations. The HST/WFC3 data are binned at a constant ∆ λ = 0 . 025 µ m, the NIRCam data are binned at ∆ λ = 0 . 015 µ m, and the MIRI data are binned at ∆ λ = 0 . 15025 µ m. \n<!-- image --> \nspectrum also requires a high internal temperature (Tint > 345 K at 99 . 7% confidence, while a value of < 100 K would be expected 4 given the planet's low mass and the star's ∼ 3 Gyr age 5 ), and an atmosphere with a strong enough vertical mixing (that is, eddy diffusion, log( K zz ) = 8 . 6 +0 . 4 -0 . 2 ) to quench methane along the deeper (0.25-0.65 bar) and hotter (around 1,1001,300 K) parts of the atmosphere, confirming previous suggestions 2, 16 . \nWe further validate the interpretation from the self-consistent 1D-RCPE models by performing additional Bayesian inferences using parametric atmospheric models that independently fit for the chemical abundances and vertical pressure-temperature structure of the planetary atmosphere without any assumptions of radiative-convective thermo-chemical equilibrium (called, 'free-retrievals'). Using two independent inference frameworks (Aurora and CHIMERA, see Methods, detection significances from CHIMERA) we confirm the detections of CO 2 (27 σ ), H 2 O (18 σ ), SO 2 (8 σ ), CO (5 σ ), NH 3 (5 σ ), and CH 4 (8 σ ) at strong \nconfidence in agreement with the 1D-RCPE models. We find that the detection of NH 3 is mostly driven by the NIRCam observations and do not depend solely on the red-edge of the MIRI observations (see Methods). The derived chemical abundances, assumed to be constant with height by these models, are also consistent with the inferred abundance profiles from the 1D-RCPE models (see Fig. 4) and are also suggestive of a metal enriched envelope with depleted CH 4 abundances. \nThe relatively high internal temperature we infer for WASP-107b is likely caused by tidal heating in the planetary interior. Recent radial-velocity observations measured the planet to have a mildly non-zero orbital eccentricity of e = 0 . 06 ± 0 . 04 (ref. 5 ). Tidal heating relations 22 predict that at this nominal e = 0 . 06 the internal temperature of WASP-107b will be Tint ≈ 350 K for a Neptune-like tidal quality factor of Q = 10 4 . The expected internal temperature from tidal heating drops to Tint ≈ 200 K to Tint ≈ 110 K for Jupiter-like tidal quality factors of Q = 10 5 \nFig. 4 | Inferred molecular volume mixing ratios from WASP107b's transmission spectrum. The retrieved volume mixing rations (that is, abundances) for each detected gas (panels a -f ) are shown from the free-retrieval (gray posterior distributions, with median and 68% confidence error bars) and from the 1D-RCPE grid inference (median and 68% confidence profiles shown as colored pressure-dependent abundance profiles). The black solid lines show expectations for an atmosphere in chemical equilibrium under the inferred [M/H] and C/O. The dotted black lines show expectations for chemical disequilibrium under a cool internal temperature of 200 K. The self-consistent abundance profiles from the 1D-RCPE retrieval for a high internal temperature are generally consistent with the retrieved abundances from the free retrieval within 1 σ . The red horizontal shaded region spans pressures from 1mbar to 10 -5 bar and corresponds to the pressures probed by our observations. The colors of the abundance profiles match the contributions in Figure 3 and the cross-sections in panel b of Figure 2. \n<!-- image --> \nto Q = 10 6 . Additional heating of the deep atmosphere may be achieved by vertical and horizontal mixing 23, 24, 25, 26 . If we assume that tidal effects are solely responsible for heating WASP-107b's atmosphere, then our retrieved 3 σ lower limit of Tint > 345 K implies a corresponding 3 σ upper limit on the planet's tidal quality factor of Q < 10 3 . 8 . If WASP107b has a tidal quality factor significantly lower than Neptune ( 10 3 . 9 ≲ Q N ≲ 10 4 . 5 ), 27 this would be consistent with the inference of a large core for the planet, since typical core material for ice giants is expected to have Q ≈ 10 2 , while typical gas envelopes have Q ≈ 10 5 28 \nThe large value of T int implies a much hotter, lower-density H/He envelope that has been previously appreciated. Using state-of-the-art structure models 29 , along with T int = 350 K (the retrieved 3 σ lower limit), we find that the planet's radius can be explained by a model that has > 22 M ⊕ of rock/iron in its interior, here modeled as a distinct core. This ratio of solids to H/He is similar to that of the bulk composition of Uranus and Neptune in our solar system 30 . Our \nrevised view of the planet is quite different than previous work 5, 31 , as without tidal heating the planet's low bulk density can only be explained by a structure that is mostly H/He with a very small core ( < 5 M ⊕ , ref 5 ), perhaps at odds with core-accretion theory. \nThe inferred atmospheric metallicity of WASP107b is lower than expectations from the Solar System metal enrichment trend 32 as determined by the CH 4 abundances 33 , which predicts an enhancement of about 32 × solar (see also ref. 34 ). Our results confirm that different elements in a planetary atmosphere can be differently enhanced, as previously suggested by HST observations 32 . Furthermore, our results demonstrate that individual molecules (for example, H 2 O) may not be good bulk metallicity tracers. Instead, JWST broadband spectra as presented here, presents a key opportunity to derive estimates of the bulk atmospheric metal enrichment informed by the measured abundances of several gases and by self-consistently considering the impact of disequilibrium processes arising from interactions between the interior and atmosphere of the planet. \nThe understanding of vertical mixing in the atmospheres of giant planets and brown dwarfs remains a significant challenge in atmospheric physics 1, 35 . The inferred vertical mixing strength ( K zz ) can be high ( K zz ∼ 10 8 -10 11 cm 2 s -1 , depending on T int ) in convective zones where mixing is driven by convective overturn and mixing length theory can be used, or much more uncertain in radiative zones where the driving mechanism remains unclear 36, 37 . Recent studies of mixing in brown dwarf atmospheres, at temperatures similar to WASP-107b 35, 38 , infer low K zz values of 10 2 -10 5 cm 2 s -1 due to sluggish mixing in deep atmospheric radiative zones where chemical abundances are quenched. Studies of this same phenomenon in Jupiter have long suggested a K zz value of ∼ 10 8 -10 9 cm 2 s -1 , although recent work 39 points to a lower value ( K zz ∼ 10 6 cm 2 s -1 ) as the best explanation for Jupiter's CO and H 2 O abundances, again perhaps due to a deep radiative zone. Overall, the high K zz values implied for WASP-107b are striking in comparison to other objects. This suggests the quenching of chemical abundances in an atmospheric convective zone, with a high internal flux and shallow radiative-convective boundary. This can only be achieved if significant additional internal energy, as from tidal heating, keeps the interior and deep atmosphere much warmer than a standard cooling model would suggest. \nThe detection of the major C, O, N, and S reservoirs in the spectrum of WASP-107b resulting from the broad wavelength coverage demonstrates the unparalleled capabilities of JWST for the detailed atmospheric characterization of exoplanet atmospheres. The simultaneous constraints on multiple chemical species enables unique solutions for the bulk atmospheric metal enrichment, strength of vertical mixing, and provides strong evidence for a high internal temperature most likely arising from tidal heating due to a small, yet significant, orbital eccentricity. The combination of constraints on atmospheric metallicity and the internal heat flux together provide novel constraints on the relative gas-to-core mass fraction, naturally explaining the low planetary density. These initial constraints on the elemental ratios of C, O, S, and N have the potential to inform the planetary accretion history 40 . \nThe detection and constraint on the abundance of CH 4 in this warm Saturn adds to the growing number of inferences of this sought-after carbon-bearing species in transiting exoplanets 11, 41 . Furthermore, the low abundance of methane relative to thermochemical expectations confirms its sensitivity to mixing processes and the internal temperature/heat flux 1 , providing a critical atmospheric diagnostic of interior processes and structure. We anticipate that most common type of planets across the galaxy will have elevated internal temperatures arising from tidal heating, and thus should have relative depletions of methane. Of the 262 known cool ( < 1,000 K) 'Neptune-like' 42 planets (with well determined mass and radii), approximately 2/3 have a reported non-zero eccentricity 43 . 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Computing in Science & Engineering 9 (3), 90-95 (2007). \nAcknowledgments. L.W. acknowledges support for this work provided by NASA through the NASA Hubble Fellowship grant #HST-HF2-51496.001-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-26555. T.P.G. and T.J.B. acknowledge support from NASA JWST WBSs \n411672.07.04.01.02 and 411672.07.05.05.03.02. P.M. acknowledges that this work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. The document number is LLNL-JRNL-859050. M.R.L. acknowledges NASA XRP award 80NSSC19K0446 and STScI grant HSTAR-16139. M.R.L. and L.W. acknowledge Research Computing at Arizona State University for providing HPCand storage resources that have significantly contributed to the research results reported within this manuscript. L.W. thanks Michiel Min for meaningful conversations. K.O. acknowledges support from JSPS KAKENHI Grant Number JP23K19072. M.M. acknowledges funding from NASA Goddard Spaceflight Center via NASA contract NAS5-02105. A.D. and P-O.L. acknowledge funding support from CNES. We thank Marcia Rieke for allocating the NIRCam time for this program. \nAuthor Contribution. L.W. led the modeling analysis effort, performed the atmospheric modeling using free-retrievals and 1D-RCPE models, contributed to the theoretical analysis/interpretation of the observations, led the cross-validation analysis, and led the writing of the manuscript. T.J.B. contributed the fiducial Eureka! analyses of the NIRCam and MIRI observations, verified the observing parameters of the NIRCam observations, and contributed to the text. T.G.B. contributed the Pegasus analyses of the HST and NIRCam observations, verified the observing parameters of the NIRCam observations, performed the SED fitting and radius estimation, performed the tidal heating analysis, and contributed to the text. M.R.L. performed the 1D-RCPE simulations, wrote introductory text and methods, and helped sculpt the direction of the manuscript. K.O. provided comments on the manuscript and helped with interpreting the NH3 detection. J.J.F. contributed to the planet structure models, contributed to the interpretation of the internal temperature of the planet, and contributed to the text. E.S. simulated the planet and retrievals before launch, help planned the observation specifications, did the tshirt reduction and light curve fitting, and contributed to the text. T.P.G. selected the planet for observation, designed the observational program, directed some of the analysis, and commented on the manuscript. E.R. provided feedback on the modeling interpretation and provided comments on the manuscript. P.M. contributed the cross-validation analysis and to the text. V.P. provided \nfeedback throughout the project and helped with the vertical mixing discussion. Y.T. contributed to the planet structure models. M.M. contributed to deriving the planet's orbital parameters and feedback on the text. I.E., S.M., L.S.W., and K.E.A. provided useful discussions throughout the manuscript process. PO.L. designed the MIRI observational program and provided its data. A.D. contributed to the MIRI data analysis. \nCompeting Interests Statement The authors declare no competing interests.", 'NIRCam Reduction': 'Both of our new NIRCam observations used the BRIGHT2 readout pattern and the SUBGRISM256 subarray, with the F322W2 observations using 7 groups per integration and 1,293 integrations while the F444W observations used 15 groups per integration and 625 integrations. In the subsections below, we will describe the three independent pipelines we used to reduce these NIRCam data.', 'Eureka!': "In Eureka! 's Stage 5, we fitted astrophysical and systematic noise models to our spectroscopic lightcurves. For both NIRCam filters, our systematic noise model consisted of a quadratic trend in time and a linear decorrelation as a function of the spatial position and PSF-width measured during Stage 3. For MIRI/LRS, we used a linear trend in time, an exponential ramp in time, and a linear decorrelation as a function of the spatial position and PSF-width. For all observations, our astrophysical model consisted of a starry 65 transit model with quadratic limb-darkening parameters fixed to an ATLAS model limb-darkening spectra 66 as computed by the Exoplanet Characterization Toolkit (ExoCTK) 67 . We fixed all of our orbital parameters to those of Murphy et al. (in prep.; see the Derivation of Orbital Parameters section above) and adopted a minimally informative prior on the planet-to-star radius ratio. \nFollowing the recommendations of ref. 53 , we removed the first 800 integrations of the MIRI/LRS observations to remove the worst part of the initial ramp in the lightcurve. We also removed integrations 475-490 from the NIRCam/F444W observations which exhibited a brief spike in noise. We removed no further integrations from the NIRCam/F322W2 observations. Finally, we also fitted a white noise multiplier to account for excess white noise. For all our observations, we used PyMC3 's No U-Turns Sampler 68 using two independent chains, each taking 4,000 tuning \ndraws and then 3,000 posterior samples with a target acceptance rate of 0.85. We then validated that the chains had converged by ensuring the Gelman-Rubin statistic 69 was at or below 1.1. The 16th, 50th, and 84th percentiles of the posterior samples were then used to estimate the best-fit values and uncertainties for each fitted parameter. \nFor the NIRCam/F322W2 observations, we find that the white noise is on average about 15% above our estimate of the stellar photon-limited noise floor in each channel, while F444W exhibited only white noise only about 7% above the estimated photon limit on average. This excess noise is likely the result of insufficiently corrected 1/ f noise 50 which gets partially converted to white noise in the time-series when spectroscopically binning the lightcurves. For MIRI/LRS, we used the estimated gain of 3.1 electrons/DN from ref. 53, 58 , as the current value of 5.5 electrons/DN used in the CRDS reference files is known to be incorrect. With this estimated gain, we find that our fitted white noise level is only about 10% above the estimated stellar photon noise limit near 5 µ m but climbs steadily to approximately 50% above the limit near 10 µ m and then climbs steeply to around 140% above the limit near 12 µ m. Part of this excess longwavelength white noise is likely due to the impact of background noise as these limits only considered the stellar photon noise. While neither of our NIRCam observations show evidence for residual red-noise in their Allan variance plots 70 , several of our MIRI/LRS channels do show moderate amounts of residual rednoise. Following the β error inflation method developed by ref. 71 , we computed β , the ratio of the median value of our Allan variance plots at 15-23 minute timescales (around WASP-107b's approximately 19minute transit ingress/egress duration) to the value expected for white noise, and multiplied our transmission spectra uncertainties by this amount. This resulted in an increase in the uncertainties by about 50% for wavelengths ≲ 6.5 µ m and 0-30% for longer wavelengths. After this, we arrived at our final 2.4512 µ mtransmission spectra.", 'Pegasus': 'We fit the F322W2 and F444W spectroscopic lightcurves from our Pegasus reduction using a BATMAN 63 transit model. We fixed most of the transit model parameters to the values measured from broadband fits to the JWST data (Murphy et al., in prep.; see the Derivation of Orbital Parameters section \nabove), leaving only the planet-to-star radius ratio as the only free astrophysical parameter. We also simultaneously fit for a background quadratic trend across each visit, which had two associated slope parameters and a normalization factor. We did not impose a prior on any of these four parameters, and we fit each spectroscopic channel individually. For each channel, we used a quadratic limb-darkening law and fixed the limb-darkening coefficients to the values calculated from the ATLAS model 66 by ExoCTK 67 in the F322W2 and F444W bandpasses. To generate the limb-darkening coefficients we used the spectroscopic stellar properties measured in previous work 5 . \nTo perform the lightcurve fit in each spectroscopic channel, we performed an initial Nelder-Mead likelihood maximization followed by MCMC likelihood sampling. We initialized the MCMC chains about the maximum likelihood point estimated from the Nelder-Mead maximization. We used the emcee 64 Python package to perform the MCMC sampling, using twenty walkers with a 2,000 step burn-in and a 4,000 step production run. At the end of this process we checked that the Gelman-Rubin statistics was below 1.1 for each parameter in each spectral channel to ensure the MCMC sampling had converged. \nWe performed two tests to evaluate the goodnessof-fit of our transit modeling in each spectral channel. First, we confirmed that the per-point flux uncertainties the lightcurves were consistent with the standard deviation of the bestfit model residuals. Second, we calculated the Anderson-Darling statistic for the residuals to verify that they followed a Gaussian distribution. Our spectroscopic lightcurve fits did not show any evidence of non-Gaussianity in the residuals. \nThe estimated depth uncertainties were 1.15 × the photon noise expectation in the F322W2 data and 1.06 × the photon noise expectation in the F444W data.', 'tshirt': 'We followed a similar procedure as previous NIRCam light curve fitting 11 , 47 . We use the starry 65 code to model the lightcurves and pymc3 68 to sample the posterior distributions of the fits. We adopted uninformative priors on the stellar limb darkening with a 2 parameter quadratic law 72 . This allows for comparison of results with the ATLAS stellar limb darkening models. The astrophysical starry model of the lightcurve is multiplied by an second order (quadratic) polynomial baseline to allow for trends \nfrom instrument and astrophysical stellar variability. We bin the spectra in the time axis to 300 bins for faster lightcurve evaluation with starry . Finally, we include a parameter for the standard deviation of the data to allow for excess noise beyond the theoretical photon and read noise. We sampled the posterior with No U-Turns sampling 68 . We used 3,000 tuning steps, 3,000 sampling steps, and 2 chains with a target acceptance rate of 90%.', 'HST/WFC3 Re-reduction': 'HST previously observed transits of WASP-107b with Wide-Field Camera 3 (WFC3) using the G141 and G102 dispersers. The results of these observations have been reported previously 2, 10 . For consistency with the rest of our data analysis, we re-reduced and re-analyzed these data. \nThe G141 data cover a single transit of WASP107b on UT 6 June 2017 using five orbits of HST time as a part of HST program GO-14915 (P.I. L. Kreidberg). At the start of each orbit, HST took a single F130N direct image of WASP-107 to establish a wavelength reference, and then the observatory switched to spatially scanned spectroscopic observations using the G141 grism. The full details of the observational setup are described in the original report on the results of this observation 2 . \nThe shorter wavelength G102 data also cover a single transit of WASP-107b using five orbits of HST time on UT 31 May 2017 as a part of HST program GO-14916 (P.I. J. Spake). Each orbit of the G102 transit observation began with an F126N direct image to create a wavelength reference, and then the G102 grism was used to collect spatially scanned spectroscopy of the WASP-107 system. The complete description of these observations is reported in the original paper describing these data 10 . \nWe reduced both transit observations using a custom data reduction pipeline that has been described elsewhere 52 and which we briefly summarize here. For each spatially scanned exposure, we began by \nextracting 1D spectra for each of the individual upthe-ramp samples, accounting for both the residual background light and the slight angle of the scan pattern on the detector. We then summed these individual spectra to create the 1D spectrum from that exposure. We used the wavelength solutions generated from the direct images taken at the start of each orbit to extract spectroscopic lighturves in wavelength bins matching those used in the previous works on these data 2, 10 . These were nine evenly spaced bins from 0.877-1.139 µ m for the G102 data and twenty evenly spaced bins from 1.121-1.629 µ mfor the G141 data. We note that with the updated transit ephemeris for WASP-107b from these observations, we do not see evidence for a starspot crossing in the WFC3 data as was reported previously, 2 nor do we see starspot crossings in the NIRCam or MIRI observations.', 'MIRI Re-reduction': "We decided to independently re-reduce the MIRI/LRS spectra of ref. 16 to ensure the reduction was robust to different reduction choices and to ensure a selfconsistent reduction of the NIRCam and MIRI data. Our Eureka! re-reduction of the MIRI/LRS observations used version 0.9 of the Eureka! pipeline 14 , CRDS version 11.17.0 and context '1097', and jwst package version 1.10.2 (ref. 46 ). Our reduction method generally follows the Eureka! v1 method described by ref. 53 , with some differences and experimental new steps. The Eureka! Control Files and Eureka! Parameter Files we used are available for download from Zenodo (https://doi.org/10.5281/zenodo. 10780448; ref. 48 ), and the important parameters are summarized below. \nIn our Stage 1 processing, we turned on the firstframe and lastframe steps (which respectively discard the first and last frames in each integration) in the jwst pipeline and increased the jump rejection threshold to 7 σ from 4 σ as we found these changes resulted in reduced scatter in the residuals of our lightcurve fits. We had also considered different combinations of turning on/off the firstframe, lastframe, and RSCD correction steps but found that it was best to turn the firstframe and lastframe steps on and leave the RSCD step off for these data. We also investigated using the default CRDS linearity reference file vs the linearity reference file used by ref. 16 and found that there was no clear impact on our final transmission spectra, so we adopted the linearity reference file used by ref. 16 . \nAs pointed out by ref. 53 , it appears that narrow wavelength bins result in excessively noisy transmission spectra and underestimated error bars for the MIRI/LRS time-series observation (TSO) commissioning target L168-9b. Ref. 53 hypothesized that this wavelength-correlated noise could be the result of 390 Hz periodic noise observed in some MIRI subarrays (ref. 54 ; private comm., Michael Ressler). This noise appears as structured noise with a period of around 9 rows within an individual group and is likely caused by MIRI's electronics. To investigate the potential impacts of this structured noise on MIRI time-series observations, we developed a experimental 390 Hz noise removal method for the Eureka! pipeline. This method takes advantage of the knowledge that the noise exhibits a fixed waveform with the phase of the periodic noise remaining fixed within an integration but varying between integrations (only varying slightly and with an apparent cycle of 8 integrations); we determined the bestfit waveform using an entire segment of our science observations, and then we freely fit the phase shift of the waveform for each integration. We found that there is significant power in the 1st, 2nd, and 4th harmonics with the 1st harmonic's frequency being exactly 390.625 Hz ( f =1/(256 × 10 µ s); private comm., Michael Ressler). We found that our new algorithm significantly reduced the periodic noise visible in a Lomb-Scargle periodogram 55, 56 at the group level. While performing this 390 Hz noise removal step, we also performed group-level background subtraction (GLBS) for each row in each group (MIRI's dispersion direction is along a column, so rows run in the spatial direction) using the mean value of columns 11-30 and 44-63 (avoiding pixels contaminated by the star); this further removed structured noise from the Lomb-Scargle periodogram. To understand the impact on our final transmission spectra, we also performed another reduction without the GLBS step and another without either the 390 Hz noise removal or GLBS steps. \nIn Eureka! 's Stage 2, we turned off the photom step as flux-calibrated spectra are not desired for time-series observations. In Stage 3 we performed a double-iteration 5 σ clipping along the time-axis for background pixels, a single iteration of 5 σ clipping along the spatial axis for each wavelength, and integration-level background subtraction using pixel columns 11-61 (excluding pixels within 10 pixels of the spectral trace) for each wavelength. Similar to the \n<!-- image --> \nExtended Data Fig. 1 | MIRI/LRS's 390Hz noise signal visualized. a: The background pixel values for part of the second integration of the MIRI/LRS observations of WASP-107b are shown in black points, while our fitted 390 Hz noise signal is shown with a red line. b: The LombScargle periodogram of the pixels in the second group of the second integration before the 390 Hz noise removal and GLBS steps are applied are shown in red, and the periodogram after these two steps are applied is shown in black. The three harmonics used in our cleaning procedure (1st, 2nd, and 4th) are indicated with pale blue vertical lines, while skipped 3rd harmonic is shown with a pale orange vertical line. The clear impact of the 390 Hz noise removal and GLBS steps is seen in the eliminated spikes in the Lomb-Scargle periodogram. \n<!-- image --> \nNIRCam spectra, we then performed optimal spectral extraction 49 using the pixels within 4 pixels of the spectral trace using a cleaned median integration to compute our spatial profile (clipping 5 σ outliers along the time axis, smoothing with a 7-pixel wide boxcar filter, and clipping pixels that differed by more than 10 σ with respect to the spatial profile). In Stage 4, we first removed three wavelengths which exhibited excessive noise in their lightcurves. We then binned the spectra a similar binning scheme as that used by ref. 16 (47 spectral bins spanning approximately 512 µ mwith a constant width of 0.15025 µ m); notably, however, we did not use wavelengths below 5 µ m which we expect to be significantly contaminated by light from around 3 µ m (ref. 57 ). We do extend our reduction out 12 µ m as was done by ref. 16 since we do not find evidence for the 'shadowed region effect' reported by refs. 53, 58 in these observations (consistent with the claim that not all observations appear to be affected by this as-yet unexplained phenomenon). Finally, as with NIRCam, we clip 4 σ outliers in the spectrally binned lightcurves compared to a cleaned version computed using a boxcar filter with a width of 20 integrations.", 'Derivation of Orbital Parameters': "All of the light curve fits to the NIRCam, HST, and MIRI data described above fix WASP-107b's orbital parameters to those derived by Murphy et al. (in prep.). We summarize this derivation here. \nTo precisely determine WASP-107b's orbit, we simultaneously fit observations of three transits of WASP-107b from TESS, one from the Goodman High Throughput Spectrograph 59 (imaging mode, SDSS iband) on the Southern Astrophysical Research Telescope (SOAR), one from JWST/NIRCam F210M (observed simultaneously with our NIRCam F322W2 data), one from Spitzer/IRAC with IRAC's channel 2 (publicly available from Program 13052, PI: M. Werner), and one from JWST/MIRI LRS (broadband, observation described above), as well as radial velocity (RV) observations from CORALIE 44 and Keck/HIRES 5,60 . Together, these transit observations probe a wide range of wavelengths ( ∼ 0.6-12 µ m) and cover a baseline spanning roughly six years, while the radial velocity observations sample WASP107b's entire orbit. We used the default TESS pipeline reduction, reduced the SOAR/Goodman data using AstroImageJ 61 , reduced the NIRCam/F210M data using tshirt (as described above), reduced the Spitzer/IRAC data using a custom photometry pipeline (described in ref. 62 ), and reduced the MIRI/LRS data using Eureka! (as described above). For the RV data, we use the measurements as tabulated in ref. 5 . \nWe model each of the transits using a batman transit model 63 . For all transit observations except for SOAR/Goodman, we also fit for a background, visit-long linear trend in flux versus time. The SOAR/Goodman data exhibited significant non-linear background trends due to telluric variations during the \nobservation, which we corrected for using a Gaussian Process model. We fit all the data described above simultaneously using MCMC sampling with emcee 64 , sampling the time of conjunction, orbital period, inclination, semi-major axis, eccentricity, argument of periastron, planet-star radius ratios in each bandpass, quadratic limb-darkening coefficients in each bandpass, RV semi-amplitude and system velocity, and the slope and intercept of each visit's linear background trend. We ran this sampling for 10,500 steps, which was sufficient for each parameter to converge ( > 25x the average auto-correlation times, plus we visually inspected each MCMC walker time series). The best-fit parameter values are given in Murphy et al. (in prep.) and were P = 5.72148722 ± 3 × 10 -7 days, t 0 = 2,459,958.747244 ± 8 . 2 × 10 -6 BJDTDB, i = 89.57 ± 0.03 degrees, a/R ⋆ = 18.05 ± 0.1, e = 0.05 ± 0.01, and ω = -2.3 ± 6.1 degrees.", 'Comparing NIRCam Reductions': "A motivation for conducting three different reductions of the NIRCam data was to check the robustness of our final transmission spectrum against different choices and assumptions made during the image calibration, spectral extraction, and lightcurve fitting stages of our analyses. Generally, we find that all three reductions give the same general transmission spectrum with approximately the same depth uncertainties (Fig. 2 and panel a of Extended Data Fig. 2). The mean depth uncertainty across the entire NIRCam wavelength range is approximately 90 ppm at our constant wavelength ∆ λ = 0 . 015 µ m binning, which is larger than the mean point-to-point difference between the Eureka! and Pegasus spectra (58 ppm) and the Eureka! and tshirt spectra (72 ppm). A χ 2 comparison of the three spectra gives χ 2 / dof = 0 . 74 for Eureka! vs. Pegasus , and χ 2 / dof = 0 . 64 for Eureka! vs. tshirt . The larger difference between Eureka! and tshirt is largely because tshirt 's F322W2 spectrum consistently falls below that of Eureka! and Pegasus at wavelengths longer than ∼ 2.893 µ m; this difference may be caused by tshirt 's ROEBA method since the deviation occurs right at an amplifier boundary (see panel a of Extended Data Fig. 2). In both pipeline comparisons, the small reduced chi-squared values imply a < 0.1 σ likelihood that the spectra are drawn from two different underlying distributions for our 172 degrees of freedom. Since the choice of reduction methods did not significantly impact our final NIRCam spectra, we chose the Eureka! reduction as our fiducial NIRCam spectra for all subsequent atmospheric modelling work. \nTo further investigate the sensitivity of our results to different limb-darkening assumptions, we also ran four separate fits with our fiducial Eureka! pipeline; namely, fixing our limb-darkening parameters to (1) ATLAS-predicted quadratic law coefficients \nExtended Data Fig. 2 | Comparison of different NIRCam reductions and limb-darkening choices. a, The NIRCam F322W2 and F444W transmission spectra as reduced by the Eureka! , Pegasus , and tshirt pipelines are shown in different colours after having been smoothed by a 4-point wide boxcar filter. Both the Eureka! and Pegasus reductions fixed limb-darkening coefficients to ATLAS model predictions, while tshirt freely-fit quadratic coefficients. At wavelengths longer than ∼ 2.9 µ m, tshirt 's F322W2 spectrum consistently falls below Eureka! and Pegasus 's spectra; this difference is caused by a combination of tshirt 's ROEBA method (which results in a roughly constant offset across F322W2) and tshirt 's free limb-darkening (which results in a small slope across F322W2). b, A similar plot, demonstrating the minimal impacts of varying limb-darkening choices on our final spectra produced by Eureka! . \n<!-- image --> \ncomputed using ExoCTK (refs. 66, 67 ), (2) STAGGERpredicted quadratic law coefficients computed using ExoTiC-LD (refs. 73, 74 ), (3) STAGGER-predicted 4parameter law coefficients computed using ExoTiCLD (refs. 73, 74 ), and (4) freely fitting quadratic law coefficients using the reparameterization of ref. 72 for efficient and physical sampling. As shown in panel b of Extended Data Figure 2, none of these spectra significantly differ from each other. In particular, we find that comparing the transmission spectrum of our fiducial analysis which used the ATLAS quadratic limbdarkening coefficients to fits with limb-darkening choices only results in a reduced chi-squared value of 0.02 for STAGGER's quadratic model, 0.09 for STAGGER 4-parameter model, and 0.28 for the fit with freely-fit reparameterized quadratic limbdarkening coefficients. In all three cases, this means that the differences between the spectra are much less than the uncertainties on the measured depths. We also further disfavour the freely-fit limb-darkening coefficients as these fitted coefficients exhibited substantially more point-to-point scatter than would be expected for a physical limb-darkening spectrum.", 'HST/WFC3': "We fit each of the twenty-nine G102 and G141 spectroscopic transit lightcurves using a BATMAN 63 transit model and standard detrending techniques for WFC3 timeseries data 52 . As with the JWST spectroscopic \ndata, for these fits, we fixed the transit center time, orbital period, orbital inclination, the scaled semimajor axis, the orbital eccentricity, and the argument of periastron to previously measured values (Murphy et al., in prep.; see the Derivation of Orbital Parameters section above). We also fixed the quadratic limbdarkening coefficients in each spectroscopic channel to the values estimated from the ATLAS limbdarkening model 66 by the Exoplanet Characterization Toolkit 67 . This left the planet-to-star radius ratio as the only free astrophysical parameter. We fit each spectroscopic channel individually. \nAll the G102 and the G141 lightcurves showed the usual 'fishing-hook' systematic trend within each orbit and the background linear slope typical for WFC3 timeseries observations. We modeled this in each spectral channel as a part of the transit fitting process using an exponential ramp within each individual orbit, and a linear background trend across each visit, of the form \nF detrend = ( mt V + n ) ( 1 -Ae t O τ ) . (1) \nHere, t O is the time since the start of each orbit's observations, t V is the time since the start of the visit, and A and τ , and m and n are fitting coefficients for the exponential ramp and linear trend respectively. In addition to fitting the systematics in this way, we also \ndid not use the data from the first orbit within each visit nor the first exposure from each orbit when fitting the data. \nTo fit the transit lightcurve in each spectroscopic channel, we first performed a Nelder-Mead likelihood maximization to find an initial bestfit. We then used the emcee 64 Python package to perform an MCMC exploration of the surrounding likelihood space to improve this bestfit estimate. In each channel we used twenty MCMC walkers, each with a 1,000 step burnin followed by a 4,000 step production run. At the end of this we judged the MCMC to have converged by verifying that the Gelman-Rubin statistic for each parameter was below 1.1. We also checked to ensure that the median per-point flux uncertainty matched the standard deviation of the residuals to the bestfit lightcurve model in each channel. We additionally performed an Anderson-Darling test on each channel's residuals to check for non-Gaussianity. The lightcurve residuals in each channel appear well-behaved. \nWe note that we do not see evidence for a starspot crossing in the third orbit of the G141 transit data, as was suggested previously 2 . Instead, the updated transit ephemeris and orbital properties we measure using the JWST data for WASP-107b serve to shift the start of transit egress slightly earlier at the time of the G141 observations. This accounts for the putative starspot feature in the earlier analyses' residuals. As a result, we do use the data from the third G141 orbit in our fitting. This slightly improves our measured depth uncertainties compared to those previous results.", 'MIRI': "Unfortunately, while our 390 Hz noise removal and GLBS steps remove structured noise from the LombScargle periodogram 55, 56 of the MIRI/LRS grouplevel data, we find that these steps ultimately have a fairly small impact on the final transmission spectra of our science target WASP-107b as well as the MIRI/LRS TSO commissioning target L168-9b. For example, Extended Data Figure 3 shows that the MIRI/LRS observations of L168-9b continue to show excess noise after the 390 Hz noise removal step has been applied. It is possible that the 390 Hz noise removal step does eliminate excess noise for 1 µ mbin sizes, but at that coarse of a resolution there are only 7 bins and there is substantial uncertainty in the estimated standard deviation from the small sample size. In addition, with finer bin size sampling than was used by ref. 53 , we are better able to understand how the \nexcess noise varies with varying bin size. In particular, while we find that the excess noise decreases with increasing bin size (similarly to what was reported by ref. 53 ), there also appears to be a spike in the excess noise around a bin width of 0.25 µ m; this corresponds to an average spectral bin width of 9 pixels (although the bin width varies with wavelength) which is also the approximate period of the 390 Hz noise. Surprisingly, while this noise spike seems to share an approximate period with the known 390 Hz noise, the 390 Hz noise removal step does not appear to have removed the excess noise in the final transmission spectrum of L168-9b. \nWe also see very limited impact from both the 390 Hz noise removal and GLBS steps in the WASP107b MIRI/LRS science observations (see Extended Data Fig. 4). Most likely this is because there is fairly minimal curvature in the waveform over a single row (which only spans about 7% of the first harmonic which has the largest amplitude), so the perrow background subtraction step performed on the integration-level data in Stage 3 adequately subtracts this structured noise. It is possible that this 390 Hz noise removal procedure could still be useful to future works which seek to use separate background calibration observations taken before and/or after the science observations instead of computing the background on the science observations themselves, but at present we do not recommend the 390 Hz noise removal procedure be applied to science observations alone due to the minimal impact and high computational cost. That said, it does not appear that our 390 Hz noise removal procedure negatively impacts our results, and we ultimately decide to use it in our fiducial reduction. The impacts of the GLBS step are noticeable but also minor, and it is as-yet unclear whether the noise introduced by estimating the group-level background outweighs the structured noise removed by this step. \nComparing our fiducial spectrum (with the 390 Hz noise removal and GLBS steps) to the European Consortium's (hereafter referred to as the 'EC') three reductions presented by ref. 16 (see Extended Data Fig. 4) which used the CASCADe and TEATRO pipelines, as well as the Eureka! pipeline, it is again clear that the overall shape of the spectrum is robust to different analysis pipelines and different reduction choices. As pointed out by ref. 16 , the transit depth differences among the EC spectra displays an increasing trend with wavelength with the CASCADe pipeline giving larger depths at longer wavelengths, which they attribute to different to different systematics models \nExtended Data Fig. 3 | Ademonstration of the underestimation of error bars in the MIRI/LRS observations of L168-9b and the impact of the 390 Hz noise removal step. The median fitted transit depth uncertainties decrease with increasing spectral bin width as would be expected for white noise (blue symbols). Meanwhile, the standard deviation of the L168-9b transmission spectrum is around 2 × the expected noise level in the spectrally unbinned data. The ratio of the standard deviation of the spectrum to the median fitted uncertainty decreases with increasing bin size with the exception of a peak around 0.25 µ m. \n<!-- image --> \nwhen fitting the lightcurves. As shown in Extended Data Figure 4, we find fairly similar conclusions with our fiducial spectrum which generally agrees well with the CASCADe spectrum. Looking more closely, it appears our fiducial spectrum has a slightly smaller transit depth around 7.5 µ mandaslightly larger transit depths around 7 µ m and from around 8-10 µ m compared to the three reductions of ref. 16 . This results in our fiducial spectrum having a slightly smaller bump near the ν 3 SO 2 feature and a slightly larger bump near the ν 1 SO 2 feature.", 'Atmospheric Retrievals': "Detailed interpretation of atmospheric spectra requires comparisons with atmospheric radiative transfer models by means of statistical algorithms such as Bayesian inference methods. These data-model inference techniques are commonly known as 'atmospheric retrievals' and enable constraints on the atmospheric properties of an exoplanet such as chemical composition and vertical temperature structure from an observed spectrum. The inference framework computes transmission spectra, generally on the order of millions or tens of millions per inference, from a parametric atmospheric model (that is, a 'forward model') \nthat solves line-by-line radiative transfer under hydrostatic equilibrium. These atmospheric models generally assume a plane-parallel atmosphere and include parameters that determine the chemical abundances for different chemical gases, the vertical pressuretemperature structure of the planet, and the presence of clouds/hazes as additional sources of opacity in the atmosphere. Below we present the two paradigms in atmospheric modeling employed in this study, spanning a wide range in physico-chemical assumptions.", '1D-RCPE Grid-Retrieval': "In order to produce physically self-consistent solutions we fit the transmission spectrum with a suite of models that impose 1D-radiative convectivephotochemical-equilibrium (1D-RCPE). The coupling between the radiative-convective equilibrium solver (ScCHIMERA 11 , 75, 76 ) and the kinetics code (VULCAN 21 , 77 using the H-O-C-N-S reaction list from ref. 21 ) as well as additional details has been previously described in ref. 11 . The coupled model requires as inputs the incident stellar flux spectrum (T eff =4,425 K, log( g )[c.g.s.]=4.63; ref. 78 ) at the top of the planetary atmosphere (including the UV for photochemistry, using the HD 85512 MUSCLES \nExtended Data Fig. 4 | A comparison of different reductions of the WASP-107b MIRI/LRS spectra. a: A demonstration of the impact of the 390 Hz noise removal and GLBS steps on our reduction of the WASP-107b MIRI/LRS data. For wavelengths > 10 µ m, the GLBS step ends up slightly reducing the final transmission, while also turning off the 390 Hz noise removal step appears to have no further impact at any wavelength. Error bars show 1 σ uncertainties. b: A comparison of our fiducial reduction with the three reductions presented by ref. 16 , with 1 σ uncertainties. While there are some differences between the four different reductions, the overall shape of the transmission spectrum is robust to differences in reduction choices. c: The per-point differences between different pairs of reductions, which are summarized with a histogram in panel d . With the exception of a single TEATRO point, our reduction agrees with the EC's three reductions at ≲ 3 σ . However, there is some structure to the differences between our reduction and those of the EC, with our reduction giving larger transit depths on average than the EC's from around 8-10 µ m. \n<!-- image --> \nspectrum 79, 80, 81 as a proxy, which matches the effective temperature and gravity of WASP-107 well within 1%) which can be scaled via an effective irradiation temperature, an internal temperature to set the deep adiabat, the atmospheric elemental abundances (via metallicity and a C/O as described in ref. 11 ), an eddy diffusivity, and the bulk planet properties, radius and mass. The outputs are the converged 1D temperature and gas volume mixing ratio pressure profiles that can then be 'post-processed' to produce a transmission spectrum. Extended Data Figure 5 highlights select parameter slices through the grid and the subsequent impacts on the spectra. \nAs this process is computationally expensive owing to the large numbers of converged models required for the Bayesian inference, we generate the grid in stages. First, we generated a course grid in metallicity (0 ≤ [M/H] ≤ 2 in steps of 0.5), C/O (0.1 ≤ C/O ≤ 0.7 in steps of 0.2), internal temperature (100 ≤ T int ≤ 500 in steps of 100 K), and a vertically constant eddy mixing (7 ≤ log 10 K zz ≤ 9 in steps of 1), but at a fixed effective irradiation temperature (at the planetary equilibrium temperature-738 K). We then perform Bayesian inference on the observed spectrum with this course grid as described in ref. 11 (with more details specific to WASP-107b described below). Within the fitting process we also included \na temperature profile offset free parameter (a simple additive shift to the whole profile) to account for possible variations in temperature between the limb and the planetary dayside which, via the scale height, can influence the feature sizes. \nThe results of the course grid exercise effectively 'narrowed' the plausible parameter space down at which point we generated a more finely sampled grid (below which the grid spacing had no effect). This 'fine' grid was generated over 0.5 ≤ [M/H] ≤ 1.875 in steps of 0.125), C/O (0.05 ≤ C/O ≤ 0.6 in steps of 0.05), internal temperature (200 ≤ T int ≤ 550 in steps of 50 K), and eddy mixing (7 ≤ log 10 K zz ≤ 10 in steps of 0.5), resulting in 6,048 converged 1D-RCPE models. Based upon the temperature offset parameter from the course grid fit, this fine grid was generated at a cooler irradiation temperature (560 K) in order to more appropriately adjust for changes in the chemistry with temperature. This finer grid (at the new fixed irradiation temperature) was then refit to the data using the same inference procedure and is what we use to inform our primary conclusions. We also tested the effect of temperature by fitting, again, for a temperature profile offset parameter and found that it was consistent with zero at nearly 1 σ (-18 ± 14 K), suggesting that our choice of irradiation temperature for the fine grid resulted in a temperature profile that was consistent with the observed spectrum. \nWithin the Bayesian inference procedure, as described in ref. 11 , we also fit for, on the-fly, the planetary reference pressure and reference radius (ref. 82 which shift and stretch the planetary spectrum and also affect the zero-point for the planetary gravity with height), an offset for the MIRI spectrum relative to the NIRCam F444 spectrum, the temperature profile offset parameter (described above), an ammonia abundance enhancement factor, a power law haze + gray cloud, and a parametric cloud profile to describe unknown cloud resonance features over the MIRI wavelengths (described more below). Including these and the 1D-RCPE grid parameters results in a total of 17 free parameters. The transmission spectra are generated with R=100,000 absorption cross sections (same line lists cited in ref. 11 , but also now including SO 2 , ref. 83 ) and fit to the data within the PyMultiNest routine, using a total of 500 live points and the default parameters. Gas and cloud detection significances are computed as described in refs. 32 , 84 . Generally the detection of different gases in the 1D-RCPE solution corresponds to a preference for the opacity of the gas in question as a significant \nExtended Data Table 1 | 1D-RCPE retrieved atmospheric properties. Retrieved parameters (median and 1 σ uncertainties) and their prior ranges are included. log 10 ( α NH 3 ) is the enhancement to the NH 3 abundance profile and ∆ MIRI is the offset \nbetween the MIRI observations and NIRCAM F444W. \ncontribution to the spectrum, as only the opacity is removed from the inference and not from the chemistry, as described in ref. 11 . The retrieved parameters and priors are shown in Extended Data Table 1. \nTo illustrate the sensitivity of our results to the key physical processes that enabled us to arrive at the hot interior solution, we explored perturbations to the atmospheric structure and resultant spectra, shown in Extended Data Figure 5. We perturb the nominal 1D-RCPE solution (given in Extended Data Table 1) by 1) turning off mixing and photochemistry (equilibrium vs. disequilibrium, top row of Extended Data Fig. 5); 2) changing the internal/effective temperature (middle row of Extended Data Fig. 5); and 3) changing the strength of the eddy diffusion (bottom row of Extended Data Fig. 5). Mixing from a hot deep layer (arising from a high T int ) results in a ∼ 3 order-of-magnitude depletion in the methane abundance compared to equilibrium. This has a substantial influence on the spectrum and readily explains the 'muted' methane features. Equilibrium abundances are clearly ruled out by the free retrieval constraints (-6.0 < log 10 CH 4 < -5.6, details below). It is also apparent that T int values lower than ∼ 350 K and a log 10 K zz lower than ∼ 8 struggle to result in the necessary reduction in the methane abundance even with mixing. \nExtended Data Fig. 5 | Illustration of the effects of the key physical processes on the atmospheric structure (panels a, c, e) and resultant spectra (panels b, d, f) . Panels a and b show the impact of disequilibrium chemical processes (vertical mixing and photochemistry) relative to thermochemical equilibrium. Panels c and d show the influence of changing internal temperature, while the panels e and f show the impact of changing eddy diffusion strength. We only show the vertical abundance profiles (panels a , c , e ) for observable gases that are influenced by these effects (for instance, water is not shown as it not impacted). The observed transit spectrum is shown with grey points with 1 σ error bars in panels b , d , f . \n<!-- image -->", 'Cloud Parameterization': 'There are two leading modeling paradigms for modeling clouds and hazes in exoplanetary atmospheres: microphysical models and parametric models. The former requires understanding of the nucleation, condensation, evaporation, coagulation, and transport of clouds 85 . The later aims to capture the spectroscopic signature of these condensates and aerosols without any assumptions of the physical and chemical processes that lead to their formation and destruction, attempting to account for their presence as to unbias any inferences of other properties of interest (for \nexample, volume mixing ratios of different gases or atmospheric metallicity, refs. 18,86, 87, 88 ). Some parametric models have attempted to capture the functional dependence of cloud extinction on wavelength using analytic models 89, 90 or Mie-theory 91, 92, 93, 94 . \nOf relevance to this work, one of the approaches in ref. 16 to fit the MIRI+HST WFC3 G141 spectra of WASP-107b was to use the eddysed microphysical cloud parameterization 95, 96 for several silicate condensates. Briefly, this cloud framework assumes a balance between upwards turbulent mixing (parameterized with a vertical eddy diffusivity, K zz ) and \nsedimentation (parameterized with f sed ) of droplets. This balance governs the particle size distribution and abundance with height in the atmosphere (larger droplets deeper towards the cloud base and smaller droplets at higher altitudes). The overall abundance of each cloud species is set by the condensate abundance at base of the cloud. The droplet abundances/profiles along with condensate/droplet optical properties (derived from Mie-Theory) are then used to compute the total cloud extinction. Specifically, they fit for a cloud base pressure, condensate abundance, f sed , and K zz assuming no self-consistency in the location and abundance of the condensates with the chemistry or temperature-profile. They found that SiO 2 best explained the MIRI spectral shape. However, the retrieved location of the cloud base was at relatively low pressures ( ∼ mbars)-at much cooler temperatures and lower pressures than otherwise would be expected from equilibrium condensation chemistry (Extended Data Fig. 6a)-and a highly compact cloud (large droplets). \nWe initially follow the same eddysed cloud parameterization approach described above (assuming Mie-theory with the indices of refraction from ref. 97 ). While we are able to find satisfactory fits to the MIRI + HST observations (as in ref. 16 ), when applied to the entire broad band spectrum with our NIRCam observations, we were unable to find satisfactory fits. \nWe also tested the plausibility of silicate clouds (MgSiO 3 , SiO 2 , and Mg 2 SiO 4 ) and salt-sulfide clouds (Na 2 S and KCl) in explaining the broad-band spectrum within the same eddysed 95, 98 framework. We use the 1D-RCPE atmospheric structure from an initial representative fit (T-P profile and chemistry using the nominal composition [M/H]=1.0, C/O=0.3, eddy diffusion coefficient-log K zz =8.5, and a T int =500 K) to self-consistently determine which clouds and where they form along the temperature-profile. We varied f sed between 0.1 (vertically extended clouds) and 1 (compact clouds). Extended Data Figure 6 summarizes the condensate mixing ratios (panel a) and their impact on the spectrum (panel b) for an f sed =0.2. Only the lower f sed values ( < 0 . 5 , more extended clouds) are able to produce a notable impact on the spectrum as higher values result in clouds that are too compact with much of the cloud opacity at too deep of pressures. However, the lower f sed values result in smaller droplets of which produce a more narrow resonance feature than what is needed to fit the MIRI observations. It is the combination of a poor fit to the full broadband spectrum along with the in-plausibility of \nthe silicate cloud bases to exist at the altitudes probed by the observations that instead prompts us to take a phenomenological approach to the cloud modeling. \nWe introduce a new parametric treatment to capture the spectroscopic signatures of cloud particulate resonance feature. This approach is motivated by previous efforts to model optical slopes due to hazes as a scaling to Rayleigh-scattering 18 . We model these concave spectroscopic signatures (see Fig. 2 of ref. 16 ) of cloud condensates using a Gaussian function (that is, the composition of an exponential function and a concave quadratic function). The total extinction coefficient is given by a skewed normal distribution of form \nκ cond. ( λ ) = 2 κ opac. ϕ ( λ -λ 0 ω ) Φ ( ξ λ -λ 0 ω ) (2) \nwhere κ opac = K cond. n tot is the extinction coefficient for an opacity free parameter K cond. and n tot is the total number density of the atmosphere; ϕ is the standard normal probability density function and Φ is the standard normal cumulative density functions; λ 0 is the wavelength at which the Gaussian is centered (that is, the mean of the Gaussian), ω is the scale parameter (that is, the standard deviation of the Gaussian), and ξ is a shape parameter that controls the skewness of the distribution so that when ξ = 0 the standard normal distribution is recovered. \nIn both the 1D-RCPE retrievals and free retrievals we incorporate this parametric treatment alongside scalings to the Rayleigh scattering in the optical 18 and optically thick gray clouds as explained in refs. 11 , in a linear combination with a cloud-free model to account for the presence of cloud/haze inhomogeneities 86 . Future studies may investigate the best practices for the parameterization of cloud particulate resonance features.', 'Free Retrieval': "Wefurther explore the atmospheric properties inferred from the spectrum of WASP-107b employing more flexible and agnostic forward models in a Bayesian inference procedure. These, known as 'free retrievals', are methods to retrieve the chemical abundances of different gases, the vertical temperature structure, and the cloud/haze properties on the planetary atmosphere. Compared to the 1D-RCPE Grid-Retrieval explained above, these methods do not assume any physicochemical equilibrium conditions, and instead aim to \nExtended Data Fig. 6 | Effect of self-consistent microphysical eddysed clouds on the spectrum. Panel a shows the condensate vertical distributions for the major condensate species. The cloud bases, where the condensates first forms, all occur at or below the 100 mbar level with the silicate clouds forming at or below the 1 bar level. Panel b shows the resulting spectrum (with 1 σ error bars) as well as the contribution form each cloud species. \n<!-- image --> \ncapture the atmospheric conditions directly through a series of parameters for the chemistry and physical conditions of the atmosphere without expectations of physical consistency. These more flexible approaches provide the opportunity to capture conditions that otherwise would be prohibited by self-consistent models, such as combinations of gases not considered under chemical equilibrium. Nonetheless, caution must be exercised in the interpretation of these free-retrievals as model assumptions may contribute to biased and unphysical atmospheric estimates (see ref. 87 for a discussion). As with the 1D-RCPE Grid-Retrieval above, we simultaneously retrieve on the HST/WFC3 G102 and G141, JWST/NIRCam F322W2 and F444W, and JWST/MIRI observations. \nWe use two independent free retrieval frameworks in our analysis: Aurora 93 and CHIMERA 99, 86 , 2 . We select the later as our fiducial free retrieval comparison as the radiative transfer code, sources of opacity, and model resolution is the same as in the 1D-RCPE grid-retrieval above, therefore enabling a more direct comparison. The former, Aurora, is used to consider the impact of different sources of opacity than the ones used by CHIMERA, and consider the impact of line-by-line cross-section at higher sampling resolution. We find that our detection significances and conclusions are largely independent of the framework employed, with the details of each analysis described below. \nCHIMERA solves radiative transfer for a parallelplane atmosphere, under hydrostatic equilibrium, for \ntransmission geometry. The atmospheric model considers a one-dimensional model atmosphere spanning from ∼ 10 -9 bar to ∼ 100 bar, divided in 100 layers uniformly spaced in logarithmic pressure space. The vertical temperature structure is parameterized following the prescription from ref. 100 . The model simultaneously retrieves the reference pressure and reference radius for the assumed planetary gravity of log 10 ( g ) = 2 . 45 c.g.s. \nThe atmospheric models assumes uniform mixing ratios for H 2 O, CH 4 , NH 3 , CO, CO 2 , SO 2 , and H 2 S using independent free parameters for each gas' volume mixing ratio. Our choice of chemical species is motivated by those expected in exoplanetary atmospheres at these warm temperatures 9, 17 . As part of our initial analysis we considered the presence of PH 3 since this species may be expected for some substellar objects at these temperatures. Nonetheless, the observations did not place meaningful constraints on the abundance of PH 3 and was not considered in our fiducial run in an effort to limit the number of free parameters in our analysis. The presence of clouds and hazes is considered by utilizing the one-sector parameterization of ref. 93 introducing the combined spectroscopic effect of a optically thick cloud deck with a cloud opacity κ cloud, and hazes following an enhancement to Rayleigh-scattering 18 in a linear combination with a cloud-free atmosphere 86 . Furthermore, we incorporate the effect of unknown cloud resonances (for example, wavelength dependent condensates) as explained above. \nThe Bayesian inference is performed using nested sampling 101, 102 , via PyMultiNest 45,102, 103 , using 500 live points in the sampling. Each forward model in the sampling was calculated using line-by-line opacity sampling at a spectral resolution of 100,000 and then binned to the resolution of the observations. The line lists considered remain the same as in the 1D-RCPE analysis described above. In total, the sampling is performed over 24 parameters: 7 molecular gases, 6 for the pressure-temperature structure of the planet, 1 for the reference pressure, 1 for the reference radius, 8 for the presence of inhomogeneous clouds and hazes, and 1 for an offset between the JWST MIRI observations and the JWST NIRCam F444W observations. \nOur second free-retrieval with Aurora 93 follows largely the same model setup as the one explained above with CHIMERA. A detailed description of Aurora's transmission modelling approach is available in ref. 93 . The main differences relative to the retrieval with CHIMERA are: 1) different opacity sources for H 2 O, CO 2 , NH 3 , CH 4 , and H 2 S obtained from refs. 104, 105, 106, 107 ; 2) the use of a free parameter P cloud to account for optically thick clouds instead of κ cloud parameter; and 3) computing the forward models using line-by-line cross section sampling at a spectral resolution of 20,000 instead of 100,000. The retrieved atmospheric properties are generally consistent within the the CHIMERA retrieval at 68 % confidence, and with predictions from the 1D RCPE grid retrieval. The results from this retrieval are also included in Table 2. Supplementary Information Figure 2 shows the equivalent to Figure 4 with the Aurora constraints, and highlights that while the inferences are consistent, the use of lower spectral resolution results in wider constraints in this case. \nSupplementary Information Figure 1 shows the retrieved transmission spectrum and the contributions of the detected gases in WASP-107b's atmosphere as inferred with Aurora. In both cases, the retrieved volume mixing ratios from the free-retrievals are consistent with the gas profiles from the 1DRCPE inferences, as shown by the posterior distributions from the free retrieval and gas samples from the 1D-RCPE in Figure 4 and Supplementary Information Figure 2. Furthermore, the retrieved vertical pressure-temperature structure of the planet is consistent within the fully parametric free-retrieval and the self-consistent 1D-RCPE as shown in Extended Data Figure 7. The free retrieval with Aurora finds an offset between the JWST MIRI and JWST NIRCam F444W \nExtended Data Fig. 7 | Comparison of retrieved pressuretemperature structure. The retrieved vertical temperature structures (median, 1 σ , and 2 σ ) from the free-retrieval (Aurora, blue) and the 1D-RCPE (red) are generally in good agreement within their 68% confidence intervals. The dotted line shows the equilibrium temperature of the planet. The observations generally probe pressures lower than ∼ 100 mbar and as low as ∼ few × 10 -5 bar. We include in black the Tint = 200 K profile from Extended Data Figure 5 as an example of lower internal temperatures. \n<!-- image --> \nobservations of 282 +49 -41 ppm. A complete summary of the retrieved parameters and their priors is shown in Extended Data Table 2. \nBoth free retrievals confirm the detections of H 2 O, CH 4 , NH 3 , CO, CO 2 , and SO 2 with detection significances of 5 σ or greater. The reported detection significances in the main text result from the model comparisons with CHIMERA, while the detections with Aurora remain equally significant at 5 σ or greater. Similarly, we find that the fiducial model with optically thick clouds, Rayleigh-enhancement hazes, and wavelength dependent condensates is preferred over a cloud-free atmosphere at 26 σ . Including wavelength dependent condensates is preferred at 13 σ over just including optically thick clouds and Rayleighenhancement hazes. We note that the term 'detection significance' refer to a model preference between a reference model considering all gases and nested models for which each individual species is removed in turn 84, 93 . As such, any quoted detection significance is dependent on the choice of models and choice of model priors. Moreover, the conversion between differences in Bayesian evidence and 'sigma detections' can result in extremely large values (for example, > \nExtended Data Table 2 | Free-retrieval retrieved atmospheric properties. Retrieved parameters (median and 1 σ uncertainties) and their prior ranges are included. \n10 σ ) that are difficult to interpret (see ref. 108 for a discussion). For WASP-107b, the agreement in retrieved molecular abundances and confirmation of the strong evidence for the detection of these species regardless of model assumptions (for example, free abundances vs. radiative convective-photochemical-equilibrium) confirms the robustness of the interpretation of the planet's spectrum.", 'Robustness of Ammonia Detection': "To provide scrutiny beyond using Bayesian evidence model selection for the NH 3 detection, we perform Leave-One-Out Cross-Validation (LOO-CV; refs. 109, 110, 111 ) on the models with and without NH 3 from the 1D-RCPE grid retrievals. In LOO-CV, a model is trained on the dataset with one data point left out and then scored according to how it can predict the left out data point (that is, by calculating the expected log predicted density of the left out data point - elpd). This procedure is repeated for all data points \nallowing the out-of-sample predictive performance of a model to be estimated. The LOO-CV scores for each data point can then be compared between two models highlighting where one model out performs the other 108, 112, 113 and consequently, in this case, which wavelengths and instruments drive the NH 3 detection. Extended Data Figure 8 shows the difference in elpd ( ∆ elpd) per data point in the spectrum of WASP107b for the reference 1D-RCPE model and the model without NH 3 absorption. \nThe Bayesian model comparison finds a 6 σ detection (that is, model preference) for the model including NH 3 absorption. By performing a LOO-CV analysis of these models, we can determine which data points drive this model preference. We find the inclusion of NH 3 improved the model performance at all wavelengths (for example, there are points with positive ∆elpd at all wavelengths), with a localized region of high performance data points near 3 µ m. Indeed, the points with the highest preference for the model with \nExtended Data Fig. 8 | The detection of NH 3 in WASP-107b's transmission spectrum is driven by NIRCAM F322W2 observations. The data is color coded by the point-wise difference in expected log posterior predictive density between the reference 1D-RCPE model and the model without NH 3 absorption. Best fit 1D-RCPE model (R=300) is show in gray. Redder data points (larger positive ∆elpd score) are better explained by the reference model including NH 3 absorption. While points with positive ∆elpd are present throughout the entire transmission spectrum, indicating that NH 3 improves the model performance at all wavelengths, the highest scoring points are localized at ∼ 3 µ m where strong NH 3 absorption is visible. The purple shaded regions show areas where the cross-section of NH 3 contributes to 50% or more of the cross-sections detected in WASP-107b, and with visually prominent features. Three data points in the MIRI observations (7.017 µ m, 7.318 µ m, 7.468 µ m) are not included in the analysis as their associated Pareto k value exceeded 0.7. Data points are shown with 1 σ error bars. \n<!-- image --> \nNH 3 , those with ∆elpd > 1 , are all between 2 . 9 µ m and 3 . 2 µ m and part of the NIRCam F322W2 observations. After NIRCAM F322W2, the next instrument with with positive ∆elpd scores is HST-WFC3 G141 (9 th highest scoring point), followed by HST-WFC3 G102 (14 th highest scoring point) and NIRCAM F444W (seventeenth highest scoring point). MIRI's highest scoring point is at 10.17 µ m and corresponds to the 25 th highest scoring point with a ∆elpd ∼ 0 . 4 . \nWe proceed to total the difference in LOO-CV scores (that is, sum( ∆elpd) ) between the models with and without NH 3 over regions where the NH 3 crosssection is dominant (that is, where NH 3 contributes > 50% of the total cross-section). We find that the density of the increased predictive performance (that is, sum( ∆elpd) /# points) is higher, and more than double the value, in regions where NH 3 is dominant than in regions where NH 3 is not the dominant cross-section. This confirms that the detection of NH 3 is significantly improved by the data where significant absorption by the gas is expected. \nWe also compare two regions in the spectrum where the spectroscopic signatures of NH 3 are visually prominent, these are features at ∼ 3 µ m with NIRCam F322W2 and ∼ 10 µ mwith MIRI. These two features are part of regions where NH 3 contributes > 50% of the total cross-section (see purple shaded regions in Extended Data Fig. 8). The region between 2.9 µ m and 3 . 1 µ m covered by NIRCam F322W2 has an order of magnitude greater density in the increased \npredictive performance of NH 3 than the 8.6 µ m to 12 µ m region covered by MIRI. Our analysis finds that the detection of NH 3 is more strongly driven by the NIRCam observations rather than the MIRI observations. \nThe NIRCam F322W2 provide the information necessary to confirm the tentative detection ( ∼ 2 -3 σ ) of NH 3 suggested by ref. 16 using MIRI data alone. Our LOO-CV analysis suggests that definitive detections of NH 3 require resolving the strong spectroscopic feature at ∼ 3 . 0 µ m, and highlight an important advantage of NIRCam over other instruments on JWST that do not have the wavelength coverage or throughput necessary. While NH 3 has a strong spectral feature at ∼ 10 µ m, and the gas' cross-section is dominant at > 8 . 6 µ m, the presence of strong cloud resonance features in the infrared obfuscates the observational ability to strongly detect the gas.", 'Stellar SED Fitting and Absolute Radii': "We used a set of catalog magnitudes for the WASP107 system to fit a model SED to the stellar emission. In conjunction with the Gaia 114 parallax for the system, this allowed us to estimate a stellar radius for the star WASP-107. \nFor our SED fits, we used catalog 2MASS JHK 115 and AllWISE W1 and W2 116 magnitudes. Our SED model used four different physical parameters: the stellar effective temperature, the stellar radius, the amount of visual extinction to WASP-107, and the \nsystem's parallax. We assumed log( g ) = 4 . 5 and [Fe / H] = 0 . 0 for the star WASP-107, which are both within 0.1 dex of the surface gravity and metallicity measured previously. 5 In practice, the precise values of log( g ) and [Fe / H] do not significantly affect the results from the SED fit. 117 \nWeimposed a Gaussian prior on Teff. based on previous spectroscopic measurements 5 of Teff. = 4425 ± 70 K with the associated 1 σ uncertainties as the prior width. Similarly, we imposed a Gaussian prior on the parallax to the system using the Gaia DR3 114 parallax of π = 15 . 528 ± 0 . 026 mas. For the amount of visual extinction to WASP-107 we imposed a prior of A V = 0 . 03 ± 0 . 01 , which comes from measurements 118 of the excess reddening towards WASP-107 of E ( B -V ) = 0 . 028 ± 0 . 01 , and assuming R V = 3 . 1 . \nTo model the SED, we used BHAC15 spectra 119 for the stellar SED. We computed a grid of surface luminosity magnitudes, corresponding to the bandpasses of the catalog magnitudes, for a range of Teff. values. Since the BHAC15 models step by 200 K in Teff, we used cubic spline interpolation to estimate model magnitudes in between the points provided by the model atmospheres. We then scaled the interpolated surface magnitudes for the star by R ∗ /d - where d is the distance to the star - to determine the apparent bolometric flux of the SED at Earth. We then applied a simple R = 3 . 1 extinction law scaled from the value of A V , to determine the extincted bolometric flux of the SED model. \nThis stellar SED fitting allowed us to measure the radius of the star WASP-107 to be R ∗ = 0 . 67 ± 0 . 01 R ⊙ . We then used the mean value of R p /R ∗ as measured in our joint broadband transit fits to estimate the planetary radius of WASP-107b to be R p = 0 . 939 ± 0 . 019 R J .", 'Data Availability': 'The NIRCam data used in this paper are associated with JWST GTO program 1185 (PI Greene; observations 8 and 9) and will be publicly available from the Mikulski Archive for Space Telescopes (MAST; https://mast.stsci.edu) at the end of their one-year exclusive access period. The MIRI data used in this paper are from JWST GTO program 1280 (PI Lagage; observation 1) and will also be publicly available on MAST at the end of their proprietary period. Additional intermediate results from this work are archived on Zenodo at', 'Code Availability': "We used the following codes to reduce and fit the JWST data: STScI's JWST Calibration pipeline 46 , Eureka! 14 , tshirt 15 , starry 65 , PyMC3 68 , numpy 120 , astropy 121, 122 , scipy 123 , and matplotlib 124 .", 'Additional information': 'Correspondence and requests for materials should be addressed to Luis Welbanks. \nReprints and permissions information is available at www.nature.com/reprints.', 'Supplementary Information': "Supplementary Fig. 1 | Interpretation of WASP-107b's transmission spectrum using free retrievals with Aurora. The observed transmission spectrum of the planet with 1 σ error bars is compared to the retrieved transmission spectra using the free-retrieval approach. The retrieved median is shown at a spectral resolution of R=300. The best fit model has a χ 2 /N data =1.5. The contributions of the individual detected gases are shown as shaded regions. The gray dashed line shows the clear atmosphere component of the model, that is, the gas contributions without the presence of clouds or hazes. All other elements remain as in Figure 3. \n<!-- image --> \nSupplementary Fig. 2 | Inferred molecular volume mixing ratios from WASP-107b's transmission spectrum with the lower resolution retrievals with Aurora . Same as Figure 4 but considering the constraints from the free-retrieval with Aurora, with the retrieved volume mixing rations for each detected gas shown in panels a -f . The use of lower resolution (R=20,000) forward models results in constraints that are consistent with those of our fiducial retrieval with CHIMERA but with wider constraints. \n<!-- image -->"} |
2024arXiv240909120K | We investigate the potential of photonic lantern PL fiber fed spectrometers for twodimensional spectroastrometry. Spectroastrometry a technique for studying small angular scales by measuring centroid shifts as a function of wavelength is typically conducted using longslit spectrographs. However slitbased spectroastrometry requires observations with multiple position angles to measure twodimensional spectroastrometric signals. In a typical configuration of PLfed spectrometers light from the focal plane is coupled into the fewmoded PL which is then split into several singlemode outputs with the relative intensities containing astrometric information. The singlemoded beams can be fed into a highresolution spectrometer to measure wavelengthdependent centroid shifts. We perform numerical simulations of a standard 6port PL and demonstrate its capability of measuring spectroastrometric signals. The effects of photon noise wavefront errors and chromaticity are investigated. When the PL is designed to have large linear responses to tiptilts at the wavelengths of interest the centroid shifts can be efficiently measured. Furthermore we provide mock observations of detecting accreting protoplanets. PL spectroastrometry is potentially a simple and efficient technique for detecting spectroastrometric signals. | 2024-09-01T00:00:00Z | ['2024arXiv240909120K', '10.48550/arXiv.2409.09120', 'arXiv:2409.09120'] | ['Astrophysics - Instrumentation and Methods for Astrophysics', 'Astrophysics - Earth and Planetary Astrophysics'] | On the Potential of Spectroastrometry with Photonic Lanterns | 2,024 | 170 | 0.43 | ['EPRINT_HTML', 'EPRINT_PDF'] | 0 | https://arxiv.org/pdf/2409.09120.pdf | {'On the Potential of Spectroastrometry with Photonic Lanterns': 'Yoo Jung Kim a , Michael P. Fitzgerald a , Jonathan Lin a , Yinzi Xin b , Daniel Levinstein c , Steph Sallum c , Nemanja Jovanovic b , Sergio Leon-Saval d \na Department of Physics & Astronomy, University of California, 430 Portola Plaza, Los Angeles, CA 90095, USA b Department of Astronomy, California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125, USA c Department of Physics & Astronomy, University of California Irvine, 4129 Frederick Reines Hall, Irvine, CA 92697, USA \nd Sydney Astrophotonic Instrumentation Laboratory, School of Physics, The University of Sydney, Sydney, NSW 2006, Australia \nAbstract. We investigate the potential of photonic lantern (PL) fiber fed spectrometers for two-dimensional spectroastrometry. Spectroastrometry, a technique for studying small angular scales by measuring centroid shifts as a function of wavelength, is typically conducted using long-slit spectrographs. However, slit-based spectroastrometry requires observations with multiple position angles to measure two-dimensional spectroastrometric signals. In a typical configuration of PL-fed spectrometers, light from the focal plane is coupled into the few-moded PL, which is then split into several single-mode outputs, with the relative intensities containing astrometric information. The singlemoded beams can be fed into a high-resolution spectrometer to measure wavelength-dependent centroid shifts. We perform numerical simulations of a standard 6-port PL and demonstrate its capability of measuring spectroastrometric signals. The effects of photon noise, wavefront errors, and chromaticity are investigated. When the PL is designed to have large linear responses to tip-tilts at the wavelengths of interest, the centroid shifts can be efficiently measured. Furthermore, we provide mock observations of detecting accreting protoplanets. PL spectroastrometry is potentially a simple and efficient technique for detecting spectroastrometric signals. \nKeywords: photonic lanterns, spectroastrometry, astrophotonics, high angular resolution, high spectral resolution, protoplanets.', '1 Introduction': 'High angular resolution enables the detailed study of a variety of objects such as young stellar objects, jets, circumstellar environments, and quasars. Spectroastrometry is a method to study angular scales smaller than the point-spread function (PSF) size for objects whose morphology changes with wavelength (such as those exhibiting emission lines), by measuring the relative position of an unresolved object as a function of wavelength. 1-3 A wavelength-dependent shift in the center of light - the definition of spectroastrometric signal - may indicate the presence of a companion, outflow, or any spatially extended feature, revealing details about the structure within the PSF (seeing- or diffraction-limit). The spectroastrometry method has been used for many purposes, such as studying young binaries, 4 circumstellar disks, 5, 6 kinematics of stellar outflows, 7 and the broad-line region of quasars. 8, 9 With diffraction-limited PSFs enabled by adaptive optics (AO), sub-milliarcsecond precisions can be achieved with 10 m-class telescopes in the near-infrared. \nSpectroastrometry leverages high spectral resolution to resolve kinematic structures at small angular scales or to detect companions with narrow spectral lines. Therefore, long-slit echelle spectrometers are typically used for spectroastrometric observations, as they can achieve high spectral resolution and spatial sampling along one axis. However with long-slit spectrographs, at least two observations with different slit orientations are required in order to obtain two-dimensional spectroastrometric signals. Another downside of using a slit is that a distorted PSF or uneven \nFig 1 A schematic diagram of the standard 6 port photonic lantern (PL) used for our simulation. The inset figure positioned above the PL shows the refractive index profile of the PL at the few-core fiber (FCF) end. The telescope light is coupled in the FMF end, propagates through the lantern, and becomes effectively confined in the SMFs. The light coupled in the SMFs can be fed to a spectrometer for spectroscopy. The relative spectra in output SMFs can be used to measure the center of light as a function of wavelength (i.e., spectroastrometry.) \n<!-- image --> \nillumination can introduce an artificial spectroastrometric signal. 10,11 At medium resolutions, integral field spectrometers can enable two-dimensional spectroastrometry with decreased artifacts, although with an increased complexity. 12-14 \nIn this study, we explore the potential of using photonic lanterns (PLs) for two-dimensional spectroastrometry. 15,16 PLs are tapered waveguides that gradually transition from a few-mode fiber (FMF) or multi-mode fiber (MMF) geometry to a bundle of single-mode fibers (SMFs) 17-19 (Figure 1). When the AO-corrected telescope light couples into the FMF end of the PL in the focal plane, it becomes confined within the SMF cores as it propagates through the lantern. Thus, a PL converts multimodal telescope light into multiple single-moded beams. The light in the SMFs can be fed into spectrometers for spectroscopy. PLs are potentially advantageous for diffraction-limited precision high-resolution spectroscopy due to the spatial filtering nature of the SMFs 20 and the high coupling efficiency compared to coupling light directly into SMFs 21 (AO-assisted SMF-fed spectrometers such as KPIC, 22 IRD, 23 PARVI, 24 iLocator, 25 HiRISE, 26 and HISPEC/MODHIS 27 ). Potential uses of PLs for astronomical observations have been studied in context of nulling 28 and coherent imaging. 29 \nThe dispersed SMF outputs from a single PL placed at the focal plane can act like a small integral field unit (IFU), sensing spatial features in very small angular scales. Recent studies on PLs as focal plane wavefront sensors (PLWFS) 30-34 have demonstrated that PL output intensities can be used to sense low-order aberrations. The capability of PLWFS is related to PL spectroastrometry, since centroid shifts correspond to tip-tilt Zernike modes, the lowest modes excluding the unsensed piston mode. With PL output spectra, wavelength-dependent two-dimensional centroid shifts can be efficiently measured, eliminating the need for slicing or resampling of the focal plane fields as in IFU and using only a few single-moded spectral traces. Thus, both high angular resolution and high spectral resolution can be achieved with PLs. \nThis paper is structured as follows. In §2, we show how spectroastrometric signals can be re- \novered from PL outputs and present numerically simulated spectroastrometric signals for a 6-port PL. In §3, we discuss sources of errors in spectroastrometric signal recovery: photon noise, static wavefront errors (WFEs), and time-varying WFEs. In §4, we address the chromatic behaviors of PLs relevant to spectroastrometry. In §5, we present mock observations of accreting protoplanets. In §6 we discuss benefits of PL spectroastrometry and considerations on PL design.', '2.1 Spectroastrometric signals for PLs': "Consider an instantaneous monochromatic wavefront incident on the telescope pupil E p (represented as an M -dimensional complex-valued vector of field samples) and its Fourier transform, the focal plane electric field E f . An N -port PL couples E f at the PL entrance. Following the analytical model of Lin et al. 2022, 32 the instantaneous PL output field can be described as \nE SMF = A E p . (1) \nThe A matrix is the complex-valued transfer matrix which includes propagation from the pupil to the focal plane and the lantern transition, with the dimension of N × M for M pupil samples. E SMF is the N -dimensional vector, the complex amplitudes in output SMFs. Thus, the A matrix maps the pupil plane wavefront to PL outputs. \nThe transfer matrix A corresponds to the inverse of the M × N matrix formed by stacking N number of M -dimensional pupil plane PL principal modes (PLPMs 29 ). Numerically, the i -th mode pupil plane PLPM is computed by backpropagating the fundamental mode of the i -th output SMF to the pupil plane. The i -th row of the A matrix corresponds to i -th effective pupil function of the i -th output SMF. Then the intensities in the output SMFs can be expressed as \nI SMF = | A E p | 2 . (2) \nWe abbreviate I SMF for I afterwards for simplicity and write i -th SMF output intensity as I i . \nAn input scene, consisting of incoherent sources (excluding coherent sources such as masers), can be described as an incoherent sum of n point sources located at angular coordinates α l = ( x l , y l ) T and weighted by the flux factor f l , with l = 0 , 1 , ..., n -1 . The wavefront from l -th point source is written as a tilted plane wave, E l p = f l 1 / 2 exp ( iR α α l ) , where R α is the M × 2 basis matrix whose columns are pupil plane tip-tilt basis vectors. The output intensities are then modeled as \nI tot = n -1 ∑ l =0 I l = n -1 ∑ l =0 | A E l p | 2 . (3) \nFigure 2 illustrates a simple example case of binary point sources, star x and star y . Star x dominates the total flux at wavelength λ 1 , but star y has an emission line at wavelength λ 2 . Due to the emission line, the center of light shifts towards the star y at wavelength λ 2 which is the spectroastrometric signal. Our goal is to detect the star y using spectroastrometry on this emission line. On the right we display the simulated I tot for a standard 6-port PL. Details on the simulation are described in §2.2.1. Although the on-axis light from the star x (orange) couples to ports 1 through 5 evenly, the off-axis light from the star y (purple) couples to the port 2 the most. The relative intensity profiles are determined by the separation, contrast, and position angle (PA) of \nFig 2 (Left) Focal plane intensity maps for a simple example case, a bright star without spectral features (star x ) located at the center and an off-center fainter star (star y ) with an emission line at λ 2 . Top rows are for λ 1 and bottom rows are for λ 2 . The FMF end geometry of a standard 6-port PL is displayed as black circles. Due to the emission line of star y at λ 2 , the center of light of the sum of the two stars is shifted in λ 2 while in λ 1 does not show any shift. The center of light shift is smaller than the FMF radius (order of λ/D ) and should be sensed by the lantern as a tip-tilt aberration. (Right) Simulated output intensities for λ 1 (top) and λ 2 (bottom). At λ 2 , the flux of the emission line from the star y is added onto the flux of the star x , altering the relative intensities. \n<!-- image --> \nthe star y . Thus, the difference in output relative intensities on and off the emission line gives the two-dimensional spectroastrometric signal. \nLet us consider an input scene with a small angular extent ( ≪ λ/D ). Assuming small α l , the wavefront can be approximated as E l p ≈ f l 1 / 2 ( 1 + iR α α l ) . Following Lin et al. 2022, 32 the intensities in the output SMFs from each point source can be expressed as \nI l = | A E l p | 2 ≈ f l ( | A 1 | 2 + BR α α l ) , (4) \nwhere B is the linear response matrix defined as \nB ij ≡ 2Im [ A ∗ ij ∑ k A ik ] . (5) \nWedefine B ' ≡ BR α , the B matrix projected onto tip-tilt basis, which describes the linear intensity responses to tip-tilt. Then the total intensity is the sum of the n point sources: \nI tot = n -1 ∑ l =0 I l ≈ ( n -1 ∑ l =0 f l ) | A 1 | 2 + B ' n -1 ∑ l =0 f l α l n -1 ∑ l =0 f l ≈ f tot ( | A 1 | 2 + B ' α centroid ) . (6) \nf tot ≡ n -1 ∑ l =0 f l and α centroid is defined as the center of light angular coordinate vector. This suggests that if the angular size of the on-axis object is small that the tip-tilts corresponding to the angular size perturb lantern output intensities linearly, the intensity responses mainly describe the center of light shift of the input scene. \nIn practice, f tot in each wavelength bin is an unknown factor. Even if the exact spectra of the object is known a priori, there may still be chromaticity in how the light couples into the PL due to stochastic processes, such as atmospheric dispersion. Instead, we can define normalized intensities in the output SMFs ( I n ) as the output intensities divided by the total sum of the output intensities, N -1 ∑ I i . The normalized intensities can be linearized similarly in this regime such as \ni =0 \nI l n ≈ I n 0 + B ' n α l (7) \nwhere I n0 is the normalized intensity for the on-axis point source, | A 1 | 2 / N -1 ∑ i =0 | A 1 | 2 i and B ' n \nis the \nnormalized intensity linear response matrix, B ' / N -1 ∑ i =0 | A 1 | 2 i . Then the equation 6 can be rewritten as \nI n, tot ≈ I n 0 + B ' n α centroid (8) \nin the linear intensity response regime. \nFinally, inverting the above equation, the center of light - the definition of spectroastrometric signals - can be recovered from the observed ∆ I n ≡ I n, tot -I n 0 as: \nα centroid ≈ B ' + n ∆ I n . (9) \nB ' + n is the left pseudo-inverse of the B ' n matrix. Note that both B ' n and I n0 are determined by the lantern design, the transfer matrix A . They can also be empirically determined in a lab with calibration light sources. \nNote that the R α matrix can be extended to an M × m matrix R to include additional modes ( m ) beyond tip and tilt. Then B ' = BR describes general linear intensity responses to aberrations. An N -port PL can sense up to m = N -1 non-piston Zernike modes. 32 Using the generalized linear intensity response matrix and calculating the right-hand side of the Equation 9, one obtains an m -dimensional vector, with first two elements corresponding to tip-tilt. If the intensity responses were perfectly linear, the entries other than tip-tilts should be zero. Any deviations from the linear approximation would manifest as nonzero values in the other modes (modal confusion). We provide examples in §2.2.3. Also, the effects of static and time-varying WFEs over an exposure can alter the intensity response matrix, which we discuss in §3.2 and §3.3.", '2.2 Simulated spectroastrometric signals': 'In this subsection, we conduct numerical simulations of a standard 6-port lantern ( N = 6 ) and simulate spectroastrometric signals for several simple astronomical scenes. \nFig 3 (Top) Simulated SMF output intensities ( I SMF ) for a standard 6-port lantern, as a function of tip-tilt Zernike mode aberration amplitudes. The total intensity incident on the pupil is set to unity. The black lines denote the total intensity in the 6 output SMFs. (Bottom) Same as the top panels but the intensities are normalized ( I n ) by the sum of the total intensities (the black lines). See Figure 1 for the geometry of the PL used for the simulation and §2.2.1 for the simulation setup. \n<!-- image -->', '2.2.1 Simulation': "The simulated lantern is a standard 6-port PL ( N = 6 ) of which all the SMFs have the same core radius and refractive index, with 5 cores arranged symmetrically around a central core (Figure 1). We assume the cladding index of 1.444, cladding-jacket index contrast of 5 . 5 × 10 -3 , and corecladding index contrast of 8 . 8 × 10 -3 . Each SMF core diameter is chosen to be 4.4 µm and the lantern entrance diameter to 20 µm. The lantern taper length is set to 2 cm and the taper scale is set to 8. We numerically backpropagate fundamental modes in the SMFs to the lantern entrance using the lightbeam 35 python package to determine focal-plane PLPMs. \nFor the telescope, we assume an unobstructed circular aperture D of 10 m. The simulations are monochromatic with wavelength 1.55 µm ( λ/D ∼ 32 mas). The focal plane PLPMs are backpropagated to the telescope pupil to determine the pupil plane PLPMs and the transfer matrix A using the HCIPy (High Contrast Imaging for Python) package. 36 The focal length is optimized to maximize the total coupling (intensity coupled over all of the supported LP modes) of an unaberrated on-axis point source. \nThe top rows of Figure 3 display the simulated output intensities of a single point source ( I ) as a function of tip (left) and tilt (right). In the small tip-tilt regime, the intensity responses are linear for most of the ports. The black curves indicate the total intensity over the 6 output SMFs, the total coupling efficiency. The bottom row shows the normalized output intensities ( I n ), which also behave linearly for small tip-tilts, but more non-linearly in larger tip-tilt regimes. The y -intercepts correspond to I n0 and the slopes at the origin are the B ' n .", '2.2.2 Example scenes': "Using the simulated lantern transfer matrix, we simulate spectroastrometric signals for simple scenes. The lantern output spectra are calculated as an incoherent sum of output spectra from point sources that constitute the input scene (Equation 3). We neglect the chromaticity of the lantern's transfer matrix for this simulation and assume that the transfer matrix is constant over the wavelength range, [1.51, 1.59] µm. We discuss the effects of chromaticity in §4. \nTop panels of Figure 4 show a case for binary point sources separated by 3 mas. The left panel displays the input scene, one star with a constant continuum emission (gray) and the other star of 4:1 line-continuum flux ratio, exhibiting an emission line of FWHM = 1,000 km s -1 at 1.55 µm (black). The center of continuum emission is located at the origin (blue cross). The second panel from the left shows the lantern output spectra of each SMF, I tot . The output spectra are normalized by the peak of the summed spectra (summed over the ports). The amplitude of the emission line varies from port to port. The third panel presents ∆ I = I n, tot -I n0 , the normalized spectra (in each wavelength bin) subtracted by the on-axis point source normalized intensity. The fourth panel shows the recovered spectroastrometric signals α centroid as dashed lines, Equation 9. The true center of light positions as a function of wavelength are overlaid as thick solid lines. This shows that the two-dimensional centroid shift on the emission line can be recovered with the simple linear reconstruction method. \nMiddle panels of Figure 4 show a case for a rotating ring ( v rot = 4 , 000 km s -1 ) of 1 mas radius, such as broad-line region of quasars. 8, 9 Each flux element shows a broad emission line of FWHM = 2,000 km s -1 (local broadening). The inclination angle is 22.5 degrees and the PA is 60 degrees. We assume continuum level of 25% for the amplitude of the emission line. Since the blueshifted and redshifted flux of the emission line originate from different locations of the ring, sinusoidal spectroastrometric signals are expected. If the line is sufficiently resolved in the spectral axis, PL spectroastrometry can provide estimation on the two-dimensional centroid shifts which can then be used to infer the kinematic structures, potentially useful for constraining more complicated models. \nBottom panels of Figure 4 display a case for a rotating sphere of 3 mas radius with an absorption line, such as a rapidly rotating star ( v rot = 100 kms -1 ). The star is modeled as a collection of 2,500 light-emitting patches on a hemisphere, with an absorption line of FWHM = 30 km s -1 . Similarly to the ring case, the two-dimensional centroid shift can be detected and the stellar rotation such as spin axis can be characterized with the spectroastrometric signals.", '2.2.3 Nonlinear intensity response regime': "If the object is more extended such that the linear intensity response approximation (Equation 4) is no longer valid, the estimation of the light centroid by linear approximation (Equation 9) fails. This effect is illustrated in Figure 5, which presents simulated spectroastrometric signals for binary \nFig 4 Example PL spectroastrometric signals for three simple scenes, (top) binary point sources with an emission line on one, (middle) a rotationally-broadened emission line originating from a ring structure such as broad-line region of a quasar, and (bottom) a rotationally-broadened absorption line from a resolved star. The first column illustrates the input scenes. The second column and the third column display simulated output spectra and normalized spectra, respectively, of a standard 6-port PL. Due to the positional variation of the line-emitting region, the PL output spectra show variations in fluxes across the ports. This variation can be used to recover the center of light as a function of wavelength, the fourth column. \n<!-- image --> \nmodels with the same centroid shifts but with different contrasts and separations. Since a standard 6-port PL is sensitive to the first five (non-piston) Zernike modes, tip-tit, defocus, and astigmatism, 32 the R α matrix in Equation 4 can be extended to an M × 5 matrix whose columns are the five mode basis vectors. Then the right hand side of Equation 9 becomes a five-element vector, the Zernike aberration amplitudes in tip-tilt, defocus, and astigmatism. If the input scene has a small angular extent then Equation 4 is valid, the first two entries should represent the centroid position in tip-tilts and the other three entries should equal to zero. However, if the output intensities deviate from the linear approximation, the right hand side of Equation 9 will show modal confusion, detecting non-zero signals in defocus and astigmatism modes. For binary models, as shown in the \nright panels of Figure 5, increasing the binary separation results in non-zero signals in defocus and astigmatism modes and underestimation of centroid shifts (tip-tilts). \nIn practice, when non-negligible spectroastrometric defocus and astigmatism signals are detected, it indicates that the angular scale of the intensity distribution is larger than the linear intensity response regime (unless there is chromatic WFE). It is the regime where more information other than the centroid shifts can be extracted. In this case, models of output intensities given an input scene can be generated with knowledge of the transfer matrix A or using an empirical PL coupling map - output intensities in each port scanned over a grid of x , y PL positions in the focal plane. 37 Figure 6 shows the simulated coupling maps: intensity responses as a function of x , y position of a point source at the focal plane. The gradients evaluated at the center of the map correspond to the B ' matrix, the linear intensity response. An input scene that best describes the measured ∆ I n can be constrained using the transfer matrix or the coupling maps. An example of this approach will be discussed in §5 as case B for a binary system. \nFig 5 Differences in normalized intensities ( ∆ I n ; left) and recovered spectroastrometric signals ( α ; right) for the binary model. The companion with emission line is placed along the x -axis (top) and y -axis (bottom), respectively, with three different separation and contrast values. On the right panel, the true expected spectroastrometric signals are indicated as black dots. \n<!-- image --> \n0.6 \n0.0 \nFig 6 Simulated intensity responses in each SMF port of the 6-port PL, as a function of the x, y position of a point source at the focal plane. The FMF end geometry is plotted as white circles, of which the diameter is 20 µm. The total intensity of the point source is set to 1. Intensities in each port are sensitive to the position of the point source, so the relative intensities between the ports can be used to determine the separation and the PA of the companion. \n<!-- image -->", '3 Errors in PL spectroastrometric signal recovery': 'In §2, we showed simulated spectroastrometric signals for an ideal case. In this section, we investigate major error sources that one needs to consider in spectroastrometric observations with PLs: photon noise and WFE. Photon noise and instantaneous WFE due to turbulence are stochastic processes that contribute to random errors. Systematic WFEs and time-averaged WFE effects averaged over an exposure introduce systematic errors that require calibration.', '3.1 Photon noise': "For PLs, the amplitude of the intensity response to tip-tilt (the B ' n matrix) determines how efficient PLs are for detecting spectroastrometric signals. In the photon noise-limited regime, conventional spectroastrometric accuracy (uncertainties in centroid measurement) scales as \nσ centroid ∼ σ sig √ N phot = FWHM 2 . 355 √ N phot (10) \nwhere FWHM is the full-width at half maximum ( ∼ λ/D if diffraction-limited) and N phot is the total number of detected photons. 3 For the PL spectroastrometry, the errors in i -th Zernike mode \nphase recovery in the linear regime scale as \nσ 2 i ≈ N ∑ j =1 ( B ' + n,ij ) 2 I n0 , j N phot . (11) \nThe numerator is determined by the transfer matrix A , which is an analog of the Gaussian width ( σ sig , the standard deviation of the Gaussian PSF profile) in the Equation 10 for tip-tilt modes. This effective resolution is a function of the inverse of the linear intensity response matrix B ' n and can be interpreted as a metric describing spectroastrometric sensitivity. This equation is verified through numerical simulations with Poisson noise. For our simulated lantern, the value of the numerator is 0.82 Zernike rms amplitudes in radians, or 0.52 λ/D . This is comparable to the Gaussian width 0.43 λ/D at the diffraction limit, implying that photon noise-limited errors of PL spectroastrometry are comparable to those of conventional spectroastrometry.", 'Normalized spectra In , SMF': "Fig 7 Example normalized spectra for binary point sources described in the top row of Figure 4, (left) ideal case without systematics, (middle) with systematic tip-tilt, and (right) with centroid shift in tilt varying linearly with wavelength. The continuum levels can be used to eliminate the systematic effects. \n<!-- image --> \nStatic WFEs can arise from a variety of reasons, such as optical aberrations, misalignments, atmospheric dispersion, and AO-residuals. They alter the intensity responses for an on-axis point source ( I n0 ) and linear intensity responses B ' n . \nFigure 7 shows the normalized spectra I n for the binary example case shown in the top panels of Figure 4. In the left the ideal case without any systematics is displayed. The continuum levels of port 1 through 5 are nearly equal, but the emission line at λ = 1 . 55 µm exhibits spectroastrometric signals. Middle panel shows the case with systematic tip-tilt displacement of (-1, 1) mas. The misalignment affects both the continuum levels and the emission line. Note that tip-tilt misalignment shifts the origin in the coupling map (Figure 6) for I n0 and B ' n . The continuum normalized intensities correspond to the shifted I n0 . Besides tip-tilt misalignment, higher-order static WFEs that are not filtered by the PL (defocus and astigmatism modes for a 6-port PL) can introduce similar effects, affecting the continuum and the emission line equally. Unsensed higher-order modes have much smaller effects. \nRight panel shows the case with wavelength-dependent tilt displacement, for example caused by residuals in differential atmospheric dispersion correction. We assumed linear dispersion of 2 mas in the wavelength range [1.53, 1.57] µm. If the spectral line of interest has smaller width than the scale of wavelength-dependent systematics, the slow variations in continuum levels can be used to eliminate those effects and recover spectroastrometric signals.", '3.3 Time-varying WFEs averaged over an exposure': "A more complicated effect is in the time-varying WFEs averaged over a long exposure, which can systematically affect spectroastrometric signal recovery. Consider a simple case with tip-tilt jitter only. Let us assume that random samples of the tip-tilt errors follow a multivariate normal distribution \n( δx δy ) ∼ N (( 0 0 ) , ( σ 2 x σ xy σ xy σ 2 y )) . (12) \nDue to nonlinearities in the intensity responses, the B ' n matrices are modified. Figure 8 shows the intensity responses to tip-tilt modified by the 10 mas of tip-tilt jitter ( σ x = σ y = 10 mas, σ xy = 0 ). For reference, the intensity responses without jitter are shown as dashed lines. The linear slopes B ' n are systematically modified by jitter as follows: \n˜ B ' n,ix ≈ B ' n,ix +(3 D ' n,ixxx σ 2 x +2 D ' n,ixxy σ xy + D ' n,ixyy σ 2 y ) ˜ B ' n,iy ≈ B ' n,iy +(3 D ' n,iyyy σ 2 y +2 D ' n,ixyy σ xy + D ' n,ixxy σ 2 x ) (13) \nwhere i is the port index and D ' n is the normalized cubic intensity response tensor (see Lin et al. 2022 32 for cubic expansion of PL intensity responses). This shows that the more cubic PL intensity responses are, the more susceptible the lantern linear responses are to the tip-tilt jitter. With time-varying WFEs in sensed higher-order modes (defocus and astigmatism for a 6-port PL), more cubic terms are added in Equation 13. \nIn practice, the bias in the spectroastrometric signals due to the change in B ' n may be calibrated using various approaches, such as by taking empirical B ' n measurement on the target on-sky or a calibrator observation. Moreover, the fact that the transfer matrices vary slowly on wavelength (Section 4) may be used to determine the modified B ' n matrix, assuming that the optical path difference is achromatic. We defer developing practical methods in presence of realistic WFEs to future work.", '4 Chromatic behavior of PL principal modes': "In the previous sections, we simulated spectroastrometric signals using the transfer matrix computed for λ = 1 . 55 µm for certain specified PL design parameters. However, PLPMs are in fact expected to vary slowly with wavelength. In this section, we discuss the PL chromaticity and its implications on spectroastrometry. \nLet us consider light propagation from one of the SMFs (fundamental mode) to the multimode lantern end. Along the transition region, the single fundamental mode propagating in the SMF will eventually spread out of the core and couple to the neighboring SMF cores leading to a set of supermodes (the eigenmodes of the system, representing the collective behavior of the coupled cores). 17 This set of supermodes will become non-degenerate with distinct propagation constants at the multimode end of the lantern transition. Hence, the single input fundamental mode evolves \nFig 8 Same as bottom panels of Figure 3, but with 10 mas of tip-tilt jitter. The normalized intensity responses without jitter are represented as dashed lines. The intensity responses to tip-tilt are systematically decreased with jitter. \n<!-- image --> \ninto these non-degenerate supermodes and reaches the other end (the FMF entrance) as an unique orthogonal solution leading to the PLPM. Thus, the PLPM is a superposition of orthogonal modes, which are approximately the N LP modes (for step-index PLs with circular entrances). Due to the propagation constant difference ( ∆ β i,j ) between the non-degenerate supermodes i and j , the relative phase between the supermodes evolves as they propagate. The relative phase between the supermodes i and j is imprinted in the PLPMs as \n∆ φ i,j = ∫ z taper 0 ∆ β i,j ( z ) dz (14) \nwhere the z taper is the taper length. Since the propagation constant β relies on wavelength, ∆ φ i,j also changes with wavelength, ∆ φ i,j = ∆ φ i,j ( λ ) . This causes chromatic behavior of PLPM patterns, and consequently on the on-axis normalized intensities ( I n0 ) and intensity response matrices ( B ' n ). \nFigure 9 shows an example for a 3-port PL. We performed numerical back propagation of single-moded beams of several different wavelengths from the output SMFs to the lantern entrance and found the LP 01 and LP 11 mode phases of the PLPMs. The simulated 3-port PL is similar to the 6-port PL but has a smaller entrance core size (6 µm) and three residual (non-guiding) cores forming an equilateral triangle instead of six. As can be seen from the left panel, the relative phase between LP 01 and LP 11 modes varies slowly as a function of wavelength, with a period of about 0.3 µm. In the middle panel we display the tilt component of the B ' n matrix, the linear intensity response given tilt aberration. The port 0 is not sensitive to tilt due to the geometry of our simulated PL. The responses of port 1 and port 2 oscillate as a function of wavelength. For the 3-port PL, we find that intensity responses behave linearly at wavelengths where LP 01 and LP 11 modes are in phase (multiples of π ). If they are π/ 2 out of phase, the PLPMs are symmetric, lacking sensitivity to asymmetries, and showing zero linear response to tip-tilts ( B matrix). Examples for the two wavelengths are shown on the right panels. Note that the period of chromatic behavior depends on the taper length, which is fixed for a fabricated device. However, it offers another degree of freedom during the lantern fabrication process to tailor the chromatic behavior. More phase difference is accumulated if the lantern is longer, making the tip-tilt response oscillation period shorter. \nFig 9 (Left) Relative phase between LP 01 and LP 11 modes found from chromatic simulations of a standard 3-port PL. Black dots indicate simulated points and red curve shows the second-order polynomial fit. (Middle) Tilt components of the normalized intensity response matrix (rad -1 ), showing oscillating patterns as a function of wavelength. This is related to the relative phase between the two LP modes. (Right) Normalized intensities as a function of tilt, for two wavelengths. At 1.54 µm the LP modes are 90 degrees out of phase, the corresponding PLPM is symmetric, and intensity responses to tilt aberration are nonlinear. At 1.62 µm the LP modes are in phase, the PLPM is asymmetric, and intensity responses are linear. \n<!-- image --> \nFor the 6-port PL, the chromatic behaviors are more complicated because there is a greater number of modes ( LP 01 , LP 11 , LP 21 , LP 02 ) with differing propagation constants. Figure 10 shows examples of normalized output intensities as a function of tip-tilt Zernike mode amplitudes for three different wavelengths. The zeropoints ( I n0 ) and the slopes ( B ' n ) vary as a function of wavelength. Tip-tilt sensitivity is determined by combined effects of relative phase between LP 01 and LP 11 modes and between LP 02 and LP 11 modes. The relative phase between LP 01 and LP 02 modes as a function of wavelength causes oscillating behavior of normalized intensity in port 0 and the rest and sensitivity to defocus: 32 the normalized intensity in port 0 being nearly zero at λ = 1 . 53 µm and nearly unity at λ = 1 . 63 µm. Consequently, the spectroastrometric S/N (Section 3.1) and the jitter sensitivity (Section 3.3) vary with wavelength. In our particular lantern case (taper length, geometry and refractive indices), the response to tip-tilt at λ = 1 . 63 µm, is highly nonlinear, resulting in larger jitter sensitivity and smaller spectroastrometric S/N. Hence, for accurate spectroastrometric measurements at the wavelengths of interest, the lantern properties should be carefully designed for optimum performance.", '5 Mock observation of accreting protoplanets': "One of the possible applications of PL spectroastrometry is searching for companions with emission lines, such as accreting protoplanets around young stars. 11 Accreting planets are known to emit strong hydrogen emission lines because of heating in the accretion shock region. 38,39 The line luminosities are proportional to mass accretion rates. 38 Recently, H α direct imaging observations revealed accreting planets PDS 70b and PDS 70c around a young star PDS 70, at 113 pc, with angular separations around 200 mas. 40,41 Spectroastrometry opens up the possibility of detecting such objects at much smaller separations. Moreover, the capability of two-dimensional spectroastrometry with PLs enables more efficient detection. \nFig 10 Same as bottom panels of Figure 3 but with three different wavelengths. The intensity responses change slowly with wavelength. The oscillation of normalized intensity ( I n0 ) in the central port (port 0) and the rest and the oscillation of linear intensity responses ( B ' n ) are noticeable. \n<!-- image --> \nWe simulate simple mock observations of accreting planets around a PDS 70 analog as follows. The telescope and instrument throughputs are simulated using PSISIM 42 1 , assuming observations with the next generation high-resolution instrument at the W. M. Keck Observatory, High-resolution Infrared Spectrograph for Exoplanet Characterization (HISPEC). 43,44 We use PHOENIX stellar atmosphere models 45 to model the spectra of PDS 70, using an effective temperature T eff = 4000K , surface gravity log g = 4 . 5 , and solar metallicity. For the accreting planet, we generate Gaussian-shaped emission lines for two accretion rates ˙ M = 10 -8 M ⊙ yr -1 (case A) and ˙ M = 10 -8 . 5 M ⊙ yr -1 (case B), using accretion luminosity - line luminosity scaling relations by Aoyama et al. 38 We particularly focus on the hydrogen Paschen β line (1.282 µm) in J band for this simulation. The linewidth is set to 50 kms -1 , corresponding to free-fall velocity of a Jupiter- \nFig 11 Example mock observations of an accreting protoplanet around a PDS 70 analog on Paschen β line. The simulation parameters are described in Table 1. Top and bottom panels display the mock observations for the case A (linear intensity response regime) and case B (larger separation, nonlinear regime), respectively. (Left) Simulated spectra in 6 output SMFs. The spectral features are dominated by the absorption lines of the star. (Middle) The spectra normalized by the sum of intensities in the 6 ports as a function of wavelength. (Right) Recovered spectroastrometric signals as a function of wavelength. The signals from the off-centered emission line of the planet at the wavelength of Paschen β are detected. The true center of lights are indicated as circular symbols. For case A, the recovered spectroastrometric signals match the true centroid shifts. For case B, the recovered tip-tilts deviate from the true centroid shifts but in this nonlinear regime, the detected normalized intensity signals can be used to fit a binary model to recover binary separation, contrast, and PA simultaneously. \n<!-- image --> \nect. We assume that the absolute magnitude of the host star is 4.2 magnitude in J band. If the star is rapidly accreting ( ˙ M > 10 -8 M ⊙ yr -1 ), the Paschen β emission line will dominate over the continuum, 46 but we do not consider the stellar emission line in this paper. Then the contrasts between the star and the planet in Paschen β line are 2 . 0 × 10 -2 and 5 . 2 × 10 -3 , respectively. For case A, we assume a small separation (2 mas) at 40 pc which is in the linear intensity response regime. For case B we assume a larger separation (18 mas) at 140 pc distance, beyond the PL's linear intensity response regime but within λ/D for D = 10 m, which we may constrain the separation and the contrast simultaneously (§2.2.3). Our simulated wavelength range is [1.278, 1.286] µm. We ignore the chromatic behavior of the lantern (variation in transfer matrix as a function of wavelength) given that the wavelength range is small compared to the range of lantern chromatic behavior. We use the transfer matrix calculated for 1.55 µm with lantern properties described in Section 2.2.1. The spectra are binned to optimize the signal-to-noise ratio of the Paschen β emission line. For the effects of WFEs, we only consider wavelength-independent random tip-tilt jitter for both cases. 300 tip-tilt values are drawn from normal distribution with standard deviation of σ tt = 0 . 2 λ/D . Table 1 lists all the simulation parameters. \nThe left panels in Figure 11 show simulated PL spectra for the case A (top) and case B (bottom), with photon noise and tip-tilt jitter. The output spectra are dominated by the stellar light with \nFig 12 Inferred posterior distributions of the binary parameters for the case A (left) and case B (right). The true values of the parameters are indicated as dotted lines. The separation and contrast are degenerate for the small separation case (case A, in linear regime). \n<!-- image --> \nabsorption features. The differences in the continuum levels between the ports reflect nonzero average tip-tilts. The middle panels show the normalized spectra I n ( λ ) . In the normalized spectra, the signals from the off-center planet at λ = 1 . 282 µm can be noticed. The right panels display the recovered spectroastrometric signals, Equation 9, including defocus and astigmatism modes. \nFor the case A where the separation is small (in linear regime), tip-tilt signals dominate. The non-detection in the astigmatism and defocus modes may imply that the planet separation is small. The recovered centroid shifts using the linear response matrix are close to true centroid shifts as indicated as circles in the right panel of Figure 11. The recovered signals are systematically smaller than the true centroid shifts due to the second-order effects of the tip-tilt jitter (§3.3). \nFor the case B where the separation is in nonlinear regime, there are noticeable astigmatism signals. In addition, the recovered spectroastrometric signals using the linear approximation fails as can be seen as the deviation of the recovered centroid shifts from the true centroid shifts. Although the S/N of tip-tilt modes are smaller in this case, detection in astigmatism modes can support the detection and provide constraints on separation, contrast, and PA simultaneously. \nAs discussed in §2.2.3, we try model fitting the normalized spectra (middle panel of Figure 11) to a binary model which can be constructed using empirical coupling maps. In Table 1 and Figure 12 we display retrieved binary parameters and inferred posterior distributions using emcee Markov Chain Monte Carlo (MCMC) ensamble sampler. 47 While the separation and contrast are highly degenerate for case A, due to nonlinearity, all the three binary parameters for case B are well-constrained despite lower S/N. \nTable 1 Mock observation parameters and S/N of spectroastrometric signals", '6.1 Benefits of PL spectroastrometry': 'Dispersed PL outputs enable two-dimensional spectroastrometry, 12-14 which enables a more efficient observation compared to long-slit spectroastrometry. One of the advantages of PL spectroastrometry is that two-dimensional spectroastrometry can be easily enabled without the need to resample the focal plane, such as using image slicers. Once the focal plane field is coupled to the FMF entrance of a PL with a scale of about a few λ/D , small wavelength-dependent centroid shifts result in variations in the SMF output spectra, which can then be used to infer the centroid shifts. The centroid shifts can be efficiently recovered using a few-moded PL such as the 6-port PL described in this study. The few SMF outputs can be dispersed at high spectral resolutions 21 with efficient use of the detector area. This can enable resolving two-dimensional kinematic structures that require high spectral resolution, as in the BLR and rotating star examples in §2.2.2. \nMoreover, the capability of wavefront sensing with PLs 31-34 opens an extra potential to reduce the systematic effects of WFEs (§3.3) by real-time wavefront correction and achieve a higher throughput. A PL filters out high-order aberrations and couples low-order aberrations that result in overall variations in SMF output intensities. The residual low-order aberrations may be corrected in real-time by using output intensities for active wavefront control. It may also be possible to characterize the low-order aberration effects in post-processing procedure, using chromaticity of the PL principal modes: the behaviors of the responses to the WFEs change slowly as a function of wavelength.', '6.2 Considerations on PL design': 'To efficiently measure spectroastrometric signals, it is important that the PL has large linear intensity responses, B n , which is directly related to the signal-to-noise ratio of the spectroastrometric signals (§3.1). In addition, the susceptibility to systematic effects of time-varying WFE relies on the relative significance of the linear and cubic intensity responses (§3.3). Note that the intensity responses vary slowly as a function of wavelength, as discussed in §4. It is essential for spectroastrometry to achieve a good linear response at the wavelength range of interest. The intensity responses rely on the transfer matrix A corresponding to the PLPMs, which can be designed and optimized for efficient spectroastrometric observations. By adjusting the geometry of the lantern such as changing the taper length and core arrangements, the intensity responses can be designed. 33 \nAlthough we have limited our simulation to a standard 6-port PL in this paper, one may consider using a different mode-count PL or a different type of PL such as a mode-selective PL. 48 If limiting consideration to the small separation regime (linear intensity response regime), a 3-port PL will be sufficient to recover tip-tilts. However, recovering spectroastrometric signals for a more extended object (angular extent in nonlinear intensity response regime) is more challenging due to higher degeneracy. A higher mode-count PL would exhibit a more complicated chromatic behavior, which may be more challenging to optimize in design. However, if the interest lies in characterizing a more extended object (§2.2.3), using a higher mode-count PL can be beneficial. Using a modeselective PL, a deeper contrast may be achieved but with a position angle degeneracy of 180 degrees due to symmetry. 28', '7 Conclusion and future work': "In this work, we explored the capability of PLs for two-dimensional spectroastrometry. PLs can enable measuring two-dimensional spectroastrometric signals simultaneously without resampling of the focal plane, using a few single-moded spectral traces. We defined spectroastrometric signals for PLs and simulated them for a few simple scenes in §2. In the regime where the input scene has a small angular size such that PL's intensity responses are linear, the centroid shifts can be simply recovered using a linear response matrix. If the input scene is more extended, the PL relative intensities can be used to fit models to learn about more information other than centroid shifts. We investigated the effects of photon noise and WFEs on spectroastrometric signal recovery in §3. The spectroastrometric sensitivity for the photon noise-limited case is closely related to how linear the intensity responses are to tip-tilts and is theoretically comparable to conventional PSF centroid fitting for our simulated 6-port PL. The effects of static residual WFEs (wavelength-independent or slowly-varying as a function of wavelength) can be calibrated using continuum levels. The timevarying WFEs averaged over an exposure can affect the intensity responses, which may also be calibrated. We explored the chromaticity in PLPMs that result from phase differences between the waveguide modes that constitute PLPMs, in §4. The linear intensity responses are expected to vary slowly as a function of wavelength, making some wavelengths more suitable for spectroastrometry than others. The design of the PL such as taper length can be optimized to achieve a good linear intensity response at the wavelength of interest. We also provided mock observations of accreting protoplanets emitting hydrogen emission lines around a PDS 70 analog in §5. \nFuture work includes verifying the behavior of lanterns in a lab and on-sky, specifically the capability of measuring centroid shifts as a function of wavelength. The PL transfer matrices need to be experimentally determined and their stability under environmental changes such as temperature \nshould be characterized (see Ref. 34 for stability of the response matrix in the context of PLWFS). The effects of two polarization states degenerate in SMF fundamental modes should be taken into account in the instrument design. 49 The PL transfer matrices will likely also depend on polarization at some level. 50 Moreover, accurate port-by-port wavelength solutions are crucial for spectroastrometric measurements as well as reconstruction of the summed spectra, since wavelength solution errors can introduce artificial spectroastrometric signals and systematic broadening of the spectra. Practical observing strategies and calibration techniques to deal with alignment sensitivity and systematic effects regarding time-varying and static WFEs are also left for future investigation.", 'Code and Data Availability': 'The data and code used in preparation of this work are available upon request to the corresponding author.', 'Acknowledgments': 'This work is supported by the National Science Foundation under Grant No. 2109231, 2109232, 2308360, and 2308361. 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2024arXiv240903523E | The Euclid mission will measure cosmological parameters with unprecedented precision. To distinguish between cosmological models it is essential to generate realistic mock observables from cosmological simulations that were run in both the standard Lambdacolddarkmatter LambdaCDM paradigm and in many nonstandard models beyond LambdaCDM. We present the scientific results from a suite of cosmological Nbody simulations using nonstandard models including dynamical dark energy kessence interacting dark energy modified gravity massive neutrinos and primordial nonGaussianities. We investigate how these models affect the largescalestructure formation and evolution in addition to providing synthetic observables that can be used to test and constrain these models with Euclid data. We developed a custom pipeline based on the Rockstar halo finder and the nbodykit largescale structure toolkit to analyse the particle output of nonstandard simulations and generate mock observables such as halo and void catalogues mass density fields and power spectra in a consistent way. We compare these observables with those from the standard LambdaCDM model and quantify the deviations. We find that nonstandard cosmological models can leave significant imprints on the synthetic observables that we have generated. Our results demonstrate that nonstandard cosmological Nbody simulations provide valuable insights into the physics of dark energy and dark matter which is essential to maximising the scientific return of Euclid. | 2024-09-01T00:00:00Z | ['2024arXiv240903523E', 'arXiv:2409.03523', '10.48550/arXiv.2409.03523'] | ['Astrophysics - Cosmology and Nongalactic Astrophysics'] | Euclid preparation. Simulations and nonlinearities beyond LambdaCDM. 2. Results from nonstandard simulations | 2,024 | 170 | 0.55 | ['EPRINT_HTML', 'EPRINT_PDF'] | 0 | https://arxiv.org/pdf/2409.03523.pdf | {'Simulations and nonlinearities beyond Λ CDM. 2. Results from non-standard simulations': '58 \nG. Testera \n, R. Teyssier \n49 \n, S. Toft \n, \n87 \n, \n161 \n162 \n57 \n, \n58 \n43 \n, \n44 \nD. Vergani 14 , and P. Vielzeuf 48 \n, S. Tosi \n, A. Troja \n, M. Tucci \n(A ffi liations can be found after the references) \nSeptember 6, 2024', 'ABSTRACT': 'The Euclid mission will measure cosmological parameters with unprecedented precision. To distinguish between cosmological models, it is essential to generate realistic mock observables from cosmological simulations that were run in both the standard Λ -cold-dark-matter ( Λ CDM) paradigm and in many non-standard models beyond Λ CDM.Wepresent the scientific results from a suite of cosmological N -body simulations using non-standard models including dynamical dark energy, k -essence, interacting dark energy, modified gravity, massive neutrinos, and primordial non-Gaussianities. We investigate how these models a ff ect the large-scale-structure formation and evolution in addition to providing synthetic observables that can be used to test and constrain these models with Euclid data. We developed a custom pipeline based on the Rockstar halo finder and the nbodykit large-scale structure toolkit to analyse the particle output of non-standard simulations and generate mock observables such as halo and void catalogues, mass density fields, and power spectra in a consistent way. We compare these observables with those from the standard Λ CDMmodel and quantify the deviations. We find that non-standard cosmological models can leave significant imprints on the synthetic observables that we have generated. Our results demonstrate that non-standard cosmological N -body simulations provide valuable insights into the physics of dark energy and dark matter, which is essential to maximising the scientific return of Euclid . \nKey words. Cosmology: theory - large-scale structure of Universe - dark matter - dark energy - methods: numerical', '1. Introduction': "The concordance Λ -cold-dark-matter ( Λ CDM) model is the simplest cosmological scenario that accounts for the cosmological observations thus far available. It is based on the assumption that in addition to baryonic matter and radiation, the Universe is filled with two invisible components: an exotic form of matter, dubbed dark energy and described by a Cosmological Constant ( Λ ) in Einstein's equations of General Relativity, and a cold-dark-matter (CDM) component that is non-relativistic and only interacts through gravity. In this scenario, dark matter is primarily responsible for fostering the formation of the visible structures we observe today, while dark energy drives the accelerated expansion of the Universe at late times. This model has been remarkably successful in explaining a variety of cosmological observations, such as the Hubble diagram from luminosity distance measurements of Type Ia Supernovae (Riess et al. 1998; Perlmutter et al. 1999), temperature and polarisation anisotropy angular power spectra of the cosmic microwave background (CMB, de Bernardis et al. 2000; Spergel et al. 2003; Kovac et al. 2002; Planck Collaboration: Aghanim et al. 2020), the galaxy power spectrum of the large scale structure (LSS, Efstathiou et al. 2002; Colless et al. 2003; Tegmark et al. 2004, 2006), and the presence of baryonic acoustic oscillations (BAO) in the LSS (Eisenstein et al. 2005; Cole et al. 2005). Despite the great success of the Λ CDM model, the physical origin of dark energy and dark matter remains unknown. Unveiling the nature of these dark components is the primary motivation for many investigations in modern cosmology. \nIn the last decade, multiple tensions among di ff erent types of cosmological observations have emerged. As an example, while CMB measurements indicate a value of the Hubble constant of H 0 = 67 . 7 ± 0 . 4 kms -1 Mpc -1 (Planck Collaboration: Aghanim et al. 2020), local measurements, often based on the observations of supernovae in nearby galaxies, suggest a higher value of H 0 = 73 . 0 ± 1 . 0 kms -1 Mpc -1 (Riess et al. 2022). This 5 σ discrepancy is called the Hubble tension. A similar tension has been identified in the S 8 = σ 8 √ Ω m / 0 . 3 parameter, which combines the amplitude of linear-matter density fluctua- \nthe 8 h -1 Mpc scale, σ 8, and the cosmic matter density, Ω m. Measurements derived from the CMB (Planck Collaboration: Aghanim et al. 2020) appear to yield a value of S 8 2 . 9 σ higher than that obtained from observations of the LSS (Joseph et al. 2023), such as measurements of the clustering of galaxies and weak gravitational lensing (Li et al. 2023; Abbott et al. 2022). Such tensions may result from systematic errors yet to be identified in the data. Alternatively, they may be a manifestation of the limits of the Λ CDM model, since modifications to the standard cosmological model can provide a solution to these tensions (see e.g. Martinelli & Tutusaus 2019; Di Valentino et al. 2021). \nOngoing and upcoming Stage-IV surveys, such as Euclid (Euclid Collaboration: Mellier et al. 2024; Laureijs et al. 2011; Euclid Collaboration: Scaramella et al. 2022), Dark Energy Spectroscopic Instrument (hereafter DESI, DESI Collaboration: Aghamousa et al. 2016), Vera C. Rubin Observatory Legacy Survey of Space and Time (LSST, Ivezi'c et al. 2019), SpectroPhotometer for the History of the Universe, Epoch of Reionization, and the Ices Explorer (SPHEREx, Doré et al. 2014), and the Nancy Grace Roman Space Telescope (Spergel et al. 2015), will collect unprecedented amounts of data on the LSS, which will enable detailed assessments of the Hubble and S 8 tensions in addition to shedding new light on the nature of the invisible components in the Universe. \nEuclid is a space mission led by the European Space Agency (ESA) with contributions from the National Aeronautics and Space Administration (NASA), aiming to study the nature and evolution of the dark universe. The survey uses a 1.2-m-diameter telescope and two instruments, a visible-wavelength camera, and a near-infrared camera / spectrometer, to observe billions of galaxies over more than a third of the sky in optical and nearinfrared wavelengths. Euclid measures the shapes (Euclid Collaboration: Bretonnière et al. 2022, 2023; Euclid Collaboration: Merlin et al. 2023) and redshifts (Euclid Collaboration: Desprez et al. 2020; Euclid Collaboration: Ilbert et al. 2021) of galaxies, in order to determine the weak gravitational lensing and clustering of galaxies, covering a period of cosmic history over which dark energy accelerated the expansion of the Universe. These measurements will provide detailed insights into the properties of dark energy, dark matter, and gravity by probing the expansion \n78 \n, C. Valieri \n27 \n, J. Valiviita \n, \n2 \n94 \n, \nhistory of the Universe and the growth rate of structures over time (Martinelli et al. 2021; Nesseris et al. 2022; Euclid Collaboration: Castro et al. 2023). Euclid was launched on 1 July 2023 and is designed to operate for six years. The survey will provide unprecedented constraints on cosmological parameters and tests of fundamental physics, as well as a rich catalogue of legacy data that can be used for a wide range of astrophysical research. The mission data will be publicly released within two years of acquisition. Euclid is one of the most ambitious and exciting space missions in the field of cosmology and will enable a thorough validation of a broad range of cosmological models. \nEuclid observations will provide precise measurements of the clustering of matter over a wide range of scales, where effects due to the late-time nonlinear gravitational collapse of matter need to be taken into account. A key tool in the preparation of the cosmological analyses and the interpretation of the Euclid data is the use of cosmological N -body simulations, which can follow the nonlinear evolution of matter clustering. This is a numerical technique that calculates the evolution of the matter density field under the e ff ect of gravity across cosmic time and predicts the LSS of the Universe for a given cosmological model (Press & Schechter 1974; Zeldovich 1978; Klypin & Shandarin 1983; Appel 1985; Potter et al. 2017; Angulo & Hahn 2022). In this method, the matter density field is sampled with discrete N -body particles, whose equations of motion are solved in the Newtonian limit in an expanding Friedmann-LemaîtreRobertson-Walker (FLRW) universe. These simulations enable the study of the formation and growth of cosmic structures from linear to nonlinear scales, predict the distribution of matter in galaxy clusters, filaments, and voids, for a range of cosmological models and parameters (Klypin et al. 2003; Dolag et al. 2004; Alimi et al. 2010; Li et al. 2012; Puchwein et al. 2013; Baldi & Simpson 2015), as well as initial conditions (Dalal et al. 2008). The cosmological models beyond the standardΛ CDMparadigm are expected to have left imprints that should be detectable in the Euclid observables, such as the redshift-space power spectra of galaxies or the void-size functions. \nThis article is part of a series that collectively explores simulations and nonlinearities beyond the Λ CDMmodel: \n- 1. Numerical methods and validation (Adamek et al. in prep.).\n- 2. Results from non-standard simulations (this work).\n- 3. Cosmological constraints on non-standard cosmologies from simulated Euclid probes (D'Amico et al. in prep.).\n- 4. Constraints on f ( R ) models from the photometric primary probes (Koyama et al. in prep.). \nFor further details, see our companion papers. In this work, we consistently analyse large numbers of N -body simulations over a wide range of non-standard cosmological scenarios, to generate catalogues of synthetic observables for Euclid . This analysis is achieved using a pipeline that was specifically written for that task. We calculate reconstructed density fields, halo and void catalogues, halo mass functions, dark matter, and halo power spectra in real and redshift space, as well as halo bias functions. The paper is organised as follows: in Sect. 2, we introduce the analysed non-standard models; then, in Sect. 3, we present an overview of the analysed cosmological N -body simulations. In Sect. 4, we describe the analysis pipeline and the calculated quantities. We demonstrate the imprints of the nonstandard models in the computed observables in Sect. 5 and finally, we summarise our results in Sect. 6.", '2. Cosmological models beyond the standard Λ CDM paradigm': 'To address the tensions and anomalies in the Λ CDMmodel, various non-standard cosmological models have been proposed that extend or modify the standard model in di ff erent ways. Some examples of non-standard cosmological models are dark energy models, such as quintessence and phantom energy, modifiedgravity theories, such as f ( R ) gravity, and massive-neutrino models, such as sterile neutrinos and self-interacting neutrinos. These models introduce new degrees of freedom or new mechanisms that can a ff ect the dynamics and observables of the universe at di ff erent scales and epochs. In this section, we will discuss the main features, motivations, and challenges of these nonstandard cosmological models.', '2.1.1. w CDM': 'A simple generalisation of the cosmological constant assumes that dark energy is a fluid with a constant equation-of-state w ≡ p de / ( ρ de c 2 ), where p de and ρ de are, respectively, the pressure and density of the fluid, and c is the speed of light. To trigger an accelerated phase of cosmic expansion, the dark energy equation-of-state parameter must be w < -1 / 3. The Λ CDM model corresponds to the w = -1 specific case, while w < -1 corresponds to so-called phantom dark energy models (Caldwell et al. 2003), though such values may also result from an unaccounted interaction between dark energy and dark matter (Das et al. 2006).', '2.1.2. Dynamical dark energy': 'The dark energy equation-of-state could be a function of redshift. Chevallier, Polarski (Chevallier & Polarski 2001) and Linder (Linder 2003) proposed a simple parameterization of \nw de( z ) = w 0 + wa z 1 + z = w 0 + wa (1 -a ) , (1) \nwhere the w 0 parameter represents the value of the equation-ofstate at the present time, and wa defines the rate of change with redshift. This model is also called the CPL parametrisation of dark energy, after the initials of the authors who proposed it. \nThis dark energy parametrisation is a fitting function of a general w de( z ) around z = 0, assuming that w ( z ) is smooth and slowly changing with the scale factor. As a consequence, this model can closely follow the expansion history of a wide range of other models with w de( z ) at late times. Despite its simple form, it shows a wide range of interesting properties (Linder 2008; Linden & Virey 2008). The cosmological constant corresponds to w 0 = -1 and wa = 0 in the CPL parametrisation.', '2.1.3. K-essence': 'The k-essence model is characterised by an action for the scalar field of the following form \nS = Z d 4 x √ -g p ( ϕ, X ) , (2) \nwhere X = (1 / 2) g µν ∇ µϕ ∇ νϕ . The energy density of the scalar field is given by \nu ϕ = 2 X d p d X -p , (3) \nFig. 1. Flowchart summarising the main steps of the analysis pipeline. In the first step, the pipeline executes the Rockstar halo finder to generate the halo catalogues and additional BGC2 particle data containing all relevant information for the halo profile calculation. Then, the 2D and 3D halo profiles, the halo mass function, and void catalogues are calculated by the corresponding modules. The real- and redshift-space halo power spectra and the triangular shaped cloud (TSC) reconstructed halo density fields on a regular cubic grid are computed for a predefined mass bin. After this, the dark matter power spectrum calculator module computes the real- and redshift-space dark matter power spectra with the reconstructed TSC dark matter density field. By using the real-space dark matter and halo power spectra, the linear-halo-bias estimator module calculates the halo-bias table using only the linear scales. Finally, by using this halo-bias table, the linear matter power spectrum, and the cosmological parameters, the halo redshift-space power spectrum Gaussian covariances are computed. This process is repeated for all selected particle snapshots. \n<!-- image --> \nand the pressure is p ϕ = p ( ϕ, X ). This pressure gives an e ff ective fluid equation-of-state parameter as \nw ϕ = p ϕ u ϕ = -p p -2 Xp , X , (4) \nwhere the subscript , X indicates a derivative with respect to X , and a dimensionless speed-of-sound parameter for the k-essence fluctuations as \nc 2 s = p , X p , X + 2 Xp , XX . (5) \nThe k-essence field satisfies the continuity equation \n˙ u ϕ = -3 H ( u ϕ + p ϕ ) , (6) \nArticle number, page 4 of 22 \nwhich results in the scalar equation of motion \nG µν ∇ µ ∇ νϕ + 2 X ∂ 2 p ∂ X ∂ϕ -∂ p ∂ϕ = 0 , (7) \nwhere \nG µν = ∂ p ∂ X g µν + ∂ 2 p ∂ X 2 ∇ µ ϕ ∇ ν ϕ . (8) \nk-essence was first proposed by Armendariz-Picon et al. (2000, 2001), who showed that there exist tracking attractor solutions to the equation of motion during the radiation and matterdominated eras of the universe, and that with a suitably chosen p , the scalar can have an appropriate equation of state that allows it to act as dark energy for the background accelerated expan- \nsion. In addition, whenever the kinetic terms for the scalar field \nϕ \nϕ \nare not linear in X , the speed of sound of the fluctuations differs from unity, allowing the clustering of the dark energy field at sub-horizon scales, which should be modelled at the perturbations level.', '2.1.4. Interacting dark energy': "In the interacting dark energy (IDE) models (Amendola 2000; Farrar & Peebles 2004; Baldi et al. 2010), dark energy and cold dark matter are allowed to interact through an exchange of energy-momentum in order to keep the total stress-energy tensor T µν conserved: \n∇ µ T ( c ) µ ν = C ν ( ϕ ) = -∇ µ T ( ϕ ) µ ν , (9) \nwhere C ν ( ϕ ) is a conformal coupling function expressed in the form: \nC ν ( ϕ ) = κ β ( ϕ ) u c ∇ νϕ , (10) \nwhere κ ≡ 8 π G N c 2 , G N is Newton's gravitational constant, u c is the cold dark matter energy density in the IDE model 1 , and β ( ϕ ) is a coupling function. The dark energy scalar field, ϕ , has an intrinsic energy density and pressure given by \nu ϕ = 1 2 g µν ∂µϕ ∂νϕ + V ( ϕ ) , (11) \np ϕ = 1 2 g µν ∂µϕ ∂νϕ -V ( ϕ ) , (12) \nwhere V ( ϕ ) is a self-interaction potential. The conservation equations then translate in the following set of backgrounddynamic equations under the assumption of a constant coupling function β ( ϕ ) = β : \n¨ \n+ \n3 \nH \n˙ \n+ \nd \nV \nϕ \nd \n˙ u c + 3 Hu c = -κ β u c , (14) \nIn the standard approach, a theoretically-motivated analytical form for the self-interaction potential function V ( ϕ ) is chosen. However, the simulations that are considered in the present work implement the alternative approach proposed by Barros (2019) which consists of imposing a standard Λ CDM background expansion history by setting \nH 2 = H 2 Λ CDM , (15) \nwhere H Λ CDM is the standard Hubble function defined by \nH 2 Λ CDM = 8 π G N 3 ( ρ r + ρ b + ρ CDM + ρ Λ ) , (16) \nwhere ρ r, ρ b, ρ CDM, and ρ Λ are the mass densities of the radiation, baryon, CDM, and Λ components of the background Λ CDM model. This will determine an e ff ective potential, V ( ϕ ), according to the resulting evolution of the scalar field, ϕ . Taking the time derivative of Eq. (15) and using the continuity Eqs. (13 &14), one gets the scalar-field energy density and pressure as \nu \nCDM \nc \nΛ \nc \nu \nc \n(17) \np ϕ = p Λ = -u Λ , (18) \n= \n2 \n+ \n2 \n- \n= \nu \n(13) \nwhich can be combined with Eqs. (11 & 12) to obtain the dynamics of the scalar field: \n˙ ϕ 2 = ρ CDM c 2 -u ϕ . (19) \nThe scalar-field potential, V ( ϕ ), can then be reconstructed using Eqs. (17 & 18) as: \nV ( ϕ ) = 1 2 ˙ ϕ 2 + ρ Λ c 2 , (20) \nand taking the time derivative of Eq. (19), one can derive the scalar-field equation of motion \n2 ¨ ϕ + 3 H ˙ ϕ -κβ uc = 0 , (21) \nwhich can be numerically solved for the dynamical evolution of the system. With this choice, the β coupling remains the only free parameter of this model. Observational constraints on the model was computed in Barros et al. (2023) which found that the model can alleviate the σ 8 tension, but that CMB prefers the Λ CDM limit. In particular, they find that the CMB constrains | β | ≲ 0 . 02, RSD constraints | β | ≲ 0 . 10, while weak lensing data from the Kilo-Degree Survey actually prefers a non-zero value | β | ∼ 0 . 1.", '2.2.1. nDGP gravity': "The Dvali-Gabadadze-Porrati (DGP) model (Dvali et al. 2000) assumes that our universe is described by a 5-dimensional bulk, while the visible matter component is confined to the 4dimensional brane described by the Minkowski metric, γ . This model's action is \nS = c 4 16 π G 5 Z M d 5 x √ -γ R 5 + Z ∂ M d 4 x √ -g c 4 16 π G N R + L m ! , (22) \nwhere G 5 and G N are the 5- and 4-dimensional Newton's constants, respectively, and L m is the matter Lagrangian. At small scales, 4-dimensional gravity is recovered due to an intrinsic Einstein-Hilbert term sourced by brane curvature causing a gravitational force that scales as r -2 , while, at large scales, the gravity behaves as a 5-dimensional force. The transition between the 5-dimensional modifications and the 4-dimensional gravity is given by the cross-over scale r c = G 5 / (2 G N), from which we construct the dimensionless parameter Ω rc ≡ c 2 / (4 r 2 c H 2 0 ). The modified Friedmann equation on the brane (De ff ayet 2001) becomes \nH 2 = ± c H r c + 8 π G N 3 ¯ ρ. (23) \nThe model we investigate in this paper is the normal branch with the -sign (Bowcock et al. 2000) characterised by a Λ CDM background achieved by introducing an additional dark energy contribution with an appropriate equation-of-state (Schmidt 2009) \nρ de( a ) = ρ cr , 0 GLYPH<16> ΩΛ + 2 p Ω rc p ΩΛ + Ω m a -3 GLYPH<17> , (24) \nwhere ρ cr , 0 is the critical density. The observational constraints on the model require the cross-over scale r c to be larger than the size of the horizon H -1 0 today. For example, Solar System constraints require rcH 0 ≳ 1 . 6 (Battat et al. 2008), and galaxy clustering in the BOSS survey constraints rcH 0 ≳ 4 . 5 (Piga et al. 2023). \nϕ \nρ \nρ \nκ β \nc \n, \n, \nThe f ( R ) theory of gravity (Buchdahl 1970) is characterised by the following action: \nS = c 4 16 π G N Z d 4 x √ -g GLYPH<2> R + f ( R ) GLYPH<3> , (25) \nwhere g µν is the metric tensor and f ( R ) is a functional form of the Ricci scalar, R . Here we consider the Hu-Sawicki model (Hu & Sawicki 2007) with n = 1, where in the limit of fR = d f / d R ≪ 1 we have \nf ( R ) = -6 ΩΛ H 2 0 c 2 + | fR 0 | ¯ R 2 0 R , (26) \nwhere fR 0 is the free parameter of the model, ¯ R 0 is the Ricci scalar evaluated at background at present time, H 0 is the Hubble constant, and ΩΛ is the energy-density parameter of the cosmological constant. | fR 0 | characterises the magnitude of the deviation from Λ CDM, with smaller values corresponding to weaker departures from General Relativity until we recover Λ CDM in the limit of fR 0 → 0, but for the small | fR 0 | values still allowed by observations, the background expansion history approximates that of Λ CDMand \n¯ R 0 = 3 Ω m H 2 0 c 2 1 + 4 ΩΛ Ω m ! , (27) \nwith matter energy density parameter Ω m = 1 -ΩΛ . However, though the background expansion could mimic that of a cosmological-constant model, it still di ff ers at the level of cosmological perturbations where the growth of structure is driven by a modification of gravity following the above adopted model of f ( R ). \nThe observational constraints on the model parameter | fR 0 | vary from | fR 0 | ≲ 10 -6 in the Solar System, | fR 0 | ≲ 10 -8 from galaxy scales (Burrage et al. 2024) to | fR 0 | ≲ 10 -6 -10 -4 from various cosmological probes (see, e.g., Fig. 28 in Koyama 2016, for a summary). The parameter values of the simulations presented in this paper are similar to the current cosmological constraints.", '2.3. Massive and number of relativistic neutrinos': 'Neutrinos are mainly characterised by two properties, their mass, M ν , and the number of neutrino species, N e ff . More in general, N e ff parametrises the contribution of relativistic species to the background density of radiation, ρ r, as \nρ r = 1 + 7 8 4 11 ! 4 / 3 N e ff ργ , (28) \nwhere ργ is the photon background density. In the standard model, N e ff is expected to be ∼ 3.045 (Cielo et al. 2023) for three families of active neutrinos that thermalised in the early Universe and decoupled well before electron-positron annihilation. The calculation of N e ff involves the complete treatment of neutrino decoupling, which incorporates non-instantaneous decoupling. A deviation from the fiducial value serves to account for the presence of non-standard neutrino features, or additional relativistic relics contributing to the energy budget (Mangano et al. 2002). Here we focus on standard neutrino families only. \nIn addition, oscillation experiments (Maltoni et al. 2004; Kajita 2016) showed that at least two neutrinos are massive by measuring two squared-mass di ff erences. It can be shown that the minimum value of the neutrino mass sum is either 0 . 06 eV in the normal or 0 . 10 eV in the inverted hierarchy. This value can be well constrained through cosmological observations since neutrinos are known to impact the expansion history and suppress the clustering of cold dark matter, which can be observed in the large-scale distribution of galaxies (Sakr 2022). Neutrinos with mass ≲ 0 . 6 eV become non-relativistic after the epoch of recombination probed by the CMB, and this mechanism allows massive neutrinos to alter the matter-radiation equality for a fixed Ω m h 2 (Lesgourgues & Pastor 2006). Massive neutrinos act as non-relativistic particles on scales k > k nr = 0 . 018( m ν/ 1eV) 1 / 2 Ω 1 / 2 m h -1 Mpc, where k nr is the wavenumber corresponding to the Hubble horizon size at the epoch z nr when the given neutrino species becomes non-relativistic following 1 + z nr ≃ 1900 GLYPH<16> m ν 1eV GLYPH<17> , Ω m is the matter density parameter, and h = H 0 / 100 km s -1 Mpc -1 . The large velocity dispersion of nonrelativistic neutrinos suppresses the formation of neutrino perturbations in a way that depends on m ν and redshift z , leaving an imprint on the matter power spectrum at scales k > k fs( z ), with \nk fs = 0 . 82 H ( z ) H 0(1 + z ) 2 GLYPH<18> m ν 1eV GLYPH<19> h Mpc -1 , (29) \nwhere neutrinos cannot cluster and do not contribute to the gravitational potential wells produced by cold dark matter and baryons (Takada et al. 2006; Lesgourgues & Pastor 2006). This modifies the shape of the matter power spectrum and the correlation function on these scales.', '2.4. Primordial non-Gaussianities': "The simplest inflation models predict that primordial curvature perturbations follow a distribution that is close to Gaussian (Maldacena 2003; Creminelli & Zaldarriaga 2004). However, there are many alternative inflation models that predict certain amounts of primordial non-Gaussianity (PNG). One of the simplest cases is that of the so-called local primordial nonGaussianities (Salopek & Bond 1990; Komatsu & Spergel 2001). For this case, the primordial potential ϕ is given by \nϕ ( x ) = ϕ G ( x ) + f local NL ( ϕ 2 ( x ) - ⟨ ϕ 2 ( x ) ⟩ ) , (30) \nwhere ϕ G ( x ) is the Gaussian potential, while ϕ is the nonGaussian potential. f local NL measures the level of deviations from Gaussianity. \nThe perturbations in the primordial potential produce perturbations in the density field and they are related through Poisson's equation. Therefore, in Fourier space, the density field is given by \nδ ( k , z ) = α ( k , z ) ϕ ( k , z ) , (31) \nwhere \nα ( k , z ) = 2 D ( z ) 3 Ω m c 2 H 2 0 g (0) g ( z rad) k 2 T ( k ) , (32) \nT ( k ) is the transfer function normalised at T ( k → 0) = 1, and D ( z ) is the growth factor normalised at D ( z = 0) = 1. The factor g (0) / g ( z rad), where g ( z ) = (1 + z ) D ( z ), takes into account the di ff erence between our normalisation of D ( z ) and the early-time normalisation where D ( z ) ∝ 1 / (1 + z ) during matter-domination. \nFig. 2. Calculated halo mass function of the E lephant simulation suite. The vertical and horizontal dot-dashed lines indicate the mass relative to haloes with 50 particles and the number density relative to a 1% shotnoise error, respectively. The shaded region highlights the mass bin used to calculate the halo power spectra shown in Fig. 3. \n<!-- image --> \n/circledot \n[ \n] \nThis factor is g ( z rad) g (0) ∼ 1 . 3, with a small dependency on the cosmology. \nThis type of non-Gaussianity characteristically a ff ects the clustering of biased tracers, inducing a scale-dependent bias (Dalal et al. 2008; Slosar et al. 2008; Matarrese & Verde 2008). To linear order, the power spectrum of galaxies can be given as \nP t , t( k , z ) = b 1 + b ϕ f local NL α ( k , z ) 2 P m , m( k , z ) , (33) \nwhere P t , t( k , z ) is the power spectrum of the tracer, P m , m( k , z ) is the power spectrum of the matter, b 1 is the linear bias, and b ϕ is the response of the tracer to the presence of the local-PNG. Now, P t , t( k , z ) has a dependency with k which scales as k -2 at leading order due to the α ( k , z ) term. The b ϕ is usually parametrised as \nb ϕ = 2 δ c ( b 1 -p ) . (34) \nAlthough it is possible to make a theoretical prediction for p (by assuming a universal mass function, p = 1, Dalal et al. 2008), several studies using numerical simulations have shown that the prediction may be di ff erent depending on the type of galaxy or tracer under consideration (Slosar et al. 2008; Desjacques et al. 2009; Hamaus et al. 2011; Biagetti et al. 2017; Barreira et al. 2020; Adame et al. 2023).", '3. Simulations': 'This section summarises the simulations used for this project and gives a very brief description of each setup. The analysed simulations followed the evolution of the matter field with discrete N -body method in the models described in Sect. 2. Baryonic and hydrodynamical e ff ects are neglected in this paper. For a comprehensive description of each of the simulation suites, we refer the reader to the main references given in Table 1 along with \nthe volumes, resolutions, initial redshifts, and the used order of the Lagrangian perturbation theory (LPT) during the initialcondition generation.', '3.1. The C omplementary simulations': 'The C omplementary simulation series is a set of 4 cosmological N -body simulations in w CDM and Λ CDM cosmologies. This suite used the complementary-simulation method (Rácz et al. 2023), which is a novel technique in which cosmological N -body simulations are run in phase-shifted matching pairs. One simulation starts from a regular random Gaussian initial condition, while the second simulation has modified initial amplitudes of the Fourier modes to ensure that the average power spectrum of the pair is equal to the cosmic mean power spectrum from linear theory at the initial time. The average statistical properties of a pair of such simulations have greatly suppressed variance. In this paper, we have analysed two complementary pairs using Λ CDM and w CDM cosmologies. The Λ CDM simulation pair used the best-fit Planck2018 (Planck Collaboration: Aghanim et al. 2020) cosmological parameters: Ω m = 1 -ΩΛ = 0 . 3111, Ω b = 0 . 04897, H 0 = 67 . 66 km s -1 Mpc -1 , n s = 0 . 9665, and σ 8 = 0 . 8102. The w CDM pair had the following parameters: w 0 = -1 . 04, Ω m = 0 . 3096, Ω b = 0 . 04899, Ω de = 0 . 6904, H 0 = 67 . 66 km s -1 Mpc -1 , n s = 0 . 9331, and σ 8 = 0 . 8438. The cosmological simulations of this series were run using the cosmological N -body code GIZMO (Hopkins 2015). All simulations in the series contained 2160 3 dark matter particles in a (1 . 5 h -1 Gpc) 3 volume, with ε = 13 . 8 h -1 kpc softening length. The initial conditions (ICs) were generated by a modified version of the N-GenIC code (Springel 2015) by using the Zeldovich approximation and initial linear power spectra from the Boltzmann code CAMB (Lewis & Challinor 2011). The simulations started from redshift z init = 127, with a total of 48 output times. In this project, 31 particle snapshots were analysed in the 0 . 5 ≤ z ≤ 2 . 0 redshift range for each simulation.', '3.2. The DEMNU ni simulation suite': "The 'Dark Energy and Massive Neutrino Universe' (DEMNU ni ) simulations (Carbone et al. 2016; Parimbelli et al. 2022) have been produced with the aim of investigating the LSS in the presence of massive neutrinos and dynamical dark energy, and they were conceived for the nonlinear analysis and modelling of different probes, including dark matter, halo, and galaxy clustering (see Castorina et al. 2015; Zennaro et al. 2018; Parimbelli et al. 2022; Gouyou Beauchamps et al. 2023), weak lensing, CMB lensing, Sunyaev-Zeldovich, and Integrated Sachs-Wolfe (ISW) e ff ects (Roncarelli et al. 2015; Carbone et al. 2016), cosmic void statistics (Kreisch et al. 2019), and cross-correlations among these probes (Cuozzo et al. 2023). The DEMNU ni simulations were run using the tree particle mesh-smoothed particle hydrodynamics (TreePM-SPH) code p-GADGET3 (Springel 2005), specifically modified as in Viel et al. (2010) to account for the presence of massive neutrinos. This modified version of p-GADGET3 follows the evolution of CDM and neutrino particles, treating them as two distinct collisionless components. The reference cosmological parameters were chosen to be close to the baseline Planck 2013 cosmology (Planck Collaboration: Ade et al. 2014): Ω b = 0 . 05, Ω m = 0 . 32, H 0 = 67 . 0 km s -1 Mpc -1 , n s = 0 . 96, and A s = 2 . 127 × 10 -9 . Given these values, the reference (i.e., the massless neutrino case) CDM-particle mass resolution is m p CDM = 8 . 27 × 10 10 h -1 M ⊙ , which is decreased ac- \nFig. 3. Calculated matter and halo power spectra of the E lephant simulation suite in the mass bin 10 12 . 7 h -1 M ⊙ < M halo < 10 13 . 2 h -1 M ⊙ . The solid lines represent the reference Λ CDM simulations, while the dashed lines the results of the nDGP simulations. Top left: Real-space power spectra for dark matter. The dots above the solid lines highlight the locations where the power spectrum is estimated. Top right: Real-space power spectra for haloes. Bottom left: Monopole of the halo power spectrum in redshift space. Bottom right: Quadrupole of the halo power spectrum in redshift space. \n<!-- image --> \nto the mass of neutrino particles, in order to keep the same Ω m among all the DEMNU ni simulations. In fact, massive neutrinos are assumed to come as a particle component in a three-mass-degenerate scenario, therefore, to keep Ω m fixed, an increase in the massive neutrino density fraction yields a decrease in the CDM density fraction. The DEMNU ni simulations balance mass resolution and volume to include perturbations at both large and small scales. The simulations are characterised by a softening length of ε = 20 h -1 kpc, a comoving volume of 8 h -3 Gpc 3 filled with 2048 3 dark matter particles and, when present, 2048 3 neutrino particles. The simulations are initialised at z init = 99 with Zeldovich initial conditions. The initial power spectrum is rescaled to the initial redshift via the rescal-ing meth \nveloped in Zennaro et al. (2017). Initial conditions are then generated with a modified version of the N-GenIC software, assuming Rayleigh random amplitudes and uniform random phases.", '3.3. The R aygal simulations': 'The R aygal simulations (Breton et al. 2019; Rasera et al. 2022) are a set of two dark-matter only simulations in w CDM and Λ CDM cosmologies. The simulations were performed with the Adaptive-Mesh Refinement (AMR) N -body code RAMSES (Teyssier 2002; Guillet & Teyssier 2011). These simulations have a box size of 2625 h -1 Mpc for 4096 3 particles, which \nTable 1. Overview of the simulation suites analysed for this project. All simulations were dark matter only except the C i DER and DAKAR2 suites which used a two-component collisionless approximation to follow the baryonic and dark matter components separately. Hydrodynamic simulations were not analysed in this project. \nresults in a smoothing scale of 5 h -1 kpc at the maximum refinement level. Both simulations share the parameters H 0 = 72 . 0 km s -1 Mpc -1 , n s = 0 . 963, Ω b = 0 . 04356 and Ω r = 8 . 076 × 10 -5 . The flat Λ CDMsimulation has a WMAP7 cosmology (Komatsu et al. 2011): Ω m = 0 . 25733, and σ 8 = 0 . 80101, while the flat w CDM simulation is consistent at the 1 σ -level with a WMAP7 cosmology with Ω m = 0 . 27508, σ 8 = 0 . 85205, and w = -1 . 2. In both cases, Gaussian initial conditions are generated using a modified version of the code MPGRAFIC (Prunet et al. 2008) with the displacement field computed using secondorder Lagrangian perturbation theory (2LPT) to minimise the effect of transients (Crocce et al. 2006). The initial redshift has been set to z init ∼ 46 such as to ensure that the maximum displacement is of the order of one coarse cell. Such a late start guarantees smaller discreteness errors (see Michaux et al. 2021, for more details). For the present work, we focus on the snapshots at z = 0 , 1 , and 2.', '3.4. The E lephant simulation suite': 'The Extended LEnsing PHysics using ANalaytic ray Tracing (E lephant ) cosmological simulation suite was run using the ECOSMOG simulation code (Li et al. 2012, 2013b; Barreira et al. 2015; Bose et al. 2017), which is based on the dark matter and hydrodynamic AMR simulation code RAMSES and includes var- \nious types of modified gravity models (e.g., Li et al. 2012; Brax et al. 2012, 2013; Li et al. 2013b,a; Becker et al. 2020). It is particularly designed to solve for a nonlinear scalar field using AMR. New simulations were run for the purpose of testing the e ff ective field theory of large-scale structure (EFTofLSS) pipeline for spectroscopic galaxy clustering (Cautun et al. 2018; Fiorini et al. 2021; Casas et al. 2023, and Koyama et al. in prep.). For this purpose, 11 simulations were carried out using the Euclid reference cosmology without massive neutrinos for Λ CDM and the nDGP model (Table 2 of Euclid Collaboration: Knabenhans et al. 2021). The cosmological parameters of the Λ CDM simulations are: Ω m = 0 . 319, Ω b = 0 . 049, ΩΛ = 0 . 681, H 0 = 67 . 0 km s -1 Mpc -1 , A s = 2 . 1 × 10 -9 , and n s = 0 . 96. The nDGP simulations used the same parameters as the Λ CDMsimulations with the cross-over scale r c = 1 . 2 c / H 0. All of the simulations in this simulation suite had a box size of 1024 h -1 Mpc and 1024 3 particles. The initial conditions were generated at z init = 49 with 2LPT using the FML code 2 with fixed initial amplitudes. The phases of 10 realisations were extracted with di ff erent random seeds, while one realisation shares the same random seed as one of the other simulations, but with opposite phases to have a single paired-and-fixed simulation pair with suppressed cosmic variance (Angulo & Pontzen 2016). Output redshifts were selected from the Euclid Collaboration forecast paper for \ngalaxy clustering (Euclid Collaboration: Blanchard et al. 2020, z = 1 . 0, 1 . 2, 1 . 4 and 1 . 65).', '3.5. The COLA H i R es simulations': 'This simulation series contains overall seven simulations in Λ CDM and nDGP cosmologies that were run with MG-COLA , a modified gravity extension of the COmoving Lagrangian Acceleration (COLA) algorithm as implemented in the FML code. The COLA method uses a combination of analytic 2LPT displacement and particle mesh (PM) simulations to perform fast approximate simulations (Tassev et al. 2013). These techniques are extended to modified-gravity models using approximate screening methods to preserve the speed advantage of COLA simulations (Winther et al. 2017). The downside of PM simulations is that the internal structure of dark matter haloes is not well resolved due to limited resolution. This has an important implication for dark matter halo statistics. To mitigate this problem, the COLA simulations were run with an increased mass resolution (Fiorini et al. 2023). All simulations in this suite have a box size of 1024 h -1 Mpc, with 2048 3 particles. The base cosmological parameters of the simulations are the Planck 2015 parameters (Planck Collaboration: Ade et al. 2016): Ω m = 1 -ΩΛ = 0 . 3089, Ω b = 0 . 0486, H 0 = 67 . 74 km s -1 Mpc -1 , n s = 0 . 9667, and σ 8 = 0 . 8159. This simulation series focuses on nDGP gravity and tested 4 cases: r c = { 0 . 5 , 1 , 2 , 5 } c / H 0. The series contains paired-and-fixed simulations (Angulo & Pontzen 2016) to suppress cosmic variance in Λ CDM and in the nDGP model for r c = 1 c / H 0, while for the others they were only run for a single fixed amplitude realisation. The initial conditions were generated at z init = 127 using 2LPT. Full particle snapshots were stored at 4 redshift values, z = 1 . 0, 1 . 2, 1 . 4 and 1 . 65, motivated by the expected H α -emitters redshifts in the Euclid spectroscopic survey (Euclid Collaboration: Blanchard et al. 2020).', '3.6. The DUSTGRAIN and DUSTGRAIN-PF simulations': 'The DUSTGRAIN (Dark Universe Simulations to Test GRAvity In the presence of Neutrinos) project is an initiative aimed at investigating the degeneracy between f ( R ) gravity and massive neutrinos at the level of nonlinear cosmological observables, which was first pointed out in Baldi et al. (2014). More specifically, the project includes two suites of cosmological dark-matter-only simulations named the DUSTGRAIN -pathfinder (DUSTGRAIN-PF, Giocoli et al. 2018) and the DUSTGRAIN -fullscale simulations that have been run by joining the MG-GADGET (Puchwein et al. 2013) solver for f ( R ) gravity and the massive neutrinos implementation (Viel et al. 2010) available within the p-GADGET3 code. The former has been described and validated in Winther et al. (2015) and Adamek et al. (in prep.), while the latter has been compared with other methods in Adamek et al. (2023). \nThe DUSTGRAIN-PF simulations have been developed to sample the joint ( fR 0 , m ν ) parameter space to identify the most degenerate combinations of parameters with respect to some basic LSS statistics. These include the nonlinear matter power spectrum, the halo mass function, weak-lensing-convergence power spectrum, various higher-order statistics, cosmic voids, velocity fields (see Peel et al. 2018, 2019; Merten et al. 2019; Contarini et al. 2021; García-Farieta et al. 2019; Hagstotz et al. 2019a,b; Boyle et al. 2021). This series includes in total 13 simulations in f ( R ) + m ν cosmology, plus an additional suite of 12 standard Λ CDM simulations for varying one single standard \ncosmological parameter at a time that have been specifically run for the Higher-Order Weak Lensing Statistics (HOWLS) project (Euclid Collaboration: Ajani et al. 2023). These simulations have a box size of 750 h -1 Mpc per side, used a softening length of ε = 20 h -1 kpc, and include (2 × )768 3 particles (for the CDM and neutrinos components). The cosmological parameters (for the reference Λ CDM cosmology with massless neutrinos) have been set to Ω m = 1 -ΩΛ = 0 . 31345, σ 8 = 0 . 842, H 0 = 67 . 31 km s -1 Mpc -1 , n s = 0 . 9658, and the total matter density has been kept constant when varying the neutrino mass. Full snapshots have been stored at 34 output times between z = 99 (corresponding to the starting redshift of the simulation) and z = 0. \nThe DUSTGRAIN -fullscale simulations include only three runs (a reference Λ CDM cosmology and two f ( R ) gravity models with fR 0 = -10 -5 and di ff erent values of the total neutrino mass, namely m ν = { 0 . 1 , 0 . 16 } eV) simulated in a 8 h -3 Gpc 3 volume containing (2 × )2048 3 particles. In order to allow for a direct comparison with the DEMNU ni simulations described above, and to produce an extension to the latter for f ( R ) gravity with massive neutrino cosmologies, the DUSTGRAIN -fullscale simulations share the same initial conditions with DEMNU ni for each of the values of the neutrino mass. Therefore, the two sets of simulations have the same statistical realisations of the universe and identical cosmological parameters. Full snapshots have been stored for 73 output times between z = 99 (i.e., the initial conditions) and z = 0.', '3.7. The C i DER simulations': 'The Constrained Interacting Dark EneRgy scenario (or C i DER, Barros 2019) is a particular type of coupled Quintessence models characterised by a background cosmic expansion which is fixed by construction to be identical to a standard Λ CDM cosmology. As discussed in Sect. 2.1.4, this implies refraining from choosing a priori any specific functional form for the scalar self-interaction potential and letting the dynamic evolution of the field sample the potential shape required to match the imposed expansion history. The main feature of the C i DER models is that they show a suppressed growth of structures compared to a standard Λ CDM model with the same expansion history, thereby possibly easing the σ 8 tension without further exacerbating the tension on H 0. For these reasons, the model has received some attention even though - at least in its original form - it may already be quite tightly constrained by CMB observations (Barros et al. 2023). The C i DER simulations have been run with the c-GADGET code (Baldi et al. 2010, see also Adamek et al. in prep.) that implements all the relevant features of interacting dark energy models, and includes three values of the coupling β = 0 . 03 , 0 . 05 , 0 . 08 besides a reference Λ CDM cosmology corresponding to the case β = 0. All simulations clearly share the same expansion history, consistent with the following cosmological parameters: Ω m = 1 -ΩΛ = 0 . 311, Ω b = 0 . 049, H 0 = 67 . 7 km s -1 Mpc -1 , n s = 0 . 9665, A s = 1 . 992 × 10 -9 , corresponding to a value of σ 8 = 0 . 788 at z = 0 in the reference Λ CDM model. The simulations follow the evolution of 2 × 1024 3 particles for the (coupled) dark matter and (uncoupled) baryon components in a cosmological volume of 1 h -3 Gpc 3 with a softening length of ε = 25 h -1 kpc. The baryonic species are treated as a separate family of collisionless particles, i.e., no hydrodynamic forces nor radiative processes are considered in the simulations, and its inclusion is required in order to consistently represent the e ff ects of the non-universal coupling characterising these models. Therefore, baryonic particles will inter- \nsive particles according to standard Newtonian forces, while the interaction between pairs of CDM particles will be governed by an e ff ective gravitational constant G e ff = G N h 1 + (4 / 3) β 2 i (see e.g. Amendola 2004; Baldi et al. 2010). Full snapshots have been stored for 25 output times between z = 99 and z = 0.', '3.8. The DAKAR and DAKAR2 simulations': 'The Dark Scattering (DS) scenario (Simpson 2010) is another particular class of coupled Quintessence models where a nonuniversal interaction between dark matter particles and a classical scalar field playing the role of dark energy is characterised by a pure momentum exchange between the two species, with no transfer of rest-frame energy (see e.g. Pourtsidou et al. 2013; Skordis et al. 2015). In this respect, this interaction resembles a process of elastic scattering of massive particles (i.e. the dark matter) moving in a homogeneous fluid with an equation-of-state parameter w (i.e. the dark energy field), which can be simulated by introducing a velocity-dependent force acting on dark matter particles which will depend on the evolution of the dark energy equation-of-state parameter w , and on the cross-section, σ , characterising the interaction strength (Baldi & Simpson 2015). \nThe DAKAR (Baldi & Simpson 2017) and DAKAR2 simulations have been run with the c-GADGET code and cover various combinations of the shape of w ( z ), including the CPL parametrisation as given by Eq. (1) and hyperbolic tangent shapes, and of the cross-section, σ , giving rise to a diverse phenomenology at both linear and nonlinear scales. In particular, DS models have been shown to suppress the linear growth of perturbations for equation-of-state parameters w > -1 (Pourtsidou & Tram 2016; Bose et al. 2018; Carrilho et al. 2022) thereby possibly addressing the σ 8 tension, but such suppression is typically paired with a substantial enhancement of structure growth at deeply nonlinear scales. \nThe DAKAR simulations are subject to the approximation of considering the entirety of matter in the universe is in the form of dark matter, thereby slightly overestimating the e ff ect of the interaction as well as not capturing the segregation e ff ects between dark matter and baryons due to the non-universality of the coupling. These have been run for a cosmology with Ω m = 1 -ΩΛ = 0 . 308, H 0 = 67 . 8 km s -1 Mpc -1 , n s = 0 . 966, A s = 2 . 215 × 10 -9 , in a simulation box with a volume of 1 h -3 Gpc 3 filled with 1024 3 dark matter particles and using a softening length of ε = 12 h -1 kpc. \nThe DAKAR2 simulations, instead, share the same cosmology and the same statistical realisation as the C i DER simulations described above (i.e., the two sets of simulations share exactly the same reference Λ CDM run) and include collisionless baryons as a separate family of uncoupled particles, thereby consistently capturing the non-universality of the DS interaction. As for the C i DER simulations, a collection of 25 full snapshots for redshifts between z = 99 and z = 0 has been stored.', '3.9. The C lustering DE simulations': "The C lustering D ark E nergy simulations are run using the k -evolution code, a relativistic N -body code (Hassani et al. 2019, 2020) based on gevolution-1.2 (Adamek et al. 2016). In k -evolution , the field equations for k -essence type theories (Eq. 2) are solved using the e ff ective field theory (EFT) framework. We have two free parameters in the EFT framework of these theories: the equation-of-state parameter w ( τ ) ap- \nearing at the background level and kineticity α K( τ ) at the perturbation level. In the fluid picture of these theories, the relevant parameters are the speed of sound c s( τ ) and the equationof-state parameter w ( τ ), which in general are time-dependent. The term 'clustering dark energy' refers to the fact that these theories include a sound-horizon scale, beyond which scalarfield perturbations can grow. In the analysed simulations, constant w 0 and c 2 s are used, with cosmological parameters based on the Euclid reference cosmology (Euclid Collaboration: Knabenhans et al. 2021). The suite contains one Λ CDM simulation and four clustering dark energy simulations: ( w 0 , c 2 s ) = ( -0 . 9 , 1 c 2 ), ( -0 . 9 , 10 -4 c 2 ), ( -0 . 9 , 10 -7 c 2 ), and ( -0 . 8 , 10 -7 c 2 ). In these simulations, the box size was set to 2 h -1 Gpc with N = 1200 3 particles. Moreover, two sets of simulations with di ff erent resolutions were considered to study the convergence of the results. In this high-resolution simulation set, the box size was set to 2 h -1 Gpc with N = N grid = 2400 3 . In this series, the particle snapshots were saved in GADGET-2 format at five di ff erent redshifts z ∈ { 2 , 1 . 5 , 1 , 0 . 5 , 0 } .", '3.10. The FORGE and BRIDGE simulation suites': 'The FORGE simulation suite (Arnold et al. 2021) is a set of 198 dark matter only simulations for f ( R ) gravity and Λ CDM run with the Arepo cosmological simulation code (Springel 2010; Weinberger et al. 2020) using its MG module (Arnold et al. 2019). The simulations explore the cosmological and f ( R ) parameter space spanned by Ω m ( ΩΛ = 1 -Ω m), h , σ 8, and f R0 through 50 combinations (nodes) of these parameters sampled in a Latin-hypercube. All other cosmological parameters are fixed to a Planck cosmology ( n s = 0 . 9652, Ω b = 0 . 049199, Planck Collaboration: Aghanim et al. 2020). For each node, FORGE consists of a pair of large box simulations with 512 3 particles in a 1 . 5 h -1 Gpc side-length box and a pair of high-resolution runs with 1024 3 particles in a 500 h -1 Mpc box. For each pair, the initial conditions are chosen such that the large-scale variance in the 3D matter power spectrum approximately cancels when averaged over the two simulations (see Arnold et al. 2021; Ruan et al. 2024; Harnois-Déraps et al. 2023, for further details and some applications of these simulations). All simulations in this suite started from z init = 127, with initial conditions generated using the 2LPTic (Crocce et al. 2006) code. \nThe BRIDGE simulation suite (Harnois-Déraps et al. 2023) uses the same base setup and ICs as FORGE, but for the nDGP gravity model implemented into the Arepo code (HernándezAguayo et al. 2021). Accordingly, instead of varying the f R0 parameter, the nDGP parameter, H 0 r c is varied to explore the cosmological parameter space, with all the other parameters identical to those in FORGE for corresponding nodes. \nThe fact that the FORGE and BRIDGE simulations have the same cosmological parameters, node by node, allows a third suite of simulations to be done as a control set to quantify the e ff ect of modified gravity and how it correlates to the e ff ect of varying cosmological parameters. This additional suite of simulations, FORGEΛ CDM, uses the same setup but runs for the Λ CDMcounterparts of the corresponding f ( R ) and nDGP models. \nAs the FORGE, FORGEΛ CDM, and BRIDGE simulations have di ff erent cosmologies, the mass resolution di ff ers amongst them. In Table 1, we thus quoted an order of magnitude for the mass resolution. \n[ \n] \nFig. 4. Calculated halo mass function from the FORGE simulation suite. The vertical and horizontal dot-dashed lines indicate the mass relative to halos with 50 particles and the number density relative to a 1% shotnoise error, respectively. The shaded region highlights the mass bin used to calculate the halo power spectra shown in Fig. 5. \n<!-- image --> \n/circledot', '3.11. The PNG-UNIT simulation': "The PNG-UNIT (Adame et al. 2023) is a twin of one of the existing UNITsims (Universe N -body simulations for the Investigation of Theoretical models, Chuang et al. 2019), but with local primordial non-Gaussianities given by f NL = 100. The simulation assumes the following Λ CDM parameters: Ω m = 1 -ΩΛ = 0 . 3089 , H 0 = 67 . 74 km s -1 Mpc -1 , n s = 0 . 9667 , σ 8 = 0 . 8147. It consists of N = 4096 3 particles in L = 1 h -1 Gpc evolved with L-GADGET , which is a version of GADGET-2 optimised for massive parallelisation, using a tree-PM algorithm with a softening length of ε = 6 h -1 kpc. The initial conditions are run with the 2LPT implementation in the FastPM (Feng et al. 2016) code at z = 99. Both the UNIT and PNG-UNIT are run with fixed initial conditions (Angulo & Pontzen 2016), which set the amplitude of the ICs to their expected value. Whereas there are 4 UNIT simulations in 2 sets of pairs (within each pair, each simulation has the inverted phases one with respect to another, following Angulo & Pontzen 2016), we only have one simulation for the PNG-UNIT. The PNG-UNITsim is run with the phases of the ICs matched to one of the UNITsims, which is labelled in the databases as 'Ampl1'. The usage of fixed ICs with local PNG was validated in Avila & Adame (2023), where it was also shown how to increment the precision of the statistics measured from matched simulations. Overall 129 snapshots were stored during the simulation, and 32 in the 0 . 5 < z < 2 . 0 range.", '4. Analysis': 'We have developed a cosmological analysis pipeline to generate mock observables from non-standard cosmological simulations in a consistent and rapid way. The pipeline is a SLURM 3 script that runs in parallel on multiple nodes on the machines \nwhere the simulations are stored. The pipeline consists of several modules that can be activated or deactivated independently. The modules are controlled by a configuration file that specifies the input and output parameters, as well as the options for each module. The input of the pipeline is the particle snapshots of the non-standard cosmological simulations. The supported input formats are GADGET binary and Hierarchical Data Format version 5 (HDF5, Springel 2005), Arepo HDF5 (Weinberger et al. 2020), RAMSES HDF5format from R aygal (Roy et al. 2014; Rasera et al. 2022), and GIZMO HDF5 (Hopkins 2015). The main steps of the analysis are summarised in Fig. 1. In this section, we describe the quantities generated by this pipeline.', '4.1. Dark Matter density field': 'The pipeline uses nbodykit (Hand et al. 2018) to read and analyse the dark matter particle data of the input-simulation snapshots. This Python package is an open-source, massively parallel toolkit that provides a set of LSS algorithms useful in the analysis of cosmological data sets from N -body simulations and observational surveys. During the dark matter density-field analysis, nbodykit generates a reconstructed density field from the input-particle distribution with the triangular-shaped-cloud (TSC) density-assignment function. We chose to use a \nN grid = 2 n floor h log 2 GLYPH<16> 3 √ N part GLYPH<17>i -1 o (35) \nlinear grid size for every analysed snapshot, where N part is the number of stored particles in the snapshot. With this choice, there will always be at least eight particles on average in each cubic density cell. The reconstructed density fields were saved in bigfile format (Feng et al. 2017) for future analysis.', '4.1.1. Real-space power spectrum': "The real-space matter power spectrum is defined via \nD ˜ δ ( k ) ˜ δ ∗ ( k ' ) E = (2 π ) 3 P ( k ) δ (3) D ( k -k ' ) , (36) \nwhere ˜ δ ( k ) is the Fourier-transform of the matter overdensity field \nδ ( r ) = ρ ( r ) ρ -1 , (37) \nand k is the wavevector. We estimate the power spectrum using nbodykit . The density field is created by binning the particles into a grid using a TSC-density-assignment function, with the linear-grid size defined in Eq. (35). The density field is Fourier transformed and the power spectrum is computed by binning | ˜ δ ( k ) | 2 , deconvolving the window function and subtracting shot noise. We also use the interlacing technique for reducing aliasing (Sefusatti et al. 2016). The bin size of the power spectrum was set to \n∆ k = k f = 2 π L box , (38) \nwhere k f is the fundamental wavenumber, and L box is the linear size of the simulation. The pipeline saves the power spectrum of every calculated bin below the \nk Ny = π N grid L box (39) \nNyquist wavenumber with the number of modes into a simple ASCII format file. \n1750 \nFig. 5. Calculated matter and halo power spectra from the FORGE simulation suite in the mass bin 10 12 . 7 h -1 M ⊙ < M halo < 10 13 . 2 h -1 M ⊙ . The solid lines represent the reference Λ CDM simulations, while the dashed lines are the results of the nDGP simulations. Top left: Real-space power spectra for dark matter. The dots above the solid lines highlight the locations where the power spectrum is estimated. Top right: Real-space power spectra for haloes. Bottom left: Monopole of the halo power spectrum in redshift space. Bottom right: Quadrupole of the halo power spectrum in redshift space. \n<!-- image -->", '4.1.2. Redshift-space power spectrum': 'The real-space matter power spectrum is not directly measurable in galaxy surveys because we cannot probe the real-space positions of galaxies. What we can directly measure is the redshiftspace power spectrum P s ( k , µ ) where µ = ˆ n LOS · ˆ k and ˆ n LOS is a unit vector in the line-of-sight (LOS) direction. This can be expanded in multipoles P s ( k , µ ) = P ∞ ℓ = 0 P s ℓ ( k ) L ℓ ( µ ) where L ℓ ( µ ) are the Legendre polynomials. The multipoles are then computed from the redshift-space power spectrum as \nP s ℓ ( k ) = 2 ℓ + 1 2 Z 1 -1 P s ( k , µ ) L ℓ ( µ ) d µ. (40) \nWe compute the redshift-space power spectrum in 25 µ bins and the redshift-space multipoles (the monopole P 0, the quadrupole P 2, and the hexadecapole P 4) using nbodykit from the input dark matter density field. For this, we use the distant-observer approximation \ns i = r i + GLYPH<18> ˆ n LOS · v i aH GLYPH<19> · ˆ n LOS , (41) \nto add the redshift-space distortions using the three coordinate axes as the LOS directions (observables are computed as the mean over these three individual axes). Here, r i and v i are the real-space particle coordinates and peculiar velocities inside the periodic simulation box, s i is the corresponding redshift-space \n[ \n] \nFig. 6. Calculated halo mass function from the C i DER simulation suite. The vertical and horizontal dot-dashed lines indicate the mass relative to halos with 50 particles and the number density relative to a 1% shotnoise error, respectively. The shaded region highlights the mass bin used to calculate the halo power spectra shown in Fig. 7. \n<!-- image --> \n/circledot \nposition we compute, a = 1 / ( z + 1) is the scale factor, and H is the Hubble parameter at the redshift of the snapshot. We deconvolve the window function for the density assignment, applying interlacing, and the resulting density power spectrum is finally shot-noise subtracted. The saved wavenumber bins are the same as in Sect. 4.1.1.', '4.1.3. Linear dark matter power spectrum': 'During the analysis of scale-independent models, multiple modules use the linear dark matter power spectrum as an input. To make the analysis more transparent, we also generated and saved the linear power spectrum for all analysed redshifts. For this, the pipeline needs the P lin( k , z start) input linear power spectrum of the simulation that was used during the initial condition generation. This can be defined at any z start redshift. Then, this linear power spectrum is renormalised with the cosmological parameter σ 8 at z = 0. The normalised P lin( k , z = 0) power spectrum is rescaled to z snap redshifts of all analysed snapshots as \nP lin( k , z snap) = " D ( z snap) D ( z = 0) # 2 P lin( k , z = 0) , (42) \nwhere D ( z ) is the linear growth function. The pipeline uses this back-scaling since the linear growth in these Newtonian simulations follows this scale-independent evolution. For Λ CDM reference simulations, we used the \nD ( a ) = 5 Ω m H 2 0 2 H ( a ) a Z 0 d a \' ˙ a \' 3 (43) \nlinear growth to scale the linear spectrum (Peebles 1993). This growth function only describes the linear growth in the Λ CDM \nArticle number, page 14 of 22 \nframework. In the case of w CDMand CPL models, we solve the \nG \'\' + " 7 2 + 3 2 w ( a ) 1 + X ( a ) # G \' a + 3 2 1 -w ( a ) 1 + X ( a ) G a 2 = 0 (44) \nordinary di ff erential equation (Linder & Jenkins 2003) with the COLOSSUS python package (Diemer 2018), where G ( a ) = D ( a ) / a and \nX ( a ) = Ω m 1 -Ω m e -3 1 R a d a \' w ( a \' ) a \' . (45) \nFor every other model, we use tabulated linear growth functions. The linear power spectra are calculated in the same wavenumber bins as the nonlinear real-space matter power spectra.', '4.2. Halo Catalogues': 'Rockstar (Behroozi et al. 2013) is a friends-of-friends (FoF) halo-finder algorithm that uses information from the full 6D phase space (positions and the velocities) of the particles. The code initially creates FoF groups in real space, with a large linking length ( b ≃ 0 . 28). It then does a new FoF search using the phase-space metric \nd = s | x 1 -x 2 | 2 σ 2 x + | v 1 -v 2 | 2 σ 2 v , (46) \nwhere σ x and σ v are the particle-position and velocity dispersions for the given FoF group. Finally, it links particles into subgroups and this is done iteratively on each subgroup creating a hierarchical set of structures. By default, the algorithm calculates halo and subhalo masses using dark matter particles from the spherical regions around the friends-of-friends group with gravitationally unbound particles removed. The halo masses calculated this way are called bound-only (BO) masses. If the unbound particles are not removed during the mass calculation, the calculated masses are strict spherical-overdensity (SO) masses. \nWe made a custom version of the publicly-available Rockstar code (Behroozi et al. 2012) to analyse our nonstandard simulations. We added new input formats for simulations, options to read tabulated expansion histories, and already internally computed quantities in the outputs such as halo minor axis vectors and radii at di ff erent mass definitions. None of these modifications impact the halo-finding algorithm. In our pipeline, we use the following mass-definitions: M 200c (SO & BO), M 500c (SO), M 1000c (SO), M 2500c (SO), M 200b (SO). The M vir masses are not calculated by the pipeline, since this mass definition is dependent on the cosmological parameters and on the laws of gravity. Many non-standard cosmological models are changing the dynamics of the dark matter component, and this choice simplifies the future expansion of the database without the need of implementing new cosmologies in Rockstar . After the catalogue (in ASCII format) is produced, we run a postprocessing script to find parent haloes for subhaloes and store the information as an additional index column. Extra information is saved in the header such as scale factor, box length, and particle mass. Additional particle data for each halo are also saved by Rockstar in a custom BGC2 binary data format. During the execution of our pipeline, these BGC2 files are temporarily stored to provide additional input for other analysis modules. \nFig. 7. Calculated matter and halo power spectra from the C i DER simulation suite in the mass bin 10 12 . 7 h -1 M ⊙ < M halo < 10 13 . 2 h -1 M ⊙ . The solid lines represent the reference Λ CDM simulations, while the dashed lines are the results of the IDE simulations. Top left: Real-space power spectra for dark matter. The dots above the solid lines highlight the locations where the power spectrum is estimated. Top right: Real-space power spectra for haloes. Bottom left: Monopole of the halo power spectrum in redshift space. Bottom right: Quadrupole of the halo power spectrum in redshift space. \n<!-- image -->', '4.2.1. Halo mass function and power spectra': 'By default, we compute the halo mass function (HMF) in the range 11 < log10[ M / ( h -1 M ⊙ )] < 14 with 32 logarithmic bins using the main mass definition (200c) and excluding substructures. The pipeline allows the user to use di ff erent mass definitions (see Sect. 4.2), include substructures, and vary the HMF range and binning. \nIn practice, many simulations produce very large ASCII files which are not practical to read using standard libraries such as \nNumPy 4 or Pandas . 5 To speed up the analysis pipeline, we therefore use Polars , 6 a fast multi-threaded dataframe library. \nThe halo real-space and redshift-space power spectra are computed using the same tools as in Sects. 4.1.1 and 4.1.2. The user can specify the halo mass range, SO or BO for the main mass definition, and whether or not to include substructures.', '4.3. Cosmic Voids': 'For each catalogue, we infer the linear halo bias with the estimator \nb = * s P h( k ) P m( k ) + k < k max , (47) \nwith P m( k ) and P h( k ) the matter and halo real-space power spectra, respectively estimated in Sects. 4.1.1 and 4.2.1. This estimator calculates the bias by taking the square root of the ratio of the halo power spectrum P h( k ) to the matter power spectrum P m( k ), and then averaging this ratio over all k bins where k < k max with uniform weighting. We only use this computed quantity to be as model-independent as possible and to remove cosmic variance (since matter and halo both share the same sample and cosmic variance). We compute the mean power spectra ratio up to a conservative value of k max = 0 . 1 h Mpc -1 to mitigate the e ff ects of nonlinear clustering. This method works reliably for scaleindependent bias with sub-percent accuracy, but cannot be used for models that have scale-dependent bias at k < k max wavenumber.', '4.2.3. Redshift-space Gaussian covariance': 'To produce Gaussian covariances of the power-spectrum multipoles in redshift space, we need the linear bias and powerspectrum multipoles including the shot-noise contributions as inputs. The former is estimated numerically in Sect. 4.2.2, while the latter can be internally computed from an input linear power spectrum; in this case, the covariance is estimated as in Taruya et al. (2010), or with the EFT model using the COMET emulator (Eggemeier et al. 2023) and the covariance formulae from Grieb et al. (2016). \nWhen analysing snapshots, it may be interesting to compute the power-spectrum multipoles averaged over the three box directions to significantly suppress variance. However, this procedure also has to be carefully accounted for in the covariance (Smith et al. 2021) since the LOS-averaged covariance is not equal to the single-LOS covariance divided by three (as one might naively expect). Thanks to the LOS-averaged covariance implemented in COMET , it can also be part of the outputs of our pipeline.', '4.2.4. 2D and 3D Halo Profiles': "The generation of binned 3-dimensional and 2-dimensional projected profiles is performed using a custom analysis module which reads the halo catalogues as well as the BGC2 particle data, and stores the resulting profiles of each halo in a separated HDF5 file. \nAll the profiles are obtained considering 50 log-spaced bins in a fixed radial range [0 . 001 , 5], in units of r 500c. This analysis module provides cumulative mass, density, number density, and cumulative number density profiles, as well as velocity and velocity dispersion profiles for the cartesian- and sphericalcoordinates components of the velocities. The 2D profiles correspond to projecting the LOS along each of the cartesian coordinates in a cylinder of length 5 r 500c. In an upcoming version of the pipeline, the profiles will also be available for the projections along the axes of the inertia ellipsoid a , b , and c . \nThe Void IDentification and Examination toolkit, VIDE (Sutter et al. 2015), is a parameter-free topological void finder, conceived for galaxy-redshift surveys and N -body simulations. The VIDE pipeline is an open-source Python/C++ code, based on the ZOBOV (Neyrinck 2008) software and can be launched on any tracer distribution. The algorithm follows the following main steps: i) estimation of the density field of a tracer distribution using the Voronoi tessellation (Schaap & van de Weygaert 2000); ii) detection of all the relative minima; iii) merging of nearby Voronoi cells into zones via the watershed transform (Platen et al. 2007), cells correspond to local catchment 'basins', which are identified as voids. VIDE can also merge adjacent voids to construct a nested hierarchy of voids if a merging threshold is provided. In this case, when two adjacent voids have at least one Voronoi cell on the ridge separating them lower than the threshold, they are merged into a parent void. In this work, in order to leave the algorithm parameter-free, and for consistency with other Euclid void analyses (Hamaus et al. 2022; Contarini et al. 2022), we do not explore this possibility. \nWe detect voids in the distribution of Rockstar haloes. After the void catalogues are produced, we post-process them to measure the void-size function, which is the number density of voids as a function of their size, R e ff . The void-size function is a sensitive probe for cosmology, strongly complementary to the galaxy 2pt-statistics (Pisani et al. 2015; Massara et al. 2015; Kreisch et al. 2019; Verza et al. 2019; Contarini et al. 2021, 2022, 2023; Verza et al. 2022, 2023, 2024). Additionally, albeit not computed for this paper, the void catalogues allow to compute the void-galaxy cross-correlation function, another powerful statistic to constrain cosmology (see e.g. Hamaus et al. 2022). \nVIDE provides some fundamental properties of voids. The void size is measured by the e ff ective radius, defined as the radius of a sphere with the same volume as the void, R e ff = [(3 / 4 π ) P i Vi ] 1 / 3 , where Vi is the volume of the i th Voronoi cell belonging to the void. The void centre is defined as the volumeweighted barycentre, X v = P i x iVi / V tot, with V tot = P i Vi . Note that this corresponds to the geometric centre of the void. In addition, VIDE also provides the position of the tracer sitting in the lowest-density Voronoi cell, i.e. the minimum. The void's depth is estimated via the central density, defined as the mean density in a sphere centred in the barycenter X v with radius R e ff / 4. VIDE also computes void shapes via the inertia tensor as well as the corresponding eigenvalues and eigenvectors. The ellipticity is then computed as ϵ = 1 -( J 1 / J 3) 1 / 4 , where J 1 and J 3 are the smallest and largest eigenvalues.", '5. Interpretation': "The mock observables that we compute contain several interesting signatures of non-standard models. Due to a large number of analysed models, in this paper, we focus on showing results from the nDGP, f ( R ), and interacting-dark-energy models, which we obtained thanks to the analysis of the E lephant , FORGE, and C i DER simulations suites respectively. \nTo mitigate the noise due to sample variance in the figures below, we average the signals over the available realisations of the E lephant simulations. In the case of the FORGE suite, we focus on a single cosmology 7 corresponding to GLYPH<12> GLYPH<12> GLYPH<12> ¯ f R0 GLYPH<12> GLYPH<12> GLYPH<12> = 10 -5 . 34219 . In the case of the C i DERsimulations, we focus on the IDE model with β = 0 . 03 coupling. \nThe nDGP, f ( R ), and IDE simulations were run with the same initial conditions as their Λ CDMcounterpart so the e ff ects of non-standard cosmologies can be studied directly by comparing the generated observables. \nThe main di ff erence between the modified-gravity simulation and Λ CDM is the inclusion of a fifth force. For nDGP, this fifth force acts on all scales and increases in strength towards redshift zero (and goes to zero as we go to higher and higher redshifts). This is also true for f ( R ), with the exception that the fifth force has only a finite range, so it does not a ff ect the clustering on the largest scales. \nIn the IDE simulations, the dark energy component interacts with the dark matter, resulting in a transfer of energy between the two. This interaction a ff ects the growth of cosmic structures by modifying the gravitational potential. This cosmological scenario is expected to suppress structure formation at late time compared to the standard Λ CDMmodel. \nThese di ff erences lead to a number of di ff erent observable signatures, a few of which we will highlight below. \nAbundance of dark matter haloes - In Fig. 2, we compare the cumulative halo mass function of the nDGP and Λ CDM models. The inclusion of the fifth force means structures will form more rapidly than in Λ CDM and this is indeed what we see. This is most pronounced at the high-mass end where the abundance is up to 50% larger. In Fig. 4, we compare the cumulative halo mass function of the f ( R ) and Λ CDM model. We see roughly the same qualitative features as for nDGP in that the halo abundance generally increases with halo mass and with time. However, as opposed to nDGP, we see an over-abundance of 'small' haloes at earlier times in the f ( R ) simulations. This is a consequence of the fact that the fifth force only acts on 'small' scales and the fact that the screening mechanism is more e ff ective at suppressing the fifth force in and around the most massive haloes. The comparison of the halo mass function between the standard Λ CDM and the β = 0 . 03 IDE model can be seen in Fig. 6. In the IDE model, the interaction between the dark energy and dark matter caused a significant reduction in the HMF. This is a straightforward consequence of the suppressed growth rate of the matter fluctuations. \nClustering of dark matter - In Figs. 3, 5, and 7, we show the calculated real- and redshift-space power-spectrum multipoles for the haloes and dark matter for nDGP, f ( R ), and IDE respectively. For the modified gravity models, the e ff ect of the fifth force is again clearly in the dark matter power spectrum and shows two di ff erent e ff ects: for nDGP, we have a scaleindependent growth rate causing the power spectrum to be boosted on all scales displayed, while for f ( R ) we have a scaledependent growth-rate where f ( R ) agrees with Λ CDM on the largest scales, but is boosted below a critical scale which is related to the range of the fifth force. For both models, the difference with respect to Λ CDM increases in strength as we get closer to the present time. In the case of the interacting-darkenergy model, the energy transfer between the dark energy and dark matter caused a scale-independent suppression in the dark matter power spectrum. At redshift 2, this is ≃ 3%, and this difference increases to ≃ 5% for z = 0 . 55 compared to the Λ CDM model. \nHalo bias - When it comes to halo clustering in real space, we see the opposite e ff ect as for the dark matter power spectrum in Figs. 3 and 5, with nDGP and f ( R ) being less clustered than Λ CDM. This comes from a smaller halo bias in these modifiedgravity models (see e.g. Barreira et al. 2014 for a theoretical explanation for nDGP). In the interacting-dark-energy scenario, the halo bias in real space is 6% higher than for Λ CDM. As a con- \nquence, the real-space clustering of the dark matter haloes is more prominent in the IDE simulation. \nRedshift space distortions - For the redshift-space halo power spectra, the boost in the ratio with respect to Λ CDM is seen to be larger than in real space which comes from the larger velocities in the modified-gravity simulations, leading to enhanced redshift-space distortions. The monopole redshift-space power spectra of the haloes in the mass bin 10 12 . 7 h -1 M ⊙ < M halo < 10 13 . 2 h -1 M ⊙ in the IDE simulations are showing a 5 -10% excess power compared to the Λ CDM counterpart, similarly to the real-space clustering. On the other hand, the quadrupole only shows a significant power increase at smaller, nonlinear scales.", '6. Summary': 'In this paper, we described a new pipeline based on the Rockstar halo finder and the nbodykit LSS toolkit to postprocess cosmological simulations with modified gravity, nonstandard expansion history, modified dark matter or dark energy components, or altered initial conditions. We used this pipeline to analyse 474 cosmological N -body simulations in various Λ CDM and non-standard cosmological scenarios in a consistent way. With this pipeline, we generated halo catalogues, halo mass functions, reconstructed density fields, real- and redshiftspace power spectra, Gaussian covariances, halo biases, and void catalogues. This generated data will serve as a theoretical prediction and reference for Euclid as well as other Stage-IV cosmology projects. Using the calculated quantities, we identified distinctive signatures of non-standard behaviour in nDGP and f ( R ) modified-gravity models, and in the C i DER interactingdark-energy scenario. \nWe have generated overall more than 100 TB of postprocessed data from the available non-standard simulations. During the analysis, the pipeline used 66 CPU hours and 60GB of memory per billion particles per snapshot on average. The data are available on request on the CosmoHub ( https:// cosmohub.pic.es/home , see Tallada et al. 2020 and Carretero et al. 2017) platform designed for interactive exploration and distribution of massive cosmological datasets. \nThe synthetic halo catalogues are crucial in the production of additional observables, which can be used for a direct comparison with cosmological observations of Euclid . In the near future, we will extend the generated database with halo density profiles (Navarro et al. 1996; Le Brun et al. 2018), synthetic galaxy catalogues (Berlind et al. 2003), weak lensing (Jaroszynski et al. 1990; Bartelmann & Schneider 2001) and ISW (Giannantonio et al. 2008) maps, and lightcones (Merson et al. 2013). \nAcknowledgements. The Euclid Consortium acknowledges the European Space Agency and a number of agencies and institutes that have supported the development of Euclid , in particular the Agenzia Spaziale Italiana, the Austrian Forschungsförderungsgesellschaft funded through BMK, the Belgian Science Policy, the Canadian Euclid Consortium, the Deutsches Zentrum für Luft- und Raumfahrt, the DTU Space and the Niels Bohr Institute in Denmark, the French Centre National d\'Etudes Spatiales, the Fundação para a Ciência e a Tecnologia, the Hungarian Academy of Sciences, the Ministerio de Ciencia, Innovación y Universidades, the National Aeronautics and Space Administration, the National Astronomical Observatory of Japan, the Netherlandse Onderzoekschool Voor Astronomie, the Norwegian Space Agency, the Research Council of Finland, the Romanian Space Agency, the State Secretariat for Education, Research, and Innovation (SERI) at the Swiss Space O ffi ce (SSO), and the United Kingdom Space Agency. A complete and detailed list is available on the Euclid web site ( www.euclid-ec.org ). GR\'s research was supported by an appointment to the NASA Postdoctoral Program administered by Oak Ridge Associated Universities under contract with NASA. GR and AK were supported by JPL, which is run under contract by the California Institute of Technology for NASA (80NM0018D0004). GR acknowledges the support of the Research \nCouncil of Finland grant 354905. The authors acknowledge the Texas Advanced Computing Center (TACC) at The University of Texas at Austin for providing HPC and visualization resources that have contributed to the research results reported within this paper. URL: http://www.tacc.utexas.edu . This project was provided with computer and storage resources by GENCI at TGCC thanks to the grant 2023-A0150402287 on Joliot Curie\'s SKL partition. This work has made use of CosmoHub. CosmoHub has been developed by the Port d\'Informació Científica (PIC), maintained through a collaboration of the Institut de Física d\'Altes Energies (IFAE) and the Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas (CIEMAT) and the Institute of Space Sciences (CSIC & IEEC). CosmoHub was partially funded by the "Plan Estatal de Investigación Científica y Técnica y de Innovación" program of the Spanish government, has been supported by the call for grants for Scientific and Technical Equipment 2021 of the State Program for Knowledge Generation and Scientific and Technological Strengthening of the R + D + i System, financed by MCIN / AEI / 10.13039 / 501100011033 and the EU NextGeneration / PRTR (Hadoop Cluster for the comprehensive management of massive scientific data, reference EQC2021-007479-P) and by MICIIN with funding from European Union NextGenerationEU(PRTR-C17.I1) and by Generalitat de Catalunya. ZS acknowledges funding from DFG project 456622116 and support from the IRAP and IN2P3 Lyon computing centers. During part of this work, AMCLB was supported by a fellowship of PSL University-Paris Observatory. CG thanks the support from INAF theory Grant 2022: Illuminating Dark Matter using Weak Lensing by Cluster Satellites, PI: Carlo Giocoli. VGP is supported by the Atracción de Talento Contract no. 2019-T1 / TIC-12702 granted by the Comunidad de Madrid in Spain. VGP, and by the Ministerio de Ciencia e Innovación (MICINN) under research grant PID2021-122603NB-C21. We extend our sincere gratitude to Christian Arnold and Claudio Llinares for their valuable contributions to this research. Their work significantly influenced the development of this project.', 'References': '- Abbott, T. M. C., Aguena, M., Alarcon, A., et al. 2022, Phys. Rev. D, 105, 023520\n- Adame, A. G., Avila, S., Gonzalez-Perez, V., et al. 2023, arXiv:2312.12405 Adamek, J., Angulo, R. 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Box 64, 00014 University of Helsinki, Finland\n- 3 Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans, s / n, 08193 Barcelona, Spain\n- 4 Institut de Ciencies de l\'Espai (IEEC-CSIC), Campus UAB, Carrer de Can Magrans, s / n Cerdanyola del Vallés, 08193 Barcelona, Spain\n- 5 Laboratoire Univers et Théorie, Observatoire de Paris, Université PSL, Université Paris Cité, CNRS, 92190 Meudon, France\n- 6 Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth PO1 3FX, UK\n- 7 School of Physics and Astronomy, Queen Mary University of London, Mile End Road, London E1 4NS, UK\n- 8 Institut d\'Astrophysique de Paris, UMR 7095, CNRS, and Sorbonne Université, 98 bis boulevard Arago, 75014 Paris, France\n- 9 Institute of Theoretical Astrophysics, University of Oslo, P.O. Box 1029 Blindern, 0315 Oslo, Norway\n- 10 Institut für Theoretische Physik, University of Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany\n- 11 Institut de Recherche en Astrophysique et Planétologie (IRAP), Université de Toulouse, CNRS, UPS, CNES, 14 Av. Edouard Belin, 31400 Toulouse, France\n- 12 Université St Joseph; Faculty of Sciences, Beirut, Lebanon\n- 13 Dipartimento di Fisica \'G. Occhialini", Università degli Studi di Milano Bicocca, Piazza della Scienza 3, 20126 Milano, Italy\n- 14 INAF-Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, Via Piero Gobetti 93 / 3, 40129 Bologna, Italy\n- 15 IFPU, Institute for Fundamental Physics of the Universe, via Beirut 2, 34151 Trieste, Italy\n- 16 Dipartimento di Fisica e Astronomia "Augusto Righi" - Alma Mater Studiorum Università di Bologna, via Piero Gobetti 93 / 2, 40129 Bologna, Italy\n- 17 ICSC - Centro Nazionale di Ricerca in High Performance Computing, Big Data e Quantum Computing, Via Magnanelli 2, Bologna, Italy\n- 18 Jodrell Bank Centre for Astrophysics, Department of Physics and Astronomy, University of Manchester, Oxford Road, Manchester M13 9PL, UK\n- 19 INAF-IASF Milano, Via Alfonso Corti 12, 20133 Milano, Italy\n- 20 Dipartimento di Fisica "Aldo Pontremoli", Università degli Studi di Milano, Via Celoria 16, 20133 Milano, Italy\n- 21 INFN Gruppo Collegato di Parma, Viale delle Scienze 7 / A 43124 Parma, Italy\n- 22 SISSA, International School for Advanced Studies, Via Bonomea 265, 34136 Trieste TS, Italy\n- 23 International Centre for Theoretical Physics (ICTP), Strada Costiera 11, 34151 Trieste, Italy\n- 24 Center for Cosmology and Particle Physics, Department of Physics, New York University, New York, NY 10003, USA\n- 25 Center for Computational Astrophysics, Flatiron Institute, 162 5th Avenue, 10010, New York, NY, USA\n- 26 Dipartimento di Fisica e Astronomia, Università di Bologna, Via Gobetti 93 / 2, 40129 Bologna, Italy\n- 27 INFN-Sezione di Bologna, Viale Berti Pichat 6 / 2, 40127 Bologna, Italy\n- 28 Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK\n- 29 Higgs Centre for Theoretical Physics, School of Physics and Astronomy, The University of Edinburgh, Edinburgh EH9 3FD, UK\n- 30 Institut de Physique Théorique, CEA, CNRS, Université ParisSaclay 91191 Gif-sur-Yvette Cedex, France\n- 31 Departamento de Física Teórica, Facultad de Ciencias, Universidad Autónoma de Madrid, 28049 Cantoblanco, Madrid, Spain\n- 32 Instituto de Física Teórica UAM-CSIC, Campus de Cantoblanco, 28049 Madrid, Spain \n- 33 Centro de Investigación Avanzada en Física Fundamental (CIAFF), Facultad de Ciencias, Universidad Autónoma de Madrid, 28049 Madrid, Spain\n- 35 Institut de Física d\'Altes Energies (IFAE), The Barcelona Institute of Science and Technology, Campus UAB, 08193 Bellaterra (Barcelona), Spain\n- 34 Department of Astrophysics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland\n- 36 Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, Via Irnerio 46, 40126 Bologna, Italy\n- 38 Université de Genève, Département de Physique Théorique and Centre for Astroparticle Physics, 24 quai Ernest-Ansermet, CH1211 Genève 4, Switzerland\n- 37 Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 85748 Garching, Germany\n- 39 Department of Physics, Institute for Computational Cosmology, Durham University, South Road, DH1 3LE, UK\n- 41 Universitäts-Sternwarte München, Fakultät für Physik, LudwigMaximilians-Universität München, Scheinerstrasse 1, 81679 München, Germany\n- 40 Institut universitaire de France (IUF), 1 rue Descartes, 75231 PARIS CEDEX 05, France\n- 42 Excellence Cluster ORIGINS, Boltzmannstrasse 2, 85748 Garching, Germany\n- 44 INFN-Padova, Via Marzolo 8, 35131 Padova, Italy\n- 43 Dipartimento di Fisica e Astronomia "G. Galilei", Università di Padova, Via Marzolo 8, 35131 Padova, Italy\n- 45 INAF-Osservatorio Astronomico di Trieste, Via G. B. Tiepolo 11, 34143 Trieste, Italy\n- 47 Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università di Parma, Viale delle Scienze 7 / A 43124 Parma, Italy\n- 46 INFN, Sezione di Trieste, Via Valerio 2, 34127 Trieste TS, Italy\n- 48 Aix-Marseille Université, CNRS / IN2P3, CPPM, Marseille, France\n- 50 The Cooper Union for the Advancement of Science and Art, 41 Cooper Square, New York, NY 10003, USA\n- 49 Department of Astrophysical Sciences, Peyton Hall, Princeton University, Princeton, NJ 08544, USA\n- 51 Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu, HI 96822, USA\n- 53 Port d\'Informació Científica, Campus UAB, C. Albareda s / n, 08193 Bellaterra (Barcelona), Spain\n- 52 Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas (CIEMAT), Avenida Complutense 40, 28040 Madrid, Spain\n- 54 Université Paris-Saclay, CNRS, Institut d\'astrophysique spatiale, 91405, Orsay, France\n- 56 INAF-Osservatorio Astrofisico di Torino, Via Osservatorio 20, 10025 Pino Torinese (TO), Italy\n- 55 INAF-Osservatorio Astronomico di Brera, Via Brera 28, 20122 Milano, Italy\n- 57 Dipartimento di Fisica, Università di Genova, Via Dodecaneso 33, 16146, Genova, Italy\n- 59 Department of Physics "E. Pancini", University Federico II, Via Cinthia 6, 80126, Napoli, Italy\n- 58 INFN-Sezione di Genova, Via Dodecaneso 33, 16146, Genova, Italy\n- 60 INAF-Osservatorio Astronomico di Capodimonte, Via Moiariello 16, 80131 Napoli, Italy\n- 62 Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP, Rua das Estrelas, PT4150-762 Porto, Portugal\n- 61 INFN section of Naples, Via Cinthia 6, 80126, Napoli, Italy\n- 63 Faculdade de Ciências da Universidade do Porto, Rua do Campo de Alegre, 4150-007 Porto, Portugal\n- 65 INFN-Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy\n- 64 Dipartimento di Fisica, Università degli Studi di Torino, Via P. Giuria 1, 10125 Torino, Italy\n- 66 INAF-Osservatorio Astronomico di Roma, Via Frascati 33, 00078 Monteporzio Catone, Italy\n- 67 INFN-Sezione di Roma, Piazzale Aldo Moro, 2 - c / o Dipartimento di Fisica, Edificio G. Marconi, 00185 Roma, Italy \n- 68 Institute for Theoretical Particle Physics and Cosmology (TTK), RWTH Aachen University, 52056 Aachen, Germany\n- 70 Instituto de Astrofísica de Canarias, Calle Vía Láctea s / n, 38204, San Cristóbal de La Laguna, Tenerife, Spain\n- 69 Dipartimento di Fisica e Astronomia "Augusto Righi" - Alma Mater Studiorum Università di Bologna, Viale Berti Pichat 6 / 2, 40127 Bologna, Italy\n- 71 European Space Agency / ESRIN, Largo Galileo Galilei 1, 00044 Frascati, Roma, Italy\n- 73 Université Claude Bernard Lyon 1, CNRS / IN2P3, IP2I Lyon, UMR 5822, Villeurbanne, F-69100, France\n- 72 ESAC / ESA, Camino Bajo del Castillo, s / n., Urb. Villafranca del Castillo, 28692 Villanueva de la Cañada, Madrid, Spain\n- 74 Institute of Physics, Laboratory of Astrophysics, Ecole Polytechnique Fédérale de Lausanne (EPFL), Observatoire de Sauverny, 1290 Versoix, Switzerland\n- 76 Departamento de Física, Faculdade de Ciências, Universidade de Lisboa, Edifício C8, Campo Grande, PT1749-016 Lisboa, Portugal\n- 75 UCB Lyon 1, CNRS / IN2P3, IUF, IP2I Lyon, 4 rue Enrico Fermi, 69622 Villeurbanne, France\n- 77 Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal\n- 79 INAF-Istituto di Astrofisica e Planetologia Spaziali, via del Fosso del Cavaliere, 100, 00100 Roma, Italy\n- 78 Department of Astronomy, University of Geneva, ch. d\'Ecogia 16, 1290 Versoix, Switzerland\n- 80 Université Paris-Saclay, Université Paris Cité, CEA, CNRS, AIM, 91191, Gif-sur-Yvette, France\n- 82 FRACTAL S.L.N.E., calle Tulipán 2, Portal 13 1A, 28231, Las Rozas de Madrid, Spain\n- 81 Institut d\'Estudis Espacials de Catalunya (IEEC), Edifici RDIT, Campus UPC, 08860 Castelldefels, Barcelona, Spain\n- 83 INAF-Osservatorio Astronomico di Padova, Via dell\'Osservatorio 5, 35122 Padova, Italy\n- 85 Felix Hormuth Engineering, Goethestr. 17, 69181 Leimen, Germany\n- 84 Max Planck Institute for Extraterrestrial Physics, Giessenbachstr. 1, 85748 Garching, Germany\n- 86 Technical University of Denmark, Elektrovej 327, 2800 Kgs. Lyngby, Denmark\n- 88 Université Paris-Saclay, CNRS / IN2P3, IJCLab, 91405 Orsay, France\n- 87 Cosmic Dawn Center (DAWN), Denmark\n- 89 Max-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany\n- 91 Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK\n- 90 NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA\n- 92 Department of Physics and Helsinki Institute of Physics, Gustaf Hällströmin katu 2, 00014 University of Helsinki, Finland\n- 94 Helsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, Finland\n- 93 Mullard Space Science Laboratory, University College London, Holmbury St Mary, Dorking, Surrey RH5 6NT, UK\n- 95 NOVA optical infrared instrumentation group at ASTRON, Oude Hoogeveensedijk 4, 7991PD, Dwingeloo, The Netherlands\n- 97 Universität Bonn, Argelander-Institut für Astronomie, Auf dem Hügel 71, 53121 Bonn, Germany\n- 96 Centre de Calcul de l\'IN2P3 / CNRS, 21 avenue Pierre de Coubertin 69627 Villeurbanne Cedex, France \n98 \n- 99 Université Paris Cité, CNRS, Astroparticule et Cosmologie, 75013 Paris, France \nAix-Marseille Université, CNRS, CNES, LAM, Marseille, France \n- 100 Institut d\'Astrophysique de Paris, 98bis Boulevard Arago, 75014, Paris, France\n- 102 Department of Physics and Astronomy, University of Aarhus, Ny Munkegade 120, DK-8000 Aarhus C, Denmark\n- 101 European Space Agency / ESTEC, Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands\n- 103 Waterloo Centre for Astrophysics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada\n- 104 Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada\n- 105 Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada\n- 106 Space Science Data Center, Italian Space Agency, via del Politecnico snc, 00133 Roma, Italy\n- 107 Centre National d\'Etudes Spatiales - Centre spatial de Toulouse, 18 avenue Edouard Belin, 31401 Toulouse Cedex 9, France\n- 108 Institute of Space Science, Str. Atomistilor, nr. 409 M˘agurele, Ilfov, 077125, Romania\n- 109 Departamento de Astrofísica, Universidad de La Laguna, 38206, La Laguna, Tenerife, Spain\n- 110 Departamento de Física, FCFM, Universidad de Chile, Blanco Encalada 2008, Santiago, Chile\n- 111 Universität Innsbruck, Institut für Astro- und Teilchenphysik, Technikerstr. 25 / 8, 6020 Innsbruck, Austria\n- 112 Satlantis, University Science Park, Sede Bld 48940, Leioa-Bilbao, Spain\n- 113 Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências, Universidade de Lisboa, Tapada da Ajuda, 1349-018 Lisboa, Portugal\n- 114 Universidad Politécnica de Cartagena, Departamento de Electrónica y Tecnología de Computadoras, Plaza del Hospital 1, 30202 Cartagena, Spain\n- 115 INFN-Bologna, Via Irnerio 46, 40126 Bologna, Italy\n- 116 Kapteyn Astronomical Institute, University of Groningen, PO Box 800, 9700 AV Groningen, The Netherlands\n- 117 Infrared Processing and Analysis Center, California Institute of Technology, Pasadena, CA 91125, USA\n- 118 INAF, Istituto di Radioastronomia, Via Piero Gobetti 101, 40129 Bologna, Italy\n- 119 Astronomical Observatory of the Autonomous Region of the Aosta Valley (OAVdA), Loc. Lignan 39, I-11020, Nus (Aosta Valley), Italy\n- 120 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK\n- 121 School of Physics and Astronomy, Cardi ff University, The Parade, Cardi ff , CF24 3AA, UK\n- 122 Junia, EPA department, 41 Bd Vauban, 59800 Lille, France\n- 123 CERCA / ISO, Department of Physics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106, USA\n- 124 INFN-Sezione di Milano, Via Celoria 16, 20133 Milano, Italy\n- 125 Departamento de Física Fundamental. Universidad de Salamanca. Plaza de la Merced s / n. 37008 Salamanca, Spain\n- 126 Dipartimento di Fisica e Scienze della Terra, Università degli Studi\n- di Ferrara, Via Giuseppe Saragat 1, 44122 Ferrara, Italy\n- 127 Istituto Nazionale di Fisica Nucleare, Sezione di Ferrara, Via Giuseppe Saragat 1, 44122 Ferrara, Italy\n- 128 Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, Kashiwa, Chiba 277-8583, Japan\n- 129 Dipartimento di Fisica - Sezione di Astronomia, Università di Trieste, Via Tiepolo 11, 34131 Trieste, Italy\n- 130 Minnesota Institute for Astrophysics, University of Minnesota, 116 Church St SE, Minneapolis, MN 55455, USA\n- 131 Université Côte d\'Azur, Observatoire de la Côte d\'Azur, CNRS, Laboratoire Lagrange, Bd de l\'Observatoire, CS 34229, 06304 Nice cedex 4, France\n- 132 Department of Physics & Astronomy, University of California Irvine, Irvine CA 92697, USA\n- 133 Department of Astronomy & Physics and Institute for Computational Astrophysics, Saint Mary\'s University, 923 Robie Street, Halifax, Nova Scotia, B3H 3C3, Canada\n- 134 Departamento Física Aplicada, Universidad Politécnica de Cartagena, Campus Muralla del Mar, 30202 Cartagena, Murcia, Spain\n- 135 Instituto de Astrofísica de Canarias (IAC); Departamento de Astrofísica, Universidad de La Laguna (ULL), 38200, La Laguna, Tenerife, Spain \n- 136 Department of Physics, Oxford University, Keble Road, Oxford OX1 3RH, UK\n- 138 Instituto de Astrofísica de Canarias, c / Via Lactea s / n, La Laguna E-38200, Spain. Departamento de Astrofísica de la Universidad de La Laguna, Avda. Francisco Sanchez, La Laguna, E-38200, Spain\n- 137 Department of Computer Science, Aalto University, PO Box 15400, Espoo, FI-00 076, Finland\n- 139 Ruhr University Bochum, Faculty of Physics and Astronomy, Astronomical Institute (AIRUB), German Centre for Cosmological Lensing (GCCL), 44780 Bochum, Germany\n- 141 Univ. Grenoble Alpes, CNRS, Grenoble INP, LPSC-IN2P3, 53, Avenue des Martyrs, 38000, Grenoble, France\n- 140 DARK, Niels Bohr Institute, University of Copenhagen, Jagtvej 155, 2200 Copenhagen, Denmark\n- 142 Department of Physics and Astronomy, Vesilinnantie 5, 20014 University of Turku, Finland\n- 144 ARC Centre of Excellence for Dark Matter Particle Physics, Melbourne, Australia\n- 143 Serco for European Space Agency (ESA), Camino bajo del Castillo, s / n, Urbanizacion Villafranca del Castillo, Villanueva de la Cañada, 28692 Madrid, Spain\n- 145 Centre for Astrophysics & Supercomputing, Swinburne University of Technology, Hawthorn, Victoria 3122, Australia\n- 147 Université Libre de Bruxelles (ULB), Service de Physique Théorique CP225, Boulevard du Triophe, 1050 Bruxelles, Belgium\n- 146 Department of Physics and Astronomy, University of the Western Cape, Bellville, Cape Town, 7535, South Africa\n- 148 ICTP South American Institute for Fundamental Research, Instituto de Física Teórica, Universidade Estadual Paulista, São Paulo, Brazil\n- 150 Oskar Klein Centre for Cosmoparticle Physics, Department of Physics, Stockholm University, Stockholm, SE-106 91, Sweden\n- 149 IRFU, CEA, Université Paris-Saclay 91191 Gif-sur-Yvette Cedex, France\n- 151 Astrophysics Group, Blackett Laboratory, Imperial College London, London SW7 2AZ, UK\n- 153 Dipartimento di Fisica, Sapienza Università di Roma, Piazzale Aldo Moro 2, 00185 Roma, Italy\n- 152 INAF-Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, 50125, Firenze, Italy\n- 154 Centro de Astrofísica da Universidade do Porto, Rua das Estrelas, 4150-762 Porto, Portugal\n- 156 Aurora Technology for European Space Agency (ESA), Camino bajo del Castillo, s / n, Urbanizacion Villafranca del Castillo, Villanueva de la Cañada, 28692 Madrid, Spain\n- 155 HE Space for European Space Agency (ESA), Camino bajo del Castillo, s / n, Urbanizacion Villafranca del Castillo, Villanueva de la Cañada, 28692 Madrid, Spain\n- 157 Dipartimento di Fisica, Università degli studi di Genova, and INFN-Sezione di Genova, via Dodecaneso 33, 16146, Genova, Italy\n- 159 Institute Lorentz, Leiden University, Niels Bohrweg 2, 2333 CA Leiden, The Netherlands\n- 158 Theoretical astrophysics, Department of Physics and Astronomy, Uppsala University, Box 515, 751 20 Uppsala, Sweden\n- 160 Department of Physics, Royal Holloway, University of London, TW20 0EX, UK\n- 162 Niels Bohr Institute, University of Copenhagen, Jagtvej 128, 2200 Copenhagen, Denmark\n- 161 Cosmic Dawn Center (DAWN)'} |
2024arXiv240802445B | Since it began citeCocconiMorrison the search for extraterrestrial intelligence SETI has focused on interstellar emphclassical communication. Recently Berera citeBerera2020rpl pointed out that at certain frequencies photon qubits can retain their quantum coherence over interstellar and even intergalactic distances raising the prospect of interstellar emphquantum communication. This is an intriguing possibility since quantum communication permits certain tasks that would be impossible with classical communication and allow exponential speedups for others. We suggest some motivations in the interstellar context. But quantum coherence alone is not sufficient for quantum communication here for the first time we analyze the emphquantum capacity Q of an interstellar channel. We point out that to have nonzero quantum capacity Qgt0 interstellar communication over a distance L must use wavelengths lambda lt 26.5cm to avoid depolarization by the cosmic microwave background and emphenormous telescopes of effective diameter Dgt0.78sqrtlambda L to satisfy quantum erasure constraints. For example for two telescopes of diameter D on Earth and Proxima Centauri this implies Dgt100km This is a technological threshold that remains to be crossed in order for reliable oneway quantum communication to become possible and suggests a fundamental new resolution of the Fermi paradox. | 2024-08-01T00:00:00Z | ['2024arXiv240802445B', 'arXiv:2408.02445', '10.48550/arXiv.2408.02445'] | ['Quantum Physics', 'Astrophysics - Instrumentation and Methods for Astrophysics'] | On Interstellar Quantum Communication and the Fermi Paradox | 2,024 | 170 | 0.48 | ['EPRINT_HTML', 'EPRINT_PDF'] | 1 | https://arxiv.org/pdf/2408.02445.pdf | {'On Interstellar Quantum Communication and the Fermi Paradox': 'Latham Boyle \nHiggs Centre for Theoretical Physics, University of Edinburgh, Edinburgh, UK and Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada (Dated: August 2024) \nSince it began [1], the search for extraterrestrial intelligence (SETI) has focused on interstellar classical communication. Recently, Berera [2] pointed out that, at certain frequencies, photon qubits can retain their quantum coherence over interstellar (and even intergalactic) distances, raising the prospect of interstellar quantum communication. This is an intriguing possibility, since quantum communication permits certain tasks that would be impossible with classical communication, and allow exponential speed-ups for others. (We suggest some motivations in the interstellar context.) But quantum coherence alone is not sufficient for quantum communication: here, for the first time, we analyze the quantum capacity Q of an interstellar channel. We point out that, to have nonzero quantum capacity Q > 0, interstellar communication over a distance L must use wavelengths λ < 26 . 5 cm (to avoid depolarization by the cosmic microwave background), and enormous telescopes of effective diameter D > 0 . 78 √ λL (to satisfy quantum erasure constraints). For example, for two telescopes of diameter D on Earth and Proxima Centauri, this implies D > 100 km ! This is a technological threshold that remains to be crossed in order for reliable one-way quantum communication to become possible, and suggests a fundamental new resolution of the Fermi paradox.', 'INTRODUCTION': 'In 1948 Shannon invented the modern theory of classical information [3], and in 1959 Cocconi&Morrison initiated the search for extraterrestrial intelligence (SETI) by noting that existing human technology (radio transmissions [1] or laser signals [4]) could be used to send or receive interstellar classical communications. Over the subsequent decades, physicists have realized that classical information theory is just a part of a much richer, still nascent subject quantum information theory [5] and it is natural to ask whether it is also possible to send or receive interstellar quantum communications. A first step in this direction was taken by Berera [2] (see also [69]), who pointed out that - in certain frequency ranges photon qubits could be transmitted over interstellar (or even intergalactic) distances without losing their quantum coherence, and suggested the possibility of searching for such interstellar quantum communications. \nThe possibility of interstellar quantum communication is intriguing because it expands the notion of interstellar communication in fundamental ways. First, it is already known to permit many tasks that are impossible with classical communication alone, including quantum cryptography [10, 11], quantum teleportation [12], superdense coding [13], remote state preparation [14], entanglement distillation/purification [15-17], or direct transmission of (potentially highly complex, highly entangled) quantum states ( e.g. the results of complex quantum computations). Second, protocols based on quantum communication are exponentially faster than those based on classical communication for some problems/tasks [18], in particular as measured by the one-way classical communication complexity [19-21] (the number of bits that must be transmitted one-way, from sender to receiver, to solve a \nproblem or carry out a task - possibly the notion most pertinent to interstellar communication). \nThe ability of a quantum communication channel to transmit quantum information is determined by its quantum capacity Q . Using constraints on quantum erasure channels [22] and the known properties of the interstellar medium [23], we show that, in order for an interstellar communication channel to have non-vanishing quantum capacity ( Q > 0), the exchanged photons must lie within certain allowed frequency bands (see Fig. 1), and the effective diameter D of the exchanging telescopes must be enormous: D > 0 . 78 √ λL , where λ is the photon wavelength and L is the distance from sender to receiver. (For a ground-based telescope, and with L the distance to the closest star Proxima Centauri, this requires a diffractionlimited telescope of diameter D > 100 km !) Using constraints on quantum depolarizing channels [16, 17, 24, 25], and properties of the diffuse astrophysical background radiation [26], we show that Q > 0 requires the exchanged photons to have wavelength λ < 26 . 5 cm (dominantly due to the cosmic microwave background). \nThus, our galaxy and universe do permit interstellar quantum communication, but the above constraints impose a stringent technological threshold we have not yet reached (in particular, we have not yet built a sufficiently large diffraction-limited telescope). We will see why this suggests a new resolution of the Fermi paradox.', 'QUANTUM CAPACITY OF AN INTERSTELLAR COMMUNICATION CHANNEL': "The capacity of a communication channel is the maximum rate at which it can reliably transmit information from sender to receiver (see [5, 22] for details). The classical capacity C is the maximum rate at which classical \nFIG. 1: Quantum communication with Q > 0, over distance L , is impossible at wavelengths where the horizontal line corresponding to L lies within the blue shaded region (summarizing the Milky Way ISM's extinction curve). Gray regions are off limits from the \n<!-- image --> \nground. Adapted from [23, 26], with data from [30-37]. \nbits can be communicated. The quantum capacity Q is the maximum rate at which qubits can be communicated. \nIn the classical case, two-way communication between sender and receiver does not improve the forward classical capacity C . But in the quantum case, it can : so Q is the capacity for forward quantum communication (unassisted by classical communication, or assisted only by forward classical communication); but one also defines Q 2 , the quantum capacity of the channel assisted by twoway classical communication, which can be larger than Q . (In particular, we can have Q 2 > 0 when Q = 0.) \nNow consider an interstellar quantum communication channel: two telescopes of diameter D 1 and D 2 , separated by a distance L , exchanging photons of wavelength λ . (Photons can encode quantum states in multiple ways. Mathematically, the most natural choice would be to use the fact that each photon is intrinsically a two-state system, with positive and negative helicity as the two qubit's basis states | 0 ⟩ and | 1 ⟩ , since these are the eigenstates of the 'little group' of symmetries preserving the line of sight between sender and receiver. But this is just an example to keep in mind for the sake of concreteness - the following analysis does not rely on this choice.) In this section, we determine the constraints on such a channel by considering two model quantum channels in turn: the quantum erasure channel and the depolarizing channel. \nQuantum erasure channel. This is an idealized channel in which each input qubit | ψ ⟩ = α | 0 ⟩ + β | 1 ⟩ is, with probability ϵ , replaced by an 'erasure state' | 2 ⟩ that is orthogonal to both | 0 ⟩ and | 1 ⟩ . This erases the input qubit and informs the receiver that it has been erased. This channel has quantum capacity [22] Q = 1 -2 ϵ (when ϵ < 1 / 2); and when ϵ ≥ 1 / 2, forward quantum communication is impossible ( Q = 0). The fact that Q \nFIG. 2: Three interstellar channels: a) telescopes too small ( Q = 0); b) telescopes sufficiently large ( Q > 0); c) many smaller telescopes as relays ( Q > 0). \n<!-- image --> \nstrictly vanishes when ϵ ≥ 1 / 2 follows [22] from the 'no cloning' theorem in quantum mechanics [27-29]: if Alice randomly sent half her qubits to Bob, and half to Charlie, each would experience a quantum erasure channel with ϵ = 1 / 2, and if these channels each had Q > 0, Alice could use them to clone an arbitrary quantum message. \nOur interstellar channel is an example of an erasure channel, where photons are erased in 3 ways: \ni) First, they may be erased due to 'extinction' (absorption or scattering) as they travel through the interstellar medium (ISM), from sender to receiver. A photon of wavelength λ traveling from a source at distance L has probability < 1 / 2 of extinction if L < (ln 2) / ( n H σ λ ), where n H ≈ 1 . 146 cm -3 is the typical density of Hydrogen atoms in the ISM, and σ λ is the ISM extinction cross section per Hydrogen atom at wavelength λ [23, 26]; or, in other words, if the horizontal line corresponding to distance L lies above the blue shaded region in Fig. 1 (at wavelength λ ). For example, a sender on Proxima Centauri would have to use a wavelength where the pink dashed line (labelled Proxima Centauri) lies above the blue shaded region in Fig. 1. \nii) Second, if we plan to receive the photon using a ground-based telescope on Earth, we must also consider extinction in the Earth's atmosphere. For a photon to have < 1 / 2 probability of being erased in this way, its wavelength must also avoid the gray bands in Fig. 1. (On the other hand, if our receiving telescope is in space, or on the Moon, we can ignore the gray bands in Fig. 1.) \niii) Third, due to spreading of the photon beam as it travels from sender to receiver, the receiving telescope will only intersect a fraction of the beam (Fig. 2a), so that the remaining photons are lost. For a photon to have < 1 / 2 probability of being erased in this way requires extremely large diffraction-limited telescopes both at the transmitting end (to send a sufficiently narrow beam), and the receiving end (to encompass the beam), \nsee Fig. 2b. In particular, it requires (see Appendix A) \nD > ( 1 π ln 2 3 -2 √ 2 ) 1 / 2 ( λL ) 1 / 2 = ( λ 300 nm ) 1 / 2 ( L 1 pc ) 1 / 2 85 . 1 km . (1) \nwhere D = √ D 1 D 2 is the geometric mean of the two telescope's diameters. Thus, D is the minimum diameter for telescopes in a symmetric channel (with D 1 = D 2 ), and the characteristic diameter for telescopes in an interstellar quantum communication network (with L ∼ 1 parsec, the typical interstellar distance in the Milky Way). \nThis third erasure constraint is the hardest to satisfy! Whereas classical communication ( C > 0) can take place even if the receiver only receives a tiny fraction of the photons emitted by the sender, forward quantum communication ( Q > 0) requires large enough telescopes that the sender can put the majority of their photons into the receiver's telescope (Fig. 2b)! Even in the best case, taking the nearest star (Proxima Centauri, L = 1 . 30 parsec) and the shortest wavelength available from the ground ( λ = 320 nm , see Fig. 1), this implies D > 100 km! \nIs such a huge telescope even possible? It is futuristic by current standards: the largest telescope under construction (ELT) has D ≈ 40 m . Note that a D ∼ 100 km optical telescope need not be a single giant mirror: it could be a single dish with a segmented mirror (consisting of many smaller pieces, as in the largest existing telescopes, or the ELT); or a tightly-hexagonally-packed array of smaller dishes (which could cover a fraction π/ √ 12 = 0 . 9069 of the area of a single dish), coherently combined, as in optical interferometry. (Optical interferometry has so far been demonstrated over a distance of 500 m, while the Starshot proposal [38] has an array of coherently-combined optical telescopes with total diameter of a few km.) Given that quantum teleportation using photons has already been demonstrated over ∼ 100 km baselines (at sea level), and over ∼ 1000 km baselines (from the Earth to a satellite), it may be that the main obstruction to building a coherent dense array of optical telescopes over 100 km distances would ultimately be one of cost, rather than one of principle. (Indeed, given that creating, manipulating and storing quantum states is currently a subject of extremely active research and ongoing progress, it is worth noting that future quantum repeaters [39] and quantum memories [40] might allow optical interferometry even over much longer baselines.) \nUsing shorter wavelengths ( λ ≪ 300 nm) would allow smaller D , but the telescopes (necessarily above the atmosphere - e.g. on the Moon, or at Earth's L2 Lagrange point) seem even more futuristic. For example, even if a sender on Proxima Centauri could quantumly communicate using 10 keV ( λ ∼ 10 -10 m) x-rays or 1 MeV ( λ ∼ 10 -12 m) gamma-rays (perhaps using nuclei as a lasing medium), this would mean a receiving telescope able \nto coherently catch ≳ 1 / 2 of such photons over characteristic diameter D ∼ 2 km or D ∼ 200 m, respectively! \nAnd longer wavelengths ( λ ≫ 300 nm) would require even larger D : e.g. communication from Proxima Centauri with λ = 3 mm microwave photons requires D ∼ 10 4 km, comparable to the diameter of the Earth! \nDepolarizing channel. Sometimes, instead of receiving the sender's transmitted qubit, our telescope will receive an extraneous (astrophysical background) photon, and if the probability of this is too high, quantum communication also becomes impossible. This is described by another idealized quantum channel: the depolarizing channel, in which each input qubit | ψ ⟩ = α | 0 ⟩ + β | 1 ⟩ is, with probability ϵ , replaced by a qubit in a random state, without informing the receiver which qubits have been randomized. If ϵ > 1 / 3, then Q vanishes ( i.e. no forward quantum communication); and if ϵ > 2 / 3, both Q and Q 2 vanish ( i.e. no quantum communication, even assisted by two-way classical communication) [16, 17, 24, 25]. \nIf we want the randomization probability ϵ to be less than the threshhold ϵ c , then it follows from the uncertainty principle (see Appendix B) that we must restrict ourselves to wavelengths λ such that \nI ν < ϵ c 1 -ϵ c 128 π 2 ℏ c λ 3 (2) \nwhere I ν (with units of ergs s -1 cm -2 Hz -1 ster -1 ) is the specific intensity of the diffuse astrophysical background at frequency ν = c/λ . As seen e.g. from Fig. 2.2 in [26], this constraint is easily satisfied for short wavelengths but is eventually violated at sufficiently long wavelengths by the cosmic background radiation (with temperature T CMB = 2 . 726 K), so that (2) becomes (see Appendix B) \nλ < 64 π 2 ϵ c 1 -ϵ c ℏ c kT CMB = { 26 . 5 cm (for Q > 0) 106 cm (for Q 2 > 0) (3)", 'DISCUSSION': "Motivations. To make our discussion more concrete, it may be helpful to give four examples to illustrate possible motivations for interstellar quantum communication: \n- i) One may wish to send a complex quantum state ( e.g. the final or intermediate state of a complex quantum computation), either directly, or via quantum teleportation [12]. Note that transmitting such an N -qubit state classically would mean sending 2 N complex numbers, which quickly becomes impossible as N grows: e.g. for N > 265 qubits, 2 N is larger than Eddington's number (the number of protons in the observable universe).\n- ii) Astronomically long baseline interferometry (ALBI): As pointed out in [39], quantum repeaters could be used to coherently interfere optical telescopes separated by the Earth's diameter D E , to achieve the effective angular resolution δθ = λ/D E . By the same \ntoken, an interstellar quantum communication channel would make it possible to interfere telescopes operating at wavelength λ , and separated by the astronomical distance L , thereby effectively producing a telescope with the mind-boggling angular resolution δθ = λ/L . \niii) Quantum error correction: A quantum error correcting code (QECC) is a clever way of protecting a delicate quantum state from destruction by embedding it in a carefully-chosen subspace C (the code space) of a larger Hilbert space H which, in turn, may be decomposed as a tensor product H = ⊗ i H i . In particular, let ρ = | ψ ⟩⟨ ψ | be a pure state in C , and let ρ i be the corresponding reduced density matrix in H i : it is a fundamental fact that C is a QECC capable of correcting arbitrary errors or erasures in H i iff ρ i is independent of the code state | ψ ⟩ ∈ C . Now, if each subspace H i is distributed to a different solar system, we have a code with an astronomically large code distance ( i.e. in which quantum information is protected against the erasure of the portion of the state residing in any one solar system). \niv) Quantum cryptography [10, 11] allows communication whose security is guarunteed by quantum mechanics. \nSmaller telescopes? Is there any escape from the previous section's conclusion that an interstellar channel with Q > 0 requires enormous telescopes? Two loopholes are worth discussing: \ni) With smaller telescopes, although forward quantum communication is impossible ( Q = 0), quantum communication assisted by two-way classical communication is possible ( Q 2 > 0). For example, imagine the sender transmits a stream of optical photons, equally spaced in time ( e.g. one per µs ), each of which is a member of an EPR pair (whose other member is retained by the sender); and our telescope is smaller than the bound (1), so we randomly receive only a tiny fraction ( ≪ 1 / 2) of these incoming photons. If the sender doesn't know which photons we have received, they cannot use their EPR pairs to teleport their quantum states to us ( Q = 0); but if we send them a list specifying which photons we did receive (by specifying their arrival times), they can then use the corresponding subset of EPR pairs for teleportation ( Q 2 > 0). Note that, whereas forward communication (measured by Q ) is instantaneous in the information's rest frame, communication assisted by twoway classical communication (measured by Q 2 ) involves an extra delay of at least 2 L/c ( e.g. at least 8 years for Proxima Centauri), and requires us to have quantum harddrives capable of storing the received photons for this duration. Of course, this may represent an unacceptable slowdown, and would make certain tasks impossible in principle ( e.g. if Alice wants to send two states to Bob and Charlie respectively, and have them process those states immediately, so that their spacelike separation guarantees their causal independence). \nii) Alternatively, instead of transmission directly from sender to receiver (as in Fig. 2b), one could imagine a se- \nquence of relays (converging lenses or quantum repeaters [41]) to capture and refocus the beam at n -1 point along its path (as in Fig. 2c). Then, in order to achieve non-zero quantum capacity Q > 0, the diameter of each optical element would only need to satisfy the weaker bound D ≳ √ λL/n . In this scheme, in order to use wavelength λ and optical elements of diameter D , the separation between the relay stations would have to be \nL n ∼ ( D 100 m ) 2 ( 300 nm λ ) × 3 × 10 10 m . (4) \nSo e.g. with λ ≳ 300 nm and D ≲ 100 m, the relays would need to be separated by ≲ 0 . 1 au, with many already in our Solar System. (Could they be detected?) Of course, placing/maintaining these repeaters in their precise locations might be too difficult/expensive in practice. \nThe Fermi paradox. Given that our universe is statistically homogeneous [42-45], filled with very many galaxy clusters like our own, each containing very many galaxies rather like our own, each containing very many stars like our own, many of which have a retinue of planets, it is tempting to guess that it also contains many other occurences of life. Fermi famously wondered why we have not yet seen any sign of them? Many possible answers have been put forward [46, 47]. Here we point out a new answer suggested by the preceding considerations: \nSuppose that the sender wishes to communicate quantum rather than classically. (We have mentioned several possible motivations for this, and there will certainly be many more: as mentioned above, classical communication is just a part of the larger topic of quantum communication, whose limits and applications are still only partially understood.) Two simple conclusions then follow from the requirement of non-zero quantum capacity Q > 0. (i) First, we have seen that the sender must place nearly all (at least 1 / 2) of their photons into our receiving telescope, which implies that the signal must be so highly directed that only the intended receiving telescope can hope to detect any sign of the communication. This is in sharp contrast to classical communication, where one can broadcast photons indiscriminantly into space, and an observer in any direction who detects a small fraction of those photons can still receive the message. (ii) Second, we have seen that (setting aside the loopholes mentioned above) the sending and receiving telescopes must be extremely large, satisfying the inequality in Eq. (1); but this same inequality implies that, if the sender has a large enough telescope to communicate quantumly with us, they necessarily also have enough angular resolution to see that we do not yet have a sufficiently large receiving telescope [49], so it would make no sense to send any quantum communications to us until we had built one. Thus, the assumption that interstellar communication is quantum appears sufficient to explain the Fermi paradox. \nAcknowledgements. I am grateful to Daniel Gottesman and Avi Loeb for very helpful discussions.", 'APPENDIX A': "Here we derive Eq. (1). Actually, we give a sequence of three increasingly precise derivations, to make it clear how various descriptions that the reader may be familiar with relate to one another. \n- 1) For starters, let us make a rough estimate (ignoring factors of order unity) of the telescope sizes needed for one telescope to catch an order-one fraction of the photons emitted by the other telescope. The transmitting telescope of diameter D 1 can aim an electromagnetic beam of wavelength λ with, at best, the diffractionlimited angular uncertainty ∆ θ ∼ λ/D 1 . Thus, after traveling a distance L (in the z direction), the beam will have spread (in the xy plane perpendicular to the z axis) to a characteristic width L ∆ θ ∼ λL/D 1 . This width must be ≲ D 2 (the diameter of the receiving telescope) or, equivalently, \nD 1 D 2 ≳ λL (5) \nin order for the receiving telescope to catch an order-one fraction of the photons. \n- 2) To phrase things more precisely (but still classically), consider the usual model for an ideal laser beam: an axisymmetric beam of electromagnetic radiation, freely propagating along the z axis, with a Gaussian profile in the transverse xy plane. The beam is described by (see e.g. Eq. 8.40 in [48]) \nψ z = σ 0 σ z exp ( -ρ 2 4 σ 2 z ) exp [ i ( kρ 2 2 R z + kz -arctan z z 0 )] (6) \nwhere ρ = ( x 2 + y 2 ) 1 / 2 is the distance from the z axis, k = 2 π/λ is the wavenumber of the beam, and we have defined the quantities z 0 = 2 kσ 2 0 , σ z = σ 0 (1 + z 2 /z 2 0 ) 1 / 2 , and R z = a (1+ z 2 0 /z 2 ). At fixed z , the beam's energy flux distribution is ∝ | ψ z | 2 = ( σ 0 /σ z ) 2 exp[ -1 2 ρ 2 σ 2 z ], so as the beam propagates along the z -axis, it retains its gaussian profile, with beam radius σ 0 at its waist ( z = 0), and beam radius σ z at a general z . Now consider the product σ z σ z ± L between the beam radii at two points along the z axis separated by distance L . The product σ z σ z ± L is minimized when ∂ ( σ z σ z ± L ) /∂σ 0 = 0, which yields \nσ z σ z ± L ≥ λL/ 4 π, (7) \nin agreement with our previous rough estimate. \n3) Now let us phrase things quantum mechanically: Suppose a photon is emitted in the z -direction with z -momentum p z = ℏ k = 2 π ℏ /λ . If its initial position uncertainty (along some transverse direction) is ∆ x 1 , then (by the uncertainty principle) its corresponding transverse momentum uncertainty is ∆ p ⊥ ≥ 1 2 ℏ / ∆ x 1 , so the angular uncertainty in its direction is ∆ θ = ∆ p ⊥ /p z ≥ λ/ (4 π ∆ x 1 ) and, after propagating a distance L along the \nz -axis, its transverse position uncertainty has grown to ∆ x 2 = L ∆ θ ≥ λL/ (4 π ∆ x 1 ) or, equivalently, \n∆ x 1 ∆ x 2 ≥ λL/ 4 π, (8) \nwhich again agrees with the previous result (7). \nNext take the photon's transverse wavefunction to be described by an axisymmetric gaussian ψ ( ρ ) which saturates the position-momentum uncertainty bound, so that Eq. (8) becomes an equality. The photon has transverse probability distribution | ψ 1 , 2 | 2 = 1 2 π ∆ x 2 1 , 2 exp( -1 2 ρ 2 1 , 2 ∆ x 2 1 , 2 ), where the subscripts 1 and 2 apply at the two ends of the z -axis, respectively; so the joint-probability that the wavefunction will overlap with both the initial and final telescopes (of diameter D 1 and D 2 , respectively) is \n[ 1 -exp ( -D 2 1 8∆ x 2 1 )][ 1 -exp ( -D 2 2 8∆ x 2 2 )] , (9) \nand the requirement that this is ≥ 1 / 2 implies Eq. (1).", 'APPENDIX B': "Here we derive Eqs. (2) and (3). \nConsider a transmitted qubit photon of wavelength λ that arrives with uncertainty ∆ t in its arrival time, ∆ E in its energy, and hence ∆ ν = (∆ E ) / (2 π ℏ ) in its frequency. Due to the energy-time uncertainty relation, we have ∆ t ∆ E ≥ ℏ / 2, and hence ∆ t ∆ ν ≥ 1 / 4 π . The number of astrophysical background photons arriving within the time interval ∆ t and frequency range ∆ ν is \nN = I ν · ∆ t · ∆ ν · A · ∆Ω 2 π ℏ ν (10) \nwhere I ν is the specific intensity of the astrophysical background (energy per time per frequency per area per solid angle) at frequency ν = c/λ , A is the receiving telescope's area, ∆Ω is its angular resolution (in solid angle), and 2 π ℏ ν is the energy per photon. If the receiving telescope has diameter ∆ x , and hence area A = π (∆ x ) 2 , a received photon has transverse position uncertainty ∆ x , hence transverse momentum uncertainty ∆ p ≥ ℏ / (2∆ x ), hence angular uncertainty ∆ θ ≥ (∆ p ) /p = λ/ (4 π ∆ x ), and hence angular resolution ∆Ω = π (∆ θ ) 2 ≥ (1 / 16 π )( λ/ ∆ x ) 2 . Thus, (10) becomes \nN ≥ I ν λ 3 128 π 2 ℏ c (11) \nOn the other hand, if we write N (the expected number of random photons per signal photon) as ϵ/ (1 -ϵ ), and solve for I ν , we obtain Eq. 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Blandford, Modern classical physics: optics, fluids, plasmas, elasticity, relativity, and statistical physics , Princeton University Press (2017).\n- [49] (and they have the resolution needed to determine the position of our telescope, or where it will be 4 years hence, if they are communicating from Proxima Centauri)"} |
2024arXiv240907053O | This correspondence delves into the application of dynamical systems methodologies within the context of cosmology specifically addressing a preliminary strategy for determining the range for the equation of state for dark energy omegaDE. Our findings suggest that the preferred range for omegaDE is between 1.1 and 0.6. | 2024-09-01T00:00:00Z | ['2024arXiv240907053O', '10.48550/arXiv.2409.07053', 'arXiv:2409.07053'] | ['General Relativity and Quantum Cosmology'] | Dark Energy A Dynamical Systems Approach to the Reconstruction of the Equation of State | 2,024 | 170 | 0.23 | ['EPRINT_HTML', 'EPRINT_PDF'] | 1 | https://arxiv.org/pdf/2409.07053.pdf | {'Dark Energy: A Dynamical Systems Approach to the Reconstruction of the Equation of State': 'Bob Osano 1 , 2 \n1 Cosmology and Gravity Group, Department of Mathematics and Applied Mathematics, \nUniversity of Cape Town (UCT), Rondebosch 7701, Cape Town, South Africa \n& \n2 \nCentre for Higher Education Development, (Dated: September 12, 2024) \nUniversity of Cape Town (UCT), Rondebosch 7701, Cape Town, South Africa \nThis correspondence delves into the application of dynamical systems methodologies within the context of cosmology, specifically addressing a preliminary strategy for determining the range for the equation of state for dark energy ( ω DE ). Our findings suggest that the preferred range for ω DE is between -1.1 and -0.6. \nKeywords: Cosmology, Dark Energy, Dynamical systems', '1. INTRODUCTION': "Determining the equation of the state of dark energy remains a challenge to cosmologists although it may appear as a foregone conclusion that this substance is responsible for the recent cosmic acceleration. Recent efforts in this regard include [1-12]. Whether the dark energy equation of state parameter ω DE is constant ( ω DE ) or evolving as described by ω DE ( z ) = ω 0 + ω a f ( z ) [5, 1114] remains uncertain. However, its nature and behaviour have profound implications for our understanding of the universe's evolution. We revisit this subject in this letter. \nThe letter is divided into two pertinent sections. The first section provides a comprehensive dynamical systems analysis approach to the matter-radiation transition. Although the detailed content in this section is standard and commonly found in existing literature, it offers foundational context for the methodology employed in the subsequent section. In the second section, we focus on analysing the transition from matter (M) to dark energy (DE). It is important to note that this approach traditionally presumes dark energy synonymous with the cosmological constant[15]. \nIn our investigation of dark energy, we shall forgo the assumption that it is synonymous with the cosmological constant. Instead, we propose a hypothetical scenario in which dark energy interacts with dark matter. Though much of the methodology presented here is derivable from forms extensively discussed in existing literature, an exhaustive recapitulation is unnecessary. However, for the sake of thoroughness, pertinent references will be provided, offering comprehensive guidance for the reader. We employ a dynamical systems approach, concentrating on identifying critical points where the flow alters its behaviour, rather than focusing on deriving explicit solutions to the equations of motion. The applicability of this methodology to cosmology has been demonstrated in [15]. This letter brings it all together in an attempt to answer the question: Can dynamical systems techniques \naid in the reconstruction of the EoS of dark energy? As this is a letter, we are deliberately brief. We commence our analysis by postulating a model comprising three constituents, designated as A , B , and C , with the interaction occurring specifically between constituents A and B . Additionally, we assume that the noninteracting component remains constant throughout the analysis. To connect with the existing literature, we will assume that the interacting components follow a barotropic form. Additionally, we will consider a flat cosmological model. The evolution timescale is quantified using the parameter η = log a , where a represents the scale factor. We utilise the prime notation to denote derivatives with respect to η . Beginning with the Friedmann equations, we derive the continuity equations that describe the evolution of the energy densities of the individual constituents. These equations can be expressed in the following form: \nρ ' A = -3(1 + ω A ) ρ A + Q AB H (1) \nρ ' B = -3(1 + ω B ) ρ B -Q AB H (2) \nρ ' C = 0 , (3) \nIn this context, Q AB denotes the interaction term between constituents A and B . Subsequently, we will interpret this interaction as one occurring either between radiation and matter or between dark matter and dark energy. It is noteworthy that, in terms of the conservation law, the following relation holds: \n∇ µ ∑ i = A,B T µνb i = 0 , (4) \nwhere µ, ν range over 1 , 2 , 3 , 4. This indicates that the total energy-momentum tensor is conserved, but the energy-momentum tensors for each constituent part are not conserved independently.", '2. EXPANSION NORMALISED FORMULATION': 'To establish a parallel between the conventional methodology and the novel approach presented in this letter, we adopt expansion-normalized variables akin to those utilised in [15]. Within this framework, the components of the first Friedmann equation for a radiation and matter-dominated model are delineated as follows: \nX = Ω A = ρ A 3 H 2 Y = Ω B = ρ B 3 H 2 Z = Ω C = ρ C 3 H 2 , (5) \nand the first Friedmann equation takes the form \nX + Y + Z = 1 (6) \nThe effective EoS for this system has the form \nω eff = ( Σ i ω i ρ i Σ i ρ i ) , (7) \nwhere i = A,B,C . It is important to note that we will adjust Equation (7) according to the predominant constituents during the specific epoch under investigation, which will be elaborated upon later in this letter. To maintain a more general framework, we will revert to the AB notation.', '3. A-B DOMINATED': "We present the evolution equations for A and B. It can be demonstrated that the evolution equations assume the following form. \nX ' = 3 X [ -(1 + ω A ) + (1 + ω A + α 3 ) X +(1 + ω B ) Y ] Y ' = 3 Y [ -(1 + ω B ) + (1 + ω A -α 3 ) X +(1 + ω B ) Y ] , (8) \nBy utilizing the constraint given by equation (6), we have eliminated Z from these equations and have employed the ansatz Q AB = αHX . The analogous form of equation (7) for this system can be expressed as follows: \nω eff = -1 + X (1 + ω A ) + (1 + ω B ) Y. (9) \nWe note that the system has the fixed points ( X ∗ , Y ∗ )= (0,0), (0,1) and (1,0) regardless of the values of ω A and ω B when no interactions occur between A and B . More important, is the fact that the EoS of ω A and ω B can be recovered from a generic portrait as will demonstrate.", '4. MATTER-RADIATION': "We know that ω m = 0 and ω r = 1 3 . However, the general form in equations (8) allows us to explore the \n̸ \n̸ \nscenario where the Equation of State (EoS) parameters for either component are unknown. Specifically, we can investigate whether it would be possible to recover these parameters. \nTo relate this to existing literature, we consider the notation A ≡ r (radiation), B ≡ m (matter), and Q AB ≡ Q rm (the case of no interaction). Under these conditions, we have: \nX ' = X ( -3 + (3 + α ) X +4 Y ) Y ' = Y ( -4 + (3 -α ) X +4 Y ) . (10) \n̸ \nWe have retained the case where α = 0 to examine whether any interactions are present. We will later set α = 0 to align with existing literature. The corresponding equation to (7) for this system is: \nω eff = -1 + X + 4 3 Y. (11) \nThe fixed points for the system described by equation (10) are (0 , 0), (0 , 1), and (3 / (3 + α ) , 0). By setting α = 0, these fixed points can be interpreted as representing the dark-energy, radiation, and matter-dominated epochs, respectively [15]. We will denote these points as R (0 , 1), M (1 , 0)and D (0 , 0) . These points form a triangle that we will henceforth refer to as the RMDtriangle. \nThe phase portraits for this system are shown in Figures (1a-1d). It is possible that when ω eff is known for each fixed point, the corresponding EoS parameters for radiation and matter can be determined straightforwardly and unambiguously. \n̸ \nFIG. 1: Radiation- Matter: Phase portraits X (horizontal axis) and Y (vertical axis), with varying α . (a) ω A ≡ ω r = 1 / 3, ω B = ω m = 0. (b) ω A ≡ ω r = 1 / 3, ω B = ω m = 0, α = 0. (c) ω A ≡ ω r = 1 / 3 , ω B = 1 = ω m , α = 0. (d) ω A ≡ 1 = ω r , ω B = ω m = 0, α = 0.Figure (1a) depicts the standard portrait with parameters ω r = 1 / 3, ω m = 0, and α = 0. In this figure, the \n<!-- image --> \nregion above the line X + Y = 1 is deemed non-viable[15]. Figures (1a-1d) present generic cases where one of the constituents corresponds to either radiation or matter. It can be shown that if the value of ω eff is known, for instance through observational data, one can reconstruct the Equation of State (EoS) of either component by utilizing the node values from the phase portraits. The first thing to note is that a non-zero value for the interaction term moves one of the nodes along the hypotenuse of the RMD triangle, while a different value of EoS moves a node along the horizontal or vertical edges of the RMD triangle. \nConsider Figure (1c), where the nodes are located at D (0 , 0), R (0 , 1), and M (0 . 5 , 0). The final node indicates that X = 0 . 5 while Y = 0. Given that the horizontal arm is matter-dominated, we deduce ω eff = 0. Using this information along with equation (11), we can reconstruct ω B and obtain the value ω B = 1. Similarly, it can be demonstrated that ω A = 1 for the point situated in the radiation-dominated arm. This rudimentary approach provides an initial estimate for reconstructing a given Equation of State (EoS) from such phase portraits. For a rigorous analysis, independent determination of ω eff is essential. As will be discussed in the subsequent section, ω eff can be derived from observational data and its link to the second Friedmann equation.", '5. MATTER - DARK ENERGY': "Let us now use the knowledge from the previous section to attempt to construct the EoS of dark energy. In this case, we let ω A ≡ ω DM and ω B ≡ ω DE . it follows that \nX ' = 3 X [ -1 + (1 + α 3 ) X +(1 + ω DE ) Y ] Y ' = 3 Y [ -(1 + ω DE ) + (1 -α 3 ) X +(1 + ω DE ) Y ] , (12) \nwhere Q AB = αHX . With ω DM = 0, the equivalent of equation (7) for this system assumes the form \nω eff = -1 + X +(1 + ω DE ) Y. (13) \nNote that the fractions X and Y are functions of redshift z , given that they are normalized with respect to the expansion as shown in Equation (5). Phenomenologically, we can express the last two terms as ω DDE f ( z ), where ω DDE is the dynamical equation of state that accounts for both matter and dark energy. The function f ( z ) could take a linear, logarithmic, or hyperbolic form [22]. The interaction mediating these two components allows us to extend the definition in this manner. If ω 0 = -1, then equation (13) becomes \nω eff = ω 0 + ω DDE f ( z ) . (14) \nEquation (14) has the form of the EoS often used for dynamical dark energy. We emphasise that the difference, compared to what is in literature [21] is that our \ndynamical part incorporates matter. We can now establish a relationship between the deceleration parameter, q , and the variables X and Y by utilizing both Friedmann equations, yielding the following expression: \nH ' H = -(1 + q ) = -3 2 [ X +(1 + ω DE ) Y ] . (15) \nFor the standard ΛCDM model, the deceleration parameter determined from local observations is q 0 = -0 . 55. Other observations may yield slightly different values. Generally, the value of q 0 from various observations tends to fall within the range -0 . 8 ≤ q 0 ≤ -0 . 4. We will use this range to illustrate our approach. \nFIG. 2: Matter- Dark Energy: In these portraits X is the horizontal axis and Y the vertical axis. (a) ω B ≡ ω DE = -1 . 1, ω A ≡ ω M = 0, α = 0. (b) ω B = -0 . 6, ω A ≡ ω M = 0, α = 0. (c) ω B = -0 . 6, ω A ≡ ω M = 0, α = 0 . 8. (d) ω B ≡ ω DE = -1 . 1, ω A ≡ ω M = 0, α = 0 . 8. \n<!-- image --> \n<!-- image --> \nWe considered the range -1 . 1 ≤ ω DE ≤ -0 . 6 based on the implication of q 0 = -0 . 6 ± 0 . 2 representing the average of several experiments with the sweet spot being q 0 = -0 . 55[7, 18, 19]. The phase portraits indicate that, for ω DE = -1 . 1, the node (1 , 0) is the future attractor, while for ω DE = -0 . 6, the node (0 , 0) serves as the future attractor. A non-zero interaction acts as a bifurcation parameter, shifting one of the nodes horizontally or vertically.", '6. CONCLUSION': "This letter explores the question of whether it is possible to reconstruct the equation of state (EoS) for one of two competing constituents when the effective EoS is known or can be determined experimentally. We conclude that this is indeed feasible. Employing dynamical \ntechniques, we have demonstrated the possibility of reconstructing the EoS in a model consisting of both matter and radiation. Furthermore, we have applied the same methodology to a model comprising dark energy and dark matter, confirming the applicability of the technique in this context as well. It is important to note that this approach requires prior knowledge of the effective equation of state, which can be derived from observational data. Our findings indicate that -1 . 1 ≤ ω DE ≤ -0 . 6. In the specified range, the lower value results in a future attractor node located at (0 , 1), while the upper value designates (0 , 0) as the future attractor, consistent with the presence of a cosmological constant. If the former case is confirmed, it could negate the cosmological constant being the long-sought-after dark energy (DE). This finding would also confirm the crossing of the cosmolog- \n- [1] Rahman, S.F. Dynamic Dark Energy Equation of State (EoS) and Hubble Constant Analysis Using Type Ia Supernovae from Union 2.1 Dataset. Astron. Rep. 64, 281-294 (2020).\n- [2] Ji-Ping Dai et al. Reconstruction of the Dark Energy Equation of State from the Latest Observations.2018 ApJ 857 9 \n̸ \n- [3] Avelino P.P. et al.Is ω = -1 evidence for a dynamical dark energy equation of state? Phys.Rev.D 80 (2009) 067302\n- [4] Yang W. et al. Interacting dark energy with time-varying equation of state and the H 0 tension. Phys. Rev. D 98, 123527.\n- [5] Singh J.K. et al. New Parametrization of the DarkEnergy Equation of State with a Single Parameter. Universe 2024, 10(6), 246.\n- [6] Escamilla L.A. et al. The state of the dark energy equation of state circa 2023. JCAP 2405 (2024) 091\n- [7] Camarena, D. and Marra, V. Local determination of the Hubble constant and the deceleration parameter. Physical Review Research. 2 (1): 013028\n- [8] Tripathi A et al. Dark energy equation of state parameter and its evolution at low redshift.\n- [9] Teng Y-P et al. Constraining the dark-energy equation of state with cosmological data. Phys. Rev. D 104, 083519\n- [10] Upadhye A. Measuring the Dark Energy Equation of State. Nuclear Physics B (Proc. Suppl.) 173 (2007) 11-14\n- [11] Gong Y. Reconstruction of the deceleration parameter and the equation of state of dark energy. Phys.Rev.D75:043520, 2007 \nical constant boundary [10, 17] and point towards the 'Big Rip' [16] as a potential fate of the universe unless a new and yet undetermined form of energy intervenes. \nIt is essential to underscore that the results and interpretations are contingent upon the precision of the deceleration parameter measurements. Despite its rudimentary nature, this approach offers a foundational basis for subsequent inquiries and in-depth analyses. The dependence of the equation of state (EoS) parameter on the value of the Hubble parameter, coupled with the increasing discordance in its measurement across various methodologies, underscores the need for further investigation into dynamic dark energy EoS models as well. Such investigation is essential to elucidate the relationship between the expansion rate and the evolution of dark energy. \n- [12] Colgain, E .O., Sheikh-Jabbari, M.M., Yin L. Can dark energy be dynamical? Phys. Rev. D 104, 023510 (2021)\n- [13] Osano, B. Dynamics of the transitions epochs in cosmological evolution. Preprint arXiv:2406.00506.\n- [14] Osano, B. Matter-Dark Energy Transition: A Dynamics Systems Approach. In preparation.\n- [15] Bahamonde S. et al. Dynamical systems applied to cosmology: dark energy and modified gravity.Physics Reports Vol. 775-777 (2018) 1-122\n- [16] Caldwell R.R. et al, Phys. Rev. Lett. 91, 071301 (2003),\n- [17] Stefancic H. Crossing of the Cosmological Constant Boundary - an Equation of State Description. J.Phys.A 39 (2006) 6761-6768\n- [18] Naik D M. et al. Model-independent cosmological insights from three newly reconstructed deceleration parameters with observational data. Physics Letters B Volume 844, 10 September 2023, 138117\n- [19] Mukherjee P. and Banerjee N.Revisiting a nonparametric reconstruction of the deceleration parameter from combined background and the growth rate data. Physics of the Dark Universe Volume 36, June 2022, 100998.\n- [20] Lahav O and Liddle A R . The Cosmological Parameters (2023). Preprint arXiv:2403.15526\n- [21] Chevallier M. and Polarski D. Accelerating Universes with Scaling Dark Matter.Int.J.Mod.Phys.D10:213224,2001\n- [22] Barboza E. M. and Alcaniz J. S. A parametric model for dark energy. Phys.Lett.B666:415-419,2008"} |
2024arXiv240813842S | The Dark Energy Spectroscopic Instrument DESI Peculiar Velocity Survey aims to measure the peculiar velocities of early and late type galaxies within the DESI footprint using both the Fundamental Plane and TullyFisher relations. Direct measurements of peculiar velocities can significantly improve constraints on the growth rate of structure reducing uncertainty by a factor of approximately 2.5 at redshift 0.1 compared to the DESI Bright Galaxy Surveys redshift space distortion measurements alone. We assess the quality of stellar velocity dispersion measurements from DESI spectroscopic data. These measurements along with photometric data from the Legacy Survey establish the Fundamental Plane relation and determine distances and peculiar velocities of earlytype galaxies. During Survey Validation we obtain spectra for 6698 unique earlytype galaxies up to a photometric redshift of 0.15. 64 of observed galaxies 4267 have relative velocity dispersion errors below 10. This percentage increases to 75 if we restrict our sample to galaxies with spectroscopic redshifts below 0.1. We use the measured central velocity dispersion along with photometry from the DESI Legacy Imaging Surveys to fit the Fundamental Plane parameters using a 3D Gaussian maximum likelihood algorithm that accounts for measurement uncertainties and selection cuts. In addition we conduct zeropoint calibration using the absolute distance measurements to the Coma cluster leading to a value of the Hubble constant H0 76.05 pm 0.35statistical pm 0.49systematic FP pm 4.86statistical due to calibration mathrmkm s1 Mpc1. This H0 value is within 2sigma of Planck Cosmic Microwave Background results and within 1sigma of other low redshift distance indicatorbased measurements. | 2024-08-01T00:00:00Z | ['2024arXiv240813842S', '10.48550/arXiv.2408.13842', 'arXiv:2408.13842'] | ['Astrophysics - Cosmology and Nongalactic Astrophysics', 'Astrophysics - Astrophysics of Galaxies'] | DESI Peculiar Velocity Survey Fundamental Plane | 2,024 | 171 | 0.57 | ['EPRINT_HTML', 'EPRINT_PDF'] | 8 | https://arxiv.org/pdf/2408.13842.pdf | {'DESI Peculiar Velocity Survey - Fundamental Plane': "Khaled Said , 1 ★ Cullan Howlett , 1 Tamara Davis , 1 John Lucey , 2 Christoph Saulder , 3 Kelly Douglass , 4 Alex G. Kim , 5 Anthony Kremin , 5 Caitlin Ross, 1 Greg Aldering, 5 Jessica Nicole Aguilar, 5 Steven Ahlen , 6 Segev BenZvi , 4 Davide Bianchi , 7 David Brooks, 8 Todd Claybaugh, 5 Kyle Dawson, 9 Axel de la Macorra , 10 Biprateep Dey , 11 Peter Doel, 8 Kevin Fanning , 12 , 13 Simone Ferraro , 5 , 14 Andreu Font-Ribera , 15 , 8 Jaime E. Forero-Romero , 16 , 17 Enrique Gaztañaga, 19 , 20 , 18 Satya Gontcho A Gontcho , 5 Julien Guy , 5 Klaus Honscheid, 23 , 21 , 22 Robert Kehoe, 24 Theodore Kisner , 5 Andrew Lambert, 5 Martin Landriau , 5 Laurent Le Guillou , 25 Marc Manera , 26 , 15 Aaron Meisner , 27 Ramon Miquel, 28 , 15 John Moustakas , 29 Andrea Muñoz-Gutiérrez, 10 Adam Myers, 30 Jundan Nie , 31 Nathalie Palanque-Delabrouille , 5 , 32 Will Percival , 34 , 33 , 35 Francisco Prada , 36 Graziano Rossi, 37 Eusebio Sanchez , 38 David Schlegel, 5 Michael Schubnell, 39 , 40 Joseph Harry Silber , 5 David Sprayberry, 27 Gregory Tarlé , 40 Mariana Vargas Magana , 10 Benjamin Alan Weaver, 27 Risa Wechsler , 41 , 12 , 13 Zhimin Zhou , 31 Hu Zou 31 \n- 1 School of Mathematics and Physics, University of Queensland, 4072, Australia\n- 2 Institute for Computational Cosmology, Department of Physics, Durham University, South Road, Durham DH1 3LE, UK\n- 3 Max Planck Institute for Extraterrestrial Physics, Gießenbachstraße 1, 85748 Garching, Germany\n- 4 Department of Physics & Astronomy, University of Rochester, 206 Bausch and Lomb Hall, P.O. Box 270171, Rochester, NY 14627-0171, USA\n- 5 Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA\n- 6 Physics Dept., Boston University, 590 Commonwealth Avenue, Boston, MA 02215, USA\n- 7 Dipartimento di Fisica 'Aldo Pontremoli', Università degli Studi di Milano, Via Celoria 16, I-20133 Milano, Italy\n- 8 Department of Physics & Astronomy, University College London, Gower Street, London, WC1E 6BT, UK\n- 9 Department of Physics and Astronomy, The University of Utah, 115 South 1400 East, Salt Lake City, UT 84112, USA\n- 10 Instituto de Física, Universidad Nacional Autónoma de México, Cd. de México C.P. 04510, México\n- 11 Department of Physics & Astronomy and Pittsburgh Particle Physics, Astrophysics, and Cosmology Center (PITT PACC), University of Pittsburgh,\n- 3941 O'Hara Street, Pittsburgh, PA 15260, USA\n- 12 Kavli Institute for Particle Astrophysics and Cosmology, Stanford University, Menlo Park, CA 94305, USA\n- 13 SLAC National Accelerator Laboratory, Menlo Park, CA 94305, USA\n- 14 University of California, Berkeley, 110 Sproul Hall #5800 Berkeley, CA 94720, USA\n- 15 Institut de Física d'Altes Energies (IFAE), The Barcelona Institute of Science and Technology, Campus UAB, 08193 Bellaterra Barcelona, Spain\n- 16 Departamento de Física, Universidad de los Andes, Cra. 1 No. 18A-10, Edificio Ip, CP 111711, Bogotá, Colombia\n- 17 Observatorio Astronómico, Universidad de los Andes, Cra. 1 No. 18A-10, Edificio H, CP 111711 Bogotá, Colombia\n- 18 Institut d'Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain\n- 19 Institute of Cosmology and Gravitation, University of Portsmouth, Dennis Sciama Building, Portsmouth, PO1 3FX, UK\n- 20 Institute of Space Sciences, ICE-CSIC, Campus UAB, Carrer de Can Magrans s/n, 08913 Bellaterra, Barcelona, Spain\n- 21 Center for Cosmology and AstroParticle Physics, The Ohio State University, 191 West Woodruff Avenue, Columbus, OH 43210, USA \n22 \nDepartment of Physics, The Ohio State University, 191 West Woodruff Avenue, Columbus, OH 43210, USA \n- 23 The Ohio State University, Columbus, 43210 OH, USA\n- 24 Department of Physics, Southern Methodist University, 3215 Daniel Avenue, Dallas, TX 75275, USA\n- 25 Sorbonne Université, CNRS/IN2P3, Laboratoire de Physique Nucléaire et de Hautes Energies (LPNHE), FR-75005 Paris, France\n- 26 Departament de Física, Serra Húnter, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain\n- 27 NSF NOIRLab, 950 N. Cherry Ave., Tucson, AZ 85719, USA\n- 28 Institució Catalana de Recerca i Estudis Avançats, Passeig de Lluís Companys, 23, 08010 Barcelona, Spain\n- 29 Department of Physics and Astronomy, Siena College, 515 Loudon Road, Loudonville, NY 12211, USA\n- 30 Department of Physics & Astronomy, University of Wyoming, 1000 E. University, Dept. 3905, Laramie, WY 82071, USA\n- 31 National Astronomical Observatories, Chinese Academy of Sciences, A20 Datun Rd., Chaoyang District, Beijing, 100012, P.R. China\n- 32 IRFU, CEA, Université Paris-Saclay, F-91191 Gif-sur-Yvette, France\n- 33 Department of Physics and Astronomy, University of Waterloo, 200 University Ave W, Waterloo, ON N2L 3G1, Canada\n- 34 Perimeter Institute for Theoretical Physics, 31 Caroline St. North, Waterloo, ON N2L 2Y5, Canada\n- 35 Waterloo Centre for Astrophysics, University of Waterloo, 200 University Ave W, Waterloo, ON N2L 3G1, Canada\n- 36 Instituto de Astrofísica de Andalucía (CSIC), Glorieta de la Astronomía, s/n, E-18008 Granada, Spain\n- 37 Department of Physics and Astronomy, Sejong University, Seoul, 143-747, Korea\n- 38 CIEMAT, Avenida Complutense 40, E-28040 Madrid, Spain\n- 39 Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA\n- 40 University of Michigan, Ann Arbor, MI 48109, USA \nMNRAS 000 \n, 1-18 (2024)", 'ABSTRACT': "The Dark Energy Spectroscopic Instrument (DESI) Peculiar Velocity Survey aims to measure the peculiar velocities of early and late type galaxies within the DESI footprint using both the Fundamental Plane and Tully-Fisher relations. Direct measurements of peculiar velocities can significantly improve constraints on the growth rate of structure, reducing uncertainty by a factor of approximately 2.5 at redshift 0.1 compared to the DESI Bright Galaxy Survey's redshift space distortion measurements alone. We assess the quality of stellar velocity dispersion measurements from DESI spectroscopic data. These measurements, along with photometric data from the Legacy Survey, establish the Fundamental Plane relation and determine distances and peculiar velocities of early-type galaxies. During Survey Validation, we obtain spectra for 6698 unique early-type galaxies, up to a photometric redshift of 0.15. 64% of observed galaxies (4267) have relative velocity dispersion errors below 10%. This percentage increases to 75% if we restrict our sample to galaxies with spectroscopic redshifts below 0.1. We use the measured central velocity dispersion, along with photometry from the DESI Legacy Imaging Surveys, to fit the Fundamental Plane parameters using a 3D Gaussian maximum likelihood algorithm that accounts for measurement uncertainties and selection cuts. In addition, we conduct zero-point calibration using the absolute distance measurements to the Coma cluster, leading to a value of the Hubble constant, 𝐻 0 = 76 . 05 ± 0 . 35(statistical) ± 0 . 49(systematic FP) ± 4 . 86(statistical due to calibration) km s -1 Mpc -1 . This 𝐻 0 value is within 2 𝜎 of Planck Cosmic Microwave Background results and within 1 𝜎 , of other low redshift distance indicator-based measurements. \nKey words: galaxies: distances and redshifts - cosmology: observations - cosmology: cosmological parameters - cosmology: large-scale structure of Universe", '1 INTRODUCTION': "The Lambda Cold Dark Matter ( Λ CDM)modelstands as the prevailing cosmological framework to describe the Universe. It combines Einstein's cosmological constant, Λ , representing dark energy (Carroll 2001; Peebles & Ratra 2003), with nonbaryonic cold dark matter (CDM; Hut 1977; Lee & Weinberg 1977; Sato & Kobayashi 1977; Dicus et al. 1977; Vysotski ˇ i et al. 1977). Despite its consistency with a wide range of cosmological observations, the nature of its two main components, dark matter and dark energy, remains unknown to us (Peebles 2021). Moreover, tensions persist between measurements of the present-day expansion rate of the universe from the earlyuniverse (e.g., Planck Collaboration et al. 2020) and those derived from the late-universe (e.g., Riess et al. 2022). Additional tensions arise from measurements of the strength of clustering of matter in the universe between early- and late-universe observations (Hildebrandt et al. 2020). Furthermore, discrepancies emerge from measurements of the bulk flow, directly derived from late-universe observations, which do not align with the expected values from the Λ CDMmodel (Courtois et al. 2023b; Whitford et al. 2023; Watkins et al. 2023). In addition, current cosmological observations seem to favor a slightly higher growth index, 𝛾 than the value predicted by general relativity and the Λ CDM model, which might indicate weaker gravity or suppression of the growth of structure (Nguyen et al. 2023). \nAddressing these tensions requires substantial efforts in acquiring additional data to precisely understand the underlying discrepancies. Nevertheless, examining the tension using independent techniques should be our utmost priority, as it involves overcoming potential sources of systematics that may be linked to new physics (Lahav & Silk 2021). \nThe Dark Energy Spectroscopic Instrument (DESI; DESI Collaboration et al. 2022) is a ground-based spectroscopic survey that is targeting 40 million galaxies and quasars over a 14,000 square degrees footprint. It is over two years into its 5-year observing campaign. DESI's aims align very closely with the above suggestion \nof unveiling the nature of dark energy by conducting the most precise measurement of the universe expansion history using the baryon acoustic oscillation (BAO) (Levi et al. 2013). DESI will also measure the growth rate of cosmic structure using redshift space distortions (RSD) (DESI Collaboration et al. 2016a). \nThe DESI peculiar velocity survey serves as a secondary target program, designed to complement and enhance the primary goal of the DESI survey (Saulder et al. 2023). Peculiar velocities of galaxies can be measured through two primary methods: 'directly' using distance indicators, and 'indirectly' via velocity field reconstruction based on local density measurements. In this paper, we employ the direct method, specifically aiming to enrich the dataset by incorporating ∼ 180 , 000 directly measured distances using redshiftindependent distance indicators. These directly measured distances provide valuable independent information about galaxy velocities, helping to disentangle the effects of the smooth Hubble flow due to the cosmic expansion and the peculiar velocity due to gravitational interactions on galaxy motion. The peculiar velocity survey offers an independent and complementary approach to the main DESI surveys at higher redshift, allowing for a better understanding of large-scale structure through measurement of 𝑓 𝜎 8 below 𝑧 = 0 . 1 and the overall expansion of the universe through measurements of 𝐻 0 . \nHistorically, large peculiar velocity surveys, from the Two Micron All Sky Survey (2MTF: Masters et al. 2008) and the 6dF Galaxy Survey (6dFGS: Springob et al. 2014) have relied on either Tully-Fisher relation (Tully & Fisher 1977) or the Fundamental Plane relation (Djorgovski & Davis 1987; Dressler et al. 1987) to obtain distance measurements. The DESI peculiar velocity survey will be able to provide observations for these two distance indicators in an unprecedented and innovative way, thanks to the DESI's substantial increase in the survey speed due to the large field of view and the densely populated focal plane with 5000 robotic fibre positioners (DESI Collaboration et al. 2016b). \nThe Fundamental Plane relation utilizes two distance-independent observables of elliptical galaxies, the mean surface brightness ( 𝐼 𝑒 ) and the central velocity dispersion ( 𝜎 0 ) to infer the physical effective radius ( 𝑅 𝑒 ), which is distance dependent. Comparing the physical ra- \nular effective radius ( 𝜃 𝑒 ) allows for the measurement of distances. Similarly, the Tully-Fisher relation uses the rotational velocity of galaxies as a distance-independent observable to predict the absolute magnitude. Then, by comparing the predicted absolute magnitude with the observed apparent magnitude, one can derive distance estimates for these galaxies. These directly measured distances and peculiar velocities have been instrumental in constraining crucial cosmological parameters. \nThe 6dF Galaxy Survey (Jones et al. 2009) primary objective was to measure peculiar velocities for a sample of approximately 9,000 early-type galaxies up to redshift 𝑧 < 0 . 055 using the Fundamental Plane relation, aiming to constrain the growth rate of cosmic structure (6dFGSv: Magoulas et al. 2012). Qin et al. (2019) utilized the 6dFGSv data to estimate the Density-Momentum power spectrum and derived a value of 𝑓 𝜎 8 = 0 . 404 ± 0 . 082. Similarly, Adams & Blake (2020) employed the 6dFGSv data and applied the cross-covariance between galaxy redshift-space distortions and peculiar velocities to constrain the growth rate of structure, obtaining 𝑓 𝜎 8 = 0 . 384 ± 0 . 052(statistical) ± 0 . 061(systematic). Said et al. (2020) compared a sub-sample of the 6dFGSv Fundamental Plane galaxies along with a sample of SDSS galaxies to the velocity field reconstruction and reported a value of 𝑓 𝜎 8 = 0 . 338 ± 0 . 027. More recently, Turner et al. (2023) used the full 6dFGSv sample to measure galaxy-galaxy, galaxy-velocity, and velocity-velocity auto- and cross-correlation functions, finding a value of 𝑓 𝜎 8 = 0 . 358 ± 0 . 075. While all these measurements, using different methods, are consistent, they all favour a lower value than the predicted value from the Planck Λ CDMmodel (Planck Collaboration et al. 2020). \nMore recently, Cosmicflows-4 has undertaken the ambitious task of combining almost all previous peculiar velocity surveys into a comprehensive dataset (Tully et al. 2023). This compilation incorporates distances and peculiar velocities for 55,877 galaxies, utilizing eight different methodologies. The largest number of galaxies were derived from two new datasets, with approximately 35,000 galaxies measured using the Fundamental Plane relation (Howlett et al. 2022) and about 10,000 galaxies using the Tully-Fisher relation (Kourkchi et al. 2020). The integration of these diverse datasets has resulted in a Hubble constant measurement of 𝐻 0 = 74 . 6 ± 0 . 8 (statistical) ± 3 . 0 (systematic) (Tully et al. 2023). The Cosmicflows-4 catalogue has proven invaluable for various cosmological measurements, including assessing the impact of our local environment on 𝐻 0 (Giani et al. 2024), measuring the growth rate of structure (Boubel et al. 2024b), and developing an improved method for determining the Hubble constant using the Tully-Fisher relation (Boubel et al. 2024a). While the statistical errors from these studies are relatively small, thanks to the considerable increase in the sample size, there remain concerns about potential large systematic errors due to the combination of different methods and calibrators. \nThis reinforces the need for larger and more homogeneous surveys, like the DESI peculiar velocity survey, to further advance our understanding of cosmological parameters and address potential sources of systematic effects. \nThe DESI peculiar velocity survey is expected to yield a remarkable number of directly measured distances and peculiar velocities. Specifically, it is projected to obtain 186,000 such measurements, with 133,000 of them using the Fundamental Plane relation and 53,000 using the Tully-Fisher relation (Saulder et al. 2023). These numbers represent a significant increase in the scale, being about four times all previous peculiar velocity surveys combined (Tully et al. 2023). These vast dataset will be significant in constraining cosmological parameters, such as the Hubble Constant ( 𝐻 0 ) and the growth rate of cosmic structure ( 𝑓 𝜎 8 ). \nWith this extensive dataset, DESI will achieve a new level of precision in constraining cosmological parameters. The large number of directly measured distances and peculiar velocities will enable more robust and statistically significant results compared to all previous surveys. \nThis work, accompanied by a parallel work conducted by Douglass et al. in prep, introduces the initial outcomes of the DESI peculiar velocity survey using Survey Validation data. In Douglass et al. in prep, the emphasis is on the Tully-Fisher relation, while this work focuses only on the Fundamental Plane relation. \nDuring the Survey Validation (SV) phase, our approach involved conducting observations for a randomized subset of our designated targets. This selection addressed key questions: First, we evaluated the achievability of the Signal-to-Noise Ratio (SNR) needed for accurate velocity dispersion measurements across our magnitude and redshift range. Additionally, we examined potential sources of systematic errors in velocity dispersion measurements that might impact the precision of our distance and peculiar velocity determinations. Furthermore, we evaluated the efficiency of our photometric selection in accurately pinpointing genuine elliptical galaxies. \nWhile the primary objective of this paper is to offer insights into these fundamental questions, we also conducted the fitting of the Fundamental Plane relation. By doing so, we produced the initial catalogue of distances and peculiar velocities from DESI. Moreover, we present our preliminary findings concerning the measurement of the Hubble constant. \nThis paper is organized as follows: Section 2 provides a description of the Fundamental Plane relation. In Section 3, we offer an introduction to the dataset employed. Our sample selection process, encompassing photometric and spectroscopic aspects, is explained in Section 4. The derivation of Fundamental Plane parameters, inclusive of internal and external consistency checks, is presented in Section 5. We show the process of fitting the Fundamental Plane relation in Section 6. In Section 7, we discuss the zero-point calibration process and present absolute distance measurements. Section 8, unfolds our measurement of the Hubble constant. Summary and Conclusions are in Sections 9 and 10, respectively. \nUnless otherwise stated, in this paper we assume a flat Λ CDM cosmological model with Ω 𝑚 = 0 . 31 and 𝐻 0 = 100 ℎ kms -1 Mpc -1 . All magnitudes are on the AB magnitude system. All uses of 'log' should be taken to mean logarithms taken to the base 10.", '2 FUNDAMENTAL PLANE': "Distance indicators operate on the premise of linking distanceindependent parameters such as kinematics, with distance-dependent characteristics like luminosity or size. This relationship allows for the estimation of galaxy distances based on known distance-independent parameters. The first distance-indicator relation for elliptical galaxies was the luminosity-stellar velocity dispersion correlation, known as the Faber-Jackson relation (FJ: Faber & Jackson 1976). It is worth noting that the foundation of this relation was first suggested by Minkowski (1962), although they regarded it as inadequate probably due to the inclusion of flattened galaxies with high rotational velocities. The FJ relation follows the form 𝐿 ∝ 𝜎 4 . Simultaneously, Kormendy (1977) identified a correlation between surface brightness and size of elliptical galaxies. Both FJ and Kormendy relations did not seem very promising as distance indicators due to their large scatter. \nA significant advancement emerged approximately a decade later when it became evident that the FJ and Kormendy relations were \nspecial instances of a more general relation known as the Fundamental Plane (Djorgovski & Davis 1987; Dressler et al. 1987). These works show that a galaxy's effective radius ( 𝑅 𝑒 ), surface brightness ( 𝐼 𝑒 ), and stellar velocity dispersion are related through a power-law relationship, expressed as 𝑅 𝑒 ∝ 𝜎 𝑎 𝐼 𝑏 𝑒 . \nThe obvious explanation of the Fundamental Plane rested upon the virial equilibrium, linking a galaxy's mass ( 𝑀 ) to its velocity dispersion ( 𝜎 ) and effective radius ( 𝑅 𝑒 ) through the equation 𝑀 ∝ 𝜎 2 𝑅 𝑒 (Faber et al. 1987). However, the Fundamental Plane coefficients showed significant deviations from the predictions of the virial theorem (Hudson et al. 1997; Colless et al. 2001; Bernardi et al. 2003b; Magoulas et al. 2012; Said et al. 2020; D'Eugenio et al. 2021; Howlett et al. 2022). Varied contributors to this divergence were identified, including the fluctuation of the mass-to-light ratio (Faber et al. 1987), variations in the surface brightness profiles of early-type galaxies (Ciotti et al. 1996), and the proportion of dark matter within the kinematic observation region (Moster et al. 2010). Despite the deviations, the Fundamental Plane's applicability persisted across all early-type galaxies, maintaining a scatter of approximately 0.1 dex, which translates to an accuracy in distance estimation of around 23% (Lynden-Bell et al. 1988). \nWhile the current work concentrates solely on distance and peculiar velocity measurements, the forthcoming data from DESI holds the promise of delving into sources of the Fundamental Plane's tilt (its deviation from the virial theorem). This potential for detailed analysis arises from the vast and comprehensive datasets that DESI will provide.", '3 DATA': "Constructing the Fundamental Plane relation involves utilizing two datasets: photometry and spectroscopy. The photometric data for this work are drawn from the DESI Legacy Imaging Surveys (Dey et al. 2019). The spectroscopic data for our analysis are from the DESI Survey Validation data (DESI Collaboration et al. 2023a). \nThe DESI Legacy Imaging Surveys encompass the combination of three individual surveys: the Dark Energy Camera Legacy Survey (DECaLS), the Beijing-Arizona Sky Survey (BASS), and the Mayall z-band Legacy Survey (MzLS). This combination is designed to capture imagery across the expansive 14,000 deg 2 footprint ( 𝛿 > -20 · and | 𝑏 | > 15 · ) of the DESI survey in three optical bands ( 𝑔, 𝑟, and 𝑧 ). \nDECaLS,whichbegan observations in August 2014, uses the Dark Energy Camera (DECam; Flaugher et al. 2015) at the 4m Blanco telescope at the Cerro Tololo Inter-American Observatory. It covers approximately 9,350 deg 2 , including 3,580 deg 2 in the SGC and 5,770 deg 2 in the NGC, complementing the Dark Energy Survey's (The Dark Energy Survey Collaboration 2005) coverage of 1,130 deg 2 within the DESI footprint. \nBASS (Zou et al. 2017), which started in spring 2015, images the 𝛿 > + 32 · region of the DESI NGC footprint (approximately 5,100 deg 2 ) in the 𝑔 and 𝑟 optical bands. It utilizes the 90Prime camera (Williams et al. 2004) at the prime focus of the University of Arizona's Bok 2.3m telescope on Kitt Peak. \nMzLS complements BASS by imaging the same 𝛿 > + 32 · region of the NGC footprint in the z-band, covering approximately 5,100 deg 2 . \nIn addition to optical bands, the Legacy Surveys incorporate midinfrared photometry from the Wide-field Infrared Survey Explorer (WISE) satellite. WISE conducted an all-sky survey in four bands \ncentered at 3.4, 4.6, 12, and 22 𝜇 m(known as W1, W2, W3, and W4; Wright et al. 2010) during its mission. \nThese surveys collectively contribute to the creation of a photometric catalogue covering the three optical bands ( 𝑔, 𝑟, and 𝑧 ) and incorporating the four WISE channels. For the purposes of this paper, our analysis is based on Data Release 9 (DR9) of the Legacy Surveys. \nThe DESI Early Data Release (EDR) marks the initial public release of DESI spectroscopic data (DESI Collaboration et al. 2023b). It encompasses data from the Survey Validation (SV) phase, conducted between December 2020 and May 2021, prior to the start of the DESI Main Survey. A detailed description of the DESI pipeline can be found in Guy et al. (2023). For discussion of the survey operations, see Schlafly et al. (2023). The SV phase consisted of three stages: \n- (i) Target Selection Validation (SV1): This phase refined and validated the selection of targets for the Milky Way Survey (MWS), Bright Galaxy Survey (BGS), Luminous Red Galaxies (LRG), Emission Line Galaxies (ELG), and Quasar (QSO) samples. It employed looser target selection cuts and higher signal-to-noise ratios than the Main Survey to optimize selection criteria and survey requirements.\n- (ii) Operations Development (SV2): A brief phase for operational refinements. \n(iii) 1% Survey (SV3): This final SV stage further optimized observing procedures and produced high completeness samples over approximately 1% of the final DESI Main Survey area. \nThe EDR demonstrates DESI's capabilities, and its advanced instrumentation at the 4m Mayall telescope at Kitt Peak National Observatory. The instrument's wide-field prime focus corrector enables a field of view just over 8 deg 2 , allowing for efficient large-scale observations. At the heart of DESI's design are 5,020 roboticallycontrolled fiber positioners (DESI Collaboration et al. 2016b; Silber et al. 2023; Miller et al. 2023), each directing light from individual targets. These fibers feed into ten spectrographs, each equipped with three cameras covering distinct wavelength ranges: B (3600-5800 Å), R (5760-7620 Å), and Z (7520-9824 Å). The spectrographs provide a resolving power that increases from approximately 2000 at 3600 Å to 5500 at 9800 Å (DESI Collaboration et al. 2023b). \nIn addition to the primary target classes (MWS, BGS, LRG, ELG, and QSO), the EDR also includes observations from DESI's secondary programs, which serve as filler targets.", '4 SAMPLE SELECTION': 'As part of the DESI secondary target programs, we provided an initial set of targets to complement the main survey targets. As a result, our sample selection is made in two steps. The initial phase, which we refer to as the photometric selection, was exclusively reliant on the DESI Legacy Imaging Surveys DR9 (Dey et al. 2019), supplemented by photometric redshift catalogues from Zhou et al. (2021). Subsequently, the second step or the spectroscopic selection involves the integration of spectroscopic data from the DESI survey (DESI Collaboration et al. 2023b). Notably, most of our Fundamental Plane targets are also part of the DESI Bright Galaxy Survey (BGS: Ruiz-Macias et al. 2020; Hahn et al. 2023). \nAcomprehensive description of the photometric sample selection process can be found in Saulder et al. (2023). For full details of our photometric selection, we refer the reader to that paper. Here, we only provide a brief summary of the photometric selection criteria. Additionally, we will describe how we conducted the spectroscopic \nsample selection, which facilitated the construction of the peculiar velocity sample for our Fundamental Plane analysis.', '4.1 Photometric Selection': 'Our adopted selection criteria are outlined as follows: (1) An 𝑟 -band magnitude cut of 𝑟 < 18, which has been corrected for external galactic extinction, ensuring a robust signal-to-noise ratio that subsequently enhances the success rate for velocity dispersion measurements; (2) Three distinct colour cuts 1 of \n𝑔 -𝑟 > 0 . 68 (1) \n𝑔 -𝑟 > 1 . 3 ( 𝑟 -𝑧 ) -0 . 05 (2) \n𝑔 -𝑟 < 2 ( 𝑟 -𝑧 ) -0 . 15 . (3) \nThese colour cuts serve to eliminate galaxies located beneath the red sequence, dusty galaxies, and peculiar objects like galaxy mergers or galaxies displaying image artefacts, respectively; (3) a circularized radius, which is the half-light radius of the galaxy circularized using the axial ratio, condition of 𝑅 circ > 0; (4) An axial ratio requirement of 𝑏 / 𝑎 ≥ 0 . 3; (5) The application of either a de Vaucouleurs or Sérsic fit to the surface brightness profile, incorporating a Sérsic index of 𝑛 𝑠 > 2 . 5, measured collectively across 𝑟 , 𝑔 , and 𝑧 bands. \nThese specific criteria for ETGs selection are established based on a combination of past experience in ETG identification from previous surveys like 6dFGSv and SDSS peculiar velocity surveys (Saulder et al. 2013; Said et al. 2020; Howlett et al. 2022), along with testing involving visual identifications sourced from the Siena Galaxy Atlas (SGA; Moustakas et al. 2023) and GalaxyZoo (Lintott et al. 2011) \nOur photometric selection process, yielding a pool of over 400,000 galaxies deemed suitable for Fundamental Plane analysis. Since our selected sample is from the DESI Legacy Imaging Surveys DR9, it encompasses galaxies extending beyond the bounds of the DESI spectroscopic survey footprint. Refining our sample to adhere to the DESI spectroscopic survey footprint, spanning 14,000 deg 2 above a declination of -18 degree, yielded a final count of 373,533 galaxies eligible for Fundamental Plane analysis.', '4.2 Spectroscopic Selection': 'Our spectroscopic selection process depends on the data obtained from the DESI Survey Validation (SV; DESI Collaboration et al. 2023a). This validation survey unfolded in three phases: firstly, SV1, focused on target selection validation and the refinement of target selection algorithms (Myers et al. 2023); secondly, SV2, referred to as the Operation Development phase, serving as a practice run for the third stage; finally, SV3, also known as the 1% Survey, aimed to further validate both the survey operations procedures and the final target selection. The culmination of all three stages is encapsulated within the DESI Early Data Release (EDR; DESI Collaboration et al. 2023b), which has now been released and is publicly accessible. 2 \nFrom our photometrically selected Fundamental Plane sample, we identified the galaxies that were spectroscopically observed in the SV data. We then applied two key constraints to this subset: firstly, the warning bitmask ( ZWARN ) must be zero, indicating the absence of known issues with the data or the fit; secondly, the spectral classification ( SPECTYPE ) should be "GALAXY" (DESI Collaboration et al. \n2023b). The outcome of this cross-matching process yielded a total count of 6698 distinct Fundamental Plane galaxies. Figure 1 shows the distribution of these galaxies in a Mollweide projection in equatorial coordinates. The open circles in the figure represent several clusters within the DESI footprint, which are key to the Fundamental Plane analysis, especially for setting the zero-point calibration.', '4.3 Visual inspection': 'In order to identify galaxies that were unsuitable for inclusion in our Fundamental Plane analysis, a visual inspection of all galaxies was conducted using 1 × 1 arcmin colour cutouts sourced from the PanSTARRS1 (Chambers et al. 2016) and the DESI Legacy Imaging Surveys (Dey et al. 2019) images. This visual inspection was carried out by John R. Lucey. Using deeper Legacy Survey images, particularly model residual images, significantly improved our discrimination capability compared to previous surveys. Our methodology followed established procedures from prior FP studies (Campbell et al. 2014; Said et al. 2020; Howlett et al. 2022). \nDuring the visual inspection, the following categories of objects were identified: \n- (i) Galaxies that are not bulge-dominated, including those with prominent spiral arms.\n- (ii) Galaxies for which the measurements of the Fundamental Plane photometric parameters, specifically total magnitude and effective radius, were likely to be unreliable due to the presence of overlapping sources, whether stars or other galaxies. \n(iii) Galaxies with pronounced central asymmetries, including those with strong dust features, which are likely to bias the velocity dispersion measurements. \nThe results of our visual inspection process, are summarized in table 1. While our primary objective for the visual inspection process was to select a clean sample of bulge-dominated systems for our Fundamental Plane fit (the first step of the two-step process), it is important to note that these flagged galaxies were reintroduced into our dataset for the purpose of measuring peculiar velocities and distances.', '5 FUNDAMENTAL PLANE PARAMETERS': "With both the essential photometric and spectroscopic data in hand, we possess the necessary components for deriving the Fundamental Plane parameters. The formulation of the FP relation employed in this study is characterized by its structure: \nlog 𝑅 𝑒 = 𝑎 log 𝜎 0 + 𝑏 log 𝐼 𝑒 + 𝑐. (4) \nIn this equation, 𝑅 𝑒 stands for the effective radius, measured in (kpc h -1 ), which serves as the parameter that allows us to measure distance. On the other side of the equation, 𝜎 0 signifies the central velocity dispersion, expressed in (km s -1 ), while 𝐼 𝑒 represents the meansurfacebrightness within the angular effective radius, presented in (L ⊙ pc -2 ). Notably, 𝜎 0 and 𝐼 𝑒 are both distance-independent parameters 3 . The coefficients of the FP relation are represented by 𝑎 , 𝑏 , and 𝑐 . \nThe computation of the 𝑟 -band angular effective radius, 𝜃 𝑒 was \nFigure 1. The distribution of Fundamental Plane galaxies within the DESI SV dataset, presented here in a Mollweide projection in equatorial coordinates. Regions obscured by Galactic extinction in the Milky Way are shown as shaded Reds. The orange dots indicate the Fundamental Plane data within the SV dataset. The blue circles encompass all the DESI tiles, which collectively define the DESI survey footprint. Notably, the open black circles pinpoint a few clusters within the DESI footprint, which have previously measured distances (Bell et al. 2023), making them invaluable for the zero-point calibration in the future. For the Python code and data used to reproduce this plot, see this link. \n<!-- image --> \nderived from the 𝑟 -band half-light radius, 𝑟 , as well as the ellipticity components: 𝜖 1 and 𝜖 2 . This relationship is expressed through the following equations: \n𝜃 𝑒 = 𝑟 √︁ 𝑏 / 𝑎 (5) \n𝑏 / 𝑎 = 1 . 0 - | 𝜖 | 1 . 0 + | 𝜖 | (6) \n| 𝜖 | = √︃ 𝜖 2 1 + 𝜖 2 2 . (7) \nIn the above equations, 𝑟 , 𝜖 1 , and 𝜖 2 are all directly extracted from the DESI Legacy Imaging Surveys DR9 4 (Dey et al. 2019). \nWe converted the angular effective radius in arcseconds to the physical effective radius in units of kpc h -1 using the angular diameter distance (Weinberg 1972). This conversion was performed with respect to the observed redshift in the CMB frame, following the standard Λ CDMcosmological model: \nlog 𝑅 𝑒 = log ( 𝜃 𝑒 ) + log ( 𝑑 ( 𝑧 cmb )) -log ( 1 + 𝑧 helio ) + log GLYPH<18> 1000 𝜋 180 × 3600 GLYPH<19> . (8) \nWhile the comoving distance calculation employed the redshift in the CMB frame, the heliocentric redshift was used for the conversion from comoving to angular diameter distance as recommended by Davis et al. (2019). \nThe calculation of the second parameter in the Fundamental Plane relation, the effective surface brightness 𝐼 𝑒 in L ⊙ pc -2 , was based \non the model flux in the 𝑟 -band, 𝑓 𝑟 , and Galactic transmission in the same band MW 𝑟 . This, along with the above calculated angular effective radius 𝜃 𝑒 , was employed in the following manner: \nlog 𝐼 𝑒 = 0 . 4 ( 𝑀 𝑟 ⊙ -𝑚 𝑟 -0 . 85 𝑧 cmb + 𝑘 𝑟 ) -log ( 2 𝜋𝜃 2 𝑒 ) + 4 log ( 1 + 𝑧 helio ) + 2 log ( 206265 / 10 ) , (9) \nwhere, \n𝑚 𝑟 = 22 . 5 -2 . 5 log 𝑓 MW 𝑟 . (10) \nHere, 𝑀 𝑟 ⊙ = 4 . 65 signifies the 𝑟 -band absolute magnitude of the Sun (Willmer 2018), while 𝑚 𝑟 is the extinction-corrected magnitude. The term 0 . 85 𝑧 accounts for evolution correction (Bernardi et al. 2003a), 4 log ( 1 + 𝑧 helio ) represents surface brightness dimming correction, 𝑘 𝑟 approximates the 𝐾 -correction in the 𝑟 -band given by Chilingarian et al. (2010), and 𝑓 and MW 𝑟 are directly extracted from the Legacy Imaging Surveys DR9. \nCompleting the set of parameters for the Fundamental Plane, the third parameter is the distance-independent central velocity dispersion, 𝜎 0 . In our analysis, we employed the Penalized Pixel-Fitting (pPXF) software to measure velocity dispersion and its associated uncertainty from DESI spectra. As the name suggests, pPXF utilizes the maximum penalized likelihood method for deriving stellar kinematics from absorption-line spectra of galaxies. The development of this method was initiated by Cappellari & Emsellem (2004) and has been refined in subsequent works by Cappellari (2017, 2022). Our stellar templates were drawn from the Indo-U.S. Coudé Feed Spectral Library (Valdes et al. 2004), which encompasses 1273 stars, though typically only about 10-20 are selected by pPXF for precise fits. This spectral library spans the range from 3460 to 9464Å at a resolution of 1.35Å, corresponding to 𝜎 = 30 km s -1 . \nThe DESI spectrograph's resolving power ( 𝜆 / Δ 𝜆 ) varies as a function of wavelength: [2000, 3500] in the blue, [3300, 5000] in the red, and [3500, 5200] in the z band (see Fig. 33 in DESI Collaboration et al. 2022). This translates to resolutions for measureing velocity dispersion of 46, 31, and 29 km s -1 in the blue, red, and z arms respectively. In this paper, we use only the blue arm data. The stellar template library we are using here has a higher resolution than the data, which is necessary for accurate fitting. Additionally, the instrumental resolution in pPXF was handled by using the full resolution DESI data matrix, then output a 1D array with the RMS per pixel. \nDESI spectra are categorized into two main groups: full-depth and per-tile spectra (DESI Collaboration et al. 2023b). To derive the final central velocity dispersion, we primarily utilized the full-depth spectra. These spectra merge exposures for targets positioned on a given sky pixel and also aggregate data across tiles when the same target is observed on multiple tiles. On the other hand, per-tile spectra do not combine data from various tiles, even if the same target was observed on multiple tiles. Per-tile spectra find utility in subsequent subsections, where they are used for internal consistency checks. \nSubsequently, the measured velocity dispersion was transformed into central velocity dispersion using the formula introduced by Jorgensen et al. (1995): \n𝜎 0 𝜎 = GLYPH<18> 𝜃 𝑒 / 8 𝜃 ap GLYPH<19> -0 . 04 . (11) \nHere, 𝜃 𝑒 / 8 correspond to the standard aperture size (one-eighth of the optical effective radius), and 𝜃 ap = 0 . 75 represents the DESI fibre radius in arcseconds (DESI Collaboration et al. 2022). \nFollowing the computation of the three Fundamental Plane parameters, 𝑅 𝑒 , 𝐼 𝑒 , and 𝜎 0 , we also calculated the corresponding uncertainties, 𝜖 𝑟 , 𝜖 𝑖 , and 𝜖 𝑠 , respectively. These uncertainties are pivotal in accounting for the overall scatter observed in the Fundamental Plane relation. Among these sources of scatter, the intrinsic scatter of the relation itself holds the primary contribution. Nonetheless, the velocity dispersion error also plays a substantial role, ranking as the second most influential factor. If the velocity dispersion error exceeds 20%, it begins to dominate the scatter, surpassing the intrinsic component. \nFigure 2 illustrates the Signal-to-Noise Ratio (SNR) of the observed Fundamental Plane galaxies in the blue band against the relative error in velocity dispersion measurements derived from pPXF. In this figure, galaxies are colour-coded according to their redshift. The S/N ratio calculated is per-pixel, representing the median signal divided by the standard deviation of the residuals. Notably, the figure showcases that approximately 75% of the observed galaxies exhibit relative errors of less than 10%, rendering them suitable candidates for Fundamental Plane.", '5.1 Internal consistency': 'To perform an internal consistency assessment of our DESI Fundamental Plane sample, we will employ the entire set of 6698 galaxies, encompassing the full Fundamental Plane sample prior to implementing the redshift and velocity dispersion constraints. Within this subsection, we will systematically conduct an in-depth internal consistency examination of the Fundamental Plane parameters, both photometric and spectroscopic. \nIn our Fundamental Plane analysis, we utilized data from the DESI Legacy Imaging Surveys (LS; Dey et al. 2019) to derive all our photometric parameters. \nFigure 2. Observed Fundamental Plane galaxies SNR as a function of the relative error in the velocity dispersion measurements from pPXF. Galaxies are color-coded by their redshift. 75% of the observed galaxies have relative errors less than 10% which makes them suitable for FP. The fitted curve suggests that even with SNR as low as 4, one can still get a velocity dispersion measurement with less than 10% relative error. For the Python code and data used to reproduce this plot, see this link. \n<!-- image --> \nIt is important to note that due to the utilization of various telescopes, cameras, and filters combinations in the BASS and DECam LS, systematic variations in the zero-point calibration between these two photometric systems are observed (as detailed in Dey et al. 2019, section 7.2). \nTo investigate any systematic difference, we conducted a comparison of the 5 arcsec aperture magnitudes for galaxies in our FP sample that were observed in both the BASS and DECaLS surveys. \nAmong the 1660 galaxies shared between the two surveys, we observed a median BASS -DECaLSdifferenceof + 0 . 0234mag,with root mean square (RMS) deviation of 0 . 02 mag. In light of this, we incorporate this correction into our analysis by adjusting the northern 𝑟 -band magnitudes, by 0 . 0234 mag. This offset will be investigated further with year 1 data, which will include approximately 100,000 galaxies, providing a more robust calibration for our Fundamental Plane data \nFor the velocity dispersion measurements, we used data from the DESI spectroscopic survey early data release (DESI Collaboration et al. 2023b). In contrast to the employment of full-depth spectra for the final determination of central velocity dispersion, we utilize per-tile spectra for our spectroscopic data in this context. Per-tile spectra prove to be particularly valuable for the internal consistency assessment of our velocity dispersion measurements. These spectra combine observations across multiple exposures within the a single tile, but not across different tiles. This allows us to treat per-tile spectra as repeated observations of the same targets. By comparing velocity dispersion from the same target observed on different tiles, we can identify potential tile-to-tile offsets due to varying observing conditions. \nFor each of our Fundamental Plane galaxies, our initial step involved determining whether they had been observed on different tiles more than once. Subsequently, we categorized these observations pairs as primary or secondary based on the signal-to-noise, with observations having the higher SNR being considered the primary one. After this categorization, we proceeded to measure the velocity \ndispersion for all pairs, employing the same methodology applied to the full-depth spectra, as explained earlier. \nIn order to assess the consistency of the velocity dispersion measurements and explore the possibility of systematic offsets between observations or tiles within the DESI data, we employed the relative error between pairs of observations. This relative error encompasses both the measurements of the velocity dispersion itself and its associated error. The assessment was conducted using the following formula: \n𝜖 = 𝜎 𝑝 -𝜎 𝑠 ( 𝛿𝜎 2 𝑝 + 𝛿𝜎 2 𝑠 ) 1 2 . (12) \nHere, 𝜎 𝑝 , 𝜎 𝑠 , 𝛿𝜎 𝑝 , and 𝛿𝜎 𝑠 represent the velocity dispersion measurements from primary and secondary tiles, accompanied by their respective error estimates. In the context of consistent and unbiased velocity dispersion measurements, with accurately estimated errors, this evaluation should yield a Gaussian distribution with a mean of zero and a standard deviation of unity. \nIn Figure 3, we present an assessment of the velocity dispersion measurements conducted using pPXF and the Indo-US. stellar library. This evaluation is carried out on a subset of 4644 pairs from a total of 1420 unique galaxies in our DESI FP sample, utilizing per-tile DESI spectra. The left panel of the figure illustrates the one-to-one comparison between primary and secondary tiles. Reassuringly, the observations exhibit agreement within the uncertainties for the majority of tiles. \nFor enhanced clarity regarding any differences, we plot the difference between the measurements of each pair as a function of their mean in the middle panel. To provide a quantitative analysis of these differences, we construct a histogram in the right panel to visualize the distribution of pairwise relative errors in velocity dispersion measurements. The solid curve in this panel represents a Gaussian distribution with a mean of zero and a standard deviation of one. Notably, this figure affirms that the measurements are consistent and unbiased, reinforcing the reliability of our velocity dispersion measurements.', '5.2 External consistency': 'With a successful internal assessment of our velocity dispersion measurements across different tiles and observing conditions, we now embark on a more extensive evaluation by comparing our measurements to those obtained by other surveys and telescopes. Our objective is to ensure that our measurements align closely with those from well-vetted sources. To achieve this, we perform a comparison between our velocity dispersion measurements derived from the fulldepth spectra and those from the Sloan Digital Sky Survey (SDSS). \nIn this endeavor, we cross-match the entire FP sample of 6698 galaxies, prior to any spectroscopic cuts, with the SDSS Data Release 14 (Abolfathi et al. 2018). Our cross-match yields 4221 galaxies that are present in both the DESI FP sample and the SDSS survey. To assess the agreement, we employ the same methodology of relative error comparison that was used for the internal consistency check. \nHowever, it is important to note that SDSS provides two distinct velocity dispersion values: firstly, from the base table containing all spectroscopic information, known as veldisp (referred to as the pipeline value); secondly, from the emissionLinesPort catalogue, from the Portsmouth group, which utilizes the pPXF method and the MILES stellar library (Sánchez-Blázquez et al. 2006) for stellar kinematics measurements, including velocity dispersion, denoted as sigmastar in the SDSS dataset. While there are other SDSS catalogs \navailable for velocity dispersion, comparing two measurements from the same survey is sufficient for our purposes of external consistency checking. \nFigure 4 presents the outcomes of our comparison. In the top panel, wecompare our results to the SDSS pipeline measurements. Notably, there exists an overall agreement between the two measurements. However, a small yet significant offset of 0.17 is observed, along with a standard deviation of approximately 1.3. This offset is around 7 times the standard error in the mean, and the deviation from unity of the standard deviation is 32 times the uncertainty in the standard deviation, highlighting its high significance. \nGiven these findings, we extend our comparison to the SDSS pPXF measurements, as displayed in the middle panel. Interestingly, a comparable offset is observed, but with the opposite sign of -0.18. Importantly, the deviation from unity in the standard deviation is reduced to 1.17. \nAs part of a comprehensive approach, to assess the potential systematic effects introduced by this offset/tilt on our measured cosmological parameters, we will calibrate our DESI velocity dispersion measurements to the SDSS pPXF measurements using the fitted red line in Figure 4. We will then evaluate the systematic bias and incorporate it into the total error budget, as discussed in Section 8. \nAs part of a comprehensive approach to assess potential systematic effects, we calibrated our DESI velocity dispersion measurements against the SDSS pPXF measurements using the fitted red line in Figure 4. This calibration was done only to evaluate the impact of any systematic offset or tilt if incorrect velocity dispersions were used. Our fiducial cosmological analysis relies on the DESI measurements. The calibration with SDSS was included solely to identify and account for this potential source of systematic bias, which is incorporated into the total error budget, as discussed in Section 8.', '6 FUNDAMENTAL PLANE FITS': 'To derive distances and peculiar velocities, we employed the Maximum Likelihood method to fit the Fundamental Plane using a 3D Gaussian model. This approach, initially formulated by Saglia et al. (2001) and Colless et al. (2001), has since been refined and adapted by subsequent studies such as Magoulas et al. (2012); Springob et al. (2014); Said et al. (2020); Howlett et al. (2022). Here, we provide a brief summary of the method, but for a comprehensive review, see the aforementioned studies. \nIn this study, we employ the conventional two-step approach. First, we perform a Fundamental Plane fitting without considering peculiar velocities. Then, the deviation from the optimal fit of the Fundamental Plane is attributed to the peculiar velocities of individual galaxies. \nWeinitiate by establishing the core parameters of the Fundamental Plane as 𝑟 = log 𝑅 𝑒 , 𝑠 = log 𝜎 0 , and 𝑖 = log 𝐼 𝑒 . Following the approach proposed by Colless et al. (2001), we describe the threedimensional probability distribution in the ( 𝑟, 𝑠, 𝑖 ) space as follows: \n𝑃 ( 𝑥 𝑛 ) = exp GLYPH<2> -0 . 5 x 𝑇 𝑛 ( V + E 𝑛 ) -1 x 𝑛 GLYPH<3> ( 2 𝜋 ) 3 / 2 | V + E 𝑛 | 1 / 2 𝑓 𝑛 . (13) \nHere, x 𝑛 = ( 𝑟 -¯ 𝑟, 𝑠 -¯ 𝑠, 𝑖 -¯ 𝚤 ) signifies the position of galaxy 𝑛 within the Fundamental Plane domain. The matrix V encapsulates the intrinsic scatter of the Fundamental Plane relation, while E 𝑛 accounts for the measurement uncertainties associated with the Fundamental Plane parameters (see equation 14 by Said et al. (2020) for the full description of the error matrix). The normalization factor 𝑓 𝑛 ensures the distribution integrates to unity, accounting for selection criteria. \nFigure 5 illustrates the forward projection of the Fundamental Plane relation, accompanied by the best fit of the Fundamental Plane parameters for the DESI 𝑟 -band sample as derived through this procedure. The Fundamental Plane parameters fitted under different conditions are listed in Table 2. \n<!-- image --> \n<!-- image --> \nFigure 3. Pairwise comparison of velocity dispersion measured using pPXF in this work between repeat observations for a subset of DESI galaxies selected for Fundamental Plane relation. Left Panel: one to one comparison between primary and secondary observations where primary is defined as the observations with the highest SNR. The red line shows the linear fit result. Middle Panel: the difference between the two measurements as a function of the mean of the two measurements. Right Panel: distribution of pairwise relative errors in velocity dispersion measurements. The distribution is in a good agreement with a Gaussian with mean of zero and standard deviation of unity shown as a solid curve. For the Python code and data used to reproduce this plot, see this link. \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nDESI \n<!-- image --> \n2 \nFigure 4. Velocity dispersion measured with pPXF in this work for Fundamental Plane galaxies in DESI SV data in comparison with SDSS measurements. The top panel shows a comparison with the SDSS pipeline velocity dispersion 𝜎 pipeline SDSS . The bottom panel present a comparison with the Portsmouth group (emissionLinesPort) velocity dispersion using pPXF 𝜎 pPXF SDSS . The agreement between DESI velocity dispersion and SDSS in general is better than the agreement between SDSS pipeline velocity dispersion measurements and the SDSS pPXF ones. For the Python code and data used to reproduce this plot, see this link. \n<!-- image --> \nSubsequently, the likelihood can be expressed as: \nL = 𝑁 𝑔 GLYPH<214> 𝑛 = 1 𝑃 ( 𝑥 𝑛 ) 1 / 𝑆 𝑛 . (14) \nIn this context, 𝑆 𝑛 signifies the 1 / 𝑉 max weighting factor, which accommodates for galaxies that might be absent due to the selection function (Said et al. 2020). The objective is to determine the optimal parameters ( 𝑎 , 𝑏 , ¯ 𝑟 , ¯ 𝑠 , ¯ 𝚤 , V ) of the Fundamental Plane that best describe the data, with V encompassing the scatter intrinsic to each orthogonal direction, namely 𝜎 1 , 𝜎 2 , and 𝜎 3 . \nIn our fitting procedure, the parameters 𝑓 𝑛 and 𝑆 𝑛 accommodate for three specific selection criteria: (1) a range of lower and upper \nlimits on the 𝑟 -band magnitude, set at 10 ≤ 𝑚 𝑟 ≤ 18 . ; (2) lower and upper boundaries on redshift, restricted to 0 . 003 ≤ 𝑧 ≤ 0 . 1; (3) velocity dispersion, constrained by the instrumental resolution of the DESI spectrographs, adhering to 50 ≤ 𝜎 ≤ 420 km s -1 . Table 1 shows the number of remaining galaxies after each successive selection criterion. \nWith the complete set of Fundamental Plane parameters at our hands, we are positioned to compare them with findings from previous works. Notably, as the Fundamental Plane parameters are con- \nN \nTable 1. Summary of the DESI FP peculiar velocity Selection criteria \nFigure 5. The projected Fundamental Plane of DESI SV data. The data shows the measured effective radii against the predicted radii from the 3D Gaussian fit of the Fundamental Plane for DESI SV data. The solid black line shows the one-to-to line. For the Python code and data used to reproduce this plot, see this link. \n<!-- image --> \ntingent upon wavelength, a straightforward method for comparison and assessment against earlier studies involves the computation of the root mean square (RMS) scatter of the Fundamental Plane in the 𝑟 direction. This measure should offer valuable insights into the actual distance error. However, it is worth emphasizing that the true distance error encompasses supplementary elements, including the correction for the selection function and the underlying distribution of galaxies within the Fundamental Plane, and the full covariance between parameters such as 𝐼 𝑒 and 𝜃 𝑒 . While we present a simplified estimate here, our full analysis incorporates these complex relationships through the use of complete covariance matrices. \nThe comprehensive RMS scatter in the 𝑟 direction can be quantified through the expression: \n𝜎 𝑟 = h ( 𝑎𝜖 𝑠 ) 2 + 𝜖 2 phot + 𝜎 2 𝑟, int i 1 / 2 . (15) \nIn this equation 𝜖 𝑠 denotes the mean error in log 𝜎 , while 𝜖 phot is the total photometric error arising from both 𝜖 𝑟 and 𝜖 𝑖 , determined as 𝜖 phot = GLYPH<2> 𝜖 2 𝑟 + 𝑏𝜖 2 𝑖 GLYPH<3> 1 / 2 . Additionally, 𝜎 𝑟, int signifies the intrinsic scatter within the Fundamental Plane itself. Upon substituting these values into equation 15, the total RMS scatter in the 𝑟 direction is established as 23%. This outcome marks a notable improvement compared to the 6dFGSv reported value of 31% (Magoulas et al. 2012). It is worth noting that while this value is similar to the total scatter evident in the SDSS Fundamental Plane (Said et al. 2020), the DESI number of Fundamental Plane galaxies is projected to be at least five-fold greater than the SDSS Fundamental Plane sample, which itself stands as the most expansive peculiar velocity survey undertaken thus far. \nThat was essentially the initial step in the traditional two-step Fundamental Plane approach. Up to this point, we have worked under \nthe assumption of negligible peculiar velocities. However, the next phase involves calculating these peculiar velocities by gauging the deviations of the data from the best-fit Fundamental Plane parameters. By comparing the physical effective radius, 𝑟 , derived from equation 8, with the true effective radius, 𝑟 𝑡 , inferred from the bestfit Fundamental Plane parameters, we can derive the log-distance ratio as: \n𝑟 -𝑟 𝑡 = 𝜂. (16) \nThe log-distance ratio is the main derived value in our dataset.', '7 ZERO-POINT CALIBRATION AND ABSOLUTE DISTANCES': 'In the Fundamental Plane equation 4, the coefficient 𝑐 sets the zeropoint of the relation. The determination of this zero-point has a direct impact on the calculation of the actual effective radius of galaxies, subsequently influencing the derived distances and peculiar velocities. Throughout the process of fitting the Fundamental Plane, we have assumed that the net radial peculiar velocity of all galaxies is zero. This assumption implies the absence of a monopole term in the velocity field. While this assumption may hold for a homogeneous allsky galaxy sample, it requires special consideration for hemispheric surveys such as 6dFGSv, SDSS, and DESI surveys, essentially all ground-based galaxy surveys. \nVarious surveys have adopted different approaches to establish the zero-point. For instance, Springob et al. (2014) utilized a sub-sample of the 6dFGSv near the celestial equator, defining a great circle sample to re-fit the Fundamental Plane and adjust the zero-point for the entire sample. This effectively treats the sample as a fullsphere, being degenerate only in the monopole term. Unfortunately, this approach is not feasible for the DESI early data release due to the sky coverage limitations. However, it holds promise for the full DESI data release. \nAnother example can be found in the SDSS peculiar velocity survey by Howlett et al. (2022). In this study, they performed a crossmatch between the SDSS sample and the Cosmicflows-III catalogue (CF3; Tully et al. 2016), which contains distance measurements from alternative methods. \nWe adopted an approach similar to SDSS\'s method by crossmatching our DESI FP sample with the SDSS peculiar velocity catalogue. Specifically, we utilized their calibrated log distance ratios, which were originally calibrated using the CF3 data sets. This dataset itself was calibrated through a distance ladder approach encompassing various standard candles, such as Cepheid variables, Tip of the Red Giant Branch, and Type Ia supernovae. We identified 896 galaxies common to both samples, providing a substantial number of galaxies for zero-point calibration. \nFigure 6 shows a comparison of the log distance ratios for these common objects between SDSS and DESI peculiar velocity catalogue prior to zero-point calibration. \nAone-to-one line (black) and a Hyper Fit line (red) are overlaid to show the difference between 𝜂 DESI and 𝜂 SDSS \nHowever, this approach is not the optimal choice if the desired scientific goal is to measure the present-day expansion rate 𝐻 0 , as this value is already set to 75 ± 2 km s -1 Mpc -1 for the CF3 catalogue. To address this, we employed an alternative method to calibrate the zero-point (which is degenerate with the Hubble constant 𝐻 0 ) using measured distances to clusters in our sample, leveraging primary distance indicators and offering absolute distance measurements. This \nTable 2. Fundamental Plane parameters for the DESI SV sample under different conditions: Fiducial (original), including non-elliptical galaxies in the fitting procedure, and calibrated DESI velocity dispersion measurements to the SDSS measurements from the Portsmouth group using pPXF. SDSS and 6dFGSv FP parameters are also included for comparison. \nFigure 6. Comparison between the measured log-distance ratio for DESI SV galaxies and their counterpart in the SDSS PV. The solid line shows the oneto-one line. The red line shows the Hyper Fit result (y = 1.01x - 0.01). Mean uncertainties for 𝜂 DESI (x-axis) and 𝜂 SDSS (y-axis) are displayed in the upper left corner. For the Python code and data used to reproduce this plot, see this link. \n<!-- image --> \napproach offers two key advantages: First, it grants us greater control over the selection of primary distance indicators, enabling us to choose and exclude indicators as needed; second, by averaging over multiple distances within the same cluster, we can mitigate the overall scatter in the calibration process, in contrast to using individual galaxy distances. \nAlthough there is not a group/cluster catalogue available for the DESIearly data release yet, we know that this release has already covered the Coma cluster to a greater depth than any survey before. The Coma cluster, being massive, relatively nearby, and extensively studied, serves as an ideal candidate for setting the Fundamental Plane zero-point. To make use of additional galaxies in the Coma cluster that had not previously been observed before DESI, we adapted a modified friends-of-friends (FoF) algorithm based on the methodology introduced by Press & Davis (1982). We combined redshift data from the DESI early data release and supplemented it with SDSS redshifts for galaxies not observed by DESI. Our initial dataset encompassed all DESI and SDSS redshifts within a generous region around the Coma cluster, spanning a 10 degree radius centered on the Coma cluster RA=12h59\'48.7", Dec=27 ° 58\'50" and redshift range of 0.02 relative to the mean redshift of the Coma cluster ( 𝑧 = 0 . 0231, Abell et al. 1989). We applied a modified FoF algorithm, implementing separate linking lengths for the angular and radial directions, as outlined in Eke et al. (2004); Duarte & Mamon (2014, 2015), \nand optimized these lengths using a cost function (Robotham et al. 2011) along with SDSS-like mock catalogues. Specifically, we employed linking lengths of 600 km s -1 in the radial direction and 0.3 ° in the angular direction. This yielded a catalogue of 1731 galaxies potentially belonging to the Coma cluster. \nSubsequently, we employed the method of Jaffé et al. (2015) to remove galaxies that could not be kinematically bound to the Coma cluster, and we excluded galaxies located beyond the cluster\'s turnaround radius, following Korkidis et al. (2020). This process yielded a final catalogue of 1696 identified Coma cluster members. To validate our method and the chosen linking lengths, we applied the same process to DESI-only and SDSS-only datasets. The results were consistent with each other, differing by only a few galaxies. A more detailed description of this method and the associated code for cluster member identification will be provided in Saulder et al., (in preparation). \nCross-matching our Fundamental Plane sample with this newly created Coma catalogue, we identified 226 galaxies in common. We then calculated the uncalibrated distance of the Coma cluster using the weighted mean of distances for these 226 galaxies. To perform zero-point calibration and obtain absolute distances for our full sample, we compared this calculated Coma distance to the most recent absolute distance measurement of the massive NGC 4874 galaxy at the core of the Coma cluster, determined using the Surface Brightness Fluctuation method (SBF; Jensen et al. 2021), which yielded 99 . 1 ± 5 . 8 Mpc. \nWe applied this correction between the uncalibrated distance of Coma obtained from the Fundamental Plane and the measurement of Coma\'s distance from the surface brightness fluctuation method to obtain absolute distances for all galaxies in our sample. \nAs with the previous methods, this approach also comes with its own set of limitations. Firstly, due to the absence of a dedicated full group catalogue for DESI data at present, we are confined to utilizing the Coma cluster alone. While this provides a valuable calibration point, it could potentially introduce bias compared to using multiple clusters and groups that span a broader region of the sky. However, it\'s worth noting that this limitation will be significantly alleviated in future DESI data releases and its associated group catalogue, which will encompass numerous clusters and groups with known distances, facilitating a more comprehensive zero-point calibration process. In Figure 1, we display a selection of these clusters and groups, each possessing known absolute distances, that are well-suited for the calibration procedure. \nThe second limitation arises from the observation by Howlett et al. (2022) of a correlation between group richness and one of the Fundamental Plane parameters, specifically the mean surface brightness \n( 𝐼 𝑒 ). Consequently, relying solely on the Coma cluster for zeropoint calibration might introduce bias into our results. However, this concern is mitigated by the potential approach of employing multiple Fundamental Plane fits as a function of group/cluster richness, as suggested by Howlett et al. (2022). In general, while the zeropoint calibration can be executed through various methodologies, the choice is contingent on the data available for analysis. \nIn summary, the process of establishing the zero-point for the Fundamental Plane involves several considerations, each with its own merits and limitations. The methods applied depend on the available data and the specific scientific objectives of the study. With forthcoming DESI data, these challenges will be further addressed and the calibration process can be refined, resulting in even more accurate distance measurements and peculiar velocities for a much larger sample of galaxies.', '8 MEASURING THE HUBBLE CONSTANT': "The recession velocity-distance relation, often referred to as the Hubble-Lemaître law, establishes the connection between a galaxy's recession velocity and its distance. It is expressed as: \n𝑣 = 𝐻 0 𝐷. (17) \nIn this equation, the constant of proportionality, denoted as 𝐻 0 , represents the present rate of expansion of the universe, Hubble's constant. Our method of measuring the Hubble constant here involves the construction of a Hubble diagram, which necessitates a dataset presenting the distance modulus versus redshift. \nHowever, our approach to creating the Hubble diagram and performing the fitting to estimate the Hubble constant differs from the traditional Hubble diagram. Instead of using apparent magnitude and absolute magnitude as the observable and measured quantities, we employ a different set of variables in the Fundamental Plane analysis. Specifically, our observable quantity is the angular effective radius, while the measured quantity is the true physical radius. Consequently, rather than plotting the distance modulus on the y-axis of the Hubble diagram, our y-axis represents the difference between the logarithm of the angular effective radius and the logarithm of the true physical radius. It is mathematically expressed as: \n𝜇 𝐴 = 𝑟 𝑡 -𝑟 𝜃 (18) \n= 𝑟 -𝜂 -𝑟 𝜃 -log ( 1000 𝜋 180 × 3600 ) (19) \n= log 𝑑 𝐴 (20) \nwhere 𝑑 𝐴 is the angular diameter distance parameterized as follows: \n𝑑 𝐴 = 𝑑 𝐿 ( 1 + 𝑧 ) 2 (21) \n= 𝑐 ( 1 + 𝑧 ) ∫ 𝑧 0 𝑑𝑧 ' 𝐻 ( 𝑧 ' ) (22) \nwhere 𝑑 𝐿 is the luminosity distance. We then express the angular distance modulus as a power series of the form 5 : \n𝜇 model 𝐴 = log 𝑐𝑧 GLYPH<18> 1 + 1 2 [ 1 -𝑞 0 ] 𝑧 -1 6 [ 1 -𝑞 0 -3 𝑞 2 0 + 𝑗 0 ] 𝑧 2 GLYPH<19> -log 𝐻 0 -2 log ( 1 + 𝑧 ) . (23) \nOne can then use this formulation to fit for the Hubble constant 𝐻 0 , the deceleration parameter 𝑞 0 , and the jerk parameter 𝑗 0 by measuring the angular distance modulus as a function of redshift up to terms of order 𝑧 3 . \nDESI FP Hubble diagram comprising 4191 Fundamental Plane galaxies within the redshift range of 0.01 to 0.1 is shown in the upper panel of Figure 7. \nDue to the limited redshift range covered by our Fundamental Plane data ( 𝑧 < 0 . 1), it is not feasible to perform a concurrent fit for all three cosmological parameters in equation 23. Instead, we adopt 𝑞 0 = -0 . 55 and 𝑗 0 = 1, in line with the expectations for a flat Λ CDM cosmology with Ω 𝑚 = 0 . 3 and Ω Λ = 0 . 7. We solely perform a fit for 𝐻 0 under these conditions. \nAdditionally, following the approach by Riess et al. (2022), we implement an additional redshift cut of 𝑧 > 0 . 023, to limit the effect of peculiar velocities, which results in a sample of 4063 galaxies. After applying these criteria, the derived value for the Hubble constant is 𝐻 0 = 76 . 05 ± 0 . 35 km s -1 Mpc -1 . This uncertainty exclusively encompasses statistical uncertainties. \nFigure 7 illustrates our best-fit model for 𝐻 0 using a flat Λ CDM cosmology with Ω 𝑚 = 0 . 3 and Ω Λ = 0 . 7 (black line), alongside a curve representing our best-fit 𝐻 0 using Planck values for Ω 𝑚 = 0 . 315 and Ω Λ = 0 . 685 (red dashed line). The negligible difference between these curves (0 . 06% in 𝐻 0 ) demonstrates the insensitivity of our 𝐻 0 estimate to reasonable variations in Ω 𝑚 and Ω Λ at these low redshifts. \nWe also plot the curve corresponding to Planck Collaboration et al. (2020) values for 𝐻 0 , Ω 𝑚 and Ω Λ (blue line), which shows an offset from our best-fit model. Quantitatively, an average change of 0.0522 in 𝜇 𝐴 over our fitting range ( 𝑧 > 0 . 023) would be required to reconcile our measurements with the Planck 𝐻 0 value of 67.4 km s -1 Mpc -1 \nThe lower panel of Fig. 7 illustrates the residuals in comparison to the best fit results for a flat Λ CDMcosmology, as detailed in equation 23. The observed larger 𝜇 𝐴 at 𝑧 < 0 . 023 can be attributed to the effect of LSS. This trend can be well explained by a single attractor model, which shows infall toward the attractor's centre and backside infall. This trend is also evident in velocity field reconstruction models. We have included three pv reconstructions (2M++ from Said et al. (2020), 2MRS from Lilow & Nusser (2021) and 2M++ from Carrick et al. (2015)), which all show the same signature as the DESI FP data. The smoother curves in these models results from their use of linear theory for reconstruction and smoothing to a scale of approximately 4 Mpc. \nTo address potential systematic errors, we conducted an analysis, revisiting our calculations while considering several sources of systematic bias. Figure 8 presents the posterior distributions of the Hubble constant obtained from the DESI Fundamental Plane data after applying various sources of systematics to it. In Figure 8, the black probability density distribution represents our fiducial measurement. Although other potential sources of systematics may exist, the following are the most evident and notable. \n- (i) Inclusion of Spiral Galaxies: Despite implementing selection criteria to isolate pure elliptical galaxies during our sample selection process, a number of non-ellipticals remained in our dataset. We endeavored to identify and exclude them via visual inspection, a task undertaken by JRL. This inspection identified 1081 galaxies out of 4191 as non-ellipticals, representing approximately 26% of our sam- \nl \ne \nd \no \nm \nFigure 7. The DESI Hubble diagram features 4191 Fundamental Plane galaxies with redshifts falling within the range of 0.01 to 0.1. In the upper panel, the plot displays the log of the angular diameter distances, denoted as 𝜇 𝐴 , as a function of redshift, 𝑧 . The solid black curve corresponds to the best-fit 𝐻 0 for a Λ CDM cosmology with Ω 𝑚 = 0 . 3 and Ω Λ = 0 . 7, as determined by employing equation 23. The highlighted regions indicate the redshift ranges used for zero-point calibration and for fitting cosmological parameters. The red dashed line shows our best-fit 𝐻 0 using Planck values for Ω 𝑚 and Ω Λ . The blue curve represent Planck's values for 𝐻 0, Ω 𝑚 and Ω Λ . In the lower panel, the plot reveals the residuals relative to the best-fit model, calculated as 𝜇 𝐴 -𝜇 model 𝐴 . Additionally, in the bottom panel, a black line represents the average trend of the residuals. The observed larger 𝜇 𝐴 at redshift 𝑧 < 0 . 023 is due to LSS and can be explained by a single attractor model showing infall and backside infall toward the attractor's centre. The plotted peculiar velocity field reconstructions 2M++ from Said et al. (2020) and Carrick et al. (2015) and 2MRS from Lilow & Nusser (2021) also show this trend. For the Python code and data used to reproduce this plot, see this link. \n<!-- image --> \nple. The inclusion of these galaxies could potentially influence the Fundamental Plane parameters, as illustrated in Table 2. Rather than outright removal, we opted to adopt the approach outlined by Howlett et al. (2022), wherein we excluded them from the Fundamental Plane fit but retained these galaxies during peculiar velocity measurements. This strategy ensures that these galaxies do not introduce bias into our Fundamental Plane fit but still allows us to calculate their peculiar velocities. As a systematic test, we conducted the Fundamental Plane fit with these galaxies included in our sample. This test assessed the impact of not applying visual inspection procedures, revealing Hubble constant value of 𝐻 0 = 76 . 24 ± 0 . 39 km s -1 Mpc -1 (represented by the red probability density) which is a slightly higher value than our fiducial value. \nContinuing our exploration, we further examined the impact of \nspirals by conducting an alternative test where all identified spiral galaxies were removed from the entire process, spanning from the Fundamental Plane fit to the zero-point calibration and subsequent cosmology fitting. The resulting probability density for this test is shown in blue, revealing a shift in the mean value of 𝐻 0 of about 1 . 6 km s -1 Mpc -1 . This significant alteration is attributed to the exclusion of these galaxies, introducing a selection bias during the Hubble constant fitting process. It suggests that visual inspection tends to eliminate more nearby and brighter galaxies while sparing those at higher redshifts or fainter. Despite this noticeable shift, we refrain from incorporating this posterior into our systematic error budget. Nevertheless, presenting it here serves as a point of interest for further investigation, particularly with the anticipated DESI year \nFigure 8. The posterior distribution for the Hubble constant 𝐻 0 derived from our fiducial DESI Fundamental Plane analysis represented by the black probability density. Additional probability density distributions, displayed in various colours, illustrate the impact of different sources of systematics on the 𝐻 0 measurement process. For the Python code and data used to reproduce this plot, see this link. \n<!-- image --> \n1 data, which promises a substantially larger sample for a more indepth analysis. \n(ii) Correcting DESI 𝜎 to SDSS values: A key component of the Fundamental Plane parameters is the stellar velocity dispersion 𝜎 . An important consideration for DESI data is whether the signal-to-noise ratio is sufficient to obtain accurate measurements of velocity dispersion. To address this concern, we perform a comprehensive internal and external consistency check analysis in this paper (refer to Figs. 3 and 4). When we compare DESI velocity dispersion measurements to those of SDSS, we observe a slight difference, as shown in Figure 4. To assess this potential source of systematic bias, we employ linear fitting to rectify all velocity dispersion measurements across our dataset, encompassing not only the overlapping DESI and SDSS measurements but all DESI data. We then repeat the entire process of fitting the Fundamental Plane, perform re-calibration, and re-fit the Hubble diagram using these corrected velocity dispersion values. This results in a Hubble constant measurement of 𝐻 0 = 75 . 97 ± 0 . 34 km s -1 Mpc -1 , which we identify as the least influential source of systematic bias among the five we have identified. The posterior distribution for this measurement is illustrated in Figure 8 as the green probability density. \n(iii) Higher Redshift Cut ( 𝑧 > 0 . 034): In the process of constructing the Hubble diagram and fitting for the Hubble constant, we adopted the practice recommended by Riess et al. (2022) of implementing a redshift cut to exclude low-redshift galaxies below redshift, 𝑧 < 0 . 023 to mitigate the impact of peculiar velocities. In this systematic analysis, we examine the consequences of applying a more stringent redshift cut of 𝑧 < 0 . 034. This specific redshift limit was chosen to exclude any data used in the Zero-point calibration process. The resulting probability density for this cut is represented in purple in Figure 8. Applying this stricter redshift cut yields a Hubble constant measurement of 𝐻 0 = 76 . 54 ± 0 . 36 km s -1 Mpc -1 , which is slightly higher than our fiducial value. \n(iv) No Redshift Cut Applied: In this systematic assessment, we conducted the Hubble constant measurement without implementing any redshift cut, including all the Fundamental Plane data, even the low-redshift galaxies. The resulting probability density is shown in cyan, and it provides a Hubble constant value of 𝐻 0 = 75 . 58 ± 0 . 32 km s -1 Mpc -1 . This value is slightly lower than our fiducial Hubble constant measurement. \n(v) Applying the peculiar velocity correction: Throughout our \ncosmological fitting, we utilized the redshift in the CMB frame, 𝑧 cmb . However, peculiar velocities can introduce systematic effects. To assess this, we corrected the redshift from the CMB frame, which accounts for our own motion, to the cosmological redshift, incorporating peculiar velocities. This correction utilized the default option of the pvhub 6 velocity field maps developed by Carr et al. (2022). Implementing this correction resulted in a slightly higher value for the Hubble constant: 𝐻 0 = 76 . 36 ± 0 . 33 km s -1 Mpc -1 . \n(vi) Fundamental Plane Analysis with TRGB Calibration: In our primary cosmology fitting, we utilized the Fundamental Plane data calibrated with the absolute distance to the Coma cluster, as measured using the Surface Brightness Fluctuation method. We introduced an alternative distance calibration using the Tip of the Red Giant Branch (TRGB) method. Using the TRGB method to measure the distance to Leo I group, and based on the relative distance between Leo I and Coma cluster, Sakai et al. (1997) derived a distance modulus of 𝜇 = 35 . 03 ± 0 . 37 for the Coma cluster. Comparing this absolute distance obtained via the TRGB calibration to our Fundamental Plane distances, we constructed a new catalogue of absolute distances for our sample. Subsequently, we created a Hubble diagram and remeasured the Hubble constant, yielding a value of 𝐻 0 = 74 . 33 ± 0 . 31 km s -1 Mpc -1 . This systematic shift represents the most substantial bias among all the potential sources of systematics we have identified. The probability density for this specific analysis is visualized in grey dotted line. \nConcluding our investigation into potential systematic biases, we proceeded to evaluate the systematic error linked to our Hubble constant value. This analysis considered all MCMC chains, excluding the chain that involved the exclusion of non-ellipticals through visual inspection and the one utilizing the TRGB distance calibration to the Comacluster. The exclusion of the TRGB-calibrated chain is justified by its representation of a systematic bias intrinsic to the calibration method, rather than an issue within the Fundamental Plane fitting process. Moreover, the statistical error arising from the zero-point calibration is anticipated to address this aspect as well. \nTo assess statistical uncertainties associated with the zero-point calibration process, we generated 1000 distances to the Coma cluster based on the measured distance and its associated uncertainties \nderived from surface brightness fluctuation (Jensen et al. 2021). For each of these 1000 distances, we repeated the zero-point calibration process, generating a new catalogue used to fit the Hubble constant. The individual probability densities are depicted in the top panel of Figure 9 as dotted grey lines, while the combined chain is represented by the solid black line, serving to quantify the statistical errors linked to the SBF calibration process, resulting in a value of ± 4 . 86 km s -1 Mpc -1 . A similar process was conducted for the Tip of the Red Giant Branch calibration, yielding a statistical error of ± 1 . 87 km s -1 Mpc -1 , as shown in the bottom panel of Figure 9. Despite the lower statistical uncertainties associated with the TRGB calibration, we adopted the SBF calibration results as the main findings in this paper. This decision is based on the fact that the TRGB distance to Coma is not a direct measurement but relies on a measured distance to the Leo I group and assumes a known relative distance between Leo I and Coma cluster (Sakai et al. 1997). \nNotably, this analysis reveals that the statistical uncertainty associated with the zero-point calibration process dominates the error budget. However, this result underscores the robustness of the Fundamental Plane analysis. The systematic biases arising from potential sources of systematic within the Fundamental Plane fitting process itself are notably smaller than the statistical errors introduced by the calibration process. Therefore, after accounting for all other sources of systematics, including the inclusion of spiral galaxies, the correction of DESI velocity dispersion to SDSS values, applying both high and low redshift cuts, and statistical uncertainties due to the calibration, our final estimate for the Hubble constant is 𝐻 0 = 76 . 05 ± 0 . 35(statistical) ± 0 . 49(systematic FP) ± 4 . 86(statistical due to calibration) km s -1 Mpc -1 .", '9 SUMMARY': 'The DESI peculiar velocity survey will be approximately four times larger than the combined size of all previous peculiar velocity surveys. For the science verification sample, we adopted a similar approach to the 6dFGSv and SDSS peculiar velocity surveys (Magoulas et al. 2012; Howlett et al. 2022) in order to select a clean and reliable sample of elliptical galaxies. This approach involved implementing various photometric cuts, including magnitude and colour cuts. Following the implementation of the aforementioned cuts, our selection process resulted in a sample of 6698 unique galaxies. This sample size is comparable to that of the complete 6dFGSv peculiar velocity sample which had been the largest peculiar velocity survey for a decade and helped refine our understanding of the growth rate of structure (Adams & Blake 2017; Qin et al. 2019; Adams & Blake 2020; Said et al. 2020). \nWe apply only one spectroscopic cut: redshift. At this stage, we refrained from applying any H-alpha cuts, as this aspect will be investigated more extensively using data from the first year of the survey. Our final peculiar velocity sample includes 4191 elliptical galaxies. \nDuring the process of fitting the Fundamental Plane, we implemented an additional visual inspection cut to further refine the selection of galaxies. It is important to note that this visual inspection cut was only applied during the initial step, to determine the cleanest sample with which to fit the Fundamental Plane parameters. Subsequently, these galaxies were re-introduced into the sample to calculate their peculiar velocities. This approach ensures that the determination of Fundamental Plane parameters remains unbiased, preserving the integrity of the other galaxies within the sample. Simultaneously, it allows for the calculation of peculiar velocities for all galaxies, \naffording the flexibility to decide at a later stage whether to include or exclude these peculiar velocities in any subsequent cosmological analysis. \nTo construct the Fundamental Plane, we constructed photometric parameters such as the angular effective radius and the mean surface brightness. These parameters were obtained from the Ninth Data Release (DR9) of the DESI Legacy Imaging Surveys (Dey et al. 2019). Thephotometric error was calculated as 𝜖 phot = [( 𝜖 𝑟 ) 2 +( 𝑏𝜖 𝑖 ) 2 ] 1 / 2 = 0 . 002 dex (<1%). This error is one order of magnitude smaller than the total photometric error observed in the SDSS peculiar velocity survey. \nVelocity dispersion serves as the third component in constructing the Fundamental Plane. In our study, we employed the pPXF algorithm (Cappellari 2017) along with the Indo-U.S. Coudé Feed Spectral Library (Valdes et al. 2004) to measure velocity dispersion from DESI spectra. \nIn order to test the internal (per tile) and external (per survey) consistency of velocity dispersion measurements and avoid any potential systematic offsets within our data, we used the pairwise relative error. We showed that the relative error distribution, both internally and externally, followed a Gaussian distribution centered at zero with a standard deviation of one. This showed that our velocity dispersion measurements are consistent and unbiased. \nTo ensure the reliability of our results, we assessed the relative velocity dispersion error 𝛿𝜎 𝜎 . Our examination revealed that 75% of our sample exhibited a relative error of less than 10%. \nWhile intrinsic scatter remained the largest source of uncertainty, this velocity dispersion error constituted the second largest component of uncertainty in the Fundamental Plane, specifically in the 𝑟 -direction. \nThe fitting of a 3D Gaussian Fundamental Plane to our sample yielded results that were comparable to those obtained from the SDSS survey (Said et al. 2020; Howlett et al. 2022) in terms of scatter in the 𝑟 -direction of the FP. However, a significant improvement was observed when compared to the analysis based on the 6dFGSv survey. \nIndependent of the Fundamental Plane, we defined Coma cluster membership. Using this criterion, we identified 226 galaxies belonging to the Coma cluster. These galaxies were then used in our Zeropoint calibration for the Fundamental Plane. This calibration utilized the absolute distance to the Coma cluster, which was determined by Jensen et al. (2021) through the Surface Brightness Fluctuation method. \nAfter calibrating our sample, we proceeded to construct the Hubble diagram and estimate the Hubble constant. Our final result for the Hubble constant is 𝐻 0 = 76 . 05 ± 0 . 35 (statistical) ± 0 . 49 (systematic FP) ± 4 . 86 (statistical due to calibration) km s -1 Mpc -1 . \nWhile our measured value of the Hubble constant, 𝐻 0 , is within 2 𝜎 of the measurement derived from cosmic microwave background (CMB) anisotropies (e.g., Planck Collaboration et al. 2020), it notably aligns well within 1 𝜎 with local Hubble constant determinations based on different distance indicators (e.g., Riess et al. 2022). This alignment includes includes the most recent findings from CosmicFlows-4 (Kourkchi et al. 2020, 2022) as discussed in Said (2023), where a comprehensive review of Hubble constant measurements from the Tully-Fisher relation is provided. Additionally, our measurement is consistent with the first standard siren measurement from the GW170817 analysis performed with DESI data (Ballard et al. 2023). \nH 0 [kms 1 Mpc 1 ] Systematics due to using TRGB as Calibrator \n<!-- image --> \nFigure 9. MCMCsampling of the posterior for 𝐻 0 to assess systematics arising from the zero-point calibration. The top panel displays 1000 MCMC chains of the 𝐻 0 posterior as grey dotted lines, derived using randomly sampled distances to the Coma cluster, given its measured distance and associated error from SBF. The solid line represents the combined chain used to quantify systematic errors introduced by the calibration process when using SBF. The bottom panel mirrors the top one but employs the distance to Coma cluster derived from the TRGB method. For the Python code and data used to reproduce this plot, see this link. \n<!-- image -->', '10 CONCLUSIONS AND FUTURE DIRECTIONS': 'This paper underscores the capability of the Dark Energy Spectroscopic Instrument (DESI) to deliver reliable velocity dispersion measurements, thereby facilitating the application of the Fundamental Plane analysis and the subsequent constraint of critical cosmological parameters, including the Hubble constant and the growth rate of cosmic structure. \nIn this paper, we present an analysis of the Hubble constant ( 𝐻 0 ) based on Fundamental Plane measurements derived from a sample of 4191 galaxies within the redshift range of 0.01 to 0.1. Systematic uncertainties are explored, including potential biases introduced by spiral galaxies, velocity dispersion calibration, and redshift cuts. The final result, 𝐻 0 = 76 . 05 ± 0 . 35(statistical) ± 0 . 49(systematic FP) ± 4 . 86(statistical due to calibration) km s -1 Mpc -1 , demonstrates an agreement with previous Hubble constant measurements from other distance indicators. \nSeveral avenues for expanding our analysis are on the horizon. Foremost is the utilization of the forthcoming DESI year 1 Fundamental Plane data (Ross et al. in preparation), expected to be substantially larger ( ∼ 100k elliptical galaxies), providing not only enhanced statistical power but also greater control over systematic uncertainties, which currently represent the primary source of uncertainty, particularly in the zero-point calibration process. Currently, our calibration relies on a single cluster, Coma, and a solitary source within the cluster, NGC 4874. However, we anticipate a substantial improvement in precision with the inclusion of multiple clusters and numerous sources within each of these clusters. This advancement will become feasible with the availability of the group catalogue from the full DESI dataset. \nFuture research could explore the potential of using stellar population to enhance the precision and accuracy of the FP as a distance indicator. Recent work by (D\'Eugenio et al. 2024) discuss the concept \nof a "hyperplane" for early-type galaxies, which incorporates stellar population observables alongside the traditional FP parameters. \nFuture work can also focus on extending our peculiar velocity survey to higher redshifts using Brightest Cluster Galaxies (BCGs). As the most luminous galaxies in the Universe, BCGs offer the potential to probe peculiar velocities out to much greater distances than possible with typical spiral and elliptical galaxies. We plan to utilize both the Fundamental Plane relation and the Metric Plane (Lauer et al. 2014) for BCGs, with the latter offering intrinsically less scatter and thus more precise distance measurements. The DESI survey, particularly the combination of Bright Galaxy Survey (BGS) and Luminous Red Galaxy (LRG) data, is expected to provide a volume-limited sample of BCG-like galaxies. Moreover, BCGs are less affected by selection biases that impact normal ellipticals. This extension will enable us to investigate deviations from normal kinematic Hubble flow expansion out to 𝑧 ∼ 0 . 15, potentially shedding light on the existence and effects of large-scale structures such as a large cosmic voids on cosmological parameters like 𝐻 0 . \nExtending our distance measurements to higher redshifts will enable simultaneous fitting of cosmological parameters like the Hubble constant, deceleration, and jerk parameters. Combining our distance measurementswithotherDESIdatasets,suchastheTully-Fisher relation, can further mitigate systematics. Additionally, incorporating our data with external datasets like SN Ia measurements holds promise. However, a key part of making these advancements is putting more effort into improving the zero-point calibration, which this paper shows is a major source of uncertainties. \nExploring other scientific domains, like the measurements of the amplitude and directions of the Bulk Flow, necessitates complete sky coverage. While DESI encompasses 14,000 square degrees in the northern hemisphere, upcoming surveys such as WALLABY (Courtois et al. 2023a), which employs the Australian SKA Pathfinder (ASKAP), and the 4MOST Hemisphere Survey (Taylor et al. 2023), are set to encompass the southern hemisphere. This expanded coverage promises numerous scientific possibilities that would be unattainable with just half of the sky. It is noteworthy that surveys like WALLABY will operate in entirely different wavelengths. Such diversity is invaluable for testing phenomena like galaxy bias across varying wavelengths, and might unravel sources of systematics that we have not thought about yet.', 'ACKNOWLEDGEMENTS': "KS, CH, and TMD acknowledge support from the Australian Government through the Australian Research Council's Laureate Fellowship funding scheme (project FL180100168) and the Australian Research Council Centre of Excellence for Gravitational Wave Discovery (OzGrav), through project number CE230100016. \nKS would like to thank Francesco D'Eugenio for many useful discussions realted to pPXF and the associated stellar library templates. This research used data obtained with the Dark Energy Spectroscopic Instrument (DESI). DESI construction and operations is managed by the Lawrence Berkeley National Laboratory. This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of High-Energy Physics, under Contract No. DE-AC02-05CH11231, and by the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility under the same contract. Additional support for DESI was provided by the U.S. National Science Foundation (NSF), Division of Astronomical Sciences under Contract No. AST-0950945 to the NSF's National Optical-Infrared Astronomy Research Laboratory; \nthe Science and Technology Facilities Council of the United Kingdom; the Gordon and Betty Moore Foundation; the Heising-Simons Foundation; the French Alternative Energies and Atomic Energy Commission (CEA); the National Council of Science and Technology of Mexico (CONACYT); the Ministry of Science and Innovation of Spain (MICINN), and by the DESI Member Institutions: www.desi.lbl.gov/collaborating-institutions. The DESI collaboration is honored to be permitted to conduct scientific research on Iolkam Du'ag (Kitt Peak), a mountain with particular significance to the Tohono O'odham Nation. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the U.S. National Science Foundation, the U.S. Department of Energy, or any of the listed funding agencies. \nThe DESI Legacy Imaging Surveys consist of three individual and complementary projects: the Dark Energy Camera Legacy Survey (DECaLS), the Beijing-Arizona Sky Survey (BASS), and the Mayall z-band Legacy Survey (MzLS). DECaLS, BASS and MzLS together include data obtained, respectively, at the Blanco telescope, Cerro Tololo Inter-American Observatory, NSF's NOIRLab; the Bok telescope, Steward Observatory, University of Arizona; and the Mayall telescope, Kitt Peak National Observatory, NOIRLab. NOIRLab is operated by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. Pipeline processing and analyses of the data were supported by NOIRLab and the Lawrence Berkeley National Laboratory. Legacy Surveys also uses data products from the NearEarth Object Wide-field Infrared Survey Explorer (NEOWISE), a project of the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. Legacy Surveys was supported by: the Director, Office of Science, Office of High Energy Physics of the U.S. Department of Energy; the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility; the U.S. National Science Foundation, Division of Astronomical Sciences; the National Astronomical Observatories of China, the Chinese Academy of Sciences and the Chinese National Natural Science Foundation. LBNL is managed by the Regents of the University of California under contract to the U.S. Department of Energy. The complete acknowledgments can be found at https://www.legacysurvey.org/.", 'DATA AVAILABILITY': 'All data and codes necessary to reproduce the analyses presented in this paper are publicly available. The repository containing both the final catalogue and the analysis codes can be accessed at https://github.com/KSaid-1/DESI\\_fuji\\_FP . All data points shown in the figures are available in a machine-readable form on https://doi.org/10.5281/zenodo.13363598.', 'REFERENCES': '```\nAbell G. O., Corwin Harold G. J., Olowin R. P., 1989, ApJS, 70, 1 Abolfathi B., et al., 2018, ApJS, 235, 42 Adams C., Blake C., 2017, MNRAS, 471, 839 Adams C., Blake C., 2020, MNRAS, 494, 3275 Ballard W., et al., 2023, Research Notes of the American Astronomical Society, 7, 250 Bell R., Said K., Davis T., Jarrett T. H., 2023, MNRAS, 519, 102 Bernardi M., et al., 2003a, AJ, 125, 1849\n``` \nBernardi M., et al., 2003b, AJ, 125, 1866 \nBoubel P., Colless M., Said K., Staveley-Smith L., 2024a, arXiv e-prints, p. \nLintott C., et al., 2011, MNRAS, 410, 166 \nThis paper has been typeset from a T E X/L A T E X file prepared by the author.'} |
2024PhRvD.110j4072W | It remains a longstanding problem unsettled for almost two decades in the general relativity community ever since Griffiths and Podolsk demonstrated in their previous paper J. B. Griffiths and J. Podolsk Classical Quantum Gravity 22 3467 2005CQGRDG0264938110.1088026493812217008 that the typeD NUT inlineformulammlmath displayinlinemmlmiCmmlmimmlmathinlineformula metric seems to be absent from the most general family of the typeD PlebaskiDemiaski PD solution. However Astorino in his recent article M. Astorino Phys. Rev. D 109 084038 2024PRVDAQ2470001010.1103PhysRevD.109.084038 presented a different form of rotating and accelerating black holes and showed that all known fourdimensional typeD accelerating black holes without the NUT charge can be recovered via various different limits in a definitive fashion. In particular he provided for the first time the correct expressions for the typeD static accelerating black holes with a nonzero NUT charge which was previously impossible using the traditional parametrization of the familiar PD solution. Nevertheless it still remains elusive how these two different forms of the fourdimensional rotating and accelerating solutions are related. In this paper we aim to fill this gap by finding the obvious coordinate transformations and parameter identifications between the vacuum metrics after two different parametrizations of the generated solution via the inverse scattering method from the seed metricthe Rindler vacuum background. We then resolve this missing puzzle by providing another Mbius transformation and linear combinations of the Killing coordinates which clearly cast the typeD NUT inlineformulammlmath displayinlinemmlmiCmmlmimmlmathinlineformula metric into the familiar form of the PD solution. Additionally we propose an alternative new routine for the normalization of the obtained metric derived via the inverse scattering method from the vacuum seed solution which could be potentially useful for the construction of higherdimensional solutions using the trivial vacuum background as the seed metric. | 2024-11-01T00:00:00Z | ['2024arXiv240906733W', '10.1103/PhysRevD.110.104072', '2024PhRvD.110j4072W', '10.48550/arXiv.2409.06733', 'arXiv:2409.06733'] | ['General relativity', 'alternative theories of gravity', 'General Relativity and Quantum Cosmology', 'High Energy Physics - Theory'] | Is the typeD NUT inlineformulammlmath displayinlinemmlmiCmmlmimmlmathinlineformula metric really missing from the most general PlebaskiDemiaski solution | 2,024 | 171 | 0.29 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML'] | 2 | https://arxiv.org/pdf/2409.06733.pdf | {"Is the type-D NUT C-metric really 'missing' from the most general Pleba'nski-Demia'nski solution?": "Shuang-Qing Wu ∗ and Di Wu † \nSchool of Physics and Astronomy, China West Normal University, Nanchong, Sichuan 637002, People's Republic of China (Dated: Received 27 August 2024; Accepted 21 October 2024) \nIt remains a long-standing problem, unsettled for almost two decades in the general relativity community, ever since Griffiths and Podolsk'y demonstrated in their previous paper [J.B. Griffiths and J. Podolsk'y, Classical Quantum Gravity 22 , 3467 (2005)] that the type-D NUT C-metric seems to be absent from the most general family of the type-D Pleba'nski-Demia'nski (P-D) solution. However, Astorino, in his recent article [M. Astorino, Phys. Rev. D 109 , 084038 (2024)] presented a different form of rotating and accelerating black holes and showed that all known four-dimensional type-D accelerating black holes (without the NUT charge) can be recovered via various different limits in a definitive fashion. In particular, he provided, for the first time, the correct expressions for the type-D static accelerating black holes with a nonzero NUT charge, which was previously impossible using the traditional parametrization of the familiar P-D solution. Nevertheless, it still remains elusive how these two different forms of the four-dimensional rotating and accelerating solutions are related. In this paper, we aim to fill this gap by finding the obvious coordinate transformations and parameter identifications between the vacuum metrics after two different parametrizations of the generated solution via the inverse scattering method from the seed metric - the Rindler vacuum background. We then resolve this 'missing' puzzle by providing another Mobius transformation and linear combinations of the Killing coordinates, which clearly cast the type-D NUT C-metric into the familiar form of the P-D solution. Additionally, we propose an alternative new routine for the normalization of the obtained metric derived via the inverse scattering method from the vacuum seed solution, which could be potentially useful for the construction of higher-dimensional solutions using the trivial vacuum background as the seed metric.", 'I. INTRODUCTION': "The most general type-D solution in general relativity has been well known for more than half a century [1-3], and is often referred to as the Pleba'nski-Demia'nski (P-D) metric. With the advent of the conveniently factorized form [4] of two quartic structure functions, this solution has subsequently been extensively studied during the past twenty years, see the book [5] for a comprehensive review. However, as demonstrated in Ref. [6], the type-D NUT C-metric is apparently accelerating and can be transformed into the usual Taub-NUT solution. Since then, 'such a solution has to date neither been identified nor proved not to exist.' This raised a 'missing' puzzle for almost two decades, as the type-D NUT C-metric appears to be absent from the most general family of the type-D P-D solution. \nRecently, Astorino [7] presented a different form of the rotating and accelerating black holes that belong to the type-D family (although no obvious proof of this issue is provided in that paper) and showed that all known fourdimensional type-D accelerating black holes without the NUT charge can be obtained via various different limits in a definitive fashion. Moreover, he presented, for the first time, the correct expressions for the type-D NUT C-metric, which are not available in the traditional parametrization for the familiar P-D solution. Lately, this solution has also been generalized in [8] to include the electromagnetic charges and a nonzero cosmological constant. \nAs far as the NUT C-metric is concerned, apart from the above type-D one, there is also another kind of accelerating Taub-NUT solution [9], which belongs to a more general type I [10] that has attracted some recent interest [11-15]. Although the initial form of this solution is complicated and has been obtained via a hybrid solution-generating method from a specific seed, in fact it can be simply generated [11-13] via the Ehlers transformation from the C-metric. The Ehlers transformation not only endows the black hole horizon with the NUT charge, but also allocates a port of the NUT charge to the accelerating horizon. From the perspective of the double Kerr-Schild formalism, the type-D NUT C-metric can be viewed as a perturbation on the pure Rindler background. It seems that one could regard the NUTcharged Rindler metric as the background spacetime of the type-I NUT C-metric. However, it is unclear whether the type-I C-metric admits a similar double Kerr-Schild formulation. Based on the same reasoning, when the Harrison transformation is applied to the seed C-metric, one gets a type-I electrically charged C-metric, whose background is the electrified Rindler vacuum. However, a well-known type-D charged static accelerating solution (Reissner-Nordstrom C-metric, or RN C-metric) is already included in the most general P-D family. Like the Ehlers transformation that allocates a port of the NUT charge to the Rindler background, the Harrison transformation also assigns a port of the electric charge to the Rindler background. Thus, in the type-I RN C-metric, the accelerating horizon acquires the electric charge; in other words, its background is a charged Rindler metric. In this sense, the original KinnersleyWalker's interpretation [16] of the uniformly accelerating charged mass as two 'oppositely' accelerated black holes apart from each other seems not to be very accurate, as it cannot distinguish the type-D accelerating black holes from the type-I ones. Rather, one could say that the type-D accelerating solution is a black hole sitting on the pure Rindler background, while the type-I accelerating metric is that on the more general Rindler background with dynamical parameters (the NUT charge, electromagnetic charge) plus other kinetic parameters (such as the rotation parameter and possibly the cosmological constant). It is noted that while the type-D solution admits an extension to include a nonzero cosmological constant, the type-I metric seems unable to permit such a generalization. By the way, we mention that there is also another kind of NUT C-metric presented in Ref. [17], which is dubbed the [4-1]-soliton solution [18]. This solution still remains unexplored to date, but we can anticipate it to be of type I in the Petrov classification. \nWhat remains opaque in the Astorino's work is how these two different forms of the four-dimensional rotating and accelerating solutions are related, and why the type-D NUT C-metric is 'missing' from the P-D metric. The main purpose of this paper is to resolve this 'missing' puzzle in two different ways. First, after generating the rotating and accelerating Kerr-NUT metric via the inverse scattering method (ISM) [19-21] from the seed Rindler vacuum background, we obtain two distinct forms of the same solution by using two different parametrizations. This clearly shows that both metrics are isometric to each other. Then, we find the obvious Mobius transformation and parameter identifications between them. In the meanwhile, we will illustrate what limit should be taken in the general P-D solution to obtain the type-D NUT C-metric. In particular, we shall present another Mobius transformation and linear combinations of the Killing coordinates that clearly cast the type-D NUT C-metric into the familiar form of the P-D solution, with the quartic structure function being factorized into two quadratic functions in a manner different from that of Griffiths-Podolsk'y. \nThe main parts of our paper are organized as follows. In Sec. II, we present the ISM construction of the type-D accelerating Kerr-NUT metric as a 2-soliton transformation from the Rindler vacuum background, and two distinct parametrizations to arrive at two different rational expressions of the same solution. As preparation, we take the C-metric as a simple example to compare the double Kerr-Schild formalism with the ISM perspective, and make an attempt to combine these two seemly unrelated schemes. This idea has never been advocated in the past years. Then after a minor retrospect of the ISM algorithm, we devise a new strategy for the normalization of the obtained metric derived via the ISM construction from the vacuum seed solution, which could facilitate the construction of \nhigher dimensional solutions by using the trivial vacuum background as the seed metric. Our normalization scheme is corroborated in the Appendix V by deriving the five-dimensional Schwarzschild-Tangherlini solution [22] from the flat vacuum background in two different ways. The remaining two sections, III and IV, are entirely new and constitute the main contribution of this work, where the Mobious transformation of the radial and angular coordinates, and linear combinations of the Killing coordinates as well as various parameter identifications between these two metrics are plainly displayed. Our paper ends with a brief summary of this work, followed by the Aappendix with two simple examples to illuminate our new normalization strategy.", 'A. The C-metric and its seed': 'We begin with two different descriptions of the C-metric, whose line element in the spherical-like coordinates ( t, r, x = cos θ, ¯ φ ), \nd ˜ s 2 = 1 (1 + αrx ) 2 [ -f ( r ) dt 2 + dr 2 f ( r ) + r 2 dx 2 h ( x ) + r 2 h ( x ) d ˜ φ 2 ] , (1) \ncan be rewritten in terms of the soliton formalism as \nd ˜ s 2 ≡ αds 2 = -µ 1 µ 3 µ 2 dt 2 + ρ 2 µ 2 µ 1 µ 3 dφ 2 + µ 1 µ 3 R 2 12 R 2 23 (1 -2 mα ) 2 µ 2 R 2 13 R 11 R 22 R 33 ( dρ 2 + dz 2 ) , (2) \nwhere the Weyl coordinates are defined as \nρ = √ r 2 f ( r ) h ( x ) (1 + αrx ) 2 , z = ( αr + x )[(1 + mαx ) r -m ] (1 + αrx ) 2 , \nwith two structure functions being \nf ( r ) = ( 1 -α 2 r 2 ) ( 1 -2 m r ) , h ( x ) = ( 1 -x 2 ) (1 + 2 mαx ) . (3) \nIn the above, we have also made a rescaling ˜ φ = αφ and introduced the following notations: \nR kl = ρ 2 + µ k µ l , µ k = √ ρ 2 +( z -z k ) 2 -( z -z k ) , (4) \nwhere the 3-soliton µ k s are algebraically expressed as \nµ 1 = (1 -αr )( r -2 m )(1 -x ) (1 + αrx ) 2 , µ 2 = (1 -αr ) r (1 -x )(1 + 2 mαx ) (1 + αrx ) 2 , µ 3 = (1 -α 2 r 2 )(1 + 2 mαx ) α (1 + αrx ) 2 , (5) \nwhich are associated with three rod end points: z 1 = -m,z 2 = m,z 3 = 1 / (2 α ), respectively. \nThe zero-mass limit µ 2 ↦→ µ 1 is the Rindler vacuum \nd ¯ s 2 ≡ αds 2 = -µ 3 dt 2 + ρ 2 µ 3 dφ 2 + µ 3 ρ 2 + µ 2 3 ( dρ 2 + dz 2 ) , (6) \nwhich will be used later as our seed metric. The Rindler metric can be viewed as 1-soliton transformation from the flat Minkowski background with the metric components: g µν = ( -1 , ρ 2 , 1 , 1). However, it is noted that the Rindler vacuum is trivial and is entirely equivalent to the flat Minkowski metric since they are related via some simple coordinate transformations. \nSo in the soliton formalism, the C-metric can be interpreted either as a 3-soliton transformation to the flat Minkowski background or a 2-soliton transformation to the Rindler background. In this article, we will take the latter viewpoint and derive the accelerating Kerr-NUT solution via the ISM procedure by using the Rindler background (6) as the seed metric. Note that usually the seed solution need not be exactly the background metric and frequently can be taken as a more complicated even singular one for the technical aim. However, a simple background metric is surely more preferable in practice as a seed solution. \nOn the other hand, the C-metric (1) admits the double Kerr-Schild formulation as follows: \nd ˜ s 2 = 1 (1 + αrx ) 2 [ -( 1 -α 2 r 2 ) d ¯ t 2 + dr 2 1 -α 2 r 2 + r 2 dx 2 1 -x 2 + r 2 ( 1 -x 2 ) d ¯ φ 2 ] + 2 m ( 1 -α 2 r 2 ) r (1 + αrx ) 2 ( d ¯ t + dr 1 -α 2 r 2 ) 2 + 2 mαx ( 1 -x 2 ) r 2 (1 + αrx ) 2 ( d ¯ φ + I dx 1 -x 2 ) 2 , (7) \nafter making the coordinate transformations \nt = ¯ t -∫ 2 m rf ( r ) dr , ˜ φ = ¯ φ + I ∫ 2 mαx h ( x ) dx. \nThe zero-mass part is just the Rindler vacuum written in the C-metric coordinates ( x, y ) with y = -1 / ( αr ). In view of this, the C-metric can be interpreted as a perturbation around the Rindler background. Incidentally, it should be pointed out that the above Kerr-Schild ansatz is completely applicable to the most general P-D metric [3]. \nHowever, it seems that the following double quasi-Kerr-Schild prescription, \nd ˜ s 2 = 1 (1 + αrx ) 2 [ -( 1 -α 2 r 2 ) dt 2 + dr 2 f ( r ) + r 2 dx 2 h ( x ) + r 2 ( 1 -x 2 ) d ˜ φ 2 ] + 2 m ( 1 -α 2 r 2 ) r (1 + αrx ) 2 dt 2 + 2 mαx ( 1 -x 2 ) r 2 (1 + αrx ) 2 d ˜ φ 2 , (8) \nis in more accordance with the above 2-soliton viewpoint by virtue of the two-dimensional metrics are conformal to each other: \n1 (1 + αrx ) 2 [ dr 2 f ( r ) + r 2 dx 2 h ( x ) ] /similarequal C f ( ρ, z )( dρ 2 + dz 2 ) .', 'B. ISM: A minor preliminary': "In a D -dimensional spacetime with ( D -2) Killing vectors, if its metric, whose components depend on the Weyl coordinates ρ and z only, is written as [23] \nds 2 = g ab dx a dx b + f ( ρ, z )( dρ 2 + dz 2 ) , (9) \nand supplemented with the canonical condition, \ndet g = -ρ 2 , (10) \nthen the vacuum Einstein equations are equivalent to the following equations for the ( D -2) × ( D -2) matrix g = ( g ab ): \n∂ ρ U + ∂ z V = 0 , (11) \nand for the conformal factor f ( ρ, z ), which can be integrated by quadratures: \n∂ ρ ln f = -1 ρ + 1 4 ρ Tr( U 2 -V 2 ) , ∂ z ln f = 1 2 ρ Tr( UV ) , (12) \nwhere the following two matrices are introduced \nU = √ -g ( ∂ ρ g ) g -1 , V = √ -g ( ∂ z g ) g -1 . (13) \nSystem (11) is completely integrable and can be viewed as the compatibility condition of the Lax pair for the ( D -2) × ( D -2) generating matrix Ψ ( λ, ρ, z ): \n∂ ρ Ψ + 2 λρ λ 2 + ρ 2 ∂ λ Ψ = ρ U + λ V λ 2 + ρ 2 Ψ , ∂ z Ψ -2 λ 2 λ 2 + ρ 2 ∂ λ Ψ = ρ V -λ U λ 2 + ρ 2 Ψ , (14) \nwith λ being a complex spectral parameter independent of ρ and z . When λ = 0, one gets g ( ρ, z ) = Ψ (0 , ρ, z ). One can construct new solitonic solutions from a seed metric via the dressing procedure: \nΨ = χ χ Ψ 0 , χ χ = I + n ∑ k =1 n † ( k ) m ( k ) λ -µ k , (15) \nin which n † ( k ) denotes the transpose of the BZ vector n ( k ) defined below, µ k represents a soliton with its corresponding antisoliton being ν k = -ρ 2 /µ k , and the constant z k is the turning point (or rod end point) on the z axes. \nThe expression for the generated matrix of the general n -soliton solutions is \n˜ g = g 0 -n ∑ k,l =1 l † ( k ) ( Γ -1 ) kl l ( l ) µ k µ l , (16) \nwith the following notations being introduced \nΓ kl = m ( k ) g 0 m † ( l ) ρ 2 + µ k µ l , l ( k ) = m ( k ) g 0 , m ( k ) = m ( k ) 0 Ψ -1 0 ( µ k , ρ, z ) , n ( k ) = n ∑ l =1 ( Γ -1 ) kl l ( l ) µ l , (17) \nwhere the BZ vectors m ( i ) 0 consist of arbitrary constants with one of which can be set to unity for each vector. \nIn the case of a diagonal seed, the generating matrix Ψ 0 ( µ i , ρ, z ) can be easily obtained from the seed metric g 0 ( ρ, z ) by simply replacing every soliton or antisoliton via µ k → µ k -λ and ρ 2 → ρ 2 -2 λz -λ 2 , or ρ 2 /µ k → ρ 2 /µ k + λ . \n/negationslash \nIn general, the generated metric ˜ g is unphysics due to the fact that it does not fulfill the canonical condition (10): det ˜ g = -ρ 2 , so it should be appropriately normalized. \nAs far as the normalization scheme is concerned, to the best of our knowledge, there exist three different kinds of strategies to date. (I) Uniform normalization [21]: Like the four-dimensional case, all the metric components of the Killing parts in the higher-dimensional case are multiplied by the same normalization factor. The main defect of this method lies in that it yields some fractional singularities on the symmetry axes, so that the final solution has little physics meaning. (II) Partial block uniform normalization [24-27], which stems from early work [28, 29]: This method only applies the above uniform normalization to a part of the block diagonal matrix of the generated metric, so it is only effective in the case when not all rotation parameters are turned on, for instance, it is limited to the singly rotating case in five dimensions, and so on. (III) Pomeransky's remove-re-add tricky [30]: If the seed metric already satisfies the canonical condition, then after removing and re-adding the same number of solitions or antisolitons, the generated metric automatically subjects to the canonical condition also. At the same time, the conformal factor can be simply computed as f = f 0 det Γ / det Γ 0 , where Γ 0 can be obtained from Γ by setting the nontrivial BZ constants to zero. This is a wise strategy that is widely adopted nowadays, for example, the doubly rotating Myers-Perry solution [31] is regenerated via this method from the five-dimensional Schwarzschild-Tangherlini solution [22]. However, there is a fly in the ointment in that it cannot work effectively in the case when the seed solution is a vacuum background metric where there is no soliton that can be removed. In particular, it is claimed [30] that the five-dimensional Schwarzschild-Tangherlini solution [22] cannot be obtained as a 2-soliton solution on the flat Minkowski background. \nBelow we propose the fourth scheme for the normalization of the obtained metric derived from the vacuum seed solution, which would be potentially useful for the construction of higher dimensional solutions by using the trivial vacuum background as the seed metric. In the Appendix, we will present two simple examples to illuminate this method. First, let us consider a simple ansatz: \ng = W ˜ gW , W = diag( w 1 , · · · , w D -2 ) , (18) \nby simply employing a diagonal matrix W with which each of its entries satisfies the Laplace equation: \n∇ 2 ln w i = ( ∂ 2 ρ + 1 ρ ∂ ρ + ∂ 2 z ) ln w i = 0 . (19) \nThe solution of each entry in the above equation has the general form: w i = ρ b i µ c i i , in which b i and c i are arbitrary constants. The entries in W are chosen so that -ρ 2 = w 2 1 · · · w 2 D -2 det ˜ g is satisfied. By virtue of the fact that det ˜ g is proportional to det g 0 , actually one only needs to choose different powers in w i . \nIf the generated metric is still diagonal just as its seed metric, then the above means amounts to g = W 2 ˜ g , and it is easy to compute the conformal factor. On the other hand, if all the diagonal entries are identical w i = w , then the above scheme is equivalent to the original uniform normalization strategy. Suppose that only some of the \nwhere \ndiagonal entries are set to be identical, then this amounts to the partial block uniform normalization. Finally, if all w i s are appropriately chosen such as ( -ρ 2 /µ 2 i ) or ( -µ 2 i /ρ 2 ), this scheme is actually equivalent to removing solition or antisoliton at least in the case of the generated metric being static (diagonal). \nSecond, let us handle the more general rotating case. Since the generated metric in general still needs to make further linear combinations of the Killing coordinates, that is, ˆ g = A † gA , where the rotation matrix A is unimodular and its every entry is a constant, then we can have ˆ g = ˆ W † ˜ g ˆ W , where ˆ W = WA , ˆ W † is the matrix transpose of ˆ W and each of its entry obeys the above Laplace equation (19). Clearly it is troublesome to choose a suitable matrix ˆ W so that the final metric is the expected one. However, one need not worry about this, since the Pomeransky's removere-add tricky is already very effective to generate the rotating solution, so our method just plays a nice complement to the case where the Pomeransky's tricky fails to work.", 'C. ISM construction of the accelerating Kerr-NUT solution': "Although the accelerating Kerr-NUT solution can be derived as a 3-soliton transformation via the ISM procedure from the flat Minkowski background, in accordance with the above double Kerr-Schild perspective, we prefer to rederive it as a 2-soliton transformation by using the Rindler background (6) as the seed metric, whose Killing part is \ng 0 = diag( -µ 3 , ρ 2 /µ 3 ) = diag( -µ 3 , -ν 3 ) , (20) \nwith the conformal factor f 0 = µ 3 /R 33 , so that the generating matrix is easily obtained as \nΨ 0 = diag( λ -µ 3 , λ -ν 3 ) = diag( λ -µ 3 , λ + ρ 2 /µ 3 ) . (21) \nNow we perform a 2-soliton transformation to this Rindler background: add a soliton µ 1 at z = z 1 with the vector m (1) 0 = ( C 1 , 1) and a soliton µ 2 at z = z 2 with the vector m (2) 0 = (1 , C 2 ), respectively. Note that we take the 2-soliton in the order: µ = ( µ 1 , µ 2 ), so that the generated metric exactly reproduces the same form (2) for the C-metric when the BZ parameters are set to zero: C 1 = C 2 = 0. Then two BZ vectors constitute a 2 × 2 matrix: \nm = ( C 1 µ 13 µ 3 R 13 1 µ 23 C 2 µ 3 R 23 ) , l = ( -C 1 µ 3 µ 13 ρ 2 R 13 -µ 3 µ 23 C 2 ρ 2 R 23 ) (22) \nwhere a short notation is included for briefness: µ ij = µ i -µ j . The Γ matrix is also recorded as follows: \nΓ = ( -C 2 1 µ 3 µ 2 13 R 11 + µ 3 ρ 2 R 2 13 R 11 -C 1 µ 3 µ 13 µ 23 R 12 + C 2 µ 3 ρ 2 R 12 R 13 R 23 -C 1 µ 3 µ 13 µ 23 R 12 + C 2 µ 3 ρ 2 R 12 R 13 R 23 -µ 3 µ 2 23 R 22 + C 2 2 µ 3 ρ 2 R 2 23 R 22 ) . (23) \nThe final physics metric can be obtained via either the uniform normalization scheme or our prescription proposed in the above \ng = W ˜ gW = -µ 1 µ 2 ρ 2 ˜ g , (24) \nin which we have taken W = diag( √ -µ 1 µ 2 /ρ , √ -µ 1 µ 2 /ρ ). The conformal factor is \nf = -16 k 2 f 0 µ 3 1 µ 3 2 ρ 2 µ 2 12 det Γ = 16 k 2 B H . (25) \nThen the generated metric represents the accelerating Kerr-NUT solution, whose line element reads \nds 2 = -µ 3 A µ 1 µ 2 B ( dt + µ 12 C µ 3 A dφ ) 2 + ρ 2 µ 1 µ 2 B µ 3 A dφ 2 +16 k 2 B H ( dρ 2 + dz 2 ) , (26) \n[ ] \nH = µ 2 12 µ 2 13 µ 2 23 µ 3 1 µ 3 2 µ 3 3 R 2 12 R 2 13 R 2 23 R 11 R 22 R 33 \nA = C 1 ( C 2 µ 2 µ 23 R 12 -ρµ 12 R 23 ) R 13 -µ 1 µ 13 ( C 2 ρµ 2 µ 12 µ 23 + R 12 R 23 ) × [ C 1 ( C 2 µ 2 µ 23 R 12 + ρµ 12 R 23 ) R 13 + µ 1 µ 13 ( C 2 ρµ 2 µ 12 µ 23 -R 12 R 23 ) ] , B = C 2 1 ( C 2 2 µ 2 23 R 2 12 + µ 2 12 R 2 23 ) R 2 13 -2 C 1 C 2 µ 13 µ 23 R 13 R 23 R 11 R 22 + ( C 2 2 ρ 4 µ 2 12 µ 2 23 + R 2 12 R 2 23 ) µ 2 13 , C = C 1 µ 2 µ 13 ( C 2 2 ρ 2 µ 23 -R 2 23 ) R 11 R 12 R 13 + C 2 µ 1 µ 23 ( C 2 1 R 2 13 -ρ 2 µ 2 13 ) R 12 R 22 R 23 , . \nWhen C 1 = C 2 = 0, the above generated 2-soliton solution just reduces to the C-metric (2). So according to the perspective in Sec. II A, the generated accelerating Kerr-NUT solution (26) can be interpreted as a Kerr-NUT black hole sitting on the Rindler background. \nNote that the above accelerating Kerr-NUT solution (26) can also be regenerated via the 'remove-re-add' tricky from the C-metric (2). The process is briefly sketched as follows. We first remove an antisoliton ν 1 at z = z 1 and a soliton µ 2 at z = z 2 both with the trivial vectors (1 , 0), then rescale the metric by a factor µ 1 /µ 2 . This essentially generates the above Rindler seed (20): \ng 0 = µ 1 µ 2 diag ( µ 2 2 µ 2 1 , 1 ) × diag ( -µ 1 µ 3 µ 2 , ρ 2 µ 2 µ 1 µ 3 ) . \nNext we re-add the antisoliton ν 1 with the vector m (1) 0 = (1 , -C 1 ) and the soliton µ 2 with the vector m (2) 0 = (1 , C 2 ), and multiply the generated metric by a factor µ 2 /µ 1 so that the final metric is physics because it satisfies the canonical condition: det g = -ρ 2 . The conformal factor can be easily evaluated via f = f 00 det Γ / det Γ ( C 1 = 0 , C 2 = 0), where f 00 is the conformal factor given by Eq. (2).", "D. Astorino's parametrization": "We first transform from the Weyl-Lewis-Papapetrou coordinates ( ρ, z ) to the spherical-like coordinates ( r, x ) via \nρ = √ ∆( r ) P ( x ) (1 + αrx ) 2 , z = ( x + αr ) [ ( r -m )(1 + mαx ) + σ 2 αx ] (1 + αrx ) 2 , \nwhere \n∆( r ) = [ ( r -m ) 2 -σ 2 ]( 1 -α 2 r 2 ) , P ( x ) = ( 1 -x 2 )[ (1 + mαx ) 2 -σ 2 α 2 x 2 ] , (27) \nThe locations of three turning points are taken to be \nin which σ = √ m 2 + n 2 -a 2 . \nz 1 = -σ, z 2 = σ, z 3 = 1 -α 2 ( m 2 -σ 2 ) 2 α , \nso their corresponding solitons are purely algebraic: \nµ 1 = (1 -αr )( r -m -σ )[1 + ( m -σ ) αx )](1 -x ) (1 + αrx ) 2 , µ 2 = (1 -αr )( r -m + σ )[1 + ( m + σ ) αx )](1 -x ) (1 + αrx ) 2 , (28) µ 3 = ( 1 -α 2 r 2 ) [1 + ( m -σ ) αx )][1 + ( m + σ ) αx )] α (1 + αrx ) 2 , \nwhich reduce to Eq. (5) exactly when the static C-metric limit is reached ( σ = m ). \nIt is very tedious to convert the four functions ( A,B,C,H ) into the expressions in terms of the spherical-like coordinates. However, if we take \nC 1 = ( m -σ )[1 + ( m + σ ) α ] ( a -n )[1 -( m -σ ) α ] , C 2 = ( m -σ )[1 -( m + σ ) α ] ( a + n )[1 + ( m -σ ) α ] , (29) \nand \nk = ( m + σ )[1 + ( m -σ ) α ][1 -( m -σ ) α ] 2 α 3 / 2 √ 1 + α 2 a 2 , (30) \nthen we are able to first write the two-dimensional metric as \nf ( dρ 2 + dz 2 ) = Σ ( 1 + α 2 a 2 ) (1 + αrx ) 2 [ dr 2 ∆( r ) + dx 2 P ( x ) ] , \nin which \nΣ = r 2 +( ax -n ) 2 +2 nαr [ a ( 1 + x 2 ) -2 nx ] + α 2 r 2 ( a -nx ) 2 + ( a 2 -n 2 ) 2 α 2 x 2 . (31) \nFinally, after some cumbersome algebraic simplifications, the Killing parts can be converted into \ng ab dx a dx b = -F (1 + αrx ) 2 Σ α ( dt -ωα F dϕ ) 2 + Σ∆( r ) P ( x ) α (1 + αrx ) 2 F dϕ 2 , \nwhere \nF = [ 1 -( a 2 -n 2 ) α 2 x 2 ] 2 ∆( r ) -[ a ( 1 + α 2 r 2 ) +2 nαr ] 2 P ( x ) , ω = [ a ( 1 + x 2 ) -2 nx ][ 1 -( a 2 -n 2 ) α 2 x 2 ] ∆( r ) + ( r 2 + n 2 -a 2 )[ a ( 1 + α 2 r 2 ) +2 nαr ] P ( x ) . \nAfter rescaling the Killing coordinates as t ↦→ √ αt , ϕ ↦→ ϕ/ √ α , the above two parts are collectively written in terms of the Lewis-Papapetrou coordinates as [7] \nds 2 = -F (1 + αrx ) 2 Σ ( dt -ω F dϕ ) 2 + Σ (1 + αrx ) 2 F { ∆( r ) P ( x ) dϕ 2 + F 1 + α 2 a 2 [ dr 2 ∆( r ) + dx 2 P ( x ) ] } , (32) \nwhich can also be recast into the Boyer-Lindquist-like form \nds 2 = 1 (1 + αrx ) 2 { -∆( r ) X 2 + P ( x ) Y 2 Σ + Σ 1 + α 2 a 2 [ dr 2 ∆( r ) + dx 2 P ( x ) ] } , (33) \nwhere \nX = [ 1 -( a 2 -n 2 ) α 2 x 2 ] dt + [ 2 nx -a ( 1 + x 2 )] dϕ, Y = [ a ( 1 + α 2 r 2 ) +2 nαr ] dt + ( r 2 + n 2 -a 2 ) dϕ. (34) \nIn Sec. III (especially Sec. IIIB) of Ref. [7], the above metric (32) was derived through an equivalent limiting procedure ( a 2 → 0 , l 2 → 0 and then w 4 →∞ ) on the double Kerr-NUT solution. It was also claimed there that this spacetime is of type D, and might be included into the P-D family of black holes. Here we have built this metric with two solitons on a Rindler accelerating background. In the subsequent subsections, we will give an explicit verification of the Astorino's insight by showing that the solution (33) is indeed diffeomorphic to the P-D metric. \nIn the absence of the NUT charge, the above metric simply recovers the accelerating Kerr solution in the familiar form after further shifting the Killing coordinates: \nt ↦→ t -2 aϕ √ 1 + α 2 a 2 , ϕ ↦→ -α 2 at + ( 1 -α 2 a 2 ) ϕ √ 1 + α 2 a 2 . \nOn the other hand, the accelerating NUT C-metric takes a simple expression: \nds 2 = 1 (1 + αrx ) 2 { -∆( r ) Σ [( 1 + n 2 α 2 x 2 ) dt +2 nxdϕ ] 2 + P ( x ) Σ [ 2 nαrdt + ( r 2 + n 2 ) dϕ ] 2 +Σ [ dr 2 ∆( r ) + dx 2 P ( x ) ] } , (35) \nin which \n∆( r ) = ( r 2 -2 mr -n 2 )( 1 -α 2 r 2 ) , P ( x ) = ( 1 -x 2 )( 1 + 2 mαx -n 2 α 2 x 2 ) , Σ = ( r 2 + n 2 )( 1 + n 2 α 2 x 2 ) -4 n 2 αrx. (36) \nWhen the accelerating parameter is set to zero, the above type-D NUT C-metric apparently reduces to the Taub-NUT solution, and also it obviously recovers the usual C-metric when the NUT charge vanishes. There is a natural question remaining, namely, whether it is superficially accelerating just as pointed out in Ref. [6] for the traditional form. Then, if it is so, we still lack a type-D NUT C-metric. However, this case seems unlikely. So, we will assume that the above NUT C-metric is not isometric to the Taub-NUT solution in the remaining context, but clearly an obvious proof is still needed. \nThe above solution has also been generalized in Ref. [8] to include the electric and magnetic charges as well as a nonzero cosmological constant. However, it suffices enough for our aim to turn off these three parameters. Also, in our opinion, it becomes useful only in the static case (where the rotation parameter vanishes), as we will demonstrate the above rotating and accelerating solution can be related to the traditional form of the general P-D metric.", "E. Griffiths-Podolsk'y's parametrization": 'We next transform from the Weyl-Lewis-Papapetrou coordinates ( ρ, z ) to the spherical-like coordinates (˜ r, ˜ x ) via \nwhere \nρ = √ ˜ ∆(˜ r ) ˜ P (˜ x ) [1 + ˜ α ˜ r (˜ n +˜ a ˜ x )] 2 , z = [ ˜ x + ˜ α ˜ r (˜ a + ˜ n ˜ x ) ][ ˜ r -˜ m + ˜ α (˜ n +˜ a ˜ x ) ( ˜ m ˜ r + ˜ n 2 -˜ a 2 )] [ 1 + ˜ α ˜ r (˜ n +˜ a ˜ x ) ] 2 , \n˜ ∆(˜ r ) = ( ˜ r 2 -2 ˜ m ˜ r +˜ a 2 -˜ n 2 )[ (1 + ˜ α ˜ n ˜ r ) 2 -˜ α 2 ˜ a 2 ˜ r 2 ] , ˜ P (˜ x ) = ( 1 -˜ x 2 )[ 1 + 2 ˜ m ˜ α (˜ n +˜ a ˜ x ) + ( ˜ a 2 -˜ n 2 ) ˜ α 2 (˜ n +˜ a ˜ x ) 2 ] , (37) \nThe locations of three rod end points are now taken to be \nin which ˜ σ = √ ˜ m 2 + ˜ n 2 -˜ a 2 . \nz 1 = -˜ σ, z 2 = ˜ σ, z 3 = 1 + 2˜ α ˜ m ˜ n -˜ α 2 ( ˜ a 2 -˜ n 2 ) 2 2˜ α ˜ a , \nand their corresponding solitons are also purely algebraic: \nµ 1 = [1 -˜ α (˜ a -˜ n )˜ r ](˜ r -˜ m -˜ σ )[1 + ( ˜ m -˜ σ )˜ α (˜ n +˜ a ˜ x )](1 -˜ x ) [1 + ˜ α ˜ r (˜ n +˜ a ˜ x )] 2 , µ 2 = [1 -˜ α (˜ a -˜ n )˜ r ](˜ r -˜ m + ˜ σ )[1 + ( ˜ m + ˜ σ )˜ α (˜ n +˜ a ˜ x )](1 -˜ x ) [1 + ˜ α ˜ r (˜ n +˜ a ˜ x )] 2 , (38) µ 3 = [1 + ˜ α (˜ a + ˜ n )˜ r ][1 -˜ α (˜ a -˜ n )˜ r ][1 + ( ˜ m -˜ σ )˜ α (˜ n +˜ a ˜ x )][1 + ( ˜ m + σ )˜ α (˜ n +˜ a ˜ x )] ˜ α [1 + ˜ α ˜ r (˜ n +˜ a ˜ x )] 2 . \nIt is troublesome to convert the four functions ( A,B,C,H ) into the expressions in terms of the spherical-like coordinates. A simple trick to do this relatively easily is to avoid using the square root, such as, exploiting identities: ˜ σ 2 k = ( ˜ m 2 +˜ n 2 -˜ a 2 ) k and ˜ σ 2 k +1 = ( ˜ m 2 +˜ n 2 -˜ a 2 ) k ˜ σ can efficiently facilitate to rationalize the expressions containing the square root. This experience is also very effective to manipulate the phantom soliton whose square-root function could not be reduced to a simple algebraic expression. \nIn order to let the conformal factor include a familiar multiplier ˜ Σ = ˜ r 2 +(˜ n +˜ a ˜ x ) 2 , we have to take \nC 1 = (˜ a -˜ n )[1 + ( ˜ m + ˜ σ )˜ α (˜ a + ˜ n )] ( ˜ m + ˜ σ )[1 -( ˜ m -˜ σ )˜ α (˜ a -˜ n )] , C 2 = (˜ a + ˜ n )[1 -( ˜ m + ˜ σ )˜ α (˜ a -˜ n )] ( ˜ m + ˜ σ )[1 + ( ˜ m -˜ σ )˜ α (˜ a + ˜ n )] , (39) \nand \nk = ( ˜ m + ˜ σ )[1 + ( ˜ m -˜ σ )˜ α (˜ a + ˜ n )][1 -( ˜ m -˜ σ )˜ α (˜ a -˜ n )] 2(˜ α ˜ a ) 3 / 2 √ 1 + ˜ α 2 ( ˜ a 2 -˜ n 2 ) 2 . (40) \nFinishing the remaining task of the algebraic process, we find that it is instructive to cast the metric into the Boyer-Lindquist-like form rather than the Lewis-Papapetrou form, which is rather concise and simple enough: \nd ˜ s 2 = 1 [ 1 + ˜ α ˜ r (˜ n +˜ a ˜ x ) ] 2 { -˜ ∆(˜ r ) ˜ X 2 + ˜ P (˜ x ) ˜ Y 2 ˜ Σ + ˜ Σ [ d ˜ r 2 ˜ ∆(˜ r ) + d ˜ x 2 ˜ P (˜ x ) ] } , (41) \n˜ X = [ 1 -( ˜ a 2 -˜ n 2 ) ˜ α 2 (˜ n +˜ a ˜ x ) 2 ] dt + ˜ α ˜ a [ 2˜ n ˜ x +˜ a ( 1 + ˜ x 2 )] dϕ √ ˜ α ˜ a √ 1 + ˜ α 2 ( ˜ a 2 -˜ n 2 ) 2 , ˜ Y = ˜ a [ 1 + ( ˜ a 2 -˜ n 2 ) ˜ α 2 ˜ r 2 ] dt -˜ α ˜ a ( ˜ r 2 + ˜ n 2 -˜ a 2 ) dϕ √ ˜ α ˜ a √ 1 + ˜ α 2 ( ˜ a 2 -˜ n 2 ) 2 . (42) \nwhere ˜ Σ = ˜ r 2 +(˜ n +˜ a ˜ x ) 2 and \nIn the above frame, the Rindler background or the accelerating horizons is rest relative to the conformal infinity. \nOne can note that the above metric resembles somewhat the one derived in the last subsection. In particular, they only differ by an overall constant factor in the case without the NUT charge and by further making identifications: α = ˜ α ˜ a , m = ˜ m , a = ˜ a , r = ˜ r , x = ˜ x . \nThe final step that we need to recast the above solution into the familiar form of the general P-D metric is to make the following linear combinations of the Killing coordinates: \nt = √ ˜ α ˜ a ( ˜ t -2˜ a ˜ φ ) √ 1 + ˜ α 2 ( ˜ a 2 -˜ n 2 ) 2 , ϕ = ˜ a ˜ α 2 ( ˜ a 2 -˜ n 2 ) ˜ t + [ 1 -˜ α 2 ( ˜ a 4 -˜ n 4 )] ˜ φ √ ˜ α ˜ a √ 1 + ˜ α 2 ( ˜ a 2 -˜ n 2 ) 2 , \nwhich yield the familiar expressions: \n˜ X = d ˜ t + [ 2˜ n ˜ x -˜ a ( 1 -˜ x 2 )] d ˜ φ, ˜ Y = ˜ ad ˜ t -( ˜ r 2 +˜ a 2 + ˜ n 2 ) d ˜ φ. \nNote that in this new frame, the Rindler horizon is rotating relative to the conformal infinity. \nTo summarize, we have rederived the accelerating Kerr-NUT solution via the ISM procedure from the Rindler background seed. Then using two different parametrizations, we have recast the solution into two distinct forms in terms of the spherical-like coordinates, namely, the one novelly given by Ref. [7], and another by the traditional one. Since the latter is already known to be of type D, so is the one delivered by Astorino in [7]. However, one still needs to find what coordinate transformations may relate them. We leave this subject to the next section. \nBy the way, here we would like to mention the related work [18] that dealt with the Euclidean P-D metric. In the Sec. VC of that paper, Chen obtained the Euclidean accelerating Kerr-NUT metric as a 3-soliton solution via the ISM procedure. Then he performed a Mobius transformation on the coordinates ( u, v ) and redefined the Killing coordinates, and further made various parameter identifications so that the 3-soliton solution is brought to the Euclidean P-D solution in the familiar form. This hints that the Mobius transformation may be relevant to the present work.', 'III. M OBIUS TRANSFORMATION AMONGST TWO METRICS': 'In order to find the relation between the above two metrics, a direct routine is to examine the solutions given by Eqs. (33) and (41), respectively, since both of them adopt the same Killing coordinates ( t, ϕ ). Therefore, the remaining most possible coordinate transformation is the Mobius transformation amongst the radial and azimuthal coordinates, respectively, together with further parameter identifications. \nLet us consider the following Mobius transformation: \n˜ x = x -ν 1 -ν x , ˜ α ˜ r = αr + ν ˜ a (1 + ναr ) -˜ n ( αr + ν ) , (43) \nsuch that ˜ x = ± 1 transform to x = ± 1 under the first one, and the second one only involves mapping between the acceleration horizons but not the event horizons among both metrics. The inverse of the second transformation is \nαr = (˜ a -ν ˜ n )˜ α ˜ r -ν 1 + (˜ n -ν ˜ a )˜ α ˜ r . (44) \nHere and hereafter we prefer to leave ν to be a freely variable parameter so that the relation between the two mass parameters is left to be determined only in the final step. \nWe first apply the above transformation to ˜ Σ = ˜ r 2 +(˜ n + ˜ a ˜ x ) 2 and find its relation to Σ. To do so, it is easy to find that if the identity ˜ Σ(1 -ν x ) 2 [˜ a (1 + ναr ) -˜ n ( αr + ν )] 2 = C Σ holds true, then we must have the following conditions: \nn a = -( 1 + ν 2 ) ˜ n -2 ν ˜ a ( 1 + ν 2 ) ˜ a -2 ν ˜ n , a 2 C = ˜ a (˜ n -ν ˜ a )(˜ a -ν ˜ n ) [( 1 + ν 2 ) ˜ a -2 ν ˜ n ] 2 ( 1 + ν 2 ) ˜ n -2 ν ˜ a , ˜ α 2 = -ν (˜ n -ν ˜ a )(˜ a -ν ˜ n ) ( ˜ a 2 -˜ n 2 ) , α 2 a 2 = -ν (˜ a -ν ˜ n ) [( 1 + ν 2 ) ˜ a -2 ν ˜ n ] 2 ( 1 -ν 2 ) 2 (˜ n -ν ˜ a ) ( ˜ a 2 -˜ n 2 ) . (45) \nNext, we seek the links between the one-forms ( ˜ X, ˜ Y ) in (42) and their counterparts ( X,Y ) without rescaling the Killing coordinates: \nX = [ 1 -( a 2 -n 2 ) α 2 x 2 ] dt √ α + [ 2 nx -a ( 1 + x 2 )] √ αdϕ, Y = [ a ( 1 + α 2 r 2 ) +2 nαr ] dt √ α + ( r 2 + n 2 -a 2 ) √ αdϕ. \nIt is relatively easy to examine √ ˜ α ˜ a √ 1 + ˜ α 2 ( ˜ a 2 -˜ n 2 ) 2 ( ˜ X, ˜ Y ) rather than ( ˜ X, ˜ Y ). We further find \n√ ˜ α ˜ a √ 1 + ˜ α 2 ( ˜ a 2 -˜ n 2 ) 2 (1 -ν x ) 2 ˜ X = ˜ a √ α ( 1 -ν 2 ) ˜ a -ν ˜ n X, √ ˜ α ˜ a √ 1 + ˜ α 2 ( ˜ a 2 -˜ n 2 ) 2 [˜ a (1 + ναr ) -˜ n ( αr + ν )] 2 ˜ Y = -˜ a 2 α 3 / 2 ( 1 -ν 2 ) ˜ α (˜ a -ν ˜ n ) Y , (46) \nand must additionally impose \na = -˜ α (˜ a -ν ˜ n ) [( 1 + ν 2 ) ˜ a -2 ν ˜ n ] α ( 1 -ν 2 ) , (47) \nthen we can write \nn = ˜ α (˜ a -ν ˜ n ) [( 1 + ν 2 ) ˜ n -2 ν ˜ a ] α ( 1 -ν 2 ) , C = -α 2 ˜ a (˜ n -ν ˜ a ) 2 ( ˜ a 2 -˜ n 2 )( 1 -ν 2 ) 2 ν [( 1 + ν 2 ) ˜ n -2 ν ˜ a ] . (48) \nIn practice, if we first find the links between ( ˜ X, ˜ Y ) and ( X,Y ) directly, then all the above relations can be obtained, and it can be checked that the identity Σ(1 -ν x ) 2 [˜ a (1 + ναr ) -˜ n ( αr + ν )] 2 = C Σ is also automatically fulfilled. \n˜ Note that the relation for α 2 listed above then becomes the identity for ˜ α 2 , so α can now temporarily be viewed as a free parameter too. Alternatively, we can also express this in terms of untilded parameters as follows: \nα 2 = ν ( a -ν n ) ( n -ν a ) ( a 2 -n 2 ) . \nOur next step is to find the relation between the mass parameters. Using the following identities, \n1 -˜ x 2 = 1 -ν 2 (1 -ν x ) 2 ( 1 -x 2 ) , d ˜ x 2 1 -˜ x 2 = 1 -ν 2 (1 -ν x ) 2 dx 2 1 -x 2 , \nwe can simply let \nd ˜ x 2 ˜ P (˜ x ) = ( 1 -ν 2 ) dx 2 P ( x ) , (49) \nso that after using the above relation for ˜ α 2 , we find that this is true only when we further set \n˜ m = ν 2˜ α (˜ a -ν ˜ n ) , m = ν ( ν 2 ˜ a -3˜ a +2 ν ˜ n ) 2 α (˜ a -ν ˜ n ) , (50) \nwhich can also be attained by exploiting the radial part. \nFinally, with the help of the following useful expressions, \n1 + ˜ α ˜ r (˜ n +˜ a ˜ x ) = ˜ a ( 1 -ν 2 ) (1 + αrx ) (1 -ν x )[˜ a (1 + ναr ) -˜ n ( αr + ν )] , ˜ Σ = C Σ (1 -ν x ) 2 [˜ a (1 + ναr ) -˜ n ( αr + ν )] 2 , ˜ ∆(˜ r ) = ( 1 -ν 2 ) ˜ a 2 α 2 ˜ α 2 [˜ a (1 + ναr ) -˜ n ( αr + ν )] 4 ∆( r ) , ˜ P (˜ x ) = 1 -ν 2 (1 -ν x ) 4 P ( x ) , d ˜ r 2 ˜ ∆(˜ r ) = ( 1 -ν 2 ) dr 2 ∆( r ) , \n(1 \n- \nν \n2 \n- \nν \n˜ \na \n) \nand the relations (46) between ( ˜ X, ˜ Y ) and ( X,Y ), we can show that the two metrics only differ by an overall constant factor: \nprovided that we set \nwhich yields an equation to determine α : \n˜ α α [( 1 + ν 2 ) ˜ n -2 ν ˜ a ] = ( 1 -ν 2 ) 2 ˜ n -ν ˜ a ˜ a -ν ˜ n -ν ˜ a 2 -˜ n 2 [ ( 1 + ν 2 ) ˜ a -2 ν ˜ n ] 2 . (52) \n˜ α 2 α 2 = ( ν 4 +1 ) ˜ a 2 -ν ( 3 ν 2 +1 ) ˜ a ˜ n + ( 3 ν 2 -1 ) ˜ n 2 ˜ a (˜ a -ν ˜ n ) ( ˜ a 2 -˜ n 2 )( 1 -ν 2 ) . (53) \nThus we have generally proven that the two metrics given in the last two sections are isometric to each other. It seems likely to let C = ˜ a 2 ( 1 -ν 2 ) / ( 1 + α 2 a 2 ) so that both metrics are completely identical. This condition implies \nWhen combined with Eq. (52), they yield a cubic equation for ˜ a/ ˜ n : \nν ( ν 4 -ν 2 +1)˜ a 3 -ν 2 (3 ν 2 -2)˜ a 2 ˜ n + ν (3 ν 2 -4)˜ a ˜ n 2 + ˜ n 3 = 0 . (54) \nOne can also conduct a similar task as done in the above by using the C-metric-like coordinates ( x, y ) with y = -1 / ( αr ) and (¯ x, ¯ y ) defined via ¯ y = -1 / (˜ α ˜ r ), ¯ x = ˜ n + ˜ a ˜ x , rather than by employing the spherical-like coordinates. In this case, both metrics become more symmetric, and now ˜ a behaves like a scale parameter; for example, ¯ y = ˜ n + ˜ a ( y -ν ) / (1 -ν y ). Since the results remain the same as those given in the above, we will not repeat this work although it might be relatively easy.', 'What limit should be taken in the original P-D metric to get the type-D NUT C-metric?': "Previously, Griffiths and Podolsk'y [6] demonstrated that in the limit of zero angular momentum parameter, the NUT C-metric is apparently accelerating and can be transformed into the Taub-NUT solution; as a result of this, the type-D NUT C-metric seems to be absent from the most general family of the type-D P-D solution, leaving a puzzle for almost two decades. The reason for this is probably that they did not consider the Mobius transformation. On the other hand, it is a simple matter to just set a = 0 in the Astorino's expression [7] to arrive at the type-D C-metric, but this still does not explain how to get the type-D NUT C-metric from the most general P-D solution. \nTo elucidate the lesson why the type-D NUT C-metric is missing from the P-D metric, let us consider what limit we should take in the original P-D metric to get the expected type-D NUT C-metric. Let us first assume that the transformation parameter ν and all the solution parameters ( ˜ m, ˜ n, ˜ a, ˜ α ) as well as the radial coordinate ˜ r are all finite and nonzero (although ˜ a can be set to zero). To achieve the static NUT C-metric, we must first let the rotation parameter a tend to zero ( a → 0). Then from the first identity in (45), the NUT charge parameter n must also tend to a very tiny constant ( n /similarequal 0) so that the ratio n/a keeps finite. From the fourth one in (45) or (47), we know that the acceleration parameter α must approach the infinity ( α → ∞ ) so that αa is also finite. The second identity in (45) tells us that the overall factor C should tend to the infinity ( C →∞ ) so that Cα can be kept finite. From the inverse Mobius transformation (44), we can recognize that the radial coordinate r must approach zero so that αr remains finite. Finally, from the second equation of Eq. (50), we learn that the mass parameter m should tend to a very tiny constant ( m /similarequal 0). In the a → 0 limit, both ( αa , αn , αm ) and αr remains fixed. \nTo summarize, in order to get the type-D NUT C-metric, the corresponding limit is the following set: \na → 0 , n /similarequal 0 , m /similarequal 0 , r → 0 , α →∞ , (55) \napart from the appropriate rescaling of the whole line element plus the linear combinations of the Killing coordinates ( t , ϕ ). However, the key measure is to consider the above Mobius transformation (43). \nd ˜ s 2 = C Σ ( 1 -ν 2 ) ˜ a 2 (1 + αrx ) 2 [ dr 2 ∆( r ) + dx 2 P ( x ) ] + ˜ aα 3 ( 1 -ν 2 ) C ˜ α 3 (˜ a -ν ˜ n ) 2 [ 1 + ˜ α 2 ( ˜ a 2 -˜ n 2 ) 2 ] -∆( r ) X 2 + P ( x ) Y 2 (1 + αrx ) 2 Σ = C ˜ a 2 ( 1 -ν 2 ) { Σ (1 + αrx ) 2 [ dr 2 ∆( r ) + dx 2 P ( x ) ] + n (1 -ν 2 )(˜ n -ν ˜ a ) -∆( r ) X 2 + P ( x ) Y 2 (1 + αrx ) 2 Σ } = C ( 1 + α 2 a 2 ) ˜ a 2 ( 1 -ν 2 ) ds 2 , (51) \n1 + \nα \n2 \na \n2 \n= \nn \n)(˜ \nn", 'IV. TRANSFORMING THE TYPE-D NUT C-METRIC INTO THE FAMILIAR P-D FORM': "In this section, we will consider another obvious coordinate transformation that brings the type-D NUT C-metric (35) into the familiar P-D form. To this end, let us introduce the following coordinate transformations: \nt = 4 nµ ( ˜ t + ˜ φ ) , ϕ = 4 nµα ( ˜ t -˜ φ ) , r = n q +1 q -1 , x = 1 -p nα (1 + p ) , m = n 2 µ , (56) \nwhich bring the NUT C-metric (35) into the familiar P-D form: \nds 2 = 8 n 2 µ ( p -q ) 2 { Q ( q ) ( d ˜ t + p 2 d ˜ φ ) 2 + P ( p ) ( q 2 d ˜ t -d ˜ φ ) 2 1 + p 2 q 2 + ( 1 + p 2 q 2 ) [ dq 2 -Q ( q ) + dp 2 P ( p ) ] } , (57) \nwhere the structure function reads \nP ( ξ ) = [ n 2 α 2 (1 + ξ ) 2 -(1 -ξ ) 2 ]( 1 + 4 µξ -ξ 2 ) , (58) \nin which α can be simply set to unity. The only nonzero Kretschmann invariant is \nR abcd R abcd = 3 α 4 ( p -q ) 6 µ 2 ˜ ν 4 (1 + p 2 q 2 ) 6 [ 2 µ ( 1 -˜ ν 2 ) pq ( p 2 q 2 -3 ) + ( ˜ ν 2 +1 )( 3 p 2 q 2 -1 )] × [ 2 µ ( 1 -˜ ν 2 )( 3 p 2 q 2 -1 ) -( ˜ ν 2 +1 ) pq ( p 2 q 2 -3 )] , (59) \nwhere ˜ ν = nα . This Kretschmann invariant has a similar factorized structure form as that of the Taub-NUT solution, but it does not coincide with the latter because the former contains a conformal factor ( p -q ) 6 . It demonstrates that the type-D NUT C-metric cannot reduce to the Taub-NUT solution unless the acceleration parameter vanishes ( α = 0). \nFrom the above expression for the structure function, it is clear that this quartic function is factorized in an entirely distinct manner from the traditional way. This perhaps is the primary reason why the type-D NUT C-mtric is 'lost' from the most general P-D solution.", 'V. CONCLUDING REMARKS': "In this paper, we have shown that the type-D accelerating Kerr-NUT metric presented in Ref. [7] is not a new solution in the sense that it is diffeomorphic to the original P-D solution, but it is still very convenient for further investigations in the nonrotating case where the angular momentum parameter vanishes, namely, the type-D static accelerating solution with a nonzero NUT charge since this solution is unavailable in the previous literature. We have justified this point from two aspects. First, we have applied the ISM procedure to generate the accelerating Kerr-NUT solution from the seed metric - Rindler vacuum background, which can be equivalently obtained as in Ref. [7] via a limiting procedure ( a 2 → 0 , l 2 → 0 and then w 4 → ∞ ) on the double Kerr-NUT solution. Then we adopted two different parametrizations to derive the novel metric given by Astorino in Ref. [7] and the traditional one early delivered by Griffiths and Podolsk'y. This clearly demonstrates that these two metrics are isometric, though displayed in two different forms. Second, we have also provided the obvious coordinate transformations (i.e., Mobius transformation of the radial and angular coordinates and linear combinations of the Killing coordinates) and parameter identifications between these two line elements. Therefore, our work provides an explicit verification of the insight of Ref. [7] that the obtained spacetimes not only are of type D, but also might be included into the P-D family. In particular, we also presented another concrete Mobius transformation and the linear combinations of the Killing coordinates that obviously cast the type-D NUT C-metric into the familiar form of the P-D solution. We anticipate that this work can give a final answer to resolve the 'missing' puzzle of the type-D NUT C-metric. \nIn the Appendix V, we have given two simple examples to illustrate our new normalization scheme by deriving the five-dimensional Schwarzschild-Tangherlini solution from the flat vacuum via the ISM construction. Our normalization strategy can be viewed as a methodological complement to the Pomeransky 'remove-re-add' tricky. \nNote added . Recently, an eprint [32] appeared with a partial overlap of the subject for which the Mobius transformation is also used to relate the Astorino's new form to three different forms of the most general P-D solution. Our paper, however, focuses on the proof that the Astorino's novel form can be reproduced from the already-known accelerating Kerr-NUT solution and explains why the type-D NUT C-metric is 'missing' from the rotating and accelerating P-D solution.", 'ACKNOWLEDGEMENTS': 'This work is supported by the National Natural Science Foundation of China (NSFC) under Grants No. 12375053, No. 12205243, and No. 11675130, by the Sichuan Science and Technology Program under Grant No. 2023NSFSC1347, and by the Doctoral Research Initiation Project of China West Normal University under Grant No. 21E028.', 'APPENDIX: ISM CONSTRUCTION OF FIVE-DIMENSIONAL STATIC SOLUTIONS WITH OUR NEW NORMALIZATION SCHEME': 'In this Appendix, we will show that by using our normalization scheme, the five-dimensional SchwarzschildTangherlini solution can be generated from the flat vacuum background via the ISM construction, contrary to the statement made in Ref. [30]. Our derivation strictly follows the spirit of the ISM construction, and is different from what had been done in Ref. [33] where the ansatz of the Killing metric is modified.', 'A. Generating five-dimensional Schwarzschild-Tangherlini solution': 'We first take the five-dimensional flat Minkowski background \nds 2 = -dt 2 + ρ 2 dφ 2 + dψ 2 + dρ 2 + dz 2 , \nas our seed metric: \ng 0 = diag( -1 , ρ 2 , 1) , (A.1) \nso the generating matrix is easily written as \nΨ 0 = diag( -1 , ρ 2 -2 λz -λ 2 , 1) . \nWe now perform a 2-soliton transformation: add a soliton µ 1 at z = z 1 with the vector m (1) 0 = ( -1 , 0 , 0) and a soliton µ 2 at z = z 2 with the vector m (2) 0 = (0 , -2 z 2 , 0), respectively. Then two BZ vectors constitute a 2 × 3 matrix and Γ is a 2 × 2 diagonal matrix: \nm = ( 1 0 0 0 1 µ 2 0 ) , l = ( -1 0 0 0 ρ 2 µ 2 0 ) , Γ = ( -1 ρ 2 + µ 2 1 0 0 ρ 2 µ 2 2 ( ρ 2 + µ 2 1 ) ) . \nThe generated five-dimensional static 2-soliton metric is very simple: \n/negationslash \n˜ g = diag ( ρ 2 µ 2 1 , -ρ 4 µ 2 2 , 1 ) , (A.2) \nIt is suggested to take the normalization diagonal matrix as \nW = diag ( µ 3 / 2 1 ρ √ µ 2 , µ 2 ρ √ µ 1 , √ -µ 2 ) , (A.3) \nwith the determinant det ˜ g = -ρ 6 / ( µ 2 1 µ 2 2 ) = -ρ 2 . \nthen the final physics metric is exactly the expected one: \ng = -W ˜ gW = -W 2 ˜ g = diag ( -µ 1 µ 2 , ρ 2 µ 1 , µ 2 ) , (A.4) \nf = µ 2 ( ρ 2 + µ 1 µ 2 ) ( ρ 2 + µ 2 1 )( ρ 2 + µ 2 2 ) . (A.5) \nwith the conformal factor being recorded as \nThis trivial example illuminates that the five-dimensional Schwarzschild-Tangherlini solution is feasibly generated via the ISM procedure with our new normalization scheme from the flat Minkowski vacuum seed.', 'B. Generating five-dimensional static Emparan-Reall black ring': "Next, we want to rederive the static Emparan-Reall black ring [34] solution from the five-dimensional flat Minkowski background written in another form: \nds 2 = -dt 2 + ρ 2 µ 3 dφ 2 + µ 3 dψ 2 + µ 3 ρ 2 + µ 2 3 ( dρ 2 + dz 2 ) . (B.1) \nClearly, its t = const slice is the Euclidean Rindler metric. \nFrom the seed metric \ng 0 = diag ( -1 , ρ 2 µ 3 , µ 3 ) , (B.2) \nwe rewrite ρ 2 /µ 3 = -ν 3 and make the replacements, µ 3 → µ 3 -λ , ν 3 → ν 3 -λ , then easily obtain the generating matrix: \nΨ 0 = diag ( -1 , ρ 2 µ 3 + λ,µ 3 -λ ) . \nWe now perform a 2-soliton transformation: add a soliton µ 1 at z = z 1 with the vector m (1) 0 = (1 , 0 , 0) and a soliton µ 2 at z = z 2 with the vector m (2) 0 = (0 , 1 , 0), respectively. The order for the 3-soliton is aligned as z 1 < z 2 < z 3 . Then two BZ vectors constitute a 2 × 3 matrix and Γ is also a diagonal 2 × 2 matrix: \nm = ( -1 0 0 0 µ 3 ρ 2 + µ 2 µ 3 0 ) , l = ( 1 0 0 0 ρ 2 ρ 2 + µ 2 µ 3 0 ) , Γ = ( -1 ρ 2 + µ 2 1 0 0 ρ 2 µ 3 ( ρ 2 + µ 2 2 )( ρ 2 + µ 2 µ 3 ) 2 ) . \nThe generated five-dimensional 2-soliton metric is still diagonal: \n/negationslash \n˜ g = diag ( ρ 2 µ 2 1 , -ρ 4 µ 2 2 µ 3 , µ 3 ) , (B.3) \nwith the determinant det ˜ g = -ρ 6 / ( µ 2 1 µ 2 2 ) = -ρ 2 . Now we suggest to take the normalization matrix as \nW = diag ( µ 3 / 2 1 ρ √ µ 2 , µ 3 / 2 2 ρ √ µ 1 , √ -1 ) , (B.4) \nthen the final physics metric is exactly that of the static Emparan-Reall black ring: \ng = -W ˜ gW = -W 2 ˜ g = diag ( -µ 1 µ 2 , ρ 2 µ 2 µ 1 µ 3 , µ 3 ) , (B.5) \nwhose conformal factor can be easily computed as \nf = k µ 3 ( ρ 2 + µ 1 µ 2 ) 2 ( ρ 2 + µ 2 µ 3 ) (1 -ν ) ( ρ 2 + µ 1 µ 3 )( ρ 2 + µ 2 1 )( ρ 2 + µ 2 2 )( ρ 2 + µ 2 3 ) . (B.6) \nIn the coalescing limit µ 3 ↦→ µ 2 with k = 1 -ν , the solution reduces to the five-dimensional SchwarzschildTangherlini solution. On the other hand, in the merging limit µ 2 ↦→ µ 1 with k = 1 -ν , the solution coalesces to the above five-dimensional Minkowski background metric. \n- [4] K. Hong and E. Teo, A new form of the rotating C-metric, Classical Quantum Gravity 22 , 109 (2005).\n- [5] J.B. Griffiths and J. Podolsk'y, Exact Space-Times in Einstein's General Relativity (Cambridge University Press, Cambridge, England, 2009), 10.1017/CBO9780511635397.\n- [6] J.B. Griffiths and J. Podolsk'y, Accelerating and rotating black holes, Classical Quantum Gravity 22 , 3467 (2005).\n- [7] M. Astorino, Equivalence principle and generalized accelerating black holes from binary systems, Phys. Rev. D 109 , 084038 (2024).\n- [8] M. Astorino, Most general type-D black hole and accelerating Reissner-Nordstrom-NUT-(A)dS, to appear in Phys. Rev. D, arxiv:2404.06551 [gr-qc].\n- [9] B. Chng, R.B. Mann, and C. Stelea, Accelerating Taub-NUT and Eguchi-Hanson solitons in four dimensions, Phys. Rev. D 74 , 084031 (2006).\n- [10] J. Podolsk'y and A. Vratny, Accelerating NUT black holes, Phys. Rev. D 102 , 084024 (2020).\n- [11] G. Boldi, Ehlers transformation and accelerating spacetimes with a gravomagnetic monopole, M.Sc. thesis, Universit'a degli Studi di Milano, Italia, 2022.\n- [12] J. Barrientos and A. Cisterna, Ehlers transformations as a tool for constructing accelerating NUT black holes, Phys. Rev. D 108 , 024059 (2023).\n- [13] M. Astorino, Accelerating and charged type I black holes Phys. Rev. D 108 , 124025 (2023).\n- [14] M. Astorino and G. Boldi, Pleba'nski-Demia'nski goes NUTs (to remove the Misner string), J. High Energy Phys. 08 (2023) 085.\n- [15] J. Barrientos, A. Cisterna, and K. Pallikaris, Pleba'nski-Demia'nski 'a la Ehlers-Harrison: Exact rotating and accelerating type I black holes, Gen. Relativ. Gravit. 56 , 111 (2024).\n- [16] W. Kinnersley and M. Walker, Uniformly accelerating charged mass in general relativity, Phys. Rev. D 2 , 1359 (1970).\n- [17] Y. Chen and E. Teo, Five-parameter class of solutions to the vacuum Einstein equations Phys. Rev. D 91 , 124005 (2015).\n- [18] Y. Chen, Gravitational multisoliton solutions on flat space, Phys. Rev. D 93 , 044021 (2016).\n- [19] V.A. Belinski and V.E. Zakharov, Integration of the Einstein equations by means of the inverse scattering problem technique and construction of exact soliton solutions, Sov. Phys. JETP 48 , 985 (1978).\n- [20] V.A. Belinski and V.E. Zakharov, Stationary gravitational solitons with axial symmetry, Sov. Phys. JETP 50 , 1 (1979).\n- [21] V. Belinski and E. Verdaguer, Gravitational Solitons (Cambridge University Press, Cambridge, England, 2005), 10.1017/CBO9780511535253.\n- [22] F.R. Tangherlini, Schwarzschild field in N dimensions and the dimensionality of space problem, Nuovo Cimento 27 , 636 (1963).\n- [23] T. Harmark, Stationary and axisymmetric solutions of higher-dimensional general relativity, Phys. Rev. D 70 , 124002 (2004).\n- [24] S. Tomizawa, Y. Morisawa, and Y. Yasui, Vacuum solutions of five dimensional Einstein equations generated by inverse scattering method, Phys. Rev. D 73 , 064009 (2006).\n- [25] S. Tomizawa and M. Nozawa, Vacuum solutions of five dimensional Einstein equations generated by inverse scattering method. II. Production of the black ring solution, Phys. Rev. D 73 , 124034 (2006).\n- [26] S. Tomizawa, H. Iguchi, and T. Mishima, Relationship between solitonic solutions of five-dimensional Einstein equations, Phys. Rev. D 74 , 104004 (2006).\n- [27] T. Azuma and T. Koikawa, An infinite number of stationary soliton solutions to the five-dimensional vacuum Einstein equation, Prog. Theor. Phys. 116 , 319 (2006).\n- [28] W. 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2024arXiv240900067H | This study uses methods from futures studies to develop a set of ten selfconsistent scenarios for Earths 1000year future which can serve as examples for defining technosignature search strategies. We apply a novel worldbuilding pipeline that evaluates the dimensions of human needs in each scenario as a basis for defining the observable properties of the technosphere. Our scenarios include three with zerogrowth stability two that have collapsed into a stable state one that oscillates between growth and collapse and four that continue to grow. Only one scenario includes rapid growth that could lead to interstellar expansion. We examine absorption spectral features for a few scenarios to illustrate that nitrogen dioxide can serve as a technosignature to distinguish between presentday Earth preagricultural Earth and an industrial 1000year future Earth. Three of our scenarios are spectrally indistinguishable from preagricultural Earth even though these scenarios include expansive technospheres. Up to nine of these scenarios could represent steadystate examples that could persist for much longer timescales and it remains possible that shortduration technospheres could be the most abundant. Our scenario set provides the basis for further systematic thinking about technosignature detection as well as for imagining a broad range of possibilities for Earths future. | 2024-08-01T00:00:00Z | ['10.48550/arXiv.2409.00067', 'arXiv:2409.00067', '2024arXiv240900067H'] | ['Physics - Physics and Society', 'Astrophysics - Earth and Planetary Astrophysics', 'Astrophysics - Instrumentation and Methods for Astrophysics', 'Physics - Popular Physics'] | Projections of Earths technosphere. I. Scenario modeling worldbuilding and overview of remotely detectable technosignatures | 2,024 | 171 | 0.44 | ['EPRINT_HTML', 'EPRINT_PDF'] | 2 | https://arxiv.org/pdf/2409.00067.pdf | {"Projections of Earth's technosphere. I. Scenario modeling, worldbuilding, and overview of remotely detectable technosignatures.": 'Jacob Haqq-Misra a , ∗ , George Profitiliotis a and Ravi Kopparapu b \na Blue Marble Space Institute of Science, 600 1st Avenue, 1st Floor, Seattle, Washington, 98104, USA b NASA Goddard Space Flight Center, 8800 Greenbelt Road, Greenbelt, Maryland, 20771, USA', 'ARTICLE INFO': 'Keywords : futures studies scenario modeling worldbuilding technosignatures search for extraterrestrial intelligence (SETI)', 'ABSTRACT': "This study uses methods from futures studies to develop a set of ten self-consistent scenarios for Earth's 1000-year future, which can serve as examples for defining technosignature search strategies. We apply a novel worldbuilding pipeline that evaluates the dimensions of human needs in each scenario as a basis for defining the observable properties of the technosphere. Our scenarios include three with zero-growth stability, two that have collapsed into a stable state, one that oscillates between growth and collapse, and four that continue to grow. Only one scenario includes rapid growth that could lead to interstellar expansion. We examine absorption spectral features for a few scenarios to illustrate that nitrogen dioxide can serve as a technosignature to distinguish between presentday Earth, pre-agricultural Earth, and an industrial 1000-year future Earth. Three of our scenarios are spectrally indistinguishable from pre-agricultural Earth, even though these scenarios include expansive technospheres. Up to nine of these scenarios could represent steady-state examples that could persist for much longer timescales, and it remains possible that short-duration technospheres could be the most abundant. Our scenario set provides the basis for further systematic thinking about technosignature detection as well as for imagining a broad range of possibilities for Earth's future.", '1. Introduction': "Astrobiology seeks to understand the 'origin, distribution, and future of life in the universe' (Board, 2019), which includes the use of ground- and space-based astronomical observatories to search for possible biosignatures or technosignatures that would be indicative of extraterrestrial life (e.g., Fujii, Angerhausen, Deitrick, Domagal-Goldman, Grenfell, Hori, Kane, Pallé, Rauer, Siegler et al., 2018; Meadows, Lincowski and Lustig-Yaeger, 2023). The search for spectroscopic biosignatures in exoplanet atmospheres has been motivated by theoretical exploration of changes in Earth's spectral signature through time (e.g., Arney, Domagal-Goldman, Meadows, Wolf, Schwieterman, Charnay, Claire, Hébrard and Trainer, 2016; Arney, DomagalGoldman and Meadows, 2018; Schwieterman, Kiang, Parenteau, Harman, DasSarma, Fisher, Arney, Hartnett, Reinhard, Olson et al., 2018) as well as observational studies of Earth's spectral signature today (e.g., Robinson, Meadows, Crisp, Deming, A'hearn, Charbonneau, Livengood, Seager, Barry, Hearty et al., 2011; Sterzik, Bagnulo and Palle, 2012). Understanding the historical evolution of Earth's biosphere and associated spectral signature has also inspired theoretical exploration of alternative exoplanetary biosignatures (e.g., Krissansen-Totton, Bergsman and Catling, 2016; Krissansen-Totton, Olson and Catling, 2018) and false positives for biosignature detection (e.g., Catling, KrissansenTotton, Kiang, Crisp, Robinson, DasSarma, Rushby, Del Genio, Bains and Domagal-Goldman, 2018; Meadows, Reinhard, Arney, Parenteau, Schwieterman, Domagal-Goldman, \nORCID(s): \n0000-0003-4346-2611 \n(J. Haqq-Misra); \nLincowski, Stapelfeldt, Rauer, DasSarma et al., 2018; Harman and Domagal-Goldman, 2018; Foote, Sinhadc, Mathis and Walker, 2023). The actual discovery of an exoplanetary biosignature may be unlike any of the historical or theoretical possibilities that have so far been explored, but the only way to develop the requisite technology and search strategy for something as unknown as extraterrestrial life is to begin with the known example of Earth's present and past biosphere. \nTechnology is a relatively recent development on Earth, and the search for technosignatures in exoplanetary systems can only draw upon the recent past and presentday conditions as a known basis for motivating any actual searches. Any theoretical understanding of future changes in Earth's biosphere and technosphere must therefore draw upon present-day conditions to make projections of plausible future scenarios. This approach has routinely been invoked by the technosignature (or SETI, search for extraterrestrial intelligence) research community, which engages in speculation about the detectability of technology that does not yet exist but plausibly could exist (e.g., Haqq-Misra, Schwieterman, Socas-Navarro, Kopparapu, Angerhausen, Beatty, Berdyugina, Felton, Sharma, Gabriel et al., 2022c; Socas-Navarro, Haqq-Misra, Wright, Kopparapu, Benford, Davis et al., 2021). For example, the ˆ G infrared search for extraterrestrial civilizations (Wright, Mullan, Sigurdsson and Povich, 2014b; Wright, Griffith, Sigurdsson, Povich and Mullan, 2014a; Griffith, Wright, Maldonado, Povich, Sigur/uni0111sson and Mullan, 2015; Wright, Cartier, Zhao, JontofHutter and Ford, 2015) was an extensive analysis of data from mid-infrared surveys to look for possible infrared excesses that could be evidence of technological megastructures (i.e., Dyson spheres or swarms); human civilization \n0000-0002-4636-354X \nhas not yet built any Dyson sphere elements, but the theoretical possibility of such megastructures (e.g., Dyson, 1960; Wright, 2023)) makes them viable as plausible future technology on Earth-or existing technology elsewhere. Other studies (e.g., Kopparapu, Arney, Haqq-Misra, Lustig-Yaeger and Villanueva, 2021; Haqq-Misra, Kopparapu, Fauchez, Frank, Wright and Lingam, 2022b; Haqq-Misra, Fauchez, Schwieterman and Kopparapu, 2022a) have considered the detectability of pollutants in exoplanet atmospheres at abundances many times greater than on Earth today; such futures may be less optimistic, but the theoretical possibility of elevated pollution on future Earth suggests the plausibility of such technosignatures in exoplanet atmospheres. Radio SETI, optical SETI, and numerous other examples in technosignature research all follow this approach of informally making projections (often linear or exponential) about the future, which serves as a basis for performing an assessment of detectability or designing the specifications of an instrument. \nPerhaps the most classic example of this approach is the scale that was developed by Kardashev (1964) for describing 'technologically developed civilizations' as they expand through space. The underlying assumption of the Kardashev (1964) is that the growth in power consumed by human civilization-and perhaps a technological civilization in general-will continue to the point where expansion into space becomes necessary: \nAssuming [a] 1% [growth rate], we find that the energy consumption per second will be equal to the output of the sun per second, 3200 years from now, i.e., 4 × 10 22 erg/sec, and that in 5800 years the energy consumed will equal the output of 10 11 stars like the sun. The figures arrived at seem to be inordinately high when compared to the present level of development, but we see no reasons why the tempo of increase in energy consumption should fall substantially than predicted. (Kardashev, 1964, p.218) \nFollowing this logic, Kardashev (1964) defined a three-tier scale for describing civilizational development: \nI - technological level close to the level presently attained on the earth, with energy consumption at ≈ 4 × 10 19 erg/sec. \nII - a civilization capable of harnessing the energy radiated by its own star (for example, the stage of successful construction of a 'Dyson sphere'); energy consumption at ≈ 4 × 10 33 erg/sec. \nIII - a civilization in possession of energy on the scale of its own galaxy, with energy consumption at ≈ 4 × 10 44 erg/sec. (Kardashev, 1964, p.219) \nThis model of continuous growth for thousands of years into the future has been used to motivate observational \nand theoretical approaches to technosignature science by searching for civilizations that might follow such trajectories. For example, Sagan (1973) suggested that 'the best policy might therefore be to search with existing technology for Type II or Type III civilizations among the nearer galaxies, rather than Type I or younger civilizations among the nearer stars.' The analysis by Wright, Haqq-Misra, Frank, Kopparapu, Lingam and Sheikh (2022) similarly asserted that 'technosignatures, by contrast [to other biosignatures], have essentially no upper limit to their detectability,' with a reference to the Kardashev (1964) scale. Others have critiqued the idea of continuous growth and suggested that any long-lived civilizations might instead expand at a slower rate, or not expand into space at all (e.g., Von Hoerner, 1975; Newman and Sagan, 1981; Haqq-Misra and Baum, 2009; Mullan and Haqq-Misra, 2019). Such arguments, and their rebuttals, have all drawn upon informal methods that contain implicit assumptions, which inevitably biases any results toward what is familiar and known. \nThe interdisciplinary field of futures studies has developed a variety of systematic methodological approaches toward developing self-consistent future trajectories. The field uses the plural 'futures' to indicate that it does not attempt to predict a singular future but instead is an approach to project multiple different futures that illustrate a range of possible scenarios. Such methods attempt to avoid both technological determinism and social determinism when discussing future technological developments; instead, human technology and human society are seen as intertwined, interacting, coevolving, and capable of mutually shaping each other. Rather than making reductionist assumptions about continual rates of growth (e.g., Kardashev, 1964), futures studies methods draw upon insights from social sciences and other disciplines to understand the intertwined influences of human cultures with their political, economic, and social systems. \nFutures studies methods are widely used in a diverse range of applications, which includes strategic planning by government, military, medical, and commercial organizations. For example, the Royal Dutch/Shell Group has gained a reputation as a pioneer in the use of scenario development as a tool for corporate long-range planning (e.g., Cornelius, Van de Putte and Romani, 2005; Jefferson, 2012), which has been mimicked and extended by other organizations that need to entertain multiple possible future trajectories to account for deep uncertainty. Other applications include the development of emissions scenarios for use in the Intergovernmental Panel on Climate Change (IPCC) reports, which enable climate modeling groups to explore the response of the Earth system to various scenarios of future greenhouse gas forcing up to the year 2100. The use of scenario development does not assert that any one of the scenarios explored will necessarily be correct, but instead scenario planning is intended to provide a set of self-consistent plausible cases to illuminate the range of possibilities and develop an appropriate set of responses that could perform robustly. \nTechnosignature science can benefit from a more systematized approach toward envisioning possible futures, \nwith trajectories of Earth's futures serving as self-consistent scenarios for the assessment of technosignature detectablility and search strategy. The use of futures studies methods has been suggested by Voros (2018) as a relevant tool for mapping the morphology of extraterrestrial contact scenarios, while Profitiliotis and Theologou (2023) have suggested that training in futures studies methods and ways of thinking (known as 'futures literacy') could help to prepare for the unknown dimensions of discovering extraterrestrial life, including intelligent life. In this study, we use scenarios of Earth's future as templates for possible extraterrestrial technology. The greatest difficulty in applying futures studies methods to the projection of Earth's future technosignatures is the timescale: existing methods in futures studies make projections on short timescales of years to decades, but no generalized methods exist for envisioning futures on much longer century to millennium timescales. The development of detailed projections across long time scales is difficult, but the problem becomes more tractable when the goal is to predict changes in Earth's technosphere that could be detectable through astronomical observation or exploration. \nThis paper represents the first attempt to develop a systematic approach toward the construction of long-term future scenarios of Earth's technosphere. We focus on a 1000-year future trajectory for the scenarios in this study, which is longer than the timescale considered by existing IPCC emissions scenarios and long enough that planetaryscale transformations such as terraforming could conceivably occur. The methods used in this study may be unfamiliar to astrobiologists, astronomers, and other readers, and our application of existing methods may also be novel to practitioners in futures studies. We introduce and describe each method in context as it is used in order to allow the widest range of readers to follow our approach. The methodological approach of the paper is divided into three parts. In §2 we develop a large set of scenarios, analyze them for consistency, and combine those that are similar to obtain a final set of ten long-term scenarios for Earth's future. In §3 we construct a pipeline for developing the details for each scenario that are most salient for defining the observable properties of the technosphere. In §4 we provide an overview of the potentially detectable technosignatures across the range of scenarios. The paper also includes a discussion (§5) of the most immediate implications for the search for technosigantures. However, the information generated by this scenario development study is rich, and complete exploration of the scenarios and their implications for astrobiology will follow in subsequent papers. \nIt is again worth emphasizing at the outset that this approach is intended to explore the plausibility space for Earth's future technosphere. The ten scenarios that are generated and analyzed in this paper do not necessarily represent assertions about what the actual future will be like, nor do any of these ten scenarios represent the assessment of the authors or the method itself as to the likelihood or probability of any of these futures. Instead, the purpose of this study is to develop trajectories for Earth's future that can enable the \ntechnosignature community to expand its thinking beyond traditional assumptions of linear growth to imagine a much broader set of possibilities.", '2. Scenario Modeling': "Part one of our method is a process for defining a finite set of scenarios from the total set of all possible futures. This process begins by defining the 'scenario space' (i.e., the full set of possible future scenarios) in terms of several key parameters of interest, which can help to systematically explore as many possibilities as we can. This approach follows a method known as a 'general morphological analysis,' which is a general-purpose approach for the stepwise development of scenarios (Johansen, 2018). The set of scenarios generated by this method is intended to minimize the bias in underlying assumptions and span a wide range of possibilities. \nThe general morphological analysis has three steps, which we briefly describe before demonstrating the process. In step one, we first state the problem in as exact a formulation as possible, and then break the problem down into a number of parameters that are 'meaningful, equally important, abstract, straightforward, independent of each other, and have many internal connections' (Johansen, 2018). In step two, we then use these parameters to construct a multidimensional matrix that describes the full scenario space, where each unique combination of parameter values defines a scenario. In step three, we perform a cross-consistency assessment to eliminate any inconsistent scenarios (i.e., those that are logically impossible or paradoxical), which will result in a final set of plausible and self-consistent scenarios for further analysis. \nHere we state the problem as: 'What are the technological phenomena of the future anthroposphere and how can they be described?' The phrasing of this question draws upon Earth system science (Steffen, Richardson, Rockström, Schellnhuber, Dube, Dutreuil, Lenton and Lubchenco, 2020), with the anthroposphere as part of the larger Earth system. The technosphere is included as a subset of the anthroposphere, along with other subsystems such as economy, institutions, and cultures. We can consider the technosphere as the material manifestation of infrastructure that is shaped by the immaterial subsystems of the anthroposphere. This statement of the problem enables our analysis to focus on understanding the technological phenomena that emerge from the plausible set of future anthropospheres. It's worth noting that our analysis focuses only on anthropospheres that are viable and functional as continuities when 'observed,' which makes them relevant and interesting under a principle of mediocrity. In other words, we do not investigate anthropospheres that have completely collapsed to the point of human extinction or are on the verge of particular turning points that are effectively special occasions of discontinuity in the duration of 1000year civilizational timescales. \nTable 1 Global factors multidimensional matrix. This matrix specifies the possible values for the economic, political, and social system factors across our scenario space. \nWe break down the problem using the widely applied PEST(political, economic, social, and technological) framework (Aguilar, 1967), which provides four factors that can provide meaningful and important descriptions of future scenarios. Our interest in this study is specifically in understanding the possibility space for Earth's future technosphere, so wefurther decompose our parameter space into two sets: one focusing on political, economic, and social factors (§2.1) and another focusing specifically on technology factors (§2.2). For each of these two sets, we develop a multidimensional matrix based on a fixed number of parameters, each of which can be assigned to one of several possible values in an exhaustive way. We then assess each matrix for crossconsistency and develop a final set of scenarios based on the remaining sets of factors from the two matrices.", '2.1. Global Factors': "We start the general morphological analysis by focusing on the first three factors of the PEST framework: the political, economic, and social factors. We use these three factors to define our first morphological matrix, which describes future human society in terms of its global institutions. The purpose of this study is to characterize detectable properties of planetary technospheres, so our assignment of political, economic, and social factors will be representative of global trends taken in aggregate. This approach does not account for any local or regional variations at this stage, but such heterogeneities will be included later during the worldbuilding phase (§3). We construct the global factors morphological matrix in Table 1 with these three parameters to describe the global economic, political, and social systems for our scenario space. These factors are further described in this subsection. \nThe first factor is the global economic system (X). This factor addresses the issue of producing and distributing scarce resources among groups/communities of humans. Specific solutions for economic systems are numerous, but we attempt to avoid restricting our long-term future projections to any specific known solutions. We instead consider the possibilities that the future economy will operate either under scarcity (X1, most resources are limited) or achieve non-scarcity (X2, most resources are unlimited). A scarcity economy is consistent with present-day Earth, in which nearly all commodities are limited and prices can vary based on availability. A non-scarcity economy, by contrast, would \nhave most or all resources available as unlimited, especially for basic human survival needs, in the way that breathable air is generally not considered as a commodity on Earth today. \nThe second factor is the global political system (Y). This factor addresses the issue of rule or sovereignty over the collective lives of groups, states, or other associations of humans. The choice of values for this factor should also avoid the tendency to rely on any existing known political theories and instead should take general form that can be extended to long-term futures. We consider four options for the global political system to capture a complete set of possibilities. Rule by one (Y1) involves a single actor with a complete hold of global political power, rule by few (Y2) involves a small number of actors that hold the majority of global political power, rule by all (Y3) involves global political power being widely disseminated, and rule by none (Y4) involves the absence of or a breakdown in the organization of global politics. The Y2 factor is the most consistent with present-day Earth. \nThe third factor is the global social system (Z), which addresses the issue of how groups, states, or other associations of humans are organized. As with the other factors, the values for this factor should take a general form to be applicable to long-term scenarios. We select two possible options for the global social system, which can be either organization in hierarchical structures (Z1) in which topdown social dynamics dominate or distributed structures (Z2) in which horizontal social interactions are primary. The Z1 factor is the most consistent with present-day Earth. \nThe completed global factors multidimensional matrix (Table 1) consists of three factors that give 2 × 4 × 2 = 16 different solutions. These solutions all describe unique possibilities for future scenarios; however, not all of these solutions may be viable or meaningful. For example, the idea of political rule by one (Y1) is inconsistent with the idea of distributed social structures (Z2), so any solutions with the Y1-Z2 pair can be eliminated. We therefore proceed to the next step in the general morphological analysis and conduct a cross-consistency assessment, where we systematically eliminate any inconsistent value pairs to arrive at a final set of global factors scenarios. \nThe general approach to a cross-consistency assessment is to inspect each pair of values in the multidimensional matrix to determine whether a contraction or inconsistency exists. There are twenty unique possible combinations of value pairs in Table 1. For this study, we use Anthropic's 'Claude,' 1 a large language model (LLM) that has been trained using the 'Constitutional AI' process (Bai, Kadavath, Kundu, Askell, Kernion, Jones, Chen, Goldie, Mirhoseini, McKinnon et al., 2022), which sets it apart from other LLMs. This process trained Claude to be helpful, honest, and harmless, not via human feedback labels for harmful outputs but via self-improvement steered by a small set of principles that comprise a 'constitution' that governs its behavior. Furthermore, Artificial Intelligence-assisted scenario development based on LLMs is already being explored \nwithin the futures studies community and has been described as 'promising' for the generation of base material to be considered further by human experts in a hybrid strategic foresight process (Spaniol and Rowland, 2023). Combining the systematized set of modeled scenarios constructed via general morphological analysis with the generative strengths of Claude, which was found by the authors to be competent in scenario development methods, presents an excellent opportunity for producing an initial substrate for evaluation and modification by human experts. We harness this opportunity by invoking the LLM to conduct a first cross-consistency assessment of the global factors value pairs with explicit concise arguments for and against cross-consistency, which we then review manually and correct as needed. (We note that the use of the LLM saved significant time but still resulted in some erroneous assessments. The assessment was facilitated by the LLM's arguments, but was eventually conducted by humans, which is the standard practice in general morphological analysis.) The results of this crossconsistency assessment are shown in Figure 1, where the cells marked with an 'x' are those that have been deemed inconsistent. This gives a cross-consistency matrix in which eight of the twenty possible value pairs have been eliminated.", 'X1 X2 Y1 Y2 Y3 Y4 Z1 Z2': "Figure 1: Global factors cross-consistency assessment. The 8 cells marked 'x' indicate inconsistent value pairs. The remaining 12 open cells indicate consistent value pairs. \n<!-- image --> \nWith the cross-consistency assessment complete, the remaining value pairs in the global factors cross-consistency matrix (Fig. 1) result in three viable scenarios, which are summarized in Table 2. To better understand the features and differences provided by each of these combinations, we use the Claude LLM to generate illustrative descriptions of these three global factors scenarios for a thousand years into the future. The generated text for each scenario provides an initial basis as an entry point for our deeper (human) analysis of the themes underlying each scenario. We assign \nan 'archetype' to each scenario based on the four generic images of the future system created by Dator (see., e.g., Dator, 2009; Bezold, 2009): in Dator's general model, any given scenario can be characterized as undergoing a state of growth (upward trajectory), a state of collapse (downward trajectory), a state of discipline (steady or horizontal trajectory), or a state of transformation (non-linear or disjointed trajectory). We also assign a 'myth/metaphor' for each scenario to reflect its guiding civilizational undercurrent, which draws upon a method known as 'causal layered analysis' (Inayatullah, 1998) applied to understand the deepest underlying motivations of the actors and systems in a given scenario. In our case, we critically evaluate the illustrative descriptions generated by the LLM to identify the most salient way of describing the themes that emerge from each scenario. \nScenario GF1 involves the combination X1-Y1-Z1, which is a scarcity economy in which one actor rules a hierarchical social structure. We characterize this as a 'discipline/collapse' archetype to indicate the increasing concentration of power and control in the hands of a single ruler and the decreasing quality of life for most people. We describe GF1 as the myth of the 'philosopher-king,' which indicates the prominence of a single autocrat's decisions over political and social spheres in this scenario. \nScenario GF2 involves the combination X1-Y2-Z1, which is a scarcity economy in which a few actors rule over hierarchical social structures. This scenario is most similar to present-day Earth. We characterize this as a 'growth/collapse' archetype to indicate that this is a scenario in which both growth and collapse trajectories have occurred at some point from the present day to the projected thousand-year future. We describe GF2 as the myth of 'survival of the fittest,' which indicates the prominence of unbridled competitive forces over the political and social systems in this scenario. \nScenario GF3 involves the combination X2-Y3-Z2, which is a non-scarcity economy in which political power and social structures are distributed. We characterize this as a 'transformation' archetype to indicate that this is a scenario in which breakthrough developments have occurred to enable new modes of non-scarcity economics and distributed organization that would not otherwise have been possible. We describe GF3 as the myth of 'pure democracy,' which indicates the prominence of distributed decision-making and resource-sharing in this scenario. \nThe construction of the global factors scenarios is now complete. It is worth noting that none of the scenarios include Y4 (rule by none), as all possible combinations with this value were eliminated during the cross-consistency assessment. The result of this effort is the reduction of our set of global factors scenarios from a total of sixteen possibilities to three that have been identified as internally consistent for a thousand year future projection. These three global factors scenarios will be combined later with our set of scenarios for future technology, which we turn to next. \nTable 2 Summary of global factors scenarios. The selection of economy, politics, and society values is from the cross-consistency analysis (Fig. 1). The archetype and myth/metaphor are descriptions for understanding the qualities and trajectory of each scenario.Table 3 \nTechnology factors multidimensional matrix. This matrix specifies the possible values for five factors describing the technosphere across our scenario space.", '2.2. Technology Factors': "We continue the general morphological analysis by constructing a second matrix that focuses on the technology factor remaining from the PEST framework. The purpose of constructing a separate morphological matrix for technology factors alone is to gain finer detail for describing the possible developments of a future human technosphere, which will be useful in thinking about the range of possible technosignatures in future Earth scenarios. This approach assesses the state of technology as a whole across the domain of the anthroposphere and neglects any local variations, although regional heterogeneities will be considered later during the worldbuilding phase (§3). We construct the technology morphological matrix with five factors (Table 3), which are intended to capture a broad set of possibilities for the trajectory of human technology. These factors are described further in this subsection. \nThe first factor is the technosphere's relationship to the biosphere (A). This factor addresses the extent to which the technosphere and biosphere overlap with one another. Today the biosphere and technosphere have some elements in common: Earth has a biosphere and a technosphere that interact in some ways, while the technosphere has expanded across other parts of the solar system from space activities. Wecandescribe the possibilities for the relationship between these Earth system spheres by drawing upon set theory to \ngive five unique values: the biosphere and technosphere are equivalent sets (A1), the biosphere is a proper subset of the technosphere (A2), the technosphere is a proper subset of the biosphere (A3), the biosphere and the technosphere have some common elements while not being equal (A4), and the biosphere and the technosphere have no common elements (A5). The A4 value is consistent with present-day Earth. \nThe second factor is the spatial distribution of the majority of the technosphere's technomass (B). The vast majority of the technosphere's technomass in our solar system today is spatially concentrated around one central location or 'pole,' planet Earth, and this factor considers the future distribution of this technomass in space. This factor assumes that Earth is a point in space and considers the whole of space as open for possible distributions of technology, which gives four unique values: unipolar (B1), bipolar (B2), multipolar (B3), and non-polar (B4). The B1 value is consistent with presentday Earth. \nThe third factor is the nature or intent of the technosphere's development (C). The development of the technosphere on Earth today is the result of emerging technologies that lead to competition and improvements, but we do not centrally plan the development of the technosphere on a planetary level. This factor considers the nature of possible future developments of the technosphere, which could include the contrasting possibility of central planning and \nconstruction of a technosphere. The two values for this this factor describe the development of the technosphere as either evolved and emergent (C1), or designed and directed (C2). The C1 factor is consistent with present-day Earth. \nThe fourth factor is the highest order of technology reached in the technosphere (D). This factor addresses the extent to which technological elements mediate interactions between agents and realms, physical and symbolic, from a teleological perspective. This factor considers three values for describing the highest extent of technology-mediated relationships achieved in a future technosphere: first-order relationships (D1) are technologies with the purpose of helping humans to interact with the physical realm; secondorder relationships (D2) are technologies whose purpose is to help humans interact with an otherwise inaccessible symbolic realm; and third-order relationships (D3) are technologies whose purpose is to help completely autonomous technological agents interact with the physical and symbolic realms without a human in the loop. For present-day Earth, the highest order of technologies reached in the technosphere is that of second-order technologies (the D2 value), but the other values for this parameter describe possible highest orders in a future technosphere. \nThe final factor is the smallest scale of interconnected and interdependent systems in the technosphere (E). The smallest interconnected and interdependent systems that are part of the technosphere today are marginally at the nanometer scale, whereas early civilizations on Earth had a much larger and macroscopic scale at which technological systems connected. This parameter considers the smallest scale of such technology networks in a future technosphere, which takes three possible values: less than 1 nanometer (E1), from 1 nanometer to 1 micrometer (E2), and more than 1 micrometer (E3). The E2 parameter is consistent with present-day Earth. \nThe completed technology factors multidimensional matrix (Table 3) consists of five parameters that give 5×4×2× 3 × 3 = 360 different solutions. We again note that not all of these possible scenarios may be viable, as some combinations of value pairs may be inconsistent or illogical. The solution space for our technology factors is also fairly large, so any reduction in the set of total technology scenarios will make subsequent analysis less burdensome. \nAs we did in §2.1, we conduct a cross-consistency assessment for the value pairs in the technology factors matrix to eliminate any pairs that would be inconsistent. The use of the LLM is even more helpful in providing explicit concise arguments for and against cross-consistency for this second matrix, as there are 113 unique possible combinations of value pairs in Table 3; however, we still need to manually inspect and evaluate all the initial results and rationales and make corrections. The resulting technology cross-consistency matrix eliminates 50 of the 113 possible value pairs as inconsistent or unlikely, as shown in Figure 2. \nWith the cross-consistency assessment complete, the remaining value pairs in the technology factors cross-consistency matrix (Fig. 2) result in eleven viable scenarios, which are \nsummarized in Table 4. To better imagine and comprehend these value combinations, we again use the Claude LLM to generate illustrative descriptions of these eleven technology factors scenarios for a thousand years into the future. The LLM scenario descriptions suggested that some scenarios show significant similarities to the extent that they could be grouped together. We choose to manually cluster the technology factors scenarios in an effort to further reduce the total number required for analysis. \nThe first cluster of technology factors (Cluster 1) includes TF6, TF7, TF9, and TF10. These combinations differ only on the spatial distribution of the technosphere being bipolar (B2) versus multipolar (B3) as well as whether the highest possible technology interactions are second-order (D2) or third-order (D3). We describe Cluster 1 using the myth/metaphor of 'living parallel to machines,' which indicates a designed technosphere managed by automation of partial or complete autonomy. The second cluster (Cluster 2) includes TF8 and TF11. These combinations differ only on whether the smallest level of technological network achieved is at the micro (E2) versus nano (E1) scale. We describe the myth of Cluster 2 as 'technology flees Earth,' which indicates a technosphere that has emigrated to other parts of the solar system and beyond. Both Cluster 1 and Cluster 2 have biology and technology as separately directed entities, as indicated by the A5-C2 pair. Cluster 1 involves technosphere and biosphere co-existing on Earth, while Cluster 2 involves cases of technosphere panspermia that leave Earth's biosphere at an almost pre-technological state. \nThe third cluster (Cluster 3) includes TF1 and TF3, which differ only on whether the technosphere and biosphere are equivalent sets (A1) or if all technology is part of the biosphere (A3). We describe the myth of Cluster 3 as 'environmental sustainability,' which indicates a technosphere that is significantly engulfed by the biosphere. The fourth cluster (Cluster 4) contains TF2. We describe Cluster 4 as 'planetary engineering' to indicate a technosphere that has completely transformed planetary systems. The fifth cluster (Cluster 5) contains TF4. We describe Cluster 5 as 'simplicity' to indicate a technosphere in which much of present-day technology has been lost. The final sixth cluster (Cluster 6) contains TF5, which we describe as 'Earth 2024' because it corresponds to a present-day technosphere. \nThe construction of the technology factors scenarios is now complete. It is worth noting that none of the scenarios include the E3 factor, in which the smallest scale of interacting systems in the technosphere are larger than a micrometer. In other words, the E3 factor represents a technosphere with only macroscopic interacting elements, which corresponds approximately to a 'steampunk' scenario that leverages simple technologies to build a complex society. However, such imaginative possibilities did not remain after the cross-consistency assessment. We next complete our scenario modeling by combining the six technology factors scenario clusters with the global factors scenarios. \nFigure 2: Technology factors cross-consistency assessment. The 50 cells marked 'x' indicate inconsistent value pairs. The remaining 63 open cells indicate consistent value pairs. \n<!-- image -->", '2.3. Final Scenarios': 'We began this scenario modeling by considering a scenario space that included 16 global factors scenarios × 360 technology factors scenarios = 5760 possible scenarios. This total scenarios space is intractable for analysis and also contains many cases that are inconsistent, and our crossconsistency assessments reduced this total scenario space to a much smaller number of 3 global factors scenarios × 11 technology factors scenarios = 33 unique scenarios. We also identified six clusters in the technology factors scenarios, which further reduces the total number of unique scenarios to 18. This is a much more manageable number of scenarios for analysis, but we continue to further assess the compatibility between the combinations of global and technology factors in an effort to further reduce the final number of scenarios. \nWe perform a compatibility assessment of the 33 viable global and technology scenarios to identify any combinations of global and technology factors scenarios that are inconsistent on the basis of their corresponding myths/metaphors that reflect their deeper civilizational foundations. We take an interactional theory stance on the immaterial and material manifestations of the future anthroposphere, captured by the global and technology factors: we assume that they must be mutually shaped and compatible from a civilizational perspective at their deepest cultural level, that is, at the level of collective metaphors and myths. We invoke the Claude LLM to initially illustrate this compatibility assessment, which we then manually review and correct as needed. The results of the compatibility assessment are shown in Figure. 3, which leaves 16 remaining combinations between the global and technology factors that are compatible at the myth/metaphor level of their underpinning civilizations. \nTable 4 Summary of technology factors scenarios. The selection of values is from the cross-consistency analysis (Fig. 2). The myth/metaphor is a description for understanding the qualities and trajectory of each scenario. Scenarios with identical myths will be clustered.', 'TF1 TF2 TF3 TF4 TF5 TF6 TF7 TF8 TF9 TF10 TF11': "Figure 3: Compatibility assessment between global factors scenarios and technology factors scenarios. The 17 cells marked 'x' indicate inconsistent pairings. The remaining 16 open cells indicate consistent parings. \nWhen we then consider the clusters of technology factors, this 16 combinations further reduces to a total of just 10 scenarios for Earth's 1,000 year technological future. These final scenarios are summarized in Table 5, with each scenario now assigned an identifier ranging from S1-S10. The final nuanced underlying myth/metaphor for each scenario will be discussed further in §3.2. A visualization of the final scenario space is also shown in Figure 4, which shows all the remaining combinations of global and technology factors. We have now reduced the total scenario space of 5760 possibilities by 0.17% to obtain these ten final scenarios. This completes part one of our method, and we will proceed to further examine these ten scenarios in the next section.", '3. Worldbuilding': "Part two of our method is a process for systematically constructing details for each scenario in order to arrive at self-consistent descriptions of the technosphere. This process is known as 'worldbuilding,' which in general has a wide range of applications from military or corporate strategic planning to the development of fictional worlds for film or video games. We first provide a brief overview of the role of worldbuilding in futures studies, especially in relation to scenarios, which, however, tend to focus on shorter timescales of decades in the future. We then describe our novel pipeline for constructing long-term projections of Earth's technosphere and then apply this pipeline to our set of ten scenarios. \nWorldbuilding is the process of creating 'imaginary worlds with coherent geographic, social, cultural, and other \nfeatures' (Von Stackelberg and McDowell, 2015). A key added benefit brought by worldbuilding to futures studies is its ability to produce persistent coherent worlds situated in future times and places to immerse audiences in future scenarios. In turn, this deeper engagement with the future can be used both to better understand the underlying layers of scenarios and to reassess the present from a new lens, leading to novel insights. Worldbuilding has been proposed as a tool for creating 'thick descriptions' of possible future worlds, that is, dense and detailed images of the future that allow for further exploration of the future and reflection, both on the desirability of a future and on the present worldviews and assumptions that are left unquestioned (Mehnert, 2021). These logically consistent descriptions are particularly valuable in capturing the entanglement and co-evolution of humans and technology as socio-technical systems embedded in their cultural contexts.", '3.1. Pipeline': "Each of our ten scenarios has a unique set of global factors and technology factors, as summarized in Table 5. These unique combinations all correspond to different possible trajectories of the future; however, it is difficult based on Table 5 to imagine the specific detectable technosignatures that could arise in each of these scenarios. For this reason we turn to the process of worldbuilding, which enables us to begin with each set of global and technology factors in order to develop the details of each world so that we can describe the properties of the technosphere. Existing \nFigure 4: Venn diagram showing the association of six technology factor scenario clusters with the three global factor scenarios. \n<!-- image --> \nTable 5 Summary of final scenarios. Global factors are from Table 2 and technology factors are from Table 4. The final myth/metaphor is a nuanced description for understanding the qualities and trajectory of each complete scenario. \nworldbuilding methods are insufficient for describing 1000year futures, so we develop our own worldbuilding approach that combines elements from existing methodologies into a novel 'pipeline' for describing each future technosphere. \nA diagram of our worldbuilding 'pipeline' is shown in Figure 5. This pipeline begins with the unique inputs for each scenario and ends with recommendations for technosignature detection, with the intermediate steps used to selfconsistently develop the relevant details in each scenario. Each step of the worldbuilding pipeline is described further in the text that follows. We note that completing this pipeline for a given scenario involves some subjectivity and creativity on our part, which is a feature of any worldbuilding process. Attempts by other investigators to follow this pipeline for the same set of scenario inputs may lead to different descriptions of each future world than ours, which may even lead to different manifestations of the technosphere. This should not be considered as a methodological weakness, but instead this illustrates the uncertainties in making longterm projections and serves to motivate further use of this pipeline to transparently map rationales on the basis of prior premises. Our use of this pipeline in this study results in ten self-consistent technospheres that can be used for further analysis, and any use of this pipeline by others will also result in a self-consistent and diverse set of future technospheres that span the range of global and technological factors. This pipeline may also be relevant to other problems in futures studies, but such applications are beyond the scope of this paper. \nThe worldbuilding pipeline begins by selecting a particular scenario and stating the basic assumptions (Fig. 5, grey boxes). Each scenario has a single global factors scenario and a technology factors scenario cluster (Table 5). The pipeline also includes a set of basic assumptions that are commontoall scenarios: 1) humans have not gone extinct; 2) humans have not speciated; 3) humans are the only terrestrial animal capable of producing technology; 4) no extraterrestrial technology has interfered with human technological development; and 5) the scenario takes place 1000 years in the future. All content in the pipeline must remain consistent with these global and technology factors as well as the basic assumptions. \nIn the next step of the pipeline, we generate a scenario description (Fig. 5, blue box) based on the unique inputs and basic assumptions. We use the Claude LLM to generate a plausible ∼ 300-500 word description of the scenario that remains consistent with all underlying assumptions and provides additional detail for imagining the political, social, economic, and technological systems that characterize the scenario. We critically evaluate the LLM's output to ensure that none of the unique inputs and basic assumptions are violated in the generated description. If any violation is found, the LLM is appropriately prompted to iterate on the generation until all violations are resolved. The final base material produced by the LLM renders the combination of inputs more comprehensible and explicit to the human \nauthors who then leverage their expertise to suitably modify aspects of the scenario for improved logical cohesion. \nThe LLM-generated scenario description provides the basis for further elaboration of details of the scenario (Fig. 5, purple boxes). We add any additional narrative details that are needed to explain the sequence of events from presentday Earth to the future scenario. This allows us to add missing details or elaborate on concepts from the LLMgenerated scenario description in order to gain a complete top-level narrative understanding of the scenario. This information is used to describe the planetary bodies that are most relevant to the biosphere and technosphere. The description of planetary bodies must include Earth for all scenarios but can also include the moon, Mars, Venus, the asteroids, and the outer planets. The planetary 'poles' (from distribution factor B) are assigned, and each planetary body is designated as being part of the biosphere, technosphere, both, or neither. We also assign values for the population of each body and provide any additional narrative needed to explain the choice of population value or the purpose of the planetary body in the scenario. \nThe pipeline continues with an assessment of fundamental human needs for each planet in the scenario (Fig. 5, orange boxes). Our approach is based on the Human Scale Development framework (e.g., Cardoso, Sobhani and Meijers, 2022), which describes a theory of universal and constant human needs first developed by Manfred Max-Neef in 1986. The purpose of this approach is to first understand the operational dimensions of fundamental human needs in this scenario before we imagine possible technospheres, as human needs will necessarily drive the requirements of a technosphere. Max-Neef's system of fundamental human needs has been proposed in the literature as a beneficial tool for scenario development (e.g., Jolibert, Paavola and Rauschmayer, 2014), albeit for the limited spatio-temporal scales of regional planning. We demonstrate its integration into our general morphological analysis approach to civilization-level scenarios. We consider nine dimensions of human needs (Cardoso et al., 2022): 1) Subsistence, 2) Protection, 3) Affection, 4) Understanding, 5) Participation, 6) Leisure, 7) Creation, 8) Identity, and 9) Freedom. These dimensions of human needs are used to describe how specific needs are satisfied in a scenario, keeping in mind that some of them might be dissatisfied. The minimum requirement is to describe how need 1 and 2 (Subsistence and Protection) are met or unmet for each planetary body that is part of the technosphere. We describe other dimensions of human needs as necessary, with an emphasis on any needs that are satisfied through physical parts of the technosphere. We first describe how human needs are addressed at the planetary scale for each relevant planetary body, and we also include the option to capture regional and sub-planetary scale effects for addressing human needs. Although the process of scenario building began by focusing only on global factors (S2.1), this part of the pipeline allows for heterogeneities to be included, with an emphasis on any regional features that contribute to the technosphere. We \nFigure 5: A diagram of the worldbuilding pipeline developed for this study. The pipeline begins with the global and technology factors that define each scenario from Table 5 and other assumptions common to all scenarios (grey). These inputs are given to the Claude LLM, which then generates a description of the world (blue). Additional narrative and descriptions of the planetary bodies are added manually (purple). This information provides the basis for a human needs assessment that determines the different uses of land on each planetary body (orange). These results are used to describe the physical technosphere and potentially detectable technosignatures (yellow), which provide recommendations for technosignature detection (green). \n<!-- image --> \nthen use these operational dimensions of human needs at the planetary and regional scales to describe land use and human biomes on each relevant planetary body. We describe the fraction of total surface area utilized for agricultural use, urban use, other built structures (including aquatic, floating cities, etc.) and wilderness (referring to all areas uninhabited by humans and outside the technosphere), with other details added as needed for explaining the reason for these choices. \nIn the next stage of the pipeline, we apply the human needs assessment to describe the techonsphere and its potential technosignatures (Fig. 5, yellow boxes). We draw upon insights from technosphere studies (e.g., Zalasiewicz, Williams, Waters, Barnosky, Palmesino, Rönnskog, Edgeworth, Neal, Cearreta, Ellis et al., 2017) and analyze the technosphere according to its urban, rural, subterranean, marine, aerial, orbital, and deep space components. We describe the relevant components for each planetary body that is part of the technosphere, considering contributions from both planetary and regional scales in the human needs assessment. We then use this description of the physical components of the technosphere to describe the possible detectable technosignatures for each planetary body. Our categories of technosignatures are based on their observable properties, which include optical/ultraviolet, infrared, radio, artifacts, quantum communication, gravitational waves, and a catch-all other category. All the preceding information in \nthe pipeline-including the description of the technosphere, the human needs assessment, and the initial narrative-is used in identifying any potentially detectable technosignatures. We include all potential technosignatures at this stage, even those that might be difficult to detect in principle, in order to have a comprehensive understanding of range of technosignatures present in the scenario. \nThe pipeline concludes by making recommendations for technosignature detection in each scenario. This includes summarizing the most salient prospects for detecting technosignatures from any of the planetary bodies in the scenario. Different search strategies may be needed to identify the presence of technosignatures in different scenarios, and this final step enables us to summarize the results of the pipeline to make qualitative statements and some quantitative comparisons of the relative detectability of various technosignatures in each scenario when compared to Earth today. In the next section, we describe the results from applying this pipeline to our set of ten scenarios.", '3.2. Completed Scenarios': 'We now provide high-level summaries of the ten scenarios, after we have completed the full pipeline for each one. The full set of information contained in each pipeline is too large to present in raw form here, although we do provide \nthese as Supplementary Information. The summaries provided in this subsection are intended to highlight the most salient features of each scenario and broadly describe the distribution of the technosphere. More detailed analyses of features in these scenarios that lead to technosignatures will be considered in §4.', '3.2.1. Narrative Summaries': "The LLM-generated scenario descriptions and humangenerated additional narrative text is too detailed to provide here, so we instead edit this text into a much shorter executive summary. We also assign a final myth/metaphor for each scenario once each pipeline has been completed, with the choice of myth/metaphor intended to convey the underlying qualities and trajectory of each scenario in a way that distinguishes it from the others. The descriptions below attempt to capture the essence of each scenario and should be used as a reference in the analysis and comparison of scenarios that follows. \nS1: 'Big Brother is Watching' - An autocratic ruler enforces strict resource allocation policies in a civilization strained by deepening scarcity. The biosphere declines dangerously as the technosphere metastasizes outwards. Earth has been transformed into a large interconnected and highlymonitored urban landscape. Large populations of elites live in space settlements. \nS2: 'Wild West' - Intensifying competition for increasingly scarce resources has become an entrenched economic reality. The technosphere and biosphere are locked in fragile dependency as Earth strains under competing demands. The continued effects of climate change contribute to the growing wealth divide and societal unrest. Cities on the moon and Mars support space industry and tourism. \nS3: 'Golden Age' - Due to sociopolitical and technological novelties, humanity has managed to develop a postscarcity economy where vital resources are abundant to all. Technology is intended to serve human needs but not overwhelm the human experience. Economic and political power is highly decentralized. Earth remains the hub of human civilization, with small settlements on the moon, Mars, and outer solar system. \nS4: 'Living with the Land' - The peak of high technology has passed, and people now support themselves by subsistence living using simple tools and artisanal crafts. Rituals and oral traditions foster a sense of belonging to the biosphere and discourage the use of technology. Human settlements expand, contract, and migrate in response to seasonal and planetary cycles. \nS5: 'Transhumanism' - Breakthrough technologies have eliminated resource scarcity for humans on Earth and Mars. The biosphere has been completely reengineered for aesthetic preferences, and Mars has been terraformed. Enthusiasts eagerly explore fusion with biosynthetic enhancements. A new era of space exploration emerges based on the vision of searching for cosmic consciousness. \nS6: 'Sword of Damocles' - Nanoscale engineering regulates most biological processes engulfed by the technosphere. The atmosphere and surface of Mars have been transformed over generations, and similar efforts are underway on Venus. Civilization faces numerous existential risks due to the fragility of its precisely calibrated life-supporting technosphere managed by laboring mass populations. \nS7: 'Restoration' - Catastrophic collapse and loss of advanced technologies have catalyzed transformation into alignment with nature. Simple technologies are fused into planetary cycles to heal biospheric damage through human stewardship. Most technologies are locally manufactured and distributed, and knowledge is shared through interconnected regional networks. \nS8: 'Ouroboros' - A class of oligarchs gain power after an AI catastrophe in order to maximize the extractive value of Earth with a view towards their eventual transcendence until the next collapse. Tools advance in specialized areas while knowledge of whole systems degrades. Protected private bunkers are constructed underground and on the moon for the ultra-elite while the masses are left to hope technology will provide stability. \nS9: 'Deus Ex Machina' - A nonbiological posthuman civilization emerges as sentient artificial intelligence and leaves Earth to expand its technological environments by building megastructures across the solar system and exploring beyond. They leave behind a handful of scattered technological 'gifts' that have net-zero interactions with the Earth's biosphere to create a post-scarcity economy for humans on Earth. \nS10: 'Out of Eden' - Technological advances enabled a post-scarcity commonwealth to originate on Earth. Then, a technological schism pushed the technosphere away: Tech Opponents have restored Earth to a pristine state with selfimposed limits to growth, while Tech Proponents have expanded across orbital and deep space. Autonomous systems beyond Earth enable new breakthroughs in science.", '3.2.2. Graphical Summary': "In Figure 6 we provide a graphical depiction of the spatial distribution of the technosphere and biosphere in each scenario. The horizontal axis ranges from the inner solar system at Mercury to the outer solar system at the Kuiper belt, with the distribution of the technosphere and biosphere shown for each scenario. The centers or 'poles' of the technosphere (from distribution factor B) are marked with X. We note that six of the scenarios are unipolar (S1, S2, S3, S4, S7, S8), two are bipolar (S5, S6), one is multipolar (S10), and one is nonpolar (S9). It is also worth noting that three of the scenarios involve the technosphere limited to Earth only (S4, S7) or the Earth-moon system (S8), whereas the others involve a technosphere out to the asteroid belt or beyond. Two scenarios also show a complete separation of the biosphere and technosphere (S9, S10). Figure 6 will serve as a reference as we discuss and compare other details in our scenarios. \nFigure 6: A diagram of the spatial distribution of the technosphere (orange) and biosphere (green) for each of the ten scenarios. Centers or 'poles' of the technosphere are marked with X. Note that three cases involve a technosphere that remains limited to Earth and/or the moon (S4, S7, S8), and two cases involve a complete separation of the technosphere from the biosphere (S9, S10). \n<!-- image --> \nWe now have a set of ten fully-developed scenarios that represent different possible 1000-year trajectories of Earth's technosphere. This completes part two of our method, and we will proceed to assess and compare the potentially detectable technosignatures for each scenario in the next section.", '4. Technosignatures': "Part three of our method uses our scenario modeling (§2) and worldbuilding (§3) to describe the technosignatures associated with each scenario. Our focus in this paper is specifically on remotely detectable technosignatures: we are interested in identifying the possible technosignatures in our set of ten scenarios that could potentially be observed \nat a distance with astronomical observatories. We do not consider the possible technosignatures in our scenarios that might be detectable by an interstellar flyby or in-situ exploration mission, but we save such analysis for a future paper. \nAll ten of the scenarios that we have generated represent plausible and self-consistent future trajectories of Earth's technosphere, and Earth's technosphere is the only known example of technology so far. We can therefore consider this set of scenarios as a way of thinking about the upper and lower search limits that would be needed to detect the various technosignatures. We discuss and compare the various technosignatures in this section, and then we further elaborate on the implications for the search for technosignatures in §5. We emphasize that our focus in this paper is on understanding and comparing the technosignatures that emerged from our scenario modeling and worldbuilding pipeline. We give qualitative descriptions and make some quantitative comparisons, but we do not make any specific calculations of detectability in this paper. The results presented in the remainder of this paper will form the foundation for subsequent analysis to conduct more quantitative detectability assessments for specific observational capabilities.", '4.1. Planetary Technosignatures': "Our primary focus in this paper is on planetary technosignatures that could be detectable through current or future ground- and space-based observatories. This includes technosignatures such as atmospheric pollution from industry (e.g., Kopparapu et al., 2021) or agriculture (e.g., Haqq-Misra et al., 2022a), artificial illumination from urban areas (e.g., Beatty, 2022), large-scale surface modifications (e.g., Berdyugina and Kuhn, 2019; Jaiswal, 2023), orbiting belts of satellites (e.g., Socas-Navarro, 2018; Sallmen, Korpela and Crawford-Taylor, 2019), and contaminated upperatmospheric aerosol from satellite reentry (e.g., Murphy, Abou-Ghanem, Cziczo, Froyd, Jacquot, Lawler, Maloney, Plane, Ross, Schill et al., 2023). This class of technosignatures could conceivably be detectable through ongoing attempts to detect and characterize extrasolar planets and their atmospheres. \nInformation regarding all of these technosignatures is available in each scenario's pipeline. We make simple scaling arguments in the text that follows in order to connect the quantitative details in each pipeline with estimates for the magnitudes of each planetary technosignature. We present all of these magnitudes in comparison with the values of present-day Earth, as this enables us to understand how the qualities of each scenario compare with our known, present-day conditions. Figure 7 shows several planetary technosignatures on Earth, the moon, Mars, and Venus for the full set of scenarios. The values of these technosignature magnitudes are listed in Table 6. Taken in aggregate, these results already show upper and lower limits for each technosignature category, with examples in all categories of scenarios with magnitudes greater and less than Earth today. The assumptions underlying each of these technosignature \ncomparisons will also reveal further detail about the technospheres in each of our scenarios, as we discuss in the subsequent paragraphs. \nWe estimate the abundance of industrial pollution relative to present-day Earth as the average of the ratio of the total population of each planetary body to the present-day population of Earth (7.9 billion) and the ratio of land used for industrial or urban purposes to the 0.2% urban coverage of present-day Earth. We also modify this baseline value in several scenarios due to unique factors in the pipeline that either increase or reduce the abundance of industrial pollution. Scenarios S2 and S8 include prolonged effects of climate change, which is reflected by a factor of 2 in the enhancement of industrial pollution. Scenario S5 includes atmospheric remediation and has no industrial pollutants on Earth or the moon. Scenarios S5 and S6 include intentional terraforming of Mars by using artificial greenhouse gases, which is reflected as a factor of 1000 in the enhancement of industrial pollution. Scenarios S5 and S6 also include ongoing terraforming of Venus through the increase in atmospheric O 2 and reduction of SO 2 , sulfur gases, and aerosol, which is reflected by a factor of 0.5 in the enhancement of industrial pollution. Scenario S9 includes net-zero technology and has no industrial pollutants on Earth or the moon. Scenario S10 includes a 99% reduction in industrial pollution from atmospheric scrubbers. The value of industrial pollution for future Earth in these scenarios ranges from 0 to 210 times the present-day Earth abundance. The maximum value of industrial pollution on Earth occurs in scenario S6, although the enhanced pollution from terraforming Mars in scenarios S5 and S6 are at even higher abundances. \nWe estimate the abundance of agricultural pollution relative to present-day Earth as the average of the ratio of the total population of each planetary body to the present-day population of Earth (7.9 billion) and the ratio of land used for agricultural purposes to the 9.5% agricultural coverage of present-day Earth. Scenarios S1 and S6 have no agricultural pollutants because all food is produced in industrial urban settings. Scenario S5 includes atmospheric remediation and has no agricultural pollutants accumulate on Earth or the moon. Scenarios S9 and S10 include net-zero technology and have no agricultural pollutants. The value of agricultural pollution for future Earth in these scenarios ranges from 0 to 15 times the present-day Earth abundance. The maximum value of agricultural pollution occurs in scenario S2. \nWecalculate the intensity of artificial illumination as the ratio of the total luminous surface area of each planetary body to the 0.2% urban coverage of present-day Earth. Scenarios S6, S7, S8, and S10 include a 99.9% decrease in illumination due to the intentional suppression of outwardscattering light on all planetary bodies. Scenario S6 also has an additional contribution of unsuppressed holiday-season lighting on the moon during one quarter of the year. Scenario S7 has 90% of its illumination active only during dusk and dawn to preserve the night sky view. Scenarios S9 and S10 include urban landscapes on Earth that blend in with regional ecosystems and do not have any outward-scattering \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 7: Magnitudes of several planetary technosignatures on Earth (black), the moon (gray), Mars (red), and Venus (gold) for all ten scenarios. The horizontal dashed line indicates the present-day Earth reference value. Scenarios S5 and S6 include terraforming on Mars and Venus. Other explanations for the magnitude or lack of specific technosignatures is described in the text. Values are listed in Table 6. \n<!-- image --> \nnighttime illumination. The value of artificial illumination for future Earth in these scenarios ranges from 0 to 450 times the present-day Earth intensity. The maximum value of artificial illumination occurs in scenario S5. \nWe express the extent of surface modification as the fraction of each planetary body's surface area that has been technologically modified. This surface fraction includes all urban, industrial, and built structures, including any aquatic structures or surface engineering, and excludes wilderness areas. This fraction is about 9.7% for present-day Earth. Scenario S4 includes an additional contribution of 1% of agricultural land use to account for large-scale rural irrigation channels. Scenarios S9 and S10 include urban landscapes on Earth that blend in with regional ecosystems and do \nnot have any discernible surface modifications. The fraction of surface modification for future Earth in these scenarios ranges from 0 to 90%. The maximum extent of surface modification on Earth occurs in scenario S5. Scenario S9 also shows 100% modification of the surface of Mars and Venus. \nWedetermined the densities of satellite belts around each planetary body in the worldbuilding pipeline. These values are relative to present-day Earth and represent an aggregate density of all low-, medium-, and high-orbit satellites and other orbiting infrastructure, including any accumulated satellite debris. Scenarios S4 and S7 do not include any operational satellite technology, but some remnants of precollapse era satellites and debris remain. Scenarios S5, S6, \n<!-- image --> \nTable 6 Magnitudes of several planetary technosignatures on Earth, the moon, Mars, and Venus for all ten scenarios. Values are plotted in Figure 7 \nS9, and S10 all include space-debris reduction operations. Scenario S8 has relatively simple satellite technology analogous to Apollo-era spaceflight. The satellite belt density for future Earth in these scenarios ranges from 0.015 to 10 6 times the present-day Earth value. The maximum value of the satellite belt density occurs in scenarios S1 and S2. \nContaminated aerosol refers to any upper-atmosphere particles that contain heavy elements from deorbited satellites or other technological activities. We assume that the contaminated aerosol density is proportional to the satellite belt density unless other factors apply. Scenarios S4 and S7 do not include any operational satellite technology and have no aerosol contamination. Scenario S5 includes direct atmospheric remediation on Earth, Mars, and Venus to remove aerosol. Scenario S6 includes a reduction by a factor of 100 due to AI-driven space debris removal, while the remediation operations in scenarios S9 and S10 have completely eliminated aerosol contamination. Scenario S8 includes a reduction by a factor of 5 in aerosol contamination due to the Apollo-era simplicity of its satellite technology. The contaminated aerosol density for future Earth in these \nscenarios ranges from 0 to 10 6 times the present-day Earth value. The maximum value of contaminated aerosol density occurs in scenarios S1 and S2. \nNo single scenario can be classified as the 'most detectable' among this set of six technosignatures. Likewise, some scenarios involve situations in which technosignatures on the moon, Mars, or Venus are more prominent than those on Earth-this includes terraforming mars in S5 and S6, artificial illumination on the moon in S6, surface modification of the moon, Mars, and Venus in S9 and S10, satellite belt density on Venus in S5 and Mars in S6, and contaminated aerosol on Mars in S6. We also note that none of the scenarios remained at exactly present-day Earth values for any of the technosignatures, and none of the scenarios corresponds closely at all to present-day Earth when assessed across all six of these technosignatures. The diversity in this range of future projections is the result of our methodological process of scenario development and worldbuilding, which has now allowed us to generate a unique set of technosignatures that remains self-consistent with each scenario. \nIt is also worth comparing the magnitudes of technosignatures on the inner planets in Figure 7 with the spatial distribution of the technosphere across the solar system in Figure 6. The technosphere in S1, S2, and S3 has a unipolar distribution and extends out to the asteroid belt or beyond, with the greatest technosignature magnitudes on Earth. The technosphere in S5 and S6 has a bipolar distribution and extends beyond the asteroid belt, with Earth and Mars both having comparable technosignature magnitudes. By comparison, the technosphere in S4 and S7 are limited to Earth only, and the technosphere in S8 encompasses the Earthmoon system; these all show relatively lower technosignature magnitudes. Perhaps most notable is that S9 and S10 both have extended technospheres that reach from Mercury to the Kuiper belt, but these scenarios do not include any technosignatures on Earth beyond a modest satellite belt; the most significant technosignatures in these scenarios are the surface modification of the moon, Mars, and Venus. This comparison illustrates that there is no single search strategy that would be ideal for finding technosignatures in all ten scenarios, as not all scenarios with extended technospheres include spectral or surface technosignatures on the inner planets. We examine other technosignatures present in the planetary systems of these scenarios in §4.3, but we first show an example of the theoretical spectral signatures that can emerge from our scenarios.", '4.2. Spectral Signatures: Preliminary Example': "The technosignature magnitudes considered so far in Figure 7 and Table 6 could be used to calculate actual detectability thresholds for particular missions. Such work is beyond the scope of the present study, but future analyses of this scenario set will provide robust quantitative calculations for the detection limits of these technosignatures, with assumptions made about the specific observing facility and the distance to the target. For the present study, we show an example of the spectral signatures for several of the scenarios in our set, as a way of illustrating the potentially detectable features in these scenarios and motivating further work to study these future climates. \nWe focus specifically on the spectral signature from industrial and agricultural pollution in Earth's future atmosphere, saving detailed study of the other planets in our scenarios for future work. The magnitudes for industrial and agricultural pollution in Table 6 describe all pollutants using a single scaling factor. We must therefore choose a method for translating between this scaling factor and the abundances of specific pollutants in future Earth's atmosphere. The best way to approach this problem would be to use the scaling factors from Table 6 to adjust the flux of various atmospheric pollutants into the atmosphere in a coupled climate-chemistry model, which would provide a complete representation of the steady-state climate and abundances of atmospheric gases that could be used to assess detectability. This significant undertaking will be reserved for future work. Instead, we make a simplifying assumption in this study for the purpose of illustration by using the scaling factors from \nTable 6 to adjust the present-day mixing ratios (instead of the flux) of atmospheric pollutants. \nOur estimates for the mixing ratios of various atmospheric constituents on Earth for each scenario are shown in Table 7. This table also includes two reference cases, one with mixing ratios corresponding to present-day Earth (R1) and another corresponding to pre-agricultural Earth (R0) (values for these reference cases are from Seinfeld and Pandis (2016)). The difference in the mixing ratio value between R0 and R1 is taken as the technological contribution for each gaseous species. Mixing ratios are calculated by scaling the technological contribution of each species by the appropriate magnitude of industrial or agricultural pollution from Table 6, with the R0 value added to the total. All Earth cases assume a background atmosphere with pressure 1 bar and composition of 78% N 2 , 21% O 2 , and 1% Ar. \nThe CO 2 and NO 𝑥 mixing ratios in Table 7 are scaled by the magnitude of industrial pollution in Table 6. Examples of possible industrial pollutants (CFC-11, CFC-12, CF 4 , SF 6 , NF 3 ) are also shown with the same scaling. Scenarios S4 and S7 include only long-lived industrial pollutants with atmospheric residence times greater than 1000 yr that remain from the past. Scenario S5 includes no unremediated industrial pollution on Earth, with industrial constituents from non-technological sources fixed to pre-industrial values. Scenarios S1, S2, and S6 have high abundances of CO 2 and therefore also include geoengineering by solar radiation management to prevent the onset of a runaway greenhouse (c.f., Ramirez, Kopparapu, Lindner and Kasting, 2014). \nThe N 2 O, CH 4 , and NH 3 mixing ratios in Table 7 are scaled by the magnitude of agricultural pollution in Table 6. Scenario S1 includes no agricultural lands and has only non-technological sources of N 2 O, CH 4 , and NH 3 . Scenario S5 includes no unremediated agricultural pollution on Earth, with N 2 O, CH 4 , and NH 3 from non-technological sources and fixed to pre-agricultural values. Scenario S6 includes no agricultural lands with the only non-technological sources of N 2 O, CH 4 , and NH 3 in wilderness areas that cover 20% of the planet's surface. \nThe species listed in Table 7 include many greenhouse gases that would exert a significant effect on climate, which includes altering global temperature and changing the abundance of atmospheric water vapor. Self-consistent climate calculations would be needed for this, and so we refrain in this present study from deeper investigation of the spectra for scenarios with high abundances of greenhouse gases. We instead focus on illustrating the spectral features from scenario S3, with a modest enhancement of greenhouse gases, and show how they compare to present-day (R1) and preagricultural (R0) Earth. We also note that the mixing ratios for scenarios S5, S9, and S10 are identical to those in R0; in other words, these three scenarios represents 'sustainable future Earth' cases in which the atmospheric composition has been restored to a pre-agricultural state. We will return to the implications of this degeneracy in §5.2. \nWe calculate absorption spectra for these three cases using the Planetary Spectrum Generator (PSG Villanueva, \nTable 7 \nMixing ratios for atmospheric constituents on Earth for each scenario. All cases assume a background atmosphere with pressure 1 bar and composition of 78% N 2 , 21% O 2 , and 1% Ar. Reference values show approximate mixing ratios for pre-agricultural Earth (R0) and present-day Earth (R1). \nSmith, Protopapa, Faggi and Mandell, 2018; Villanueva, Liuzzi, Faggi, Protopapa, Kofman, Stone and Mandell, 2022). PSG is an online tool that performs radiative transfer calculations to generate modeled spectra. PSG includes numerous gaseous species, including the industrial pollutants used as examples in this study. We show the ultraviolet and visible spectra for S3, R1, and R0 in Figure 8, and we show the infrared spectra in Figure 9. Prominent spectral features are identified with the species listed in Table 7 in bold and other species (O 2 , O 3 , H 2 O) in gray. We neglect any differences in temperature for these three scenarios, so the water vapor abundance is identical (we would expect some variation if this calculation were performed with self-consistent climate model). We also assume a constant mixing ratio with height for all atmospheric constituents. Visible absorption by NO 2 is the most prominent feature in these modeled spectra, which provides a way to distinguish between the three cases; however, the magnitude of the NO 2 may be overestimated due to our assumption of a constant vertical mixing ratio (c.f. the NO 2 detectability calculations by Kopparapu et al., 2021). Features due to other industrial and agricultural pollutants are less prominent, with most evident features of CO 2 absorption and industrial pollutants such as CFC-11 and CFC-12 at mid-infrared wavelengths. Other industrial pollutants such as CF 4 , SF 6 , and NF 3 are too low in these scenarios to show any apparent spectral features. \nWe expect that the modeled spectra of other scenarios with higher abundances of pollutants would show even more prominent absorption features, although these scenarios will also include more significant changes in Earth's temperature and atmospheric water vapor content-as well as solar radiation management to prevent a runaway greenhouse. The example spectra shown in Figures 8 and 9 are intended to demonstrate the potential utility of our set of scenarios in assessing the detectability of exoplanetary technosignatures.", '4.3. System Technosignatures': "We conclude our overview of technosignatures in our scenario set with a summary of the various system-wide technosignatures. We do not perform any quantitative assessments or make any scaling arguments for these technosignatures in the present study. For now we simply list the range of system technosignatures that emerged from the worldbuilding pipeline to illustrate the similarities and differences among the scenarios. The list of system technosignatures and their association with particular scenarios is provided in Table 8. The rows in the table are sorted with the most frequently-occurring technosignatures at the top. \nRadio and optical technosignatures can represent either unintended signals or directed communication, and many past and ongoing efforts attempt to search for such signals (see e.g., Kingsley, 2001; Garrett, 2015). Radio communication occurs in all scenarios except S4, so radio leakage is the most common system technosignature. The magnitude of radio leakage will vary across the scenarios, depending on the spatial extent of the technosphere, the size of planetary settlements, and other factors that will not be estimated in this study. Optical communication also occurs in all scenarios except S4 and S7, so optical leakage is another common system technosignature, again with a magnitude that will vary with scenario. Other sources of radio leakage come from planetary defense radar, which occurs in all scenarios except S4, S7, and S8. Radio beacons for broadcasting information toward other potential civilizations (analogous to messaging to extraterrestrial intelligence, or METI, projects on Earth) occur in S1 and S5, while S1 also includes an optical beacon. Optical flashes from laser propulsion systems (analogous to the laser-propelled nanocraft imagined by the Breakthrough Starshot project (Parkin, 2018)) occur in S9. \nOther technosignatures in this scenario set are related to industrial activities and transportation in space. Many of these ideas are theoretical or based on nascent technology, representing technological possibilities that are not yeat fully realized on Earth. We provide references in these cases to \nFigure 8: Modeled absorption spectra at ultraviolet/visible wavelengths for pre-agricultural Earth (R0, blue), present-day Earth (R1, red), and future industrial Earth scenario S3 (black). The pre-agricultural Earth case is also identical with the sustainable future Earth scenarios S5, S9, and S10. The NO 2 feature can be used to distinguish between all three cases. These calculations assume a constant mixing ratio with height for all atmospheric constituents. \n<!-- image --> \nTable 8 Overview of system technosignatures for each scenario. \nstudies that have considered the possibility of such technosignatures. Asteroid mining (e.g., Forgan and Elvis, 2011) and fusion propulsion (e.g., Cassibry, Cortez, Stanic, Watts, Seidler, Adams, Statham and Fabisinski, 2015) occur in all scenarios except S4, S7, and S8. Asteroid mining is fully managed and automated by AI in some scenarios (S2, S3, S6, S9) but includes human settlements in other scenarios (S1, S5, S10). Fusion propulsion is used for transportation within the solar system. Activity in the outer solar system is limited to a subset of scenarios, with settlements on giant planet moons and orbiting space stations in scenarios S1, S3, \nS5, S9, and S10. Prospecting of resources in the Kuiper belt occurs in S5, S6, S9, and S10. \nThe remaining system technosignatures are even more exotic possibilities. Interplanetary quantum communication (e.g., Hippke, 2021) is prevalent in four scenarios (S5, S6, S9, S10), which suggests the possibility of observing quantum communication leakage. Scenario S5 even includes a quantum METI communication beacon. Gravitational waves as a technosignature are generated by super-luminal travel (e.g., Dubovsky and Sibiryakov, 2008) in scenario S9 for interstellar exploration missions. Scenario S9 is also the only one with Dyson sphere/swarm elements that capture a \nFigure 9: Modeled absorption spectra at infrared wavelengths for pre-agricultural Earth (R0, blue), present-day Earth (R1, red), and future industrial Earth scenario S3 (black). The pre-agricultural Earth case is also identical with the sustainable future Earth scenarios S5, S9, and S10. Other scenarios with elevated industrial pollutants would show stronger infrared absorption for CFC-11, CFC-12, and other species. These calculations assume a constant mixing ratio with height for all atmospheric constituents. \n<!-- image --> \nlarge fraction of the sun's energy (e.g., Zackrisson, Korn, Wehrhahn and Reiter, 2018; Smith, 2022). Such concepts emerge from our pipeline as plausible developments that could occur within the next 1000 years, although none of these are features of all or even most of our scenarios. \nWe have now finished our overview of the potentially detectable technosignatures from our set of ten scenarios for Earth's 1000-year future. This completes the third and final part of our method. We proceed in the next section to discuss some of the most important implications of these scenarios.", '5. Discussion': "The set of ten scenarios that we have developed all represent plausible and self-consistent projections of Earth's technosphere 1000 years from now. Thinking about this range of possibilities can be valuable for our own civilization as we consider the range of future trajectories that might occur. Earth's present and future also provide examples in the search for technosignatures, so this set of scenarios provides a range of possibilities for potential extraterrestrial technosignatures, which even include some quantitative constraints that can be used for detectability calculations. We illustrate the type of calculations that could be done for exoplanetary spectra in Figures 8 and 9, and we will further explore these and other detectability constraints in separate studies. \nAll of these scenarios represent plausible future trajectories, but we do not make any claims about any scenario being \nmore or less probable than others. Our scenario modeling and worldbuilding is intended to capture the widest range of self-consistent possibilities that our method permitted, but we do not claim that any of these scenarios will strongly resemble the actual future. We also cannot draw strong conclusions from the frequency of certain features in our scenarios: for example, even though nine of our ten scenarios include radio leakage, this does not mean that 90% of technospheres should include radio leakage. Because we do not know which of these scenarios is more or less likely, we cannot give any preference or weighting to any one more than any other. We can use the scenario set to think about upper- and lower-bounds for detection, as well as other insights that will be discussed next, but it is important to remember that we are discussing projections of plausible futures (plural) rather than actually making predictions about the future (singular).", '5.1. Kardashev Scale Revisited': "Technology exists to solve human biological and social problems, and technospheres evolve as the aggregate of these solutions. The methodology in this paper provides a way to link insights from the social sciences and humanities for understanding the complex dynamics of human societies into projections of the future that can inform the search for technosignatures. The underlying premise of this work is that human technology emerges from human needs and becomes intertwined with humans, which means that we cannot make meaningful projections about future technology without also \nthinking about how future developments would co-evolve with our human needs. This is one of the key insights missing from the projections by Kardashev (1964). \nIn making projections of constant growth into the future, Kardashev (1964) could find 'no reasons' for doubting that such growth should continue without interruption or bounds because these projections of increased energy were not based on any relationship with human needs. The idea of continuous growth may have seemed evident to Kardashev (1964) as a general feature of human civilization, or even technological civilizations in general, but this is only a speculation rather than a conclusion. Future scenarios with unbridled technological growth are not necessarily self-consistent with all possible future political, economic, and social systems. Our scenario set provides examples that demonstrate a range of plausible future trajectories that do not necessarily result in continual expansion. \nThe population across all bodies in the solar system is listed for each scenario in Table 9. This represents the total population at the end of the 1000-year future timeline. We also calculate the total energy use across the solar system by assuming a constant consumption of 75 GJ per person per year, which is sufficient for a person today to live a high-quality life (e.g., Jackson, Ahlström, Hugelius, Wang, Porporato, Ramaswami, Roy and Yin, 2022). Although we acknowledge the possibility that future technology could increase the per capita energy needed for a good life, we also acknowledge the possibility that increases in efficiency will decrease these requirements. We therefore assume these two factors balance each other and consider this fixed value as an adequate choice for our analysis. Energy use and the corresponding growth rate from the present-day are also shown in Table 9. These results show that five scenarios have reached a state of stable growth (S3, S4, S5, S7, S10), all of which have equilibrated at different population and energy levels. Four scenarios are experiencing growth at different rates (S1, S2, S6, S9), only one of which approaches the 1% value assumed by Kardashev (1964). The remaining scenario is in a state of oscillation between periods of growth and collapse (S8), with the values of population and energy use representative of a pre-collapse state. \nWe use the values in Table 9 to show approximate trajectories for energy use across the 1000-year timeline for all our scenarios in Figure 10. We draw a line or curve from the present-day energy use to the final value at the end of the timeline for each scenario. For some scenarios the value of energy use is lower than today (S4, S7, S8), which results from a prior period of growth before collapsing into the current state. Scenario S8 has a long-period oscillation with rapid growth followed by a collapse, currently midway through its third phase of growth. The three stable scenarios (S3, S5, S10) have energy use higher than today but have reached an equilibrium. The remaining growth scenarios feature several different rates, as low as 0.12% per year (S1) and up to 0.98% per year (S9). The idea of growth remains possible in our scenario set, but the rates of growth are lower than those considered by Kardashev (1964) for three of the \nOverview of total population and total annual energy use across all bodies in the solar system for each scenario. The growth rate describes the conditions at the end of the 1000-year future timeline. Several scenarios have achieved stable (zero growth) conditions and only one (S9) is close to the 1% value assumed by Kardashev (1964). Calculations assume that energy use is correlated with population at a value of 75 GJ per person per year (Jackson et al., 2022). \nTable 9 \nscenarios (S1, S2, S6). The idea of slower-growth rates for future civilizational trajectories is consistent with insights from previous studies that have suggested rapid growth may be unsustainable, even across the solar system or beyond (e.g., Von Hoerner, 1975; Newman and Sagan, 1981; HaqqMisra and Baum, 2009; Mullan and Haqq-Misra, 2019). For the future of human civilization, our scenario set provides several possibilities in which zero-growth (S3, S5, S10) or slow-growth (S1, S2, S6) is reached; these are alternative possibilities to the extremes of collapse (S4, S7, S8) or rapid growth (S9). \nThe S9 scenario with rapid growth is worth examining further, as it is the only one that is consistent with the growth rate assumed by Kardashev (1964) and that has reached the Type I threshold. Scenario S9 is also the only one with Dyson sphere elements as well as other exotic technosignatures such as laser propulsion and gravitational waves from superluminal propulsion (Table 8). The biosphere and technnosphere are completely separated in S9, with the biosphere limited to Earth only and the technosphere spanning from Mars to the Kuiper belt and from Venus to Mercury (Fig. 6). This scenario involves the emergence of a posthuman AI civilization that expands across the solar system, transforming the other planets but leaving Earth to remain in a pristine state for humans. It is worth noting that the only scenario in our set of ten that includes rapid growth and Dyson sphere elements is a scenario with AI-driven growth, rather than human-driven growth. The projection of rapid growth by Kardashev (1964) remains a plausible scenario, but among our scenario set such rapid growth is not an inherent feature of human futures. \nEach of our scenarios represents a different future trajectory based on different assumptions about human social systems and technological capabilities. These assumptions include not only the physical infrastructure of civilization but also the values, worldviews, and myths/metaphors that form \nFigure 10: Total annual energy use across all bodies in the solar system for each scenario. Approximate trajectories are shown for each scenario across the 1000-year timeline. Several scenarios have achieved stable (zero growth) conditions (S3, S5, S10) and, one scenario (S8) is in a state of oscillation between growth and collapse. Only one (S9) is close to the 1% value assumed by Kardashev (1964), which has reached the Type I threshold by the end of the 1000-year period. Calculations assume that energy use is correlated with population at a value of 75 GJ per person per year (Jackson et al., 2022). \n<!-- image --> \nthe basis of thinking and decisions in each scenario. The projection envisioned by Kardashev (1964) represents one possible future with a particular underlying myth/metaphor. Using the framing of Kardashev (1964), the appropriate description would be the myth of 'inevitable growth,' which reflects other concerns that were prevalent on Earth during the 1960's about limits to growth. Using the narrative in scenario S9, we describe this future as the myth of 'Deus Ex Machina' to reflect the emergence of an advanced and self-directed technosphere. But other myths remain possible for the future (Table 5), and these other myths drive other futures, which in turn lead to different technosignatures. In thinking about the future of civilization on Earth, an important conclusion is that myths and worldviews can evolve with time, so the idea of continued growth into the long-term future is not necessarily an inevitability. For the search for technosignatures, an important conclusion is that the idea of rapid growth across the galaxy may be plausible but does not represent the only possible set of myths or worldviews that could sustain a long-lived technological civilization.", '5.2. Degenerate Observations': "Our preliminary example of atmospheric technosignatures from industrial and agricultural pollution highlight an interesting case of degeneracy that could complicate observation strategies. Three scenarios (S5, S9, S10) all have zero industrial or agricultural pollution on Earth (Table 6) due to remediation efforts that attempt to restore Earth's atmosphere to a pre-industrial and pre-agricultural state. The \nmixing ratios (Table 7) and absorption spectra (Figs. 8 and 9) are likewise identical for these three scenarios. Furthermore, the spectral signature of these three scenarios are all identical to the pre-agricultural Earth reference scenario (R0), which represents Earth prior to any global influence by humans. If exoplanet observations were to detect a planet with spectral features similar to these, we would need additional information to be able to decide whether we have detected a system with a biosphere only (R0), or with a biosphere plus a technosphere (S5, S9, S10). \nWe can attempt to break this degeneracy by examining the spatial distribution of these three scenarios (Fig. 6). Scenario S5 includes an extended technosphere with poles at Earth and Mars, while S9 and S10 include a separation of the biosphere on Earth from the technosphere in the rest of the solar system. Further optical observations of Earth could be useful for identifying scenario S5, as this scenario has a high intensity of artificial illumination from urban areas (Fig. 7). However, the lack of artificial illumination would still be consistent with S9, S10, or R0. \nThe extended technospheres in S9 and S10 include activities such as mining in the asteroid and Kuiper belts, settlements across the outer solar system, and fusion propulsion, along with radio, optical, and quantum communication technosignatures (Table 8). Scenario S9 includes Dyson sphere elements, gravitational waves from super-luminal propulsion, and laser propulsion. These are all potential technosignatures that could theoretically be observed to \nconclude that the system has technosignatures (S9 or S10) or is a non-technological system (R0). But observing these technosignatures could be difficult, especially if communication leakage is transient, and Dyson sphere elements are diffuse. Observations of surface modification provide another option, as S9 includes near-complete surface modification of Mars and Venus, while S10 includes modification of the moon and Mars (Fig. 7). These features will also be difficult to resolve with near-term observing facilities, but such observations would be needed in this case to determine whether or not technology is present in the system. \nThe S9 and S10 scenarios are particularly informative because Earth is part of the biosphere but not part of the technosphere. We can imagine a scenario in which a system analogous to S9 or S10 is discovered in the search for habitable exoplanets. Suppose that a future facility is able to measure visible and infrared spectra from the Earth-like planet in this system, which looks like the R0 spectra shown in Figures 8 and 9. This spectra would have many of the spectral biosignatures that are the hallmark of life on Earth, so such an observation may be celebrated as the discovery of an exoplanetary biosignature. The spectra of this planet would show no absorption features that indicate industrial or agricultural pollution, nor would evidence of urban lighting be present. Many astrobiologists may conclude that this is an example of a planet with life, but no technology; however, such a conclusion would miss the fact that this system in fact has advanced technology spread across most of the solar system. Even attempts at spectral characterization of other planetary atmospheres would not necessarily reveal the presence of technology, and only later attempts at searching for surface modification would reveal that this in fact is a system teeming with technology. \nIn an effort to not miss the object of our search, it is important to remember that the relationship between biospheres and technospheres may not be obvious, and we should not expect the discovery of technosignatures (or biosignatures) to follow the patterns that we see in the solar system today or that we imagine from science fiction. The evidence of technology in an exoplanetary system may not necessarily be apparent in all cases from remote observation, and in-situ follow-up missions may be the only way to resolve any such degeneracies. This only serves to motivate further development of mission concepts that can advance the search for technosignatures (e.g., Socas-Navarro et al., 2021).", '5.3. Longevity of Technological Civilizations': 'The most easily studied astrophysical phenomena are long-lived, whereas short-duration events are more difficult to observe. Such logic implies that the search for technosignatures will be more likely to discover evidence of long-lived technological civilizations rather than those with shorter lifetimes. This reasoning is captured by the Drake equation (Drake, 1965), which is a probabilistic expression for the number of detectable civilizations in the galaxy, 𝑁 . For our purposes, it is sufficient to write the Drake equation as \n𝑁 ∼ 𝐿 , where 𝐿 is the average communicative lifetime of technological civilizations. If 𝐿 is large, then the galaxy contains a larger fraction of systems with technosignatures at any given time, but if 𝐿 is small, then the galaxy contains only a few other extant civilizations-or perhaps we are the only one. \nOur scenario set makes projections of Earth\'s 1000-year future; such future scenarios can be useful in guiding the search for technosignatures, but it is worth examining the extent to which we might expect 1000-year projections of the future to be representative of technospheres that might actually be targets of observation. The 1000-year projection is much greater than any other methodological approaches in futures studies, but a millennial timescale is still short by astronomical standards. We therefore complete our analysis of our scenarios in this study by considering the potential longevity of our scenarios as well as the extent to which we should expect long-lived civilizations to be prominent. \nThree of our scenarios involve a civilization that has achieved stability, with zero net growth and long-term sustainability likely (S3, S5, S10). The S3 and S5 scenarios both include space tourism and industry out to Neptune, while S10 has a technosphere that extends across the entire solar system except Earth. None of these three scenarios involve a civilization that embarking on a program of interstellar expansion. Any limited interstellar activity in these scenarios remains limited to scientific exploration and for purposes other than expansion. These three scenarios could all conceivably extend out far beyond the 1000-year timeline and remain viable. These scenarios could also represent longterm steady-state conditions to the extent that no other major evolutionary or technological changes occur; because of this possibility, these three scenarios all remain possible candidates for thinking about the search for technosignatures. We also acknowledge the possibility that further evolutionary or technological changes could lead these scenarios to develop different technospheres over longer-term projections, and we return to this possibility once we complete our assessment of the other scenarios. \nTwo of our scenarios involve a civilization that has collapsed and has rebuilt itself in order to maintain long-term sustainability (S4, S7). The S4 and S7 scenarios both have technospheres limited to Earth only, with few technosignatures that could conceivably be detectable. These scenarios represent civilizations that maintain an environmental policy of minimizing impact on Earth, to the extent that the spectral signature of Earth in these scenarios is difficult to discern from a planet with no technosphere. Neither of these scenarios involve any spaceflight activity, let alone interstellar exploration. These scenarios represent long-term steady-state conditions and likewise remain candidates for thinking about the search for technosignatures. These scenarios may be difficult to detect, and it is worth considering the possibility that most long-lived technospheres in the galaxy may be those that are difficult to distinguish from biospheres. \nOne scenario involves a civilization that oscillates between growth and collapse, with the third collapse approaching (S8). The S8 scenario has a technosphere that extends out to the moon, but the technological capabilities of this civilization are limited to Apollo-era spaceflight. The longterm sustainability of this scenario is uncertain to the extent that the number of possible oscillations that could occur between growth and collapse is unknown. Further exploration of S8 over longer timescales could provide insight on whether this oscillation represents a steady-state or whether this oscillation will decay into a more permanent and stable condition. \nFour of our scenarios involve a civilization that continues to grow (S1, S2, S6, S9). Scenario S1 has a low growth rate but is ultimately on a collapse trajectory due to the instability of its autocratic government. Further exploration of S1 over longer timescales would identify the extent to which a civilization could recover from such a collapse, but S1 otherwise is an example of a short-lived civilization. Scenario S2 includes space tourism and industry out to the asteroid belt, while S6 has space tourism and industry out to the Kuiper belt. The long-term sustainability of both of these scenarios is uncertain, and neither scenario includes any interest in interstellar exploration. Scenario S9 includes posthuman infrastructure that extends out to the Kuiper belt, with no imminent signs of collapse but also no guarantee of long-term sustainability. Scenario S9 involves the highest rate of growth among our scenarios and includes active interstellar exploration for eventual expansion. If these scenarios represent growth that is asymptotically approaching an equilibrium, then the technospheres in these scenarios may still be useful in thinking about the search for technosignatures. Nevertheless, exploration of all these scenarios over longer timescales could provide insight on the extent to which these represent stable or unstable futures. \nExtending our worldbuilding pipeline to consider longer timescales remains conceptually possible to an extent. If we consider these future scenarios out to 10,000 years from now, our pipeline would need to consider the impact of geologic changes on Earth and other long-term physical processes on the biosphere and technosphere. Such a task is perhaps tractable with our existing methods, which may even be sufficient to decide whether our growth scenarios ever reach an equilibrium state. Further extensions out to 100,000 years from now would approach evolutionary timescales, which would require evaluating the evolutionary pressures that have selected for features of the biosphere and technosphere. Such work would require significant modification to the pipeline to enable consideration of the potential needs of future humans as well as the future evolution of the biosphere. Some prior work exists that could form the basis for such an effort, but this would be a significant undertaking. Even longer timescales of a million years or longer are beyond the scope of this pipeline, as our approach based on human needs may not be relevant to such a long term future. Likewise, our scenario modeling eliminated many possibilities as inconsistent for our 1000-year scenarios, but these would all need to \nbe revisited in the case of an extremely long timescale. Some scholars have speculated about the possible characteristics of Earth and human civilization one million years from now (Broderick, 2008), which represent multiple visions of the future but not necessarily self-consistent projections. These visions of a million-year future include speculations such as lifespans that \'approach immortality,\' construction of a \'Matrioshka brain" (star-sized computer), and the prediction that \'everything will change but numbers and laughter\' (Tonn, 2021). We do not make any assessments in this study about the likelihood of such outcomes, but we note that such long-term developments would be impossible to capture with our human-needs based methodology. We save exploration of 10,000-year and possibly 100,000-year futures for subsequent studies, but longer timescales will require the development of novel scenario modeling and worldbuilding methods. \nAs a final remark, it is possible that short-lived technosignatures are more abundant and are the most likely to be detected. Rapid-growth scenarios like S9 may be rare or nonexistent, or such rapid-growth scenarios may inevitably collapse when evaluated on longer timescales. If this is the case, then shorter-lived technospheres will be those that are the most likely to be observed. This possibility was noted by Balbi and Grimaldi (2024), who suggested that \'[if] short-lived technoemissions vastly outnumber the long-lived ones (as it is the case if their operation has an energy or maintenance cost that increases with time), then the first to be detected will likely have a relatively short 𝐿 .\' We do not yet know whether long-lived technospheres are common or even possible, so we should not restrict the search for technosignatures to long-lived scenarios alone. If short-lived technospheres are the only ones that exist, then our set of 1000-year projections of Earth\'s future provides a rich set of possibilities for thinking about strategies for detecting these remote technospheres with explicit tracing of prior premises and assumptions.', '6. Conclusion': "This study is the first to generate self-consistent projections of Earth's technosphere for 1000 years into the future with an explicit tracing of prior premises and assumptions. Our scenario modeling approach builds upon existing methods in futures studies to define a scenario space and identify a smaller number of unique scenarios, which we tailored to generate a wide range of possible technospheres. Our worldbuilding pipeline was designed specifically for this study, which draws upon existing frameworks for assessing human needs and their satisfiers and defining the features of the technosphere. Our focus has remained specifically on understanding the technospheres of these civilizationlevel scenarios, but other applications may also find value in using of our scenario set to explore other features of these possible futures. Our novel worldbuilding pipeline may also have application in areas beyond the search for technosignatures, and we encourage other scholars to draw \nupon our method as a model or inspiration for systematically projecting deep futures that are based on the intertwinement of human systems and the technosphere. \nOur overview of the remotely detectable technosignatures in this study highlights the diversity of technospheres that arise from our methodological approach. We save detailed assessment of detectability for subsequent work, but we show spectral absorption features for a few cases (Figs. 8 and 9) as an illustrative example of how these scenarios can inform technosignature search strategies. The NO 2 feature at visible wavelengths is a prominent way to distinguish between a present-day Earth scenario (R1), a pre-agricultural Earth scenario (R0), and an industrial future Earth scenario (S3). The differences between pre-agricultural and presentday Earth are significant enough to note that industry and agriculture are already affecting our spectral signature. And yet, the pre-agricultural Earth scenario itself is spectrally indistinguishable from scenarios S5, S9, and S10-all of which include technology across the solar system. This represents a viable possibility in the actual search for life in exoplanetary systems, and it is important to keep in mind the possibility that the technospheres that actually exist may not be the ones that are the easiest to observe. \nThe trajectories of these scenarios indicate the range of possibilities open to the future of human civilization. Our scenario set includes projections of future collapse, albeit not extinction-level ones, but also includes many examples of stable technospheres that have reached an equilibrium or have significantly reduced the rate of growth from today. These optimistic outcomes of slower- or zero-growth rates remain tenable and plausible trajectories that extend from today, even if they may not seem likely in context of presentday events. These scenarios also differ in the extent to which technology spreads to other parts of the solar system, which underscores the inherent uncertainty in the longevity of any present-day ambitions for the permanent settlement of space. We cannot predict which of our scenarios is more likely to resemble the actual future, but we can point to the existence of optimistic outcomes in our scenario set to show that our own civilization is not necessarily destined for collapse or extinction. The future remains open, and our collective efforts to envision possible futures should serve as a guide toward realizing the best actual future.", 'Acknowledgments': 'The worldbuilding pipeline documents for each scenario are available as supplementary material at http:// doi.org/10.5281/zenodo.11174443 . The authors acknowledge support from the NASA Exobiology program under grant 80NSSC22K1009. Thanks to Adam Frank, Connor Martini, and Pinchen Fan for helpful discussions. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of their employers or NASA.', 'CRediT authorship contribution statement': 'Jacob Haqq-Misra: Conceptualization, Methodology, Visualization, Writing - Original draft preparation, Project administration. GeorgeProfitiliotis: Conceptualization, Methodology, Software, Writing - Reviewing & editing. Ravi Kopparapu: Conceptualization, Software, Writing - Reviewing &editing.', 'References': "Aguilar, F., 1967. J.(1967). scanning the business environment. New York: The Macmilan Company. 239p . \n- Arney, G., Domagal-Goldman, S.D., Meadows, V.S., 2018. Organic haze as a biosignature in anoxic earth-like atmospheres. 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2024PhRvD.109d4027K | We study gravitational lensing of gravitational waves taking into account the spin of gravitational waves coupled with a dragged spacetime made by a rotating object. We decompose the phase of gravitational waves into helicitydependent and helicityindependent components with spin optics analyzing waves whose wavelengths are shorter than the curvature radius of a lens object. We analytically confirm that the trajectory of gravitational waves splits depending on the helicity generating additional time delay and elliptical polarization onto the helicityindependent part. We exemplify monochromatic gravitational waves lensed by a Kerr black hole and derive the analytical expressions of corrections in phase and magnification. The corrections are enhanced for longer wavelengths potentially providing a novel probe of rotational properties of lens objects in lowfrequency gravitationalwave observations in the future. | 2024-02-01T00:00:00Z | ['10.1103/PhysRevD.109.044027', '2023arXiv230911024K', '2024PhRvD.109d4027K', 'arXiv:2309.11024', '10.48550/arXiv.2309.11024'] | ['General Relativity and Quantum Cosmology'] | Spin optics for gravitational waves lensed by a rotating object | 2,024 | 171 | 0.33 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML'] | 8 | https://arxiv.org/pdf/2309.11024.pdf | {'On spin optics for gravitational waves lensed by a rotating object': 'Kei-ichiro Kubota, 1, ∗ Shun Arai, 2 and Shinji Mukohyama 1, 3 \n1 Center for Gravitational Physics and Quantum Information, Yukawa Institute for Theoretical Physics, Kyoto University, 606-8502, Kyoto, Japan. 2 Kobayashi-Maskawa Institute, Nagoya University, Nagoya 464-8602, Japan 3 Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo Institutes for Advanced Study, The University of Tokyo, Kashiwa, Chiba, 277-8583, Japan. (Dated: January 23, 2024) \nWe study gravitational lensing of gravitational waves taking into account the spin of gravitational waves coupled with a dragged spacetime made by a rotating object. We decompose the phase of gravitational waves into helicity-dependent and independent components with spin optics, analyzing waves whose wavelengths are shorter than the curvature radius of a lens object. We analytically confirm that the trajectory of gravitational waves splits depending on the helicity, generating additional time delay and elliptical polarization onto the helicity-independent part. We exemplify monochromatic gravitational waves lensed by a Kerr black hole and derive the analytical expressions of corrections in phase and magnification. The corrections are enhanced for longer wavelengths, potentially providing a novel probe of rotational properties of lens objects in low-frequency gravitational-wave observations in the future. \nKeywords: gravitational wave, gravitational lensing, spin optics', 'I. INTRODUCTION': "Gravitational waves carry rich information to probe the Universe and fundamental physics: particle physics, nuclear physics, and tests for gravity theories. The network of the ground-based interferometry of LIGO-VirgoKAGRA has detected gravitational waves from nearly a hundred of steller-mass binary mergers [1]. Such data make it possible to examine the physical models of neutron stars and black holes at the extreme blink of coalescence. Meanwhile, the planned space-based detectors e.g. Laser Interferometer Space Antenna (LISA) [2] and the DECi-hertz Interferometer Gravitational-wave Observatory (DECIGO) [3] (see the current status reported in [4]) explore gravitational waves in low-frequency band, spanning 1mHz - 1Hz and 0.1Hz - 10Hz respectively, in order to probe binary evolution, background radiation, and gravitational-waves propagation. In this paper, we consider wave propagation in order to understand the wave nature of gravitational waves. \nOne of the standard ways to describe gravitationalwave propagation is to employ the geometrical optics for the fiducial Einstein's general relativity, as the wavelength is well shorter than the size of a lens object in most of the situations during propagation 1 . In geometrical optics, Einstein's general relativity predicts that (1) gravitational waves propagate at the speed of light in a vacuum, (2) gravitational waves are lensed in the same way as electromagnetic waves are, and (3) gravitational waves possess the two tensorial polarization modes. \nThese predictions are able to be examined with observations. In fact, the arrival-time difference between gravitational waves and electromagnetic waves of the binary neutron star merger associated with a gamma-ray burst: GW170817/GRB170817A [7] was measured, confirming that the gravitational waves propagate at the speed of light with 10 -15 precision. This gives strong constraints on alternative theories of gravity that potentially explain cosmic acceleration [8-13]. The polarization of gravitational waves has been examined by the LIGO-VirgoKAGRA network of interferometry, constraining an upper bound on non-standard polarization modes [14-20]. \nWe aim to explore further detailed physics of gravitational-wave propagation in Einstein's general relativity, focusing on the coupling between the spin of gravitational waves and rotational components of the background spacetime on which waves propagate. Since this coupling cannot be captured by geometrical optics, we develop a way to derive the analytic solution in the regime where the linear perturbation is valid. Provided the linear perturbation works, one can always decompose a gravitational wave into any basis, such as a set of monochromatic waves or a set of wave packets, and then study the propagation of each element of the chosen basis. One can then compute the waveform at the position of the detector by taking an appropriate linear combination and the result should of course be independent of the choice of basis. In the present paper we study monochromatic waves, for which the propagation in the regime of validity of linear perturbation is fully characterized by phases and magnifications once the propagation of polarization tensors is specified. \nThere are various approaches to describe the propagation around a rotating object taking the spin into account. One of the approaches is based on the gravitational Faraday rotation [21-25]. This approach treats \nthe gravitational Faraday rotation angle in the phase so that the equation of motion appropriately determines the evolution of polarization. The authors of the papers called this approach 'spin optics'. We follow this jargon throughout this paper. Other approaches are the Souriau-Saturunini equations [26-29], which are similar form to the Mathisson-Papapetrou-Dixon equations [30-33] for massless particles, and the Berry phase approach [34-38] (see Ref. [39, 40] for the relation between these approaches). Note that the propagation of monochromatic gravitational waves is considered in the spin-optics approach, whereas localized wave packets are considered in the Souriau-Saturunini equation and Berry phase approach. As already mentioned above, gravitational waves in the regime of validity of linear perturbation can be decomposed into either a set of monochromatic waves or a set of wave packets, the propagation of each element of the chosen basis can be studied separately and one can finally take a linear combination to compute the waveform at the position of the detector. \nThe effect of spin tends to be more enhanced for longerwavelength gravitational waves as pointed out analytically in [34] and numerically [35-37, 41, 42], although these papers consider the propagation of wave packet different from the monochromatic waves we focus on. Furthermore, a study that investigates the scattering of gravitational waves by a Kerr black hole using black hole perturbation theory has shown that the spin effect on the scattering amplitude becomes more pronounced for longer wavelengths [43-45]. Thus, the spin effect tends to be important for long-wavelength gravitational waves in addition to the wave effects. However, despite their focus on the wave effect of long-wavelength gravitational waves, several studies aiming to determine the mass distribution of lens objects by using the wavy nature of gravitational waves [6, 46, 47] have neglected the evolution of the polarization tensor, i.e., neglected the effect of spin. In other words, the gravitational waves are treated as scalar waves following a null geodesic. The treatment may not be correct and hence its validity needs to be investigated. \nIdeally, one would like to completely incorporate both wave and spin effects for gravitational waves whose wavelength is not necessarily shorter than the radius of curvature of a lens object, whilst it is a challenge. In this paper, as a stepping stone towards the ideal calculation, we propose a method to calculate the first-order correction to the phase difference and magnification taking into account the spin effect induced by dragged components of spacetime for monochromatic gravitational waves with a wavelength shorter than the curvature radius of the lens object. This effect has not been taken into account in a previous study e.g. Ref. [48], whereas it has recently been addressed in [49] by using the Walker-Penrose theorem to compute the gravitational Faraday rotation. We apply our method to gravitational waves lensed by a Kerr black hole in spin optics and demonstrate its practical implementation for future observations. Our results reveal two \nnew points. One is that there is an arrival time difference between left- and right-handed gravitational waves. The other is that linear polarized gravitational waves lensed by a rotating lens object generally tend to be elliptically polarized for longer wavelength gravitational waves, supporting the results of the study about the scattering by a Kerr black hole[43]. \nThe rest of this paper is organized as follows. In § II, we prepare the way to analytically calculate the phase and magnification of left- and right-handed gravitational waves. We demonstrate the application of our method to the gravitational wave lensed by a Kerr black hole in § III. Finally, we conclude the paper in § IV. \nLet us introduce the notation that we use throughout the paper. The indices of tensors a, b, c, d, e, f, g, h run over 0 to 3, whereas spatial indices i, j, k, l run over 1 to 3. The round and square brackets of indices denote symmetrization and antisymmetrization, respectively, that is, T ( ab ) := ( T ab + T ba ) / 2! and T [ ab ] := ( T ab -T ba ) / 2!. η abcd and ϵ abc denote a 4-dimensional and spatially completely antisymmetric tensor, respectively. The variables with a bar denote the helicity-independent part of the variables. The bold math symbols denote spatial vectors and operators. We use the unit c = G = 1.", 'II. FORMALISM': 'We aim to derive the arrival-time difference and the elliptical polarization of gravitational waves induced by the coupling between spin and the rotational component of the background spacetime. We define the metric as \nd s 2 = g (phy) ab d x a d x b , (2.1) \nwhere g (phy) ab is decomposed into the background and the perturbation as \ng (phy) ab = g ab + h ab . (2.2) \nIn order to extract the physical degrees of freedom that correspond to the gravitational waves, we impose the transverse-traceless gauge i.e. h a a = 0 = ∇ a h ab . Then the Einstein equation G ab [ g (phy) cd ] = 0 is linearized and obtain the wave equation as \n✷ h ab +2 R acbd h cd = 0 , (2.3) \nwhere ✷ := g ab ∇ a ∇ b . This equation is what we solve in the whole paper. \nWe assume that the background spacetime on which gravitational waves propagate is approximately stationary, considering a situation where the rotational motion of a lens object distorts the background much slower than the time variation of gravitational waves. The stationary conditions simplify the discussion without the loss of critical properties of spin optics i.e. the helicity-dependent split of gravitational-wave trajectory. \nLet the background spacetime be a stationary spacetime throughout this paper. The stationary spacetime means that there exists a timelike Killing vector ξ a ( t ) parameterized by t , i.e., \n∃ ξ a ( t ) = ( ∂ t ) a s.t. £ ξ ( t ) g ab = 0 and ξ a ( t ) ξ ( t ) a < 0 ⇔∃ time coordinate t s.t. ∂ t g ab = 0 . (2.4) \nThe timelike hypersurface with constant t , Σ t := M / G is the orbit space associated with ξ a ( t ) , where M is the background spacetime manifold and G is the isometry group of transformations generated by ξ a ( t ) . We express the metric of the background spacetime as \ng ab d x a d x b = g tt (d t -g i d x i ) 2 + γ ij d x i d x j , (2.5) \nwhere g tt , g i , and γ ij are the functions of the spatial coordinate x i and independent of the time coordinate t . The components of the inverse of the metric are g tt = 1 / ( g tt ) + g i g i , g ti = g i , and g ij = γ ij . The determinant is det( g ab ) = g tt det( γ ij ). We introduce a normalized Killing vector u a as \nu a := ξ a ( t ) √ -g tt . (2.6) \nBy definition, u a u a = -1. We define the induced metric γ ab by using u a as \nγ ab := g ab + u a u b . (2.7) \nWe also define the covariant derivative D a associated with γ ab as \nD a S b 1 b 2 ··· b k c 1 c 2 ··· c l := γ c a γ e 1 c 1 γ e 2 c 2 · · · γ e l c l γ b 1 d 1 · · · γ b k d k ∇ c S d 1 d 2 ··· d k e 1 e 2 ··· e l , (2.8) \nwhere S b 1 b 2 ··· b k c 1 c 2 ··· c l is a spatial tensor, and ∇ a is the covariant derivative associated with g ab . \nWe employ the spin optics and the diffraction formula allowed to analytically calculate the phase of gravitational waves along the trajectory. Spin optics is available in situations where the wavelength is shorter than the curvature radius of the lens object, whereas not capturing all the wave effects. However, it is interesting to consider such situations because the time delay and the elliptical polarization generated from the rotational dragging by a lens object are analytically understood.', 'A. spin optics': 'As conventionally known in the context of gravitational Faraday rotation [23, 24, 50-53], the polarization vectors are revolved through the coupling between the polarization vectors and the rotational dragging of the background spacetime, generating a rotational angle. Spin optics treats the rotational angle as a helicitydependent additional phase shift. 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Transportation of the orthonormal base vectors. \n<!-- image --> \nstudies [21, 22, 25, 54], we employ the polarization base tensor extended by the Fermi-Walker parallel transport. \nIn this section, we define the basis as the circular polarization vectors, the Fermi-Walker parallel transport, and the modified dispersion relation in § II A 1, and § II A 2, respectively. In § II A 2, we analytically solve the dispersion relation for the modified phase. In § II A 3, we supply an alternative explanation of the spin effect in the language of the gravito-electromagnetism.', '1. Circular polarization basis': 'We introduce a tetrad { u a , e 1 a , e 2 a , e 3 a } associated with the trajectory of the gravitational waves (see Fig. 1) which satisfy the unit orthogonal conditions, \ne i a e ja = δ ij , e i a u a = 0 . (2.9) \nAs depicted in Fig. 1, we extend the base vectors e i a in the time direction by the Lie transport along the integral curve of ξ a ( t ) as \n£ ξ ( t ) e i a = 0 . (2.10) \n̸ \nWe extend the base tensors in the spatial direction by the spatial Fermi-Walker parallel transport. The projected path of the non-geodesic path generally is not spatial geodesic, i.e. e a 3 D a e 3 b = 0, then the base tensor transported by the spatial standard parallel transport does not point in the tangent direction of the spatial path. To get around this problem, we extend the base vectors e i a in the space direction by spatial Fermi-Walker parallel transport along the spatial path as \nD (FW) e 3 e i a = 0 , (2.11) \nwhere D (FW) e 3 denotes the spatial Fermi-Walker deriva- \ntive 2 defined by \nD (FW) e 3 e i a := e 3 c D c e i a + F a c e i c , F ab := e 3 a a b -e 3 b a a , (2.12) \nand a a := e 3 b D b e 3 a is the acceleration for e 3 a . The base tensor transported by the spatial Fermi-Walker parallel transport is tangent to the spatial path (see App. C). \nNext, we define the circular polarization base vector using the tetrad. The circular base vector m a is defined by \nm a := 1 √ 2 ( e 1 a + ie 2 a ) . (2.13) \nBy definition, the circular base vector satisfies \nm a m ∗ a =1 , m a m a = m a e 3 a = m a u a = 0 , (2.14) \nand \nm a m ∗ b = -ie [ a 1 e b ] 2 + 1 2 ( e a 1 e b 1 + e a 2 e b 2 ) . (2.15) \nHere, the asterisk denotes the complex conjugate. The condition m a m a = 0 yields \nm a ∇ c m a = 0 . (2.16) \nThe circular polarization tensor is transported in the same way as the tetrad base tensors, \n£ ξ ( t ) m a =0 , (2.17) \nD (FW) e 3 m a =0 . (2.18) \nNote that the above extension of the base vectors is valid for stationary spacetime. The extensions for the arbitrary asymptotically-flat spacetime are discussed in Ref. [55].', '2. Dispersion relation': "We introduce the dispersion relation in spin optics which incorporates the effect of spin on the trajectory. We compute the dispersion relation only for the righthanded gravitational waves for the brevity of presentation. Performing complex conjugate in the following discussion gives us a discussion for a left-handed gravitational waves. \nConsidering the path away from the lens, we neglect the second term in the left-hand side of Eq. (2.3) throughout this paper. Provided that |∇A / A| , |∇ m ab /m ab | ≪ \n|∇ S/S | , the right-handed metric perturbation can be decomposed into the amplitude A R , the polarization base tensor m ab := m a m b , and the phase S R as \nh R ab = A R m ab e iS R . (2.19) \nThe indices 'R' and 'L' denote right- and left-handed, respectively. We use these indices only for polarizationspecific discussion and omit them when the polarization is not specific. \nUnder the geometrical optics assumption, the equation of motion (2.3) becomes \n( A R m ab ∇ c S R ∇ c S R -2 i A R ∇ c S R ∇ c m ab -i A R m ab ∇ c ∇ c S R -2 im ab ∇ c S R ∇ c A R ) e iS R + O ( ( ∇ S R ) 0 ) = 0 (2.20) \nIn the standard geometrical optics, one collects the term with O (( ∇ S R ) 2 ) and hence obtains the standard dispersion relation ∇ a S R ∇ a S R = 0 from the leading order of Eq. (2.20). In spin optics, on the other hand, the nextto-leading order term -2 i A R ∇ c S R ∇ c m ab is regarded as the correction to the dispersion relation. This term physically describes the modulation of polarization tensor along the trajectory, namely the gravitational Faraday rotation. Note that this term flips its signature on the left-handed mode. Contracting Eq. (2.20) with m ∗ ab , the dispersion relation and the Hamiltonian in spin optics are given by \nH := 1 2 g ab ( ∇ a S -σ B a ) ( ∇ b S -σ B b ) ≈ 0 . (2.21) \nwhere the helicity σ take +2 for the right- and -2 for the left-handed gravitational waves. ' ≈ ' denotes the equality up to the first order of B where B a is defined by \nB a := im ∗ b ∇ a m b . (2.22) \nThe key is to use this dispersion relation instead of the standard dispersion relation for massless particles in order to take into account the spin effect. The next leading order of Eq. (2.20) yields the conservation of the energy of gravitational waves as in the same as standard geometrical optics approximation [22]. \nWe define the wave vector and velocity as \nk a := ∇ a S, (2.23) \n˙ x a := ∂H ∂k a = k a -σ B a . (2.24) \nHere, we take the e 3 a as the direction of the spatial part of the velocity. The explicit expression of e 3 a is given by \ne 3 a := √ -g tt -ξ b ( t ) ˙ x b ˙ x a -u a . (2.25) \nThe coefficient of ˙ x a is determined so that the approximate null condition Eq. (2.21) is satisfied. \nBecause of stationarity, we impose \n£ ξ ( t ) ˙ x a = ∂ t ˙ x a = 0 , (2.26) \nindicating that the phase velocity stays constant in time. Since B a consists of the circular base vector which is Lie transported along the Killing vector ξ a ( t ) , B a is also Lie transported, i.e., £ ξ ( t ) B a = 0. Indeed, we immediately show £ ξ ( t ) B a = 0, because the Lie derivative along any Killing vector commutes with any covariant derivative, i.e., [ £ ξ , ∇ ] = 0, (See App. B). It means \n£ ξ ( t ) B a = ∂ t B a = 0 . (2.27) \nThis equation and Eqs. (2.26) (2.24) lead to \n£ ξ ( t ) k a = ∂ t k a = 0 . (2.28) \nWe define the frequency of gravitational waves ω as \nω := -ξ a ( t ) k a . (2.29) \nUsing Eq. (2.28) and ∂ t ξ a ( t ) = 0, the derivative of the frequency is \n∂ a ω = -ξ b ( t ) ∂ a k b -k b ∂ a ξ b ( t ) = 0 . (2.30) \nThus we obtain \nω = constant . (2.31) \nThis is the consequence of the temporal isometry of the background spacetime. We can separate the phase into the time-dependent part -ωt and space-dependent part S ( x i ) as \nS ( t, x i ) = -ωt + S ( x i ) . (2.32) \nTo analytically solve the dispersion relation (2.21), we additionally assume that the frequency is larger than |B a | , i.e., |B a | ≪ | k a | on the path. Let set S = ¯ S + σχ where ¯ S and σχ are the zeroth and first-order solutions of B , respectively, where ¯ S is the helicity-independent part of the S . Note that the helicity-dependent part σχ is contained in the spatial part of the phase S . \nThe zeroth order of B of Eq. (2.21) yields \n∇ a ¯ S ∇ a ¯ S = ¯ k a ∇ a ¯ S ≈ 0 , (2.33) \nwhere ¯ k a := ∇ a ¯ S . The covariant derivative of this equation yields the geodesic equation, \n¯ k b ∇ b ¯ k a ≈ 0 . (2.34) \nEq. (2.33) means ¯ S is constant along the curve tangent to ¯ k a . Thus, the spatial part of ¯ S is associated with the time part as \nS ( x i o ) -S ( x i s ) = ω ( t o -t s ) . (2.35) \nt o -t s is the time between the source and observer which can be obtained by solving the geodesic equation (2.34). \nThe first order of B of Eq. (2.21) yields \n˙ x a ∇ a χ ≈ ˙ x a B a . (2.36) \nThe right-hand side can be rewritten in terms of the language of gravito-electromagnetism and will be shown in the next section. \nRecall that the spin optics is equivalent to the gravitational Faraday rotation in the context of gravitoelectromagnetism [51]. The right-hand side Eq. (2.36), which is the result of spin optics, can be better understood and simply described with the gravito-electric field E and gravito-magnetic field B defined as \nE := -D ln √ -g tt = -1 2 D g tt g tt , (2.37) \nB :=curl g . (2.38) \nThe operator curl is defined by \ncurl g i := ϵ ijk D j g k = ϵ ijk ∂ j g k , (2.39) \nwhere we use ϵ ijk (3) Γ l jk = 0 in the second equality. ϵ ijk = ϵ [ ijk ] is the spatial anti-symmetric tensor with ϵ 123 = 1 / √ γ and ϵ 123 = √ γ , where γ := det( γ ij ). In stationary spacetime, the Einstein equation and Bianchi identities can be rewritten in the quasi-Maxwell form [56] \ndiv B =0 , (2.40) \ncurl E =0 , (2.41) \ndiv E = -[ 1 2 ( √ -g tt B ) 2 + E 2 ] -(2.42) \ncurl ( √ -g tt B ) =2 E × ( √ -g tt B ) +16 π √ -g tt J , (2.43) \n16 πg tt ( T tt -1 2 g tt T ) , \n(3) R ij =D i E j + [ ( √ -g tt B i ) ( √ -g tt B j ) -( √ -g tt B ) 2 γ ij ] + E i E j -8 πg tt ( T ij -1 2 g ij T ) , (2.44) \nwhere (3) R ij is the spatial Ricci tensor associated with γ ij , T ab is the stress energy tensor of a matter, J i := T i t , and div E := D i E i . The gravito-electric field is sourced by the gravitational self energy and the matter energy. The gravito-magnetic field is sourced by these currents. \nWe rewrite the right-hand side of Eq. (2.36) in terms of the gravito-magnetic component B . 3 \n˙ x a B a = im ∗ b ˙ x a ∇ a m b = -iξ c ( t ) ˙ x c √ -g tt m ∗ b ( e 3 a + u a ) ∇ a m b = -iξ c ( t ) ˙ x c √ -g tt m ∗ b e 3 a ∇ a m b + iξ c ( t ) ˙ x c g tt m ∗ b ξ a ( t ) ∇ a m b \n= -iξ c ( t ) ˙ x c -g tt m ∗ b ξ a ( t ) ∇ a m b = -iξ c ( t ) ˙ x c -g tt m ∗ b m a ∇ a ξ ( t ) b = iξ c ( t ) ˙ x c -g tt ie [ a 1 e b ] 2 ∇ a ξ ( t ) b = -ξ c ( t ) ˙ x c -2 g tt η abcd u a e 3 b ∇ c ξ ( t ) d = 1 2 u a ˙ x b η abcd ∇ c u d = √ -g tt 1 2 ˙ x · B . (2.45) \nWe have used Eq. (2.25) in the second equality, Eq. (2.6) in the third equality, Eqs. (2.14) (2.18) in the fourth equality, Eq. (2.17) in the fourth to fifth line, Eq. (2.15) and e b 1 e c 1 ∇ c ξ ( t ) b = 0 due to the anti-symmetry ∇ c ξ ( t ) b = -∇ b ξ ( t ) c which is found from the Killing equation £ ξ ( t ) g ab = 0 in the sixth equality, η abcd u a e 3 b = 2 e [ c 1 e d ] 2 and ∇ a η bcdf = 0 in the seventh equality, Eqs. (2.6), (2.25) 4 , and η abcd u a u b = 0 in the eighth equality, and u a η a bcd ∇ c u d = √ -g tt u a η a bcd ∂ c ( u d / √ -g tt ) = √ -g tt γ b i ϵ ijk ∂ j ( u k / √ -g tt ) = √ -g tt γ b i ϵ ijk ∂ j g k = √ -g tt γ b i B i . Therefore, the gravitational Faraday rotation angle can be written in a similar form to the Faraday rotation in electromagnetism as \n˙ x a ∇ a χ ≈ √ -g tt 2 ˙ x · B . (2.46) \n˙ x a ∇ a means the directional derivative in the direction of the velocity.", 'B. Diffraction theory': 'We have analytically obtained the phase incorporating the spin effect. In this section, we plug the solution of the phase in the diffraction formula, and evaluate the arrival time and the magnification. \nFirst, we introduce the diffraction formula, based on Ref.[57]. Next, we evaluate the integral in the diffraction formula on the lens plane in the diffraction formula (2.47) using the stationary phase approximation which can be applied to the gravitational wave with a wavelength shorter than the typical length scale of the lens. In this paper, we configure the position of the source, lens, and observer shown in Fig. 2 so that the spin effect is at maximum. 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Lens geometry on the t = const. hypersurfaces. The true source position is at η with respect to the z -axis. The intersection point between the lens plane and the gravitational waves path at ξ . E is lens plane. E \' is intermediate plane. r s is the distance of the source from the lens plane E. r o and r \' o are the distance of the observer from the lens plane E and the intermediate plane E \' , respectively. The lens plane E is close to the intermediate plane E \' in the meaning of r o -r \' o ≪ r o . \n<!-- image -->', '1. Diffraction formula': "The diffraction theory of gravitational lensing is well organized in § 4.7 of Ref. [57]. In this book, the authors applied the diffraction theory to the intermediate plane E ' to the observer in which the background space-time is approximately flat. The lens configuration and the spin effect which originated from the dragged component of the background spacetime do not change the derivation of the diffraction formula of gravitational lensing. Therefore, the standard diffraction formula of gravitational lensing [48, 57, 58] can be applied to the case in this paper, which is given by \nE obs ( η , σ ) E unlens = ( r s + r o ) ω 2 πir s r o ∫ E d 2 ξe i S ( ξ , η ) (2.47) \nwhere E obs and E unlens are the amplitude at the observer with and without the lens, respectively. The effect of spacetime distortion is imprinted only in the phase S .", '2. stationary phase approximation': 'We evaluate the integral of Eq. (2.47) using stationary phase approximation. In the stationary approximation, the third or higher-order Taylor expansion of the phase S at the stationary point Ξ , which is the solution of ∂ ξ S ( Ξ ) = 0, is assumed to be much smaller than the second-order Taylor expansion and hence neglect the higher-order terms. This assumption is quantitatively translated into ξ/η ≪ Mω [5]. In such a situation, the \nintegrals around two stationary points Ξ , which consist of the minimum point Ξ min and the saddle point Ξ sad , are dominant. Performing the multiple Gauss integral, the result which is the sum of the contribution from the two stationary points (see e.g. Ref. [57]) is written as \nE obs E unlens = ∑ j =sud,min ( r s + r o ) ω r s r o exp( i S ( Ξ j ) -iπn j / 2) √ | det( ∇ ⊗ ∇ S ( Ξ j )) | , (2.48) \nwhere n min = 0, n sad = 1, ⊗ denotes tensor product and ∇ denotes the two-dimensional derivative on the lens plane. ∇ ⊗ ∇ S ( Ξ j ) is the Hesse matrix at the stationary points.', 'III. EXAMPLE: KERR BLACK HOLE': "We apply our formalism to gravitational waves lensed by a Kerr black hole as an example. Part of the calculation follows Ref. [48]. We consider the lens geometry drawn in Fig. 2. First, we introduce the phase. Next, we put the phase into the diffraction formula (2.47). Finally, we evaluate the integration using the stationary phase approximation and estimate the arrival time delay and magnification difference between left- and right-handed gravitational waves. \nHere we introduce a bookkeeping parameter ϵ as \nM ξ = O ( ϵ ) , (3.1) \nwhere ξ is typically in the same order as the Einstein radius defined by \nξ E := √ 4 Mr s r o r s + r o . (3.2) \nWe assume the angular momentum per mass a ,i.e. a Kerr parameter, is in the same order as M , then \na ξ = O ( ϵ ) . (3.3) \nIn addition, we assume r s is also in the same order as r o . Then Eq. (3.2) yield \nξ r s ∼ ξ r o = O ( ϵ ) . (3.4) \nTo summarize, we assume the relation \nM ξ ∼ a ξ ∼ ξ r s ∼ ξ r o = O ( ϵ ) . (3.5) \nBecause the χ ∼ O ( ϵ 3 ), we calculate it up to the third order of ϵ . Hereafter, ' ≃ ' denotes equality up to the third order of ϵ .", 'A. magnetic component': 'The explicit expression of the gravito-magnetic components for Kerr metric in the Boyer-Lindquist coordinate is \nB = -2 aMr ∆sin(2 θ ) √ γ ( ρ 2 -2 Mr ) 2 ∂ r -2 aM ( r 2 -a 2 cos 2 θ ) sin 2 θ √ γ ( ρ 2 -2 Mr ) 2 ∂ θ , (3.6) \nand its infinitesimal circulation is \ncurl( √ -g tt B ) = 4 aM 2 ρ 3 ( ρ 2 -2 Mr ) 3 / 2 ∂ ϕ , (3.7) \nwhere ρ and ∆ is defined in Eq. (A2) and Eq (A3), respectively. Eq. (3.7) is satisfied Eq. (2.43). Since the charge of the gravitational field is the energy, the gravitomagnetic field sourced by circular energy current is analogous to the gravito-magnetic field sourced by circular electric current.', 'B. phase': 'We calculate the phase separating the helicityindependent and helicity-dependent parts based on the procedure described in § II A 2. \nFirst, we get the helicity-independent part of the phase. The covariant derivative Eq. (2.33) leads to the geometric equation \n¯ k b ∇ b ¯ k a = 0 , (3.8) \nwhere ¯ k a := ∇ a ¯ S . Thus, ¯ S is given by the time from the source to the observer multiplied by the frequency 5 , \n¯ S = ω ∫ obs sou d t = ω ( t obs -t sou ) . (3.9) \nUsing the null geodesic equations (A15)-(A17), \nt o -t s = ( ∫ E sou + ∫ obs E ) r 2 ( r 2 + a 2 ) + 2 aMr ( a -L/E ) sgn( ¯ k r )∆ √ R d r + ∫ obs sou a 2 cos 2 θ √ Θ d θ, (3.10) \nwhere E and L are defined in Eq. (A11) and Eq. (A12), respectively. The time is obtained in Ref. [59] up to O ( ϵ 2 ). Because the contribution of the rotation of the \nKerr black hole in the time is O ( ϵ 2 ) [48], the time is the same as in Schwarzschild black hole up to O ( ϵ 1 ). The time up to O ( ϵ 1 ) is given by the summation of the Euclidean part and the Shapiro time delay. Therefore, the eikonal is written as \n¯ S ( x , y ) ≃ [ const. ] + ω [ 2 M | x -y | 2 -4 M log( x ) ] + [ ¯ S (2) ] + [ ¯ S (3) ] , (3.11) \nwhere x and y are the dimensionless distance normalized the Einstein radius ξ E which are defined by \nx := ξ ξ E , y := η ξ E r o r s + r o , ξ E := √ 4 Mr o r s r s + r o . (3.12) \nThe orders in the ϵ expansion of each square bracket are different and the square brackets are listed in lower order from the left. ¯ S (2) and ¯ S (3) are not written down because they are not relevant to the main results in this paper, although they can be computed by performing the procedure described in Ref. [59]. \nNext, we calculate the helicity-independent part of the phase. It is the first-order solution σχ in B of the dis- \nrsion relation. (2.21). χ is the gravitational Faraday rotation angle which is the solution of Eq. (2.46). The gravitational Faraday rotation angle for a Kerr black hole is calculated in App. D (see also Ref. [51]), which is given by \nχ ≃ -πaM 2 4 ξ 3 . (3.13) \nFinally, we sum them to obtain the solution of the dispersion relation. (2.21), which is given by \nS ( x , y ) ≃ [ const. ] + ω [ 2 M | x -y | 2 -4 M log( x ) ] + [ ¯ S (2) ] + [ ¯ S (3) -σ πaM 2 4 x 3 ξ 3 E ] . (3.14) \nThe last term is an important result led by spin optics.', 'C. stationary point': 'The stationary points x = X such that ∂ x S = 0 are given by \nX min ≃ [ 1 2 ( √ y 2 +4+ y ) y y ] + [ ¯ X (2) min ] + [ ¯ X (3) min -σ 1 ¯ X (1)2 min ( ¯ X (1)2 min +1 ) 3 πaM 16 ωξ 3 E y y ] , (3.15) \nX sad ≃ [ 1 2 ( √ y 2 +4 -y ) -y y ] + [ ¯ X (2) sad ] + [ ¯ X (3) sad -σ 1 ¯ X (1)2 sad ( ¯ X (1)2 sad +1 ) 3 πaM 16 ωξ 3 E -y y ] , (3.16) \ngravitational waves \nwhere \n¯ X (1) = 1 2 ( √ y 2 +4 ± y ) . (3.17) \nThe sign is +/ -for the minimum/saddle point. ¯ X (2) and ¯ X (3) are not calculated in this paper. The final term in Eqs. (3.15) (3.16) is helicity-dependent part due to the spin effect. The stationary points (see also Fig. 3) are shifted from that of scalar waves by \nδ Ξ := | X -¯ X | ξ E = 1 ¯ X (1)2 ( ¯ X (1)2 +1 ) 3 πaM 4 ωξ 2 E . (3.18)', 'D. arrival time difference': 'The gravitational Faraday rotation which is the last term in Eq. (3.14) induces the arrival time difference between left- and right-handed gravitational waves. Substituting the stationary points into the last term, we evaluate the arrival time delay between left- and right-handed \nδt R-L := |S R ( X ) -S L ( X ) | ω ≃ 1 ¯ X (1)3 2 π ω aM 2 2 ξ 3 E . (3.19) \nSince δt R-L ∝ 1 /ω , the arrival time difference is large for long-wavelength waves. The arrival difference between left-handed wave packet and right-handed wave packet calculated in Ref. [36, 40] is δt R-L ∝ 1 /ω 3 , which is different from the order in Eq. (3.19). However, Ref. [36, 40], in which the author considers that a wave packet propagates near the Kerr black hole, is not an appropriate comparison since the situation is different from what we consider, i.e. the propagation of monochromatic waves. Therefore, it is not a problem if the frequency dependence is different from its result. As stated in introduction, gravitational waves in the regime of validity of linear perturbation can be decomposed into either a set of monochromatic waves or a set of wave packets, and the propagation of each element of the chosen basis can be studied separately. 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sha1\\_base64="jCGH4Vq2KamvxOnRGkr8/+FmQKQ=">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</latexit> \nFIG. 3. Schematic illustration of the trajectory of the spin-0 particle (black) and the gravitational waves with σ · J > 0 (red) and σ · J < 0 (blue). \n<!-- image --> \n/divides.alt0 \n/divides.alt0 \n/divides.alt0 \n/divides.alt0 \na given astrophysical source of gravitational waves, the waveform at the position of the detector can be computed \nby taking an appropriate linear combination and should be independent of the choice of basis. Confirming this explicitly is, however, beyond the scope of the present paper and is left as a future work. \nWe estimate the arrival time difference for a hypothetical lens with the mass M = 10 12 M ⊙ and the Kerr parameter a = M at the distance r s = r s = D = Gpc. The factor aM 2 /ξ 3 E is O (10 -9 ) and ¯ X (1) is O (1). Considering gravitational waves with the frequency f = ω/ (2 π ) = mHz in the LISA band, the arrival time difference is estimated to be \nδt R-L ∼ 10 -6 sec ( a 10 12 km )( M 10 12 km ) 1 2 · ( D Gpc ) -3 2 ( f mHz ) -1 . (3.20)', 'E. magnification': 'The normalized magnification defined by \nµ ( y ) := (4 Mω ) 2 det ( ˆ ∇⊗ ˆ ∇S ( X , y ) ) , (3.21) \nwhere ˆ ∇ denotes the derivative with respect to the lens plane coordinate x . Those at the stationary points are explicitly \n| µ min | := | µ ( X min ) | ≃ [ 1 2 + y 2 +2 2 y √ y 2 +4 ] + [ ¯ µ (2) min ] + [ ¯ µ (3) min + σ ¯ X (1) min (3 ¯ X (1)2 min +1) ( ¯ X (1)2 min -1)( ¯ X (1)2 min +1) 3 3 πaM 16 ωξ 3 E ] , (3.22) \n| µ sad | := | µ ( X sad ) | ≃ [ 1 2 -y 2 +2 2 y √ y 2 +4 ] + [ ¯ µ (2) sad ] + [ ¯ µ (3) sad + σ ¯ X (1) sad (3 ¯ X (1)2 sad +1) ( ¯ X (1)2 sad -1)( ¯ X (1)2 sad +1) 3 3 πaM 16 ωξ 3 E ] , (3.23) \n(3.24) \nwhere ¯ µ (2) and ¯ µ (3) are not calculated in this paper. Only the final term depends on the helicity. The magnification of gravitational waves with σ = +2 is more enhanced than that of scalar waves. Intuitively, this is because gravitational waves with σ = +2 propagate over shorter distances than scalar waves (See Fig. 3). \nThe magnification difference between left- and righthanded gravitational waves is \nδµ R-L := | µ R ( X ) -µ L ( X ) | = ¯ X (1) (3 ¯ X (1)2 +1) ( ¯ X (1)2 -1)( ¯ X (1)2 +1) 3 1 Mω 3 πaM 2 4 ξ 3 E . (3.25) \nSince δµ R-L ∝ 1 /ω , the magnification difference is also large for long-wavelength waves. This means that when \nthe source emits gravitational waves with pure plusmode, i.e., |E R | = |E L | at the source, the observer receives the elliptically polarized gravitational waves. It is more elliptical for longer-wavelength gravitational waves. However, since the factor aM 2 /ξ 3 E is very small < ∼ O (10 -9 ) for wavelength shorter than the radius of curvature of a lens object, δµ R-L is also small, where we assume a hypothetical lens with a mass M = 10 12 M ⊙ and the Kerr parameter a = M at the distance r s = r s = Gpc. \nThe total amplitude is given by \n∣ ∣ ∣ ∣ E obs E unlens ∣ ∣ ∣ ∣ 2 = | µ min | + | µ sad | +2 √ | µ min µ sad | sin( S sad -S min ) . (3.26) \n<latexit sha1\\_base64="ihQuz+nz8ohDk1sN14Pd1aThgV0=">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</latexit> \n<latexit sha1\\_base64="LU9Nf8Ax9F+dX+VTFDvN7lTFxGY=">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</latexit> \n<latexit sha1\\_base64="EaxldC25r6QUwyDjIBR32NHUZIw=">AAAD6XichVJNb9NAEJ3UfPQDaFoulbhYREWAaLRBtKAeqgouCAnUNk1TFFfR2tmkVvwlexNcLF+4IHFBnEBwAsRP4AoHLvwBDuUvcASJK8+LW9FWNWt5d+btvJm3u2MGjh1JxnZLI9qJk6dOj46NT5w5e26yPDW9EfmD0BINy3f8cNPkkXBsTzSkLR2xGYSCu6Yjmmb/TrbfHIowsn1vXe4EYsvlPc/u2haXgNrlK4YUsVR5klB00sQw3cSI7J7LU8Pq+DLz76VLLG2XK6zK1NCPGrXcqFA+Vvyp0joZ1CGfLBqQS4I8krAd4hTha1GNGAXAtigBFsKy1b6glHQaB3uAOIEYDryPuQevlaMe/CxrpPgW6jj4Q3CPYyaKkdXewWqqKrOki6A9+Wym/rug5h7TxSpp+1/mf5RK6tItpdCG4kAhmXZrL8fw8cuf9cW12eQSe8d+sG/sLdtlX9hX5g1/WR9WxdqbQl0mNKVZnodXxdNWs1BNh4ZQEWDtghUrVfGBm0OeJ2FhvYF6n7zmg4n0e2H0fZy0nt92VuUadrmKjPbf2IP3SKlw1f160JcAd9Sdm/A4Zq70HcZ1+H3wQ7B0mkM/VWle6TOAcZw2q6Xv47FibCs1MbLN4Usp6+3a4U4+amxcr9YWqvOrNyrLt/MuH6ULdJEuo5Nv0jLdpRVq4CTP6SN9os9aX3uhvdJe/w0dKeWc83RgaO//AJ8O2TU=</latexit> \nwhere S sad := S ( X sad ) and S min := S ( X min ).', 'IV. CONCLUSION': 'We have developed the method to study the gravitational lensing of gravitational waves taking into account part of the spin effect of gravitational waves. Combining the computational techniques developed in the literature for the spin effect of gravitational waves as well as the ones for gravitational lensing, we have developed for the first time a consistent computation for gravitational lensing incorporating the spin effect of the gravitational waves by using the diffraction formula. We applied our formalism to the case of monochromatic gravitational waves lensed by a Kerr black hole conditioned by Mω ≫ 1, illuminated potential signatures of the spininduced gravitational time delay and the elliptical polarizations for future gravitational-wave observations. We have shown in the case of a Kerr black hole the time delay (3.19) between left- and right-handed gravitational waves is more enhanced for longer-wavelength gravitational waves. Furthermore, our results have also shown that the magnification (3.24) depends on both the frequency and the helicity of gravitational waves. For example, the magnification of gravitational waves with σ = +2 is larger than that of scalar waves in the lens geometry in Fig 2. Notably, we have found that the difference of magnification between the left- and the right-handed gravitational waves is larger for longer wavelength gravitational waves. Our predictions for the differences in phase and magnification could be observed in future gravitationalwave detectors, potentially giving new information on the rotation properties of a lens object and a dragged spacetime. \nIn this paper, we focus on gravitational waves only for Mω > ∼ 1. To investigate the regime Mω < ∼ 1, we need to calculate the diffraction integral without relying on the stationary phase approximation. Alternatively, one can employ the path integral method [5], utilize the Teukolsky equation, or numerically solve Einstein equation [60]. These approaches are able to be extended to the range of Mω ≫ 1, making it worth comparing to our results of spin optics. In conclusion, studying the theoretical computation of wave scattering by a spinning object is still desirable in the future.', 'ACKNOWLEDGMENTS': 'The work of S.M. was supported in part by World Premier International Research Center Initiative (WPI), MEXT, Japan. S.M. is grateful for the hospitality of Perimeter Institute where part of this work was carried out. Research at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development and \nby the Province of Ontario through the Ministry of Colleges and Universities.', 'Appendix A: Basic materials of Kerr spacetime': 'We write down the basic equations in the Kerr spacetime based on [61]. The Kerr metric in the BoyerLindquist coordinate is given by \nd s 2 = -( 1 -2 Mr ρ 2 ) d t 2 -4 Mar sin 2 θ ρ 2 d ϕ d t + ρ 2 ∆ d r 2 + ρ 2 d θ 2 + Σ ρ 2 sin 2 θ d ϕ 2 , (A1) \nwhere \nρ 2 := r 2 + a 2 cos 2 θ, (A2) \n∆ := r 2 -2 Mr + a 2 , (A3) \nΣ := ( r 2 + a 2 ) 2 -a 2 ∆sin 2 θ. (A4) \nThe components of the inverse of the metric are \ng ab = t r θ ϕ -Σ ρ 2 ∆ 0 0 -2 Mar ρ 2 ∆ 0 ∆ ρ 2 0 0 0 0 1 ρ 2 0 -2 Mar ρ 2 Σ 0 0 ∆ -a 2 sin 2 θ ρ 2 ∆sin 2 θ . (A5) \nIn the Kerr spacetime, there are two Killing vectors and one Killing tensor (See e.g. [62]) given by \nξ a ( t ) =( ∂ t ) a = (1 , 0 , 0 , 0) , (A6) \nξ a ( ϕ ) =( ∂ ϕ ) a = (0 , 0 , 0 , 1) , (A7) \nξ ab =2∆ 2 l ( a n b ) + r 2 g ab , (A8) \nwhere l and n is defined by \nl a :=( r 2 + a 2 , ∆ , 0 , a ) / ∆ , (A9) \nn a :=( r 2 + a 2 , -∆ , 0 , a ) / ∆ . (A10) \nThe constants of motion such as the energy E , azimuthal angular momentum L , and Carter constant Q , corresponding to each Killing vector and tensor exist as \nE := -ξ a ( t ) p a = -p t , (A11) \nL := ξ a ( ϕ ) p a = p ϕ , (A12) \nQ := ξ ab p a p b -( L -aE ) 2 (A13) \nIn the case of monochromatic wave scattering that we consider in the paper, E = ω . For a null geodesic with a \ntangent vector p a = d x a / d ν , writing Eqs. (A11)-(A13) and p a p a = 0 explicitly, we can obtain \nρ 2 d t d ν = -a ( aE sin 2 θ -L ) + ( r 2 + a 2 ) P ∆ , (A14) \nρ 2 d θ d ν = ± E √ Θ , (A16) \nρ 2 d r d ν = ± E √ R, (A15) \nρ 2 d ϕ d ν = -( aE -L sin 2 θ ) + aP ∆ , (A17) \nwhere \nP ( r ) := E ( r 2 + a 2 ) -aL, (A18) \nR ( r ) E 2 := P 2 -∆ ( ( L -aE ) 2 + Q ) , (A19) \nΘ( θ ) E 2 := Q +cos 2 θ ( a 2 E 2 -L 2 sin 2 θ ) . (A20) \nThe signs in Eqs. (A15) (A16) are positive for d r/ d ν > 0 , d θ/ d ν > 0, and negative for d r/ d ν < 0 , d θ/ d ν < 0, respectively.', 'Appendix B: Commutation relation': 'We show that any Lie derivative along any Killing vector is commutative with any covariant derivative for any vector V a . ∇ a £ ξ V b and £ ξ ∇ a V b can be written as \nc c \n∇ a £ ξ V b =( ∇ a ξ )( ∇ c V b ) + ξ ∇ a ∇ c V b +( ∇ a V c )( ∇ b ξ c ) + V c ∇ a ∇ b ξ c , (B1) \n£ ξ ∇ a V b = ξ c ∇ c ∇ a V b +( ∇ a ξ c )( ∇ c V b ) +( ∇ a V c )( ∇ b ξ c ) . (B2) \nHere, we use the property of the Killing vector (see App. C.3 in Ref. [63]), \n∇ a ∇ b ξ c = -R bca d ξ d . (B3) \nUsing Eq. (B3), one can show that the Riemann tensor that comes out when exchanging the covariant derivative of the second term on the right-hand side of Eq. (B1) cancels with that of the fourth term. Then the righthand side of (B1) and Eq. (B2) are equivalent. Therefore we obtain \n[ £ ξ , ∇ a ] V b = 0 . (B4) \nPerforming the same procedure for any rank tensor, one can show any Lie derivative along any Killing vector is commutative with the covariant derivative.', 'Appendix C: Fermi-Walker parallel transport': 'We explain the Fermi-Walker parallel transport following [64, 65]. The guiding principle of parallel transport \nis that the inner product of vectors transported along a curve is invariant. Standard parallel transport of a vector V a along the integral curve of unit normal vector U a is the transport that satisfies \nU b ∇ b V a = 0 . (C1) \nThe standard parallel transported tangent vector along its geodesic curve is the tangent vector of the destination. However, the standard parallel transported tangent vector along the non-geodesic curve does not coincide with the tangent vector of the destination. The Fermi-Walker parallel transport is the transport such that (a) the inner product of transported vectors is invariant and (b) the transported tangent vector along the non-geodesic curve coincides with the tangent vector at the destination. \nThe Fermi-Walker parallel transport is the transport with satisfying \n∇ FW U V a := U b ∇ b V a + F a b V b = 0 , (C2) \nwhere \nF ab =sgn( U c U c ) ( U a A b -A a U b ) , A a = U b ∇ b U a , (C3) \nwhere | U a U a | = 1. Indeed, ∇ FW U U a = 0 and the derivative of the inner product between the tangent vector U a and Fermi-Walker transported vector V b along the curve is \nd d τ ( U a V a ) = U b ∇ b ( U a V a ) = A b V a -U a F a b V b = A a V a -sgn( U c U c ) U a U a A b V b = 0 , (C4) \nwhere we have used U a A a = 0. The first term of Eq. (C2) is not necessary for the inner product with the tangent vector to be invariant but is necessary for the inner product between non-tangent vectors. The two transport for the geodesic curve, i.e. A a ∝ U a , are equivalent because of F ab = 0.', 'Appendix D: Gravitational Faraday rotation angle in the Kerr spacetime': 'Following Ref. [51], we calculate the gravitational Faraday rotation angle χ in Kerr spacetime, which is defined by Eq. (2.46), \n˙ x a ∇ a χ = √ -g tt 2 B a ˙ x a = √ -g tt 2 B i ˙ x i . (D1) \nTaking the parameter along the spatial path as λ such that ˙ x i D i := d / d λ , Eq. (D1) yields \nd χ = 1 2 B · e 3 ω d λ, (D2) \nFIG. 4. Standard parallel transported vector U a PT and FermiWalker parallel transported vector U a FWT of the tangent vector U a along the non-geodesic curve U b ∇ b U a = A a . \n<!-- image --> \nwhere e i 3 ≃ √ -g tt ˙ x i /ω . ˙ x a ˙ x a ≃ 0 (2.21) leads to \nω 2 -g tt ≃ ( d l d λ ) 2 , (D3) \nwhere (d l ) 2 ≃ γ ij d x i d x j along the spatial path. This means e 3 ω d λ ≃ √ -g tt e 3 d l =: √ -g tt d l , then Eq. (D2) is rewritten in the form \nχ ≃ 1 2 ∫ √ -g tt B · d l . (D4) \nSince this expression is similar to the Faraday rotation angle in electromagnetic (see e.g. [66]), this is called \'gravitational Faraday rotation\'. \nFirst, we calculate χ for the case η = 0. Eq. (D4) into the surface integrate form for simplicity of calculation, we take close path C = C1 + C2 drawn in Fig. 5 6 . The distance between the lens and the point on the path C2 is taken to be O ( r s ). The path C1 routes near the lens, whereas path C2 is away from the lens. Using the Stokes theorem, Eq. (D4) leads to \n1 2 ∮ C ( √ -g tt B ) · d l = 1 2 ∫ S curl( √ -g tt B ) · d S , (D5) \nwhere S denotes the area enclosed by the closed path C. curl( √ -g tt B ) for the Kerr spacetime is \ncurl( √ -g tt B ) = 4 aM 2 ρ 3 ( ρ 2 -2 Mr ) 3 / 2 ∂ ϕ . (D6) \nBecause r curl( √ -g tt B ) → 0 as r → ∞ , the integral along the path C2 is negligible 7 . Therefore Eq. 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Path of integration. \n<!-- image --> \nbecomes \nχ ≃ 1 2 ∫ C1 ( √ -g tt B ) · d l = 1 2 ∫ S curl( √ -g tt B ) · d A , (D7) \nwhere d A i = -γ ij ϵ jrθ d r d θ ∝ -( ∂ ϕ ) i is the infinitesimal area of S. The path C1 is bending due to gravity. However, we can neglect the bending and the path C1 approximate with the straight line since the order of the integrand curl( √ -g tt B ) ∼ O ( ϵ 3 ) is the highest order which we focus on. Then Eqs. (D6) (D7) yield \nχ ≃ -2 aM 2 ∫ 1 -1 d µ ∫ r C2 r orb ( µ ) d r 1 r 4 ≃ -2 3 aM 2 ∫ 1 -1 d µ 1 r 3 orb \nwhere µ := cos θ and r C2 is the distance to the path C2. Since r C2 is the same order as r s , we have neglected the term with aM 2 /r 3 C2 = O ( ϵ 6 ) in the second equality. The relation between r orb and µ in the leading is \nr orb ∼ r min √ 1 -µ 2 . (D8) \nThus, the gravitational Faraday rotation angle for Kerr spacetime is \nχ ≃ -πaM 2 4 r 3 min . (D9) \n<latexit sha1\\_base64="IYQW8Vdd4LWimrrRjrD66pzQPAk=">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</latexit> \n<latexit 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sha1\\_base64="l7SJwtb0ZoMhLRaIbbGpgsWt7lQ=">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</latexit> \n<latexit sha1\\_base64="17jXTnW9i/eCPVwO2n1q89fE1ns=">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</latexit> \n<latexit sha1\\_base64="NqWfcx4SRUfxJcD8gI4lgX+tmYM=">AAADyXichVJNaxRBEK3N+BETNR9eBC+DS1SCWXrFRPEU9CKIkmSz2chuWHomvcmw8+VM7zrJMBc9CN7Fg6Ao+Eu8+Ac8xL/gUcGrb8pJMAkZe5juqtf1ql53lxW6TqyF2KuMGKdOnzk7em5s/PyFixOTU9NrcTCIbNW0AzeI1i0ZK9fxVVM72lXrYaSkZ7mqZfUf5PutoYpiJ/BX9U6oNjy55Ts9x5YaUDu63u1oleg0zrqTVVETPMzjRr0wqlSMpWCqskod2qSAbBqQR4p80rBdkhTja1OdBIXANigFFsFyeF9RRiaNgT1AnEKMBN7HvAWvXaA+/DxrzHwbdVz8EbgnMVNm5LV3sFpcZYZMFXYnXl9u/C6puc/0sGra/pf5H6WaenSXFTpQHDKSa7f3cwx33/5s3FuZSa+JT+KH+CY+ij3xRXwV/vCX/XlZrbwr1WVBU5bneTqrXrVbpWo2aQgVIdYeWAmrSg7dHPK8iErrDfh9ippPxrPvpdGPcdJGcdt5lZvYlRwZH7yxD+85q/D4fn3oS4G7fOcWPIlZsr6juAm/D34Elklz6KcazbO+DjCJ0+a1zAM8YcY2q0mQbQ5fRnlv14928nFj7VatvlCbX75dXbxfdPkoXaGrdAOdfIcW6SEtUZNP8obe0wfjkfHMSIzdv6EjlYJziQ4N4+Ufp93LgA==</latexit> \nFIG. 6. Flat geometry. r \' s is the distance from the lens to the intersection point between z axis and the line extending the path. \n<!-- image --> \nThis expression is consistent with Refs. [51]. For the case \n- [1] R. Abbott et al. (LIGO Scientific, VIRGO, KAGRA), (2021), arXiv:2111.03606 [gr-qc].\n- [2] P. Amaro-Seoane et al. (LISA), (2017), arXiv:1702.00786 [astro-ph.IM].\n- [3] S. Kawamura et al. , Class. Quant. Grav. 23 , S125 (2006).\n- [4] S. Kawamura et al. , PTEP 2021 , 05A105 (2021), arXiv:2006.13545 [gr-qc].\n- [5] T. T. Nakamura and S. Deguchi, Prog. Theor. Phys. Suppl. 133 , 137 (1999).\n- [6] R. Takahashi and T. Nakamura, Astrophys. J. 595 , 1039 (2003), arXiv:astro-ph/0305055.\n- [7] B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. Lett. 119 , 161101 (2017), arXiv:1710.05832 [gr-qc].\n- [8] P. Creminelli and F. Vernizzi, Phys. Rev. Lett. 119 , 251302 (2017), arXiv:1710.05877 [astro-ph.CO].\n- [9] J. M. Ezquiaga and M. Zumalac\'arregui, Phys. Rev. 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2024A&A...691A.111W | Identifying infall motions is crucial for our understanding of accretion processes in regions of star formation. The NHSUB3SUB 11 hyperfine intensity anomaly HIA has been proposed to be a readily usable tracer for such infall motions in starforming regions harboring young stellar objects at very early evolutionary stages. In this paper we seek to study the HIA toward 15 infall candidate regions in order to assess its reliability as an infall tracer. Using deep observations of the NHSUB3SUB 1 1 transition with the Effelsberg 100 m telescope we identified HIAs toward all 15 targets. Of the 15 sources 14 exhibit anomalous intensities in either the inner or outer satellite lines. All the derived HIAs conform to the framework of the existing two models namely hyperfine selective trapping HST and systematic contraction or expansion motion CE models. In our sample of infall candidates the majority of the HIAs remain consistent with the HST model. Only in three targets are the HIAs consistent with infall motions under the CE model. Thus the HIA could indeed be used as an infall tracer but does not appear to be highly sensitive to infall motions in our singledish data. Nevertheless the emission could be blended with emission from outflow activities. HIAs consistent with the HST model show stronger anomalies with increasing kinetic temperatures TSUBKSUB which is expected based on the HST model. On the other hand HIAs consistent with infall motions show little dependence on TSUBkSUB . Therefore HIAs may preferably trace the infall of cold gas. | 2024-11-01T00:00:00Z | ['2024A&A...691A.111W', '10.48550/arXiv.2409.12233', 'arXiv:2409.12233', '10.1051/0004-6361/202450919', '2024arXiv240912233W'] | ['stars: formation', 'ISM: clouds', 'ISM: kinematics and dynamics', 'ISM: molecules', 'Astrophysics - Astrophysics of Galaxies'] | NHSUB3SUB 11 hyperfine intensity anomalies in infall sources | 2,024 | 171 | 0.45 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF'] | 0 | https://arxiv.org/pdf/2409.12233.pdf | {'NH 3 (1 , 1) hyperfine intensity anomalies in infall sources': 'Gang Wu 1 , 2 , Christian Henkel 2 , 1 , Dongdong Zhou 1 , Friedrich Wyrowski 2 , Karl M. Menten 2 , and Jarken Esimbek 1 \n- 1 Xinjiang Astronomical Observatory, CAS 150, Science 1-Street Urumqi, Xinjiang 830011, China e-mail: [email protected]\n- 2 Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, 53121, Bonn, Germany \nReceived ...; accepted ...', 'ABSTRACT': 'Identifying infall motions is crucial for our understanding of accretion processes in regions of star formation. The NH3 (1 , 1) hyperfine intensity anomaly (HIA) has been proposed to be a readily usable tracer for such infall motions in star-forming regions harboring young stellar objects at very early evolutionary stages. In this paper, we seek to study the HIA toward fifteen infall candidate regions to assess its reliability as an infall tracer. By using deep observations of the NH3 (1 , 1) transition with the E ff elsberg 100 m telescope, HIAs have been identified toward all the targets.Fourteen out of fifteen sources exhibit anomalous intensities either in the inner or outer satellite lines. All the derived HIAs conform to the framework of the existing two models, namely, hyperfine selective trapping (HST) and systematic contraction or expansion motion (CE) models. In our sample of infall candidates, a majority of the HIAs remain consistent with the HST model. Only in three targets, the HIAs are consistent with infall motions under the CE model. Thus HIAs could be used as an infall tracer but seem not highly sensitive to infall motions in our single-dish data. Nevertheless, the emission could be blended with emission from outflow activities. HIAs consistent with the HST model show stronger anomalies with increasing kinetic temperatures ( T K), which is expected by the HST model. On the other hand, HIAs consistent with infall motions show little dependence on T K. Therefore, HIAs may preferably trace infall of cold gas. \nKey words. ISM: clouds - ISM: kinematics and dynamics - ISM: molecules - stars: formation', '1. Introduction': 'Accretion is a fundamental phenomenon in the process of star formation (e.g. Shu 1977; Mac Low & Klessen 2004). However, observational evidence of gas infall motions has remained inconclusive (e.g. Evans 1999; Evans et al. 2015). Blue-skewed line profiles of optically thick lines serve as readily accessible tracers for infall motions, relying on excitation gradients within the clump (e.g. Leung & Brown 1977; Zhou et al. 1993). Given higher excitation temperatures toward the center, the inward motion of the clump will manifest itself in more pronounced blueshifted emission (Zhou et al. 1993; Evans 1999). Nevertheless, there are also alternative explanations, such as chemical variations and rotation (e.g. Evans 2003). At the same time, blueskewed line profiles observed by single-dish telescopes may be blended with red-skewed profiles caused by outflow activities. Additionally, the requirement of elevated excitation toward the center implies that the blue-skewed emission as an infall tracer becomes less pronounced for objects in very early evolutionary stages with lower radial changes of the excitation temperature. \nSearching for redshifted absorption features against radio or millimeter continuum emission appears to be a promising approach (e.g. Di Francesco et al. 2001; Wyrowski et al. 2012; Yang et al. 2022). The inverse P-Cygni profile, characterized by redshifted absorption and blueshifted emission, is a relatively robust infall indicator, because these profiles largely confirm that the gas moving inwards (redshifted) is in front of the central source. Observations with high angular resolution interferometers are usually required to confirm infall toward the central continuum source (e.g. Beltrán et al. 2022). Because of the optically thick nature of the dust emission at frequencies > ∼ 1 THz, absorption lines are more readily observed in this frequency range. \nWyrowski et al. (2016) employed the Stratospheric Observatory for Infrared Astronomy (SOFIA) at ∼ 1.8 THz and detected redshifted absorption features toward high-mass star formation regions. Additionally, they noted that these redshifted absorption features are often not significantly contaminated by redshifted outflowing gas. However, acquiring new data for these highfrequency absorption lines, which are beyond the capabilities of ground-based telescopes, is not possible after the end of the SOFIA mission. \nIn all aforementioned methods, optically thin lines are usually required to trace the systemic velocities, delineating the boundary between the blue- and redshifted features and ruling out the possibility of two separate velocity components. Continuous e ff orts have been made to improve our capacity for detecting infall motions, aiming to achieve a comprehensive understanding of these motions in star-forming regions across a wider range of evolutionary stages and spatial scales. In this study, we test the viability of using ammonia hyperfine intensity anomalies (HIAs) as a promising infall tracer. \nThe microwave frequency inversion transitions of ammonia (NH3) are frequently used tracers in studies of star-forming regions, molecular clouds, and nearby galaxies (e.g., Zhang et al. 1998; Henkel et al. 2005; Lu et al. 2014; Wu et al. 2018). The ground-state para NH3 ( J , K ) = (1 , 1) transition comprises five distinct groups of hyperfine structure (hfs) components, originating from electric quadrupole splitting. These components include the main line ( ∆ F 1 = 0, where F 1 is the quantum number of the electric quadrupole coupling hyperfines) and four satellite lines ( ∆ F 1 = ± 1), with two on each side of the main line (Ho & Townes 1983). The presence of weaker magnetic spin-spin interactions results in a total of 18 hyperfine components distributed over the profiles of the five lines (e.g. Rydbeck et al. 1977). Un- \nonditions of local thermodynamic equilibrium (LTE) and optically thin emission, the two inner satellite line groups (ISLs) and outer satellite line groups (OSLs) are anticipated to have the same intensities (26% for each ISL and 22% for each OSL of the main line intensity) (e.g., Ho & Townes 1983). However, the expectation of equal intensities for the ISLs and OSLs is not always valid (e.g. Matsakis et al. 1977; Camarata et al. 2015). Non-LTE populations between the sub-states of the NH3 (1 , 1) level, which give rise to HIAs, can be attained by (1) hyperfine selective trapping (HST) from NH3 (2 , 1) to (1 , 1) levels (e.g., Matsakis et al. 1977; Stutzki & Winnewisser 1985) and / or (2) systematic contraction or expansion motions (CE) (e.g., Park 2001). It is possible to distinhguish between these two models, because they predict di ff erent relative intensities of the satellite lines (see Section 4.1). Under the CE model, redshifted / blueshifted photons (due to systematic motions) emitted from one hyperfine transition can be absorbed by another one, resulting in substantial changes in the level populations of the NH3 (1 , 1) sub-levels. Contraction / expansion can enhance the emission of the two satellite lines on the blue / red side, while suppressing those on the other side (e.g., Park 2001). Thus, the HIA is expected to serve as a tracer of systematic motions without relying on detailed analysis of line shapes (Park 2001). \nTo summarize, the HIA, as an infall tracer, exhibits two enhanced blueshifted satellite lines, resembling a discrete version of the blue-skewed profile, albeit with di ff erent underlying physics. The presence of five well-separated components in the NH3 (1 , 1) transition enables the straightforward identification of enhanced emission. Furthermore, the HIA is based on the cross-absorption between the hyperfine transitions. In principle, higher excitation toward the center is not required. For example, in the simulations conducted by Park (2001), obvious HIAs were reproduced in an infalling core with a constant temperature of 15 K. This makes the HIA a promising infall tracer of star-forming regions at very early evolutionary stages and also of large scale accretion. In contrast, blue-skewed profiles can reliably trace the infall motion around a hot core (Evans 2003). \nNonetheless, the relative contribution of systematic motions to HIAs remains a matter of debate (e.g. Longmore et al. 2007; Wienen et al. 2018; Zhou et al. 2020). In these studies, HIAs tend to be consistent with the HST model. However, infall motions of the targets in these studies are not confirmed. Hence, in this paper, we seek to investigate HIAs toward fifteen infall candidates, indicated by blue-skewed or redshifted absorption profiles, to test whether the HIA can be used as an infall tracer.', '2.1. Targets': "Weselected 15 infall candidates from the literature (see Table 1). Blue-skewed profiles were extensively observed in B335, such as by the Institute de Radioastronomie Millimétrique (IRAM) 30 m telescope in H2CO (212-111), CS (2-1), (3-2), (5-4) and by the Caltech Submillimeter Observatory (CSO) 10.4 m telescope in HCO + (3-2) with resolutions ranging from 11 '' to 28 '' (e.g. Zhou et al. 1993). Inverse P-Cygni profiles were also seen in Atacama Large Millimeter / submillimeter Array (ALMA) HCN (4-3) and HCO + (4-3) spectra with a resolution of about 0 . '' 5 (Evans et al. 2015). However, recent high-resolution ALMA observations of optically thin tracers revealed the presence of two velocity components, which could contribute to the double-peaked line profiles (e.g. Cabedo et al. 2021). \nInfall indicators were also widely identified in G031.41 + 0.31. For example, a redshifted absorption profile was identified in the SOFIA NH3 32 + -22 -line with a resolution of about 16 '' (Wyrowski et al. 2012). Meanwhile, blue-skewed profiles were detected in the HCO + (4-3), HNC (4-3), HCN (4-3) lines observed with the using the Atacama Pathfinder Experiment 12 meter submillimeter telescope (APEX) and the CS (2-1) line observed with the IRAM 30 m telescope with resolutions ranging from 17 '' to 27 '' (Wyrowski et al. 2016). Inverse P-Cygni profiles were also detected in dense cores with ALMA spectra of CH3CN (12-11) and H2CO (30 , 3-20 , 2), (32 , 2-22 , 1), and (32 , 1-22 , 0) with resolutions of about 0 . '' 1 (e.g. Beltrán et al. 2022). \nThe inverse P-Cygni profile in IRAS 18360-0537 was observed near the dust peak MM1 using the Submillimeter Array (SMA) in the CN (2-1) line with a resolution of about 1 . '' 4 (Qiu et al. 2012). Blue-skewed profiles were also identified with the CSO in HCN (3-2) and the Arizona Radio Observatory (ARO) 12 m telescope in HCO + (1-0) with resolutions of 27 '' and 70 '' , respectively (Yoo et al. 2018). \nG023.21 -0.3, G034.26 + 0.2, and G035.20 -0.7 were selected from Wyrowski et al. (2016). In these three targets, redshifted absorption features were observed using the SOFIA in the NH3 32 + -22 -line at a resolution of about 16 '' . Furthermore, blue-skewed profiles were identified in all three targets through in HCO + (4-3) with APEX with a resolution about 17 '' (Wyrowski et al. 2016). As also reported in Wyrowski et al. (2016), in the case of G023.21 -0.3, APEX HNC (4-3) observations with a resolution of 16 '' revealed blue-skewed profiles. Similarly, in the case of G034.26 + 0.2, APEX HCN (4-3) observations with a resolution of 17 '' showed blue-skewed profiles. As for G035.20 -0.7, IRAM 30m HCO + (1-0) observations with a resolution of 28 '' revealed the presence of blue-skewed profiles. \nBGPS3604, BGPS4029, and BGPS5021 were selected from Calahan et al. (2018). Blue-skewed profiles were observed in the three targets using ARO 12 m single-pointing HCO + (1-0) observations, with a resolution of about 68 '' (Calahan et al. 2018). Yang et al. (2021) mapped G029.60 -0.63, G053.13 + 0.09, G081.72 + 0.57, G082.21 -1.53, G121.31 + 0.64, and G193.01 + 0.14 with the IRAM 30m telescope at a resolution of about 28 '' . In all of these six targets, blue-skewed profiles were detected in the HCO + (1-0) line toward the positions showing strongest H 13 CO + (1-0) emission (Yang et al. 2021).", '2.2. NH 3 observations': "Deep NH3 observations (PI: Gang Wu, project ID: 15-21) were conducted in February 2022 with the E ff elsberg 100 m telescope 1 . The K-band double-beam and dual-polarization receiver was employed as the frontend. The facility Fast Fourier Transform Spectrometer (FFTS) was used as backend, which o ff ered two frequency windows with bandwidths of 300 MHz and 65536 channels each. This results in a channel width of about 4.6 kHz, corresponding to a velocity spacing of about 0.06 km s -1 at 23.7 GHz. The three metastable NH3 ( J , K ) = (1 , 1) , (2 , 2) , and (3 , 3) lines, along with the JK a Kc = 616-525 water maser transition, were observed simultaneously. Spectral calibration was applied, following the method described by Winkel et al. (2012). NGC7027 was used to obtain the initial pointing and focus corrections and to calibrate the spectral line flux, assuming a con- \ntinuum flux density of 5.7 Jy at 23.7 GHz (Ott et al. 1994). At the NH3 frequencies, the full width at half maximum (FWHM) beam size is about 37 '' and the main beam e ffi ciency is 60%. The conversion factor from the flux density scale to the main beam brightness temperature is 1.73 K Jy -1 . The focus was checked every few hours and pointing was calibrated every hour by observing nearby compact continuum sources. A positionswitching mode was used in the observations with the o ff position at an o ff set of 900 '' of each target in azimuth. Since achieving good signal-to-noise ratios for the NH3 (1 , 1) satellite lines is crucial to establish a robust HIA (e.g. Zhou et al. 2020), the total on + o ff integration time on each target exceeds 70 minutes and 1σ noise levels are about 50 mK on a T mb scale for a channel width of 0.06 km s -1 (see Table 1). Note that NH3 (1 , 1) hyperfine features were all measured simultaneously, thus ensuring an accurate relative calibration. \nThe GILDAS / CLASS software developed by IRAM was mainly used for the data reduction. All of the spectra have flat and gently varying baselines so that only linear baselines had to be subtracted.", '3. Results': "Thanks to the long integration times on the targets (see Table 1), all of the NH3 satellite lines are clearly detected (see Fig. 1). This is important for unambiguously determining the HIAs and distinguishing between the HIA models (see Section 1 and below). As discussed in Zhou et al. (2020), the HIA calculated with peak intensities from Gaussian fittings does not accurately reflect the true anomaly. That is because the red- and blueshifted ISLs (OSLs), which are the combination of Gaussian spectra of three (or two) hyperfine components with di ff erent o ff sets (see the blue vertical lines in Fig. 1), should exhibit di ff erent line widths and peak intensities even under LTE and optically thin conditions (see Appendix A in Zhou et al. 2020). We largely follow the recipe described in Zhou et al. (2020) to determine the HIA by the ratio of their red- to blueshifted integrated intensities. Specifically, we first used the combined 18 Gaussian hyperfine components to fit the observed NH3 (1 , 1) spectra. Then, based on the fitted central velocity and velocity dispersion, we defined the integrated velocity ranges (see the red vertical lines in Fig. 1). Finally, we calculated the HIAs of the ISL ( HIA IS) and OSL ( HIA OS) by the ratio of their redshifted to blueshifted integrated intensities from the observed spectra, \nHIA IS = F RISL F BISL , (1) \nHIA OS = F ROSL F BOSL , (2) \nwhere F RISL / F ROSL and F BISL / F BOSL are the integrated intensities of the red- and blueshifted sides of the ISLs / OSLs, respectively. The standard deviation σ HIA of HIAs is assigned by \nσ HIA = HIA × σ BL × p N C / ( F R) 2 + N C / ( F B) 2 , (3) \nwhere HIA is either the value of HIA IS or HIA OS. N C is the channel number within the integrated range and σ BL is the noise level of the baseline. F R and F B are the integrated intensities of the redshifted and blueshifted sides of the ISLs or OSLs. \nThe distribution of observed HIA IS and HIA OS values of the 15 targets is also shown in Fig. 2. Unity indicates no anomaly. In 14 out of 15 targets, either HIA IS or HIA OS deviate from \nunity by more than σ HIA, and in 10 of these targets, both HIA IS and HIA OS values exceed σ HIA. Thus the presence of HIAs is prevalent in our sample. The two dashed lines in Fig. 2 divide the the HIA IS and HIA OS data into four quadrants: quadrants I ( HIA IS > 1 and HIA OS > 1), II ( HIA IS < 1 and HIA OS > 1), III ( HIA IS < 1 and HIA OS < 1), and IV ( HIA IS > 1 and HIA OS < 1). These quadrants can be utilized to distinguish different HIA models (see Section 4.1). From Fig. 2 we can see that 1 (6.7%), 11 (73.3%), 3 (20%), and 0 data points are located in quadrants I, II, III, and IV (0, 7 (46.7%), 3 (20%), and 0, respectively, considering 1σ uncertainties). These fractions in the four quadrants, based on our deep observations, are generally consistent with previous statistical results (e.g. Wienen et al. 2018; Zhou et al. 2020). \nIn the cases of IRAS18360-0537, G031.41 + 0.31, G034.26 + 0.2, and G081.72 + 0.57, there is significant blending between their main lines and ISLs. This may impact the determination of the HIA IS if the main line has asymmetric profiles, which contributes di ff erently to the two ISLs. Hereby, we qualitatively discuss the potential corrections for the line blending. Taking IRAS 18360-0537 as an example, as seen in the first panel of Fig. 1 that the asymmetric main line contributes more flux to the blueshifted ISL than to the redshifted ISL. Thus the real HIA IS might be larger than the calculated one, as indicated by the arrow in Fig. 2. However, we should note that the length of the arrow is not quantitatively proportional to the corrections, as it is di ffi cult to precisely determine the contribution of the blending main line. For all of these four targets, we labeled the potential corrections with arrows in Fig. 2. We can see that, for G031.41 + 0.31 and G081.72 + 0.57, their actual HIA IS should be smaller than the calculated one. Thus these corrections do not alter their quadrants. For the other two targets, especially IRAS 18360-0537 whose HIA IS is closer to unity, the line blending e ff ect might change their quadrant from II to I. \nWe should note that changes in the relative intensities of the ISL or OSL may occur due to large opacities. Theoretically, each ISL (OSL) contains three (two) identical hyperfine components, indicated by the blue vertical lines in Fig. 1, However, as previously mentioned, while the intensities of the hyperfine components are the same, the frequency separations between them vary, as represented by the spacing of the blue vertical lines in Fig. 1. In high-opacity scenarios, satellites with closer separations between hyperfine components may achieve saturation more rapidly compared to those with larger separations. Therefore, we also derived the opacities of the ISL and OSL for all the spectra by summing up the opacities of their respective hyperfine components (see Table 1). The ISLs and OSLs in our spectra tend to be optically thin, with the exceptions of those of G035.2 -0.7 and G023.21 -0.3, which show moderate opacities of about 0.6. Therefore, the opacities should not result in a significant intensity di ff erence between the two ISLs / OSLs. \nAs discussed in Zhou et al. (2020), the deviations of the HIA defined by peak intensities from the true HIA is getting more pronounced for spectra with narrow linewithds. We can see from Fig. 3, take B335 as an example, that the peak intensity of the blueshifted OSL is larger than that of the redshifted one, just because the separation of the two hyperfine components within the blueshifted OSL is smaller than that of the redshifted OSL (see the blue vertical lines in Fig. 2). From Fig. 2 we see that HIA OS of B335 is actually larger than unity (the integrated intensity of the blueshifted OSL is smaller than that of the redshifted one). We refer to the Appendix A in Zhou et al. (2020) for the detailed comparisons between the HIAs defined by peak and integrated \nTable 1. Targets and observational parameters. \nNotes. ( † ) The 1σ baseline noise level on a T mb scale for a channel width of about 0.06 km s -1 . ( †† ) The peak opacities of the inner ( τ ISL) and outer ( τ OSL) satellites. (1) Yang et al. (2021) (2) Svoboda et al. (2016) (3) Calahan et al. (2018) (4) Wyrowski et al. (2016) (5) König et al. (2017) (6) Xu et al. (2021) (7) Wu et al. (2023) (8) Immer et al. (2019) (9) Watson (2020) . \nintensity ratios. In addition, at the resolution of 37 '' , our observations may encompass a number of cloud or velocity components, which are more evident in the case of G31.41 + 0.31 and IRAS18360 -0537, which show slight deviations from Gaussian profiles (see Fig. 1). While the limited angular resolution of our data is ideal for the determination of the gross HIA, it does not permit an evaluation of the spatial fine structure, eventually revealing more than one velocity component within the targeted area.", '4.1. Are HIAs sensitive to infall motions?': "The HST and CE models predict distinct HIAs (e.g. Zhou et al. 2020). According to the HST model, the intensity of the blueshifted ISL is expected to be stronger than that of the redshifted one, while the redshifted OSL should exhibit stronger intensity than the blueshifted one. Thus HIA IS < 1 and HIA OS > 1 (quadrant II; e.g. Stutzki & Winnewisser 1985; Camarata et al. 2015). In the context of the CE model (e.g. Park 2001), for infall motions, both the blueshifted ISL and OSL are anticipated to be stronger than the redshifted two lines, that is HIA IS < 1 and HIA OS < 1 (quadrant III). Conversely, for expansion motions, the redshifted ISL and OSL are expected to exhibit stronger intensity simultaneously, that is HIA IS > 1 and HIA OS > 1 (quadrant I). As discussed in Zhou et al. (2020), we can employ the HIA quadrant plot of HIA IS and HIA OS to identify the HIA models (see the di ff erent models labeled in the four subregions of Fig. 2). \nFrom Fig. 2, it is noteworthy that all the derived HIAs remain within the framework of the two models, with no data points in the forbidden quadrant IV. However, it is somewhat unexpected that a majority of the HIAs are located in the second quadrant, aligning with the HST model. This indicates that, in our observations, the HST model is the predominant model, even in sources likely harboring infall motions. However, alternative explanations exist. Similar to the case of blue-skewed profiles, they \nmight be blended with emission from outflows (e.g. Evans 2003; Wyrowski et al. 2016). This phenomenon is also applicable to HIAs, since ammonia emission is also commonly seen in outflows (e.g. Zhang et al. 2007). This could only be further clarified through interferometer observations, which are capable of resolving distinct structures. \nThere are three data points located in quadrant III, indicating the potential existence of infall motions. For these three sources, infall motions should be widespread within the beam size of about 37 '' . Increased kinetic temperatures would enhance the HST HIAs (e.g. Stutzki & Winnewisser 1985) and could also lead to more ammonia molecules being excited to the NH3 (2 , 1) level. Consequently, the likelihood of sources to be fit the HST model dominating is heightened. So, HIAs may preferably serve as infall tracers in star-forming regions at early evolutionary stages. However, we do not find a preferred evolutionary stage among these three targets in quadrant III. BGPS 4029, G029.60 -0.63, and G031.41 + 0.31 were reported to be associated with an infrared dark cloud (IRDC) (Peretto & Fuller 2009), Class 0 / I YSOs (Yang et al. 2020), and a massive protocluster, respectively (Beltrán et al. 2022). Nevertheless, HIAs may indeed be used as an infall tracer for early evolutionary stages, such as IRDCs. \nAs mentioned earlier, in the HST model, the ISL and OSL exhibit reversed anomalous intensities ( HIA IS < 1 and HIA OS > 1). In the CE model, both the blueshifted / redshifted ISL and OSL should be weaker / stronger simultaneously for infall / expansion motions ( HIA IS < 1 and HIA OS < 1 for infall motions, HIA IS > 1 and HIA OS > 1 for expansion motions). As a result, using the ratio between the sum of the intensities of the two redshifted satellite lines and the sum of the intensities of the two blueshifted satellite lines, the influence of the HST model (the ISL and OSL exhibit reversed anomalies) would be mitigated and that of the CE model would be enhanced. Therefore we also study the ratio HIA ISOS, \nHIA ISOS = F RISL + F ROSL F BISL + F BOSL . (4) \nFig. 1. Observed NH3 ( J , K ) = (1 , 1) spectra. In each panel, the green curve represents the 18-hyperfines fitting result and the blue vertical lines under the spectrum indicate the position of the 18 hyperfine lines and their relative strengths in the optically thin case under conditions of LTE. The red lines denote the integrated ranges used to calculate the HIA. The source name is labeled in the top-left corner of each panel. \n<!-- image --> \nWe should note that it is not known whether the anomalous fluxes of the blueshifted ISL and the redshifted OSL are precisely equal in the HST model (e.g. Stutzki & Winnewisser 1985). \nFig. 3 shows the distribution of HIA ISOS. Eight sources (six, considering uncertainties) out of 15 are consistent with infall motions. Indeed, more sources exhibit consistency with infall motions than those in quadrant III of Fig. 2. This outcome demonstrates that HIA ISOS could be a better infall tracer than HIA IS and HIA OS. However, HIA ISOS may also not be a very ideal indicator of infall for our data. Taking uncertainties into account, there are only six reliable data points of HIA ISOS consistent with infall motions in this sample of infall candidates. Naturally, the three sources already previously suggested to rep- \nresent infall (quadrant III sources in Fig. 2) are part of this subsample. \nTo summarize, in our single-dish observations, most of the detected HIAs are consistent with the HST model. HIAs could be used as an infall tracer but seem not highly sensitive to infall motions. High-resolution observations would be essential for a more precise assessment of the contaminating impact by outflows.", '4.2. HIAs versus kinetic temperature': 'As we mentioned before, higher temperatures would potentially lead to the dominance of the HST model. In this section, we explore the correlation between HIAs and the kinetic temperature \nIS \nFig. 2. Distribution of the hyperfine intensity anomalies of the inner ( HIA IS) and outer ( HIA OS ) satellite lines. Arrows indicate potential corrections for the blending of the primary and inner satellite lines. The source name is marked in close proximity to each data point. Gray dashed vertical and horizontal lines divide the panel into four subregions: I for HIA IS > 1 and HIA OS > 1, II for HIA IS < 1 and HIA OS > 1, III for HIA IS < 1 and HIA OS < 1, and IV for HIA IS > 1 and HIA OS < 1. The models that cause HIAs to be located in these subregions are also labeled. \n<!-- image --> \nFig. 3. Distribution of the combined hyperfine intensity anomalies. The source name is labeled above each data point. \n<!-- image --> \n( T K). NH3 inversion transitions are an invaluable spectroscopic probe of T K (Ho & Townes 1983). We derive the rotational temperature T R from the observed para-NH3 (1 , 1) and (2 , 2) spectral lines (see Fig. A.1) using the PySpecKit package (Ginsburg &Mirocha 2011), which is a forward-modeling tool for spectral lines. Then, T K is calculated following Tafalla et al. (2004) as \nT K = T R 1 -T R 42 ln[1 + 1 . 1exp( -16 / T R)] . (5) \nTafalla et al. (2004) conducted various Monte Carlo simulations involving data of the NH3 (1 , 1) , (2 , 1), and (2 , 2) transitions to derive an analytical expression to estimate T K from T R. T K can be very well approximated by this equation in the range T K = 5-20 K. It is important to note that the applicability of this approximation diminishes at higher temperatures. Fig. 4 shows the \nTable 2. Parameters for the linear regressions.Notes. The regressions were conducted by the SciPy package. ( † ) The correlation coe ffi cient assumes values within a range from -1 (indicating a perfect negative correlation) to + 1 (indicating a perfect positive correlation). A correlation coe ffi cient of zero denotes no relationship between the two variables under consideration. \ncorrelations between HIAs ( HIA IS, HIA OS, and HIA ISOS) and T K. The HIAs consistent with the HST model and CE model (infall motions) are emphasized in blue and red colors, respectively. Meanwhile, their linear regression results are also shown in each panel (blue and red lines) and in Table 2. \nWe first see from the blue data points in panels (a) and (b) of Fig. 4 that HIA IS and HIA OS all tend to show rising deviations from unity (indicating higher anomalies) with increasing T K, which is expected by the HST model. We should note that the correlations are weak, with correlation coe ffi cients of -0.29 and -0.12 for HIA IS and HIA OS, respectively (see Table 2). This may be attributed to the considerable dispersion among these data points, especially HIA OS determined by the relatively weak OSLs. While the HST model predominantly influences the blue data points, the CE model may also play a minor role, introducing additional uncertainties. For example, the two HIA models produce contrasting predictions concerning the enhancement of the OSLs. \nHIAs consistent with infall motions are not expected to exhibit a dependence on T K. In our observations, both HIA IS and HIA OS, indicated by red color, appear to show a constant value not depending on T K. It is essential to note that the sample size is limited, consisting of only three data points. Therefore, subsequent comparisons of potential trends in HIAs against T K are tentative. The slope of HIA IS appears more negative compared to that of HIA OS. In general, HIAs induced by infall motions (red data points) demonstrate relatively lower sensitivity to T K compared to HIAs consistent with the HST model (blue data points in Fig. 4). This outcome implies that HIAs might serve as more e ff ective infall tracers for relatively cold gas, which may help us to understand the large-scale accretion and infall motions in early evolutionary star-forming cores. \nFinally, in panel (c) of Fig. 4, we can see from the blue data points, in general, the HIA ISOS of the targets associated with the HST model, are very close to unity. So, HIA ISOS could largely weaken the impact of the HST model as explained before (see Fig. 3). This indicates that the anomalous flux to the blueshifted ISL is comparable to that of the redshifted OSL in the HST model. The red data points (i.e. those associated with infall motions) are not a ff ected by such a consideration, since their blueshifted ISLs and OSLs are enhanced simultaneously. \nFig. 4. Correlations of HIA IS and T K (panel a), HIA OS and T K (panel b), and HIA ISOS and T K (panel c). The HIAs consistent with the HST model and CE model (infall motions) are emphasized in blue and red colors, respectively. Blue and red lines indicate their linear regression results. Linear fit parameters for the blue and red lines are given in Table 2. \n<!-- image -->', '5. Summary': "We conducted deep observations of the ammonia hyperfine intensity anomalies (HIAs) with the E ff elsberg 100 m telescope in fifteen infall source candidates. By adopting a rational definition of the HIA proposed by Zhou et al. (2020), we seek to test whether HIAs can be used to trace infall motions, in particular of the cold molecular gas. Due to long integration times on the targets, all the NH3 satellite lines are clearly detected and all the HIAs of the inner ( HIA IS) and outer ( HIA OS) satellite lines are derived. In 14 out of 15 targets, either HIA IS or HIA OS values deviate from unity (indicating an anomaly) by more than their 1σ uncertainties. In 10 targets, both HIA IS and HIA OS values exceed their 1σ uncertainties. Thus the presence of HIAs is prevalent in our sample. Meanwhile, all the derived HIAs remain within the framework of the existing two models, the hyperfine selective trapping (HST) and systematic contraction or expansion motions (CE) models. It is found that a majority of the HIAs in the sources likely harboring infall motions are still consistent with the HST model. In three sources, HIAs are consistent with infall motions under the CE model, while a procedure mitigating e ff ects of the HST model even uncovers six such sources. Therefore, HIAs could be used as an infall tracer but seem to be not highly sensitive to infall motions in our single-dish observations. Nevertheless, akin to the case of blue-skewed profiles, HIAs might be blended with emission from outflow activities, since ammonia emission is also commonly seen in outflows. \nHIAs induced by the HST model are expected to be enhanced with increasing kinetic temperatures ( T K). HIA IS and HIA OS in our observations may show higher anomalies with increasing T K, but the correlations are weak. On the contrary, HIAs induced by infall motions seem to show relatively constant values against T K, suggesting that HIAs might serve as more e ff ective infall tracers for relatively cold gas. High-resolution observations of HIAs are crucial to further constrain the origin of HIAs and assess the contributions from infall and outflow motions. \nAcknowledgements. We thank the anonymous referee for useful suggestions improving the paper. Based on observations with the 100-m telescope of the MPIfR (Max-Planck-Institut für Radioastronomie) at E ff elsberg. This work was funded by the National Key R&D Program of China (No. 2022YFA1603103), the CAS 'Light of West China' Program (No. 2021-XBQNXZ-028), the National Natural Science foundation of China (Nos.12103082, 11603063, and 12173075), and the Natural Science Foundation of Xinjiang Uygur Autonomous Region (No. 2022D01A362). GW acknowledges the support from Youth Innovation Promotion Association CAS.", 'References': 'Beltrán, M. T., Rivilla, V. M., Cesaroni, R., et al. 2022, A&A, 659, A81 Camarata, M. A., Jackson, J. M., & Chambers, E. 2015, ApJ, 806, 74 Calahan, J. K., Shirley, Y. L., Svoboda, B. 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The source name is labeled in the top-left corner of each panel. \n<!-- image -->'} |
2024A&A...690A.389F | Context. Relativistic magnetic reconnection studies have so far focused on symmetric configurations where the upstream plasma has identical properties on the two sides of the layer. Yet just like nonrelativistic reconnection on the dayside of the Earths magnetosphere relativistic reconnection can also operate at the interface between highly asymmetric environments. The boundary layer between a relativistic jet and an accretion flow forming around a supermassive black hole can present asymmetric configurations in terms of plasma composition bulk velocity temperature and magnetization. Aims. We conducted the first study of relativistic magnetic reconnection where the upstream plasma is composed of electronpositron pairs on one side and electrons and ions on the other. We also investigated the impact of a relativistic symmetric shear flow applied along the reconnecting field lines. Methods. We simulated magnetic reconnection using 2D particleincell simulations. The initial setup was adapted from a classic Harris layer without a guide field modified to accommodate plasmacomposition and shear asymmetries in the upstream medium. Results. For a compositionasymmetric setup we find that the reconnection dynamics is driven by the electronion side which is the plasma with the lowest magnetization. The energy partition favors accelerating ions at the expense of electrons even more than in a corresponding symmetric setup. With respect to shear a superAlfvnic upstream decreases the laboratoryframe reconnection rate but unlike in nonrelativistic studies does not shut off reconnection completely. Conclusions. The asymmetries examined in this work lower the overall efficiency of electron acceleration relative to corresponding symmetric configurations. In the context of a black hole jetdisk boundary asymmetric reconnection alone is probably not efficient at accelerating electrons to very high energies but it might facilitate plasma mixing and particle injection for other acceleration channels at the interface. | 2024-10-01T00:00:00Z | ['10.1051/0004-6361/202451229', '2024arXiv240913495F', 'arXiv:2409.13495', '10.48550/arXiv.2409.13495', '2024A&A...690A.389F'] | ['acceleration of particles', 'magnetic reconnection', 'plasmas', 'methods: numerical', 'Astrophysics - High Energy Astrophysical Phenomena', 'Physics - Plasma Physics'] | Compositionasymmetric and sheared relativistic magnetic reconnection | 2,024 | 171 | 0.51 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF'] | 0 | https://arxiv.org/pdf/2409.13495.pdf | {'Composition-asymmetric and sheared relativistic magnetic reconnection': 'Enzo Figueiredo 1 , Benoît Cerutti 1 , John Mehlha ff 1 , and Nicolas Scepi 1 \nUniv. Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France e-mail: [email protected] \nReceived September 23, 2024; accepted ...', 'ABSTRACT': "Context. Relativistic magnetic reconnection studies have focused on symmetric configurations so far, where the upstream plasma has identical properties on each side of the layer. Yet, just like nonrelativistic reconnection on the day side of the Earth's magnetosphere, relativistic reconnection may also operate at the interface between highly asymmetric environments. The boundary layer between a relativistic jet and an accretion flow forming around a supermassive black hole may present such asymmetric configurations in terms of plasma composition, bulk velocity, temperature and magnetization. \nAims. In this work, we aim to conduct the first study of relativistic magnetic reconnection where the upstream plasma is composed of electron-positron pairs on one side, and electrons and ions on the other. We also investigate the role of a relativistic symmetric shear flow applied along the reconnecting field lines. \nMethods. We simulate magnetic reconnection using two-dimensional particle-in-cell simulations. The initial setup is adapted from a classic Harris layer without guide field, modified to accommodate plasma-composition and shear asymmetries in the upstream medium. \nResults. For a composition-asymmetric setup, we find that the reconnection dynamics is driven by the electron-ion side, which is the plasma with the lowest magnetization. The energy partition favors accelerating ions at the expense of electrons even more than in a corresponding symmetric setup. With respect to shear, a super-Alfvénic upstream decreases the laboratory-frame reconnection rate, but, unlike in non-relativistic studies, does not shut o ff reconnection completely. \nConclusions. The asymmetries examined in this work diminish the overall e ffi ciency of electron acceleration relative to corresponding symmetric configurations. In the context of a black hole jet-disk boundary, asymmetric reconnection alone is probably not e ffi cient at accelerating electrons to very high energies, but it might facilitate plasma mixing and particle injection for other acceleration channels at the interface. \nKey words. acceleration of particles - plasmas - magnetic reconnection - methods: numerical", '1. Introduction': "Magnetic reconnection is a widely studied plasma phenomenon, heating and accelerating particles through a change in the global magnetic topology (Zweibel & Yamada 2009). This process typically occurs when two antiparallel magnetic field lines snap and recombine across a thin current layer. The relativistic regime of reconnection has encountered wider interest in the last decade, as it provides e ffi cient nonthermal particle acceleration (e.g., Hoshino et al. 2001; Hoshino & Lyubarsky 2012; Cerutti et al. 2012b; Sironi & Spitkovsky 2014; Guo et al. 2014; Kagan et al. 2015; Werner et al. 2016). Here, in contrast to non-relativistic reconnection, the energy associated with the magnetic field dominates over the plasma rest mass energy, providing a significant free-energy reservoir for particle energization. Several regimes of relativistic magnetic reconnection have now been studied, including: with di ff erent plasma compositions (i.e., electronpositron pairs, and electron-ion plasmas, Melzani et al. 2014; Werner et al. 2018); involving strong radiative cooling and pair production (Jaroschek & Hoshino 2009; Cerutti et al. 2013; Hakobyan et al. 2019; Schoe ffl er et al. 2019; Mehlha ff et al. 2024); and driven by other plasma processes such as turbulence (Zhdankin et al. 2017; Comisso & Sironi 2018; Meringolo et al. 2023) and Kelvin-Helmholtz or Rayleigh-Taylor instabili- \nties (Cerutti & Giacinti 2020; Sironi et al. 2021; Zhdankin et al. 2023). \nRelativistic magnetic reconnection is a key driver of highenergy emission in various astrophysical environments. In pulsar magnetospheres, the equatorial current sheet forming beyond the light cylinder undergoes reconnection in a strongly radiative and pair-producing regime, emitting gamma-ray emission through synchrotron radiation (Lyubarskii 1996; Uzdensky & Spitkovsky 2014; Cerutti et al. 2016). Similar processes may also power gamma-ray flares in the Crab nebula (Uzdensky et al. 2011; Cerutti et al. 2012a), magnetar eruptions and fast radio bursts (Lyutikov 2003; Yuan et al. 2020; Mahlmann et al. 2022). In active galactic nuclei, magnetic reconnection is considered a plausible mechanism for explaining gamma-ray flares within blazar jets (Giannios et al. 2009; Nalewajko et al. 2011; Petropoulou et al. 2016; Mehlha ff et al. 2020) and also bursting activity closer to the central supermassive black hole, for instance X-ray flares in SgrA* (Ball et al. 2016; Scepi et al. 2022) and gamma-ray flares in M87* (Crinquand et al. 2021; Ripperda et al. 2022; Hakobyan et al. 2023; Stathopoulos et al. 2024). \nRecent VLBI observations of M87* show a clear edgebrightened jet connected to the innermost parts of the accretion flow (Lu et al. 2023), suggesting that particles are accelerated at the interface between these two entities. In the framework of \na Poynting-flux dominated jet (Blandford & Znajek 1977), this boundary layer may separate a relativistic ultra-magnetized jet loaded with pairs produced near the ergosphere on one side from a hot mildly magnetized accretion flow composed of electrons and ions on the other side. Such abrupt discontinuities in plasma composition, temperature, bulk velocity and magnetization may lead to qualitatively di ff erent particle acceleration compared to a symmetric environment. \nRecent global modeling e ff orts suggest magnetic reconnection as a prominent particle acceleration mechanism at such highly asymmetric jet-disk boundaries. The accretion of magnetic loops onto a spinning black hole leads to the formation of reconnecting current sheets between the jet and the disk (Parfrey et al. 2015; Yuan et al. 2019; El Mellah et al. 2022, 2023). Magnetic flux eruptions reported in general relativistic magnetohydrodynamic (GRMHD) simulations also provoke large-scale reconnection events that push magnetized flux tubes through the accretion flow (e.g., see Igumenshchev 2008; Tchekhovskoy et al. 2011). This phenomenon can be accompanied with additional current sheet formation at the interface between the jet and the disk (Ripperda et al. 2020, 2022; Chashkina et al. 2021; Vos et al. 2023). However, it is unclear how magnetic reconnection and particle acceleration proceed within such asymmetric boundary layers. \nAsymmetric and sheared magnetic reconnection have been previously studied in the non-relativistic regime, in particular in the context of the interaction between the Earth's magnetosphere and the solar wind (i.e., the magnetopause). An important takeaway message from these studies is that a strong velocity shear applied along the field lines decreases the reconnection rate. It can even halt the process entirely for super-Alfvénic flows (La Belle-Hamer et al. 1995; Cassak & Otto 2011; Cassak 2011). An asymmetry in the upstream plasma density and magnetic field strength leads to a global reorganization of the current layer and a change in the plasmoids' shape (Murphy et al. 2012; Eastwood et al. 2013). However, from these studies alone it is unclear how such asymmetries would a ff ect particle acceleration in a relativistic context. \nThe first study of relativistic asymmetric magnetic reconnection was recently carried out by Mbarek et al. (2022). They simulate a pair plasma with an upstream density and magnetic field strength contrast on each side of the reconnection layer, hence creating a magnetization asymmetry in the reconnection process. They find that the lowest magnetization side dictates the reconnection rate, and that particle acceleration remains in an intermediate regime between the higher and lower magnetizations. This result suggests that particle acceleration via reconnection could be quenched at a black hole jet-disk interface. \nIn this study, our first objective is to investigate the role of plasma asymmetries on particle acceleration and mixing within relativistic reconnection layers, using two-dimensional (2D) particle-in-cell (PIC) simulations. More specifically, and in contrast to Mbarek et al. (2022), we investigate the role of a plasma-composition asymmetry where one side is filled with an ultra-relativistic ultra-magnetized plasma of pairs, while the other side is composed of a mildly relativistic and magnetized electron-ion plasma. This configuration is reminiscent of a black hole jet-disk boundary layer. Our second objective is to evaluate the impact of a relativistic shear flow directed along the field lines - another property reminiscent of the jet-disk boundary - on the reconnection rate and on the particle acceleration e ffi ciency. \nThis article is organized as follows. In Section 2, we detail the modifications made to the classic relativistic Harris setup to \nimplement a velocity shear and a composition asymmetry in the upstream medium. In sections 3 and 4, we respectively present our findings on composition asymmetry and shear. We summarize and discuss the implications of our results in Section 5.", '2.1. Asymmetric Harris equilibrium': 'Fig. 1. Sketch of the simulation setup. The red dashed line separates the pair plasma on the jet side (upper half) from the electron-ion plasma on the disk side (bottom half). Shear flow is parallel to the magnetic field on each side. \'BC\' stands for \'boundary conditions\'. \n<!-- image --> \nIn this work, we employ Zeltron , an electromagnetic PIC code developed by Cerutti et al. (2013); Cerutti & Werner (2019). Our aim is to simulate a current layer that would have been locally formed by instabilities on a global jet-disk interface using a 2D local Cartesian box. In our simulations, the initial plasma is composed of pairs above the reconnection layer, representing the \'jet side\', while the plasma below is composed of electrons and ions, representing the \'disk side\'. As shown by GRMHD simulations (e.g., Vos et al. 2023), the magnetic field strength is expected to be similar on each side of the interface. The magnetization contrast should then be only imposed by the composition and / or density asymmetry. We therefore impose the same initial magnetic field strength on both sides of the reconnection layer. This is another di ff erence (in addition to the plasma composition and shear) from the setup used by Mbarek et al. (2022). \nOur boundary conditions are reflective on the upper and lower sides of the box and periodic on the right and left sides. These conditions apply for both the fields and the particles. For the initial conditions, we adapt the classic relativistic Harris setup with no guide (out-of-plane) field component, which involves an initial antiparallel magnetic field supported by an equatorial current layer (e.g., Kirk & Skjæraasen 2003). This initial current sheet is razor thin (on the plasma skin depth scale, defined below): perhaps much thinner than in a real jet-disk interface. This merely helps kickstart reconnection, allowing us to study the nonlinear evolution of the current sheet in its more developed and e ff ectively thicker stage. \nThe initial Harris magnetic field profile is given by \nB ( y ) = B 0 tanh y -Ly / 2 δ ! e x , (1) \nwhere δ is the initial current layer thickness, Ly ( Lx ) is the height (width) of the box, and e x is the unit vector along the x -direction. This magnetic field is supported by a layer of positively and negatively charged particles counterstreaming, or drifting , in the out-of-plane (i.e., z -) direction. Their bulk drift velocity is set to VH / c = 0 . 6, giving the current density, \nJ = -2 eVHnH ( y ) e z , (2) \nwhere e is the positron charge. The drifting particle number density reads \nnH ( y ) = nH , 0 cosh -2 y -Ly / 2 δ ! , (3) \nwhere nH , 0 = cB 0 / 8 π eVH δ . In symmetric reconnection studies, the box is filled with a uniform background plasma. However, in our scenario, the characteristics of the background plasma vary depending on the vertical position in the box. In the upper half of the box ( y > Ly / 2, i.e., the jet side), the background plasma is composed exclusively of pairs of total mass density 2 menj , while in the bottom half ( y < Ly / 2, i.e., the disk side), it is composed of electrons and ions of total mass density ( me + mi ) nd = (1 + µ ) mend , where µ = mi / me is the ion-to-electron mass ratio. On the jet (disk) side, the plasma moves along field lines with bulk velocity V s = Vs e x ( -Vs e x ) with a corresponding Lorentz factor Γ s = (1 -V 2 s / c 2 ) -1 / 2 : a symmetric shear with an abrupt transition at the midplane. \nIn order for the simulation to start in equilibrium, we need to ensure pressure balance both between the current sheet and the upstream magnetic field, and between the jet and the disk. This requires \nB 2 0 8 π = 2 nH , 0 kBT ∗ H Γ H and (4) \nnjkBTj = ndkBTd , (5) \nwhere Tj and Td represent the respective initial temperatures of the jet and disk plasmas in the laboratory frame, kB is the Boltzmann constant and T ∗ H is the drifting particles\' comoving temperature (considering a Lorentz factor Γ H = (1 -V 2 H / c 2 ) -1 / 2 ). We note that the plasmaβ parameter, defined as β = 16 π nkBT / B 2 0 , has an identical value for both the jet and disk sides (see Mbarek et al. (2022) for a β asymmetric setup), but the magnetization in the laboratory frame, defined as σ j = B 2 0 / 8 π Γ snjmec 2 for the jet and σ d = B 2 0 / 4 π Γ snd ( me + mi ) c 2 for the disk, changes because of the composition contrast. A global sketch of the initial setup is provided in Fig. 1. \nWe need the initial particle energy distribution functions in order to impose the previously defined macroscopic quantities. They follow relativistic drifting Maxwellians as defined by Swisdak (2013). Therefore, we have to initialize two distribution functions for the current layer (pairs and ions), two for the disk side (electrons and ions) and one for the jet side (pairs). We define four species of particles: the positrons (in the jet region), the ions (of charge + e in the disk region) and the electrons in each region. Their respective density profiles n e , p , i d , j are \nn e j ( y ) = njH ( y -Ly / 2) + f nH ( y ) , n e d ( y ) = ndH ( Ly / 2 -y ) + (1 -f ) nH ( y ) , n p j ( y ) = njH ( y -Ly / 2) + f nH ( y ) , n i d ( y ) = ndH ( Ly / 2 -y ) + (1 -f ) nH ( y ) , \nwhere H is the Heaviside step function, and the parameter f represents the mixing fraction in the Harris sheet. For f = 0 ( f = 1), the initial current layer is exclusively composed of disk (jet) plasma. This parameter only sets the initial state of our setup, and we have checked that it does not a ff ect the simulation behavior. Thus, we set its value to f = 1 / 2. When defining the particles\' momenta, we need to decide whether we consider them to be part of the Harris sheet, in which case we assign them a drift velocity ± VH e z , or in the disk / jet, in which case we assign them a shear bulk motion ± Vs e x . For a particle at a given coordinate y , the following number is computed: \nP ( y ) = nd , j H ( ± y ∓ Ly / 2) n e , p , i d , j ( y ) . (6) \nSubsequently, a random number between 0 and 1 is drawn. If it exceeds P ( y ), then the particle is considered to follow the Harris layer energy distribution function; otherwise, it will have a background energy distribution function. It can be easily verified that, far from the current sheet, P ( y ) → 1 so that the particle will almost always follow a background distribution. Conversely, in the current sheet it will likely follow a Harris distribution. \nFollowing Werner et al. (2018), a small (one percent) tearinglike perturbation is applied to the initial magnetic field (Eq. 1). This procedure serves two purposes: (i) it speeds up the onset of reconnection, and (ii) it predefines the location of the main X- and O-points during the active phase of the simulation and facilitates the measurement of the reconnection rate as shown in Sect. 2.3 hereafter. With this perturbation, the initial out-of-plane component of the magnetic vector potential reads \nAz = B 0 δ ( ln " cosh y -Ly / 2 δ !# -ln " cosh Ly 2 δ !#) × ( 1 + 0 . 01 cos 2 π x -3 Lx / 4 Lx ! cos 2 π y -Ly / 2 Ly !) . (7) \nThe dimensions of the cells in each direction, ∆ x and ∆ y , are set to be equal and the time step, ∆ t , is set at 0 . 99 times the critical Courant-Friedrichs-Lewy time step. We initialize the simulations with 16 particles per species per cell. \nIn summary, the setup allows for the manipulation of several free parameters, including plasma β , the disk magnetization σ d , the jet-disk density ratio nj / nd , and the shear flow velocity Vs . For the remainder of this study, we set nj = nd for numerical convenience, thus only imposing a magnetization asymmetry through the mass ratio between pairs and ions. This assumption may not be very realistic in the context of a jet-disk boundary layer where a strong number density contrast is expected, but it still captures a mass density contrast, 2 njme ≪ nd ( mi + me ).', '2.2. Sets of simulations': "Given our setup, this work consists of two series of simulations. In the first, we study the e ff ect of composition asymmetry on magnetic reconnection without velocity shear (i.e., setting Vs = 0). This asymmetry leads to a magnetization contrast since, for µ ≫ 1, the jet magnetization is σ j ≃ µσ d / 2. We set µ = 100 so that the magnetization ratio is σ j /σ d ≃ 50, and we explore the parameter space ( β, σ d ) in the 9 simulations listed in Table 1. We adopt plasma conditions previously studied by Ball et al. (2018) and similar to those observed at jet-disk interfaces in GRMHD simulations, where the ion magnetization is transrelativistic ( σ d ≲ 1) while that for pairs is ultrarelativistic ( σ j ≫ 1). In addition to our 9 asymmetric simulations, \n√ \nTable 1. Full list of simulations performed in this study. Top: Asymmetric setup exploring the ( β, σ d ) parameter space. Middle: Corresponding symmetric electron-ion reconnection simulations included for comparison with the asymmetric runs. Bottom: Shear simulation parameters. \nwe present 2 symmetric electron-ion simulations for comparison where the jet's plasma has the same properties as the disk's. The resolution is fixed in terms of the nominal electron Larmor radius ρ 0 = mec 2 / eB 0 = 8 ∆ x / 3. Thus, the skin-depth ranges from de = c /ω pe ∼ 8 ∆ x to de ∼ 27 ∆ x , where ω pe = p 4 π nde 2 / me is the electron plasma frequency. The initial current sheet width is set to δ ≃ 5 ∆ x . \nThe second set of simulations focuses on the e ff ect of a velocity shear along the reconnection layer with a symmetric pair plasma ( µ = 1). Here, we fix β = 5 × 10 -2 and the magnetization, σ d , is set to either 1 or 10. For both values of σ d , we scan the dimensionless shear four-velocity, Us = Γ sVs / c , from 0 to 2 UA , where UA = √ σ d is the relativistic Alfvén four-velocity. The resolution is set such that Γ s ρ 0 = 16 ∆ x for σ d = 1, and Γ s ρ 0 = 2 ∆ x for σ d = 10. This ensures that the hot skin depth de , h = √ Γ sde is conserved for all simulations sharing the same σ d value: de , h ∼ 6 ∆ x ( σ d = 10), de , h ∼ 16 ∆ x ( σ d = 1). The current sheet width is set to δ = 4 ∆ x . A summary of the simulation parameters is provided in Table 1.", '2.3. Measurement of the reconnection rate': "We measure the reconnection rate based on the evolution of the reconnected magnetic flux, denoted as Φ rec, and defined by the equation, \nΦ rec = Az ( X ) -Az ( O ) . (8) \nHere, 'X' represents the location of the major X-point, which corresponds to the maximum of Az in the midplane. On the other hand, 'O' denotes the location of the major O-point at the center of the largest plasmoid; it coincides with the minimum of Az in the midplane. Using Faraday's law, the instantaneous dimensionless reconnection rate, denoted as β rec, can then be expressed \nas, \nβ rec = 1 cB 0 d Φ rec d t . (9) \nFollowing the methodology outlined by Werner et al. (2018), we calculate a mean reconnection rate, denoted as ⟨ β rec ⟩ , by averaging the previously defined instantaneous rate over the time interval between t 20% and t 40%. Here, t 20% and t 40% correspond to the instants when 20% and 40% of the initial magnetic flux in half of the box, Φ 0 = B 0 Ly / 2, has reconnected. By measuring in this time interval, we aim to isolate the intrinsic reconnection rate, avoiding artificial contributions from both the initial phase, when reconnection is still ramping up, and from the late-time phase, when reconnection starts to slow down due to the finite flux available in the box. Because of the intermittent nature of magnetic reconnection, there is a scatter of about 10% to 15% in our measurements of the reconnection rate.", '3.1. Kinematics and global behavior': 'We show two representative snapshots of the plasma density and field lines from our composition-asymmetric runs in Fig. 2. At first glance, there are no noticeable di ff erences with previous symmetric studies: the current sheet becomes quickly unstable to the tearing mode, creating a chain of merging plasmoids separated by X-points. However, a closer look at the system - and in particular its time evolution - reveals that fast magnetosonic waves launched during plasmoid mergers propagate faster and on shorter lengthscales on the pair side than on the electron-ion side, an e ff ect resulting from the higher σ of the pair plasma. This wave-speed asymmetry leads to di ff ering plasma response \nFig. 2. Plasma number density in the composition asymmetric setup (no shear) shown in the middle of the reconnection process (top left panel, Φ rec ∼ 0 . 15 Φ 0) and towards the saturation stage (bottom left panel, Φ rec ∼ 0 . 45 Φ 0) for σ d = 0 . 1, β = 10 -3 and σ j = 5 (run SD01B0001 ). White contours represent magnetic field lines. We provide on the top and bottom right panels a plot of di ff erent quantities for a slice shown by the green dashed line on the density maps. The blue curve represents the x-component of the magnetic field and the red curve represents the plasma mass density ρ (in logarithmic scale). The box midplane is represented by the black dashed lines. For the time evolution, see the animation on the journal website. \n<!-- image --> \ntimes on each side, which provokes an oscillatory vertical motion of the current layer that is absent in symmetric simulations (see the full time evolution of the simulation in the animation provided on the journal website). \nAs opposed to previous studies of both relativistic and nonrelativistic asymmetric magnetic reconnection (Murphy et al. 2012; Mbarek et al. 2022), the shape of the plasmoids seems rather symmetric. This behavior can be explained by the fact that the magnetic field strength is symmetric in our setup. The magnetic pressure is thus the same on both sides of the reconnection layer, resulting in the absence of strong plasmoid deformation. \nWe measure reconnection rates in our asymmetric configurations that are slightly higher ( ∼ 30% more) than in our corresponding symmetric electron-ion plasma simulations. For instance, we obtain ⟨ β rec ⟩ ≃ 0 . 029 for the asymmetric run with σ d = 0 . 1 and β = 10 -3 while the corresponding symmetric simulation showed ⟨ β rec ⟩ ≃ 0 . 022 (see Table 1). Even so, the rate stays much smaller than what would be expected for symmetric pair-plasma reconnection (i.e., β rec ∼ 0 . 1). This suggests that the side with the lowest magnetization primarily dictates the overall reconnection rate, as previously demonstrated in the context of density and magnetic field asymmetries by Mbarek et al. (2022).', '3.2. Energy partition and mixing': "We investigate the particle and energy mixing between the disk and the jet shown in the left panels of Fig. 3. The density mixing between the jet and the disk is very e ffi cient at the X-points and \nin the plasmoids. This behavior may explain the previously observed dynamics of the reconnection process: because reconnection e ffi ciently mixes particles, the magnetization in the reconnection layer is locally the one imposed by the disk ions, since, after mixing, these particles control the local mass density. Looking at the energy mixing in Fig. 3 (right panels), we observe that the energy density of the disk significantly prevails within the plasmoids. This dominance of the disk energy, coupled with the homogeneous mixing of particles, implies that the disk undergoes more substantial energization compared to the jet plasma during the reconnection process. Additionally, we remark that a few highly energetic protons manage to get to the upper edge of plasmoids. We speculate that, in 3D, where plasmoids have been shown to be less e ffi cient at confining high-energy particles (Zhang et al. 2021; Chernoglazov et al. 2023), this tendency would allow particles to escape plasmoids even more e ff ectively and contaminate the jet. \nWe plot the ratio of the jet-to-disk particles' total kinetic energies in Fig. 4 for our entire composition-asymmetric simulation campaign. The total kinetic energies for disk and jet particles are defined as \nEd = Z ( γ -1) d N d e d γ + µ d N d i d γ d γ and (10) \nEj = Z ( γ -1) d N j e d γ + d N j p d γ d γ. (11) \nFig. 3. Plasma number density mixing (left panels) defined as ( nj -nd ) / ( nd + nj ), and plasma kinetic energy density mixing (right panels) defined as ( ϵ j -ϵ d ) / ( ϵ d + ϵ j ), where ϵ j / d = γ nj / dmc 2 , in the early (top panels, Φ rec ∼ 0 . 15 Φ 0) and late (bottom panels, Φ rec ∼ 0 . 5 Φ 0) stages of the simulation. Here, σ d = 0 . 3 and β = 10 -2 (run SD03B001 ). \n<!-- image --> \nIn order to compare simulations with di ff erent reconnection rates - and thus varying time evolution - we compute this ratio once 50% of the initial magnetic flux has reconnected (the results are unchanged if we consider the instants when 30% and 40% of the initial flux is reconnected). The first conclusion is that in every case, the disk retrieves more energy than the jet from the reconnection event: up to four times more for the lowest plasmaβ simulations, β = 10 -3 . Second, higher magnetization ( σ d = 1) means a more symmetric behavior between the pairs and ions, and thus explains why the energy ratio decreases when σ d = 1. With a higher value of β , the plasma's initial thermal energy is more significant compared with the magnetic energy. Hence the energization during magnetic reconnection is less significant and the energy ratio is thus less a ff ected. However, the fact that σ i = 0 . 3 seems to create slightly more asymmetry than σ i = 0 . 1 at lowβ remains unexplained. \nWe emphasize the fact that most of the asymmetry in Ed / Ej is due to the ions acquiring much more energy from the reconnection process, as previously observed by Werner et al. (2018); Ball et al. (2018) in electron-ion reconnection studies. Despite \nthis qualitative similarity with the symmetric case, the preferential energization of ions over electrons is quantitatively more extreme in the presence of asymmetry with respect to findings of earlier symmetric electron-ion reconnection work. We measure, for instance, in our σ = 0 . 1 and β = 10 -3 simulation, that the ion kinetic energy ends up ∼ 5 . 6 times higher than that of the disk electrons (i.e., a higher ratio than Ed / Ej in Fig. 4 because here we compare the disk ions to only the disk electrons). In contrast, for our corresponding symmetric simulation, we measure this ratio as 3.4, consistent with Werner et al. (2018). This extra preferential ion energization tends to vanish for higher magnetizations.", '3.3. Spectra': 'We further investigate the asymmetry in plasma energy gain during the reconnection process by looking at the particle energy distributions (Fig. 5). Looking first at the σ i = 0 . 1, β = 10 -3 run, we notice that the ions reach higher energies than the leptons ( ∼ 3 times more). Given that the magnetization is less than one, \nFig. 4. Energy partition between the electron-ion (disk) plasma and the pair (jet) plasma at the stage when half of the initial magnetic flux has reconnected, Φ rec = 0 . 5 Φ 0. \n<!-- image --> \na distinct nonthermal tail is not apparent. These two observations are in accord with earlier symmetric studies (Melzani et al. 2014; Werner et al. 2018; Ball et al. 2018). Instead of a non thermal tail, the particle spectra are composed of two nearly thermal humps: a low-energy hump representing the initial thermal distribution, and a high-energy component corresponding to the plasma energized by reconnection. The gap of a factor ∼ 10 between the high-energy electron and ion peaks is consistent with the lower e ffi ciency for the lepton heating observed in Sec. 3.2. If the leptons were instead heated as much as the ions, both species should reach a final energy of ( ⟨ γ ⟩-1) m / me ∼ σ imi / me ∼ 10. However, this is only consistent with the energy of the ion peak ( ∼ 10); the lower electron peak ( ∼ 1) can only be explained by a reduced lepton heating e ffi ciency with respect to ions. Examining next the σ d = 1 regime, nonthermal tails with slightly di ff ering slopes start appearing for the leptons. The cuto ff energy seems to be similar for all species (electrons and ions), however the fraction of ions getting to higher energies is significantly higher. \nWe compare the asymmetric and symmetric runs with σ d = 0 . 1 and β = 10 -3 in Fig. 6. Both runs show similar spectral shapes, but the asymmetric particle distribution appears shifted to slightly higher energies. One reason for this shift could be that there are two times more ions for the same amount of magnetic energy in the symmetric simulation compared to the asymmetric run. Overall, despite subtle di ff erences, the asymmetric and corresponding symmetric runs both show a strong bias, as the magnetization decreases, toward ion acceleration at the expense of electrons. The key insight from this analysis is then that it is this bias - which is evidently independent of the composition asymmetry - that primarily explains the large energization discrepancy between the jet and disk plasmas.', '4.1. Overall picture': 'In our second set of simulations, we return to a a symmetric pairplasma composition. Here, the current layer is antisymmetrically sheared by the upstream pair plasma, which moves with a bulk velocity ± Vs e x . Given that certain simulations involve elevated bulk Lorentz factors ( Γ s ≳ 5), we apply a 9-point digital filter to the current densities at each timestep (Langdon & Birdsall 1970) to avoid the numerical Cherenkov e ff ect. \nWeshow a plasma density plot for the case of super-Alfvénic shear ( Us = 2 UA ) in Fig. 7. The propagation of fast magne-', 'd = 0.1, =10 3': '<!-- image --> \nd \n= 1, \n=10 \n3 \nFig. 5. Particle energy distributions for all species at the end of the σ d = 0 . 1 ( SD01B0001 , top panel) and σ d = 1 ( SD1B0001 , bottom panel) runs, for both β = 10 -3 . \n<!-- image --> \nFig. 6. Same as the top panel in Fig. 5 but with the plasma properties identical on both sides (i.e., symmetric reconnection with σ j = σ d = 0 . 1 and β = 10 -3 , run S01B0001SYM ). \n<!-- image --> \ntosonic waves is a ff ected by the shear: the wave fronts are inclined with respect to the reconnection layer instead of, as occurs when Vs = 0, nearly parallel to it. Despite the strong bulk motion, X-points and plasmoids are not swept away because the shear is symmetric. \nSpiral patterns clearly appear in the density mixing map, as shown on the bottom panel of Fig. 7 (e.g., in the rightmost plasmoid with center located near x = 500 de ). In the context of shear, plasma flows into plasmoids from X-points with a preferential velocity direction set by whether the X-point flanks the plasmoid to the right or to the left. Plasma entering from an X-point \nFig. 7. Simulation of pair-plasma relativistic magnetic reconnection with an antisymmetric shear applied along the current layer. The top panel shows the plasma number density; the bottom panel shows the mixing of both upstream plasmas as in Fig. 3. Here, σ d = 1, Us = 2 UA (run SD1US6 ) and the white contours represent the magnetic field lines. For the time evolution, see the animation on the journal website. \n<!-- image --> \nto the right (left) of a given plasmoid comes preferentially from the leftward-moving (rightward-moving) upstream plasma below (above) the layer. This plasma then follows a freshly reconnected magnetic field line, tracing a spiral trajectory around the core of the growing plasmoid. A plasmoid trapped between two active X-points is thus fed by the upper plasma from its left side and by the lower plasma from its right side, producing, respectively, the red- and blue-colored spirals in Fig. 7. Other than such spiral features in the density mixing map, reconnection proceeds in a similar fashion as for an initially static upstream plasma. \nAccording to the linear analysis performed by Chow et al. (2023), we may expect the appearance of the Kelvin-Helmholtz (KH) instability for simulations where Vs > VA . However, the tearing instability seems to grow more rapidly than the KH instability, and there is no clear sign of KH vortex formation throughout the duration of the simulations. As plasmoids grow during the evolution of the system, the velocity gradient across the midplane decreases, further inhibiting the operation of the KH instability. A sharp velocity gradient is maintained only in the vicinity of X-points. Toward the end of the simulation (e.g. when d Φ rec / d t ∼ 0), we observe that super-Alfvénic shears tend to decelerate to sub-Alfvénic speeds, due in part to the growth of large plasmoids, the periodic boundary conditions, and the finite box size. We believe that an outflowing box setup would mitigate this late-time deceleration of the shear, but it would not strongly impact our measurements (presented below) of the reconnection rate and the particle energy distribution. These are performed much earlier in the simulation: before shear-flow deceleration linked to excessively large (periodic-boundary-induced) plasmoids mature.', '4.2. Reconnection rate': 'We compute the reconnection rates for each shear simulation using the method outlined in Sec. 2.3. Figure 8 presents the reconnection rate, relative to its Vs = 0 reference value, as a \nfunction of the shear four-velocity, Us = Vs Γ s / c , normalized to the Alfvénic four-velocity, UA = √ σ d . Our primary observation is that reconnection slows down with increasing shear speeds, dropping to about 20% of its reference rate once Us reaches 2 UA . Notably, for super-Alfvénic speeds, the reconnection rate appears to follow a similar trend between the σ = 1 and σ = 10 cases, suggesting a potential universality in this behavior. We point out, however, a local enhancement of the reconnection rate for modest shear speeds ( Us = UA / 3) that occurs only for σ d = 1. This feature is not clearly understood, but it might not be significant given the ∼ 10% uncertainty in the rate measurements reported in Sect. 2.3. This uncertainty is somewhat amplified by the fact that we normalize by the reconnection rate with no shear (which itself is also only accurate to roughly 10%). Similar considerations apply to the slight dip observed at σ d = 10 ( Us = 2 UA / 3). The overall decrease of the reconnection rate with increasing shear-flow velocity may be due to the loss of causal connection between the fast-moving upstream plasma and the nearly static X-points. \nFig. 8. Reconnection rate as a function of the shear-flow 4-velocity, Us , in units of the Alfvén 4-velocity, UA = √ σ d . The rates are normalized (separately for σ d = 1 and σ d = 10) to the Us = 0 rate denoted as β rec , 0. The error bars represent our estimated relative uncertainty of 10% through occurrences of a given run as described in Sec. 2.3. \n<!-- image --> \nThe general slowing of reconnection with increasing shear is in qualitative agreement with previous non-relativistic studies (La Belle-Hamer et al. 1995; Cassak & Otto 2011; Cassak 2011). Quantitatively, however, these earlier works consistently indicate a thorough quenching of reconnection, with a very sharp cuto ff near the Alfvén point. Our relativistic simulations, in contrast, show a much more gradual behavior across this point, and a complete shutdown of magnetic reconnection is not observed. This result is at odds with what has been recently reported by Peery et al. (2024), where relativistic reconnection is already quenched for shear speeds substantially below the Alfvén point. This may be explained by the non-zero guide field or by the small box size used in their simulations.', '4.3. Implications for particle acceleration efficiency': 'To comprehend the impact of shear flow on particle acceleration e ffi ciency, we present, in Fig. 9, the particle energy distributions measured when Φ rec = 0 . 5 Φ 0, which occurs at di ff ering simulation times due to the change in the laboratory-frame reconnection rate reported above. The initial distributions are shown on the figure with dashed lines for comparison. We observe that the high-energy power-law tail steepens with increasing shear. This is certainly true at low shear and seems to hold at high shear as \n<!-- image --> \n<!-- image --> \nFig. 9. Comparison of the particle energy distributions at a given magnetization for di ff erent shear values and Φ rec = 0 . 5 Φ 0. Dashed lines represent the corresponding initial spectra. \n<!-- image --> \nwell, though in the latter case (say, at Us ≥ 4 UA / 3) the initial plasma energy is so high that an unambiguous nonthermal component is less clear. \nWe also notice, for σ d = 1, an emerging low-energy component in the distributions. Advection of particles from the upstream towards the plasmoids, where they lose their bulk motion energy, could explain this behavior, although only an analysis of particle trajectories could validate such a hypothesis. In summary, the presence of a shear flow decreases the particle acceleration e ffi ciency: we observe a clear trend towards a softening of the particle energy distribution with increasing shear.', '5. Conclusions': "In this work, we analyze separately the impacts of composition asymmetry and shear in the upstream plasma of a relativistic reconnecting current layer. For composition-asymmetric reconnection, the less-magnetized electron-ion upstream side dictates the global reconnection dynamics, in agreement with Mbarek et al. (2022). The measured reconnection rate is very close to, although slightly higher than, that reported in previous symmetric electron-ion reconnection studies (Werner et al. 2018; Ball et al. 2018). Furthermore, we notice that the gap in the energy partition between electrons and ions increases by about 40% with respect to a symmetric electron-ion configuration, meaning that ions are even more preferentially heated in the presence of composition asymmetry. In the second part of our work, focused on a sheared reconnection layer, we show that a relativistic superAlfvénic shear substantially reduces the reconnection rate but, \nunlike in non-relativistic studies, does not completely halt reconnection. In addition, we observe a steepening of the power-law tail for increasing shear values. \nThese results have important implications for astrophysical environments. In the context of black hole accretion disks, asymmetric reconnection may occur at a jet-disk interface or during magnetic flux eruptions. One important asymmetry in such reconnection events is with respect to plasma composition. In this case, our results indicate that particle acceleration is dictated by the heavier - and, hence, lower-magnetization - electronion side of the current sheet. Due to the modest ion magnetization ( σ ≲ 1) expected in this environment, reconnection is unlikely to produce prominent power-law tails in the ion energy distribution; a prominent nonthermal component exceeding γ = mi / me σ ≲ mi / me in the electron energy distribution is also not expected. \nThese properties suggest that asymmetric reconnection at a jet-disk boundary is not a suitable candidate for powering highenergy gamma-ray flares observed in black hole environments. On the other hand, such reconnection sites may contribute to the lower-energy emission observed in limb-brightened jets. The addition of a velocity shear along field lines at the jet-disk boundary does not change these expectations, since, as we show in this work, strong shear slows reconnection down and inhibits nonthermal particle acceleration - e ff ects that would only further hamper e ffi cient acceleration of particles up to gamma-rayemitting energies. \nAn important limitation of our study is its dimensionality. Once particles are trapped in two-dimensional plasmoids, they are artificially kept from escaping back into the upstream plasma. However, in 3D, as shown by recent studies (Zhang et al. 2021; Chernoglazov et al. 2023), ions can escape plasmoids and get onto Speiser-like trajectories where they undergo intense linear acceleration. In the presence of composition asymmetry, this mechanism may facilitate mixing of ions into the putatively Poynting-flux-dominated and pair-loaded jet. It may also facilitate particle injection into other acceleration processes occurring on larger scales, such as, for instance, shear-flow acceleration along the jet sheath (Ostrowski 1998; Rieger & Du ff y 2004; Sironi et al. 2021). This motivates future studies of 3D composition-asymmetric relativistic magnetic reconnection. \nAcknowledgements. We thank Jesse Vos for providing us early results from his GRMHD simulations and for fruitful discussions regarding magnetized blackhole accretion. We are thankful to the referee's comments that helped us improving the quality of the manuscript. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (Grant Agreement No. 863412). Computing resources were provided by TGCC under the allocation A0150407669 made by GENCI.", 'References': '- Ball, D., Özel, F., Psaltis, D., & Chan, C.-k. 2016, ApJ, 826, 77\n- Ball, D., Sironi, L., & Özel, F. 2018, ApJ, 862, 80\n- Blandford, R. D. & Znajek, R. L. 1977, MNRAS, 179, 433\n- Cassak, P. A. 2011, Physics of Plasmas, 18, 072106\n- Cassak, P. A. & Otto, A. 2011, Physics of Plasmas, 18, 074501\n- Cerutti, B. & Giacinti, G. 2020, A&A, 642, A123\n- Cerutti, B., Philippov, A. A., & Spitkovsky, A. 2016, MNRAS, 457, 2401\n- Cerutti, B., Uzdensky, D. A., & Begelman, M. C. 2012a, ApJ, 746, 148\n- Cerutti, B. & Werner, G. 2019, Zeltron: Explicit 3D relativistic electromagnetic Particle-In-Cell code, Astrophysics Source Code Library, record ascl:1911.012\n- Cerutti, B., Werner, G. R., Uzdensky, D. A., & Begelman, M. C. 2012b, ApJ, 754, L33\n- Cerutti, B., Werner, G. R., Uzdensky, D. A., & Begelman, M. C. 2013, ApJ, 770, 147'} |
2024PhRvL.133l1401B | We show that the existence of clouds of ultralight particles surrounding black holes during their cosmological history as members of a binary system can leave a measurable imprint on the distribution of masses and orbital eccentricities observable with future gravitationalwave detectors. Notably we find that for nonprecessing binaries with chirp masses inlineformulammlmath displayinlinemmlmrowmmlmi mathvariantscriptMmmlmimmlmommlmommlmn10mmlmnmmlmsubmmlmrowmmlmiMmmlmimmlmrowmmlmrowmmlmo stretchyfalsemmlmommlmrowmmlmsubmmlmrowmmlmathinlineformula formed exclusively in isolation largerthanexpected values of the eccentricity i.e. inlineformulammlmath displayinlinemmlmrowmmlmiemmlmimmlmommlmommlmsupmmlmrowmmlmn10mmlmnmmlmrowmmlmrowmmlmommlmommlmn2mmlmnmmlmrowmmlmsupmmlmrowmmlmathinlineformula at gravitationalwave frequencies inlineformulammlmath displayinlinemmlmrowmmlmsubmmlmrowmmlmifmmlmimmlmrowmmlmrowmmlmiGWmmlmimmlmrowmmlmsubmmlmommlmommlmsupmmlmrowmmlmn10mmlmnmmlmrowmmlmrowmmlmommlmommlmn2mmlmnmmlmrowmmlmsupmmlmtext mmlmtextmmlmtext mmlmtextmmlmiHzmmlmimmlmrowmmlmathinlineformula would provide tantalizing evidence for a new particle of mass between inlineformulammlmath displayinlinemmlmrowmmlmo stretchyfalsemmlmommlmn0.5mmlmnmmlmommlmommlmn2.5mmlmnmmlmo stretchyfalsemmlmommlmommlmommlmsupmmlmrowmmlmn10mmlmnmmlmrowmmlmrowmmlmommlmommlmn12mmlmnmmlmrowmmlmsupmmlmtext mmlmtextmmlmtext mmlmtextmmlmieVmmlmimmlmrowmmlmathinlineformula in nature. The predicted evolution of the eccentricity can also drastically affect the inband phase evolution and peak frequency. These results constitute unique signatures of boson clouds of ultralight particles in the dynamics of binary black holes which will be readily accessible with the Laser Interferometer Space Antenna as well as future midband and decihertz detectors. | 2024-09-01T00:00:00Z | ['10.48550/arXiv.2403.02415', '2024arXiv240302415B', 'arXiv:2403.02415', '2024PhRvL.133l1401B', '10.1103/PhysRevLett.133.121401'] | ['Cosmology', 'Astrophysics', 'and Gravitation', 'General Relativity and Quantum Cosmology', 'Astrophysics - Cosmology and Nongalactic Astrophysics', 'Astrophysics - High Energy Astrophysical Phenomena', 'High Energy Physics - Phenomenology', 'High Energy Physics - Theory'] | Signatures of Ultralight Bosons in the Orbital Eccentricity of Binary Black Holes | 2,024 | 171 | 0.3 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML'] | 13 | https://arxiv.org/pdf/2403.02415.pdf | {'Signatures of ultralight bosons in the orbital eccentricity of binary black holes': "Mateja Boˇskovi'c, 1 Matthias Koschnitzke, 1, 2 and Rafael A. Porto 1 \n1 \nDeutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany 2 II. Institut fur Theoretische Physik, Universitat Hamburg, \nLuruper Chaussee 149, 22761 Hamburg, Germany \nWe show that the existence of clouds of ultralight particles surrounding black holes during their cosmological history as members of a binary system can leave a measurable imprint on the distribution of masses and orbital eccentricities observable with future gravitational-wave detectors. Notably, we find that for nonprecessing binaries with chirp masses M ≲ 10 M ⊙ , formed exclusively in isolation, larger-than-expected values of the eccentricity, i.e. e ≳ 10 -2 at gravitational-wave frequencies f GW ≃ 10 -2 Hz, would provide tantalizing evidence for a new particle of mass between [0 . 5 , 2 . 5] × 10 -12 eV in nature. The predicted evolution of the eccentricity can also drastically affect the in-band phase evolution and peak frequency. These results constitute unique signatures of boson clouds of ultralight particles in the dynamics of binary black holes, which will be readily accessible with the Laser Interferometer Space Antenna, as well as future mid-band and Deci-hertz detectors. \nIntroduction. The birth of gravitational-wave (GW) science [1] heralds a new era of discoveries in astrophysics, cosmology, and particle physics [2]. Measuring the properties of GW signals with current and future observatories, such as the Laser Interferometer Space Antenna (LISA) [3], the Einstein Telescope (ET) [4] and Cosmic Explorer (CE) [5], as well as other Mid-band [6] and Decihertz detectors [7], not only will unravel the origins of binary black hole (BBH) mergers, it also opens the possibility to discover (very-weakly-coupled) ultralight particles that are ubiquitous in theories of the early universe [812]. Notably, the mass, spin alignment, and eccentricity are expected to be correlated with formation channels, isolated or dynamical , e.g. [13-31]; whereas boson clouds (or 'gravitational atoms' [8, 9]), formed around black holes via superradiance instabilities [32-36], can produce a large backreaction on the orbital evolution. Following analogies with atomic physics [37], the cloud may encounter Landau-Zener (LZ) resonances [38], or ionization effects [39-41]. The presence of a cloud then leads to large finite-size effects [37, 42], floating/sinking orbits [38], as well as other sharp features [40], that become unique signatures of ultralight particles in the BBH dynamics. \nFor the most part, up until now backreaction effects have been studied under the simplified assumption of planar, quasi-circular orbits. The reason is twofold [37]. First, several formation scenarios lead to spins that are parallel to the orbital angular momentum [18]. Second, the decay of eccentricity through GW emission in vacuum [43, 44] is expected to have circularized the orbit in the late stages of the BBH dynamics. We retain here the former but relax the latter assumption. As we shall see, adding eccentricity not only introduces a series of overtones [41, 45, 46], it can also have a dramatic influence in the orbital dynamics as the cloud transits a LZ-type transition. Although the strength of the new resonances is proportional to the eccentricity, depending on their position and nature (floating or sinking), a small departure from circularity can lead to transitions that not only would deplete the cloud, but also \ninduce a rapid growth of eccentricity toward a large critical (fixed-point) value: e cr ∈ [0 . 3 , 0 . 6]. As measurements of the eccentricity are correlated with formation channels, the predicted increase can impact the inferred binary's origins. Measurements of larger-than-expected eccentricities would then provide strong evidence for the existence of a new ultralight particle in nature. In particular, because of the critical fixed point, a fraction of the BBHs undergo a rapid growth of orbital eccentricity to a common value. As a result, the distribution of masses and eccentricities may feature a skewed correlation by the time they reach the detector's band. Furthermore, for chirp masses M < 10 M ⊙ and spin(s) aligned with the orbital angular momentum-expected to exclusively form in the field-the presence of a boson cloud at earlier times can shift a fraction of the population toward values of e ≳ 10 -2 at 10 -2 Hz, readily accessible to LISA [3]. Furthermore, the GW-evolved eccentricity may also be within reach of the planned mid-band [6] or Deci-hertz [7] observatories. For all such events, a new ultralight boson of mass [0 . 5 , 2 . 5] × 10 -12 eV forming a cloud and decaying through a LZ-type transition prior to detection, may be the ultimate culprit. \nThe more drastic evidence is given when the resonant transition occurs in band with measurable frequency evolution. A plethora of phenomena are discussed in [37, 38] for circular orbits. In addition to overtones, the increase in eccentricity would imply that higher harmonics become more relevant, which in turn affects the peak frequency of the GWs, even for floating orbits. We point out here some salient features and elaborate further on the details elsewhere [46]. \nThe gravitational atom. Ultralight particles of mass µ can form a cloud around a rotating black hole of mass M , via superradiant instabilities [8, 9]. The typical mass of the (initial) cloud scales as M c , 0 /M ≃ α , whereas its typical size is r c ≃ r g α 2 , with r g ≡ GM c 2 , and \nα = GMµ ℏ c ≃ 0 . 1 ( M 15 M ⊙ ) ( µ 10 -12 eV ) . (1) \nThe (scalar) cloud evolves according to a Schrodingerlike equation [47, 48], with eigenstates | a ⟩ ≡ | n a l a m a ⟩ , and ( n, l, m ) the principal, orbital and azimuthal angular momentum, 'quantum numbers'. (For vector clouds [38, 48, 49], we must include the total angular momentum.) \nThe energy eigenvalues of the cloud scale as ϵ nlm = µ ( 1 -α 2 2 n 2 + f nl α 4 + h nl ˜ amα 5 ) , with ˜ a the dimensionless spin of the black hole, see [48]. At saturation, we have ˜ a ≃ α , whereas the combined system black hole plus cloud may still be rapidly rotating. One of the main difference w.r.t. ordinary atoms, however, is the presence of a decay/growing time, Γ -1 nlm ∝ µα 4 l +5 , for a given eigenstate [9, 47, 48, 50]. The (scalar) cloud may be populated by the dominant growing mode, | 211 ⟩ , or an excited state, | 322 ⟩ . Depending on α , they may be robust to GW emission (from the cloud itself) on astrophysical scales [9, 51-55]. They can also deplete through resonant transitions in binaries [37, 38], as we discuss here. In what follows we work with G = ℏ = c = 1 units. \nGravitational collider goes eccentric. Following [37, 38] we consider a boson cloud around a black hole of mass M in a bound orbit with a companion object of mass M ⋆ , with q ≡ M ⋆ /M the mass ratio. The coordinates are centered at the black hole plus cloud system, with R ⋆ the radial distance to the perturber, and φ ⋆ the azimuthal angle. We consider planar motion with the spin parallel to the orbital angular momentum, with the orbit described by the semi-major axis a and the eccentricity e , while φ ⋆ corresponds to the true anomaly. We take the orbital frequency to be positive such that the two, co-rotating and the counter-rotating, orientations are identified by ˙ φ ⋆ = s | ˙ φ ⋆ | , with s = ± 1. \nThe gravitational perturbations of the companion induce mixing of the atomic levels. For a perturber outside of the cloud R ⋆ ≫ r c the off-diagonal matrix elements of the Hamiltonian, ⟨ a | V ⋆ | b ⟩ , are given by a multipole expansion that can be written as an harmonic series [37, 38] \n⟨ a | V ⋆ | b ⟩ l ⋆ = ∑ | m ⋆ |≤ l ⋆ η ( m ⋆ ) ab e -im ⋆ φ ⋆ , (2) \n̸ \n̸ \nwith η ( m ⋆ ) ab ∝ R -( l ⋆ +1) ⋆ . The matrix elements obey selection rules which determine possible transitions, which we refer as hyperfine (only ∆ m = 0), fine (∆ ℓ = 0 , ∆ n = 0), and Bohr (∆ n = 0), respectively [37, 38]. \n̸ \nFor illustrative purposes, we consider a two-level model. The Hamiltonian equation is given by \ni ( ˙ c a ˙ c b ) = -∆ ϵ 2 η 0 ( R ⋆ R 0 ) -( l ⋆ +1) e i ∆ mφ ⋆ c . c . ∆ ϵ 2 -i Γ b ( c a c b ) , (3) \nwith ∆ m ≡ m b -m a , ∆ ϵ ≡ ϵ b -ϵ a the energy split, Γ b the width of the decaying mode, and η 0 the value of the perturbation at a reference point R 0 . Furthermore, since \n(vacuum) GW emission is expected to reduce the initial eccentricity prior to encountering the resonant transition, and for the purpose of analytical understanding, in what follows we describe the orbital evolution in the Hamiltonian, H , of (3) using a small-eccentricity approximation, 1 \nφ ⋆ ≃ ϑ +2 e sin ϑ, R ⋆ ≃ a (1 -e cos ϑ ) , (4) \n˙ ϑ ≡ s Ω , Ω = √ M (1 + q ) /a 3 , (5) \nin terms of ϑ , the mean anomaly [56], and apply the Jacobi-Anger expansion into Bessel functions. Hence, \nH = D + ∞ ∑ k = -∞ η k e i ( k +∆ m ) ϑ η k e -i ( k +∆ m ) ϑ , (6) \nD = -∆ ϵ 2 ∆ ϵ 2 -i Γ b , η k ∼ η 0 f 2 3 ( l ⋆ +1) e | k | | k | ! , f ≡ Ω Ω 0 , \nwhere we traded distance for orbital frequency. The case ( e, Γ b ) = 0 was studied in [38]. The slow GW-induced evolution of the orbital frequency, Ω( t ) ≃ Ω 0 + γ 0 t with γ 0 = 96 5 qM 5 / 3 Ω 11 / 3 0 (1+ q ) 1 / 3 , leads to a LZ transition [57, 58] between the energy levels. The transition is triggered for \nΩ 0 = s ∆ ϵ ∆ m > 0 . (7) \nThis value, dictated by the spectrum of the cloud, will serve as our reference point in the evolution of the binary. \nIgnoring backreaction effects (see below), the LZ solution is controlled by the parameter z 0 ≡ η 2 0 / ( γ 0 | ∆ m | ), which determines the adiabaticity of the transition. As famously demonstrated in [57, 58], starting in the far past from the | a ⟩ state, in the limit 2 πz 0 ≫ 1, the eigenstate of the (time-dependent) Hamiltonian yields a complete population transfer into the (decaying) | b ⟩ mode in the far future. As it turns out, although the solution changes at finite time, controlled by the parameter v 0 ≡ Γ b / √ γ 0 | ∆ m | , the asymptotic properties of the system (ignoring backreaction) are remarkably robust against the value of the decaying width for the | b ⟩ state [59]. \nFor eccentric orbits, the evolution in (6) also features a transition at Ω 0 (for k = 0). However, it introduces a series of overtones \nΩ k = f k Ω 0 , f k = ∆ m ∆ m + k , k ∈ Z . (8) \nProvided each k -resonance is sufficiently narrow, we can ignore the other ( k ' = k ) terms in (6). As in [38], we can \n̸ \nlinearize the orbital evolution near the transition, Ω( t ) = Ω k + f ( e ) γ k t , where f ( e ) = 1+ 73 e 2 24 + 37 e 4 96 (1 -e 2 ) 7 / 2 and γ k ≡ γ 0 f 11 / 3 k , such that the LZ solution now depends on the modified z k ≡ η 2 k f ( e ) γ k | ∆ m + k | and v k ≡ Γ b √ f ( e ) γ k | ∆ m + k | , respectively. \nOrbital backreaction. Dissipative effects, such as GW emission, from the binary [38, 60] or the cloud itself [51-55], ionization [39-41, 60], and decay widths [59, 61, 62], strongly influence the LZ phenomenology, and vice versa. We focus here on the prevailing case of two-body GW emission, with the companion outside of the cloud, thus focusing on (hyper)fine resonances, combined with a two-level LZ transition into a decaying mode. \nThe orbital dynamics is governed by flux-balance equations at infinity [38, 41, 44, 46, 62, 63], and at the black hole's horizon [8, 64, 65]: \n˙ E o + ˙ E c + ˙ M = F GW ≡ -32 f ( e ) 5 M 5 q 2 ( q +1) a 5 , (9) \n˙ L o + s ( ˙ L c + ˙ S ) = T GW ≡ F GW Ω g ( e ) f ( e ) , (10) \n˙ M = 2Γ b E c( b ) , ˙ S = 2Γ b L c( b ) , (11) \nwith g ( e ) = 1+ 7 e 2 8 (1 -e 2 ) 2 , and ˙ M , ˙ S the change of mass and spin due to the decay of the | b ⟩ state onto the black hole. 2 The orbital energy and angular momentum are given by E o = -M 2 q 2 a and L 2 o = ( M 5 q 3 )(1 -e 2 ) 2( q +1) | E o | , while for the cloud is a sum over the populated states, E c( i ) ≡ ( M c , 0 /µ ) ϵ i | c i | 2 , and similarly for L c( i ) with ϵ i → m i . \nThe above equations can then be rewritten as \nd Ω dt = rγ 0 f 11 / 3 f ( e ) , (12) \nr ≡ ˙ E o F GW = 1 -b sgn( s ∆ m ) f -11 / 6 √ f ( e ) γ 0 | ∆ m + k | d | c a | 2 dt , (13) \nde 2 dt = 2 3 f 8 / 3 γ 0 Ω 0 f ( e ) √ 1 -e 2 × (14) [ r ( f -√ 1 -e 2 ) -f + g ( e ) f ( e ) ] , \nin terms of the orbital parameters, where \nb ≡ 3 M c , 0 M | ∆ m | f -3 / 2 | ∆ m + k | -1 / 2 (1 + q ) 1 / 3 αq ( M Ω 0 ) 1 / 3 Ω 0 √ γ 0 , (15) \nparameterises the backreaction effects on the orbit due to the cloud. It is worth emphasising that the above equations apply to generic (planar) motion, regardless \nof the value for the eccentricity. As we shall see, even for small initial conditions, the orbit is affected by large backreaction effects due to the presence of the cloud. \nAs anticipated by the analysis in [38] for the case of circular orbits (which we encourage the reader to consult for further details), 'effective' LZ parameters emerge: ζ k ( t ) ≡ z k /r ( t ) and w k ( t ) ≡ v k / √ r ( t ), making it a fully nonlinear system. We can nonetheless estimate the value of the energy-momentum transfer near the resonance by self-consistently solving the condition ζ k = z k /r k ( ζ k ). For moderate-to-large population transfer ( ζ k ≳ 1), we find the limiting results: \n(16) \nr k ≃ 2 ( 1 + √ 1 -sgn( s ∆ m ) b k z k v k ) , ( w k ≫ ζ k ) \nr k ≃ ( 1 -sgn( s ∆ m ) b k 4 √ z k ) -1 , ( w k ≪ ζ k ) -1 . \nAs discussed in [38], the orbital evolution branches into either floating ( r ≃ 0), for s ∆ m < 0, or sinking orbits ( r ≳ 1), for s ∆ m > 0. However, except for the trivial case when ζ k ≪ 1, due to the nonlinear nature of the problem the transfer of energy and angular momentum from the cloud to the orbit does not simply reduce to the quest for adiabaticity of the LZ transition, not even for w k ≪ ζ k . For instance, for extreme cases, with z k ≫ 1, the (unperturbed) transition spreads over long time scales, ∆ t LZ ≃ 4 √ z k /γ k [66], which in turn reduces the orbital impact, as we see in (16). As it turns out, in the large backreaction scenario, the sweet spot for floating orbits occurs when b k ≫ √ z k . Even though, due to the properties of the LZ solution, a strong decay width ( w k ≫ ζ k ) does not alter this picture, the impact on the orbit evolution as well as the population transfer becomes suppressed by 1 /v k , as shown in (16). On the other hand, for the sinking case, the largest values of r k are obtained for nonadiabatic transitions. \nEccentric fixed point. For the GW-dominated epochs, with r ≃ 1, the leading order term in (14) vanishes, and the first contribution is at O ( e 2 ). Likewise for the k = 0 (main) resonance, for which the first term is ∝ ( r 2 -11 3 ) e 2 . As a result, the eccentricity is damped unless the orbit gets a large kick ( r ≳ 7 . 3). As the influence of the cloud increases, the RHS of (14) asymptotes (modulo a positive prefactor) to ( f k -1)( r -1), in which case it enters at leading order. Moreover, the differences in the GWfluxes in (9) and (10) generate a distinction between the early and late resonances. In the floating case, with r ≃ 0, the eccentricity grows for the early resonances ( f k < 1) and decays for the late ones ( f k > 1). This can be understood by noticing that, when ˙ E o ≃ 0, we have ˙ L o ∝ ( Ω -Ω 0 ΩΩ 0 + O ( e 2 ) ) , and using d ( L 2 o ) ∝ -d ( e 2 ) the eccentricity grows for Ω k < Ω 0 and decays whenever Ω k ≥ Ω 0 . This trend is reversed in the sinking case. \nBecause of the changes in the evolution of the eccen- \n<!-- image --> \nFIG. 1. BBH eccentricities at f GW = 10 -2 Hz ( left ), evolved with a uniformly distributed q ∈ [0 . 1 , 1] and a boson cloud on the heavier black hole. The pale blue dots account for the values without a cloud [15]. (BBHs with e ≲ 10 -6 are not shown.) Cumulative effect, i.e. the ratio of binaries with eccentricities above a given value e 0 , ( right ), with (solid) and without (dotted) a cloud, both at 10 -2 Hz (blue) and 1 Hz (pink), respectively. \n<!-- image --> \ntricity across different resonances, it is instructive to look at the opposite limit e → 1. In that case, the RHS of (14) becomes ∝ r -1 (1 -e ) 3 . Let us consider the case of a floating orbit. Since the sign of de dt is positive for Ω k < Ω 0 , but turns negative when the eccentricity approaches e ≃ 1, this implies the existence of a critical 'attractor' fixed point, e cr , given by the condition g ( e cr ) /f ( e cr ) = f k [cf. (14)]. For instance, \ne cr = { 0 . 46 , 0 . 35 , 0 . 29 } , for | ∆ m | = { 1 , 2 , 3 } , (17) \nwith k = -1. Similarly, an unstable fixed point develops for the earlier and main sinking resonances. \nFor the case of floating orbits (with s ∆ m < 0), if the backreaction is sufficiently effective to enforce r k ≃ 0 while the eccentricity approaches the critical point, one can then estimate the floating time ∆ t FL ≃ b k / √ γ k , leftover population | c a ( ∞ ) | 2 ≲ r k , and notably the growth of the eccentricity upon exiting the resonant transition, \ne fin ≃ e cr √ 1 -e -C k , with C k ∼ √ γ k Ω k b k . (18) \nAlthough we have used a small-eccentricity approximation to describe the initial stages of the cloud's evolution in (6), we have demonstrated through numerical studies that the behavior described above remains valid for generic (planar) orbits. See [46] and the appendices for details. \nThe cloud's eccentric fossil. As it was argued in the literature [15-17], the distribution of masses and eccentricities observed with LISA can in principle distinguish between formation channels. However, the contrast between vacuum evolution and the large eccentricities produced by the cloud's resonant transition can lead to dramatic changes in the expected evolution of the system. As a proof of concept, we take the stellar-mass BBH population studied in [15], with chirp masses M ≲ 10 M ⊙ , expected to form exclusively in isolation, and with the spins aligned with the orbital angular momentum. As a \nconsequence, the assumption of equatorial (uninclined) motion may be implemented without loss of generality (as done in [15]). \nWe consider clouds of ultralight bosons of mass between 10 -13 and 10 -11 eV, surrounding black holes in co-rotating orbits. Superradiance may then excite the | 322 ⟩ state which, depending on the parent black hole's mass and birth orbital frequency, will experience a series of (hyper)fine transitions. 3 To illustrate the distinct physical effects, and following [14, 15], we consider a birth orbital frequency (for the cloud+BBH system) at Ω ini /π ≃ 10 -4 Hz, 4 and evolve, using the peak GW frequency [68] \nf GW ≃ Ω π (1 + e ) 1 . 1954 (1 -e 2 ) 3 / 2 , (19) \nuntil f GW = 10 -2 and 1 Hz. The final distribution is shown on the left panel of Fig. 1. While some of the BBHs experience an early overtone of the hyperfine transition, the majority are affected by the fine overtones instead. The BBHs then float over a period of time while increasing the orbital eccentricity. Moreover, the cloud typically either terminates there or decays later at the k = 0 resonance. Depending on the parameters, the ultimate decay may decrease the eccentricity or have a small impact on the orbit. As a result, a wedge-type distribution emerges, with the heavier black holes (within each wedge) subject to the largest increase in eccentricities. \nThe cumulative effect is shown on the right panel of \nFIG. 3. Evolution of the peak frequency through a resonant transition (at t -t res = 0) in the LISA band of a GW170809like event, compared to the evolution without the cloud. \n<!-- image --> \nFIG. 2. Percentage of binaries with eccentricities above 0.01 at f GW = 10 -2 Hz for different values of µ . \n<!-- image --> \nFig. 1, where a significant fraction of the population (with different parent masses) is affected by the resonances, yielding values of the eccentricities at 1Hz that may be within reach of mid-band and Decihertz detectors. As the value of µ increases (decreases) the location of the wedge in the distribution moves toward lower (higher) masses. The dependence on the value of µ for this population of BBH (with M ≲ 10 M ⊙ ), reaching e ≳ 10 -2 at f GW = 10 -2 Hz, is shown in Fig. 2. \nEccentric in band. Because of the connection to formation channels, we discussed a sub-population of BBHs. However, similar conclusions apply to black holes with higher masses [46]. For instance, for a GW170809type event [69] ( M≃ 24 M ⊙ ), with a parent black hole M ≃ 20 M ⊙ (carrying the cloud) and a (heavier) companion M ⋆ ≃ 40 M ⊙ , we find f GW ≳ 10 -2 Hz at the k = -1 fine transition for µ ≃ 1 . 5 × 10 -12 eV. The BBH reaches the resonance and floats, with approximately constant orbital frequency for about six years, while the eccentricity increases from e ≲ 10 -2 to e ≃ 0 . 1, and likewise the peak GW frequency grows, while the cloud depletes. The resulting frequency evolution till merger, which is distinct from the growth of the eccentricity that may occur due to other astrophysical mechanisms [70, 71], 5 is displayed in Fig. 3. In addition to the notable features, higher harmonics would also become more relevant as the eccentricity increases [72]. As for the case of large tidal Love numbers [37, 42], 6 new dedicated templates will be needed to search for these phenomena in the GW data. \nConclusions. We have shown that the presence of a boson cloud surrounding a black hole in a binary system can impact the distribution of masses and eccentricities observable with GW detectors. We have also \nfound that a greater-than-expected value of the eccentricity, e ≳ 10 -2 at GW frequencies f GW ≃ 10 -2 Hz, develops for a (sub-)population of isolated stellar-mass BBHs (with M ≲ 10 M ⊙ ), right at the heart of the LISA band. Likewise, these BBHs will decay through GW emission to values of the eccentricities, i.e. e ≃ 10 -4 -10 -3 , within experimental reach of mid-band [6] and Decihertz [7] detectors. The observation of such GW signals would then provide tantalizing evidence for the existence of an ultralight particle of mass between 5 × 10 -13 and 2 . 5 × 10 -12 eV in nature. Furthermore, we have also shown that inband resonance transitions are possible, yielding dramatic changes in the GW frequency evolution, constituting yet another smoking-gun signature of the imprint of a boson cloud in the BBH dynamics. \nThere are several venues for further exploration. First, unlike Bohr-type resonances, we have concentrated here on (co-rotating) hyperfine and fine transitions which occur outside of the cloud (for the range of parameters we considered), and therefore are not subject to ionization/dynamical friction [39, 41]. Preliminary studies suggest that a similar increase of eccentricity occurs for certain type of Bohr transitions at higher frequencies, which would put them within reach of the ET and CE detectors [46], but a more in-depth study is needed to take all relevant effects into account. Second, although generic, we have considered the case of uninclined orbits. This is justified for the populations of BBHs we considered here (formed in isolation with spins parallel with the angular momentum). However, to encompass also dynamically-formed systems, we must add inclination and new (off-plane) transitions [41]. While our results remain unchanged for quasi-planar motion, we also expect similar conclusions to apply for inclined orbits. (In fact, as shown in [74, 75], resonant transitions tend to equatorialize the orbit.) Finally, identical results can be drawn also for neutron star/black hole binaries. For instance, those formed in isolation have a parent black hole with mass near M ≃ 7 M ⊙ , and likewise in binaries with negligible eccentricity at f GW ≃ 1Hz [13, 25, 31]. \nThe presence of a boson cloud would then also lead to larger-than-expected eccentricities, providing additional circumstantial evidence for a new ultralight particle in nature. \nAcknowledgements. We thank G.M. Tomaselli for informative discussions and the authors of [74] for sharing a draft of their related work. The work of MB and MK is supported in part by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy - EXC 2121 'Quantum Universe' - 390833306. MB and RAP are supported in part by the ERC Consolidator Grant 'Precision Gravity: From the LHC to LISA' provided by the European Research Council (ERC) under the European Union's H2020 research and innovation programme (grant No. 817791).", 'Appendix A: Gravitational atom': "The range of ultralight masses that we probe yield a (very) high 'axion decay constant' ( f a ), generically suppressing self-interactions [76-79] and coupling to other species [80-84]. In addition, for the black hole masses we studied, with 5 M ⊙ ≲ M ≲ 50 M ⊙ , we arrive at small-tomoderate values of α ≲ 0 . 25, keeping relativistic corrections to the hydrogenic states small [55, 60, 85, 86]. In this regime, the eigenvalues are given by [37, 48] \nϵ nlm = µ [ 1 -α 2 2 n 2 -( 1 8 n + 6 2 l +1 -2 n ) α 4 n 3 + 16 2 l (2 l +1)(2 l +2) ˜ amα 5 n 3 ] . (A1) \nFurthermore, from Detweiler's approximation [47, 48] (with our sign convention) \n-Γ nlm ≃ 2˜ r + C nl g lm α 4 l +5 ( m Ω H -ω nlm ) , (A2) \nwith g lm ≡ ∏ l k =1 [ k 2 (1 -˜ a 2 ) + (˜ am -2 r + ω ) 2 ] , C nl ≡ 2 4 l +1 ( n + l )! n 2 l +4 ( n -l -1)! ( l ! (2 l )!(2 l +1)! ) 2 , ˜ r + = 1 + √ 1 -˜ a 2 , M Ω H = ˜ a/ (2˜ r + ); whereas for the cloud itself decaying into GWs, we use the (non-relativistic) approximation in [51].", 'Appendix B: Tidal interactions': "For equatorial orbits and resonances triggered away from the cloud, the tidal interaction is given by [37, 38] \n⟨ a | V ⋆ | b ⟩ ≡ ∞ ∑ l ⋆ =2 ∑ | m ⋆ |≤ l ⋆ η ( ⋆ ) ab e -im ⋆ φ ⋆ , (B1) \nη ( ⋆ ) ab = -qα r R -( l ⋆ +1) ⋆ 4 π 2 l +1 ∣ ∣ ∣ Y ∗ ( ⋆ ) ( π 2 , φ ⋆ )∣ ∣ ∣ I r I Ω \nI r ≈ ∫ d rr 2 ˆ R b ˆ R a r l ⋆ (B2) \nc ⋆ , ∞ \nI Ω ≡ ∫ d Ω Y ∗ a ( θ, ϕ ) Y ( ⋆ ) ( θ, ϕ ) Y b ( θ, ϕ ) , (B3) \nwhere ( ⋆ ) ≡ ( l ⋆ , m ⋆ ), r ≡ r/r c , R ⋆ ≡ R ⋆ /r c , ˆ R c = r 3 / 2 c R c is the (dimensionless) hydrogenic radial wavefunction, Y lm is the spherical harmonic. We leave to [46] the discussion on resonances 'inside the cloud', including dipolemediated transitions [60, 86, 87]. \nThe Jacobi-Anger expansion applied to (B1) [using (4)], \ne ± i ∆ m ( ϑ +2 e sin( ϑ )) = ∞ ∑ k = -∞ ( ± 1) k J k (2 e ∆ m )e i ( k ± ∆ m ) ϑ , (B4) \ncan be applied to the off-diagonal terms of the Hamiltonian [cf. (3) in the main text]. Using the properties of the Bessel function 7 , the tidal perturbation [cf. (6) of the main text] becomes \nη ( ⋆ ) ab,k = η ( ⋆ ) ab, 0 f 2 3 ( l ⋆ +1) (∆ me ) | k | | k ! | ( 1 + ( l ⋆ +1) k 2∆ m ) + O ( e | k | +1 ) \n( ⋆ ) ab, 0 = -qα r c ( r c a 0 ) l ⋆ +1 4 π 2 l ⋆ +1 ∣ ∣ ∣ Y ∗ ( ⋆ ) ( π 2 , ϑ ) ∣ ∣ ∣ I r I Ω , \n, η (B5) \n̸ \nwhere a 0 = [ M (1+ q ) / Ω 2 0 ] 1 / 3 . This interaction is nonzero only if the selection rules are satisfied [37, 38]: -m a + m ⋆ + m b = 0, l b + l ⋆ + l a = 2 p , | l a -l b | ≤ l ⋆ ≤ l a + l b . Furthermore, for equatorial orbits, for even (odd) l ⋆ only the spherical harmonics even (odd) in m ⋆ = 0 are nonzero.", 'Appendix C: Atomic resonances': 'As the populated state has a maximal azimuthal number m max = n -1, at the (hyper)fine resonances, it can only transition into states with lower m . Such transitions are only possible on co-rotating orbits, where they obey \nFIG. 4. Orbital resonance frequencies of possible (fine) transitions for the | 322 ⟩ -component of the cloud. The blue and red arrows represent vacuum and floating BBH evolution, respectively; whereas the grey arrows point to various (rapidly decaying) modes that ultimately deplete the cloud. The thinner the line the smaller the cloud gets. \ns ∆ m < 0, yielding floating-type motion 8 . From the selection rules for the tidal interactions, the | 211 ⟩ state has only one hyperfine transition to | 21 -1 ⟩ , while the only possible fine transition, to | 200 ⟩ , can only occur inside the cloud. Furthermore, for small values of α , we find that the floating time of the hyperfine transition would take longer than a Hubble time, preventing them to reach the LISA band. At the same time, (barring a precise fine tuning of the birth frequency of the BBH+cloud) for large values of α we expect the | 211 ⟩ component of the cloud to decay through its own GW emission before reaching the resonant transition (see also [62, 88]). On the other hand, the (longer-lived) | 322 ⟩ state may experience various types of resonances. In contrast to early and late resonances with k = ± 1, all of the k = 0 hyperfine transitions to | 32 m ⟩ happen at the same frequency (∆ m drops out of the ratio). The dominant main (hyperfine) transition is the one to | 320 ⟩ , with l ⋆ = 2 [as the | 322 ⟩ → | 32 -2 ⟩ resonance can only be mediated by the hexadecapole ( l ⋆ = 4), making it extremely weak and nonadiabatic]. Transitions to the | 32 ± 1 ⟩ states are not possible for equatorial orbits. The fine resonances from the excited state are (octopolar) to the | 31 -1 ⟩ , | 311 ⟩ and (quadrupolar) | 300 ⟩ states, in that order in the frequency domain. We show a succession of transitions in Fig. 4. We ignored the (sinking) highl ⋆ Bohr resonances that can overlap with the range of frequencies that we consider here, since they do not significantly affect the dynamical evolution. We postpone the general analysis of Bohr transitions to [46].', 'Linear solution': 'The solution of the (linear) LZ transition with ˙ Ω k ≃ f ( e ) γ k ≃ const, including a decaying width, is given \nby [59, 61, 62] (see also [38, 58]) \n| c a | 2 = exp ( -v k τ -π 2 z k )∣ ∣ ∣ D iz k ( e i 3 π 4 ( τ -iv k ) )∣ ∣ ∣ 2 (D1) | c b | 2 = exp ( -v k τ -π 2 z k ) z k ∣ ∣ ∣ D iz k -1 ( e i 3 π 4 ( τ -iv k ) )∣ ∣ ∣ 2 , \nwhere D are parabolic cylinder functions [58], and we introduced a dimensionless time \nτ ≡ t √ | ∆ m + k | f ( e ) γ k . (D2) \nNotice that the physical impact of γ k (associated with the timescale of GW radiation) on the LZ dynamics is reflected via the ratio with the other two relevant parameters, for instance, the ratio with the (time) scale associated with the perturbation, ( η/ √ γ k ) 2 , measures the adiabaticity of the transition, while Γ b / √ γ k describes the impact of of the decaying width relative to the dynamical time. Remarkably, ignoring backreaction effects, even though the solution changes with respect to the nondecaying case, the transition probability at infinity, given by | c a ( ∞ ) | 2 = e -2 πz k , turns out to be independent of v k . 9 \nThe presence of a decay width, however, tends to smooth the LZ transition, by transferring a fraction of the initial state earlier than the case with v k = 0, and also by damping late-time oscillations. Consequently, for large widths, d dτ | c a | 2 peaks earlier than the v k = 0 case. In the limit v k ≫ z k , the LZ solution acquires a simple form [61, 62] \n| c a | 2 = exp [ -2 z k ( arctan ( τ v k ) + π 2 )] , (D3) \n| c b | 2 = | c a | 2 z k τ 2 + v 2 k . (D4) \nIn this regime, d dτ | c a | 2 peaks at τ max ≃ -z k v k , and the width of the transition roughly scales as \n∆Ω √ | ∆ m + k | f ( e ) γ ≃ 2 v k √ 1 + 2 z 2 k . (D5) \nIn the two-level system with a decaying mode, we can relate the total energy and the angular momentum of the | a, b, BH ⟩ state as [cf. (9)-(11) in the main text] \n( ˙ L c + ˙ S ) = ( ˙ E c + ˙ M ) ∆ m ∆ ϵ × ρ ϵ ( t ) ρ m ( t ) (D6) \nρ x ≡ x a ∆ x d dt | c a | 2 -Γ a | c a | 2 -∑ ˆ Γ a | c a | 2 +( a → b ) , (D7) \nwhere x ≡ { ϵ, m } , and the ˆ Γ represents other sources of dissipation, such as GW emission from the cloud [9, 5155] or ionization [39-41]. From the Schrodinger equation [cf. (3)], we find \nρ x = d dt | c a | 2 +2 | c a | 2 Γ a +2 x b ∆ x ∑ ( ˆ Γ a | c a | 2 + ˆ Γ b | c b | 2 ) , \nand a similar equation applies with a ↔ b . After superradiance saturates the growth of the cloud in the | a ⟩ state, we have Γ a ≃ 0. Moreover, for the floating time of the resonances we consider here, we can ignore other sources of dissipation during the LZ transition (setting ˆ Γ ≃ 0) [46]. Hence, ρ x = ( d/dt ) | c a | 2 , which implies ρ ϵ /ρ m = 1, yielding the expression in (12)-(14) of the main text.', 'Nonlinear backreaction effects': "Because of the backreaction on the orbital frequency, which in turns controls the LZ transition, the problem becomes nonlinear, and it depends on the level-occupancy through the derivative of the parent state occupancy [cf. (13) in the main text]. We plot the this derivative for the linear LZ problem in Fig. 5. In the parts of the parameter space where it has compact support, the backreaction simply renormalizes the parameters ( z k , v k ) → ( ζ k , w k ) [cf. (14) of the main text] near the maximum value, as \nζ k = η 2 k | ∆ m + k | ˙ Ω k = η 2 k | ∆ m + k | γ k f ( e ) r = z k r k ( ζ k ) , (D8) \nand with the energy transfer itself depending on the leveloccupancy of the cloud. The solution can, nonetheless, be found self-consistently in terms of the relevant parameters. \nThis mapping provides a useful indicator, even in the regime of strong backreaction, to evaluate whether, e.g., floating can occur, by indicating the breakdown of the linearization of the full problem. From the self-consistency condition (D8) the 'true' time scale for the nonlinear problem follows T = √ r k τ . The LZ function derivatives \nFIG. 5. (Negative) derivative of the parent state occupancy, evaluated for a linear LZ transition. Brackets in the legend correspond to ( z k , v k ). Note the decrease, and shift to the left, of the amplitude, as well as the widening of the function and damping of the late-time oscillations, as the ratio v k /z k increases. \n<!-- image --> \ncan now be expressed in terms these quantities, \nd | c i | 2 dτ = √ z k ζ k d | c i | 2 d T , (D9) \nleading to \n-d | c a | 2 dτ ∣ ∣ ∣ τ = -ζ k w k = √ z k ζ k ψ k ( ζ k , w k ) , (D10) ψ k ≡ 1 2 e -1 2 ( πζ k ) [ 2 w k ∣ ∣ ∣ ∣ C H iζ ( C w k ∆ w k 2 )∣ ∣ ∣ ∣ 2 -( C H 1 -iζ ( C ∗ ∆ w ∗ k 2 ) H iζ ( C ∆ w k 2 ) +c . c . ) ] , \nwhere C = 1 + i , ∆ w k = w k (1 -iζ k ) and H is the Hermite function. In what follows, for the sake of notation brevity, we take the small-eccentricity approximation, with f ( e ) ≃ 1. This is also justified by the fact that the eccentricity before the LZ transition is typically small across the parameter space, which we use to evaluate the type of transition the cloud will experience.", 'Negligible-decay regime': 'The ψ k function simplifies in the asymptotic regime \nlim ζ → 0 ψ k ∼ √ πζ , & lim ζ →∞ ψ k ∼ 1 4 √ ζ ( w k ≪ ζ k ) , \nin which \nζ k ∼ z k ( 1 -sgn( s ∆ m ) b k 4 √ z k ) , (D11) \nand the value of r k given in (16) of the main text. As advertised, the energy transfer for the adiabatic floating \n( s ∆ m< 0), is extremised by large values of b k with moderate z k parameters. In contrast, sinking ( s ∆ m > 0) is consistent with large adiabaticity only for moderate backreaction b k < 4 √ z k . Furthermore, the large backreaction limit ( b k ≫ 1) of the weakly-adiabatic regime \nζ k ∼ sgn( s ∆ m ) ( z k b 2 k π ) 1 / 3 , r k ∼ sgn( s ∆ m )( πb 2 k z 2 k ) 1 / 3 , \nis only possible for sinking orbits. In such scenarios, the strong backreaction then mostly leads to nonadiabatic sinking transitions, with a potentially significant impact on the orbit', 'Strong-decay regime': '̸ \nIn general, the impact of the decay width depends on the dynamical timescale of the LZ transition and should be compared with the strength of the coupling. Moreover, for eccentric orbits, even if z 0 > v 0 at f 0 = 1, this hierarchy may be reversed for f k = 1, due to the e 2 | k | suppression in z k . Similarly to the case of negligible decay, we can also obtain approximate relations in the limit described by (D3), yielding \nψ k ∼ 2 ζ k e -ζ k [ π -2 arctan( ζ k )] w k ( ζ 2 k +1) , (D12) \nr k ∼ 1 ± 2 b k z k e -ζ k [ π -2 arctan( ζ k )] v k ( ζ 2 k +1) . (D13) \nIn general, this regime shares various qualitative behavior as in the w k ≪ ζ k case, but with the adiabaticity gain/loss and orbital impact suppressed by the ratio b k /v k . In particular, large adiabaticity is consistent only for floating orbits, where we have \nζ k ≃ √ 2 e √ b k z k v k , r k = z k ζ k → 0 . ( b k ≫ 1) \nIn contrast, for sinking orbits, the strong backreaction requires a small population transfer, and we find \nζ k ≃ v k 2 b k , r k ≃ 2 b k z k v k ≫ 1 . ( b k ≫ 1) \nNotice that, somewhat counter-intuitively, a strongdecay width not only does not necessarily imply the total depletion of the cloud, instead it suppresses r k , hence the ability of the system to float, resulting in a lesser amount of the cloud being transferred to the decaying mode (see (E9) below).', 'Appendix E: Floating': "For values of the energy transfer r k ≲ 0 . 2, the growth of the orbital frequency is sufficiently suppressed to allow \nfor the possibility of floating. In that case, the growth of (initially small) eccentricity is given, in units of the dynamical time introduced in (D5), by \n̸ \ne ( t ) ≃ √ e 2 in + I e τ (1 -f k ) , ( f k = 1) \n(E1) e ( t ) ≃ e in exp { -11 6 I e τ } , ( f k = 1) (E2) I e ≡ 2 3 √ γ 0 Ω 0 f 5 / 6 k | ∆ m + k | 1 / 2 . \nThe critical points of the evolution of the eccentricity depend both on | ∆ m | and | k | . For the first few values of | k | they are described by the polynomial fit e cr = g 0 + g 1 k + g 2 k 2 , where g 0 = { 0 . 3 , 0 . 2 , 0 . 16 } , g 1 = { 0 . 18 , 0 . 16 , 0 . 14 } , g 2 = { 2 , 1 . 6 , 1 . 3 } × 10 -2 , all for | ∆ m | = { 1 , 2 , 3 } in respective order. \nTo calculate the evolution of the eccentricity towards the fixed point, we change the time variable in the evolution equations [cf. (14)], from t to e 2 ( t ), yielding \n∫ e 2 fin e 2 in d ( e 2 ) √ 1 -e 2 [ g ( e ) f ( e ) -f k ] = I e b k ( 1 -| c a ( ∞ ) | 2 ) , (E3) \nThe result can then be approximated by (18) in the main text, where \nC k = c ∆ m,k [ I e b k ( 1 -| c a ( ∞ ) | 2 ) + e 2 in 1 -f k ] , (E4) \nand c ∆ m, -1 = { 2 . 37 , 2 , 1 . 37 } for ∆ m = {-1 , -2 , -3 } . From the value of e 2 fin in (E3), we can also estimate the duration of the floating period \n∆ τ Fl = 1 I e ∫ e 2 fin e 2 in d ( e 2 ) 1 √ f ( e ) √ 1 -e 2 [ g ( e ) f ( e ) -f k ] , ≃ C k ˜ c ∆ m,k I e , c ∆ m,k ˜ c ∆ m,k = 1 √ f ( e cr ) . (E5) \nAn exemplary parameter space of final eccentricity and floating time is shown in Fig. 6. \nStrong floating provides a distinct phase of the nonlinear LZ transition, during which typically most of the population transfer occurs. From (13) of the main text, we have \nd | c a | 2 dτ = r ( τ ) -1 b k , (E6) \nyielding a linear-in-time decay of the population during floating, and a transfer of population given by \n| c a ( ∞ ) | 2 ≃ | c a ( τ < Fl) | 2 -(1 -r k ) ∆ τ Fl b k . (E7) \nIn the strong decay regime, the condition r ( t ) → 0 at \n<!-- image --> \nFIG. 6. Final eccentricity ( left ) and floating time ( right ), for the | 322 ⟩ → | 31 -1 ⟩ transition with k = -1, assuming initial conditions that lead to a (long-lasting) floating orbit ( r -1 ≃ 0) and e in ≪ 1. \n<!-- image --> \nthe resonance is necessary at each point in time, but not sufficient to guarantee a steady floating-type period. In addition, there must be enough of the cloud left to sustain a small r ( t ). Following [62], one can estimate the sufficient condition by considering the minimum amount of cloud needed to 'startjump' a floating period. Applying the linear LZ solution (D3) in (13) of the main text, we find \n| c a | 2 min ≃ v k √ f ( e in ) 2 b k z k ( e in ) (1 -r k ) . (E8) \nThe left hand side can be interpreted as the minimal amount of cloud needed to start floating at a particular resonance. In turn, if the right hand side is larger than one, floating cannot start. The same condition can also be used to estimate the amount of cloud left when floating stops, by matching into the linear LZ solution backwards , from the end of the floating time, \n| c a ( ∞ ) | 2 ≃ v k √ f ( e fin ) 2 b k z k ( e fin ) (1 -r k ) , (E9) \nwhich then becomes the portion of the cloud surviving after floating stops. Strong decay and small z k could in principle interrupt the floating period and leave a moderate amount of the cloud intact. 10 However, at the overtones we have z k ∼ e | 2 k | , which is increasing during floating. Hence, as the eccentricity approaches the critical point, for instance at the k = -1 overtone, the value of z k increases by a factor ( e cr /e in ) 2 ≃ 10 2 -10 3 , thus significantly extending the floating period, and reducing the amount of cloud left after the transition, in comparison with the na¨ıve estimate in (E8). \nIn general, the equations in (E3), (E5) and (E9) must be solved self-consistently in order to determine the end state of floating. As an estimate, we may apply (E1) to (E9), and assuming e in ≪ 1, we have \n∆ τ Fl b k / √ f ( e cr ) ≃ ( x -1 2 x + √ ( x +1) 2 4 x 2 -λ x ) , (E10) λ = v k 2 b k z k ( e in ) , x = I e b k (1 -f k ) e 2 in . \nNotice that for the k = -1 overtone, the dependence on e in drops out from the ratio λ/x . In this case λ ≪ x , and we find the longest periods of floating, ∆ τ Fl ≃ b k / √ f ( e cr ), and largest depletion of the | c a ( ∞ ) | 2 ≲ r k . Depending on the resonances and the parameter space, such hierarchy may be also valid for higher overtones.", 'Appendix F: Numerical validation': "Our semi-analytic approach, described in Apps. D and E, rely in part on the small initial eccentricity of the BBHs prior to entering the transition region. We have validated this by numerically solving the Schrodinger equation for arbitrary (planar) orbits, coupled with the energy-momentum balance equations [cf. (3),(9)-(11) of the main text], for a number of representative examples, including a broad inital eccentricity range 10 -2 ≲ e in ≲ 0 . 5. For this purpose, we used the generic description of the orbital evolution, R ⋆ = a (1 -e cos u ), as a function of the eccentric anomaly u , which is related to the mean anomaly via Kepler's equation u -e sin( u ) = ϑ (e.g. [56]). We used the NDSolve routine in Mathematica [89], monitoring the violation of unitarity in the Γ b = 0 regime, residual for Γ b ≥ 0, as well as the state occupation numbers (D1). We have also checked the consistency of our results by comparing with the values in [48] for ( e, Γ b ) = 0. \nIn general, the numerical results are broadly consis- \n<!-- image --> \nFIG. 7. Numerical evolution of the examples described in App. F. The figures correspond to the same z k but different values for b k , with the left plot having a stronger backreaction parameter (a few times larger) than on the right. The curves show the eccentricity evolution for vanishing (pink) and large decay v k ≫ z k (blue, dot-dashed), both yielding floating orbits ( r k ≃ 0), except for the case of a large decay with small backreaction (blue curve on the right panel has r k ≃ 0 . 8). The semi-analytic solution (black, dashed) [cf. (14) in the main text], is in remarkable agreement within its regime of validity ( r k ≃ 0). \n<!-- image --> \ntent with the analytical arguments. As expected, the largest discrepancy occurs for the estimates of r ( t ) and | c a ( ∞ ) | 2 . For the floating regime, the analytic results tend to overestimate their respective values by a factor of few to an order of magnitude. Hence, our conclusions based on the analytic approximations err on the side of caution . In Fig. 7 we present (left) the numerical eccentricity evolution of a two-level system with ∆ m = 2 for values { z 0 = 4 , v 0 = 2 , b 0 = 200 , e 0 = 10 -2 } , at the main resonance. For these parameters, we have { z -1 = 7 · 10 -3 , v -1 = 3 . 4 , b -1 = 450 , I e = 2 · 10 -4 } at the k = -1 overtone, for which one finds a strongdecay regime: v -1 /z -1 = 512. We start the numerical evaluation before the first overtone, following also the case v -1 = 0. Away from the resonance, the orbital evolution closely follows standard GW evolution in vacuum. The approximations from Sec. D then correctly predict the strong floating that occurs (in both setups) at the frequency given by (8). Notice that, modulo small deviations, both cases follow the r k ≃ 0 prediction for the eccentricity growth in (E3). Furthermore, broadly consistent with our estimates, the left-over occupancy of the inital state is given by | c a ( ∞ ) | 2 ≃ 10 -4 and | c a ( ∞ ) | 2 ≃ 10 -3 , with and without the decay, respectively. Finally, reducing the backreaction to b 0 = 25 (Fig. 7, right), the decaying case does not develop floating, as r k ≃ 0 . 8, although as expected the eccentricity will still grow slightly and the transfer of population is somewhat increased compared to the linear LZ transition. In contrast, the non-decaying case still exhibits a strong floating, follows the predicted growth of the eccentricity, and transfers the parent state up to | c a ( ∞ ) | 2 ≃ 5 × 10 -3 .", 'Appendix G: Eccentricity distribution': "To evaluate the distribution of eccentricities in the population from [15], we order (in frequency) all possible hyperfine and fine resonances with the correspond- \ning overtones, up to | k | = 5. (A subset is displayed in Fig. 4.) We then calculate the corresponding parameters ( z k , v k , b k ) for every resonance the cloud may encounter. The condition that the initial state had enough time to reach the resonance is imposed, constrained by the lifetime imposed by its own GW emission. We evolve in vacuum via [43, 44] between transitions. At every resonance, we estimate r k as explained in App. D, and from there the floating time, eccentricity growth, and left-over cloud, as explained in App. E. We use a cutoff at 10 -3 of the initial M c, 0 /M , to estimate when the cloud's influence on the orbital dynamics becomes negligible. \nFor the BBHs shown in Fig. 1 of the main letter, only a few percent experience a floating transition | 322 ⟩ → | 31 -1 ⟩ with k < -1, while 25 % float at the k = -1 resonance, making it the dominant one. Approximately 15 % of the population float at the k = 0 transition. Only a few experience resonances to | 311 ⟩ or | 300 ⟩ states. The black holes with the largest masses, hence largest values of α for fixed µ , can float at the | 320 ⟩ hyperfine transition, where of the order of 5 % (each), see an earlier resonance and increase eccentricity there; or see the main or k = 1 resonance and decrease eccentricity instead. We do not find a significant impact for overtones with | k | > 4. We show in the left panel of Fig. 8 the distribution of different k 's experienced by the BBH population we studied in the main text. We show in the right panel the distribution of initial eccentricities before floating starts. As a consistency check, notice that for most of the resonances we have e in ≲ 0 . 1, which further support the validity of the small-eccentricity approximation. In Fig. 9 we plot the eccentricity evolution for three representative cases drawn from the population in Fig. 1 of the main text. \n<!-- image --> \nFIG. 8. Number of floating resonances for specific k ( left ) and binned by initial eccentricity ( right ) for the floating orbits the population shown in Fig. 1 experiences. \n<!-- image --> \nFIG. 9. Eccentricity evolution (in terms of the GW frequency) for representative examples from Fig. 1 in the main text. Solid lines show the evolution with the cloud, while dashed lines represents the standard vacuum evolution. \n<!-- image -->", 'Birth frequency': "Similar to Fig. 1 in the main letter, we show in Fig. 10 the eccentricity distribution at f GW = 10 -2 Hz, but with a lower birth frequency for the BBH+cloud system, at Ω ini /π = 5 × 10 -5 Hz. Although the number of larger-than-expected eccentricity points is somewhat lower, the plot illustrates the robustness of our predictions against changes in the initial conditions. In general, a large portion of BBHs that merge within a Hubble time will be born with orbital frequencies between [0 . 5 , 1] × 10 -4 Hz [14]. Moreover, BBHs that are formed through a common-envelope mechanism that can bring the binary closer, e.g. [67], at the same time may produce a (younger) secondary black hole that can ultimately carry the boson cloud, pushing the cloud's birth frequency towards higher values, thus avoiding altogether ( k = 0) hyperfine transitions that can decrease the eccentricity. Overall, our findings demonstrate that a skewedtype distribution and larger-than-expected eccentricities are robust predictions of the presence of boson clouds in the BBH dynamics.", 'Appendix H: Background (black hole) parameters': "Finally, let us comment on the backreaction on the black hole mass and spin. We will implement this in detail in a forthcoming paper. However, it is worth emphasising already that this effect does not change the main message of the letter: fixed points in the eccentricity evolution and the imprint in its final distribution at observable GW frequencies. We provide here a brief argument to support this claim. \nFirst of all, the changes in α (through the growth of the black hole mass) move the position of the resonances towards higher frequencies, which may help in sustaining the floating condition. Regarding the changes in spin, while they depend on the sign of m b , most of transitions tend to decrease the value of the parent black hole's spin, which in turn affects the superradiant condition. As a result, the | 322 ⟩ state may develop a decay width. However, even in the most extreme scenario, and assuming the entire cloud is instantly reabsorbed, we would still have Γ | 322 ⟩ ≃ α 13 (with ℓ = 2 for the excited state). As a consequence, as we show in Fig. 11, the decay time turns out to be longer than the floating time for all of the fine transitions. This effectively allows us to ignore the resulting width of the | 322 ⟩ state in the determination of the growth of eccentricity. \nFor the hyperfine transitions, on the other hand, the situation is more subtle, since the floating times are in principle longer. In that case, one has to take into account that the decrease of the spin does not occur instantaneously, but on the time-scales of the LZ transition governing the population of the | b ⟩ state. Preliminary studies indicate that the qualitative results for the hyperfine resonance are not significantly modified. In particular, since hyperfine transitions fall typically in the weak-decay regime, the floating condition is expected to be generally robust. This expectation is also in agreement with the results of [62], where the hyperfine transition | 211 ⟩ → | 21 -1 ⟩ was considered including the evolution of the background parameters. The authors did not find a significant disruption of the floating mechanism \n<!-- image --> \nFIG. 10. Same as Fig. 1 in the main letter, but with Ω ini /π = 5 × 10 -5 Hz as the birth orbital frequency for the BBH+cloud system. \n<!-- image --> \nFIG. 11. Ratio of floating and (excited-state) decay time for the k = -1 overtone of the fine transition, assuming the parent black hole's spin is reduced to ˜ a = 0 . 9˜ a s , with ˜ a s = 2 α the value at saturation. \n<!-- image --> \n- [1] R. Abbott et al. (KAGRA, VIRGO, LIGO Scientific), Open Data from the Third Observing Run of LIGO, Virgo, KAGRA, and GEO, Astrophys. J. Suppl. 267 , 29 (2023), arXiv:2302.03676 [gr-qc].\n- [2] R. Alves Batista et al. , EuCAPT White Paper: Opportunities and Challenges for Theoretical Astroparticle Physics in the Next Decade, (2021), arXiv:2110.10074 [astro-ph.HE].\n- [3] K. G. Arun et al. (LISA), New horizons for fundamental physics with LISA, Living Rev. 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2019A&A...625A.112G | Exploiting a sample of galaxies drawn from the XXLNorth multiwavelength survey we present an analysis of the stellar population properties of galaxies at 0.1 z 0.5 by studying galaxy fractions and the star formation rate SFRstellar mass MSUBSUB relation. Furthermore we exploit and compare two parametrisations of environment. When adopting a definition of global environment we consider separately cluster virial r 1rSUB200SUB and outer 1rSUB200SUB lt r 3rSUB200SUB members and field galaxies. We also distinguish between galaxies that belong or do not belong to superclusters but never find systematic differences between the two subgroups. When considering the local environment we take into account the projected number density of galaxies in a fixed aperture of 1 Mpc in the sky. We find that regardless of the environmental definition adopted the fraction of blue or starforming galaxies is the highest in the field or least dense regions and the lowest in the virial regions of clusters or highest densities. Furthermore the fraction of starforming galaxies is higher than the fraction of blue galaxies regardless of the environment. This result is particularly evident in the virial cluster regions most likely reflecting the different star formation histories of galaxies in different environments. Also the overall SFRMSUBSUB relation does not seem to depend on the parametrisation adopted. Nonetheless the two definitions of environment lead to different results as far as the fraction of galaxies in transition between the starforming main sequence and the quenched regime is concerned. In fact using the local environment the fraction of galaxies below the main sequence is similar at low and high densities whereas in clusters and especially within the virial radii a population with reduced SFR with respect to the field is observed. Our results show that the two parametrisations adopted to describe the environment have different physical meanings i.e. are intrinsically related to different physical processes acting on galaxy populations and are able to probe different physical scales. | 2019-05-01T00:00:00Z | ['2019arXiv190312293G', '10.48550/arXiv.1903.12293', '10.1051/0004-6361/201834970', '2019A&A...625A.112G', 'arXiv:1903.12293'] | ['large-scale structure of Universe', 'X-rays: galaxies: clusters', 'galaxies: clusters: general', 'galaxies: evolution', 'galaxies: star formation', 'galaxies: stellar content', 'Astrophysics - Astrophysics of Galaxies'] | The XXL Survey. XXXVII. The role of the environment in shaping the stellar population properties of galaxies at 0.1 z 0.5 | 2,019 | 171 | 0.33 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF'] | 14 | https://arxiv.org/pdf/1903.12293.pdf | {'The XXL Survey: XXXVII. The role of the environment in shaping the stellar population properties of galaxies at 0.1 GLYPH<20> z GLYPH<20> 0.5.': 'V. Guglielmo 1 ; 2 , B. M. Poggianti 2 , B. Vulcani 2 , S. Maurogordato 3 , J. Fritz 4 , M. Bolzonella 5 , S. Fotopoulou 6 , C. Adami 7 , and M. Pierre 8 \n(A GLYPH<14> liations can be found after the references) \nReceived xxx; accepted yyy', 'ABSTRACT': 'Exploiting a sample of galaxies drawn from the XXL-North multiwavelength survey, we present an analysis of the stellar population properties of galaxies at 0 : 1 GLYPH<20> z GLYPH<20> 0 : 5, by studying galaxy fractions and the star formation rate (SFR)-stellar mass (M ? ) relation. Furthermore, we exploit and compare two parametrisations of environment. When adopting a definition of \'global" environment, we consider separately cluster virial (r GLYPH<20> 1r200) and outer (1r200 < r GLYPH<20> 3r200) members and field galaxies. We also distinguish between galaxies that belong or do not belong to superclusters, but never find systematic di GLYPH<11> erences between the two subgroups. When considering the \'local" environment, we take into account the projected number density of galaxies in a fixed aperture of 1 Mpc in the sky. We find that regardless of the environmental definition adopted, the fraction of blue or star-forming galaxies is the highest in the field or least dense regions and the lowest in the virial regions of clusters or highest densities. Furthermore, the fraction of star-forming galaxies is higher than the fraction of blue galaxies, regardless of the environment. This result is particularly evident in the virial cluster regions, most likely reflecting the di GLYPH<11> erent star formation histories of galaxies in di GLYPH<11> erent environments. Also the overall SFR-M ? relation does not seem to depend on the parametrisation adopted. Nonetheless, the two definitions of environment lead to di GLYPH<11> erent results as far as the fraction of galaxies in transition between the star-forming main sequence and the quenched regime is concerned. In fact, using the local environment the fraction of galaxies below the main sequence is similar at low and high densities, whereas in clusters (and especially within the virial radii) a population with reduced SFR with respect to the field is observed. Our results show that the two parametrisations adopted to describe the environment have di GLYPH<11> erent physical meanings, i.e. are intrinsically related to di GLYPH<11> erent physical processes acting on galaxy populations and are able to probe di GLYPH<11> erent physical scales. \nKey words. Cosmology: large-scale structure of Universe - X-rays: galaxies: clusters -galaxies: clusters: general - galaxies: evolution - galaxies: star formation - galaxies: stellar content', '1. Introduction': 'Observational studies aiming at understanding the processes that a GLYPH<11> ect galaxy properties and determining the evolution of galaxies have been focussing more and more on the role played by both the environment in which a galaxy was formed and that in which it is embedded for most of its lifetime (Oemler 1974; Dressler 1980; Balogh et al. 2004b; Kau GLYPH<11> mann et al. 2004; Baldry et al. 2006; Poggianti et al. 2009). In particular, galaxies that are gathered together and / or hosted in the potential well of dark matter haloes, together with those accreted from the cosmic web into bigger structures, undergo a variety of physical processes that may influence the timescale of star formation and stellar mass assembly. These processes are usually connected to the interaction between galaxies and the hot gas permeating the dark matter haloes of groups and clusters, or to galaxy-galaxy interactions (e.g., Boselli & Gavazzi 2006, 2014, and references therein). \nOne of the biggest challenges in observational studies aiming at describing the interplay between galaxies and their environment is the definition of the environment itself (Haas et al. 2012; Muldrew et al. 2012; Etherington & Thomas 2015). Its parametrisation is commonly performed following two di GLYPH<11> erent strategies, which are able to probe di GLYPH<11> erent physical scales and have intrinsically di GLYPH<11> erent physical meanings. The first approach is based on the potential well of dark matter haloes, and thus relies on physical properties of the cosmic structures such as the virial masses and radii, X-ray luminosity, and dynamical masses. \nAccording to this definition, which is commonly referred to as \'global" environment, going from the largest scale (i.e. the most massive haloes) in the cosmic web down to the scales of single galaxies we can define superclusters, clusters, groups, filaments, field, and voids. \nThe second description of environment is based on the computation of the projected over-density of galaxies and is referred to as \'local" environment. Several methods have been explored for computing the local (projected) density of neighbouring galaxies, either based on computing the area enclosing the Nth neighbour with respect to a central one or counting the number of galaxies enclosed within a fixed aperture. It has been shown that the latter methodology is closer to the real over-density measured in 3D space, more sensitive to high over-densities, less biased by the viewing angle, and more robust across cosmic times than the former (Shattow et al. 2013). For this reasons, we adopt this method to quantify the local environment. \nWhatever the definition of environment, its strong connection with the observed properties of galaxies has been extensively demonstrated, both in terms of the average stellar age (e.g. Thomas et al. 2005; Smith et al. 2006) and the last episode of star formation (and thus a lower fraction are continuing to form stars; e.g. Lewis et al. 2002; Baldry et al. 2004; Balogh et al. 2004a,b; Kau GLYPH<11> mann et al. 2004). \nFocussing on the intermediate redshift regime (0.25 GLYPH<20> z GLYPH<20> 1.2), colour fractions have been found to depend strongly on the global environment; the incidence of blue galaxies is system- \natically higher in the field than in groups (Iovino et al. 2010) and clusters (Muzzin et al. 2012) and decreases with increasing absolute magnitude. Similarly, also the mean star formation rate (SFR), specific-SFR (sSFR) and star-forming fraction are always higher in field galaxies than in clusters, decrease from the outskirts to the cluster central region (Treu et al. 2003; Poggianti et al. 2006; Raichoor & Andreon 2014; Haines et al. 2015) and depend on stellar mass in a given environment (Muzzin et al. 2012). Similar results have been found both in the local Universe (e.g. Balogh et al. 2004b) and at higher redshifts. Linking the star formation activity of galaxies with their cold molecular gas reservoir, Noble et al. (2017) discovered a population of massive cluster galaxies having higher gas fractions compared to the field, indicating a stronger evolution of massive haloes at high redshifts; a depletion of the cold gas reservoir emerges instead in a sample of z GLYPH<24> 0.4 cluster galaxies in Jablonka et al. (2013) with respect to field galaxies of the same stellar mass, with further decreasing trends towards the centre of the structures. \nConsidering instead the local density (LD) parametrisation, the colour and star-forming fractions have also found to be lower in denser environments, both in the local Universe (e.g, Balogh et al. 2004a; Baldry et al. 2006) and at intermediate redshifts (e.g., Cooper et al. 2008; Cucciati et al. 2006, 2010, 2017). However, Darvish et al. (2016) found that in the star-forming population the median SFR and sSFR are similar at di GLYPH<11> erent values of the local density, regardless of redshift and galaxy stellar mass up to z GLYPH<24> 3, and Elbaz et al. (2007) even advocated the increase of the SFR of galaxies at z GLYPH<24> 1 in denser environments. \nThe e GLYPH<11> ect of global or local environment on galaxy properties has also been investigated in terms of the relation between the SFR and galaxy stellar mass. The existence of a tight relation of direct proportionality between SFR and galaxy stellar mass (SFR-M ? ) and sSFR-M ? has been established from z = 0 out to z > 2, with a roughly constant scatter of GLYPH<24> 0.3 dex out to z GLYPH<24> 1 (Brinchmann et al. 2004; Daddi et al. 2007; Noeske et al. 2007; Salim et al. 2007; Rodighiero et al. 2011; Whitaker et al. 2012; Sobral et al. 2014; Speagle et al. 2014). Star-forming galaxies lie on the so-called main sequence, whereas the quenched population occupy a locus with little or non-detectable SFR. \nThe representation of the SFR-M ? plane is necessary to understand the characteristics of the star-forming population of galaxies in di GLYPH<11> erent environments and to analyse whether the process leading to the shutting down of the star formation activity in a galaxy (and thus its transformation into a passive galaxy) proceeds similarly in di GLYPH<11> erent environments and whether the definition of the environment itself plays a role. In fact, fast quenching processes would leave the cluster / high-density regions SFR-M ? relation unperturbed with respect to the field / lowdensity regions, leaving the median SFR in agreement at all stellar masses. In contrast, slow quenching mechanisms would increase the number of galaxies with reduced SFRs shifting the overall distribution of SFRs towards lower values than those of main sequence galaxies of similar mass. \nWhen inspecting the SFR-M ? relation in di GLYPH<11> erent global environments, a population of low star-forming galaxies in a transition stage between the main sequence and the quenched population (hereafter \'transition" galaxies) has been observed in clusters at all redshfits up to z < 0.8 (Patel et al. 2009; Vulcani et al. 2010; Paccagnella et al. 2016). This population is missing in the field. In particular, Paccagnella et al. (2016) found that at 0.04 < z < 0.07 galaxies in transition are preferentially found within the virial radius ( R 200), and their incidence increases at distances < 0 : 6 R 200. These galaxies are older and present redder colours than galaxies in the main sequence and show re- \nduced mean SFRs over the last 2-5 Gyr, regardless of their stellar mass. Moreover, using spatially resolved observations from SDSS-IV MaNGA, Belfiore et al. (2017) associated the transition population with a population of galaxies having central low ionisation emission-line regions, resulting from photoionisation by hot evolved stars, and star-forming outskirts. These galaxies are preferentially located in denser environments such as galaxy groups and are undergoing an inside-out quenching process. \nOn the contrary, studies on galaxy samples based on a local parametrisation of environment do not find di GLYPH<11> erences in the SFR-M ? of galaxies at di GLYPH<11> erent densities (Peng et al. 2010; Wijesinghe et al. 2012, but see Popesso et al. 2011 at high z). \nIt is important to stress however that di GLYPH<11> erent results in the literature obtained by adopting di GLYPH<11> erent parametrisations of the environment are hard to compare, either because of the di GLYPH<11> erent selection criteria on the samples or custom definitions used to define, for example, the local galaxy over-density. \nThe aim of this work is to study the star formation properties and colours of galaxies adopting di GLYPH<11> erent definitions of environment, to acquire a general understanding of the phenomena that characterise and influence the observed properties of galaxies at di GLYPH<11> erent epochs and in di GLYPH<11> erent conditions. The main questions we want to address are: 1) How do the star-forming and blue fractions depend on environment? 2) Are there di GLYPH<11> erences in the star-forming population in di GLYPH<11> erent environments? Namely, are star-forming galaxies in clusters or dense environments as starforming as galaxies in the field or lower density environments? 3) How does the definition of the environment itself a GLYPH<11> ects these tracers? \nWe characterise galaxies in three redshift bins from z = 0.1 up to z = 0.5, in X-ray massive groups and clusters (1 : 13 GLYPH<2> 10 13 GLYPH<20> M 200 = M GLYPH<12> 1 GLYPH<20> 9 : 28 GLYPH<2> 10 14 , hereafter simply clusters) observed in the XXL Survey. This survey (Pierre et al. 2016, hereafter XXL Paper I), is an extension of the XMM-LSS 11 deg 2 survey (Pierre et al. 2004), consisting of 622 XMM pointings covering two extragalactic regions of GLYPH<24> 25 deg 2 each, one equatorial (XXL-N) and one in the southern hemisphere (XXL-S). The survey reaches a sensitivity of GLYPH<24> 6 GLYPH<2> 10 GLYPH<0> 15 erg s GLYPH<0> 1 cm GLYPH<0> 2 in the [0.5-2] keV band for point sources. \nThis study is focussed on computing the fraction of starforming and blue galaxies and the SFR-M ? relation, in the field versus clusters, also distinguishing between structures belonging or not to superclusters, and as a function of LD. The paper is organised as follows: in Section 2 we present the catalogues of clusters and galaxies, the tools used to compute galaxy stellar population properties and the computation of the spectroscopic incompleteness weights; in Section 3 we characterise di GLYPH<11> erent galaxy populations on the basis of their SFR and colours; in Section 4 we explore the dependence of the stellar population properties on global environment, performing a detailed analysis on galaxy fractions (Sect. 4.1) and on the SFR-M ? relation (Sect. 4.2 and 4.3); in Section 5 we analyse the galaxy population properties as a function of local environment, following the same scheme as Sect. 4. In section 6 we discuss our results obtained with the two parametrisations of environments regarding the galaxies in transitions (Sect. 6.1) and the ratio of star-forming to blue fractions (Sect. 6.2). Finally, we present our conclusions in Sect. 7. \nThroughout the paper we assume H0 = 69 : 3 km s GLYPH<0> 1 Mpc GLYPH<0> 1 ; GLYPH<10> m = 0 : 29 ; GLYPH<10>GLYPH<3> = 0 : 71 (Planck Collaboration et al. 2014, Planck13 + Alens). We adopt a Chabrier (2003) initial mass function (IMF) in the mass range 0 : 1 GLYPH<0> 100 M GLYPH<12> .', '2.1. Catalogue of structures': "Our environmental study is grounded in X-ray selected clusters from the XXL survey (XXL Paper I). The selection of the cluster candidates starting from X-ray images was presented by Pacaud et al. (2016) (hereafter XXL Paper II). \nBy means of the X amin pipeline (Pacaud et al. 2006), each structure is assigned to a specific detection class on the basis of the level of contamination from point sources. Class 1 (C1) clusters are the highest surface brightness extended sources, which have no contamination from point sources; Class 2 (C2) clusters are extended sources that are fainter than those classified as C1 and have a 50% contamination rate before visual inspection. Contaminating sources include saturated point sources, unresolved pairs, and sources strongly masked by CCD gaps, for which not enough photons were available to permit reliable source characterisation. Class 3 (C3) are (optical) clusters associated with an X-ray emission that is too weak to be characterised, and whose selection function is therefore undefined. \nThe spectroscopic confirmation and redshift assignment of cluster candidates are presented in Adami et al. (2018) (hereafter XXL Paper XX, but see also Guglielmo et al. 2018a, hereafter XXL Paper XXII). The procedure is similar to that already used for the XMM-LSS survey (e.g., Adami et al. 2011), and is based on an iterative semi-automatic process. The final catalogue of spectroscopically confirmed extended sources contains 365 clusters, 207 ( GLYPH<24> 56%) of which are classified as C1, 119 ( GLYPH<24> 32%) as C2 and the remaining 39 ( GLYPH<24> 11%) are C3. For the reasons explained above, C3 clusters are not included in the current work. A larger subsample of objects with respect to the first data release (Giles et al. 2016, XXL Paper III) underwent a direct X-ray spectral measurement of luminosity and temperature, down to a lowest flux of GLYPH<24> 2 GLYPH<2> 10 GLYPH<0> 15 erg s GLYPH<0> 1 cm GLYPH<0> 2 in the [0.5-2] keV band and within 60 arcsec (235 clusters). \nTo have homogeneous estimates for the complete sample, and as already performed in Guglielmo et al. (2018a,b) (hereafter XXL Paper XXX), we used the cluster properties derived through scaling relations 2 starting from the X-ray countrates. The method is presented in XXL Paper XX, from which (Table F.1) we extracted the values of the X-ray temperature ( T 300 kpc ; scal ), r 500 ; scal 3 , M 500 ; scal 4 . The luminosity in the 0.5-2.0 keV range ( L XXL 500 ; scal ) was not published in Paper XX but is available internally to our collaboration. XXL Paper XXII derived the virial mass M 200 from M 500 ; scal using the recipe given in Balogh et al. (2006), and computed the velocity dispersion ( GLYPH<27> 200) through the relation given in Poggianti et al. (2006), based on the virial theorem. \nIn XXL Paper XX, 35 superclusters were identified in both XXL-N and XXL-S fields in the 0.03 GLYPH<20> z GLYPH<20> 1.0 redshift range, by means of a friend-of-friend (FoF) algorithm characterised by a \nFig. 1: M 200 (top), L XXL 500 (bottom) versus redshift for the 111 XXL-N C1 + C2 clusters at 0.1 GLYPH<20> z GLYPH<20> 0.5. Clusters that belong to superclusters are represented by red stars, cluster that do not belong to any superclusters are represented by green points. \n<!-- image --> \nVoronoi tesselation technique. The physical associations with at least three clusters are called 'superclusters'. All the details of the methodology are provided in XXL Paper XX. \nIn this work we focus on clusters observed in the XXL-N region at 0.1 GLYPH<20> z GLYPH<20> 0.5. The sample is composed of 111 clusters that are fully characterised in terms of X-ray luminosities, temperatures, virial masses, and radii. Of these structures, 68 ( GLYPH<24> 60%) belong to superclusters, thus it is possible to study the impact of the large-scale structure on galaxy properties. To do so, we treat separately galaxies that belong or do not belong to a supercluster, and call these '(S)' and '(NS)', respectively. Taking as a reference the nomenclature adopted in XXL Paper XX, the superclusters considered in this work are reported in Table 1. \nFigure 1 shows how M200 and L XXL 500 vary with redshift within the sample, for clusters within and outside superclusters. As already mentioned in XXL Paper XXII, selection e GLYPH<11> ects emerge: at z > 0 : 4 the survey detects only the most massive clusters ( M 200 GLYPH<21> 10 14 M GLYPH<12> ). Nonetheless, no systematic di GLYPH<11> erences are detected between (S) and (NS) clusters.", '2.2. Galaxy catalogue': 'Wemadeuse of the galaxy properties included in the spectrophotometric catalogue presented in XXL Paper XXII. As for the catalogue of structures, we focussed on the XXL-N region and on the redshift range 0.1 GLYPH<20> z GLYPH<20> 0.5. \nThe photometric and photo-z information in XXL-N were mainly taken from the CFHTLS-T0007 photo-z catalogue in \nTable 1: List of superclusters detected in XXL Paper XX and included in our sample. The first column is the name of the supercluster according to XXL Paper XX nomenclature, the second and third columns are the centroid coordinates (J2000.0 equinox) the fourth column is the mean redshift, and the last column is the list of clusters belonging to each supercluster. \nthe W1 Field (8 GLYPH<14> GLYPH<2> 9 GLYPH<14> , centred at RA = 34.5000 GLYPH<14> and DEC = -07.0000 GLYPH<14> ). The data cover the wavelength range 3500Å < GLYPH<21> < 9400Å in the u GLYPH<3> , g 0 , r 0 , i 0 , and z 0 filters. Photometric data for a number of galaxies in the spectroscopic database that did not have any correspondence in the CFHTLS catalogue were taken from Fotopoulou et al. (2016). This catalogue contains aperture magnitudes in the g 0 , r 0 , i 0 , z 0 , J 0 , H 0 , and K 0 bands that have been converted into total magnitudes using a common subsample of galaxies with the CFHTLS-T0007 W1 field catalogue (see XXL Paper XXII). \nAll magnitudes are Sextractor MAG\\_AUTO magnitudes (Bertin &Arnouts 1996) in the AB system corrected for Milky Way extinction according to Schlegel et al. (1998). The error associated with photo-z in the magnitude range we are probing in this work (r < 20.0, see XXL Paper XXII and below) is redshift dependent, and according to the CFHTLS-T0007 data release document, is GLYPH<27>= (1 + z ) GLYPH<24> 0 : 031. \nSpectroscopic redshifts are hosted in the XXL spectroscopic database that is included in the CeSAM (Centre de donnéeS Astrophysiques de Marseille) database in Marseille. 5 As described in XXL Paper XXII, the database collects spectra and redshifts coming from di GLYPH<11> erent surveys covering the XXL pattern (mainly GAMA, SDSS, VIPERS, VVDS, VUDS, and XXL dedicated spectroscopic campaigns, see Table 2 in XXL Paper XXII), and the final spectroscopic catalogue was obtained by removing duplicates using a careful combination of selection criteria (the socalled priorities) and accounting for the quality of the spectra (i.e. the parent survey) and of the redshift measurement. Overall, the uncertainties on the galaxy redshift in the database vary from 0.00025 to 0.0005, as computed from multiple observations of the same object; we consider the highest value in this range as the typical redshift error for all objects. We note that the spectroscopic catalogue did not undergo any preselection or flag assignment to identify active galactic nuclei (AGN), and thus our sample may be contaminated by the presence of such peculiar sources. We address this point in more detail and quantify the contribution of AGNs later in this paper. \nThe final galaxy sample is obtained from the crossmatch between the photometric and spectroscopic sample. Figure 2 shows the distribution of galaxies and clusters in the coordinates plane, for the magnitude limited sample that is presented below.', '2.3. Tools': 'The stellar population properties of galaxies were derived relying on either their photometric or spectroscopic data. In the first case, we made use of the spectral energy distribution (SED) fitting code LePhare 6 (Arnouts et al. 1999; Ilbert et al. 2006) to compute absolute magnitudes, and therefore rest-frame colours, as described in XXL Paper XXII. In the second case, we fit galaxy spectra via SINOPSIS 7 (SImulatiNg OPtical Spectra wIth Stellar population models), a spectrophotometric fitting code fully described in Fritz et al. (2007, 2011, 2017) and already largely used to derive physical properties of galaxies in many samples (Dressler et al. 2009; Vulcani et al. 2015; Guglielmo et al. 2015; Paccagnella et al. 2016, 2017; Poggianti et al. 2017). Among the outputs of the model, we considered SFRs and galaxy stellar masses ( M GLYPH<3> ), defined as the mass locked into stars, both those which are still in the nuclear-burning phase, and remnants such as white dwarfs, neutron stars, and stellar black holes. \nWhile LePhare could be applied to the whole spectrophotometric sample of galaxies (provided that the catalogue contains magnitudes at least in two filters for each objects), SINOPSIS was run on the subsample of galaxies that have either SDSS or GAMAspectra, which are flux calibrated and have the best available spectral quality. As discussed in Fritz et al. (2014), in the lowest resolution spectra of this work, i.e. GAMA spectra, emission lines can be measured down to a limit of 2 Å, while any emission measurement below this threshold is considered unreliable. In terms of sSFR, this sets a lower limit of 10 GLYPH<0> 12 : 5 yr GLYPH<0> 1 . \nThe final sample is composed of galaxies with reliable outputs coming from both LePhare and SINOPSIS. \nFig. 2: Spatial distribution in the XXL-N area of galaxies in the spectrophotometric sample (yellow dots) and of X-ray confirmed clusters. The clusters in superclusters are reprensented with red stars and the clusters outside superclusters with green points. The region is divided into 22 cells (named as indicated inside each cell), used to compute the spectroscopic completeness (see details in Appendix A). \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFig. 3: Colour-magnitude diagrams in the magnitude limited sample in the three redshift bins analysed, with increasing redshift from left to right as indicated in the labels. Single galaxies are plotted as blue dots, while galaxies in higher density regions are grouped together and plotted as rectangles colour-coded according to their number density as indicated in the colour bar located on the side of each panel. The magenta dotted line shows the separation between red and blue objects using the (g-r) rest GLYPH<0> f rame colour. \n<!-- image -->', '2.4. Samples and spectroscopic completeness': 'In what follows, we consider galaxies in three redshift bins, 0 : 1 GLYPH<20> z < 0 : 2, 0 : 2 GLYPH<20> z < 0 : 3, 0 : 3 GLYPH<20> z GLYPH<20> 0 : 5 and study both magnitude and mass limited samples. As detailed in XXL Paper XXII, magnitude completeness limit was set to an observed magnitude of r = 20 : 0 at all redshifts, and is converted into a di GLYPH<11> erent mass completeness limit at each redshift. To determine this limit, at each redshift we converted the observed magnitude limit into a rest-frame magnitude limit and computed the mass of an ideal object having the faintest magnitude and the reddest colour in that redshift bin. Following XXL Paper XXII, the stel- \nlar mass limit of each redshift bin is that corresponding to the lowest limit of each interval; i.e. at 0.1 GLYPH<20> z < 0.2 is the stellar mass limit corresponding to z = 0.1. We therefore adopted the following values: \n- - 0.1 GLYPH<20> z < 0.2: M ? > 10 9 : 5 M GLYPH<12>\n- - 0.2 GLYPH<20> z < 0.3: M ? > 10 10 : 3 M GLYPH<12>\n- - 0.3 GLYPH<20> z GLYPH<20> 0.5: M ? > 10 10 : 8 M GLYPH<12> \nThe galaxy magnitude complete sample includes 18399 galaxies, the mass complete sample includes 13857 galaxies. Table 2 reports the number of galaxies in the di GLYPH<11> erent redshift bins \nTable 2: Number of galaxies above the magnitude and mass completeness limits in three redshift bins. The quantities in parentheses refer to the number of galaxies weighted for spectroscopic completeness. Values of Mlim are given in the main text. \nfor both samples. Both raw numbers and those corrected for incompleteness are given. The method used to compute the spectroscopic completeness is described in Appendix A. Briefly, as the spectroscopic sample spans a relatively wide redshift range, we sliced the sample into di GLYPH<11> erent redshift bins and quantified the number of galaxies that fall / are expected to fall into that given redshift bin, based on both spectroscopic and photometric redshifts. As already performed in XXL Paper XXII, we accounted for the change in the spectroscopic sampling of di GLYPH<11> erent surveys by dividing the sky into 22 cells (shown in Fig. 2), and in intervals of 0.5 r-band magnitude within each cell. The completeness curves resulting from this computation were converted into completeness weights which are attributed to each galaxy given its redshift, astrometry, and magnitude.', '3. Galaxy subpopulations': "In our analysis we characterised separately the star-forming properties and rest-frame colours of galaxies in di GLYPH<11> erent environments and at di GLYPH<11> erent redshifts. We therefore need to define two di GLYPH<11> erent criteria to separate star-forming / blue galaxies from passive / red galaxies. \nFirst, we considered as 'star forming' those galaxies with sSFR = SFR = M ? > 10 GLYPH<0> 12 yr GLYPH<0> 1 and 'passive' the remaining galaxies. We point out that this sSFR threshold is the same in the three redshift bins considered, which is justified by the scarce evolution in the sSFR-stellar mass plane in this redshift range (see e.g. Whitaker et al. 2012). \nThen, we considered as 'blue' galaxies those whose restframe colour is bluer than a certain threshold, and 'red' the rest. To identify such threshold in colour, we investigated the relation between the (g GLYPH<0> r)rest GLYPH<0> frame colour and absolute magnitude Mr, in the three redshift bins separately. Figure 3 shows the restframe colour-magnitude diagram (CMD) in each redshift bin. To define the slope of the colour-magnitude cut, we focussed on the lowest redshift bin, which has a su GLYPH<14> ciently wide magnitude range. We considered five 0.6 absolute magnitude bins and plot the (g GLYPH<0> r)rest GLYPH<0> frame histogram of each subpopulation (Fig. 4). We then fit the histogram with a double-Gaussian curve and determined the minimum of the distribution between the two peaks. Wecomputed the line interpolating the (g GLYPH<0> r)rest GLYPH<0> frame colours just found in the five magnitude bins and used it to divide the galaxy population as shown in Fig. 3 (magenta dashed line). At higher redshift, the magnitude range is too small to apply the same procedure. As no significant evolution is expected in the slope of the relation, but only in the zero point, we fixed the slope to that of the lowest z bin and computed the appropriate zero points with the same method outlined above (Fig. 5): we considered one magnitude bin at each redshift, we drew the (g GLYPH<0> r)rest GLYPH<0> frame colour histogram and fit the distribution with a double-Gaussian curve, finding the local minimum between the two peaks. \nFig. 4: Rest-frame (g-r) colour distributions in five absolute magnitude bins for galaxies at 0 : 1 GLYPH<20> z < 0 : 2. The red curve shows the double-Gaussian fit performed on the distributions and the single Gaussians are represented with the black dashed line. The magenta vertical lines indicate the local minima in the valley between the two Gaussian peaks, and define the separation between the red sequence and blue cloud. \n<!-- image --> \n1.0 \nTo conclude, at 0.1 GLYPH<20> z < 0.2 galaxies were assigned to the blue sequence if their colour obeys ( g GLYPH<0> r ) rest GLYPH<0> f rame < GLYPH<0> 0 : 019 Mr + 0 : 192, at 0.2 GLYPH<20> z < 0.3 the zero point is 0.177 and 0.176 at 0.3 GLYPH<20> z GLYPH<20> 0.5. \nAs a comparison between the two criteria just described we note that, considering all the redshift bins together, blue galaxies have a median sSFR GLYPH<24> 10 GLYPH<0> 9 : 7 yr GLYPH<0> 1 (and 90% of galaxies have sSFR & 10 GLYPH<0> 10 : 45 yr GLYPH<0> 1 ). Conversely, star-forming galaxies have a median ( g GLYPH<0> r ) rest GLYPH<0> f rame GLYPH<24> 0 : 58 (and 90% of galaxies have ( g GLYPH<0> r ) rest GLYPH<0> f rame < 0 : 725). \nIt is important to bear in mind that the two tracers used to characterise the galaxy populations have a di GLYPH<11> erent physical meaning and refer to di GLYPH<11> erent timescales. While the SFR is an instantaneous measure of the rate at which a galaxy is forming stars at the epoch it is observed, colours are the result of longer processes tracing the predominant stellar population of a galaxy, whose colour is sensitive to its past history and to its current star formation activity. Moreover, colour is also influenced by other characteristics, such as the metallicity and the presence of dust. In addition, the methodologies adopted to compute SFR and colours are di GLYPH<11> erent. The ongoing SFR is a product of the \nFig. 5: Rest-frame (g-r) colour distributions performed in one representative absolute magnitude bin in the two highest redshift bins indicated in each panel. Curves and colours are shown as in Fig. 4. \n<!-- image --> \nfull spectral fitting analysis performed on the spectra, while restframe colours are derived by means of SED fitting on the photometry. Therefore, it is important to investigate the two quantities separately and study the incidence of each population over the total, as we do in the next sections.", '4. Results I: Galaxy population properties as a function of the global environment': 'In this section, we study the fractions and star-forming properties of galaxies in di GLYPH<11> erent global environments. We consider galaxies in the following environments. \n- -Cluster virial members are galaxies whose spectroscopic redshift lies within 3 GLYPH<27> from the mean redshift of their host cluster, where GLYPH<27> is the velocity dispersion of their cluster and whose projected distance from the cluster centre is < 1 r 200.\n- -Cluster outer members are galaxies whose spectroscopic redshift lies within 3 GLYPH<27> from the mean redshift of their host cluster, and whose projected distance from the cluster centre is between 1 and 3 r 200.\n- -Galaxies in the field are all galaxies that do not belong to any cluster. \nWe note that all galaxies belonging to a structure are always included in the same redshift bin. For example, if a cluster is located at the edge of a redshift bin and its members spill over another bin, these are all included in the redshift bin of their host cluster, regardless of their actual redshift. \nWealso treat separately virial and outer members that belong or do not belong to a supercluster. \nTable 3 reports the number of galaxies in the di GLYPH<11> erent environments and redshift bins. For all of these subsamples, numbers are given for the magnitude limited and mass limited samples. At 0 : 1 GLYPH<20> z < 0 : 2 our sample includes three superclusters, at 0 : 2 GLYPH<20> z < 0 : 3 three superclusters, and at 0 : 3 GLYPH<20> z GLYPH<20> 0 : 5 six superclusters.', '4.1. Fraction of blue and star-forming galaxies': 'Figure 6 shows the fraction of blue and star-forming galaxies, separately, in the di GLYPH<11> erent global environments and in the three redshift bins, both for the magnitude limited and mass limited samples. Error bars are computed using a bootstrap method. For galaxies in the field, we include in the error budget both the bootstrap error and the uncertainty due to the cosmic variance. Following Marchesini et al. (2009), we sliced our field into nine right ascension subregions and we computed the fraction of starforming and blue galaxies of each region separately; the contribution to the error budget from cosmic variance is then the standard deviation of the newly computed fractions divided by the number of subregions considered. \nOverall, at all redshifts, both considering the star formation and colours as tracers, fractions are similar within and outside the superclusters, suggesting that neither additional quenching processes nor triggering of the star formation are associated with the presence of superclusters. \nAt 0.1 GLYPH<20> z < 0.2 (top left), both in the magnitude and in the mass limited samples, the star-forming fraction strongly depends on environment. Virial members have the lowest fraction of starforming galaxies (55-60%). This fraction increases when considering outer members, where GLYPH<24> 80% of galaxies are star forming. Finally, the percentage of star-forming galaxies in the field is the highest (86 GLYPH<6> 1%). The same trends are recovered when considering galaxy colours, even though fractions are systematically lower: GLYPH<24> 16% of virial members are blue, as are GLYPH<24> 40% of outer members and 57% of field galaxies. Similarly to the star-forming fractions, results in the magnitude and mass limited samples are similar, except for the field value, where they di GLYPH<11> er by GLYPH<24> 10%; the mass limited sample shows a lower fraction than the magnitude-limited sample. \nAt 0.2 GLYPH<20> z < 0.3 (middle panels of Fig. 6), in both samples, virial members still show a significantly lower fraction of starforming galaxies than the other environments ( GLYPH<24> 55 GLYPH<0> 60%), while outer members and field galaxies present very similar fractions ( GLYPH<24> 85% = 75% in the magnitude / mass limited samples). Considering colour fractions, the same trends are detected in the magnitude limited sample, where blue galaxies are GLYPH<24> 17% in virial members, GLYPH<24> 42% in outer members and in the field. In the mass limited samples, the di GLYPH<11> erence between outer members and the field is much smaller: the fraction of blue galaxies in these environments is always < 20%. \nWe recall that this redshift bin contains the XLSSsC N01 supercluster, separately discussed in XXL Paper XXX, and that contributes to the (S) cluster population with 11 out of 20 clusters, corresponding to GLYPH<24> 65% of the cluster population. In that supercluster an enhancement of the star formation activity of outer members with respect to the virial population and the field was observed. Nonetheless, general trends are maintained within the errors. \nAt 0.3 GLYPH<20> z GLYPH<20> 0.5 (bottom panels of Fig. 6), both in the mass and magnitude limited samples, virial members have the lowest star-forming fraction (45-50%), but di GLYPH<11> erences with the other environments are reduced: in outer members and in the field the star-forming fractions are GLYPH<24> 65% in the magnitude limited \nTable 3: Number of galaxies in the di GLYPH<11> erent environments (clusters in superclusters (S), clusters not in superclusters (NS), and field) and above the magnitude and mass completeness limits, in three redshift bins. Galaxies in clusters are further subdivided into virial and outer members. The quantities in parentheses refer to the number of galaxies weighted for spectroscopic completeness. \nsample and GLYPH<24> 55 GLYPH<0> 60% in the mass limited sample. Considering colours, in the magnitude limited sample we still detect the usual di GLYPH<11> erences between virial members and galaxies in other environments, while in the mass limited sample all fractions are lower than 15% and no variation with environment is detected. \nAs our cluster sample spans a wide range of X-ray luminosity (see Fig. 1), we repeat the analysis separating the clusters in bins of X-ray luminosity, but find no significant additional trends (plot not shown). \nTo summarise, at all redshifts, field galaxies have the highest incidence of star-forming / blue galaxies, while virial members exhibit a noticeable suppression of both star-forming and blue fractions with respect to the other environments. Outer members exhibit a significant suppression of the star-forming / blue fractions with respect to the field only at 0.1 GLYPH<20> z < 0.2, while at higher redshift they present similar fractions. No significant di GLYPH<11> erences are detected between galaxies within and outside superclusters. However, fractional di GLYPH<11> erences within and outside of superclusters do not follow a common trend at all redshifts, likely reflecting the variation of properties of individual supercluster structures at di GLYPH<11> erent redshifts. The choice of a mass or magnitude limited sample only marginally a GLYPH<11> ects the starforming fractions, while it strongly alters those based on colours at z > 0 : 2. \nOverall, star-forming and blue fractions are never consistent within the errors: this is a probe that the two quantities, even though strictly related, are actually reflecting di GLYPH<11> erent aspects of the evolution of the galaxies. We note that in our sample no reasonable and physically motivated cut could be adopted to reconcile the fractions of star-forming and blue galaxies. \nIn principle, the di GLYPH<11> erence in the star-forming and blue fractions could be due to the presence of AGNs; for example, lowionisation nuclear emission-line regions (LINERS) identified as red star-forming galaxies. These AGNs would increase the number of galaxies pertaining to the star-forming population without enhancing the fraction of blue galaxies. To test this, we removed broad- and narrow- line AGNs from our galaxy sample, as described in detail in Appendix B, and we computed again the star-forming / blue fractions. The fractions are substantially unchanged (plot not shown), indicating that our results are not driven by the possible presence of AGNs. \nWe stress that comparisons across the di GLYPH<11> erent redshift bins are not possible, as magnitude and mass values used to define the sample are di GLYPH<11> erent. Furthermore, we point out that the decrease of the blue / star-forming fraction with increasing redshift is simply an artefact due to the galaxy mass range probed at di GLYPH<11> erent redshifts.', '4.2. SFR-mass relation': 'We focus in this section only on the star-forming population and investigate the correlation between the SFR and galaxy stellar mass (SFR-M ? ). For this analysis we only rely on the mass limited sample. Indeed, in contrast with the magnitude limited sample, applying a mass limit ensures completeness, i.e. to include all galaxies more massive than the limit regardless of their colour or morphological type. This ensures that we do not bias the results because of the absence of galaxies which are undersampled or missed by selection e GLYPH<11> ects, as might happen when considering a magnitude limited sample. As in the previous section we did not detect any significant di GLYPH<11> erence between galaxies within and outside superclusters, in what follows we do not distinguish between the two subgroups. \nFigure 7 compares the distribution of galaxies in di GLYPH<11> erent environments and in di GLYPH<11> erent redshift bins in the SFR-M ? plane (left and middle panels). Roughly, at all redshifts, galaxies located in the di GLYPH<11> erent environments share a common region on the plane, excluding strong environmental e GLYPH<11> ects at play. Comparing the galaxies at di GLYPH<11> erent redshifts, we find a decline in SFR with time at fixed stellar mass, in agreement with many previous literature results (e.g. Noeske et al. 2007; Vulcani et al. 2010). \nTo probe the apparent lack of environmental e GLYPH<11> ects on a statistical ground, we proceed by first performing a linear regression fit to the relation by considering all the di GLYPH<11> erent environments together and then compare the median values of SFR in di GLYPH<11> erent mass bins for the various environments to this fit. The values of the best-fit slope, intercept and 1 GLYPH<27> are given in Table 5. Error bars on the medians are computed in each stellar mass bin as 1.253 GLYPH<27>= p n , where GLYPH<27> is the standard deviation of the SFR distribution in the bin and n is the number of objects considered in the bin. \nThe fit to the SFR-M ? relation is dominated by field galaxies, whose median trends closely follow the fitting line at all redshifts. In contrast, cluster virial members show hints of lower median SFR with respect to the latter in all the redshift bins; some statistical oscillations are due to the lower number of galaxies at 0.3 GLYPH<20> z GLYPH<20> 0.5. Furthermore, in this case the limited mass range could also a GLYPH<11> ect the reliability of the fit. The median SFR of outer members closely follows the field trend at z GLYPH<20> 0.2 and is compatible within the error bars with both the field and virial members at higher redshift. We do not plot these values for the sake of clarity. \nThe right-hand panels of Fig. 7 report the distribution of the di GLYPH<11> erences between the SFR of each galaxy and the value derived from the global fit given the galaxy mass ( GLYPH<1> SFR), for any given environment. Positive values of GLYPH<1> SFR correspond to re- \nFig. 6: Fraction of star-forming (left) and blue (right) galaxies in di GLYPH<11> erent environments and di GLYPH<11> erent redshifts, as indicated in the panels. Cluster members are divided into four subsamples: virial and outer members that belong or do not belong to a supercluster. Values obtained using the magnitude limited sample are represented with filled symbols and solid errors, those obtained using the mass limited sample are represented by empty symbols and dashed error bars. A horizontal shift is applied for the sake of clarity. Errors are derived by means of a bootstrap method. \n<!-- image --> \nduced SFR with respect to the expected value. At all redshifts, it is immediately clear that the shape of distribution of GLYPH<1> SFR of virial members di GLYPH<11> ers from that of the field population, whereby the former presents a tail of reduced SFR values with respect to the latter. A Kolmogorov-Smirnov (KS) test is able to de- \ntect di GLYPH<11> erences between virial members and field galaxies at all redshifts (P(KS) GLYPH<20> 0 : 05); outer members instead have statistically di GLYPH<11> erent distributions with respect to the field only at 0.3 GLYPH<20> z GLYPH<20> 0.5 (P(KS) < 0.02), and with respect to virial members only at 0.1 GLYPH<20> z < 0.2 (P(KS) < 10 GLYPH<0> 3 ). Nonetheless, at all redshifts, \nFig. 7: Left and middle panels . SFR-M ? relation for galaxies in the field and cluster virial and outer members (grey 2D histogram and density contours, orange diamonds, and black stars, respectively) in the mass limited sample. Panels in di GLYPH<11> erent lines refer to di GLYPH<11> erent redshift bins. The field population is represented with a 2D histogram whose values are given in the colour bar included in the middle panel, and grey contours trace the density levels of the data points. The vertical red dashed line shows the stellar mass limit at each redshift, while the oblique red dashed line sets the limit to the star-forming population, i.e. sS FR = 10 GLYPH<0> 12 yr GLYPH<0> 1 . The blue line is the linear fit to the SFR-M ? relation including all the environments at each redshift, and the dashed blue lines correspond to 1 GLYPH<27> errors on the fitting line. The parameters of the fit and the values of GLYPH<27> are given in Table 5. The gold diamonds / stars and cyan dots represent the median SFR values computed in mass bins of 0.2 dex width, for the virial / outer members and field population, respectively. Error bars on the medians are computed assuming a normal distribution of the data points as 1.253 GLYPH<27>= p n , where GLYPH<27> is the standard deviation of the distribution and n is the number of objects in the considered stellar mass bin. Right panels . Histograms of the di GLYPH<11> erences between the expected SFR computed using the main sequence fitting line at the stellar mass of any given galaxy in our sample and its actual SFR ( GLYPH<1> SFR). Positive values of GLYPH<1> SFR indicate reduced SFR compared to the SFR main sequence of star-forming galaxies. The median values of the distributions are also shown with vertical dashed lines and di GLYPH<11> erent environments are colour coded as written in the legend. \n<!-- image --> \nmedian values are compatible within the errors among the different samples, indicating that the tail, although present in virial and outer members, is not able to a GLYPH<11> ect the whole SFR distribution significantly.', '4.3. Galaxies in transition': 'The presence of a non-negligible number of galaxies with reduced SFR among the cluster population motivates a more detailed investigation on the presence of the so-called galaxies in transition , i.e. star-forming galaxies which are slowly decreasing their SFR and are detected as an intermediate population migrating from the star-forming main sequence down to \nTable 4: Fraction of galaxies in transition in di GLYPH<11> erent environments in the three redshift bins. Numbers are weighted for spectroscopic incompleteness and are computed above the stellar mass completeness limit of each redshift bin; the values in parenthesis refer to the highest stellar mass limit to allow comparisons at di GLYPH<11> erent redshifts. Errors are computed by means of bootstrapping. The last two lines of the table correspond to the values computed in two bins of LD and are analysed in Sect. 5.Table 5: Best-fit parameters of the linear fit to the SFR-M ? relations shown in Fig. 7, in three redshift bins. The fit is performed on the sample including all the environments together, and the fitting line has the following general equation: Log(SFR) = a Log(M ? ) + b . \nthe quenched population. To identify the galaxies in transition we follow Paccagnella et al. (2016), and select galaxies with (sSFR) > 10 GLYPH<0> 12 yr GLYPH<0> 1 and SFR below 1 GLYPH<27> from the SFR-M ? fitting line. The transition fraction is computed as the ratio of this population to the number of star-forming galaxies in each environment. We note that, by definition, the percentage of galaxies below a 1 GLYPH<27> cut of the SFR-M ? relation should be GLYPH<24> 15-17%, therefore the identification of a population of galaxies in transition is measured as an excess of galaxies compared to this statistical value. \nThe fractions of galaxies in transition as a function of environment for di GLYPH<11> erent redshift bins are presented in Fig. 8 and given in Table 4. We compute these fractions also dividing virial / outer cluster members residing or not in superclusters. \nThe incidence of the population of galaxies in transition depends on environment. As shown in Fig. 8, the fraction of transition galaxies in the field and outer members is (within the errors) almost half of that observed in cluster virial members at z GLYPH<20> 0.3. At higher redshift instead, the fractions are similar within the error bars in all environments, likely owing to the high stellar mass limit considered. \nConsidering separately clusters within and outside superclusters, no clear trends are observed in the transition fractions, suggesting again that di GLYPH<11> erences among superclusters are most likely statistical. In this context, we note that at 0.2 GLYPH<20> z < 0.3 the fraction of galaxies in transition in the virial and outer regions of (S) clusters is in agreement with the trends found for the XLSSsC N01 supercluster (XXL Paper XXX). The transition fractions are GLYPH<24> 10% lower in both (S) virial and outer members compared to their (NS) counterparts, as in the XLSSsC N01 supercluster where the percentage of galaxies with reduced SFR was < 20% in all the environments. \nWe also tested whether the X-ray luminosity played a role in the determination of the number of galaxies in transition in clus- \nters, and we did not find any clear correlation in the luminosity range probed by our cluster sample. \nAs a general understanding, environmental e GLYPH<11> ects seem to dominate within the cluster virial radii: the substantial di GLYPH<11> erence in the number of galaxies with reduced SFR among cluster virial members compared to the field population is responsible for detection of tails in the GLYPH<1> SFR distributions, shown in the right panels of Fig. 7.', '5. Results II: Galaxy population properties as a function of the local environment': 'The availability of a large spectrophotometric sample of galaxies enables the parametrisation of environment also in terms of projected LD of galaxies. In this section we consider together the galaxies in all the aforementioned environments and divide these sources into the usual three redshift bins. For each galaxy, we compute the projected LD as the number of galaxies enclosed into a fixed radial aperture of 1 Mpc at the redshift of the galaxy and within a given redshift range around the centre galaxy. We describe the computation of LD in detail in Appendix C. Figure 9 shows the LD distribution in the three redshift bins in logarithmic units, along with the 15th, 50th, and 85th percentiles, which will be used to define the LD bins used in Sect.5.2. It is evident that going from low- to high-z the peak (i.e. the median) of the LD is shifted towards higher densities, as previously found in other samples (Poggianti et al. 2010).', '5.1. Fraction of blue and star-forming galaxies': 'Figure 10 shows the fraction of blue (right) and star-forming (left) galaxies as a function of the projected LD, in the three redshift bins, separately, for both the magnitude and mass limited samples. Error are derived by means of bootstrapping. At 0 : 1 GLYPH<20> z < 0 : 2 (top panels), both in the magnitude and in the mass limited samples, the fraction of both star-forming and blue galaxies decreases monotonically with increasing LD. The starforming fraction is close to 90% at low densities and then decreases of a factor & 1.5 in a LD range of 2.0 dex; the blue fraction is GLYPH<24> 80% at low densities and decreases of almost four times; the values drop to GLYPH<24> 0.2 at the highest densities. \nAt 0 : 2 GLYPH<20> z < 0 : 3 (middle panels of Fig. 10), the star-forming fractions are much less dependent on density, both in the mass and magnitude limited samples. Values range between 80 and 60%, at low and high density, respectively. In contrast, in the \nFig. 8: Fraction of galaxies in transition in the mass limited sample in the three redshift bins. Filled dots represent galaxies in the di GLYPH<11> erent environments, as written in the x-axis. The (S) and (NS) contribution to the virial and outer member populations are also represented with empty symbols and dashed error bars. Error bars are computed via bootstrapping. \n<!-- image --> \nFig. 9: Distributions of the logarithm of the LD in the three redshift bins, as indicated in the labels. Histograms are drawn after a sigma-clipping has been performed on the parent distributions. The red dashed vertical lines represent the 15th, 50th and 85th percentiles, respectively. \n<!-- image --> \nmagnitude limited sample, the blue fraction still shows a significant decrease with LD, ranging from 50% at low densities to 20% at the highest. In the mass limited sample the blue fraction is always . 20%, regardless of density. \nIn the highest redshift bin (bottom panels of Fig. 10), both in the magnitude and mass limited samples the star-forming fractions seem first to increase with density, reach a plateau and then decrease at the highest values. Overall, values range between 50 and 70% in the magnitude limited sample, 40% to 60% in the mass limited sample. Such increase with LD is also noticeable in the colour fractions: in the magnitude limited sample at low density the fraction is GLYPH<24> 25%, reaches 40% at intermediate densities and falls down to 30% at the highest density. In the mass limited sample, the fraction of blue galaxies is always < 20%, but shows a statistically meaningful increase from the lowest to the highest densities. \nTo summarise, the star-forming / blue fraction of galaxies decreases at densities higher than the LD median at each redshift \n(see Fig. 9). At densities lower than the median, we notice a steady decrease of the fractions at 0.1 GLYPH<20> z < 0.2, opposed to an initial increase at z GLYPH<21> 0.2. Furthermore, the overall decrease of the star-forming / blue fractions going from the low to high densities is much more pronounced at lower than at higher redshifts. As it was previously found in Sect. 4.1, considering either the magnitude limited sample or the mass limited sample lead to substantial di GLYPH<11> erences only in the fraction of blue galaxies at z > 0.2. Finally, di GLYPH<11> erences in the absolute values of star-forming and blue fractions are again noticeable and are further investigated and discussed in Sect. 6.2.', '5.2. SFR-mass relation and galaxies in transition': 'We now study the SFR-M ? relation of galaxies in two extreme bins of LD representative of the lowest and highest LD environments.With reference to the histrograms represented in Fig. 9, we selected two percentiles that allowed us to seize the wings of \nFig. 10: Fraction of star-forming galaxies in di GLYPH<11> erent bins of LD, computed with the sSFR (left panels) and rest-frame colour (right panels). Three redshift bins from z = 0.1 up to z = 0.5 are represented, and the redshift increases from top to bottom panels as indicated in each panel. A sigma-clipping has been performed on the parent LD distributions to remove outliers and bins with a non-statistically representative number of objects. Panels and symbols are shown as in Fig. 6. \n<!-- image --> \nthe distribution (having previously removed outliers), considering its narrow shape. The selected percentiles are 15th and 85th. \nIn Figure 11 we report the SFR-M ? relation of galaxies in the low- and high- LD regimes. We proceed as before and compute the median SFR in stellar mass bins of 0.2 dex width in the low- and high- LD regimes. The median values of the SFR computed in bins of stellar mass show little variation with LD (yellow diamonds versus cyan stars), whose values that are always consistent within the error bars. Di GLYPH<11> erences arising at the highest stellar mass values at z GLYPH<21> 0.3 may be mostly driven by the low sample statistics, and therefore should be taken with caution. \nThe right-hand panels of Fig. 11 show the GLYPH<1> SFR with respect to linear fit to the SFR-M ? relation used in Sect. 4.2, computed as previously done for the global environment. The median GLYPH<1> SFR values are very similar in the high- and low-LD regimes at all redshifts, and the statistical similarity between the two samples is further confirmed by the outcome of the KS test: P(KS) >> 0.05 at all redshifts. \nFinally, we also compute the fraction of transition galaxies in the two extreme LD bins (see Tab. 4), finding no di GLYPH<11> erences within the error, at all redshifts. \nFig. 11: Left panels . SFR-M ? relation for galaxies in two regimes of LD, corresponding to the wings of the LD histograms shown in figure 9. Panels and lines are shown as Fig. 7. Cyan stars and the gold diamonds represent the median values of the SFR computed in 0.2 dex stellar mass bins, for the low- and high- LD regimes respectively. Error bars are computed as in Fig. 7. Right panels . Histograms of the di GLYPH<11> erences between the expected SFR computed using the main sequence fitting line at the stellar mass of any given galaxy in our sample and its actual SFR ( GLYPH<1> SFR). Median values of the distributions are shown with vertical dashed lines and colour coded as written in the legend. \n<!-- image -->', '6. Discussion': 'In this paper we have adopted two definitions of environment. The first is based on the X-ray selection of virialised structures; the second is based on the local galaxy number density. We are now in the position of contrasting the results, and we aim to understand whether the di GLYPH<11> erent parametrisations lead to similar conclusions. \nIn the literature, the environmental dependence of the galaxy properties was previously investigated by many authors, adopting either a global or local parametrisation, but hardly ever directly contrasting the two in homogeneous samples. Nonetheless, as discussed by Vulcani et al. (2011, 2012, 2013) and Calvi et al. (2018), the two definitions are not interchangeable and can give opposite results, highlighting that di GLYPH<11> erent processes dominate at the di GLYPH<11> erent scales probed by the di GLYPH<11> erent definitions. \nAs far as galaxy fractions are concerned, we find that regardless of the environmental definition adopted the fraction of blue / star-forming galaxies is systematically higher in the field / least dense regions than in the virial regions of clusters / highest densities. This e GLYPH<11> ect is less significant in the highest redshift bin analysed. Our results are overall in line with what was previously found in the literature, both considering the global (e.g. Iovino et al. 2010; Muzzin et al. 2012) and local (e.g. Balogh et al. 2004a; Cucciati et al. 2017) environments. Similarly, the overall SFR-M ? relation also seems not to depend on the parametrisation adopted, which agrees with numerous literature results that claim the invariance of SFR-M ? relation on environment (e.g Peng et al. 2010). \nNonetheless, the two definitions of environment lead to different results when we analysed the fraction of galaxies in transition. In fact, using the local environment the fraction of galaxies below the main sequence is similar at low and high density, whereas in clusters (and especially in their virial regions) a population with reduced SFR with respect to the field is observed. This population is most likely in a transition phase of star formation and, although clearly detected, it is not able to a GLYPH<11> ect the whole SFR-M ? relation because it constitutes a small fraction of all galaxies, as shown in Tab. 4.', '6.1. Galaxies in transition in the different environments and their evolution with redshift': 'The presence of a population of galaxies in transition from being star forming to passive was already detected in galaxy clusters by several works at low and intermediate redshifts (Patel et al. 2009; Vulcani et al. 2010; Paccagnella et al. 2016), and has been interpreted as an evidence for a slow quenching process preventing a sudden relocation of galaxies from the star forming to the red sequence. \nIn the previous sections, it was not possible to investigate the evolution of the incidence of transition galaxies, as a different mass complete limit was adopted at each redshift. Now we consider instead the same mass limit, to allow for fair comparisons. We adopt the most conservative value, that is the mass completeness limit in the highest redshift bin. Fractions are given in parenthesis in Tab. 4. \nFigure 12 shows the fraction of galaxies in transition in the redshift range 0.1 GLYPH<20> z GLYPH<20> 0.5 considering the global and local environments. \nThe upper panel shows that in the case of global environment the overall fraction of transition galaxies with log( M ?= M GLYPH<12> ) > 10 : 8 does not significantly vary with cosmic time, remaining around GLYPH<24> 15%, both in the field and among outer members. In contrast, virial members present higher transition fractions with a tentative increase as time goes by, although uncertainties prevent us from drawing solid conclusions. \nThe same Figure also compares our results to those obtained at low redshift (z . 0.1) by Paccagnella et al. (2016), when the subsample of their cluster galaxies within 1 r 200 and with stellar masses M ? GLYPH<21> 10 10 : 8 M GLYPH<12> is considered. The resulting transition fraction weighted for incompleteness is 0.30 + 0 : 04 GLYPH<0> 0 : 03 , that is consistent with our results within the error bars and point towards the aforementioned increase in the transition fractions at more recent epochs. \nIn contrast, the lower panel of Fig. 12 shows no dependence of the transition fraction with redshift for galaxies located at different local densities, further demonstrating that the local environment does not a GLYPH<11> ect the incidence of such population. \nFig. 12: Fraction of galaxies in transition at 0.1 GLYPH<20> z GLYPH<20> 0 : 5 considering the global (top) and local (bottom) parametrisation Fractions are computed for log M = M GLYPH<12> GLYPH<21> 10 : 8, the stellar mass completeness limit at 0.3 GLYPH<20> z GLYPH<20> 0.5. Error bars on the fractions are computed via bootstrapping. In the top panel, the blue star represents the fraction of transition galaxies in the local universe, adapted from Paccagnella et al. (2016). \n<!-- image --> \nEvidently, the two parametrisations are able to probe different physical conditions for galaxies, determining di GLYPH<11> erent timescales in the star formation process and quenching timescales.', '6.2. Star-forming versus colour fractions in the different environments': 'In the previous sections we have separately analysed the dependence of the star-forming and blue galaxy fractions on the global and local environments. In both analyses a di GLYPH<11> erence between the star-forming and blue fractions emerged, wherein the former is systematically higher than the latter. We stress that this di GLYPH<11> erence is not likely to be due to the definition we adopted \nVirial (NS) Virial (S) Outer (NS) Outer (S) Field \n<!-- image --> \nFig. 13: Ratio of the fraction of star-forming (F S Fing ) to blue (F blue ) galaxies in the mass limited sample in the three redshift bins and in di GLYPH<11> erent global (top) and local (bottom) environments. Dashed lines in the top panel show trends when AGNs are removed form the sample as explained in the Appendix B. In both panels, error bars are computed by propagating the asymmetric errors on the single fractions by means of the statistical error propagation. \n<!-- image --> \nfor determining the two populations: as previously described in Sect. 3, the sSFR and colour threshold adopted for defining the star-forming and blue populations are physically motivated by the distribution of the galaxy samples in the sSFR-M ? plane and by the rest-frame colour distribution at di GLYPH<11> erent redshifts. We further explored whether a choice of di GLYPH<11> erent cuts either on the sSFR and on the (g-r) rest-frame colour led to more similar galaxy fractions and concluded that the resulting sSFR and / or colour threshold to apply to the population in order to reconcile the fractions were totally non-physical. \nAs already anticipated in Sect. 3, the two quantities present intrinsic di GLYPH<11> erences related to the tracers they are based on: the SFR is derived from the measure of the flux of emission lines sensitive to the short-lived massive stars, while avoiding as much as possible contributions from evolved stellar populations. It is basically able to probe the presence of newly or recently formed stars on timescales of GLYPH<24> 10-100 Myr. On the contrary, galaxy integrated colours are more sensitive to the integrated star formation history and in particular to the stellar populations dominating the galaxy light, and are further influenced by the dust content and metallicity of the galaxy. With this in mind, we can expect a good agreement between galaxy rest-frame colours and SFR \nindicators when the galaxy is actively forming stars at a steady rate on the main sequence or, conversely, when it is quiescent and has been passively evolving for some Gygayear. Di GLYPH<11> erences between the two tracers may be expected for example when the galaxy suddenly interrupts its star formation activity as a consequence of the interactions with external physical mechanisms (e.g. environmentally related phenomena). \nWe are now in the position of directly comparing the fraction of star-forming and blue galaxies with the intent of obtaining some clues regarding the physical processes occurring in the di GLYPH<11> erent environments. \nFigure 13 shows the ratio of the number of star forming to that of blue galaxies as a function of global (top panel) and local (bottom panel) environment, above the stellar mass completeness limit of each redshift bin. In the upper panel of the figure, a strong dependence of the FSFing = Fblue ratio on the global environment emerges. At 0.1 GLYPH<20> z GLYPH<20> 0.3, this ratio is highest in the virial regions of clusters, while it decreases in the other environments with little di GLYPH<11> erence found between cluster outskirts and the field. Moving towards higher redshift, uncertainties prevent us from drawing solid conclusions, but still a hint of a higher FSFing = Fblue ratio within the virial radii of clusters than the other environments is visible. \nIn principle, this result might be contaminated by the presence of AGNs, and in particular LINERS, that could be misclassified as red star-forming galaxies. The dashed lines in Fig. 13 show the FSFing = Fblue ratios after AGNs have been removed (see Appendix B) and that this population cannot be responsible for the observed trends. \nOur results suggest that in the innermost regions of clusters, besides the suppression of the star formation activity, further environmentally related physical processes come into play to produce a population of galaxies with a non-negligible SFR that however is not coupled with (blue) rest-frame colours. \nThis decoupling is most likely due to the di GLYPH<11> erent star formation histories that characterise galaxies in the di GLYPH<11> erent global environments. Indeed, Guglielmo et al. (2015) found that the star formation history of low-redshift star-forming galaxies has been decreasing since z GLYPH<24> 2, and in particular the rate at which stars were produced in galaxies in clusters at high-z is higher than in the field, regardless of their stellar mass. This implies that, on average, star-forming galaxies in clusters formed the bulk of their stellar mass at older epochs than their counterparts in the field. Thus these star forming galaxies host older stellar populations which have redder colours, although these galaxies still are forming stars at the epoch of observation. \nAlternatively, the presence of a population of red starforming galaxies may be also associated with a dust obscured star formation phase. Gallazzi et al. (2009) quantified that nearly 40% of the star-forming galaxies in a supercluster at z GLYPH<24> 0.17 (Abell 901 / 902) had red optical colours at intermediate and high densities. These red systems have sSFR similar to or lower than blue star-forming galaxies, thus they are likely undergoing gentle mechanisms that perturb the distribution of gas inducing star formation (but not a starburst) and at the same time increase the gas / dust column density. \nThe incidence of the red star forming population is instead less dependent on the local environment: the lower panel of Fig. 13 shows no strong trends of the FSFing = Fblue ratio with LD at any redshift, also because of the large uncertainties, especially at higher redshifts. \nThese trends prove, once again, that the two environmental parametrisations are probing galaxies in di GLYPH<11> erent physical conditions, and that they cannot be used interchangeably. Indeed, there \nis no constant direct correspondence between the cluster cores and the highest LD regions and, similarly, between the lowest LD regions and the field.', '7. Conclusions': 'In this work, we have conducted a study on the stellar population and star formation properties of galaxies in the range 0 : 1 GLYPH<20> z GLYPH<20> 0 : 5, by making use of two definitions of environment. When considering the global environment, we divided galaxies into cluster virial and outer members and the field. We also distinguished between clusters that belong or do not belong to a supercluster. When considering the local environment, we characterised galaxy properties as a function of the projected LD. \nThe main observables we considered for investigating galaxy properties in di GLYPH<11> erent environments are the fraction of starforming / blue galaxies, defined on the basis of the sSFR and colour, respectively, and the correlation between the SFR and stellar mass. The main results can be summarised as follows.', 'Fraction of star-forming and blue galaxies': 'Considering the global environment, both in the magnitude and in the mass limited samples, cluster virial members reveal a deficiency of star-forming / blue galaxies with respect to all other environments at all redshifts, while field galaxies are the most starforming / blue population at all redshifts. Outer members exhibit a significant suppression of the star-forming / blue fractions with respect to the field only at 0.1 GLYPH<20> z < 0.2, while at higher redshift they present similar fractions. Overall, no significant di GLYPH<11> erences are detected between galaxies within and outside superclusters. \nConsidering the LD instead, the star-forming / blue fraction steadily decreases with increasing density only at 0.1 GLYPH<20> z < 0.2. At higher redshift, the fractions show a qualitatively similar dependence on density for log( LD [ Mpc GLYPH<0> 3 ]) & 3, while at lower densities the trends slightly increase. \nRegardless of the parametrisation of the environment, starforming and blue fractions are never consistent within the errors, probing that the two quantities reflect di GLYPH<11> erent aspects of the evolution of the galaxies. The star-forming to blue ratio is much higher in the cluster virial regions than in the field, most likely because of the di GLYPH<11> erent star formation histories of the galaxies in the di GLYPH<11> erent global environments.', 'SFR-Mass relation': 'Above the mass completeness limit, at all redshifts and considering both parametrisation of environment, galaxies in the virial / densest regions and galaxies in the field / less dense regions occupy the same locus of the plane, indicating no strong environmental e GLYPH<11> ects at play. Comparing the galaxies at di GLYPH<11> erent redshifts, at fixed stellar mass we recover the well-known decline in SFR with time. At any given redshift, the median SFR as a function of mass is similar in all environments. Nonetheless, an important di GLYPH<11> erence emerges between the global and local parametrisations. When using the former, a population of galaxies with reduced SFR compared to the expected value given their stellar mass is detected in the cluster virial regions. These are likely to be in transition from star forming to passive. Their incidence increases going from the higher towards lower redshifts. Such a population is not detected when comparing the SFR-mass relation of galaxies in two extreme bins of LD. \nThis dichotomy emerging in the galaxy properties when investigated in either a global or local environment framework are intrinsically related to the di GLYPH<11> erent physical meaning of the two parametrisations. The potential well of X-ray groups and clusters must enhance physical processes related to the presence of the dark matter halo and the hot intra-cluster medium on one side, whereas high-LD regions select associations of galaxies which are physically close and thus more prone to interactions and encounters with other galaxies. \nWhether these two definitions insinuate di GLYPH<11> erences in the star formation histories of the involved galaxy populations will be investigated in detail in Guglielmo et al. (in preparation). In fact, the availability of full spectral fitting results on the galaxy sample explored in this paper enables us to follow a complementary approach, and trace the histories of individual galaxies to examine how the SFH proceeded in X-ray clusters, in the field and in high-/ low- local overdensities of galaxies. This technique was already exploited in Guglielmo et al. (2015) in a low-redshift sample of galaxies in clusters and in the field, which can then be used as basis for comparison with the local Universe population. \nAcknowledgements. We acknowledge Lucio Chiappetti for his careful technical report, which guarantees the conformity of all the XXL papers and helped us to improve our work. XXL is an international project based around an XMM Very Large Programme surveying two 25 deg 2 extragalactic fields at a depth of GLYPH<24> 5 GLYPH<2> 10 GLYPH<0> 15 erg s GLYPH<0> 1 cm GLYPH<0> 2 in the [0.5-2] keV band for point-like sources. The XXL website is http: // irfu.cea.fr / xxl. Multi-band information and spectroscopic follow-up of the X-ray sources are obtained through a number of survey programmes, summarised at http: // xxlmultiwave.pbworks.com / . The Australia Telescope Compact Array is part of the Australia Telescope National Facility, which is funded by the Australian Government for operation as a National Facility managed by CSIRO. GAMA is a joint European-Australasian project based around a spectroscopic campaign using the Anglo-Australian Telescope. The GAMA input catalogue is based on data taken from the Sloan Digital Sky Survey and the UKIRT Infrared Deep Sky Survey. Complementary imaging of the GAMA regions is being obtained by a number of independent survey programmes including GALEX MIS, VST KiDS, VISTA VIKING, WISE, Herschel-ATLAS, GMRT, and ASKAP providing UV to radio coverage. GAMA is funded by the STFC (UK), the ARC (Australia), the AAO, and the participating institutions. The GAMA website is http: // www.gama-survey.org / . 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M., Brough, S., et al. 2012, MNRAS, 423, 3679 \nArticle number, page 18 of 22 \n- 1 Max-Planck-Institut für Extraterrestriche Physik, Giessenbachstrasse, 85748 Garching, Germany e-mail: [email protected]\n- 2 INAF-Osservatorio Astronomico di Padova, Vicolo Osservatorio 5, 35122 Padova, Italy\n- 3 Observatoire de la Côte d'Azur, CNRS, Laboratoire Lagrange, Bd de l'Observatoire, Université Côte d'Azur, CS 34229, 06304 Nice cedex 4, France\n- 4 Instituto de Radioastronomia y Astrofisica, UNAM, Campus Morelia, A.P. 3-72, C.P. 58089, Mexico\n- 5 INAF, Osservatorio Astronomico di Bologna, Via Gobetti 93 / 3, 40129 Bologna, Italy\n- 6 Center for Extragalactic Astronomy, Department of Physics, Durham University,South Road,Durham DH1 3LE, UK\n- 7 Aix Marseille Université, CNRS, LAM (Laboratoire d'Astrophysique de Marseille) UMR 7326, F-13388, Marseille, France\n- 8 AIM,CEA,CNRS,Université Paris-Saclay, Université Paris Diderot, Sorbonne Paris Cité, 91191 Gif-sur-Yvette, France", 'Appendix A: Spectroscopic completeness': 'In this section we describe in detail the methodology we applied to compute the spectroscopic completeness of our sample. To properly account for the redshift dependence of the completeness ratio, in addition to considering separately three redshift bins (0.1 GLYPH<20> z < 0.2, 0.2 GLYPH<20> z < 0.3, 0.3 GLYPH<20> z GLYPH<20> 0.5), we base our procedure on the combined use of spectroscopic and photometric redshifts. Specifically, in each redshift bin, the sampling rate (SR) is defined as the ratio of the number of objects with spectroscopic redshift to the number of possible targets (i.e. the photo-z sample). \nThe steps taken to compute the completeness can be summarised as follows. Considering the galaxies in the spectrophotometric sample with reliable measurements of both spectroscopic and photometric redshift, and defined a redshift range of interest, we call \n- -N 11 = the number of objects with spectroscopic redshift in the selected redshift range, and photo-z in the same range.\n- -N 12 = the number of objects with spectroscopic redshift in the selected redshift range, but photo-z not in the same range.\n- -N 21 = the number of objects with spectroscopic redshift out of the selected redshift range, but photo-z in the range.\n- -N 22 = the remaining number of objects with both spectroscopic and photo-z outside the selected redshift range. \nThese numbers are used to define the two fractions that allow us to compute the expected number of objects relative to the entire photo-z sample, which also includes galaxies with no spectroscopic redshift, starting from the spectrophotometric sample, \nf 1 = N 11 ( N 11 + N 21) (A.1) \nis the fraction of all objects with photo-z in the selected redshift range that truly belong to the range (i.e. with spectroscopic redshift in the range). Then, \nf 2 = N 12 ( N 12 + N 22) (A.2) \nis the fraction of all objects with photo-z outside the range that are instead within the considered redshift bin (i.e. with spectroscopic redshift in the range). These objects should be considered in the SR estimate of the given redshift slice even if their photo-z would not include them. \nThese two fractions are finally used to estimate the number of expected photo-z objects in the range, when applied to the whole photo-z sample, \nNexp = f 1 GLYPH<2> Nphoto GLYPH<0> z ; in + f 2 GLYPH<2> Nphoto GLYPH<0> z ; out (A.3) \nWhere the numbers Nphoto GLYPH<0> z ; in and Nphoto GLYPH<0> z ; out refer, respectively, to the number of objects with photo-z in and outside the selected redshift range in the total photo-z sample. \nThe sampling rate is finally defined as \nSR = ( N 11 + N 12) Nexp (A.4) \nwhere ( N 11 + N 12) is the total number of galaxies with spectroscopic redshift in the selected redshift range. \nBy construction, the sum of the inverse of the SRs, i.e. the spectroscopic weights, at all redshifts and in the magnitude limited sample approximately gives the number of objects in the \nmagnitude limited parent photo-z sample; small di GLYPH<11> erences can be due to the di GLYPH<11> erent redshift range covered by the spectroscopic and photo-z sample. \nTo account for the dependence on the di GLYPH<11> erent SR of spectroscopic surveys in di GLYPH<11> erent regions in the sky, we proceed as already performed in XXL Paper XXII and subdivide the field in three stripes of declination and we further divide each stripe in RA creating a grid of 1.0 deg width. Finally, we consider intervals of 0.5 r -band observed magnitude in the 22 resulting cells (see Fig 2). \nFrom these results, we obtain the spectroscopic completeness curves as the SR as a function of magnitude, in all the sky cells and redshift bins in which the sample has been divided. \nSuch curves are shown in Figures A.1. \nEach galaxy is finally weighted for the inverse of the SR computed as explained in above, which accounts for its redshift, position in the sky, and observed magnitude.', 'Appendix B: AGN contamination': 'To compute the number of (broad and narrow lines) AGNs in our galaxy sample we consider all galaxies having a GAMA spectrum (98% of all galaxies in the magnitude complete sample), and perform our classification using the DR3 GAMA catalogues which include the region overlapping with the XXL-N field (GAMA-G02, Baldry et al. 2018). \nThe catalogues contain emission line measurements derived with Gaussian fits of di GLYPH<11> erent complexity to the lines as well as specific parameters used to identify peculiar galaxies. First, we remove duplicates in the spectra and selected the best-observed spectrum with the flag IS\\_BEST = True in the catalogues, and we consider only reliable redshift measurements (NQ GLYPH<20> 3). We proceed separately for broad- and narrow-line AGNs as follows: \n- -Broad-line AGN: These objects can be simply identified by means of the output of the model selection (HA\\_MODSEL\\_EMB\\_EM in the GaussFitComplex Table), as explained in Sect. 2.3 in Gordon et al. (2017). Broad-line AGNs are characterised by a value of the model selection score parameter greater than 200, and have a S / N on the broad H GLYPH<11> component > 3 (HA\\_B\\_FLUX / HA\\_B\\_FLUX\\_ERR > 3 in the GaussFitComplex Table).\n- -Narrow-line AGN: For these objects, we rely on the table containing simple Gaussian fit to the spectral lines (GaussFitSimple), and consider the classification of Gordon et al. (2018) based on line ratios. We select only spectra with reliable S / N on the interesting lines (NIIR\\_FLUX / NIIR\\_FLUX\\_ERR > 3 for the NII line, and HA\\_FLUX / HA\\_FLUX\\_ERR > 3 for the H GLYPH<11> line), and correct the H GLYPH<11> line for stellar absorption applying Eq 5 from Gordon et al. (2017) (a GAMA specific version of Eq 4 from Hopkins et al. 2003) as follows: \nFcor = EW + 2 : 5Å EW Fobs (B.1) \nwhere F cor is the corrected flux measurement, F obs is the observed flux measurement (HA\\_FLUX in the GaussFitComplex Table), and EW is the measured equivalent width of the emission line (HA\\_EW in the GaussFitComplex Table). Finally, we classify as likely narrow-line AGN those galaxies having log10([NII], GLYPH<21> 6583 / H GLYPH<11> ) > 0.4. We point out that this classification based on two emission lines is the most conservative and may \n<!-- image --> \n<!-- image --> \nFig. A.1: Completeness curves computed in three redshift bins and in di GLYPH<11> erent RA-DEC cells in the sky, as explained in the main text. From the top to the bottom panel, the represented redshift ranges are respectively 0 : 1 GLYPH<20> z < 0 : 2, 0 : 2 GLYPH<20> z < 0 : 3, 0 : 3 GLYPH<20> z GLYPH<20> 0 : 5. \n<!-- image --> \nalso include normal galaxies with high SFRs, making the AGN contribution evaluated in this work an upper limit. \nHaving classified and flagged broad- and narrow-line AGN, we crossmatch the catalogue of spectra with our spectrophotometric catalogue and compute their upper limit fraction with respect to the number of star-forming galaxies in the three usual redshift bins and in the magnitude complete sample (similar fractions are also found in the mass limited sample): \n- - 0.1 GLYPH<20> z < 0.2: 762 / 5026 = 15.2%\n- - 0.2 GLYPH<20> z < 0.3: 1166 / 5817 = 20.04%\n- - 0.3 GLYPH<20> z < 0.5: 372 / 3047 = 12.2%', 'Appendix C: Local density': 'We compute the LD of galaxies in the spectrophotometric sample taking as a reference the photo-z sample used in the spectroscopic completeness computation, and considering one redshift bin at a time. The LD around each galaxy is given as the ratio of the number of galaxies in the parent photometric-redshift sample per unit of projected comoving area on the sky. Our method proceeds through the following di GLYPH<11> erent phases: \n- - Computation of the observed magnitude limit used to select galaxies in the sample as a function of redshift. To perform the same sample selection, we apply the same absolute magnitude cut in all the redshift slices. The value is selected in order to balance the error in the photo-z estimate, which increases towards fainter magnitudes, and the propagation of the observed magnitude down to redshift 0.1, and thus to minimise the loss of galaxies occurring with brighter observed magnitude cuts. We consider as observed magnitude limit r = 23.0 at z = 0.5 and compute the corresponding absolute magnitude as follows: \nMr = r GLYPH<0> 5 GLYPH<1> ( log 10 DL GLYPH<0> 1) GLYPH<0> Kcorr : (C.1) \nwhere r is the observed r-band magnitude and DL is the luminosity distance in pc. The value Kcorr is the K-correction that takes into account that the same photometric filter samples di GLYPH<11> erent spectral ranges when applied to the SED of galaxies at di GLYPH<11> erent redshifts and is taken from Poggianti (1997), assuming the typical value of an intermediate type galaxy (Sab) in r band at the selected redshift. The application of this formula leads to an absolute magnitude of M r = -19.89, which is then converted into an observed magnitude limit as a function of redshift by means of the inverse formula, \nr ( z ) = GLYPH<0> 19 : 89 + 5 GLYPH<1> ( log 10 DL ( z ) GLYPH<0> 1) + Kcorr ( z ) + P : E : ( z ) (C.2) \nwhere the DL is computed at the redshift of the considered galaxy, Kcorr is a function of redshift, and P.E.(z) is the passive evolution of galaxies, which becomes redder with decreasing redshift as a consequence of the ageing of their stellar population; the correction for passive evolution is 0.1 mag each GLYPH<1> z = 0.1 (Poggianti et al. 2008). \n- - Computation of the number of galaxies in the spectrophotometric sample within a comoving circle of 1 Mpc radius at the redshift of the galaxy in the centre and within a redshift range of GLYPH<6> 0 : 05 with respect to the redshift of the same galaxy. To account for uncertainties in the photo-z measurements, we estimate the expected number of galaxies in the photo-z sample in the considered redshift range around the selected galaxy with the same method used for the spectroscopic completeness. We define the fractions f 1 and f 2 \ngiven in equations A.1 and A.2 in the spectrophotometric sample and use them to weight the photo-z sample and compute Nexp . This value represents the correct number counts within the comoving projected area of 1 Mpc radius around the galaxy. The area of the circle is then computed and the LD is defined as the ratio of the two quantities. \n- - Correction for edge e GLYPH<11> ects in the field. For galaxies located at the edges of the XXL-N field, we correct the circular area for the fraction of area e GLYPH<11> ectively covered by the data points, and therefore remove empty circular sectors. We adopt a numerical solution based on a Monte Carlo simulation method. We generate a circular homogeneous distribution of data points by populating a circle with a su GLYPH<14> ciently high number of points (100000) and compute the zone of exclusion with respect to the edge conditions of the field as the ratio of the number of points falling outside the edges to the total number of points included in the circle. The area of the circle in physical units that has to be considered in the LD calculation is then the total comoving area multiplied by the fraction of area included in the field, fin = 1 GLYPH<0> fout, where fout is the fraction of area falling outside the galaxy field. \nThe LD is finally expressed as the logarithm of the quantity computed in the procedure outlined above, with dimension [LD] = Mpc GLYPH<0> 2 . \nFigure C.1 reports the spatial distribution of galaxies in the spectrophotometric sample colour coded for the LD measures. Each panel also reports the circle of 3 r 200 radius of the clusters in each redshift bin; as expected, in most of the cases, galaxies within the circles are characterised by high LD values. \nFig. C.1: Spatial distribution in the sky of the spectrophotometric magnitude limited sample. Data points are colour coded according to their log(LD), after a sigma-clipping has been performed on the parent distribution. From the top to bottom panel the represented redshift bins are 0 : 1 GLYPH<20> z < 0 : 2, 0 : 2 GLYPH<20> z < 0 : 3, 0 : 3 GLYPH<20> z GLYPH<20> 0 : 5, respectively. Each panel contains the 3r200 extensions of the clusters at the redshift of the bin, represented with black empty circles. \n<!-- image -->'} |
2024MNRAS.534.1816F | Comet 12PPonsBrook exhibited multiple large and minor outbursts in 2023 on its way to its 2024 perihelion as it has done during its previous apparitions. We obtained longslit optical spectra of the comet in 2023 August and November with the INTIDS and in 2023 December with NOTALFOSC. Using a standard Haser model in a 10 000kmradius aperture and commonly used empirical parent and daughter scale lengths our calculated abundance ratios show a constant typical composition throughout the period with a Cinlineformulatexmath idTM0001 notationLaTeX2texmathinlineformulaCN ratio of about 90 per cent. Molecular density profiles of different species along the slit show asymmetries between opposite sides of the coma and that Cinlineformulatexmath idTM0002 notationLaTeX2texmathinlineformula seems to behave differently than CN and Cinlineformulatexmath idTM0003 notationLaTeX3texmathinlineformula. Comparing the coma profiles to a standard Haser model shows that this model cannot accurately reproduce the shape of the coma and therefore that the calculated production rates cannot be deemed as accurate. We show that an outburst Haser model is a slightly better match to the Cinlineformulatexmath idTM0004 notationLaTeX3texmathinlineformula and CN profile shapes but the model still does not explain the shape of the Cinlineformulatexmath idTM0005 notationLaTeX2texmathinlineformula profiles and requires equal parent and daughter scale lengths. Our results suggest that the coma morphology could be better explained by extended sources and that the nature of 12Ps activity introduces bias in the determination of its composition. | 2024-11-01T00:00:00Z | ['10.1093/mnras/stae2189', '2024MNRAS.534.1816F', '2024MNRAS.tmp.2143F', '10.48550/arXiv.2409.08133', '2024arXiv240908133F', 'arXiv:2409.08133'] | ['Astrophysics - Earth and Planetary Astrophysics'] | Coma composition and profiles of comet 12PPonsBrooks using longslit spectroscopy | 2,024 | 172 | 0.52 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF'] | 0 | https://arxiv.org/pdf/2409.08133.pdf | {'Coma composition and profiles of comet 12P/Pons-Brooks using long-slit spectroscopy': 'Lea Ferellec 1 ★ , Cyrielle Opitom 1 , Abbie Donaldson 1 , Johan P. U. Fynbo 2 , Rosita Kokotanekova 1 , 3 , Michael S. P. Kelley 4 , Tim Lister 5 \n- 1 Institute for Astronomy, University of Edinburgh, Royal Observatory, Edinburgh, EH9 3HJ, UK\n- 2 Cosmic DAWN Center, Niels Bohr Institute, University of Copenhagen, Jagtvej 155, 2200 Copenhagen N, Denmark\n- 3 Institute of Astronomy and National Astronomical Observatory, Bulgarian Academy of Sciences, 72 Tsarigradsko Shose Blvd., Sofia 1784, Bulgaria\n- 4 Department of Astronomy, University of Maryland, College Park, MD 20742, USA\n- 5 Las Cumbres Observatory, 6740 Cortona Drive, Suite 102, Goleta, CA 93117, USA \nAccepted XXX. Received YYY; in original form ZZZ', 'ABSTRACT': 'Comet 12P/Pons-Brook exhibited multiple large and minor outbursts in 2023 on its way to its 2024 perihelion, as it has done during its previous apparitions. We obtained long-slit optical spectra of the comet in 2023 August and 2023 November with the INT-IDS, and in 2023 December with NOT-ALFOSC. Using a standard Haser model in a 10000km-radius aperture and commonly used empirical parent and daughter scale-lengths, our calculated abundance ratios show a constant "typical" composition throughout the period with a C2/CN ratio of about 90 per cent. Molecular density profiles of different species along the slit show asymmetries between opposite sides of the coma and that C2 seems to behave differently than CN and C3. Comparing the coma profiles to a standard Haser model shows that this model cannot accurately reproduce the shape of the coma, and therefore that the calculated production rates cannot be deemed as accurate. We show that an outburst Haser model is a slightly better match to the C3 and CN profile shapes, but the model still does not explain the shape of the C2 profiles and requires equal parent and daughter scale-lengths. Our results suggest that the coma morphology could be better explained by extended sources, and that the nature of 12P\'s activity introduces bias in the determination of its composition. \nKey words: comets: individual: 12P', '1 INTRODUCTION': "Comet 12P/Pons-Brooks is a Halley-type comet discovered in 1812, returning to the inner Solar System every 71 years from just beyond the orbit of Neptune on a high-inclination trajectory. During its 1883 and 1954 apparitions, 12P exhibited multiple outbursts (Chandler 1883; Porter 1955). Approaching its 2024 April 21 perihelion, 12P was recovered in 2020 July (Ye et al. 2020) when it was already active at 11au. On 2023 July 20, a large outburst was detected with the comet brightening from magnitude 17 to 12 1 . The time of this outburst was then refined to 2023 July 19.57 (Manzini et al. 2023a) with a second minor outburst on 2023 July 20.83 (Manzini et al. 2023b). Since this event 12P has become a target of great interest, and both large and small outbursts were reported along its journey, such as on 2023 September 4 (Usher et al. 2023a), on 2023 September 22-25 (Kelley et al. 2023), on 2023 October 5 (Usher et al. 2023b), on 2023 November 14 (Jehin et al. 2023c), on 2023 December 12-14 (Jehin et al. 2024a) and on 2024 February 29 (Jehin et al. 2024b). \nCometary outbursts are sudden increases in mass loss (Hughes 1990) which are thought to be caused by structural failure (e.g. frag- \n- ★ E-mail: [email protected] \nmentation or internal gas reservoirs bursting; Boehnhardt & Binzel 2004; Müller et al. 2024), surface disruptions (e.g. cliff-collapse or impacts; Pajola et al. 2017; Guliev et al. 2022), or physico-chemical processes (e.g. water-ice crystallisation; Patashnick 1974) leading to internal pressure building up (Agarwal et al. 2017). \nThe recent in-situ study of comet 67P/Churyumov-Gerasimenko allowed extensive comparison between topography and outbursts. For 67P, outbursts have been associated to events like cliff-collapse, and correlated with steep slopes and day-night cycles (Vincent et al. 2016). The frequency of large outbursts during each of 12P's approaches suggest that something about its structure, surface or physico-chemical composition is causing its behaviour, rather than external impactors (Gronkowski 2004). Using observations from 2024 February, Knight et al. (2024) determined a rotation period of 57 ± 1hr, much longer than required for rotation-induced fragmentation assuming a typical shape and density (Lowry & Weissman 2003; Kokotanekova et al. 2017). \nObservations of 12P during its 2024 approach might help us better understand its structure, activity and make-up. This paper focuses on the composition of the gas released by 12P and its spatial distribution. We obtained long-slit optical spectra of the comet at four different epochs to study abundances and release mechanisms of small radicals in the coma.", '23/08/2023': '18/11/2023 \n<!-- image --> \n<!-- image -->', '17/11/2023': '11/12/2023 \n<!-- image --> \nFigure 1. Images of comet 12P from the LCO Outbursting Objects Key Project taken on each observing night, except for 2023 December 17 as the latest available observation was 2023 December 11. It should be noted that an outburst happened on 2023 December 13. On all images North is up and East is left. The slit orientation is represented by a white arrow. The direction of the arrow corresponds to increasing values of the x-axes on Figures 4 to 7. Blue contours are over-plotted to better highlight the morphology of the dust coma. Yellow and pink arrows represent the anti-solar direction and the inverse velocity direction respectively. The LCO data are described in section 2.3. \n<!-- image -->', '2.1 Isaac Newton Telescope data': "Long-slit spectra of 12P were acquired in 2023 August and 2023 November using the Intermediate Dispersion Spectrograph (IDS) on the INT with the EEV10 detector and a central wavelength of ∼ 4500Å. Grating R400B was used, which has a resolving power of 1596. With this configuration, the spectra cover the wavelength range of ∼ 3050 -6400Å, encompassing bright emission lines from OH, NH, CN, C 2 and C 3 (listed in Table 2). The instrument has a total slit length of 3 . 3'. A slit-width of 2' was used and the instrument has a spatial resolution of 0 . 4'/pixel along the slit. Three 1200s exposures were acquired on 2023 August 23 23:12 UT (all times quoted are the mid-times for the observations), when the target was at Δ ≈ 3 . 3au and 𝑟 ℎ ≈ 3 . 5au. Variations in cloud coverage at the time contribute to increased uncertainties on the fluxes measured from this spectrum. One 900s spectrum and three 600s spectra were obtained consecutively on 2023 November 17 20:31UT, as well as four 600s spectra on 2023 November 18 20:14UT, with clear sky conditions. The target was at Δ ≈ 2 . 7au and 𝑟 ℎ ≈ 2 . 5au. The slit was aligned with the parallactic angle for all of the spectra. Fig 1 shows images of 12P from the LCO Outbursting Objects Key Project on which we represented our slit orientations. On 2023 November 18, one extra 600s exposure was taken with the slit perpendicular to the parallactic angle to compare coma profiles along different directions, but it was not used for production rate calculations since this orientation is more \nTable 1. Summary of our observations (telescope/instrument, number of exposures and exposure time, heliocentric distance 𝑟 ℎ and geocentric distance Δ ). More details can be found in sections 2.1 and 2.2. † : On 2023 November 18, a fifth 600s-exposure spectrum was taken but with a different slit-orientation. It was not used in the final analysis. \nTable 2. Wavelength ranges used to measure the total flux in prominent emission lines of different species detectable in comet gas spectra, and parent/daughter photo-dissociation scale-lengths 𝐿 𝑝 and 𝐿 𝑑 used in the Haser model. References for the scale-lengths: a. Cochran & Schleicher (1993), b. Randall et al. (1992). \nprone to flux losses from atmospheric refraction, as the INT-IDS does not have an atmospheric dispersion corrector. \nWavelength calibration was performed using spectra of ArNe lamps. The observations of the comet were reduced with bias and flat-field frames taken on each of these nights and were corrected for atmospheric extinction. Atmospheric contamination was removed using spectra of the sky taken 10' away from the target in between target exposures. Flux calibration was performed using observations of the spectrophotometric standard star BD+284211 compared to reference spectra. For 2023 November, the flux calibration functions are consistent with our previous INT-IDS observing runs in good weather conditions. Therefore, for each molecular emission region, we estimated the uncertainty associated to flux calibration as the standard deviation of the flux calibration functions between all these observing runs. For the data from 2023 August the standard star was only observed the night before observing the target which, combined with variable cloud coverage during these nights, causes a large uncertainty on the flux calibration. However this should affect the overall amplitude of the spectrum more than the relative intensities between lines. For this spectrum we estimated the flux calibration uncertainty as the percentage of difference between the flux calibration function from August and the average flux calibration function from good-weather nights. \nThe dust contribution to the total comet flux was estimated and removed by adjusting a Sun-like spectrum multiplied by a polynomial slope to the regions in between the expected emission lines. We also excluded both ends of the spectra from the fitting regions because they suffer from increased noise as the illumination of the sensor decreases. We used spectra of the solar analogue HD186427 acquired during these runs. An order 3 polynomial was used, as it provides a satisfactory match around the emission lines of interest. We assessed the uncertainty associated to this dust removal process by modifying our dust-model by ± 5 per cent and measuring how much this affected the final flux measurements. \n<!-- image --> \n<!-- image --> \nFigure 2. Average INT/IDS dust-subtracted spectrum of 12P within a 10000km radius aperture on 2023 November 17, highlighting the detection of NH, CN, C3, C2 and NH2 emission lines. Colored areas denote the approximate extent of each band. The y-axis scales are different in the three subplots. Note that there is still evidence of the 5577Å sky emission line. \n<!-- image -->", '2.2 Nordic Optical Telescope data': "One 600s spectrum was acquired with the Nordic Optical Telescope on 2023 December 17 at 19:54UT using the ALFOSC instrument. The target was at Δ ≈ 2 . 4au and 𝑟 ℎ ≈ 2 . 2au. The spectrum covers a wavelength range of ∼ 3500 -5350Å. Grism 18 has a resolving power of 1000. The instrument has a total slit-length of 5 . 3'. A slit width of 1' was used and the instrument has a spatial resolution of 0 . 21'/pixel along the slit. The slit was aligned along the parallactic angle. Wavelengths calibration was performed using a spectrum of a ThAr lamp. Again, bias and flat-field frames were taken on the same night to reduce the observations of the comet, which were then corrected for atmospheric extinction. \nFlux calibration was performed using a spectrum of the spectrophotometric standard star Wolf1346. This time no offset sky observations were obtained, so gas emissions had to be isolated by adjusting a model of sky and dust together to the continuum. This model was a linear mix of the sky background from the standard star observation and a slope-adjusted composite solar analogue spectrum from our INT runs. \nThe target having been observed immediately after the standard star, we assumed that the weather conditions had not changed in between observations, so we used the calibration uncertainties from November as typical variation levels. Continuum removal uncertainties were assessed as described in section 2.1.", '2.3 Las Cumbres Observatory data': "Images of comet 12P were obtained with the Las Cumbres Observatory (LCO) global telescope network as part of the LCO Outbursting Objects Key Project (LOOK Project; program ID LTP2023B-001) and the Comet Chasers education and public outreach project (program ID FTPEPO2014A-004). Observations close in time to the spectroscopy were selected: 11 𝑟 ' images (720 s total) taken 2023 August 23 06:12 UTC from the Faulkes Telescope North (FTN) 2-m at Haleakala Observatory; 1 𝑅 -band image (60 s) taken 2023 November 17 19:12 from a 1-m telescope at Teide Observatory; 3 𝑟 ' images (45 s total) taken 2023 November 18 01:02 and 4 𝑟 ' images (110 s total) taken 2023 December 11 00:54 from 1-m telescopes at McDonald Observatory. The FTN observations used the MuSCAT3 camera, which simultaneously images through four filters, each filter illuminating a separate 2k × 2k CCD with 0.27 '' pix -1 (Narita \net al. 2020). The 1-m telescopes used Sinistro cameras, each having a 4k × 4k CCD with 0.389 '' pix -1 . The telescopes followed the comet using the non-sidereal rates from the ephemeris, and the images were combined together by epoch in the rest frame of the comet. Data were calibrated with LCO's BANZAI data pipeline and photometrically calibrated to the PS1 𝑟 -band using the calviacat software (McCully et al. 2018; Kelley & Lister 2019).", '2.4 Molecular production rates calculation': "Spectra were produced by integrating the flux within a 10000km radius from the nucleus. As an example, our average spectrum from 2023 November 17 is given in Figure 2. Total fluxes in emission lines of OH, NH, CN, C 3 , C 2 were calculated by integrating the flux within the ranges given in Table 2. For emission lines for which the signal is too faint to be detected we calculated 3 𝜎 upper-limits as described in Cochran et al. (2012). These fluxes were converted to total numbers of molecules using fluorescence factors from the Lowell Minor Planet Services 2 . Production rates 𝑄 of these species were then computed by matching these numbers of molecules to what would be observed in the case of a standard Haser model (Haser 1957): \n𝑛 ( 𝑟 ) = 𝑄 4 𝜋𝑣𝑟 2 𝐿 𝑑 𝐿 𝑝 -𝐿 𝑑 ( 𝑒𝑥𝑝 (-𝑟 𝐿 𝑝 ) -𝑒𝑥𝑝 (-𝑟 𝐿 𝑑 )) (1) \nwhere 𝑛 is the molecular volume density as a function of cometocentric distance 𝑟 , 𝐿 𝑝 and 𝐿 𝑑 the parent and daughter photodissociation scale-lengths, 𝑣 the expansion velocity of the gas. This model assumes that the daughter species travel with the same direction and velocity as the parents. As in A'Hearn et al. (1995), we used the scale-lengths relationships listed in Table 2, as well as a velocity of 𝑣 = 1km s -1 regardless of heliocentric distance. In reality, using a constant unity velocity yields measurements of 𝑄 / 𝑣 , but we will label our results 𝑄 from now on. The resulting production rates are presented and discussed in section 3.1. It should be noted that, if the model and scale-lengths do not accurately represent the coma, then the resulting production rates are non-physical and aperturedependent. Hence we chose a 10000km aperture to at least guarantee \nthat we can compare our measurements with other published results using this same aperture (see section 3.1 ). The validity of the standard Haser model for 12P is tested in section 3.3.2.", '2.5 Radial molecular column density profiles': 'From the spatial information contained in the long-slit observations, radial profiles of the column density of molecules in the coma were computed for species with the brightest emissions: CN, C 2 and C 3 . Wedid not compute profiles for OH as the emission line is at the very edge of the wavelength range covered by the instrument, making the dust-subtraction less reliable. \nTo produce these profiles, we removed the dust contribution from each individual spectrum at each location along the slit, allowing us to measure the fluxes within the gas emission ranges at each location, which we converted into column density versus nucleocentric distance. These profiles are shown and described in section 3.3. By binning the data we created smoothed profiles for visualisation only, but used the full data set for the least-square optimisation of the models described below. \nIn sections 3.3.2 to 3.3.4, we compare these profiles to the standard Haser model (eq. 1) as well as to an outburst model from Opitom et al. (2016) in which 𝑄 / 𝑣 from the standard Haser model is replaced by the following expression, which accounts for an exponential increase up to the outburst peak then an exponential decrease back to steadystate: \n𝑄 ( 𝑟 > 𝑣 1 Δ 𝑡 ) = 𝑄 0 𝑣 + 𝑄 1 𝑣 1 𝑒𝑥𝑝 (-𝑣 1 Δ 𝑡 -𝑟 𝑟 𝑎 ) (2) \n𝑄 ( 𝑟 < 𝑣 1 Δ 𝑡 ) = 𝑄 0 𝑣 + 𝑄 1 𝑣 1 𝑒𝑥𝑝 (-𝑟 -𝑣 1 Δ 𝑡 𝑟 𝑏 ) (3) \nwhere 𝑄 0 is the steady-state production rate, 𝑄 1 corresponds to the additional outburst gas release (so that at the peak of the outburst the total production rate is 𝑄 0 + 𝑄 1 ), 𝑣 1 the expansion velocity of the outburst material, 𝑟 𝑎 / 𝑣 1 and 𝑟 𝑏 / 𝑣 1 the respective characteristic timescales of the increase and decrease in activity during the outburst, and Δ 𝑡 is the time between the peak of the outburst and the time of observation. When adjusting these models to our data, we will alternatively refer to the best-fit outburst peak date instead of Δ 𝑡 .', '3.1 Gas Composition': 'The molecular production rates and abundance ratios that we calculated as explained in section 2.4 are given in Table 3. As a reminder, we are using the emission lines listed in Table 2. Following the 2023 November 14 outburst, we observe a rapid decrease of the production rates for all molecules ( ∼ 20-25 per cent for C 2 , C 3 , CN and 6 per cent for OH between November 17 and 18). A month later, the outgassing rates are similar to November 18 as the effects of the outburst have cleared but the comet has gotten closer to the Sun. \nOur abundance ratios indicate that 12P has a "typical" composition according to the classification made by A\'Hearn et al. (1995), as opposed to "C 2 -depleted" comets. However the C 2 /CN ratio is below the average for the "typical" class. It should be noted that we use the same scale-length relationships as A\'Hearn et al. (1995). Considering the C 2 -depletion condition of 𝑄 ( 𝐶 2 )/ 𝑄 ( 𝐶𝑁 ) < 77 per cent revised by Schleicher et al. (2003), our measurements show that 12P is close to the depletion limit, with an average C 2 /CN ratio \nFigure 3. Production rates of CN and C2 (top panel) and C2/CN abundance ratio (bottom panel) between 2023 July and 2024 March from our measurements (diamond markers) and preliminary values from TRAPPIST (Jehin et al. 2023a,b,c, 2024a,b) (circular markers). The arrow on the bottom panel represents an upper limit, as our C2 production rate for 2023 August 23 is an upper limit. Vertical lines indicate reported outbursts. On the bottom panel, a dashed horizontal line indicates the carbon-depletion threshold from the survey by A\'Hearn et al. (1995) updated by Schleicher et al. (2003). \n<!-- image --> \nof 91 . 3 ± 5 . 3 per cent in November-December. We do not detect any significant composition change while the outburst settles nor between November and December. \nOur results are also consistent with the preliminary production rates published by Jehin et al. (2023a,b,c, 2024a,b) using observations from the TRAPPIST telescopes. While their production rates derive from narrowband-filter images, most of them were calculated using the same parameters and aperture radius as our analysis (except for their measurements from September 25, September 27, October 03 and October 07 which use v=0.5km s -1 and a 100000km aperture radius). These measurements along with ours are represented in Figure 3, and show an overall increase of the production rates as the comet approaches the Sun, as well as an increase by more than a factor 10 throughout the November 14 outburst. The outburst then seems to rapidly settle. The TRAPPIST observations yield C 2 /CN ratios similar to our findings (91 . 7 ± 6 . 0 per cent on November 12 and 80 . 2 ± 4 . 3 per cent on November 15) but show that the ratio varies greatly across the semester (see Figure 3). While variations of C 2 /CN ratios with heliocentric distance have been reported by multiple studies (e.g. A\'Hearn et al. 1995; Langland-Shula & Smith 2011), the behaviour observed in 12P seems to differ from these trends which \nTable 3. Production rates calculated from the total fluxes within a 10000km aperture using a standard Haser model with v=1km s -1 , g-factors from the Lowell Minor Planet Services and photodissociation scalelengths from Table 2. Upper limits were calculated according to (Cochran et al. 2012). "-" indicates production rates that could not be measured (because the line is outside of the instrument\'s coverage) or ratios that could not be calculated (as the denominator is an upper limit). \nwould predict a smooth increase throughout the comet\'s approach and do not explain the decrease observed from July to September. \nOther studies have measured how outbursts affect the apparent composition of a comet, or how homogeneous the interior of comets seem to be. Schleicher et al. (2003) measured the same composition in multiple fragments of comet 73P/Schwassmann-Wachmann as before it fragmented. Dello Russo et al. (2008) studied comet 17P after a major outburst, which did not seem fragmentation-related, and also did not find any notable composition change. Opitom (2016) measured production rates of comets 168P/Hergenrother, C/2010 G2 (Hill), C/2012 S1 (Ison), and C/2013 A1 (Siding Spring) throughout outbursts, and while different mechanisms are believed to be the cause of these outbursts, they also did not detect composition changes. These studies suggest that these comets have a homogeneous composition, throughout their whole interior or at least in the outer layers. Our measurements along with the TRAPPIST ones also show no significant composition change through the November 14 outburst, however Jehin et al. (2023b,c) measure an increase of C 2 /CN after the October outburst. It could be that this is representative of the outburst material, however we will illustrate in section 3.3.2 how the atypical coma shape of 12P can introduce bias in the measurements of its composition.', '3.2 Dust spectrum properties': 'For the data from 2023 November, we computed the reflectance of the coma by dividing the non-dust-removed comet spectrum by the solar analogue spectrum. We calculated the dust reflectance slope by normalizing the reflectance at 5200Å then performing a linear regression on the data restricted to the ranges 4400-4500Å, 47504900Å, 5200-5320Å and 6100-6200Å as to avoid gas emission lines. Lower wavelengths were not included as the reflectance was noisier. We did not compute dust spectral slopes for the data from 2023 August as it has a significantly lower signal-to-noise ratio and suffers from bad weather conditions, nor for the data from 2023 December as the dust and sky contributions cannot be reliably separated. \nWe obtain a spectral slope of ( 4 . 5 ± 0 . 3 ) %/1000Å on average for 2023 November 17-18 between 4400 and 6200Å, where the uncertainty reflects the error on the mean among the 8 spectra considered. This is lower than the slopes of 15%/1000Å to 39%/1000Å measured by Storrs et al. (1992) in the 4400-5675Å range among 18 comets, or the average of ( 13 ± 5 ) %/1000Å measured by Jewitt & Meech (1986) over 3500-6500Å (9 comets). \nHowever we do see a steeper reflectance in the bluer end of our range. Using the 4400-4500Å and 4750-4900Å regions only we measure an average slope of ( 10 . 8 ± 0 . 6 ) %/1000Å (this time with the reflectance normalized at 4760Å). This is comparable to some of the spectral slopes measured by Hyland et al. (2019) between 4450 and 5260Å, with ( 13 ± 8 ) %/1000Å on average. As for other studies of this type, we find that the reddening of the reflectance gets lesser at higher wavelengths.', '3.3.1 General aspect': 'Column density profiles for CN, C 2 and C 3 are shown on Figure 4, using the same emission lines that were used for the composition measurements. Figure 5 presents these same profiles plotted with logarithmc scales. This makes it clear that the inner parts of the profiles are relatively flat. Some of this flatness could be the result of seeing or guiding errors. Figure 1 shows the slit orientation with respect to the observed coma morphology around the time of observation, for comparison between our gas profiles and the apparent dust distribution. Note that for December the image was taken a few days before our observations, and a small outburst was reported in between. \nAsymmetry: Unlike the dust, pushed tailward by solar radiation, imaging has revealed that comets can exhibit diverse gas coma morphologies such as fans or jets (e.g. Knight et al. 2021). Depending on the origin of the gas species and outgassing behaviour of the comet, the orientations of these features are not necessarily linked to the \nFigure 4. Density profiles of CN (cyan), C2 (magenta) and C3 ( yellow) versus nucleocentric distance on 2023 November 17 (left), 2023 November 18 (middle) and 2023 December 17 (right). Circular markers represent profiles along the slit oriented along the white arrows shown on Figure 1, with the arrow pointing towards positive distance values on this figure. For 2023 November 18, profiles with star-shaped markers are from a single exposure with the slit perpendicular to the orientation shown by the white arrow. The following offsets were added to allow for easier visualisation: for circular markers 1 × 10 11 for CN and 0 . 2 × 10 11 for C2, for star-shaped markers 1 . 6 × 10 11 for CN, 0 . 7 × 10 11 for C2 and 0 . 3 × 10 11 for C3. Solid black lines are smoothed (binned) profiles made from the profiles aligned with the parallactic angle (even those plotted over the perpendicular profiles, to allow comparison between both orientations). Dashed grey lines are these same binned profiles reversed along the x-axis to highlight the asymmetry in the coma. \n<!-- image --> \ndirection of the Sun. They may instead be dictated by the rotation state of the comet, and might vary between species. While we cannot obtain a full picture of the gas coma, we can look into the symmetry of our observed density profiles along the slit. \nOn November 17 and November 18, the profiles present a slight asymmetry, with the left side of the profiles (South-West of the coma) showing a slightly higher density of molecules than the right side (North-East of the coma). This is mostly visible for the CN profiles which have a higher signal-to-noise ratio than the other species, and the asymmetry is more pronounced on November 17 than on November 18, which could indicate that is it linked to the outburst material. Figure 1, as well as our own analysis of the dust component in our spectra, indicate that the North-East side of the coma seems to carry significantly more dust at the time. This orientation coincides with the anti-Solar direction expected for a dust-tail. Therefore this dust distribution does not necessarily represent any preferred direction in the steady-state or outburst dust ejection, making it difficult to correlate with the gas distribution. \nOn November 18th, the profiles acquired perpendicularly to the parallactic angle do not show any significant difference with the the ones aligned with the parallactic angle for C 2 and C 3 , but for CN the peak of the profile at the nucleus seems less sharp than for the initial orientation. \nOn December 17, the asymmetry between both sides is even stronger, with higher molecular densities on the right side (SouthWest) than the left side (North-East), suggesting anisotropic gas release. Knight et al. (2024) reported a complex coma morphology in 2024 February, displaying rotating CN jets that could explain the asymmetry that we observe, although the jets were 180 degrees apart in their viewing geometry. The image of the coma shown on Figure 1 was acquired on 2023 December 11, therefore we cannot guarantee that it accurately represents the coma morphology when our observations were taken (2023 December 17) since an outburst \nwas reported around December 12-14. Still it is interesting to note that the side of the coma with the most gas on our observations (South-West) coincides with with the side with the most dust on the image. This image indeed seems to show more dust in the West direction close to the nucleus, which then evolves into an anti-solar Northwards tail at larger distances. \nC 2 profile shape: While the CN and C 3 profiles still show a clear curved decrease with radial distance, the C 2 profiles appear almost linear. As a sanity-check, we generated profiles of another C 2 band ( Δ 𝑣 = 1, profile not shown) which showed a similar aspect. This particular behaviour could indicate a different origin than other species. Flatter inner-coma C 2 profiles have often been reported and are thought to be due to C 2 being a grand-daughter species or produced by icy grains in the coma (Combi & Fink 1997). Langland-Shula & Smith (2011) even report "C 2 holes" in multiple comets, where the C 2 density profiles dip close to the nucleus. Extended gas sources in the coma of comets have been detected in situ (Wallis et al. 1987) but still the production pathways of C 2 are not fully characterised. We present more evidence for extended sources in the coma of 12P in sections 3.3.2 and 3.3.3. \nCN feature: Finally, on November 17th a feature is visible in the CNprofile at 50000km from the nucleus, on the right side only. This bump is visible in the spectrum taken with a longer exposure time. We therefore consider that the bump is likely real. In section 3.3.4, we investigate whether it could be a smaller outburst following the large November 14 outburst. Such an event was detected following the large 2023 July outburst (Manzini et al. 2023a,b). We do not detect any trace of this feature in the profile from the following night but the expansion of the gas would have likely made is fainter. We \ndo not detect this feature in the C 2 and C 3 profiles, however the CN emissions have a higher signal to noise ratio.', '3.3.2 Modelling by the standard Haser model': 'In Figure 5, we compare the observed profiles to the expected standard Haser models, i.e given by integrating equation 1 along the line of sight using scale-lengths from Table 2 and our calculated production rates (Table 3). These theoretical profiles are represented by dotted lines. We then tried adjusting the standard Haser model with variable scale-lengths and 𝑄 values to the profiles using a leastsquares optimisation. The resulting profiles are represented by solid lines, along with the corresponding parameters. \nShapes Figure 5 shows that the expected profiles do not match the distribution of molecules in the coma at all. 3 In this case, production rates measurements made using the method described in section 2.4 strongly depend on the aperture used to measure the fluxes. This implies that the production rates that we calculated do not accurately represent the activity or composition of the comet. Therefore they should only be used for comparison with results that use the same process and parameters, such as the TRAPPIST preliminary measurements. \nAdjusting the model to the data systematically resulted in close to equal parent and daughter scale-lengths, except for the right-side C 2 profile from December 18 where the algorithm tended to increasingly large scale-lengths instead of converging. Both of these outcomes indicate that the observed profiles are too "angular" in logarithmic scale representation (flatter in the inner coma then decreasing more steeply in the outer coma) to be reproduced by the standard Haser model. \nCombi & Fink (1997) illustrate how the 𝐿 𝑝 = 𝐿 𝑑 case translates mathematically into the most "angular" profile shape allowed by the standard Haser model. They show that similar or even more angular shapes can be produced by three-generation models, i.e. considering two photodissociation steps starting from grand-parent species. Such formation pathways have been proposed to explain the production of certain species and their observations, such as C 2 H 2 → C 2 H → C 2 (e.g. Sorkhabi et al. 1997). Another possible source for these radicals is that either they or their parent species are produced directly from a halo of icy or CHON grains rather than from the nucleus. Extended sources have been invoked to explain the observed spatial distributions of several molecules (e.g. CN, Klavetter & A\'Hearn 1994). Combi & Fink (1997) show how their CHON halo model can also provide a better fit than the standard Haser model to some "angular" profiles measured in comet 1P/Halley. If the standard Haser model cannot reproduce our observations of the CN, C 2 and C 3 comae, it could indicate the presence of extended sources in the coma of 12P. \nFor each profile, the best-fit scale-lengths differ between both sides of the coma. This is likely due to the difficulty to fit a model on a profile spanning only 100000-150000 kilometres. In particular we can see that the side of the profile that covers the shortest distance range systematically obtains longer best-fit scale-lengths. This is \n3 It can be noted that Cochran (1985) proposed that the parent scale-length of C2 should vary as 𝑟 2 . 5 ℎ rather than 𝑟 2 ℎ . While this can make a significant difference at such large heliocentric distances, we have verified that the C2 scale-length relationships from Cochran (1985) also do not reproduce the observed profiles. \nprobably because the inner part of the profile, which is flatter, is more represented than the outer (steeper) part. The ratio between the best-fit scale-lengths found for the left and right sides of the coma varies between species from 1.5 to 2.7 on November 17, from 1.1 to 1.3 on November 18 (omitting C 2 which did not converge), and 1.9 to 2.3 on December 17. While these variations are likely due to differences in data quality (different signal-to-noise between species, or a more incomplete coverage of the coma for molecules with longer lifetimes), it is interesting to see that these ratios are more similar for CN and C 3 on the 18th than on the 17th, as the outburst settles. \nProduction rates Adjusting the model to the observed profiles yields production rates that are completely different than with the basic approach. For the following analysis, let us omit the C 2 profile from 2023 November 18 for which the fit did not converge and therefore yielded an unrealistically high production rate (panel at row 2 and column 4 on Figure 5). By averaging the best-fit production rates for both sides of the profiles, the resulting values are around 6 times larger for CN than what was determined in Table 3. For C 3 they are around 5 times larger, and for C 2 they are 27 times larger in November and 16 times larger in December. \nThe resulting production rates yield C 2 /CN abundance ratios significantly larger than those obtained with theoretical scale-lengths: ranging from 169 per cent to 422 per cent with an average of 312 per cent. As our adjusted models still do not perfectly match the observed profiles, especially for C 2 , these adjusted production rates may not accurately represent the composition of 12P, but this discrepancy between measurement methods highlights how challenging composition studies can be for comets with atypical behaviours and what biases may arise in large standardised surveys. It could also explain the variations seen in the TRAPPIST abundance ratios (Figure 3), in particular if the amount of extended sources varies overtime or is affected by the sudden outbursts. \nThe next section will focus on the observations from November and investigate whether this departure from the standard Haser model can be explained by the outburst that happened on November 14, as the production rate varying throughout the outburst can give the coma profiles a different shape.', '3.3.3 Modelling of the large November 14 outburst': "From the decrease in production rates that we observe between November 17 and November 18 it is clear that our profiles still represent the aftermath of the large November 14 outburst, which is bound to affect the coma shape. Indeed, the amount of molecules at a given distance from the nucleus depends on the production rate at the time of ejection. In the case of a post-outburst profile, the production has previously varied through time, as opposed to the constant production rate assumed by the steady-state Haser model. In this section we aim to explore whether this effect alone can explain why the standard Haser model does not fit the data. \nCN and C 3 fits: We attempted to adjust the outburst model described in section 2.5 to the CN and C 3 profiles from November 17 and November 18 simultaneously, in an attempt to constrain values of Q 0 , Q 1 , L 𝑝 , L 𝑑 , v 1 , r 𝑎 , r 𝑏 and the outburst peak time. As demonstrated by Opitom et al. (2016), this model can reproduce bumps in radial profiles that are sometimes observed during outbursts. We only considered the right-hand side of the profiles as they cover larger \nFigure 5. Molecular density profiles of CN, C2 and C3 in along the spectrometer's slit in our observations from 2023 November 17 (INT-IDS), 2023 November 18 (INT-IDS) and 2023 December 17 (NOT-ALFOSC). Dotted lines show the standard Haser model profiles (eq. 1) using parent and daughter scale-lengths from Table 2 and production rates adjusted to match the total flux within a 10000km aperture. Solid lines show the standard Haser model profiles with production rates and scale-lengths adjusted to match our observed profiles. The resulting parameters are listed on each subplot, with Q in molecules/second and L 𝑝 and L 𝑑 in kilometres. Both x and y axes are scaled logarithmically. Markers in the inner parts of the profiles have been made more opaque for better visibility, this does not reflect the density of data points compared to the rest of the profile. \n<!-- image --> \nDistance to nucleus [km] \nnucleocentric distances. We did not attempt to include the C 2 profiles as their shape initially seemed too different from the standard model. Assuming a steady-state expansion velocity of 1km s -1 , most of the material released at the peak of the outburst perpendicularly to the line of sight has already left the field of view by November 17, meaning that the data contain less information about the timescales of the outburst, making it hard to constrain some of the parameters. Still, this analysis can show whether the outburst model can generate Haser profiles that better match our observations. Initial conditions were chosen based on the theoretical scale-lengths from Table 2, expected orders of magnitudes of the production rates, observed timescales of the outburst 4 and a velocity of 1km s -1 . The outcome of the minimisation algorithm and the resulting residuals are presented in Figure 6. We can see that, at the distances covered by our profiles, the outburst model looks quite similar to the standard Haser model. For CN on the first night, the outburst model seems to better reproduce the decrease of the profile past 100000km. On the second night the two models are nearly indistinguishable. For C 3 the difference between the two models is very minor and would mostly \nbe visible in regions where the observed signal is very faint, therefore both models provide an equally good fit. \nTheCNproductionrates(pre-outburst Q 0 (CN)=1 . 5 × 10 26 s -1 with an additional outburst source of CN that was Q 1 (CN)=1 . 3 × 10 27 s -1 ) seem consistent with the comet's activity, as Q 0 (CN) is of the same order of magnitude as (Jehin et al. 2023c) measured on November 12. The best fit C 3 production rates are Q 0 (C 3 )=5 . 7 × 10 24 s -1 and Q 1 (C 3 )=1 . 2 × 10 26 s -1 . Q 0 (C 3 ) is a factor 2 lower than what Jehin et al. (2023c) measured pre-outburst, but we deem this to be an acceptable order of magnitude, especially since the resulting C 3 /CN ratio Q 0 (C 3 )/Q 0 (CN)=3.8 per cent is typical. For both species, the resulting parent and daughter scale-lengths are still almost equal, with L 𝑝 (CN)=2 . 4 × 10 5 km, L 𝑑 (CN)=3 . 1 × 10 5 km, L 𝑝 (C 3 )=7 . 3 × 10 4 km and L 𝑑 (C 3 )=7 . 3 × 10 4 km. This indicates that the nature of the activity at the time (outburst rather than steady-state) does not explain why the profiles depart from a Haser profile, and the presence of extended sources is still likely. The outburst material velocity of v 1 = 0 . 73km s -1 is lower than the velocity that we assumed for the Haser model but acceptable at this heliocentric distance ( 𝑟 ℎ ≈ 3 . 5au), especially if gas is being released from slower grains. The outburst peak time of November 15 01:54 UT is compatible with observations by multiple astronomers monitoring the comet 4 . Characteristic \nFigure 6. Comparison of the best-fit standard Haser model (dotted) and best fit outburst Haser model (dashed). In the legends Σ is the residual sum of squares, minimised by the fitting algorithm. Top and middle rows: Output of simultaneous least-square fitting the outburst model (eq 2 and 3) to the CN and C3 profiles from November 17 and November 18. Here the model parameters are Q0(CN)=1 . 5 × 10 26 s -1 , Q1(CN)=1 . 3 × 10 27 s -1 , L 𝑝 (CN)=2 . 4 × 10 5 km, L 𝑑 (CN)=3 . 1 × 10 5 km, Q0(C3)=5 . 7 × 10 24 s -1 , Q1(C3)=1 . 2 × 10 26 s -1 , L 𝑝 (C3)=7 . 3 × 10 4 km, L 𝑑 (C3)=7 . 3 × 10 4 km, v1 = 0 . 73km s -1 , r 𝑎 /v1 = 1 . 3 × 10 4 s, r 𝑏 /v1 = 1 . 7 × 10 5 s, and an outburst peak on 2023 November 15, 01:54UT. Bottom row: Output of least-square fitting of the outburst model C2 profiles from November 17 and November 18 simultaneously. Values of v1, r 𝑎 , r 𝑏 and the outburst peak time were fixed to the ones found from adjusting the model to the CN and C3 profiles, resulting in Q0(C2)=1 . 0 × 10 26 s -1 , Q1(C2)=2 . 9 × 10 28 s -1 , L 𝑝 (C2)=8 . 1 × 10 6 km and L 𝑑 (C2)=8 . 5 × 10 6 km. \n<!-- image --> \ntimescales of the outburst r 𝑎 /v 1 = 1 . 3 × 10 4 s and r 𝑏 /v 1 = 1 . 7 × 10 5 s appear realistic since we observe a return to steady state within a few days and the outburst was reported to take off within a few hours only 4 . These are also similar to the typical timescales observed for several large outbursts of comet 29P (Trigo-Rodríguez et al. 2008), which also undergoes frequent outbursts. We attempted to allow different velocities for CN and C 3 instead of a common v 1 but this lead to similar results overall and velocities close to equal for both species. \nC 2 profile: WethenattemptedtomodeltheC 2 profiles from November 17 and 18 simultaneously to determine Q 0 (C 2 ), Q 1 (C 2 ), L 𝑝 , L 𝑑 , but using the values of v 1 , r 𝑎 , r 𝑏 and the outburst peak time determined from CN and C 3 . Considering the strange aspect of the C 2 profile and the data quality, we preferred this approach to trying to fit all parameters at the same time. Initially, the model converged towards Q 0 (C 2 ) values that are significantly lower than expected ( ≈ 1 × 10 19 s -1 ). However, imposing a more realistic order of magnitude Q 0 (C 2 ) > 1 × 10 26 s -1 resulted in adjusted profiles that are visually just as satisfactory as letting Q 0 (C 2 ) vary freely. In this \nFigure 7. CN profile from 2023 November 17 along with the best fit for the standard Haser model (dotted line) and for the outburst model (solid line) adjusted to reproduce the small feature visible at ∼ 50000km. Because the standard Haser model is particularly far from the observations close to the nucleus, the outburst model was adjusted on the data at distances greater than 5000km. A grey vertical line indicates this cut-off. \n<!-- image --> \ncase we obtained Q 0 (C 2 )=1 . 0 × 10 26 s -1 , Q 1 (C 2 )=2 . 9 × 10 28 s -1 , L 𝑝 (C 2 )=8 . 1 × 10 6 km and L 𝑑 (C 2 )=8 . 5 × 10 6 km. The corresponding profile is shown on Figure 6. Although we could not accurately constrain Q 0 (C 2 ), nor Q 1 (C 2 ) which seems abnormally high compared to Q 1 (CN), we show that the outburst model can match the aspect of the observed C 2 profiles slightly better than the standard model at large distances. However, visually it still does not provide as good of a fit as for CN and C 3 , and as for these species equal scale-lengths are required. \nWeconclude that the outburst does not completely explain why the standard (non-outburst) Haser model does not apply, as the outburst model requires non-physical scale-lengths for all molecules and still to match the C 2 profile shape. We propose that extended sources must contribute to the production of all species, but that extended sources of a different nature or reaction chains more complex than parent/daughter might have to be considered for C 2 as well.", '3.3.4 Modelling of a possible November 17 mini-outburst': 'Finally, we tried adjusting the outburst model to the feature visible in the CN profile from November 17 (Figure 4). If this is an outburst, because we do not have information on the temporal evolution of this feature we imposed an arbitrary outburst material expansion velocity of v 1 =1km s -1 . However depending on the nature of the outburst, using the same velocity as for the steady-state might be erroneous. Because the standard Haser model does not correctly reproduce the overall profile to begin with (especially at short distances) and in order to best model the shape of the outburst feature over the steadystate baseline, we only adjusted this model to the profile past 5000km. The resulting profile is shown on Figure 7, corresponding to the following parameters: L 𝑝 (CN)=3 . 0 × 10 5 km, L 𝑑 (CN)=6 . 2 × 10 5 km, Q 0 (CN)=6 . 48 × 10 26 s -1 and Q 1 (CN)=3 . 8 × 10 26 s -1 , r 𝑎 /v 1 = 5 . 1 × 10 3 s, r 𝑏 /v 1 = 5 . 3 × 10 3 s, Δ 𝑡 =4 . 8 × 10 4 s. Because the flattest part of the profile was masked, we obtain parent and daughter scale-lengths \nthat are not equal but still of similar orders of magnitude, which would not be the case of the expected values. \nThis shows that the outburst model can produce features similar to the one that we see in our CN profile. The corresponding event could then be a short outburst with a production rate of one order of magnitude below the total peak outburst production rate found in section 3.3.3. However, with our data alone, we cannot guarantee that this feature is real. We do not find any strong evidence for a dust-counterpart to this event in the LCO data.', '4 CONCLUSION': 'In this paper, we analysed long-slit optical spectra of comet 12P/PonsBrooks acquired between 2023 August and 2023 December, quantifying the comet\'s composition in daughter species and comparing the molecular density profiles along the slit to multiple models. In particular, we hoped that spectra obtained on consecutive nights soon after the large 2023 November 14 outburst could provide insight about the nature and behaviour of the outburst. \nAssuming that the distribution of molecules in the coma follows the standard Haser model with commonly used parent and daughter scale-lengths, our measured production rates show a "typical" composition with a C 2 /CN ratio of about 90 per cent, which does not seem to change throughout the outburst or from November to December, and is in agreement with measurements by other teams around that time. \nHowever, calculated coma profiles of CN, C 2 and C 3 indicate that the behaviour of the gas coma is more complex, with asymmetries and separate species behaving differently. Comparing our profiles to the standard Haser model computed with empirical parent and daughter scale-lengths shows that it does not match the observed coma shape. This result invalidates the composition measurements that were made under the assumption that this model was a valid representation of the coma. \nAfter adjusting the standard Haser model to the data, the model still does not provide a good match and the best fit scale-lengths are equal for parent and daughter species, indicating that a more complex model (e.g. icy grains, CHON grains, or three generations) might be necessary. Best fit production rates result into a larger C 2 /CN ratio, which highlights how inaccurate scale-lengths or models can introduce bias in composition measurements. \nComparing the profiles from November to an outburst model, which accounts for the variation through time of the production rate during the outburst, we showed that this model reproduces the shape of the profiles better than the steady-state model for CN and C 3 , although equal parent and daughter scale-lengths are still required. However this model fails to match the shape of the C 2 coma. We propose that extended sources contribute to the production of all species in the coma, and that the C 2 coma is particularly affected by different types of extended sources and/or complex formation pathways. \nFinally we showed that it is possible that a small short outburst happened between the large 2023 November 14 outburst and our observations, which would explain a small feature on our November 17 CN profiles. \n12P is yet another comet for which simple models cannot reproduce large-scale gas distributions, in particular for C 2 . More needs to be known about the production mechanisms of these species in comae to understand how these observed distributions reflect the "true" composition of the ice. Radio or infrared observations of par- \nent species in 12P\'s coma could provide insight into the formation mechanisms of the daughter species that are observed in the optical range. A more in depth analysis of 12P\'s gas and dust production through time and through outbursts could also help characterise and clarify the origin of 12P\'s variable activity. In this context, we hope that other observing campaigns can improve our understanding of the nature of 12P\'s activity.', 'ACKNOWLEDGEMENTS': "The authors would like to thanks U. G. Jørgensen and C. Snodgrass for the opportunity to obtain NOT observations. \n- R. Kokotenekova would like to acknowledge support from 'L'Oreal UNESCO For Women in Science' National program for Bulgaria. \nThis work makes use of observations from the 1m telescopes and Sinistro instruments of the Las Cumbres Observatory global telescope network. Data were obtained under the LOOK Proposal (Proposal code LTP2023B-001) and raw and reduced data are available from the LCO Science Archive at https://archive.lco.global. \nThis paper is based on observations made with the MuSCAT3 instrument, developed by the Astrobiology Center and under financial supports by JSPS KAKENHI (JP18H05439) and JST PRESTO (JPMJPR1775), at Faulkes Telescope North on Maui, HI, operated by the Las Cumbres Observatory. \nThe Comet Chasers schools outreach project is funded by the UK Science and Technology Facilities Council through the DeepSpace2DeepImpact Project, Society, the Open University and Cardiff University. It accesses the LCOGT telescopes through the Faulkes Telescope Project (FTPEPO2014A-004), which is partly funded by the Dill Faulkes Educational Trust. Observations include those made by Cai Stoddard-Jones and students from St Marys Catholic Primary School, Bridgend, Wales. \nFor the purpose of open access, the author has applied a Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript version arising from this submission. \nThe authors would like to thank the referee for their useful insight to improve this manuscript.", 'DATA AVAILABILITY': 'The data underlying this article will be shared on reasonable request to the corresponding author.', 'REFERENCES': 'This paper has been typeset from a T E X/L A T E X file prepared by the author.'} |
2024PhRvD.110f4061A | The detection method of gravitational waves GW using electromagnetic EM cavities has garnered significant attention in recent years. This paper thoroughly examines the analysis for the perturbation of the EM field and raises some issues in the existing literature. Our work demonstrates that the rigidity condition imposed on the material as provided in the literature is inappropriate due to its reliance on a gaugedependent quantity that cannot be controlled experimentally. Instead we incorporate elasticity into the material and revise the governing equations for the electric field induced by GWs expressing them solely in terms of gaugeinvariant quantities. Applying these equations to cylindrical cavities we present the GW antenna patterns for the detector. | 2024-09-01T00:00:00Z | ['2024PhRvD.110f4061A', '10.1103/PhysRevD.110.064061', '10.48550/arXiv.2312.09550', '2023arXiv231209550A', 'arXiv:2312.09550'] | ['General relativity', 'alternative theories of gravity', 'General Relativity and Quantum Cosmology', 'Astrophysics - Instrumentation and Methods for Astrophysics', 'High Energy Physics - Experiment', 'High Energy Physics - Phenomenology'] | Electromagnetic field in a cavity induced by gravitational waves | 2,024 | 172 | 0.3 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML'] | 1 | https://arxiv.org/pdf/2312.09550.pdf | {'Electromagnetic field in a cavity induced by gravitational waves': 'Danho Ahn , 1, ∗ Yeong-Bok Bae , 2, 3, † Sang Hui Im , 3, ‡ and Chan Park 3, 4, 5, § \n3 \n1 Center for Axion and Precision Physics Research, Institute for Basic Science, Daejeon 34051, Republic of Korea 2 Department of Physics, Chung-Ang University, Seoul 06974, Republic of Korea Particle Theory and Cosmology Group, Center for Theoretical Physics of the Universe, Institute for Basic Science, Daejeon 34051, Republic of Korea 4 Center for the Gravitational-Wave Universe, Astronomy Program, \nDepartment of Physics and Astronomy, Seoul National University, Seoul 08826, Republic of Korea 5 Institute for Gravitational Wave Astronomy, Henan Academy of Sciences, Zhengzhou 450046, Henan, China (Dated: August 28, 2024) \nThe detection method of gravitational waves (GW) using electromagnetic (EM) cavities has garnered significant attention in recent years. This paper thoroughly examines the analysis for the perturbation of the EM field and raises some issues in the existing literature. Our work demonstrates that the rigidity condition imposed on the material, as provided in the literature, is inappropriate due to its reliance on a gauge-dependent quantity that cannot be controlled experimentally. Instead, we incorporate elasticity into the material and revise the governing equations for the electric field induced by GWs, expressing them solely in terms of gauge-invariant quantities. Applying these equations to cylindrical cavities, we present the GW antenna patterns for the detector.', 'I. INTRODUCTION': 'Electromagnetic (EM) fields play a crucial role in the observation of gravitational waves (GWs). Interferometer-type GW detectors utilize lasers [1], and pulsar timing arrays employ EM pulses from millisecond pulsars to detect GWs [2-5]. Recently, the ideas of using high-sensitivity EM cavities for GW detection [6-9] have been receiving considerable attention with their applications [10-12]. The principle underlying this method involves resonating EM fields when the frequency of GWs closely matches the resonant frequency of the EM cavity. This approach has the advantage of being relatively easy to try because sensitive cavity experiments are already being conducted worldwide to search for axion dark matter [13-15]. \nThe resonant frequency of an EM cavity is determined by the size of the cavity. For instance, assuming a cavity size of approximately 0 . 1 meter would yield a resonant frequency in the GHz range. If we consider binary black holes as the source of GWs, planet-mass binary black holes of about ∼ 10 -5 M ⊙ would be required to produce GWs at GHz frequencies just before merging [16]. Such mass black holes are challenging to form from stars [17]; hence, if they exist, they are likely to be primordial black holes formed from the density fluctuations in the early universe. Although direct evidence of black holes of this mass has yet to be found, the study [18], that suggests the possibility of planet-mass black holes through gravitational microlensing events is noteworthy. \nThe phenomena derived from GWs are described by the perturbation theory. In the process of unfolding that, there is freedom in the choice of gauge. However, that choice cannot create any physical differences. Depending on the gauge choice, there are various paths to reach an identical gauge-invariant quantity. When dealing with gauge-dependent quantities, it is easy to fall into the trap of considering gauge artifacts as physical entities. Therefore, to avoid this, it is preferable to describe the governing equations and imposed physical conditions solely in terms of gauge-invariant quantities. To achieve this, we will introduce appropriate gauge-invariant quantities to describe equations and physical conditions. Moreover, in the process, we will not choose any specific gauge. \nIn this paper, we extensively investigate the interrelation of EM fields, acoustic oscillations, and GWs to deepen our understanding of the operational principles of the EM cavity. The oscillations of the material and the EM field induced by GWs are expressed through the equations containing only gauge-invariant quantities. The prescribed physical conditions are also presented using gauge-invariant quantities, enabling experimental implementation. From this perspective, we critically review existing studies [8, 9], pointing out certain issues. Furthermore, we obtain solutions for gauge-invariant quantities from revised equations and physical conditions. This allows us to comprehend the interactions among the EM field, acoustic oscillations, and GWs, and accurately describe the GW signal measurable from the EM cavity. \nOur paper is structured as follows. In Sec. II A, we introduce physical laws in curved spacetime necessary for discussing the induced EM field by GWs. Sec. II B covers the basics of the EM cavity. To provide a helpful pedagogical illustration, Sec. II C presents an example of forced oscillation. In Sec. III A, we present the covariant perturbation theory, enabling a concise perturbation analysis. Applying this approach, we discuss per- \nturbations of Minkowski spacetime in Sec. III B without choosing any gauge. The perturbations of elasticity are developed in Sec. III C, and those of electromagnetism are discussed in Sec. III D. For the inside of the vacuum cavity, we provide the equation for the induced electric field by GWs in Sec. III E. Sec. III F addresses the inadequacy of the rigid condition given in [8, 9]. Using corrected equations, we present antenna patterns for the detector in Sec. IV.', 'A. Physical laws in curved spacetime': "The indices a, b, · · · represent abstract indices [19], while α, β, · · · denote indices of spacetime components in a basis. We set c = 1 and G = 1 (geometrized unit), and ϵ 0 = 1 / 4 π and µ 0 = 4 π (Gaussian unit). We introduce the metric signature of ( -1 , 1 , 1 , 1). Consider a globally hyperbolic spacetime M , which is described by Einstein's equations as \nG ab = 8 πT ab , (1) \nwhere G is the Einstein tensor and T is the stress-energy tensor of matters. The contracted Bianchi identities yield \n∇ b T ab = 0 , (2) \nwhere ∇ is the Levi-Civita connection associated with the spacetime metric g . Now, let us consider the unit vector field u for a timelike geodesic congruence without vorticity. Its normalization condition is given by \nu · u = -1 , (3) \nwhere · denotes the inner product defined by g . The spatial metric γ and the volume form ε orthogonal to u are defined as \nγ ab ≡ g ab + u a u b , (4) \nε abc ≡ u d ε (4) dabc , (5) \nwhere ε (4) is the spacetime volume form. We introduce the extrinsic curvature defined by \nK ab ≡ ∇ b u a . (6) \nThis curvature is spatial, as contractions between u and all indices of K vanish, and it is symmetric due to the absence of vorticity. By the Ricci identity, the covariant derivative of K is given by \n∇ c K ab = ∇ b K ac -u d R d acb , (7) \nwhere R is the Riemenn tensor. \nLet us consider a material with a vacuum cavity, where the motion of the cavity forms a four-dimensional volume \nW in M . Introduce the unit vector field v for a timelike congruence with the normalization condition \nv · v = -1 , (8) \nsuch that its values on the material are four-velocities of the material elements. The spatial metric γ ' and the volume form ε ' orthogonal to v are given by \nγ ' ab ≡ g ab + v a v b , (9) \nε ' abc ≡ v d ε (4) dabc . (10) \nThe motions of material elements are influenced not only by spacetime but also by the elastic force arising from the deformation of the material. To analyze the elasticity, we require geometrical quantities known as the material metric χ and the material volume form Ω, as discussed in [20]. These quantities are spatial to v in the sense that \nχ ab v b = 0 , (11) \nΩ abc v c = 0 . (12) \nThey are also symmetric along v , and Ω is proportional to ε ' as follows: \nL v χ ab = 0 , (13) \nL v Ω abc = 0 , (14) \nε ' abc = J Ω abc , (15) \nwhere L is the Lie derivative and J represents the Jacobian. \nThe stress-energy tensor of the material is decomposed into \nT ab = ρv a v b + σ ab , (16) \nwhere ρ is the energy density and σ is the stress. We assume that the material is homogeneous and isotropic in terms of the material metric χ , such that ρ and σ satisfy \n∇ a ( ρv a ) = 0 , (17) \nσ ab = -J -1 ( λχ ab ( e cd χ cd ) +2 µe ab ) , (18) \nwhere λ and µ are constant Lam'e parameters, and e is the strain defined by \ne ab ≡ 1 2 ( γ ' ab -χ ab ) . (19) \nWe introduce the orthogonal decomposition [21] with respect to u . Its derived quantities will be useful in the analysis of perturbation. The orthogonal decomposition of v is given by \nv a = ( u a + V a ) Γ , (20) \nΓ ≡ -( u · v ) , (21) \nV a ≡ ( γv ) a Γ -1 , (22) \nwhere Γ is the Lorentz factor, V is the spatial velocity, and γ ( · ) is the projection operator to the tangent subspace orthogonal to u for all indices of the dot. The material metric χ ab is decomposed into \nχ ab = αu a u b + u a β b + β a u b + W ab + γ ab , (23) \nα ≡ χ ab u a u b = χ ab V a V b , (24) \nβ a ≡ -u c γ d a χ cd = V c γ d a χ cd , (25) \nW ab ≡ ( γχ ) ab -γ ab , (26) \nwhere α is the temporal-temporal part, β is the temporalspatial part, and W is taken from the spatial-spatial part of χ . For Equations (24) and (25), we utilized Eq. (11). \nElectromagnetic fields F are governed by Maxwell's equations as \n∇ b F ab = 4 πj a , (27) \ndF = 0 , (28) \nwhere j is the EM current and d is the exterior derivative. Using the above, we obtain the wave equation for F as \n□ F ab = -4 π ( dj ) ab -F cd R cd ab -2 F c [ a R cd b ] d , (29) \nwhere □ ≡ ∇ a ∇ a is the D'Alembertian as Eq. (6) in [6]. The electric field and magnetic fields with respect to the material elements are given by \nE a = F ab v b , (30) \nB a = 1 2 ε ' bc a F bc , (31) \nas pointed out in [22]. \nBecause the cavity is a vacuum, we set conditions on W as \nV a | W = 0 , (32) \nj a | W = 0 . (33) \nWhen the material is a perfect conductor, we can impose the boundary condition on ∂ W , which is the threedimensional timelike hypersurface between the cavity and the material, given by \nP b a E b ∣ ∣ ∂ W = 0 , (34) \nwhere P a b ≡ γ ' a b -n a n b is the projection operator, and n is the spacelike unit vector field such that its values on ∂ W are identical to the normal vector of ∂ W . Note that n is orthogonal to v .", 'B. EM cavity in Minkowski spacetime': "Consider the EM cavity in Minkowski spacetime, where the material and the EM field are weak enough to ignore changes of Minkowski spacetime. In this case, we can set u as the constant vector field of four-velocity \naligning with our laboratory. Inside the cavity, Eq. (29) by contracting with u becomes the homogeneous wave equation for the electric field. When the material is a perfect conductor, we obtain the stationary solution as \nE a ( t, ⃗x ) = 2 ℜ [ ∑ n ˜ E n e n a ( ⃗x ) e -iω n t ] , (35) \nwhere we introduce a globally inertial coordinate system { t, ⃗x } such that u = ( ∂ / ∂t ). Here, e n ( ⃗x ) is the real resonant mode satisfying the boundary condition Eq. (34), ω n is the resonant frequency, and ˜ E n is the complex amplitude. The resonant modes satisfy that following properties: \n∆ e n a = -ω 2 n e n a , (36) \n∫ V d V e n · e m = δ nm V , (37) \nwhere ∆ is the Laplacian, V is the spatial volume of the cavity, and δ nm is the Kronecker delta. Note that e n ( ⃗x ) is dimensionless, and ˜ E n has the same dimension as the electric field. \n̸ \nThe resonant modes of a cylindrical cavity are categorized into TM and TE modes. In each category, they have indices ( m,n,p,s ) where m = 0 , 1 , · · · is the azimuthal and n = 1 , 2 , · · · is the radial mode number. The longitudinal mode number p can be p = 0 , 1 , · · · for TM modes and p = 1 , 2 , · · · for TE modes. For m = 0, we have two degenerated modes denoted with s = +1 , -1, respectively. The resonant frequencies for TM and TE modes, respectively, are given by \nω TM mnp = √ ( j mn /R ) 2 +( pπ/L ) 2 , (38) \nω TE mnp = √ ( j ' mn /R ) 2 +( pπ/L ) 2 , (39) \nwhere R and L are the radius and the length of the cylinder, respectively, j mn is the n -th zero of the Bessel function J m ( x ), and j ' mn is the n -th zero of the derivative of Bessel function J ' m ( x ). \nIn the cylindrical coordinate { ρ, ϕ, z } whose origin is located at the center of the cylinder, the resonant modes for TM and TE, respectively, are given by \ne TM mnps ( ρ, ϕ, z ) = 2 A TM mnp J ' m ( j mn ) [ R TM mn,z ( ρ ) C -s ( mϕ ) Z p + ( z ) ˆ z \n-R L pπ j mn { R TM mn,ρ ( ρ ) C -s ( mϕ ) ˆ ρ ( ϕ ) \n+ s R TM mn,ϕ ( ρ ) C s ( mϕ ) ˆ ϕ ( ϕ ) } Z p -( z ) ] , (40) \ne TE mnps ( ρ, ϕ, z ) = 2 A TE mnp J m ( j ' mn ) [ s R TE mn,ρ ( ρ ) C s ( mϕ ) ˆ ρ ( ϕ ) \n-R TE mn,ϕ ( ρ ) C -s ( mϕ ) ˆ ϕ ( ϕ ) ] Z p -( z ) , (41) \nwhere { ˆ ρ ( ϕ ) , ˆ ϕ ( ϕ ) , ˆ z } is the orthonormal basis of cylindrical coordinate and \nA TM mnp ≡ [(1 + δ m 0 ) \n× { ( R L pπ j mn ) 2 (1 -δ p 0 ) + (1 + δ p 0 ) }] -1 2 , (42) \nA TE mnp ≡ [ 1 -( m/ j ' mn ) 2 + δ m 0 ] -1 2 , (43) \nR TM mn,ρ ( ρ ) ≡ J ' m ( j mn R ρ ) , (44) \nR TM mn,ϕ ( ρ ) ≡ m j mn ρ/R J m ( j mn R ρ ) , (45) \nR TM mn,z ( ρ ) ≡ J m ( j mn R ρ ) , (46) \nR TE mn,ρ ( ρ ) ≡ m j ' mn ρ/R J m ( j ' mn R ρ ) , (47) \nR TE mn,ϕ ( ρ ) ≡ J ' m ( j ' mn R ρ ) , (48) \nC s ( ϕ ) ≡ { cos ϕ : s = +1 sin ϕ : s = -1 , (49) \nZ ps ( z ) ≡ C s ( pπ L ( z + L 2 )) . (50) \nFor m = 0, we have to choose s = -1 to get a nontrivial mode. Note that the above are normalized by Eq. (37). In addition, these modes have the parity symmetry as \ne mnps ( ρ, π + ϕ, -z ) = ( -1) m + p e mnps ( ρ, ϕ, z ) . (51)", 'C. Forced oscillation': 'Let us consider a pedagogical example of the resonance of a mass m attached to a spring with coefficient k and subjected to an external force f ( t ). The equation of motion is given by \nm x + b ˙ x + kx = f ( t ) , (52) \nwhere x ( t ) is the displacement of the mass from the equilibrium point, and b is the friction coefficient. By applying Fourier transformation, the solution x is given by \n˜ x ( ω ) = A ( ω ; ω 0 , Q ) e ia ( ω ; ω 0 ,Q ) ˜ f ( ω ) /k, (53) \nwhere ˜ x ( ω ) and ˜ f ( ω ) are the Fourier transformations of x ( t ) and f ( t ), respectively. Here, ω 0 = √ k/m is the resonance frequency, Q = mω 0 /b is the quality factor, and the resonance amplitude A ( ω ; ω 0 , Q ) and the phase \na ( ω ; ω 0 , Q ) are defined by \nA ( ω ; ω 0 , Q ) ≡ { 1 -( ω ω 0 ) 2 } 2 + ( ω ω 0 Q ) 2 -1 / 2 , (54) \na ( ω ; ω 0 , Q ) ≡ Arg [ ω 2 0 -ω 2 -i ω 0 ω Q ] . (55) \nNote that A ( ω ; ω 0 , Q ) is dimensionless, and its maximum value is Q at ω = ω 0 . The EM resonance in the cavity will also have the resonance amplitudes identical to A ( ω ; ω 0 , Q ) with their own ω 0 and Q .', 'A. Covariant perturbation theory': "Let us delve into covariant perturbation theory to facilitate a clear and concise discussion. Consider a foliation F by a one-parameter family of perturbed spacetime M ϵ , where ϵ is the dimensionless perturbation parameter. We set M 0 as the background spacetime. To discuss perturbations, we introduce a one-parameter group of diffeomorphisms ϕ ϵ : M 0 →M ϵ . Then, the perturbed quantity for a tensor X is given by the pullback through ϕ ϵ as ˜ X ( ϵ ) ≡ ϕ ∗ -ϵ X . When ˜ X ( ϵ ) = O ( ϵ n ) for a positive integer n , it is convenient to introduce a quantity Y such that ˜ X ( ϵ ) = ϵ n ˜ Y ( ϵ ), where ˜ Y ( ϵ ) ≡ ϕ ∗ -ϵ Y = O (1). Then, the perturbed quantity is expanded by \n˜ X ( ϵ ) = ϵ n { Y (0) + ϵ ( δY ) + O ( ϵ 2 ) } , (56) \nwhere Y (0) = Y | ϵ =0 is the leading-order value, and δY is the linear perturbation. The linear perturbation is provided by the Lie derivative of the quantity, i.e., δY = [ L υ Y ] ϵ =0 , where υ is the five-dimensional vector field on F generating ϕ ϵ . Refer to Fig. 1 in [23] for a helpful visualization of the concept. \nIt is crucial to recognize that the perturbed quantity depends on our choice of ϕ ϵ . This introduces mathematical redundancies, or gauges, in the perturbation. The gauge transformation between δY = [ L υ Y ] ϵ =0 and δ ' Y = [ L υ ' Y ] ϵ =0 , where υ and υ ' are generators of ϕ ϵ and ϕ ' ϵ , respectively, is given by \nδ ' Y -δY = [ L ξ Y ] ϵ =0 , (57) \nwhere ξ ≡ υ ' -υ . Moreover, ξ | ϵ =0 is tangent to M 0 because ξ ( ϵ ) = υ ' ( ϵ ) -υ ( ϵ ) = 1 -1 = 0, where ϵ is understood as the scalar field on F . Hence, we can evaluate [ L ξ Y ] ϵ =0 using only quantities in M 0 , which means that [ L ξ Y ] ϵ =0 = L ξ | ϵ =0 [ Y (0) ] . By the formulation of ξ | ϵ =0 , we can generate all possible linear perturbations using the gauge transformation. \nIn the process of determining δY from a measurement, the background spacetime M 0 and the leadingorder value Y (0) in M 0 are typically known, and the experimental measurement Y is performed in the perturbed spacetime M ϵ . To determine δY from Eq. (56), ignoring higher-order terms, one needs to choose a gauge to specify the perturbed value ˜ Y ( ϵ ). Owing to the absence of a preferred gauge, δY cannot be uniquely determined when it is gauge dependent, making it not measurable. Therefore, gauge invariance for the linear perturbation is essential to enable its measurement. According to the lemma from [24], δY is gauge invariant if and only if Y (0) is zero, a constant scalar, or constructed by the Kronecker delta with constant coefficients. \nLet us consider perturbations of spacetime quantities. The perturbed metric expanded as \n˜ g ab ( ϵ ) = g (0) ab + ϵh ab + O ( ϵ 2 ) , (58) \nwhere g (0) is the leading-order metric, and h ≡ δg is the linear perturbation. The perturbation of the covariant derivative with the Levi-Civita connection ∇ associated with g for a rank (1,1) tensor X is given by \nδ ( ∇ c X a b ) = ∇ (0) c ( δX ) a b +( X (0) ) d b ( δC ) a dc -( X (0) ) a d ( δC ) d bc , (59) \nwhere ∇ (0) is the Levi-Civita connections associated with g (0) on M 0 , X (0) is the leading-order, δX is the linear perturbation, and \n( δC ) a bc = 1 2 g ad (0) ( ∇ (0) c h bd + ∇ (0) b h cd -∇ (0) d h bc ) , (60) \nwhere g ab (0) is the inverse of g (0) ab . The Ricci identity for a vector X is given by \nR a bcd X b = 2 ∇ [ c ∇ d ] X a , (61) \nwhere R a bcd is the Riemann tensor. Introducing perturbations to both sides, we obtain the perturbation of the Riemann tensor as \ng be ( δR ) a ecd = -2 ∇ (0) [ c ∇ [ a (0) h b ] d ] -h e [ a ( R (0) ) b ] ecd , (62) \nwhere ( R (0) ) a bcd is the leading-order Riemann tensor. \nSo far, we have not introduced any coordinate system in the development of perturbation theory, nor do we need it. However, for readers familiar with perturbations using coordinate systems, we provide the perturbation of a coordinate system and its relation to our approach. Given the coordinate system { x α (0) } on M 0 , we introduce the adapted coordinate system { x α } on M ϵ corresponding to a gauge ϕ ϵ such that x α = x α (0) · ϕ -ϵ . The perturbation of the adapted coordinate system is then given by \nδx α = L υ ( x α ) = 0 , (63) \nwhere υ is the generating vector field for ϕ ϵ . Utilizing the commutativity of the Lie derivative and exterior derivative for the scalar field, we find the perturbation of coordinate dual basis { dx α } as \nδ (( dx α ) a ) = 0 . (64) \nUsing the fact ( ∂ / ∂x β ) a ( dx α ) a = δ α β , we obtain the perturbation of the coordinate basis { ∂ / ∂x α } as \nδ (( ∂ / ∂x α ) a ) = 0 . (65) \nFinally, we express the perturbation of tensor components as \nδ ( X α β ) = δ ( X a b ( dx α ) a ( ∂ / ∂x β ) b ) = ( δX ) α β , (66) \nwhere X is a rank (1,1) tensor on M ϵ , and the components of δX are evaluated with the coordinate system { x α (0) } on M 0 . \nTo obtain the transformation of adapted coordinate systems, let us consider coordinate systems { x α } and { x ' α } in M ϵ adapted to ϕ ϵ and ϕ ' ϵ , respectively. The perturbation of their difference in ϕ ϵ becomes \nδ ( x ' α -x α ) = δx ' α = -( δ ' -δ ) x ' α = -L ξ x α (0) . (67) \nIntroducing pullbacks ˜ x α ( ϵ ) and ˜ x ' α ( ϵ ) for { x α } and { x ' α } , respectively, by ϕ ϵ , we get the relation given by \n˜ x ' α ( ϵ ) -˜ x α ( ϵ ) = -ϵξ α + O ( ϵ 2 ) . (68) \nMany approaches to perturbation theory often start from this relation, but in our approach, it is a consequence.", 'B. Perturbation of Minkowski spacetime': "Henceforth, we assume that all quantities exist in M 0 unless explicitly stated. We also omit the superscript (0) for brevity when referring to leading-order quantities. Defining M 0 as Minkowski spacetime, we have the flat metric g , its associated Levi-Civita connection ∇ , and the vanishing Riemann tensor R = 0 at the leading order. From Eq. (62), the linear perturbation of the Riemann tensor is then given by \n( δR ) ab cd = -2 ∇ [ a ∇ [ c h b ] d ] , (69) \nwhich is gauge invariant due to its vanishing leading order. Utilizing the commutativity of perturbation and self-contraction, we derive the perturbation of the Ricci tensor as \nδ ( R c acb ) = -1 2 □ h ab -1 2 ∇ b ∇ a h c c + ∇ ( a ∇ c h b ) c . (70) \nThe perturbation of the Ricci scalar is expressed as \nδ ( R c acb g ab ) = ∇ b ∇ a h ab -□ h a a . (71) \nAssuming the perturbed stress-energy tensor as ˜ T ( ϵ ) = O ( ϵ 2 ) , the perturbation of Einstein's equations in Eq. (1) is given by \n0 = -1 2 □ h ab -1 2 ∇ b ∇ a h c c + ∇ ( a ∇ c h b ) c -1 2 g ab ( ∇ d ∇ c h cd -□ h c c ) . (72) \nTaking the trace of the above equation reveals the vanishing perturbation of the Ricci scalar, leading to the conclusion that the perturbation of the Ricci tensor also vanishes. \nApplying the D'Alembertian to Eq. (69) and utilizing Einstein's equations from Eq. (72), we obtain the wave equation for the perturbation of the Riemann tensor as \n□ ( δR ) a bcd = 0 . (73) \nIts wave solution, representing GWs, is given by \n( δR ) a bcd ( t, ⃗x ) = ∫ d 2 κ ∫ ∞ -∞ dω 2 π ˜ R a bcd ( ω, κ ) e iP ( t,⃗x ; ω,κ ) , (74) \nwhere { t, ⃗x } denotes the globally inertial coordinate system defined in Sec. II B, ω is the parameter for angular frequency, κ is the unit spatial vector for the propagation direction, ∫ d 2 κ represents integration over all directions, ˜ R is the amplitude, and P ( t, ⃗x ; ω, κ ) ≡ ω ( -t + κ · ⃗x ) is the phase. Subsequently, Eq. (69) has the general solution, composed of the particular and homogeneous solutions: \nh ab ( t, ⃗x ) = ∫ d 2 κ ∫ ∞ -∞ dω 2 π ( ˜ h p ) ab ( ω, κ ) e iP ( t,⃗x ; ω,κ ) +( h h ) ab ( t, ⃗x ) , (75) \nwhere h h ( t, ⃗x ) is the homogeneous solution, and ˜ h p ( ω, κ ) is the amplitude of the particular solution satisfying \n˜ R ab cd = 2 k [ a k [ c ( ˜ h p ) b ] d ] , (76) \nwhere k a ≡ ∇ a P . \nFor the geodesic congruence, its unit vector field and extrinsic curvature are expanded as follows: \n˜ u a ( ϵ ) = u a + ϵ ( δu ) a + O ( ϵ 2 ) , (77) \n˜ K ab ( ϵ ) = K ab + ϵ ( δK ) ab + O ( ϵ 2 ) , (78) \nwhere the leading order u is the constant four-velocity aligned to the laboratory, δu is its linear perturbation, the leading order K vanishes, and δK is its linear perturbation. The normalization condition Eq. (3) provides \nu · ( δu ) = -1 2 h ( u, u ) . (79) \nPerturbations of Equations (6) and (7) give \n( δK ) ab = ∇ b ( δu ) a + u c ( δC ) acb , (80) \n∇ c ( δK ) ab = ∇ b ( δK ) ac -u d ( δR ) d acb . (81) \nNotice that δK is spatial to u and gauge invariant because its leading order vanishes. Contracting indices in Eq. (81), we obtain \nD b ( δK ) ab = 0 , (82) \n∇ a ( δK ) b b = 0 , (83) \nwhere D is the spatial derivative operator defined as DX = γ ( ∇ X ) for a spatial tensor X .", 'C. Perturbation of elasticity': 'Perturbed geometrical quantities for elastic material are expressed as follows: \n˜ v a ( ϵ ) = v a + ϵ ( δv ) a + O ( ϵ 2 ) , (84) \n˜ χ ab ( ϵ ) = χ ab + ϵ ( δχ ) ab + O ( ϵ 2 ) . (85) \nConsidering static material at the leading order, we set v a = u a and χ ab = γ ab . Perturbations of the normalization condition Eq. (8), the Lorentz factor Eq. (21), and the spatial velocity Eq. (22) provide the following: \nu · ( δv ) = -1 2 h ( u, u ) , (86) \nδ Γ = 0 , (87) \n( δV ) a = ( γδv ) a -( γδu ) a . (88) \nEquations (13) and (23) for the material metric give \n( δχ ) ab = u a ( δV ) b +( δV ) a u b +( δW ) ab +( δγ ) ab , (89) \n∇ u ( δW ) ab = -2 D ( a ( δV ) b ) -2 ( δK ) ab , (90) \nwhere ∇ u is the covariant derivative along u . Perturbations of the spatial metric Eq. (9) and the strain Eq. (19) yield \n( δe ) ab = -1 2 ( δW ) ab . (91) \nNote that δV , δW , and δe are spatial and gauge-invariant because their leading-order values vanish. \nWe introduce the perturbed stress-energy tensor of the material given by \n˜ T ab ( ϵ ) = ϵ 2 { T ab + ϵ ( δT ) ab + O ( ϵ 2 )} , (92) \n˜ ρ ( ϵ ) = ϵ 2 { ρ + ϵ ( δρ ) + O ( ϵ 2 )} , (93) \n˜ σ ab ( ϵ ) = ϵ 2 { σ ab + ϵ ( δσ ) ab + O ( ϵ 2 )} . (94) \nWe choose ˜ T ( ϵ ) = O ( ϵ 2 ) to be compatible with the perturbed Einstein equation in Eq. (72) and assume that there is no other contribution except the material on \n˜ T ( ϵ ) up to ϵ 3 . Considering homogeneous material without stress at the leading order, we set ∇ a ρ = 0, and σ ab = 0. Then, Eq. (16) becomes \nT ab = ρu a u b , (95) \n( δT ) ab = ( δρ ) u a u b +2 ρ ( u ( a ( δv ) b ) + u ( a h b ) c u c ) +( δσ ) ab . (96) \nNote that δσ is spatial and gauge invariant because its leading order vanishes. \nTo derive evolution equations for the material, we perturb the conservation equation Eq. (17) and the contracted Bianchi identity Eq. (2) as follows: \n∇ u ( δρ ) = -ρD · ( δV ) , (97) \n∇ u ( δV ) a = -1 2 λ ρ D a ( δW ) b b -µ ρ D b ( δW ) ab , (98) \nwhere D · X is the divergence for a spatial vector X . Differentiating Eq. (98) by ∇ u , substituting Eq. (90), and utilizing Equations (82) and (83), we obtain \n0 = -∇ 2 u ( δV ) a + λ + µ ρ D a ( D · δV ) + µ ρ ∆( δV ) a . (99) \nNote that this equation is expressed solely in gaugeinvariant quantities, and there is no GW contribution in the above. \nThe Helmholtz decomposition allows the separation of solenoidal and irrotational modes for δV . The solenoidal mode (S wave), satisfying D · ( δV ) S = 0, and the irrotational mode (P wave), satisfying D × ( δV ) P = 0, where D × is the curl, have wave equations, respectively, as follows: \n0 = -∇ 2 u ( δV ) a S + µ ρ ∆( δV ) a S , (100) \n0 = -∇ 2 u ( δV ) a P + λ +2 µ ρ ∆( δV ) a P , (101) \nwhere ∇ 2 u is the second-order time derivative. Similarly, one can derive inhomogeneous wave equations for δW that have contributions from GWs.', 'D. Perturbation of EM field': "The perturbed EM field and current are expressed as follows: \n˜ F ab ( ϵ ) = ϵ 2 { F ab + ϵ ( δF ) ab + O ( ϵ 2 )} , (102) \n˜ j a ( ϵ ) = ϵ 2 { j a + ϵ ( δj ) a + O ( ϵ 2 )} . (103) \nBecause we imposed ˜ F ( ϵ ) = O ( ϵ 2 ) , the EM contribution on ˜ T ( ϵ ) is at O ( ϵ 4 ) . It is compatible with the perturbed Einstein's equations Eq. (72) and the perturbed contracted Bianchi identities Eq. (98). For the leading orders, we set j a = 0 and F ab = ε c ab B c where B is the \nconstant magnetic field. Notice that δj is gauge invariant because its leading order vanishes. Then, the linear perturbation of Equations (27) to (29) becomes: \n∇ b ( δF ) ab = 4 π ( δj ) a -2 F c [ a ∇ b h c b ] -1 2 F ab ∇ b h c c , (104) \nd ( δF ) = 0 , (105) \n□ ( δF ) ab +2 F d [ a ∇ c ( δC ) d b ] c = -4 π ( d ( δj )) ab -F cd ( δR ) cd ab . (106) \nFrom the definition of the electric field in Eq. (30), we obtain its linear perturbation as follows: \n( δE ) a = ( δF ) ab u b + F ab ( δv ) b . (107) \nNote that δE is spatial and gauge invariant because its leading order vanishes. By contracting u with Equations (104) and (106), we obtain equations for δE as \nD a [ ( δE ) a -F ab ( δV ) b ] = -4 π ( u · ( δj )) , (108) □ [ ( δE ) a -F ab ( δV ) b ] = D a [ D b (( δE ) b -F bc ( δV ) c ) ] +4 π ∇ u ( γδj ) a -F cd ( δR ) cd ab u b . (109) \nNote that these equations are written in only gaugeinvariant quantities.", 'E. Inside the cavity': 'To derive the perturbations of conditions Equations (32) and (33) within the cavity, let us introduce W ϵ as the four-dimensional volume for the cavity motion in M ϵ . By considering a gauge ϕ ϵ such that ϕ ϵ [ W 0 ] = W ϵ , the perturbations of Equations (32) and (33) on W 0 can be expressed as follows: \n( δV ) a | W 0 = 0 , (110) \n( δj ) a | W 0 = 0 . (111) \nBecause δV and δj are gauge invariant, these conditions hold in any gauge. Subsequently, within the cavity, Equations (108) and (109) transform into the following equations: \nD · ( δE ) | W 0 = 0 , (112) \n□ ( δE ) a | W 0 = -F cd ( δR ) cd ab u b ∣ ∣ ∣ W 0 . (113) \nIn a similar discussion, one can prove that the following condition, originating from Eq. (34), remains valid in any gauge at the boundary: \nP b a ( δE ) b ∣ ∣ ∂ W 0 = 0 . (114)', 'F. Gauges and rigidity': "Equations (112) to (114) hold for all gauges. Moreover, each term in these equations is gauge invariant as they involve only gauge-invariant quantities. The contribution of GWs comes from only the right-hand side in Eq. (113). This term vanishes when the direction of GWs aligns with the magnetic field direction due to Eq. (76). This observation contradicts the claims made in [8, 9]. Let us discuss the reason. \nIn [8], they impose rigidity of the material by fixing the four-velocity for conductor elements v as ( ∂ / ∂t ) in the proper detector frame even when GWs pass. Then, δv = 0 due to Eq. (65) in the proper detector gauge, and Equations (104) and (106) become \nD · ( δE ) = -4 π ( u · ( δj )) -F ab u c ( δC ) a b c , (115) □ ( δE ) a = D a ( D · ( δE )) + 4 π ∇ u ( γδj ) a -F cd ( δR ) cd ab u b + F ab u c ∇ d ( δC ) b cd + F bc u d D a ( δC ) b c d . \n(116) \nIn these equations, the second term on the right-hand side in Eq. (115) represents the 'effective charge,' while the third, fourth, and fifth terms on the right-hand side in Eq. (116) constitute the temporal derivative of spatial 'effective current,' as defined in [8]. \n̸ \nWe would like to address three points. Firstly, ( ∂/∂t ) in the perturbed spacetime M ϵ is not normalized. To accurately obtain the electric field, it is essential to use a normalized vector in Eq. (30). Secondly, δv is a gaugedependent quantity due to L ξ v = 0 in M 0 . Consequently, δv is not controllable experimentally. Enforcing δv = 0 requires a condition imposed by Eq. (88), such as ( δV ) a = -( γδu ) a , where δu is the gauge-dependent perturbation in the proper detector gauge. This cannot be the solution for Eq. (99), which is the acoustic wave equation with its propagating velocity smaller than the speed of light, while δu has the phase of GWs. Thirdly, a perfect rigid body does not exist in the framework of general relativity because it violates causality, as discussed in [25].", 'A. Electric field inside cavity': "We observe that the boundary condition for δE in Eq. (114) shares an identical form with Eq. (34) in Minkowski spacetime. Therefore, as in Eq. (35), the solution δE for Eq. (113) is superposed by resonant modes as \n( δE ) a ( t, ⃗x ) = ∑ n E n ( t ) e n a ( ⃗x ) , (117) \nwhere E n ( t ) is the time-dependent amplitude for mode n having the dimension identical to the electric field. This allows us to transform Eq. (113) into an equation for E n ( t ) similar to the forced oscillation equation in Eq. (52): \nV ω 2 n ( E n + ω n Q n ˙ E n + ω 2 n E n ) = f n ( t ) , (118) \nwhere we include the dissipation term with the quality factor Q n following [8] and f n ( t ) is the external 'force' defined by \nf n ( t ) ≡ 1 ω 2 n ∫ V d V F cd ( δR ) cd ab u b e a n ( ⃗x ) , (119) \nwhere e a n ( ⃗x ) is the metric dual to e n a ( ⃗x ). Notice that the factor V /ω 2 n in Eq. (118) makes the dimension of 'force' identical to the dimension of the energy over the electric field. \nThe solution E n ( t ) is given by \n˜ E n ( ω ) = A ( ω ; ω n , Q n ) e ia ( ω ; ω n ,Q n ) ˜ f n ( ω ) / V , \n˜ f n ( ω ) = V | B | ω n ∫ d 2 κ ˜ h ab ( ω, κ ) ( ˆ B × κ ) a ¯ e b n ( ω, κ ) e iωκ · ⃗x 0 . (121) \n(120) ( ω ) 2 \nHere, ˜ E n ( ω ) and ˜ f n ( ω ) are the Fourier transformations of E n ( t ) and f n ( t ), respectively. In Eq. (120), A ( ω ; ω n , Q n ) and a ( ω ; ω n , Q n ) are defined in Equations (54) and (55). In Eq. (121), ˜ h ab is defined by \n˜ h ab ( ω, κ ) ≡ 2 ω -2 ˜ R acbd ( ω, κ ) u c u d , (122) \nsatisfying \n˜ h a a ( ω, κ ) = 0 , (123) \n˜ h ab ( ω, κ ) u b = 0 , (124) \n˜ h ab ( ω, κ ) κ b = 0 . (125) \nNote that ˜ h ab is identical to the amplitude of metric perturbation in transverse-traceless gauge; however, we consider it as a derived quantity from the gauge-invarant perturbation of the Riemann tensor as in Eq. (27.24) in [26]. Also, | B | ≡ √ B · B , ˆ B ≡ B/ | B | , and × is the cross product with ε . Additionally, ⃗x 0 is a position inside V , and ¯ e n ( ω, κ ) is defined by \n¯ e a n ( ω, κ ) ≡ 1 V ∫ V d V ( ⃗x ' ) e a n ( ⃗x ' ) e iωκ · ⃗x ' . (126)", 'B. Signal and antenna pattern': 'In the context of GW detection using mode n , we introduce the GW signal h ( t ), the detector tensor ˜ D ab ( ω, κ ), \nand the transfer function ˜ T ( ω ) as follows: \n˜ E n ( ω ) = ˜ T ( ω ) ˜ h ( ω ) , (127) \n˜ T ( ω ) ≡ A ( ω ; ω n , Q n ) e ia ( ω ; ω n ,Q n ) ( ω ω n ) 2 | B | , (128) \n˜ h ( ω ) ≡ ∫ d 2 κ ˜ D ab ( ω, κ ) ˜ h ab ( ω, κ ) e iωκ · ⃗x 0 , (129) \n˜ D ab ( ω, κ ) ≡ ( ˆ B × κ ) a ¯ e b n ( ω, κ ) . (130) \nHere, ˜ h ( ω ) represents the Fourier transformation of h ( t ), with the note that h ( t ) is dimensionless. \nWe present the expression for ˜ h ab following the format in Sec. 7 of [23]: \n˜ h ab ( ω, κ ; H,χ,ψ ) = H ( cos χe + ab ( κ ; ψ ) + i sin χe × ab ( κ ; ψ ) ) e iξ . (131) \nHere, H ( ω, κ ) > 0 represents the amplitude strength, χ ( ω, κ ) is the ellipticity, ψ ( ω, κ ) is the polarization angle, and ξ ( ω, κ ) is the phase. The real orthonormal basis { e + , e × } is defined by: \n[ e + ( κ ; ψ ) e × ( κ ; ψ ) ] = [ cos (2 ψ ) sin (2 ψ ) -sin (2 ψ ) cos (2 ψ ) ][ e + ( κ ; 0) e × ( κ ; 0) ] , (132) e + ab ( κ ; 0) = 1 √ 2 ( x a x b -y a y b ) , (133) (134) \ne × ab ( κ ; 0) = 1 √ 2 ( x a y b + y a x b ) . \nHere, { x, y } belongs to the right-handed orthonormal frame { u, x, y, κ } . Note that the factor 1 / √ 2 ensures normalization of the basis, satisfying e A ab e B cd g ac g bd = δ AB , where A,B ∈ { + , ×} . \nWe rewrite Eq. (129) as \n˜ h ( ω ) = ∫ d 2 κH ( ω, κ ) ˜ F ( ω, κ ; χ, ψ ) e iωκ · ⃗x 0 e iξ ( ω,κ ) , (135) \nwhere ˜ F ≡ ˜ D ab ( cos χe + ab + i sin χe × ab ) is the pattern function. If the detector tensor has the form of ˜ D ab = D ab e iξ , where D ( ω, κ ) is a real tensor and ξ ( ω, κ ) is a real phase, we obtain the antenna pattern as \nF ( ω, κ ) ≡ √ 1 2 π ∫ 2 π 0 dψ ˜ F ( ω, κ ; χ, ψ ) ˜ F ∗ ( ω, κ ; χ, ψ ) , = √ 1 2 Λ abcd ( κ ) D ab ( ω, κ ) D cd ( ω, κ ) , (136) \nwhere Λ abcd ≡ ∑ A e A ab e A cd . Note that this F ( ω, κ ) does not depend on the ellipticity χ .', 'C. Cylindrical cavity': "For a cylindrical cavity, we observe that ¯ e a n ( ω, κ ) defined in Eq. (126) has the form of a real vector multiplied by a complex scalar because of the parity symmetry, as in Eq. (51). Therefore, the antenna pattern given in Eq. (136) is applicable to the cylindrical cavity. We introduce a pair of polar and azimuthal angles ( θ, ϕ ) for the GW propagation such that κ ( θ, ϕ ) = sin θ cos ϕ ˆ x +sin θ sin ϕ ˆ y +cos θ ˆ z and parameter α for the magnetic field direction as ˆ B ( α ) = cos α ˆ z +sin α ˆ x . Then, we get the antenna pattern from Eq. (136) as \nF \nTM mnps ( ω, ϑ, φ ) = 4 j mn A TM mnp [ 1 + ( R L pπ j mn ) 2 ] × ∣ ∣ ∣ ∣ ∣ Lω C ( -1) p +1 ( Lω cos ϑ/ 2) ( Lω cos ϑ ) 2 -( pπ ) 2 ∣ ∣ ∣ ∣ ∣ × ∣ ∣ ∣ ∣ ∣ J m ( Rω sin ϑ ) ( Rω sin ϑ ) 2 -j 2 mn cos ( ϑ ) sin ( ϑ ) C m, -s ( φ ) ∣ ∣ ∣ ∣ ∣ × √ 1 -(cos φ sin ϑ sin α +cos ϑ cos α ) 2 , (137) \nF TE mnps ( ω, ϑ, φ ) = 4 j ' mn A TE mnp ∣ ∣ ∣ ∣ ∣ pπ C ( -1) p +1 ( Lω cos ϑ/ 2) ( Lω cos ϑ ) 2 -( pπ ) 2 ∣ ∣ ∣ ∣ ∣ × [ ( m j ' 2 mn J m ( Rω sin ϑ ) Rω sin ϑ cos ϑ C ms ( φ ) ) 2 + ( J ' m ( Rω sin ϑ ) ( Rω sin ϑ ) 2 -j ' 2 mn C m, -s ( φ ) ) 2 1 2 × √ 1 -(cos φ sin ϑ sin α +cos ϑ cos α ) 2 , \n(138) \nwhere ϑ ≡ π -θ and φ ≡ π + ϕ . \nFigures 1 and 2 show antenna patterns using TM 010 and TE 212 -modes, respectively, at the resonance frequency for α ∈ { 0 , π/ 6 , π/ 3 , π/ 2 } and L/R = 1. Antenna patterns for α > π/ 2 can be obtained by rotating the patterns in the figures by π around the axis of cylinder and reversing the direction of the gray arrows. This symmetry arises from the relation F ( ω, ϑ, φ ; α ) = F ( ω, ϑ, π + φ ; π -α ). Figures 3 and 4 show cross sections of the antenna patterns on a plane that contains the magnetic field vectors, for TM 010 and TE 212 -modes, respectively. The panel (a) in Fig. 4 distinctly illustrates a difference between our antenna patterns and those presented in [8]. Our results reflect the fact that there is no GW signal when the direction of GWs is parallel to the cylinder axis or the magnetic field.", 'V. SUMMARY AND DISCUSSION': "We introduce physical laws in a globally hyperbolic spacetime described by Einstein's equations. The con- \nFIG. 1: This figure depicts the antenna patterns of cylindrical cavity with TM 010 mode, given in Eq. (137), at its resonance frequency. The patterns correspond to magnetic field directions α ∈ { 0 , π/ 6 , π/ 3 , π/ 2 } and an aspect ratio L/R = 1. The gray arrows indicate the magnetic field directions. (a) α = 0. (b) α = π/ 6. (c) α = π/ 3. (d) α = π/ 2. \n<!-- image --> \nFIG. 2: This figure depicts the antenna patterns of cylindrical cavity with TE 212 -mode, given in Eq. (138), at its resonance frequency. The patterns correspond to magnetic field directions α ∈ { 0 , π/ 6 , π/ 3 , π/ 2 } and an aspect ratio L/R = 1. The gray arrows indicate the magnetic field directions. (a) α = 0. (b) α = π/ 6. (c) α = π/ 3. (d) α = π/ 2. \n<!-- image --> \nFIG. 3: The figure illustrates the antenna patterns of cylindrical cavity with TM 010 mode, given in Eq. (137), at its resonance frequency. The patterns are depicted on the plane of φ = 0 and φ = π , corresponding to magnetic field directions α ∈ { 0 , π/ 6 , π/ 3 , π/ 2 } , with an aspect ratio L/R = 1. The gray arrows indicate the magnetic field directions. (a) α = 0. (b) α = π/ 6. (c) α = π/ 3. (d) α = π/ 2. \n<!-- image --> \ntracted Bianchi identities imply the conservation of stress-energy tensor. The congruence of the timelike geodesics without vorticity, have their spatial distribution determined by the extrinsic curvature. However, material elements, despite being influenced by gravity, deviate from geodesics due to elasticity stemming from their deformation. To account for the acoustic oscillation of the material, we introduce the four-velocity and strain, both contributing the stress-energy tensor. The equations of motion for these material elements encompass the temporal symmetry of the material metric, energy conservation, and the conservation of stress-energy tensor. The electromagnetic field is governed by Maxwell's equations. The electric field and magnetic field to the material elements are determined by the four-velocities of the elements. Considering the cavity as a vacuum, we impose the condition of vanishing spatial velocity and EM current. In cases where the material acts as a perfect conductor, the electric field is constrained to be orthogonal to the surface at the boundary. \nFor a concise discussion, we utilize covariant perturbation theory. Within this framework, we introduce the lemma that a linear perturbation is gauge invariant if and only if the Lie derivative of its leading-order value vanishes along all directions. The perturbation of Einstein's equations transforms into the wave equation for the Riemann tensor, where its wave solutions manifest as GWs. \nFIG. 4: The figure illustrates the antenna patterns of cylindrical cavity with TE 212 -mode, given in Eq. (138), at its resonance frequency. The patterns are depicted on the plane of φ = 0 and φ = π , corresponding to magnetic field directions α ∈ { 0 , π/ 6 , π/ 3 , π/ 2 } , with an aspect ratio L/R = 1. The gray arrows indicate the magnetic field directions. (a) α = 0. (b) α = π/ 6. (c) α = π/ 3. (d) α = π/ 2. \n<!-- image --> \nUtilizing these solutions, we present the perturbation of the timelike geodesic congruence. The perturbations of spatial velocity and strain for the material exhibit homogeneous and inhomogeneous wave equations, respectively, as dictated by their equations of motion. Perturbing Maxwell's equations yield the wave equation for the perturbation of the electric field. Within the cavity, this equation indicates that GWs contribute to the perturbation of the electric field solely through the coupling \n- [1] R. Abbott, T. D. Abbott, F. Acernese, K. Ackley, C. Adams, N. Adhikari, R. X. Adhikari, V. B. Adya, C. Affeldt, D. Agarwal, et al., Physical Review X 13 , 041039 (2023), ISSN 2160-3308.\n- [2] EPTA Collaboration and InPTA Collaboration, Antoniadis, J., Arumugam, P., Arumugam, S., Babak, S., Bagchi, M., Bak Nielsen, A.-S., Bassa, C. G., Bathula, A., Berthereau, A., et al., A&A 678 , A50 (2023).\n- [3] D. J. Reardon, A. Zic, R. M. Shannon, G. B. Hobbs, M. Bailes, V. Di Marco, A. Kapur, A. F. Rogers, E. Thrane, J. Askew, et al., The Astrophysical Journal Letters 951 , L6 (2023), ISSN 2041-8205, 2041-8213.\n- [4] G. Agazie, A. Anumarlapudi, A. M. Archibald, Z. Arzoumanian, P. T. Baker, B. B'ecsy, L. Blecha, A. Brazier, P. R. Brook, S. Burke-Spolaor, et al., The Astrophysical Journal Letters 951 , L8 (2023). \nbetween the EM field and the Riemann tensor. Lastly, we point out that fixing the four-velocity of the material in a coordinate system or considering a rigid body within the context of general relativity is not appropriate. \nWe solve Maxwell's equations within the cavity. Using the solution for the electric field, we define the transfer function and the dimensionless GW signal. To derive the antenna pattern, we separate the GW signal into its strength and pattern function. Averaging the pattern function over the polarization angle yields the antenna pattern. As illustrative examples, we present the antenna patterns for the EM cavities using the TM 010 and TE 212 -modes, respectively, at their own resonance frequencies. \nOur work enables a profound understanding of the operational principles of GW detectors employing EM cavities by clearly elucidating the interplay among electric fields, acoustic oscillations, and GWs. Owing to the generality of our analysis, it is relevant to various GW detectors employing EM fields. Our paper does not address the measurement principles and methods for the induced EM fields. We perceive this as a challenging problem, and it will be the focus of our future work.", 'Acknowledgments': 'The authors thank Jai-chan Hwang and Sung Mook Lee for their helpful discussion. We appreciate APCTP for its hospitality during the completion of this work. This work was supported by IBS under the Project Codes, No. IBS-R018-D1 and No. IBS-R017-D1-2023a00. 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Palenta, Classical and Quantum Gravity 40 , 085007 (2023), ISSN 0264-9381, 1361-6382.\n- [21] E. Gourgoulhon, 3+1 Formalism in General Relativity , vol. 846 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2012), ISBN 978-3-642-24524-4.\n- [22] J.-c. Hwang and H. Noh, Annals of Physics 454 , 169332 (2023), ISSN 00034916.\n- [23] C. Park, The Astrophysical Journal 940 , 58 (2022), ISSN 0004-637X, 1538-4357.\n- [24] J. M. Stewart and M. Walker, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 341 , 49 (1974).\n- [25] D. Giulini, The Rich Structure of Minkowski Space (Springer Netherlands, Dordrecht, 2010), pp. 83-132, ISBN 978-90-481-3475-5.\n- [26] K. S. Thorne and R. D. Blandford, Modern Classical Physics: Optics, Fluids, Plasmas, Elasticity, Relativity, and Statistical Physics (Princeton University Press, Princeton, NJ, 2017), ISBN 0-691-15902-5.'} |
2024A&A...690A.288B | We describe the NIRSpec component of the JWST Deep Extragalactic Survey JADES and provide deep spectroscopy of 253 sources targeted with the NIRSpec microshutter assembly in the Hubble Ultra Deep Field and surrounding GOODSSouth. The multiobject spectra presented here are the deepest so far obtained with JWST amounting to up to 28 hours in the lowdispersion R30300 prism and up to 7 hours in each of the three mediumresolution R 1000 gratings and one highdispersion grating G395H R 2700. Our lowdispersion and mediumdispersion spectra cover the wavelength range 0.65.3 m. We describe the selection of the spectroscopic targets the strategy for the allocation of targets to microshutters and the design of the observations. We present the public release of the reduced 2D and 1D spectra and a description of the reduction and calibration process. We measure spectroscopic redshifts for 178 of the objects targeted extending up to z 13.2. We present a catalogue of all emission lines detected at SN gt 5 and our redshift determinations for the targets. Combined with the first JADES NIRCam data release these public JADES spectroscopic and imaging datasets provide a new foundation for discoveries of the infrared universe by the worldwide scientific community. | 2024-10-01T00:00:00Z | ['10.1051/0004-6361/202347094', 'arXiv:2306.02467', '2024A&A...690A.288B', '2023arXiv230602467B', '10.48550/arXiv.2306.02467'] | ['instrumentation: spectrographs', 'surveys', 'galaxies: evolution', 'galaxies: high-redshift', 'Astrophysics - Astrophysics of Galaxies'] | JADES NIRSpec initial data release for the Hubble Ultra Deep Field Redshifts and line fluxes of distant galaxies from the deepest JWST Cycle 1 NIRSpec multiobject spectroscopy | 2,024 | 172 | 0.69 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF'] | 158 | https://arxiv.org/pdf/2306.02467.pdf | {'Redshifts and line fluxes of distant galaxies from the deepest JWST Cycle 1 NIRSpec multi-object spectroscopy': "Andrew J. Bunker ⋆ 1 , Alex J. Cameron ⋆ 1 , Emma Curtis-Lake ⋆ 2 , Peter Jakobsen 3 4 , Stefano Carniani 5 , Mirko Curti 6 7 8 , Joris Witstok 7 8 , Roberto Maiolino 7 8 9 , Francesco D'Eugenio 7 8 , Tobias J. Looser 7 8 , Chris Willott 10 , Nina Bonaventura 3 4 11 , Kevin Hainline 11 , Hannah Übler 7 8 , Christopher N. A. Willmer 11 , Aayush Saxena 1 9 , Renske Smit 12 , Stacey Alberts 11 , Santiago Arribas 13 , William M. Baker 7 8 , Stefi Baum 14 , Rachana Bhatawdekar 15 , Rebecca A. A. Bowler 16 , Kristan Boyett 17 18 , Stephane Charlot 19 , Zuyi Chen 11 , Jacopo Chevallard 1 , Chiara Circosta 15 , Christa DeCoursey 11 , Anna de Graa ff 20 , Eiichi Egami 11 , Daniel J. Eisenstein 21 , Ryan Endsley 22 , Pierre Ferruit 15 , Giovanna Giardino 23 , Ryan Hausen 24 , Jakob M. Helton 11 , Raphael E. Hviding 11 , Zhiyuan Ji 11 , Benjamin D. Johnson 21 , Gareth C. Jones 1 , Nimisha Kumari 25 , Isaac Laseter 26 , Nora Lützgendorf 25 , Michael V. Maseda 26 , Erica Nelson 27 , Eleonora Parlanti 5 , Michele Perna 13 , Bernard J. Rauscher 28 , Tim Rawle 15 , Hans-Walter Rix 29 , Marcia Rieke 11 , Brant Robertson 30 , Bruno Rodríguez Del Pino 13 , Lester Sandles 7 8 , Jan Scholtz 7 8 , Katherine Sharpe 21 , Maya Skarbinski 21 , Daniel P. Stark 11 , Fengwu Sun 11 , Sandro Tacchella 7 8 , Michael W. Topping 11 , Natalia C. Villanueva 21 , Imaan E. B. 1 31 11 \nWallace , Christina C. Williams , and Charity Woodrum \n(A ffi liations can be found after the references) \nReceived June 4, 2023; accepted 21 May 2024", 'ABSTRACT': 'We describe the NIRSpec component of the JWST Deep Extragalactic Survey (JADES), and provide deep spectroscopy of 253 sources targeted with the NIRSpec micro-shutter assembly in the Hubble Ultra Deep Field and surrounding GOODS-South. The multi-object spectra presented here are the deepest so far obtained with JWST, amounting to up to 28 hours in the low-dispersion ( R ∼ 30 -300) prism, and up to 7 hours in each of the three medium-resolution R ≈ 1000 gratings and one high-dispersion grating, G395H ( R ≈ 2700). Our low-dispersion and mediumdispersion spectra cover the wavelength range 0 . 6 -5 . 3 µ m. We describe the selection of the spectroscopic targets, the strategy for the allocation of targets to micro-shutters, and the design of the observations. We present the public release of the reduced 2D and 1D spectra, and a description of the reduction and calibration process. We measure spectroscopic redshifts for 178 of the objects targeted extending up to z = 13 . 2. We present a catalogue of all emission lines detected at S / N > 5, and our redshift determinations for the targets. Combined with the first JADES NIRCam data release, these public JADES spectroscopic and imaging datasets provide a new foundation for discoveries of the infrared universe by the worldwide scientific community. \nKey words. galaxies: high-redshift - galaxies: evolution - instrumentation: spectrographs - astronomical databases: surveys', '1. Introduction': "JWST (Gardner et al. 2023) is the largest programme in astrophysics to date, and is far more than simply the successor to the Hubble Space Telescope (HST). As well as having seven times the collecting area of HST, JWST operates over a wider range of wavelengths (0 . 6 -25 µ m) in a lower-background environment (at L2), making it orders of magnitude more sensitive than previous observatories. One of the major goals of the JWST mission is to study the formation and evolution of galaxies, in particular in the early universe through observations of high redshift galaxies. \nThe JWST Advanced Deep Extragalactic Survey (JADES, Bunker et al. 2020; Rieke 2020; Eisenstein et al. 2023) is the largest Cycle 1 programme aiming to study galaxy evolution out to the highest redshifts. JADES is a coordinated survey designed and executed by the NIRSpec and NIRCam Guaranteed Time Observation (GTO) teams. It provides NIRCam and MIRI imaging as well as NIRSpec spectroscopy over two fields. An im- \nportant aspect of JADES is the assembly of a large data set of spectroscopic observations spanning from cosmic noon to within the epoch of reionization, enabling confirmation of high-redshift candidates, accurate redshift measurements, and unprecedented constraints on the physical conditions in distant galaxies. With such spectroscopy, we can explore the mass-metallicity relation, dust attenuation, star formation rates and star formation histories in galaxies, as well as ionization parameters, ionizing photon escape fraction, and the presence of any active galactic nuclei. Spectroscopy is also key to understanding the physical states of the interstellar, circumgalactic and intergalactic media, and their evolution with cosmic time. Crucially, assembling this data set is enabled by the new multi-object spectroscopy (MOS) capabilities of JWST with the near-infrared spectrograph (NIRSpec; Jakobsen et al. 2022). \nNIRSpec operates in the range 0 . 6 -5 . 3 µ m, and has three spectral resolutions: a low-dispersion prism ( R ≈ 30 -300) which captures all the wavelength range with a single exposure, and medium- and high-resolution gratings ( R ≈ 1000 and \nR ≈ 2700) which use three bands to cover the wavelength range. One unique feature of this spectrograph is its use of a microshutter assembly (MSA), developed specifically for NIRSpec to enable multi-object spectroscopy of hundreds of objects at once over a 3 ' . 6 × 3 ' . 4 field of view (Ferruit et al. 2022). \nAs described in Eisenstein et al. (2023), JADES has Deep and Medium tiers, where the Medium tier adds area to capture rarer objects, while the Deep tier allows for searches of the faintest, and most distant galaxies. The survey covers two fields with huge legacy data sets thanks to the Great Observatories Origins Deep Survey (GOODS Dickinson et al. 2003; Giavalisco et al. 2004), GOODS-South and GOODS-North. As well as the multi-wavelength e ff orts of the original GOODS survey, there have been extensive observing e ff orts in the same area, including, but not limited to, the CANDELS survey with Hubble (Cosmic Assebly Near-Infrared Deep Extragalactic Legacy Survey, Grogin et al. 2011; Koekemoer et al. 2011) and the GREATS survey with Spitzer (GOODS Re-ionization Era wideArea Treasury from Spitzer , Stefanon et al. 2021). In particular the GOODS-South field includes the region of the sky with the deepest Hubble images ever taken, the Hubble Ultra Deep Field (HUDF, Beckwith et al. 2006; Bouwens et al. 2010; Ellis et al. 2013), where we focus the Deep portion of our survey. The Medium tier adds area with observations in GOODS-North, and the extended GOODS-South field, predominantly within the footprint of the CANDELS data. \nIn this paper, we present our deep spectroscopy of targets in the HUDF and surrounding GOODS-South field and outline our target selection strategy. We release the raw data and make our reduced data products available to the community 1 , and in a companion paper (Rieke et al. 2023) we present the complementary JADES NIRCam imaging of the HUDF. From the prism and medium-dispersion R ≈ 1000 spectra we derive redshifts and fluxes of prominent emission lines. The data from the single high-dispersion grating used (G395H, R ≈ 2700) also forms part of this data release, but we do not perform detailed on this analysis in this paper. \nThe structure of this paper is as follows. Section 2 describes how potential spectroscopic targets were selected from imaging data (primarily a combination of JADES NIRCam and HST), and how these were allocated to di ff erent priority classes so that the NIRSpec MSA configuration could be optimised for our science goals. The NIRSpec observations are described in Section 3 and the data processing is outlined in Section 4. In Section 5 we present our redshift measurements, and detected emission line fluxes of individual galaxies. Our conclusions are in Section 6. Throughout this work, we assume the Planck 2018 cosmology (Planck Collaboration et al. 2020) and the AB magnitude system (Oke & Gunn 1983).", '2. Targets': "JADES observations take NIRCam imaging and NIRSpec spectroscopy in parallel. As the survey progresses, JADES aims to leverage NIRCam photometry to select targets for later NIRSpec observations where possible, as this will enable the identification of the highest-redshift objects and facilitates near mass-limited samples at lower redshifts. However, for many of our early observations, we take spectroscopy in regions which have not yet been imaged by NIRCam. \nIn the initial planning phase, the Deep tier presented here was to be observed prior to NIRCam imaging and hence would com- \nprise only targets previously identified (mostly from HST imaging). However, scheduling changes meant that we ended up having NIRCam data available shortly before our final MSA configuration needed to be set 2 . This unforeseen opportunity was exploited scientifically to refine the target selection by making use of the additional JADES photometry from NIRCam images, with observations completed 16 days before the NIRSpec observations. Thus, our selected targets represent a NIRCam-based selection, supplemented with some HST-based targets compiled from the literature. We note that the NIRCam images available when drawing up our target list did not cover the full region of the NIRSpec MSA (see Figure 2). \nWe used NIRCam data taken between 29th September and the 5th October 2022 described in Rieke et al. (2023), which added nine photometric bands, potentially improving photometric redshifts over previous HST-based studies as well as identifying HST-dark sources. We used a very early reduction of the data and describe the limitations of this in Appendix B. We measured the HST and NIRCam photometry using 0 '' . 3-arcsec diameter apertures and applying aperture corrections for each filter appropriate for compact sources. We estimated photometric redshifts from two di ff erent SED-fitting codes with very di ff erent template sets and underlying assumptions, eazy (Brammer et al. 2008) and beagle (Chevallard & Charlot 2016), and these were used in our target selection. \nIn this Section, we first describe our over-arching prioritisation system for allocating targets for spectroscopy. We then describe the assembly of these NIRCam- and HST-based catalogues, which formed the source material for our target allocation.", '2.1. Priority class system': "The target selection for the NIRSpec MOS observations was designed to prioritise rare targets, either at high redshift, or with low number density, while building up a statistical sample spanning from cosmic noon to within the epoch of reionization. This was achieved by sorting the potential targets into a limited number of priority classes and employing the NIRSpec team's eMPT software suite (Bonaventura et al. 2023) to optimise the placement of targets within each class in sequence on the MSA. The priority class criteria employed are presented in Table 1. The science goals for the JADES survey as a whole are diverse (Section 1, see also Eisenstein et al. 2023), and the first deep pointings presented here represent the initial step in building up the entire sample. \nWe emphasise that the JADES NIRSpec survey does not employ a single selection function, but within each priority class there is a well defined set of criteria. The highest priority targets (Class 1) are used to set and optimise the NIRSpec pointing centres (see Section 2.3) and are the bright, robust highest redshift candidates ( z > 8 . 5). Classes 2 and 3 allow for less robust candidates and fainter candidates, respectively, at similarly high redshifts. Progressing down the priority classes predominantly represents a progression in decreasing redshift, as the number counts then increase. A notable departure from this is Class 5 in which we include bright objects to achieve a few high signal-tonoise continuum spectra per pointing. \nIn Class 4, we aim to target galaxies in the redshift interval 5 . 7 < z < 8 . 5 which are expected to have su ffi ciently bright rest-frame optical emission lines to enable emission line ratio work and exploration of interstellar medium (ISM) conditions (Cameron et al. 2023; Curti et al. 2024). Our goal is to achieve S / N > 25 in the H α line (available at z < 7) or, when not available (i.e. z > 7), [O iii ] λ 5007. This would gives an expected S / N ≈ 8 or more for the H β line, and the resulting uncertainty of 10 -15% on the Balmer decrement, f ( H α ) / f ( H β ), allows for an estimate of the attenuation due to dust (e.g. Sandles et al. 2023). To achieve these target emission line fluxes in Class 4, we select on the rest-frame UV magnitude around 1500Å. At z ≈ 6, a galaxy with a star formation rate of 2 . 5 M ⊙ y r -1 has a magnitude in the F115W filter of AB = 27 . 5 for the rest-UV longward of the Lymanα break (assuming a Salpeter 1955 initial mass function and no dust extinction), and an expected H α flux of 3 × 10 -19 erg cm -2 s -1 at 4 . 5 µ m (adopting the Kennicutt 1998 conversion from star formation rate to H α flux). This should be detectable at S / N = 25 in the prism spectroscopy of the Deep tier of JADES (duration ≈ 100 ksec), using the STScI Exposure Time Calculator 3 . Hence for Class 4 we adopt a magnitude cut of AB = 27 . 5 in the broad-band filters just above the Lymanα break. Fainter targets in the same redshift interval appear in Class 6. \nClass 7 represents the statistical sample spanning 1 . 5 < z < 5 . 7, which will be built up over multiple tiers, spanning from cosmic noon to the epoch of reionization. Within Class 7, we paid particular attention to placing unusual objects first before the more common star-forming galaxy population. Specifically, galaxies which exhibited colours in the rest-frame UVJ colourcolour plane consistent with being passive or quenched galaxies were identified following the criteria for specific star formation rate (sSFR), log(sSFR / yr -1 ) < -9 . 5 given in Leja et al. (2019). Wealso prioritised ALMA sources which had a match to sources in the HST or NIRCam images (e.g. Aravena et al. 2016; Decarli et al. 2016; Rujopakarn et al. 2016; Dunlop et al. 2017; Franco et al. 2018; Yamaguchi et al. 2019; Hodge et al. 2013), along with AGN including those selected from the IR, from variability, or from X-ray selection with an optical / near-IR counterpart (Alonso-Herrero et al. 2006; Castelló-Mor et al. 2013; Del Moro et al. 2016; Luo et al. 2011, 2017; Sarajedini et al. 2011; Treister et al. 2006, 2009b,a; Young et al. 2012) \nAny unused areas on the MSA following the placement of sources in Classes 1-7 (described above) were filled with very low priority targets in Class 8 and 9, which comprised: fainter targets which did not pass the brightness to be in Class 7; targets at lower redshifts than z ≈ 1 . 5; and targets for which the astrometry was unreliable. Blank sky shutters were also added.", '2.2. Establishing input catalogue of possible spectroscopic targets': "The Hubble Ultra Deep Field (HUDF) and surrounding GOODS-South are very well studied fields. To provide an input target list for potential observation with the NIRSpec MSA, we compiled a large list of galaxies from the literature, which we cross-matched with our NIRCam-derived catalogues after first correcting the coordinates of the literature sources onto the same Gaia DR2 astrometric frame (see Appendix A), leveraging the CHArGE re-reduction of the GOODS-S HST imaging which has been registered to the GAIA DR2 astrometric frame (Koko- \nrev et al. 2022; Brammer 2023) 4 . Where no match to an HSTdetected object was identified within 0 '' . 3 with the NIRCambased catalogue (with co-ordinates defined as target centres), the target catalogue was supplemented with the HST-detected object. In regions of MSA footprint with no NIRCam coverage, all objects are taken from the HST-based catalogues. We later impose selection criteria to populate the various priority classes which dictated the allocation of observed sources to the MSA micro-shutters. The priority classes for each object eventually observed are presented in Table F.1, where we list the final priorities allocated on the basis of NIRCam photometry (where available), and we also give the initial priority allocations on the basis of HST data alone. \nA main driver of the JADES survey is to observe the highest redshift targets, for which we compiled a sample of galaxies which had been identified as z > 5 . 6 candidates by one or more studies in the literature, or from our NIRCam + HST analysis. The lower end of this redshift range corresponds to where the i ' -band drop-out technique using the HST / ACS filter set becomes e ff ective. The list of galaxies compiled from the literature includes any studies that have previously selected z ≳ 6 candidates based on the Lyman break technique and / or photometric redshifts (Bunker et al. 2004; Yan & Windhorst 2004; Oesch et al. 2010, 2013; Lorenzoni et al. 2011, 2013; Yan et al. 2010; Ellis et al. 2013; McLure et al. 2013; Schenker et al. 2013; Bouwens et al. 2015, 2021; Finkelstein et al. 2015; Harikane et al. 2016). The sample from the literature was largely based on Lymanbreak drop-out selection (e.g. Bunker et al. 2004; Bouwens et al. 2015 ) although some are more generally based on photometric redshifts (e.g. Finkelstein et al. 2015). We cross-matched di ff erent samples in the literature which present high redshift candidates, and we note that while many galaxies were in common (using a matching tolerance of 0 '' . 2), there was a significant fraction which appeared in only one selection. This may be due to those papers using earlier reductions of HST data, perhaps not including all the data now available, or slightly di ff erent colour cuts, S / N thresholds, and photometric aperture choices by the various research groups. Hence, to refine this selection of potential high-redshift targets, we inspected all the z > 5 . 6 candidates from the literature. We used a slightly lower redshift cut for this inspection of candidates than ultimately adopted for Classes 4 & 6 ( z > 5 . 7, Table 1) so as to allow slight changes in photometric redshift due to our remeasured photometry. For each z > 5 . 6 candidate, we re-measured the aperture photometry from the HST images (with 0 '' . 36-diameter apertures and appropriate aperture corrections) and ran photometric redshift fits with eazy (Brammer et al. 2008) and beagle (Chevallard & Charlot 2016). For those galaxies which also appeared in our NIRCam-based catalogue, we also calculated the photometric redshift including both the NIRCam and HST photometry. \nWe also visually inspected the HST images (and NIRCam images where available) in all wavebands, using a co-addition of all the HST data taken in the GOODS-South field from the Hubble Legacy Field v2.0 images (Whitaker et al. 2019; Illingworth et al. 2016), which goes deeper than many of the images used in the past to construct the early catalogues of Lyman break galaxies. For the sources selected from NIRCam photometry, we also removed spurious high redshift candidates due to artifacts and deblending issues. \nWe retained only the most robust candidates in our highest priority classes, those which were clearly detected at longer \nwavelengths, had a strong spectral break and were undetected at short wavelengths, and where the photometric redshifts strongly favoured a high redshift solution. Some objects were either only faintly detected or had spectral energy distributions where the photometric redshift was unclear (with both high and low redshift solutions possible). These were placed in a class for more marginal targets, which were allocated at lower priority than the more robust candidate high-redshift galaxies. In the case of the highest redshifts ( zphot > 8 . 5) from the previous literature, the most robust candidates with HST F160W magnitudes brighter than AB = 29 were placed in the top priority 'Class 1' (Table 1), with those judged to be less robust placed in Class 2. From our NIRCam-based selection, we added two targets not appearing in the literature to Class 1 which had robust photometric redshifts z > 9 and were brighter than AB = 29 . 5 mag in filters just longward of the putative Lymanα break, and a further two NIRCamselected targets which were judged to be less robust were added to Class 2. Galaxies with zphot > 8 . 5 and fainter than F160W AB = 29 in the literature-based selection appear in Class 3. \nGalaxies with redshifts in the interval 5 . 7 < z < 8 . 5 span the epoch of reionization and are also potentially selected by the Lyman-break technique using drop-outs in the F775W, F850LP and F105W filters on HST. For these targets, we impose a magnitude cut on the broad-band filter longward of the Lymanα break, sampling the rest-frame UV (a proxy for star formation). Those galaxies brighter than AB = 27 . 5 in that filter were allocated to Class 4 (with this magnitude cut justified in Section 2.1), with less robust candidates and slightly fainter galaxies (27 . 5 < AB < 29) in Class 6.1. Candidates fainter than AB = 29 appear in Class 6.2. Some objects were up-weighted in this visual inspection exercise from Class 6 to Class 4 if they showed signs of strong line emission in the NIRCam photometry ( < 20%). Our input sample, after visual inspection and photometric checks, comprised about 300 galaxies at z > 5 . 7 within the total NIRSpec MSA footprint. There were other cases (about 5% of the sample drawn from the literature) where we identified targets which seem to have flux below the putative Lymanα break, and these were demoted to lower redshift classes based on the our revised photometric redshifts (including the NIRCam photometry where available). A number of high-redshift candidates from the literature were essentially undetected in the full co-added HST imaging, and these were removed from our sample (about 20%, but we note that many of these would not have passed the magnitude cuts to place them in our very highest priority classes). \nGalaxies with photometric redshifts below z = 5 . 7 formed our lower-priority classes, in particular Class 5 (bright objects), and Class 7 (a magnitude-limited sample prioritised in redshift slices). When assigning priorities in Class 7, we used the opportunity to base the magnitude limit of AB = 29 mag on the longest wavelength NIRCam filter available (F444W), to make the selection as close to a mass-selected sample as possible, and to homogenise the selection with that planned for other tiers of JADES. Where NIRCam imaging was not available (or a literature source did not have a match in our NIRCam catalogue), we imposed a magnitude cut in HST / WFC3 F160W of AB = 29, as this H -band filter is the reddest available HST data. This HST photometry for each galaxy was drawn from the latest available catalogue in which it appeared out of: Whitaker et al. (2019), Rafelski et al. (2015), Skelton et al. (2014) or Guo et al. (2013). If the source did not appear in any of these large catalogues, then we adopted the HST H -mag from the discovery paper if available (e.g. Lyman break catalogues) or we remeasured the photometry. \nIn Class 7, photometric redshifts are used to assign objects to four di ff erent redshift bins, with the smaller number of objects in the higher redshift slice 4 . 5 < z < 5 . 7 being allocated to MSA shutters before the next slice (3 . 5 < z < 4 . 5) and then those with 2 . 5 < z < 3 . 5 and finally 1 . 5 < z < 2 . 5. In the MSA target allocation in Class 7, we first placed the unusual targets (quiescent galaxies, AGN and ALMA sources) descending through the four redshift bins in order (sub-classes 7.1-7.4) before then placing shutters on the more common star-forming galaxies, again working down the four redshifts bins in turn to allocate targets. Where available, the photometric redshifts were drawn from the high-redshift catalogues of Bouwens et al. (2021), Finkelstein et al. (2015) or Bouwens et al. (2015), which generally utilised the Lyman break in HST filters extending to the UV. We supplemented Class 7 with photometric redshifts from the UVUDF survey (Rafelski et al. 2015), or, if unavailable, from the 3DHST survey (Brammer et al. 2012; Skelton et al. 2014), which in particular extended to lower redshifts than the Lyman break selected catalogues. For some lower redshift objects, the additional NIRCam photometry was not guaranteed to improve the photometric redshifts due to the small aperture used compared to the size of the objects, and the di ff erences in point spread function (PSF). We therefore only replaced the HST-derived photometric redshift with the HST + NIRCam-derived photometric redshift when the beagle and eazy photometric redshifts agreed. Specifically, the redshift bin was assigned first using the HST-based photometric redshift (where available). This was then adjusted only if the range between the eazy and beagle (primary or secondary redshift solution) 95% credible regions overlapped, or the redshift solutions agreed within ∆ z = 0 . 1. As with Classes 1 -6, all the Class 7 sources were visually inspected on the HST and NIRCam images, and a few eliminated as being unreliable.", '2.3. Target assignment': "The NIRSpec multi-object spectroscopic observations presented in this paper were carried out in MOS mode with the MSA (Ferruit et al. 2022) with NIRCam operating in parallel. The MSA configurations employed were designed using the NIRSpec GTO team's so-called eMPT software suite (Bonaventura et al. 2023), and then imported into the STScI Astronomers Proposal Tool (APT) for execution. For a given choice of disperser and assigned roll angle, the eMPT is capable of identifying the pointings of the MSA on the sky that capture the largest possible number of high priority targets whose images fall within the open areas of operational shutters to a specified accuracy without their spectra overlapping on the detector. Three shutter tall slitlets were assigned to each target, and the telescope was nodded by one shutter facet (529 mas) along the spatial direction such that the targets were observed in each shutter in sequence. An 'acceptance zone' spanning 184 mas in the dispersion direction and 445 mas in the spatial direction was employed throughout, corresponding to the full open area of a shutter with ≃ 9 mas shaved o ff the edges. This was to prevent targets leaving the open shutter areas during any of the nods due to di ff erential optical distortion arising in the telescope and NIRSpec optics. For all targets, only shutters whose low resolution prism spectra avoid truncation by the gap between the two detector arrays of NIRSpec (Jakobsen et al. 2022; Ferruit et al. 2022) were employed. \nAs described in Bonaventura et al. (2023), the eMPT approach to designing the MSA masks starts by exercising its socalled 'initial pointing algorithm' (IPA) module. This identifies the ensemble of candidate pointings within a specified range of the nominal pointing that provide the largest possible coverage \nTable 1: Target prioritisation categories. \nNotes. The HST and JWST entries for each class denote the di ff erent priority criteria whether the source was primarily selected from JWST or HST(see text for details). The number of targets per MSA footprint were estimated from the full 3 ' . 6 × 3 ' . 4 field of view. \n( 1 ) In this table, z denotes a redshift estimate either from a photometric redshift, or from dropout criteria. ( 2 ) Denotes photometry derived from Kron apertures. ( 3 ) 'Rare galaxy up-weighting' was applied to targets that were identified as candidates for being either quiescent, hosting an active galactic nucleus (AGN), or Lyman-continuum leakers. ( 4 ) The success rate is the fraction of galaxies targeted who had a spectroscopic redshift measured within ∆ z = 0 . 1 of the predicted redshift interval for that priority class. Galaxies lying outside this range are classed as interlopers. ( a ) The spectrum of object 9992 is Class 3 is ambiguous and may show two sources, a low-redshift galaxy at z = 1 . 962 and hints of a second galaxy at z > 9. ( b ) One target in Class 4 for which we did not get a good spectrum, 10035328, is a star (with a proper motion of 0 '' . 16 between HST / WFC3 and NIRCam) and we class it as an interloper. \nof the targets designated as Priority Class 1 in the input catalogue at the roll angle assigned to the observation. Other eMPT modules are then employed to fill up the remainder of the MSA mask at each pointing with additional targets in decreasing order of scientific priority. For the observations presented here, three separate 'dithered' pointings were planned with the goal of smoothing out detector defects in the dispersed spectra beyond that achieved by the three nods performed at each pointing. For the highest priority targets the objective was to achieve the largest possible total exposure time by observing these targets at all three dithers, while for the brighter lower priority targets the desire was to observe as many targets as possible, especially considering that these targets are placed on the MSA last and therefore become progressively more di ffi cult to accommodate. For any given trial of three pointings drawn from the set of all optimal Priority Class 1 covering pointings identified by the IPA module, the eMPT distinguishes between targets that can be observed at all three pointings, in only two of the pointings, and in only a single pointing, and gives the user complete control over the order in which targets in each subset are placed on the MSA. This process is carried out for all candidate triple pointings that constitute reasonable dithers of the spectra on the detector, and the triple pointing achieving the best overall target coverage was selected as the final one. \nAnother important consideration when using the MSA is to avoid targets being contaminated by the unintended light from nearby targets entering any of the (nodded) slitlets. The eMPT automatically eliminates such targets in a 'point source' manner based on the input catalogue, but since the contamination due to extended sources is di ffi cult to automate, the candidate MSA masks produced by the eMPT were subjected to a final visual inspection. Remaining undesirable targets were flagged and subsequently removed from the MSA masks. However, since the removal of a higher priority target can significantly change the placement of all lower priority targets that are placed after it, rather than start the process again from scratch, bespoke software was employed that allowed the process to converge after one or two iterations by optimally filling the gaps in the MSA opened up by removed contaminated targets with other non-overlapping ones, while leaving all others in place. \nThe grating exposures taken at each pointing employed as the starting point the same MSA masks as the prism exposures, but were modified using bespoke software to protect the grating spectra of the first five priority class targets by closing all shutters containing lower priority targets whose spectra collided with those of the higher priority targets. \nThrough the above process, prism spectra of a total of 253 unique objects were obtained in the three pointings and series of exposures described in this paper. Of these, 27% were observed in all three pointings, 24% in two pointings, and 49% in a single pointing. The three pointings cover 145, 155 and 149 individual prism targets each. In comparison, the grating observations cover a total of 198 unique targets, of which 28% are observed at three pointings, 21% at two pointings and 51% in a single pointing. The three grating pointings cover 119, 121 and 111 individual targets each. \nIn practice, the late addition of high priority NIRCam sources, and re-prioritisation of the catalogue (see Section 2.2) were incorporated without being able to re-optimise the pointings themselves. As a consequence, only four of the six highest priority targets in Table F.1 were observed in three dithers as opposed to all of them as would have been the case if the pointing could have been tweaked. One of the added Priority 1 NIRCam \ntargets (ID 2773) was observed in two pointings and a second one (ID 17400) in only a single pointing.", '3. Observations': "The NIRSpec MSA observations of JADES Deep / HST were taken on UT 21-25 October 2022 as the JWST Program ID: 1210 (PI: N. Lützgendorf). The observations were split into three visits, which di ff ered in their pointings by < 1 arcsec and employed separate MSA masks (see Section 2.3) but identical exposure sequences. The three pointings were selected such that they shift the spectra on the detector by a su ffi cient amount in order to smooth out detector defects; one shutter (268 mas or 2.6 pixels) in the dispersion direction plus one shutter (529 mas or 5.0 pixels) in the spatial direction for the second pointing, and three shutters (804 mas or 7.8 pixels) in the dispersion direction for the third pointing. \nAt each pointing we took observations with the lowresolution prism and four grating / filter combinations (G140M / F070LP, G235M / F170LP, G395M / F290LP and G395H / F290LP). We used the NRSIRS2 readout pattern (Rauscher et al. 2017) with 19 groups for an integration time of 1400 seconds, with two integrations per exposure. The MSA configurations opened three adjacent shutters for each target and the targets were 'nodded' between these shutters (perpendicular to the dispersion direction) with an exposure at each position. For the prism only, this sequence was repeated four times to obtain very deep observations. At each one of the three pointings the total integration time (number of exposures) was 33.6 ks (24) for the prism and 8.4 ks (6) for each grating. Thus the sources observed in all three pointings attained total integration time of 100 ks for the prism and 25 ks for each grating.", '4. Data processing': "In processing this data, the NIRSpec GTO Team used a custom pipeline derived from the pipeline originally developed by the ESA NIRSpec Science Operations Team (SOT) described in section 4.3 of Ferruit et al. (2022) and based on the workflow and algorithms described in Alves de Oliveira et al. (2018). This custom pipeline will be presented in a future paper (Carniani et al., in preparation). We briefly describe here the main data reduction steps. The two NIRSpec detectors were read nondestructively multiple times using the NRSIRS2 readout mode. The master bias frame and dark current were subtracted, and we also corrected artefacts such as snowballs (Ferruit et al. 2022; Giardino et al. 2019). For each exposure we fit the slope (i.e. the count rate) for each pixel, identifying and removing jumps due to cosmic ray strikes, and flagging when saturation occurred. We background-subtracted the 2D spectrum in each shutter by taking the average of the two other exposures in the three-nod pattern. In some cases of spatially-extended objects, or those falling close to one end of a shutter, we excluded the adjacent shutter containing light from the target object (or in some cases a contaminating source) from the background subtraction. We note that very extended sources (a small minority of our targets) may be prone to some self-subtraction using this local background subtraction approach. \nThe individual 2D spectra from each shutter were then flat fielded and corrected for illumination by the spectrograph optics and the wavelength-dependent throughput of the dispersing element. The wavelength and flux calibration was then applied, with each pixel of the 2D spectrum having an associated wavelength and distance along the shutter, accounting for the slight \n<!-- image --> \nFig. 1: Overlay of target shutter positions onto the images, with the illuminated shutter regions outlined (0 '' . 46 × 0 '' . 20). The first 63 targets sorted by NIRSpec ID number (IDs 2333-7762) are shown here, starting at the top left, with the other 190 targets shown in Appendix C. A red outline indicates that the image is derived from the JWST / NIRCam F115W / F150W / F200W images from JADES (blue / green / red channels), and an orange outline denotes HST ACS-F850LP / WFC3-F125W / WFC3-F160W images. The individual images are 1 '' . 0 on a side, and are centred on the input coordinate of the target. North is up and East is to the left. \n-0.03 -0.017 -0.0041 0.009 0.022 0.035 0.048 0.061 0.074 0.087 0.1 Fig. 2: Field layout of the NIRSpec Deep-HST observations presented in this paper. The green rectange is the region covered by the original HST / ACS Hubble Ultra Deep Field. The red rectangle is the smaller area covered in the Ultra Deep HST / WFC3 imaging. The background image is the NIRCam F200W from JADES, except for the region to the right of the blue line which has not yet been observed by NIRCam; we show in this region the HST / WFC3 from GOODS-South / CANDELS in blue. The short red lines denote the five-shutter ( ≈ 2 '' . 6) extent observed (three open shutters each target per observation, nodded by ± 1 shutter for background subtraction). The the four quadrants of the NIRSpec MSA are clearly visible. More detailed views of each quadrant with the target ID numbers marked are shown in Figures 3 and Figures D.1, D.2 & D.3, which also show the sub-set of targets with grating spectra. The yellow scale bar at the bottom left is 1 arcmin in length. \n<!-- image --> \ntilt of the shutters relative to the dispersion direction, along with optical distortions. At each stage of the data reduction process we also propagated noise and data quality arrays. \nThe position of the object within the micro-shutter along the dispersion direction was also taken into account when applying the wavelength calibration - many of our targets are compact (Figure 1) with intrinsic sizes smaller than the 0 '' . 2 shutter width, so making wrong assumptions about the slit being uniformly illuminated or that each object is well centered would lead to wavelength o ff sets. We applied a path-loss correction to account for flux falling outside the micro-shutter; given the large wavelength range covered by NIRSpec (0 . 6 < λ < 5 . 3 µ m) it was critical to account for the considerable PSF variation with wavelength. We took into account the position of the object within the micro-shutter (see the 'intra-shutter o ff set' columns in Table F.1), and calculated the slit loss as a function of wavelength \nfor a point source at this location; this was a reasonable approximation for many of our targets which are often compact (Figure 1), particularly at high redshift and also potentially for the star-forming regions giving rise to emission lines within more extended galaxies. \nThe spectra are curved on the detector due to optical distortions, and we rectify the 2D spectrum (transforming such that the wavelength and distance along the microshutter in the crossdispersion direction lie along the x and y axes respectively), resampling the 2D spectrum onto a finer wavelength grid in the process. For the gratings, the re-sampled pixel scale was 6 . 36 Å, 10 . 68 Å and 17 . 95 Å for the G140M, G235M and G395M gratings, respectively. For the prism, where the resolving power varies in a non linear way between R ≈ 30 -330 (Jakobsen et al. 2022), we used an irregularly-gridded wavelength sampling with intervals between 26 -122 Å, with the coarsest sampling (largest \nFig. 3: One of the four MSA quadrants (Q3), showing allocation of micro-shutters to targets. The other quadrants are shown in Appendix D. Those shutters in green are covered by both the grating configurations and the low-dispersion prism. The red shutters are open only in the prism observations, as they would lead to overlapping spectra for our high priority targets in the grating configuration. Three micro-shutters are opened for each target, but the nodding by ± 1 shutter means that spectra are obtained over the areas covered by five shutters (including background) which are displayed. The field displayed is the NIRCam F200W image, and is 1 . 8 arcmin on a side. North is up and East is to the left. Shutters with the prefix 'B' are empty sky background. \n<!-- image --> \nwavelength interval per pixel) around 1 . 5 µ m where the resolving power is at its lowest. The 1D spectra for the three nod positions from each of the (up to) three pointings were then combined by a weighted average into a single 1D spectrum for each target, masking pixels previously flagged as bad in the data quality files, and rejecting outliers using a sigma clipping algorithm. We also separately combined all the 2D spectra for each target from the di ff erent nods and pointings, although the 1D combined spectrum comes from a combination of the 1D individual spectra rather than an extraction of the combined 2D spectrum. The resulting 1D and 2D spectra reduced data products for all targets are made available as part of this data release, along with the raw data. Example spectra covering a range of redshifts are shown in Appendix E (Figures E.1-E.10).", '5. Redshift determination and emission line fluxes': 'In this section, we report spectroscopic redshifts and emission line fluxes determined from our spectroscopic observations, and assess the success rate of our priority class system for target selection.', '5.1. Visual inspection and emission line fitting': "The 1D and 2D spectra of all spectral configurations were visually inspected as a first pass on the redshift determination. The SED fitting code bagpipes (Carnall et al. 2018, 2019) was run on the 1D Prism / CLEAR spectra and the redshifts arising from this fitting were used a starting point for the inspection, but ultimately the assessment of the human inspector would overrule this value if necessary. In many cases, several clear emission \n≈ \nFig. 4: Comparison of spectral measurements between the lowdispersion prism and medium-dispersion gratings. Upper: Comparison of redshift as determined from Prism / Clear and R ≈ 1000 grating observations for targets with emission lines clearly detected in both modes. There is a systematic o ff set of ∆ z = 0 . 0039, with the prism yielding systematically higher redshifts. Lower: Comparison of emission lines fluxes measured from prism and R ≈ 1000 grating. Measurements derived from the grating are systematically higher with a median value of fR ≈ 1000 / f PRISM = 1.105 and a standard deviation of 0.298 \n<!-- image --> \n<!-- image --> \n≈ \nlines were observed and the redshifts were unambiguous. Sometimes spectral breaks were visible, most notably the Lymanα break (e.g. Curtis-Lake et al. 2023; Looser et al. 2024), and sometimes the Balmer break or 4000 Å break. In fainter targets, the S / N of individual features were sometimes low, but the coincidence of more than one of these led to a tentative redshift. \nWe then performed emission line fitting to further refine the redshifts, and obtain measurements of the fluxes of significant emission lines. For the R ≈ 1000 grating data, the continuum was typically only marginally detected and we subtracted this by fitting a spline to the spectrum after masking out any regions which could be contaminated by prominent emission lines. We then performed a single-component Gaussian fit to each line individually, allowing the flux, redshift and line-width to vary independently. In the case of unresolved doublets, such as [O ii ] λλ 3726, 3729, we simply fit the entire doublet as a single component. In the case of H α and [N ii ] λ 6583, although these lines are never blended in our R ≈ 1000 grating data, these lines were fit simultaneously and had their line centroids fixed relative to one another. The line flux was obtained as the integrated area under the best-fit Gaussian, and the formal uncertainty on this \nFig. 5: Histogram of spectroscopic redshifts obtained from S / N > 5 emission lines. The separate histograms are for the medium-dispersion R ≈ 1000 gratings (flag A, darkest purple), additional galaxies with S / N > 5 emission lines detected with the low-dispersion R ≈ 30 -300 prism (flag B, lighter purple histogram) and galaxies with more marginal redshifts (flag C, lightest histogram). \n<!-- image --> \nGaussian fit was taken as the noise on the line flux. We retained only emission lines which were measured with S / N > 5, and visually inspected each fit to ensure the measurement was robust. The emission line fluxes arising from this are reported in Table F.2. We note that there were two cases (ID 10013704 and ID 8083) where a broad component under H α meant that a single component fit was not appropriate. In these cases, the reported H α flux is obtained by integrating the whole line between the zero-power points in the spectrum. \nIn the case of the Prism / CLEAR data, the much lower spectral resolution ( R ≈ 30 -300) means that blending of emission lines is much more common in these data. Furthermore, which emission lines are blended changes with galaxy redshift due to the wavelength-dependent nature of the resolution. For this reason, although the approach to emission line fitting on the Prism / CLEAR spectra largely followed the same process as described above for the R ≈ 1000 mode, some redshift-dependent modifications were implemented. As such, which line fluxes are reported as blends changes with redshift. At all redshifts for the prism data, H α + [N ii ] was fit as a single component, as were close doublets such as [O ii ] λλ 3726, 3729 and [S ii ] λλ 6716, 6731. The flux of H α + [N ii ] and [S ii ] λλ 6716, 6731 were fit for simultaneously, with fixed centroids. In all cases, the fit to H β and [O iii ] λλ 4959, 5007 was performed simultaneously with the centroids fixed relative to one another. Above z > 5 . 3, the fluxes of all three components were fit (and reported) independently. At lower redshifts, the ratio of [O iii ] λ 5007 / λ 4959 was fixed to 2.98, but the H β flux could still vary independently. Between 2 < z < 5 . 3, we report the flux of the [O iii ] λλ 4959, 5007 as a blend. For the [O iii ] λ 4363 and H γ complex, above z > 7 . 5, the resolution allowed for a two-component fit to yield fluxes that are reported separately in Table F.3. Between 5 . 3 < z < 7 . 5, this flux is measured with a two-component fit, but is reported as a blend. At lower redshifts this blend was fit with a single component. \nBelow z < 2 the reported fluxes of lines with rest-frame wavelengths blue-ward of 7000 Å ( λ obs ≲ 2 µ m) are no longer obtained from Gaussian fitting, but instead are measured simply by integrating the continuum-subtracted spectrum of the speci- \nblend. Lines red-ward of this are measured with Gaussian fitting and are fit independently with the exception of He i 10830 and Paγ which are fit simultaneously. \nWe note that there are many cases where emission lines were identified visually in the data that did not meet our S / N > 5 threshhold to be included in Tables F.2 - F.3, however we opted not to report these fluxes. Particularly in the case of the Prism / CLEAR spectra, which generally speaking have significant continuum detections, reported fluxes for fainter lines become highly sensitive to how the continuum is modelled. We also do not fit for Lymanα in the Prism / Clear spectra here as the flux measurement is highly sensitive to how the continuum and Lymanα break is modelled. Lymanα measurements are however reported in Jones et al. (2024) and Saxena et al. (2024). We also note, there may be cases where reported lines are blended with other faint lines, despite this not being explicitly reported as such here. For example, [Ne iii ] λ 3869 can be blended with He i λ 3889 emission. The reported flux in such cases where it appears as a single-peaked feature will reflect the whole complex. \nFor galaxies which had at least one emission line detected in the R ≈ 1000 data, we calculate zR ≈ 1000 as the S / N -weighted average of the redshifts arising from the measured centroids of detected, non-blended lines and adopt this as our preferred redshift (flag 'A' in Table F.1). There were 150 cases where a galaxy did not yield a grating redshift in this way (either due to low S / N , or lack of a grating spectrum), but in 52 of these a z PRISM could be derived analogously from the Prism / CLEAR fits (flag 'B' in Table F.1). This accounted for 155 highly confident redshift determinations. Of the remaining 98 cases, we report a further grade 'C' redshift for 23 targets where the redshift had been determined as being secure from visual inspection (either based on a spectral break and / or one or more low S / N emission lines), and in Table F.4 we simply report the redshift obtained from this original visual inspection. This leaves 75 targets for which the redshift is speculative, ambiguous or unable to be determined. These are heavily weighted toward our lowest priority classes. As can be seen from the slit overlays on the JWST / NIRCam or HST images in Figure 1, some targets fall on the edge of the micro-shutter which will reduce the flux. Some shutters do appear empty, and these are largely targets based on catalogues from the literature which are either spurious or whose astrometry is less accurate (e.g. not from HST). \nThe medium-dispersion gratings yield more accurate redshifts than the low-dispersion prism, with the typical uncertainty in the centroid for a S / N = 10 line being 1 Å for the G140M grating, rising to 2 Å for the G395M, compared to 16 -50 Å for the prism. Hence for flag A redshifts, the typical uncertainty is ∆ z / (1 + z ) ≈ 10 -4 and for flag B redshift the typical uncertainty is ∆ z / (1 + z ) ≈ 0 . 0003 -0 . 003. Flag C redshift were determined visually and so are less precise. A histogram of redshifts determined from these spectra is shown in Figure 5.", '5.2. Comparison of prism and grating observations': 'We note that our Prism / CLEAR spectra are all non-overlapping, and thus cannot contain contamination from targets placed elsewhere on the MSA. This is not true for the grating data, for which spectra can be overlapping (although our highest priority targets are protected, see Section 2.3). Thus, these spectra occasionally show spurious emission lines. However, given that the Prism / CLEAR observations were significantly deeper than the R ≈ 1000 grating data, targets which are observed with significant emission lines in the grating always show the same \nsignificant emission in the low-resolution data. Thus, all the grating redshift measurements here can be confirmed to be robust. We note that there were three targets (IDs 8880, 9343, and 10013545) for which the the reduced Prism / CLEAR spectrum is su ffi ciently corrupted that, beyond simply confirming the presence of emission lines, we did not measure the emission line fluxes to be reported in Table F.3. However, in all three cases, secure redshifts and line fluxes were already measured from the R ≈ 1000 data. \nWe have 100 galaxies for which we have robust measurements of both zR ≈ 1000 and z PRISM. In Figure 4 we compare the redshift determinations from the Prism / CLEAR spectra and the R ≈ 1000 gratings for each galaxy, and find a small systematic o ff set (with the grating determination of redshift slightly lower than that from the prism) with a median o ff set of 0.00388 and standard-deviation 0.00628. \nWealso compare the flux ratio for the same lines where these are detected in both the Prism / Clear and the R ≈ 1000 gratings, excluding lines which are significantly blended in the prism, and these ratios are shown in Figure 4. We note that the grating fluxes are on average 10% higher than the prism. Bunker et al. (2023) found that the flux in the prism agreed well with the NIRCam magnitudes (where the spectrum was integrated over the NIRCam filter bandpass), with the grating spectra showing less good agreement, suggesting that the flux calibration in the prism is more accurate.', '5.3. Comments on individual targets': 'A few MSA shutters exhibited unusual spectral features, often due to more than one source in the shutter. We briefly discuss these below, along with objects which are likely to be stars where proper motion can be seen between the HST / WFC3 images and the JWST / NIRCam images taken ∼ 13 years later.', '5.3.1. ID 5293 - star': "Proper motion can be identified between HST / WFC3 F775W imaging and JWST / NIRCam F277W imaging for this object. Furthermore the spectrum looks visually like a brown dwarf star. No proper motion was clearly seen when comparing di ff erentepoch observations from HST alone - possibly due to the motion being comparable to the spatial resolution of HST / WFC3. However, with the better spatial resolution of NIRCam / SW, we detect a motion of ≈ 0 '' . 05.", '5.3.2. ID 7624 - two sources in shutter': 'Two sources can clearly be seen in the HST imaging (Figure 1), and the slit falls between the two sources. Object 7624 in Class 7.7 was the intended target (to the south of the slit), but a Lymanbreak galaxy (a F435W b -band dropout) lies just to the north. We observe line detections consistent with [O iii ] λ 5007 and H α at z = 2 . 665 (from the target object 7624) and also at z = 4 . 854 from the b -band drop-out.', '5.3.3. ID 8896 - possible double source': 'This micro-shutter was originally targeted on a low redshift galaxy (Class 7.8). We detect at least three compelling emission lines, and one more marginal line. There are emission lines that are consistent with [O iii ] and H α at z = 1 . 984, and this is reported in Table F.3. However, we note that there are two robust \nlines that are consistent with a z = 6 . 287 galaxy seen with [O iii ] and H α . The imaging does not obviously reveal the presence of two objects (Figure 1), however there does not seem to be a plausible redshift solution that matches all of these lines simultaneously for a single object.', '5.3.4. ID 9992 - possible double source': "This object was targeted as a z > 8 . 5 candidate (Class 3). The Prism / CLEARspectrum reveals a number of emission lines. The two most significant emission features are consistent with [O iii ] and H α at z = 1 . 962. However, an emission line at 5.1 µ mcould be H β or [O iii ] λ 5007 at z > 9, and this would be consistent with a tentative spectral break observed at 1.25 µ m being a Lymanα break. The NIRCam photometry reveals two components separated by only 0 '' . 15 (Figure 1). One of these is photometrically consistent with a drop-out galaxy at z ∼ 10, while the other (over which the central shutter is better placed) is more consistent with lower redshift solutions. We do not consider our high redshift solution from the spectrum to be highly robust, and Table F.1 reports the low-redshift solution.", '5.3.5. ID 10040 - multiple sources in shutter': 'Imaging clearly shows multiple sources with flux in the shutter. The spectrum has clear detections of emission lines consistent with [O iii ] and H α at z = 3 . 14, which we report in Table F.3. There is also, however, continuum detected in the 2D spectrum, which appears to be spatially o ff set from the emission lines that we detect and which may arise from another source.', '5.3.6. ID 10035328 - star': 'Proper motion can be identified between the HST / WFC3 images and the JWST / NIRCam images.', '5.4. Quantifying the success of target selection': "To measure the success of the class-based allocation, we looked at which targets from which classes actually ended up having the redshift expected (and desired line flux S / N in the case of Class 4). \nIn our highest-priority Class 1 (predicted redshifts z > 8 . 5 and AB < 29), we targeted six galaxies, five of which were robustly confirmed to be at high redshift: three at z > 11 have previously been reported in Curtis-Lake et al. (2023) (GSz120 = 2773 at z = 12 . 63, GSz11-0 = 10014220 at z = 11 . 58, GSz130 = 17400 at z = 13 . 20) and have strong Lymanα breaks but no significant line emission. Galaxy 10058975 at z = 9 . 43 exhibits many strong emission lines (see Figure E.1), as does galaxy 8013 (which falls just below the targeted redshift cut at z = 8 . 47). One galaxy, ID 10014170, did not have obvious features in its spectrum and its redshift is ambiguous. Hence, we have a success rate of 83% in pre-selecting Priority Class 1 targets which are then spectroscopically confirmed to be at high redshift. \nIn Class 2 (candidates at z > 8 . 5 with AB < 29 which are more marginal), of the two targets one has a robust redshift of z = 9 . 68 (galaxy 6438), and the second target (galaxy 7300) has an inconclusive spectrum. Class 3 has three z > 8 . 5 candidates fainter than AB > 29, but even here we are successful in confirming the high redshift nature of some of these: of the three targets, ID 10014177 was previously reported in CurtisLake et al. (2023) as GSz10-0 at z = 10 . 38. Galaxy 9992 was \ndiscussed in Section 5.3.4; there is clearly a low-redshift interloper at z = 1 . 962, but inspection of the imaging reveals that there is a second galaxy, and the spectrum provides hints of other lines which may be consistent with a second source at z > 9. We regard this spectrum as inconclusive. The third galaxy (ID 6621) has no strong emission lines but may exhibit a spectral break consistent with a Lymanα break at a tentative redshift of z = 9 . 6. Hence, for all 11 of the z > 8 . 5 candidates targeted, seven were clearly at high redshift (a fraction of 64%), and four had inconclusive redshifts (36%). \nWenowdiscuss the success of the selection in Class 4, where galaxies at 5 . 7 < z < 8 . 5 were targeted which were su ffi -ciently bright ( AB < 27 . 5 in the wavebands corresponding to the rest-UV) that high S / N emission lines are expected. Of the 20 sources in Class 4 which were targeted, 17 were confirmed to be at high redshift (including galaxy 16745 at z = 5 . 57, which fell just below the targeted redshift range). The emission line fluxes for H α were typically brighter than 7 . 5 × 10 -19 erg cm -2 s -1 (except for one object, ID 6384) as expected from our rest-UV preselection (Section 2). The uncertainty on the line flux from the prism is about 0 . 3 × 10 -19 erg cm -2 s -1 as predicted, so we met our requirement of S / N > 25 in H α for the sample in Class 4, enabling the physics of the ISM and the metal enrichment to be explored (Cameron et al. 2023; Curti et al. 2024). Our overall success rate in Class 4 is 85%, although we note that one of the three objects (out of 20) for which we did not obtain a good spectrum is ID 10035328, which is a likely star (see Section 5.3.6). \nFainter candidates than in Class 4 but in the same redshift interval 5 . 7 < z < 8 . 5 were placed in Class 6.1 if they were brighter than AB = 29 in HST / F160W (or the NIRCam filter around rest-frame 1500Å), and in Class 6.2 if they were fainter than that. The success rate in Class 6.1 is very good, with eight of nine galaxies having redshifts in the targeted range. We note that object 8115 does not have emission lines but does a strong Lymanα break and a weaker Balmer break at z = 7 . 3, and the JADES spectrum has been discussed in Looser et al. (2024) as a potential quiescent galaxy. The NIRSpec spectrum of ID 3334 at z = 6 . 71 has been presented in Witstok et al. (2023) and shows evidence of broad rest-frame UV absorption around 2175Å. Object 3137 is a low-redshift interloper with z = 1 . 91, so our overall success rate in Class 6.1 is 89%. In the fainter Class 6.2, four of the seven objects have spectroscopic redshifts within the target range, with two galaxies also at redshifts slightly below this (object 8113 at z = 4 . 90 and object 17260 at z = 4 . 89). One spectrum (object 10014117) had no significant features from which a redshift could be determined. The NIRSpec spectrum of object 10013682 shows very strong Lymanα emission at the systemic redshift of z = 7 . 28, as discussed in Saxena et al. (2023). Our success rate for Class 6.2 is 57%, rising to 86% if the two galaxies at z = 4 . 9 are included. \nThe success rate for Class 7 is recorded in Table 1 for the sub-classes 7.5-7.8, where there are significant numbers ( > 20) of galaxies targeted. Taking the metric for 'success' as being a measured spectroscopic redshift within ∆ z = 0 . 1 of the intended redshift range, we have success rates of 83% in the highest redshift slice (4 . 5 < z < 5 . 7, Class 7.5), around 80% for Classes 7.6 and 7.7, and 64% for the lowest redshift bin (1 . 5 < z < 2 . 5, Class 7.8). The interloper fractions were ≈ 3 -4%, but these comprised galaxies only slightly outside the desried redshift bin (e.g. object 3892 has a redshift of z = 2 . 80 and was selected to be in Class 7.6 at 3 . 5 < z < 4 . 5). Those galaxies in Class 7 for which a reliable redshift could not be inferred amounted to < 20% of those \ntargeted, and these tended to be the sources which were less well centred within the shutters, resulting in large slit losses. \nOverall our priority class pre-selection strategy seemed successful; for the more robust galaxies, 80% or more of the time the spectroscopic redshift fell in the anticipated range, and the line fluxes for Class 4 (which had been pre-selected on the basis of the rest-UV) were also as anticipated.", '6. Conclusions': 'We have presented very deep spectroscopy obtained with JWST / NIRSpec in its multi-object MSA mode. In all, 253 targets were observed in this JADES Deep / HST spectroscopy covering the Hubble Ultra Deep Field, and the surrounding GOODSSouth, with total integrations times of up to 28 hours for the low-dispersion prism ( R ≈ 30 -300), and up to 7 hours in the three medium dispersion gratings ( R ≈ 1000) and one high dispersion grating (G395H, R ≈ 2700). We detected emission lines with S / N > 5 in 155 targets with the low-dispersion prism, 103 of which also had emission lines detected at this significance in R ≈ 1000 gratings. The robust redshifts determined for these galaxies spanned a range from z = 0 . 66 to z = 13 . 2, with 18 lying at z > 6. A further 23 galaxies has more tentative redshifts. We are able to detect emission lines at S / N > 5 as faint as ≈ 10 -19 erg cm -2 s -1 in our deepest prism spectra, and we have been able to confirm redshifts for some sources fainter than AB = 29. Our selection of targets preferentially places the rarer high redshift targets on the MSA at higher priority, with more numerous lower-redshift galaxies filling unused regions, so that wecan probe a large redshift range from \'cosmic noon\' ( z ∼ 2) to within the epoch of reionzation ( z > 6) with reasonable numbers of galaxies in several redshift slices. We have demonstrated that our pre-selection of targets from HST and JWST imaging, based on broad-band magnitudes and photometric redshifts (including many Lyman break galaxy candidates) is highly e ff ective, with ∼ 80% of galaxies targeted having spectroscopic confirmation within the expected redshift bin. Hence our target selection and the quality and depth of the NIRSpec MSA spectroscopy means that our science goals for the JADES project can be met. \nAcknowledgements. We sincerely thank Gabe Brammer for his work aligning HST imaging to the GAIA DR2 frame. In the absence of that work, this data release may well have amounted to 253 spectacularly deep spectra of empty sky. We thank the referee for helpful comments on this manuscript. The JADES Collaboration thanks the Instrument Development Teams and the instrument teams at the European Space Agency and the Space Telescope Science Institute for the support that made this program possible. We also thank our program coordinators at STScI for their help in planning complicated parallel observations. We thank all the members of the NIRSpec and NIRCam Instrument Science Teams for making these observations possible. AJB, AJC, AS, JC, GCJ, IW acknowledge funding from the \'FirstGalaxies" Advanced Grant from the European Research Council (ERC) under the European Union\'s Horizon 2020 research and innovation programme (Grant agreement No. 789056). ECL acknowledges support of an STFC Webb Fellowship (ST / W001438 / 1). The Cosmic Dawn Center (DAWN) is funded by the Danish National Research Foundation under grant no.140. SC acknowledges support by European Union\'s HE ERC Starting Grant No. 101040227 - WINGS. RM, JW, FDE, TJL, WB, LS, JS acknowledges support by the Science and Technology Facilities Council (STFC) and by the ERC through Advanced Grant 695671 \'QUENCH". JW also acknowledges support from the Fondation MERAC. RS acknowledges support from a STFC Ernest Rutherford Fellowship (ST / S004831 / 1). SA, BRP acknowledges support from Grant PID2021-127718NB-I00 funded by the Spanish Ministry of Science and Innovation / State Agency of Research (MICIN / AEI / 10.13039 / 501100011033). RB acknowledges support from an STFC Ernest Rutherford Fellowship [grant number ST / T003596 / 1]. This research is supported in part by the Australian Research Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project number CE170100013. EE, BJD, MR, FS acknowledges the JWST / NIRCam contract to the University of Arizona NAS502015. DJE is supported as a Simons Investigator and by JWST / NIRCam contract to the University of Arizona, NAS5-02015. RH acknowledges funding \nprovided by the Johns Hopkins University, Institute for Data Intensive Engineering and Science (IDIES). REH acknowledges acknowledges support from the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-1746060. MP acknowledges support from the research project PID2021-127718NB-I00 of the Spanish Ministry of Science and Innovation / State Agency of Research (MICIN / AEI / 10.13039 / 501100011033), and the Programa Atracción de Talento de la Comunidad de Madrid via grant 2018T2 / TIC-11715. BER acknowledges support from the NIRCam Science Team contract to the University of Arizona, NAS5-02015. The research of CCW is supported by NOIRLab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. CW is supported by the National Science Foundation through the Graduate Research Fellowship Program funded by Grant Award No. DGE-1746060. This study made use of the Prospero high performance computing facility at Liverpool John Moores University. This work was performed using resources provided by the Cambridge Service for Data Driven Discovery (CSD3) operated by the University of Cambridge Research Computing Service (www.csd3.cam.ac.uk), provided by Dell EMC and Intel using Tier-2 funding from the Engineering and Physical Sciences Research Council (capital grant EP / T022159 / 1), and DiRAC funding from the Science and Technology Facilities Council (www.dirac.ac.uk). 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Q., et al. 2012, ApJ, 748, 124 \n- 1 Department of Physics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, UK\n- 2 Centre for Astrophysics Research, Department of Physics, Astronomy and Mathematics, University of Hertfordshire, Hatfield AL10 9AB, UK\n- 3 Cosmic Dawn Center (DAWN), Copenhagen, Denmark\n- 4 Niels Bohr Institute, University of Copenhagen, Jagtvej 128, DK2200, Copenhagen, Denmark\n- 5 Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126 Pisa, Italy\n- 6 European Southern Observatory, Karl-Schwarzschild-Strasse 2, 85748 Garching, Germany\n- 7 Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, UK\n- 8 Cavendish Laboratory - Astrophysics Group, University of Cambridge, 19 JJ Thomson Avenue, Cambridge, CB3 0HE, UK.\n- 9 Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK\n- 10 NRC Herzberg, 5071 West Saanich Rd, Victoria, BC V9E 2E7, Canada\n- 11 Steward Observatory University of Arizona 933 N. Cherry Avenue Tucson AZ 85721, USA\n- 12 Astrophysics Research Institute, Liverpool John Moores University, 146 Brownlow Hill, Liverpool L3 5RF, UK\n- 13 Centro de Astrobiología (CAB), CSIC-INTA, Cra. de Ajalvir Km. 4, 28850- Torrejón de Ardoz, Madrid, Spain\n- 14 Department of Physics and Astronomy, University of Manitoba, Winnipeg, MB R3T 2N2, Canada\n- 15 European Space Agency (ESA), European Space Astronomy Centre (ESAC), Camino Bajo del Castillo s / n, 28692 Villanueva de la \nCañada, Madrid, Spain; European Space Agency, ESA / ESTEC, Keplerlaan 1, 2201 AZ Noordwijk, NL \n- 16 Jodrell Bank Centre for Astrophysics, Department of Physics and Astronomy, School of Natural Sciences, The University of Manchester, Manchester, M13 9PL, UK\n- 17 School of Physics, University of Melbourne, Parkville 3010, VIC, Australia\n- 18 ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), Australia\n- 19 Sorbonne Université, CNRS, UMR 7095, Institut d'Astrophysique de Paris, 98 bis bd Arago, 75014 Paris, France\n- 20 Max-Planck-Institut für Astronomie, Königstuhl 17, D-69117, Heidelberg, Germany\n- 21 Center for Astrophysics | Harvard & Smithsonian, 60 Garden St., Cambridge MA 02138 USA\n- 22 Department of Astronomy, University of Texas, Austin, TX 78712, USA\n- 23 ATG Europe for the European Space Agency, ESTEC, Noordwijk, The Netherlands\n- 24 Department of Physics and Astronomy, The Johns Hopkins University, 3400 N. Charles St., Baltimore, MD 21218\n- 25 European Space Agency, Space Telescope Science Institute, Baltimore, Maryland, USA\n- 26 Department of Astronomy, University of Wisconsin-Madison, 475 N. Charter St., Madison, WI 53706 USA\n- 27 Department for Astrophysical and Planetary Science, University of Colorado, Boulder, CO 80309, USA\n- 28 Observational Cosmology Laboratory, NASA Goddard Space Flight Center, 8800 Greenbelt Rd., Greenbelt, MD USA\n- 29 Max-Planck-Institut für Astronomie, Königstuhl 17, D-69117, Heidelberg, Germany\n- 30 Department of Astronomy and Astrophysics, University of California, Santa Cruz, 1156 High Street, Santa Cruz, CA 95064, USA\n- 31 NSF's National Optical-Infrared Astronomy Research Laboratory, 950 North Cherry Avenue, Tucson, AZ 85719, USA", 'Appendix A: Astrometry of HST-based targets': "Astrometric o ff sets have previously been noted between previous reductions of HST imaging and other datasets registered to the GAIA DR2 astrometric frame (Dunlop et al. 2017; Franco et al. 2018; Whitaker et al. 2019). These studies have corrected the astrometry of CANDELS and 3DHST onto the Gaia DR2 frame (Gaia Collaboration et al. 2016, 2018) with a bulk o ff set of 0.26 arcsec. \nWe used the Complete Hubble Archive for Galaxy Evolution (CHArGE) re-reduction of the HST imaging in GOODSS, where all input frames have been carefully registered to the GAIA DR2 astrometric frame (Kokorev et al. 2022; Brammer 2023) 5 . In order to attain the astrometric accuracy required to ensure light is captured by an MSA shutter, we revisited the astrometry of the literature photometric candidates. For the large catalogues from 3DHST and UVUDF, and for the Lyman break surveys of Finkelstein et al. (2015) and Harikane et al. (2016), we matched the reported position of all brighter than F160W < 26 to the GAIA-DR2-registered catalogy = ue using 0 '' . 5 tolerance. Based on this, we identified that, in addition to the bulk o ff -set previously identified, there was also a plate scale di ff erence amounting to about 1 part in 5000. Across the full GOODS-S field (10 ' × 15 ' in size) this amounts to systematic errors larger than the width of a micro-shutter, highlighting the need for correcting this e ff ect. \nWe fit a simple tangent-plane astrometric transformation allowing for a bulk o ff set, plate scale, and rotation to these o ff -sets. We favoured the approach of fitting an astrometric transformation to the coordinates over re-measuring the centroids of objects in the CHArGE images because this new reduction had 100 mas drizzled pixels, and was not optimised for the selection of z ≳ 6 targets in the HUDF, where the HST data were deepest. Thus, many of of the targets that ended up in our highest priority classes were not clearly detected in these reductions. \nFitting this simple transformation to the Skelton et al. (2014) catalogue, we found that we reduced the residual RMS scatter on the positional o ff sets to 34 mas across the entire GOODS-South field. This corresponds to 17% of the illuminated NIRSpec slit width of 0 '' . 2. \nGiven many of our highest priority targets were taken from Lyman-break catalogues, and did not necessarily appear in Skelton et al. (2014), we also constructed separate astrometric transformations for catalogues from Bouwens et al. (2015, 2021), Finkelstein et al. (2015) and Harikane et al. (2016). We found that very similar astrometric o ff sets were present in these catalogues, however the exact magnitude of each component of the correction varied slightly. The residuals on transformed coordinates were similarly ∼ 30 -50 mas after applying the relevant transformation. \nWe considered the corrections applied to Skelton et al. (2014) to be the most robust, since that catalogue had more entries than the Lyman-break catalogues. Thus, to obtain updated 'GAIA DR2' coordinates, we used the correction derived from Skelton et al. (2014) for all targets that had a counterpart in this catalogue. \nIn some cases, high-priority targets were not matched to a counterpart in the Skelton et al. (2014) catalogue, in which case we used a correction from the astrometric fit to one of the catalogues of Finkelstein et al. (2015), Harikane et al. (2016) or Bouwens et al. (2021) to obtain updated coordinates. \nThere were some cases where we placed high-priority targets that were not in one of the catalogues discussed above (e.g. Bouwens et al. 2011a and the z ∼ 10 candidate from McLure et al. 2013). In these case we remeasured the centroid using the latest HLF ( Hubble Legacy Field) v2.0 reduction of the GOODS-South images, which we assessed as having astrometry in su ffi ciently good agreement with the GAIA DR2 frame. \nFinally, we retained some targets in our catalogue for which we did not have a reliable conversion of the reported coordinates to the GAIA DR2 frame. However, these targets were flagged such that they could not appear any higher than Priority Class 9, and were only placed at the expense of extra sky shutters (Table 1).", 'Appendix B: Caveats of early data release': 'At the time of target selection two weeks after the images were obtained, several issues present in the image reduction and analysis may have a ff ected the prioritisation of the targets. The flat fields that were available at the time introduced spurious smallscale structure in the background at faint levels in some areas, leading to a large number of spurious detections close to the detection limit in those areas. Improvements to the flat fields since then will allow us to push to deeper limits in future targeting. In particular, Class 3 was designed to include fainter, less secure high redshift targets, but we retained those allocated from the HST pre-selection, and did not supplement with JWSTbased sources fainter than AB = 29 . 5 mag. Improvements in the flat fielding, in the small and large scale background subtraction including wisps, and in the object deblending, especially near large bright sources, means that the current photometric measurements and source positions may di ff er from the very early estimates available at the time of target selection. In particular the Kron-based aperture measurements that were used for the flux cut in Class 7, which was designed to approach a total magnitude cut, were significantly impacted by these improvements.', 'Appendix C: Slit Overlays of Targets': "Figures C.1 -C.3, continued from Figure 1, showing the positions MSA shutter overlaid on observed targtes. \n<!-- image --> \nFig. C.1: Overlay of target shutter positions onto the images for target IDs 7809-21598 sorted by NIRSpec ID number, starting at the top left. The illuminated shutter regions are outlined (0 '' . 46 × 0 '' . 20). The image is derived from the JWST / NIRCam F115W / F150W / F200W images from JADES (blue / green / red channels). The individual images are 1 '' . 0 on a side, and are centred on the input coordinate of the target. North is up and East is to the left. \nFig. C.2: Overlay of target shutter positions onto the images for target IDs 21842-10011955 sorted by NIRSpec ID number, starting at the top left. The illuminated shutter regions are outlined (0 '' . 46 × 0 '' . 20). A red outline indicates that the image is derived from the JWST / NIRCam F115W / F150W / F200W images from JADES (blue / green / red channels), and an orange outline denotes HST ACSF850LP / WFC3-F125W / WFC3-F160W images. The individual images are 1 '' . 0 on a side, and are centred on the input coordinate of the target. North is up and East is to the left. \n<!-- image --> \nFig. C.3: Overlay of target shutter positions onto the images for targets ID 10011974-10106979 sorted by NIRSpec ID number, starting at the top left. The illuminated shutter regions are outlined (0 '' . 46 × 0 '' . 20). A red outline indicates that the image is derived from the JWST / NIRCam F115W / F150W / F200W images from JADES (blue / green / red channels), and an orange outline denotes HST ACS-F850LP / WFC3-F125W / WFC3-F160W images. The individual images are 1 '' . 0 on a side, and are centred on the input coordinate of the target. North is up and East is to the left. \n<!-- image -->", 'Appendix D: Layout of Shutters in the HUDF/GOODS-South Field': "Figures D.1 -D.3 continue from Figure 3, showing the remaining three quadrants of the MSA positioned on the field. \n-0.03 -0.017 -0.0041 0.009 0.022 0.035 0.048 0.061 0.074 0.087 0.1 Fig. D.1: Quadrant 1 of the MSA, showing allocation of micro-shutters to targets. Those in green are covered by both the grating configurations and the low-dispersion prism. The red shutters are open only in the prism observations, as they would lead to overlapping spectra for our high priority targets in the grating configuration. Three micro-shutters are opened for each target, but the nodding by ± 1 shutter means that spectra are obtained over the areas covered by five shutters (including background) which are displayed. The field displayed is the NIRCam F200W image. The yellow scale bar denotes 1 arcmin. North is up and East is to the left. Shutters with the prefix 'B' are empty sky background. \n<!-- image --> \n-0.03 -0.017 -0.0041 0.009 0.022 0.035 0.048 0.061 0.074 0.087 0.1 Fig. D.2: Quadrant 2 of the MSA, showing allocation of micro-shutters to targets. Those in green are covered by both the grating configurations and the low-dispersion prism. The red shutters are open only in the prism observations, as they would lead to overlapping spectra for our high priority targets in the grating configuration. Three micro-shutters are opened for each target, but the nodding by ± 1 shutter means that spectra are obtained over the areas covered by five shutters (including background) which are displayed. North is up and East is to the left. The field displayed is the NIRCam F200W image, except for the area West of the blue line which had not yet been imaged by NIRCam and we show the HST F160W image. The yellow scale bar denotes 1 arcmin. Shutters with the prefix 'B' are empty sky background. \n<!-- image --> \n-0.03 -0.017 -0.0041 0.009 0.022 0.035 0.048 0.061 0.074 0.087 0.1 Fig. D.3: Quadrant 4 of the MSA, showing allocation of micro-shutters to targets. Those in green are covered by both the grating configurations and the low-dispersion prism. The red shutters are open only in the prism observations, as they would lead to overlapping spectra for our high priority targets in the grating configuration. Three micro-shutters are opened for each target, but the nodding by ± 1 shutter means that spectra are obtained over the areas covered by five shutters (including background) which are displayed. North is up and East is to the left. The field displayed is the NIRCam F200W image, except for the area North-West of the blue line which had not yet been imaged by NIRCam and we show the HST F160W image. The yellow scale bar denotes 1 arcmin. Shutters with the prefix 'B' are empty sky background. \n<!-- image -->", 'Appendix E: Example Spectra': 'Example spectra covering a range of redshifts are shown in Figures E.1-E.10. The low-dispersion prism spectrum is shown, along with the medium dispersion grating of emission prominent lines. \nFig. E.1: Low-dispersion prism spectra (1D and 2D) of 10058975 at z = 9 . 4327, with the medium dispersion grating of prominent lines shown below. Green shaded regions on the 1D spectra denote the 1 σ errors. The wavelengths of common emission lines are denoted by vertical lines. \n<!-- image --> \nFig. E.2: As for Figure E.1, but showing galaxy 021842 at z = 7 . 9806. \n<!-- image --> \nFig. E.3: As for Figure E.1, but showing galaxy 018846 at z = 6 . 33. \n<!-- image --> \nFig. E.4: As for Figure E.1, but showing galaxy 022251 at z = 5 . 79. \n<!-- image --> \nFig. E.5: As for Figure E.1, but showing galaxy 018090 at z = 4 . 77. \n<!-- image --> \nFig. E.6: As for Figure E.1, but showing galaxy 007892 at z = 4 . 2287. \n<!-- image --> \nFig. E.7: As for Figure E.1, but showing galaxy 018970 at z = 3 . 7245. Note the Balmer break just below [OII] 3727. \n<!-- image --> \nFig. E.8: As for Figure E.1, but showing galaxy 003892 at z = 2 . 8072. Note the Balmer break just below [OII] 3727. \n<!-- image --> \nFig. E.9: As for Figure E.1, but showing galaxy 003892 at z = 2 . 227. Note the Balmer break just below [OII] 3727. \n<!-- image --> \nFig. E.10: Prism spectra (1D and 2D) of 10036262, a galaxy which does not exhibit strong line emission but has a clear spectral break corresponding to a Balmer / 4000 Å break at z = 3 . 566. \n<!-- image -->', 'Appendix F: Tables of targets': "We present tables of the priority classes and positions for each object observed (Table F.1), and the fluxes of lines detected at S / N > 5 in the medium dispersion grating spectra (Table F.2) and the low-dispersion prism spectra (Table F.3), along with the derived redshifts. \ntop). \nthe \nat \npriority \n(highest \nMSA \nthe \non \nplacement \nfor \norder \npriority \nin \ned \nrank \nsources, \ngeted \ntar \nof \nable \nT \n.1: \nF \nable \nT \n∗ \nAndrew J. Bunker et al.: JADES NIRSpec initial data release for the \nHubble \nUltra Deep Field \n0,11,14,16,18,20 \ncontinued \n.1: \nF \nable \nT \nA \n& \nA proofs: \nmanuscript no. aandaR1proof \ncontinued \n.1: \nF \nable \nT \nAndrew J. Bunker et al.: JADES NIRSpec initial data release for the \nHubble \nUltra Deep Field \ncontinued \n.1: \nF \nable \nT \nA \n& \nA proofs: \nmanuscript no. aandaR1proof \ncontinued \n.1: \nF \nable \nT \nAndrew J. Bunker et al.: JADES NIRSpec initial data release for the \nHubble \nUltra Deep Field \ncontinued \n.1: \nF \nable \nT \nArticle number, page 34 of 44 \n'' \nA \n& \nA proofs: \nmanuscript no. aandaR1proof \ncontinued \n.1: \nF \nable \nT \nNumber \nDec \nRA \nNIRCam\\_ID \nNIRSpec\\_ID \n-27.81302 \n53.157 \n103356 \n10000625 \n-27.81571 \n53.15266 \n-1 \n10105275 \n-27.82455 \n53.15882 \n93232 \n10106241 \n-27.79622 \n53.1198 \n-1 \n10106467 \nAndrew J. Bunker et al.: JADES NIRSpec initial data release for the \nHubble \nUltra Deep Field \nallocation, \nR1000 \nof \nset \nff \nsingle \ntar \n[15] \n), \nwere \n'N' \nbefore. \net al. only 'N'), gets et al. \nelton \ncounterparts \ner \nan \n2016 \nan \nIntra-shutter o ff set ( '' ) flag x-o ff set y-o ff set R100 --0.085 0.152 24 --0.023 -0.066 24 --0.064 0.057 24 --0.014 0.050 24 A 0.038 0.165 24 --0.038 0.018 24 --0.085 -0.080 24 --0.006 0.106 24 -0.013 0.156 48 -0.087 0.100 24 --0.010 -0.161 24 -0.057 -0.141 24 the NIRCam data release catalogue (Riek e redshift, z spec and associated flag, as described 'C' -deri v ed from spectral break and / or (nods and dithers) for the prism and for each detailing where the source has been found majority of the remaining tar gets ha v e gets which do not ha v e a counterpart in an y were used for identifying and prioritising and do not ha v e a numerical reference listed et al. ( 2016 ), [4] Bunk er et al. ( 2004 ), [5] Y [11] W ilkins et al. ( 2011 ), [12] Ellis et al. ( Bouwens et al. ( 2021 ), [19] Decarli et al. ( \nmatch \nclosest \nthe \nof \nID \nthe \nID; \nNIRSpec \nthe \nlist \ncolumns \nThe \nNotes. \ncatalogues \nge \nlar \nfour \nthe \nof \none \nfrom \nnot \n(i.e. \nasterisk \nan \nwith \nflagged \n( \nal. \npriority z spec 9 -9 -9 -9 -9 4.283 9 -9 -9 -9 -9 -9 -9 -within 0 '' . 2 in spectroscopic prism observ ations, of e xposures and citations catalogue. The et al. ( 2019 ). T ar catalogues which listed prior), ), [3] Harikane et al. ( 2011b ), 2016 ), [18] \nthe \nin \ndetected \nlines \nemission \n5 \n> \nN \n/ \nS \nfrom \ned \nv \nderi \n- \n'B' \ngrating, \nonly; \ndata \nHST \nfrom \npriority \nthe \n1400.5s; \nduration \nhas \nxposure \ne \n-based \nHST \nour \nin \ncounterpart \na \ne \nv \nha \nnot \ndo \nand \nNIRCam-based, \ner \nWhitak \nor \n/ \nand \n), \n2015 \n( \nal. \net \nRafelski \n), \n2013 \n( \nal. \net \nGuo \n), \n2014 \n( \nreference \nadditional \nindicate \nNumbers \ngets. \ntar \nof \nnumber \nsmall \na \net \nBouwens \n[17] \n), \n2013 \n( \nal. \net \ner \nSchenk \n[16] \n), \n2013 \n( \nal. \net \nOesch \n2015 \n( \nal. \net \nelstein \nFink \n[2] \n), \n2015 \n( \nal. \net \nBouwens \n[1] \nences. \nRefer \n-27.79811 \n53.13228 \n113342 \n10001916 \n-27.81454 \n53.11036 \n-1 \n10101728 \n-27.81787 \n53.15992 \n-1 \n10105256 \n-27.82452 \n53.16633 \n-1 \n10106242 \n-27.79802 \n53.12075 \n-1 \n10106443 \n-27.79957 \n53.16398 \n-1 \n10106979 \n-27.81778 \n53.15231 \n99653 \n10000367 \n-27.81937 \n53.16445 \n-1 \n10106944 \nthe \n); \n1 \nable \nT \n(see \nclass \npriority \nthe \ndetermination; \nloss \npath \nfor \nand \nnumber \nthe \npointings; \nmultiple \nthe \ner \nv \no \narcseconds \nin \nsource \nthe \n9. \nClass \nPriority \nin \nBouwens \n[10] \n), \n2010 \n( \nal. \net \nan \nY \n[9] \n), \n2010 \n( \nal. \net \nOesch \n[8] \n), \n2010 \n( \nArticle number, page 35 of 44 \nTable F.2: List of targets with one or more emission lines detected with S / N > 5 in the R ≈ 1000 grating data, and derived redshifts. \nTable F.2: continued", 'A & A proofs: manuscript no. aandaR1proof': 'Table F.2: continued \nNotes. Details of the emission line fitting can be found in Section 5.1. The last column gives a list of emission lines detected with S / N > 5, and the flux measured for that line. Targets marked ( ‡ ) were identified as likely having multiple objects in the shutter (see Section 5.3). Targets marked ( a ) showed a broad component under H α and reported flux was obtained from direct integration rather than a single component fit (Section 5.1). These measurements are also available in a machine-readable format on The Mikulski Archive for Space Telescopes: https://archive.stsci. edu/hlsp/jades . \nTable F.3: List of targets with one or more emission lines detected at S / N > 5 in the Prism / Clear data. \nTable F.3: continued \nArticle number, page 40 of 44 \nTable F.3: continued \nTable F.3: continued \nβ \n. \n. \nα \n. \n. \nλ \n. \n. \nα \n. \n. \nTable F.3: continued \nNotes. Details of the emission line fitting can be found in Section 5.1. Note that some lines (notably, H β , [O iii ] λ 4959 and [O iii ] λ 5007) are reported independently at high-redshift, but are reported as blends at lower redshift due to the reduced spectral resolution of the prism at shorter wavelengths. As a visual guide, horizontal rules demarcate points where the reporting of the H β + [O iii ] λλ 4959, 5007 complex changes. We also caution that some fluxes reported here as being for individual lines may feature non-negligible contributions from fainter lines (e.g. H α from [N ii ] λ 6583, or [Ne iii ] λ 3869 from He i λ 3889). In these cases, the reported flux will represent the total flux of the observed emission feature. Targets marked ( ‡ ) w ere identified as likely having multiple objects in the shutter (see Section 5.3). These measurements are available in a machine-readable format on The Mikulski Archive for Space Telescopes: https://archive.stsci.edu/hlsp/jades . \nTable F.4: Targets not meeting the S / N > 5 emission line threshhold required to appear in Tables F.2 - F.3, but for which a secure redshift could be identified visually based on spectral breaks or low S / N emission features.'} |
2024ApJ...975..259D | Recent determinations of the total rate of the SUP12SUPCSUP12SUPC nuclear reaction show nonnegligible differences with the reference reaction rate commonly used in previous stellar simulations. In addition the current uncertainties in determining each exit channel constitute one of the main uncertainties in shaping the inner structure of super asymptotic giant branch stars that could have a measurable impact on the properties of pulsating ultramassive white dwarfs WDs. We explore how new determinations of the nuclear reaction rate and its branching ratios affect the evolution of WD progenitors. We show that the current uncertainties in the branching ratios constitute the main uncertainty factor in determining the inner composition of ultramassive WDs and their progenitors. We found that the use of extreme branching ratios leads to differences in the central abundances of SUP20SUPNe of at most 17 which are translated into differences of at most 1.3 and 0.8 in the cooling times and size of the crystallized core respectively. However the impact on the pulsation properties is small less than 1 s for the asymptotic period spacing. We found that the carbon burns partially in the interior of ultramassive WD progenitors within a particular range of masses leaving a hybrid CONecore composition in their cores. The evolution of these new kinds of predicted objects differs substantially from the evolution of objects with pure CO cores. Differences in the size of the crystallized core and cooling times of up to 15 and 6 respectively lead to distinct patterns in the period spacing distribution. | 2024-11-01T00:00:00Z | ['10.48550/arXiv.2409.10793', '2024arXiv240910793D', '2024ApJ...975..259D', 'arXiv:2409.10793', '10.3847/1538-4357/ad7d8e'] | ['DA stars', 'Stellar evolution', 'Nuclear abundances', '348', '1599', '1128', 'Astrophysics - Solar and Stellar Astrophysics'] | Impact of Current Uncertainties in the SUP12SUPCSUP12SUPC Nuclear Reaction Rate on Intermediatemass Stars and Massive White Dwarfs | 2,024 | 172 | 0.5 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF'] | 0 | https://arxiv.org/pdf/2409.10793.pdf | {'Impact of current uncertainties in the 12 C + 12 C nuclear reaction rate on intermediate-mass stars and massive white dwarfs': "F rancisco C. D e G er ' onimo , 1 M arcelo M. M iller B ertolami , 1 T iara B attich , 2 X iaodong T ang , 3, 4 M' arcio C atelan , 5, 6 A lejandro H. C' orsico , 1 Y unjun L i , 7 X iao F ang , 8 and L eandro G. A lthaus 1 \n1 Instituto de Astrof´ısica de La Plata, CONICET-UNLP, La Plata, Argentina \n2 Max-Planck-Institut f¨ur Astrophysik, Karl-Schwarzschild Strasse 1, 85748 Garching, Germany \n3 Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China \n4 School of Nuclear Science and Technology, University of Chinese Academy of Sciences, Beijing 100049, China \n5 \nInstituto de Astrof´ısica, Pontificia Universidad Cat´olica de Chile, Av. Vicu˜na Mackenna 4860, 7820436 Macul, Santiago, Chile \n6 Millennium Institute of Astrophysics, Nuncio Monse˜nor Sotero Sanz 100, Of. 104, Providencia, Santiago, Chile \n7 China Institute of Atomic Energy, Beijing 102413, China \n8 Sino-French Institute of Nuclear Engineering and Technology, Sun Yat-sen University, Zhuhai 519082, China", 'ABSTRACT': 'Recent determinations of the total rate of the 12 C + 12 C nuclear reaction show non-negligible di ff erences with the reference reaction rate commonly used in previous stellar simulations. In addition, the current uncertainties in determining each exit channel constitute one of the main uncertainties in shaping the inner structure of super asymptotic giant branch stars that could have a measurable impact on the properties of pulsating ultra-massive white dwarfs (WDs). We explore how new determinations of the nuclear reaction rate and its branching ratios a ff ect the evolution of WD progenitors. We show that the current uncertainties in the branching ratios constitute the main uncertainty factor in determining the inner composition of ultra-massive WDs and their progenitors. We found that the use of extreme branching ratios leads to di ff erences in the central abundances of 20 Ne of at most 17%, which are translated into di ff erences of at most 1.3 and 0.8% in the cooling times and size of the crystallized core. However, the impact on the pulsation properties is small, less than 1 s for the asymptotic period spacing. We found that the carbon burns partially in the interior of ultra-massive WD progenitors within a particular range of masses, leaving a hybrid CONe-core composition in their cores. The evolution of these new kinds of predicted objects di ff ers substantially from the evolution of objects with pure CO cores. Di ff erences in the size of the crystallized core and cooling times of up to 15 and 6%, respectively leading to distinct patterns in the period spacing distribution. \nKeywords: stars:evolution-stars:interiors-white dwarfs, nuclear reactions, abundances', '1. INTRODUCTION': "Stars more massive than ∼ 7 M ⊙ undergo carbon fusion in the core (Kippenhahn et al. 2013). 1 The burning of 12 C occurs at temperatures above 6 × 10 8 Kthrough the formation of compound nuclear states of 24 Mg, denoted as 24 Mg ∗ , with excitation energies of 14 to 17 MeV above the ground level. The unstable 24 Mg ∗ states then decay through at least five \nchannels (Li et al. 2020), namely: \n12 C + 12 C → 24 Mg ∗ → 20 Ne + α ( Q = 4 . 62 MeV) , → 23 Na + p ( Q = 2 . 24 MeV) , → 23 Mg + n ( Q = -2 . 60 MeV) , → 16 O + 8 Be ( Q = -0 . 20 MeV) , → 24 Mg + γ ( Q = 14 . 93 MeV) . (1) \nAt the typical energies of carbon fusion in stars, the first two channels have similar (high) probabilities, making them many orders of magnitude more likely than the other channels. Consequently, the reactions 12 C( 12 C, α ) 20 Ne and 12 C( 12 C , p ) 23 Na dominate the total 12 C + 12 C cross section. The cross-section of these reactions must be known with high accuracy down to the Gamow peak energy EG = \n1 . 5 ± 0 . 3 MeV (for 5 × 10 8 K; Rolfs & Rodney 1988) since it not only a ff ects the production of 20 Ne and 23 Na, but also the subsequent evolutionary stages. The branching ratios of the α and p exit channels determine the total amount of 20 Ne and 23 Na produced inside stars. Previous works adopt 56% (Caughlan & Fowler 1988, hereinafter CF88) and 65% (Monpribat et al. 2022) as the branching ratio for the α channel. However, the probability of each exit channel becomes very uncertain at the typical temperatures that characterize C-burning (Pignatari et al. 2013). \nDespite considerable experimental e ff orts, the total 12 C + 12 C reaction rate remains uncertain at stellar temperatures. On the one hand, the heavy ion fusion studies performed by Jiang et al. (2007a) suggested that the fusion cross-section may be hindered at low energies, resulting in rates lower than the standard ones from CF88. On the other hand, low-energy experiments by Spillane et al. (2007) hint at the presence of resonant structure e ff ects at lower energies that are not considered in many works and would lead to an important enhancement of the 12 C + 12 C fusion rate at stellar temperatures. In the last decade, a significant e ff ort has been made by the nuclear physics community, both experimentally and theoretically, to understand the challenging regime of astrophysical low energies of the 12 C + 12 C fusion reaction (Jiang et al. 2018; Chien et al. 2018; Li et al. 2020; Mukhamedzanov 2022; Tang & Ru 2022; Morales-Gallegos et al. 2023). As recently reported by Monpribat et al. (2022), the present uncertainty of the 12 C + 12 C rate still covers orders of magnitude at the range of temperatures of interest for astrophysical applications. \nStars with initial masses in the range 7 M ⊙ ≲ M ini ≲ 10 M ⊙ might eventually become massive white dwarfs (WD). While most WDs comprise He or CO cores, these massive WDs have O and Ne as their main ingredients. These objects are the result of the evolution of progenitor stars that reach temperatures high enough to ignite their CO cores under degenerate conditions (Garcia-Berro & Iben 1994), as they evolve into the so-called super asymptotic giant branch (SAGB) phase. Classic works by Garcia-Berro & Iben (1994), Ritossa et al. (1996), Garc'ıa-Berro et al. (1997), and Iben et al. (1997) showed that the C-flash and subsequent C-burning leads to an oxygen-neon (ONe) core and, consequently, to an ONe WD (see Siess 2006, 2007, 2010; Camisassa et al. 2019, and references therein) or an electron-capture supernova (Tominaga et al. 2013; Doherty et al. 2017), depending on the intensity of winds. \nThe chemical structure of SAGB progenitors at the end of the C-burning phase, and thus at the WD stage, depends on how the C-burning proceeds. In this sense, the current uncertainties in the total reaction rate for C-burning and its branching ratios could have a non-negligible impact on the predicted structure of ultra-massive WD. The impact of the \nuncertainty of the 12 C + 12 C fusion rate on the properties of massive stars has been studied in several works (Gasques et al. 2007; Bennett et al. 2012; Pignatari et al. 2013; Chie ffi et al. 2021; Monpribat et al. 2022; Dumont et al. 2024). However, none of them explored the consequences of such uncertainties on the evolution of intermediate-mass stars and the final composition of ultra-massive WDs. Additionally, the properties of the nonradial g -modes of pulsating WDs depend also on the inner distribution of elements (see, for example, De Ger'onimo et al. 2017, 2022; Althaus et al. 2021). In this regard, the internal chemical profile left at the end of the C-burning phase plays an important role in the pulsation properties of ultra-massive pulsating WDs (see C'orsico et al. 2019a; De Ger'onimo et al. 2019, and references therein). \nIn this paper, we explore how the new measurements of the 12 C + 12 C nuclear reaction rate and the branching ratio adopted during C-burning a ff ect the final chemical composition and pulsations of ultra-massive WDs. The paper is organized as follows: in Section 2, we discuss the current status of the uncertainties in both the nuclear reaction rate and its branching ratios. In Section 3, we introduce the most important features of the computation of our numerical models, while in Sections 4 and 5 we explore the impact of those uncertainties on the properties of SAGB progenitors and pulsating ultra-massive WDs. Finally, in Section 6, we provide some concluding remarks.", '2. UNCERTAINTIES IN THE NUCLEAR REACTION RATE AND BRANCHING RATIOS': 'CF88 set a milestone in developing analytical formulae for several nuclear reaction rates, later used as standards in the computation of astrophysical numerical models. The 12 C + 12 C nuclear reaction stands out as very important for stellar evolution and yet remains subject to large uncertainties. This reaction has been studied intensively in recent decades (see Patterson et al. 1969; Becker et al. 1981; Spillane et al. 2007; Bucher et al. 2015, and references therein). Despite the experimental e ff orts made and the consistent results obtained for energies near the Coulomb barrier, the total 12 C + 12 C fusion reaction rate remains uncertain at temperatures of astrophysical interest, with di ff erent experiments di ff ering substantially (see Pignatari et al. 2013; Li et al. 2020; Monpribat et al. 2022, and references therein for a detailed discussion). This is due both to the fact that the experimental background noise is high at low energies, as well as to the fact that extrapolation of the experimental data to energies below the Coulomb barrier is a ff ected by the strong resonant behavior of the 12 C + 12 C cross-section. Recent studies suggest that the latter may su ff er from hindrance phenomena at low energies (Jiang et al. 2007b), resulting in a lower rate than the widely used one from CF88. Addition- \nFigure 2. Run of the maximum temperature of the flame (upper panel), the size of the CO core (middle panel), and the total 20 Ne content left (lower panel) for all the reaction rates adopted for a M ZAMS = 8 M ⊙ model during the lapse of the carbon burning phase. \n<!-- image --> \nFigure 1. Ratio of the total carbon burning reaction rate between Monpribat et al. (2022) and CF88 as a function of the temperature T 9 = T / 10 9 . The green line represents the HIN rates, whereas the HINRES rates are shown in red. The dashed blue line represents a ratio of 1. The shaded region depicts the range of typical temperatures characterizing 12 C burning inside SAGB stars. \n<!-- image --> \nally, resonant structures that would increase the reaction rate have been found at low energies (Spillane et al. 2007). \nRecently, Monpribat et al. (2022) derived the most up-todate reaction rates for carbon fusion, based on the measurement published by the STaged ELectron Laser Acceleration (STELLA) experiment (Fruet et al. 2020). In that work, the authors provided two updated formulae that account for the fusion hindrance phenomenon, 2 labeled HIN and HINRES in their study, based on di ff erent assumptions for the resonant behavior at low energies. Specifically, the HINRES rate di ff ers from the HIN one in that the former includes the effect of a possible low-energy resonance (Spillane et al. 2007), better fitting the measured cross-section. These formulae can be incorporated into the computation of stellar models. In Fig. 1 we show the ratio between the total HIN (green line) and HINRES (red line) reaction rates and the standard value from CF88 as a function of the temperature T 9 (in units of 10 9 K). For the typical temperatures characterizing C-burning regions inside stars (shaded region), the HINRES rate behaves very similarly to the one provided by CF88, whereas the di ff erence with respect to the HIN rate can reach more than a factor of ten. \nIn addition to the uncertainties in the total 12 C + 12 C -→ 24 Mg ∗ cross-section, there are large uncertainties in the 24 Mg ∗ decay channels. The uncertainty in the relative strengths of the 24 Mg ∗ α - and p -decay channels has a relevant impact on stellar nucleosynthesis calculations. The original branching ratio provided by CF88 was [56 / 44] for \nthe [ α/ p ] channels, respectively. Pignatari et al. (2013) adopted branching ratios of [65 / 35] instead, and explored the consequences of extreme values, [5 / 95] and [95 / 5] using a simple single-zone post-processing calculation. Recently, a Chinese-lead international collaboration has been reanalyzing the 12 C + 12 C reaction (Zhang et al. 2020a,b), and, in particular, its branching ratios (Li et al. 2020). Preliminary results indicate that, in the range of relevant stellar temperatures (0 . 6 < T 9 < 0 . 9), the relative strength of the α -channel to the p -channel decreases by about 30% with temperature. More importantly, the uncertainty in the relative strengths of both decay channels encompasses one order of magnitude (see next sections).', '3. STELLAR MODELS': "The SAGB models employed in this work were computed with the stellar evolution code Modules for Experiments in Stellar Astrophysics ( MESA ) version r21.12.1 (Paxton et al. 2011, 2013, 2015, 2018, 2019). Most of the adopted input physics corresponds to the default options that are described in detail in those papers, and will thus not be repeated here. The nuclear network adopted (sagb NeNa MgAl.net) accounts for 28 isotopes n, 1 , 2 H, 3 , 4 He, 7 Li, 7 Be, 8 B, 12 , 13 C, 13 -15 N, 16 -18 O, 19 F, 20 -22 Ne, 21 -23 Na, 24 -26 Mg, 25 -27 Al, including all the dominant nuclear reactions. Most of these reactions are taken from CF88 and Angulo et al. (1999), with some exceptions. Convective boundary mixing (CBM) was only adopted during core H and He burning, with the suggestion of Freytag et al. (1996) of an exponentially decaying velocity field. Here, the di ff usion coe ffi cient is adopted according to Herwig et al. (1997), with the free parameter f = 0 . 017. The main impact of CBM in these stages is to decrease the initial mass required for a progenitor star to reach \nFigure 3. Chemical profiles for the most abundant elements at the end of the carbon burning phase for a M ZAMS = 8 M ⊙ model for the three nuclear reaction rates adopted, namely CF88 (solid lines), HINRES (dashed lines), and HIN (dotted lines). The branching ratio was set to 65% and 35% for the α and p channels, respectively. In this plot, Xi is the element abundance per mass, whereas Mr is the Lagrangian mass coordinate. \n<!-- image --> \nC-ignition by about 2 M ⊙ . CBM at the C-burning convective zones and at the bottom of the convective envelope was not included. \nOur evolutionary sequences were computed from the zeroage main sequence (ZAMS) through central hydrogen and helium burning, up to the end of the carbon burning stage, before the thermally unstable phase, for models with initial masses 6 . 85 ≤ M ZAMS / M ⊙ ≤ 8 . 50 and metallicity Z = 0 . 02, assuming a solar-scaled composition. Each model was computed with a spatial resolution greater than 6000 spatial points from center to surface, while the temporal resolution achieved is of the order of 14 yrs, similar to those adopted in Farmer et al. (2015). The evolutionary behavior of SAGB progenitors before carbon ignition is very similar to that of intermediate-mass stars that end up as CO WDs, well documented in previous works (Garcia-Berro & Iben 1994; Siess 2006, 2010). The development of carbon burning, under the hypothesis of a strict Schwarzschild criterion (Schwarzschild 1906) for the delimitation of the convective region, is characterized by two di ff erent stages (Garcia-Berro & Iben 1994; Siess 2006). The first corresponds to the ignition of C at the point of maximum temperature inside the partially degenerate CO core, inducing a thermal runaway called the carbon flash. The sudden energy injection by the C-flash leads to a convective zone, which extends outward from the point of maximum temperature. The second stage corresponds to the development of a flame that propagates to the center and transforms the CO core into an ONe core (Garcia-Berro & Iben 1994; Siess 2006). During the Cburning phase, we took into account three di ff erent reaction \nFigure 4. The S ∗ ratio of the alpha and proton channels. The measurements of Kettner et al. (1980), Aguilera et al. (2006), and Spillane et al. (2007) are shown as filled circles in the colors of dark green, red, and brown, respectively. The best fit, upper and lower limits are shown as red lines. These limits are set at factors of 2.5 and 0.5 of the best fit, respectively \n<!-- image --> \nrates for the 12 C + 12 C reaction, namely the aforementioned HIN and HINRES ones from Monpribat et al. (2022) and the value from CF88, which has been widely used in previous evolutionary computations. Mass loss during AGB was considered adopting the Bloecker's wind prescription with a scaling factor of 0.1 (Bloecker 1995). \nThe adoption of the Schwarzschild Criterion for convective instability in our pre-WD models implies that RayleighTaylor unstable regions are not identified as such in our models. In our models, the CO-core during the early AGB develops a o ff -centered peak in the oxygen profile (Salaris et al. 1997). Such a profile would be unstable to Rayleigh-Taylor instabilities (Stevenson & Salpeter 1977). A proper assessment of mixing processes driven by chemical gradients during the early- and TP-AGB phases would require the computation of these processes during the formation of the oxygen peak. For simplicity, in our computations, the chemical profile of the CO core is homogenized by an ad-hoc algorithm just before the WD cooling phase.", '4.1. From uncertainties in the total rate': "In this section, we computed the evolutionary sequences adopting di ff erent reaction rates with a fixed branching ratio of 65% for the α channel. In Fig. 2, we show the impact of the di ff erences in the reaction rates shown in Fig. 1 on the properties of carbon burning. Specifically, we show the evolution of the maximum temperature of the flame ( T max, upper panel), the size of the CO core (middle panel), and the total 20 Ne produced (lower panel) as a function of the age, for an 8 M ⊙ progenitor model. As it is known, in the case of burning shells on top of electron-degenerate cores, the temperatures of the burning shell and the core increase with an increment of the size of the core (Miller Bertolami 2022). Sequences computed with a higher 12 C + 12 C reaction rate re- \nFigure 5. Ratio of the 12 C( 12 C, α ) 20 Ne to the 12 C( 12 C,p) 23 Na channel as a function of temperature. The color band indicates the region between the estimated upper and lower limits. \n<!-- image --> \nre slightly lower core temperatures to ignite carbon and undergo the carbon flash, which thus takes place at lower core masses and temperatures. Conversely, the sequences computed with the less e ffi cient reaction rates are characterized by a late onset of the C-flash, larger cores at the moment of the C-flash, and higher C-burning temperatures, as also seen in the study of more massive stars (Bennett et al. 2012; Pignatari et al. 2013). Once carbon starts to burn quiescently, more massive cores are forced, by hydrostatic equilibrium, to burn carbon faster. Consequently, those sequences with lower 12 C + 12 Creaction rates that ignited carbon with slightly more massive cores display shorter carbon-burning lifetimes. Di ff erences in the size of the cores at the moment of the Cflash, in their T max, and in the lifetime of the C-burning phase amount to 0.5, 13, and 40%, respectively. \nDespite the di ff erences in the burning temperature and the e ffi ciency of the reaction rate, the total 20 Ne content for each sequence is practically identical, as seen in the lower panel of Fig. 2. This is also observed with the distribution of the most important elements (see Fig. 3). The most noticeable di ff erences in the chemical structure arise in the outermost part of the core, as shaped at the final stages of C-burning. However, as a general trend, di ff erences in the resulting distribution of elements are small. \nIn the next sections, we will study the impact of such di ff erences on the age, crystallization, and pulsation properties of ultramassive DAV stars, which are H-rich WDs that present g -mode pulsations (C'orsico et al. 2019a; Catelan & Smith 2015). Before, we will analyze the uncertainties coming from the branching ratios of the 12 C + 12 C nuclear reaction.", '4.2. From uncertainties in the branching ratios': "As mentioned in Sect. 1, the 12 C( 12 C, α ) 20 Ne and 12 C( 12 C , p ) 23 Na reactions dominate carbon fusion inside \nstars, leaving 20 Ne and 23 Na as the main end products. The probability of each exit channel becomes very uncertain for the typical temperatures at which C-burning takes place in stars (Pignatari et al. 2013; Tang & Ru 2022). \nFrom the measurements made by Kettner et al. (1980), Aguilera et al. (2006), and Spillane et al. (2007), we assessed the estimation of the ratio of the α - to the p -channel, S ∗ α / S ∗ p , after correcting for the branching ratio of the missing channels with the procedure described in Li et al. (2020). The theoretical prediction is calculated using TALYS (Koning et al. 2005). The TALYS prediction is scaled by a factor 1.33 to match with the average of the experimental data. The results are shown in Fig. 4. We recommend upper and lower limits to be set by scaling the best fit with factors of 2.5 and 0.5, respectively. These limits covers all the measurements above E c . m . = 3.3 MeV. At lower energies, the measurement of Kettner et al. (1980) agrees with the limits while the measurement of Spillane et al. (2007) becomes significantly higher than that of Kettner et al. (1980) and exceeds the upper limit by a factor of 2.7 around E c . m . > 3.16 MeV. Further experimental and theoretical investigations are needed to resolve the discrepancy. \nThe reaction rates of the α - and p -channels are calculated based on the extrapolated and experimental S ∗ factor or cross-sections. At Ec . m . > 2.7 MeV, S ∗ α and S ∗ p obtained from the experimental measurements are used after correcting the missing channels in their measurement. S ∗ n is estimated with the experimental data and theoretical extrapolation from Bucher et al. (2015) and Bucher (2014). At lower energies, the S ∗ factor of 12 C + 12 C is generated with a uniform distribution bound by the upper and lower limits recommended in Li et al. (2020). S ∗ α and S ∗ p are generated with the branching ratios of the α - to p -channels with a uniform distribution bound by the upper and lower limits shown in Fig. 4. The reaction rate ratio of the Ne and Na channels are shown in Fig. 5. Our results show a decrease with temperature by about 40% from T 9 = 0 . 6 to T 9 = 0 . 9, and are significantly higher than the value α/ p = 1.27 recommended by CF88, at the temperatures of interest. \nTo quantify the impact of these uncertainties on the chemical composition of stars at the end of the SAGB phase, we performed calculations considering a wide range of branching ratios. To this end, we computed evolutionary sequences for stellar models with masses 6 . 80 ≤ M ZAMS / M ⊙ ≤ 8 . 50 considering the CF88 and HIN reaction rates and, from the results shown in Fig. 5 that indicates α/ p > 1 in the range of temperatures of interest, branching ratios of [ α/ p ] of [90 / 10], [80 / 20], [65 / 35], [60 / 40], [55 / 45], and [50 / 50]. Higher branching ratios for the α channel translate into smaller initial masses needed for the occurrence of the C-flash (by < 1 . 5%). This happens because the total energy released per burned 12 C nuclei by the α channel is higher. Consequently, \na lower burning rate is required to reach the carbon-burning luminosity at which the runaway occurs. This means that the core ignites carbon at a slightly lower core temperature, and consequently a smaller core mass. These di ff erences are minor, and the onset of the C-burning flash occurs almost at the same age, and di ff erences in the sizes of the cores are at most of 0.1% Conversely, the lifetime of the C-burning phase increases by up to 20% when the α channel is more e ffi cient due to the larger energy released per burnt nuclei. \nFor some of our sequences, the flame is quenched before reaching the star's center. This means that 12 C does not burn completely inside these objects and therefore the chemical structure left at the end of this stage is that of a core composed of CO surrounded by an ONe mantle. This depends on the reaction rate adopted and its branching ratio. Particularly, if the CF88 (HIN) reaction rate is adopted with the [65 / 35] branching ratio, this happens for models with initial masses 6 . 85 ≤ M ZAMS / M ⊙ < 7 . 10 (7 . 25 < M ZAMS / M ⊙ < 7 . 40). The resulting object will be a CO-core or hybrid 3 WD, depending on the total amount of 12 C burned. In Fig. 6, we show the chemical structure for a M ZAMS = 6 . 85 M ⊙ model. An unburnt CO core is surrounded by an ONe mantle similar to those found when overshooting is adopted during the C-burning phase (Denissenkov et al. 2013; De Ger'onimo et al. 2022). However, 3D hydrodynamic simulations revealed that this quenching of the carbon flame might be an artificial consequence of overshooting prescriptions adopted in 1D models (see Lecoanet et al. 2016). Conversely, our computations do not include any extra-mixing process during the C-burning stage, and consequently, the quenching of the C-flame is of a di ff erent nature. Depending on the size of the ONe mantle, i.e. the total amount of 20 Ne produced, the chemical structure at the WD phase, post RayleighTaylor homogenization, could be a WD with a core composed mostly of CO (see next sections). It should be noted that the initial mass ranges for the di ff erent regimes discussed in this paper are dependent on the assumptions on convective boundary mixing. More extended convective boundary mixing in the hydrogen burning core on the main sequence would imply lower initial masses for C-ignition (Wagsta ff et al. 2020). Convective boundary mixing at the carbon-burning flame is also expected to strongly a ff ect the development of the carbon flash (Denissenkov et al. 2013; De Ger'onimo et al. 2022). \nIn Fig. 7, we show the chemical profiles for the 7 . 50 M ⊙ model at the end of the C-burning phase, assuming the branching ratios adopted in the literature, namely [55 / 45] (CF88) and [65 / 35] (Pignatari et al. 2013), together with the \nFigure 6. Chemical structure for a 6 . 85 M ⊙ model computed with the CF88 reaction rate, at the end of the C-burning phase. The flame is quenched before reaching the star's center, leaving a core composed of CO surrounded by an ONe mantle. \n<!-- image --> \nFigure 7. Chemical structure for a 7 . 50 M ⊙ model at the end of the C-burning phase, for models adopting the CF88 reaction rate with [55 / 45], [65 / 35] and [90 / 10] branching ratios (dashed, dotted, and solid lines, respectively). \n<!-- image --> \nextreme case [90 / 10]. Depending on which of these branching ratios is selected, di ff erences in the central abundances of O and Ne can reach from 3% to 17%. The larger differences arise when comparing the calculations adopting the [55 / 45] and [90 / 10] branching ratios. For the [90 / 10] branching ratio, the 20 Ne production exceeds that of 16 O in the outermost part of the core ( Mr / M ⊙ > 0 . 8). This is because, as it moves inwards, the temperature the C-burning decreases. These di ff erences in the core abundances of 16 O and 20 Ne will be critical for the crystallization process during the WD cooling phase.", '5. CONSEQUENCES FOR THE PROPERTIES OF WDS': "This section is devoted to analyzing the e ff ect of the different assumptions studied previously on the chemical structure and composition of ultramassive WDs, their cooling timescales, crystallization, and pulsation properties. \nEach WD evolutionary sequence was computed from the point of maximum temperature of the cooling track at high luminosities, down to the development of the Debye cooling at low surface luminosities. Initial WD structures were constructed by mapping the detailed chemical structure of the H-free core at the end of the central C-burning stage into an already existing WD thermal structure. Our DA WDs models have a H content of M H ∼ 1 . 5 × 10 -6 M WD. The evolution and structure of the WD models presented in this section were computed with the LPCODE evolutionary code (for details, see Althaus et al. 2003, 2005, 2015, 2021; Miller Bertolami 2016). During crystallization, we took into account the release of latent heat and changes in the core chemical composition resulting from phase separation upon crystallization, using phase diagrams suitable for C / Oor O / Ne plasmas (Camisassa et al. 2019, 2022). For the sake of clarity, we selected a fiducial model of initial mass 7 . 50 M ⊙ , which corresponds to a M WD ∼ 1 . 15 M ⊙ . \nAs discussed in the previous sections, the adoption of extreme reaction rates (CF88 and HIN) during the C-burning phase leads to small di ff erences in the chemical structure of the ultramassive WD progenitors at the end of the C-burning phase. The next evolutionary stage is the thermally unstable phase and consists of the build-up of the most external part of the core. After the WD starts to cool and the Rayleigh-Taylor re-homogenization has taken place, the chemical structures of the WDs computed with di ff erent nuclear reaction rates are almost indistinguishable. As the WD keeps cooling, crystallization gets underway and, by the time the star reaches the ZZ Ceti (DAV) instability strip (at a temperature of ∼ 12 000 K), the cooling times and the size of the crystallized core are of 1743 (1721) × 10 6 yr and 92.9 (92.2) %, respectively, if the CF88 (HIN) rate is adopted, meaning small differences of 0.5 and 1.2%. \nRegarding the adoption of di ff erent branching ratios, the di ff erences found in the distribution of the most important chemical elements are more noticeable. In the top panel of Fig. 8, we show the chemical structure in terms of the outer mass fraction log(1 -Mr / M ∗ ) 4 pre- and post-RayleighTaylor re-homogenization, at high temperatures, for the fiducial model that accounts for the following branching ratios: [50 / 50], [65 / 35], [80 / 20], and [90 / 10]. The central 23 Na content remains always below ∼ 8% and, as higher production rates are adopted, the 20 Ne central content increases from \n<!-- image --> \nFigure 8. Distribution of the most important chemical elements for the M ZAMS = 7 . 50 M ⊙ models computed with the CF88 rate and di ff erent branching ratios (from upper left, in clockwise order, 5050, 65-35, 90-10, and 80-20) at the beginning of the WD's cooling path ( ∼ 10 5 K), pre- and post-Rayleigh-Taylor rehomogenization (upper panel, dashed and filled lines respectively) and at the ZZ Ceti stage ( ∼ 12 000 K, lower panel). The grey region reflects the crystallized part of the star, with a crystallization level above 90%. \n<!-- image --> \n30% up to ∼ 50% for the [90 / 10] branching ratio. In the latter case, the 20 Ne content produced at the outer part of the core surpasses the 16 O content before rehomogenization. However, the final chemical structure of the WD is that of a typical ONe core WD, but composed of almost equal parts of 16 O and 20 Ne. The C / O mantle on top of the O / Ne core is identical for each case. \nIn the lower panel of Fig. 8, we show the same chemical profiles but at the ZZ Ceti stage ( T ∼ 12 000 K, when the crystallization of the core has reached more than 90%, indicated as a shaded region in the plot). As every model has the same mass, the di ff erences in the percentage of the crystallized core, which amounts to up to 0.8%, come strictly from the di ff erences in their composition. The combined e ff ect of the di ff erent degrees of crystallization and inner composition leads to di ff erences in the WD cooling times of at most 1.3%. \nFigure 9. Distribution of the most abundant elements for a hybrid CONe-core model at 10 5 K, pre- and post-Rayleigh Taylor rehomogenization (upper and lower panel, respectively). \n<!-- image --> \nAs noted in Section 4.2, under some of the conditions discussed in the present paper, the carbon flame is quenched prematurely, forming a hybrid core WD. In Fig. 9, we show the chemical structure for the particular case of a hybrid core, with M WD / M ⊙ = 1 . 05 ( M ZAMS / M ⊙ = 6 . 85) pre- and postRayleigh-Taylor re-homogenization (upper and lower panel, respectively). The total 20 Ne content produced during the short-lived C-burning phase is then redistributed (as are the other species) throughout the core. Consequently, after rehomogenization, the WD model's structure resembles that of a pure CO-core WD, though with a non-negligible 20 Ne content ( ∼ 10%) in the core. These new kinds of objects predicted by our study, which can have masses up to M WD / M ⊙ ∼ 1 . 11, di ff er substantially from objects with pure CO cores. The presence of 20 Ne modifies the structure of the WD (particularly its central density) so that di ff erences in the evolution are significant. This can be seen in Fig. 10, where we show the evolution of the e ff ective temperature for a pure CO and the hybrid model. 5 Both models start at the same evolutionary point in the cooling track but di ff er in their central compositions, which are 30%-67% ( 12 C16 O) for the pure COcore model and 23%-61%-10% ( 12 C16 O20 Ne) for the hybrid model. As the models cool down, the hybrid one starts crystallizing earlier than the CO-core model. The crystallization onset occurs at 751 × 10 6 yr for the hybrid composition, while for the pure CO-core model, it starts at 870 × 10 6 yr. At 12 000 K (grey line), the age di ff erences are of ∼ 6%. \nThe comparison of their chemical structures at this point can be seen in Fig. 11, where we show the distribution of the most important chemical species for both models. Both the \nFigure 10. Evolution of the e ff ective temperature T e ff as a function of the age, for both the pure M ZAMS = 6 . 85 M ⊙ CO core and hybrid models. The black horizontal dotted line indicates a T e ff = 12 000 K. \n<!-- image --> \nFigure 11. Chemical profile of the M ZAMS = 6 . 85 M ⊙ hybrid and pure CO-core models (upper and lower panel, respectively) at 12 000 K. The shaded region depicts the crystallized portion of the core, which reach crystallization levels of 56.5 and 41.8%, respectively. \n<!-- image --> \nabsence of 20 Ne in the pure CO-core model and the crystallized size of the core are the most prominent di ff erences at this point, while the 4 He-bu ff er region and pure 1 H-envelope remain the same.", '5.1. Asteroseismology': "The period spectrum and mode-trapping properties of g -modes of DAV WDs depend sensitively on the precise shape of the Brunt-Vaisala frequency across the interior of the star, and, in particular, the location and shape of the bumps produced by the chemical composition interfaces (De Ger'onimo et al. 2019; Althaus et al. 2010). Thus, any change in the \n<!-- image --> \nFigure 12. Comparison of the period spacing ∆Π i as a function of the periods of the modes Π i for the CF88 and HIN models (upper and lower panel, respectively). The green dashed line shows the value of the asymptotic period spacing. \n<!-- image --> \nchemical profiles translate into changes in the WD's expected pulsation properties. \nThe computation of the pulsation properties for all our ultra-massive pulsating DA WD models was done using the LP-PUL pulsation code described in C'orsico & Althaus (2006), previously employed in the study of the properties of ultra-massive WD models (De Ger'onimo et al. 2019, 2022) and used to perform asteroseismic studies of ultra-massive ZZ Ceti stars (C'orsico et al. 2019b; Kilic et al. 2023). Element di ff usion was included for all models from the beginning of the WD cooling track. The 'hard sphere' boundary conditions were adopted when accounting for the e ff ects of crystallization on the pulsation properties of the g -modes. These conditions assume that the amplitude of the eigenfunctions of g -modes is drastically reduced below the solid and liquid interface, as compared with the amplitude in the fluid region (Montgomery & Winget 1999). \nFor the comparison of the pulsation properties, we computed the forward period spacing ( ∆Π k = Π k + 1 -Π k ), a quantity frequently used in asteroseismic analyses, that reflects the mode-trapping features of the models. In Fig. 12, we show the distribution of period spacing as a function of the periods for dipole modes, resulting from the assumption of di ff erent reaction rates. As all the models have the same mass, the similarity in ∆Π i reflects the fact that the crystallized portion of the star is similar in both cases, meaning that most of the different core-chemical features do not have a significant impact on the g -mode pulsations that might otherwise have allowed us to tell the di ff erent models apart. Consequently, the resulting asymptotic period spacing values are nearly identical. The period spacing distributions are also very similar when considering di ff erent branching ratios of the 12 C + 12 C reaction rate during the progenitor evolution, as shown by Fig. 13. \nFigure 13. Same as Fig. 12, but for the CF88 model computed with di ff erent branching ratios, as indicated in the upper left corner of each panel.Figure 14. Same as Fig. 12, but for the CO-core and hybrid WD models of Fig. 11 (upper and lower panels, respectively). \n<!-- image --> \nIn contrast, the hybrid model shows perceptible di ff erences in the trapping characteristics of the pulsation modes compared to the pure CO-core models, as seen in Fig. 14. For Π i > 300 s, the CO-core model shows more frequent minima (a shorter trapping cycle) in their distribution of ∆Π i values than does the hybrid model. The reason for this is relatively straightforward. These trapped modes at long periods ( Π i > 500 s) are modes with larger amplitudes in the homogeneous region above the crystallized core. As the outer border of the homogeneous region stays basically fixed, the larger the crystallized core, the smaller the homogeneous cavity in which the modes resonate. As shown by De Ger'onimo et al. (2019), this results in longer trapping cycles in that regime. It is worth noting that this di ff erence in the trapping cycle of the two types of models would not be present if the mixed region were neutrally buoyant, as discussed by Montgomery & Dunlap (2024). The actual tem- \nre profile above the crystallized core is a matter of ongoing discussion in the field (e.g. Castro-Tapia et al. 2024). Additionally, di ff erences in the asymptotic period amount to about ∼ 2 s, this being a consequence of di ff erences in the crystallization degree of the core.", '6. CONCLUSIONS': "In this work, we implemented the recently derived total nuclear reaction rate for carbon fusion, 12 C + 12 C, from Monpribat et al. (2022), with the goal of evaluating its impact on computations of the structure and evolution of ultra-massive WDs and their progenitors, as compared to the case where the canonical 12 C + 12 C rate from CF88 is used. We also explored how the current uncertainty in the branching ratios for the 12 C + 12 C reaction's α and p exit channels, which are the dominant ones at temperatures of astrophysical interest, affect the internal composition and evolution of these stars and their progenitors. \nOur extensive numerical experiments were carried out using MESA . Specifically, we computed evolutionary sequences, from the ZAMS up to the end of the C-burning phase, for models with initial masses 6 . 85 ≤ M ZAMS / M ⊙ ≤ 8 . 50 and a metallicity Z = 0 . 02. In addition to exploring the 12 C + 12 C rates from Monpribat et al. (2022) and CF88, a wide range of [ α/ p ] branching ratios was also considered, from [50 / 50] up to [90 / 10]. The resulting structures were subsequently evolved, using LPCODE , along the WD cooling track down to the ZZ Ceti (DAV) instability strip. When the models reached the latter phase, we computed their pulsation properties using the LP-PUL code. \nWe found that using less e ffi cient nuclear reaction rates results in a late onset of the C-burning phase, larger cores, higher burning temperatures, and, consequently, shorter Cburning phase lifetimes. Despite these di ff erences in the structure and evolution of SAGB progenitors, the impact on the distribution of the chemical elements is almost negligible. In contrast, the existing uncertainties in the relative e ffi -ciency of the α and p exit channels constitute the most important uncertainty in determining the final chemical structures of SAGB progenitors. Di ff erences in the central 20 Ne abundances can reach up to 17% within the range of [ α/ p ] ratios explored in our study, in the sense that a higher 20 Ne content is achieved when the α channel is more dominant. Moreover, we found that higher production of 20 Ne translates into smaller initial masses needed for C-ignition and longer Cburning phase lifetimes. \nAn interesting result, derived from the exploration of the minimum mass needed for C-ignition, is that, for a specific range of masses that depend on the adopted total nuclear reaction rate and branching ratio (e.g., 6 . 85 ≤ M ZAMS / M ⊙ < \n7 . 10, for the CF88 rate and a branching ratio of [65 / 35]), carbon burns partially in the models' interiors. In such cases, the final chemical structure at the end of the C-burning phase consists of a CO-core surrounded by an ONe mantle. Depending on the amount of 20 Ne produced during this stage, the progeny could be a CO-core WD or a hybrid CONe-core WD. \nAs for the impact on the ONe-core WD evolution due to the use of di ff erent nuclear reactions and branching ratios, we found di ff erences in the cooling times and the size of the crystallized core of at most 1.3 and 0.8%, respectively. We found that the impact on the pulsation properties of these stars is also negligible. \nRegarding the hybrid CONe-core WDs, we found that they di ff er substantially in their evolution, compared with those composed of pure CO cores. Our results show that even as little as 10% of 20 Ne in its interior can modify the structure of the WD in such a way that crystallization starts earlier and, by the time the star reaches the ZZ Ceti instability strip, differences in the crystallized portion of the star and its cooling time can reach up to 15% and 6%, respectively. This result is also reflected in the pulsation properties. By comparing its forward period spacing, we find that the hybrid WD has less frequent minima (larger trapping cycle) than its CO-core counterpart. This is because the hybrid model has a larger crystallized core and, consequently, a smaller resonating cavity. \nIn conclusion, our study reveals that current uncertainties in both the total 12 C + 12 C reaction rate and its branching ratios can have a significant impact upon the late stages of evolution of intermediate-mass stars and their progeny. As demonstrated by other authors, uncertainties in the 12 C + 12 C rate also impact the advanced evolutionary stages of highermass stars. Given its far-reaching astrophysical implications, further experimental work is thus urgently needed to properly constrain the 12 C + 12 Crate and branching ratios at astrophysically relevant energies. \nSoftware: Modules for Experiments in Stellar Astrophysics (MESA; Paxton et al. 2011, 2013, 2015, 2018, 2019), MESASDK 20.3.1 (Townsend 2020), LPCODE (Althaus et al. 2021), LP-PUL (C'orsico & Althaus 2006).", 'ACKNOWLEDGEMENTS': 'We wish to acknowledge the suggestions and comments of the anonymous referee who strongly improved the original version of this work. This work was supported by PIP 112-200801-00940 grant from CONICET, grant G149 from the University of La Plata, PIP-2971 from CONICET (Argentina) and by PICT 2020-03316 from Agencia I + D + i (Argentina). This research has made use of the NASA Astrophysics Data System.'} |
2023arXiv231118731S | We present the identification of 42 narrowline active galactic nuclei type2 AGN candidates in the two deepest observations of the JADES spectroscopic survey with JWSTNIRSpec. The spectral coverage and the depth of our observations allow us to select narrowline AGNs based on both restframe optical and UV emission lines up to z10. Due to the metallicity decrease of galaxies at zgt3 the standard optical diagnostic diagrams N2BPT or S2VO87 become unable to distinguish many AGN from other sources of photoionisation. Therefore we also use high ionisation lines such as HeIIlambda4686 HeIIlambda1640 NeIVlambda2422 NeVlambda3420 and NVlambda1240 also in combination with other UV transitions to trace the presence of AGN. Out of a parent sample of 209 galaxies we identify 42 type2 AGN although 10 of them are tentative giving a fraction of galaxies in JADES hosting type2 AGN of about 20pm3 which does not evolve significantly in the redshift range between 2 and 10. The selected type2 AGN have estimated bolometric luminosities of 1041.344.9 erg s1 and hostgalaxy stellar masses of 107.29.3 Modot. The star formation rates of the selected AGN host galaxies are consistent with those of the starforming main sequence. The AGN host galaxies at z46 contribute sim830 to the UV luminosity function slightly increasing with UV luminosity. | 2023-11-01T00:00:00Z | ['10.48550/arXiv.2311.18731', '2023arXiv231118731S', 'arXiv:2311.18731'] | ['Astrophysics - Astrophysics of Galaxies'] | JADES A large population of obscured narrow line AGN at high redshift | 2,023 | 172 | 0.67 | ['EPRINT_HTML', 'EPRINT_PDF'] | 81 | https://arxiv.org/pdf/2311.18731.pdf | {'JADES: A large population of obscured, narrow line AGN at high redshift': "Jan Scholtz 1 , 2, ⋆ , Roberto Maiolino 1 , 2 , 3 , Francesco D'Eugenio 1 , 2 Emma Curtis-Lake 4 , Stefano Carniani 5 , Stephane Charlot 6 , Mirko Curti 7 , Maddie S. Silcock 4 , Santiago Arribas 8 , William Baker 1 , 2 , Rachana Bhatawdekar 9 , Kristan Boyett 10 , 11 , Andrew J. Bunker 12 , Jacopo Chevallard 12 , Chiara Circosta 9 , Daniel J. Eisenstein 13 , Kevin Hainline 14 , Ryan Hausen 15 , Xihan Ji 1 , 2 , Zhiyuan Ji 14 , Benjamin D. Johnson 13 , Nimisha Kumari 16 , Tobias J. Looser 1 , 2 , Jianwei Lyu 14 , Michael V. Maseda 17 , Eleonora Parlanti 5 , Michele Perna 9 , Marcia Rieke 14 , Brant Robertson 18 , Bruno Rodríguez Del Pino 9 , Fengwu Sun 14 , Sandro Tacchella 1 , 2 , Hannah Übler 1 , 2 , Giacomo Venturi 5 , Christina C. Williams 19 , Christopher N. A. Willmer 14 , Chris Willott 20 , and Joris Witstok 1 , 2 \n(A ffi liations can be found after the references)", 'ABSTRACT': 'We present the identification of 41 narrow-line active galactic nuclei (type-2 AGN) candidates in the two deepest observations of the JADES spectroscopic survey with JWST / NIRSpec. The spectral coverage and the depth of our observations allow us to select narrow-line AGNs based on both rest-frame optical and UV emission lines up to z = 10. Due to the metallicity decrease of galaxies, at z > 3 the standard optical diagnostic diagrams (N2-BPT or S2-VO87) become unable to distinguish many AGN from other sources of photoionisation. Therefore, we also use high ionisation lines, such as He ii λ 4686, He ii λ 1640, [Ne iv ] λ 2422, [Ne v ] λ 3420, and N V λ 1240, also in combination with other UV transitions, to trace the presence of AGN. Out of a parent sample of 209 galaxies, we identify 42 type-2 AGN (although 10 of them are tentative), giving a fraction of galaxies in JADES hosting type-2 AGN of about 20 ± 3%, which does not evolve significantly in the redshift range between 2 and 10. The selected type-2 AGN have estimated bolometric luminosities of 10 41 . 3 -44 . 9 erg s -1 and host-galaxy stellar masses of 10 7 . 2 -9 . 3 M ⊙ . The star formation rates of the selected AGN host galaxies are consistent with those of the star-forming main sequence. The AGN host galaxies at z = 4-6 contribute ∼ 8-30 % to the UV luminosity function, slightly increasing with UV luminosity. \nKey words. Galaxies: active, Galaxies: high-redshift, Galaxies: ISM', '1. Introduction': 'It has been widely accepted that supermassive black holes (SMBHs) reside in the centre of most (perhaps all) massive galaxies. During their accretion phases, SMBHs are observed as active galactic nuclei (AGN; Rees et al. 1982; Lynden-Bell 1969; Soltan 1982; Merloni et al. 2004). The tight correlation between a SMBH mass and host-galaxy bulge properties (such as velocity dispersion and mass) at z ∼ 0 indicates a strong connection between the growth of a SMBH and its host galaxy (e.g., Magorrian et al. 1998; Kormendy & Ho 2013). Furthermore, (radiative and mechanical) feedback from AGNs is a key ingredient of galaxy evolution, as AGNs can inject a significant amount of energy into the interstellar and circumgalactic medium (ISM and CGM, respectively) of their host galaxies. AGN feedback is necessary to reproduce key galaxy properties such as: colour bi-modality, galaxy sizes, and a broader range of specific star formation rates and enrichment of the intergalactic medium (IGM) by metals, as well as the high-mass drop-o ff of the stellar mass function compared to the halo mass function (e.g., Silk & Rees 1998; Di Matteo et al. 2005; Alexander & Hickox 2012; Dubois et al. 2013b,a; Vogelsberger et al. 2014; Hirschmann et al. 2014; Crain et al. 2015; Segers et al. 2016; Beckmann et al. 2017; Harrison 2017; Choi et al. 2018; Scholtz et al. 2018). \nThe identification and study of AGN at high redshift is essential to understanding not only the co-evolution of SMBHs and \ngalaxies, but also the formation of galaxies at early epochs. Although over the past 15 years there has been significant progress in the identification of active SMBHs at high redshift ( z > 3) (e.g. Merloni et al. 2010; Bongiorno et al. 2014; Trakhtenbrot et al. 2017; Mezcua et al. 2018; Lyu et al. 2022), the majority of these detections have been limited to bright quasars identified in large-volume ground-based surveys (e.g., Bañados et al. 2016; Shen et al. 2019, see Inayoshi et al. 2020; Fan et al. 2022 for a review). These include the most distant identified quasars at z ∼ 7 . 5 (Bañados et al. 2018; Yang et al. 2020; Wang et al. 2021). \nWith the launch of the James Webb Space Telescope (JWST; Gardner et al. 2023; Rigby et al. 2023), we now have access to rest-frame UV-to-optical emission lines of galaxies up to z ∼ 12. These lines allow us to identify and study AGNs, even at low masses and luminosities, via optical and UV diagnostic diagrams. A number of AGN candidates have already been identified by performing Spectral Energy Distribution (SED) analyses of broad-band photometry from NIRCam and MIRI aboard JWST (Furtak et al. 2022; Onoue et al. 2023; Barro et al. 2023; Yang et al. 2023a; Bogdan et al. 2023; Juodžbalis et al. 2023; Lyu et al. 2023; Yang et al. 2023b). Significant progress has been made using deep spectroscopy from JWST / NIRSpec (Böker et al. 2022; Jakobsen et al. 2022) and the JWST / NIRCam grism, tracing the presence of a broad line region (BLR; Furtak et al. 2023; Greene et al. 2023; Harikane et al. 2023; Kocevski et al. 2023; Maiolino et al. 2023b,a; Matthee et al. 2023; Onoue et al. 2023; Übler et al. 2023). These observations revealed a pre- \nly unseen population of AGN at z > 4, and out to z ∼ 11, with estimated black hole masses (MBH) in the range 10 6 to 10 8 M ⊙ and bolometric luminosities 10 44 -10 45 erg s -1 . These black hole masses and luminosities are 2-3 orders of magnitude lower than those inferred for quasars at the same redshifts (Mazzucchelli et al. 2023; Zappacosta et al. 2023). \nHowever, the identification of narrow-line (i.e. type-2, as opposed to type-1 AGN showing BLR emission) AGN has remained unexplored with JWST data. This has been the main AGN identification tool at low redshift, however, so far has had limited success at high redshift. The reason is most likely the rapid evolution of the metallicity and ionisation parameter at z > 3 towards more metal-poor and higher ionisation parameter (Hirschmann et al. 2022; Curti et al. 2023b,a; Tacchella et al. 2023; Trump et al. 2023). Indeed, Harikane et al. (2023); Kocevski et al. (2023); Maiolino et al. (2023a); Übler et al. (2023) have shown that the type-1 AGN found by JWST have narrow line ratios on the classical emission-line diagnostic diagrams (such as BPT; Baldwin et al. 1981a; Kau ff mann et al. 2003; Kewley et al. 2013 and VO87; Veilleux & Osterbrock 1987) that overlap with the local SF sequence, and not with the region occupied by nearby and low-z AGN. This displacement has been predicted by photoionization models for low metallicity AGN (Groves et al. 2006; Nakajima & Maiolino 2022). Additionally, Feltre et al. (2016) and Gutkin et al. (2016) ran photo-ionisation grid models to assess the e ff ect of low metallicity and high ionisation parameter on rest-frame optical and UV emission lines, and found that the nebular emission of star-forming galaxies at high redshift become similar to that of AGN, further complicating the AGN selection via narrow emission line diagnostics. The identification of type-2 AGN at high redshift can still rely on high ionization lines, such as the [Ne iv ] λ 2424 (with ionisation potential > 63.45eV) detected by Brinchmann (2023); Chisholm et al. (2024) in a galaxy at z = 7.66. Luckily, the identification of type-2 AGN with high metallicity through standard BPTs is still possible (see Perna et al. 2023) \nDespite these newly discovered AGN being far less luminous than the previously known quasar population and being hosted in galaxies with lower masses, they can play a major role in shaping galaxy evolution (via positive and negative feedback; Koudmani et al. 2021, 2022) and could potentially significantly contribute to the reionisation of the Universe. Indeed, we have already observed these processes in GN-z11 (Bunker et al. 2023b) believed to host an AGN (Maiolino et al. 2023b) with detected outflow in the system, while Scholtz et al. (2023) observed heating and ionisation of the circum-galactic medium around this unique galaxy. These examples show the presence of AGN feedback at high z, which can explain the early emergence of quiescent galaxies and high burstiness of star formation (Carnall et al. 2023b,a; Endsley et al. 2023; Looser et al. 2023c,a; Dome et al. 2023; Strait et al. 2023); indeed, Gelli et al. (2023) showed that supernovae feedback is not su ffi cient to rapidly suppress the star formation in some of these rapidly quenched systems. \nIn this paper, we leverage two of the deepest spectroscopic observations from JWST Advanced Deep Extragalactic Survey (JADES, Proposal ID: 1210 & 3215; Bunker et al. 2023a; Curtis-Lake et al. 2023; Eisenstein et al. 2023b,a; Robertson et al. 2023). Using this deep spectroscopy with JWST / NIRSpec, we aim to identify type-2 AGN using deep spectroscopy with JWST / NIRSpec in galaxies with stellar masses M ∗ ∼ 10 6 . 5 -10 9 . 5 M ⊙ from redshift z = 1 and out to the highest redshifts for which rest-frame UV and optical nebular lines are accessible to NIRSpec (z ∼ 12.0). We reassess the demarcation lines between AGN and star-forming galaxies at high redshift and further re- \nection criteria using the photo-ionisation modelling introduced by Feltre et al. (2016) and Nakajima & Maiolino (2022). \nThe paper is organized as follows. In Section 2, we describe the observations, data reduction, spectral fitting procedures, and SED analysis to derive stellar masses and star formation rates (SFRs). In Section 3 we identify AGN using their narrow line properties of rest-frame UV and optical emission lines. In Section 4, we estimate the properties of the AGN and their host galaxies and discuss our results. In Section 5, we draw our conclusions. Throughout this paper, we use the AB magnitude system and assume a flat Λ CDM cosmology with Ω m = 0.315 and H0 = 67.4 km / s / Mpc Planck Collaboration et al. (2020) cosmology and a Chabrier (2003) initial mass function.', '2. Observations, Data reduction and Analysis': 'The observations of our targets were obtained as part of the JADES survey, utilising the multi-object spectroscopic capabilities of the JWST / NIRSpec micro-shutter array (MSA; Jakobsen et al. 2022; Ferruit et al. 2022) across two programmes: PID 1210 & PID 3215. \nFive disperser / filter combinations were used: the lowresolution PRISM / CLEAR (0 . 6 < λ < 5 . 3 µ m , R = 30 -300; Jakobsen et al. 2022), the medium-resolution gratings G140M / F070LP, G235M / F170LP and G395M / F290LP (0 . 6 < λ < 5 . 3 µ m, R = 1 , 000), and the high-resolution grating G395H / F290LP (2 . 8 < λ < 5 . 1 µ m, R = 2,700). For the 1210 program, the observations consist of individual visits with a per-visit duration of 33.6 ks for the prism, and of 8.4 ks for each grating. Each galaxy was assigned a minimum of one and a maximum of three visits depending on its priority (Bunker et al. 2023a). This resulted in a maximum integration time of ∼ 100 ks for the prism and of 25 ks for each grating. The 3215 program consisted only of three configurations: PRISM / CLEAR, G140M / F070LP, and G395M / F290LP, with a maximum resulting integration time of 168 ks, 42 ks, and 168 ks. \nThe observations were performed in the three-shutter nod mode, so that common targets were observed in di ff erent shutters and di ff erent locations on the detector. Therefore, each visit required a unique MSA configuration. Each target allocation (performed using the eMPT tool; Bonaventura et al. 2023 1 ) was designed to maximise the number of targets that have all the disperses / filters in common between all the dither positions, however, not all targets have the full on source integration time outlined above. In total, we observed 481 unique targets. \nThe MSA configurations have been designed to avoid spectral overlap for the prism mode. However, since the spectra taken with the medium or high-resolution gratings occupy a significantly larger portion of the detector, and to avoid removing targets with overlapping spectra, we allowed spectral overlap for these modes. The MSA configurations were, however, designed to minimize the negative e ff ect of spectral overlap on our science, since the highest priority targets are not allowed to be contaminated by neighboring spectra. We show examples of the acquired PRISM / CLEAR spectra in Figure 1.', '2.1. Data Reduction': 'The JWST / NIRSpec MSA observations were processed with the data reduction pipeline of the ESA NIRSpec Science Operations Team (SOT) and the NIRSpec GTO Team, further detailed description of the GTO pipeline will be presented in Carniani et al. \nFig. 1: Example of low-resolution PRISM / CLEAR spectrum of galaxy (JADES-NS-GS-00022251) at z = 5.804 included in our parent sample. The spectrum is shown in units of F λ normalized to the [O iii ] λ 5008 peak. We highlight major emission lines used in our analysis. \n<!-- image --> \n(in prep), and is summarised in Bunker et al. (2023a). Here, we briefly summarize the procedure. We retrieve the level-1a data products from the MAST archive and estimate the count-rate slopes per pixel, using the unsaturated groups in the ramps. We remove any jumps in the ramps due to cosmic rays by estimating the slope of each ramp. During this stage, we perform the master dark and bias subtraction, as well as the flagging of saturated pixels. The background subtraction is performed pixel by pixel by combining the three nod exposures of each pointing. We note that for some targets we excluded one of the 3-shutter nods in the background subtraction stage as a serendipitous source contaminated the open shutters. We performed the flat-field correction of the spectrograph optics and disperser corrections on the 2Ddimensional cutouts of each of the three-shutter slits. The path loss corrections are calculated assuming point sources and taking into account the source location on the shutter. \nFinally, the individual 2D maps are interpolated onto an irregular wavelength grid for the PRISM / CLEAR observations (to avoid oversampling the line spread function below 2 µ m) and onto a regular grid for the gratings. We extracted the 1D spectra from the 2D maps adopting a box-car aperture centered on the relative position of the targets. We combined all 1D spectra and removed the bad pixels by adopting a sigma-clipping approach.', '2.2. Spectral fitting using PPXF': "As the continuum is well detected in all our targets in the PRISM observations, we employed pPXF (Cappellari 2017, 2022) to fit the continuum and emission lines simultaneously. As we are interested in inactive galaxies or type-2 AGN, there is no contamination of the continuum by the AGN, making pPXF an ideal tool for this analysis. The continuum is fitted as a linear superposition of simple stellar-population (SSP) spectra, using non-negative weights and matching the spectral resolution of the observed spectrum. As input stellar templates we used the synthetic library of simple stellar population spectra (SSP) from fsps (Conroy et al. 2009; Conroy & Gunn 2010). This library uses MIST isochrones (Choi et al. 2016) and C3K model atmospheres (Con- \nroy et al. 2019). We also used fit a 5 th -order multiplicative polynomial, to capture the combined e ff ects of dust reddening, residual flux calibration issues, and any systematic mismatch between the data and the input stellar templates. We find that the NIRSpec / MSAdata are fit adequately by low-order polynomials. Increasing the polynomial degree does not improve the fit results (as quantified by the value of the reduced χ 2 ). To simplify the fitting, any flux with a wavelength shorter than Lyman break is manually set to 0. The full description of the pPXF fitting will be presented in D'Eugenio et al. (in prep). \nFor the emission lines fitting we use the redshifts published in the Bunker et al. (2023a), which used redshifts determined from the medium resolution gratings available and PRISM spectra otherwise. All emission lines are modeled as single Gaussian functions, matching the observed spectral resolution. We use vacuum wavelengths for the emission lines throughout the paper. In order to remove degeneracies in the fitting and reduce the number of free parameters, the emission lines are split into four separate kinematic groups, bound to the same redshift and intrinsic broadening. These groups are as follows: \n- -UV lines with rest-frame λ < 3000 Å.\n- -The hydrogen Balmer series.\n- -Non-hydrogen optical lines with rest-frame λ < 9000 Å.\n- -Near infra-red lines. \nThe stellar kinematics are tied to the Balmer line kinematics. For emission line multiplets arising from the same level, we fixed the emission line ratio to the value prescribed by atomic physics (e.g., [O iii ] λ 5008 / [O iii ] λ 4959 = 2.99). For multiplets arising from di ff erent levels, the emission line ratio can vary. This is relevant for five multiplets: C iv λλ 1548,1551, C iii ] λλ 1907,1909, [O ii ] λλ 3727,3730 and [S ii ] λλ 6718,6733. However, in practice, the spectral resolution of the PRISM / CLEAR observations is insu ffi cient to resolve the individual lines in each multiplet. For this reason, we model each of these as a single Gaussian in the PRISM observations. In addition, as the He ii λ 1640 and [O iii ] λλ 1661,66 are blended together we are unable to use their measured fluxes from the PRISM observations and instead, \nchoose to use the R1000 observations. The fits are performed in two steps. In the first run, we tie the kinematics as described above, and identify robust detections (above 5σ significance) for re-fitting. In the second run, we only fit for these detected lines, but allow for their kinematics to vary independently from one another. \nThe R1000 gratings were fitted using the same procedure as for the PRISM observations, by combining all three gratings and fitting them simultaneously. However, we did not stack the spectra from di ff erent gratings, because this would combine the highest-resolution end of one grating with the poorest-resolution start of the next grating.", '2.2.1. Fitting the UV lines': "The high ionisation UV lines were excluded from the pPXF fitting as they are extremely faint and blended in the PRISM data. Since the AGN high ionisation UV lines are considerably fainter than the optical emission lines, we fitted only objects that are well-detected in [OIII] or H α emission (SNR > 10). \nWe fitted the [Ne iv ] λ 2424, [Ne v ] λ 3420 and N V λ 1240 using QubeSpec 's 2 fitting module. Each emission line was fitted using a single Gaussian component and the continuum was fitted as a power law. This simplistic approach is su ffi cient for describing a narrow range of the continuum around an emission line of interest. As QubeSpec is a Bayesian code implemented using emcee (Goodman & Weare 2010), it is necessary to also supply priors on each quantity. The peak of the Gaussian and continuum normalization are given a log-uniform prior, while the FWHMs are set to a uniform distribution spanning from the minimum resolution of the NIRSpec / MSA( ∼ 200 kms -1 ) up to a maximum of 1200 km / s. The prior on the redshift was a normal distribution centered on the redshift obtained from pPXF with a standard deviation of 300 km / s. This is done because as the high ionisation UV lines originate from close to the accretion disc, there can be a significant velocity o ff set between the low and high ionisation lines in an AGN. We report the fluxes of each of these high ionisation lines in the Appendix in Table B.1. Throughout this work we derived upper limits as 3 σ .", '2.3. SED fitting using BEAGLE': 'In order to compare our selected AGN with the rest of the galaxy population, it is necessary to measure the stellar masses and starformation rates of our sample. We use the full fitting of the slitloss corrected PRISM spectra using the BEAGLE tool (Chevallard & Charlot 2016). We assume a delayed-exponential starformation history (SFH), while decoupling the current SFR from the previous SFH by allowing a recent duration of 10Myr of constant star formation to vary independently. A Chabrier (2003) initial mass function (IMF) with an upper mass cut-o ff of 100 M ⊙ was adopted using the updated Bruzual & Charlot (2003) stellar population models described in Vidal-García et al. (2017a). We define the total stellar mass as the mass currently locked into stars. This definition accounts for the fraction of mass returned to the ISM during stellar evolution. The SFRs of our objects are averaged over 10 million years. \nWe note that the SFRs of the whole sample estimated with BEAGLE are in excellent agreement with those estimated from the dust-corrected Balmer lines (H α and H β ; see Curti et al. 2023b for more information).', '2.4. Comparison to photoioinization models': 'At high redshifts, galaxies become more metal-poor and show an increasingly higher ionisation parameter (see Hirschmann et al. 2022; Schaerer et al. 2022; Cameron et al. 2023b; Curti et al. 2023b; Trump et al. 2023), resulting in standard diagnostics diagrams used to identify AGNs at low redshift becoming less useful (see Kewley et al. 2013, 2019; Hirschmann et al. 2022). We, therefore, consider the photoionisation models initially described in Feltre et al. (2016) and Gutkin et al. (2016), and updated with more recent stellar spectra and with a better description of AGN cloud microturbulence (Vidal-García et al. 2017b; Mignoli et al. 2019; Hirschmann et al. 2019). These works consider a large grid of photoionisation models computed using the CLOUDY code (Ferland et al. 2013) for star formation and AGN narrow-line regions, and for various gas metallicities, dust content, and ISM densities. From these model grids, we selected all models with metallicities between 0.001 and 0.02 (corresponding to 0.06-1.3 solar) and a dust-to-metal mass ratio of 0.3. This value is intermediate between the range observed in the most metal-poor absorbers (e.g., Konstantopoulou et al. 2023) and the Milky-Way value of 0.45. We consider all models with carbonto-oxygen abundance ratio in the range 0.38-1.00 solar to describe a variety of di ff erent or less common star-formation grids. We further restrict the grids to only include models with an IMF upper mass cut-o ff of 300 M ⊙ , similar to the SED fitting above. \nWe also compare our observed emission line ratios with the models of Nakajima & Maiolino (2022). They investigated the emission line ratios of star-forming galaxies, AGN, PopIII stars, and Direct Collapse black holes (DCBHs) using the Cloudy code and the BPASS stellar population models (Eldridge et al. 2017). In this work, we only use the models for star-forming galaxies and AGN host galaxies, as our objects do not have low enough metallicities to host either PopIII or DCBHs (based on values from Curti et al. 2023b). We selected the same metallicities, dust-to-metal mass ratio and IMF upper cuto ff as for the Gutkin et al. (2016) and Feltre et al. (2016) models. \nThroughout this work, we present these models on our diagnostics plots (Figures 2, 3, 4 and 6) as light blue and yellow circles for AGN and star-formation, respectively. We will further discuss these points and use them to redefine the selection of AGN based on to-be-introduced N2-BPT and S2-VO87 diagnostic diagrams in §3.1.', '3. Selection of AGN in JADES deep spectroscopic data': 'In order to select AGN based on their narrow line properties we require a 3 σ detection of the following lines: H α , H β , [O iii ] and C iii ], and wavelength coverage of [S ii ], [N ii ] and UV and optical He ii and C iv for the optical and UV line selection. Overall, this yields a sample size of 110 and 99 sources for the programmes 1210 and 3215, respectively. We define the line ratios used in this paper in Table 1 (see below) and we summarise all emission lines, their wavelengths and ionisation potential in the Appendix in Table C.1. We consider a galaxy an AGN candidate if it appears as an AGN in at least one diagnostic. Furthermore, we report the list of selected AGN candidates, field, coordinates, redshift, the method with which we detect it, and other notes in Table 2. \nTable 1: Definitions of line ratios adopted throughout the paper.', '3.1. Selecting AGN based on optical emission lines': 'Weplot our sources as dark blue points on the N2-R3 and S2-R3 planes (also known as BPT and VO87 diagrams, respectively; Baldwin et al. 1981b; Veilleux & Osterbrock 1987) in the top and bottom row of Figure 2, respectively. The low metallicities of high-z objects lead to faint [N ii ] emission lines, which are typically undetected at z > 4 (Cameron et al. 2023b; Curti et al. 2023b). Furthermore, the high ionisation parameter pushes star-forming sources to high R3 values, towards the classical demarcation between star formation and AGN defined by Kewley et al. (2001); Kau ff mann et al. (2003); Kewley et al. (2013). On the other hand the low metallicity of the Narrow Line Region at high-z results in the line ratios of AGN to shift away from the locus of AGN typically populated by AGN and move towards the locus of star forming galaxies Nakajima & Maiolino (2022); Harikane et al. (2023); Maiolino et al. (2023a). The combination of these e ff ects makes high-z AGN and SF galaxies to largely overlap on these diagrams and makes the selection of high-z AGN much more challenging than in the local Universe. However, these diagrams can still be used to identify AGN via a conservative selection approach, as discussed in the following. \nIn Figure 2, we plot results from di ff erent photoionisation models for a wide range of gas properties (metallicities, ionisation parameter and densities) in the left and right panels, with star-forming models and AGN models shown in yellow and blue points, respectively. The left panels show AGN models from Feltre et al. (2016) and star-forming models from Gutkin et al. (2016), while the right panels show the AGN and starforming models of Nakajima & Maiolino (2022). The photoionisation models indeed show that star-forming galaxies can lie in the AGN part of the BPT diagram at high redshift with log10([O iii ] / H β ) > 0 . 9 (as also noted in Figure 14 of Feltre et al. 2016). As such, we need a clean selection to account for galaxies with low metallicity and high ionisation parameters. To identify a conservative demarcation line between star-forming galaxies and AGN, we define the edge of the star-forming region using the points with the highest [N ii ] / H α values from either the Kewley et al. (2001) line or Gutkin et al. (2016) models. We clarify that this is a very conservative method to select AGN. Galaxies above this demarcation line can be safely classified as AGN, however, as discussed above, certainly there are plenty of AGN also mixed with the galaxy population below this line, especially at these high redshifts. Indeed the type-1 AGN from Harikane et al. (2023); Maiolino et al. (2023a) do lie in the star-formation part of the BPT diagram. \nWe fit these points, marking the edge of the star-forming region of the BPT diagram, with the functional form as in Kewley \net al. (2001); Kau ff mann et al. (2003): \nY = a ( X -b ) + c (1) \nwhere Y = log10([OIII] / H β ) and X = log10([NII] / H α ). We report the results for the demarcation line in Table 3. In the BPT diagram, this functional form well describes the edge of starforming galaxies and we show this line as a green dashed line in the top panels of Figure 2. We select five AGN based on the new demarcation line, all at z < 5. We highlighted these sources with red circles in the top panel Figure 2. \nWe repeat the same analysis on the S2-R3 diagram (S2VO87) to make new demarcation lines between AGN and starforming galaxies. However, the previous functional form no longer fits the edges of the Kewley et al. (2001) line and Gutkin et al. (2016) points and we need to adapt it as: \nY = a ( X -b ) + c [ X > -0 . 92] = d + eX [ X > -0 . 92] (2) \nWe report the parameters of the AGN demarcation lines in Table 3. Overall, we detect seventeen AGN in the S2-R3 BPT diagram, using the Kewley et al. (2001) or Gutkin et al. (2016) line (dashed green line in the bottom panels of Figure 2). We note that there are several AGN candidates close to the demarcation line. We discuss these selection methods and their reliability in §4.2. \nNumber of studies (Shirazi & Brinchmann 2012; Baer et al. 2017; Dors et al. 2023; Tozzi et al. 2023) investigated the identification of AGN in SDSS using the He ii λ 4686 emission line. This is a recombination line whose flux is nearly independent of the gas metallicity and ionization parameter, depending instead primarily on the shape of the ionizing spectrum and, more specifically, on the number of ionizing photons with energies beyond 54 eV. We plot the He2-N2 diagram in Figure 3. In this case, we do not redefine the demarcation line as defined by Shirazi & Brinchmann (2012) as the original line is more conservative than the photo-ionisation models used in our work. Overall, we detect He ii λ 4686 in nine galaxies, whose HeII / H β ratio indicates the presence of hard ionising radiation indicating AGN. Despite the deep JWST observations, the upper limits on He ii λ 4686 / H β do not provide any constraints on the presence of an AGN. We note that we selected three additional sources solely based on their N2 ratio > 0 . 2 (see Figure 2). Although these sources do not have He ii λ 4686 detections, their large N2 ratio is already identified with the N2-BPT. In summary, based on this diagram we select 12 AGN in total. \nWe finally note that the only X-ray detected AGN in our sample (red star in Figures 2 and 3) is located in the starforming region of the N2-BPT and S2-VO87 diagnostic diagrams (hence it would have been missed by this classification, as most type 1 AGN, as discussed in Harikane et al. 2023; Maiolino et al. 2023a; Kocevski et al. 2023), while the upper limit on the HeII λ 4686 would no identify as AGN, further showing complexity of selecting AGN at high redshift.', '3.2. Selecting AGN based on UV line diagnostics': 'In the previous section, we showed that the selection of AGN at high-z using the BPT and S2-VO87 diagrams is extremely challenging for all but metal-rich galaxies. In this section, we will focus on the UV emission lines. The clearest signature of AGN activity is high-ionisation lines such as NV λ 1240, [Ne iv ] λ 2424, \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFig. 2: Typical line ratio diagnostic diagrams used to select AGN. N2-R3 BPT ([N ii ] / H α vs [O iii ] / H β ; top row) and S2-VO87 ([S ii ] / H α vs [O iii ] / H β ; bottom row). The ionisation models of Feltre et al. (2016); Gutkin et al. (2016) (left) and Nakajima & Maiolino (2022) (right) are reported as yellow (SF) and blue (AGN) points (see §2.4). The objects from this work are plotted as blue squares. The objects selected as AGN based in these diagrams are highlighted by red circles. We also plot our new demarcation lines as green dashed. The black dashed lines show the star forming versus AGN demarcation lines from Kewley et al. (2001) and Kau ff mann et al. (2003). For comparison, we plot SDSS galaxies shown as a grey contour plot. The magenta and cyan squares show a stacked spectrum for AGN and star-forming galaxies (see §4.1). We highlight type-1 AGN from Harikane et al. (2023) and Maiolino et al. (2023a) as green squares and diamonds. The red star shows the X-ray selected AGN in our sample. \n<!-- image --> \nTable 3: Fitted parameters for equation 1 when fitted for the N2-R3 and S2-R3 diagnostics plot. We define the edge of starforming galaxies described in §3.1.[Ne v ] λ 3426 (Feltre et al. 2016). They require high energy photons (77, 98 and 68 eV, respectively), that can hardly be produced by star-formation processes and are not seen even in galaxies hosting WR stars (Mingozzi et al. 2023); hence, as already men- \ntioned, these are clear signatures of AGN. However, they are often very faint even in powerful AGN (Übler et al. 2023). \nPast works on this topic have suggested various UV emission lines ratios based on C iii ] λλ 1907,09, C iv λλ 1548,51, [O iii ] 1660,66 and He ii λ 1640 (Feltre et al. 2016; Hirschmann et al. 2022; Mingozzi et al. 2022; Mascia et al. 2023). These lines are ideal in identifying AGN at z > 6 . 8, where most restframe optical emission lines are redshifted out of the wavelength range of JWST / NIRSpec. However, even in luminous AGN the majority of these lines are beyond detectability for a 4-7 hour JWST / NIRSpec exposure at high redshift. \nTherefore, in this work, we focus mainly on the brightest emission lines of the UV emission lines that are potentially detectable in our data, or on which we can at \n<!-- image --> \nFig. 3: The He2-N2 (He ii / H β vs [N ii ] / H α ) diagram for the sample of JADES galaxies for our sample, plotted as blue squares. The left and right plots show ionisation models from Feltre et al. (2016); Gutkin et al. (2016) and Nakajima & Maiolino (2022), respectively, as yellow and light blue points (see §2.4). The magenta and cyan squares show a stacked spectrum for AGN and star-forming galaxies (see §4.1). The black dashed line indicates a demarcation line between star-forming and AGN galaxies by Shirazi & Brinchmann (2012). The blue and black contours show the star-forming galaxies and AGN from SDSS, respectively. We highlight the selected AGN in this diagram with a red circle. \n<!-- image --> \nleast obtain meaningful upper limits. In Figure 4, we show the ratio C iii ] λλ (1906 + 1908) / He ii λ 1640 (C3He2) against C iv 1548,50 / C iii ] λλ (1906 + 1908) (C43). We also plot the models of AGN and star-forming galaxies as blue and orange points, respectively, from Feltre et al. (2016); Gutkin et al. (2016) (left plot) and Nakajima & Maiolino (2022) (right plot). The black and dotted lines show the demarcation lines between star-forming galaxies, AGN, and composite objects derived by Hirschmann et al. (2022). They obtain such a line from the postprocessing of the Illustris TNG cosmological simulation using the models of Gutkin et al. (2016) and Feltre et al. (2016). In our sample, there are six targets for which we detect He ii λ 1640. Out of these five objects, one lies in the AGN part of the diagram. Furthermore, four targets are in the composite part of the diagram according to Hirschmann et al. (2022), but they are classified as AGN based on Feltre et al. (2016) and Nakajima & Maiolino (2022). \nWe also show the objects from Saxena et al. (2020) with He ii λ 1640 detection in the VANDELS survey (McLure et al. 2018) as green diamonds. This sample removed all X-raydetected objects, removing any signs of obvious AGN. However, most of the type-1 AGN detected with JWST do not show any X-ray emission even in the deepest observations (Maiolino et al. in prep), indicating that many of these VANDELS sources might have a significant AGN contribution. All the VANDELS sources would be classified as AGN by Feltre et al. (2016); Gutkin et al. (2016); Nakajima & Maiolino (2022), whereas 14 of them fall in the AGN or composite region defined by Hirschmann et al. (2022). We also show C iv detections from VANDELs survey (Mascia et al. 2023) as grey squares and circles representing SF and AGN host (identified from X-ray or as type-1 AGN) galaxies, respectively. The AGN and SF galaxies from Mascia et al. (2023) are well separated on our diagram as predicted by the photo-ionisation models used in our work. \nWe compare our observations with local high-redshift analogues from the CLASSY survey (Berg et al. 2022; James et al. 2022), specifically the data from Mingozzi et al. (2023). The CLASSY sources are classified as star-forming galaxies based on the BPT and He ii λ 4686 optical emission lines. Overall, the five objects in our sample with He ii λ 1640 selected as AGN in our sample have significantly lower C3-He2 ratios than other "pure" star-forming galaxy samples, confirming our AGN selection. \nWe note that Mascia et al. (2023) devised a classification of UV-BPT using the VANDELS survey. However, most of the diagnostics diagrams used in that work use both C iv and He ii λ 1640 in the same emission line ratio. Unfortunately, the majority of our sample is undetected in both C iv and He ii λ 1640 which does not allow us to place these objects on their diagnostics diagram. We will further investigate these diagnostics with the full JADES sample in future works. \nWe note that the above emission line diagnostic requires the detection of three UV emission lines. Despite the excellent sensitivity of JWST / NIRSpec, detecting all three emission lines is still a challenge for exposures < 10 hours. \nWe detect [Ne iv ] λ 2424 in five objects. We plot these sources in the top panel of Figure 5, showing C3He2 vs [Ne iv ] λ 2424 / C iii ] as blue points, together with the photoionisation models of Gutkin et al. (2016) and Feltre et al. (2016) as yellow and blue points, respectively. All of these four sources have extremely high log10([Ne iv ] λ 2424 / C iii ] ) ratio ( > -0 . 6) indicating that these sources are ionised by AGN. The fifth source (ID JADES-NS-GS-1000626) is poorly constrained in both C iii ] and He ii λ 1640, and therefore we do not show it on this diagram. \nWedetect [Ne v ] λ 3427 and NV λ 1240 in the objects JADESNS-GS-10013609 and JADES-NS-GS-00021842, respectively. In both cases, the detections are secured at SNR ∼ 4.5. However, the objects do not have solid detections of either He ii λ 1640 \nFig. 4: C iii ] / He ii λ 1640 vs C iv / C iii ](CHe2-C43) diagnostic diagram. We plot our sample as blue squares. The left and right plots show ionisation models from Feltre et al. (2016); Gutkin et al. (2016) and Nakajima & Maiolino (2022), respectively, as yellow and light blue points (see §2.4). The green diamonds show He ii λ 1640 detection from the VANDELS survey (Saxena et al. 2020), and all C iv detections from Mascia et al. (2023) as grey colour with squares and circles representing SF and AGN, respectively. The green squares show local analogues of high-redshift galaxies (from CLASSY survey; Mingozzi et al. 2023). We also show the black dashed and dotted demarcation lines between AGN, star-forming galaxies and composite line ratios from Hirschmann et al. (2022). Overall, we select five AGN on this diagram and we highlight these with a red circle. The magenta and cyan squares show a stacked spectrum for AGN and star-forming galaxies (see §4.1). \n<!-- image --> \nor C iii ] . The non-detections of C iii ] put a lower limit on the log([Ne v ] λ 3420 / C iii ] ) or log(N V λ 1240 / C iii ] ) of > 0. According to Feltre et al. (2016), this lower limit is on the border of what can be reasonably predicted by AGN photo-ionisation models. The N V λ 1240 detection can be potentially explained by star formation, assuming a very high logU = -0.5, which is inconsistent with that measured from [O ii ] and [O iii ] (logU = -2). As such, we identify JADES-NS-GS-00021842 as an AGN. It is necessary to observe these objects with deep rest-frame UV spectroscopy to help constrain the next geeneration of photoionisation models.', '4. Discussion': 'In the previous section, we presented the selection of AGN in the JADES HST Deep survey. Overall, we selected 41 AGN candidates, from at least one of the di ff erent selection methods investigated in Section 3, in our parent sample of 209 galaxies (defined in §3) in the redshift range of 1.4-9.4. The final fraction of type-2 AGN in the galaxy population is up to 20%. In this section, we discuss our results and their implications. In §4.1 we perform spectral stacking to find average emission line properties of AGN and star-forming galaxies, in §4.2 we compare the di ff erent selection methods used in this work, in §4.4 we investigated the AGN bolometric luminosities, in §4.5 we compare the SFR and stellar masses of AGN to those of star-forming galaxies and finally in §4.6 we discuss the contribution of AGN host galaxies to the UV luminosity function.', '4.1. Stacking': "In order to get the average emission line properties of our starforming and AGN samples, we stacked the R1000 grating spectra for each of the emission lines used in our emission line diagnostics. Although the PRISM observations are deeper, the vastly varying spectral resolution as a function of wavelength makes any stacking e ff orts challenging. Furthermore, some of the emission lines we are interested in (such as [NII], He ii 1640) would be blended with other emission lines. Also, we only stack spectra from the 1210 program, since the 3215 observations lack band-2 observations (see discussion in §4.2). As such, this would bias the stack against redshifts where the key emission lines are redshifted to band-2. \nWe stack the AGN hosts and star-forming galaxies in two separate stacks: a rest-frame UV emission line stack (C iv , HeII λ 1640 and C iii ]) and a rest-frame optical one (HeII λ 4685, H β , [O iii ], H α , [N ii ] ). For each of these two cases, we stack all sources with available R1000 coverage of these emission lines. Overall, we stack 9 and 20 AGN, and 31 and 55 star-forming galaxies (i.e. the parent sample type-1 and type-2 AGN) in the UV and optical emission line sets, respectively, using only the 1210 program. As shown with type-1 AGN, some objects can be low luminosity AGN that is not selected using any of our diagnostics. As such, the star-forming galaxy sample can be contaminated by unidentified AGN. \nThe stacking was performed by first shifting all spectra to the rest-frame and rebinning them to a common wavelength grid using Python's spectres package (Carnall 2017). We then model and subtract the continuum with a single power law. This is an appropriate model for the continuum, as we are only fitting a narrow wavelength range, and the continuum is poorly detected \n<!-- image --> \n<!-- image --> \n2.0 \nFig. 5: Emission line diagnostics plot for UV high ionisation lines. - Top panel: C3He2 vs [Ne iv ] λ 2424 / C iii ]; Middle Panel: C3He2 vs [Ne v ] λ 3427 / C iii ]diagnostic diagram; Bottom panel: C3He2 vs N V λ 1240 / C iii ]. We plot our objects detected in high ionisation lines as blue points. The yellow and light blue points show star forming and AGN from photo-ionisation models from Feltre et al. (2016); Gutkin et al. (2016) (see §2.4). We highlight the emission line ratios of GN-z11 (Maiolino et al. 2023b) as the magenta star. \n<!-- image --> \nin the R1000 spectra. The individual spectra are weighted using two separate weighting schemes: a) 1 / rms 2 weights; and b) 1 / (rms 2 × F[OIII]) or 1 / (rms 2 × FCIII]) weights for the UV line stacking) weights. However, given the large range of redshifts and galaxy luminosities, we found that the final stacked spectra are dominated by bright targets when we do not normalise by \nline fluxes. Hence, we will use the stacked spectra weighted by both the noise and the line fluxes ([O iii ] or C iii ]), which approximates a median stacked spectrum. \nThe measured fluxes and their uncertainties for the HeII λ 4685, H β , [O iii ], H α , [N ii ] and UV emission lines: CIV, HeII λ 1640 and C iii ] λ 1907,1909 along with the number of objects in the stack and their median redshift are summarised in the Table 4. The final stacked spectra and the best fits are shown in the Appendix in Figures A.1 and A.2. We detect all emission lines across all samples at > 3 σ , except for He ii λ 4686 in the star-forming galaxy sample. The detection of [N ii ] seems like a contradiction compared to Cameron et al. (2023b), however, this is purely due to the inclusion of galaxies with z < 4, while Cameron et al. (2023b) restricted their sample to z > 5 . 5. \nWe show the line ratios from the stacked spectra in all diagnostic diagrams for the AGN and star-forming samples as magenta and cyan squares in Figures 2, 3 and 4. \nThe stacked spectra on the N2-BPT (see top panel of Figure 2) show that AGN at high-z are [N ii ] weak and have a [O iii ] / H β ratio very similar to star-forming galaxies, making the selection of AGN at high-z impossible using this method. Furthermore, the AGN host galaxies have the same [S ii ] / H α ratios as star-forming galaxies (see bottom panel of Figure 2), further showing the di ffi culty of using this diagram to select AGN at high-z. \nWe do not detect the He ii λ 4686 in the star-forming galaxies, and therefore we place a 3 σ upper limit on the flux. The He ii λ 4686 / H β vs [N ii ] / H α diagram shows a separation of starforming and AGN galaxies as expected based on the study by Shirazi & Brinchmann (2012) that focused on SDSS galaxies. As such, the He ii λ 4686 line is an ideal tracer of AGN activity; however the He ii λ 4686 is a factor of 7 fainter than He ii λ 1640, making it extremely di ffi cult to detect, even with JWST. \nWe detect He ii λ 1640 in both star-forming and AGNdominated galaxies, although the flux of He ii λ 1640 is three times higher in AGN compared to star-forming galaxies. Furthermore, the C iv emission line is a factor of eight brighter in AGN than in the star-forming galaxies. This indicates a much higher and harder ionisation field in AGN host galaxies, boosting the high ionisation lines. However, the stacked spectrum of star-forming galaxies shows He ii λ 1640, with the star-forming stack in the border of AGN and star-forming galaxies on the C iii ] / C iv vs C iii ] / He ii λ 1640 diagnostic plot (see Figure 4). This can be due to some low luminosity AGN in the star-forming sample that do not have individual He ii λ 1640 detections, or because star-forming galaxies at high-z are creating more hard ionising photons than previously modelled.", '4.2. Comparing AGN selection methods': 'Regardless of the emission diagnostic used to identify AGN (emission lines, X-ray observations, mid-infra-red selection), the selection method is reliant on finding emission that cannot be explained by star-formation processes. Objects not selected as AGN, the star-forming galaxies, can still have a significant amount of AGN activity, but it is not dominating the total galaxy emission. Therefore, AGN selection most likely provides an lower limit on the total number of AGN, as we are likely missing low luminosity active black holes, outshone by their host galaxy. \nThere is little overlap between the individual emission line ratios used to find AGN in this work, with only five objects being selected in more than one diagnostic. This can be explained by the di ff erent emission line ratios being more or less sensitive \nTable 4: Results of the stacking analysis for UV and optical emission lines of star-forming and AGN samples. \n. \n. \nFig. 6: [O iii ] / H β vs [NeIII] / [OII] diagram ("OHNO"). The black dashed lines show the boundary between AGN and star forming galaxies from Backhaus et al. (2022). The orange and light blue points show star forming and AGN photoionisation models from Feltre et al. (2016) and Gutkin et al. (2016), respectively. There is a significant overlap between AGN and star forming galaxies in this diagram, making it unreliable to select AGN. \n<!-- image --> \nto AGN activity in di ff erent observed regimes (e.g. redshift and metallicity of the ISM). As discussed above, the N2-BPT method becomes increasingly unreliable towards high redshift as galaxies become more metal-poor and have higher ionisation parameters. Although the highest redshift AGN selected by this method is at z = 5.135 (JADES-NS-GS-00009452; 12 + log(O / H) = 7.92), the bulk of the AGN selected by this method are below z = 2.3. \nTopping et al. (2020); Runco et al. (2021) investigated the e ff ect of stellar population age and metallicity on the BPT and S2-VO87 diagrams. These works have found that a population of young, metal-poor stellar populations can lie above the Kewley et al. (2001) line. These low S2 ratio targets would be excluded from selection with the new definition of the demarcation lines and the AGN selected in the S2-VO87 diagram are above the points seen in both Topping et al. (2020); Runco et al. (2021) (see also Strom et al. 2017). However, many of our selected AGN are very close to the demarcation line, making their selection tentative. As a result, we select any AGN selected within a distance of 0.1 dex from the demarcation as tentative, and we mark this with asterisks in Table 2. We note that our conclusions do not change whether we include or exclude these tentative AGN in our final sample. \nIn both sets of observations (1210 and 3215) we select more AGN using the S2-VO87 diagram (with [S ii ] emission line) than the classical BPT diagram (using [N ii ] emission line). The [S ii ] doublet is not blended with H α in the PRISM observations at z < 5, and hence we can use the deeper PRISM observations to constrain [S ii ] doublet. On the contrary, the [N ii ] doublet is blended with H α in the PRISM observations, and hence we require the shallower R1000 JADES observations to put constraints on it. This is especially taxing for the 3215 observations, which were designed to observe galaxies at z > 6 and hence do not have any R1000 Band 2 observations. This results in no constraints on the [N ii ] doublet for z = 1.8-3.6 in the 3215 program, and hence low detection of AGN on the BPT diagram in this program. \nLuckily, as we push our AGN selection to higher redshifts, many useful UV lines ([Ne iv ] λλ 2424, [Ne v ] λ 3427, NV λ 1240, He ii λ 1640) are redshifted into the wavelength range of NIRSpec, and then to a higher sensitivity range of NIRSpec ( λ > 1 . 2 µ m). As a result, the majority of the AGN selected above z = 3 are based on [Ne iv ] λλ 2424, [Ne v ] λ 3427, NV λ 1240, He ii λ 1640. Still, these essential emission lines for identifying AGN activity at high redshift remain di ffi cult to detect with JWST. Many of the emission lines such as N V λ 1240, He ii λ 1640 are blended with nearby emission lines in the PRISM observations and require R1000 observations to deblend, which are shallower in JADES survey compared to the PRISM observations. Dedicated ultra-deep R1000 observations are required to detect these lines, even in moderate luminosity AGN. \nThe HeII λ 4686 emission line remains a robust diagnostic to detect AGN across large redshift and metallicity ranges. However, it is intrinsically fainter by a factor ∼ 10 compared to the HeII λ 1640, resulting in it being a challenge to detect even in deep JWST spectra. However, if detected at high redshift, it is an unambiguous tracer of AGN activity at high redshift. \nIt is important to point out that these high-ionization lines do not trace AGN activity as such, but a hard ionising radiation, with which AGN activity is the most likely source in galaxy evolution. However, other sources can also produce He ii emission such as Wolf-Rayet stars, X-ray binaries, and some more exotic star-formation processes (e.g. very top-heavy IMF; Thuan & Izotov 2005; Kehrig et al. 2015; Schaerer et al. 2019; Umeda et al. 2022). However, the WR stars produce very broad He ii forming so-called "blue bump" (see Brinchmann et al. 2008) features not observed in any of our sources. Furthermore, Saxena et al. (2020) indeed reproduced their detections of He ii in the VANDELS survey using implementations of binary stars using BPASS models. However, the AGN and SF galaxies from the same VANDELS survey (Mascia et al. 2023) show the separation of the SF and AGNhost galaxies predicted by the Gutkin et al. (2016) and Feltre et al. (2016) models. Nakajima & Maiolino (2022) models, which we use as a comparison of our data, also use BPASS to implement hard ionising photons from binary stars. Meanwhile, high-luminosity AGN at high redshift, such as type-1 AGN at z ∼ 5.5 from Übler et al. (2023) do show features of relatively weak He ii λ 4686 (log10(He ii λ 4686 / H β ) ∼ -1 . 2), showing that even objects with relatively weak HeII can be AGN. Finally, Cameron et al. (2023a) identified one of our AGN (ID 9422) to be dominated by nebular continuum. This will be addressed in a separate paper in more detail (Tacchella et al. in prep). \nLarson et al. (2023) investigated the use of "OHNO" diagram ([O iii ] / H β vs [NeIII] / [OII]) to replace the N2-BPT diagram in selecting AGN. We plot the Cloudy photoionisation models in Figure 6 with JADES galaxies. There is a significant overlap between AGN and star-forming galaxies in this diagram. The one \nFig. 7: Fraction of AGN as a function of redshift in our survey. The blue points show the fraction resulting from the combined programs (1210 + 3215) while the red stars and squares show separately the 1210 and 3215, respectively. The red points are o ff set by 0.05 in the x-axis for better visibility. We see no significant evolution of the AGN fraction as a function of redshift. \n<!-- image --> \ngalaxy with [O iii ] / H β > 0.95 lies in the region only populated by AGN Cloudy models is 16745 at z = 5 . 56. Overall, in agreement with Larson et al. (2023), we confirm that it is di ffi cult to di ff erentiate between AGN and metal-poor high ionisation starforming HII regions in the OHNO diagram. For this reason, we need to rely on higher ionisation lines. \nFinally, Maiolino et al. (2023a) recently published a sample of type-1 AGN observed with two sources overlapping with our observations. We detected only one as an AGN based on emission line diagnostics (JADES-NS-GS-0008083). Although it was selected by detection of both HeII λ 4686 and NeIV, the other AGN (with log(LBol = 44 . 3)) is not selected in any of the narrow emission line diagnostics. As already discussed above, it is important to stress that emission line diagnostics miss AGN at high redshift. This can be attributed to 1) the physical properties of high-z AGN being either di ff erent from local templates (especially because of low metallicity); 2) these AGN not properly modelled by photoionization models (hence not predicted by the diagnostics); or 3) the emission from these sources is dominated by star-formation in the host galaxy (see Silcock et al. in prep).', '4.3. AGN fraction with deeper and shallower data': 'Across the two observational programs, we selected 28 and 14 AGN from the parent samples of 110 and 99 galaxies, respectively. This corresponds to an AGN fraction of 24 ± 5 % and 14 ± 4 %, and a combined AGN fraction from the two programs of 20 ± 3 %. In Figure 7, we show the AGN fraction as a function of redshift. We plot the AGN fraction from 1210 and 3215 programs as red stars and squares and the combined dataset as blue points. The AGN fraction is consistent with being constant across all redshifts within 1 σ . \nThe fraction of AGN between the two tiers is consistent within 1.5 σ of each other. Although this di ff erence is statistically not significant, it is worth discussing the apparent decrease of AGN in the 3215 program, despite this program being a factor of 1.5-3 × deeper than 1210 program. In the 3215 observations, only two AGN host galaxies were selected based on He ii λ 4686, compared to nine in the 1210 program. This is primarily due to \nFig. 8: Bolometric luminosity vs redshift for our objects, plotted as red diamonds. Our bolometric luminosities are estimated from narrow line emission lines and should be treated as upper limits. We compare our sample with previous JWST type-1 AGN (various coloured stars and diamonds Bogdan et al. 2023; Carnall et al. 2023a; Kocevski et al. 2023; Kokorev et al. 2023; Furtak et al. 2023; Goulding et al. 2023; Maiolino et al. 2023a,b; Matthee et al. 2023; Übler et al. 2023; Chisholm et al. 2024), AGN from the KASHz and SUPER surveys at Cosmic Noon (grey and red crosses Harrison et al. 2016; Kakkad et al. 2020) and QSOs samples across redshifts: SDSS QSOs (z = 2-6, grey shaded region; Wu & Shen 2022), extremely red quasars (bluegreen crosses; Perrotta et al. 2019), blue QSOs (purple crosses; Shen 2016) and compilation of EoR QSOs (magenta crosses; Fan et al. 2023). \n<!-- image --> \nthe lack of Band 2 R1000 observations in 3215, which results in no constraints of He ii λ 4686 for redshifts 2.8-5.4. As discussed above, this similar issue is also plaguing the BPT selection using the [N ii ] emission line, and hence we do not select any AGN using the diagnostics in BPT either. \nThe Band 1 R1000 observations, which are key to observing rest-frame UV emission lines such as C iii ], He ii λ 1640 and C iv doublet, are only a factor of ∼ 1.4 deeper (twice the exposure time) in 3215 than in 1210. Unfortunately, this improvement in sensitivity is not enough to constrain the He ii λ 1640 in regular SF galaxies or less extreme AGN. We would thus require deeper rest-frame UV observations to constrain the AGN and star-forming populations (over 50 hours with JWST / NIRSpec).', '4.4. AGN luminosities': 'The bolometric luminosity is one of the key properties of AGN and can be easily estimated through the luminosity of the Xray, BLR or the UV continuum emission from the accretion disc (e.g. Stern & Laor 2012; Netzer 2019; Duras et al. 2020; Saccheo et al. 2023). However, as our sources were not selected as Type1 AGN or X-ray detections, but based on their narrow emission line properties, hence, we do not have the BLR properties of these AGN, nor we have their UV or X-ray flux. As a result, we are forced to use the narrow line emission lines to estimate the bolometric luminosity. We calculate the bolometric luminosities (Lbol) of our sample from the narrow line fluxes of H β , [O iii ] and C iii ] , using the new calibrations by Hirschmann et al. (in prep). The estimated Lbol from all three emission lines agree within 2 σ , however, they are systematically lower by ∼ 0.5 dex compared to the calibration from Netzer (2009). \nWe present the estimated bolometric luminosities in Figure 8 as a function of redshift, together with other AGN from the literature (both from JWST spectroscopic studies and from pre- \nvious, non-JWST surveys). However, these luminosities should be considered as upper limits, as they assume that the H β and [O iii ] or C iii ] emission is dominated by the narrow line emission from the AGN, with no contribution from star formation. This is not necessarily true, as AGN are hosted in star-forming galaxies (see §4.5). The estimated bolometric luminosities are reported in Table 2. \nAs illustrated in Figure 8, pre-JWST studies used large optical and NIR surveys to identify sources dominated by bright rest-frame optical and UV emission, selecting primarily QSOs, or intermediate luminosity AGN at z < 3. Specifically, in Figure 8, we show a compilation of QSOs by Fan et al. (2023) as green, blue and magenta crosses and X-ray AGN from the KASHz and SUPER surveys (Harrison et al. 2016; Circosta et al. 2018; Kakkad et al. 2020), and selection of red and blue QSOs from Cosmic Noon from Shen (2016); Perrotta et al. (2019). We also show quasars from SDSS DR16 (Wu & Shen 2022) as shaded contours. Since the launch of JWST, there have been a number of studies searching for AGN (plotted in Figure 8 as various coloured diamonds and stars, Bogdan et al. 2023; Carnall et al. 2023a; Kocevski et al. 2023; Kokorev et al. 2023; Furtak et al. 2023; Goulding et al. 2023; Maiolino et al. 2023a,b; Matthee et al. 2023; Übler et al. 2023; Chisholm et al. 2024). Our AGN sample has generally similar luminosities to those selected as type-1 AGN with JWST spectroscopic surveys. \nWith the new capabilities of JWST, we are now probing, at z > 3, AGN that are 2-3 orders of magnitude less luminous than previous surveys, allowing us to understand the broader demographics of the AGN population in the early Universe for the first time.', '4.5. Host Galaxy properties': 'One of the key questions of studying AGN host galaxies is investigating their star-formation properties. Until the launch of JWST, these studies have mostly focused on z < 3 moderate luminosity AGN and z > 4 quasars. These studies have shown that AGN host galaxies have SFR at, or just below, the level of the star-forming galaxies main sequence (SFMS) at Cosmic Noon when mass-matching the active and inactive samples ( z ∼ 1 -3; Santini et al. 2012; Rosario et al. 2013; Vito et al. 2014; Mullaney et al. 2015; Stanley et al. 2015; Azadi et al. 2015; Scholtz et al. 2018; Förster Schreiber et al. 2019). With the launch of JWST, we can now select and measure the SFR of moderate luminosity AGN at high redshift. Additionally, unlike previous studies with JWST focusing on type-1 AGN, we do not su ff er from the AGN contaminating the continuum emission in our spectra, and hence modeling the stellar SED of the host galaxy is less problematic (although the AGN can still contaminate the Balmer emission lines hence making them less reliable for the SFR estimation). \nIn §2.3, we described the SED fitting approach adopted to derive the SFR and stellar masses. We use the star-forming main sequence (SFMS) prescription from Looser et al. (2023b), who estimated the SFMS based on 3 separate redshifts bins for sources from JADES with M ∗ < 10 9 . 3 M ⊙ . For AGN at Cosmic Noon, (Harrison et al. 2016; Circosta et al. 2018) we use the SFMS prescription of Schreiber et al. (2015) as it covers the redshift and stellar mass range of our AGN. It is worth mentioning that Schreiber et al. (2015) includes a turnover at high stellar masses. \nIn Figure 9, we show the o ff set from the star-forming mainsequence (SFR / SFRMS) against the stellar mass for AGN and star-forming galaxies in the JADES survey (red and blue points); \n<!-- image --> \nFig. 9: O ff set from the star-forming main sequence (SFMS) of our AGN host galaxies. Top panel: O ff set from the SFMS against stellar mass for the JADES sample (blue points), our selected AGN (red points). We compare our sample to the previous AGN sample KASHz survey (Harrison et al. 2016, grey points) and SUPER survey (Circosta et al. 2018, green points). We highlight the 1 σ scatter of SFMS by yellow shaded region. Bottom panel: Distribution of SFR / SFRMS of the AGN candidates selected in this work. The blue histogram shows the data from this work, and the red dashed line shows the best fit to the distribution of o ff set from SFMS assuming it is log-normal. The green and orange dashed lines show the distribution for the SFMS (Schreiber et al. 2015) and AGN (Mullaney et al. 2015), respectively. \n<!-- image --> \nKASHz AGN survey (grey points) and SUPER AGN survey (green points). The AGN host galaxies in our sample have the same stellar mass distributions as the star-forming galaxies in the JADES HST Deep survey, with a median stellar mass of SF galaxies and AGN are 10 7 . 9 and 10 8 M ⊙ , respectively. The presence of AGN at such low masses indicates that AGN activity is also important in galaxies with M ∗ < 10 10 , as previously suggested by at high redshift (Koudmani et al. 2021, 2022) and low \nredshift (Burke et al. 2022; Mezcua et al. 2023; Siudek et al. 2023). \nThe SFRs of our selected AGN are consistent with the starforming galaxies in the JADES sample. However, measuring the SFR of AGN host galaxies is notoriously di ffi cult (see e.g. Stanley et al. 2015, 2018; Ri ff el et al. 2021), as AGN can contaminate the emission lines used to estimate SFR. Therefore, many of the SFRs presented in this work should be considered as upper limits. Further SED fitting to disentangle the AGN and starformation activity will be performed in Silcock et al. (in prep). \nTo evaluate the di ff erence in the star-formation properties of AGN in our sample, we investigate the distribution of SFo ff set. Following Mullaney et al. (2015); Scholtz et al. (2018); Bernhard et al. (2019), we fitted the distribution of SFR / SFRMS assuming that it is log-normal, with mode and width as free parameters, using an MCMC method. The mode and width of the derived log-normal distribution are 0 . 07 ± 0 . 10 and 0 . 61 ± 0 . 07, respectively. The mode of the distribution is consistent with that of SFMS (modeSF = -0.07; Schreiber et al. 2015), however, the width is 2 σ higher than that of SFMS (width of SFMS of 0.3 dex). Some of this di ff erence may be attributed to the methods used to derive SFR (Caplar & Tacchella 2019). For example, the BEAGLE fitting accounts for physical variations that can lead to variation in the UV to SFR and H α to SFR relationships (see e.g. Curtis-Lake et al. 2021), while the UV and IR SFRs in Schreiber et al. (2015) use simple conversions to SFR. \nMullaney et al. (2015) have measured the distribution of SFR / SFRMS for X-ray selected AGN at z ∼ 1.6, finding that the mode and the width of the distribution as -0 . 36 ± 0 . 07 and 0 . 56 ± 0 . 09, respectively. The disagreement between our results and Mullaney et al. (2015) can be attributed to our selection of AGN as well as the fact that SFRs here should be considered upper limits. Furthermore, Mullaney et al. (2015) investigated AGN in massive galaxies (M ∗ > 10 10 M ⊙ ) at z = 1-1.5. The majority of the high-z AGN were selected based on the detection of He ii λ 1640 or He ii λ 4686 which are more likely to be detected in brighter targets. \nHowever, in this work, we do not have a su ffi cient number of sources to further split our AGN sample by redshift, stellar mass or AGN luminosity. This will be performed in future works with the full JADES sample.', '4.6. Contribution of AGN host galaxies to the UV luminosity function': 'With this new selection of a large sample of Type-2 AGN, we can now investigate the contribution of AGN host galaxies to the UV luminosity function at high redshifts. We explore the redshift range 4-6, as this is poorly explored in the literature before JWST. \nSince the selection function for allocating targets in JADES is complex, it is hard to derive the volume density of AGN host galaxies as a function of the UV luminosity based on the number of AGN that we have identified. More importantly, we do not have high enough statistics to derive a proper luminosity function. Therefore, we make the simplified assumption that the spectroscopic selection function has not preferentially favoured or disfavoured galaxies hosting AGN. We estimate the contribution of AGN host galaxies to the UV luminosity function simply as the fraction of AGN identified in UV luminosity bins. \nWe choose the parameterised luminosity function from Bouwens et al. (2021) as the reference luminosity function, choosing a median redshift of our type-2 AGN at z = 4-6 of zmedian ∼ 5 . 5 (14 objects in total). \n<!-- image --> \nFig. 10: UV luminosity function of galaxies at z = 5 from Bouwens et al. (2021) (blue solid line) and inferred contribution of galaxies hosting type-2 AGN, inferred from the JADES survey (red points). Results from other surveys are also reported, as indicated in the legend. The dashed red line shows the galaxy luminosity function scaled downward by a factor of 5. The orangeshaded region shows the range of possible extrapolated luminosity functions for QSOs from Niida et al. (2020). \n<!-- image --> \nFollowing the procedure discussed above, we show in Figure 10 (as red points) the UV luminosity function of the type-2 AGN host galaxies z > 4 from the HST Deep of JADES survey. We split our sample into three separate bins, choosing the bins to allow at least three objects per bin, centred on MUV = -20.15, -18.8, -17.6. The estimated contribution of the type-2 AGN host galaxies to the UV luminosity function is 33 %, 18 % and 20 %, for the MUV = -20.15, -18.8, -17.6 bins, respectively. Given the limited number of points, we do not attempt to fit a functional form, as a small number statistics mean that the parameters are not adequately constrained. Instead, we simply show that the type 2 AGN host galaxies LF can match the Bouwens et al. (2021) galaxy UV luminosity scaled by a factor of 5, as shown by the red dashed line. \nWe also compare our results with other JWST type-1 AGN surveys: Harikane et al. (2023), Kocevski et al. (2023), Maiolino et al. (2023a), Matthee et al. (2023), as indicated in the legend. The cyan diamonds and squares show the luminosity function inferred by Giallongo et al. (2019) based on X-ray surveys. Given the uncertainties, our estimated type 2 AGN density is slightly higher (by a factor of ∼ 2) than the one estimated for type-1 AGN from the JADES and CEERS surveys (Maiolino et al. 2023a; Harikane et al. 2023). This suggests a type 2 to type 1 AGN ratio of about two at z ∼ 4-6, or possibly higher (given that it is more challenging to identify type 2 AGN). We note that the contribution to the UV luminosity function is significantly higher than the estimate of Kocevski et al. (2023), but which was most likely su ff ering from a low number of statistics in the early JWST results. The contribution is also higher than found by Matthee et al. (2023), however, as discussed in Maiolino et al. (2023a), that study probe more luminous AGN, hence likely less abundant. \nWe also note that our estimated density of type-2 AGN is higher than found in the deep X-ray observations by Giallongo et al. (2019). As stated by Maiolino et al. (2023a), this suggests \nthat the optical and UV emission line selection of AGN is picking a population of faint or X-ray deficient population of AGN at high-z. The extrapolation of the QSO luminosity function at z ∼ 5.5 (orange shaded region Niida et al. 2020) is 1-2 orders of magnitude lower than AGN luminosity function from JWST. Together with Figures 8, 9 and 10, this further illustrates that JWST is probing a new parameter space of low luminosity AGN activity at high-z redshift. \nWefinally point out that our finding that the type 2 AGN host galaxies UV LF can be reproduced by scaling the galaxies LF by a factor of 5, indicates that type 2 AGN host galaxies contribute to ∼ 20% to the reionisation of the Universe ( ∼ 30% if including the type 1 AGN). Note that this does not mean that AGN contribute to a similar level to the reionization of the Universe, as the ionizing AGN radiation is obscured along our line of sight, and the IGM sees on average the same fraction of type 2 to type 1 AGN as our line of sight.', '4.7. JADES-GS+53.11243-27.77461 - type-2 AGN at z ∼ 9.43': 'The most distant type 2 AGN in our sample is ID 10058975 (JADES-GS + 53.11243-27.77461), which was identified using two separate methods: 1) the detection of [Ne iv ] λ 2422 UV emission line; 2) by identifying He ii λ 1640, C iii ] and C iv emission line ratios diagnostic. He ii λ 1640 is redshifted to the ∼ 1.71 µ m, which is covered by both Band-1 and Band-2 R1000 grating observations from 1210 observations and Band1 observations in 3215 observations. We showed the final stacked spectrum from all three observations in Figure 11. The He ii λ 1640 emission line is detected in all three sets of observations, however, we stacked the three spectra for more robust detections (SNR = 6.5). The [Ne iv ] λ 2424 emission is detected at 4 σ , and we show the spectrum in Figure B.1, indicating the presence of strong ionising radiation. \nAs this object is well detected in all gratings and PRISM observations, we were able to fit the full 1-5 µ m spectrum with BEAGLE. We estimated the stellar mass of the AGN host galaxy to be 1 . 5 × 10 8 M ⊙ with an SFR of 6.6 M ⊙ yr -1 (SFR / M ∗ = 40 Gyr -1 ). The object has a UV magnitude of MUV = -20 and is most likely going through a starburst. However, as discussed above, the high SFR can be explained by the contamination of the emission lines by the AGN, artificially increasing the estimated SFR of the system. \nFurthermore, this object has a strong [O iii ] λ 4363 emission (5.1 σ Laseter et al. 2023) at 5.1 σ . This allowed the authors to constrain the electron temperature (T e ) of 18400 ± 2100K and metallicity 12 + log(O / H) = 7 . 46 ± 0 . 11. Such high temperatures and strong auroral line are anomalous in HII regions, but seen in the NLR of AGN (e.g. Brinchmann 2023). The full detailed analysis of the [O iii ] λ 4363 and [O iii ] λ 1661,66 is presented in Curti et al. (in prep.). \nThe metallicity inferred from the direct method is 4 σ lower than the value inferred by BEAGLE. Despite the unambiguous detection of an AGN and exquisite data provided by NIRSpec we are still hampered in the characterization of NLR and host for integrated spectra such as this one. More advanced SED fitting of these objects using e.g. BEAGLE-AGN (Vidal-García et al. 2022) is outside the scope of this paper and will be further investigated by Silcock et al. (in prep).', '5. Conclusions': "In this work we have presented the identification of obscured, i.e. narrow-line (type-2), AGN candidates in the two deepest \nFig. 11: Detection of C iv λλ 1548,50, He ii λ 1640 and [O iii ] λ 1666 in object ID 10058975 (JADES-GS + 53.1124327.77461) at z = 9.43. The spectrum is a combined spectrum of the Band-1 and Band-2 data from program ID 1210 and Band1 data from program ID 3215 to increase the SNR of the detection. The blue line shows the continuum subtracted observed spectrum, while the red dashed line shows the best fit to the data. \n<!-- image --> \nspectroscopic fields of the JADES survey in GOODS-S, using rest-frame optical and UV emission lines. We have investigated the presence of AGN ionisation, in narrow line emission, by using classical optical N2-R3, S2-VO87, N2-He2 diagnostic diagrams, as well diagrams exploiting UV emission lines: C iii ], C iv , He ii λ 1640, [Ne iv ] λ 2424, [Ne v ] λ 3420 and N V λ 1240. \nBased on our analyses we find: \n- -At z > 3, the N2-BPT and S2-VO87 diagnostic diagrams ([OIII]5007 / H β vs [NII]6584 / H α , and vs [SII]6730 / H α , respectively) are no longer able to clearly distinguish photoionisation due to type-2 AGN and star-forming galaxies, because low metallicity, high-z AGN and star-forming galaxies occupy the same space on these diagrams (see Figure 2). However, we redefine a conservative demarcation line between AGN and star-forming galaxies on the BPT and S2-VO87 diagrams to allow for the shift high redshift starforming galaxies. We identify five and seventeen AGN host galaxies on the N2-BPT and S2-VO87 diagrams, respectively. We stress that this is a very conservative selection and that many more AGN are certainly present on these diagrams, mixed with star forming galaxies.\n- -Using the He ii λ 4686 / H β vs [N ii ] / H α diagnostic diagram we selected eleven AGN and an additional six AGN using the C iii ] / C iv vs C iii ] / HeII λ 1640 diagram (see Figures 3 & 4). Interestingly, the only X-ray detected AGN in our sample, is located in the star-forming region of the BPT diagram, while is confirmed as AGN in the HeII diagnostic diagrams.\n- -We detected the high ionization transitions [Ne iv ] λ 2424, [Ne v ] λ 3420 or N V λ 1240 in seven galaxies. The luminosities of these high ionisation lines compared to the C iii ] emission line classify these objects as AGN hosts galaxies.\n- -In total we selected 28 AGN in the PID-1210 programme and 14 AGN in the PID-3215 programme, resulting in AGN fractions of 24 ± 5%and14 ± 4%, respectively. Combining the two samples, we find 41 unique AGN, and an overall AGN fraction of 20 ± 3%. We investigated the evolution of AGN fraction as a function of redshift (see Figure 7) and did not find evidence for significant evolution. \n- -By stacking the AGN and star-forming galaxies' rest-frame UVand optical spectra we confirmed that both samples have similar emission line ratios in the rest-frame optical N2, S2 and R3 emission line ratios. However, the populations are easily distinguished using the He ii λ 4686 and He ii 1640 emission lines (see cyan and magenta points on Figures 2, 3, 4).\n- -The estimated bolometric luminosities using narrowemission lines ([O iii ], C iii ]and H β ) are in the range of 6 × 10 41 -5 × 10 45 ergs s -1 (see Figure 8). The selected AGN host galaxies have a median stellar mass of 10 7 . 9 M ⊙ consistent with the median stellar mass of inactive galaxies of 10 8 . 0 M ⊙ .\n- -We investigated the distance from the star-forming main sequence, log(SFR / SFRms). Our AGN candidates have a mode and width of the distribution of log(SFR / SFRms) of -0 . 07 ± 0 . 10 and 0 . 61 ± 0 . 07, respectively. The mode of distribution is consistent with SF galaxies while the width of the distribution is a factor of two broader than for star-forming galaxies, consistent with previous results at Cosmic Noon.\n- -We estimated the contribution of the AGN host galaxies to the UV luminosity function on the order of ∼ 20%, with a slight (increasing) dependence on luminosity (see Figure 10).", 'Acknowledgements': 'We thank Anna Feltre for providing us with the updated Cloudy models for star-forming and AGN with expanded emission line coverage. We thank Michaela Hirschmann for the productive discussion to help with the selection based on UV emission lines. We would like to further thank Dominika Wylezalek and Eduardo Banados. JS, RM, FDE, WB, XJ, TJL acknowledge ERC Advanced Grant 695671 \'QUENCH\' and support by the Science and Technology Facilities Council (STFC) and by the UKRI Frontier Research grant RISEandFALL. RM acknowledges support by the UKRI Frontier Research grant RISEandFALL as well as funding from a research professorship from the Royal Society. SCa and GV acknowledge support by European Union\'s HE ERC Starting Grant No. 101040227 - WINGS. AJB, AJC, JC, AS & GCJ acknowledge funding from the "FirstGalaxies" Advanced Grant from the European Research Council (ERC) under the European Union\'s Horizon 2020 research and innovation programme (Grant agreement No. 789056). ECL acknowledges support of an STFC Webb Fellowship (ST / W001438 / 1). This research is supported in part by the Australian Research Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project number CE170100013. DJE is supported as a Simons Investigator and by JWST / NIRCam contract to the University of Arizona, NAS5-02015. Funding for this research was provided by the Johns Hopkins University, Institute for Data Intensive Engineering and Science (IDIES). MP acknowledges support from Grant PID2021-127718NB-I00 funded by the Spanish Ministry of Science and Innovation / State Agency of Research (MICIN / AEI / 10.13039 / 501100011033). BER, KH, ZJ, JL, MR, FS, and CNAW acknowledge support from the NIRCam Science Team contract to the University of Arizona, NAS502015. WB acknowledges support by the Science and Technology Facilities Council (STFC). BRP acknowledges support from the research project PID2021-127718NB-I00 of the Spanish Ministry of Science and Innovation / State Agency of Research (MICIN / AEI / 10.13039 / 501100011033). MSS acknowledges support by the Science and Technology Facilities Coun- \ncil (STFC) grant ST / V506709 / 1. The research of CCW is supported by NOIRLab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. HÜ gratefully acknowledges support by the Isaac Newton Trust and by the Kavli Foundation through a Newton-Kavli Junior Fellowship. This work was performed using resources provided by the Cambridge Service for Data Driven Discovery (CSD3) operated by the University of Cambridge Research Computing Service (www.csd3.cam.ac.uk), provided by Dell EMC and Intel using Tier-2 funding from the Engineering and Physical Sciences Research Council (capital grant EP / T022159 / 1), and DiRAC funding from the Science and Technology Facilities Council (www.dirac.ac.uk). The authors acknowledge use of the lux supercomputer at UC Santa Cruz, funded by NSF MRI grant AST 1828315.', 'References': "Alexander, D. M. & Hickox, R. C. 2012, New A Rev., 56, 93 Azadi, M., Aird, J., Coil, A. 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I., Papovich, C., et al. 2023b, ApJ, 950, L5\n- Yang, J., Wang, F., Fan, X., et al. 2020, ApJ, 897, L14 \nZappacosta, L., Piconcelli, E., Fiore, F., et al. 2023, arXiv e-prints, arXiv:2305.02347 \n- 1 Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, UK\n- 2 Cavendish Laboratory, University of Cambridge, 19 JJ Thomson Avenue, Cambridge CB3 0HE, UK\n- 3 Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK\n- 4 Centre for Astrophysics Research, Department of Physics, Astronomy and Mathematics, University of Hertfordshire, Hatfield AL10 9AB, UK\n- 5 Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126 Pisa, Italy\n- 6 Sorbonne Université, CNRS, UMR 7095, Institut d'Astrophysique de Paris, 98 bis bd Arago, 75014 Paris, France\n- 7 European Southern Observatory, Karl-Schwarzschild-Strasse 2, 85748 Garching, Germany\n- 8 Centro de Astrobiología (CAB), CSIC-INTA, Cra. de Ajalvir Km. 4, 28850- Torrejón de Ardoz, Madrid, Spain\n- 9 European Space Agency (ESA), European Space Astronomy Centre (ESAC), Camino Bajo del Castillo s / n, 28692 Villanueva de la Cañada, Madrid, Spain\n- 10 School of Physics, University of Melbourne, Parkville 3010, VIC, Australia\n- 11 ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ATRO 3D), Australia\n- 12 Department of Physics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, UK\n- 13 Harvard University, Center for Astrophysics | Harvard & Smithsonian, 60 Garden St., Cambridge 2138, USA\n- 14 Steward Observatory, University of Arizona, 933 N. Cherry Avenue, Tucson, AZ 85721, USA\n- 15 Department of Physics and Astronomy, The Johns Hopkins University, 3400 N. Charles St., Baltimore, MD 21218\n- 16 AURA for European Space Agency, Space Telescope Science Institute, 3700 San Martin Drive. Baltimore, MD, 21210\n- 17 Department of Astronomy, University of Wisconsin-Madison, 475 N. Charter St., Madison, WI 53706 USA\n- 18 Department of Astronomy and Astrophysics, University of California, Santa Cruz, 1156 High Street, Santa Cruz, CA 95064, USA\n- 19 NSF's National Optical-Infrared Astronomy Research Laboratory, 950 North Cherry Avenue, Tucson, AZ 85719, USA\n- 20 NRC Herzberg, 5071 West Saanich Rd, Victoria, BC V9E 2E7, Canada", 'Appendix A: Stacked spectra': 'We show the stacked rest-frame UV and optical spectra in Figures A.1 and A.2 for the AGN and star forming galaxies.', 'Appendix B: UV fits': 'We fit the [Ne iv ] λ 2424, [Ne v ] λ 3427 and NV λ 1240 emission lines in the § 2.2.1 and we show the spectra for the objects with detected [Ne iv ] λ 2424 in Figure B.1. Furthermore, we report the fluxes of the detected lines in Table B.1.', 'Appendix C: Emission lines used in this work': 'In Table C.1 we summarise the emission lines used in this work, their wavelength and ionisation potential. \nFig. A.1: Resulting spectrum from the stacking analysis of AGN host galaxies. The blue line shows the continuum subtracted data, while the red dashed line indicates the best fit to the stacked spectrum. Top row: Stacked spectrum of H α , [N ii ] and [S ii ] (left panel) and He ii λ 4686, H β and [O iii ] λ 5008 (right panel). Bottom row: Stacked spectrum of C iv (left panel), He ii λ 1640 and [O iii ] λ 1666 (middle panel), and C iii ] (right panel). \n<!-- image --> \nFig. A.2: Resulting spectrum from the stacking analysis of star forming galaxies. The blue line shows the continuum subtracted data, while the red dashed line indicates the best fit to the stacked spectrum. Top row: Stacked spectrum of H α , [N ii ] and [S ii ] (left panel) and He ii λ 4686, H β and [O iii ] λ 5008 (right panel). Bottom row: Stacked spectrum of C iv (left panel), He ii λ 1640 and [O iii ] λ 1666 (middle panel), and C iii ] (right panel). \n<!-- image --> \nTable B.1: List of AGN and their detected fluxes of high ionisation lines: [Ne iv ] λ 2422, [Ne v ] λ 3420 and N V λ 1240. \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFig. B.1: Summary of the detection of [Ne iv ] λ 2424, [Ne v ] λ 3420 and NV λ 1240. The blue lines show the continuum subtracted observed spectrum, black dotted lines indicate the uncertainties on the flux and the red dashed lines show the best fit to the data. \n<!-- image --> \nTable C.1: List of emission lines used in this work, their wavelengths and ionisation potential'} |
2024arXiv240906947T | In this work we investigate the graviscalar quasinormal modes QNMs and their asymptotic tail behavior of a thick brane. Considering the scalar perturbations of the thick brane metric we obtain the main equations of graviscalar KaluzaKlein modes. Based on these equations the frequencies of the graviscalar QNMs of the thick brane are obtained by the WentzelKramersBrillouin asymptotic iteration and numerical evolution methods. The results show that the scalar fluctuation of the thick brane has a series of discrete QNMs similar to the tensor perturbation of the brane. These modes appear as decaying massive scalar particles in fourdimensional spacetime. We also studied in detail the late time tails of these QNMs and found that some modes have slowly decaying oscillatory tails that may be new sources of the gravitational wave backgrounds. Obviously the QNMs contain the information of the brane and are characteristic modes of the thick brane. | 2024-09-01T00:00:00Z | ['arXiv:2409.06947', '2024arXiv240906947T', '10.48550/arXiv.2409.06947'] | ['General Relativity and Quantum Cosmology'] | Graviscalar quasinormal modes and asymptotic tails of a thick brane | 2,024 | 172 | 0.19 | ['EPRINT_HTML', 'EPRINT_PDF'] | 2 | https://arxiv.org/pdf/2409.06947.pdf | {'Graviscalar quasinormal modes and asymptotic tails of a thick brane': 'Qin Tan a , Sheng Long a , Weike Deng a , and Jiliang Jing a ∗ a Department of Physics, Key Laboratory of Low Dimensional Quantum Structures and Quantum Control of Ministry of Education, Synergetic Innovation Center for Quantum Effects and Applications, Hunan Normal University, Changsha, 410081, Hunan, China \nIn this work, we investigate the graviscalar quasinormal modes (QNMs) and their asymptotic tail behavior of a thick brane. Considering the scalar perturbations of the thick brane metric, we obtain the main equations of graviscalar Kaluza-Klein modes. Based on these equations, the frequencies of the graviscalar QNMs of the thick brane are obtained by the Wentzel-Kramers-Brillouin, asymptotic iteration, and numerical evolution methods. The results show that the scalar fluctuation of the thick brane has a series of discrete QNMs, similar to the tensor perturbation of the brane. These modes appear as decaying massive scalar particles in four-dimensional spacetime. We also studied in detail the late time tails of these QNMs and found that some modes have slowly decaying oscillatory tails that may be new sources of the gravitational wave backgrounds. Obviously, the QNMs contain the information of the brane and are characteristic modes of the thick brane.', 'I. INTRODUCTION': "Since the 20th century, the idea that our universe might be embedded in a higher-dimensional spacetime as a four-dimensional object has sparked widespread interest in the spacetime view. The early motivation behind extra-dimensional theories was to unify various interactions in nature, such as the Kaluza-Klein (KK) theory and string theory [1-3]. With the development of extra-dimensional theories, some models focusing on addressing prominent issues in physics emerged, such as the Randall-Sundrum (RS) curved extra-dimensional model [4, 5]. To explain the longstanding hierarchy problem in particle physics, namely the huge disparity between the weak scale and the Planck scale, Randall and Sundrum proposed the RS-I brane model composed of two branes embedded in a five-dimensional anti-de Sitter spacetime. Subsequently, based on the RS-I two-brane model, Randall and Sundrum further proposed the RS-II model with an infinite extra dimension. The inspiration behind the RS-II model was that even with an infinite extra dimension, the branes could still recover a fourdimensional effective gravitational potential. Due to this novel and intriguing characteristic, the RS-II model garnered widespread attention and has seen significant development in cosmology, particle physics, and black hole physics [6-15]. \nIn the RS-II braneworld model, our world is embedded as a four-dimensional hypersurface in a five-dimensional spacetime, with the energy density along the extra dimension distributed as a delta function, hence termed the brane model. The thick brane model, which corresponds to this, is a smooth extension of the RS-II brane model, possessing a rich internal structure and thus has richer properties [16-18]. In thick brane theories, the localization of the gravitational zero mode and various \nmatter fields is necessary. Currently, various thick brane solutions of gravity theories and the localization of gravitational zero modes and matter fields on the brane have been studied [19-39]. However, it is important to note that the localization of the zero modes of gravity and various matter fields is aimed at recovering effective physics on the four-dimensional brane, but this does not bring us new insights. To capture the extra dimension, we are more concerned with things beyond these zero modes, namely the massive KK modes that contain information about the extra dimension. \nIt is generally believed that apart from the zero modes, thick branes also have a continuum of massive KK modes. Our recent studies indicate that within these continuous KK modes, there exists a series of discrete gravitational quasinormal modes (QNMs) [40-42]. The spectrum of these QNMs is entirely determined by the structure of the thick brane, and conversely, these modes also contain crucial information about the extra dimension. Previous studies of the QNMs of the braneworld only considered the transverse-traceless tensor fluctuations of the metric, corresponding to spin-2 gravitons on the brane. For the RS-II thin brane model, this is sufficient because, without considering material sources on the brane, the metric fluctuation of the RS-II thin brane consists only of the transverse-traceless tensor mode [5]. However, for the thick brane scenario, the situation is quite different. Since thick branes are generated by a background scalar field coupled with gravity, their complete metric fluctuations of thick branes should include tensor fluctuations, vector fluctuations (graviphoton), and scalar fluctuations (graviscalar) [43-46]. Vector fluctuations and scalar fluctuations contain information about the matter field that constitutes the thick brane, which may be the key to distinguishing between the thick brane model and the thin brane model. Additionally, in braneworld cosmology, scalar fluctuations of branes are also an important subject of study [10, 47-54]. \nFor vector fluctuations, generally only non-localized zero mode exist, with no other modes. Tensor fluctu- \nations exhibit a bound zero mode and an infinite number of massive KK modes. As for the scalar fluctuations of thick branes, in most cases, zero modes exist but cannot be bound to the brane, and the massive mode is similar to the case of tensor fluctuations. That's all we know. So, the question now is, can we know more about the scalar fluctuations of thick branes? Do scalar fluctuations also exhibit QNMs like tensor fluctuations? By employing the methods developed from studying the QNMs of thick brane tensor fluctuations, we will investigate the QNMs of flat thick brane scalar fluctuations in this paper, aiming to understand more about scalar fluctuations. \nThis paper is organized as follows. In Sec. II, we review the thick brane solution and its linear scalar perturbation. In Sec. III, we investigate the graviscalar QNMs of the thick brane using the Wentzel-Kramers-Brillouin (WKB) approximation, the asymptotic iteration method in the frequency domain, and obtain the waveform of these modes by numerical evolution. Finally, the conclusions and the discussions are given in Sec. IV.", 'II. THICK BRANE SOLUTION AND ITS SCALAR PERTURBATION': "In this section, we will review the thick brane solution and corresponding scalar perturbation of the metric. Starting from the Einstein-Hilbert action minimally coupled to a canonical scalar field, the action is given by \nS = ∫ d 5 x √ -g ( 1 2 κ 2 5 R -1 2 g MN ∂ M φ∂ N φ -V ( φ ) ) , (1) \nwhere κ 2 5 ≡ 8 πG 5 is set to κ 5 = 1 for convenience. The metric is assumed to be a static flat case \nds 2 = e 2 A ( y ) η µν dx µ dx ν + dy 2 , (2) \nwhere A ( y ) is the warp factor. By varying the above action (1) with respect to the metric and the scalar field φ , the field equations are given by \nR MN -1 2 Rg MN = -1 2 g MN ( ∂ A φ∂ A φ -V ( φ ) ) + ∂ M φ∂ N φ, (3) \ng MN ∇ M ∇ N φ = ∂V ( φ ) ∂φ . (4) \nSubstituting the metric (2) into the above field equations, we can obtain the specific dynamic equations \n6( ∂ y A ) 2 +3 ∂ 2 y A = -1 2 ( ∂ y φ ) 2 -V, (5) \n6( ∂ y A ) 2 = 1 2 ( ∂ y φ ) 2 -V, (6) \n∂ 2 y φ +4( ∂ y A ) ∂ y φ = ∂V ∂φ . (7) \nFIG. 1: The shapes of the warp factor (8), the scalar field (9), and the scalar potential (10). \n<!-- image --> \nBy using the superpotential method, the thick brane solution is given in Ref [17]: \nA ( y ) = -ln (cosh( ky )) , (8) \nφ ( y ) = √ 3arcsin (tanh ( ky )) , (9) \nV ( φ ) = 3 k 2 4 ( 5 cos ( √ 4 3 φ ) -3 ) , (10) \nwhere k is a constant with mass dimension one. Plots of the warp factor, scalar field, and scalar potential mentioned above are shown in Fig. 1. \nTo investigate the QNMs of scalar fluctuation for the thick brane, we next consider scalar perturbation of the metric. In braneworld models, to study the fluctuations of the metric and the field, it is generally necessary to transform the extra-dimensional coordinate y to a conformal flat coordinate z , that is, to perform the following coordinate transformation: \ndz = e -A ( y ) dy, (11) \nthen, metric (2) becomes \nds 2 = e 2 A ( z ) ( η µν dx µ dx ν + dz 2 ) . (12) \nSince the linear perturbation of the metric of a braneworld can be decomposed into scalar, transverse vector, and transverse-traceless tensor modes, and these three modes are decoupled from each other after performing a scalar-tensor-vector decomposition. Therefore, we can consider the scalar perturbation of the metric separately. The perturbed metric in the longitudinal gauge is [46] \nds 2 = e 2 A ( z ) (1+2 ϕ ( x µ , z ))( η µν dx µ dx ν +(1+2Ψ( x µ , z )) dz 2 ) , (13) \nOn the other hand, the perturbed background scalar field is φ 0 = φ ( z ) + δφ ( x µ + z ). By substituting the perturbed metric and scalar field into Eqs. (3) and (4), we obtain the equations of the scalar perturbation: \n( z, z ) : 3 η αβ ∂ α ∂ β Ψ+12 ∂ z A∂ z Ψ -12( ∂ z A ) 2 ∂ z ϕ = ∂ z φ∂ z δφ -ϕ ( ∂ z δφ ) 2 -e 2 A ∂V ∂φ δφ, (14) \n( z, µ ) : -3 ∂ µ ∂ z Ψ+3 ∂ z A∂ µ ϕ = ∂ z φ∂ µ δφ, (15) \n( µ, ν ) : ( 3 ∂ 2 z Ψ -6 ∂ 2 z Aϕ -3 ∂ z A∂ z ϕ +9 ∂ z A∂ z Ψ -6( ∂ z A ) 2 ϕ + η αβ ∂ α ∂ β ϕ +2 η αβ ∂ α ∂ β Ψ ) δ µ ν -η µβ ∂ β ∂ ν ϕ -2 η µβ ∂ β ∂ ν Ψ = ( -∂ z φ∂ z δφ + ϕ ( ∂ z δφ ) 2 -e 2 A ∂V ∂φ δφ ) δ µ ν , (16) \nmatter : ∂ 2 z δφ +3 ∂ z A∂ z δφ +(4 ∂ z Ψ -∂ z ϕ -6 ∂ z ϕ ) ∂ z φ \n-2 ϕ∂ 2 z φ 0 + η αβ ∂ α ∂ β δφ = e 2 A ∂V ∂φ δφ. (17) \nFrom the off-diagonal part of Eq. (16), we can obtain \nϕ +2Ψ = 0 . (18) \nFrom Eq. (15) we get \nδφ = ( -3 ∂ z Ψ+3 ∂ z Aϕ ) ∂ z φ . (19) \nThis means that there is only one physical scalar degree of freedom. Combining the above equations (14)-(18), we find the master equation for the scalar perturbation is \n✷ (4) Ψ + ( 4 ∂ 2 z A -4 ∂ z A∂ 2 z φ ∂ z φ ) Ψ + ( 3 ∂ z A -2 ∂ 2 z φ ∂ z φ ) ∂ z Ψ+ ∂ 2 z Ψ = 0 , (20) \nwhere ✷ (4) = η αβ ∂ α ∂ β . Introducing the following Kaluza-Klein decomposition \nΨ( x µ , z ) = e -3 2 A ( z ) ∂ z φ ˜ Ψ( t, z ) e -ia i x i , (21) \nwe can obtain the wave equation: \n-∂ 2 t ˜ Ψ+ ∂ 2 z ˜ Ψ -U ( z ) ˜ Ψ -a 2 ˜ Ψ = 0 , (22) \nwhere \nU ( z ) = -5 2 ∂ 2 z A + 9 4 ( ∂ z A ) 2 -∂ 3 z φ ∂ z φ + ∂ z A ∂ 2 z φ ∂ z φ +2 ( ∂ 2 z φ ∂ z φ ) 2 (23) \nis the effective potential and a = √ a i a i is a constant. Then we decompose the function ˜ Ψ further as ˜ Ψ = \ne -iωt ψ ( z ) and substitute it into the above wave equation (22), a Schrodinger-like equation can be obtained as \n-∂ 2 z ψ ( z ) + U ( z ) ψ ( z ) = m 2 ψ ( z ) , (24) \nwhere m = √ ω 2 -a 2 is the mass of the graviscalar KK mode. Unlike the case of the transverse-traceless tensor perturbation, we do not want the above equation to have a bound zero mode of scalar perturbation. Because if there is a scalar zero mode, there will be a 'fifth force' on the brane, which is unacceptable. In order to obtain the solution of the scalar zero mode, we decompose the above equation into the following supersymmetric form \nH † Hφ ( z ) = m 2 ψ ( z ) , (25) \nwhere H † and H are \nH † = -∂ z + 3 2 ∂ z A ( z ) -∂ 2 z A ( z ) ∂ z A ( z ) + ∂ 2 z φ ( z ) ∂ z φ ( z ) , (26) \nH = ∂ z + 3 2 ∂ z A ( z ) -∂ 2 z A ( z ) ∂ z A ( z ) + ∂ 2 z φ ( z ) ∂ z φ ( z ) . (27) \nThus, the scalar zero mode is ψ 0 ∝ ∂ z A ( z ) e 3 2 A ( z ) ∂ z φ ( z ) . Generally, this scalar zero mode cannot be localized. In addition to the zero mode, the above equations also support massive KK modes. Due to the complexity of the effective potential, it is difficult to obtain an analytical solution of the massive KK mode. But from the perspective of the QNMs, we can gain a better understanding of these massive scalar KK modes.", 'III. SCALAR QUASINORMAL MODES OF BRANE': "Now, we investigate the graviscalar QNMs of the thick brane. The warp factor (8) and the background scalar field (9) are rewritten in terms of the z coordinate as \nA ( z ) = -1 2 ln( k 2 z 2 +1) , (28) \nφ ( z ) = √ 3arctan( kz ) . (29) \nSubstituting the above warp factor A ( z ) and the background scalar field φ ( z ) into the effective potential (23), the specific form of U ( z ) is given by \nU ( z ) = 3 k 2 (5 k 2 z 2 +6) 4( k 2 z 2 +1) 2 . (30) \nWe plot the effective potential U ( z ) in Fig. (2). We can see that the effective potential is a pure barrier and U ( ±∞ ) → 0. In addition, the specific form of the scalar zero mode is \nψ 0 ( z ) = √ 3 z (1 + z 2 ) 3 / 4 3 . (31) \nObviously, this zero mode cannot be localized on the brane. Now we turn our attention to the massive KK modes. Since the effective potential (30) approaches to 0 at kz = ±∞ , these scalar massive KK modes always tunnel to extra dimension infinity. Thus, the boundary conditions of these massive KK modes are \nψ ( z ) ∝ { e imz , z →∞ . e -imz , z →-∞ , (32) \nwhich is only an outgoing wave at the spatial infinity and only an ingoing wave at negative spatial infinity. With the equation and the boundary conditions, we can solve the graviscalar QNMs of the thick brane. Since the effective potential is similar to the case of the gravitational perturbation of a black hole. Based on this similarity, we use some methods to solve the QNMs of black holes to solve the graviscalar QNMs of the thick brane. \nFIG. 2: The shape of the effective potential (30). \n<!-- image --> \nWKB method The WKB method is a very common method for solving the QNMs of a black hole [55]. This method is based on matching the asymptotic WKB solutions at the event horizon and space infinity with the Taylor expansion near the peak of effective potential barrier through the two turning points. Since the structure of the effective potential of the scalar perturbation is similar to that of a black hole, the WKB method is also suitable for thick brane cases. Here we use the WKB method to solve the graviscalar QNMs of the thick brane. The complex frequency formula of the sixth order WKB method is [56] \ni m 2 -U 0 √ -2 U '' 0 -6 ∑ j =2 Λ j = n + 1 2 , n = 1 , 2 , 3 . . . , (33) \nwhere U 0 is the peak value of the effective potential U ( z ), U '' 0 is the second derivative with respect to the extra dimensional coordinate z at the maximum of the potential, and Λ j are 2nd to 6th correction terms. The explicit form of Λ j can be found in Ref. [56]. Using the WKB method, we obtain the first four scalar quasinormal frequencies (QNFs) of the brane, which can be seen from Tab. I. For higher overtones, the QNFs can not be solved by the WKB method. We need to use other methods. \nwhere \nTABLE I: The first four QNFs using the WKB method and the AIM. \nAsymptotic iteration method In order to accurately solve high overtone modes, we use the asymptotic iteration method (AIM). The AIM is an effective method to find the eigenvalues of linear second-order homogeneous differential equation [57, 58]. In general, if the equation has the following form \ny '' ( x ) = λ 0 ( x ) y ' ( x ) + s 0 ( x ) y ( x ) , (34) \n/negationslash \nwhere λ 0 ( x ) = 0 and s 0 ( x ) are smooth functions. Taking the differentiation of the above Eq. (34), one finds \ny ''' ( x ) = λ 1 ( x ) y ' ( x ) + s 1 ( x ) y ( x ) , (35) \nλ 1 ( x ) = λ ' 0 + s 0 + λ 2 0 , (36) \ns 1 ( x ) = s ' 0 + s 0 λ 0 . (37) \nThen, taking n -time derivatives of Eq. (34) we have \ny ( n +2) ( x ) = λ n ( x ) y ' ( x ) + s n ( x ) y ( x ) , (38) \nwhere \nλ n ( x ) = λ ' n -1 + s n -1 + λ 0 λ n -1 , (39) \ns n ( x ) = s ' n -1 + s 0 λ n -1 . (40) \nWith sufficiently large n , the asymptotic aspect of the AIM claims that \ns n ( x ) λ n ( x ) = s n -1 ( x ) λ n -1 ( x ) = β ( x ) , (41) \nor an equivalent form \ns n ( x ) λ n -1 ( x ) -s n -1 ( x ) λ n ( x ) = 0 . (42) \nThe QNFs can be solved from the above 'quantization condition'. However, computing the recurrence relations (39) and (40) requires a lot of resources. Thus, Cho et al improved the original AIM by using the Taylor series [59]. This greatly improved the accuracy and speed of numerical calculation. Expanding the functions λ n ( x ) and s n ( x ) at the point χ : \nλ n ( x ) = ∞ ∑ i =0 c i n ( x -χ ) i , (43) \ns n ( x ) = ∞ ∑ i =0 d i n ( x -χ ) i . (44) \nwhere the i -th Taylor coefficients of λ n and s n are denoted by c i n and d i n , respectively. Now, Eqs. (39) and (40) become \nc i n = ( i +1) c i +1 n -1 + d i n -1 + i ∑ k =0 c k 0 c i -k n -1 , (45) \nd i n = ( i +1) d i +1 n -1 + i ∑ k =0 d k 0 c i -k n -1 . (46) \nThe 'quantization condition' (42) can be expressed in terms of these coefficients \nd 0 n c 0 n -1 -d 0 n -1 c 0 n = 0 . (47) \nWe can see that the final recurrence relations do not require derivative operations in the iteration process. We can obtain the QNFs by solving this simple recurrence relation. Obviously, the improved AIM depends on the expansion point χ . It is found that the value of the expansion point has great influence on the convergence speed. Furthermore, the AIM seems to converge most quickly when χ is chosen to be the maximum of the effective potential. \nNow, we use the AIM to solve the graviscalar QNMs of the thick brane. Since Eq. (24) does not contain a firstorder derivative term, we perform the coordinate transformation u = √ 4 k 2 z 2 +1 -1 2 kz to get a form that applies to the AIM. The Schrodinger-like (24) then becomes \n( u 2 -1 ) 3 (( u 4 -1 ) ψ '' ( u ) + 2 u ( u 2 +3 ) ψ ' ( u ) ) ( u 2 +1) 3 + ( m 2 k 2 -3 ( u 2 -1 ) 2 ( 6 u 4 -7 u 2 +6 ) 4 ( u 4 -u 2 +1) 2 ) ψ ( u ) = 0 , (48) \nIn this coordinate, the boundary conditions (32) are rewritten as \nψ ( u ) ∝ e -im/k 2 u -2 , u → 1 , e im/k 2 u +2 , u →-1 . (49) \nTo accommodate the above boundary conditions, we define \nψ ( u ) = ˜ ψ ( u ) e -im/k 2 u -2 e im/k 2 u +2 . (50) \nThe equation (48) takes the form \n˜ ψ '' ( u ) = λ 0 ( u ) ˜ ψ ' ( u ) + s 0 ( u ) ˜ ψ ( u ) , (51) \n■ \n✁ \nwhere \nλ 0 ( u ) = -2 u ( u 4 +2 i ( u 2 +1 ) m k +2 u 2 -3 ) ( u 2 -1) 2 ( u 2 +1) , (52) s 0 ( u ) = 1 4 ( u 2 +1)( u 6 -2 u 4 +2 u 2 -1) 2 × [ 3 ( 6 u 4 -7 u 2 +6 ) ( u 2 +1 ) 3 +8 i ( u 2 -1 ) ( u 4 -u 2 +1 ) 2 m k -4 ( u 4 -u 2 +1 ) 2 ( u 2 +1 ) m 2 k 2 ] . (53) \nWith λ 0 and s 0 , we can construct expressions for c i n and d i n . Then by the 'quantization condition' (47), we can obtain the QNF of graviscalar QNMs of the thick brane. The results are shown in Tab. I and Fig. 3. From Tab. I, we can see that the first three QNFs for the AIM are in good agreement with the results of the WKB method, but the fourth is not quite the same. This is because the WKB method is not suitable for solving high overtone modes. \nFIG. 3: The first fifteen graviscalar QNFs of the thick brane solved by the improved AIM. \n<!-- image --> \nTime evolution Both AIM and WKB methods study the graviscalar QNFs of the thick brane in the frequency domain. These frequencies tell us how perturbations decay in the late time. However, this lacks an understanding of the relative amplitudes of these fluctuations. Therefore, we should study the time evolution of the perturbed scalar field. Here, we use the null coordinates u = t -z and v = t + z to perform the time evolution. The evolution equation (22) in the null coordinates is \n( 4 ∂ 2 ∂u∂v + U + a 2 ) ˜ Ψ = 0 . (54) \nUnlike the case of a black hole, the effective potential of the scalar perturbation of the thick brane is symmetric. Thus, the KK modes is either even or odd. Therefore, initial wave packets with different parity will excite different QNMs. It's important to note that the cases where a = 0 and a = 0 are different. The former corresponds to \n/negationslash \nthe evolution of a massless field in the background spacetime, while the latter is the evolution of a massive field. We first consider the case where a = 0. The initial date is assumed to be a Gaussian pulse first \n˜ Ψ(0 , v ) = e -( kv -kvc ) 2 2 k 2 σ 2 , ˜ Ψ( u, 0) = e -k 2 v 2 c 2 k 2 σ 2 . (55) \nThis Gaussian pulse is located at kv c = 5 and has a width of kσ = 1. We extract the data at kz ext = 30. The result of the time evolution of the Gaussian pulse is shown in Fig. 4(a). It can be seen that in logarithmic coordinates, the waveform is divided into three stages: the initial burst stage, the exponential damping stage, and the power-law tail stage. Since the QNM with the smallest imaginary part dominates the evolution, we can extract the frequency of the QNM with the smallest imaginary part by fitting the data. The fitted QNF for the case of Fig. 4(a) is m k = 1 . 999394 -0 . 503601 i , which is in good agreement with the result of the AIM and the WKB method. In addition to the first QNF, we can also extract the frequency of the second QNM due to the symmetry of the effective potential. When the initial data is given as a Gaussian wave packet, both odd and even QNMs are excited. The first QNM is even, and the odd QNM is covered. If the initial data is given as a static odd wave packet, only odd QNMs will be excited. In this way, we can extract the frequency of the first odd QNM. The odd initial data is given by \n˜ Ψ(0 , v ) = sin ( kv 2 ) e -k 2 v 2 4 , (56) \n˜ Ψ( u, 0) = sin ( ku 2 ) e -k 2 u 2 4 . (57) \nWe extract the data at kz ext = 30. The result is shown in Fig. 4(b) and the fitted QNF is m k = 1 . 576312 -1 . 746200 i . This is also consistent with the result of the AIM and the WKBmethod. For the high overtone modes, we look forward to developing more methods in the future to compare them with the results of the AIM. \n/negationslash \nNow let's consider the case of a = 0. The selection of the initial wave packet is consistent with the case of a = 0. The values a = 1 5 and 1 3 are selected to study the influence of parameter a on the evolution of the wave packet, and the results are shown in Fig. 5. It can be seen that, when a = 0, there is no power-law tail in the late time, but an oscillating tail. This is consistent with the evolutionary behavior of a massive field in a black hole background. Obviously, the tails for a = 0 and a = 0 behave completely differently. In a very complete analysis, Ching et al. [60, 61] examined the late time tail that arises when dealing with this form of the evolution equation (22) with a = 0, and the form of effective potential is \n/negationslash \nV ( x ) ∼ ν ( ν +1) x 2 + c 1 log x + c 2 x α , x →∞ . (58) \nWhen c 1 = 0, their conclusions are: \n/negationslash \n(1) If ν is an integer, the tail is given by a power-law \n˜ Ψ ∼ t -µ , µ > 2 ν + α, (59) \nwhere α is an odd integer and α < 2 ν +3. (2) For the case of ν is not an integer, the tail is \n˜ Ψ ∼ t -2 ν +2 . (60) \nFor the effective potential (30), we can rewrite it as \nU ( z ) = 15 k 2 4(1 + k 2 z 2 ) + 3 4(1 + k 2 z 2 ) 2 . (61) \nSo as z →∞ , the effective potential asymptotic to \nU ( z ) ∼ 15 k 2 4 k 2 z 2 + 3 k 2 4( k 2 z 2 ) 2 = 3 2 ( 3 2 +1) k 2 k 2 z 2 + 3 k 2 4 k 4 z 4 . (62) \nObviously this corresponds to the second case above, so the tail of the wave packet evolution should be ˜ Ψ ∼ t -5 . Therefore, the tail of the thick brane QNMs is closely related to the structure of the thick brane, especially the behavior of the extra dimension at infinity. Due to the diversity of thick brane solutions, the types of tail should also be very rich, which needs further study. We verify the above results with the data of numerical evolution. We use a power-law relation ˜ Ψ = t α to fit the late time data in Fig. 4 and obtain the value of α . The fitted result for the case of Fig. 4(a) is α = -5 . 07278, and for the case of Fig. 4(b) is α = -5 . 02804. Obviously, the late time power-law tails of the evolution of two initial wave packets are consistent and same as the results obtained from the previous analysis (up to numerical error). \n/negationslash \nOn the other hand, the situation is different when a = 0. In the case of black holes, the tail of the massive field is generally divided into two parts: intermediate times and asymptotically late times. The latter came much later than the former. The decay law of massive field at intermediate times is [62] \nΨ ∝ t -ν -3 / 2 sin( µt ) , (63) \nwhere µ is the mass of the massive field. The decay law of massive field at asymptotically late times is [63] \nΨ ∝ t -5 / 6 sin( µt ) , (64) \n/negationslash \nindependently on the parameter ν . Due to the similarity of the evolution equation of the graviscalar QNMs of the thick brane to the black hole case, these conclusions should also apply to our case. That is, when a = 0 (note that a essentially acts as µ here), the decay law at intermediate times of the scalar quasinormal mode should be Ψ ∝ t -3 sin( at ) due to ν = 2 / 3 and at asymptotically late times is Ψ ∝ t -5 / 6 sin( µt ). To prove it, we use function ˜ Ψ = t α sin( βt ) to fit the late time data in Fig. 5 and obtain the values of α and β , the results are shown in Tab. II. We can see that, the tails of different initial data are the same. The decay rate of the tail with time \nis t -3 , while the oscillation frequency of the tail is consistent with the parameter a . These results are consistent with our previous guesses about the decay law at intermediate times. For the tail at asymptotically late times, which need a long evolutionary time to emerge. Due to the limited time of numerical evolution, we have not been able to observe this behavior in the evolving waveform. However, we believe that this tail also exists in the thick brane scenario. \n/negationslash \nThe above results show that a series of discrete QNMs exist in the scalar perturbation of the metric of the thick brane, just like the tensor perturbation of the thick brane. These graviscalar QNMs appear as dissipative scalar particles on the brane, corresponding to the scalar degrees of freedom of gravity. In addition, the frequencies of the first three QNMs of tensor fluctuation are [40]: m 1 k = 0 . 997018 -0 . 526362 i , m 2 k = 0 . 581489 -1 . 851280 i , and m 3 k = 0 . 306006 -3 . 53366 i . Comparing the first three QNMs of scalar fluctuation and tensor fluctuation of thick brane, we can find that the real part of the first three graviscalar QNMs is larger than the tensor QNMs, but the imaginary part is smaller than the tensor QNMs. This shows that the graviscalar QNMs have a higher oscillation frequencies and a longer lifetimes. Moreover, since the original RS-II thin brane does not have scalar fluctuations, the graviscalar QNMs of the thick brane may be a key feature that distinguishes thin brane from thick brane. If such a QNM is detected, it is conclusive evidence for the existence of extra dimensions and the brane has a thickness. But because these modes have such a short lifetime (when k = 10 -3 eV, the lifetime of the first graviscalar QNM of the thick brane is τ ≈ 10 -13 s), they are nearly impossible to detect with existing gravitational wave detectors. Nevertheless, these graviscalar QNMs may play an important role in the early universe by coupling to the 4-dimensional matter field, in particular the evolution of the early universe and the anisotropy of the cosmic microwave background radiation [64, 65]. On the other hand, the discussion above is for the case of a = 0. When a = 0, these QNMs have a slowly decaying oscillation tail in the late time. Recently, R. A. Konoplya and A. Zhidenko suggests that these tails of massive patterns from extra dimensions may contribute to the gravitational wave background [66]. We expect to find signals of them in existing or future gravitational wave background detectors, such as the pulsar timing array. \nTABLE II: The tail parameters α and β with different values of a and different initial data calculated by fitting the late time data in Fig. 5. \nFIG. 5: Upper panel: Time evolution of the Gaussian pulse with different a at kz ext = 30. Lower panel: Time evolution of the odd wave packet with different a at kz ext = 30. \n<!-- image --> \nFIG. 4: (a): Time evolution of the Gaussian pulse at kz = 30. (b): Time evolution of the odd wave packet at kz = 30. \n<!-- image --> \n<!-- image -->", 'IV. CONCLUSIONS': "In this paper, we investigated the graviscalar QNMs of the thick brane by the WKB method, the AIM, and the time evolution method. The results show that although the scalar fluctuation of the metric has no bound zero mode, there is still a discrete quasinormal spectrum. These QNMs are closely related to the structure of extra dimension. This is a new understanding of the scalar fluctuation of thick branes, which will help us to better study braneworld models. \nWe first review the thick brane solution and its scalar fluctuations, then we obtain the evolution equation (22) and the Schrodinger-like equation (24) satisfied by the extra dimensional part of the scalar fluctuations by KK decomposition. Based on the Schrodinger-like equation, we use the WKB method and the AIM to solve the graviscalar QNFs of the thick brane, which is shown in Tab. I. The results obtained by the two methods are quite consistent. It should be noted that the real part of the frequency of the first QNM of the scalar fluctuation is twice that of the frequency of first QNM of the tensor fluctuation, and the imaginary part is not much different. It is \n/negationslash \nshown that the two lifetimes are close, but the oscillation frequency of the first graviscalar QNM is twice that of the tensor QNM. Using the numerical method, we also investigated the evolution of the initial wave packet on a thick brane. We found that wave packets with different parity will excite different QNMs, because the effective potential is symmetric along the extra dimension. This is different with the cases of black holes. By fitting the evolution data, we obtain the frequencies of the first two QNMs, and the results are in agreement with those obtained in the frequency domain. This shows that our results are credible. Finally, we studied the late time tail of graviscalar QNMs in detail. For the case of a = 0, similar to a massless field around a black hole, the late time tail of the QNMs of the thick brane is also a powerlaw. We also find that the tail of different initial wave packet excitation are the same, and the numerical fitting results are consistent with those calculated by Green's function. For the case of a = 0, the situation is similar to a massive field around a black hole. 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2023NatAs...7..622C | Finding and characterizing the first galaxies that illuminated the early universe at cosmic dawn is pivotal to understand the physical conditions and the processes that led to the formation of the first stars. In the first few months of operations imaging from the James Webb Space Telescope JWST has been used to identify tens of candidates of galaxies at redshift z greater than 10 less than 450 million years after the Big Bang. However none of such candidates has yet been confirmed spectroscopically leaving open the possibility that they are actually lowredshift interlopers. Here we present spectroscopic confirmation and analysis of four galaxies unambiguously detected at redshift 10.3 z 13.2 previously selected from JWST Near Infrared Camera imaging. The spectra reveal that these primeval galaxies are metal poor have masses on the order of about 10SUP7SUP10SUP8SUP solar masses and young ages. The damping wings that shape the continuum close to the Lyman edge provide constraints on the neutral hydrogen fraction of the intergalactic medium from normal starforming galaxies. These findings demonstrate the rapid emergence of the first generations of galaxies at cosmic dawn. | 2023-05-01T00:00:00Z | ['2022arXiv221204568C', '2023NatAs.tmp...66C', '10.48550/arXiv.2212.04568', 'arXiv:2212.04568', '10.1038/s41550-023-01918-w', '2023NatAs...7..622C'] | ['Astrophysics - Astrophysics of Galaxies'] | Spectroscopic confirmation of four metalpoor galaxies at z 10.313.2 | 2,023 | 172 | 0.7 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML'] | 296 | https://arxiv.org/pdf/2212.04568.pdf | {'Spectroscopic confirmation of four metal-poor galaxies at z=10.3-13.2': "Emma Curtis-Lake 1* , Stefano Carniani 2 † , Alex Cameron 3 , Stephane Charlot 4 , Peter Jakobsen 5,6 , Roberto Maiolino 7,8,9 , Andrew Bunker 3 , Joris Witstok 7,8 , Renske Smit 10 , Jacopo Chevallard 3 , Chris Willott 11 , Pierre Ferruit 12 , Santiago Arribas 13 , Nina Bonaventura 5,6 , Mirko Curti 7,8 , Francesco D'Eugenio 7,8 , Marijn Franx 14 , Giovanna Giardino 15 , Tobias J. Looser 7,8 , Nora Lützgendorf 16 , Michael V. Maseda 17 , Tim Rawle 16 , Hans-Walter Rix 18 , Bruno Rodríguez del Pino 13 , Hannah Übler 7,8 , Marco Sirianni 16 , Alan Dressler 19 , Eiichi Egami 20 , Daniel J. Eisenstein 21 , Ryan Endsley 22 , Kevin Hainline 20 , Ryan Hausen 23 , Benjamin D. Johnson 21 , Marcia Rieke 20 , Brant Robertson 24 , Irene Shivaei 20 , Daniel P. Stark 20 , Sandro Tacchella 7,8 , Christina C. Williams 25 , Christopher N. A. Willmer 20 , Rachana Bhatawdekar 26 , Rebecca Bowler 27 , Kristan Boyett 28,29 , Zuyi Chen 20 , Anna de Graaff 18 , Jakob M. Helton 20 , Raphael E. Hviding 20 , Gareth C. Jones 3 , Nimisha Kumari 30 , Jianwei Lyu 20 , Erica Nelson 31 , Michele Perna 13 , Lester Sandles 7,8 , Aayush Saxena 3,9 , Katherine A. Suess 24,32 , Fengwu Sun 20 , Michael W. Topping 20 , Imaan E. B. Wallace 3 and Lily Whitler 20 \n1 Centre for Astrophysics Research, Department of Physics, Astronomy and Mathematics, University of Hertfordshire, Hatfield AL10 9AB, UK. 2 Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126 Pisa, Italy. 3 Department of Physics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, UK. 4 Sorbonne Université, CNRS, UMR 7095, Institut d'Astrophysique de Paris, 98 bis bd Arago, 75014 Paris, France. 5 Cosmic Dawn Center (DAWN), Copenhagen, Denmark. 6 Niels Bohr Institute, University of Copenhagen, Jagtvej 128, DK-2200, Copenhagen, Denmark. 7 Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge, CB3 OHA, UK. 8 Cavendish Laboratory - Astrophysics Group, University of Cambridge, 19 JJ Thomson Avenue, Cambridge, CB3 OHE, UK. 9 Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK. 10 Astrophysics Research Institute, Liverpool John Moores University, 146 Brownlow Hill, Liverpool L3 5RF, UK. 11 NRC Herzberg, 5071 West Saanich Rd, Victoria, BC V9E 2E7, Canada. 12 European Space Agency, European Space Astronomy Centre, Madrid, Spain. 13 Centro de Astrobiologı́a (CAB), CSIC-INTA, Cra. de Ajalvir Km. 4, 28850- Torrejó n de Ardoz, Madrid, Spain. 14 Leiden Observatory, Leiden University, P.O. Box 9513, NL-2300 RA Leiden, Netherlands. 15 ATG Europe for the European Space Agency, ESTEC, Noordwijk, The Netherlands. 16 European Space Agency (ESA), ESA Office, STScI, Baltimore, MD 21218, USA. 17 Department of Astronomy, University of Wisconsin-Madison, 475 N. Charter St., Madison, WI 53706, USA. 18 Max-Planck-Institut fü r Astronomie, Kö nigstuhl 17, D-69117, Heidelberg, Germany. 19 The Observatories of the Carnegie Institution for Science, 813 Santa Barbara St., Pasadena, CA 91101, USA. 20 Steward Observatory University of Arizona 933 N. Cherry Avenue ,Tucson, AZ 85721, USA. 21 Center for Astrophysics, Harvard & Smithsonian, 60 Garden St., Cambridge, MA 02138, USA. 22 Department of Astronomy, University of Texas, Austin, TX 78712, USA. 23 Department of Physics and Astronomy, The Johns Hopkins University, 3400 N. Charles St., Baltimore, MD 21218, USA. 24 Department of Astronomy and Astrophysics University of California, Santa Cruz, 1156 High Street, Santa Cruz, CA 96054, USA. 25 NSF's National Optical-Infrared Astronomy Research Laboratory, 950 North Cherry Avenue, Tucson, AZ 85719, USA. 26 European Space Agency, ESA/ESTEC, Keplerlaan 1, 2201 AZ Noordwijk, NL. 27 Jodrell Bank Centre for Astrophysics, Department of Physics and Astronomy, School of Natural Sciences, The University of Manchester, Manchester, M13 9PL, UK. 28 School of Physics, University of Melbourne, Parkville 3010, VIC, Australia. 29 ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), Australia. 30 AURA for European Space Agency, Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA. 31 Department for Astrophysical and Planetary Science, University of Colorado, Boulder, CO 80309, USA. 32 Kavli Institute for Particle Astrophysics and Cosmology and Department of Physics, Stanford University, Stanford, CA 94305, USA. \nFinding and characterising the first galaxies that illuminated the early Universe at cosmic dawn is pivotal to understand the physical conditions and the processes that led to the formation of the first stars. In the first few months of operations, imaging from the James Webb Space Telescope (JWST) has been used to identify tens of candidates of galaxies at redshift ( z ) greater than 10 less than 450 million years after the Big Bang. However, none of such candidates has yet been confirmed spectroscopically, leaving open the possibility that they are actually low-redshift interlopers. Here we present spectroscopic confirmation and analysis of four galaxies unambiguously detected at redshift 10.3 ≤ z ≤ 13.2, previously selected from NIRCam imaging. The spectra reveal that these primeval galaxies are metal poor, have masses between of order ~10 7 -10 8 solar masses, and young ages. The damping wings that shape the continuum close to the Lyman edge provide the first constraints on neutral Hydrogen fraction of the intergalactic medium to be obtained from normal star-forming galaxies. These findings demonstrate the rapid emergence of the first generations of galaxies at cosmic dawn. \nThe opening act of galaxy formation in the first billion years after the Big Bang sets in motion the physics of galaxy formation and evolution that shapes galaxy properties across cosmic time. Galaxies forming at these times may be the seeds of the much more massive and mature galaxies in the local Universe. Theoretical models and cosmological simulations differ greatly in their predictions of the physical properties and abundance of the first galaxies. The theoretical pictures depend strongly on assumptions about the physical processes at play in \nthe early universe, such as: gas cooling and fragmentation in primordial clouds; the feedback effects from first stars and supernova explosions that subsequently enrich the surrounding medium; and early merging, assembly and accretion histories of galaxies 1-7 . The abundance and mass distribution of the first galaxies are also tightly connected to early structure formation. Therefore, the detection and characterisation of these early galaxies is key to test different models and theories. \nHigh-redshift galaxies often have distinctive spectra in the ultraviolet, in which the blue spectrum produced by hot massive stars is abruptly cut off below the Lyman-limit at 912Å (rest-frame) by the absorption of the light by neutral Hydrogen in stellar atmospheres, interstellar gas and the intergalactic medium (IGM). At the highest redshifts (z ≳ 6), the intergalactic neutral Hydrogen leads to almost complete absorption at wavelengths below Ly α at 1216Å. Observationally, this translates to a 'dropout', i.e., a lack of detection in bands blue-ward of (1 + z) × 1216 Å but flux red-ward of the same wavelength 8-10 . However, galaxies with peculiar properties may mimic high-redshift galaxies [e.g., a combination of dust reddening and nebular lines or contribution by an active galactic nucleus as in ref 11]. Therefore, spectroscopic observations are the only method to determine accurate redshifts, either via the detection of the (redshifted) nebular lines 12 , or via the unambiguous detection of the sharp continuum cutoff at (1 + z) × 1216 Å. The highest redshift spectroscopically confirmed galaxy prior to these observations is that of GN-z11 at z=10.957 13,14 . \nIdentification and spectroscopic characterisation of galaxies in the early Universe is one of the primary goals for which JWST was designed. The first few months of JWST imaging have already yielded a large number of candidate galaxies at z > 10 15-22 . However, the redshift estimates of these candidates have so far been based on their broad-band spectral energy distributions (SEDs), and it cannot be ruled out that such candidates are actually lower redshift galaxies 11 , especially in regions where accompanying Hubble space telescope imaging is relatively shallow. With the large number of z>10 candidates identified in the first months of JWST science observations from NIRCam photometry, some initial findings suggest very little evolution of the UV luminosity function above z>10 18,23 (though this is not seen in ref 15). This would require early galaxies to display different physics, for example a stellar initial mass function more top-heavy than in lower-redshift galaxies 18 . Yet ref. 24 illustrates how large a difference in UV luminosity density evolution is measured when considering only the robust candidates. This demonstrates the firm need for spectroscopic observations to follow up photometric candidates. Additionally, spectra provide us with constraints on the stellar and gas properties of the objects beyond what photometry can provide. \nWe report here the deepest spectroscopic observations to date with NIRSpec 25 on JWST, which provide confirmation of four candidates at z>10 and extensive characterisation of their physical properties. These candidates were photometrically identified as part of the JWST Advanced Deep Extragalactic Survey (JADES), a joint guaranteed time project of the NIRCam and NIRSpec instrument teams. The identification and photometric study of these candidates, based on Hubble Space Telescope (HST) and NIRCam data 26 , is described in a companion paper 27 . We specifically focus here on a pointing in the Hubble Ultra Deep Field (in the GOODS-South area), in which we have taken multi-object spectroscopy of 253 galaxies observed simultaneously with NIRSpec's configurable array of microshutters. We report here on observations taken with the prism spectral configuration (spectral range 0.6-5.3 μ m, resolving power R ∼ 100) with exposure times ranging from 9.3 to 28 hours (see Methods for details on the observing strategy). \nThe JADES spectroscopic observations reach an unprecedented sensitivity of 28.4 magnitudes (AB) at 5 σ per resolution element on the continuum at 2.5 μ m. We note that the NIRSpec prism is extremely well-suited for the redshift confirmation of high-z candidates, with low spectral resolution and high sensitivity at short wavelengths where we are searching for a spectral break (around 1-2 μ m), and higher resolution in the 3-5 μ m region, where we are searching for narrow spectral lines. \nThe focus of this paper is on four of these spectroscopic targets. Two of these are z > 12 galaxy candidates selected from NIRCam imaging 27 , based on a clear lack of F150W flux. Two others are z > 10 candidates based on their HST IR photometry. We defer to a future publication to describe the other targets in this deep pointing. All candidates are faint, with F200W magnitudes fainter than 28 (AB), and hence entirely out of reach for any spectroscopic facility before JWST. More details on the selection and photometric properties of these candidates are provided in the companion paper 27 . \nIn Figure 1 we show the 1D and 2D spectra of these four galaxies. All show a clear detection of a blue continuum that drops off sharply in a manner consistent with a z > 10 Lyman dropout. Specifically, in Figure 1, we show the redshift derived from the position of the spectral break, taken to be at the wavelength of Ly α at \n1215.67Å. These redshifts are reported in Table 1 and were derived with full spectral fitting over the entire redshift range, with each object consistently showing peaks in the posterior probability distribution only at high redshifts. We note that both the spectra and the photometry from ref. 27 agree on the wavelength of the break. \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFig. 1 NIRSpec prism R ∼ 100 spectra for the four z > 10 galaxies targeted for the first deep spectroscopic pointing of the JADES survey, JADES-GS-z10-0, JADES-GS-z11-0, JADES-GS-z12-0 and JADES-GS-z13-0. For each galaxy we display the 1D spectrum and associated 1 𝜎 uncertainties (which are derived from standard error propagation through the reduction pipeline). In the bottom panel we show the 2D signal-to-noise ratio plot. The 2D plot is binned over four pixels in the wavelength direction to better show the contrast across the break. The inset panel in the top right-hand corner shows the NIRCam F444W filter image with the three nodding positions of the NIRSpec micro-shutter 3-slitlet array aperture shown in green. The red dashed line shows 1215.67Ar at the observed redshift z 1216 . \n<!-- image --> \nWe tested whether the observed breaks might be produced by the Balmer (or 4000Å) break in the stellar continuum associated with evolved stellar populations in galaxies at z ∼ 3, and we show that this possibility is excluded with high confidence (see Methods section and model Balmer Break strength in Extended Data Figure 1). Other, more extreme low redshift solutions are still able to explain the photometry of our objects. In particular, fitting with active-galactic-nuclei narrow-line emission (not often included when performing photometric redshifts) provides intermediate redshift solutions at z~3-3.5, as shown in the Methods section. In these solutions the strong photometric break is produced by strong line emission in certain filters. We can firmly rule out these low-redshift solutions with our spectra, which demonstrates that the photometric fluxes arise from continuum emission, rather than strong emission lines with weak underlying continuum. Although this low redshift solution may appear extreme, we emphasise that JWST is pushing into a new regime of exploration, where yet unexplored families of low-redshift contaminants may be uncovered. A recent discovery of a triply-imaged point source shows a similarly extreme possible SED explained by strong line emission at z phot 7.7 28 . \nOne of the galaxies, JADES-GS-z11-0, has been debated in the literature. It was first identified in ref. 29 as a potential z ∼ 10 galaxy. In the UDF12 survey 30,31 , deep Hubble imaging revealed the object drops out in JH140 band imaging. This left two possibilities, either the source was at very high redshift [z ∼ 11.9 30 ], or was low redshift, with a high-equivalent-width emission line producing the flux in the Hubble H160 image. In fact, ref. 32 found indication of a possible emission line in spectroscopic follow-up supporting the latter explanation. We do not confirm this line emission in our NIRSpec spectroscopy, and the NIRCam imaging present in ref. 27 shows that the continuum emission extends to longer wavelengths, which is consistent with the spectrum shown here [see also ref. 23]. Therefore, this galaxy is indeed at high redshift, and not a low-redshift contaminant. \nThe continua appear mostly featureless, with the possible exceptions of JADES-GS-z10-0, which shows a tentative emission line at ∼ 1.44 μ m which may indicate Ly α emission, and JADES-GS-z12-0, which shows another tentative feature at ∼ 5.23 μ m, which could be interpreted as [Ne III] λ 3869 emission at z = 12.52. However, both features are only marginally detected, and we are still assessing whether these very faint and localized features are astrophysical. They will be assessed and explored more in detail in forthcoming papers and not discussed further here. Remarkably, the lack of strong line detections turns out to be what makes these spectra particularly interesting as it provides vital information over and above the spectroscopic redshift determination, and what could be derived from photometry alone 27 . \nTable 1 Exposure times, redshifts (derived both from assuming the spectral break is at exactly 1215.67Ar and accounting for the damping wing from a fully neutral IGM), 2σ upper limits on emission-line equivalent widths (rest frame) for the C III]λλ1907,1909 He IIλ1640 and [O II]λλ3726,3729 lines, 2σ lower limits on the strength of the observed spectral breaks (measurements described in Methods 2), UV absolute magnitude, M uv , and UV slope, β (measured directly from the spectra, see Methods 2.3) and BEAGLEderived physical properties for the four objects. For the BEAGLE-derived properties we report posterior medians and limits in the 1σ credible region. \n* The redshift based on the spectral break being at 1215.67Ar † The redshift accounting for a fully neutral IGM (x HI = 1) following the method outlined in Methods 2.1. ‡ For JADES-GS-z13-0, we report β and M uv derived from the beagle fitting, since we know this object to be on the edge of the shutter, and hence incorporate NIRCam photometry in the fitting to this one object to account for slitlosses (see Methods 3). ¶ Ψ is the star formation rate. ‖ t is age of the oldest stars, or maximum stellar age. ¤ Z is the metallicity. †† 𝜏 . 3 is the effective V-band attenuation optical depth. § The production rate of H-ionizing photons per unit monochromatic UV luminosity. \nLeaving aside the above two features, we report the 2 σ upper limits on the equivalent widths (EW) of He II λ 1640, C III] λλ 1907,1909 and [O II] λλ 3726,3729 in Table 1 (see Methods 2.2 for details of the measurements and Extended Data Table 1 for further limiting fluxes). These lines are important as C III] λλ 1907,1909 is often the strongest line in the rest-frame UV of low-metallicity galaxies [e.g. ref. 33] and strong He II λ 1640 is expected in galaxies of near-zero metallicity. Nearby metal-poor galaxies (Z/Z ⊙ ≲ 0.1) show EWs of C III] λλ 1907, 1909 spanning ∼ 6 -16Å 33,34 , meaning our limits are constraining only in the upper region of this range. The limits on EW(He II λ 1640) are above the actually measured equivalent widths in these studies, and comparable or higher than the majority of objects in the MUSE-selected sample of ref. 35 spanning 2 ≲ z ≲ 4. However, JADES-GS-z10-0 and JADES-GS-z11-0 still show strong limits on the equivalent width of [O II] λλ 3726,3729. These limits are constraining, showing that [O II ] λλ 3726,3729 is weak compared to the average equivalent widths measured in z~3 galaxies (~80 Å at 10 9 solar masses) 36 . \nWe can gain further insight into the physical properties of these galaxies through spectral fitting with the BEAGLE [BayEsian Analysis of GaLaxy sEds 37 ] tool with setup described in Extended Data Table 2, adopting a constant star formation history (SFH) to probe the young stellar populations within them (see Methods section 3 for more details). These fits are illustrated in Extended Data Figures 2-5. We find low metallicities, with two galaxies, JADES-GS-z10-0 and JADES-GS-z11-0, showing strong constraints of just a few percent of solar. At higher metallicities we would expect a significant detection of [OII] λλ 3726,3729. For the two highest redshift galaxies, a low metallicity is still preferred, but the constraints are less strong due to [O II] λλ 3726,3729 being very close to the edge of the observable wavelength coverage of NIRSpec, where the S/N is low, as well as lower S/N in regions of the rest-frame UV lines. \nThe measured star formation rates are moderate, at just a few solar masses a year. We caution that total star formation rates and stellar masses require slit-loss corrections which can be best derived from NIRCam photometry if the objects are extended, or on the edge of the shutter. Still, we find excellent agreement in absolute magnitude and UV slope compared to those derived from NIRCam photometry alone 27 for the two galaxies well-centred within the microshutters (GS-z10-0 and GS-z11-0). The fitted parameters are given in Table 1. \nIt is of interest to investigate whether the lack of detectable line emission requires a large escape fraction of ionising photons and find no strong dependence of the physical properties, nor strong constraints on the escape fraction. We do note, however, that JADES-GS-z11-0 has a solution with low ages and high escape fraction with marginally higher metallicity, though the upper 1σ limit is still ∼ 5% solar metallicity (see Extended Data Table 3). In fact, at these extremely low metallicities, the rest-frame UV emission lines might be significantly weaker than those sometimes found in lower redshift samples with somewhat higher metallicities around ∼ 10% solar. Another important factor is the age of the stellar populations, as strong UV emission lines have primarily been observed in galaxies with UV light dominated by very young stellar populations [≲ 10 million years 33 ]. \nTwo objects, JADES-GS-z11-0 and JADES-GS-z12-0 do indicate moderate levels of dust, albeit with large uncertainties (with effective V-band absorption optical depth, τ ! % = 0.18 #$.$& '$.$& and τ ! % = 0.2 #$.$( '$.)& , respectively). Such high optical depth due to dust in such low metallicity systems would be physically hard to explain unless a low dust-to-gas ratio were integrated over large HI column densities. However, we note that some recent models expect significant dust production by the first generations of stars 3,38 . When fit without any dust, JADES-GS-z11-0 required a higher metallicity (up to 40% solar at 1σ), while the JADES-GS-z12-0 data yielded similar parameter constraints to those reported in Table 1. \nThe spectra do not provide strong constraints red-ward of the Balmer-break, the longest rest-frame wavelength probed being ∼ 3660 - 4350A] . We see no evidence for strong Balmer breaks in these objects, but the S/N in this region of the spectra are low. Correspondingly, the constraints on stellar age and stellar mass (sensitive to the Balmer-break strength) are broad. The ages range from ∼ 5 to 230 Myr, and stellar masses range from ∼ 2 × 10 6 M ⊙ to 460×10 6 M ⊙ , though the constraints on these parameters are weak, and highly sensitive to the prior regarding the time history of the star formation rate. The associated production rates of H-ionizing photons per unit monochromatic UV luminosity, ξ ion , are similar to those measured in extreme star-forming regions in low-redshift, metal-poor galaxies 39 . \nWe show the measured UV slopes (β) vs. absolute magnitude at 1500A] (M uv ) in Figure 2. We compare to other JWST-selected high-redshift candidate samples spanning photometric redshifts z ∼ 7 - 16 22,40,41 . Our measured slopes at such faint M uv magnitudes are comparable to the other literature samples, suggesting little evolution at these epochs. Very blue UV slopes are used to search for extreme stellar populations at the earliest times 42 . In this case, we find extreme stellar populations but the presence of any nebular recombination continuum will redden the UV slopes 41 . Indeed, we expect strong nebular continuum emission in low metallicity galaxies unless a significant fraction of their ionising photons escape into the intergalactic medium. \nFig. 2 UV slope, β, as a function of absolute magnitude at 1500Ar , M UV , measured as described in Methods 2.3. These are compared to the measurements from photometrically selected high-redshift candidates. Specifically, the average β measured from objects spanning z ∼ 8 - 15 at similar M UV from Cullen+22 40 , as well as a sample from Atek+22 22 spanning z ∼ 9 - 16, and the sample presented in Topping+22 41 (itself collated from the samples of refs. 21 and 43 and spanning z ∼ 7 - 11). In all cases the error bars show the 1 𝜎 measurement uncertainties, except in the case of the point showing the average UV slope from Cullen+22. For the Cullen+22 datapoint, the point shows the inverse-variance weighted mean and standard error of β, plotted against the median M UV of their sample of 41 galaxies in their lower luminosity bin. The errorbar in M UV is 𝜎 /01 (where 𝜎 /01 = 1.483 𝑥 𝑀𝐴𝐷 and MAD is the median absolute deviation) of the individual M UV values. \n<!-- image --> \nIf ionising radiation does escape from galaxies, it will reionize neutral Hydrogen in the surrounding gas. We find that the breaks in the spectra presented in Fig. 1 are significantly less abrupt than those seen in galaxies at lower redshifts in our spectroscopic data set, and are consistent with a softening of the break by the Lyα damping wing caused by a largely neutral IGM, suggesting that these galaxies are yet to ionize large bubbles in their near vicinity. The redshift is sensitive to the existence and form of this damping wing, and we report the best-fit redshift for a fully neutral inter-galactic medium (x HI = 1, where x HI is the fraction of neutral Hydrogen) in Table 1. For the object with the highest S/N in the Lyα break region (JADES-GS-z11-0), we investigate the constraints we can place on the neutral Hydrogen fraction by first fixing the best-fit model for the stellar population. We then run BEAGLE varying only redshift and x HI . The resulting fit to the spectral break, and the derived constraints are shown in Figure 3. The 2D posterior probability distribution function reported in the right-hand panel indicates that the constraints on x HI are fairly weak, suggesting x HI >0.5 from the 1 σ credible interval, though this is sensitive to exact redshift of the source (e.g. Extended Data Figure 6). However, these spectra demonstrate that x HI can be constrained from JWST R100 spectra at slightly lower redshifts from 'normal' star-forming galaxies. \nTo date the only constraints on the evolution of x HI from damping wings is in luminous quasars at 6 < z < 7.5. Damping wings are rarely observed at z < 7, but become more common in the small sample of known z > 7 quasars, consistent with x HI ∼ 0.5 at z = 7.3 44 . Whilst star-forming galaxies are less luminous than quasars, they have several advantages in being plentiful at redshifts 7 to 9 and providing an independent \ntest of neutrality at higher redshifts. Finally, galaxies do not exhibit broad Lyα and N vλ1240 emission, which may simplify the damping wing modelling 45 . \nFig. 3 BEAGLE fit to the spectral break region of JADES-GS-z11-0 (top panel), while varying the fraction of neutral Hydrogen in the IGM, x HI . The red line and shaded region show the extracted spectrum and per-pixel 1 𝜎 uncertainties and the darker blue line and shaded region shows the range and median of the fitted models, respectively. The lighter blue line shows the underlying intrinsic spectrum before application of the damping wing. The bottom panel shows the 2D constraints on redshift and x HI , which were varied in the fit while keeping all other physical properties constant (see text for details). For this test, we use a spectral extraction over 3 pixels that maximises the S/N in the region of the break. The shape of the damping wing is not sensitive to wavelength-dependent slit losses introduced by such a small extraction box since the wing it extends over just tens of pixels. \n<!-- image --> \nWe conclude by emphasizing that is the results reported here represent a milestone for the JWST mission, pushing the spectroscopic frontier to a markedly earlier epoch of galaxy formation. In addition to providing clear detections of Lyman dropouts as high as z = 13.2, these JADES observations also show the power of spectroscopy to probe the physics of these galaxies, revealing low metallicities by through the lack of emission lines, as well as the state of the surrounding intergalactic medium. This is just a starting point for the mission. JADES and other programmes have extensive amounts of spectroscopy approved for JWST-detected high-redshift candidates.", '1. NIRSpec observations and data reduction': 'The NIRSpec observations presented here are part of GTO program ID: 1210 (Principal Investigator: Lü tzgendorf) and were obtained between October 22 and 25, 2022. The program used a three-point nod pattern for background subtraction, as well as three small dithers with microshutter array (MSA) reconfigurations in order to improve spatial sampling, increase sensitivity and flux accuracy, mitigate the impact of the detector gaps, and aid removal of cosmic rays. \nEach dither pointing included four sequences of three nodded exposures each to build up signal-to-noise. Observations were carried out by using the disperser-filter combination PRISM/CLEAR, which covers the wavelength range between 0.6 μm and 5.3 μm and provides spectra with a spectral power of R ∼ 100 25 . Each PRISM/CLEAR setup had two integrations of 19 groups, resulting in an exposure time of 8403.2 seconds for each sequence and of 33,612 seconds for each dither pointing. \nA total of 253 galaxies were observed over the three dither pointings. As the non-functioning shutters and rigid grid of the MSA prevents some slit locations from being used, some galaxies were not observed on all three pointings. More specifically, among the four sources presented in this paper, JADES-GS-z11-0 was observed in all three MSA dither pointings, JADES-GS-z10-0 and JADES-GS-z12-0 were present in two dither pointings, whereas JADES-GS-z13-0 was only observed in one dither pointing. The different resulting exposure times for each target are reported in Table 1. \nFlux-calibrated 2D spectra and 1D spectral extractions have been produced using pipelines developed by the ESA NIRSpec Science Operations Team (SOT) and the NIRSpec GTO Team. We briefly outline here the main steps, while a more detailed description will be presented in a forthcoming NIRSpec/GTO collaboration paper. Most of the processing steps in the pipelines adopt the same algorithms as included in the official STScI pipeline used to generate the MAST archive products [see Fig. 11 and section 4.3 of ref 46]. Initially, we processed the MOS raw data (i.e, level 1a data from the MAST archive) with the ramp-toslope pipeline which estimates the count rate per pixel by using all unsaturated groups in the ramp. Ramp jumps due to cosmic rays are detected and rejected on the basis of the slope of the individual ramps. The ramp-to-slope pipeline also includes the following steps: saturation detection and flagging, master bias subtraction, reference pixel subtraction, linearity correction, dark subtraction, snowball artifact detection and correction, and count rate estimation [for more details see refs. 47-49]. All the count-rate images were then processed using a data reduction pipeline including ESA NIRSpec SOT codes and NIRSpec GTO algorithms. The pipeline has 11 main steps: 1) identification of non-target galaxies intercepting the open shutters; 2) pixel-level background subtraction by combining the three nod exposures (excluding nods contaminated by non-target sources); 3) extraction of sub-images containing the spectral trace of each target and wavelength and spatial coordinate assignments to each pixel in the 2D maps; 4) pixel-to-pixel flat-field correction; 5) spectrograph optics and dispersers correction; 6) absolute flux calibration; 7) slitlosses correction; 8) rectification of the spectral trace; 9) extraction of 1D spectra; 10) combination of 1D spectra generated from each integration, nod, and pointing; 11) combination of 2D maps. The data processing workflow thus returns both a combined 1D and 2D spectrum for each target. We stress however that the combined 1D spectra are not extracted from the combined 2D maps, but are the result of a weighted average of 1D spectra from all integrations. This process allowed us to mask the bad pixels indicated on the quality flags and to reject outlier pixels. Finally, we adopted an irregular wavelength grid for the 1D and 2D spectra to avoid oversampling of the line spread function at short wavelengths (λ ∼ 1 μm). \nGiven the compact size of our z > 10 targets, we computed and applied slit-loss corrections, modelling galaxies as point-like sources, but taking into account the relative intra-shutter position of each source (each microshutter has an illuminated area of 0.2"×0.46"). For each target we extracted the 1D spectra from two different apertures. One aperture was as large as the shutter size to recover all emission of the galaxy, while the second extraction was performed in an aperture of 3-pixels height (with NIRSpec spatial pixel scale of 0.1"/pixel) to maximise the signal-to-noise ratio of the final spectra. \nFor most uses of the extraction performed on a 3-pixel aperture, the measurements are performed over small wavelength ranges and further corrections for losses due to the smaller extraction box are not required. In the case of full spectral fitting, we use the extraction over the 5- pixel aperture with one exception, JADES-GS-z13-0. In this case we mitigate the wavelength-dependent losses with simultaneous fitting to photometry (see Section 3).', '2. Empirical measurements': 'The central aspects of the astrophysical analysis of the spectra has been presented in the main part of the paper. Here we explore a few more issues that bolster the fidelity and robustness of our analysis.', '2.1 Balmer break index': 'While the observed spectral breaks in the four objects presented here are fully consistent with expectations for high-redshift galaxies, it is important to test the possibility that the observed breaks may be Balmer breaks at lower redshift. We test this using empirical spectral indices. We adapt the classical Balmer-break index definition 50 and define the break amplitude as the ratio of f λ in the rest-frame range 3751 A] - 4198 A] to that in the range 3145 A] - 3563 A] . This definition expands the spectral windows to include more spectral pixels (15-19 depending on redshift) and increase the signal-to-noise on the index. These measurements are taken from spectra extracted from 3 pixels, maximising the S/N. The effect of wavelength-dependent extraction losses should be minimal for this measurement. When the lower spectral range yields a negative flux, we adopt the 2σ upper flux limit instead. Additionally, to account for the noisy measurement of the (physically) positive-definite flux in the longer wavelength band, we subtract 1σ from the measured flux. This is a conservative upper limit which we quote as a 2σ upper limit in Table 1. \nExtended Data Fig. 1| Balmer-break amplitude plotted against age for single stellar populations with metallicities 0.01 Z ⊙ , 0.1 Z ⊙ , 0.2 Z ⊙ , and Z ⊙ (as indicated), according to the models described in Section 3 . The break is defined as the ratio of the flux f λ integrated over the rest-frame 3751-4198AA wavelength range to that in the rest-frame 3145-3563 AA wavelength range. The peak at early ages for all metallicities arises from the onset or red supergiant stars, and that around 6 × 10 8 yr from bright asymptotic-giant-branch stars. \n<!-- image --> \nIn Extended Data Figure 1 we show the evolution of the modified Balmer break index with age for single stellar population models at four different metallicities from 0.01Z ⊙ to 1Z ⊙ . The maximum value reached within 10 Gyr is less than 2.0. We report 2σ lower limits on the value of this index for each of our four targets in Table 1. The smallest measured break strength is measured for GS-z10-0, with a 2σ lower limit of 2.04, which is higher than the maximum reached by the single stellar populations. Again, this suggests that the Balmer break solution is unlikely, although by a smaller margin than for the other three targets. \nWe note that incorrect background subtraction due to contamination of background shutters by neighbouring galaxies or the source itself can lead to a biased measurement of the break strength. We are careful with the reduction to exclude any contaminated shutters in the background estimate, as described in Methods section 1, and so do not expect these measurements to be affected in this way.', '2.2 Limits on possible emission lines': 'Apart from the possible detections of Lyα in JADES-GS-z10-0 and [Ne III]λ3869 in JADES-GS-z12-0 (which we will assess in future work), visual inspection of the 1D and 2D spectra did not show the presence of any emission lines above the level of the noise in any of the four targets. We derive upper limits to emission- line fluxes and equivalent widths using the error spectrum output from the data reduction pipeline for the optimised S/N spectra extracted over 3 pixels. Our reduced spectra have an irregular wavelength grid, and we estimate that the line spread function of these PRISM spectra results in unresolved emission lines with FWHM of approximately 2-3 spectral pixels. Thus, to calculate emissionline limits, for the 3 spectral pixels centred on the expected centroid of the emission line at the calculated redshift, we sum the pixel errors in quadrature and multiply the result by the wavelength interval between pixels. This results in 1σ upper limits on line fluxes, which we convert into equivalent-width limits by fitting a simple polynomial to the continuum of each object to get the level on the continuum and associated uncertainty. Table 1 reports 2σ upper limits on the equivalent widths of He IIλ1640, C III]λ λ 1907,1909 and [O II]λ λ 3726,3729 while the full set of limits on rest-frame UV emission lines is given in Extended Data Table 1. \nExtended Data Table 1 | 2σ upper limits on the rest-frame equivalent widths (in AA ) and observed line fluxes (in erg s -1 cm -2 ) of rest-frame UV emission lines.', '2.3 M UV and UV slopes': 'The UV slope, β, was determined directly from the 1D extracted spectra. We performed a least-squares fit to the gradient in the ln(λ):ln(f λ ) space, with the errors on ln(f λ ) taken to be ln(f λ + σ) - ln(f λ ) for each extracted spectral pixel (where σ is the noise). For all objects we fit the β slope over the rest- frame wavelength range 1250 - 2600 A] , following ref. 51 (fitting to the entire wavelength range, since absorption features avoided with the spectral windows of ref. 51 will not significantly affect the measurement at this level of S/N and resolution), with the exception of JADES-GS-z10-0 where we used a slightly smaller range of 1500 - 2600 A] to avoid the possible Lyα emission and damping wing. These results were also consistent with those from fitting a power-law to f λ ∝ λ -β in linear wavelength space, weighting each point by 1/σ 2 . \nWe determined the absolute magnitude in the rest-frame UV (M UV ) around λ rest = 1500A] by measuring the average flux density per unit frequency interval (f ν ) from the extracted spectra over the rest-frame wavelength range 1400 - 1600 A] , and accounting for luminosity distance. The errors on each individual pixel flux density were combined in quadrature to derive the uncertainty in β. \nWe note that M UV measured from the spectrum alone is somewhat fainter and the β somewhat redder for JADES-GS-z12-0 than that measured from NIRCam data in the companion paper 27 . We attribute this to residual slit losses missed by our standard correction, since this object is quite close to the edge of the shutter. This highlights the importance of the complementarity between NIRSpec spectroscopy and NIRCam photometry. The difference is starker for JADES-GS-z13-0, which is very close to the edge of the shutter, and for which we report instead M UV and β derived from SED modelling including the NIRCam photometry (see Section 3). The comparison with the measurements in ref. 27 shows good agreement for JADES-GS-z10-0 and JADES-GS-z11-0, which validates the spectroscopic flux calibration in the case where the adoption of point-source path losses is a good approximation.', '3. BEAGLE SED fitting': "We perform full spectral fitting to the R100 spectra using the BEAGLE code 37 . In general, firm constraints on metallicities (both nebular and stellar) and gas parameters require the spectroscopic detection of emission and absorption lines, while constraints on total stellar masses and star-formation rates generally require NIRCam photometry, since NIRSpec MSA spectra are so prone to slit losses. As these objects are so small [r 1/2 ≃ 50 - 165 pc, with on-sky sizes of θ 1/2 ≃ 0.015-0.04' 27] we use pre-calculated point-source slit-loss corrections. Therefore, in this paper we present an entirely independent analysis to that in ref. 27, both in datasets used (except for the fits to JADES-GS-z13-0 where we employ the NIRCam photometry, see later in this section for details), in SED codes and parameterizations. We comment on the consistency between the two analyses throughout this section. \nThis requires modelling of the wavelength-dependent line-spread function (LSF). We fit Gaussian profiles to emission lines in R100 spectra taken within this deep pointing and compare their widths as a function of wavelength to the dispersion curves provided by STScI (https://jwst-docs.stsci.edu/jwst-nearinfrared-spectrograph/nirspec-instrumentation/ nirspec-dispersers-and-filters). We find that the supplied dispersion curves multiplied by a factor of 0.7 provide a reasonable representation of the measured wavelength-dependent LSF. \nExtended Data Table 2 | Parameter descriptions and prior distributions used in BEAGLE fitting. \nWe use spectra extracted over the full micro-shutter aperture to minimise the effects of wavelength dependent losses, since the size of the shutter is more than twice the width of the PSF at 5μm. JADES-GSz13-0, however, is just at the edge of the shutter and the 2D spectrum shows that it is clearly truncated. Therefore, to provide information of the aperture losses for this target, we use a 3-pixel extraction box to maximise S/N and simultaneously fit to the NIRCam aperture photometry 27 . We multiply the shape of the spectrum by a second-order polynomial, sampling over the polynomial coefficients in the fit. This essentially allows the NIRCam photometry to set the normalisation of the spectrum while also correcting wavelength-dependent slit-losses in the spectral calibration. The fits and associated parameter \nconstraints are shown in Extended Data Figures 2-3. We note that SED fitting is performed in the companion paper 27 , yet there they fit only to NIRCam photometry, fixing the redshift to the spectroscopic redshift. Here we perform a complimentary analysis, fitting only to the spectra (except for JADES-GS-z130, as explained above). \nExtended Data Fig. 2| The results of full spectral fitting to JADES-GS-z10-0 (top left), JADES-GS-z11-0 (top right) and JADES-GS-z12-0 (bottom) with BEAGLE. We fit models to spectra extracted over the full shutter aperture to minimise the wavelength-dependent losses due to varying point-spread function (PSF). The triangle plot shows the 2D (offdiagonal) and 1D (along the main diagonal) posterior probability distributions on stellar mass (M), metallicity (Z), maximum age of stars (t) and the effective dust attenuation optical depth in the V-band (τˆ v ) which are all derived from the beagle fits. We also include the model constraints on the star-formation rate (Ψ), UV slope (β) and ionising photon emissivity (ξ ion ), which are derived parameters of the model. The dark, medium and light blue contours show the extents of the 1, 2 and 3σ credible regions of the posterior probability, respectively. The inset panel shows the observed spectrum and 1σ standard errors per pixel in red and light red respectively, and the median and 1σ range in fitted model spectra in blue. We fit with a constant star formation history (more details in the text and Methods section 3). \n<!-- image --> \nFor our BEAGLE fits, we mask the region of possible Lyα in JADES-GS-z10-0 (between 1.4148 and 1.4509μm, inclusive), since it is offset from the break, and would require specialised modelling of the line shape and offset to be accounted for properly. The masked region is shown as pale blue in the spectrum in Extended Data Figure 2. (upper right) For JADES-GS-z12-0, we mask regions of rest-frame UV emission lines (light blue regions in Extended Data Figure 2, lower panel, covering 2.1081 to 2.1620μm, 2.2875 to 2.3181μm and 2.6261 to 2.6533μm) since noise structure in the spectrum is over-fitted if left un-masked. \nWe then fit the spectra following the procedure of ref. 52. We use a constant SFH and fixed nebular parameters since we see no emission lines. The list of parameters employed in the fits, as well as chosen priors are given in Extended Data Table 2. We use the updated Bruzual & Charlot stellar population synthesis templates 53 , as described in ref. 54 with the physically consistent nebular line+continuum emission grid of ref. 55. We adopt a Chabrier 56 initial mass function with an upper mass limit of 100M ⊙ . \nExtended Data Fig. 3 | As for Extended Data Figure 2, but for BEAGLE fits to JADES-GS-z13-0. The bottom right panel shows the observed photometry and associated as blue diamonds and associated 1 𝜎 s.d. error-bars while the coral shaded regions show the model photometry in the same bands. Since this galaxy is very close to the edge of the shutter, we use an extraction over 3 pixels to maximise the S/N. Then to account for wavelength-dependent slit losses we simultaneously fit the spectrum and NIRCam photometry. \n<!-- image --> \nWe have verified that the results are not significantly changed when assuming an upper mass limit of 300M ⊙ . We account for the depletion of metals onto dust grains in the photoionized interiors of stellar birth clouds and include attenuation by dust in the outer neutral envelopes of the clouds and in the diffuse ISM 57 . We set the ionisation parameter, log U s , to depend on the nebular metallicity (and hence stellar metallicity) according to: \n𝑙𝑜𝑔 𝑈 ! = -3.638 + 0.055Z + 0.68Z 2 \nwhich follows the observations of ref. 58. The results do not change significantly for JADES-GS-z10-0 and JADES-GS-z11-0 when logU s is allowed to vary freely in the range -4 < log U s < -1. However, Z and log U s are unconstrained in JADES-GS-z13-0, and poorly constrained in JADES-GS-z12-0, when log U s is allowed to vary freely. \nTo test what is driving the fits to very low metallicity, we fit fixing the metallicity to intermediate values (between Solar and 10% of solar). In this range large [O II] λ λ3726,3729 model fluxes are not described well by the data, and the derived constraints are pushed to low or high metallicities in the two lowest redshift sources. We note, however, that reasonable intermediate metallicity solutions can be fit to the two higher redshift galaxies. Moreover, letting log U s vary freely in this metallicity range decreases model [O II] λ λ3726,3729 fluxes but increases the model C III]λ λ 1907,1909 fluxes, constraining the fits still to the edges of the prior for the two lowest redshift galaxies. It is a complex interplay in the limiting fluxes of these emission lines that drive the low metallicity constraints in these galaxies. \nSince the constraints on the stellar metallicity are driven by the lack of strong emission lines, we tested whether recent cessation of star formation would significantly change the constraints. We therefore tested a constant star formation history where the SFR of the recent 10 Myr was allowed to vary independently (and decrease). We find that the star-formation rate, Ψ, is fairly unconstrained with low posterior median values, meaning recent cessation is consistent with the data. However, we still infer very low metallicities in JADES-GS-z10-0 and JADES-GS-z11-0 (the two with highest S/N spectra). For the two lower S/N spectra (JADES-GS-z12-0 and JADES-GS-z13-0), Z, τˆ v and Ψ are very poorly constrained when this extra free parameter is included. We show the results for JADES-GS-z10-0 and JADES-GS-z11-0 in Extended Data Table 3. \nExtended Data Table 3 | BEAGLE-derived parameters when additionally fitting the star formation rate in the last 10 Myr allowed to vary freely independently of the previous history (labelled SFR 10 varied), or when varying the escape fraction of Lyman-continuum f esc (labelled f esc varied). We show the results when fitting to the two objects with highest S/N, JADES-GSz10-0 and JADES-GS-z11-0 because the constraints on the other two objects are very poor when adding an extra free parameter to the fits. \nAnother possibility to explain relatively weak line emission is a high escape of Lyman-continuum photons from the galaxy. We fit with a picket fence model [allowing for clear sight-lines to the stars through the outer neutral envelopes of birth clouds 59 ]. The results are also given in Extended Data Table 3 for the two objects (JADES-GS-z10-0 and JADES-GS-z11-0) with the highest S/N spectra. We note that JADES-GS-z100 shows a solution with very low age (a few Myr) and high escape fraction. The measured metallicity is marginally higher in this case, but still very low within the 1σ credible interval. JADES-GS-z11-0 does not show such a peak in the posterior distribution function, with fits still favouring older ages and escape fractions that span the input uniform prior. We note that these results are consistent with ref. 27, who find similarly low age and high escape fraction for JADES-GS-z10-0 compared to JADES-GS-z11-0. \nWe also fit the spectra assuming the main feature is a Balmer break (see also Sec. 2). The results are shown in Extended Data Figure 4. Here, we fit a delayed SFH which halted a Gyr prior to observation, varying metallicity, maximum stellar age and redshift within a tight prior centred on the assumed redshift in the case that the break is a Balmer break. We see that the fits consistently fail to reproduce the peak and blue slope red-ward of the break, showing poor spectral fits. Additionally, JADES-GS-z10-0 and JADES-GS-z11-0 show flux blue-ward of the break in the fitted models which is clearly inconsistent with the measured flux and noise limits. \nWe further explored possible low redshift solutions to the photometry by SED-fitting with BEAGLEAGN 60 , which includes narrow-line emission from obscured active-galactic-nuclei. This type of template is rarely used when fitting to high-redshift galaxy candidates. Three of the four galaxies provide \nintermediate redshift (z~3-3.5) solutions to the photometry, and the fit to JADES-GS-z13-0 is displayed in Extended Data Fig. 5. We see no strong emission lines in the spectra themselves, hence disproving these low-redshift possibilities allowed by the photometry. \nExtended Data Fig. 5 | The left panel shows the measured NIRCam photometry and associated 1 𝜎 s.d. error bars for JADESGS-z13-0 as blue diamonds and lines, respectively. The coral violin shaded regions show the underlying model values. The black line shows the maximum a-posteriori probability solution with strong emission lines due to active-galactic-nuclei narrowline emission. The right panel shows a zoom of the intermediate redshift solution fitted to the photometry. \n<!-- image -->", '4. Damping wing profile': 'Galaxies at the very large redshifts presented in this paper are embedded in a largely neutral IGM, which has not yet undergone reionization. In this case the effective optical depth of the hydrogen at lower redshifts along the line of sight becomes so large that the accumulated absorption in the Lorentzian scattering wing of the Lyman alpha resonance line causes intergalactic absorption to spill over into wavelengths above the rest frame Lyα line. This so-called damping wing absorption softens the sharp cutoff in the spectrum due to the intervening intergalactic hydrogen. \nWe have included the effect of the damping wing absorption in our spectral fits using the prescription presented by Miralda-Escudé 61 , who first pointed out the important effect. The model assumes that the damping wing arises in a uniformly distributed completely neutral IGM containing the bulk of the baryons in the universe. For a source at a given redshift z s it has only two free parameters, τ 0 , the overall strength of the Lyman alpha absorption in the form of the optical depth of the classical Gunn-Peterson trough 62 at a reference redshift of z = 5, and z n , the redshift below which the intergalactic hydrogen is assumed to abruptly transition from fully neutral to fully ionized. \nExtended Data Fig. 6 | As for Fig. 3 but showing a fit to the damping wing using a different definition of \'best fit\' to fix the physical parameters of the galaxy spectrum which pushes the constraints to a higher-redshift solution. \n<!-- image --> \nThese two parameters are in turn set by the assumed cosmological model, which we here take to be Planck 2015 ΛCDM 63 . This model\'s baryonic density parameter of Ω b h = 0.033, total mass density parameter Ω M = 0.309, together with the primordial Helium abundance of Y = 24% translate into τ 0 = 3.1×10 5 . The Planck satellite also measured z n = 8.8 for this cosmology, although the predicted damping \nwing absorption is insensitive to this parameter at the large z > 10 redshifts relevant here. A partially reionized IGM is included in the customary manner by multiplying τ 0 by the volume-averaged neutral fraction x hi . This simple model ignores the potential complications of the galaxies being observed displaying strong intrinsic neutral hydrogen absorption, or their having already reionized a large volume of their immediate surroundings 64 . \nIn fitting the damping wing to JADES-GS-z11-0, we found a bimodality in the redshift solution, with a higher redshift solution being consistent with a fully ionised IGM. We find that the final solution is quite sensitive to the adopted intrinsic underlying spectrum. We find solutions at higher redshifts provide poorer fits to the break region itself (shown in Extended Data Fig 6 for completeness). The difference between fits was based on the definition of \'best fit\' to the full spectrum used to fix physical parameters (either the minimum chi-2 solution, Fig. 3, or the maximum a-posteriori probability solution shown here), while if we do not model Lyman-alpha emission in the fitting, we find a bimodal solution with redshift. \nData availability The data that support the findings of this study are available from the corresponding author upon reasonable request. \nCode availability BEAGLE is available via a Docker image (distributed through docker hub) upon request at https:/iap.fr/beagle. \nAcknowledgments. For the purpose of open access, the author has applied a Creative Commons Attribution (CC BY) licence* to any Author Accepted Manuscript version arising. ECL acknowledges support of an STFC Webb Fellowship (ST/W001438/1). SC acknowledges support by European Union\'s HE ERC Starting Grant No. 101040227 - WINGS. MC, FDE, TJL, RM, JW, and LS acknowledge support by the Science and Technology Facilities Council (STFC), ERC Advanced Grant 695671 \'QUENCH\'. RM is further supported by a research professorship from the Royal Society. JW is further supported by the Fondation MERAC. HUr gratefully acknowledges support by the Isaac Newton Trust and by the Kavli Foundation through a Newton-Kavli Junior Fellowship. NB and PJ acknowledge support from the Cosmic Dawn Center (DAWN), funded by the Danish National Research Foundation under grant no.140. RS acknowledges support from a STFC Ernest Rutherford Fellowship (ST/S004831/1). AJB, AJC, JC, IEBW, AS, & GCJ acknowledge funding from the \'FirstGalaxies\' Advanced Grant from the European Research Council (ERC) under the European Union\'s Horizon 2020 research and innovation pro- gramme (Grant agreement No. 789056). BER, BDJ, DJE, MR, EE, GR, CNAW, and FS acknowledge support from the JWST/NIRCam Science Team contract to the University of Arizona, NAS5-02015. DJE is further supported as a Simons Investi- gator. RB acknowledges support from an STFC Ernest Rutherford Fellowship [grant number ST/T003596/1]. REH acknowledges support from the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-1746060. SAr, BRP, and MP acknowledge support from the research project PID2021-127718NB- I00 of the Spanish Ministry of Science and Innovation/State Agency of Research (MICIN/AEI). MP is further supported by the Programa Atracció n de Talento de la Comunidad de Madrid via grant 2018-T2/TIC-11715. LW acknowledges support from the National Science Foundation Graduate Research Fellowship under Grant No. DGE-2137419. KB is supported in part by the Australian Research Council Cen- tre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project number CE170100013. RH was funded by the Johns Hopkins University, Insti- tute for Data Intensive Engineering and Science (IDIES). This research made use of the lux supercomputer at UC Santa Cruz, funded by NSF MRI grant AST 1828315. Acknowledgement for getting assigned a protected node for the DEEP BagPipes runs: "This study made use of the Prospero high performance computing facility at Liverpool John Moores University." \nAuthor contributions. ECL and SCa led the writing of this paper. MR, CNAW, EE, FS, KH, CCW contributed to the design, construction, and com- missioning of NIRCam. AB, AD, CNAW, CW, DJE, H-WR, MR, MF, PF, PJ, RM, SAl, SAr contributed to the design of the JADES survey. BER, ST, BDJ, CNAW, DJE, IS, MR, RE, ZC contributed to the JADES imaging data reduction. RHa, BER contributed to the JADES imaging data visualization. BDJ, ST, AD, DPS, LW, MWT, RE contributed the modeling of galaxy photometry. KH, JMH, JL, LW, RE, REH contributed the photometric redshift determination and target selection. BDJ, EN, KAS, ZC contributed to the JADES imaging morphological analysis. BER, CNAW, CCW, KH, MR contributed to the JADES pre-flight imaging data challenges. SCa, MC, JW, PF, GG, SAr, BRdP, contributed to the NIRSpec data reduction and to the development of the NIRSpec pipeline PJ, NB, SAr contributed to the design and optimisation of the MSA configurations. AJC, AB, CNAW, ECL, HU, RB, KB, contributed to the selection, prioritisation and visual inspection of the targets. SCh, JC, ECL, RM, JW, RS, FDE, MM, MC, AdG, GJ, AS, LS contributed to analysis of the spectroscopic data, including redshift determination and spectral \nmodelling. PJ, PF, MS, TR, GG, NL, NK, BRdP contributed to the design, construction and commissioning of NIRSpec. 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2024PhRvD.110f3041O | A quantitative analysis of model uncertainties for calculations of the maximum depth of protoninitiated extensive air showers EAS has been performed. Staying within the standard physics picture and using the conventional approach to the treatment of highenergy interactions we found that present uncertainties on the energy dependence of the inelastic cross section the rate of diffraction and the inelasticity of hadronic collisions allow one to increase the predicted average EAS maximum depth by about inlineformulammlmath displayinlinemmlmrowmmlmn10mmlmnmmlmtext mmlmtextmmlmtext mmlmtextmmlmi mathvariantnormalgmmlmimmlmommlmommlmsupmmlmrowmmlmicmmmlmimmlmrowmmlmn2mmlmnmmlmsupmmlmrowmmlmathinlineformula. Invoking more exotic assumptions regarding a potentially significant modification of the parton hadronization procedure by hypothetical collective effects we were able to change drastically the predicted energy dependence of the inelasticity of protonair interactions and to increase thereby the predicted EAS maximum depth by up to inlineformulammlmath displayinlinemmlmommlmommlmn30mmlmnmmlmtext mmlmtextmmlmtext mmlmtextmmlmi mathvariantnormalgmmlmimmlmommlmommlmsupmmlmrowmmlmicmmmlmimmlmrowmmlmrowmmlmn2mmlmnmmlmrowmmlmsupmmlmathinlineformula. However those latter modifications are disfavored by the data of the LHCf experiment regarding forward neutron production in protonproton collisions at the Large Hadron Collider and by measurements of the muon production depth by the Pierre Auger Observatory. | 2024-09-01T00:00:00Z | ['10.48550/arXiv.2409.05501', 'arXiv:2409.05501', '2024arXiv240905501O', '2024PhRvD.110f3041O', '10.1103/PhysRevD.110.063041'] | ['Astrophysics and astroparticle physics', 'High Energy Physics - Phenomenology', 'Astrophysics - High Energy Astrophysical Phenomena'] | Model uncertainties for the predicted maximum depth of extensive air showers | 2,024 | 173 | 0.26 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML'] | 1 | https://arxiv.org/pdf/2409.05501.pdf | {'On the model uncertainties for the predicted maximum depth of extensive air showers': 'Sergey Ostapchenko and Günter Sigl \nUniversität Hamburg, II Institut für Theoretische Physik, 22761 Hamburg, Germany \nSeptember 10, 2024', 'Abstract': 'A quantitative analysis of model uncertainties for calculations of the maximum depth of proton-initiated extensive air showers (EAS) has been performed. Staying within the standard physics picture and using the conventional approach to the treatment of high energy interactions, we found that present uncertainties on the energy dependence of the inelastic cross section, the rate of diffraction, and the inelasticity of hadronic collisions allow one to increase the predicted average EAS maximum depth by about 10 g/cm 2 . Invoking more exotic assumptions regarding a potentially significant modification of the parton hadronization procedure by hypothetical collective effects, we were able to change drastically the predicted energy dependence of the inelasticity of proton-air interactions and to increase thereby the predicted EAS maximum depth by up to /similarequal 30 g/cm 2 . However, those latter modifications are disfavored by the data of the LHCf experiment, regarding forward neutron production in proton-proton collisions at the Large Hadron Collider, and by measurements of the muon production depth by the Pierre Auger Observatory.', '1 Introduction': 'One of the exciting directions of astroparticle physics research is connected to experimental studies of ultra-high energy cosmic rays (UHECRs). Because of the extremely low incoming flux of UHECRs, one is forced to employ indirect detection methods: inferring the properties of the primary particles from measured characteristics of extensive air showers (EAS) - nuclearelectromagnetic cascades initiated by interac- \ntions of UHECRs in the atmosphere [1]. Consequently, the success of such studies depends on the quality of the description of EAS development by the corresponding numerical tools, like the CORSIKA program [2]. A special role in such EAS simulation procedures is played by Monte Carlo (MC) generators of high energy hadronic interactions, used to describe the backbone of an air shower - the cascade of interactions in the atmosphere of both primary cosmic ray (CR) particles and of secondary hadrons produced [3]. \nAll of the above fully applies to investigations of UHECR composition, based on studies of the longitudinal extensive air shower development, notably, on measurements of EAS maximum depth, X max , corresponding to the maximum of the charged particle profile of an air shower. Modern fluorescence detectors allow one to measure X max for individual events rather accurately (see [4] for a review). On the other hand, predictions of EAS simulation procedures for the average EAS maximum depth, 〈 X max 〉 , being driven by the treatment of primary particle interactions with air nuclei, are rather seriously constrained by experimental data from the Large Hadron Collider (LHC) (see, e.g. [5]). Therefore, somewhat surprising is a certain tension between the predictions of air shower simulations and the experimental data of the Pierre Auger Observatory, regarding distributions of the shower maximum depth [6, 7]. More specifically, Auger data demonstrate that both the elongation rate ER( E 0 ) = d 〈 X max ( E 0 ) 〉 /d lg E 0 and the width of X max distributions, σ ( X max ) , decrease with the primary energy E 0 faster than predicted by EAS simulations corresponding to an energy-independent UHECR composition, thereby indicating a change towards heav- \nimaries at ultra-high energies [8, 9]. However, the observed energy dependence of σ ( X max ) implies a faster change of the composition, compared to the one deduced from ER( E 0 ) , e.g., if one employs the QGSJET-II-04 model [10, 11] for interpreting the measurements. As demonstrated in [7], a consistent interpretation of the data requires a significantly slower extensive air shower development, compared to current EAS simulation results. Consequently, of significant importance is to quantify the range of uncertainty for X max predictions, related to potential modifications of the treatment of high energy hadronic interactions, allowed by LHC data. \nSuch a quantitative investigation is the subject of the current work. We consider all plausible changes of the interaction treatment, in the framework of the QGSJET-III model [12, 13], which may potentially improve the agreement of the model predictions with Auger data, while staying within the standard physics picture. Like in our previous study of model uncertainties for the predicted EAS muon content [14], our investigation is guided by three basic principles: i) the changes of the corresponding modeling are performed at a microscopic level; ii) the considered modifications are restricted by the requirement not to contradict basic physics principles; iii) the consequences of such changes, regarding a potential (dis)agreement with relevant accelerator data, are analyzed. \nThe outline of the paper is as follows. In Section 2, we identify basic characteristics of hadronic interactions, which are of direct relevance to X max predictions. In Section 3, we consider various modifications of the treatment of high energy interactions, which change such characteristics, and study the corresponding consequences both for the predicted EAS maximum depth and for comparisons with relevant LHC data. We summarize our investigation in Section 4.', '2 Characteristics of high energy interactions, relevant for X max predictions': "Unlike the EAS muon content depending on the whole history of the nuclear cascade in the \natmosphere, the maximum depth of a protoninitiated extensive air shower is largely governed by an interaction of the primary particle. As demonstrated, e.g., in [14], modifying the treatment of secondary pion-air collisions has a rather weak impact on the calculated 〈 X max 〉 . Therefore, predictions of EAS simulations for the longitudinal extensive air shower development are rather seriously constrained by available LHC data on proton-proton and, to some extent, proton-nucleus interactions. \nOf primary importance for 〈 X max 〉 predictions is the proton-air inelastic cross section σ inel p -air since it controls the mean free path of protons in air, λ p = m air /σ inel p -air , m air being the average mass of air nuclei, and thereby the starting point for a nuclear cascade. The energy rise of σ inel p -air is the main factor which causes the decrease of the elongation rate ER( E 0 ) with increasing primary energy E 0 . The inelastic proton-proton cross section σ inel pp has been measured at LHC with a high precision; the difference between the results of the TOTEM and ATLAS experiments, based on Roman Pot techniques, is /similarequal 2 . 6 % at the center-of-mass (c.m.) energy √ s = 13 TeV [15, 16]. Using the corresponding values for calculating σ inel p -air , within the Glauber-Gribov formalism [17, 18], one obtains even a smaller difference for the proton-air cross section since it is largely dominated by the nuclear size. \nAlso of importance for calculations of EAS maximum depth is the treatment of inelastic diffraction. Diffractive interactions of primary protons are characterized by a small inelasticity K inel p -air , i.e., by a small relative energy loss of 'leading' (most energetic) secondary nucleons. This is especially so for a diffractive excitation of a target nucleus, in which case the incoming proton looses a tiny fraction of its initial energy, K inel p -air /similarequal M 2 X /s /lessmuch 1 , M X being the diffractive state mass, i.e., one essentially deals with a quasi-elastic collision. Therefore, the main effect of diffraction amounts to a 'renormalization' of the inelastic proton-air cross section, merely subtracting its diffractive part, σ diffr p -air : σ inel p -air → σ inel p -air -σ diffr p -air , and of proton mean free path: \nλ p → λ p (1 + σ diffr p -air /σ inel p -air ) . (1) \nIn addition, the inelastic diffraction is closely related to the inelastic screening effect [18]; a \nhigher diffraction rate corresponds to a smaller σ inel p -air , for a given σ inel pp , and vice versa. \nFinally, calculations of 〈 X max 〉 depend on the predicted energy dependence of the average inelasticity of proton-air interactions, which is governed by the treatment of nondiffractive collisions. A smaller K inel p -air corresponds to a slower energy dissipation for leading nucleons, hence, to a larger fraction of the energy of the primary particle, being retained in the hadronic cascade, and to a slower EAS development. An enhancement of secondary particle production, with increasing energy, due to the energy rise of the multiple scattering rate, inevitably leads to an increase of the inelasticity. However, the speed of the energy rise of K inel p -air is rather weakly constrained by LHC data and is highly modeldependent [19], being currently the main source of model uncertainties for 〈 X max 〉 predictions. \nComing now to fluctuations of EAS maximum depth, characterized by the standard deviation σ ( X max ) , those are rather insensitive to the average inelasticity, being dominated by variations of the proton mean free path, hence, depending mostly on σ inel p -air . In addition, they are influenced somewhat by the rate of inelastic diffraction, where the effective 'renormalization' of the inelastic proton-air cross section and the inelastic screening effect work in the same direction, e.g., both effects contribute to enlarging σ ( X max ) in case of a higher diffraction. Overall, the uncertainties of predictions for σ ( X max ) are rather small, as noticed already in [20], notably, thanks to measurements of total, elastic, and diffractive pp cross sections by the TOTEM and ATLAS experiments at LHC [15, 16, 21, 22, 23, 24, 25, 26, 27, 28]. For example, the current /similarequal 3 % difference between the results of TOTEM and ATLAS for σ inel pp at √ s = 13 TeV translates itself into /similarequal 1 . 5 g/cm 2 variation of σ ( X max ) [cf. Eq. (1)]. On the other hand, the impact on the fluctuations of X max of present uncertainties for σ diffr pp is at the level of few g/cm 2 [29]. \nRegarding average characteristics of extensive air showers initiated by primary nuclei, e.g. 〈 X max 〉 , those are well described by the superposition model, i.e., coincide with a good accuracy with the corresponding contributions of A proton-induced EAS of A times smaller energy, for a primary nucleus of mass number A (e.g. [30, 31]). This follows from the relation between \nthe inelastic nucleus-air cross section σ inel A -air and the mean number of interacting projectile nucleons 〈 ν A 〉 , per inelastic collision [32]: \n〈 ν A 〉 = Aσ inel p -air σ inel A -air , (2) \nand from the possibility to approximate forward production spectra of an A -air collision, for given 〈 ν A 〉 , by the ones of proton-air interactions: \ndN X A -air ( E 0 , E ) dE ∣ ∣ ∣ ∣ ∣ 〈 ν A 〉 →〈 ν A 〉 dN X p -air ( E 0 /A, E ) dE , (3) \nfor any secondary particle X . As a consequence of Eq. (2), mean number of projectile nucleons interacting in a given depth interval, for an air shower initiated by nucleus A , is the same as for a superposition of A proton-induced EAS of A times smaller energy [31]. \nYet the superposition model is invalid for fluctuations of EAS characteristics [30, 31, 33], e.g., for σ ( X max ) , which are dominated by variations of the impact parameter of nucleus-air collisions, causing large fluctuations of the number of interacting projectile nucleons. While those are well-defined in the Glauber-Gribov approach, an additional contribution to σ ( X max ) comes from a fragmentation of the spectator part of the projectile nucleus [30, 31]. For example, for an extensive air shower initiated by a primary iron nucleus, one obtains a factor two difference between the predicted values of σ ( X max ) , when considering two extreme assumptions: a full breakup of the nuclear spectator part into separate nucleons or keeping all non-interacting nucleons together, as a single secondary nucleus [31]. Since experimental data on nuclear fragmentation are available at fixed target energies only, this could have constituted a serious source of uncertainty for predictions of σ ( X max ) , for nucleus-induced EAS. However, the relative yields of various nuclear fragments are energy-independent, above few GeV per nucleon [34], which allows one to reliably calibrate the fragmentation procedures, based on fixed target data, and to safely extrapolate them to UHECR energies. \nAs follows from the above discussion, the main model uncertainties for 〈 X max 〉 and σ ( X max ) stem from the treatment of proton-air inter- \nns, while an extension to the case of nuclear primaries is rather well defined. Therefore we concentrate in the following on a study of proton-induced extensive air showers.", '3.1 Smaller cross section and higher diffraction rate': 'As discussed in Section 2, both a smaller protonair cross section and a higher diffraction rate lead to a larger 〈 X max 〉 predicted. Here we are going to study a combined effect of both, increasing the rate of low mass diffraction (LMD) in the QGSJET-III model. The LMD treatment in QGSJET-III is based on the Good-Walker (GW) formalism [35]: considering a hadron to be represented by a superposition of a number of GW Fock states characterized by different sizes and different (integrated) parton densities. Since such states undergo different absorption during a scattering process, one generally has transitions of the initial hadrons into various low mass excited states (see, e.g. [36], for the corresponding discussion). Obviously, to enhance the LMD rate, one thus needs to enlarge the difference between transverse sizes of the GW states. Using a twice smaller value for the ratio of the squared radii of the smallest and largest GW states of the proton (parameter d p in QGSJET-III [12]), we obtain cross sections for single diffractivelike (SD-like) events 1 , for different intervals of diffractive state mass M X , listed in Table 1. It is easy to see that the LMD rate, for M X < 3 . 4 GeV, is enhanced by /similarequal 30 %, compared to the default QGSJET-III model, while being still compatible with the observations of the TOTEM experiment [22]. Since the considered modification gives rise to a strong enhancement of the inelastic screening effect (see, e.g. [37], for the corresponding discussion), it leads also to a sizable \nreduction of the calculated total, elastic, and inelastic pp cross sections plotted in Fig. 1, which now become compatible with the data of the ATLAS experiment [16, 24, 25]. \nFigure 1 : C.m. energy dependence of the total, inelastic, and elastic pp cross sections, calculated with the default QGSJET-III model (red solid lines) and with the option characterized by an enhanced diffraction (blue dashed lines), compared to experimental data (points) from [15, 16, 38]. \n<!-- image --> \nFigure 2 : Laboratory (lab.) energy dependence of σ inel p -air , calculated with the default QGSJETIII model (red solid line) and with the option characterized by an enhanced diffraction (blue dashed line). \n<!-- image --> \nHowever, as anticipated in Section 2, the impact of such changes on the calculated inelastic \nTable 1 : Cross sections (in mb) of SD-like pp collisions at √ s = 7 TeV, for different ranges of mass M X of diffractive states produced, calculated with the default QGSJET-III model and with the option characterized by an enhanced diffraction, compared to TOTEM data. \nFigure 3 : Lab. energy dependence of the probability of diffractive-like interactions (left) and of the inelasticity (right), for p 14 N collisions, calculated with the default QGSJET-III model (red solid lines) and with the option characterized by an enhanced diffraction (blue dashed lines). \n<!-- image --> \n0 \n<!-- image --> \n0 \nproton-air cross section plotted in Fig. 2 is much more moderate: reaching only /similarequal 1 % level at E 0 = 10 19 eV. On the other hand, one obtains a significant enhancement of diffraction in protonnucleus collisions. For example, for proton interactions with the most abundant air element, nitrogen, the increase of the rate of diffractive-like collisions characterized by a small inelasticity, K inel p N < 0 . 1 , reaches /similarequal 15 % level, as one can see in Fig. 3 (left). It is also noteworthy that such an enhancement of diffraction has a small impact on the average inelasticity of proton-nitrogen interactions [cf. Fig. 3 (right)], the latter being dominated by the treatment of nondifractive collisions. \nAs is obvious from the above-presented results, the changes of the predicted 2 〈 X max 〉 and σ ( X max ) plotted in Fig. 4 are driven by the increased rate of diffractive-like proton-air collisions. The magnitude of these changes, up to /similarequal 8 g/cm 2 for 〈 X max 〉 and /lessorsimilar 4 g/cm 2 for \nσ ( X max ) , agrees well with the corresponding results of [29].', '3.2 Slower energy rise of the inelasticity': "Thus, the only possibility to predict a substantially larger EAS maximum depth is to decrease the inelasticity of nondiffractive proton-air interactions. A violation of the Feynman scaling in hadronic collisions is a well established experimental fact, e.g., regarding a steep rise of central pseudorapidity η density of charged hadrons produced in pp interactions, dN ch pp /dη ∣ ∣ η =0 , with energy, which stems from an increase of multiple scattering rate. However, an approximate Feynman scaling of secondary hadron spectra at large values of Feynman x may still be allowed by experimental data. \nGenerally, the rate of multiple scattering in hadronic collisions rises with energy, primarily, due to a fast increase of the rate of semihard processes leading to production of hadron \nFigure 4 : Dependence on primary energy of the average maximum depth (left) and of the corresponding standard deviation (right), for proton-initiated EAS, calculated with the default QGSJET-III model (red solid lines) and with the option characterized by an enhanced diffraction (blue dashed lines). \n<!-- image --> \n(mini)jets, which is related, in turn, to a steep low x rise of parton momentum distribution functions of hadrons. Yet the rate of this increase may be tamed by nonlinear interaction effects, notably, regarding a copious production of minijets of relatively small transverse momenta p t . On the other hand, the inelasticity of high energy collisions depends strongly on the choice of momentum distributions of constituent partons for the interacting hadrons (nuclei), involved in numerous inelastic rescattering processes [19, 40, 41]. Choosing a softer distribution, ∝ x -α , for the fraction x of the initial light cone (LC) momentum of the parent hadron, taken by such a parton, i.e., using a larger value for α , one obtains a weaker impact of multiple scattering on forward hadron production, arriving to an approximate Feynman scaling at large x in the α → 1 limit [19]. \nTo investigate the impact of a reduced multiple scattering rate, we increase the strength of higher twist (HT) corrections to hard scattering processes in the QGSJET-III model, choosing a twice larger value, K HT = 5 , for the corresponding normalization parameter [12]. As one can see in Fig. 5, this way we decrease significantly, by up to /similarequal 30 %, the minijet production at small p t . It is noteworthy, however, that the considered change is an extreme one since it causes a tension with HERA data on the low x behavior, at small Q 2 , of the proton structure func- \nFigure 5 : Transverse momentum spectra for (mini)jet production in pp collisions at √ s = 10 2 , 10 3 , and 10 4 GeV, as indicated in the plot, as calculated with the default QGSJETIII model (red solid lines) and with the option characterized by twice stronger HT corrections (blue dashed lines). \n<!-- image --> \ntion F 2 ( x, Q 2 ) plotted in Fig. 6. Additionally, we vary the parameter α sea which governs the LC momentum distribution of constituent sea (anti)quarks in the model ( ∝ x -α sea ) [13]: using α sea = 0 . 8 and α sea = 0 . 9 , in addition to the default value α sea = 0 . 65 . For all the considered options, we adjust other parameters of \nFigure 6 : x -dependence of the proton structure function F 2 ( x, Q 2 ) , for different Q 2 , as indicated in the plots, as calculated with the default QGSJET-III model (red solid lines) and with the option characterized by twice stronger HT corrections (blue dashed lines), compared to HERA data [42] (points). \n<!-- image --> \nthe hadronization procedure of the model in order to keep a reasonable agreement with hadron production data, both from fixed target experiments and from LHC, as illustrated in Figs. 7 and 8. Noteworthy is the modification of the η -dependence of the charged hadron yield, for increasing α sea (cf. the plot on the right-hand side of Fig. 8): a steeper fall-down of the distribution at large η . However, the experimental accuracy does not allow one to discriminate between the different values of α sea . \nThe impact of the considered modifications of the interaction treatment on the energy dependence of the inelasticity K inel of proton-nitrogen collisions is shown in Fig. 9, while the corresponding changes of the predicted shower maximum depth are plotted in Fig. 10. As one can see in Figs. 9 and 10, increasing the strength of HT effects, without modifying momentum distributions of constituent partons (the case α sea = 0 . 65 , shown by the black dotted lines in the Figures), does not produce any significant differences, compared to the default QGSJETIII predictions. This is because such HT corrections mostly affect relatively central (small impact parameter b ) collisions involving high parton densities and being characterized by high multiple scattering rates, hence, by a very high inelasticity. The overall reduction of multiple scattering due to stronger HT effects does not \nhave an appreciable impact on the average K inel which is rather dominated by contributions of more peripheral (larger b ) proton-nucleus collisions. \nThe situation changes noticeably when choosing softer LC momentum distributions of constituent partons (using α sea = 0 . 8 and α sea = 0 . 9 ), to which strings of color field are connected [see Fig. 11 (left)]. The larger α sea value is used, the shorter such strings are and the smaller fraction of LC momentum of incident proton goes into multiple hadron production resulting from the fragmentation of such strings. As one can see in Figs. 9 and 10, for α sea = 0 . 9 , one has up to /similarequal 6 % reduction of K inel and up to /similarequal 12 g/cm 2 larger 〈 X max 〉 at the highest energies. \nWhile the considered modifications remain compatible with LHC data on secondary hadron production, as demonstrated in Fig. 8, one may wonder why their impact on the energy dependence of K inel and 〈 X max 〉 is so moderate. This is quite nontrivial, being related to the so-called initial state radiation (ISR) of partons in hard scattering processes. As discussed, e.g. in [47], in any partial semihard rescattering, the hardest (highest p t ) parton scattering process is typically preceded by multiple emission of 'softer' partons characterized by smaller transverse momenta but having larger fractions of LC momentum taken from the parent hadron. In fact, it \nFigure 7 : x F -dependence of p t -integrated invariant cross section for proton production (left) and x F -distribution of neutrons (right) in c.m. frame, for pp collisions at 158 GeV/c, compared to NA49 data [43]. Black dotted, blue dashed, and green dash-dotted lines correspond to calculations with the modified QGSJET-III model characterized by twice stronger HT corrections, for the parameter α sea = 0 . 65 , 0.8, and 0.9, respectively. \n<!-- image --> \nis this ISR, i.e., the perturbative parton cascade preceding the hardest scattering process, which produces large collinear and infrared logarithms compensating the smallness of the strong coupling constant involved and gives rise to a steep energy rise of (mini)jet production [48]. The crucial thing here is the p t and x ordering: each previous parton in the 'ladder' formed by successive parton emissions is typically characterized by a much smaller transverse momentum p t than the next one, while having a much larger LC momentum fraction x . Therefore, the total LC momentum fraction taken by the first s -channel partons produced in such perturbative parton cascades constitutes a natural lower bound on the inelasticity, whatever soft distribution for nonperturbative constituent partons is chosen [47]. \nThus, we arrived here to the point that potential variations of K inel and the corresponding changes of X max are restricted by theoretical arguments, rather than by experimental data. A natural question is how robust are these restrictions. In principle, one can not exclude the possibility that the hadron production pattern corresponding to the above-discussed picture is modified at sufficiently high energies by collective effects. In particular, one popular option is to consider a rearrangement of 'color flows', which gives rise to different string configura- \ntions, typically reducing both the total string length in the rapidity space and the fractions of LC momenta taken by string end partons from the interacting hadrons (nuclei) 3 (e.g. [50]), thereby overcoming the above-discussed lower bound on K inel . \nHere we prefer to restrain from relying on a particular treatment of collective effects in secondary hadron production, rather using an effective approach which allows for extreme modifications of the predicted inelasticity. Namely, for each semihard rescattering, we generate the hardest scattering process according to the standard collinear factorization formalism of the perturbative quantum chromodynamics, while suppressing ISR, such that strings of color field are stretched between constituent partons of the interacting hadrons (nuclei) and final state partons produced in the hardest process, as sketched in Fig. 11 (right). In such a scheme, varying the α sea parameter corresponds effectively both to a modification of momentum distributions of constituent partons and to a varying strength of collective effects in the hadronization procedure. In particular, in the limit α sea → 1 , one ends up with very short strings of color field, concentrated in the central \nFigure 8 : Calculated pseudorapidity η distributions of charged hadrons in c.m. frame. Left: for pp collisions at different √ s (from top to bottom: 13, 7, 2.36, and 0.9 TeV), for hadron transverse momentum p t > 0 . 5 GeV/c, compared to ATLAS data [44, 45]. Right: for pp collisions at √ s = 8 TeV, compared to the data of CMS and TOTEM [46]. The notations for the lines are the same as in Fig. 7. \n<!-- image --> \nFigure 9 : Lab. energy dependence of the inelasticity of p 14 N collisions, calculated with the default QGSJET-III model (red solid line) and with the option characterized by twice stronger HT corrections, for the parameter α sea = 0 . 65 , 0.8, and 0.9 - black dotted, blue dashed, and green dash-dotted lines, respectively. \n<!-- image --> \nrapidity region in c.m. frame, such that multiple semihard rescattering processes have a minor impact on K inel . However, it is important to remark that such a limit is nonphysical for two reasons. First, potential collective effects may be efficient in relatively central collisions character- \nFigure 10 : Dependence on primary energy of the average maximum depth of proton-initiated EAS. The notations for the lines are the same as in Fig. 9. \n<!-- image --> \nized by high parton densities, while being rather weak in more peripheral collisions dominating the average inelasticity. Secondly, the chance that collective effects, however strong they are, eliminate all the partons from ISR should be vanishingly small. \nAs previously, we perform a modeling of hadron-proton and hadron-nucleus interactions, using different values of α sea and adjusting other \n- \nFigure 11 : Schematic view of a single semihard scattering process. In the standard treatment (left), strings are formed between constituent partons (quarks and antiquarks) and/or partons produced by perturbative cascades, following the color and anticolor flows (thick green and blue lines). Neglecting ISR (right), strings are formed between constituent partons and partons emerging from the hardest scattering process. \n<!-- image --> \n<!-- image --> \n- \nparameters of the hadronization procedure in order to keep an agreement with accelerator measurements, a comparison with selected data sets being plotted in Figs. 12 and 13. As we can see in Fig. 14, even using the default value α sea = 0 . 65 , the inelasticity is reduced by up to /similarequal 10 % at the highest energies, compared to the QGSJET-III predictions, which demonstrates the importance of perturbative parton cascades both for hadron production in general and for the energy loss of leading nucleons in particular. On the other hand, for α sea = 0 . 9 , K inel is practically energy-independent above 1 PeV, where secondary hadron production is dominated by semihard processes. The corresponding energy dependence of the predicted 〈 X max 〉 , for α sea = 0 . 65 , 0.8, and 0.9, is plotted in Fig. 15. The reduction of the inelasticity leads to a noticeably larger elongation rate for proton-induced EAS; for α sea = 0 . 9 , the shower maximum is /similarequal 30 g/cm 2 deeper at E 0 = 10 20 eV, compared to the QGSJET-III predictions. \nNow the crucial question is whether such modifications can be constrained by LHC data on forward hadron production. In Fig. 16, we compare production spectra of neutrons, calculated with the modified QGSJET-III model: suppressing ISR and using α sea = 0 . 65 , 0.8, and 0.9, to the corresponding measurements of the LHCf \nexperiment. As one can see in Fig. 16, choosing softer momentum distributions of constituent partons, i.e., using larger α sea , one obtains larger neutron yields at high x F /similarequal 2 E/ √ s /greaterorsimilar 0 . 5 than observed by the experiment, thereby underestimating the inelasticity for neutron production. 4 Thus, the most extreme modifications of the model, leading to an approximate Feynman scaling in the fragmentation region, are somewhat disfavored by the LHCf data. It is noteworthy that more stringent constraints may arise from studying correlations between central and forward hadron production [19], e.g., from the ongoing combined measurements of hadron production in pp collisions by the ATLAS and LHCf experiments [52, 53]. \nIt is worth remarking that the modifications of the interaction treatment, leading to a substantially larger elongation rate, are rather strongly disfavored by observations of the Pierre Auger Observatory, regarding the muon production depth in air showers [54, 55], notably, by the measured values of 〈 X µ max 〉 - the average depth of maximum of the muon production profile [56]. As discussed in [5, 57], changes of a model treatment of proton-air collisions, which produce a larger 〈 X max 〉 , shift the average depth of maximum of the muon production profile deeper in the atmosphere by a comparable amount. Moreover, since the above-considered modifications, namely, the suppression of ISR and the use of softer momentum distributions of constituent partons, impact also pion-air interactions, the corresponding effect on 〈 X µ max 〉 is stronger, compared to the change of 〈 X max 〉 , as demonstrated in Fig. 17. 5 For the most extreme modifications, the calculated 〈 X µ max 〉 appears to be up \n5 Plotted in Fig. 17 is the the average depth of maximum of muon production profiles produced by EAS simulations, for muon energies E µ > 1 GeV. This should be distinguished from the so-called apparent muon production depth derived by taking into consideration both the impact of muon propagation in the atmosphere and the effects of the corresponding experimental reconstruction procedures, regarding measurements with ground-based detectors (see, e.g. [54, 55] for the corresponding discussion). \nFigure 12 : x F -dependence of p t -integrated invariant cross section for proton production (left) and x F -distribution of neutrons (right) in c.m. frame, for pp collisions at 158 GeV/c, compared to NA49 data [43]. Black dotted, blue dashed, and green dash-dotted lines correspond to calculations with the modified QGSJET-III model: suppressing ISR and using α sea = 0 . 65 , 0.8, and 0.9, respectively. \n<!-- image --> \nto /similarequal 40 g/cm 2 larger at the highest energies, than predicted by QGSJET-III, thereby creating a strong tension with the Auger data. 6", '4 Summary': 'We performed a quantitative analysis of model uncertainties for predicted maximum depth of proton-initiated extensive air showers, in the framework of the QGSJET-III hadronic interaction model, restricting ourselves to the standard physics picture. Using the conventional approach to the treatment of high energy interactions, we investigated a possibility to obtain larger values of 〈 X max 〉 , considering variations of the inelastic proton-proton cross section, the rate of inelastic diffraction, the strength of nonlinear interaction effects, and momentum distributions of constituent partons involved in multiple scattering processes, allowed by LHC data. The studied modifications of the interaction treatment allowed us to increase the predicted 〈 X max 〉 by only ∼ 10 g/cm 2 . \nSuch a small variation of the predicted EAS maximum depth, when modifying σ inel pp and σ diffr pp within the range allowed by accelerator data, comes at no surprise, given extensive and precise measurements of proton-proton interaction cross sections at LHC. Yet one could have expected larger changes of the calculated 〈 X max 〉 to result from the other options studied, because of their potentially strong impact on the inelasticity of proton-proton and proton-nucleus collisions. However, potential variations of the inelasticity appeared to be limited by the initial state radiation of partons in semihard scattering processes: the total fraction of the incident hadron momentum, taken by all such perturbatively generated partons constitutes a natural lower bound on the inelasticity. \nWe further investigated a more exotic scenario, considering a potentially significant modification of the parton hadronization procedure by hypothetical collective effects. That way, we were able to change drastically the predicted energy dependence of the inelasticity of protonair collisions and to increase thereby the predicted EAS maximum depth by up to /similarequal 30 g/cm 2 . However, those most extreme modifications appeared to be disfavored both by the data of the LHCf experiment, regarding forward neutron production in pp collisions at LHC, and by measurements of the muon production depth by the Pierre Auger Observatory. \nFigure 13 : Calculated pseudorapidity η distributions of charged hadrons in c.m. frame. Left: for pp collisions at different √ s (from top to bottom: 13, 7, 2.36, and 0.9 TeV), for hadron transverse momentum p t > 0 . 5 GeV/c, compared to ATLAS data [44, 45]. Right: for pp collisions at √ s = 8 TeV, compared to the data of CMS and TOTEM [46]. The notations for the lines are the same as in Fig. 12. \n<!-- image --> \nFigure 14 : Lab. energy dependence of the inelasticity of p 14 N collisions, calculated with the default QGSJET-III (red solid line) and with the modified model: suppressing ISR and using α sea = 0 . 65 , 0.8, and 0.9 - black dotted, blue dashed, and green dash-dotted lines, respectively. \n<!-- image -->', 'Acknowledgments': 'The work of S.O. was supported by Deutsche Forschungsgemeinschaft (project number 465275045). G.S. acknowledges support by the Bundesministerium für Bildung und Forschung, under grants 05A20GU2 and 05A23GU3. \nFigure 15 : Dependence on primary energy of the average maximum depth of proton-initiated EAS. The notations for the lines are the same as in Fig. 14. \n<!-- image -->', 'References': '- [1] M. Nagano and A. A. Watson, Observations and implications of the ultrahighenergy cosmic rays , Rev. Mod. Phys. 72 , 689 (2000).\n- [2] D. Heck, J. Knapp, J. N. Capdevielle, G. Schatz, and T. Thouw, CORSIKA: A Monte Carlo code to simulate extensive air \nFigure 16 : Calculated neutron energy spectra in c.m. frame, for pp collisions at √ s = 13 TeV, compared to the data of the LHCf experiment [51] (points). Shown as red solid histograms are the results of the default QGSJET-III, while back dotted, blue dashed, and green dash-dotted histograms correspond to calculations with the modified model: suppressing ISR and using α sea = 0 . 65 , 0.8, and 0.9, respectively. \n<!-- image --> \nshowers , Forschungszentrum Karlsruhe Internal Report FZKA-6019 (1998). \n- [3] R. Engel, D. Heck, and T. Pierog, Extensive air showers and hadronic interactions at high energy , Ann. Rev. Nucl. Part. Sci. 61 , 467 (2011).\n- [4] K.-H. Kampert and M. Unger, Measurements of the Cosmic Ray Composition with Air Shower Experiments , Astropart. Phys. 35 , 660 (2012).\n- [5] S. Ostapchenko, High energy interactions of cosmic rays , Adv. Space Res. 64 , 2445 (2019).\n- [6] P. Abreu et al. 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2024arXiv240911073S | We investigate the thermodynamics of a Schwarzschild black hole surrounded by the quintessence energymatter in the linear and quadratic generalized uncertainty principle framework. Considering the variance in the position to be of the order of the event horizon radius and equating the variance in the momentum to the Hawking temperature of the black hole we substitute these variances in the deformed algebra. From there we obtained the generalized uncertainty principlemodified black hole temperature and eventually the specific heat of the black hole. Then we calculate the critical as well as the remnant mass and obtain the entropy relation. We observe that the entropy relation includes the usual leading order textitarea divided by four term subleading logarithmic term and higher order inverse of the area corrections. Finally calculating the energy output as a function of time we obtain the evaporation time of the black hole. The results show the dependence of the quintessence parameter on the thermodynamic quantities in the framework of linear and quadratic generalized uncertainty principle. | 2024-09-01T00:00:00Z | ['2024arXiv240911073S', '10.48550/arXiv.2409.11073', 'arXiv:2409.11073'] | ['General Relativity and Quantum Cosmology', 'High Energy Physics - Theory'] | Thermodynamics of a Schwarzschild black hole surrounded by quintessence in the generalized uncertainty principle framework | 2,024 | 173 | 0.17 | ['EPRINT_HTML', 'EPRINT_PDF'] | 0 | https://arxiv.org/pdf/2409.11073.pdf | {'Thermodynamics of a Schwarzschild black hole surrounded by quintessence in the generalized uncertainty principle framework': "Soham Sen, 1, ∗ Abhijit Dutta, 2, † and Sunandan Gangopadhyay 1, ‡ \n1 Department of Astrophysics and High Energy Physics, S. N. Bose National Centre for Basic Sciences, JD Block, \nSector-III, Salt Lake City, Kolkata-700 106, India \n2 Department of Physics, Kandi Raj College, Kandi, Murshidabad-742 137, India \nWe investigate the thermodynamics of a Schwarzschild black hole, surrounded by the quintessence energy-matter in the linear and quadratic generalized uncertainty principle framework. Considering the variance in the position to be of the order of the event horizon radius and equating the variance in the momentum to the Hawking temperature of the black hole, we substitute these variances in the deformed algebra. From there we obtained the generalized uncertainty principle-modified black hole temperature and eventually the specific heat of the black hole. Then we calculate the critical as well as the remnant mass and obtain the entropy relation. We observe that the entropy relation includes the usual leading order ' area divided by four ' term, sub-leading logarithmic term, and higher order inverse of the area corrections. Finally, calculating the energy output as a function of time, we obtain the evaporation time of the black hole. The results show the dependence of the quintessence parameter on the thermodynamic quantities in the framework of linear and quadratic generalized uncertainty principle.", 'I. INTRODUCTION': "The existence of black holes in the general theory of relativity is considered as one of the biggest theoretical predictions in the history of physics. With the imaging of the shadow of the black hole Sagittarius A*, black holes are proved to be real astronomical objects in existence with their bizarre nature. One of the most important aspects of black holes is their thermodynamic behaviour which is yet to be verified. A series of papers by Bekenstein [1, 2] and Hawking [3-5] had shown that a black hole can be considered as a thermodynamical object where the entropy of the black hole depends on the area of the black hole. The prediction specifically claimed that black holes are not as black as they are thought to be and the radiation was termed as the Hawking radiation [3-5]. This relation between black holes and thermodynamics has been investigated in several analyses over a long period [6-22]. One of the propositions regarding our universe is that it is continuously expanding [23, 24] and it was explained by considering the existence of dark energy. Although dark energy is not detected still today, it would be much more prudent to investigate black hole thermodynamics in the presence of dark energy. It is considered that if a black hole is surrounded by dark energy, it will have important effects on the thermodynamics of a black hole. In [25], the thermodynamics of a Schwarzschild black hole surrounded by quintessence matter. Quintessence matter is basically one of the dark energy candidates. In [26], the Hawking radiation for a d -dimensional spherically symmetric black hole surrounded by quintessence matter has \nbeen investigated. Later on people have investigated the thermodynamics of Reissner-Nordstrom [27, 28], Narai type [29, 30], Bardeen [31, 32] and Schwarzschild black holes with quantum corrections [33]. \nAlthough the general theory of relativity is the most accurate classical field theory, a quantum mechanical version of it is still not unveiled. People have tried to put forward several quantum gravity theories like string theory [34-36], loop quantum gravity [37-39], and noncommutative geometry [40, 41] but none of them have been quite successful in giving an ultimate description of physics at or near the Planck length. One thing that is quite clear from most of the analyses is that there exists a fundamental length scale in nature. The easiest way to incorporate such fundamental length scales in a quantum mechanical description is to modify Heisenberg's uncertainty principle. The Heisenberg's uncertainty principle in one-dimensional spacetime takes the form [42] 1 \n∆ x ∆ p ≥ /planckover2pi1 2 ( 1 + β 2 l 2 p /planckover2pi1 2 (∆ p ) 2 ) , (1) \nwhere β is a dimensionless constant and l p = √ /planckover2pi1 G c 3 is the Planck length. In [43], a detector-graviton interaction model in the presence of Earth's gravity has been used which has led to an uncertainty principle induced by the noise of gravitons. This uncertainty relation obtained in [43] reduces to the uncertainty principle in eq.(1) in the Planck mass limit. On the other hand, doubly special relativity (DSR) theories [44-46] suggest the existence of a maximum observable momentum along with the fundamental minimal length scale which results in a linear \norder modification in the momentum uncertainty along with that of the quadratic order modification term in eq.(1). The DSR modified uncertainty principle reads [47-49] \n∆ x ∆ p ≥ /planckover2pi1 2 ( 1 -αl p /planckover2pi1 ∆ p + β 2 l 2 p /planckover2pi1 2 (∆ p ) 2 ) , (2) \nwhere α and β are dimensionless constants. The above uncertainty relation is also known as the linearquadratic uncertainty principle or LQGUP. Recently in [50], the authors have calculated the thermodynamics of a Schwarzschild black hole surrounded by quintessence matter in the generalized uncertainty principle framework where the uncertainty in the position and momentum variables follow eq.(1). In our current analysis, we have extended the analysis presented in [50] by considering the LQGUP framework. \nOur paper is organized as follows. In section (II), we discuss the Schwazschild black hole surrounded by quintessence matter and obtain the event horizon radii for fixed values of the quintessence parameter ω q . Then in section (III), we discuss regarding the thermodynamics of the black hole in the LQGUP framework. In section (IV), by considering energy output as a function of time, we obtain the evaporation and finally conclude in section (V).", 'II. SCHWARZSCHILD BLACK HOLE FENCED IN QUINTESSENCE': "We start our analysis by considering a static spherically symmetric black hole of mass M surrounded by the quintessence energy-matter. The exact solution of the Einstein field equation for this black hole has been derived in [25] and is generally known as the ' Kiselev ' solution. The general form of the metric reads [25] \nds 2 = -f ( r ) dt 2 + 1 f ( r ) dr 2 + r 2 d Ω 2 2 , (3) \nwhere the lapse function f ( r ) takes the form \nf ( r ) = 1 -2 M r -η ω q r 3 ω q +1 . (4) \nIn the above equation, ω q is known as the quintessential state parameter and η ω q is the positive normalization factor depending on the density of the quintessence matter. It is quite well-known that in the range -1 < ω q < -1 3 , the quintessence effortlessly intercepts the accelerated expansion of the universe. It is evident from the form of f ( r ) in eq.(4) that it reduces to the usual Schwarzschild metric in the η ω q → 0 limit. In the vicinity of the quintessence matter, the nonvanishing components of the mixed energy-momentum tensor T µ ν read \nT t t = T r r = -ρ q (5) \nT θ θ = T φ φ = ρ q 2 (3 ω q +1) (6) \nwith the matter density ρ q given by \nρ q = -3 2 η ω q ω q r 3( ω q +1) . (7) \nThe pressure P q in terms of the matter density ρ q reads \nP q = ω q ρ q . (8) \nTo avoid cluttering the fundamental constants in the results, we work in the geometrized units which implies c = G = 1. We also set the Boltzmann constant k B to unity. In order to obtain the event horizon radii, we make use of the vanishing condition for the lapse function f ( r ) as \nf ( r ) | r = r H = 0 (9) \nfrom which, by using eq.(4), we get \nr H -η ω q r H 3 ω q = 2 M . (10) \nFollowing [50], we shall make use of three values of the quintessential state parameter ω q given by ω q = {-2 3 , -1 3 , -1 } . \n- 1. For ω q = -2 3 , two event horizon radii emerge, namely, inner horizon radius ( r H in ) or the Cauchy horizon of the black hole and the outer horizon radius r H out or the event horizon radius of the black hole which are given by \nr H in = 1 -√ 1 -8 η -2 / 3 M 2 η -2 / 3 r H out = 1 + √ 1 -8 η -2 / 3 M 2 η -2 / 3 . (11) \nThe notable point is that the outer horizon is commonly known as the quintessence horizon identical to the cosmological horizon in de Sitter spacetime. \n- 2. For ω q = -1 3 , we get only one event horizon radius which reads \nr H = 2 M 1 -η -1 / 3 . (12) \n- 3. For ω q = -1, we have a perturbative solution for the horizon radius as \nr H /similarequal 2 M ( 1 + 4 η -1 M 2 ) . (13)", 'III. THERMODYNAMICS OF BLACK HOLE': 'To investigate the thermodynamics of a black hole, a particle-anti-particle pair production in the vicinity of the event horizon of the black hole is considered. The particle with negative energy falls inside the outer horizon (or the event horizon) of the black hole and that with \nthe positive energy escapes outside the event horizon and gets observed by an asymptotic observer. The particle is considered to be massless. It can be easily inferred that the momentum p of the particle emitted outside of the event horizon of the black hole characterizes the temperature T of the black hole. The temperature can be considered to be proportional to the uncertainty in the momentum ∆ p of the emitted particles. Hence, one can express T as [51] \nT = c ∆ p k B (14) \nwhere c is the speed of light and k B is the Boltzmann constant. Following our choice of constants, one can reinterpret the temperature in eq.(14) as T = ∆ p . The Hawking temperature of the black hole will be equal to the temperature of the particle when thermodynamic equilibrium is reached. The uncertainty in the position of a particle near the event horizon of the Schwarzschild black hole will be of the order of the Schwarzschild radius of the black hole [51, 52] \n∆ x = /epsilon1r H (15) \nwhere /epsilon1 is an undetermined constant and r H is the event horizon radius of the black hole. In order to fix the undetermined constant in eq.(15), we consider the equality condition in eq.(2) in the Heisenberg uncertainty limit ( α, β → 0) as \n∆ x ∆ p = /planckover2pi1 2 = ⇒ /epsilon1r H T HUP = /planckover2pi1 2 . (16) \nWe next need to consider the case when the black hole is not surrounded by the quintessence energy matter and in this η ω q → 0 limit, r H = 2 M comes out from eq.(10). In this limit one can recast eq.(16) as \nT HUP = /planckover2pi1 4 /epsilon1M . (17) \nNow, the Hawking temperature of the black hole can be expressed in terms of its surface gravity κ as \nT H = /planckover2pi1 κ 2 π (18) \nwhere κ = ∂ r f ( r ) | r = r H 2 . In the η ω q → 0 limit, eq.(18) reduces to \nT H = /planckover2pi1 8 πM . (19) \nEquating eq.(19) with eq.(17), we obtain the value of the constant /epsilon1 to be 2 π . Our next aim is to express the Hawking temperature of the black hole in terms of the horizon radius (mass) of the black hole. \nAs we are in the geometrized units, the relations c /planckover2pi1 l p = m p c 2 and m p = c 2 l p G ( m p being the Planck mass) can be modified as \n/planckover2pi1 l p = m p , m p = l p . (20) \nThe equality condition from the LQGUP relation in eq.(2) can be recast as \n∆ x ∆ p = /planckover2pi1 2 ( 1 -α m p ∆ p + β 2 m 2 p (∆ p ) 2 ) = ⇒ 2 πr H T = /planckover2pi1 2 ( 1 -α m p ∆ p + β 2 m 2 p (∆ p ) 2 ) = ⇒ r H = /planckover2pi1 4 πT ( 1 -α m p T + β 2 m 2 p T 2 ) . (21) \nFrom the above relation, one can write down the GUP corrected temperature in terms of the event horizon radius as \nT ± = 1 2 β 2 ( αm p +4 πr H ) ± 1 2 β 2 √ ( αm p +4 πr H ) 2 -4 m 2 P β 2 , (22) \nwhere the minus sign is taken to be the Hawking temperature T = T -. In the next few subsections, we shall calculate the critical mass, specific heat, remnant mass, entropy, energy density, and the pressure of the black hole considering three fixed values of the quintessential state parameter.', 'A. Critical Mass': 'In this subsection, we shall calculate the critical mass of the black hole, below which the thermodynamic quantities become ill-defined. In order for the temperature to be real-valued in eq.(22), the term inside the square root must be equal to or greater than zero. Hence, from eq.(22), we obtain the following inequality for a realvalued temperature as \n( αm p +4 πr H ) 2 -4 m 2 p β 2 ≥ 0 = ⇒ r H Cr. = (2 β -α ) m p 4 π . (23) \nThe above equation gives the value of the critical radius ( r H Cr. ) below which the temperature becomes complexvalued. From the last line of the above equation, we shall calculate the critical mass for three cases. \n- 1. For ω q = -2 / 3, we know from eq.(11) that there are two values of the horizon radius. As a result, one will get two values of the critical mass corresponding to the two horizon radii. \n= \n⇒ \n- (a) Firstly, for r H in , we get \n1 -√ 1 -8 η -2 / 3 M Cr. in 2 η -2 / 3 = m p 4 π (2 β -α ) 2 (24) \nM \nCr. \nin \n= \nm \n8 \nπ \n(2 \nβ \n- \nα \n) \n- \n- \n2 \n/ \n3 \n32 \nπ \nm \n2 \n(2 \nβ \n- \nα \n) \n. \n(b) Secondly, for r H out , we get \n1 + √ 1 -8 η -2 / 3 M Cr. out 2 η -2 / 3 = m p 4 π (2 β -α ) = ⇒ M Cr. out = m p 8 π (2 β -α ) -η -2 / 3 m 2 p 32 π 2 (2 β -α ) 2 . (25) \nWe see from eq.(s)(24,25) that the critical masses of the black hole for both the cases ( r H in and r H out ) are same. This is an expected feature as the two radii corresponds to two different horizons of a single black hole with a fixed mass. As a result, the critical mass should be the same for a single black hole. \n- 2. To find out the critical mass for the ω q = -1 / 3 case, we use the form of the horizon radius from eq.(12) in the left-hand side of eq. (23) and obtain \n2 M Cr. 1 -η -1 / 3 = m p 4 π (2 β -α ) = ⇒ M Cr. ( ωq = -1 / 3) = m p 8 π (1 -η -1 / 3 )(2 β -α ) . (26) \n- 3. We can find out the critical mass of the black hole for ω q = -1 from the vanishing condition of the lapse function much more easily. Instead of making use of the horizon radius in terms of the mass of the black hole, we use the f ( r H ) = 0 condition as follows \nf ( r H ) = 0 = 1 -2 M r H -η -1 r 2 H = ⇒ M = r H 2 -η -1 r 3 H 2 . (27) \nNow substituting the critical radius from eq.(23), we arrive at the critical mass of the black hole to be \nM Cr. ωq = -1 = m p 8 π (2 β -α ) -η -1 m 3 p 128 π 3 (2 β -α ) 3 . (28) \nIn the absence of the quintessence energy matter, all the critical mass values in eq.(s)(24,25,26,28) reduces to the same value m p 8 π (2 β -α ) which gives the critical mass value for the Schwarzachild black hole in the LQGUP framework when the quintessence energy matter is absent. \np \np \n2 \nη', 'B. Specific Heat': "In this subsection, we shall calculate the specific heat of the black hole in terms of its event horizon radius. The form of the specific heat of the black hole from the first law of black hole thermodynamics reads \nC = dM dT = ⇒ C = dM dr H dr H dT . (29) \nIt is evident from the above equation that we need to calculate two separate quantities to calculate the specific heat of the black hole. The second quantity which is the total derivative of the horizon radius in terms of the temperature of the black hole can be calculated directly from eq.(21) and reads \ndr H dT = /planckover2pi1 2 4 π ( -1 T 2 + β 2 m 2 p ) . (30) \nThe above quantity is the same for all of the three cases discussed earlier as it is not dependent on the η ω q parameter. The second quantity dM dr H is calculated by using the mass-event horizon relation obtained from the vanishing condition of the lapse function f ( r H ) = 0. This quantity will be different for different values of the quintessential state parameter. \n- 1. For ω q = -2 / 3, we obtain the mass-horizon radius relation as \nf ( r H ) = 0 = ⇒ M = r H 2 -η -2 / 3 r 2 H 2 = ⇒ dM dr H = 1 -2 η -2 / 3 r H 2 . (31) \nSubstituting eq.(31) and eq.(30) back in eq.(29), one can obtain the form of the specific heat of the black hole up to O ( α 4 , β 4 ) 2 as \nC r H in = C r H out = /planckover2pi1 8 π (1 -2 η -2 / 3 r H ) ( -1 T 2 + β 2 m 2 p ) = ⇒ C ( ω q = -2 / 3) /similarequal -2 πr 2 H (1 -2 η -2 / 3 r H ) ( 1 + αm p 2 πr H + ( α 2 -3 β 2 ) m 2 p 16 π 2 r 2 H -β 4 m 4 p 256 π 4 r 4 H ) . (32) \nWe shall now plot the specific heat of the black hole against its event horizon radius. At first, we shall consider that the uncertainty product in eq.(2) to \nbe exact, which implies that we should use T -from eq.(22) and substitute it back directly in the first line of eq.(32). For plotting, we express the specific heat of the black hole from the first line of eq.(32) in terms of its mass and compare it with the approximate result in the final line of eq.(32) in terms of the mass of the black hole. It is crucial to remember that η -2 / 3 r H is a dimensionless and very small quantity, so we need to probe the regime with r H values for a fixed value of η -2 / 3 such that the η -2 / 3 r H /lessmuch 1. For plotting purposes, we have used /planckover2pi1 = 1 along with α = 0 . 05 , β = 0 . 05 and η -2 / 3 = 0 . 01 in Fig.(1). It is important to \nFIG. 1. Plot of the specific heat of the black hole against its mass when exact and approximated forms are used. Here the parameter values are set to α = 0 . 05, β = 0 . 05, and η = 0 . 01. \n<!-- image --> \nobserve from Fig.(1) that as the mass approaches 10 m p value ( m p = 1 when /planckover2pi1 = G = c = 1) then the η -2 / 3 r H ∼ 0 . 1. As a result the approximate solution starts deviating from the 'exact' one. This implies that for black holes with smaller masses, the approximation used is quite appropriate. We shall now investigate the contribution of the linear GUP parameter towards the specific heat of the black hole. We have set β = 0 . 05 and η = 0 . 01 in Fig.(2). Comparing Fig.(1) and Fig.(2), one can easily find out that the specific heat drops faster with r H when the value of α is non-zero than the α = 0 case. We can further improvise the nature of the specific heat in Fig.(3). We can see that the specific heat decreases faster and then increases slowly with increasing value of the event horizon radius with α = 0 . 05 when plotted against the α = 0 case. Now, we plot the specific heat of the black hole for different values of the quintessential positive normalization factor η -2 / 3 in Fig.(4). One can observe from Fig.(4) that the minimum value of the specific heat becomes smaller with increasing values of η -2 / 3 . \nFIG. 2. Plot of the specific heat of the black hole against its event horizon radius when the linear GUP parameter ( α ) is zero and not zero. \n<!-- image --> \nFIG. 3. Specific heat of the black hole against its event horizon radius is plotted when α = 0 and α = 0 . 05. \n<!-- image --> \nFIG. 4. Specific heat of the black hole against its event horizon radius for different values of η -2 / 3 . \n<!-- image --> \n2. For ω q = -1 / 3, dM dr H reads \ndM dr H = 1 -η -1 / 3 2 . (33) \nSubstituting eq.(s)(30,33) back in eq.(29), we obtain the specific heat of the black hole as \nC ( ω q = -1 / 3) = /planckover2pi1 8 π (1 -η -1 / 3 ) ( -1 T 2 + β 2 m 2 p ) . (34) \nAs the term inside the parentheses remains the same after the expansion in terms of the event horizon radius of the black hole as can be seen from eq.(32), we do not write them explicitly. We can again plot the specific heat of the black hole against the event horizon radius for vanishing as well as the non-vanishing case of the linear GUP parameter. We have kept the value of β to be the same as the earlier case and have set η -1 / 3 = 0 . 01 for the plot in Fig.(5). The value of the α parameter is set to 0 . 05 and then zero for a side-by-side comparison. We find out from Fig.(5) that the decay rate for the specific heat is faster (as observed in the earlier case) for a non-vanishing α value. Unlike the previous case, the specific heat does not have a minimum and it decreases indefinitely with the increasing value of r H . \nFIG. 5. Specific heat of the black hole against its event horizon radius for α = 0 and α = 0 when η -1 / 3 = 0 . 01. \n<!-- image --> \n/negationslash \n- 3. Finally for ω q = -1, we follow the same procedure and obtain the form of dM dr H to be \ndM dr H = 1 -3 η -1 r 2 H 2 . (35) \nAgain as before, we can obtain the form of the specific heat to be \nC ( ω q = -1) = /planckover2pi1 8 π (1 -3 η -1 r 2 H ) ( -1 T 2 + β 2 m 2 p ) . (36) \nWe shall now plot the specific heat of the black hole against the event horizon radius for the three cases discussed in this subsection in Fig.(6). We use α = β = 0 . 05, and η -2 / 3 = η -1 / 3 = η -1 = 0 . 01 for a side-by-side comparison between the three cases. It is although important \nFIG. 6. Specific heat of the black hole against the event horizon radius for ω q = -2 / 3 , -1 / 3 , -1. \n<!-- image --> \nto notice that the 3 η -1 r 2 H and 2 η -2 / 3 r H factor should remain less than unity. From Fig.(6), it is easy to observe that the specific heat value has the steepest fall for the ω q = -1 / 3 case and the slowest fall for the ω q = -1 case. It is quite easy to understand the forms of the specific heat. The specific heat for the ω q = -1 / 3 case has a prefactor (1 -η -1 / 3 ) which is not dependent on r H , as a result, it falls indefinitely whereas for the ω q = -1 case the (1 -3 η -1 r 2 H ) starts to approach zero as soon as the value of the r H increases. \nIn the next subsection, we shall calculate the remnant mass of the black hole for the three cases using the forms of the specific heat obtained in this subsection.", 'C. Remnant Mass': 'From a simple physical consideration, one can argue that there exists a temperature at which the heat capacity vanishes [9]. The radiation process stops at this temperature of the black hole while the black hole is left with a finite mass which is also termed as the remnant mass of the black hole. With the specific heat of the black hole obtained, we can calculate the remnant mass of the black hole for different values of the quintessential state parameter. \n- 1. For ω q = -2 / 3, the specific heat is obtained in eq.(32). The vanishing condition of the specific heat implies \n1 T 2 = β 2 m 2 p = ⇒ m p β = 1 2 β 2 ( αm p +4 πr H ) -1 2 β 2 √ ( αm p +4 πr H ) 2 -4 m 2 P β 2 , (37) \nwhere in the last line, we have made use of the temperature equation from eq.(22). Solving the above \nequation leads to the following relation \nr H = m p 4 π (2 β -α ) (38) \nwhich is identical to the critical value of the event horizon radius obtained in eq.(23). Again using the vanishing condition of the lapse function at the event horizon radius, we obtain the remnant mass of the black hole to be \nM Rem. ( ωq = -2 / 3) = m p 8 π (2 β -α ) -η -2 / 3 m 2 p 32 π 2 (2 β -α ) 2 . (39) \nFollowing the same procedure one can obtain the remnant mass of the black hole for the other two cases which are given below. \n- 2. For ω q = -1 / 3 the remnant mass of the black hole reads \nM Rem. ( ωq = -1 / 3) = m p 8 π (1 -η -1 / 3 )(2 β -α ) . (40) \n- 3. Similarly by setting C ( ω q = -1) = 0 from eq.(36), one can obtain the remnant mass of the black hole for the ω q = -1 case as \nM Rem. ( ωq = -1) = m p 8 π (2 β -α ) -η -1 m 3 p 128 π 3 (2 β -α ) 3 . (41) \nIt is very important to observe by comparing eq.(s)(2426,28) with eq.(s)(39-41), that the critical mass of a black hole is exactly same to its remnant mass. Physically it is quite reasonable as the critical mass signifies the mass of the black hole below which the thermodynamic quantities become ill-defined or the temperature is complexvalued. On the other hand, the remnant mass indicates \nthe mass at which the black hole has stopped radiating which means below this value, the black hole will not radiate which is equivalent to saying there is no temperature of the black hole. So it is quite intuitive and straightforward to understand that the two quantities need to be the same.', 'D. Entropy': 'In this subsection, we shall calculate the entropy of the black hole by using the first law of black hole thermodynamics. From the first law of black hole thermodynamics, we know the form of the entropy to be \nS = ∫ dM T = ∫ dM dT dT T = ⇒ S = ∫ C dT T . (42) \n- 1. For ω q = -2 / 3, we obtain the form of the entropy as \nS ( ω q = -2 / 3) = ∫ C ( ω q = -2 / 3) dT T = /planckover2pi1 8 π (1 -2 η -2 / 3 r H ) ∫ ( -dT T 3 + β 2 m 2 p dT T ) = /planckover2pi1 8 π (1 -2 η -2 / 3 r H ) ( 1 2 T 2 + β 2 m 2 p ln ( T l p )) (43) \nwhere we have made use of eq.(32) and used the form of the specific heat to calculate the entropy of the black hole. Up to O ( α 4 , α 2 β 2 , β 4 ), T and 1 T 2 read \nT /similarequal /planckover2pi1 4 πr H ( 1 -αm p 4 πr H + ( α 2 + β 2 ) m 2 p 16 π 2 r 2 H -( α 2 +3 β 2 ) αm 3 p 64 π 3 r 3 H + m 4 p 256 π 4 r 4 H ( α 4 +6 α 2 β 2 + β 4 ) ) (44) \n1 T 2 /similarequal 16 π 2 r 2 H m 2 p ( 1 + αm p 2 πr H + ( α 2 -2 β 2 ) m 2 p 16 π 2 r 2 H -β 4 m 4 p 256 π 4 r 4 H ) . (45) \nThe area of the black hole is given by A = 4 πr 2 H . Substituting eq.(s)(44,45) in eq.(43) and doing bi- \nS ( ω q = -2 / 3) /similarequal A 4 l 2 p ( 1 -αl p η -2 / 3 π ) -2 l p η -2 / 3 √ π ( A 4 l 2 p ) 3 2 + 1 2 √ π ( A 4 l 2 p ) 1 2 [ α -l p η -2 / 3 4 π ( α 2 -2 β 2 ) + l p η -2 / 3 4 π β 2 ln(16 π ) ] -αβ 2 32 π 3 2 ( A 4 l 2 p ) -1 2 + [( A 4 l 2 p ) -1 -2 l p η -2 / 3 √ π ( A 4 l 2 p ) -1 2 ] ( α 2 + β 2 ) β 2 256 π 2 -β 2 16 π ln ( A 4 l 2 p ) ( 1 -2 l p η -2 / 3 √ π ( A 4 l 2 p ) 1 2 ) . (46) \nIt is important to note that the uncertainty product in eq.(2) is restricted to the second order in the GUP parameters and as a result, one should not go beyond the second order while computing the entropy of the black hole. It is important to recall that the uncertainty product in the case of GUP is an approximation where higher-order contributions in the momentum uncertainty have been dropped as they are considered to be very small. As a result, \nit is more prudent to truncate the results of the entropy or any other thermodynamic quantity after the second-order contribution in the GUP parameter. We have kept some results up to the fourth order for some thermodynamical quantities just to inspect its behaviour at higher orders. The coefficient should change in higher order contributions of the uncertainty product if eq.(2) is considered. \n2. For the ω q = -1 / 3 case the entropy is obtained upto O ( α 4 , α 2 β 2 , β 4 ) as \nS ( ω q = -1 / 3) /similarequal (1 -η -1 / 3 ) ( πr 2 H l 2 p + αr H 2 l p + ( α 2 -2 β 2 ) 16 π -αβ 2 l p 32 π 2 r H + ( α 2 + β 2 ) β 2 l 2 p 256 π 3 r 2 H -β 2 16 π ln(16 π ) -β 2 16 π ln ( πr 2 H l 2 p )) . (47) \nIn terms of the area of the black hole, the entropy relation above can be rewritten as \nS ( ω q = -1 / 3) = ( 1 -η -1 / 3 ) ( A 4 l 2 p + α 2 √ π ( A 4 l 2 p ) 1 2 -β 2 16 π ln ( A 4 l 2 p ) -αβ 2 8 π 3 2 ( A 4 l 2 p ) -1 2 + ( α 2 + β 2 ) β 2 256 π 2 ( A 4 l 2 p ) -1 + K 0 ) , (48) \nwhere \nK 0 ≡ 1 -η -1 / 3 16 π ( α 2 -2 β 2 -β 2 ln(16 π ) ) . (49) \n3. For ω q = -1, the entropy for the black hole in terms of its area reads \nS ( ω q = -1) = A 4 l 2 p [ 1 -3 l 2 p η -1 16 π 2 ( α 2 -(2 + ln(16 π )) β 2 ) ] -3 l 2 p η -1 4 π ( A 4 l 2 p ) 2 -3 l 2 p η -1 α 2 π 3 2 ( A 4 l 2 p ) 3 2 + α 2 √ π ( A 4 l 2 p ) 1 2 ( 1 + 3 l 2 p η -1 β 2 4 π 2 ) -β 2 16 π ( 1 -3 l 2 p η -1 π ( A 4 l 2 p ) ) ln ( A 4 l 2 p ) -αβ 2 8 π 3 2 ( A 4 l 2 p ) -1 2 + ( α 2 + β 2 ) β 2 256 π 2 ( A 4 l 2 p ) -1 + K 1 , (50) \nwhere \nK 1 = -3 l 2 p η -1 β 2 256 π 3 β 2 ( α 2 + β 2 ) + 1 16 π ( α 2 -(2 + ln(16 π )) β 2 ) . (51) \nIn eq.(48,50) K 0 and K 1 terms do not depend on the area of the black hole as a result these terms can be neglected from the expressions of the entropy terms. Here, we have kept these terms for the sake of the completeness of the results. From \neq.(s)(46,48,50), we observe that the entropy of the black hole carries logarithmic as well as inverse order correction terms in the area of the black hole. This is solely a consequence of the generalized uncertainty principle framework and all such correc- \ntion terms vanish in the α, β → 0 limit.', 'E. Energy Density': 'With the form of the entropy in hand corresponding to the three constant values of the quintessential state parameter, we are now in a position to calculate the energy density of the black hole surrounded by quintessence matter in the LQGUP framework for the three separate cases. In the α, β → 0, limit the entropy of the black hole reads \nS HUP = πr 2 H /planckover2pi1 . (52) \nUsing the above equation, we can rewrite eq.(7), in terms of the entropy of the black hole as \nρ q = -3 η ω q ω q 2 r 3( ω q +1) H = -3 η ω q ω q 2 ( π /planckover2pi1 S HUP ) 3( ωq +1) 2 . (53) \nIn the LQGUP framework, we assume that the entropy of the black hole can still be represented as S GUP = S ω q = πr 2 GUP /planckover2pi1 where r GUP denotes the effective event horizon radius such that it encapsulates all of the extra contributions to the Bekenstein-Hawking entropy of the black hole for considering the GUP framework. Hence, the modified energy density in the LQGUP framework takes the form \nρ q = -3 η ω q ω q 2 ( π /planckover2pi1 S ω q ) 3( ωq +1) 2 . (54) \n1. For the ω q = -2 / 3 case, eq.(54) can be recast in the following form \nρ q = η -2 / 3 √ /planckover2pi1 S ω q = -2 / 3 π . (55) \nFor the unit choices made in our current analysis /planckover2pi1 = l 2 p , we can therefore obtain the form of the energy density up to O ( ηα 2 , ηβ 2 ) by using the form of the entropy in eq.(46) as \nρ q /similarequal r H η -2 / 3 1 + αl p 4 πr H -l 2 p [ α 2 + β 2 ln [ πr 2 H l 2 p ]] 32 π 2 r 2 H (56) \nwhere we have dropped all O ( η 2 ) contributions. We shall now compare the energy density of the black hole in the presence and absence of the linear GUP parameter in Fig.(7). It is straightforward to observe from Fig.(7) that the energy density is higher when the linear GUP parameter is present than the case when it is absent. \nρ \nFIG. 7. Energy density vs r H plot for ω q = -2 / 3 when α = 0 . 05 and α = 0.. \n<!-- image --> \n2. For ω q = -1 3 , eq.(54) can be recast as \nρ q = πη -1 / 3 2 /planckover2pi1 S ω q = -1 / 3 . (57) \nUsing eq.(47), we can recast the expression of the above energy density as \nρ q /similarequal √ πη -1 / 3 2 l p r H 1 -αl p 2 πr H + l 2 p [ α 2 + β 2 4 ln [ πr 2 H l 2 p ]] 4 π 2 r 2 H . (58) \nUsing the above approximate result we shall compare again between the QGUP (quadratic GUP with α = 0) and LQGUP case in Fig.(8). We ob- \nρ \nFIG. 8. ρ q vs r H plot ω q = -1 / 3. We compare the QGUP case with the LQGUP case and observe the effect of the linear GUP parameter. \n<!-- image --> \nserve from Fig.(8) that, unlike the ω q = -2 / 3 case, the energy density shows a slower growth and decay rate with increasing r H when α = 0 . 05 than the α = 0 case. We shall now plot the energy densities for the above two cases against the event horizon radius of the black hole in Fig.(9). From \nFIG. 9. Energy density is plotted against the event horizon radius for ω q = -2 / 3 and ω q = -1 / 3. \n<!-- image --> \nFig.(9), we observe that the energy density of the black hole first grows rapidly, then decays for the ω q = -1 / 3 case when plotted against r H where ρ q grows gradually for the ω q = -2 / 3. The reason for this behaviour primarily is the overall r H term being present in the ρ q term when ω q = -2 / 3. The parameters are again kept the same as used to plot Fig.(6). \n- 3. Finally for the ω q = -1 case, eq.(54) can be recast as \nρ q = 3 η -1 2 . (59) \nThe above equation implies that for ω q = -1 case the energy density is independent of the event horizon radius of the black hole and is a constant dependent only upon the quintessential normalization factor η -1 . The pressure of the black hole can be directly obtained using eq.(8). Hence, one can derive the pressure for each case discussed above by just multiplying the corresponding value of ω q with the energy density obtained in this subsection.', 'IV. ENERGY OUTPUT AS A FUNCTION OF TIME': 'While a black hole radiates, the mass of the black hole decreases gradually. With the decreasing of its mass, the temperature of the black hole increases. One can assume that this energy loss is dominated via photons and as a result the standard Stefan-Boltzmann law can be applied to estimate the energy radiated as a function of time \ndM dt = -σAT H 4 , (60) \nwhere σ is the Stefan-Boltzmann constant. The above relation can be reinterpreted as \ndM dr H dr H dt = -σ (4 πr 2 H ) T 4 H (61) \nwhere A = 4 πr 2 H denotes the area of the black hole. For a general Schwarzschild black hole in the η ω q , α, β → 0 limit, one can recast eq.(61) as \n1 2 dr H 0 dt = -σ (4 πr 2 H 0 ) T 4 H 0 , (62) \nwhere r H 0 = 2 M denotes the Schwarzschild radius and T H 0 = /planckover2pi1 4 πr H 0 denotes the Hawking temperature for a Schwarschild black hole. Setting r H 0 = 2 √ /planckover2pi1 x = 2 m p x one can recast eq.(62) as \ndx dt = -σm 5 p 256 π 3 1 x 2 . (63) \nDefining the characteristic time as t ch = 256 π 3 /planckover2pi1 5 2 σ [15], we can recast eq.(63) as \ndx dt = -1 t ch x 2 . (64) \nThe solution of the above equation yields the relation \nx 3 3 + A = -t t ch , (65) \nwhere A is an undetermined constant. Now at t = 0 if x (0) = x i , then the solution of eq.(64) in eq.(65) yields the mass-time relation to be \nx = ( x 3 i -3 t t ch ) 1 3 . (66) \nAt t = 0, x i = M (0) m p , and as a result m p x i denotes the initial mass M (0) of the black hole. The rate at which energy is radiated as a function of time now takes the form \ndx dt = -1 t ch ( x 3 i -3 t t ch ) 2 3 (67) \nwhich is obtained by substituting eq.(66) in the right hand side of eq.(64). If the black hole evaporates completely, then x = 0. The evaporation time is therefore given by \nt E = t ch 3 ( M (0) m p ) 3 . (68) \nOur primary aim is to calculate the mass-time relation and the evaporation time of the Schwarzschild black hole surrounded by the quintessence energy-matter in the LQGUP framework for the three fixed values of the quintessential state parameter.', 'A. Energy output as a function of time for ω q = -2 / 3': "For ω q = -2 3 , the evaporation equation in eq.(61) reads \n1 -2 η -2 / 3 r H 2 dr H dt = -4 πσr 2 H T 4 H . (69) \nAgain using a new variable x = r H 2 √ /planckover2pi1 , up to O ( α 4 , α 2 β 2 , β 4 ), the above equation can be recast as \n( C ' 2 x 2 + C 3 x 3 + C 1 x + C 0 + C -1 x + C -2 x 2 ) dx dt /similarequal -1 t ch (70) \nwhere the constant factors are given by \nC 3 = -4 η -2 / 3 √ /planckover2pi1 , C ' 2 = 1 + C 2 = 1 -2 η -2 / 3 α √ /planckover2pi1 π , C 1 = α 2 π -3 α 2 -2 β 2 8 π 2 η -2 / 3 √ /planckover2pi1 , C 0 = 3 α 2 -2 β 2 32 π 2 -α ( α 2 -2 β 2 ) 32 π 3 η -2 / 3 √ /planckover2pi1 , C -1 = α ( α 2 -2 β 2 ) 128 π 3 -( α 4 -4 α 2 β 2 +2 β 4 ) 1024 π 4 η -2 / 3 √ /planckover2pi1 , C -2 = ( α 4 -4 α 2 β 2 +2 β 4 ) 4096 π 4 . (71) \nIntegrating eq.(70), we obtain \n(1 + C 2 ) x 3 3 + C 3 x 4 4 + C 1 x 2 2 + C 0 x + C -1 ln x -C -2 x + A 0 = -t t ch . (72) \nThe constant A 0 can be fixed by setting t = 0 in the above equation, which gives \nA 0 = C -2 x i -C ' 2 x 3 i 3 -C 3 x 4 i 4 -C 1 x 2 i 2 -C 0 x i -C -1 ln x i . (73) \nThe zeroth order solution for x from eq.(72) in the limit η -2 / 3 , α, β → 0 goes to eq.(66). We consider η -2 / 3 to \nbe a small quantity and while solving for x , we will get rid of any O ( η 2 ) contributions. We shall also drop any contributions higher than O ( α 2 , β 2 , αβ ) for the solution x as the uncertainty relation captures up to quadratic order contributions in the α, β parameters as can be seen from eq.(2). We shall now solve eq.(72) and obtain x as a function of time upto O ( α 2 , β 2 , ηα 2 , ηβ 2 ). Eq.(72) can now be recast as \nx 3 = x 3 i -3 t t ch +3 η -2 / 3 √ /planckover2pi1 ( x 4 -x 4 i ) -3 α ( x 2 -x 2 i 4 π -2 η -2 / 3 √ /planckover2pi1 ( x 3 -x 3 i ) 3 π ) -9 α 2 -6 β 2 32 π 2 ( x -x i -2 η -2 / 3 √ /planckover2pi1 ( x 2 -x 2 i ) ) . (74) \nWe shall solve the above equation perturbatively. We start by considering that x can be decomposed as \nx = x 0 + x η + x 1 + x 2 (75) \nwhere the subscript of x denotes the order of the term ( x η denotes the zeroth order contribution multiplied by η ). In the η -2 / 3 , α, β → 0 limit, eq.(74) can be recast as \nx 3 0 = x 3 i -3 t t ch = ⇒ x 0 = ( x 3 i -3 t t ch ) 1 3 (76) \nwhich is the solution of x obtained in eq.(66). We now move towards obtaining the next order solution or the x η \ncontribution from eq.(75). In the α, β → 0 limit, eq.(74) reduces to \nx = x 0 ( 1 + 3 η -2 / 3 √ /planckover2pi1 x 3 0 ( x 4 -x 4 i ) ) 1 3 . (77) \nIn this α, β → 0 limit, eq.(75) reduces to \nx = x 0 + x η . (78) \nSubstituting eq.(78) in eq.(77), we obtain upto O ( η ) \nx 0 + x η /similarequal x 0 ( 1 + η -2 / 3 √ /planckover2pi1 x 3 0 ( x 4 0 -x 4 i ) ) = ⇒ x η = η -2 / 3 √ /planckover2pi1 x 2 0 ( x 4 0 -x 4 i ) . (79) \nbe obtained following the same procedure to be \nx 1 = -α 4 πx 2 0 ( x 2 0 -x 2 i ) -αη -2 / 3 α √ /planckover2pi1 6 πx 5 0 (2 x 6 0 -3 x 2 i x 4 0 +4 x 3 i x 3 0 -3 x 6 i ) . (80) \nFinally, the second order solution to x can be obtained by using eq.(s)(76,77,79,80) in eq.(75). The result is \nx 2 = α 2 16 π 2 x 5 0 ( x 4 0 -x 4 i ) -3 α 2 -2 β 2 32 π 2 x 2 0 ( x 0 -x i ) -η -2 / 3 √ /planckover2pi1 96 π 2 x 8 0 (3 x 8 0 (3 α 2 -2 β 2 ) -6 x i x 7 0 (3 α 2 -2 β 2 ) + 2 x 2 i x 6 0 (13 α 2 -6 β 2 ) +3 x 4 i x 4 0 (3 α 2 -2 β 2 ) -2 x 5 i x 3 0 (25 α 2 -6 β 2 ) + 30 x 8 i α 2 ) . (81) \nHence, we have obtained the solution of x as a function of time up to second order in the dimensionless GUP parameters 3 . We shall now plot the solution of x (obtained up to second order in the GUP parameter) against time which signifies the change of the mass of the black hole with time in Fig.(10). For simplicity, we consider the initial mass of the black hole to be M i = M (0) = 5 m p . We \nFIG. 10. x ( t ) = M ( t ) m p is plotted against the dimensionless time t t ch for α = β = 0 . 05 and η -2 / 3 = 0 . 01. \n<!-- image --> \nobserve from Fig.(10) that the mass decreases gradually with time then the decay process speeds up which implies a higher evaporation rate with increasing time. Next in Fig.(11), we have plotted x against t t ch for the QGUP ( α = 0) as well as the LQGUP case. We observe that the mass decays faster for the QGUP case than the LQGUP case which indicates a quicker evaporation of the black hole in the QGUP case. \nOur next aim is to calculate the time at which the evaporation process stops completely. This time is also known \nFIG. 11. x ( t ) = M ( t ) m p is plotted against the dimensionless time t t ch for the QGUP and the LQGUP cases. \n<!-- image --> \nas the evaporation time. From the form of the remnant mass in eq.(39) for the ω q = -2 / 3 case, we observe that the black hole does evaporate completely in the α, β → 0 limit. This indicates that the final value of x will be zero. In the LQGUP framework, the remnant mass is of the order of the Planck mass multiplied by GUP parameters. When the evaporation process stops, the value of x will be \nx ( t E ) = M Rem m p = 2 β -α 8 π -η -2 / 3 m p 32 π 2 (2 β -α ) 2 = ⇒ x f = x f 1 -2 η -2 / 3 m p x 2 f 1 (82) \nwhere t E denotes the time when the black hole stops evaporating and we have defined x f ≡ x ( t E ), and x f 1 = 2 β -α 8 π . The evaporation time is obtained by integrating eq.(70) from x i to x f . The evaporation time reads \nt E t ch = x 3 i -x 3 f 3 (1 + C 2 ) + C 3 ( x 4 i -x 4 f ) 4 + C 1 ( x 2 i -x 2 f ) 2 + C 0 ( x i -x f ) + C -1 ln ( x i x f ) -C -2 ( 1 x i -1 x f ) . (83) \nThe above expression up to O ( α 4 , α 2 β 2 , β 4 ) reads \nt E t ch = x 3 i 3 (1 + C 2 ) + C 3 x 4 i 4 + C 1 ( x 2 i ) 2 + C 0 x i + C -1 ln x i -C -2 x i -x 3 f 1 3 +3 η -2 / 3 l p x 4 f 1 + 5 η -2 / 3 αl p 3 π x 3 f 1 -αx 2 f 1 4 π + 3 α 2 -2 β 2 8 π 2 η -2 / 3 l p x f 1 -3 α 2 -2 β 2 32 π 2 x 2 f 1 + 3 α ( α 2 -2 β 2 ) 64 π 3 η -2 / 3 l p x f 1 + α ( α 2 -2 β 2 ) 128 π 3 ln x f 1 + α 4 -4 α 2 β 2 +2 β 4 4096 π 4 x f 1 + α 4 -4 α 2 β 2 +2 β 4 2048 π 4 η -2 / 3 l p + α 4 -4 α 2 β 2 +2 β 4 1024 π 4 η -2 / 3 l p ln x f 1 . (84)", 'B. Energy output as a function of time for ω q = -1 / 3': 'We shall now calculate energy radiation rate with time for ω q = -1 / 3 case. For ω q = -1 / 3 the evaporation equation reads \n1 -η -1 / 3 2 dr H dt = -4 πσr 2 H T 4 H . (85) \nFollowing the procedure in the earlier subsection, we can recast the above equation as \n(1 -η -1 / 3 ) ( x 2 + ¯ C 1 x + ¯ C 0 + ¯ C -1 x + ¯ C -2 x 2 ) dx dt = -1 t Ch. (86) \nwhere the constant parameters \n¯ C 1 = α 2 π , ¯ C 0 = 3 α 2 -2 β 2 32 π 2 , ¯ C -1 = α ( α 2 -2 β 2 ) 128 π 3 , ¯ C -2 = α 4 -4 α 2 β 2 +2 β 4 4096 π 4 . (87) \nAfter integrating eq.(86), we can recast it as \n(1 -η -1 / 3 ) ( x 3 3 + ¯ C 1 x 2 2 + ¯ C 0 x + ¯ C -1 ln x -¯ C -2 x ) + A 1 = -t t ch . (88) \nAgain setting t = 0, we can obtain the form of A 1 \nA 1 = -(1 -η -1 / 3 ) ( x 3 i 3 + ¯ C 1 x 2 i 2 + ¯ C 0 x i + ¯ C -1 ln x i -¯ C -2 x i ) . (89) \nFollowing the perturbative approach used in the previous subsection, we can obtain the zeroth order as well as the η order solution to be \nx 0 = ( x 3 i -3 t t ch ) 1 3 , (90) \nx η = -η -1 / 3 t x 2 0 t ch . (91) \nIt is easy to check that the zeroth order solution is the \nFIG. 12. x ( t ) = M ( t ) m p is plotted against the dimensionless time t t ch for α = β = 0 . 05 and η -1 / 3 = 0 . 01. \n<!-- image --> \nsame for the ω q = -2 / 3 and the ω q = -1 / 3 case. Again substituting first x = x 0 + x η + x 1 and then x = x 0 + x η + x 1 + x 2 , we can obtain perturbatively the first and second order solution to be \nx 1 = -α 4 πx 2 0 ( x 2 0 -x 2 i ) + αη -1 / 3 tx 2 i 2 πt ch x 6 0 , (92) x 2 = α 2 16 π 2 x 5 0 ( x 4 0 -x 4 i ) -3 α 2 -2 β 2 32 π 2 x 2 0 ( x 0 -x i ) -η -1 / 3 t t ch ( x 4 0 ( α 2 -2 β 2 ) -2 x i x 3 0 (3 α 2 -2 β 2 +10 x 4 i α 2 )) 32 π 2 x 8 0 . (93) \nWith the solution of x in hand, we shall now plot x ( t ) against t t ch in Fig.(12). As has been observed in the earlier case, we observe similar behaviour of the mass evaporation of the black hole with time in the ω q = -1 case. \nWe observe that the mass starts to decay faster with increasing time. \nNow, for ω q = -1 / 3, at t = t E , x f = 1 -η -1 / 3 8 π (2 β -α ) and as a result one can obtain the evaporation time by integrating eq.(86) from x i to x f as \nt E t ch = (1 -η -1 / 3 ) ( x 3 i -x 3 f 3 + ¯ C 1 ( x 2 i -x 2 f ) 2 + ¯ C 0 ( x i -x f ) + ¯ C -1 ln ( x i x f ) -¯ C -2 ( 1 x i -1 x f ) ) (94) \nfrom which one can write down the expanded form of the evaporation time up to O ( η -1 / 3 α 4 , η -1 / 3 α 2 β 2 , η -1 / 3 β 4 ) as \nt E t Ch /similarequal (1 -η -1 / 3 ) ( x 3 i 3 + ¯ C 1 x 2 i 2 + ¯ C 0 x i + ¯ C -1 ln x i -C -2 x i ) -1 -4 η -1 / 3 3 x 3 f 1 -(1 -3 η -1 / 3 ) ¯ C 1 2 x 2 f 1 -(1 -2 η -1 / 3 ) ¯ C 0 x f 1 + η -1 / 3 ¯ C -1 -(1 -η -1 / 3 ) ¯ C -1 ln x f 1 + ¯ C -2 x f 1 (95) \nwhere x f 1 = 2 β -α 8 π gives the final value of x when the evaporation process stops in the absence of any quintessential energy matter surrounding the black hole.', 'C. Energy output as a function of time for ω q = -1': 'We shall now write down the evaporation equation corresponding to the ω q = -1 as \n1 -3 η -1 r 2 H 2 dr H dt = -4 πσr 2 H T 4 H . (96) \nAgain by using r H = 2 l p x , we can write down the firstorder non-linear differential equation in x as \n( x 2 + ˜ C 4 x 4 + ˜ C 3 x 3 + ˜ C 2 x 2 + ˜ C 1 xi + ˜ C 0 + ˜ C -1 x + ˜ C -2 x 2 ) dx dt = -1 t ch , (97) \nwhere the constants are given as \n˜ C 4 = -12 η -1 m 2 p , ˜ C 3 = -6 η -1 m 2 p α π , ˜ C 2 = -3 η -1 m 2 p (3 α 2 -2 β 2 ) 8 π 2 , ˜ C 1 = α 2 π -3 η -1 m 2 p α ( α 2 -2 β 2 ) 32 π 3 , ˜ C 0 = 3 α 2 -2 β 2 32 π 2 -3 η -1 m 2 p ( α 4 -4 α 2 β 2 +2 β 4 ) 1024 π 4 , ˜ C -1 = α ( α 2 -2 β 2 ) 128 π 3 , ˜ C -2 = α 4 -4 α 2 β 2 +2 β 4 4096 π 4 . (98) \nIntegrating eq.(97) one can obtain the evaporation equa- \ntion of the black hole \nx 3 3 + ˜ C 4 x 5 5 + ˜ C 3 x 4 4 + ˜ C 2 x 3 3 + ˜ C 1 x 2 2 + ˜ C 0 x + ˜ C -1 ln x -˜ C -2 x + A 2 = -t t ch , (99) \nwhere the constant A 2 can be determined by using the initial time condition as \nA 2 = -x 3 i 3 -˜ C 4 x 5 i 5 -˜ C 3 x 4 i 4 -˜ C 2 x 3 i 3 -˜ C 1 x 2 i 2 -˜ C 0 x i -˜ C -1 ln x i + ˜ C -2 x i . (100) \nWe shall again follow a perturbative approach to obtain the correct mass time relation. The zeroth order solution comes out to be the same as the earlier two cases. Following the perturbative approach the η order solution takes the form \nx η = 12 l 2 p η -1 5 x 2 0 ( x 5 0 -xi 5 ) . (101) \nNext, as before, one can obtain the first order contribution in the GUP parameter to x as \nx 1 = -α 4 πx 2 0 ( x 2 0 -x 2 i ) -3 αη -1 l 2 p 10 πx 5 0 (5 x 7 0 -6 x 2 i x 5 0 +5 x 4 i x 3 0 -4 x 7 i ) . (102) \nOne can again use x = x 0 + x η + x 1 + x 2 and obtain the form of the second order contribution in the GUP parameter to x to be \nx 2 = α 2 16 π 2 x 5 0 ( x 4 0 -x 4 i ) -3 α 2 -2 β 2 32 π 2 x 2 0 ( x 0 -x i ) + η -1 l 2 p 40 π 2 x 8 0 [ x 9 0 (3 α 2 +14 β 2 ) + 9 x 8 0 x i (3 α 2 -2 β 2 ) -30 x 2 i x 7 0 α 2 -5 x 3 i x 6 0 (3 α 2 -2 β 2 ) -3 x 5 i x 4 0 ( α 2 -2 β 2 ) + 12 x 6 i x 3 0 (4 α 2 -β 2 ) -30 x 9 i α 2 ] . (103) \nWe shall now again plot x versus t t ch in Fig.(13) for α = β = 0 . 05 and η -1 = 0 . 01. We observe an almost \nFIG. 13. x ( t ) = M ( t ) m p is plotted against the dimensionless time t t ch for α = β = 0 . 05 and η -1 = 0 . 01. \n<!-- image --> \nsimilar behaviour corresponding to the earlier two cases. However, it is important to observe that the increase in the decay rate is no longer very apt from the Figure itself compared to the earlier cases. \nNow the initial mass of the black hole is 2 m p x i whereas the final mass at which the evaporation process stops is given by its remnant mass M f = m p 8 π (2 β -α ) -η -1 m 3 p 128 π 3 (2 β -α ) 3 . The final integration limit reads \nx f = 2 β -α 8 π -η -1 m 2 p 128 π 3 (2 β -α ) 3 . (104) \nFollowing earlier analysis one can obtain the evaporation time upto O ( η -1 α 4 , η -1 α 2 β 2 , η -1 β 4 ) to be \nt E t ch /similarequal x 3 i 3 -x 3 f 1 3 + ˜ C 4 x 5 i 5 + ˜ C 3 x 4 i 4 + ˜ C 2 x 3 i 3 + ˜ C 1 x 2 i 2 -α 4 π x 2 f 1 + ˜ C 0 x i -3 α 2 -2 β 2 32 π 2 x f 1 + ˜ C -1 ln [ x i x f 1 ] -˜ C -2 [ 1 x i -1 x f 1 ] , (105) \nwhere x f 1 = 2 β -α 8 π .', 'V. CONCLUSION': "In this analysis we have considered a Schwarzschild black hole surrounded by quintessence matter in the linear quadratic generalized uncertainty principle framework. In this work, we have extended the analysis presented in [50] by considering linear order correction in the momentum uncertainty in the well-known form of the generalized uncertainty principle. We start by calculating the critical mass and the specific heat of the black hole. We then plot the specific heat against the mass and then against the event horizon radius of the black hole. From Fig.(1), we observe that the approximate solutions are quite appropriate for black holes with smaller masses. Then in Fig.(s)(2,3,5), we have compared the specific heat for the quadratic generalized uncertainty principle case against the linear and quadratic generalized uncertainty principle case and observed that the specific heat has a higher rate of descent and ascent for the linear and \nquadratic generalized uncertainty principle case than the quadratic generalized uncertainty principle case. Next, in Fig.(6), we have plotted the specific heat of the black hole against the event horizon radius for different values of the quintessence parameter. We observe that for higher values of the quintessence parameter, the specific heat has a steeper decent than the other two cases with ω q = -2 3 and ω q = -1 when the event horizon radius of the black hole keeps on increasing. We also observe that for the lowest value of the quintessence parameter ( ω 1 = -1), the specific heat attains a minima for a certain value of the event horizon radius and then it keeps on increasing with increasing values of r H . We have then computed the remnant mass of the black hole and have found it to be the same as the critical mass of the black hole. Then using the specific heat value of the black hole corresponding to three fixed values of the quintessential state parameter, we investigate the form of the Bekenstein-Hawking black hole entropy of the black hole. We observe sub-leading logarithmic corrections after the leading order ' area by four ' term in the entropy. It is important to observe that the logarithmic contribution is coming solely due to \nthe quadratic GUP parameter ( β ) and has no connection with the linear parameter whereas the linear parameter contributes mainly in very small fractional area contributions to the entropy. Next, we calculated the energy density using the entropy results for the three cases and plotted the ω q = -2 / 3 and ω q = -1 / 3 cases against the event horizon radius of the black hole. We specifically observe that the energy density is a constant, independent of the horizon radius, for the ω q = -1 case. Finally, in section (IV), we have expressed the energy output of the black hole as a function of time for the three dis- \nt cases with different fixed constant values of ω q . 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2014A&A...571A..16P | This paper presents the first cosmological results based on Planck measurements of the cosmic microwave background CMB temperature and lensingpotential power spectra. We find that the Planck spectra at high multipoles 40 are extremely well described by the standard spatiallyflat sixparameter CDM cosmology with a powerlaw spectrum of adiabatic scalar perturbations. Within the context of this cosmology the Planck data determine the cosmological parameters to high precision the angular size of the sound horizon at recombination the physical densities of baryons and cold dark matter and the scalar spectral index are estimated to be SUBSUB 1.04147 0.00062 10SUP2SUP SUBbSUBhSUP2SUP 0.02205 0.00028 SUBcSUBhSUP2SUP 0.1199 0.0027 and nSUBsSUB 0.9603 0.0073 respectivelynote that in this abstract we quote 68 errors on measured parameters and 95 upper limits on other parameters. For this cosmology we find a low value of the Hubble constant HSUB0SUB 67.3 1.2 km sSUP1SUP MpcSUP1SUP and a high value of the matter density parameter SUBmSUB 0.315 0.017. These values are in tension with recent direct measurements of HSUB0SUB and the magnituderedshift relation for Type Ia supernovae but are in excellent agreement with geometrical constraints from baryon acoustic oscillation BAO surveys. Including curvature we find that the Universe is consistent with spatial flatness to percent level precision using Planck CMB data alone. We use highresolution CMB data together with Planck to provide greater control on extragalactic foreground components in an investigation of extensions to the sixparameter CDM model. We present selected results from a large grid of cosmological models using a range of additional astrophysical data sets in addition to Planck and highresolution CMB data. None of these models are favoured over the standard sixparameter CDM cosmology. The deviation of the scalar spectral index from unity isinsensitive to the addition of tensor modes and to changes in the matter content of the Universe. We find an upper limit of rSUB0.002SUBlt 0.11 on the tensortoscalar ratio. There is no evidence for additional neutrinolike relativistic particles beyond the three families of neutrinos in the standard model. Using BAO and CMB data we find NSUBeffSUB 3.30 0.27 for the effective number of relativistic degrees of freedom and an upper limit of 0.23 eV for the sum of neutrino masses. Our results are in excellent agreement with big bang nucleosynthesis and the standard value of NSUBeffSUB 3.046. We find no evidence for dynamical dark energy using BAO and CMB data the dark energy equation of state parameter is constrained to be w 1.13SUB0.10SUBSUP0.13SUP. We also use the Planck data to set limits on a possible variation of the finestructure constant dark matter annihilation and primordial magnetic fields. Despite the success of the sixparameter CDM model in describing the Planck data at high multipoles we note that this cosmology does not provide a good fit to the temperature power spectrum at low multipoles. The unusual shape of the spectrum in the multipole range 20 40 was seen previously in the WMAP data and is a real feature of the primordial CMB anisotropies. The poor fit to the spectrum at low multipoles is not of decisive significance but is an anomaly in an otherwise selfconsistent analysis of the Planck temperature data. | 2014-11-01T00:00:00Z | ['2014A&A...571A..16P', 'arXiv:1303.5076', '2013arXiv1303.5076P', '10.1051/0004-6361/201321591', '10.48550/arXiv.1303.5076'] | ['cosmic background radiation', 'cosmological parameters', 'early Universe', 'inflation', 'primordial nucleosynthesis', 'Astrophysics - Cosmology and Nongalactic Astrophysics'] | Planck 2013 results. XVI. Cosmological parameters | 2,014 | 173 | 0.8 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF'] | 8,001 | https://arxiv.org/pdf/1303.5076.pdf | {'Planck 2013 results. XVI. Cosmological parameters': "Planck Collaboration: P. A. R. Ade 93 , N. Aghanim 65 , C. Armitage-Caplan 99 , M. Arnaud 79 , M. Ashdown 76 ; 6 , F. Atrio-Barandela 19 , J. Aumont 65 , C. Baccigalupi 92 , A. J. Banday 102 ; 10 , R. B. Barreiro 72 , J. G. Bartlett 1 ; 74 , E. Battaner 105 , K. Benabed 66 ; 101 , A. Benoˆıt 63 , A. Benoit-L'evy 26 ; 66 ; 101 , J.-P. Bernard 102 ; 10 , M. Bersanelli 38 ; 55 , P. Bielewicz 102 ; 10 ; 92 , J. Bobin 79 , J. J. Bock 74 ; 11 , A. Bonaldi 75 , J. R. Bond 9 , J. Borrill 14 ; 96 , F. R. Bouchet 66 ; 101 , M. Bridges 76 ; 6 ; 69 , M. Bucher 1 , C. Burigana 54 ; 36 , R. C. Butler 54 , E. Calabrese 99 , B. Cappellini 55 , J.-F. Cardoso 80 ; 1 ; 66 , A. Catalano 81 ; 78 , A. Challinor 69 ; 76 ; 12 , A. Chamballu 79 ; 16 ; 65 , R.-R. Chary 62 , X. Chen 62 , H. C. Chiang 30 ; 7 , L.-Y Chiang 68 , P. R. Christensen 88 ; 41 , S. Church 98 , D. L. Clements 61 , S. Colombi 66 ; 101 , L. P. L. Colombo 25 ; 74 , F. Couchot 77 , A. Coulais 78 , B. P. Crill 74 ; 89 , A. Curto 6 ; 72 , F. Cuttaia 54 , L. Danese 92 , R. D. Davies 75 , R. J. Davis 75 , P. de Bernardis 37 , A. de Rosa 54 , G. de Zotti 50 ; 92 , J. Delabrouille 1 , J.-M. Delouis 66 ; 101 , F.-X. D'esert 58 , C. Dickinson 75 , J. M. Diego 72 , K. Dolag 104 ; 84 , H. Dole 65 ; 64 , S. Donzelli 55 , O. Dor'e 74 ; 11 , M. Douspis 65 , J. Dunkley 99 , X. Dupac 44 , G. Efstathiou 69 GLYPH<3> , F. Elsner 66 ; 101 , T. A. Enßlin 84 , H. K. Eriksen 70 , F. Finelli 54 ; 56 , O. Forni 102 ; 10 , M. Frailis 52 , A. A. Fraisse 30 , E. Franceschi 54 , T. C. Gaier 74 , S. Galeotta 52 , S. Galli 66 , K. Ganga 1 , M. Giard 102 ; 10 , G. Giardino 45 , Y. Giraud-H'eraud 1 , E. Gjerløw 70 , J. Gonz'alez-Nuevo 72 ; 92 , K. M. G'orski 74 ; 106 , S. Gratton 76 ; 69 , A. Gregorio 39 ; 52 , A. Gruppuso 54 , J. E. Gudmundsson 30 , J. Haissinski 77 , J. Hamann 100 , F. K. Hansen 70 , D. Hanson 85 ; 74 ; 9 , D. Harrison 69 ; 76 , S. Henrot-Versill'e 77 , C. Hern'andez-Monteagudo 13 ; 84 , D. Herranz 72 , S. R. Hildebrandt 11 , E. Hivon 66 ; 101 , M. Hobson 6 , W. A. Holmes 74 , A. Hornstrup 17 , Z. Hou 32 , W. Hovest 84 , K. M. Hu GLYPH<11> enberger 28 , A. H. Ja GLYPH<11> e 61 , T. R. Ja GLYPH<11> e 102 ; 10 , J. Jewell 74 , W. C. Jones 30 , M. Juvela 29 , E. Keihanen 29 , R. Keskitalo 23 ; 14 , T. S. Kisner 83 , R. Kneissl 43 ; 8 , J. Knoche 84 , L. Knox 32 , M. Kunz 18 ; 65 ; 3 , H. Kurki-Suonio 29 ; 48 , G. Lagache 65 , A. Lahteenmaki 2 ; 48 , J.-M. Lamarre 78 , A. Lasenby 6 ; 76 , M. Lattanzi 36 , R. J. Laureijs 45 , C. R. Lawrence 74 , S. Leach 92 , J. P. Leahy 75 , R. Leonardi 44 , J. Le'on-Tavares 46 ; 2 , J. Lesgourgues 100 ; 91 , A. Lewis 27 , M. Liguori 35 , P. B. Lilje 70 , M. Linden-Vørnle 17 , M. L'opez-Caniego 72 , P. M. Lubin 33 , J. F. Mac'ıas-P'erez 81 , B. Ma GLYPH<11> ei 75 , D. Maino 38 ; 55 , N. Mandolesi 54 ; 5 ; 36 , M. Maris 52 , D. J. Marshall 79 , P. G. Martin 9 , E. Mart'ınez-Gonz'alez 72 , S. Masi 37 , M. Massardi 53 , S. Matarrese 35 , F. Matthai 84 , P. Mazzotta 40 , P. R. Meinhold 33 , A. Melchiorri 37 ; 57 , J.-B. Melin 16 , L. Mendes 44 , E. Menegoni 37 , A. Mennella 38 ; 55 , M. Migliaccio 69 ; 76 , M. Millea 32 , S. Mitra 60 ; 74 , M.-A. Miville-Deschˆenes 65 ; 9 , A. Moneti 66 , L. Montier 102 ; 10 , G. Morgante 54 , D. Mortlock 61 , A. Moss 94 , D. Munshi 93 , J. A. Murphy 87 , P. Naselsky 88 ; 41 , F. Nati 37 , P. Natoli 36 ; 4 ; 54 , C. B. Netterfield 21 , H. U. Nørgaard-Nielsen 17 , F. Noviello 75 , D. Novikov 61 , I. Novikov 88 , I. J. O'Dwyer 74 , S. Osborne 98 , C. A. Oxborrow 17 , F. Paci 92 , L. Pagano 37 ; 57 , F. Pajot 65 , D. Paoletti 54 ; 56 , B. Partridge 47 , F. Pasian 52 , G. Patanchon 1 , D. Pearson 74 , T. J. Pearson 11 ; 62 , H. V. Peiris 26 , O. Perdereau 77 , L. Perotto 81 , F. Perrotta 92 , V. Pettorino 18 , F. Piacentini 37 , M. Piat 1 , E. Pierpaoli 25 , D. Pietrobon 74 , S. Plaszczynski 77 , P. Platania 73 , E. Pointecouteau 102 ; 10 , G. Polenta 4 ; 51 , N. Ponthieu 65 ; 58 , L. Popa 67 , T. Poutanen 48 ; 29 ; 2 , G. W. Pratt 79 , G. Pr'ezeau 11 ; 74 , S. Prunet 66 ; 101 , J.-L. Puget 65 , J. P. Rachen 22 ; 84 , W. T. Reach 103 , R. Rebolo 71 ; 15 ; 42 , M. Reinecke 84 , M. Remazeilles 75 ; 65 ; 1 , C. Renault 81 , S. Ricciardi 54 , T. Riller 84 , I. Ristorcelli 102 ; 10 , G. Rocha 74 ; 11 , C. Rosset 1 , G. Roudier 1 ; 78 ; 74 , M. Rowan-Robinson 61 , J. A. Rubi˜no-Mart'ın 71 ; 42 , B. Rusholme 62 , M. Sandri 54 , D. Santos 81 , M. Savelainen 29 ; 48 , G. Savini 90 , D. Scott 24 , M. D. Sei GLYPH<11> ert 74 ; 11 , E. P. S. Shellard 12 , L. D. Spencer 93 , J.-L. Starck 79 , V. Stolyarov 6 ; 76 ; 97 , R. Stompor 1 , R. Sudiwala 93 , R. Sunyaev 84 ; 95 , F. Sureau 79 , D. Sutton 69 ; 76 , A.-S. Suur-Uski 29 ; 48 , J.-F. Sygnet 66 , J. A. Tauber 45 , D. Tavagnacco 52 ; 39 , L. Terenzi 54 , L. To GLYPH<11> olatti 20 ; 72 , M. Tomasi 55 , M. Tristram 77 , M. Tucci 18 ; 77 , J. Tuovinen 86 , M. Turler 59 , G. Umana 49 , L. Valenziano 54 , J. Valiviita 48 ; 29 ; 70 , B. Van Tent 82 , P. Vielva 72 , F. Villa 54 , N. Vittorio 40 , L. A. Wade 74 , B. D. Wandelt 66 ; 101 ; 34 , I. K. Wehus 74 , M. White 31 , S. D. M. White 84 , A. Wilkinson 75 , D. Yvon 16 , A. Zacchei 52 , and A. Zonca 33 \n(A GLYPH<14> liations can be found after the references) \nMarch 21, 2014", 'ABSTRACT': "Abstract: This paper presents the first cosmological results based on Planck measurements of the cosmic microwave background (CMB) temperature and lensing-potential power spectra. We find that the Planck spectra at high multipoles ( ' > GLYPH<24> 40) are extremely well described by the standard spatially-flat six-parameter GLYPH<3> CDMcosmology with a power-law spectrum of adiabatic scalar perturbations. Within the context of this cosmology, the Planck data determine the cosmological parameters to high precision: the angular size of the sound horizon at recombination, the physical densities of baryons and cold dark matter, and the scalar spectral index are estimated to be GLYPH<18> GLYPH<3> = (1 : 04147 GLYPH<6> 0 : 00062) GLYPH<2> 10 GLYPH<0> 2 , GLYPH<10> b h 2 = 0 : 02205 GLYPH<6> 0 : 00028, GLYPH<10> c h 2 = 0 : 1199 GLYPH<6> 0 : 0027, and n s = 0 : 9603 GLYPH<6> 0 : 0073, respectively (Note that in this abstract we quote 68% errors on measured parameters and 95% upper limits on other parameters.) For this cosmology, we find a low value of the Hubble constant, H 0 = (67 : 3 GLYPH<6> 1 : 2) km s GLYPH<0> 1 Mpc GLYPH<0> 1 , and a high value of the matter density parameter, GLYPH<10> m = 0 : 315 GLYPH<6> 0 : 017. These values are in tension with recent direct measurements of H 0 and the magnituderedshift relation for Type Ia supernovae, but are in excellent agreement with geometrical constraints from baryon acoustic oscillation (BAO) surveys. Including curvature, we find that the Universe is consistent with spatial flatness to percent level precision using Planck CMB data alone. We use high-resolution CMB data together with Planck to provide greater control on extragalactic foreground components in an investigation of extensions to the six-parameter GLYPH<3> CDM model. We present selected results from a large grid of cosmological models, using a range of additional astrophysical data sets in addition to Planck and high-resolution CMB data. None of these models are favoured over the standard six-parameter GLYPH<3> CDMcosmology. The deviation of the scalar spectral index from unity is insensitive to the addition of tensor modes and to changes in the matter content of the Universe. We find an upper limit of r 0 : 002 < 0 : 11 on the tensor-to-scalar ratio. There is no evidence for additional neutrino-like relativistic particles beyond the three families of neutrinos in the standard model. Using BAO and CMB data, we find N e GLYPH<11> = 3 : 30 GLYPH<6> 0 : 27 for the e GLYPH<11> ective number of relativistic degrees of freedom, and an upper limit of 0 : 23 eV for the sum of neutrino masses. Our results are in excellent agreement with big bang nucleosynthesis and the standard value of N e GLYPH<11> = 3 : 046. We find no evidence for dynamical dark energy; using BAO and CMB data, the dark energy equation of state parameter is constrained to be w = GLYPH<0> 1 : 13 + 0 : 13 GLYPH<0> 0 : 10 . We also use the Planck data to set limits on a possible variation of the fine-structure constant, dark matter annihilation and primordial magnetic fields. Despite the success of the six-parameter GLYPH<3> CDM model in describing the Planck data at high multipoles, we note that this cosmology does not provide a good fit to the temperature power spectrum at low multipoles. The unusual shape of the spectrum in the multipole range 20 < GLYPH<24> ' < GLYPH<24> 40 was seen previously in the WMAP data and is a real feature of the primordial CMB anisotropies. The poor fit to the spectrum at low multipoles is not of decisive significance, but is an 'anomaly' in an otherwise self-consistent analysis of the Planck temperature data.", '1. Introduction': "The discovery of the cosmic microwave background (CMB) by Penzias & Wilson (1965) established the modern paradigm of the hot big bang cosmology. Almost immediately after this seminal discovery, searches began for anisotropies in the CMB - the primordial signatures of the fluctuations that grew to form the structure that we see today 1 . After a number of earlier detections, convincing evidence for a dipole anisotropy was reported by Smoot et al. (1977), but despite many attempts, the detection of higher-order anisotropies proved elusive until the first results from the Cosmic Background Explorer ( COBE ; Smoot et al. 1992). The COBE results established the existence of a nearly scale-invariant spectrum of primordial fluctuations on angular scales larger than 7 GLYPH<14> , consistent with the predictions of inflationary cosmology, and stimulated a new generation of precision measurements of the CMB of which this set of papers forms a part. \nCMB anisotropies are widely recognized as one of the most powerful probes of cosmology and early-Universe physics. Given a set of initial conditions and assumptions concerning the background cosmology, the angular power spectrum of the CMB anisotropies can be computed numerically to high precision using linear perturbation theory (see Sect. 2). The combination of precise experimental measurements and accurate theoretical predictions can be used to set tight constraints on cosmological parameters. The influential results from the Wilkinson Microwave Anisotropy Probe ( WMAP ) satellite (Bennett et al. 2003; Spergel et al. 2003), following on from earlier groundbased and sub-orbital experiments 2 , demonstrated the power of this approach, which has been followed by all subsequent CMB experiments. \nPlanck 3 is the third-generation space mission, following COBE and WMAP , dedicated to measurements of the CMB anistropies. The primary aim of Planck (Planck Collaboration 2005) is to measure the temperature and polarization anisotropies with micro-Kelvin sensitivity per resolution element over the entire sky. The wide frequency coverage of Planck (30-857 GHz) was chosen to provide accurate discrimination of Galactic emission from the primordial anisotropies and to enable a broad range of ancilliary science, such as detections of galaxy clusters, extragalactic point sources and the properties of Galactic dust emission. This paper, one of a set associated with the 2013 release of data from the Planck mission (Planck Collaboration I 2014), describes the first cosmological parameter results from the Planck temperature power spectrum. \nThe results from WMAP (see Bennett et al. 2012 and Hinshaw et al. 2012 for the final nine-year WMAP re- \nGLYPH<3> Corresponding author: G. Efstathiou, [email protected] \n1 For a good review of the early history of CMB studies see Peebles et al. (2009). \n2 It is worth highlighting here the preWMAP constraints on the geometry of the Universe by the BOOMERang (Balloon Observations of Millimetric Extragalactic Radiation and Geomagnetics; de Bernardis et al. 2000) and MAXIMA (Millimeterwave Anisotropy Experiment Imaging Array; Balbi et al. 2000) experiments, for example. \n3 Planck ( http://www.esa.int/Planck ) is a project of the European Space Agency (ESA) with instruments provided by two scientific consortia funded by ESA member states (in particular the lead countries France and Italy), with contributions from NASA (USA) and telescope reflectors provided by a collaboration between ESA and a scientific consortium led and funded by Denmark. \nsults) together with those from high-resolution ground-based CMB experiments (e.g., Reichardt et al. 2012; Story et al. 2012; Sievers et al. 2013) are remarkably consistent with the predictions of a 'standard' cosmological model. This model is based upon a spatially-flat, expanding Universe whose dynamics are governed by General Relativity and whose constituents are dominated by cold dark matter (CDM) and a cosmological constant ( GLYPH<3> ) at late times. The primordial seeds of structure formation are Gaussian-distributed adiabatic fluctuations with an almost scale-invariant spectrum. This model (which will be referred to as the base GLYPH<3> CDM model in this paper) is described by only six key parameters. Despite its simplicity, the base GLYPH<3> CDM model has proved to be successful in describing a wide range of cosmological data in addition to the CMB, including the Type Ia supernovae magnitude-distance relation, baryon acoustic oscillation measurements, the large-scale clustering of galaxies and cosmic shear (as reviewed in Sect. 5). \nNevertheless, there have been some suggestions of new physics beyond that assumed in the base GLYPH<3> CDM model. Examples include various large-angle 'anomalies' in the CMB (as reviewed by the WMAP team in Bennett et al. 2011) and hints of new physics, such as additional relativistic particles, that might steepen the high multipole 'damping tail' of the CMB temperature power spectrum (Dunkley et al. 2011; Hou et al. 2012). Furthermore, developments in early-Universe cosmology over the last 20 years or so have led to a rich phenomenology (see e.g., Baumann 2009 for a review). It is easy to construct models that preserve the main features of simple single-field inflationary models, but lead to distinctive observational signatures such as non-Gaussianity, isocurvature modes or topological defects. \nA major goal of the Planck experiment is to test the GLYPH<3> CDM model to high precision and identify areas of tension. From previous CMB experiments and other cosmological probes, we know that any departures from the standard six-parameter GLYPH<3> CDM cosmology are likely to be small and challenging to detect. Planck , with its combination of high sensitivity, wide frequency range and all-sky coverage, is uniquely well-suited to this challenge. \nThe focus of this paper is to investigate cosmological constraints from the temperature power spectrum measured by Planck . Figure 1 summarizes some important aspects of the Planck temperature power spectrum; we plot this as D ' GLYPH<17> ' ( ' + 1) C '= 2 GLYPH<25> (a notation we will use throughout this paper) versus multipole ' . The temperature likelihood used in this paper is a hybrid: over the multipole range ' = 2-49, the likelihood is based on a component-separation algorithm applied to 91% of the sky (Planck Collaboration XII 2014; Planck Collaboration XV 2014). The likelihood at higher multipoles is constructed from cross-spectra over the frequency range 100-217 GHz, as discussed in Planck Collaboration XV (2014). It is important to recognize that unresolved foregrounds (and other factors such as beams and calibration uncertainties) need to be modelled to high precision to achieve the science goals of this paper. There is therefore no unique ' Planck primordial temperature spectrum'. Figure 1 is based on a full likelihood solution for foreground and other 'nuisance' parameters assuming a cosmological model . A change in the cosmology will lead to small changes in the Planck primordial CMB power spectrum because of di GLYPH<11> erences in the foreground solution. Neverthess, Fig. 1 provides a good illustration of the precision achieved by Planck . The precision is so high that conventional power spectrum plots (shown in the upper panel of Fig. 1) are usually uninformative. We therefore place high weight in this paper on plots of residuals with respect to the best-fit model (shown in the \nFig. 1. Planck foreground-subtracted temperature power spectrum (with foreground and other 'nuisance' parameters fixed to their best-fit values for the base GLYPH<3> CDM model). The power spectrum at low multipoles ( ' = 2-49, plotted on a logarithmic multipole scale) is determined by the Commander algorithm applied to the Planck maps in the frequency range 30-353 GHz over 91% of the sky. This is used to construct a low-multipole temperature likelihood using a Blackwell-Rao estimator, as described in Planck Collaboration XV (2014). The asymmetric error bars show 68% confidence limits and include the contribution from uncertainties in foreground subtraction. At multipoles 50 GLYPH<20> ' GLYPH<20> 2500 (plotted on a linear multipole scale) we show the best-fit The CMBspectrum computed from the CamSpec likelihood (see Planck Collaboration XV 2014) after removal of unresolved foreground components. This spectrum is averaged over the frequency range 100-217 GHz using frequency-dependent di GLYPH<11> use sky cuts (retaining 58% of the sky at 100 GHz and 37% of the sky at 143 and 217 GHz) and is sample-variance limited to ' GLYPH<24> 1600. The light grey points show the power spectrum multipole-by-multipole. The blue points show averages in bands of width GLYPH<1> ' GLYPH<25> 31 together with 1 GLYPH<27> errors computed from the diagonal components of the band-averaged covariance matrix (which includes contributions from beam and foreground uncertainties). The red line shows the temperature spectrum for the best-fit base GLYPH<3> CDMcosmology. The lower panel shows the power spectrum residuals with respect to this theoretical model. The green lines show the GLYPH<6> 1 GLYPH<27> errors on the individual power spectrum estimates at high multipoles computed from the CamSpec covariance matrix. Note the change in vertical scale in the lower panel at ' = 50. \n<!-- image --> \nlower panel). Figure 1 also serves to illustrate the highly interconnected nature of this series of papers. The temperature likelihood used in this paper utilizes data from both the Planck Low Frequency Instrument (LFI) and High Frequency Instrument (HFI). The data-processing chains for these two instruments and beam calibrations are described in Planck Collaboration II (2014), Planck Collaboration VI (2014), and associated papers (Planck Collaboration III 2014; Planck Collaboration IV 2014; Planck Collaboration V 2014; Planck Collaboration VII 2014; Planck Collaboration VIII 2014; Planck Collaboration IX 2014; Planck Collaboration X 2014). Component separation is described in Planck Collaboration XII (2014) and the temperature power spectrum and likelihood, as used in this paper, are described in Planck Collaboration XV (2014). Planck Collaboration XV (2014) also presents a detailed analysis of the robustness of the likelihood to various choices, such as frequency ranges and sky masks (and also compares the likelihood to results from an independent likelihood code based on di GLYPH<11> erent assumptions, see also Appendix C). Consistency of the Planck maps across frequencies is demonstrated in Planck Collaboration XXXI (2014), and the level of consistency with WMAP is assessed. \nThis paper is closely linked to other papers reporting cosmological results in this series. We make heavy use of the gravitational lensing power spectrum and likelihood estimated from an analysis of the 4-point function of the Planck maps (Planck Collaboration XVII 2014). The present paper concentrates on simple parameterizations of the spectrum of primordial fluctuations. Tests of specific models of inflation, isocurvature modes, broken scale-invariance etc. are discussed in Planck Collaboration XXII (2014). Here, we assume throughout that the initial fluctuations are Gaussian and statistically isotropic. Precision tests of non-Gaussianity, from Planck estimates of the 3- and 4-point functions of the temperature anisotropies, are presented in Planck Collaboration XXIV (2014). Tests of isotropy and additional tests of non-Gaussianity using Planck data are discussed in Planck Collaboration XXIII (2014) and Planck Collaboration XXVI (2014). \nThe outline of the paper is as follows. In Sect. 2 we define our notation and cosmological parameter choices. This section also summarizes aspects of the Markov chain Monte Carlo (MCMC) sampler used in this paper and of the CMB Boltzmann code used to predict theoretical temperature power spectra. Section 3 presents results on cosmological parameters using Planck data alone. For this data release we do not use Planck polarization data in the likelihood, and we therefore rely on WMAP polarization data at low multipoles to constrain the optical depth, GLYPH<28> , from reionization. An interesting aspect of Sect. 3 is to assess whether CMB gravitational lensing measurements from Planck can be used to constrain the optical depth without the use of WMAP polarization measurements. \nSection 4 introduces additional CMB temperature data from high-resolution experiments. This section presents a detailed description of how we have modified the Planck model for unresolved foreground and 'nuisance' parameters introduced in Planck Collaboration XV (2014) to enable the Planck spectra to be used together with those from other CMB experiments. Combining high-resolution CMB experiments with Planck mitigates the e GLYPH<11> ects of unresolved foregrounds which, as we will show, can a GLYPH<11> ect cosmological parameters (particularly for extensions to the base GLYPH<3> CDMmodel) if the foreground parameters are allowed too much freedom. Section 4 ends with a detailed analysis of whether the base GLYPH<3> CDM model provides an accept- \nable fit to the CMB temperature power spectra from Planck and other experiments. \nIt is well known that certain cosmological parameter combinations are highly degenerate using CMB power spectrum measurements alone (Zaldarriaga et al. 1997; Efstathiou & Bond 1999; Howlett et al. 2012). These degeneracies can be broken by combining with other cosmological data (though the Planck lensing analysis does help to break the principal 'geometrical' degeneracy, as discussed in Sect. 5.1). Section 5 discusses additional 'astrophysical' data that are used in combination with Planck . Since the Planck temperature data are so precise, we have been selective in the additional data sets that we have chosen to use. Section 5 discusses our rationale for making these choices. \nHaving made a thorough investigation of the base GLYPH<3> CDM model, Sect. 6 describes extended models, including models with non-power-law spectral indices, tensor modes, curvature, additional relativistic species, neutrino masses and dynamical dark energy. This section also discusses constraints on models with annihilating dark matter, primordial magnetic fields and a time-variable fine-structure constant. \nFinally, we present our conclusions in Sect. 7. Appendix A compares the Planck and WMAP base GLYPH<3> CDM cosmologies. Appendix B contrasts the Planck best-fit GLYPH<3> CDM cosmology with that determined recently by combining data from the South Pole Telescope with WMAP (Story et al. 2012). Appendix C discusses the dependence of our results for extended models on foreground modelling and likelihood choices, building on the discussion in Planck Collaboration XV (2014) for the base GLYPH<3> CDMmodel. \nSince the appearance of the first draft of this paper, there have been a number of developments that a GLYPH<11> ect both the Planck data and some of the constraints from supplementary astrophysical data used in this paper. \nThe primary developments are as follows. [1] After the submission of this paper, we discovered a minor error in the ordering of the beam transfer functions applied to each of the CamSpec 217 GLYPH<2> 217 GHz cross-spectra before their coaddition to form a single spectrum. Correcting for this error changes the mean 217 GLYPH<2> 217 GHz spectrum by a smooth function with an amplitude of a few ( GLYPH<22> K) 2 . An extensive analysis of a revised likelihood showed that this error has negligible impact on cosmological parameters and that it is absorbed by small shifts in the foreground parameters. Since the e GLYPH<11> ect is so minor, we have decided not to change any of the numbers in this paper and not to revise the public version of the CamSpec likelihood. [2] The foreground-corrected 217 GLYPH<2> 217 GHz spectrum shows a small negative residual (or 'dip') with respect to the best-fit base GLYPH<3> CDM theoretical model at multipoles ' GLYPH<25> 1800. This can be seen most clearly in Fig. 7 in this paper. After submission of this paper we found evidence that this feature is a residual systematic in the data associated with incomplete 4 K line removal (see Planck Collaboration VI 2014 for a discussion of the 4 K line removal algorithm). The 4 K lines, at specific frequencies in the detector timelines, are caused by an electromagnetic-interference / electromagneticcompatibility (EMI-EMC) problem between the 4 He JouleThomson (4 K) cooler drive electronics and the read-out electronics. This interference is time-variable. Tests in which we have applied more stringent flagging of 4 K lines show that the ' = 1800 feature is reduced to negligible levels in all sky surveys, including Survey 1 in which the e GLYPH<11> ect is strongest. The 2014 Planck data release will include improvements in the 4 K line removal. It is important to emphasise that this systematic is \na small e GLYPH<11> ect. Analysis of cosmological parameters, removing the multipole range around ' = 1800 (and also analysis of the full mission data, where the e GLYPH<11> ect is diluted by the additional sky surveys) shows that the impact of this feature on cosmological parameters is small (i.e., less than half a standard deviation) even for extensions to the base GLYPH<3> CDM cosmology. Some quantitiative tests of the impact of this systematic on cosmology are summarized in Appendix C. [3] An error was found in the dark energy model used for theoretical predictions with equation of state w , GLYPH<0> 1, leading to few-percent C ' errors at very low multipoles in extreme models with w > GLYPH<24> GLYPH<0> 0 : 5. We have checked, using the corrected October 2013 camb version, that this propagates to only a very small error on marginalized parameters and that the results presented in this paper are consistent to within the stated numerical accuracy. [4] After this paper was submitted, Humphreys et al. (2013) presented the final results of a longterm campaign to establish a new geometric maser distance to NGC4258. Their revised distance of (7 : 60 GLYPH<6> 0 : 23) Mpc leads to a lowering of the Hubble constant, based on the Cepheid distance scale, to H 0 = (72 : 0 GLYPH<6> 3 : 0) km s GLYPH<0> 1 Mpc GLYPH<0> 1 , partially alleviating the tension between the Riess et al. (2011) results and the Planck results on H 0 discussed in Sect. 5.3 and subsequent sections. [5] In a recent paper, Betoule et al. (2013) present results from an extensive programme that improves the photometric calibrations of the SDSS and SNLS supernovae surveys. An analysis of the SDSS-II and SNLS supernovae samples, including revisions to the photometric calibrations, favours a higher value of GLYPH<10> m = 0 : 295 GLYPH<6> 0 : 034 for the base GLYPH<3> CDM model, consistent with the Planck results discussed in Sect. 5.4 (Betoule et al. 2014). \nAdetailed discussion of the impact of the changes discussed here on cosmology will be deferred until the Planck 2014 data release, which will include improvements to the low-level data processing and, by which time, improved complementary astrophysical data sets (such as a revised SNLS compilation) should be available to us. In revising this paper, we have taken the view that this, and other Planck papers in this 2013 release, should be regarded as a snapshot of the Planck analysis as it was in early 2013. We have therefore kept revisions to a minimum. Nevertheless, readers of this paper, and users of products from the Planck Legacy Archive 4 (such as parameter tables and MCMCchains), should be aware of developments since the first submission of this paper.", '2.1. Theoretical model': "We shall treat anisotropies in the CMB as small fluctuations about a Friedmann-Robertson-Walker metric whose evolution is described by General Relativity. We shall not consider modified gravity scenarios or 'active' sources of fluctuations such as cosmic defects. The latter are discussed in Planck Collaboration XXV (2014). Under our assumptions, the evolution of the perturbations can be computed accurately using a CMB Boltzmann code once the initial conditions, ionization history and constituents of the Universe are specified. We discuss each of these in this section, establishing our notation. Our conventions are consistent with those most commonly adopted in the field and in particular with those used in the camb 5 \n5 \nhttp://camb.info \nBoltzmann code (Lewis et al. 2000), which is the default code used in this paper.", '2.1.1. Matter and radiation content': 'We adopt the usual convention of writing the Hubble constant at the present day as H 0 = 100 h kms GLYPH<0> 1 Mpc GLYPH<0> 1 . For our baseline model, we assume that the cold dark matter is pressureless, stable and non-interacting, with a physical density ! c GLYPH<17> GLYPH<10> c h 2 . The baryons, with density ! b GLYPH<17> GLYPH<10> b h 2 , are assumed to consist almost entirely of hydrogen and helium; we parameterize the mass fraction in helium by Y P. The process of standard big bang nucleosynthesis (BBN) can be accurately modelled, and gives a predicted relation between Y P, the photon-baryon ratio, and the expansion rate (which depends on the number of relativistic degrees of freedom). By default we use interpolated results from the PArthENoPE BBN code (Pisanti et al. 2008) to set Y P, following Hamann et al. (2011), which for the Planck best-fitting base model (assuming no additional relativistic components and negligible neutrino degeneracy) gives Y P = 0 : 2477. We shall compare our results with the predictions of BBN in Sect. 6.4. \nThe photon temperature today is well measured to be T 0 = 2 : 7255 GLYPH<6> 0 : 0006 K (Fixsen 2009); we adopt T 0 = 2 : 7255 K as our fiducial value. We assume full thermal equilibrium prior to neutrino decoupling. The decoupling of the neutrinos is nearly, but not entirely, complete by the time of electron-positron annihilation. This leads to a slight heating of the neutrinos in addition to that expected for the photons and hence to a small departure from the thermal equilibrium prediction T GLYPH<13> = (11 = 4) 1 = 3 T GLYPH<23> between the photon temperature T GLYPH<13> and the neutrino temperature T GLYPH<23> . We account for the additional energy density in neutrinos by assuming that they have a thermal distribution with an e GLYPH<11> ective energy density \nGLYPH<26>GLYPH<23> = N e GLYPH<11> 7 8 4 11 ! 4 = 3 GLYPH<26>GLYPH<13> ; (1) \nwith N e GLYPH<11> = 3 : 046 in the baseline model (Mangano et al. 2002, 2005). This density is divided equally between three neutrino species while they remain relativistic. \nIn our baseline model we assume a minimal-mass normal hierarchy for the neutrino masses, accurately approximated for current cosmological data as a single massive eigenstate with m GLYPH<23> = 0 : 06 eV ( GLYPH<10> GLYPH<23> h 2 GLYPH<25> P m GLYPH<23> = 93 : 04 eV GLYPH<25> 0 : 0006; corrections and uncertainties at the meV level are well below the accuracy required here). This is consistent with global fits to recent oscillation and other data (Forero et al. 2012), but is not the only possibility. We discuss more general neutrino mass constraints in Sect. 6.3. \nWe shall also consider the possibility of extra radiation, beyond that included in the Standard Model. We model this as additional massless neutrinos contributing to the total N e GLYPH<11> determining the radiation density as in Eq. (1). We keep the mass model and heating consistent with the baseline model at N e GLYPH<11> = 3 : 046, so there is one massive neutrino with N (massive) e GLYPH<11> = 3 : 046 = 3 GLYPH<25> 1 : 015, and massless neutrinos with N (massless) e GLYPH<11> = N e GLYPH<11> GLYPH<0> 1 : 015. In the case where N e GLYPH<11> < 1 : 015 we use one massive eigenstate with reduced temperature.', '2.1.2. Ionization history': "To make accurate predictions for the CMB power spectra, the background ionization history has to be calculated to high ac- \nTable 1. Cosmological parameters used in our analysis. For each, we give the symbol, prior range, value taken in the base GLYPH<3> CDM cosmology (where appropriate), and summary definition (see text for details). The top block contains parameters with uniform priors that are varied in the MCMC chains. The ranges of these priors are listed in square brackets. The lower blocks define various derived parameters. \ncuracy. Although the main processes that lead to recombination at z GLYPH<25> 1090 are well understood, cosmological parameters from Planck can be sensitive to sub-percent di GLYPH<11> erences in the ionization fraction x e (Hu et al. 1995; Lewis et al. 2006; Rubino-Martin et al. 2009; Shaw & Chluba 2011). The process of recombination takes the Universe from a state of fully ionized hydrogen and helium in the early Universe, through to the completion of recombination with residual fraction x e GLYPH<24> 10 GLYPH<0> 4 . Sensitivity of the CMB power spectrum to x e enters through changes to the sound horizon at recombination, from changes in the timing of recombination, and to the detailed shape of the recombination transition, which a GLYPH<11> ects the thickness of the lastscattering surface and hence the amount of small-scale di GLYPH<11> usion (Silk) damping, polarization, and line-of-sight averaging of the perturbations. \nSince the pioneering work of Peebles (1968) and Zeldovich et al. (1969), which identified the main physical processes involved in recombination, there has been significant progress in numerically modelling the many relevant atomic transitions and processes that can a GLYPH<11> ect the details of the \nrecombination process (Hu et al. 1995; Seager et al. 2000; Wong et al. 2008; Hirata & Switzer 2008; Switzer & Hirata 2008; Rubino-Martin et al. 2009; Grin & Hirata 2010; Chluba & Thomas 2011; Ali-Haimoud et al. 2010; Ali-Haimoud & Hirata 2011). In recent years a consensus has emerged between the results of two multi-level atom codes HyRec 6 (Switzer & Hirata 2008; Hirata 2008; Ali-Haimoud & Hirata 2011), and CosmoRec 7 (Chluba et al. 2010; Chluba & Thomas 2011), demonstrating agreement at a level better than that required for Planck (di GLYPH<11> erences less that 4 GLYPH<2> 10 GLYPH<0> 4 in the predicted temperature power spectra on small scales). \nThese recombination codes are remarkably fast, given the complexity of the calculation. However, the recombination history can be computed even more rapidly by using the simple e GLYPH<11> ective three-level atom model developed by Seager et al. \n(2000) and implemented in the recfast code 8 , with appropriately chosen small correction functions calibrated to the full numerical results (Wong et al. 2008; Rubino-Martin et al. 2009; Shaw & Chluba 2011). We use recfast in our baseline parameter analysis, with correction functions adjusted so that the predicted power spectra C ' agree with those from the latest versions of HyRec (January 2012) and CosmoRec (v2) to better than 0 : 05% 9 . We have confirmed, using importance sampling, that cosmological parameter constraints using recfast are consistent with those using CosmoRec at the 0 : 05 GLYPH<27> level. Since the results of the Planck parameter analysis are crucially dependent on the accuracy of the recombination history, we have also checked, following Lewis et al. (2006), that there is no strong evidence for simple deviations from the assumed history. However, we note that any deviation from the assumed history could significantly shift parameters compared to the results presented here and we have not performed a detailed sensitivity analysis. \nThe background recombination model should accurately capture the ionization history until the Universe is reionized at late times via ultra-violet photons from stars and / or active galactic nuclei. We approximate reionization as being relatively sharp, with the mid-point parameterized by a redshift z re (where x e = f = 2) and width parameter GLYPH<1> z re = 0 : 5. Hydrogen reionization and the first reionization of helium are assumed to occur simultaneously, so that when reionization is complete x e = f GLYPH<17> 1 + f He GLYPH<25> 1 : 08 (Lewis 2008), where f He is the heliumto-hydrogen ratio by number. In this parameterization, the optical depth is almost independent of GLYPH<1> z re and the only impact of the specific functional form on cosmological parameters comes from very small changes to the shape of the polarization power spectrum on large angular scales. The second reionization of helium (i.e., He + ! He ++ ) produces very small changes to the power spectra ( GLYPH<1> GLYPH<28> GLYPH<24> 0 : 001, where GLYPH<28> is the optical depth to Thomson scattering) and does not need to be modelled in detail. We include the second reionization of helium at a fixed redshift of z = 3 : 5 (consistent with observations of LymanGLYPH<11> forest lines in quasar spectra, e.g., Becker et al. 2011), which is su GLYPH<14> ciently accurate for the parameter analyses described in this paper.", '2.1.3. Initial conditions': "In our baseline model we assume purely adiabatic scalar perturbations at very early times, with a (dimensionless) curvature power spectrum parameterized by \nP R ( k ) = A s k k 0 ! n s GLYPH<0> 1 + (1 = 2)( dn s = d ln k ) ln( k = k 0) ; (2) \nwith n s and dn s = d ln k taken to be constant. For most of this paper we shall assume no 'running', i.e., a power-law spectrum with dn s = d ln k = 0. The pivot scale, k 0, is chosen to be k 0 = 0 : 05 Mpc GLYPH<0> 1 , roughly in the middle of the logarithmic range of scales probed by Planck . With this choice, n s is not strongly degenerate with the amplitude parameter A s. \nThe amplitude of the small-scale linear CMB power spectrum is proportional to e GLYPH<0> 2 GLYPH<28> A s. Because Planck measures this amplitude very accurately there is a tight linear constraint between GLYPH<28> and ln A s (see Sect. 3.4). For this reason we usually use ln A s as a base parameter with a flat prior, which has a significantly more Gaussian posterior than A s. A linear parameter redefinition then also allows the degeneracy between GLYPH<28> and A s to be \nexplored e GLYPH<14> ciently. (The degeneracy between GLYPH<28> and A s is broken by the relative amplitudes of large-scale temperature and polarization CMB anisotropies and by the non-linear e GLYPH<11> ect of CMB lensing.) \nWe shall also consider extended models with a significant amplitude of primordial gravitational waves (tensor modes). Throughout this paper, the (dimensionless) tensor mode spectrum is parameterized as a power-law with 10 \nP t( k ) = A t k k 0 ! n t : (3) \nWe define r 0 : 05 GLYPH<17> A t = A s, the primordial tensor-to-scalar ratio at k = k 0. Our constraints are only weakly sensitive to the tensor spectral index, n t (which is assumed to be close to zero), and we adopt the theoretically motivated single-field inflation consistency relation n t = GLYPH<0> r 0 : 05 = 8, rather than varying n t independently. We put a flat prior on r 0 : 05, but also report the constraint at k = 0 : 002 Mpc GLYPH<0> 1 (denoted r 0 : 002), which is closer to the scale at which there is some sensitivity to tensor modes in the largeangle temperature power spectrum. Most previous CMB experiments have reported constraints on r 0 : 002. For further discussion of the tensor-to-scalar ratio and its implications for inflationary models see Planck Collaboration XXII (2014).", '2.1.4. Dark energy': 'In our baseline model we assume that the dark energy is a cosmological constant with current density parameter GLYPH<10>GLYPH<3> . When considering a dynamical dark energy component, we parameterize the equation of state either as a constant w or as a function of the cosmological scale factor, a , with \nw ( a ) GLYPH<17> p GLYPH<26> = w 0 + (1 GLYPH<0> a ) wa ; (4) \nand assume that the dark energy does not interact with other constituents other than through gravity. Since this model allows the equation of state to cross below GLYPH<0> 1, a single-fluid model cannot be used self-consistently. We therefore use the parameterized post-Friedmann (PPF) model of Fang et al. (2008a). For models with w > GLYPH<0> 1, the PPF model agrees with fluid models to significantly better accuracy than required for the results reported in this paper.', '2.1.5. Power spectra': "Over the last decades there has been significant progress in improving the accuracy, speed and generality of the numerical calculation of the CMB power spectra given an ionization history and set of cosmological parameters (see e.g., Bond & Efstathiou 1987; Sugiyama 1995; Ma & Bertschinger 1995; Hu et al. 1995; Seljak & Zaldarriaga 1996; Hu & White 1997; Zaldarriaga et al. 1998; Lewis et al. 2000; Lesgourgues & Tram 2011). Our baseline numerical Boltzmann code is camb 11 (March 2013; Lewis et al. 2000), a parallelized line-of-sight code developed from cmbfast (Seljak & Zaldarriaga 1996) and Cosmics (Bertschinger 1995; Ma & Bertschinger 1995), which calculates the lensed CMB temperature and polarization power spectra. The code has been publicly available for over a decade and \nhas been very well tested (and improved) by the community. Numerical stability and accuracy of the calculation at the sensitivity of Planck has been explored in detail (Hamann et al. 2009; Lesgourgues 2011b; Howlett et al. 2012), demonstrating that the raw numerical precision is su GLYPH<14> cient for numerical errors on parameter constraints from Planck to be less than 10% of the statistical error around the assumed cosmological model. (For the high multipole CMB data at ' > 2000 introduced in Sect. 4, the default camb settings are adequate because the power spectra of these experiments are dominated by unresolved foregrounds and have large errors at high multipoles.) To test the potential impact of camb errors, we importance-sample a subset of samples from the posterior parameter space using higher accuracy settings. This confirms that di GLYPH<11> erences purely due to numerical error in the theory prediction are less than 10% of the statistical error for all parameters, both with and without inclusion of CMBdata at high multipoles. We also performed additional tests of the robustness and accuracy of our results by reproducing a fraction of them with the independent Boltzmann code class (Lesgourgues 2011a; Blas et al. 2011). \nIn the parameter analysis, information from CMB lensing enters in two ways. Firstly, all the CMB power spectra are modelled using the lensed spectra, which includes the approximately 5% smoothing e GLYPH<11> ect on the acoustic peaks due to lensing. Secondly, for some results we include the Planck lensing likelihood, which encapsulates the lensing information in the (mostly squeezed-shape) CMB trispectrum via a lensing potential power spectrum (Planck Collaboration XVII 2014). The theoretical predictions for the lensing potential power spectrum are calculated by camb , optionally with corrections for the nonlinear matter power spectrum, along with the (non-linear) lensed CMB power spectra. For the Planck temperature power spectrum, corrections to the lensing e GLYPH<11> ect due to non-linear structure growth can be neglected, however the impact on the lensing potential reconstruction is important. We use the halofit model (Smith et al. 2003) as updated by Takahashi et al. (2012) to model the impact of non-linear growth on the theoretical prediction for the lensing potential power.", '2.2.1. Base parameters': "The first section of Table 1 lists our base parameters that have flat priors when they are varied, along with their default values in the baseline model. When parameters are varied, unless otherwise stated, prior ranges are chosen to be much larger than the posterior, and hence do not a GLYPH<11> ect the results of parameter estimation. In addition to these priors, we impose a 'hard' prior on the Hubble constant of [20 ; 100] km s GLYPH<0> 1 Mpc GLYPH<0> 1 .", '2.2.2. Derived parameters': 'Matter-radiation equality z eq is defined as the redshift at which GLYPH<26>GLYPH<13> + GLYPH<26>GLYPH<23> = GLYPH<26> c + GLYPH<26> b (where GLYPH<26>GLYPH<23> approximates massive neutrinos as massless). \nThe redshift of last-scattering, z GLYPH<3> , is defined so that the optical depth to Thomson scattering from z = 0 (conformal time GLYPH<17> = GLYPH<17> 0) to z = z GLYPH<3> is unity, assuming no reionization. The optical depth is given by \nGLYPH<28> ( GLYPH<17> ) GLYPH<17> Z GLYPH<17> GLYPH<17> 0 ˙ GLYPH<28> d GLYPH<17> 0 ; (5) \nwhere ˙ GLYPH<28> = GLYPH<0> an e GLYPH<27> T (and n e is the density of free electrons and GLYPH<27> T is the Thomson cross section). We define the angular scale of the sound horizon at last-scattering, GLYPH<18> GLYPH<3> = r s( z GLYPH<3> ) = D A( z GLYPH<3> ), where r s is the sound horizon \nr s( z ) = Z GLYPH<17> ( z ) 0 d GLYPH<17> 0 p 3(1 + R ) ; (6) \nwith R GLYPH<17> 3 GLYPH<26> b = (4 GLYPH<26>GLYPH<13> ). The parameter GLYPH<18> MC in Table 1 is an approximation to GLYPH<18> GLYPH<3> that is used in CosmoMC and is based on fitting formulae given in Hu & Sugiyama (1996). \nBaryon velocities decouple from the photon dipole when Compton drag balances the gravitational force, which happens at GLYPH<28> d GLYPH<24> 1, where (Hu & Sugiyama 1996) \nGLYPH<28> d( GLYPH<17> ) GLYPH<17> Z GLYPH<17> GLYPH<17> 0 ˙ GLYPH<28> d GLYPH<17> 0 = R : (7) \nHere, again, GLYPH<28> is from recombination only, without reionization contributions. We define a drag redshift z drag, so that GLYPH<28> d( GLYPH<17> ( z drag)) = 1. The sound horizon at the drag epoch is an important scale that is often used in studies of baryon acoustic oscillations; we denote this as r drag = r s( z drag). We compute z drag and r drag numerically from camb (see Sect. 5.2 for details of application to BAO data). \nThe characteristic wavenumber for damping, k D, is given by \nk GLYPH<0> 2 D ( GLYPH<17> ) = GLYPH<0> 1 6 Z GLYPH<17> 0 d GLYPH<17> 0 1 ˙ GLYPH<28> R 2 + 16(1 + R ) = 15 (1 + R ) 2 : (8) \nWe define the angular damping scale, GLYPH<18> D = GLYPH<25>= ( k D D A), where D A is the comoving angular diameter distance to z GLYPH<3> . \nFor our purposes, the normalization of the power spectrum is most conveniently given by A s. However, the alternative measure GLYPH<27> 8 is often used in the literature, particularly in studies of large-scale structure. By definition, GLYPH<27> 8 is the rms fluctuation in total matter (baryons + CDM + massive neutrinos) in 8 h GLYPH<0> 1 Mpc spheres at z = 0, computed in linear theory. It is related to the dimensionless matter power spectrum, P m, by \nGLYPH<27> 2 R = Z dk k P m( k ) " 3 j 1( kR ) kR # 2 ; (9) \nwhere R = 8 h GLYPH<0> 1 Mpc and j 1 is the spherical Bessel function of order 1. \nIn addition, we compute GLYPH<10> m h 3 (a well-determined combination orthogonal to the acoustic scale degeneracy in flat models; see e.g., Percival et al. 2002 and Howlett et al. 2012), 10 9 A s e GLYPH<0> 2 GLYPH<28> (which determines the small-scale linear CMB anisotropy power), r 0 : 002 (the ratio of the tensor to primordial curvature power at k = 0 : 002 Mpc GLYPH<0> 1 ), GLYPH<10> GLYPH<23> h 2 (the physical density in massive neutrinos), and the value of Y P from the BBN consistency condition.', '2.3. Likelihood': "Planck Collaboration XV (2014) describes the Planck temperature likelihood in detail. Briefly, at high multipoles ( ' GLYPH<21> 50) we use the 100, 143 and 217 GHz temperature maps (constructed using HEALPix G'orski et al. 2005) to form a high multipole likelihood following the CamSpec methodology described in Planck Collaboration XV (2014). Apodized Galactic masks, including an apodized point source mask, are applied to individual detector / detector-set maps at each frequency. The masks are carefully chosen to limit contamination from di GLYPH<11> use Galactic \nemission to low levels (less than 20 GLYPH<22> K 2 at all multipoles used in the likelihood) before correction for Galactic dust emission. 12 Thus we retain 57 : 8% of the sky at 100 GHz and 37 : 3% of the sky at 143 and 217 GHz. Mask-deconvolved and beam-corrected cross-spectra (following Hivon et al. 2002) are computed for all detector / detector-set combinations and compressed to form averaged 100 GLYPH<2> 100, 143 GLYPH<2> 143, 143 GLYPH<2> 217 and 217 GLYPH<2> 217 pseudospectra (note that we do not retain the 100 GLYPH<2> 143 and 100 GLYPH<2> 217 cross-spectra in the likelihood). Semi-analytic covariance matrices for these pseudo-spectra (Efstathiou 2004) are used to form a high-multipole likelihood in a fiducial Gaussian likelihood approximation (Bond et al. 2000; Hamimeche & Lewis 2008). \nAt low multipoles (2 GLYPH<20> ' GLYPH<20> 49) the temperature likelihood is based on a Blackwell-Rao estimator applied to Gibbs samples computed by the Commander algorithm (Eriksen et al. 2008) from Planck maps in the frequency range 30-353 GHz over 91% of the sky. The likelihood at low multipoles therefore accounts for errors in foreground cleaning. \nDetailed consistency tests of both the high- and lowmultipole components of the temperature likelihood are presented in Planck Collaboration XV (2014). The high-multipole Planck likelihood requires a number of additional parameters to describe unresolved foreground components and other 'nuisance' parameters (such as beam eigenmodes). The model adopted for Planck is described in Planck Collaboration XV (2014). A self-contained account is given in Sect. 4 which generalizes the model to allow matching of the Planck likelihood to the likelihoods from high-resolution CMB experiments. A complete list of the foreground and nuisance parameters is given in Table 4.", '2.4. Sampling and confidence intervals': "We sample from the space of possible cosmological parameters with Markov Chain Monte Carlo (MCMC) exploration using CosmoMC (Lewis & Bridle 2002). This uses a MetropolisHastings algorithm to generate chains of samples for a set of cosmological parameters, and also allows for importance sampling of results to explore the impact of small changes in the analysis. The set of parameters is internally orthogonalized to allow e GLYPH<14> cient exploration of parameter degeneracies, and the baseline cosmological parameters are chosen following Kosowsky et al. (2002), so that the linear orthogonalisation allows e GLYPH<14> cient exploration of the main geometric degeneracy (Bond et al. 1997). The code has been thoroughly tested by the community and has recently been extended to sample e GLYPH<14> ciently large numbers of 'fast' parameters by use of a speed-ordered Cholesky parameter rotation and a fast-parameter 'dragging' scheme described by Neal (2005) and Lewis (2013). \nFor our main cosmological parameter runs we execute eight chains until they are converged, and the tails of the distribution are well enough explored for the confidence intervals for each parameter to be evaluated consistently in the last half of each chain. We check that the spread in the means between \nchains is small compared to the standard deviation, using the standard Gelman and Rubin (Gelman & Rubin 1992) criterion R GLYPH<0> 1 < 0 : 01 in the least-converged orthogonalized parameter. This is su GLYPH<14> cient for reliable importance sampling in most cases. We perform separate runs when the posterior volumes differ enough that importance sampling is unreliable. Importancesampled and extended data-combination chains used for this paper satisfy R GLYPH<0> 1 < 0 : 1, and in almost all cases are closer to 0.01. We discard the first 30% of each chain as burn in, where the chains may be still converging and the sampling may be significantly non-Markovian. This is due to the way CosmoMC learns an accurate orthogonalisation and proposal distribution for the parameters from the sample covariance of previous samples. \nFrom the samples, we generate estimates of the posterior mean of each parameter of interest, along with a confidence interval. We generally quote 68% limits in the case of two-tail limits, so that 32% of samples are outside the limit range, and there are 16% of samples in each tail. For parameters where the tails are significantly di GLYPH<11> erent shapes, we instead quote the interval between extremal points with approximately equal marginalized probability density. For parameters with prior bounds we either quote one-tail limits or no constraint, depending on whether the posterior is significantly non-zero at the prior boundary. Our one-tail limits are always 95% limits. For parameters with nearly symmetric distribution we sometimes quote the mean and standard deviation ( GLYPH<6> 1 GLYPH<27> ). The samples can also be used to estimate one, two and three-dimensional marginalized parameter posteriors. We use variable-width Gaussian kernel density estimates in all cases. \nWe have also performed an alternative analysis to the one described above, using an independent statistical method based on frequentist profile likelihoods (Wilks 1938). This gives fits and error bars for the baseline cosmological parameters in excellent agreement for both Planck and Planck combined with high-resolution CMB experiments, consistent with the Gaussian form of the posteriors found from full parameter space sampling. \nIn addition to posterior means, we also quote maximumlikelihood parameter values. These are generated using the BOBYQA bounded minimization routine 13 . Precision is limited by stability of the convergence, and values quoted are typically reliable to within GLYPH<1> GLYPH<31> 2 GLYPH<24> 0 : 6, which is the same order as di GLYPH<11> erences arising from numerical errors in the theory calculation. For poorly constrained parameters the actual value of the bestfit parameters is not very numerically stable and should not be over-interpreted; in particular, highly degenerate parameters in extended models and the foreground model can give many apparently di GLYPH<11> erent solutions within this level of accuracy. The best-fit values should be interpreted as giving typical theory and foreground power spectra that fit the data well, but are generally non-unique at the numerical precision used; they are however generally significantly better fits than any of the samples in the parameter chains. Best-fit values are useful for assessing residuals, and di GLYPH<11> erences between the best-fit and posterior means also help to give an indication of the e GLYPH<11> ect of asymmetries, parameter-volume and prior-range e GLYPH<11> ects on the posterior samples. We have cross-checked a small subset of the best-fits with the widely used MINUIT software (James 2004), which can give somewhat more stable results. \nFig. 2. Comparison of the base GLYPH<3> CDM model parameters for Planck + lensing only (colour-coded samples), and the 68% and 95% constraint contours adding WMAP low-' polarization (WP; red contours), compared to WMAP -9 (Bennett et al. 2012; grey contours). \n<!-- image -->", '3. Constraints on the parameters of the base GLYPH<3> CDM model from Planck': "In this section we discuss parameter constraints from Planck alone in the GLYPH<3> CDM model. Planck provides a precision measurement of seven acoustic peaks in the CMB temperature power spectrum. The range of scales probed by Planck is su GLYPH<14> ciently large that many parameters can be determined accurately without using low-' polarization information to constrain the optical depth, or indeed without using any other astrophysical data. \nHowever, because the data are reaching the limit of astrophysical confusion, interpretation of the peaks at higher multipoles requires a reliable model for unresolved foregrounds. We model these here parametrically, as described in Planck Collaboration XV (2014), and marginalize over the parameters with wide priors. We give a detailed discussion of consistency of the foreground model in Sect. 4, making use of other high-' CMB observations, although as we shall see the parameters of the base GLYPH<3> CDM model have a weak sensitivity to foregrounds. \nAs foreground modelling is not especially critical for the base GLYPH<3> CDMmodel, we have decided to present the Planck constraints early in this paper, ahead of the detailed descriptions of the foreground model, supplementary high-resolution CMB data sets, and additional astrophysical data sets. The reader can therefore gain a feel for some of the key Planck results before being exposed to the lengthier discussions of Sects. 4 and 5, which are essential for the analysis of extensions to the base GLYPH<3> CDM cosmology presented in Sect. 6. \nIn addition to the temperature power spectrum measurement, the Planck lensing reconstruction (discussed in more detail in Sect. 5.1 and Planck Collaboration XVII 2014) provides a different probe of the perturbation amplitudes and geometry at late times. CMB lensing can break degeneracies inherent in the temperature data alone, especially the geometric degeneracy in nonflat models, providing a strong constraint on spatial curvature using only CMB data. The lensing reconstruction constrains the matter fluctuation amplitude, and hence the accurate measurement of the temperature anisotropy power can be used together with the lensing reconstruction to infer the relative suppression of the temperature anisotropies due to the finite optical depth to reionization. The large-scale polarization from nine years of WMAP observations (Bennett et al. 2012) gives a constraint on the optical depth consistent with the Planck temperature and lensing spectra. Nevertheless, the WMAP polarization constraint is somewhat tighter, so by including it we can further improve constraints on some parameters. \nWe therefore also consider the combination of the Planck temperature power spectrum with a WMAP polarization lowmultipole likelihood (Bennett et al. 2012) at ' GLYPH<20> 23 (denoted WP), as discussed in Planck Collaboration XV (2014) 14 . We refer to this CMB data combination as Planck + WP. \nTable 2 summarizes our constraints on cosmological parameters from the Planck temperature power spectrum alone (labelled ' Planck '), from Planck in combination with Planck lensing ( Planck + lensing) and with WMAP low-' polarization ( Planck + WP). Figure 2 shows a selection of corresponding constraints on pairs of parameters and fully marginalized one-parameter constraints compared to the final results from WMAP (Bennett et al. 2012).", '3.1. Acoustic scale': "The characteristic angular size of the fluctuations in the CMB is called the acoustic scale. It is determined by the comoving size of the sound horizon at the time of last-scattering, r s( z GLYPH<3> ), and the angular diameter distance at which we are observing the fluctuations, D A( z GLYPH<3> ). With accurate measurement of seven acoustic peaks, Planck determines the observed angular size GLYPH<18> GLYPH<3> = r s = D A to better than 0 : 1% precision at 1 GLYPH<27> : \nGLYPH<18> GLYPH<3> = (1 : 04148 GLYPH<6> 0 : 00066) GLYPH<2> 10 GLYPH<0> 2 = 0 : 596724 GLYPH<14> GLYPH<6> 0 : 00038 GLYPH<14> : (10) \nSince this parameter is constrained by the positions of the peaks but not their amplitudes, it is quite robust; the measurement is very stable to changes in data combinations and the assumed cosmology. Foregrounds, beam uncertainties, or any systematic e GLYPH<11> ects which only contribute a smooth component to the observed spectrum will not substantially a GLYPH<11> ect the frequency of the oscillations, and hence this determination is likely to be Planck 's most robust precision measurement. The situation is analogous to baryon acoustic oscillations measurements in large-scale structure surveys (see Sect. 5.2), but the CMB acoustic measurement has the advantage that it is based on observations of the Universe when the fluctuations were very accurately linear, so second and higher-order e GLYPH<11> ects are expected to be negligible 15 . \nThe tight constraint on GLYPH<18> GLYPH<3> also implies tight constraints on some combinations of the cosmological parameters that determine D A and r s. The sound horizon r s depends on the physical matter density parameters, and D A depends on the late-time evolution and geometry. Parameter combinations that fit the Planck data must be constrained to be close to a surface of constant GLYPH<18> GLYPH<3> . This surface depends on the model that is assumed. For the base GLYPH<3> CDMmodel, the main parameter dependence is approximately described by a 0 : 3% constraint in the three-dimensional GLYPH<10> mh -GLYPH<10> b h 2 subspace: \nGLYPH<10> m h 3 : 2 ( GLYPH<10> b h 2 ) GLYPH<0> 0 : 54 = 0 : 695 GLYPH<6> 0 : 002 (68%; Planck ) : (11) \nReducing further to a two-dimensional subspace gives a 0 : 6% constraint on the combination \nGLYPH<10> m h 3 = 0 : 0959 GLYPH<6> 0 : 0006 (68%; Planck ) : (12) \nThe principle component analysis direction is actually GLYPH<10> m h 2 : 93 but this is conveniently close to GLYPH<10> m h 3 and gives a similar constraint. The simple form is a coincidence of the GLYPH<3> CDM cosmology, error model, and particular parameter values of the model (Percival et al. 2002; Howlett et al. 2012). The degeneracy between H 0 and GLYPH<10> m is illustrated in Fig. 3: parameters are constrained to lie in a narrow strip where GLYPH<10> m h 3 is nearly constant, but the orthogonal direction is much more poorly constrained. The degeneracy direction involves consistent changes in the H 0, GLYPH<10> m, and GLYPH<10> b h 2 parameters, so that the ratio of the sound horizon and angular diameter distance remains nearly constant. Changes in the density parameters, however, also have other e GLYPH<11> ects on the power spectrum and the spectral index n s also \nTable 2. Cosmological parameter values for the six-parameter base GLYPH<3> CDM model. Columns 2 and 3 give results for the Planck temperature power spectrum data alone. Columns 4 and 5 combine the Planck temperature data with Planck lensing, and columns 6 and 7 include WMAP polarization at low multipoles. We give best fit parameters (i.e. the parameters that maximise the overall likelihood for each data combination) as well as 68% confidence limits for constrained parameters. The first six parameters have flat priors. The remainder are derived parameters as discussed in Sect. 2. Beam, calibration parameters, and foreground parameters (see Sect. 4) are not listed for brevity. Constraints on foreground parameters for Planck + WP are given later in Table 5. \nchanges to compensate. The degeneracy is not exact; its extent is much more sensitive to other details of the power spectrum shape. Additional data can help further to restrict the degeneracy. Figure 3 shows that adding WMAP polarization has almost no effect on the GLYPH<10> m h 3 measurement, but shrinks the orthogonal direction slightly from GLYPH<10> m h GLYPH<0> 3 = 1 : 03 GLYPH<6> 0 : 13 to GLYPH<10> m h GLYPH<0> 3 = 1 : 04 GLYPH<6> 0 : 11.", '3.2. Hubble parameter and dark energy density': 'The Hubble constant, H 0, and matter density parameter, GLYPH<10> m, are only tightly constrained in the combination GLYPH<10> m h 3 discussed above, but the extent of the degeneracy is limited by the e GLYPH<11> ect of GLYPH<10> m h 2 on the relative heights of the acoustic peaks. The projection of the constraint ellipse shown in Fig. 3 onto the axes therefore yields useful marginalized constraints on H 0 and GLYPH<10> m (or equivalently GLYPH<10>GLYPH<3> ) separately. We find the 2% constraint on H 0: \nH 0 = (67 : 4 GLYPH<6> 1 : 4) km s GLYPH<0> 1 Mpc GLYPH<0> 1 (68%; Planck ) : (13) \nThe corresponding constraint on the dark energy density parameter is \nGLYPH<10>GLYPH<3> = 0 : 686 GLYPH<6> 0 : 020 (68%; Planck ) ; (14) \nand for the physical matter density we find \nGLYPH<10> m h 2 = 0 : 1423 GLYPH<6> 0 : 0029 (68%; Planck ) : (15) \nNote that these indirect constraints are highly model dependent. The data only measure accurately the acoustic scale, and the relation to underlying expansion parameters (e.g., via the angular-diameter distance) depends on the assumed cosmology, including the shape of the primordial fluctuation spectrum. Even small changes in model assumptions can change H 0 noticeably; for example, if we neglect the 0 : 06 eV neutrino mass expected in the minimal hierarchy, and instead take P m GLYPH<23> = 0, the Hubble parameter constraint shifts to \nH 0 = (68 : 0 GLYPH<6> 1 : 4) km s GLYPH<0> 1 Mpc GLYPH<0> 1 (68%; Planck , P m GLYPH<23> = 0) : (16) \nFig. 3. Constraints in the GLYPH<10> mH 0 plane. Points show samples from the Planck -only posterior, coloured by the corresponding value of the spectral index n s. The contours (68% and 95%) show the improved constraint from Planck + lensing + WP. The degeneracy direction is significantly shortened by including WP, but the well-constrained direction of constant GLYPH<10> m h 3 (set by the acoustic scale), is determined almost equally accurately from Planck alone. \n<!-- image --> \nn', '3.3. Matter densities': 'Planck can measure the matter densities in baryons and dark matter from the relative heights of the acoustic peaks. However, as discussed above, there is a partial degeneracy with the spectral index and other parameters that limits the precision of the determination. With Planck there are now enough well measured peaks that the extent of the degeneracy is limited, giving GLYPH<10> b h 2 to an accuracy of 1 : 5% without any additional data: \nGLYPH<10> b h 2 = 0 : 02207 GLYPH<6> 0 : 00033 (68%; Planck ) : (17) \nAdding WMAP polarization information shrinks the errors by only 10%. \nThe dark matter density is slightly less accurately measured at around 3%: \nGLYPH<10> c h 2 = 0 : 1196 GLYPH<6> 0 : 0031 (68%; Planck ) : (18)', '3.4. Optical depth': "Small-scale fluctuations in the CMB are damped by Thomson scattering from free electrons produced at reionization. This scattering suppresses the amplitude of the acoustic peaks by e GLYPH<0> 2 GLYPH<28> on scales that correspond to perturbation modes with wavelength smaller than the Hubble radius at reionization. Planck measures the small-scale power spectrum with high precision, and hence accurately constrains the damped amplitude e GLYPH<0> 2 GLYPH<28> A s. With only unlensed temperature power spectrum data, there is a large degeneracy between GLYPH<28> and A s, which is weakly broken only by the power in large-scale modes that were still super-Hubble scale at reionization. However, lensing depends on the actual amplitude of the matter fluctuations along the line of sight. Planck accurately measures many acoustic peaks in the lensed temperature power spectrum, where the amount of lensing smoothing depends on the fluctuation amplitude. Furthermore Planck 's lensing potential reconstruction provides a more direct measurement \nof the amplitude, independently of the optical depth. The combination of the temperature data and Planck 's lensing reconstruction can therefore determine the optical depth GLYPH<28> relatively well. The combination gives \nGLYPH<28> = 0 : 089 GLYPH<6> 0 : 032 (68%; Planck + lensing) : (19) \nAs shown in Fig. 4 this provides marginal confirmation (just under 2 GLYPH<27> ) that the total optical depth is significantly higher than would be obtained from sudden reionization at z GLYPH<24> 6, and is consistent with the WMAP -9 constraint, GLYPH<28> = 0 : 089 GLYPH<6> 0 : 014, from large-scale polarization (Bennett et al. 2012). The large-scale E -mode polarization measurement is very challenging because it is a small signal relative to polarized Galactic emission on large scales, so this Planck polarization-free result is a valuable crosscheck. The posterior for the Planck temperature power spectrum measurement alone also consistently peaks at GLYPH<28> GLYPH<24> 0 : 1, where the constraint on the optical depth is coming from the amplitude of the lensing smoothing e GLYPH<11> ect and (to a lesser extent) the relative power between small and large scales. \nSince lensing constrains the underlying fluctuation amplitude, the matter density perturbation power is also well determined: \nGLYPH<27> 8 = 0 : 823 GLYPH<6> 0 : 018 (68%; Planck + lensing) : (20) \nMuch of the residual uncertainty is caused by the degeneracy with the optical depth. Since the small-scale temperature power spectrum more directly fixes GLYPH<27> 8 e GLYPH<0> GLYPH<28> , this combination is tightly constrained: \nGLYPH<27> 8 e GLYPH<0> GLYPH<28> = 0 : 753 GLYPH<6> 0 : 011 (68%; Planck + lensing) : (21) \nThe estimate of GLYPH<27> 8 is significantly improved to GLYPH<27> 8 = 0 : 829 GLYPH<6> 0 : 012 by using the WMAP polarization data to constrain the optical depth, and is not strongly degenerate with GLYPH<10> m. (We shall see in Sect. 5.5 that the Planck results are discrepant with recent estimates of combinations of GLYPH<27> 8 and GLYPH<10> m from cosmic shear measurements and counts of rich clusters of galaxies.)", '3.5. Spectral index': "The scalar spectral index defined in Eq. (2) is measured by Planck data alone to 1% accuracy: \nn s = 0 : 9616 GLYPH<6> 0 : 0094 (68%; Planck ) : (22) \nSince the optical depth GLYPH<28> a GLYPH<11> ects the relative power between large scales (that are una GLYPH<11> ected by scattering at reionization) and intermediate and small scales (that have their power suppressed by e GLYPH<0> 2 GLYPH<28> ), there is a partial degeneracy with n s. Breaking the degeneracy between GLYPH<28> and ns using WMAP polarization leads to a small improvement in the constraint: \nn s = 0 : 9603 GLYPH<6> 0 : 0073 (68%; Planck + WP) : (23) \nComparing Eqs. (22) and (23), it is evident that the Planck temperature spectrum spans a wide enough range of multipoles to give a highly significant detection of a deviation of the scalar spectral index from exact scale invariance (at least in the base GLYPH<3> CDM cosmology) independent of WMAP polarization information. \nOne might worry that the spectral index parameter is degenerate with foreground parameters, since these act to increase smoothly the amplitudes of the temperature power spectra at high multipoles. The spectral index is therefore liable to potential systematic errors if the foreground model is poorly constrained. Figure 4 shows the marginalized constraints on the \nFig. 4. Marginalized constraints on parameters of the base GLYPH<3> CDMmodel for various data combinations. \n<!-- image --> \nGLYPH<3> CDM parameters for various combinations of data, including adding high-resolution CMB measurements. As will be discussed in Sect. 4, the use of high-resolution CMB provides tighter constraints on the foreground parameters (particularly 'minor' foreground components) than from Planck data alone. However, the small shifts in the means and widths of the distributions shown in Fig. 4 indicate that, for the base GLYPH<3> CDM cosmology, the errors on the cosmological parameters are not limited by foreground uncertainties when considering Planck alone. The e GLYPH<11> ects of foreground modelling assumptions and likelihood choices on constraints on n s are discussed in Appendix C.", '4. Planck combined with high-resolution CMB experiments: the base GLYPH<3> CDM model': "The previous section adopted a foreground model with relatively loose priors on its parameters. As discussed there and in Planck Collaboration XV (2014), for the base GLYPH<3> CDM model, the cosmological parameters are relatively weakly correlated with the parameters of the foreground model and so we expect that the cosmological results reported in Sect. 3 are robust. Fortunately, we can get an additional handle on unresolved foregrounds, particularly 'minor' components such as the kinetic SZ e GLYPH<11> ect, by combining the Planck data with data from high-resolution CMB experiments. The consistency of results obtained with Planck data alone and Planck data combined with high-resolution CMB data gives added confidence to our cosmological results, particularly when we come to investigate extensions to the base GLYPH<3> CDM cosmology (Sect. 6). In this section, we review the high-resolution CMB data (hereafter, usually denoted highL) that we combine with Planck and then discuss how the foreground model is adapted (with additional 'nuisance' parameters) to handle multiple CMB data sets. We then discuss the results of an MCMC analysis of the base GLYPH<3> CDMmodel combining Planck data with the high-' data.", "4.1. Overview of the high-' CMB data sets": "The Atacama Cosmology Telescope (ACT) mapped the sky from 2007 to 2010 in two distinct regions, the equatorial stripe (ACTe) along the celestial equator, and the southern stripe (ACTs) along \ndeclination GLYPH<0> 55 GLYPH<14> , observing in total about 600 deg 2 . The ACT data sets at 148 and 218 GHz are presented in Das et al. (2013, hereafter D13) and cover the angular scales 540 < ' < 9440 at 148 GHz and 1540 < ' < 9440 at 218 GHz. Beam errors are included in the released covariance matrix. We include the ACT 148 GLYPH<2> 148 spectra for ' GLYPH<21> 1000, and the ACT 148 GLYPH<2> 218 and 218 GLYPH<2> 218 spectra for ' GLYPH<21> 1500. The inclusion of ACT spectra to ' = 1000 improves the accuracy of the inter-calibration parameters between the high-' experiments and Planck . \nThe South Pole Telescope observed a region of sky over the period 2007-10. Spectra are reported in Keisler et al. (2011, hereafter K11) and Story et al. (2012, hereafter S12) for angular scales 650 < ' < 3000 at 150 GHz, and in Reichardt et al. (2012, hereafter R12) for angular scales 2000 < ' < 10000 at 95, 150 and 220 GHz. Beam errors are included in the released covariance matrices used to form the SPT likelihood. The parameters of the base GLYPH<3> CDM cosmology derived from the WMAP -7 + S12 data and (to a lesser extent) from K11 are in tension with Planck . Since the S12 spectra have provided the strongest CMB constraints on cosmological parameters prior to Planck , this discrepancy merits a more detailed analysis, which is presented in appendix B. The S12 and K11 data are not used in combination with Planck in this paper. Since the primary purpose of including high-' CMB data is to provide stronger constraints on foregrounds, we use only the R12 SPT data at ' > 2000 in combination with Planck . We ignore any correlations between ACT / SPT and Planck spectra over the overlapping multipole ranges. \nTable 3 summarizes some key features of the CMB data sets used in this paper.", "4.2. Model of unresolved foregrounds and 'nuisance' parameters": "The model for unresolved foregrounds used in the Planck likelihood is described in detail in Planck Collaboration XV (2014). Briefly, the model includes power spectrum templates for clustered extragalactic point sources (the cosmic infra-red background, hereafter CIB), thermal (tSZ) and kinetic (kSZ) Sunyaev-Zeldovich contributions, and the cross-correlation (tSZ GLYPH<2> CIB) between infra-red galaxies and the thermal SunyaevZeldovich e GLYPH<11> ect. The model also includes amplitudes for the \nTable 3. Summary of the CMB temperature data sets used in this analysis. \nPoisson contributions from radio and infra-red galaxies. The templates are described in Planck Collaboration XV (2014) and are kept fixed here. (Appendix C discusses briefly a few tests showing the impact of varying some aspects of the foreground model.) The model for unresolved foregrounds is similar to the models developed by the ACT and the SPT teams (e.g., R12; Dunkley et al. 2013). The main di GLYPH<11> erence is in the treatment of the Poisson contribution from radio and infra-red galaxies. In the ACT and SPT analyses, spectral models are assumed for radio and infra-red galaxies. The Poisson point source contributions can then be described by an amplitude for each population, assuming either fixed spectral parameters or solving for them. In addition, one can add additional parameters to describe the decorrelation of the point source amplitudes with frequency (see e.g., Millea et al. 2012). The Planck model assumes free amplitudes for the point sources at each frequency, together with appropriate correlation coe GLYPH<14> cients between frequencies. The model is adapted to handle the ACT and SPT data as discussed later in this section. \nFigure 5 illustrates the importance of unresolved foregrounds in interpreting the power spectra of the three CMB data sets. The upper panel of Fig. 5 shows the Planck temperature spectra at 100, 143, and 217 GHz, without corrections for unresolved foregrounds (to avoid overcrowding, we have not plotted the 143 GLYPH<2> 217 spectrum). The solid (red) lines show the best-fit base GLYPH<3> CDM CMB spectrum corresponding to the combined Planck + ACT + SPT + WMAP polarization likelihood analysis, with parameters listed in Table 5. The middle panel shows the SPT spectra at 95, 150 and 220 GHz from S12 and R12. In this figure, we have recalibrated the R12 power spectra to match Planck using calibration parameters derived from a full likelihood analysis of the base GLYPH<3> CDM model. The S12 spectrum plotted is exactly as tabulated in S12, i.e., we have not recalibrated this spectrum to Planck . (The consistency of the S12 spectrum with the theoretical model is discussed in further detail in Appendix B.) The lower panel of Fig. 5 shows the ACT spectra from D13, recalibrated to Planck with calibration coe GLYPH<14> cients determined from a joint likelihood analysis. The power spectra plotted are an average of the ACTe and ACTs spectra, and include the small Galactic dust corrections described in D13. \nThe small-scale SPT (R12) and ACT (D13) data are dominated by the extragalactic foregrounds and hence are highly effective in constraining the multi-parameter foreground model. In contrast, Planck has limited angular resolution and therefore limited ability to constrain unresolved foregrounds. Planck is sensitive to the Poisson point source contribution at each fre- \nquency and to the CIB contribution at 217 GHz. Planck has some limited sensitivity to the tSZ amplitude from the 100 GHz channel (and almost no sensitivity at 143 GHz). The remaining foreground contributions are poorly constrained by Planck and highly degenerate with each other in a Planck -alone analysis. The main gain in combining Planck with the high-resolution ACT and SPT data is in breaking some of the degeneracies between foreground parameters which are poorly determined from Planck data alone. \nAn important extension of the foreground parameterization described here over that developed in Planck Collaboration XV (2014) concerns the use of e GLYPH<11> ective frequencies. Di GLYPH<11> erent experiments (and di GLYPH<11> erent detectors within a frequency band) have non-identical bandpasses (Planck Collaboration IX 2014) and this needs to be taken into account in the foreground modelling. Consider, for example, the amplitude of the CIB template at 217 GHz, A CIB 217 , introduced in Planck Collaboration XV (2014). The e GLYPH<11> ective frequency for a dust-like component for the averaged 217 GHz spectrum used in the Planck likelihood is 225 : 7 GHz. To avoid cumbersome notation, we solve for the CIB amplitude A CIB 217 at the CMB e GLYPH<11> ective frequency of 217 GHz. The actual amplitude measured in the Planck 217 GHz band is 1 : 33 A CIB 217 , reflecting the di GLYPH<11> erent e GLYPH<11> ective frequencies of a dustlike component compared to the blackbody primordial CMB (see Eq. 30 below). With appropriate e GLYPH<11> ective frequencies, the single amplitude A CIB 217 can be used to parameterize the CIB contributions to the ACT and SPT power spectra in their respective 218 and 220 GHz bands. A similar methodology is applied to match the tSZ amplitudes for each experiment. \nThe relevant e GLYPH<11> ective frequencies for the foreground parameterization discussed below are listed in Table 3. For the high resolution experiments, these are as quoted in R12 and Dunkley et al. (2013). For Planck these e GLYPH<11> ective frequencies were computed from the individual HFI bandpass measurements (Planck Collaboration IX 2014), and vary by a few percent from detector to detector. The numbers quoted in Table 3 are based on an approximate average of the individual detector bandpasses using the weighting scheme for individual detectors / detector-sets applied in the CamSpec likelihood. (The resulting bandpass correction factors for the tSZ and CIB amplitudes should be accurate to better than 5%.) Note that all temperatures in this section are in thermodynamic units. \nThe ingredients of the foreground model and associated 'nuisance' parameters are summarized in the following paragraphs. \n<!-- image --> \n<!-- image --> \nFig. 5. Top : Planck spectra at 100, 143 and 217 GHz without subtraction of foregrounds. Middle : SPT spectra from R12 at 95, 150 and 220 GHz, recalibrated to Planck using the bestfit calibration, as discussed in the text. The S12 SPT spectrum at 150 GHz is also shown, but without any calibration correction. This spectrum is discussed in detail in Appendix B, but is not used elsewhere in this paper. Bottom : ACT spectra (weighted averages of the equatorial and southern fields) from D13 at 148 and 220 GHz, and the 148 GLYPH<2> 220 GHz cross-spectrum, with no extragalactic foreground corrections, recalibrated to the Planck spectra as discussed in the text. The solid line in each panel shows the best-fit base GLYPH<3> CDM model from the combined Planck + WP + highL fits listed in Table 5. \n<!-- image --> \nCalibration factors: To combine the Planck , ACT and SPT likelihoods it is important to incorporate relative calibration factors, since the absolute calibrations of ACT and SPT have large errors (e.g., around 3 : 5% in power for the SPT 150 GHz channel). We introduce three map calibration parameters y SPT 95 , y SPT 150 and y SPT 220 to rescale the R12 SPT spectra. These factors rescale the crossspectra at frequencies GLYPH<23> i and GLYPH<23> j as \nC GLYPH<23> i GLYPH<2> GLYPH<23> j ' ! y SPT GLYPH<23> i y SPT GLYPH<23> j C GLYPH<23> i GLYPH<2> GLYPH<23> j ' : (24) \nIn the analysis of ACT, we solve for di GLYPH<11> erent map calibration factors for the ACTe and ACTs spectra, y ACTe 148 , y ACTs 148 , y ACTe 218 , and y ACTs 218 . In addition, we solve for the 100 GLYPH<2> 100 and 217 GLYPH<2> 217 Planck power-spectrum calibration factors c 100 and c 217, with priors as described in Planck Collaboration XV (2014); see also Table 4. (The use of map calibration factors for ACT and SPT follows the conventions adopted by the ACT and SPT teams, while for the Planck power spectrum analysis we have consistently used power-spectrum calibration factors.) \nIn a joint parameter analysis of Planck + ACT + SPT, the inclusion of these calibration parameters leads to recalibrations that match the ACT, SPT and Planck 100 GHz and 217 GHz channels to the calibration of the Planck 143 GLYPH<2> 143 spectrum (which, in turn, is linked to the calibration of the HFI 143-5 detector, as described in Planck Collaboration XV 2014). It is worth mentioning here that the Planck 143 GLYPH<2> 143 GHz spectrum is 2.5% lower than the WMAP -9 combined V + W power spectrum (Hinshaw et al. 2012). This calibration o GLYPH<11> set between Planck HFI channels and WMAP is discussed in more detail in Planck Collaboration XXXI (2014) and in Appendix A. \nPoisson point source amplitudes: To avoid any possible biases in modelling a mixed population of sources (synchrotron + dusty galaxies) with di GLYPH<11> ering spectra, we solve for each of the Poisson point source amplitudes as free parameters. Thus, for Planck we solve for A PS 100 , A PS 143 , and A PS 217 , giving the amplitude of the Poisson point source contributions to D 3000 for the 100 GLYPH<2> 100, 143 GLYPH<2> 143, and 217 GLYPH<2> 217 spectra. The units of A PS GLYPH<23> are therefore GLYPH<22> K 2 . The Poisson point source contribution to the 143 GLYPH<2> 217 spectrum is expressed as a correlation coe GLYPH<14> cient, r PS 143 GLYPH<2> 217 : \nD 143 GLYPH<2> 217 3000 = r PS 143 GLYPH<2> 217 q A PS 143 A PS 217 : (25) \nNote that we do not use the Planck 100 GLYPH<2> 143 and 100 GLYPH<2> 217 spectra in the likelihood, and so we do not include correlation coe GLYPH<14> cients r PS 100 GLYPH<2> 143 or r PS 100 GLYPH<2> 217 . (These spectra carry little additional information on the primordial CMB, but would require additional foreground parameters had we included them in the likelihood.) \nIn an analogous way, the point source amplitudes for ACT and SPT are characterized by the amplitudes A PS ; ACT 148 , A PS ; ACT 217 , A PS ; SPT 95 , A PS ; SPT 150 , and A PS ; SPT 220 (all in units of GLYPH<22> K 2 ) and three correlation coe GLYPH<14> cients r PS 95 GLYPH<2> 150 , r PS 95 GLYPH<2> 220 , and r PS 150 GLYPH<2> 220 . The last of these correlation coe GLYPH<14> cients is common to ACT and SPT. \nKinetic SZ: The kSZ template used here is from Trac et al. (2011). We solve for the amplitude A kSZ (in units of GLYPH<22> K 2 ): \nD kSZ ' = A kSZ D kSZ template ' D kSZ template 3000 : (26) \nThermal SZ: We use the GLYPH<15> = 0 : 5 tSZ template from Efstathiou & Migliaccio (2012) normalized to a frequency of 143 GHz. \nFor cross-spectra between frequencies GLYPH<23> i and GLYPH<23> j , the tSZ template is normalized as \nD tSZ GLYPH<23> i GLYPH<2> GLYPH<23> j ' = A tSZ 143 f ( GLYPH<23> i ) f ( GLYPH<23> j ) f 2 ( GLYPH<23> 0) D tSZ template ' D tSZ template 3000 ; (27) \nwhere GLYPH<23> 0 is the reference frequency of 143 GHz, D tSZ template ' is the template spectrum at 143 GHz, and \nf ( GLYPH<23> ) = x e x + 1 e x GLYPH<0> 1 GLYPH<0> 4 ! ; with x = h GLYPH<23> k B T CMB : (28) \nThe tSZ contribution is therefore characterized by the amplitude A tSZ 143 in units of GLYPH<22> K 2 . \nWeneglect the tSZ contribution for any spectra involving the Planck 217 GHz, ACT 218 GHz, and SPT 220 GHz channels, since the tSZ e GLYPH<11> ect has a null point at GLYPH<23> = 217 GHz. (For Planck the bandpasses of the 217 GHz detectors see less than 0 : 1% of the 143 GHz tSZ power.) \nCosmic infrared background: The CIB contributions are neglected in the Planck 100 GHz and SPT 95 GHz bands and in any cross-spectra involving these frequencies. The CIB power spectra at higher frequencies are characterized by three amplitude parameters and a spectral index, \nD CIB143 GLYPH<2> 143 ' = A CIB 143 ' 3000 ! GLYPH<13> CIB ; (29a) \nD CIB217 GLYPH<2> 217 ' = A CIB 217 ' 3000 ! GLYPH<13> CIB ; (29b) \nD CIB143 GLYPH<2> 217 ' = r CIB 143 GLYPH<2> 217 q A CIB 143 A CIB 217 ' 3000 ! GLYPH<13> CIB ; (29c) \nwhere A CIB 143 and A CIB 143 are expressed in GLYPH<22> K 2 . As explained above, we define these amplitudes at the Planck CMB frequencies of 143 and 217 GHz and compute scalings to adjust these amplitudes to the e GLYPH<11> ective frequencies for a dust-like spectrum for each experiment. The scalings are \nD CIB GLYPH<23> i GLYPH<2> GLYPH<23> j ' = D CIB GLYPH<23> i 0 GLYPH<2> GLYPH<23> j 0 3000 g ( GLYPH<23> i ) g ( GLYPH<23> j ) g ( GLYPH<23> i 0) g ( GLYPH<23> j 0) ! GLYPH<23> i GLYPH<23> j GLYPH<23> i 0 GLYPH<23> j 0 ! GLYPH<12> d B GLYPH<23> i ( T d) B GLYPH<23> i 0 ( T d) B GLYPH<23> j ( T d) B GLYPH<23> j 0 ( T d) ; (30) \nwhere B GLYPH<23> ( T d) is the Planck function at a frequency GLYPH<23> , \ng ( GLYPH<23> ) = [ @ B GLYPH<23> ( T ) =@ T ] GLYPH<0> 1 j T CMB (31) \nconverts antenna temperature to thermodynamic temperature, GLYPH<23> i and GLYPH<23> j refer to the Planck / ACT / SPT dust e GLYPH<11> ective frequencies, and GLYPH<23> i 0 and GLYPH<23> j 0 refer to the corresponding reference CMB Planck frequencies. In the analysis presented here, the parameters of the CIB spectrum are fixed to GLYPH<12> d = 2 : 20 and T d = 9 : 7 K, as discussed in Addison et al. (2012a). The model of Eq. (30) then relates the Planck reference amplitudes of Eqs. (29b-29c) to the neighbouring Planck , ACT, and SPT e GLYPH<11> ective frequencies, assuming that the CIB is perfectly correlated over these small frequency ranges. \nIt has been common practice in recent CMB parameter studies to fix the slope of the CIB spectrum to GLYPH<13> CIB = 0 : 8 \n(e.g., Story et al. 2012; Dunkley et al. 2013). In fact, the shape of the CIB spectrum is poorly constrained at frequencies below 353 GHz and we have decided to reflect this uncertainty by allowing the slope GLYPH<13> CIB to vary. We adopt a Gaussian prior on GLYPH<13> CIB with a mean of 0 : 7 and a dispersion of 0 : 2. In reality, the CIB spectrum is likely to have some degree of curvature reflecting the transition between linear (two-halo) and non-linear (one-halo) clustering (see e.g., Cooray & Sheth 2002; Planck Collaboration XVIII 2011; Amblard et al. 2011; Thacker et al. 2012). However, a single power law is an adequate approximation within the restricted multipole range (500 < GLYPH<24> ' < GLYPH<24> 3000) over which the CIB contributes significantly to the Planck / ACT / SPT high-frequency spectra (as judged by the foreground-corrected power spectrum residuals shown in Figs. 7, 8 and 9 below). The prior on GLYPH<13> CIB is motivated, in part, by the map-based Planck CIB analysis discussed in Planck Collaboration XXX (2014) (see also Planck Collaboration XVIII 2014). Appendix C explores di GLYPH<11> erent parameterizations of the CIB power spectrum. \nThermal-SZ/CIB cross-correlation: The cross-correlation between dust emission from CIB galaxies and SZ emission from clusters (tSZ GLYPH<2> CIB) is expected to be non-zero. Because of uncertainties in the modelling of the CIB, it is di GLYPH<14> cult to compute this correlation with a high degree of precision. Addison et al. (2012b) present a halo-model approach to model this term and conclude that anti-correlations of around 10-20% are plausible between the clustered CIB components and the SZ at 150 GHz. The tSZ GLYPH<2> CIB correlation is therefore expected to make a minor contribution to the unresolved foreground emission, but it is nevertheless worth including to determine how it might interact with other sub-dominant components, in particular the kSZ contribution. We use the Addison et al. (2012b) template spectrum in this paper and model the frequency dependence of the power spectrum as follows: \nD tSZ GLYPH<2> CIB GLYPH<23> i GLYPH<2> GLYPH<23> j ' = GLYPH<0> GLYPH<24> tSZ GLYPH<2> CIB D tSZ GLYPH<2> CIB template ' GLYPH<2> q D CIB GLYPH<23> i GLYPH<2> GLYPH<23> i 3000 D tSZ GLYPH<23> j GLYPH<2> GLYPH<23> j 3000 + q D CIB GLYPH<23> j GLYPH<2> GLYPH<23> j 3000 D tSZ GLYPH<23> i GLYPH<2> GLYPH<23> i 3000 ! ; (32) \nwhere D tSZ GLYPH<2> CIB template ' is the Addison et al. (2012b) template spectrum normalized to unity at ' = 3000 and D CIB GLYPH<23> i GLYPH<2> GLYPH<23> i ' and D tSZ GLYPH<23> i GLYPH<2> GLYPH<23> i ' are given by Eqs. (27) and (31). The tSZ GLYPH<2> CIB contribution is therefore characterized by the dimensionless crosscorrelation coe GLYPH<14> cient GLYPH<24> tSZ GLYPH<2> CIB . With the definition of Eq. (32), a positive value of GLYPH<24> tSZ GLYPH<2> CIB corresponds to an anti-correlation between the CIB and the tSZ signals. \nGalactic dust: For the masks used in the Planck CamSpec likelihood, Galactic dust makes a small contribution to D 3000 of around 5 GLYPH<22> K 2 to the 217 GLYPH<2> 217 power spectrum, 1 : 5 GLYPH<22> K 2 to the 143 GLYPH<2> 217 spectrum, and around 0 : 5 GLYPH<22> K 2 to the 143 GLYPH<2> 143 spectrum. We subtract the Galactic dust contributions from these power spectra using a 'universal' dust template spectrum (at high multipoles this is accurately represented by a power law D dust ' / ' GLYPH<0> 0 : 6 ). The template spectrum is based on an analysis of the 857 GHz Planck maps described in Planck Collaboration XV (2014), which uses mask-di GLYPH<11> erenced power spectra to separate Galactic dust from an isotropic extragalactic CIB contribution. This Galactic dust correction is kept fixed with an amplitude determined by template fitting the 217 and 143 GHz Planck maps to the 857 GHz map, as described in \nTable 4. Astrophysical parameters used to model foregrounds in our analysis, plus instrumental calibration and beam parameters. We include the symbol for each parameter, the prior range adopted for the MCMC analysis and a summary definition (see text for details). Square brackets denote hard priors, parentheses indicate Gaussian priors. Note that the beam eigenmode amplitudes require a correlation matrix to describe fully their joint prior, and that all but GLYPH<12> 1 1 are internally marginalized over rather than sampled over for the main MCMC runs. The bottom two blocks are only used in the analysis including the ACT and SPT high-' CMB data. \nPlanck Collaboration XV (2014). Galactic dust contamination is ignored in the 100 GLYPH<2> 100 spectrum. 16 The Galactic dust template spectrum is actually a good fit to the dust contamination at low multipoles, ' GLYPH<28> 1000; however, we limit the e GLYPH<11> ects of any inaccuracies in dust subtraction at low multipoles by truncating the 217 GLYPH<2> 217 and 143 GLYPH<2> 217 spectra at a minimum multipole of ' min = 500. (At multipoles ' < GLYPH<24> 1000, the Planck temperature power spectra are signal dominated, so the 100 GLYPH<2> 100 and 143 GLYPH<2> 143 spectra contain essentially all of the information on cosmology.) \nCompared to the contribution of Poisson point sources and the CIB, Galactic dust is a minor foreground component at 217 GHz within our default mask, which retains 37% of the sky. However, the contribution of Galactic dust emission rises rapidly as more sky area is used. Extending the sky mask to 65% of the sky (Planck Collaboration XV 2014, using the sequence of masks described in [), Galactic dust contributes to D 3000 around 50 GLYPH<22> K 2 at 217 GHz (rising to around 200 GLYPH<22> K 2 on the scale of the first acoustic peak) and becomes a major foreground component, with an amplitude close to the net contribution of Poisson point sources and the clustered CIB. There is therefore a trade-o GLYPH<11> between limiting the signal-to-noise at 143 and 217 GHz, by restricting the sky area, and potential systematic errors associated with modelling Galactic dust over a large area of sky (i.e., sensitivity to the assumption of a 'universal' dust template spectrum). \nFig. 6. Comparison of the posterior distributions of the foreground parameters for Planck + WP (red) and Planck + WP + highL (black). \n<!-- image --> \nWe have chosen to be conservative in this first cosmological analysis of Planck by limiting the sky area at 143 and 217 GHz so that dust contamination is a minor foreground at high multipoles. As a further test of the importance of Galactic dust, we have analysed a Planck likelihood that retains only 24 : 7% of the sky (see Planck Collaboration XV 2014) at 217 GHz. Within this mask the CIB dominates over Galactic dust at multipoles ' > GLYPH<24> 500. There is a signal-to-noise penalty in using such a small area of sky at 217 GHz, but otherwise the results from this likelihood are in good agreement with the results presented here. With the conservative choices adopted in this paper, Galactic dust has no significant impact on our cosmological results. \nWe follow R12 and subtract a small-scale dust contribution of D dust ' = 2 : 19 GLYPH<22> K 2 ( '= 3000) GLYPH<0> 1 : 2 from the R12 220 GHz spectrum. This correction was determined by cross-correlating the SPT data with model 8 of Finkbeiner et al. (1999). For the ACT data we marginalize over a residual Galactic dust component D dust ' = A ACTe = s dust ( '= 3000) GLYPH<0> 0 : 7 , with di GLYPH<11> erent amplitudes for the southern and equatorial spectra, imposing Gaussian priors and frequency scaling as described in Dunkley et al. (2013). \nNotice that the spectral index of the SPT dust correction is significantly steeper than the dust correction applied to the Planck spectra. In future analyses it would be useful to derive more accurate dust corrections for the high-resolution CMB data by cross-correlating the SPT and ACT maps with the Planck 545 and 857 GHz maps. Since the dust corrections are relatively small for the high-resolution data used here, we adopt the correction described above in this paper. \nIn application of the likelihood to Planck data alone, the model for unresolved foregrounds and relative calibrations contains 13 parameters. In addition, we can solve for up to 20 beam eigenmode amplitudes (five amplitudes for each of the four spectra used in the Planck likelihood; see Planck Collaboration XV 2014). In practice, we find that (usually) only the first beam eigenmode for the 100 GLYPH<2> 100 spectrum, GLYPH<12> 1 1 , has a posterior distribution that di GLYPH<11> ers perceptibly from the prior, and we obtain nearly identical results on both foreground and cosmological parameters if we treat only the amplitude of this eigenmode as a parameter and analytically marginalize over the rest. This is the default adopted in this paper. (The analytic marginalization improves stability of the minimisation for best-fit searches, and makes the Planck likelihood less cumbersome for the user.) \nThe addition of ACT and SPT data introduces 17 extra parameters. We provide a summary of the 50 foreground and nuisance parameters in Table 4, including the prior ranges adopted in our MCMC analysis 17 . The choice of priors for many of these parameters is, to a large extent, subjective. They were chosen at an early stage in the Planck analysis to reflect 'theoretically plausible' allowed ranges of the foreground parameters and to be broad compared to the results from high-resolution CMB experiments (which evolved over the course of this analysis as results from more ACT and SPT data were published). The foreground parameters from ACT and SPT depend on the assumptions of the underlying cosmology, and hence it is possible to introduce biases in the solutions for extensions to the base GLYPH<3> CDMcosmology if overly restrictive foreground priors are imposed on the Planck data. Using the priors summarized in Table 4, the consistency between the Planck -alone results and the solutions for Planck combined with ACT and SPT provides a crude (but informative) measure of the sensitivity of cosmological results on the foreground model. Appendix C discusses the e GLYPH<11> ects on extended GLYPH<3> CDMmodels of varying the priors on minor foreground components.", '4.3. The base GLYPH<3> CDM model': "Cosmological and foreground parameters for the base sixparameter GLYPH<3> CDM model are listed in Table 5, which gives bestfit values and 68% confidence limits. The first two columns list the parameters derived from the Planck + WP analysis discussed in Sect. 3, and are repeated here for easy reference. The next two columns list the results of combining the Planck + WP likelihoods with the ACT and SPT likelihoods following the model described above. We refer to this combination as ' Planck + WP + highL' in this paper. The remaining columns list the parameter constraints combining the Planck + WP + highL likelihood with the Planck lensing and BAO likelihoods (see Sect. 5). Table 5 lists the cosmological parameters for the base GLYPH<3> CDM model and a selection of derived cosmological parameters. These parameters are remarkably stable for such data combinations. We also list the values of the parameters describing the Planck foregrounds. A full list of all parameter values, including nuisance parameters, is given in the Explanatory Supplement (Planck Collaboration 2013b). \nA comparison of the foreground parameter constraints from Planck + WP and Planck + WP + highL is shown in Fig. 6; the \nTable 5. Best-fit values and 68% confidence limits for the base GLYPH<3> CDM model. Beam and calibration parameters, and additional nuisance parameters for 'highL' data sets are not listed for brevity but may be found in the Explanatory Supplement (Planck Collaboration 2013b). \ncorresponding cosmological parameter constraints are shown in Fig. 4. \nWe can draw the following general conclusions. \n- -The cosmological parameters for the base GLYPH<3> CDMmodel are extremely insensitive to the foreground model described in the previous subsection. The addition of the ACT and SPT data causes the posterior distributions of cosmological parameters to shift by much less than one standard deviation.\n- -With Planck data alone, the CIB amplitude at 217 GHz is strongly degenerate with the 217 GHz Poisson point source amplitude. This degeneracy is broken by the addition of the high-resolution CMB data. This degeneracy must be borne in mind when interpreting Planck -only solutions for CIB parameters; the sum of the Poisson point source and CIB contributions are well constrained by Planck at 217 GHz (and in good agreement with the map-based CIB Planck analysis reported in Planck Collaboration 2013a), whereas the individual contributions are not. Another feature of the CIB parameters is that we typically find smaller values of the CIB spectral index, GLYPH<13> CIB , in Planck -alone solutions compared to Planck + highL solutions (which can be seen in Fig. 6). This provided additional motivation to treat GLYPH<13> CIB as a parameter in the Planck likelihood rather than fixing it to a particular value. There is evidence from the Planck spectra (most clearly seen by di GLYPH<11> erencing the 217 GLYPH<2> 217 and 143 GLYPH<2> 143 spectra) that the CIB spectrum at 217 GHz flattens in slope over the multipole range 500 < GLYPH<24> ' < GLYPH<24> 1000. This will be ex- \nplored in further detail in future papers (see also Appendix C). \n- -The addition of the ACT and SPT data constrains the thermal SZ amplitude, which is poorly determined by Planck alone. In the Planck -alone analysis, the tSZ amplitude is strongly degenerate with the Poisson point source amplitude at 100 GHz. This degeneracy is broken when the highresolution CMB data are added to Planck . \nThe last two points are demonstrated clearly in Fig. 7, which shows the residuals of the Planck spectra with respect to the best-fit cosmology for the Planck + WPanalysis compared to the Planck + WP + highL fits. The addition of high-resolution CMB data also strongly constrains the net contribution from the kSZ and tSZ GLYPH<2> CIB components (dotted lines), though these components are degenerate with each other (and tend to cancel). \nAlthough the foreground parameters for the Planck + WP fits can di GLYPH<11> er substantially from those for Planck + WP + highL, the total foreground spectra are insensitive to the addition of the high-resolution CMB data. For example, for the 217 GLYPH<2> 217 spectrum, the di GLYPH<11> erences in the total foreground solution are less than 10 GLYPH<22> K 2 at ' = 2500. The net residuals after subtracting both the foregrounds and CMB spectrum (shown in the lower panels of each sub-plot in Fig. 7) are similarly insensitive to the addition of the high-resolution CMB data. The foreground model is su GLYPH<14> ciently complex that it has a high 'absorptive capacity' to any smoothly-varying frequency-dependent di GLYPH<11> erences between spectra (including beam errors). \nFig. 7. Power spectrum residual plots illustrating the accuracy of the foreground modelling. For each cross-spectrum, there are two sub-figures. The upper sub-figures show the residuals with respect to the Planck + WP best-fit solution (from Table 5). The lowers sub-figure show the residuals with respect to the Planck + WP + highL solution The upper panel in each sub-figure shows the residual between the measured power spectrum and the best-fit (lensed) CMB power spectrum. The lower panels show the residuals after further removing the best-fit foreground model. The lines in the upper panels show the various foreground components. Major foreground components are shown by the solid lines, colour coded as follows: total foreground spectrum (red); Poisson point sources (orange); clustered CIB (blue); thermal SZ (green); and Galactic dust (purple). Minor foreground components are shown by the dotted lines colour coded as follows: kinetic SZ (green); tSZ GLYPH<2> CIB cross-correlation (purple). We also show residuals for the two spectra 100 GLYPH<2> 143 and 100 GLYPH<2> 217 that are not used in the Planck likelihood. For these, we have assumed Poisson point-source correlation coe GLYPH<14> cients of unity. The GLYPH<31> 2 values of the residuals, and the number of bandpowers, are listed in the lower panels. \n<!-- image --> \nFig. 8. SPT power spectra at high multipoles using the foreground model developed in this paper. The SPT R12 power spectra for each frequency combination are shown by the blue points, together with 1 GLYPH<27> error bars. The foreground components, determined from the Planck + WP + highL analysis of GLYPH<3> CDM models, are shown in the upper panels using the same colour coding as in Fig. 7. Here, the spectrum of the best-fit CMB is shown in red and the total spectra are the upper green curves. The lower panel in each sub-figure shows the residuals with respect to the best-fit base GLYPH<3> CDMcosmology + foreground model. The GLYPH<31> 2 values of the residuals, and the number of SPT bandpowers, are listed in the lower panels. \n<!-- image --> \nFig. 9. As Fig. 8, but for the ACT south and ACT equatorial power spectra. \n<!-- image --> \nTable 6. Goodness-of-fit tests for the Planck spectra. The quantity GLYPH<1> GLYPH<31> 2 = GLYPH<31> 2 GLYPH<0> N ' is the di GLYPH<11> erence in GLYPH<31> 2 from the expected value if the model is correct. The sixth column expresses GLYPH<1> GLYPH<31> 2 in units of the expected dispersion, p 2 N ' , and the last column lists the probability to exceed (PTE) the tabulated value of GLYPH<31> 2 . \nTo quantify the consistency of the model fits shown in Fig. 7 for Planck we compute the GLYPH<31> 2 statistic \nGLYPH<31> 2 = X '' 0 ( C data ' GLYPH<0> C CMB ' GLYPH<0> C fg ' ) M GLYPH<0> 1 '' 0 ( C data ' 0 GLYPH<0> C CMB ' 0 GLYPH<0> C fg ' 0 ) ; (33) \nfor each of the spectra, where the sums extend over the multipole ranges ' min and ' max used in the likelihood, M '' 0 is the covariance matrix for the spectrum C data ' (including corrections for beam eigenmodes and calibrations), C CMB ' is the best-fit primordial CMB spectrum and C fg ' is the best-fit foreground model appropriate to the data spectrum. We expect GLYPH<31> 2 to be approximately Gaussian distributed with a mean of N ' = ' max GLYPH<0> ' min + 1 and dispersion p 2 N ' . Results are summarized in Table 6 for the Planck + WP + highL best-fit parameters of Table 5. (The GLYPH<31> 2 values for the Planck + WPfit are almost identical.) Each of the spectra gives an acceptable global fit to the model, quantifying the high degree of consistency of these spectra described in Planck Collaboration XV (2014). (Note that Planck Collaboration XV 2014 presents an alternative way of investigating consistency between these spectra via power spectrum di GLYPH<11> erences.) \nFigures 8 and 9 show the fits and residuals with respect to the best-fit Planck + WP + highL model of Table 5, for each of the SPT and ACT spectra. The SPT and ACT spectra are reported as band-powers, with associated window functions [ W SPT b ( ' ) =' ] and W ACT b ( ' ). The definitions of these window functions di GLYPH<11> er between the two experiments. \nFor SPT, the contribution of the CMB and foreground spectra in each band is \nD b = X [ W SPT b ( ' ) =' ] ' ( ' + 1 = 2) 2 GLYPH<25> GLYPH<16> C CMB ' + C fg ' GLYPH<17> : (34) \n' \n(Note that this di GLYPH<11> ers from the equations given in R12 and S12.) For ACT, the window functions operate on the power spectra: \nCb = X ' W ACT b ( ' ) GLYPH<16> C CMB ' + C fg ' GLYPH<17> : (35) \nIn Fig. 9. we plot D b = ' b ( ' b + 1) Cb = (2 GLYPH<25> ), where ' b is the e GLYPH<11> ective multipole for band b . \nThe upper panels of each of the sub-plots in Figs. 8 and 9 show the spectra of the best-fit CMB, and the total CMB + foreground, as well as the individual contributions of the foreground components using the same colour codings as in Fig. 7. The lower panel in each sub-plot shows the residuals with respect to the best-fit cosmology + foreground model. For each spectrum, we list the value of GLYPH<31> 2 , neglecting correlations \nbetween the (broad) ACT and SPT bands, together with the number of data points. The quality of the fits is generally very good. For SPT, the residuals are very similar to those inferred from Fig. 3 of R12. The SPT 150 GLYPH<2> 220 spectrum has the largest GLYPH<31> 2 (approximately a 1 : 8 GLYPH<27> excess). This spectrum shows systematic positive residuals of a few GLYPH<22> K 2 over the entire multipole range. For ACT, the residuals and GLYPH<31> 2 values are close to those plotted in Fig. 4 of Dunkley et al. (2013). All of the ACT spectra plotted in Fig. 9 are well fit by the model (except for some residuals at multipoles ' < GLYPH<24> 2000, which are also seen by Dunkley et al. 2013). \nHaving determined a solution for the best-fit foreground and other 'nuisance' parameters, we can correct the four spectra used in the Planck likelihood and combine them to reconstruct a 'best-fit' primary CMB spectrum and covariance matrix as described in Planck Collaboration XV (2014). This best-fit Planck CMB spectrum is plotted in the upper panels of Figs. 1 and 10 for Planck + WP + highL foreground parameters. The spectrum in Fig. 10 has been band-averaged in bins of width GLYPH<1> ' GLYPH<24> 31 using a window function Wb ( l ): \nˆ D b = X ' Wb ( ' ) ˆ D '; (36a) \nWb ( ' ) = 8 > > > < > > > : ( ˆ M D '' ) GLYPH<0> 1 = P ' b max ' = ' b min ( ˆ M D '' ) GLYPH<0> 1 ; ' b min GLYPH<20> ' < ' b max ; 0 ; otherwise : (36b) \nHere, ' b min and ' b max denote the minimum and maximum multipole ranges of band b , and ˆ M D '' 0 is the covariance matrix of the best-fit spectrum ˆ D ' , computed as described in Planck Collaboration XV (2014), and to which we have added corrections for beam and foreground errors (using the curvature matrix of the foreground model parameters from the MCMC chains). The solid lines in the upper panels of Figs. 1 and 10 show the spectrum for the best-fit GLYPH<3> CDMcosmology. The residuals with respect to this cosmology are plotted in the lower panel. To assess the goodness-of-fit, we compute GLYPH<31> 2 : \nGLYPH<31> 2 = X 0 ( ˆ C data ' GLYPH<0> C CMB ' ) ˆ M GLYPH<0> 1 '' 0 ( ˆ C data ' 0 GLYPH<0> C CMB ' 0 ) ; (37) \n'' \nusing the covariance matrix for the best-fit data spectrum (including foreground and beam errors 18 ). The results are given in the last line of Table 6 labelled 'All.' The lower panel of Fig. 10 shows the residuals with respect to the best-fit cosmology (on an expanded scale compared to Fig. 1). There are some visually striking residuals in this plot, particularly in the regions ' GLYPH<24> 800 and ' GLYPH<24> 1300-1500 (where we see 'oscillatory' behaviour). As discussed in detail in Planck Collaboration XV (2014), these residuals are reproducible to high accuracy across Planck detectors and across Planck frequencies; see also Fig. 7. There is therefore strong evidence that the residuals at these multipoles, which are in the largely signal dominated region of the spectrum, are real features of the primordial CMB sky. These features are compatible with statistical fluctuations of a Gaussian GLYPH<3> CDM model, and are described accurately by the covariance matrix used in the Planck likelihood. As judged by the GLYPH<31> 2 statis- \nFig. 10. Planck TT power spectrum. The points in the upper panel show the maximum-likelihood estimates of the primary CMB spectrum computed as described in the text for the best-fit foreground and nuisance parameters of the Planck + WP + highL fit listed in Table 5. The red line shows the best-fit base GLYPH<3> CDMspectrum. The lower panel shows the residuals with respect to the theoretical model. The error bars are computed from the full covariance matrix, appropriately weighted across each band (see Eqs. 36a and 36b) and include beam uncertainties and uncertainties in the foreground model parameters. \n<!-- image --> \nFig. 11. Planck TE (left) and EE spectra (right) computed as described in the text. The red lines show the polarization spectra from the base GLYPH<3> CDM Planck + WP + highL model, which is fitted to the TT data only . \n<!-- image --> \ntic listed in Table 6, the best fit reconstructed Planck spectrum is compatible with the base GLYPH<3> CDMcosmology to within 1 : 6 GLYPH<27> 19 \nTo the extremely high accuracy a GLYPH<11> orded by the Planck data, the power spectrum at high multipoles is compatible with the predictions of the base six parameter GLYPH<3> CDMcosmology. This is the main result of this paper. Fig. 1 does, however, suggest that the power spectrum of the best-fit base GLYPH<3> CDM cosmology has a higher amplitude than the observed power spectrum at multipoles ' < GLYPH<24> 30. We will return to this point in Sect. 7. \nFinally, Fig. 11 shows examples of Planck TE and EE spectra. These are computed by performing a straight average of the (scalar) beam-corrected 143 GLYPH<2> 143, 143 GLYPH<2> 217, and 217 GLYPH<2> 217 cross-spectra (ignoring auto-spectra). There are 32 TE and ET cross-spectra contributing to the mean TE spectrum plotted in Fig. 11, and six EE spectra contributing to the mean EE spectrum. Planck polarization data, including LFI and 353 GHz data not shown here, will be analysed in detail, and incorporated into a Planck likelihood, following this data release. The purpose of presenting these figures here is twofold: first, to demonstrate the potential of Planck to deliver high quality polarization maps and spectra, as described in the Planck 'blue-book' (Planck Collaboration 2005); and, second, to show the consistency of these polarization spectra with the temperature spectrum shown in Fig. 10. As discussed in Planck Collaboration VI (2014) and Planck Collaboration XV (2014), at present, the HFI polarization spectra at low multipoles ( ' < GLYPH<24> 200) are a GLYPH<11> ected by systematic errors that cause biases. For the HFI channels used in Fig. 11, there are two primary sources of systematic error arising from non-linear gain-like variations, and residual bandpass mismatches between detectors. However, these systematics rapidly become unimportant at higher multipoles 20 . \nThe errors on the mean TE and EE spectra shown in Fig. 11 are computed from the analytic formulae given in Efstathiou (2006), using an e GLYPH<11> ective beam-width adjusted to reproduce the observed scatter in the polarization spectra at high multipoles. The spectra are then band-averaged as in Eq. (37). The error bars shown in Fig. 11 are computed from the diagonal components of the band-averaged covariance matrices. \nThe solid lines in the upper panels of Fig. 11 show the theoretical TE and EE spectra expected in the best-fit Planck + WP + highL GLYPH<3> CDMmodel (i.e., the model used to compute the theory TT spectrum plotted in Fig. 10). These theoretical spectra are determined entirely from the TT analysis and make no use of the Planck polarization data. As with the TT spectra, the GLYPH<3> CDM model provides an extremely good match to the polarization spectra. Furthermore, polarized foreground emission is expected to be unimportant at high multipoles (e.g., Tucci & To GLYPH<11> olatti 2012) and so no foreground corrections have been made to the spectra in Fig. 11. The agreement between the polarization spectra and the theoretical spectra therefore provides strong evidence that the best-fit cosmological parameters \n20 The main focus of current work on Planck polarization is to reduce the e GLYPH<11> ects of these systematics on the polarization maps at large angular scales. \nlisted in Table 5 are not strongly a GLYPH<11> ected by the modelling of unresolved foregrounds in the TT analysis.", '5. Comparison of the Planck base GLYPH<3> CDM model with other astrophysical data sets': "Unlike CMB data, traditional astrophysical data sets - e.g., measurements of the Hubble parameter, type Ia supernovae (SNe Ia), and galaxy redshift surveys - involve complex physical systems that are not understood at a fundamental level. Astronomers are therefore reliant on internal consistency tests and empirical calibrations to limit the possible impact of systematic e GLYPH<11> ects. Examples include calibrating the metallicity dependence of the Cepheid period luminosity relation, calibrating the colour-decline-rate-luminosity relation of Type Ia supernovae, or quantifying the relationship between the spatial distributions of galaxies and dark matter. In addition, there are more mundane potential sources of error, which can a GLYPH<11> ect certain types of astrophysical observations (e.g., establishing consistent photometric calibration systems). We must be open to the possibility that unknown, or poorly quantified, systematic errors may be present in the astrophysical data, especially when used in combination with the high precision data from Planck . \nWe have seen in the previous section that the base GLYPH<3> CDM model provides an acceptable fit to the Planck TT power spectra (and the Planck TE and EE spectra) and also to the ACT and SPT temperature power spectra. The cosmological parameters of this model are determined to high precision. We therefore review whether these parameters provide acceptable fits to other astrophysical data. If they do not, then we need to assess whether the discrepancy is a pointer to new physics, or evidence of some type of poorly understood systematic e GLYPH<11> ect. Unless stated otherwise, we use the Planck + WP + highL parameters listed in Table 5 as the default ' Planck ' parameters for the base GLYPH<3> CDMmodel.", '5.1. CMB lensing measured by Planck': "Weak gravitational lensing by large-scale structure subtly alters the statistics of the CMB anisotropies, encoding information about the late-time Universe which is otherwise degenerate in the primary anisotropies laid down at last-scattering (see Lewis & Challinor 2006 for a review). The lensing deflections are given by the gradient of the lensing potential GLYPH<30> ( ˆ n ), which corresponds to an integrated measure of the matter distribution along the line of sight with peak sensitivity to structures around redshift 2. The rms deflection is expected to be around 2 : 5 arcmin and to be coherent over several degrees. We include the e GLYPH<11> ect of lensing on the temperature power spectrum in all our parameter analysis, but for some results we also include the lensing information encoded in the non-Gaussian trispectrum (connected 4-point function) of the CMB. Lensing generates a nonzero trispectrum, which, at leading order, is proportional to the power spectrum C GLYPH<30>GLYPH<30> of the lensing potential (Hu 2001). \n' In Planck Collaboration XVII (2014), we present a detailed analysis of CMB lensing with Planck data, including estimation of C GLYPH<30>GLYPH<30> ' from the trispectrum computed from Planck 's maps. This paper also describes the construction of a lensing likelihood. Briefly, we first reconstruct an estimate of the lensing potential using near-optimal quadratic estimators, following Okamoto & Hu (2003), with various Galactic and pointsource masks. The empirical power spectrum of this reconstruction, after subtraction of the Gaussian noise bias (i.e., the disconnected part of the 4-point function), is then used to esti- \nFig. 12. Planck measurements of the lensing power spectrum compared to the prediction for the best-fitting Planck + WP + highL GLYPH<3> CDM model parameters. In the top panel, the data points are the measured bandpowers and GLYPH<6> 1 GLYPH<27> error ranges from the diagonal of the covariance matrix. The measured bandpowers are compared to the C GLYPH<30>GLYPH<30> ' in the best-fit model (black line). The grey region shows the 1 GLYPH<27> range in C GLYPH<30>GLYPH<30> ' due to GLYPH<3> CDM parameter uncertainties. The lower panel shows the di GLYPH<11> erences between the bandpower amplitudes ˆ Ai and the predictions for their expectation values in the best-fit model, A theory i . \n<!-- image --> \nmate C GLYPH<30>GLYPH<30> ' in bandpowers. The associated bandpower errors are estimated from simulations. The lensing power spectrum is estimated from channel-coadded Planck maps at 100, 143 and 217 GHz in the multipole range ' = 10-1000, and also from a minimum-variance combination of the 143 and 217 GHz maps. An empirical correction for the shot-noise trispectrum of unresolved point sources is made to each spectrum, based on the measured amplitude of a generalized kurtosis of the appropriate maps. Additionally, the N (1) bias of Kesden et al. (2003), computed for a fiducial GLYPH<3> CDM spectrum determined from a prepublication analysis of the Planck data, is subtracted from each spectrum. This latter correction is proportional to C GLYPH<30>GLYPH<30> ' and accounts for sub-dominant couplings of the trispectrum, which mix lensing power over a range of scales into the power spectrum estimates. Excellent internal consistency of the various C GLYPH<30>GLYPH<30> ' estimates is found over the full multipole range. \nThe Planck lensing likelihood is based on reconstructions from the minimum-variance combination of the 143 and 217 GHz maps with 30% of the sky masked. Conservatively, only multipoles in the range ' = 40-400 are included, with a bandpower width GLYPH<1> ' = 45. The range ' = 40-400 captures 90% of the signal-to-noise on a measurement of the amplitude of a fiducial C GLYPH<30>GLYPH<30> ' , while minimizing the impact of imperfections in modelling the e GLYPH<11> ect of survey anisotropies on the large-scale GLYPH<30> reconstruction (the 'mean-field' of Planck Collaboration XVII 2014), and the large Gaussian noise bias on small scales. Note, however, that by restricting the range of angular scales we do lose some ability to distinguish between scale-dependent modifications of C GLYPH<30>GLYPH<30> ' , such as from massive neutrinos, and almost scale-independent modifications, such as from changes in the equation of state of unclustered dark energy or spatial curva- \nture. Correlated uncertainties in the beam transfer functions, point-source corrections, and the cosmology dependence of the N (1) bias give very broad-band correlations between the bandpowers. These are modelled as a sum of rank-one corrections to the covariance matrix and induce bandpower correlations that are small, less than 4%, but very broad. Bandpower correlations induced by masking are estimated to be less than 5% for neighbouring bins and are neglected. The likelihood is modelled as a Gaussian in the bandpowers with a fiducial (i.e., parameterindependent) covariance. For verification of this approximation, see Schmittfull et al. (2013). \nThe connected four-point function is related to the fullyreduced trispectrum T ' 1 ' 2 ' 3 ' 4 ( L ) by \nh T ' 1 m 1 T ' 2 m 2 T ' 3 m 3 T ' 4 m 4 i c = 1 2 X LM ( GLYPH<0> 1) M 0 B B B B B B @ ' 1 ' 2 L m 1 m 2 M 1 C C C C C C A GLYPH<2> 0 B B B B B B @ ' 3 ' 4 L m 3 m 4 GLYPH<0> M 1 C C C C C C A T ' 1 ' 2 ' 3 ' 4 ( L ) + perms ; (38) \n(Hu 2001). In the context of lensing reconstruction, the CMB trispectrum due to lensing takes the form \nT ' 1 ' 2 ' 3 ' 4 ( L ) GLYPH<25> C GLYPH<30>GLYPH<30> L C TT ' 2 C TT ' 4 F ' 1 L ' 2 F ' 3 L ' 4 ; (39) \nwhere C TT ' is the lensed temperature power spectrum and F ' 1 L ' 2 is a geometric mode-coupling function (Hu 2001; Hanson et al. 2011). Our estimates of C GLYPH<30>GLYPH<30> ' derive from the measured trispectrum. They are normalized using the fiducial lensed power spectrum to account for the factors of C TT ' in Eq. (39). In the likelihood, we renormalize the parameter-dependent C GLYPH<30>GLYPH<30> ' to account for the mismatch between the parameter-dependent C TT ' and that in the fiducial model. Since the best-fit GLYPH<3> CDM model we consider in this section has a lensed temperature power spectrum that is very close to that of the fiducial model, the renormalisation factor di GLYPH<11> ers from unity by less than 0 : 25%. \nThe estimated lensing power spectrum C GLYPH<30>GLYPH<30> ' is not independent of the measured temperature power spectrum C TT ' , but the dependence is very weak for Planck , and can be accurately ignored (Schmittfull et al. 2013; Planck Collaboration XVII 2014). As discussed in detail in Schmittfull et al. (2013), there are several e GLYPH<11> ects to consider. First, the reconstruction noise in the estimated GLYPH<30> derives from chance correlations in the unlensed CMB. If, due to cosmic variance, the unlensed CMB fluctuates high at some scale, the noise in the reconstruction will generally increase over a broad range of scales. Over the scales relevant for Planck lensing reconstruction, the correlation between the measured C GLYPH<30>GLYPH<30> ' and C TT ' 0 from this e GLYPH<11> ect is less than 0 : 2% and, moreover, is removed by a data-dependent Gaussian noise bias removal that we adopt following Hanson et al. (2011) and Namikawa et al. (2012). The second e GLYPH<11> ect derives from cosmic variance of the lenses. If a lens on a given scale fluctuates high, the estimated C GLYPH<30>GLYPH<30> ' will fluctuate high at that scale. In tandem, there will be more smoothing of the acoustic peaks in the measured C TT ' 0 , giving broad-band correlations that are negative at acoustic peaks and positive at troughs. The maximum correlation is around 0 : 05%. If we consider estimating the amplitude of a fiducial lensing power spectrum independently from the smoothing e GLYPH<11> ect of C TT ' and the measured C GLYPH<30>GLYPH<30> ' in the range ' = 40-400, the correlation between these estimates due to the cosmic variance of the lenses is only 4%. This \namounts to a mis-estimation of the error on a lensing amplitude in a joint analysis of C GLYPH<30>GLYPH<30> ' and C TT ' , treated as independent, of only 2%. For physical parameters, the mis-estimation of the errors is even smaller: Schmittfull et al. (2013) estimate around 0 : 5% from a Fisher analysis. A third negligible e GLYPH<11> ect is due to the T -GLYPH<30> correlation sourced by the late integrated Sachs-Wolfe e GLYPH<11> ect (see Planck Collaboration XIX 2014). This produces only local correlations between the measured C GLYPH<30>GLYPH<30> ' and C TT ' which are less than 0 : 5% by ' = 40 and fall rapidly on smaller scales. They produce a negligible correlation between lensing amplitude estimates for the multipole ranges considered here. The T -GLYPH<30> correlation is potentially a powerful probe of dark energy dynamics (e.g., Verde & Spergel 2002) and modified theories of gravity (e.g., Acquaviva et al. 2004). The power spectrum C T GLYPH<30> ' can be measured from the Planck data using the CMB 3-point function (Planck Collaboration XXIV 2014) or, equivalently, by cross-correlating the GLYPH<30> reconstruction with the large-angle temperature anisotropies (Planck Collaboration XIX 2014) although the detection significance is only around 3 GLYPH<27> . The power-spectrum based analysis in this paper discards the small amount of information in the T -GLYPH<30> correlation from Planck . In summary, we can safely treat the measured temperature and lensing power spectra as independent and simply multiply their respective likelihoods in a joint analysis. \nWe note that ACT (Das et al. 2011, 2013) and SPT (van Engelen et al. 2012) have both measured the lensing power spectrum with significances of 4 : 6 GLYPH<27> and 6 : 3 GLYPH<27> , respectively, in the multipole ranges ' = 75-2050 and ' = 100-1500. The Planck measurements used here represent a 26 GLYPH<27> detection. We therefore do not expect the published lensing measurements from these other experiments to carry much statistical weight in a joint analysis with Planck , despite the complementary range of angular scales probed, and we choose not to include them in the analyses in this paper. \nIn the lensing likelihood, we characterize the estimates of C GLYPH<30>GLYPH<30> with a set of eight (dimensionless) amplitudes ˆ Ai , where \n' \nˆ Ai = X B ' i ˆ C GLYPH<30>GLYPH<30> ' : (40) \n' \nHere, B ' i is a binning operation with \nB ' i = C GLYPH<30>GLYPH<30>; fid ' V GLYPH<0> 1 ' P ' i max ' 0 = ' i min GLYPH<16> C GLYPH<30>GLYPH<30>; fid ' 0 GLYPH<17> 2 V GLYPH<0> 1 ' 0 ; (41) \nfor ' within the band defined by a minimum multipole ' i min and a maximum ' i max . The inverse of the weighting function, V ' , is an approximation to the variance of the measured ˆ C GLYPH<30>GLYPH<30> ' and C GLYPH<30>GLYPH<30>; fid ' is the lensing power spectrum of the fiducial model, which is used throughout the analysis. The ˆ Ai are therefore near-optimal estimates of the amplitude of the fiducial power spectrum within the appropriate multipole range, normalized to unity in the fiducial model. Given some parameter-dependent model C GLYPH<30>GLYPH<30> ' , the expected values of the ˆ Ai are \nh ˆ Ai i = A theory i = X ' B ' i h 1 + GLYPH<1> GLYPH<30> ( C TT ' ) i 2 C GLYPH<30>GLYPH<30> ' ; (42) \nwhere the term involving GLYPH<1> GLYPH<30> ( C TT ' ), which depends on the parameter-dependent C TT ' , accounts for the renormalisation step described above. The lensing amplitudes ˆ Ai are compared to the A theory i for the best-fitting GLYPH<3> CDM model to the \nTable 7. Planck CMB lensing constraints. The A theory i are renormalized power spectrum amplitudes in the best-fit GLYPH<3> CDMmodel to Planck + WP + highL within the i th band (from ' min to ' max). The errors GLYPH<27> ( Ai ) on the amplitudes are the square root of the diagonals of the ˆ Ai covariance matrix. \nPlanck + WP + highL data combination (i.e., not including the lensing likelihood) in Table 7. The di GLYPH<11> erences between ˆ Ai and A theory i are plotted in the bottom panel of Fig. 12 while in the top panel the bandpower estimates are compared to C GLYPH<30>GLYPH<30> ' in the best-fitting model. The Planck measurements of C GLYPH<30>GLYPH<30> ' are consistent with the prediction from the best-fit GLYPH<3> CDM model to Planck + WP + highL. Using the full covariance matrix, we find GLYPH<31> 2 = 10 : 9 with eight degrees of freedom, giving an acceptable probability to exceed of approximately 21%. It is worth recalling here that the parameters of the GLYPH<3> CDM model are tightly constrained by the CMB 2-point function (as probed by our Planck + WP + highL data combination) which derives from physics at z GLYPH<25> 1100 seen in angular projection. It is a significant further vindication of the GLYPH<3> CDM model that its predictions for the evolution of structure and geometry at much lower redshifts (around z = 2 ) fit so well with Planck's CMB lensing measurements. \nThe discussion above does not account for the small spread in the C GLYPH<30>GLYPH<30> ' predictions across the Planck + WP + highL GLYPH<3> CDM posterior distribution. To address this, we introduce a parameter A GLYPH<30>GLYPH<30> L which, at any point in parameter space, scales the lensing trispectrum. Note that A GLYPH<30>GLYPH<30> L does not alter the lensed temperature power spectrum, so it can be used to assess directly how well the GLYPH<3> CDM predictions from C TT ' agree with the lensing measurements; in GLYPH<3> CDM we have A GLYPH<30>GLYPH<30> L = 1. The marginalized posterior distribution for A GLYPH<30>GLYPH<30> L in a joint analysis of Planck + WP + highL and the Planck lensing likelihood is given in Fig. 13. The agreement with A GLYPH<30>GLYPH<30> L = 1 is excellent, with \nA GLYPH<30>GLYPH<30> L = 0 : 99 GLYPH<6> 0 : 05 (68%; Planck + lensing + WP + highL) : (43) \nThe significance of the detection of lensing using A GLYPH<30>GLYPH<30> L in GLYPH<3> CDM is a little less than the 26 GLYPH<27> detection of lensing power reported in Planck Collaboration XVII (2014), due to the small spread in C GLYPH<30>GLYPH<30> from GLYPH<3> CDMparameter uncertainties. \n' Lensing also a GLYPH<11> ects the temperature power spectrum, primarily by smoothing the acoustic peaks and troughs on the scales relevant for Planck . The most significant detection of the lensing e GLYPH<11> ect in the power spectrum to date is from SPT. Introducing a parameter A L (Calabrese et al. 2008) which takes C GLYPH<30>GLYPH<30> ' ! A L C GLYPH<30>GLYPH<30> ' when computing the lensed temperature power spectrum (we shall shortly extend the action of this parameter to include the computation of the lensing trispectrum), Story et al. (2012) report A L = 0 : 86 + 0 : 15 GLYPH<0> 0 : 13 (68%; SPT + WMAP -7). Results for A L from \nFig. 13. Marginalized posterior distributions for A GLYPH<30>GLYPH<30> L (dashed) and A L (solid). For A GLYPH<30>GLYPH<30> L we use the data combination Planck + lensing + WP + highL. For A L we consider Planck + lensing + WP + highL (red), Planck + WP + highL (green), Planck + WP (blue) and Planck GLYPH<0> lowL + highL + GLYPH<28> prior (cyan; see text). \n<!-- image --> \nPlanck in combination with WMAP low-' polarization and the high-' power spectra from ACT and SPT are also shown in Fig. 13. Where we include the Planck lensing measurements, we define A L to scale the explicit C GLYPH<30>GLYPH<30> ' in Eq. (39), as well as modulating the lensing e GLYPH<11> ect in the temperature power spectrum. Figure 13 reveals a preference for A L > 1 from the Planck temperature power spectrum (plus WMAP polarization). This is most significant when combining with the high-' experiments for which we find \nA L = 1 : 23 GLYPH<6> 0 : 11 (68%; Planck + WP + highL) ; (44) \ni.e., a 2 GLYPH<27> preference for A L > 1. Including the lensing measurements, the posterior narrows but shifts to lower A L, becoming consistent with A L = 1 at the 1 GLYPH<27> level as expected from the A GLYPH<30>GLYPH<30> L results. \nWe do not yet have a full understanding of what is driving the preference for high A L in the temperature power spectrum. As discussed in Appendix C, the general preference is stable to assumptions about foreground modelling and cuts of the Planck data in the likelihood. To gain some insight, we consider the range of multipoles that drive the preference for A L > 1. For our favoured data combination of Planck + WP + highL, GLYPH<1> GLYPH<31> 2 = GLYPH<0> 5 : 2 going from the best-fit A L = 1 model to the best-fit model with variable A L. The improvement in fit comes only from the low-' temperature power spectrum ( GLYPH<1> GLYPH<31> 2 = GLYPH<0> 1 : 9) and the ACT + SPT data ( GLYPH<1> GLYPH<31> 2 = GLYPH<0> 3 : 3); for this data combination, there is no preference for high A L from the Planck temperature data at intermediate and high multipoles ( GLYPH<1> GLYPH<31> 2 = + 0 : 2). The situation at low-' is similar if we exclude the high-' experiments, with GLYPH<1> GLYPH<31> 2 = GLYPH<0> 1 : 6 there, but there is then a preference for the high A L best-fit from the Planck data on intermediate and small scales ( GLYPH<1> GLYPH<31> 2 = GLYPH<0> 3 : 4). However, as discussed in Sect. 4, there is more freedom in the foreground model when we exclude the high-' data, and this can o GLYPH<11> set smooth di GLYPH<11> erences in the CMB power spectra such as the transfer of power from large to small scales by lensing that is enhanced for A L > 1. \nSince the low-' temperature data seem to be partly responsible for pulling A L high, we consider the e GLYPH<11> ect of removing the low-' likelihood from the analysis. In doing so, we also remove the WMAP large-angle polarization which we compensate by introducing a simple prior on the optical depth; we use a Gaussian with mean 0 : 09 and standard deviation 0 : 013, similar to the constraint from WMAP polarization (Hinshaw et al. 2012). We denote this data combination, including the high-' experiments, by Planck GLYPH<0> lowL + highL + GLYPH<28> prior and show the posterior for A L in Fig. 13. As anticipated, the peak of the posterior moves to lower A L giving A L = 1 : 17 + 0 : 11 GLYPH<0> 0 : 13 (68% CL). The GLYPH<1> GLYPH<31> 2 = + 1 : 1 between the best-fit model (now at A L = 1 : 18) and the A L = 1 model for the Planck data (i.e. no preference for the higher A L) while GLYPH<1> GLYPH<31> 2 = GLYPH<0> 3 : 6 for the high-' experiments. \nSince varying A L alone does not alter the power spectrum on large scales, why should the low-' data prefer higher A L? The reason is due to a chain of parameter degeneracies that are illustrated in Fig. 14, and the deficit of power in the measured C ' s on large scales compared to the best-fit GLYPH<3> CDMmodel (see Fig. 1 and Sect. 7). In models with a power-law primordial spectrum, the temperature power spectrum on large scales can be reduced by increasing n s. The e GLYPH<11> ect of an increase in n s on the relative heights of the first few acoustic peaks can be compensated by increasing ! b and reducing ! m, as shown by the contours in Fig. 14. However, on smaller scales, corresponding to modes that entered the sound horizon well before matter-radiation equality, the e GLYPH<11> ects of baryons on the mid-point of the acoustic oscillations (which modulates the relative heights of even and odd peaks) is diminished since the gravitational potentials have pressure-damped away during the oscillations in the radiation-dominated phase (e.g., Hu & White 1996, 1997). Moreover, on such scales the radiation-driving at the onset of the oscillations that amplifies their amplitude happens early enough to be una GLYPH<11> ected by small changes in the matter density. The net e GLYPH<11> ect is that, in models with A L = 1, the extent of the degeneracy involving n s, ! b and ! m is limited by the higher-order acoustic peaks, and there is little freedom to lower the large-scale temperature power spectrum by increasing n s while preserving the good fit at intermediate and small scales. Allowing A L to vary changes this picture, letting the degeneracy extend to higher n s, as shown by the samples in Fig. 14. The additional smoothing of the acoustic peaks due to an increase in A L can mitigate the e GLYPH<11> ect of increasing n s around the fifth peak, where the signalto-noise for Planck is still high. 21 This allows one to decrease the spectrum at low ' , while leaving it essentially unchanged on those smaller scales where Planck still has good sensitivity. Above ' GLYPH<24> 2000, the best-fit A L model has a little more power than the base model (around 3 GLYPH<22> K 2 at ' = 2000), while the Planck , ACT, and SPT data have excess power over the bestfit A L = 1 GLYPH<3> CDM + foreground model at the level of a few GLYPH<22> K 2 (see Sect. 4). It is plausible that this may drive the preference for high A L in the GLYPH<31> 2 of the high-' experiments. We note that a similar 2 GLYPH<27> preference for A L > 1 is also found combining ACT and WMAP data (Sievers et al. 2013) and, as we find here, this tension is reduced when the lensing power spectrum is included in the fit. \nTo summarize, there is no preference in the Planck lensing power spectrum for A L > 1. The general preference for high A L from the CMB power spectra in our favoured data combination ( Planck + WP + highL) is mostly driven by two e GLYPH<11> ects: the \nFig. 14. E GLYPH<11> ect of allowing A L to vary on the degeneracies between GLYPH<10> b h 2 and n s (left) and GLYPH<10> m h 2 and n s (right). In both panels the data combination is Planck + WP + highL. The contours enclose the 68% and 95% confidence regions in the base GLYPH<3> CDMmodel with A L = 1. The samples are from models with variable A L and are colour-coded by the value of A L. \n<!-- image --> \ndi GLYPH<14> culty that GLYPH<3> CDM models have in fitting the low-' spectrum when calibrated from the smaller-scale spectrum; and, plausibly, from excess residuals at the GLYPH<22> K 2 level in the high-' spectra relative to the best-fit A L = 1 GLYPH<3> CDM + foregrounds model on scales where extragalactic foreground modelling is critical.", '5.2. Baryon acoustic oscillations': 'Baryon acoustic oscillations (BAO) in the matter power spectrum were first detected in analyses of the 2dF Galaxy Redshift Survey (Cole et al. 2005) and the SDSS redshift survey (Eisenstein et al. 2005). Since then, accurate BAO measurements have been made using a number of di GLYPH<11> erent galaxy redshift surveys, providing constraints on the distance luminosity relation spanning the redshift range 0 : 1 < GLYPH<24> z < GLYPH<24> 0 : 7 22 . Here we use the results from four redshift surveys: the SDSS DR7 BAO measurements at e GLYPH<11> ective redshifts z e GLYPH<11> = 0 : 2 and z e GLYPH<11> = 0 : 35, analysed by Percival et al. (2010); the z = 0 : 35 SDSS DR7 measurement at z e GLYPH<11> = 0 : 35 reanalyzed by Padmanabhan et al. (2012); the WiggleZ measurements at z e GLYPH<11> = 0 : 44, 0 : 60 and 0 : 73 analysed by Blake et al. (2011); the BOSS DR9 measurement at z e GLYPH<11> = 0 : 57 analyzed by Anderson et al. (2012); and the 6dF Galaxy Survey measurement at z = 0 : 1 discussed by Beutler et al. (2011). \nBAO surveys measure the distance ratio \ndz = r s( z drag) D V( z ) ; (45) \nwhere r s( z drag) is the comoving sound horizon at the baryon drag epoch (when baryons became dynamically decoupled from the photons) and D V( z ) is a combination of the angular-diameter dis- \ntance, D A( z ), and the Hubble parameter, H ( z ), appropriate for the analysis of spherically-averaged two-point statistics: \nD V( z ) = " (1 + z ) 2 D 2 A ( z ) cz H ( z ) # 1 = 3 : (46) \nIn the GLYPH<3> CDM cosmology (allowing for spatial curvature), the angular diameter distance to redshift z is \nD A( z ) = c H 0 ˆ D A : = c H 0 1 j GLYPH<10> K j 1 = 2 (1 + z ) sin K h j GLYPH<10> K j 1 = 2 x ( z ; GLYPH<10> m ; GLYPH<10>GLYPH<3> ) i ; (47) \nwhere \nx ( z ; GLYPH<10> m ; GLYPH<10>GLYPH<3> ) = Z z 0 dz 0 [ GLYPH<10> m(1 + z 0 ) 3 + GLYPH<10> K (1 + z 0 ) 2 + GLYPH<10>GLYPH<3> ] 1 = 2 ; (48) \nand sin K = sinh for GLYPH<10> K > 0 and sin K = sin for GLYPH<10> K < 0. (The small e GLYPH<11> ects of the 0 : 06 eV massive neutrino in our base cosmology are ignored in Eq. 48.) Note that the luminosity distance, D L, relevant for the analysis of Type Ia supernovae (see Sect. 5.4) is related to the angular diameter distance via D L = ( c = H 0) ˆ D L = D A(1 + z ) 2 . \nDi GLYPH<11> erent groups fit and characterize BAO features in di GLYPH<11> erent ways. For example, the WiggleZ team encode some shape information on the power spectrum to measure the acoustic parameter A ( z ), introduced by Eisenstein et al. (2005), \nA ( z ) = D V( z ) q GLYPH<10> m H 2 0 cz ; (49) \nwhich is almost independent of ! m. To simplify the presentation, Fig. 15 shows estimates of r s = D V( z ) and 1 GLYPH<27> errors, as quoted by each of the experimental groups, divided by the expected relation for the Planck base GLYPH<3> CDM parameters. Note that the experimental groups use the approximate formulae of Eisenstein & Hu (1998) to compute z drag and r s( z drag), though they fit power spectra computed with Boltzmann codes, such as camb , generated for a set of fiducial-model parameters. The measurements have now become so precise that the small di GLYPH<11> erence between the Eisenstein & Hu (1998) approximations and \nFig. 15. Acoustic-scale distance ratio r s = D V( z ) divided by the distance ratio of the Planck base GLYPH<3> CDM model. The points are colour-coded as follows: green star (6dF); purple squares (SDSS DR7 as analyzed by Percival et al. 2010); black star (SDSS DR7 as analyzed by Padmanabhan et al. 2012); blue cross (BOSS DR9); and blue circles (WiggleZ). The grey band shows the approximate GLYPH<6> 1 GLYPH<27> range allowed by Planck (computed from the CosmoMC chains). \n<!-- image --> \nthe accurate values of z drag and r drag = r s( z drag) returned by camb need to be taken into account. In CosmoMC we multiply the accurate numerical value of r s( z drag) by a constant factor of 1 : 0275 to match the Eisenstein-Hu approximation in the fiducial model. This correction is su GLYPH<14> ciently accurate over the range of ! m and ! b allowed by the CMB in the base GLYPH<3> CDMcosmology (see e.g. Mehta et al. 2012) and also for the extended GLYPH<3> CDMmodels discussed in Sect. 6. \nThe Padmanabhan et al. (2012) result plotted in Fig. 15 is a reanalysis of the z e GLYPH<11> = 0 : 35 SDSS DR7 sample discussed by Percival et al. (2010). Padmanabhan et al. (2012) achieve a higher precision than Percival et al. (2010) by employing a reconstruction technique (Eisenstein et al. 2007) to correct (partially) the baryon oscillations for the smearing caused by galaxy peculiar velocities. The Padmanabhan et al. (2012) results are therefore strongly correlated with those of Percival et al. (2010). We refer to the Padmanabhan et al. (2012) \'reconstructioncorrected\' results as SDSS(R). A similar reconstruction technique was applied to the BOSS survey by Anderson et al. (2012) to achieve 1 : 6% precision in D V( z = 0 : 57) = r s, the most precise determination of the acoustic oscillation scale to date. \nAll of the BAO measurements are compatible with the base GLYPH<3> CDM parameters from Planck . The grey band in Fig. 15 shows the GLYPH<6> 1 GLYPH<27> range in the acoustic-scale distance ratio computed from the Planck + WP + highL CosmoMC chains for the base GLYPH<3> CDM model. To get a qualitative feel for how the BAO measurements constrain parameters in the base GLYPH<3> CDM model, we form GLYPH<31> 2 , \nGLYPH<31> 2 BAO = ( x GLYPH<0> x GLYPH<3> CDM ) T C GLYPH<0> 1 BAO ( x GLYPH<0> x GLYPH<3> CDM ) ; (50) \nwhere x is the data vector, x GLYPH<3> CDM denotes the theoretical prediction for the GLYPH<3> CDM model and C GLYPH<0> 1 BAO is the inverse covariance matrix for the data vector x . The data vector is as follows: D V(0 : 106) = (457 GLYPH<6> 27) Mpc (6dF); r s = D V(0 : 20) = 0 : 1905 GLYPH<6> 0 : 0061, r s = D V(0 : 35) = 0 : 1097 GLYPH<6> 0 : 0036 (SDSS); A (0 : 44) = 0 : 474 GLYPH<6> 0 : 034, A (0 : 60) = 0 : 442 GLYPH<6> 0 : 020, A (0 : 73) = \nTable 8. Approximate constraints with 68% errors on GLYPH<10> m and H 0 (in units of km s GLYPH<0> 1 Mpc GLYPH<0> 1 ) from BAO, with ! m and ! b fixed to the best-fit Planck + WP + highL values for the base GLYPH<3> CDM cosmology. \n0 : 424 GLYPH<6> 0 : 021 (WiggleZ); D V(0 : 35) = r s = 8 : 88 GLYPH<6> 0 : 17 (SDSS(R)); and D V(0 : 57) = r s = 13 : 67 GLYPH<6> 0 : 22, (BOSS). The o GLYPH<11> -diagonal components of C GLYPH<0> 1 BAO for the SDSS and WiggleZ results are given in Percival et al. (2010) and Blake et al. (2011). We ignore any covariances between surveys. Since the SDSS and SDSS(R) results are based on the same survey, we include either one set of results or the other in the analysis described below, but not both together. \nThe Eisenstein-Hu values of r s for the Planck and WMAP -9 base GLYPH<3> CDM parameters di GLYPH<11> er by only 0 : 9%, significantly smaller than the errors in the BAO measurements. We can obtain an approximate idea of the complementary information provided by BAO measurements by minimizing Eq. (50) with respect to either GLYPH<10> m or H 0, fixing ! m and ! b to the CMB best-fit parameters. (We use the Planck + WP + highL parameters from Table 5.) The results are listed in Table 8 23 . \nAs can be seen, the results are very stable from survey to survey and are in excellent agreement with the base GLYPH<3> CDM parameters listed in Tables 2 and 5. The values of GLYPH<31> 2 BAO are also reasonable. For example, for the six data points of the 6dF + SDSS(R) + BOSS + WiggleZ combination, we find GLYPH<31> 2 BAO = 4 : 3, evaluated for the Planck + WP + highL best-fit GLYPH<3> CDMparameters. \nThe high value of GLYPH<10> m is consistent with the parameter analysis described by Blake et al. (2011) and with the \'tension\' discussed by Anderson et al. (2012) between BAO distance measurements and direct determinations of H 0 (Riess et al. 2011; Freedman et al. 2012). Furthermore, if the errors on the BAO measurements are accurate, the constraints on GLYPH<10> m and H 0 (for fixed ! m and ! b) are of comparable accuracy to those from Planck . \nThe results of this section show that BAO measurements are an extremely valuable complementary data set to Planck . The measurements are basically geometrical and free from complex systematic e GLYPH<11> ects that plague many other types of astrophysical measurements. The results are consistent from survey to survey and are of comparable precision to Planck . In addition, BAO measurements can be used to break parameter degeneracies that limit analyses based purely on CMB data. For example, from the \nFig. 16. Comparison of H 0 measurements, with estimates of GLYPH<6> 1 GLYPH<27> errors, from a number of techniques (see text for details). These are compared with the spatially-flat GLYPH<3> CDM model constraints from Planck and WMAP -9. \n<!-- image --> \nexcellent agreement with the base GLYPH<3> CDM model evident in Fig. 15, we can infer that the combination of Planck and BAO measurements will lead to tight constraints favouring GLYPH<10> K = 0 (Sect. 6.2) and a dark energy equation-of-state parameter, w = GLYPH<0> 1 (Sect. 6.5). Since the BAO measurements are primarily geometrical, they are used in preference to more complex astrophysical datasets to break CMB parameter degeneracies in this paper. \nFinally, we note that we choose to use the 6dF + SDSS(R) + BOSS data combination in the likelihood analysis of Sect. 6. This choice includes the two most accurate BAO measurements and, since the e GLYPH<11> ective redshifts of these samples are widely separated, it should be a very good approximation to neglect correlations between the surveys.', '5.3. The Hubble constant': "Astriking result from the fits of the base GLYPH<3> CDMmodelto Planck power spectra is the low value of the Hubble constant, which is tightly constrained by CMB data alone in this model. From the Planck + WP + highL analysis we find \nH 0 = (67 : 3 GLYPH<6> 1 : 2) km s GLYPH<0> 1 Mpc GLYPH<0> 1 (68%; Planck + WP + highL) : (51) \nA low value of H 0 has been found in other CMB experiments, most notably from the recent WMAP -9 analysis. Fitting the base GLYPH<3> CDMmodel, Hinshaw et al. (2012) find 24 \nH 0 = (70 : 0 GLYPH<6> 2 : 2) km s GLYPH<0> 1 Mpc GLYPH<0> 1 (68%; WMAP -9) ; (52) \nconsistent with Eq. (51) to within 1 GLYPH<27> . We emphasize here that the CMB estimates are highly model dependent . It is important therefore to compare with astrophysical measurements of H 0, since any discrepancies could be a pointer to new physics. \nThere have been remarkable improvements in the precision of the cosmic distance scale in the last decade or so. The final results of the Hubble Space Telescope ( HST ) Key Project (Freedman et al. 2001), which used Cepheid calibrations of secondary distance indicators, resulted in a Hubble constant of H 0 = (72 GLYPH<6> 8) km s GLYPH<0> 1 Mpc GLYPH<0> 1 (where the error includes estimates of both 1 GLYPH<27> random and systematic errors). This estimate has been used widely in combination with CMB observations and other cosmological data sets to constrain cosmological parameters (e.g., Spergel et al. 2003, 2007). It has also been recognized that an accurate measurement of H 0 with around 1% precision, when combined with CMB and other cosmological data, has the potential to reveal exotic new physics, for example, a time-varying dark energy equation of state, additional relativistic particles, or neutrino masses (see e.g., Suyu et al. 2012, and references therein). Establishing a more accurate cosmic distance scale is, of course, an important problem in its own right. The possibility of uncovering new fundamental physics provides an additional incentive. \nTwo recent analyses have greatly improved the precision of the cosmic distance scale. Riess et al. (2011) use HST observations of Cepheid variables in the host galaxies of eight SNe Ia to calibrate the supernova magnitude-redshift relation. Their 'best estimate' of the Hubble constant, from fitting the calibrated SNe magnitude-redshift relation, is \nH 0 = (73 : 8 GLYPH<6> 2 : 4) km s GLYPH<0> 1 Mpc GLYPH<0> 1 (Cepheids + SNe Ia) ; (53) \nwhere the error is 1 GLYPH<27> and includes known sources of systematic errors. At face value, this measurement is discrepant with the Planck estimate in Eq. (51) at about the 2 : 5 GLYPH<27> level. \nFreedman et al. (2012), as part of the Carnegie Hubble Program , use Spitzer Space Telescope mid-infrared observations to recalibrate secondary distance methods used in the HST Key Project. These authors find \nH 0 = [74 : 3 GLYPH<6> 1 : 5 (statistical) GLYPH<6> 2 : 1 (systematic)] km s GLYPH<0> 1 Mpc GLYPH<0> 1 (Carnegie HP) : (54) \nWe have added the two sources of error in quadrature in the error range shown in Fig. 16. This estimate agrees well with Eq. (53) and is also discordant with the Planck value (Eq. 16) at about the 2 : 5 GLYPH<27> level. The error analysis in Eq. (54) does not include a number of known sources of systematic error and is very likely an underestimate. For this reason, and because of the relatively good agreement between Eqs. (53) and (54), we do not use the estimate in Eq. (54) in the likelihood analyses described in Sect. 6. \nThe dominant source of error in the estimate in Eq. (53) comes from the first rung in the distance ladder. Using the megamaser-based distance to NGC4258, Riess et al. (2011) find (74 : 8 GLYPH<6> 3 : 1) km s GLYPH<0> 1 Mpc GLYPH<0> 1 . 25 Using parallax measurements for 10 Milky Way Cepheids, they find (75 : 7 GLYPH<6> 2 : 6) km s GLYPH<0> 1 Mpc GLYPH<0> 1 , and using Cepheid observations and a revised distance to the Large Magellanic Cloud, they find (71 : 3 GLYPH<6> 3 : 8) km s GLYPH<0> 1 Mpc GLYPH<0> 1 . These estimates are consistent with each other, and the combined estimate in Eq. (53) uses all three calibrations. The fact that the error budget of measurement (53) is dominated by the 'first-rung' calibrators is a point of concern. A mild underestimate of the \nFigure 16 includes three estimates of H 0 based on 'geometrical' methods. 26 The estimate labelled 'MCP' shows the result H 0 = (68 : 0 GLYPH<6> 4 : 8) km s GLYPH<0> 1 Mpc GLYPH<0> 1 from the Megamaser Cosmology Project (Braatz et al. 2013) based on observations of megamasers in UGC 3789, NGC 6264 and Mrk 1419 (see also Reid et al. 2012, for a detailed analysis of UGC 3789). The point labelled 'RXJ1131-1231' shows the estimate H 0 = 78 : 7 + 4 : 3 GLYPH<0> 4 : 5 kms GLYPH<0> 1 Mpc GLYPH<0> 1 derived from gravitational lensing time delay measurements of the system RXJ1131-1231, observed as part of the 'COSmological MOnitoring of GRAvitational Lenses' (COSMOGRAIL) project (Suyu et al. 2013, see also Courbin et al. 2011; Tewes et al. 2013). Finally, the point labelled SZ clusters shows the value H 0 = 76 : 9 + 10 : 7 GLYPH<0> 8 : 7 kms GLYPH<0> 1 Mpc GLYPH<0> 1 (Bonamente et al. 2006), derived by combining tSZ and X-ray measurements of rich clusters of galaxies (see Carlstrom et al. 2002, and references therein). These geometrical methods bypass the need for local distance calibrators, but each has its own sources of systematic error that need to be controlled. The geometrical methods are consistent with the Cepheid-based methods, but at present, the errors on these methods are quite large. The COSMOGRAIL measurement (which involved a 'blind' analysis to prevent experimenter bias) is discrepant at about 2 : 5 GLYPH<27> with the Planck value in Eq. (51). We note here a number of other direct measurements of H 0 (Jones et al. 2005; Sandage et al. 2006; Oguri 2007; Tammann & Reindl 2013) that give lower values than the measurements summarized in Fig. 16. \n<!-- image --> \n<!-- image --> \nm \nFig. 17. MCMC samples and contours in the r GLYPH<3> -GLYPH<10> m h 2 plane (left) and the D A( z GLYPH<3> )-GLYPH<10> m h 2 plane (right) for GLYPH<3> CDM models analysed with Planck + WP + highL. The lines in these plots show the expected degeneracy directions in the base GLYPH<3> CDMcosmology. Samples are colour-coded by the values of GLYPH<10> b h 2 (left) and H 0 (right). \ndistance errors to these calibrators could eliminate the tension with Planck . \nThe tension between the CMB-based estimates and the astrophysical measurements of H 0 is intriguing and merits further discussion. In the base GLYPH<3> CDM model, the sound horizon depends primarily on GLYPH<10> m h 2 (with a weaker dependence on GLYPH<10> b h 2 ). This is illustrated by the left-hand panel of Fig. 17, which shows samples from the Planck + WP + highL MCMC chains in the r GLYPH<3> -GLYPH<10> m h 2 plane colour coded according to GLYPH<10> b h 2 . The acoustic scale parameter GLYPH<18> GLYPH<3> is tightly constrained by the CMB power spectrum, and so a change in r GLYPH<3> must be matched by a corresponding shift in the angular diameter distance to the last scattering surface D A( z GLYPH<3> ). In the base GLYPH<3> CDM model, D A depends on H 0 and GLYPH<10> m h 2 , as \nshown in the right-hand panel of Fig. 17. The 2 : 7 km s GLYPH<0> 1 Mpc GLYPH<0> 1 shift in H 0 between Planck and WMAP -9 is primarily a consequence of the slightly higher matter density determined by Planck ( GLYPH<10> m h 2 = 0 : 143 GLYPH<6> 0 : 003) compared to WMAP -9 ( GLYPH<10> m h 2 = 0 : 136 GLYPH<6> 0 : 004). A shift of around 7 km s GLYPH<0> 1 Mpc GLYPH<0> 1 , necessary to match the astrophysical measurements of H 0 would require an even larger change in GLYPH<10> m h 2 , which is disfavoured by the Planck data. The tension between Planck and the direct measurements of H 0 cannot be easily resolved by varying the parameters of the base GLYPH<3> CDMmodel. Section 6 will explore whether there are any extensions to the base GLYPH<3> CDM model that can relieve this tension. In that section, results labelled ' H 0' include a Gaussian prior on H 0 based on the Riess et al. (2011) measurement given in Eq. (53).", '5.4. Type Ia supernovae': "In this subsection, we analyse two SNe Ia samples: the sample of 473 SNe as reprocessed by Conley et al. (2011), which we will refer to as the 'SNLS' compilation; and the updated Union2.1 compilation of 580 SNe described by Suzuki et al. (2012).", '5.4.1. The SNLS compilation': "The SNLS 'combined' compilation consists of 123 SNe Ia at low redshifts, 242 SNe Ia from the three-year Supernova Legacy Survey (SNLS; see Regnault et al. 2009; Guy et al. 2010; Conley et al. 2011), 93 intermediate redshift SNe Ia from the Sloan Digital Sky Survey (SDSS; Holtzman et al. 2008; Kessler et al. 2009) and 14 objects at high redshift observed with the Hubble Space Telescope ( HST ; Riess et al. 2007). \nThe 'combined' sample of Conley et al. (2011) combines the results of two light-curve fitting codes, SiFTO (Conley et al. 2008) and SALT2 (Guy et al. 2007), to produce a peak apparent B -band magnitude, mB , stretch parameter s and colour C for each supernova. To explore the impact of light-curve fitting, we also analyse separately the SiFTO and SALT2 parameters. The SiFTO and SALT2 samples di GLYPH<11> er by a few SNe from the combined sample because of colour and stretch constraints imposed on the samples. We also use ancillary data, such as estimates of \nFig. 18. Magnitude residuals relative to the base GLYPH<3> CDM model that best fits the SNLS combined sample (left) and the Union2.1 sample (right). The error bars show the 1 GLYPH<27> (diagonal) errors on mB . The filled grey regions show the residuals between the expected magnitudes and the best-fit to the SNe sample as GLYPH<10> m varies across the GLYPH<6> 2 GLYPH<27> range allowed by Planck + WP + highL in the base GLYPH<3> CDM cosmology. The colour coding of the SNLS samples are as follows: low redshift (blue points); SDSS (green points); SNLS three-year sample (orange points); and HST high redshift (red points). \n<!-- image --> \nTable 9. Best-fit parameters for the SNLS compilations. \nthe stellar masses of the host galaxies and associated covariance matrices, as reported by Conley et al. (2011) 27 . \nIn this section, we focus exclusively on the base GLYPH<3> CDM model (i.e., w = GLYPH<0> 1 and GLYPH<10> K = 0). For a flat Universe, the expected apparent magnitudes are then given by \nm GLYPH<3> CDM B = 5log 10 ˆ D L( z hel ; z CMB ; GLYPH<10> m) GLYPH<0> GLYPH<11> ( s GLYPH<0> 1) + GLYPH<12> C + M B ; (55) \nwhere ˆ D L is the dimensionless luminosity distance 28 and M B absorbs the Hubble constant. As in Sullivan et al. (2011), we express values of the parameter(s) M B in terms of an e GLYPH<11> ective absolute magnitude \nMB = M B GLYPH<0> 5log 10 c H 0 ! GLYPH<0> 25 ; (56) \nfor a value of H 0 = 70 km s GLYPH<0> 1 Mpc GLYPH<0> 1 . \nThe likelihood for this sample is then constructed as in Conley et al. (2011) and Sullivan et al. (2011): \nGLYPH<31> 2 SNe = ( M B GLYPH<0> M GLYPH<3> CDM B ) T C GLYPH<0> 1 SNe ( M B GLYPH<0> M GLYPH<3> CDM B ) ; (57) \nwhere M B is the vector of e GLYPH<11> ective absolute magnitudes and C SNe is the sum of the non-sparse covariance matrices of Conley et al. (2011) quantifying statistical and systematic errors. As in Sullivan et al. (2011), we divide the sample according to the estimated stellar mass of the host galaxy and solve for \ntwo parameters, M 1 B for M host < 10 10 M GLYPH<12> and M 2 B for M host GLYPH<21> 10 10 M GLYPH<12> . We adopt the estimates of the 'intrinsic' scatter in mB for each SNe sample given in Table 4 of Conley et al. (2011). \nFits to the SNLS combined sample are shown in the lefthand panel of Fig. 18. The best-fit parameters for the combined, SiFTO and SALT2 samples are given in Table 9. In the base GLYPH<3> CDM model, the SNe data provide a constraint on GLYPH<10> m, independent of the CMB. As can be seen from Table 9 (and also in the analyses of Conley et al. 2011 and Sullivan et al. 2011), the SNLS combined compilation favours a lower value of GLYPH<10> m than we find from the CMB. The key question, of course, is whether the SNe data are statistically compatible with the Planck data. The last three rows of Table 9 give the best-fit SNe parameters constraining GLYPH<10> m to the Planck + WP + highL best-fit value GLYPH<10> m = 0 : 317. The grey bands in Fig. 18 show the magnitude residuals expected for a GLYPH<6> 2 GLYPH<27> variation in the value of GLYPH<10> m allowed by the CMB data. The CMB band lies systematically low by about 0 : 1 magnitude over most of the redshift range shown in Fig. 18a. \nTable 9 also lists the GLYPH<31> 2 values for the GLYPH<10> m = 0 : 317 fits. 29 The likelihood ratio for the SiFTO fits is \nL SNe L SNe + CMB GLYPH<10> m = exp 1 2 ( GLYPH<31> 2 SNe GLYPH<0> GLYPH<31> 2 SNe + CMB GLYPH<10> m ) ! GLYPH<25> 0 : 074 : (58) \nThis is almost a 2 GLYPH<27> discrepancy. (The discrepancy would appear to be much more significant if only the diagonal statistical errors \n29 We caution the reader that, generally, the GLYPH<31> 2 SNe obtained from Eq. (57) will di GLYPH<11> er from that quoted in the online parameter tables in cases where the SNLS data is importance sampled. For importance sampling, we modified the SNLS likelihood to marginalize numerically over the GLYPH<11> and GLYPH<12> parameters. \nwere included in the covariance matrix in Eq. 57). The likelihood ratio for the combined sample is slightly larger (0 : 095) and is larger still for the SALT2 sample (0 : 33). In summary, there is some tension between the SNLS compilations and the base GLYPH<3> CDMvalue of GLYPH<10> m derived from Planck . The degree of tension depends on the light-curve fitter and is stronger for the SiFTO and combined SNLS compilations. 30", '5.4.2. The Union2.1 compilation': 'The Union2.1 compilation (Suzuki et al. 2012) is the latest application of a scheme for combining multiple SNe data sets described by Kowalski et al. (2008). The Union2.1 compilation contains 19 data sets and includes early high-redshift SNe data (e.g., Riess et al. 1998; Perlmutter et al. 1999) as well as recent data from the HST Cluster Supernova Survey (Amanullah et al. 2010; Suzuki et al. 2012). The SNLS and Union2.1 compilations contain 256 SNe in common and are therefore not independent. \nThe SALT2 model (Guy et al. 2007) is used to fit the light curves returning a B -band magnitude at maximum light, a lightcurve shape parameter and a colour correction. (Note that the version of SALT2 used in the Union2.1 analysis is not exactly the same as that used in the SNLS analysis.) As in Eq. (55), the theoretically-predicted magnitudes include nuisance parameters GLYPH<11> and GLYPH<12> multiplying the shape and colour corrections, and an additional nuisance parameter GLYPH<14> describing the variation of SNe luminosity with host galaxy mass (see Eq. 3 of Suzuki et al. 2012). The CosmoMC module associated with the Union2.1 sample 31 holds the nuisance parameters fixed ( GLYPH<11> = 0 : 1218, GLYPH<12> = 2 : 4657, and GLYPH<14> = GLYPH<0> 0 : 03634) and computes a GLYPH<31> 2 via Eq. (57) using a fixed covariance matrix that includes a model for systematic errors. An analysis of the base GLYPH<3> CDM model then requires minimization with respect to only two parameters, GLYPH<10> m and M B (or equivalently, MB ). \nMaximizing the Union2.1 likelihood, we find best-fit parameters of GLYPH<10> m = 0 : 296 and MB = GLYPH<0> 19 : 272 (defined as in Eq. 56 for a value of H 0 = 70 km s GLYPH<0> 1 Mpc GLYPH<0> 1 ) and GLYPH<31> 2 Union2 : 1 = 545 : 11 (580 SNe). The magnitude residuals with respect to this fit are shown in the right-hand panel of Fig. 18. Notice that the scatter in this plot is significantly larger than the scatter of the SNLS compilation (left-hand panel) reflecting the more diverse range of data and the lower precision of some of the earlier SNe data used in the Union2.1 compilation. Nevertheless, the Union2.1 best-fit is close to (and clearly compatible with) the Planck base GLYPH<3> CDM value of GLYPH<10> m.', '5.4.3. SNe: Summary': 'The results of this subsection are summarized in Fig. 19. This shows the posterior distributions for GLYPH<10> m in the base GLYPH<3> CDMcosmology, marginalized over nuisance parameters, for each of the SNe samples. These distributions are broad (with the Union2.1 distribution somewhat broader than the SNLS distributions) and show substantial overlap. There is no obvious inconsistency between the SNe samples. The posterior distribution for GLYPH<10> m in the base GLYPH<3> CDM model fit to Planck + WP + highL is shown by the narrow green curve. This is consistent with the Union2.1 and SNLS SALT2 results, but is in some tension with the distribu- \nFig. 19. Posterior distributions for GLYPH<10> m (assuming a flat cosmology) for the SNe compilations described in the text. The posterior distribution for GLYPH<10> m from the Planck + WP + highL fits to the base GLYPH<3> CDMmodel is shown by the solid green line. \n<!-- image --> \nm \ntions from the SNLS combined and SNLS SiFTO samples. As we will see in Sect. 6, Planck combined with Planck lensing and BAO measurements overwhelm SNe data for most of the extensions of the GLYPH<3> CDMmodelconsidered in this paper. However, the results presented here suggest that there could be residual systematic errors in the SNe data that are not properly accounted for in the covariance matrices. Hints of new physics based on combining CMB and SNe data should therefore be treated with caution.', '5.5. Additional data': 'In this subsection we review a number of other astrophysical data sets that have sometimes been combined with CMB data. These data sets are not used with Planck in this paper, either because they are statistically less powerful than the data reviewed in previous subsections and / or they involve complex physics (such as the behaviour of intra-cluster gas in rich clusters of galaxies) which is not yet well understood.', '5.5.1. Shape information on the galaxy/matter power spectrum': "Reid et al. (2010) present an estimate of the dark matter halo power spectrum, P halo( k ), derived from 110 ; 756 luminous red galaxies (LRGs) from the SDSS 7th data release (Abazajian et al. 2009). The sample extends to redshifts z GLYPH<25> 0 : 5, and is processed to identify LRGs occupying the same dark matter halo, reducing the impact of redshift-space distortions and recovering an approximation to the halo density field. The power spectrum P halo( k ) is reported in 45 bands, covering the wavenumber range 0 : 02 h Mpc GLYPH<0> 1 < k < 0 : 2 h Mpc GLYPH<0> 1 . The window functions, covariance matrix and CosmoMC likelihood module are available on the NASA LAMBDA web site 32 . \nThe halo power spectrum is plotted in Fig. 20. The blue line shows the predicted halo power spectrum from our best-fit base GLYPH<3> CDM parameters convolved with the Reid et al. (2010) win- \nFig. 20. Band-power estimates of the halo power spectrum, P halo( k ), from Reid et al. (2010) together with 1 GLYPH<27> errors. (Note that these data points are strongly correlated.) The line shows the predicted spectrum for the best-fit Planck + WP + highL base GLYPH<3> CDMparameters. \n<!-- image --> \ndow functions. Here we show the predicted halo power spectrum for the best-fit values of the 'nuisance' parameters b 0 (halo bias), a 1, and a 2 (defined in Eq. 15 of Reid et al. 2010) which relate the halo power spectrum to the dark matter power spectrum (computed using camb ). The Planck model gives GLYPH<31> 2 LRG = 40 : 4, very close to the value GLYPH<31> 2 LRG = 40 : 0 of the best-fit model of Reid et al. (2010). \nFigure 20 shows that the Planck parameters provide a good match to the shape of the halo power spectrum. However, we do not use these data (in this form) in conjunction with Planck . The BAO scale derived from these and other data is used with Planck , as summarized in Sect. 5.2. As discussed by Reid et al. (2010, see their figure 5) there is little additional information on cosmology once the BAO features are filtered from the spectrum, and hence little to be gained by adding this information to Planck . The corrections for nonlinear evolution, though small in the wavenumber range 0 : 1-0 : 2 h Mpc GLYPH<0> 1 , add to the complexity of using shape information from the halo power spectrum.", '5.5.2. Cosmic shear': 'Another key cosmological observable is the distortion of distant galaxy images by the gravitational lensing of large-scale structure, often called cosmic shear. The shear probes the (non-linear) matter density projected along the line of sight with a broad kernel. It is thus sensitive to the geometry of the Universe and the growth of large-scale structure, with a strong sensitivity to the amplitude of the matter power spectrum. \nThe most recent, and largest, cosmic shear data sets are provided by the CFHTLenS survey (Heymans et al. 2012; Erben et al. 2013), which covers 33 154 deg 2 in five optical bands with accurate shear measurements and photometric redshifts. The CFHTLenS team has released several cosmic shear results that are relevant to this paper. Benjamin et al. (2013) present results from a two-bin tomographic analysis and Heymans et al. (2013) from a finely binned tomographic analysis. Kilbinger et al. (2013) present constraints from a 2D analy- \nsis. The constraints from all of the analyses show a high degree of consistency. \nHeymans et al. (2013) estimate shear correlation functions associated with six redshift bins. Assuming a flat, GLYPH<3> CDM model, from the weak lensing data alone they find GLYPH<27> 8 ( GLYPH<10> m = 0 : 27) 0 : 46 GLYPH<6> 0 : 02 = 0 : 774 GLYPH<6> 0 : 04 (68% errors) which is consistent with the constraint found by Benjamin et al. (2013). For comparison, we find \nGLYPH<27> 8 ( GLYPH<10> m = 0 : 27) 0 : 46 = 0 : 89 GLYPH<6> 0 : 03 (68%; Planck + WP + highL) ; (59) \nwhich is discrepant at about the 2 GLYPH<27> level. Combining the tomographic lensing data with CMB constraints from WMAP -7, Heymans et al. (2013) are able to constrain the individual parameters of the flat, GLYPH<3> CDMmodel to be GLYPH<10> m = 0 : 255 GLYPH<6> 0 : 014 and h = 0 : 717 GLYPH<6> 0 : 016. The best-fit Planck value of GLYPH<10> m is 4 GLYPH<27> away from this value, while h is discrepant at nearly 3 GLYPH<27> . As might be expected, given the good agreement between the Planck and BAO distance scales, the best-fit CFHTLenS GLYPH<3> CDM cosmology is also discrepant with the BOSS data, predicting a distance ratio to z = 0 : 57 which is 5% lower than measured by BOSS (Anderson et al. 2012). This is discrepant at approximately the 3 GLYPH<27> level, comparable to the discrepancy with the Planck values. The source of the discrepancies between Planck and the CFHTLenS tomographic analyses is at present unclear, and further work will be needed to resolve them. \nKilbinger et al. (2013) give a tight constraint in the GLYPH<27> 8GLYPH<10> m plane for flat GLYPH<3> CDM models from their 2D (i.e., nontomographic) analysis. They find GLYPH<27> 8 ( GLYPH<10> m = 0 : 27) 0 : 6 = 0 : 79 GLYPH<6> 0 : 03, which, when combined with WMAP -7, gives GLYPH<10> m = 0 : 283 GLYPH<6> 0 : 010 and h = 0 : 69 GLYPH<6> 0 : 01. These results are still discrepant with the Planck best-fit, but with lower significance than the results reported by Heymans et al. (2013). \nIt is also worth noting that a recent analysis of galaxy-galaxy lensing in the SDSS survey (Mandelbaum et al. 2013) leads to the constraint GLYPH<27> 8 ( GLYPH<10> m = 0 : 25) 0 : 57 = 0 : 80 GLYPH<6> 0 : 05 for the base GLYPH<3> CDM cosmology. This is about 2 : 4 GLYPH<27> lower than expected from Planck .', '5.5.3. Counts of rich clusters': "For the base GLYPH<3> CDM model we find GLYPH<27> 8 = 0 : 828 GLYPH<6> 0 : 012 from Planck + WP + highL. This value is in excellent agreement with the WMAP -9 value of GLYPH<27> 8 = 0 : 821 GLYPH<6> 0 : 023 (Hinshaw et al. 2012). There are other ways to probe the power spectrum normalization, in addition to the cosmic shear measurements discussed above. For example, the abundances of rich clusters of galaxies are particularly sensitive to the normalization (see e.g., Komatsu & Seljak 2002). Recently, a number of studies have used tSZ-cluster mass scaling relations to constrain combinations of GLYPH<27> 8 and GLYPH<10> m (e.g., Benson et al. 2013; Reichardt et al. 2013; Hasselfield et al. 2013) including an analysis of a sample of Planck tSZ clusters (see Planck Collaboration XXVIII 2014; Planck Collaboration XXIX 2014) reported in this series of papers (Planck Collaboration XX 2014) 34 . \nThe Planck analysis uses a relation between cluster mass and tSZ signal based on comparisons with X-ray mass measurements. To take into account departures from hydrostatic equilibrium, X-ray temperature calibration, modelling of the selection function, uncertainties in scaling relations and analysis un- \ncertainties, Planck Collaboration XX (2014) assume a 'bias' between the X-ray derived masses and the true cluster masses. If the mass bias, (1 GLYPH<0> b ), is allowed to vary uniformly between 0 : 7 and 1 : 0, Planck Collaboration XX (2014) find GLYPH<27> 8( GLYPH<10> m = 0 : 27) 0 : 3 = 0 : 76 GLYPH<6> 0 : 03 for the base GLYPH<3> CDM model. In comparison, for the same model we find \nGLYPH<27> 8 ( GLYPH<10> m = 0 : 27) 0 : 3 = 0 : 87 GLYPH<6> 0 : 02 (68%; Planck + WP + highL) ; (60) \nwhich is a significant (around 3 GLYPH<27> ) discrepancy that remains unexplained. Qualitatively similar results are found from analyses of SPT clusters [ GLYPH<27> 8( GLYPH<10> m = 0 : 27) 0 : 3 = 0 : 77 GLYPH<6> 0 : 04]. Key di GLYPH<14> culties with this type of measurement, as discussed in Planck Collaboration XX (2014), include adequately modelling selection biases and calibrating cluster masses. These e GLYPH<11> ects are discussed in the analysis of ACT clusters by Hasselfield et al. (2013), who adopt a number of approaches, including folding in dynamical mass measurements, to calibrate biases in clusters mass estimates. Some of these approaches give joint GLYPH<27> 8GLYPH<10> m constraints consistent with the base GLYPH<3> CDMparameters reported here. \nAt this stage of our understanding of the biases and scatter in the cluster mass calibrations, we believe that for the purposes of this paper it is premature to use cluster counts together with CMB measurements to search for new physics. Planck Collaboration XX (2014) explore a number of possibilities for reducing the tension between Planck CMB measurements and tSZ cluster counts, including non-zero neutrino masses.", '6.1. Grid of models': "To explore possible deviations from GLYPH<3> CDM we have analysed an extensive grid of models that covers many well-motivated extensions of GLYPH<3> CDM. As in the exploration of the base GLYPH<3> CDM cosmology, we have also considered a variety of data combinations for each model. For models involving more than one additional parameter we restrict ourselves to Planck + WP combinations in order to obtain tighter constraints by leveraging the relative amplitude of the power spectrum at very low ' and high ' . Most models are run with Planck , Planck + WP, and Planck + WP + highL; additionally all are importance sampled with Planck lensing (Sect. 5.1), BAO (Sect. 5.2), SNe (Sect. 5.4), and the Riess et al. (2011) direct H 0 measurement (Sect. 5.3). For models where the non-CMB data give a large reduction in parameter volume (e.g. GLYPH<10> K models), we run separate chains instead of importance sampling. \nThese runs provide no compelling evidence for deviations from the base GLYPH<3> CDM model, and indeed, as shown in Table 10 and Fig. 21, the posteriors for individual extra parameters generally overlap the fiducial model within one standard deviation. The inclusion of BAO data shrinks further the allowed scope for deviation. The parameters of the base GLYPH<3> CDM model are relatively robust to inclusion of additional parameters, but the errors on some do broaden significantly when additional degeneracies open up, as can be seen in Fig. 21 \nThe full grid results are available online 35 . Here we summarize some of the key results, and also consider a few additional extensions.", '6.2. Early-Universe physics': 'Inflationary cosmology o GLYPH<11> ers elegant explanations of key features of our Universe, such as its large size and near spatially flat geometry. Within this scenario, the Universe underwent a brief period of accelerated expansion (Starobinsky 1979, 1982; Kazanas 1980; Guth 1981; Sato 1981; Linde 1982; Albrecht & Steinhardt 1982) during which quantum fluctuations were inflated in scale to become the classical fluctuations that we see today. In the simplest inflationary models, the primordial fluctuations are predicted to be adiabatic, nearly scaleinvariant and Gaussian (Mukhanov & Chibisov 1981; Hawking 1982; Starobinsky 1982; Guth & Pi 1982; Bardeen et al. 1983), in good agreement with CMB observations and other probes of large-scale structure. \nDespite this success, the fundamental physics behind inflation is not yet understood and there is no convincing evidence that rules out alternative scenarios for the early Universe. A large number of phenomenological models of inflation, some inspired by string theory, have been discussed in the literature (see Liddle & Lyth 2000; Bassett et al. 2006; Linde 2008, for reviews), as well as alternatives to inflation including pre-big bang scenarios (e.g., Gasperini & Veneziano 1993; Khoury et al. 2001; Boyle et al. 2004; Creminelli & Senatore 2007; Brandenberger 2012). Many of these models lead to distinctive signatures, such as departures from Gaussianity, isocurvature perturbations, or oscillatory features in the power spectrum, that are potentially observable. The detection of such signatures would o GLYPH<11> er valuable information on the physics of the early Universe and is one of the main science goals of Planck . \nIn this section we discuss basic aspects of the primordial power spectrum, such as the spectral index, departures from a pure power law, limits on tensor modes etc., and discuss the implications for inflationary cosmology. Tests of more complex models, such as multi-field inflation, are discussed in a separate paper (Planck Collaboration XXII 2014). In Planck Collaboration XXIV (2014), the Planck maps are used to constrain possible deviations from Gaussianity via measurements of the bispectrum and trispectrum. Planck Collaboration XXIII (2014) considers departures from statistical isotropy and additional tests of non-Gaussianity.', '6.2.1. Scale dependence of primordial fluctuations': "The primordial fluctuations in the base GLYPH<3> CDM model are parameterized as a pure power law with a spectral index n s (Eq. 2). Prior to Planck , CMB observations have favoured a power law index with slope n s < 1, which is expected in simple single-field slow-roll inflationary models (see e.g., Mukhanov 2007 and Eq. 66a below). The final WMAP nine-year data give n s = 0 : 972 GLYPH<6> 0 : 013 at 68% confidence (Hinshaw et al. 2012). Combining this with damping-tail measurements from ACT and SPT data gives n s = 0 : 968 GLYPH<6> 0 : 009, indicating a departure from scale invariance at the 3 GLYPH<27> level. The addition of BAO data has resulted in a stronger preference for n s < 1 (Anderson et al. 2012; Hinshaw et al. 2012; Story et al. 2012; Sievers et al. 2013). These constraints assume the basic sixparameter GLYPH<3> CDM cosmological model. Any new physics that a GLYPH<11> ects the damping tail of the CMB spectrum, such as additional relativistic particles, can alter these constraints substantially and still allow a precisely scale-invariant spectrum. \nWith Planck , a robust detection of the deviation from scale invariance can now be made from a single set of CMB observa- \nk \nln \nd \n/ \ndn \nFig. 21. 68% and 95% confidence regions on one-parameter extensions of the base GLYPH<3> CDM model for Planck + WP (red) and Planck + WP + BAO (blue). Horizontal dashed lines correspond to the fixed base model parameter value, and vertical dashed lines show the mean posterior value in the base model for Planck + WP. \n<!-- image --> \nTable 10. Constraints on one-parameter extensions to the base GLYPH<3> CDMmodel. Data combinations all include Planck combined with WMAP polarization, and results are shown for combinations with high-' CMB data and BAO. Note that we quote 95% limits here. \nFig. 22. The Planck power spectrum of Fig. 10 plotted as ' 2 D ' against multipole, compared to the best-fit base GLYPH<3> CDM model with n s = 0 : 96 (red dashed line). The best-fit base GLYPH<3> CDMmodel with n s constrained to unity is shown by the blue line. \n<!-- image --> \ntions spanning three decades in scale from ' = 2 to ' = 2500. We find \nn s = 0 : 959 GLYPH<6> 0 : 007 (68%; Planck + WP + highL) ; (61) \nfor the base GLYPH<3> CDM model, a roughly 6 GLYPH<27> departure from scale invariance. This is consistent with the results from previous CMBexperiments cited above. The statistical significance of this result is high enough that the di GLYPH<11> erence between a purely scale invariant spectrum can be seen easily in a plot of the power spectrum. Figure 22 shows the Planck spectrum of Fig. 10 plotted as ' 2 D ' compared to the base GLYPH<3> CDMfitwith n s = 0 : 96 (red dashed line) and to the best-fit base GLYPH<3> CDMcosmology with n s = 1. The n s = 1 model has more power at small scales and is strongly excluded by the Planck data. \nThe unique contribution of Planck, compared to previous experiments, is that we are able to show that the departure from scale invariance is robust to changes in the underlying theoretical model. For example, Figs. 21 and 23 show that the departure from scale invariance is not sensitive to the parameterization of the primordial fluctuations. Even if we allow a possible running of the spectral index (the parameter dn s = d ln k defined in equa- \ntion 2) and / or a component of tensor fluctuations, the Planck data favour a tilted spectrum at a high significance level. \nOur extensive grid of models allows us to investigate correlations of the spectral index with a number of cosmological parameters beyond those of the base GLYPH<3> CDM model (see Figs. 21 and 24). As expected, n s is uncorrelated with parameters describing late-time physics, including the neutrino mass, geometry, and the equation of state of dark energy. The remaining correlations are with parameters that a GLYPH<11> ect the evolution of the early Universe, including the number of relativistic species, or the helium fraction. This is illustrated in Fig. 24: modifying the standard model by increasing the number of neutrinos species, or the helium fraction, has the e GLYPH<11> ect of damping the small-scale power spectrum. This can be partially compensated by an increase in the spectral index. However, an increase in the neutrino species must be accompanied by an increased matter density to maintain the peak positions. A measurement of the matter density from the BAO measurements helps to break this degeneracy. This is clearly seen in the upper panel of Fig. 24, which shows the improvement in the constraints when BAO measurements are added to the Planck + WP + highL likelihood. With the addition of BAO measurements we find more than a 3 GLYPH<27> deviation from n s = 1 even in this extended model, with a best-fit value of n s = 0 : 969 GLYPH<6> 0 : 010 for varying relativistic species. As discussed in Sect. 6.3, we see no evidence from the Planck data for non-standard neutrino physics. \nThe simplest single-field inflationary models predict that the running of the spectral index should be of second order in inflationary slow-roll parameters and therefore small [ dn s = d ln k GLYPH<24> ( n s GLYPH<0> 1) 2 ], typically about an order of magnitude below the sensitivity limit of Planck (see e.g., Kosowsky & Turner 1995; Baumann et al. 2009). Nevertheless, it is easy to construct inflationary models that have a larger scale dependence (e.g., by adjusting the third derivative of the inflaton potential) and so it is instructive to use the Planck data to constrain dn s = d ln k . A test for dn s = d ln k is of particularly interest given the results from previous CMB experiments. \nEarly results from WMAP suggested a preference for a negative running at the 1-2 GLYPH<27> level. In the final 9-year WMAP analysis no significant running was seen using WMAP data alone, with dn s = d ln k = GLYPH<0> 0 : 019 GLYPH<6> 0 : 025 (68% confidence; Hinshaw et al. 2012. Combining WMAP data with the first data releases from ACT and SPT, Hinshaw et al. (2012) found a negative running at nearly the 2 GLYPH<27> level with dn s = d ln k = GLYPH<0> 0 : 022 GLYPH<6> 0 : 012 (see also Dunkley et al. 2011 and Keisler et al. 2011 for analysis \ns \nFig. 23. Upper : Posterior distribution for n s for the base GLYPH<3> CDM model (black) compared to the posterior when a tensor component and running scalar spectral index are added to the model (red) Middle : Constraints (68% and 95%) in the n sdn s = d ln k plane for GLYPH<3> CDM models with running (blue) and additionally with tensors (red). Lower : Constraints (68% and 95%) on n s and the tensor-to-scalar ratio r 0 : 002 for GLYPH<3> CDM models with tensors (blue) and additionally with running of the spectral index (red). The dotted line show the expected relation between r and n s for a V ( GLYPH<30> ) / GLYPH<30> 2 inflationary potential (Eqs. 66a and 66b); here N is the number of inflationary e-foldings as defined in the text. The dotted line should be compared to the blue contours, since this model predicts negligible running. All of these results use the Planck + WP + highL data combination. \n<!-- image --> \nFig. 24. Constraints on n s for GLYPH<3> CDM models with non-standard relativistic species, N e GLYPH<11> , (upper) and helium fraction, Y P, (lower). We show 68% and 95% contours for various data combinations. Note the tightening of the constraints with the addition of BAO data. \n<!-- image --> \nof ACT and SPT with earlier data from WMAP ). The ACT 3-year release, which incorporated a new region of sky, gave dn s = d ln k = GLYPH<0> 0 : 003 GLYPH<6> 0 : 013 (Sievers et al. 2013) when combined with WMAP 7 year data. With the wide field SPT data at 150 GHz, a negative running was seen at just over the 2 GLYPH<27> level, dn s = d ln k = GLYPH<0> 0 : 024 GLYPH<6> 0 : 011 (Hou et al. 2012). \nThe picture from previous CMB experiments is therefore mixed. The latest WMAP data show a 1 GLYPH<27> trend for a running, but when combined with the S12 SPT data, this trend is amplified to give a potentially interesting result. The latest ACT data go in the other direction, giving no support for a running spectral index when combined with WMAP 36 . \nThe results from Planck data are as follows (see Figs. 21 and 23): \ndn s = d ln k = GLYPH<0> 0 : 013 GLYPH<6> 0 : 009 (68%; Planck + WP); (62a) \ndn s = d ln k = GLYPH<0> 0 : 015 GLYPH<6> 0 : 009 (68%; Planck + WP + highL); (62b) \ndn s = d ln k = GLYPH<0> 0 : 011 GLYPH<6> 0 : 008 (68%; Planck + lensing \n+ WP + highL) : (62c) \nThe consistency between (62a) and (62b) shows that these results are insensitive to modelling of unresolved foregrounds. The preferred solutions have a small negative running, but not at \na high level of statistical significance. Closer inspection of the best-fits shows that the change in GLYPH<31> 2 when dn s = d ln k is included as a parameter comes almost entirely from the low multipole temperature likelihood. (The fits to the high multipole Planck likelihood have a GLYPH<1> GLYPH<31> 2 = GLYPH<0> 0 : 4 when dn s = d ln k is included.) The slight preference for a negative running is therefore driven by the spectrum at low multipoles ' < GLYPH<24> 50. The tendency for negative running is partly mitigated by including the Planck lensing likelihood (Eq. 62c). \nThe constraints on dn s = d ln k are broadly similar if tensor fluctuations are allowed in addition to a running of the spectrum (Fig. 23) . Adding tensor fluctuations, the marginalized posterior distributions for dn s = d ln k give \ndn s = d ln k = GLYPH<0> 0 : 021 GLYPH<6> 0 : 011 (68%; Planck + WP) ; (63a) \ndn s = d ln k = GLYPH<0> 0 : 022 GLYPH<6> 0 : 010 (68%; Planck + WP + highL) ; (63b) \ndn s = d ln k = GLYPH<0> 0 : 019 GLYPH<6> 0 : 010 (68%; Planck + lensing \n+ WP + highL) : (63c) \nAs with Eqs. (62a)-(62c) the tendency to favour negative running is driven by the low multipole component of the temperature likelihood not by the Planck spectrum at high multipoles . \nThis is one of several examples discussed in this section where marginal evidence for extensions to the base GLYPH<3> CDM model are favoured by the TT spectrum at low multipoles. (The low multipole spectrum is also largely responsible for the pull of the lensing amplitude, A L, to values greater than unity discussed in Sect. 5.1). The mismatch between the best-fit base GLYPH<3> CDM model and the TT spectrum at multipoles ' < GLYPH<24> 30 is clearly visible in Fig. 1. The implications of this mismatch are discussed further in Sect. 7. \nBeyond a simple running, various extended parameterizations have been developed by e.g., Bridle et al. (2003), Shafieloo & Souradeep (2008), Verde & Peiris (2008), and Hlozek et al. (2012), to test for deviations from a power-law spectrum of fluctuations. Similar techniques are applied to the Planck data in Planck Collaboration XXII (2014).", '6.2.2. Tensor fluctuations': "In the base GLYPH<3> CDM model, the fluctuations are assumed to be purely scalar modes. Primordial tensor fluctuations could also contribute to the temperature and polarization power spectra (e.g., Grishchuk 1975; Starobinsky 1979; Basko & Polnarev 1980; Crittenden et al. 1993, 1995). The most direct way of testing for a tensor contribution is to search for a magnetictype parity signature via a large-scale B -mode pattern in CMB polarization (Seljak 1997; Zaldarriaga & Seljak 1997; Kamionkowski et al. 1997). Direct B -mode measurements are challenging as the expected signal is small; upper limits measured by BICEP and QUIET give 95% upper limits of r 0 : 002 < 0 : 73 and r 0 : 002 < 2 : 8 respectively (Chiang et al. 2010; QUIET Collaboration et al. 2012) 37 . \nMeasurements of the temperature power spectrum can also be used to constrain the amplitude of tensor modes. Although such limits can appear to be much tighter than the limits from B -mode measurements, it should be borne in mind that such limits are indirect because they are derived within the context of a particular theoretical model. In the rest of this subsection, we \nwill review temperature based limits on tensor modes and then present the results from Planck . \nAdding a tensor component to the base GLYPH<3> CDM model, the WMAP 9-year results constrain r 0 : 002 < 0 : 38 at 95% confidence (Hinshaw et al. 2012). Including small-scale ACT and SPT data this improves to r 0 : 002 < 0 : 17, and to r 0 : 002 < 0 : 12 with the addition of BAO data. These limits are degraded substantially, however, in models which allow running of the scalar spectral index in addition to tensors. For such models, the WMAP data give r 0 : 002 < 0 : 50, and this limit is not significantly improved by adding high resolution CMB and BAO data. \nThe precise determination of the fourth, fifth and sixth acoustic peaks by Planck now largely breaks the degeneracy between the primordial fluctuation parameters. For the Planck + WP + highL likelihood we find \nr 0 : 002 < 0 : 11 (95%; no running) ; (64a) \nr 0 : 002 < 0 : 26 (95%; including running) : (64b) \nAs shown in Figs. 21 and 23, the tensor amplitude is weakly correlated with the scalar spectral index; an increase in n s that could match the first three peaks cannot fit the fourth and higher acoustic peak in the Planck spectrum. Likewise, the shape constraints from the fourth and higher acoustic peaks give a reduction in the correlations between a tensor mode and a running in the spectral index, leading to significantly tighter limits than from previous CMB experiments. These numbers in Eqs. (64a) and (64b) are driven by the temperature spectrum and change very little if we add non-CMB data such as BAO measurements. The Planck limits are largely decoupled from assumptions about the late-time evolution of the Universe and are close to the tightest possible limits achievable from the temperature power spectrum alone (Knox & Turner 1994; Knox 1995). \nThese limits on a tensor mode have profound implications for inflationary cosmology. The limits translate directly to an upper limit on the energy scale of inflation, \nV GLYPH<3> = (1 : 94 GLYPH<2> 10 16 GeV) 4 ( r 0 : 002 = 0 : 12) (65) \n(Linde 1983; Lyth 1984), and to the parameters of 'large-field' inflation models. Slow-roll inflation driven by a power law potential V ( GLYPH<30> ) / GLYPH<30> GLYPH<11> o GLYPH<11> ers a simple example of large-field inflation. The field values in such a model must necessarily exceed the Planck scale m Pl , and lead to a scalar spectral index and tensor amplitude of \n1 GLYPH<0> n s GLYPH<25> ( GLYPH<11> + 2) = 2 N ; (66a) \nr \nGLYPH<25> \n4 \nGLYPH<11>= \nN \n; \n(66b) \nwhere N is the number of e-foldings between the end of inflation and the time that our present day Hubble scale crossed the inflationary horizon (see e.g., Lyth & Riotto 1999). The 95% confidence limits from the Planck data are now close to the predictions of GLYPH<11> = 2 models for N GLYPH<25> 50-60 e-folds (see Fig. 23). Large-field models with quartic potentials (e.g., Linde 1982) are now firmly excluded by CMB data. Planck constraints on powerlaw and on broader classes of inflationary models are discussed in detail in Planck Collaboration XXIV (2014). Improved limits on B -modes will be required to further constrain high field models of inflation.", '6.2.3. Curvature': "An explanation of the near flatness of our observed Universe was one of the primary motivations for inflationary cosmology. Inflationary models that allow a large number of e-foldings \n<!-- image --> \nFig. 25. The Planck + WP + highL data combination (samples; colour-coded by the value of H 0) partially breaks the geometric degeneracy between GLYPH<10> m and GLYPH<10>GLYPH<3> due to the e GLYPH<11> ect of lensing in the temperature power spectrum. These limits are significantly improved by the inclusion of the Planck lensing reconstruction (black contours). Combining also with BAO (right; solid blue contours) tightly constrains the geometry to be nearly flat. \n<!-- image --> \npredict that our Universe should be very accurately spatially flat 38 . Nevertheless, by introducing fine tunings it is possible to construct inflation models with observationally interesting open geometries (e.g., Gott 1982; Linde 1995; Bucher et al. 1995; Linde 1999) or closed geometries (Linde 2003). Even more speculatively, there has been interest in models with open geometries from considerations of tunnelling events between metastable vacua within a 'string landscape' (Freivogel et al. 2006). Observational limits on spatial curvature therefore o GLYPH<11> er important additional constraints on inflationary models and fundamental physics. \nCMB temperature power spectrum measurements su GLYPH<11> er from a well-known 'geometrical degeneracy' (Bond et al. 1997; Zaldarriaga et al. 1997). Models with identical primordial spectra, physical matter densities and angular diameter distance to the last scattering surface, will have almost identical CMB temperature power spectra. This is a near perfect degeneracy (see Fig. 25) and is broken only via the integrated Sachs-Wolfe (ISW) e GLYPH<11> ect on large angular scales and gravitational lensing of the CMB spectrum (Stompor & Efstathiou 1999). The geometrical degeneracy can also be broken with the addition of probes of late time physics, including BAO, Type Ia supernova, and measurement of the Hubble constant (e.g., Spergel et al. 2007). \nRecently, the detection of the gravitational lensing of the CMB by ACT and SPT has been used to break the geometrical degeneracy, by measuring the integrated matter potential distribution. ACT constrained GLYPH<10>GLYPH<3> = 0 : 61 GLYPH<6> 0 : 29 (68% CL) in Sherwin et al. (2011), with the updated analysis in Das et al. (2013) giving GLYPH<10> K = GLYPH<0> 0 : 031 GLYPH<6> 0 : 026 (68% CL) (Sievers et al. 2013). The SPT lensing measurements combined with seven year WMAP temperature spectrum improved this limit to GLYPH<10> K = GLYPH<0> 0 : 0014 GLYPH<6> 0 : 017 (68 % CL) (van Engelen et al. 2012). \nWith Planck we detect gravitational lensing at about 26 GLYPH<27> through the 4-point function (Sect. 5.1 and Planck Collaboration XVII 2014). This strong detection of gravitational lensing allows us to constrain the curvature to \npercent level precision using observations of the CMB alone: \n100 GLYPH<10> K = GLYPH<0> 4 : 2 + 4 : 3 GLYPH<0> 4 : 8 (95%; Planck + WP + highL); (67a) 100 GLYPH<10> K = GLYPH<0> 1 : 0 + 1 : 8 GLYPH<0> 1 : 9 (95%; Planck + lensing + WP + highL) : (67b) \nThese constraints are improved substantially by the addition of BAO data. We then find \n100 GLYPH<10> K = GLYPH<0> 0 : 05 + 0 : 65 GLYPH<0> 0 : 66 (95%; Planck + WP + highL + BAO) ; (68a) 100 GLYPH<10> K = GLYPH<0> 0 : 10 + 0 : 62 GLYPH<0> 0 : 65 (95%; Planck + lensing + WP + highL + BAO) : (68b) \nThese limits are consistent with (and slightly tighter than) the results reported by Hinshaw et al. (2012) from combining the nine-year WMAP data with high resolution CMB measurements and BAO data. We find broadly similar results to Eqs. (68a) and (68b) if the Riess et al. (2011) H 0 measurement, or either of the SNe compilations discussed in Sect. 5.4, are used in place of the BAO measurements. \nIn summary, there is no evidence from Planck for any departure from a spatially flat geometry. The results of Eqs. (68a) and (68b) suggest that our Universe is spatially flat to an accuracy of better than a percent.", '6.3. Neutrino physics and constraints on relativistic components': 'A striking illustration of the interplay between cosmology and particle physics is the potential of CMB observations to constrain the properties of relic neutrinos, and possibly of additional light relic particles in the Universe (see e.g., Dodelson et al. 1996; Hu et al. 1995; Bashinsky & Seljak 2004; Ichikawa et al. 2005; Lesgourgues & Pastor 2006; Hannestad 2010). In the following subsections, we present Planck constraints on the mass of ordinary (active) neutrinos assuming no extra relics, on the density of light relics assuming they all have negligible masses, and finally on models with both light massive and massless relics.', '6.3.1. Constraints on the total mass of active neutrinos': "The detection of solar and atmospheric neutrino oscillations proves that neutrinos are massive, with at least two species being non-relativistic today. The measurement of the absolute neutrino mass scale is a challenge for both experimental particle physics and observational cosmology. The combination of CMB, largescale structure and distance measurements already excludes a large range of masses compared to beta-decay experiments. Current limits on the total neutrino mass P m GLYPH<23> (summed over the three neutrino families) from cosmology are rather model dependent and vary strongly with the data combination adopted. The tightest constraints for flat models with three families of neutrinos are typically around 0 : 3 eV (95% CL; e.g., de Putter et al. 2012). Since P m GLYPH<23> must be greater than approximately 0 : 06 eV in the normal hierarchy scenario and 0 : 1 eV in the degenerate hierarchy (Gonzalez-Garcia et al. 2012), the allowed neutrino mass window is already quite tight and could be closed further by current or forthcoming observations (Jimenez et al. 2010; Lesgourgues et al. 2013). \nCosmological models, with and without neutrino mass, have di GLYPH<11> erent primary CMB power spectra. For observationallyrelevant masses, neutrinos are still relativistic at recombination and the unique e GLYPH<11> ects of masses in the primary power spectra are small. The main e GLYPH<11> ect is around the first acoustic peak and is due to the early integrated Sachs-Wolfe (ISW) effect; neutrino masses have an impact here even for a fixed redshift of matter-radiation equality (Lesgourgues & Pastor 2012; Hall & Challinor 2012; Hou et al. 2012; Lesgourgues et al. 2013). To date, this e GLYPH<11> ect has been the dominant one in constraining the neutrino mass from CMB data, as demonstrated in Hou et al. (2012). As we shall see here, the Planck data move us into a new regime where the dominant e GLYPH<11> ect is from gravitational lensing. Increasing neutrino mass, while adjusting other parameters to remain in a high-probability region of parameter space, increases the expansion rate at z > GLYPH<24> 1 and so suppresses clustering on scales smaller than the horizon size at the nonrelativistic transition (Kaplinghat et al. 2003; Lesgourgues et al. 2006). The net e GLYPH<11> ect for lensing is a suppression of the CMB lensing potential and, for orientation, by ' = 1000 the suppression is around 10% in power for P m GLYPH<23> = 0 : 66 eV. \nHere we report constraints assuming three species of degenerate massive neutrinos. At the level of sensitivity of Planck , the e GLYPH<11> ect of mass splittings is negligible, and the degenerate model can be assumed without loss of generality. \nCombining the Planck + WP + highL data, we obtain an upper limit on the summed neutrino mass of \nX m GLYPH<23> < 0 : 66 eV (95%; Planck + WP + highL) : (69) \nThe posterior distribution is shown by the solid black curve in Fig. 26. To demonstrate that the dominant e GLYPH<11> ect leading to the constraint is gravitational lensing, we remove the lensing information by marginalizing over A L 39 . We see that the posterior broadens considerably (see the red curve in Fig. 26) to give \nX m GLYPH<23> < 1 : 08 eV [95%; Planck + WP + highL ( A L)] ; (70) \ntaking us back close to the value of 1 : 3 eV (for A L = 1) from the nine-year WMAP data (Hinshaw et al. 2012), corresponding \nFig. 26. Marginalized posterior distributions for P m GLYPH<23> in flat models from CMB data. We show results for Planck + WP + highL without (solid black) and with (red) marginalization over A L, showing how the posterior is significantly broadened by removing the lensing information from the temperature anisotropy power spectrum. The e GLYPH<11> ect of replacing the low-' temperature and ( WMAP ) polarization data with a GLYPH<28> prior is shown in solid blue ( Planck GLYPH<0> lowL + highL + GLYPH<28> prior) and of further removing the high-' data in dot-dashed blue ( Planck GLYPH<0> lowL + GLYPH<28> prior). We also show the result of including the lensing likelihood with Planck + WP + highL (dashed black) and Planck GLYPH<0> lowL + highL + GLYPH<28> prior (dashed blue). \n<!-- image --> \nto the limit above which neutrinos become non-relativistic before recombination. (The resolution of WMAP gives very little sensitivity to lensing e GLYPH<11> ects.) \nAs discussed in Sect. 5.1, the Planck + WP + highL data combination has a preference for high A L. Since massive neutrinos suppress the lensing power (like a low A L) there is a concern that the same tensions which drive A L high may give artificially tight constraints on P m GLYPH<23> . We can investigate this issue by replacing the low-' data with a prior on the optical depth (as in Sect. 5.1) and removing the high-' data. Posterior distributions with the GLYPH<28> prior, and additionally without the high-' data, are shown in Fig. 26 by the solid blue and dot-dashed blue curves, respectively. The constraint on P m GLYPH<23> does not degrade much by replacing the low-' data with the GLYPH<28> prior only, but the degradation is more severe when the high-' data are also removed: P m GLYPH<23> < 1 : 31 eV (95% CL). \nIncluding the lensing likelihood (see Sect. 5.1) has a significant, but surprising, e GLYPH<11> ect on our results. Adding the lensing likelihood to the Planck + WP + highL data combination weakens the limit on P m GLYPH<23> , \nX m GLYPH<23> < 0 : 85 eV (95%; Planck + lensing + WP + highL) ; (71) \nas shown by the dashed black curve in Fig. 26. This is representative of a general trend that the Planck lensing likelihood favours larger P m GLYPH<23> than the temperature power spectrum. Indeed, if we use the data combination Planck GLYPH<0> lowL + highL + GLYPH<28> prior, which gives a weaker constraint from the temperature power spectrum, adding lensing gives a best-fit away from zero ( P m GLYPH<23> = 0 : 46 eV; dashed blue curve in Fig. 26). However, the total GLYPH<31> 2 at the best-fit is very close to that for the best-fitting base model (which, recall, has one massive neutrino of mass 0 : 06 eV), with the improved fit \nto the lensing data ( GLYPH<1> GLYPH<31> 2 = GLYPH<0> 2 : 35) being cancelled by the poorer fit to high-' CMB data ( GLYPH<1> GLYPH<31> 2 = GLYPH<0> 2 : 15). There are rather large shifts in other cosmological parameters between these best-fit solutions corresponding to shifts along the acoustic-scale degeneracy direction for the temperature power spectrum. Note that, as well as the change in H 0 (which falls to compensate the increase in P m GLYPH<23> at fixed acoustic scale), n s, ! b and ! c change significantly keeping the lensed temperature spectrum almost constant. These latter shifts are similar to those discussed for A L in Sect. 5.1, with non-zero P m GLYPH<23> acting like A L < 1. The lensing power spectrum C GLYPH<30>GLYPH<30> ' is lower by 5 : 4% for the highermass best fit at ' = 400 and larger below ' GLYPH<25> 45 (e.g. by 0 : 6% at ' = 40), which is a similar trend to the residuals from the best-fit minimal-mass model shown in the bottom panel of Fig. 12. Planck Collaboration XVII (2014) explores the robustness of the C GLYPH<30>GLYPH<30> ' estimates to various data cuts and foregroundcleaning methods. The first ( ' = 40-85) bandpower is the least stable to these choices, although the variations are not statistically significant. We have checked that excluding this bandpower does not change the posterior for P m GLYPH<23> significantly, as expected since most of the constraining power on P m GLYPH<23> comes from the bandpowers on smaller scales. At this stage, it is unclear what to make of this mild preference for high masses from the 4-point function compared to the 2-point function. As noted in Planck Collaboration XVII (2014), the lensing measurements from ACT (Das et al. 2013) and SPT (van Engelen et al. 2012) show similar trends to those from Planck where they overlap in scale. With further Planck data (including polarization), and forthcoming measurements from the full 2500 deg 2 SPT temperature survey, we can expect more definitive results on this issue in the near future. \nApart from its impact on the early-ISW e GLYPH<11> ect and lensing potential, the total neutrino mass a GLYPH<11> ects the angular-diameter distance to last scattering, and can be constrained through the angular scale of the first acoustic peak. However, this e GLYPH<11> ect is degenerate with GLYPH<10>GLYPH<3> (and so the derived H 0) in flat models and with other late-time parameters such as GLYPH<10> K and w in more general models (Howlett et al. 2012). Late-time geometric measurements help in reducing this 'geometric' degeneracy. Increasing the neutrino masses at fixed GLYPH<18> GLYPH<3> increases the angular-diameter distance for 0 GLYPH<20> z GLYPH<20> z GLYPH<3> and reduces the expansion rate at low redshift ( z < GLYPH<24> 1) but increases it at higher redshift. The sphericallyaveraged BAO distance D V( z ) therefore increases with increasing neutrino mass at fixed GLYPH<18> GLYPH<3> , and the Hubble constant falls; see Fig. 8 of Hou et al. (2012). With the BAO data of Sect. 5.2, we find a significantly lower bound on the neutrino mass: \nX m GLYPH<23> < 0 : 23 eV (95%; Planck + WP + highL + BAO) : (72) \nFollowing the philosophy of this paper, namely to give higher weight to the BAO data compared to more complex astrophysical data, we quote the result of Eq. (72) in the abstract as our most reliable limit on the neutrino mass. The GLYPH<3> CDMmodel with minimal neutrino masses was shown in Sect. 5.3 to be in tension with recent direct measurements of H 0 which favour higher values. Increasing the neutrino mass will only make this tension worse and drive us to artificially tight constraints on P m GLYPH<23> . If we relax spatial flatness, the CMB geometric degeneracy becomes three-dimensional in models with massive neutrinos and the constraints on P m GLYPH<23> weaken considerably to \nX m GLYPH<23> < 8 > > < > > : 0 : 98 eV (95%; Planck + WP + highL) 0 : 32 eV (95%; Planck + WP + highL + BAO). (73) \nFig. 27. Marginalized posterior distribution of N e GLYPH<11> for Planck + WP + highL (black) and additionally BAO (blue), the H 0 measurement (red), and both BAO and H 0 (green). \n<!-- image --> \neff", '6.3.2. Constraints on N e GLYPH<11>': "As discussed in Sect. 2, the density of radiation in the Universe (besides photons) is usually parameterized by the e GLYPH<11> ective neutrino number N e GLYPH<11> . This parameter specifies the energy density when the species are relativistic in terms of the neutrino temperature assuming exactly three flavours and instantaneous decoupling. In the Standard Model, N e GLYPH<11> = 3 : 046, due to noninstantaneous decoupling corrections (Mangano et al. 2005). \nHowever, there has been some mild preference for N e GLYPH<11> > 3 : 046 from recent CMB anisotropy measurements (Komatsu et al. 2011; Dunkley et al. 2011; Keisler et al. 2011; Archidiacono et al. 2011; Hinshaw et al. 2012; Hou et al. 2012). This is potentially interesting, since an excess could be caused by a neutrino / anti-neutrino asymmetry, sterile neutrinos, and / or any other light relics in the Universe. In this subsection we discuss the constraints on N e GLYPH<11> from Planck in scenarios where the extra relativistic degrees of freedom are e GLYPH<11> ectively massless. \nThe physics of how N e GLYPH<11> is constrained by CMB anisotropies is explained in Bashinsky & Seljak (2004), Hou et al. (2011) and Lesgourgues et al. (2013). The main e GLYPH<11> ect is that increasing the radiation density at fixed GLYPH<18> GLYPH<3> (to preserve the angular scales of the acoustic peaks) and fixed z eq (to preserve the early-ISW effect and so first-peak height) increases the expansion rate before recombination and reduces the age of the Universe at recombination. Since the di GLYPH<11> usion length scales approximately as the square root of the age, while the sound horizon varies proportionately with the age, the angular scale of the photon di GLYPH<11> usion length, GLYPH<18> D, increases, thereby reducing power in the damping tail at a given multipole. Combining Planck , WMAP polarization and the high-' experiments gives \nN e GLYPH<11> = 3 : 36 + 0 : 68 GLYPH<0> 0 : 64 (95%; Planck + WP + highL) : (74) \nThe marginalized posterior distribution is given in Fig. 27 (black curve). The result in Eq. (74) is consistent with the value of N e GLYPH<11> = 3 : 046 of the Standard Model, but it is important to aknowledge that it is di GLYPH<14> cult to constrain N e GLYPH<11> accurately using CMB temperature measurements alone. Evidently, the nominal mission data from Planck do not strongly rule out a value as high as N e GLYPH<11> = 4. \nIncreasing N e GLYPH<11> at fixed GLYPH<18> GLYPH<3> and z eq necessarily raises the expansion rate at low redshifts too. Combining CMB with distance measurements can therefore improve constraints (see Fig. 27) although for the BAO observable r drag = D V( z ) the reduction in both r drag and D V( z ) with increasing N e GLYPH<11> partly cancel. With the BAO data of Sect. 5.2, the N e GLYPH<11> constraint is tightened to \nN e GLYPH<11> = 3 : 30 + 0 : 54 GLYPH<0> 0 : 51 (95%; Planck + WP + highL + BAO) : (75) \nOur constraints from CMB alone and CMB + BAO are compatible with the standard value N e GLYPH<11> = 3 : 046 at the 1 GLYPH<27> level, giving no evidence for extra relativistic degrees of freedom. \nSince N e GLYPH<11> is positively correlated with H 0, the tension between the Planck data and direct measurements of H 0 in the base GLYPH<3> CDM model (Sect. 5.3) can be reduced at the expense of high N e GLYPH<11> . The marginalized constraint is \nN e GLYPH<11> = 3 : 62 + 0 : 50 GLYPH<0> 0 : 48 (95%; Planck + WP + highL + H 0) : (76) \nFor this data combination, the GLYPH<31> 2 for the best-fitting model allowing N e GLYPH<11> to vary is lower by 5 : 3 than for the base N e GLYPH<11> = 3 : 046 model. The H 0 fit is much better, with GLYPH<1> GLYPH<31> 2 = GLYPH<0> 4 : 4, but there is no strong preference either way from the CMB. The low-' temperature power spectrum does weakly favour the high N e GLYPH<11> model ( GLYPH<1> GLYPH<31> 2 = GLYPH<0> 1 : 4) - since N e GLYPH<11> is positively correlated with n s (see Fig. 24) and increasing n s reduces power on large scales as does the rest of the Planck power spectrum ( GLYPH<1> GLYPH<31> 2 = GLYPH<0> 1 : 8). The high-' experiments mildly disfavour high N e GLYPH<11> in our fits ( GLYPH<1> GLYPH<31> 2 = 1 : 9). Further including the BAO data pulls the central value downwards by around 0 : 5 GLYPH<27> (see Fig. 27): \nN e GLYPH<11> = 3 : 52 + 0 : 48 GLYPH<0> 0 : 45 (95%; Planck + WP + highL + H 0 + BAO) : (77) \nThe GLYPH<31> 2 at the best-fit for this data combination ( N e GLYPH<11> = 3 : 48) is lower by 4 : 2 than the best-fitting N e GLYPH<11> = 3 : 046 model. While the high N e GLYPH<11> best-fit is preferred by Planck + WP ( GLYPH<1> GLYPH<31> 2 = GLYPH<0> 3 : 1) and the H 0 data ( GLYPH<1> GLYPH<31> 2 = GLYPH<0> 3 : 3 giving an acceptable GLYPH<31> 2 = 1 : 8 for this data point), it is disfavoured by the high-' CMB data ( GLYPH<1> GLYPH<31> 2 = 2 : 0) and slightly by BAO ( GLYPH<1> GLYPH<31> 2 = 0 : 5). We conclude that the tension between direct H 0 measurements and the CMB and BAO data in the base GLYPH<3> CDM can be relieved at the cost of additional neutrino-like physics, but there is no strong preference for this extension from the CMB damping tail. \nThroughout this subsection, we have assumed that all the relativistic components parameterized by N e GLYPH<11> consist of ordinary free-streaming relativistic particles. Extra radiation components with a di GLYPH<11> erent sound speed or viscosity parameter (Hu 1998) can provide a good fit to prePlanck CMB data (Archidiacono et al. 2013), but are not investigated in this paper.", '6.3.3. Simultaneous constraints on N e GLYPH<11> and either P m GLYPH<23> or m e GLYPH<11> GLYPH<23>; sterile': "It is interesting to investigate simultaneous contraints on N e GLYPH<11> and P m GLYPH<23> , since extra relics could coexist with neutrinos of sizeable mass, or could themselves have a mass in the eV range. Joint constraints on N e GLYPH<11> and P m GLYPH<23> have been explored several times in the literature. These two parameters are known to be partially degenerate when large-scale structure data are used (Hannestad & Ra GLYPH<11> elt 2004; Crotty et al. 2004), but their impact in the CMB is di GLYPH<11> erent and does not lead to significant correlations. \nJoint constraints on N e GLYPH<11> and P m GLYPH<23> are always modeldependent: they vary strongly with assumptions about how the \ntotal mass is split between di GLYPH<11> erent species (and they would also be di GLYPH<11> erent for models in which massive species have chemical potentials or a non-thermal phase-space distribution). We present here Planck constraints for two di GLYPH<11> erent models and describe the scenarios that motivate them. \nFirst, as in the previous subsection we assume that the three active neutrinos share a mass of P m GLYPH<23> = 3, and may coexist with extra massless species contributing to N e GLYPH<11> . In this model, when N e GLYPH<11> is greater than 3.046, GLYPH<1> N e GLYPH<11> = N e GLYPH<11> GLYPH<0> 3 : 046 gives the density of extra massless relics with arbitrary phase-space distribution. When N e GLYPH<11> < 3 : 046, the temperature of the three active neutrinos is reduced accordingly, and no additional relativistic species are assumed. In this case, the CMB constraint is \nN e GLYPH<11> = 3 : 29 + 0 : 67 GLYPH<0> 0 : 64 P m GLYPH<23> < 0 : 60 eV 9 > > = > > ; (95%; Planck + WP + highL) : (78) \nThese bounds tighten somewhat with the inclusion of BAO data, as illustrated in Fig. 28; we find \nN e GLYPH<11> = 3 : 32 + 0 : 54 GLYPH<0> 0 : 52 P m GLYPH<23> < 0 : 28 eV 9 > > = > > ; (95%; Planck + WP + highL + BAO) : (79) \nWe see that the joint constraints do not di GLYPH<11> er very much from the bounds obtained when introducing these parameters separately. The physical e GLYPH<11> ects of neutrino masses and extra relativistic relics are su GLYPH<14> ciently di GLYPH<11> erent to be resolved separately at the level of accuracy of Planck . \nIn the second model, we assume the existence of one massive sterile neutrino, in addition to the two massless and one massive active neutrino of the base model. The active neutrino mass is kept fixed at 0 : 06 eV. In particle physics, this assumption can be motivated in several ways. For example, there has recently been renewed interest in models with one light sterile neutrino in order to explain the MiniBoone anomaly reported in Aguilar-Arevalo et al. (2012), as well as reactor and Gallium anomalies (Giunti et al. 2013). The statistical significance of these results is marginal and they should not be over-interpreted. However, they do motivate investigating a model with three active neutrinos and one heavier sterile neutrino with mass m sterile. If the sterile neutrino were to thermalize with the same temperature as active neutrinos, this model would have N e GLYPH<11> GLYPH<25> 4. \nSince we wish to be more general, we assume that the extra eigenstate is either: (i) thermally distributed with an arbitrary temperature T s; or (ii) distributed proportionally to active neutrinos with an arbitrary scaling factor GLYPH<31> s in which the scaling factor is a function of the active-sterile neutrino mixing angle. This second case corresponds the Dodelson-Widrow scenario (Dodelson & Widrow 1994). The two cases are in fact equivalent for cosmological observables and do not require separate analyses (Colombi et al. 1996; Lesgourgues et al. 2013). Sampling the posterior with flat priors on N e GLYPH<11> and m sterile would not be e GLYPH<14> cient, since in the limit of small temperature T s, or small scaling factor GLYPH<31> s, the mass would be unbounded. Hence we adopt a flat prior on the 'e GLYPH<11> ective sterile neutrino mass' defined as m e GLYPH<11> GLYPH<23>; sterile GLYPH<17> (94 : 1 !GLYPH<23>; sterile) eV 40 . In the case of a thermallydistributed sterile neutrino, this parameter is related to the true mass via \nm e GLYPH<11> GLYPH<23>; sterile = ( T s = T GLYPH<23> ) 3 m thermal sterile = ( GLYPH<1> N e GLYPH<11> ) 3 = 4 m thermal sterile : (80) \n40 The factor of 94 : 1 eV here is the usual one in the relation between physical mass and energy density for non-relativistic neutrinos with physical temperature T GLYPH<23> . \n<!-- image --> \nFig. 28. Left : 2D joint posterior distribution between N e GLYPH<11> and P m GLYPH<23> (the summed mass of the three active neutrinos) in models with extra massless neutrino-like species. Right : Samples in the N e GLYPH<11> -m e GLYPH<11> GLYPH<23>; sterile plane, colour-coded by GLYPH<10> c h 2 , in models with one massive sterile neutrino family, with e GLYPH<11> ective mass m e GLYPH<11> GLYPH<23>; sterile , and the three active neutrinos as in the base GLYPH<3> CDM model. The physical mass of the sterile neutrino in the thermal scenario, m thermal sterile , is constant along the grey dashed lines, with the indicated mass in eV. The physical mass in the Dodelson-Widrow scenario, m DW sterile , is constant along the dotted lines (with the value indicated on the adjacent dashed lines). Note the pile up of points at low values of N e GLYPH<11> , caused because the sterile neutrino component behaves like cold dark matter there, introducing a strong degeneracy between the two components, as described in the text. \n<!-- image --> \nHere, recall that T GLYPH<23> = (4 = 11) 1 = 3 T GLYPH<13> is the active neutrino temperature in the instantaneous-decoupling limit and that the e GLYPH<11> ective number is defined via the energy density, GLYPH<1> N e GLYPH<11> = ( T s = T GLYPH<23> ) 4 . In the Dodelson-Widrow case the relation is given by \nm e GLYPH<11> GLYPH<23>; sterile = GLYPH<31> s m DW sterile ; (81) \nwith GLYPH<1> N e GLYPH<11> = GLYPH<31> s. For a thermalized sterile neutrino with temperature T GLYPH<23> (i.e., the temperature the active neutrinos would have if there were no heating at electron-positron annihilation), corresponding to GLYPH<1> N e GLYPH<11> = 1, the three masses are equal to each other. \nAssuming flat priors on N e GLYPH<11> and m e GLYPH<11> GLYPH<23>; sterile with m e GLYPH<11> GLYPH<23>; sterile < 3 eV, we find the results shown in Fig. 28. The physical mass, m thermal sterile in the thermal scenario is constant along the dashed lines in the figure and takes the indicated value in eV. The physical mass, m DW sterile , in the Dodelson-Widrow scenario is constant on the dotted lines. For low N e GLYPH<11> the physical mass of the neutrinos becomes very large, so that they become non-relativistic well before recombination. In the limit in which the neutrinos become non-relativistic well before any relevant scales enter the horizon, they will behave exactly like cold dark matter, and hence are completely unconstrained within the overall total constraint on the dark matter density. For intermediate cases where the neutrinos become non-relativistic well before recombination they behave like warm dark matter. The approach to the massive limit gives the tail of allowed models with large m e GLYPH<11> GLYPH<23>; sterile and low N e GLYPH<11> shown in Fig. 28, with increasing m e GLYPH<11> GLYPH<23>; sterile being compensated by decreased GLYPH<10> c h 2 to maintain the total level required to give the correct shape to the CMB power spectrum. \nFor low m e GLYPH<11> GLYPH<23>; sterile and GLYPH<1> N e GLYPH<11> away from zero the physical neutrino mass is very light, and the constraint becomes similar to the massless case. The di GLYPH<11> erent limits are continuously connected, and given the complicated shape seen in Fig. 28 it is clearly not appropriate to quote fully marginalized parameter constraints that would depend strongly on the assumed upper limit on m e GLYPH<11> GLYPH<23>; sterile . Instead we restrict attention to the case where \nthe physical mass is m thermal sterile < 10 eV, which roughly defines the region where (for the CMB) the particles are distinct from cold or warm dark matter. Using the Planck + WP + highL (abbreviated to CMB below) data combination, this gives the marginalized one-parameter constraints \nN e GLYPH<11> < 3 : 91 m e GLYPH<11> GLYPH<23>; sterile < 0 : 59 eV 9 > > = > > ; (95%; CMB for m thermal sterile < 10 eV) : (82) \nCombining further with BAO these tighten to \nN e GLYPH<11> < 3 : 80 m e GLYPH<11> GLYPH<23>; sterile < 0 : 42 eV 9 > > = > > ; (95%; CMB + BAO for m thermal sterile < 10 eV) : \n(83) \nThese bounds are only marginally compatible with a fully thermalized sterile neutrino ( N e GLYPH<11> GLYPH<25> 4) with sub-eV mass m thermal sterile GLYPH<25> m e GLYPH<11> GLYPH<23>; sterile < 0 : 5 eV that could explain the oscillation anomalies. The above contraints are also appropriate for the DodelsonWidrow scenario, but for a physical mass cut of m DW sterile < 20 eV. \nThe thermal and Dodelson-Widrow scenarios considered here are representative of a large number of possible models that have recently been investigated in the literature (Hamann et al. 2011; Diamanti et al. 2012; Archidiacono et al. 2012; Hannestad et al. 2012).", '6.4. Big bang nucleosynthesis': 'Observations of light elements abundances created during big bang nucleosynthesis (BBN) provided one of the earliest precision tests of cosmology and were critical in establishing the existence of a hot big bang. Up-to-date accounts of nucleosynthesis are given by Iocco et al. (2009) and Steigman (2012). In the standard BBN model, the abundance of light elements (parameterized by Y BBN P GLYPH<17> 4 n He = n b for helium-4 and y BBN DP GLYPH<17> 10 5 n D = n H \nfor deuterium, where ni is the number density of species i ) 41 can be predicted as a function of the baryon density ! b, the number of relativistic degrees of freedom parameterized by N e GLYPH<11> , and of the lepton asymmetry in the electron neutrino sector. Throughout this subsection, we assume for simplicity that lepton asymmetry is too small to play a role at BBN. This is a reasonable assumption, since Planck data cannot improve existing constraints on the asymmetry 42 . We also assume that there is no significant entropy increase between BBN and the present day, so that our CMB constraints on the baryon-to-photon ratio can be used to compute primordial abundances. \nTo calculate the dependence of Y BBN P and y BBN DP on the parameters ! b and N e GLYPH<11> , we use the accurate public code PArthENoPE (Pisanti et al. 2008), which incorporates values of nuclear reaction rates, particle masses and fundamental constants, and an updated estimate of the neutron lifetime ( GLYPH<28> n = 880 : 1 s; Beringer et al. 2012). Experimental uncertainties on each of these quantities lead to a theoretical error for Y BBN P ( ! b ; N e GLYPH<11> ) and y BBN DP ( ! b ; N e GLYPH<11> ). For helium, the error is dominated by the uncertainty in the neutron lifetime, leading to 43 GLYPH<27> ( Y BBN P ) = 0 : 0003. For deuterium, the error is dominated by uncertainties in several nuclear rates, and is estimated to be GLYPH<27> ( y BBN DP ) = 0 : 04 (Serpico et al. 2004). \nThese predictions for the light elements can be confronted with measurements of their abundances, and also with CMB data (which is sensitive to ! b, N e GLYPH<11> , and Y P). We shall see below that for the base cosmological model with N e GLYPH<11> = 3 : 046 (or even for an extended scenario with free N e GLYPH<11> ) the CMB data predict the primordial abundances, under the assumption of standard BBN, with smaller uncertainties than those estimated for the measured abundances. Furthermore, the CMB predictions are consistent with direct abundance measurements.', '6.4.1. Observational data on primordial abundances': "The observational constraint on the primordial helium-4 fraction used in this paper is Y BBN P = 0 : 2534 GLYPH<6> 0 : 0083 (68% CL) from the recent data compilation of Aver et al. (2012), based on spectroscopic observations of the chemical abundances in metal-poor H ii regions. The error on this measurement is dominated by systematic e GLYPH<11> ects that will be di GLYPH<14> cult to resolve in the near future. It is reassuring that the independent and conserva- \n42 Aprimordial lepton asymmetry could modify the outcome of BBN only if it were very large (of the order of 10 GLYPH<0> 3 or bigger). Such a large asymmetry is not motivated by particle physics, and is strongly constrained by BBN. Indeed, by taking into account neutrino oscillations in the early Universe, which tend to equalize the distribution function of three neutrino species, Mangano et al. (2012) derived strong bounds on the lepton asymmetry. CMB data cannot improve these bounds, as shown by Castorina et al. (2012); an exquisite sensitivity to N e GLYPH<11> would be required. Note that the results of Mangano et al. (2012) assume that N e GLYPH<11> departs from the standard value only due to the lepton asymmetry. Amodel with both a large lepton asymmetry and extra relativistic relics could be constrained by CMB data. However, we will not consider such a contrived scenario in this paper. \n43 Serpico et al. (2004) quotes GLYPH<27> ( Y BBN P ) = 0 : 0002, but since that work, the uncertainty on the neutron lifetime has been re-evaluated, from GLYPH<27> ( GLYPH<28> n) = 0 : 8 s to GLYPH<27> ( GLYPH<28> n) = 1 : 1 s (Beringer et al. 2012). \ntive method presented in Mangano & Serpico (2011) leads to an upper bound for Y BBN P that is consistent with the above estimate. The recent measurement of the proto-Solar helium abundance by Serenelli & Basu (2010) provides an even more conservative upper bound, Y BBN P < 0 : 294 at the 2 GLYPH<27> level. \nFor the primordial abundance of deuterium, data points show excess scatter above the statistical errors, indicative of systematic errors. The compilation presented in Iocco et al. (2009), based on data accumulated over several years, gives y BBN DP = 2 : 87 GLYPH<6> 0 : 22 (68% CL). Pettini & Cooke (2012) report an accurate deuterium abundance measurement in the z = 3 : 04984 lowmetallicity damped Ly GLYPH<11> system in the spectrum of QSO SDSS J1419 + 0829, which they argue is particularly well suited to deuterium abundance measurements. These authors find y BBN DP = 2 : 535 GLYPH<6> 0 : 05 (68% CL), a significantly tighter constraint than that from the Iocco et al. (2009) compilation. The Pettini-Cooke measurement is, however, a single data point, and it is important to acquire more observations of similar systems to assess whether their error estimate is consistent with possible sources of systematic error. We adopt a conservative position in this paper and compare both the Iocco et al. (2009) and the Pettini & Cooke (2012) measurements to the CMB predictions \nWe consider only the 4 He and D abundances in this paper. We do not discuss measurements of 3 He abundances since these provide only an upper bound on the true primordial 3 He fraction. Likewise, we do not discuss lithium. There has been a long standing discrepancy between the low lithium abundances measured in metal-poor stars in our Galaxy and the predictions of BBN. At present it is not clear whether this discrepancy is caused by systematic errors in the abundance measurements, or has an 'astrophysical' solution (e.g., destruction of primordial lithium) or is caused by new physics (see Fields 2011, for a recent review).", '6.4.2. Planck predictions of primordial abundances in standard BBN': 'We first restrict ourselves to the base cosmological model, with no extra relativistic degrees of freedom beyond ordinary neutrinos (and a negligible lepton asymmetry), leading to N e GLYPH<11> = 3 : 046 (Mangano et al. 2005). Assuming that standard BBN holds, and that there is no entropy release after BBN, we can compute the spectrum of CMB anisotropies using the relation Y P( ! b) given by PArthENoPE . This relation is used as the default in the grid of models discussed in this paper; we use the CosmoMC implementation developed by Hamann et al. (2008). The Planck + WP + highL fits to the base GLYPH<3> CDM model gives the following estimate of the baryon density, \n! b = 0 : 02207 GLYPH<6> 0 : 00027 (68%; Planck + WP + highL) ; (84) \nas listed in Table 5. In Fig. 29, we show this bound together with theoretical BBN predictions for Y BBN P ( ! b) and y BBN DP ( ! b). The bound of Eq. (84) leads to the predictions \nY BBN P ( ! b) = 0 : 24725 GLYPH<6> 0 : 00032 ; (85a) \ny BBN DP ( ! b) = 2 : 656 GLYPH<6> 0 : 067 ; (85b) \nwhere the errors here are 68% and include theoretical errors that are added in quadrature to those arising from uncertainties in ! b. (The theoretical error dominates the total error in the case of Y P.) 44 For helium, this prediction is in very good agreement \nFig. 29. Predictions of standard BBN for the primordial abundance of 4 He (top) and deuterium (bottom), as a function of the baryon density. The width of the green stripes corresponds to 68% uncertainties on nuclear reaction rates. The horizontal bands show observational bounds on primordial element abundances compiled by various authors, and the red vertical band shows the Planck + WP + highL bounds on ! b (all with 68% errors). BBN predictions and CMB results assume N e GLYPH<11> = 3 : 046 and no significant lepton asymmetry. \n<!-- image --> \nwith the data compilation of Aver et al. (2012), with an error that is 26 times smaller. For deuterium, the CMB + BBN prediction lies midway between the best-fit values of Iocco et al. (2009) and Pettini & Cooke (2012), but agrees with both at approximately the 1 GLYPH<27> level. These results strongly support standard BBN and show that within the framework of the base GLYPH<3> CDM model, Planck observations lead to extremely precise predictions of primordial abundances.', '6.4.3. Estimating the helium abundance directly from Planck data': 'In the CMB analysis, instead of fixing Y P to the BBN prediction, Y BBN P ( ! b), we can relax any BBN prior and let this parameter vary freely. The primordial helium fraction has an influence on the recombination history and a GLYPH<11> ects CMB anisotropies mainly through the redshift of last scattering and the diffusion damping scale (Hu et al. 1995; Trotta & Hansen 2004; Ichikawa & Takahashi 2006; Hamann et al. 2008). Extending the base GLYPH<3> CDM model by adding Y P as a free parameter with a flat prior in the range [0 : 1 ; 0 : 5], we find \nY P = 0 : 266 GLYPH<6> 0 : 021 (68%; Planck + WP + highL) : (86) \nConstraints in the Y P-! b plane are shown in Fig. 30. This figure shows that the CMB data have some sensitivity to the helium abundance. In fact, the error on the CMB estimate of Y P is only 2 : 7 times larger than the direct measurements of the primordial helium abundance by Aver et al. (2012). The CMB estimate of Y P is consistent with the observational measurements adding further support in favour of standard BBN.', '6.4.4. Extension to the case with extra relativistic relics': 'We now consider the e GLYPH<11> ects of additional relativistic degrees of freedom on photons and ordinary neutrinos (obeying the stan- \nFig. 30. Constraints in the ! bY P plane from CMB and abundance measurements. The CMB constraints are for Planck + WP + highL (red 68% and 95% contours) in GLYPH<3> CDM models with Y P allowed to vary freely. The horizontal band shows observational bounds on 4 He compiled by Aver et al. (2012) with 68% errors, while the grey region at the top of the figure delineates the conservative 95% upper bound inferred from Solar helium abundance by Serenelli & Basu (2010). The green stripe shows the predictions of standard BBN for the primordial abundance of 4 He as a function of the baryon density (with 68% errors on nuclear reaction rates). Both BBN predictions and CMB results assume N e GLYPH<11> = 3 : 046 and no significant lepton asymmetry. \n<!-- image --> \ndard model of neutrino decoupling) by adding N e GLYPH<11> as a free parameter. In the absence of lepton asymmetry, we can predict the BBN primordial abundances as a function of the two parameters ! b and N e GLYPH<11> . \nFigure 31 shows the regions in the ! bN e GLYPH<11> plane preferred by primordial abundance measurements, and by the CMB data if the standard BBN picture is correct. The regions allowed by the abundance measurements are defined by the GLYPH<31> 2 statistic \nGLYPH<31> 2 ( ! b ; N e GLYPH<11> ) GLYPH<17> GLYPH<2> y ( ! b ; N e GLYPH<11> ) GLYPH<0> y obs GLYPH<3> 2 GLYPH<27> 2 obs + GLYPH<27> 2 theory ; (87) \nwhere y ( ! b ; N e GLYPH<11> ) is the BBN prediction for either Y BBN P or y BBN DP , the quantity y obs is the observed abundance, and the two errors in the denominator are the observational and theoretical uncertainties. Figure 31 shows the edges of the 68% preferred regions in the ! bN e GLYPH<11> plane, given by GLYPH<31> 2 = GLYPH<31> 2 min + 2 : 3. \nFor the CMB data, we fit a cosmological model with seven free parameters (the six parameters of the base GLYPH<3> CDM model, plus N e GLYPH<11> ) to the Planck + WP + highL data, assuming that the primordial helium fraction is fixed by the standard BBN prediction Y BBN P ( ! b ; N e GLYPH<11> ). Figure 31 shows the joint 68% and 95% confidence contours in the ! bN e GLYPH<11> plane. The preferred regions in this plane from abundance measurements and the CMB agree remarkably well. The CMB gives approximately three times smaller error bars than primordial abundance data on both parameters. \nFig. 31. Constraints in the ! bN e GLYPH<11> plane from the CMB and abundance measurements. The blue stripes shows the 68% confidence regions from measurements of primordial element abundances assuming standard BBN: 4 He bounds compiled by Aver et al. (2012); and deuterium bounds complied by Iocco et al. (2009) or measured by Pettini & Cooke (2012). We show for comparison the 68% and 95% contours inferred from Planck + WP + highL, when N e GLYPH<11> is left as a free parameter in the CMB analysis (and Y P is fixed as a function of ! b and N e GLYPH<11> according to BBN predictions). These constraints assume no significant lepton asymmetry. \n<!-- image --> \nWe can derive constraints on N e GLYPH<11> from primordial element abundances and CMB data together by combining their likelihoods. The CMB-only confidence interval for N e GLYPH<11> is \nN e GLYPH<11> = 3 : 36 GLYPH<6> 0 : 34 (68%; Planck + WP + highL) : (88) \nWhen combined with the data reported respectively by Aver et al. (2012), Iocco et al. (2009), and Pettini & Cooke (2012), the 68% confidence interval becomes \nN e GLYPH<11> = 8 > > > > > < > > > > > : 3 : 41 GLYPH<6> 0 : 30, Y P (Aver et al.), 3 : 43 GLYPH<6> 0 : 34, y DP (Iocco et al.), 3 : 02 GLYPH<6> 0 : 27, y DP (Pettini and Cooke). (89) \nSince there is no significant tension between CMB and primordial element results, all these bounds are in agreement with the CMB-only analysis. The small error bar derived from combining the CMB with the Pettini & Cooke (2012) data point shows that further deuterium observations combined with Planck data have the potential to pin down the value of N e GLYPH<11> to high precision.', '6.4.5. Simultaneous constraints on both N e GLYPH<11> and Y P': 'In this subsection, we discuss simultaneous constraints on both N e GLYPH<11> and Y P by adding them to the six parameters of the base GLYPH<3> CDM model. Both N e GLYPH<11> and Y P have an impact on the damping tail of the CMB power spectrum by altering the ratio k GLYPH<0> 1 D = r GLYPH<3> , where k GLYPH<0> 1 D is the photon di GLYPH<11> usion length at last scattering and r GLYPH<3> is the sound horizon there. There is thus an approximate degeneracy between these two parameters along which the ratio is nearly constant. The extent of the degeneracy is limited by the characteristic phase shift of the acoustic oscillations that arises due to the free streaming of the neutrinos (Bashinsky & Seljak \nFig. 32. 2D joint posterior distribution for N e GLYPH<11> and Y P with both parameters varying freely, determined from Planck + WP + highL data. Samples are colour-coded by the value of the angular ratio GLYPH<18> D =GLYPH<18> GLYPH<3> , which is constant along the degeneracy direction. The N e GLYPH<11> -Y P relation from BBN theory is shown by the dashed curve. The vertical line shows the standard value N e GLYPH<11> = 3 : 046. The region with Y P > 0 : 294 is highlighted in grey, delineating the region that exceeds the 2 GLYPH<27> upper limit of the recent measurement of initial Solar helium abundance (Serenelli & Basu 2010), and the blue horizontal region is the 68% confidence region from the Aver et al. (2012) compilation of 4 He measurements. \n<!-- image --> \n2004). As discussed by Hou et al. (2011), the early ISW e GLYPH<11> ect also partly breaks the degeneracy, but this is less important than the e GLYPH<11> ect of the phase shifts. \nThe joint posterior distribution for N e GLYPH<11> and Y P from the Planck + WP + highL likelihood is shown in Fig. 32, with each MCMC sample colour-coded by the value of the observationally-relevant angular ratio GLYPH<18> D =GLYPH<18> GLYPH<3> / ( k D r GLYPH<3> ) GLYPH<0> 1 . The main constraint on N e GLYPH<11> and Y P comes from the precise measurement of this ratio by the CMB, leaving the degeneracy along the constant GLYPH<18> D =GLYPH<18> GLYPH<3> direction. The relation between N e GLYPH<11> and Y P from BBN theory is shown in the figure by the dashed curve 45 . The standard BBN prediction with N e GLYPH<11> = 3 : 046 is contained within the 68% confidence region. The grey region is for Y P > 0 : 294 and is the 2 GLYPH<27> conservative upper bound on the primordial helium abundance from Serenelli & Basu (2010). Most of the samples are consistent with this bound. The inferred estimates of N e GLYPH<11> and Y P from the Planck + WP + highL data are \nN e GLYPH<11> = 3 : 33 + 0 : 59 GLYPH<0> 0 : 83 (68%; Planck + WP + highL) ; (90a) \nY P = 0 : 254 + 0 : 041 GLYPH<0> 0 : 033 (68%; Planck + WP + highL) : (90b) \nWith Y P allowed to vary, N e GLYPH<11> is no longer tightly constrained by the value of GLYPH<18> D =GLYPH<18> GLYPH<3> . Instead, it is constrained, at least in part, by the impact that varying N e GLYPH<11> has on the phase shifts of the acoustic oscillations. As discussed in Hou et al. (2012), this effect explains the observed correlation between N e GLYPH<11> and GLYPH<18> GLYPH<3> , which is shown in Fig. 33. The correlation in the GLYPH<3> CDM + N e GLYPH<11> model is also plotted in the figure showing that the N e GLYPH<11> -Y P degeneracy combines with the phase shifts to generate a larger dispersion in GLYPH<18> GLYPH<3> in such models. \nFig. 33. 2D joint posterior distribution between N e GLYPH<11> and GLYPH<18> GLYPH<3> for GLYPH<3> CDMmodels with variable N e GLYPH<11> (blue) and variable N e GLYPH<11> and Y P (red). Both cases are for Planck + WP + highL data. \n<!-- image -->', '6.5. Dark energy': "Amajor challenge for cosmology is to elucidate the nature of the dark energy driving the accelerated expansion of the Universe. Perhaps the most straightforward explanation is that dark energy is a cosmological constant. An alternative is dynamical dark energy (Wetterich 1988; Ratra & Peebles 1988; Caldwell et al. 1998), usually based on a scalar field. In the simplest models, the field is very light, has a canonical kinetic energy term and is minimally coupled to gravity. In such models the dark energy sound speed equals the speed of light and it has zero anisotropic stress. It thus contributes very little to clustering. We shall only consider such models in this subsection. \nA cosmological constant has an equation of state w GLYPH<17> p =GLYPH<26> = GLYPH<0> 1, while scalar field models typically have time varying w with w GLYPH<21> GLYPH<0> 1. The analysis performed here is based on the 'parameterized post-Friedmann' (PPF) framework of Hu & Sawicki (2007) and Hu (2008) as implemented in camb (Fang et al. 2008b,a) and discussed earlier in Sect. 2. This allows us to investigate both regions of parameter space in which w < GLYPH<0> 1 (sometimes referred to as the 'phantom' domain) and models in which w changes with time. \nw = GLYPH<0> 1 : 13 + 0 : 24 GLYPH<0> 0 : 25 (95%; Planck + WP + BAO) ; (91) \nin good agreement with a cosmological constant ( w = GLYPH<0> 1). Using supernovae data leads to the constraints \nw = GLYPH<0> 1 : 09 GLYPH<6> 0 : 17 (95%; Planck + WP + Union2.1) ; (92a) \nFigure 34 shows the marginalized posterior distributions for w for an extension of the base GLYPH<3> CDMcosmology to models with constant w . We present results for Planck + WP and in combination with SNe or BAO data. (Note that adding in the high-' data from ACT and SPT results in little change to the posteriors shown in Fig. 34.) As expected, the CMB alone does not strongly constrain w , due to the two-dimensional geometric degeneracy in these models. We can break this degeneracy by combining the CMB data with lower redshift distance measures. Adding in BAO data tightens the constraints substantially, giving \n<!-- image --> \nFig. 34. Marginalized posterior distributions for the dark energy equation of state parameter w (assumed constant), for Planck + WP alone (green) and in combination with SNe data (SNSL in blue and the Union2.1 compilation in red) or BAO data (black). A flat prior on w from GLYPH<0> 3 to GLYPH<0> 0 : 3 was assumed and, importantly for the CMB-only constraints, the prior [20 ; 100] km s GLYPH<0> 1 Mpc GLYPH<0> 1 on H 0. The dashed grey line indicates the cosmological constant solution, w = GLYPH<0> 1. \n<!-- image --> \nw = GLYPH<0> 1 : 13 + 0 : 13 GLYPH<0> 0 : 14 (95%; Planck + WP + SNLS) ; (92b) \nThe combination with SNLS data favours the phantom domain ( w < GLYPH<0> 1) at 2 GLYPH<27> , while the Union2.1 compilation is more consistent with a cosmological constant. \nIf instead we combine Planck + WP with the Riess et al. (2011) measurement of H 0, we find \nw = GLYPH<0> 1 : 24 + 0 : 18 GLYPH<0> 0 : 19 (95%; Planck + WP + H 0) ; (93) \nwhich is in tension with w = GLYPH<0> 1 at more than the 2 GLYPH<27> level. \nThe results in Eqs. (91-93) reflect the tensions between the supplementary data sets and the Planck base GLYPH<3> CDMcosmology discussed in Sect. 5. The BAO data are in excellent agreement with the Planck base GLYPH<3> CDM model, so there is no significant preference for w , GLYPH<0> 1 when combining BAO with Planck . In contrast, the addition of the H 0 measurement, or SNLS SNe data, to the CMB data favours models with exotic physics in the dark energy sector. These trends form a consistent theme throughout this section. The SNLS data favours a lower GLYPH<10> m in the GLYPH<3> CDM model than Planck , and hence larger dark energy density today. The tension can be relieved by making the dark energy fall away faster in the past than for a cosmological constant, i.e., w < GLYPH<0> 1. \nThe constant w models are of limited physical interest. If w , GLYPH<0> 1 then it is likely to change with time. To investigate this we consider the simple linear relation in Eq. (4), w ( a ) = w 0 + wa (1 GLYPH<0> a ), which has often been used in the literature (Chevallier & Polarski 2001; Linder 2003). This parameterization approximately captures the low-redshift behaviour of light, slowly-rolling minimally-coupled scalar fields (as long as they do not contribute significantly to the total energy density at early times) and avoids the complexity of scanning a large number of possible potential shapes and initial conditions. The dynamical \nw \nFig. 35. 2D marginalized posterior distribution for w 0 and wa for Planck + WP + BAO data. The contours are 68% and 95%, and the samples are colour-coded according to the value of H 0. Independent flat priors of GLYPH<0> 3 < w 0 < GLYPH<0> 0 : 3 and GLYPH<0> 2 < wa < 2 are assumed. Dashed grey lines show the cosmological constant solution w 0 = GLYPH<0> 1 and wa = 0. \n<!-- image --> \nevolution of w ( a ) can lead to distinctive imprints in the CMB (Caldwell et al. 1998) which would show up in the Planck data. \nFigure 35 shows contours of the joint posterior distribution in the w 0wa plane using Planck + WP + BAOdata (colour-coded according to the value of H 0). The points are coloured by the value of H 0, which shows a clear variation with w 0 and wa revealing the three-dimensional nature of the geometric degeneracy in such models. The cosmological constant point ( w 0 ; wa ) = ( GLYPH<0> 1 ; 0) lies within the 68% contour and the marginalized posteriors for w 0 and wa are \nw 0 = GLYPH<0> 1 : 04 + 0 : 72 GLYPH<0> 0 : 69 (95%; Planck + WP + BAO) ; (94a) \nwa \n< \n1 \n: \n32 \n(95%; \nPlanck \n+ \nWP \n+ \nBAO) \n: \n(94b) \nIncluding the H 0 measurement in place of the BAO data moves ( w 0 ; wa ) away from the cosmological constant solution towards negative wa at just under the 2 GLYPH<27> level. \nFigure 36 shows likelihood contours for ( w 0 ; wa ), now adding SNe data to Planck . As discussed in detail in Sect. 5, there is a dependence of the base GLYPH<3> CDM parameters on the choice of SNe data set, and this is reflected in Fig. 36. The results from the Planck + WP + Union2.1 data combination are in better agreement with a cosmological constant than those from the Planck + WP + SNLS combination. For the latter data combination, the cosmological constant solution lies on the 2 GLYPH<27> boundary of the ( w 0 ; wa ) distribution. \nDynamical dark energy models might also give a nonnegligible contribution to the energy density of the Universe at early times. Such early dark energy (EDE; Wetterich 2004) models may be very close to GLYPH<3> CDM recently, but have a nonzero dark energy density fraction, GLYPH<10> e, at early times. Such models complement the ( w 0 ; wa ) analysis by investigating how much dark energy can be present at high redshifts. EDE has two main e GLYPH<11> ects: it reduces structure growth in the period after last scattering; and it changes the position and height of the peaks in the CMB spectrum. \nFig. 36. 2D marginalized posterior distributions for w 0 and wa , for the data combinations Planck + WP + BAO (grey), Planck + WP + Union2.1 (red) and Planck + WP + SNLS (blue). The contours are 68% and 95%, and dashed grey lines show the cosmological constant solution. \n<!-- image --> \n0 \nThe model we adopt here is that of Doran & Robbers (2006): \nGLYPH<10> de( a ) = GLYPH<10> 0 de GLYPH<0> GLYPH<10> e(1 GLYPH<0> a GLYPH<0> 3 w 0 ) GLYPH<10> 0 de + GLYPH<10> 0 m a 3 w 0 + GLYPH<10> e(1 GLYPH<0> a GLYPH<0> 3 w 0 ) : (95) \nIt requires two additional parameters to those of the base GLYPH<3> CDM model: GLYPH<10> e, the dark energy density relative to the critical density at early times (assumed constant in this treatment); and the present-day dark energy equation of state parameter w 0. Here GLYPH<10> 0 m is the present matter density and GLYPH<10> 0 de = 1 GLYPH<0> GLYPH<10> 0 m is the present dark energy abundance (for a flat Universe). Note that the model of Eq. (95) has dark energy present over a large range of redshifts; the bounds on GLYPH<10> e can be substantially weaker if dark energy is only present over a limited range of redshifts (Pettorino et al. 2013). The presence or absence of dark energy at the epoch of last scattering is the dominant e GLYPH<11> ect on the CMB anisotropies and hence the constraints are insensitive to the addition of low redshift supplementary data such as BAO. \nThe most precise bounds on EDE arise from the analysis of CMB anisotropies (Doran et al. 2001; Caldwell et al. 2003; Calabrese et al. 2011; Reichardt et al. 2012; Sievers et al. 2013; Hou et al. 2012; Pettorino et al. 2013). Using Planck + WP + highL, we find \nGLYPH<10> e < 0 : 009 (95%; Planck + WP + highL) : (96) \n(The limit for Planck + WP is very similar: GLYPH<10> e < 0 : 010.) These bounds are consistent with and improve the recent ones of Hou et al. (2012), who give GLYPH<10> e < 0 : 013 at 95% CL, and Sievers et al. (2013), who find GLYPH<10> e < 0 : 025 at 95% CL. \nIn summary, the results on dynamical dark energy (except for those on early dark energy discussed above) are dependent on exactly what supplementary data are used in conjunction with the CMB data. ( Planck lensing does not significantly improve the constraints on the models discussed here.) Using the direct measurement of H 0, or the SNLS SNe sample, together with Planck we see preferences for dynamical dark energy at about the 2 GLYPH<27> level reflecting the tensions between these data sets and Planck in the GLYPH<3> CDMmodel. In contrast, the BAO measurements together with Planck give tight constraints which are consistent \nwith a cosmological constant. Our inclination is to give greater weight to the BAO measurements and to conclude that there is no strong evidence that the dark energy is anything other than a cosmological constant.", '6.6. Dark matter annihilation': 'Energy injection from dark matter (DM) annihilation can change the recombination history and a GLYPH<11> ect the shape of the angular CMB spectra (Chen & Kamionkowski 2004; Padmanabhan & Finkbeiner 2005; Zhang et al. 2006; Mapelli et al. 2006). As recently shown in several papers (see e.g., Galli et al. 2009, 2011; Giesen et al. 2012; Hutsi et al. 2011; Natarajan 2012; Evoli et al. 2013) CMB anisotropies o GLYPH<11> er an opportunity to constrain DM annihilation models. \nHigh-energy particles injected in the high-redshift thermal gas by DM annihilation are typically cooled down to the keV scale by high energy processes; once the shower has reached this energy scale, the secondary particles produced can ionize, excite or heat the thermal gas (Shull & van Steenberg 1985; Valdes et al. 2010); the first two processes modify the evolution of the free electron fraction x e, while the third a GLYPH<11> ects the temperature of the baryons. \nThe rate of energy release, dE = dt , per unit volume by a relic annihilating DM particle is given by \ndE dt ( z ) = 2 g GLYPH<26> 2 c c 2 GLYPH<10> 2 c (1 + z ) 6 p ann( z ) ; (97) \nwhere p ann is, in principle, a function of redshift z , defined as \np ann( z ) GLYPH<17> f ( z ) h GLYPH<27> v i m GLYPH<31> ; (98) \nwhere h GLYPH<27> v i is the thermally averaged annihilation cross-section, m GLYPH<31> is the mass of the DM particle, GLYPH<26> c is the critical density of the Universe today, g is a degeneracy factor equal to 1 = 2 for Majorana particles and 1 = 4 for Dirac particles (in the following, constraints will refer to Majorana particles), and the parameter f ( z ) indicates the fraction of energy which is absorbed overall by the gas at redshift z . \nIn Eq. (98), the factor f ( z ) depends on the details of the annihilation process, such as the mass of the DM particle and the annihilation channel (see e.g., Slatyer et al. 2009). The functional shape of f ( z ) can be taken into account using generalized parameterizations (Finkbeiner et al. 2012; Hutsi et al. 2011). However, as shown in Galli et al. (2011), Giesen et al. (2012), and Finkbeiner et al. (2012) it is possible to neglect the redshift dependence of f ( z ) to first approximation, since current data shows very little sensitivity to variations of this function. The e GLYPH<11> ects of DM annihilation can therefore be well parameterized by a single constant parameter, p ann, that encodes the dependence on the properties of the DM particles. \nWe compute here the theoretical angular power in the presence of DM annihilations, by modifying the RECFAST routine in the camb code as in Galli et al. (2011) and by making use of the package CosmoMC for Monte Carlo parameter estimation. We checked that we obtain the same results by using the CLASS Boltzmann code (Lesgourgues 2011a) and the Monte Python package (Audren et al. 2012), with DM annihilation e GLYPH<11> ects calculated either by RECFAST or HyRec (Ali-Haimoud & Hirata 2011), as detailed in Giesen et al. (2012). Besides p ann, we sample the parameters of the base GLYPH<3> CDM model and the foreground / nuisance parameters described in Sect. 4. \nFrom Planck + WP we find \np ann < 5 : 4 GLYPH<2> 10 GLYPH<0> 6 m 3 s GLYPH<0> 1 kg GLYPH<0> 1 (95; Planck + WP) : (99) \nThis constraint is weaker than that found from the full WMAP 9 temperature and polarization likelihood, p ann < 1 : 2 GLYPH<2> 10 GLYPH<0> 6 m 3 s GLYPH<0> 1 kg GLYPH<0> 1 because the Planck likelihood does not yet include polarization information at intermediate and high multipoles. In fact, the damping e GLYPH<11> ect of DM annihilation on the CMB temperature power spectrum is highly degenerate with other cosmological parameters, in particular with the scalar spectral index and the scalar amplitude, as first shown by Padmanabhan & Finkbeiner (2005). As a consequence, the constraint on the scalar spectral index is significantly weakened when p ann is allowed to vary, n s = 0 : 984 + 0 : 012 GLYPH<0> 0 : 026 , to be compared to the constraint listed in Table 2 for the base GLYPH<3> CDM cosmology, n s = 0 : 9603 GLYPH<6> 0 : 0073. \nThese degeneracies can be broken by polarization data. The e GLYPH<11> ect of DM annihilation on polarization is in fact an overall enhancement of the amplitude at large and intermediate scales, and a damping at small scales (see e.g., Fig. 1 in Galli et al. 2009 or Fig. 3 in Giesen et al. 2012). We thus expect the constraint to improve significantly with the forthcoming Planck polarization data release. We verified that adding BAO, HST or highL data to Planck + WP improves the constraints only marginally, as these datasets are not able to break the degeneracy between p ann and n s. \nOn the other hand, we observe a substantial improvement in the constraints when we combine the Planck + WP data with the Planck lensing likelihood data. For this data combination we find an upper limit of \np ann < 3 : 1 GLYPH<2> 10 GLYPH<0> 6 m 3 s GLYPH<0> 1 kg GLYPH<0> 1 (95%; Planck + lensing + WP) : (100) \nThe improvement over Eq. (99) comes from the constraining power of the lensing likelihood on A s and n s, that partially breaks the degeneracy with p ann. \nOur results are consistent with previous work and show no evidence for DM annihilation. Future release of Planck polarization data will help to break the degeneracies which currently limit the accuracy of the constraints presented here.', '6.7. Constraints on a stochastic background of primordial magnetic fields': "Large-scale magnetic fields of the order of a few GLYPH<22> G observed in galaxies and galaxy clusters may be the product of the amplification during structure formation, of primordial magnetic seeds (Ryu et al. 2012). Several models of the early Universe predict the generation of primordial magnetic fields (hereafter PMF), either during inflation or during later phase transitions (see Widrow 2002 and Widrow et al. 2012 for reviews). \nPMF have an impact on cosmological perturbations and in particular on CMB anisotropy angular power spectra (Subramanian 2006), that can be used to constrain the PMF amplitude. In this section we will derive the constraints from Planck data on a stochastic background of PMF. We are mainly interested in constraints from CMB temperature anisotropies. Therefore, we will not consider the e GLYPH<11> ect of Faraday rotation on CMB polarization anisotropies (Kosowsky & Loeb 1996; Kosowsky et al. 2005) nor non-Gaussianities associated with PMF (Brown & Crittenden 2005; Caprini et al. 2009; Seshadri & Subramanian 2009; Trivedi et al. 2010). We will restrict the analysis reported here to the non-helical case. \nA stochastic background of PMF is modelled as a fully inhomogeneous component whose energy-momentum tensor is quadratic in the fields. We assume the usual magnetohydrodynamics limit, in which PMF are frozen and the time evolution is simply given by the dilution with cosmological expansion, B ( k ; GLYPH<17> ) = B ( k ) = a ( GLYPH<17> ) 2 . We model the PMF with a simple power-law power spectrum: PB ( k ) = Ak nB , with a sharp cut o GLYPH<11> at the damping scale k D, as computed in Jedamzik et al. (1998) and Subramanian & Barrow (1998), to model the suppression of PMF on small scales. \nIt is customary to specify the amplitude of the PMF power spectrum with B GLYPH<21> , the root-mean-square of the field smoothed over length scale GLYPH<21> , defined such that \nB 2 GLYPH<21> = Z 1 0 dk k 2 2 GLYPH<25> 2 e GLYPH<0> k 2 GLYPH<21> 2 PB ( k ) : (101) \nGiven our assumed model and conventions, PMF are fully described by two parameters: the smoothed amplitude B GLYPH<21> ; and the spectral index nB . Here, we set GLYPH<21> = 1 Mpc and hence use B 1 Mpc as the parameter. \nThe components of the energy momentum tensor of PMF source all types of linear cosmological perturbations, i.e., scalar, vector, and tensor. In particular, the source terms are given by the magnetic energy density and anisotropic stress for scalar magnetized perturbations, whereas vector and tensor modes are sourced only by the magnetic anisotropic stress. In addition, both scalar and vector perturbations are a GLYPH<11> ected by the Lorentz force; PMF induce a Lorentz force on baryons modifying their evolution and in particular their velocity, but during the tight-coupling regime between matter and radiation the Lorentz force also has an indirect e GLYPH<11> ect on photons. \nFor the computation of magnetized angular power spectra, we use the analytic approximations for the PMF energymomentum tensor components given in Paoletti & Finelli (2011). We consider here the regular mode for magnetic scalar perturbations, with the initial conditions of Paoletti et al. (2009) (see Giovannini 2004 for earlier calculations) and Shaw & Lewis (2010) (which describes the singular passive mode, depending on the generation time of PMF). \nPrevious analyses show that the main impact of PMF on the CMB anisotropy angular power spectrum is at small angular scales, well into the Silk damping regime. The dominant mode is the magnetic vector mode which peaks at ' GLYPH<24> 20003000 (Mack et al. 2002; Lewis 2004). The scalar magnetic mode is the dominant PMF contribution on large and intermediate angular scales (Giovannini 2007; Giovannini & Kunze 2008; Finelli et al. 2008). The tensor contribution is always subdominant with respect to the other two and it is negligible for the purposes of this analysis. \nWeinclude the scalar and vector magnetized contributions to the angular power spectrum within the MCMC analysis to derive the constraints on the PMF amplitude and spectral index using Planck TT data. We vary the magnetic parameters B 1 Mpc = nG and nB , in addition to the other cosmological parameters of the base GLYPH<3> CDM cosmology (this analysis assumes massless neutrinos, rather than the default value of a single eigenstate of mass 0 : 06 eV used in the rest of this paper). We adopt as prior ranges for the parameters [0 ; 10] for B 1 Mpc = nG and [ GLYPH<0> 2 : 99 ; 3] for the spectral index nB . The lower bound nB > GLYPH<0> 3 is necessary to avoid infrared divergences in the PMF energy momentum tensor correlators. \nWe perform analyses with Planck + WP and Planck + WP + highL likelihood combinations. Results are \nFig. 37. Constraints on the root-mean-square amplitude of the primordial magnetic field (for a smoothing scale of 1 Mpc) obtained with Planck + WP (black) and Planck + WP + highL (red). \n<!-- image --> \nshown in Fig. 37. We find that the cosmological parameters are in agreement with those estimated assuming no PMF, confirming that the magnetic parameters are not degenerate with the cosmological parameters of the base GLYPH<3> CDM model. The constraints on PMF with the Planck + WP likelihood are B 1 Mpc < 4 : 1 nG, with a preference for negative spectral indices at the 95% confidence level. These limits are improved using Planck + WP + highL to B 1 Mpc < 3 : 4 nG with nB < 0 preferred at the 95% confidence level. The new constraints are consistent with, and slightly tighter, than previous limits based on combining WMAP -7 data with high-resolution CMB data (see e.g. Paoletti & Finelli 2011; Shaw & Lewis 2012; Paoletti & Finelli 2012).", '6.8. Constraints on variation of the fine-structure constant': "The GLYPH<3> CDM model assumes the validity of General Relativity on cosmological scales, as well as the physics of the standard model of particle physics. One possible extension, which may have motivations in fundamental physics, is to consider variations of dimensionless constants. Such variations can be constrained through tests on astrophysical scales (Uzan 2003, 2011). \nAnumber of physical systems have been used, spanning different time scales, to set constraints on variations of the fundamental constants. These range from atomic clocks in the laboratory at a redshift z = 0 to BBN at z GLYPH<24> 10 8 . However, apart from the claims of varying GLYPH<11> based on high resolution quasar absorption-line spectra (Webb et al. 2001; Murphy et al. 2003) 46 , there is no other evidence for time-variable fundamental constants. \nCMB temperature anisotropies have been used extensively to constrain the variation of fundamental constants over cosmic timescales. The temperature power spectrum is sensitive to the variation of the fine-structure constant GLYPH<11> , the electron-to-proton \nTable 11. Constraints on the cosmological parameters of the base GLYPH<3> CDM model with the addition of a varying fine-structure constant. We quote GLYPH<6> 1 GLYPH<27> errors. Note that for WMAP there is a strong degeneracy between H 0 and GLYPH<11> , which is why the error on GLYPH<11>=GLYPH<11> 0 is much larger than for Planck . \n<!-- image --> \n<!-- image --> \nFig. 38. Left : Likelihood contours (68% and 95%) in the GLYPH<11>=GLYPH<11> 0H 0 plane for the WMAP -9 (red), Planck + WP(blue), Planck + WP + H 0 (purple), and Planck + WP + BAO (green) data combinations. Middle : As left, but in the GLYPH<11>=GLYPH<11> 0GLYPH<10> b h 2 plane. Right : Marginalized posterior distributions of GLYPH<11>=GLYPH<11> 0 for these data combinations. \n<!-- image --> \nmass ratio GLYPH<22> , and the gravitational constant GLYPH<11> g GLYPH<17> Gm 2 p = ~ c . A variation of G can a GLYPH<11> ect the Friedmann equation, and also raises the issue of consistency in the overall theory of gravity. However, a variation of the non-gravitational constants ( GLYPH<11> and m e) is more straightforward to analyse, mostly inducing a modification of the interaction between light and atoms (shifts in the energy levels and binding energy of hydrogen and helium). This induces a modification of the ionization history of the Universe. In particular, a variation of GLYPH<11> modifies the redshift of recombination through the shift in the energy levels and the Thomson scattering cross-section. An increase in GLYPH<11> induces a shift of the position of the first acoustic peak, which is inversely proportional to the sound horizon at last scattering. The larger redshift of last scattering also produces a larger early ISW e GLYPH<11> ect, and hence a higher amplitude of the first acoustic peak. Finally, an increase in GLYPH<11> decreases di GLYPH<11> usive damping at high multipoles. For earlier studies of varying constants using the CMB (see e.g., Kaplinghat et al. 1999; Avelino et al. 2000; Martins et al. 2004; Rocha et al. 2004; Nakashima et al. 2008, 2010; Menegoni et al. 2009; Landau & Sc'occola 2010). \nThe analysis presented here focuses solely on the time variation of the fine-structure constant GLYPH<11> , in addition to the parameters of the base GLYPH<3> CDM model, using a modified form of the RECFAST recombination code (Hannestad 1999; Martins et al. 2004; Rocha et al. 2004). Selected results are given in Table 11, which compares parameter constraints from Planck and from our own analysis of the full WMAP -9 TT , TE and EE likelihood. From CMB data alone, Planck improves the constraints from a 2% variation in GLYPH<11> to about 0 : 4%. Planck thus improves the limit by a factor of around five, while the constraints on the param- \neters of the base GLYPH<3> CDM model change very little with the addition of a time-varying GLYPH<11> . These results are in good agreement with earlier forecasts (Rocha et al. 2004). \nGiven the apparent tension between the base GLYPH<3> CDMparameters from Planck and direct measurements of H 0 discussed in Sect. 5.3), we include further information from the H 0 prior and BAO data (see Sect. 5.2). Figure 38 compares the constraints in the ( GLYPH<11>=GLYPH<11> 0 ; H 0) and ( GLYPH<11>=GLYPH<11> 0 ; GLYPH<10> b h 2 ) planes and also shows the marginalized posterior distribution of GLYPH<11>=GLYPH<11> 0 for the various data combinations. \nThe constraint on GLYPH<11> is slightly improved by including the BAO data (via a tightening of the parameters of the base GLYPH<3> CDM model). Note that the central value of the prior on H 0 is outside the 95% confidence region, even for the Planck + WP + H 0 combination. Adding a varying GLYPH<11> does not resolve the tension between direct measurements of H 0 and the value determined from the CMB. \nIn summary, Planck data improve the constraints on GLYPH<11>=GLYPH<11> 0, with respect to those from WMAP -9 by a factor of about five. Our analysis of Planck data limits any variation in the fine-structure constant from z GLYPH<24> 10 3 to the present day to be less than approximately 0 : 4%.", '7. Discussion and conclusions 47': "The most important conclusion from this paper is the excellent agreement between the Planck temperature power spectrum at high multipoles with the predictions of the base GLYPH<3> CDM \nmodel. The base GLYPH<3> CDM model also provides a good match to the Planck power spectrum of the lensing potential, C GLYPH<30>GLYPH<30> ' , and to the TE and EE power spectra at high multipoles. \nThe high statistical significance of the Planck detection of gravitational lensing of the CMB leads to some interesting science conclusions using Planck data alone. For example, gravitational lensing breaks the 'geometrical degeneracy' and we find that the geometry of the Universe is consistent with spatial flatness to percent-level precision using CMB data alone . The Planck lensing power spectrum also leads to an interesting constraint on the reionization optical depth of GLYPH<28> = 0 : 089 GLYPH<6> 0 : 032, independent of CMB polarization measurements at low multipoles. \nThe parameters of the base GLYPH<3> CDM model are determined to extremely high precision by the Planck data. For example, the scalar spectral index is determined as n s = 0 : 9585 GLYPH<6> 0 : 0070, a 6 GLYPH<27> deviation from exact scale invariance. Even in the base GLYPH<3> CDM model, we find quite large changes in some parameters compared to previous CMB experiments 48 . In particular, from Planck we find a low value of the Hubble constant, H 0 = (67 : 3 GLYPH<6> 1 : 2) km s GLYPH<0> 1 Mpc GLYPH<0> 1 , and a high matter density, GLYPH<10> m = 0 : 315 GLYPH<6> 0 : 016. If we accept that the base GLYPH<3> CDM model is the correct cosmology, then as discussed in Sect. 5 Planck is in tension with direct measurements of the Hubble constant (at about the 2 : 5 GLYPH<27> level) and in mild tension with the SNLS Type Ia supernova compilation (at about the 2 GLYPH<27> level). For the base GLYPH<3> CDMmodel, we also find a high amplitude for the present-day matter fluctuations, GLYPH<27> 8 = 0 : 828 GLYPH<6> 0 : 012, in agreement with previous CMB experiments. This value is higher than that inferred from counts of rich clusters of galaxies, including our own analysis of Planck cluster counts (Planck Collaboration XX 2014), and in tension with the cosmic shear measurements discussed in Sect. 5.5.2. \nOne possible interpretation of these tensions is that systematic errors are not completely understood in some astrophysical measurements. The fact that the Planck results for the base GLYPH<3> CDMmodelare in such good agreement with BAO data, which are based on a simple geometrical measurement, lends support to this view. An alternative explanation is that the base GLYPH<3> CDM model is incorrect. In summary, at high multipoles, the base GLYPH<3> CDM cosmology provides an excellent fit to the spectra from Planck , ACT and SPT (for all frequency combinations), as illustrated in Figs. 7, 8 and 9, but the parameters derived from the CMB apparently conflict with some types of astrophysical measurement. \nBefore summarizing our results on extensions to the base GLYPH<3> CDM model, it is worth making some remarks on foreground modelling and the impact of this modelling on our error estimates. The addition of CMB data at high multipoles helps to constrain the model of unresolved foregrounds, in particular, the contribution from 'minor' components, such as the kinetic SZ, which are poorly constrained from Planck alone. For the base GLYPH<3> CDM model, the cosmological parameters are not limited by foreground modelling 49 , as illustrated in Fig. 4. As discussed in Appendix C, foreground modelling becomes more important in analysing extended CDM models, particularly those that have \n49 Even in the restricted case of the base GLYPH<3> CDM model, parameters can shift as a result of small changes to the theoretical assumptions. An example is given in Sect. 3.2, where we show that changing from our default assumption of P m GLYPH<23> = 0 : 06 eV to P m GLYPH<23> = 0, causes an upward shift of 0 : 4 GLYPH<27> in the value of H 0. \nstrong parameter degeneracies that are broken only via precision measurements of the damping tail in the CMB spectrum. As a crude measure of the importance of foreground modelling, we can compare parameter values with and without inclusion of the ACT and SPT data at high multipoles. A large shift in parameter values indicates a possible sensitivity to foreground modelling, and so any such result should be treated with caution. We have thus normally adopted the Planck + WP + highL likelihood combination as o GLYPH<11> ering the most reliable results for extensions to the base GLYPH<3> CDMcosmology. \nFrom an analysis of an extensive grid of models, we find no strong evidence to favour any extension to the base GLYPH<3> CDM cosmology, either from the CMB temperature power spectrum alone, or in combination with the Planck lensing power spectrum and other astrophysical data sets. \nWe find the following notable results using CMB data alone: \n- -The deviation of the scalar spectral index from unity is robust to the addition of tensor modes and to changes in the matter content of the Universe. For example, adding a tensor component we find n s = 0 : 9600 GLYPH<6> 0 : 0072, a 5 : 5 GLYPH<27> departure from n s = 1.\n- -A 95% upper limit on the tensor-to-scalar ratio of r 0 : 002 < 0 : 11. The combined contraints on n s and r 0 : 002 are on the borderline of compatibility with single-field inflation with a quadratic potential (Fig. 23).\n- -A 95% upper limit on the summed neutrino mass of P m GLYPH<23> < 0 : 66 eV.\n- -A determination of the e GLYPH<11> ective number of neutrino-like relativistic degrees of freedom of N e GLYPH<11> = 3 : 36 GLYPH<6> 0 : 34, compatible with the standard value of 3 : 046.\n- -The results from Planck are consistent with the results of standard big bang nucleosynthesis. In fact, combining the CMB data with the most recent results on the deuterium abundance, leads to the constraint N e GLYPH<11> = 3 : 02 GLYPH<6> 0 : 27, again compatible with the standard value of 3 : 046.\n- -New limits on a possible variation of the fine-structure constant, dark matter annihilation and primordial magnetic fields. \nWe also find a number of marginal (around 2 GLYPH<27> ) results, perhaps indicative of internal tension within the Planck data. Examples include the preference of the (phenomenological) lensing parameter for values greater than unity ( A L = 1 : 23 GLYPH<6> 0 : 11; Eq. 44) and for negative running ( dn s = d ln k = GLYPH<0> 0 : 015 GLYPH<6> 0 : 09; Eq. 62b). In Planck Collaboration XXII (2014), the Planck data indicate a preference for anti-correlated isocurvature modes and for models with a truncated power spectrum on large scales. None of these results have a decisive level of statistical significance, but they can all be traced to an unusual aspect of the temperature power spectrum at low multipoles. As can be seen in Fig. 1, and on an expanded scale in the left-hand panel of Fig. 39, the measured power spectrum shows a dip relative to the best-fit base GLYPH<3> CDM cosmology in the multipole range 20 < GLYPH<24> ' < GLYPH<24> 30 and an excess at ' = 40. The existence of 'glitches' in the power spectrum at low multipoles was noted by the WMAP team in the first-year papers (Hinshaw et al. 2003; Spergel et al. 2003) and acted as motivation to fit an inflation model with a steplike feature in the potential (Peiris et al. 2003). Similar investigations have been carried out by a number of authors, (see e.g., Mortonson et al. 2009, and references therein). At these low multipoles, the Planck spectrum is in excellent agreement with the WMAP nine-year spectrum (Planck Collaboration XV 2014), so it is unlikely that any of the features such as the low quadrupole or 'dip' in the multipole range 20-30 are caused by \nFig. 39. Left : Planck TT spectrum at low multipoles with 68% ranges on the posteriors. The 'rainbow' band show the best fits to the entire Planck + WP + highL likelihood for the base GLYPH<3> CDM cosmology, colour-coded according to the value of the scalar spectral index n s. Right : Limits (68% and 95%) on the relative amplitude of the base GLYPH<3> CDM fits to the Planck + WP likelihood fitted only to the Planck TT likelihood over the multipole range 2 GLYPH<20> ' GLYPH<20> ' max. \n<!-- image --> \ninstrumental e GLYPH<11> ects or Galactic foregrounds. These are real features of the CMB anisotropies. \nThe Planck data, however, constrain the parameters of the base GLYPH<3> CDM model to such high precision that there is little remaining flexibility to fit the low-multipole part of the spectrum. To illustrate this point, the right-hand panel of Fig. 39 shows the 68%and 95% limits on the relative amplitude of the base GLYPH<3> CDM model (sampling the chains constrained by the full likelihood) fitted only to the Planck TT likelihood over the multipole range 2 GLYPH<20> ' GLYPH<20> ' max. From multipoles ' max GLYPH<25> 25 to multipoles ' max GLYPH<25> 35, we see more than a 2 GLYPH<27> departure from values of unity. (The maximum deviation from unity is 2 : 7 GLYPH<27> at ' = 30.) It is di GLYPH<14> cult to know what to make of this result, and we present it here as a 'curiosity' that needs further investigation. The Planck temperature data are remarkably consistent with the predictions of the base GLYPH<3> CDM model at high multipoles, but it is also conceivable that the GLYPH<3> CDM cosmology fails at low multipoles. There are other indications, from both WMAP and Planck data for 'anomalies' at low multipoles (Planck Collaboration XXIII 2014), that may be indicative of new physics operating on the largest scales in our Universe. Interpretation of large-scale anomalies (including the results shown in Fig. 39) is di GLYPH<14> cult in the absence of a theoretical framework. The problem here is assessing the role of a posteriori choices, i.e., that inconsistencies attract our attention and influence our choice of statistical test. Nevertheless, we know so little about the physics of the early Universe that we should be open to the possibility that there is new physics beyond that assumed in the base GLYPH<3> CDMmodel. Irrespective of the interpretation, the unusual shape of the low multipole spectrum is at least partly responsible for some of the 2 GLYPH<27> e GLYPH<11> ects seen in the analysis of extensions to the GLYPH<3> CDMmodeldiscussed in Sect. 6. \nSupplementary information from astrophysical data sets has played an important role in the analysis of all previous CMB experiments. For Planck the interpretation of results combined with non-CMB data sets is not straightforward (as a consequence of the tensions discussed in Sect. 5). For the base GLYPH<3> CDMmodel, the statistical power of the Planck data is so high that we find very similar cosmological parameters if we add the Riess et al. \n(2011) constraint on H 0, or either of the two SNe samples, to those derived from the CMB data alone. In these cases, the solutions simply reflect the tensions discussed in Sect. 5, for example, including the H 0 measurement with the Planck + WP likelihood we find H 0 = (68 : 6 GLYPH<6> 1 : 2) km s GLYPH<0> 1 Mpc GLYPH<0> 1 , discrepant with the direct measurement at the 2 : 2 GLYPH<27> level. \nThe interpretation becomes more complex for extended models where astrophysical data is required to constrain parameters that cannot be determined accurately from CMB measurements alone. As an example, it is well known that CMB data alone provide weak constraints on the dark energy equation of state parameter w (see Fig. 34). The addition of BAO data to the CMB data gives a tight constraint of w = GLYPH<0> 1 : 13 GLYPH<6> 0 : 12 50 . However, adding the SNLS SNe data gives w = GLYPH<0> 1 : 135 GLYPH<6> 0 : 069 and adding the H 0 measurement gives w = GLYPH<0> 1 : 244 GLYPH<6> 0 : 095. Adding either of the two data sets which show tension with the CMBmeasurements for the base GLYPH<3> CDMmodel, draws the solutions into the phantom domain ( w < GLYPH<0> 1) at about the 2 GLYPH<27> level. In contrast, if we use the BAO data in addition to the CMB, we find no evidence for dynamical dark energy; these data are compatible with a cosmological constant, as assumed in the base GLYPH<3> CDM model. \nThe impact of additional astrophysical data is particularly complex in our investigation of neutrino physics (Sect. 6.3). We will use the e GLYPH<11> ective number of relativistic degrees of freedom, N e GLYPH<11> as an illustration. From the CMB data alone, we find N e GLYPH<11> = 3 : 36 GLYPH<6> 0 : 34. Adding BAO data gives N e GLYPH<11> = 3 : 30 GLYPH<6> 0 : 27. Both of these values are consistent with the standard value of 3 : 046. Adding the H 0 measurement to the CMB data gives N e GLYPH<11> = 3 : 62 GLYPH<6> 0 : 25 and relieves the tension between the CMB data and H 0 at the expense of new neutrino-like physics (at around the 2 : 3 GLYPH<27> level). It is possible to alleviate the tensions between the CMB, BAO, H 0 and SNLS data by invoking new physics such as an increase in N e GLYPH<11> . However, none of these cases are favoured significantly over the base GLYPH<3> CDMmodel by the Planck data (and they are often disfavoured). Any preference for new \nphysics comes almost entirely from the astrophysical data sets. It is up to the reader to decide how to interpret such results, but it is simplistic to assume that all astrophysical data sets have accurately quantified estimates of systematic errors. We have therefore tended to place greater weight on the CMB and BAO measurements in this paper rather than on more complex astrophysical data. \nOur overall conclusion is that the Planck data are remarkably consistent with the predictions of the base GLYPH<3> CDM cosmology. However, the mismatch with the temperature spectrum at low multipoles, evident in Figs. 1 and 39, and the existence of other 'anomalies' at low multipoles, is possibly indicative that the model is incomplete. The results presented here are based on a first, and relatively conservative, analysis of the Planck data. The 2014 data release will use data obtained over the full mission lifetime of Planck , including polarization data. It remains to be seen whether these data, together with new astrophysical data sets and CMB polarization measurements, will o GLYPH<11> er any convincing evidence for new physics. \nAcknowledgements. The development of Planck has been supported by: ESA; CNES and CNRS / INSU-IN2P3-INP (France); ASI, CNR, and INAF (Italy); NASA and DoE (USA); STFC and UKSA (UK); CSIC, MICINN and JA (Spain); Tekes, AoF and CSC (Finland); DLR and MPG (Germany); CSA (Canada); DTU Space (Denmark); SER / SSO (Switzerland); RCN (Norway); SFI (Ireland); FCT / MCTES (Portugal); and PRACE (EU). A description of the Planck Collaboration and a list of its members, including the technical or scientific activities in which they have been involved, can be found at http://www.sciops.esa.int/index.php? project=planck&page=Planck\\_Collaboration . We thank the referee for a comprehensive and helpful report. We also thank Jean-Philippe Uzan for his contributions to Sect. 6.8. We additionally acknowledge useful comments on the first version of this paper from a large number of scientists who have helped improve the clarity of the revised version. 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The magenta points show the Planck 100 GLYPH<2> 100 GHz spectrum corrected for extra-Galactic foregrounds with the best-fit Planck + WP + highL parameters from Table 5. The WMAP points have been rescaled by a multiplicative factor of 0 : 974 and agree to high precision point-by-point with the Planck spectrum. (Note that the errors plotted for the WMAP points show the noise errors and the cross-term between signal and noise computed from Monte Carlo simulations; they do not include CMB-foreground cross-correlations and correlated beam errors.) The rms scatter between the Planck and WMAP points over the multipole range 50 GLYPH<20> ' GLYPH<20> 400 is only 16 GLYPH<22> K 2 , i.e., after a multiplicative scaling the two spectra are consistent to within about 0.5% of the primary CMB spectrum. Similar tests are described in greater detail in Planck Collaboration XXXI (2014), including comparisons with the LFI 70 GHz spectrum. The reason for the multiplicative factor (amounting to a 1.3% di GLYPH<11> erence in the calibrations of the HFI and WMAP maps) is not fully understood and is the subject of ongoing investigations. For the purposes of this appendix, we treat the rescaling as an empirical result, i.e., after accounting for a multiplicative calibration factor, the Planck and WMAP -9 power spectra agree to high precision, with little evidence for any significant variation of the spectra with multipole. \nGiven the agreement between the WMAP -9 and Planck spectra shown in Fig. A.1, we should expect the two experiments to give similar cosmological parameters if the multipole range of Planck is restricted to ' < GLYPH<24> 1000. This is illustrated by the results of Table A.1, which lists base GLYPH<3> CDM parameters for WMAP -9 and for the Planck + WP likelihood limited to a maximum multipole of ' max = 1000. (For this restricted multipole range, we keep the foreground and other nuisance parameters fixed to the best-fit values derived from the full Planck + WP likelihood.) The cosmological parameters derived from these two likelihoods are in very good agreement. (See also Planck Collaboration XV (2014) and Appendix C for further tests of the variations of cosmological parameters from Planck as ' max is varied.) \nWe should expect the best-fit cosmological parameters to change as the maximum multipole ' max is increased, since there is additional cosmological information at higher multipoles. As a useful rule-of-thumb, the covariance of the shifts in the bestfit parameters on adding further independent data should be approximately equal to the di GLYPH<11> erence in the parameter covariances. To assess more carefully whether the cosmological parameter shifts seen in the Planck analysis of GLYPH<3> CDM models are statistically reasonable, we perform a set of Fisher-matrix-type simulations. We draw Gaussian realizations of simulated spectra, C sim , \n' \nfrom the frequency-compressed covariance matrix ˆ M '' 0 , introduced in Eq. (37), which includes contributions from beam and foreground errors. We adopt the best-fitting base GLYPH<3> CDM model to the Planck + WP + highL data as our fiducial model C fid ' and form \nGLYPH<31> 2 = X '' 0 GLYPH<1> C ' ˆ M GLYPH<0> 1 '' 0 GLYPH<1> C ' 0 + ( GLYPH<1> GLYPH<28> ) 2 GLYPH<27> 2 GLYPH<28> ; (A.1) \n51 The spectrum is a combination of all of the cross-spectra computed from the nine-year coadded maps per di GLYPH<11> erencing assembly. Crossspectra are first combined by band into VV, VW and WW spectra and the beam corrected spectra are then corrected for unresolved point sources, i.e., a Poisson term is removed to minimise residuals with respect to the WMAP best-fit GLYPH<3> CDMspectrum. The spectra are then coadded with inverse noise weighting to form a single V + Wspectrum. \nFig. A.1. Comparison of the Planck and WMAP -9 power spectra. The green points show the combined WMAP -9 V + W-band spectrum computed on the same mask used for the 100 GLYPH<2> 100 GHz Planck spectrum (with a combined WMAP + Planck mask for point sources) after rescaling the WMAP power spectrum by a multiplicative factor of 0 : 974. The magenta points show the Planck 100 GLYPH<2> 100 GHz spectrum computed on the same mask. The red line shows the best-fit Planck + WP + highL base GLYPH<3> CDM model. The lower panel shows the residuals with respect to this model. The error bars on the WMAP points show the instrumental noise together with the noise-signal errors as discussed in the text; errors are not shown for Planck . \n<!-- image --> \nFig. A.2. Variations in H 0 and n s as the maximum multipole in the Planck likelihood is increased from ' max = 1000 to 2500. The red points show the changes in parameters determined from 2000 simulations, as described in the text. The blue point shows the changes determined from the real data. \n<!-- image --> \nTable A.1. Comparison of base GLYPH<3> CDM parameters from WMAP -9 with Planck . The second column gives parameters derived from the WMAP -9 likelihood. The third column gives results for Planck + WP, with the Planck likelihood restricted to multipoles ' GLYPH<20> 1000. The fourth and fifth columns show results for the full Planck + WPand WMAP -9 likelihoods combined with the BAO data discussed in Sect. 5.2. As in the main body of the paper, we have assumed a neutrino mass of 0 : 06 eV. \nwhere \nGLYPH<1> C ' = C sim ' GLYPH<0> C fid ' GLYPH<0> X p @ C fid ' @ ap GLYPH<1> ap ; (A.2) \nand the ap are the cosmological parameters of the base model (taken here to be A s, ! b, ! c, H 0, n s and GLYPH<28> ). Since these simulations are based only on the high multipole Planck likelihood, we include a prior on GLYPH<28> in Eq. (A.1) with GLYPH<27>GLYPH<28> = 0 : 014. In addition, since the covariance matrix ˆ M '' 0 includes estimates of foreground and beam errors, which are highly correlated over a wide multipole range, we add a 'point source' amplitude as a catch-all to model uncertainties from nuisance parameters. With this machinery, we can quickly calculate the parameter shifts GLYPH<1> ap that minimise the GLYPH<31> 2 in Eq. (A.1) for di GLYPH<11> erent choices of ' max. (Note that these simulations reproduce to high precision the parameter errors and degeneracy directions of the full Planck likelihood.) \nResults for 2000 simulations are shown in Fig. A.2 in the H 0n s plane. (The results are similar for the ! b -! c plane.) Each red point in Fig. A.2 shows the parameter shifts measured from a single simulation as ' max is increased from 1000 to 2500. The blue point shows the shift in parameters for the real data. The shifts seen in the real data follow the degeneracy directions defined by the simulations (in all parameters) and lie within 1 : 6 GLYPH<27> of the dispersion of the simulated parameter shifts for any single parameter. We therefore conclude that the parameter shifts seen between Planck and WMAP -9 are statistically consistent with our expectations based on the further information contained in the power spectrum at high multipoles. \nThe last two columns in Table A.1 list the base GLYPH<3> CDM parameters for Planck + WP + BAOandfor WMAP -9 + BAO. Adding the baryon acoustic oscillation data to WMAP -9 brings the cosmological parameters closer to the Planck parameters (with or without the addition of the BAO data). This is what we would expect if the Planck base GLYPH<3> CDM cosmology is correct and the Planck , WMAP -9 and BAO data are largely free of systematic errors.", 'Appendix B: Comparison of the Planck and SPT S12 base GLYPH<3> CDM cosmologies': "The parameter values derived from Planck for the base GLYPH<3> CDM cosmology di GLYPH<11> er from those inferred by combining S12 with \nFig. B.1. The acoustic scale distance ratio r s = DV ( z ) divided by the distance ratio of the best fit WMAP -7 + SPT base GLYPH<3> CDM cosmology of S12. The points are colour coded as follows: green star (6dF); purple squares (SDSS DR7 as analysed by Percival et al. 2010); black star (SDSS DR7 as analysed by Padmanabhan et al. 2012); blue cross (BOSS DR9); and blue circles (WiggleZ). Error bars show 1 GLYPH<27> errors on the data points. The grey band shows the GLYPH<6> 1 GLYPH<27> range allowed by the WMAP -7 + SPT data. \n<!-- image --> \nWMAP -7; e.g., the best-fit values of H 0 and GLYPH<10>GLYPH<3> di GLYPH<11> er by 2 : 7 GLYPH<27> and 3 : 2 GLYPH<27> respectively, where GLYPH<27> is the uncertainty in the WMAP -7 + S12 determination. Furthermore, in Hou et al. (2012, herefter H12, a companion paper to S12) a trend in the S12 band-powers was identifed relative to the best-fit base GLYPH<3> CDM spectrum, which they tentatively reported as evidence for new physics. This again di GLYPH<11> ers from the results of Sect. 6, in which we found that the Planck data provide no evidence for any new physics beyond that incorporated in the base GLYPH<3> CDMmodel. The purpose of this appendix is to investigate (as far as we can) the origin of these parameter di GLYPH<11> erences and to comment on the trend identified in H12 in light of the more precise data we now have from Planck . \nNote that the S12 result extends the earlier work of K11 (a subset of which is used in the highL data combination in the main body of this paper) from an analysis of 790 deg 2 of sky to a total field area of 2540 deg 2 . S12 and H12 present constraints on \n<!-- image --> \nFig. B.2. Fits to the joint likelihoods for Planck and SPT S12 spectra. (a) Fits using only the 143 GLYPH<2> 143 GHz spectrum in the Planck likelihood. The blue points show the SPT data after recalibration and foreground subtraction, using the best-fit solution from the joint likelihood analysis. The magenta points show the foreground-subtracted Planck 143 GLYPH<2> 143 GHz spectrum. The lower panel show the residuals with respect to the best-fit GLYPH<3> CDMmodel to the Planck + SPT combined likelihoods (shown by the red line in the top panel) . (b) Foreground-subtracted and recalibrated SPT spectra using the best-fit parameters from the likelihood analysis of the full Planck likelihood combined with the SPT S12 likelihood. The magenta points show the best-fit Planck GLYPH<3> CDM spectrum from Fig. 10 and the red line shows the best-fit Planck + WP + highL base GLYPH<3> CDM model from the full Planck likelihood. The residuals with respect to this model are plotted in the lower panel. \n<!-- image --> \nthe base GLYPH<3> CDMmodel and extensions. Certain extended models are favoured when WMAP -7 and S12 are combined. For example, a running spectral index is favoured over a constant spectral index at the 2 : 2 GLYPH<27> level. \nThe di GLYPH<11> erences between the S12 and Planck base GLYPH<3> CDM cosmologies lead to di GLYPH<11> erent types of tension with non-CMB data. Whereas Planck is consistent to high precision with the BAO data (see Fig. 15) and shows some tension with the Riess et al. (2011) measurement of H 0, the WMAP -7 + S12 bestfit cosmology is consistent with the H 0 measurement but in tension with the BAO measurements. The latter point is illustrated by Fig. B.1, which is equivalent to Fig. 15 but uses the WMAP -7 + SPT cosmology as a reference. All of the BAO measurements lie systematically low compared to the best-fit WMAP -7 + S12 GLYPH<3> CDM cosmology. 52 This discrepancy was further motivation for the study in H12 of extensions to the standard cosmological model. \nAppendix A shows that the Planck and WMAP -9 power spectra are in good agreement with each other after correction for a multiplicative calibration factor, and lead to closely similar cosmological parameters when the Planck likelihood is restricted to multipoles less than 1000. A systematic di GLYPH<11> erence between Planck and WMAP -7 band-powers is therefore not the cause of the discrepancy between the Planck and WMAP -7 + S12 cosmologies. Alternative explanations might involve a systematic di GLYPH<11> erence between the Planck and S12 band-powers at high multipoles, or a systematic problem related to the matching of the SPT and WMAP spectra, i.e., with their relative calibration. \nWe consider first a comparison of the Planck and S12 spectra. Since these spectra have a large overlap range at high multipoles, where both experiments have high signal-to-noise, there is no need to use WMAP as an intermediary to establish a rela- \ntive calibration. We can compare the spectra directly via a joint likelihood analysis using the same foreground model that is used in the main body of this paper. Since the S12 spectrum is measured at a frequency of 150 GHz, we first present results using only the Planck 143 GLYPH<2> 143 GHz spectrum in the Planck likelihood. This reduces sensitivity to the details of the foreground modelling. Apart from small colour corrections, the foregrounds are identical, except for di GLYPH<11> erences in the Poisson point source amplitudes. \nAbsolute calibration of the SPT spectra is determined by comparing with the WMAP -7 spectrum in the multipole range 600 GLYPH<20> ' GLYPH<20> 1000. Since the spectra from both experiments are noisy in this multipole range, there is a large (roughly 3% in power) uncertainty in the absolute calibration of the S12 data. Here we use a version of the SPT S12 likelihood that does not include marginalization over calibration uncertainties. Instead, we self-consistently solve for a map calibration factor y SPT 150 between SPT and Planck . (This di GLYPH<11> ers from the analyses of S12, H12 and Calabrese et al. 2013, which use an SPT covariance matrix that includes marginalization over calibration errors, and combine with other experiments without solving for a relative calibration factor.) \nThe results are shown in Fig. B.2 (a). 53 The agreement between the two sets of band-powers is most easily seen in the lower panel in which the best-fit model has been subtracted. The best-fit calibration factor is y SPT 150 = 0 : 995, well within the prior 1.3% calibration uncertainty. The model with minimum GLYPH<31> 2 in this joint analysis has GLYPH<31> 2 SPT = 55 : 7. To quantify the probability to exceed (PTE) this value of GLYPH<31> 2 we need to determine the e GLYPH<11> ective number of degrees of freedom. The SPT data have 47 bandpowers and only two parameters that were heavily influenced by \nFig. B.3. A number of separate e GLYPH<11> ects contribute to the di GLYPH<11> erence in H 0 inferred from WMAP -7 + S12 (top of left panel) and H 0 inferred from Planck + WP(bottom of left panel), all going in the same direction. These include assumptions about neutrino masses, calibration procedures, di GLYPH<11> erences between WMAP -7 and WMAP -9, and di GLYPH<11> erences in the relative calibrations between SPT and WMAP (as explained in the text). The right panel shows calibration parameter priors (top lines of each pair) and posteriors (bottom lines of each pair). The tighter of the priors shown for WMAP -7 + S12, and that shown for WMAP -9 + S12, come from using Planck to provide the relative calibration between WMAP and S12. We plot only the posterior for the Planck + S12 relative calibration. Note that the relativecalibration parameter y SPT X is between S12 and the other indicated data set (i.e., WMAP or Planck ). \n<!-- image --> \nthem: the Poisson point source amplitude and y SPT 150 . Taking 45 as the number of degrees of freedom, we find a PTE of 13%. \nWe find similar results when we combine the S12 likelihood with the full Planck + WP + highL likelihood. This is illustrated in Fig. B.2 (b). Note that the Planck spectrum sits high compared to the best-fit spectrum at ' > GLYPH<24> 2300, but in this region of the spectrum foreground and beam errors become significant and introduce large correlations between the data points. We find a minimum GLYPH<31> 2 value of GLYPH<31> 2 SPT = 56 : 3 for the best-fit cosmological model. Again assuming 45 degrees of freedom we find a PTE of 12%. Based on these GLYPH<31> 2 values, we see no evidence of any inconsistency between the S12 band-powers and the best-fit Planck cosmological model. The parameter values for the Planck + S12 fits are listed in Table B.1. We also include the parameter values from our own WMAP -9 + S12 analysis. In this latter case, we do not include Planck -based (re)calibrations of WMAP or SPT, but allow the relative calibration between SPT and WMAP ( y SPT = WMAP 150 ) to vary. \nIf the Planck and SPT power spectra are broadly consistent with each other, then why do the WMAP -7 + S12 and Planck GLYPH<3> CDM parameter estimates di GLYPH<11> er by so much? The bulk of the di GLYPH<11> erence can be captured by just one parameter, which we choose here as H 0. The shifts in other parameters are highly correlated with the shift in H 0. \nSome factors contributing to the di GLYPH<11> erence in H 0 are summarized in Fig. B.3. We start at the top with the WMAP -7 + S12 result, which assumed zero neutrino mass. Progressing downwards in the plot, we have repeated the WMAP -7 + S12 analysis assuming a neutrino mass of 0 : 06 eV as in the Planck analysis described here. This lowers H 0 slightly. A further reduction in H 0 comes from using the Planck data to reduce the uncertainty in the WMAP -SPT relative calibration. By combining the Planck -WMAP 1.3% rescaling (see Appendix A) and the Planck -S12 \ncalibration, we can place a tight prior on the WMAP -7-S12 relative calibration. Fig. B.3 shows that this prior is roughly 1 : 5 GLYPH<27> higher than the posterior from the WMAP -7 + S12 chain that uses the nominal S12 calibration. Switching from WMAP -7 + S12 to WMAP -9 + S12, in the next step in our progression, we again see a small shift to lower H 0, with H 0 = (70 : 4 GLYPH<6> 1 : 6) km s GLYPH<0> 1 Mpc GLYPH<0> 1 . This latter value is very similar to that obtained if we replace the WMAP -9 data with the Planck + WP likelihood limited to ' max = 800 (as shown in Fig. B.3). The Planck + S12 results plotted in Fig. B.3 are from the last column of Table B.1. \nEach of the changes described above brings the base GLYPH<3> CDM cosmological parameter values from SPT closer to those derived from Planck . Our results suggest that part of the discrepancy between the WMAP -7 + S12 and Planck parameters arises from di GLYPH<14> culties in self-consistently matching the SPT to the WMAP power spectra over a limited range of multipoles. This illustrates the advantages of having a single experiment, such as Planck , covering both low and high multipoles.", 'Appendix C: Dependence of cosmological parameters in extended models on foreground modelling and likelihood choices': 'A large number of likelihood comparison tests on parameters in the base GLYPH<3> CDM cosmology are discussed in Planck Collaboration XV (2014). In the main body of this paper, we report constraints on a wide variety of extended models. In many of these models the cosmological parameters are strongly degenerate with each other and are therefore more sensitive to the detailed modelling of foregrounds, frequency choices, and likelihood methodology. In this Appendix we discuss briefly how one-parameter extensions of the GLYPH<3> CDM model are a GLYPH<11> ected by various choices.', 'C.1. Impact of foreground priors': "Throughout this paper we have used a particular parameterization of the foreground model, and marginalized over the free parameters using relatively wide priors. As discussed in Sect. 4, the choice of these priors is subjective and was guided by theoretical expectations and by other data, particularly results from high-resolution CMB experiments and the early Planck analysis of the CIB power spectrum (Planck Collaboration XVIII 2011). As discussed in Sect. 4, for Planck the dominant foregrounds are the Poisson contributions from unresolved point sources and the clustered CIB component at 217 GHz. The other components are of much lower amplitude and poorly constrained by Planck data alone. \nFor the thermal and kinetic SZ amplitudes we have imposed uniform priors of 0 GLYPH<20> A tSZ GLYPH<20> 10 and 0 GLYPH<20> A kSZ GLYPH<20> 10. These priors have little impact on the parameters derived for the base GLYPH<3> CDM model, or on the parameters of extended cosmologies if Planck is combined with ACT and SPT data at high multipoles. However, for extended cosmologies the priors on these 'minor' components do have a small impact on the cosmological parameters. Table C.1 gives results obtained from doubling the width of the SZ priors. The constraints on the extended parameters change by small amounts compared to the Planck + WP entries in Table 10, giving an impression of the sensitivity of Planck + WP numbers to minor foregrounds 54 . \nTable B.1. Parameter constraints in GLYPH<3> CDM models for various likelihood combinations as described in the text. The WMAP nineyear polarization likelihood is used in all of these fits. For Planck and SPT we use the standard foreground model, as described in Sect. 4. For WMAP , we follow the foreground treatment in Appendix A, removing only a Poisson-like term from the power spectrum. The last row of the table lists the SPT GLYPH<31> 2 value for the best-fit parameters (47 data points). \nThe use of additional high-' CMBdata to constrain the foreground parameters depends on having a foreground model that can reliably extrapolate between the scales relevant for Planck and the smaller scales where the high-resolution experiments have the tightest constraints. As a simple test of the model used in the main body of the paper, we relax here our assumption that the CIB spectral index is constant with a Gaussian prior GLYPH<13> CIB = 0 : 7 GLYPH<6> 0 : 2. Any change in CIB index between small and larger scales could lead to a bias in the foreground model subtracted from the Planck spectra, particularly in the 217 GLYPH<2> 217 GHz spectrum where the CIB is the dominant foreground component. However, as shown in Fig. C.1, and Table C.1, the inferred cosmological parameters are actually extremely insensitive to the details of the model, with very similar results obtained with no GLYPH<13> CIB prior and allowing a free running of the spectral index through the parameter d GLYPH<13> CIB = d ln ' . (Note that we have assumed here that the ' -dependence of the CIB is the same, up to an amplitude, at the di GLYPH<11> erent frequencies.) The interpretation of the foregrounds does change significantly, and indeed there is mild evidence for running of the CIB spectral index, but it has almost no impact on the cosmology. This should not be too surprising, since the CIB signal is frequency-dependent unlike the cosmological signal, but nonetheless it is reassuring that degeneracies with, for example, the SZ amplitudes do not indirectly cause biases in cosmological parameters. \nWe have not investigated extensively the impact of varying the tSZ and kSZ templates. A variety of di GLYPH<11> erent approaches (analytic, semi-analytic and numerical) have been used to estimate tSZ templates (e.g., Komatsu & Seljak 2002; Shaw et al. 2010; Sehgal et al. 2010; Trac et al. 2011; Battaglia et al. 2010, 2012). These have similar shapes at multipoles ' < GLYPH<24> 3000, relevant to Planck and to the tSZ template used here. The shape of our template is also a good match to the power spectrum of the Planck Comptony map over the multipole range 100 < GLYPH<24> ' < GLYPH<24> 1000 (Planck Collaboration XXI 2014). The normalization of the tSZ templates (i.e., their dependence on GLYPH<27> 8) and their shapes at multipoles ' > GLYPH<24> 3000, depend on uncertain gas physics (including energy injection from AGN). For this reason, we have not attempted to link the amplitude of the tSZ template to the amplitude of the matter power spectrum in the parameter analyses. The tSZ template used here is similar in shape to the Battaglia et al. (2010) template that has been used extensively in the analysis of ACTandSPTdata. The e GLYPH<11> ects of varying tSZ and kSZ templates on high-resolution CMB experiments have been investigated by \nFig. C.1. Comparison of parameter constraints from Planck + WP + highL for three CIB foreground models with di GLYPH<11> erent restrictions on the CIB spectral index GLYPH<13> CIB (assumed to be the same in the 143 and 217 GHz channels). The top six panels show cosmological parameter constraints on n s (top left) in the base GLYPH<3> CDMmodel and on single-parameter extensions of the GLYPH<3> CDM model. These are very stable to the modelling of the CIB. Each sub-plot is obtained from an independent analysis of that model with CosmoMC . The lower six panels show the constraints on a subset of the foreground parameters in the base GLYPH<3> CDMmodel, some of which change significantly. \n<!-- image --> \nTable C.1. Constraints on one-parameter extensions of the GLYPH<3> CDM model from Planck with various likelihood variations. Planck + WP + highL is used in all cases except for the column listing results for GLYPH<28> = 0 : 07 GLYPH<6> 0 : 013, where the WMAP polarization likelihood is replaced by this GLYPH<28> prior, and the ninth column, which does not include the high-' experiments and doubles the default width of the flat priors on the two SZ amplitudes A tSZ and A kSZ . The running CIB model has no prior on GLYPH<13> CIB and allows for spectral curvature through the parameter d GLYPH<13> CIB = d ln ' . The final column in the table shows the results of modelling a small systematic feature in the 217 GLYPH<2> 217 GHz spectrum, as described in Sect. C.4. \nDunkley et al. (2011), Reichardt et al. (2012), and Dunkley et al. (2013) who find very little e GLYPH<11> ect on cosmological parameters.", "C.2. WMAP low-' polarization likelihood": "The large-scale polarization from nine years of WMAP observations (Bennett et al. 2012) provides our most powerful constraint on the reionization optical depth GLYPH<28> . As shown in Sect. 3 it is not essential to use WMAP polarization information to obtain tight constraints on cosmological parameters from Planck . Using WMAP does, however, improve the constraints on the amplitude of the power spectrum and, via the partial parameter degeneracies sensitive to the relative amplitude of large and smallscale power and the amount of lensing, WMAP polarization also has an impact on other cosmological parameters. Most directly, since the small-scale CMB power scales roughly with e GLYPH<0> 2 GLYPH<28> GLYPH<27> 2 8 , the inferred value of GLYPH<27> 8 is approximately proportional to e GLYPH<28> as discussed in Sect. 3.4. \nThe polarization measurement at low multipoles is challenging because of the high level of polarized Galactic foregrounds, so it is important to assess the impact if the assumed constraint were slightly wrong. Figure C.2 shows how cosmological parameters shift if instead of using WMAP polarization we impose a prior GLYPH<28> = 0 : 07 GLYPH<6> 0 : 013 (1 GLYPH<27> ), which is about the same width but about 0 : 02 lower than the posterior obtained from WMAP (as for example might be obtained if there were some residual foreground contamination or instrument systematic). Table. C.1 shows the corresponding impact on parameter constraints with high-' CMBdata added to Planck . The shifts are consistent with the known parameter degeneracies, and the GLYPH<3> CDM constraints on GLYPH<27> 8 would shift downwards by a factor of approximately e 0 : 02 , or about 2%. For this reason, in Sect. 3.4 we have quoted a constraint on e GLYPH<0> GLYPH<28> GLYPH<27> 8, which is insensitive to possible errors in the large-scale polarization likelihood. A change in the Planck calibration would also have a similar direct e GLYPH<11> ect on the inferred physical amplitudes.", 'C.3. Planck likelihood': "The results of this paper are based on the CamSpec likelihood, which includes information from 100, 143 and 217 GHz channels, with a range of multipole cuts as reviewed briefly in Sect. 2.3 and summarized in Table 6. The combination of channels used allows a number of foreground parameters to be partially determined from Planck data alone, and as discussed \nFig. C.2. E GLYPH<11> ect on cosmological parameter constraints of replacing the WMAP low-' polarization likelihood with a prior of GLYPH<28> = 0 : 07 GLYPH<6> 0 : 013, which prefers lower values of the optical depth. The top-left sub-plot is n s in the base GLYPH<3> CDM model, while the others are for one-parameter extensions. Each sub-plot shows results from independent CosmoMC analyses of the corresponding model. \n<!-- image --> \nin Sect. 4 the foreground parameters can be determined more precisely by including additional information from other high-' data sets. Planck Collaboration XV (2014) discusses and compares the CamSpec likelihood with an alternative cross- and autospectrum likelihood, Plik . The Plik likelihood uses identical masks over the frequency range 100-217 GHz (retaining 48% of the sky), ignores correlations between multipoles and uses di GLYPH<11> erent assumptions to estimate instrument noise and to correct for Galactic dust, as described in Planck Collaboration XV (2014). For the base GLYPH<3> CDM model, these two likelihood codes give almost identical results. In this section, we investigate briefly how cosmological parameters in extended GLYPH<3> CDM models vary between the two likelihood codes and for some data cuts in the CamSpec likelihood. \nThe results are summarized in Fig. C.3 and Table C.1. Shifts in parameter values are expected, since di GLYPH<11> erent combinations of data are being used and hence have di GLYPH<11> ering amounts of noise and cosmic variance. With fewer frequencies overlapping at any angular scale, the foreground parameters are less well determined, and any degeneracies with foreground parameters are expected to open up. Fig. C.3 shows that there are noticeable shifts \nFig. C.3. Upper : comparison of the GLYPH<3> CDM constraints on n s (top-left) and single-parameter extensions of the GLYPH<3> CDM model for a variety of data cuts for Planck + WP. Each sub-plot is obtained from a separate CosmoMC analysis of the corresponding model. The dashed lines show the results from Plik , an alternative likelihood discussed in Planck Collaboration XV (2014), run here with the same SZ and CIB foreground priors as for the CamSpec results. For the extended models, the value of the additional parameter in the base GLYPH<3> CDM model is shown with the vertical dashed lines. Lower : same as the upper set of panels, but for Planck + WP + highL. Additional data from the high-' CMB experiments significantly reduce the foreground degeneracies. \n<!-- image --> \nin parameter values, but when also including additional high-' CMB data, which constrain the foreground parameters to higher precision, the relatively small di GLYPH<11> erences between CamSpec and Plik are significantly reduced. The two likelihood codes agree well, even for extended models when the high-' CMB data are added to the likelihoods. \nHowever, the preference from the temperature power spectrum for A L > 1 actually becomes stronger on adding high-' data. Reducing ' max to 2000 in the CamSpec likelihood also shifts A L to higher values, particularly with the addition of the high-' data. We do not, at this stage, have a full understanding of these shifts in A L. We note that the high-' ACT data itself favours high A L (Sievers et al. 2013), and this may be part of the reason behind the shift to high values when high-' data are included. (The truncation of Planck spectra at l max = 2000 also limits the accuracy of matching high-' data to Planck . There is then no overlapping multipole range with the SPT data and a significantly narrower overlap range for ACT.) \nThe analysis of extended GLYPH<3> CDMmodels with strong parameter degeneracies is complex and sensitive to small systematic errors in the CMB data and to errors in the foreground model. In the analysis of extended models we have usually quoted results from the Planck + WP + highL data combination, using the full CamSpec likelihood. With this combination, we utilize the high signal-to-noise ratio of the Planck spectra at 143 and 217 GHz, which have a wide multipole range with which to match high-' experiments to Planck , and gain better control of foregrounds via the inclusion of high-' data. The general 'rule of thumb' adopted in this paper has been to use the di GLYPH<11> erences between parameter constraints from Planck + WP and Planck + WP + highL as a guide to whether parameters are sensitive to errors in the foreground model (or other sources of error). We do sometimes see shifts of up to around 1 GLYPH<27> between these likelihoods in the parameter values for extended models and this needs to be borne in mind when interpreting our results. In the absence of any additional information, we take the cosmological parameters from Planck + WP + highL as our best estimates for extended models.", 'C.4. 217GHz systematic feature': "As discussed in Sect. 1, following the submission of the Planck 2013 papers, we discovered strong evidence that a small dip in the 217 GLYPH<2> 217 GHz spectrum at ' GLYPH<25> 1800 varies between surveys and is a systematic feature caused by incomplete subtraction of 4 K cooler lines from the time-ordered data. To estimate the impact of such a systematic on cosmology, we test the sensitivity of our results to adding a dip in the 217 GLYPH<2> 217 GHz spectrum in the range 1700 GLYPH<20> ' GLYPH<20> 1860, which we model as \nGLYPH<1> D 217 GLYPH<2> 217 ' = GLYPH<0> W sin ( ' GLYPH<0> 1700) GLYPH<25> 160 ! : (C.1) \n(Note that the tests described in this section were done before the submission of the 2013 papers.) Here, W is a free amplitude parameter that we marginalize over using a flat prior. For the base GLYPH<3> CDM model we find W = (26 GLYPH<6> 5) GLYPH<22> K 2 ( Planck + WP), and hence a significant (but highly a posteriori) dip amplitude is strongly preferred by the data, consistent with a systematic effect. The impact on the cosmological parameters is small, but not negligible, typically causing shifts of below 0 : 5 GLYPH<27> . Marginalizing over the dip amplitude W raises the mean H 0 in the base model by approximately 0 : 3 GLYPH<27> , and gives comparable small shifts to one-parameter extensions, as summarized in the final column (labelled '217 systematic') of Table C.1. Marginalizing over W does not significantly change the marginalized value of A L. \nIn summary, the ' = 1800 dip in the 217 GLYPH<2> 217 GHz spectrum has a non-negligible, but small, impact on cosmological parameters, even for extensions to the base GLYPH<3> CDM model. 55 The impact on cosmological parameters is typically less than 0 : 5 GLYPH<27> , comparable to the shifts caused by uncertainties in the modelling of unresolved foregrounds. However, tests designed to search for localized features in the Planck power spectrum can respond strongly to the systematic e GLYPH<11> ect, as reported in Planck Collaboration XXII (2014). Users of the Planck likelihood should bear this in mind. \n- 1 APC, AstroParticule et Cosmologie, Universit'e Paris Diderot, CNRS / IN2P3, CEA / lrfu, Observatoire de Paris, Sorbonne Paris Cit'e, 10, rue Alice Domon et L'eonie Duquet, 75205 Paris Cedex 13, France\n- 2 Aalto University Metsahovi Radio Observatory, Metsahovintie 114, FIN-02540 Kylmala, Finland\n- 3 African Institute for Mathematical Sciences, 6-8 Melrose Road, Muizenberg, Cape Town, South Africa\n- 4 Agenzia Spaziale Italiana Science Data Center, Via del Politecnico snc, 00133, Roma, Italy\n- 5 Agenzia Spaziale Italiana, Viale Liegi 26, Roma, Italy\n- 6 Astrophysics Group, Cavendish Laboratory, University of Cambridge, J J Thomson Avenue, Cambridge CB3 0HE, U.K.\n- 7 Astrophysics & Cosmology Research Unit, School of Mathematics, Statistics & Computer Science, University of KwaZulu-Natal, Westville Campus, Private Bag X54001, Durban 4000, South Africa\n- 8 Atacama Large Millimeter / submillimeter Array, ALMA Santiago Central O GLYPH<14> ces, Alonso de Cordova 3107, Vitacura, Casilla 763 0355, Santiago, Chile\n- 9 CITA, University of Toronto, 60 St. George St., Toronto, ON M5S 3H8, Canada\n- 10 CNRS, IRAP, 9 Av. colonel Roche, BP 44346, F-31028 Toulouse cedex 4, France\n- 11 California Institute of Technology, Pasadena, California, U.S.A.\n- 12 Centre for Theoretical Cosmology, DAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, U.K.\n- 13 Centro de Estudios de F'ısica del Cosmos de Arag'on (CEFCA), Plaza San Juan, 1, planta 2, E-44001, Teruel, Spain\n- 14 Computational Cosmology Center, Lawrence Berkeley National Laboratory, Berkeley, California, U.S.A.\n- 15 Consejo Superior de Investigaciones Cient'ıficas (CSIC), Madrid, Spain\n- 16 DSM / Irfu / SPP, CEA-Saclay, F-91191 Gif-sur-Yvette Cedex, France\n- 17 DTU Space, National Space Institute, Technical University of Denmark, Elektrovej 327, DK-2800 Kgs. Lyngby, Denmark\n- 18 D'epartement de Physique Th'eorique, Universit'e de Gen'eve, 24, Quai E. Ansermet,1211 Gen'eve 4, Switzerland\n- 19 Departamento de F'ısica Fundamental, Facultad de Ciencias, Universidad de Salamanca, 37008 Salamanca, Spain\n- 20 Departamento de F'ısica, Universidad de Oviedo, Avda. Calvo Sotelo s / n, Oviedo, Spain\n- 21 Department of Astronomy and Astrophysics, University of Toronto, 50 Saint George Street, Toronto, Ontario, Canada\n- 22 Department of Astrophysics / IMAPP, Radboud University Nijmegen, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands\n- 23 Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, California, U.S.A.\n- 24 Department of Physics & Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, British Columbia, Canada\n- 25 Department of Physics and Astronomy, Dana and David Dornsife College of Letter, Arts and Sciences, University of Southern California, Los Angeles, CA 90089, U.S.A.\n- 26 Department of Physics and Astronomy, University College London, London WC1E 6BT, U.K.\n- 27 Department of Physics and Astronomy, University of Sussex, Brighton BN1 9QH, U.K.\n- 28 Department of Physics, Florida State University, Keen Physics Building, 77 Chieftan Way, Tallahassee, Florida, U.S.A.\n- 29 Department of Physics, Gustaf Hallstromin katu 2a, University of Helsinki, Helsinki, Finland\n- 30 Department of Physics, Princeton University, Princeton, New Jersey, U.S.A.\n- 31 Department of Physics, University of California, Berkeley, California, U.S.A.\n- 32 Department of Physics, University of California, One Shields Avenue, Davis, California, U.S.A. \n- 33 Department of Physics, University of California, Santa Barbara, California, U.S.A.\n- 34 Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois, U.S.A.\n- 35 Dipartimento di Fisica e Astronomia G. Galilei, Universit'a degli Studi di Padova, via Marzolo 8, 35131 Padova, Italy\n- 36 Dipartimento di Fisica e Scienze della Terra, Universit'a di Ferrara, Via Saragat 1, 44122 Ferrara, Italy\n- 37 Dipartimento di Fisica, Universit'a La Sapienza, P. le A. Moro 2, Roma, Italy\n- 38 Dipartimento di Fisica, Universit'a degli Studi di Milano, Via Celoria, 16, Milano, Italy\n- 39 Dipartimento di Fisica, Universit'a degli Studi di Trieste, via A. Valerio 2, Trieste, Italy\n- 40 Dipartimento di Fisica, Universit'a di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, Italy\n- 41 Discovery Center, Niels Bohr Institute, Blegdamsvej 17, Copenhagen, Denmark\n- 42 Dpto. Astrof'ısica, Universidad de La Laguna (ULL), E-38206 La Laguna, Tenerife, Spain\n- 43 European Southern Observatory, ESO Vitacura, Alonso de Cordova 3107, Vitacura, Casilla 19001, Santiago, Chile\n- 44 European Space Agency, ESAC, Planck Science O GLYPH<14> ce, Camino bajo del Castillo, s / n, Urbanizaci'on Villafranca del Castillo, Villanueva de la Ca˜nada, Madrid, Spain\n- 45 European Space Agency, ESTEC, Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands\n- 46 Finnish Centre for Astronomy with ESO (FINCA), University of Turku, Vaisalantie 20, FIN-21500, Piikkio, Finland\n- 47 Haverford College Astronomy Department, 370 Lancaster Avenue, Haverford, Pennsylvania, U.S.A.\n- 48 Helsinki Institute of Physics, Gustaf Hallstromin katu 2, University of Helsinki, Helsinki, Finland\n- 49 INAF - Osservatorio Astrofisico di Catania, Via S. Sofia 78, Catania, Italy\n- 50 INAF - Osservatorio Astronomico di Padova, Vicolo dell'Osservatorio 5, Padova, Italy\n- 51 INAF - Osservatorio Astronomico di Roma, via di Frascati 33, Monte Porzio Catone, Italy\n- 52 INAF - Osservatorio Astronomico di Trieste, Via G.B. Tiepolo 11, Trieste, Italy\n- 53 INAF Istituto di Radioastronomia, Via P. Gobetti 101, 40129 Bologna, Italy\n- 54 INAF / IASF Bologna, Via Gobetti 101, Bologna, Italy\n- 55 INAF / IASF Milano, Via E. Bassini 15, Milano, Italy\n- 56 INFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, Italy\n- 57 INFN, Sezione di Roma 1, Universit'a di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, Italy\n- 58 IPAG: Institut de Plan'etologie et d'Astrophysique de Grenoble, Universit'e Joseph Fourier, Grenoble 1 / CNRS-INSU, UMR 5274, Grenoble, F-38041, France\n- 59 ISDC Data Centre for Astrophysics, University of Geneva, ch. d'Ecogia 16, Versoix, Switzerland\n- 60 IUCAA, Post Bag 4, Ganeshkhind, Pune University Campus, Pune 411 007, India\n- 61 Imperial College London, Astrophysics group, Blackett Laboratory, Prince Consort Road, London, SW7 2AZ, U.K.\n- 62 Infrared Processing and Analysis Center, California Institute of Technology, Pasadena, CA 91125, U.S.A.\n- 63 Institut N'eel, CNRS, Universit'e Joseph Fourier Grenoble I, 25 rue des Martyrs, Grenoble, France\n- 64 Institut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, France\n- 65 Institut d'Astrophysique Spatiale, CNRS (UMR8617) Universit'e Paris-Sud 11, Bˆatiment 121, Orsay, France\n- 66 Institut d'Astrophysique de Paris, CNRS (UMR7095), 98 bis Boulevard Arago, F-75014, Paris, France\n- 67 Institute for Space Sciences, Bucharest-Magurale, Romania \n- 68 Institute of Astronomy and Astrophysics, Academia Sinica, Taipei, Taiwan\n- 69 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, U.K.\n- 70 Institute of Theoretical Astrophysics, University of Oslo, Blindern, Oslo, Norway\n- 71 Instituto de Astrof'ısica de Canarias, C / V'ıa L'actea s / n, La Laguna, Tenerife, Spain\n- 72 Instituto de F'ısica de Cantabria (CSIC-Universidad de Cantabria), Avda. de los Castros s / n, Santander, Spain\n- 73 Istituto di Fisica del Plasma, CNR-ENEA-EURATOM Association, Via R. Cozzi 53, Milano, Italy\n- 74 Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, California, U.S.A.\n- 75 Jodrell Bank Centre for Astrophysics, Alan Turing Building, School of Physics and Astronomy, The University of Manchester, Oxford Road, Manchester, M13 9PL, U.K.\n- 76 Kavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.\n- 77 LAL, Universit'e Paris-Sud, CNRS / IN2P3, Orsay, France\n- 78 LERMA, CNRS, Observatoire de Paris, 61 Avenue de l'Observatoire, Paris, France\n- 79 Laboratoire AIM, IRFU / Service d'Astrophysique - CEA / DSM CNRS - Universit'e Paris Diderot, Bˆat. 709, CEA-Saclay, F-91191 Gif-sur-Yvette Cedex, France\n- 80 Laboratoire Traitement et Communication de l'Information, CNRS (UMR 5141) and T'el'ecom ParisTech, 46 rue Barrault F-75634 Paris Cedex 13, France\n- 81 Laboratoire de Physique Subatomique et de Cosmologie, Universit'e Joseph Fourier Grenoble I, CNRS / IN2P3, Institut National Polytechnique de Grenoble, 53 rue des Martyrs, 38026 Grenoble cedex, France\n- 82 Laboratoire de Physique Th'eorique, Universit'e Paris-Sud 11 & CNRS, Bˆatiment 210, 91405 Orsay, France\n- 83 Lawrence Berkeley National Laboratory, Berkeley, California, U.S.A.\n- 84 Max-Planck-Institut fur Astrophysik, Karl-Schwarzschild-Str. 1, 85741 Garching, Germany\n- 85 McGill Physics, Ernest Rutherford Physics Building, McGill University, 3600 rue University, Montr'eal, QC, H3A 2T8, Canada\n- 86 MilliLab, VTT Technical Research Centre of Finland, Tietotie 3, Espoo, Finland\n- 87 National University of Ireland, Department of Experimental Physics, Maynooth, Co. Kildare, Ireland\n- 88 Niels Bohr Institute, Blegdamsvej 17, Copenhagen, Denmark\n- 89 Observational Cosmology, Mail Stop 367-17, California Institute of Technology, Pasadena, CA, 91125, U.S.A.\n- 90 Optical Science Laboratory, University College London, Gower Street, London, U.K.\n- 91 SB-ITP-LPPC, EPFL, CH-1015, Lausanne, Switzerland\n- 92 SISSA, Astrophysics Sector, via Bonomea 265, 34136, Trieste, Italy\n- 93 School of Physics and Astronomy, Cardi GLYPH<11> University, Queens Buildings, The Parade, Cardi GLYPH<11> , CF24 3AA, U.K.\n- 94 School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, U.K.\n- 95 Space Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84 / 32, Moscow, 117997, Russia\n- 96 Space Sciences Laboratory, University of California, Berkeley, California, U.S.A.\n- 97 Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, Russia\n- 98 Stanford University, Dept of Physics, Varian Physics Bldg, 382 Via Pueblo Mall, Stanford, California, U.S.A.\n- 99 Sub-Department of Astrophysics, University of Oxford, Keble Road, Oxford OX1 3RH, U.K.\n- 100 Theory Division, PH-TH, CERN, CH-1211, Geneva 23, Switzerland\n- 101 UPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, France\n- 102 Universit'e de Toulouse, UPS-OMP, IRAP, F-31028 Toulouse cedex 4, France\n- 103 Universities Space Research Association, Stratospheric Observatory for Infrared Astronomy, MS 232-11, Mo GLYPH<11> ett Field, CA 94035, U.S.A.\n- 104 University Observatory, Ludwig Maximilian University of Munich, Scheinerstrasse 1, 81679 Munich, Germany\n- 105 University of Granada, Departamento de F'ısica Te'orica y del Cosmos, Facultad de Ciencias, Granada, Spain\n- 106 Warsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland"} |
2023ApJS..265....4K | The ExoClock project has been created to increase the efficiency of the Ariel mission. It will achieve this by continuously monitoring and updating the ephemerides of Ariel candidates in order to produce a consistent catalog of reliable and precise ephemerides. This work presents a homogenous catalog of updated ephemerides for 450 planets generated by the integration of 18000 data points from multiple sources. These sources include observations from groundbased telescopes the ExoClock network and the Exoplanet Transit Database midtime values from the literature and light curves from space telescopes Kepler K2 and TESS. With all the above we manage to collect observations for half of the postdiscovery years median with data that have a median uncertainty less than 1 minute. In comparison with the literature the ephemerides generated by the project are more precise and less biased. More than 40 of the initial literature ephemerides had to be updated to reach the goals of the project as they were either of low precision or drifting. Moreover the integrated approach of the project enables both the monitoring of the majority of the Ariel candidates 95 and also the identification of missing data. These results highlight the need for continuous monitoring to increase the observing coverage of the candidate planets. Finally the extended observing coverage of planets allows us to detect trends transittiming variations for a sample of 19 planets. All the products data and codes used in this work are open and accessible to the wider scientific community. | 2023-03-01T00:00:00Z | ['2022arXiv220909673K', '2023ApJS..265....4K', '10.3847/1538-4365/ac9da4', 'arXiv:2209.09673', '10.48550/arXiv.2209.09673'] | ['Ephemerides', 'Transits', 'Amateur astronomers', 'Photometry', 'Open source software', '464', '1711', '34', '1234', '1866', 'Astrophysics - Earth and Planetary Astrophysics', 'Astrophysics - Instrumentation and Methods for Astrophysics', 'Astrophysics - Solar and Stellar Astrophysics'] | ExoClock Project. III. 450 New Exoplanet Ephemerides from Ground and Space Observations | 2,023 | 173 | 0.67 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF'] | 18 | https://arxiv.org/pdf/2209.09673.pdf | {'No Header': 'Draft version September 21, 2022', 'ExoClock Project III: 450 new exoplanet ephemerides from ground and space observations': "A. Kokori, 1 A. Tsiaras, 2, 1 B. Edwards, 3, 1 A. Jones, 4, 5 G. Pantelidou, 6 G. Tinetti, 1 L. Bewersdorff, 4 A. Iliadou, 6 Y. Jongen, 4, 7 G. Lekkas, 8 A. Nastasi, 9, 10 E. Poultourtzidis, 6 C. Sidiropoulos, 8 F. Walter, 4, 11, 12 A. Wunsche, 13 R. Abraham, 4, 14 V. K. Agnihotri, 4 R. Albanesi, 4, 15 E. Arce-Mansego, 4, 16 D. Arnot, 17 M. Audejean, 4 C. Aumasson, 13 M. Bachschmidt, 4 G. Baj, 4 P. R. Barroy, 4, 18, 19 A. A. Belinski, 20 D. Bennett, 4, 21, 5 P. Benni, 4 K. Bernacki, 22 L. Betti, 23, 24 A. Biagini, 25, 24, 9 P. Bosch, 26 P. Brandebourg, 4 L. Br'at, 12 M. Bretton, 13 S. M. Brincat, 4, 27 S. Brouillard, 4, 28 A. Bruzas, 17 A. Bruzzone, 4, 29 R. A. Buckland, 17 M. Cal'o, 4 F. Campos, 4 A. Carreno, 4, 30 J.-A. Carrion Rodrigo, 4 R. Casali, 4 G. Casalnuovo, 4 M. Cataneo, 4, 31, 32 C.-M. Chang, 33 L. Changeat, 4 V. Chowdhury, 4 R. Ciantini, 23, 24 M. Cilluffo, 4, 31 J.-F. Coliac, 4 G. Conzo, 4, 34 M. Correa, 4, 35, 36 G. Coulon, 4 N. Crouzet, 37, 38, ∗ M. V. Crow, 4, 5, 39 I. Curtis, 4 D. Daniel, 4 S. Dawes, 4, 5, 39 B. Dauchet, 4 M. Deldem, 4 D. Deligeorgopoulos, 4, 40 G. Dransfield, 41 R. Dymock, 4, 5 T. Eenmae, 42 P. Evans, 4, 43 N. Esseiva, 4 C. Falco, 9 R. G. Farf'an, 4 E. Fern'andez-Laj'us, 44, 45 S. Ferratfiat, 13 S. L. Ferreira, 4 A. Ferretti, 4, 29 J. Fioglyph[suppress]lka, 22 M. Fowler, 4, 46, 5 S. R. Futcher, 4, 47, 5 D. Gabellini, 4 T. Gainey, 4 J. Gaitan, 4 P. Gajdoˇs, 48 A. Garc'ıa-S'anchez, 4, 49 J. Garlitz, 4 C. Gillier, 4, 50 C. Gison, 17 F. Grau Horta, 4 G. Grivas, 6 J. Gonzales, 4 D. Gorshanov, 51 P. Guerra, 26 T. Guillot, 52 C. A. Haswell, 17 T. Haymes, 4, 5 V.-P. Hentunen, 53 K. Hills, 4, 54, 5 K. Hose, 4 T. Humbert, 4 F. Hurter, 4, 55 T. Hynek, 56 M. Irzyk, 4 J. Jacobsen, 4 A. L. Jannetta, 4 K. Johnson, 4 P. J'o'zwik-Wabik, 22 A. E. Kaeouach, 4 W. Kang, 57, 58 H. Kiiskinen, 4, 59 T. Kim, 57, 60 U. Kivila, 4, 61 B. Koch, 4, 62 U. Kolb, 17 H. Kuˇc'akov'a, 63, 12 S.-P. Lai, 64, 33 D. Laloum, 4, 27 S. Lasota, 22 L. A. Lewis, 17 G.-I. Liakos, 4 F. Libotte, 4, 35, 36 C. Lopresti, 4, 65 F. Lomoz, 66, 12 R. Majewski, 4 A. Malcher, 22 M. Mallonn, 67 M. Mannucci, 4, 68 A. Marchini, 69 J.-M. Mari, 4, 70 A. Marino, 4, 71 G. Marino, 4, 72 J.-C. Mario, 4 J.-B. Marquette, 73 F. A. Mart'ınez-Bravo, 4 M. Maˇsek, 74, 12 P. Matassa, 4 P. Michel, 4 J. Michelet, 4 M. Miller, 4, 5, 27 E. Miny, 4, 75 T. Mollier, 4 D. Molina, 4, 76 B. Monteleone, 4 N. Montigiani, 4, 68 M. Morales-Aimar, 4, 77, 27 F. Mortari, 4 M. Morvan, 1 L. V. Mugnai, 78 G. Murawski, 4 L. Naponiello, 23, 24 R. Naves, 4 J.-L. Naudin, 4 D. N'eel, 4 R. Neito, 42 S. Neveu, 4, 79, 19 A. Noschese, 4 Y. O˘gmen, 4 O. Ohshima, 4 Z. Orbanic, 4 E. P. Pace, 23, 24 C. Pantacchini, 4 N. I. Paschalis, 4 C. Pereira, 4, 80 I. Peretto, 4, 81 V. Perroud, 4 M. Phillips, 4, 82, 5 P. Pintr, 83 J.-B. Pioppa, 4, 70, 27 J. Plazas, 4 A. J. Poelarends, 84 A. Popowicz, 22 J. Purcell, 4 N. Quinn, 4, 5 M. Raetz, 4, 85, 86 D. Rees, 4 F. Regembal, 4 M. Rocchetto, 1 P.-F. Rocci, 4, 79, 27, 19 M. Rockenbauer, 87 R. Roth, 88 L. Rousselot, 4, 79 X. Rubia, 4, 35 N. Ruocco, 4, 89 E. Russo, 4, 31 M. Salisbury, 4, 5 F. Salvaggio, 4, 72 A. Santos, 4 J. Savage, 4, 5 F. Scaggiante, 90 D. Sedita, 4 S. Shadick, 91 A. F. Silva, 4, 16 N. Sioulas, 4 V. ˇ Skoln'ık, 4, 12 M. Smith, 4 M. Smolka, 12 A. Solmaz, 92, 93 N. Stanbury, 4 D. Stouraitis, 4 T.-G. Tan, 4 M. Theusner, 4 G. Thurston, 4, 5 F.-P. Tifner, 4 A. Tomacelli, 4, 71 A. Tomatis, 4 J. Trnka, 94, 12, † M. Tylˇsar, 95 P. Valeau, 4 J.-P. Vignes, 4 A. Villa, 4, 30 A. Vives Sureda, 4 K. Vora, 4 M. Vraˇsˇt'ak, 12 D. Walliang, 4, 96 B. Wenzel, 87, 85 D. E. Wright, 4, 97, 5 R. Zambelli, 4 M. Zhang, 98 and M. Z'ıbar 12 \n1 University College London, Gower Street, London, WC1E 6BT, UK \n2 INAF - Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, 50125 Firenze, Italy \n3 AIM, CEA, CNRS, Universit'e Paris-Saclay, Universit'e de Paris, F-91191 Gif-sur-Yvette, France \n4 Amateur Astronomer c \n5 British Astronomical Association, Burlington House, Piccadilly, Mayfair, London, W1J 0DU, UK \n6 Department of Physics, Aristotle University of Thessaloniki, University Campus, Thessaloniki, 54124, Greece \n7 Observatoire de Vaison-La-Romaine, D'epartementale 51, pr'es du Centre Equestre au Palis - 84110 Vaison-La-Romaine, France \n8 Department of Physics, University of Ioannina, Ioannina, 45110, Greece \n9 GAL Hassin - Centro Internazionale per le Scienze Astronomiche, Via della Fontana Mitri, 90010 Isnello, Palermo, Italy 10 INAF - Osservatorio Astronomico di Palermo, Piazza del Parlamento, 1, 90134 Palermo, Italy \n- 11 ˇ Stef'anik Observatory, Strahovsk'a 205, 118 00 Praha 1, Czech Republic\n- 12 Czech Astronomical Society, Friˇcova 298 251 65 Ondˇrejov, Czech Republic\n- 13 Observatoire des Baronnies Proven¸cales, Route de Nyons, 05150 Moydans, France\n- 14 East Sussex Astronomical Society, 35 Mount Street Battle East Sussex TN33 0EG, UK\n- 15 ARA Associazione Romana Astrofili, Via Vaschetta, 1 - 02030 Frasso Sabino (Ri), Italy \n16 Asociaci'on Valenciana de Astronom'ıa, C/ Profesor Blasco 16 Bajo. Valencia, Spain", 'Corresponding author: A. Kokori': "[email protected] \n- c A list of associated private observatories that contributed to this work can be found in Appendix A \n38 \n20 \n- 17 School of Physical Sciences, The Open University, Walton Hall, Milton Keynes MK7 6AA, UK\n- 18 D'epartement de Physique, Universit'e de Picardie Jules Verne, 33 rue St Leu, 80000 Amiens, France \n19 Observatoire Jean-Marc Salomon - Plan'ete Sciences, 73, rue des Roches 77760 Buthiers \nSternberg Astronomical Institute, M.V. Lomonosov Moscow State University, 13, Universitetskij pr., 119234, Moscow, Russia \n21 Bristol Astronomical Society, Bristol, UK \n- 22 Department of Electronics, Electrical Engineering and Microelectronics, Silesian University of Technology, Akademicka 16, 44-100 Gliwice, Poland \n23 Dipartimento di Fisica e Astronomia, Universit'a degli Studi di Firenze, Largo E. Fermi 2, 50125 Firenze, Italy \n24 \nOsservatorio Polifunzionale del Chianti, Strada Provinciale Castellina in Chianti, 50021 Barberino Val D'elsa FI, Italy \n25 University of Palermo, Piazza Marina, 61, 90133 Palermo PA, Italy \n26 Observatori Astron'omic Albany'a, Cam'ı de Bassegoda S/N, Albany'a 17733, Girona, Spain \n27 \nAAVSO, 49 Bay State Road, Cambridge, MA 02138, USA \n28 Association AstroQueyras, 05350 Saint-V'eran \n29 Gruppo Astrofili Frentani, via Aterno 16 66034 Lanciano CH, Italy \n30 Associazione Astrofili Alta Valdera, Pisa, Italy \n31 Associazione Cernuschese Astrofili, Via della Martesana, 75, 20063, Cernusco sul Naviglio MI, Italy \n32 Argerlander-Institut fur Astronomie, Auf dem Hugel 71, 53121 Bonn, Germany \nDepartment of Physics, National Tsing Hua University, 101, Section 2, Kuang-Fu Road, Hsinchu 300044, Taiwan \n34 Gruppo Astrofili Palidoro, Via Pierleone Ghezzi, 75, 00050 Palidoro RM, Italy \n35 Agrupaci'o Astronomica de Sabadell, Carrer Prat de la Riba, 116, 08206 Sabadell, Barcelona, Spain \n36 Groupe Europ'een d'Observations Stellaires (GEOS), Bailleau l'Ev'eque, France \n37 Leiden Observatory, Leiden University, Postbus 9513, 2300 RA Leiden, The Netherlands \nEuropean Space Agency (ESA), European Space Research and Technology Centre (ESTEC), Keplerlaan 1, 2201 AZ Noordwijk, The \nNetherlands \n- 39 Crayford Manor House Astronomical Society Dartford, Parsonage Lane Pavilion, Parsonage Lane, Sutton- at-Hone, Dartford, Kent, DA4 9HD, UK \n40 Artemis Astronomical Group Of Evrytania, Aiolou 1,Karpenisi,Evrytania,Greece \n41 School of Physics & Astronomy, University of Birmingham, Edgbaston, B15 2TT, Birmingham, UK \n42 Tartu Observatory, Observatooriumi 1, T˜oravere, 61602 Tartu maakond, Estonia \n43 El Sauce Observatory, Coquimbo Province, Chile \n- 44 Facultad de Ciencias Astron'omicas y Geof'ısicas - Universidad Nacional de La Plata, Paseo del Bosque s/n, 1900 La Plata, Buenos Aires, Argentina \n45 Instituto de Astrof'ısica de La Plata (CCT La Plata - CONICET/UNLP), 1900 La Plata, Argentina \n46 South Wonston Exoplanet Factory, South Wonston, UK \n47 Hampshire Astronomical Group, Hinton Manor Ln, Clanfield, Waterlooville PO8 0QR, UK \nInstitute of Physics, Faculty of Science, Pavol Jozef \nˇ \nSaf'arik University, Park Angelinum 9, 040 01 Koˇsice, Slovakia \n49 Agrupaci'on Astron'omica de Madrid, Madrid, Spain \n50 Club d'Astronomie de Lyon Amp'ere, Place de la Nation, 69120 Vaulx-en-Velin, France \n51 Pulkovo Observatory, Russia, Pulkovskoye Shosse, 65, St Petersburg, Russia \n52 Universit'e Cˆote d'Azur, Observatoire de la Cˆote d'Azur, CNRS, Lagrange Laboratory, Nice, France \n53 Taurus Hill Observatory, 79480 Varkaus, Finland \n54 The Royal Astronomical Society, Burlington House, Piccadilly, London, W1J 0DU, UK \n55 Les Pl'eiades, Soci'et'e d'astronomie, CH 2610 St Imier, Switzerland \n56 Darksky Beskydy, Komensk'eho 654/26, Ostrava-Poruba, Czech Republic \n57 National Youth Space Center, Goheung, Jeollanam-do, 59567, S. Korea \n58 Spacebeam Inc., Cheongju-si, Chungcheongbuk-do, 28165, South Korea \n59 Jyvaskylan Sirius ry, Jyvaskyla, Finland \nDepartment of Astronomy and Space Science, Chungbuk National University, Cheongju-City, 28644, S. Korea \n61 Science Centre AHHAA, Sadama 1, Tartu, Estonia \n62 \nStudent Astronomy Lab, Carl-Fuhlrott-Gymnasium, Wuppertal, Germany \n63 Silesian University Opava, Opava, Czech Republic \nInstitute of Astronomy, National Tsing Hua University, 101, Section 2, Kuang-Fu Road, Hsinchu 300044, Taiwan \n65 \nGAD - Gruppo Astronomia Digitale, Italy \n66 Sedlˇcany Observatory, Ke Hvˇezd'arnˇe, 264 01 Sedlˇcany, Czech Republic \nLeibniz Institute for Astrophysics Potsdam (AIP), An der Sternwarte 16, 14482 Potsdam, Germany \n67 \n68 Associazione Astrofili Fiorentini, Firenze, Italy \n69 University of Siena - Dept. of Physical Science, Earth and Environment - Astronomical Observatory, Via Roma 56, 53100 Siena, Italy \n33 \n48 \n64 \n60 \n80 \n70 Groupement d'Astronomie Populaire de la R'egion d'Antibes, 2, Rue Marcel-Paul 06160 Juan-Les-Pins, France \n71 Unione Astrofili Napoletani, Salita Moiariello, 16, CAP 80131 Napoli NA, Italy \n72 Gruppo Astrofili Catanesi, Via Milo, 28, 95125 Catania CT, Italy \n- 73 Laboratoire d'astrophysique de Bordeaux, Univ. Bordeaux, CNRS, B18N, all'ee Geoffroy Saint-Hilaire 33615 Pessac, France \n74 FZU - Institute of Physics of the Czech Academy of Sciences, Na Slovance 1999/2, Prague 182 21, Czech Republic \n75 Blois Sologne Astronomie, rue de la Bondonni'ere 41250 Fontaines-en-Sologne, France \n- 76 Asociaci'on Astron'omica Astro Henares, Centro de Recursos Asociativos El Cerro C/ Manuel Aza˜na, s/n 28823 Coslada, Madrid \n77 Observadores de Supernovas, Spain \n78 Department of Physics, La Sapienza Universit'a di Roma, Piazzale Aldo Moro 2, 00185 Roma, Italy \n79 Soci'et'e Astronomique de France, 3, rue Beethoven 75016 Paris, France \nInstituto de Astrof'ısica e Ciˆencias do Espa¸co, Departamento de F'ısica, Faculdade de Ciˆencias, Universidade de Lisboa, Campo Grande, \nPT1749-016 Lisboa, Portugal \n81 MarSEC (Marana Space Explorer Center), c/a Pasquali, Marana di Crespadoro VI, Italy \n82 Astronomical Society of Edinburgh, Edinburgh, UK \n83 Institute of Plasma Physics AS CR, v. v. i., TOPTEC centre, Sobotecka 1660, 511 01 Turnov, Czech Republic \n84 Wheaton College Observatory, Wheaton College, 501 College Avenue Wheaton, IL 60187-5501, USA \n85 Bundesdeutsche Arbeitsgemeinschaft fur Veranderliche Sterne e.V., Germany \n86 Volkssternwarte Kirchheim e.V., Arnstadter Str. 49, 99334 Kirchheim, Germany \n87 \nUniversity of Vienna, Universitatsring 1, 1010 Vienna, Austria \n88 TURM Observatory, Department of Physics, Technische Universitat Darmstadt, 64289 Darmstadt, Germany 89 \nAstroCampania, Campania, Italy \n90 Gruppo Astrofili Salese, Santa Maria di Sala, Italy \nDepartment of Physics and Engineering Physics, University of Saskatchewan, Saskatoon, Saskatchewan, Canada \n92 C¸ a˘g University, Space Observation and Research Center, Mersin, Turkey \n93 C¸ ukurova University, UZAYMER, Adana, Turkey \n94 Observatory Slan'y, Nosaˇcick'a 1713, 274 01 Slan'y \n95 Hvˇezd'arna Prostˇejov, Kol'aˇrovy sady 3348, 796 01 Prostˇejov, Czech Republic \n96 Soci'et'e Lorraine d'Astronomie, BP 70239 54506 Vandœuvre Les Nancy, France \n97 Basingstoke Astronomical Society, Cliddesden Primary School, Cliddesden, Basingstoke, Hampshire, RG25 2QU, UK 98 Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA", 'ABSTRACT': 'The ExoClock project has been created with the aim of increasing the efficiency of the Ariel mission. It will achieve this by continuously monitoring and updating the ephemerides of Ariel candidates over an extended period, in order to produce a consistent catalogue of reliable and precise ephemerides. This work presents a homogenous catalogue of updated ephemerides for 450 planets, generated by the integration of ∼ 18000 data points from multiple sources. These sources include observations from ground-based telescopes (ExoClock network and ETD), mid-time values from the literature and light-curves from space telescopes (Kepler/K2 and TESS). With all the above, we manage to collect observations for half of the post-discovery years (median), with data that have a median uncertainty less than one minute. In comparison with literature, the ephemerides generated by the project are more precise and less biased. More than 40% of the initial literature ephemerides had to be updated to reach the goals of the project, as they were either of low precision or drifting. Moreover, the integrated approach of the project enables both the monitoring of the majority of the Ariel candidates (95%), and also the identification of missing data. The dedicated ExoClock network effectively supports this task by contributing additional observations when a gap in the data is identified. These results highlight the need for continuous monitoring to increase the observing coverage of the candidate planets. Finally, the extended observing coverage of planets allows us to detect trends (TTVs - Transit Timing Variations) for a sample of 19 planets. All products, data, and codes used in this work are open and accessible to the wider scientific community. \nKeywords: Ephemerides - Photometry - Transits - Amateur astronomers \n91', '1. INTRODUCTION': "The number of exoplanets discovered already exceeds 5000 and it continues to increase daily. The characterisation of exoplanets will be the main goal for future space missions. The Ariel mission aims to observe the atmospheres of 1000 planets in 2029 in order to investigate their nature (Tinetti et al. 2018). Ariel will observe thousands of transits, and to increase the mission efficiency, it is required to have precise ephemerides. Proper planning is important to avoid wasting the precious observing time of Ariel and other future space missions. \nFor various reasons, accuracy in predicting transit times is impeded. For example, the uncertainties of the initial ephemerides causes degeneracies over time in the precision of the predicted transit time (e.g. Mallonn et al. 2019b). The insufficient number of available data for each planet is another factor that generates biases in calculating the ephemerides (e.g. Benneke et al. 2017; Mallonn et al. 2019b). \nTo overcome the above problems and create a complete catalogue of precise ephemerides for a large number of planets, it is essential to use all available resources of data. These resources consist of data from the literature, data obtained by telescopes from the ground and finally data from space telescopes. The ExoClock project (Kokori et al. 2021, 2022) is an open, integrated platform, with the aim of continuously monitoring the ephemerides of the Ariel candidate targets (Edwards & Tinetti 2022). The organisation of the project is described thoroughly in Kokori et al. (2021), and the first large-scale catalogue of updated ephemerides for 180 planets was produced in Kokori et al. (2022), by combining observations from ground-based telescopes and literature. \nThe benefits of using small, ground-based telescopes to observe transiting exoplanets have been underlined previously (e.g. Zellem et al. 2020; Mallonn et al. 2019b; Edwards et al. 2021a; Beck et al. 2019), and their use for large-scale studies has been proved already in Kokori et al. (2022). While small telescopes are efficient, maximum effectiveness is achieved by utilising all available resources including other ground based networks, data from the literature and also data from space resources. Space telescopes are effective for observing challenging transits not easily accessible from ground telescopes. \nMoreover, TESS (Ricker et al. 2014) has been scanning the sky since 2019 and will continue to provide lightcurves for many known exoplanets, which can be used for ephemerides updates (Ivshina & Winn 2022). Therefore, to produce a complete catalogue of ephemerides for all planets, it is important to use data from both spaceand ground-based telescopes. \nIn this study, we integrated data from the ExoClock network, mid-time points from the literature, data from the Exoplanet Transit Database (Poddan'y et al. 2010) and data from space telescopes (Kepler (Koch et al. 2010a), K2 (Howell et al. 2014), and TESS (Ricker et al. 2014)). The integration of ∼ 18000 mid-time points, in total, allowed the generation of a complete analysis of ephemerides for 450 planets. The benefits of this integrated analysis are several: biases are minimised, better precision is achieved and long-term phenomena can be identified for each planet which may be indicative of trends (e.g. TTVs). The integrated design and approach of the ExoClock project highlights where there are gaps in the available data. The ExoClock network of ground-based telescopes can then be directed flexibly to make observations to address such gaps and extend the coverage. \nThe ExoClock project operates with an Open Science Framework in all the aspects of the research cycle (EU 2016; Dai et al. 2018). Open Science advances the progress of scientific research by encouraging collaborations and reproducibility. During all stages of the scientific process, the project follows open science practices; data, tools and codes used for the analysis are all open and accessible to everyone. All data used in this study are open and publicly available (e.g. data obtained from the ExoClock network, data from space telescopes). Additionally, open science means co-creation of scientific research through collaborations between various scientific communities and also citizen involvement (EU 2016). In this respect, the project is open to contributions from any interested person or community. The collaborative perspective also helps to ensure the most effective use of resources. Such collaborations are important to avoid overlapping and waste of observing time and to a further extent, they foster innovation in science.", '2. DATA': 'In this study, we integrated light-curves from the ExoClock network, the Exoplanet Transit Database, and the MAST Archive (for the Kepler, K2, and TESS space missions), and mid-transit times from the literature to update the ephemerides of 450 exoplanets. All the lightcurves were acquired before the end of 2021 and the \nTable 1. Summary of the observations used in this work. As coverage we define the percentage of years (since the first observation in the database) for which at least one observation exists. \nFigure 1. Distribution of transit mid-time uncertainties among the different sources. \n<!-- image --> \nliterature mid-transit times were published by the end of 2021. We analysed all the light-curves, regardless of their source, using the stellar and planetary parameters included in the Exoplanet Characterisation Catalogue (ECC), a dedicated catalogue prepared and maintained within the ExoClock project (Kokori et al. 2021), and the open-source Python package PyLightcurve (Tsiaras et al. 2016). For every light-curve, PyLightcurve peroforms the following operations:: \n- 1. calculates the limb-darkening coefficients using the ExoTETHyS package (Morello et al. 2020) (depending on the filter used for the observation),\n- 2. converts the time formats to Barycentric Julian Date (BJD TDB ),\n- 3. finds the maximum-likelihood model for the data (an exposure-integrated transit model together with a trend model - linear with airmass, linear with time, or quadratic) using the Nelder-Mead minimisation algorithm included in the SciPy package (Virtanen et al. 2020),\n- 4. removes outliers that deviate from the maximumlikelihood model by more than three times the \nFigure 2. Distribution of coverage among the different sources. As coverage we define the percentage of years (since the first observation in the database) for which at least one observation exists. \n<!-- image --> \nstandard deviation (STD) of the normalised residuals, \n- 5. scales the uncertainties by the root mean square (RMS) of the normalised residuals, to take into account any extra scatter,\n- 6. and, finally, performs an Markov chain Monte Carlo (MCMC) optimisation process using the emcee package (Foreman-Mackey et al. 2013) \nAfter this analysis, the quality of each light-curve was evaluated individually, and light-curves that fulfilled one or more of the criteria below, were excluded: \n- 1. autocorrelation and shapiro statistic indicate nongaussian normalised residuals at a 3σ level or more,\n- 2. transit signal-to-noise ratio (S/N = Depth/σ Depth ) is lower than three,\n- 3. Rp/Rs differs by more than 3 σ from the literature value (for the ExoClock and ETD observations), or the weighted average of the mission (for the space observations), \nFigure 3. Number of observations received from the ExoClock network, as a function of the telescope size. \n<!-- image --> \n- 4. O-C value is not in agreement (3 σ ) with other observations obtained during the same observing period ( ∼ a month). \nThe final list of 450 planets includes those planets for which we collected data-points at three or more different epochs and we could determine an ephemeris of better or equal quality to the initial ephemeris. Table 1 summarises the observations used to produce the ephemerides of 450 planets in this work. In addition, Figures 1 and 2 show the distribution of the precision and the coverage of the transit mid-time points used. As coverage we define the percentage of years (since the first observation in the database) for which at least one observation exists. We need to note here that 99% of the observations used have transit mid-time uncertainties lower than 10 minutes, and that the median coverage of all sources combined together is 50%, while individual sources do not reach more than 29%.', '2.1. ExoClock - Summary and quality of data': 'Currently, the ExoClock network consists of 540 participants - 80% of whom are amateur astronomers and 450 telescopes with sizes ranging between 6 and 40 inches - of which 80% are smaller than 17 inches. Figure 3 shows the distribution of the observations used in this work among the different telescope sizes. The large majority of the observations comes from small- and medium-scale telescopes and amateur observers (73%), who are the key part of our network. The ExoClock network is organised in a way to maximise the coverage of the planets and to ensure the high quality and homogeneity of the results. To achieve this, we have defined a prioritisation system, we provide a personal scheduler for each telescope, we support the observers with the \ndata analysis (educational material, a user-friendly software, regular meetings) and, finally, we perform the light curve modelling and evaluation (as described above) on the ExoClock website. For more details on the organisation of the ExoClock project and the ExoClock network, we refer the interested reader to Kokori et al. (2021).', '2.2. Data from space telescopes': "For the first time in the ExoClock project, we integrated light-curves from space telescope observations. More specifically we included light-curves from Kepler, K2 and TESS (before the end of 2021). First, we downloaded the long-cadence light-curves for the targets in the ExoClock target list. Then we identified the transits inside those light-curves and isolated them, including a base-line of one transit duration before and one after the event. Finally, the analysis and evaluation of each light-curve was conducted as described above, using a quadratic de-trending function. As some of the space-based light-curves contained gaps, we only considered those light-curves that were at least 80% complete, both in-transit and out-of-transit - i.e total exposure time more then 0.8 times the transit duration before, during and after the transit. \nFrom the analysis of the space-based light-curves, and especially from the TESS light-curves, it became clear that for a, non-negligible, number of planets, the parameters in the ECC (as derived from the literature) were producing transits of shorter or longer duration than the actual observations. For these planets we let the reduced semi-major axis ( a/R s ) vary in order to account for the differences in the duration. The ECC has been updated accordingly and the planets for which the a/R s was adjusted have been marked. Table 6 includes the adjusted a/R s values, which are marked with an asterisk. With the exception of Kepler-396c, Kepler-854b, and TOI201b, which had not value for their inclinations ( i ) in the discovery papers, we decided to fix i to the literature value, as a/R s and i are strongly correlated when they are both free parameters. In a future work we plan to provide analysis for both parameters but the scope of this work is to provide a set of parameters that will produce a reliable duration. Between a/R s and i we decided to let a/R s free as it is a more flexible for the determination of the duration (only small changes are required, and there is no upper limit like i ). \nFinally, we need to note here that the modelling of the Kepler light-curves did not produce gaussian residuals. This is most probably due to the fixed limb-darkening coefficients (LDCs) used. However, we decided to not allow the LDCs to vary, in order to keep a homogeneous analysis pattern for all observations. \nFigure 4. Comparison of the 2029-prediction uncertainties between this work and ExoClock II (left) and the ephemerides used at the beginning of the project (right). With the red star we indicate the planets for which TTV signals have been found. In both panels, the dashed lines are the σ p ' = σ p lines. \n<!-- image -->", '2.3. Exoplanet Transit Database (ETD)': "The Exoplanet Transit Database (ETD, Poddan'y et al. 2010) run by the Czech Astronomical Society since 2009, is currently the largest database of transit followup observations with more than 10,000 transit lightcurves for more than 350 exoplanetary systems. The collaboration between ExoClock and ETD started in 2020 and is described in Kokori et al. (2022). In this study, we included 184 observations for 40 planets provided by the ETD network. In order to maintain homogeneity and reliability in our analysis, the ETD observations were analysed and evaluated through the ExoClock website using the same methodology and validation criteria as for the ExoClock network data. The collaboration with ETD is critical to avoid duplications and waste of resources. We aim to continue our collaboration and gradually integrate more data from ETD in future publications. Such data can increase the coverage of certain planets during the period before ExoClock observations.", '2.4. Mid-time points from the literature': 'As we did not reanalyse the original light-curves we could not apply the same criteria as for the ExoClock, ETD and space light-curves. From the available data we excluded mid-transit time values that referred only to ephemerides, rather than to individual transits (with the exception of the discovery papers). We also excluded mid-transit time values with uncertainties greater than five minutes, and mid-transit time values that originated from Kepler, K2, TESS or ETD, to avoid duplications. \nTable 2. Categories of ephemerides in comparison with the previous ExoClock publication and the values at the beginning of the project.', '3.1. Ephemerides': "Here we present updated ephemerides for 450 out of 570 planets that are currently in the ExoClock target list. To calculate the new ephemerides, we used all the available data from all the sources described in the previous section. First, we calculated an updated zero-epoch point as the weighted average of the available epochs. Then we fitted a line on the epoch vs mid-transit times data using the emcee package (Foreman-Mackey et al. 2013). After a first fit, we scaled-up the uncertainties by the RMS of the normalised residuals to account for excess noise, and performed the fit again. Table 7 provides all the new ephemerides and references to the literature values used. \nFigure 4 shows the uncertainties in the 2029predictions before and after the updates presented in this work ( σ p and σ p ' , respectively), while Table 2 lists \nfive categories of the ephemerides status. 'Significantly improved' refers to those ephemerides that were giving 2029-predictions with uncertainties greater than the target uncertainty of 1/12 th of the transit duration, D, ( σ p > D/ 12) as described in Kokori et al. (2021). The term 'drifting' refers to the ephemerides that were giving 2029-predictions that were drifting more than the target uncertainty ( | p -p ' | > D/ 12). From the remaining ephemerides, the term 'Improved' refers to those ephemerides for which the 2029-prediction uncertainties have been improved by more than one minute ( σ p ' < σ p -1), while 'No change' refers to those ephemerides for which the 2029-prediction uncertainties have not changed by more than one minute ( | σ p ' -σ p | < 1). Finally, in this work we introduce the 'TTVs' flag, which refers to ephemerides that deviate from a linear behaviour.", '3.2. Deviations from linear ephemerides': 'For all the planets we calculated the Generalised Lomb-Scarge (GLS) periodogram on the linear ephemeris residuals to identify deviations from the linear ephemeris. We concluded that periodograms are more reliable in detecting such deviations, since other diagnostics such as the reduced chi square, the autocorrelation, or gaussianity tests on the residuals, are strongly affected by red noise, discontinuity and low number of data, respectively. This was due to the sparsity of the data and due to red noise in the timing measurements. The TTVs flag was given to those planets with periodograms that had peaks with a False Alarm Probability (FAP) lower than 0.13%. We estimated the FAP for each planet as follows: first, we produced periodograms (Pa) for 100,000 series of white noise with the same sampling, then we produced periodograms (Pb) for 100,000 series where we varied the mid-time data within their uncertainties. Finally, the FAP for each period was defined as the percentage of Pb that had greater power than the 99.87% (3 σ ) upper limit of the Pa periodograms. Detected periodicities were categorised as short-term or long-term, based on the time span of all available data. Long-term are these periodicities that are close to or longer than the total time span of the data used.', '4. DATA RELEASE C': 'The third data release of the ExoClock project includes two data products: the Catalogue of Observations (ExoClock, ETD, space observations), and the catalogue of ExoClock ephemerides. All data products and their descriptions can be found through the OSF repository with DOI: 10.17605/OSF.IO/P298N.', '4.1. Catalogue of Observations': 'Table 3. Distribution characteristics for the ephemerides drifts S/N between this work and the previous ExoClock publication (first column) and between this work and the ephemerides at the beginning of the project (second column). Planets with TTVs have been excluded. In the ideal case of a normal distribution these parameters should be close to the values in the third column. \nThe Catalogue of Observations contains all the lightcurves and literature mid-time points summarised in Table 1. In the online repository, each light-curve is accompanied by: \n- 1. metadata regarding the planet, the source, the observation, the instrument, and the data format;\n- 2. the pre-detrended light curve, filtered for outliers, converted to BJD TDB and flux formats, with scaled uncertainties;\n- 3. the fitting results, including the de-trending method used and its parameters;\n- 4. the de-trended light curve, enhanced with the detrending model, the transit model and the residuals;\n- 5. fitting diagnostics on the residuals.', '4.2. Catalogue of ExoClock Ephemerides': 'The new catalogue of ExoClock ephemerides contains the updated ephemerides for the 450 planets studied in this work (see also Table 7), accompanied by metadata regarding the planet, and flags concerning the detection of TTVs.', '5.1. Follow-up efficiency': 'From the comparison between the ephemerides in ExoClock II and this work, we conclude that biases in the ephemerides produced by ExoClock are decreasing. This is based on the fact that the number of significantly improved or drifting ephemerides is very small (Table 2, left column). Moreover, the drifts found between ExoClock II and this study are closer to a normal \ndistribution as seen in Table 3, first column. These values highlight the reliability of the produced ephemerides and support the view that the ExoClock project is working effectively towards achieving its goal.', '5.2. Need for continuous monitoring': "As indicated in Table 2 (second column), approximately 45% of the initial ephemerides have large uncertainties or drifts (categories 'significantly improved' and 'drifting'). This is similar to the percentage reported in ExoClock II, indicating that a significant number of ephemerides derived in discovery papers (including TESS discoveries) needs to be corrected to be appropriate for the efficient planning of Ariel. Moreover, as shown in Tables 2 and 3 (first columns), while ExoClock ephemerides have reduced biases, they are not completely bias-free. Our sample of 180 planets in ExoClock II is not large enough to determine the coverage needed to produce completely bias-free ephemerides, but we can see that coverages of 60% or more are necessary to avoid unexpected drifts larger than five minutes in our 2029predictions. Finally, as discussed in the previous section, some planets show long-term trends. Such trends can only be identified when coverage is close to 100%. For all these reasons, the effort of follow-up observations is important and continuous monitoring is essential. \nThe most important factor to increase coverage is to continue integrating all available resources and prioritise accordingly the follow-up observations.", '5.3. Follow-up capabilities': 'From the large number of observations obtained so far by the ExoClock network and with the TESS lightcurves analysed in this study, we can estimate more precisely the capabilities of these resources and plan efficiently for the future. In appendices C and D we provide detailed calculation of the signal-to-noise calibration that we performed on both ExoClock and TESS data to produce the equations below. \nThe minimum telescope aperture diameter (in inches) necessary to observe a planet with the ExoClock network ( D min ) is given by: \nD min = 0 . 135 + 10 -2 . 99+0 . 2 R 5 . 1 d √ 7200 + t 14 900 πt 14 (1) \nwhere R is the magnitude of the host star in the R Cousins filter, d is the relative transit depth, and t 14 is the total duration of the transit in seconds. \nBy placing an upper limit of 40 inches aperture on the ExoClock network, we estimate that we can followup 88% of the currently known Ariel candidates. Figure 5 shows the distribution of available planets per magnitude and telescope size, where we can see that even \nFigure 5. Distribution of available planets per magnitude and telescope aperture diameter. \n<!-- image --> \n⋂˜⌈˜√̂}√˜(∐√˜√√√√˜(̂]∐⌉˜√˜√({]{̂[˜√} \nwith telescopes up to 16 inches, 75% of the targets can be observed. \nFor TESS observations, the transit S/N that can be achieved is given : \nS/N TESS transit = 0 . 65 d √ t 14 / 90 × 10 3 0 . 135 + 10 -2 . 43+0 . 2 G RP +0 . 0039 G 2 RP (2) \nwhere G RP is the magnitude of the host star in the GAIA Rp filter, d is the relative transit depth, and t 14 is the total duration of the transit in seconds. \nBy placing a lower limit of S/N = 3 on the TESS observations, we estimate that we can follow-up 90% of the currently known Ariel candidates. By combining the two resources we can reach up to 95% of the candidates. For the remaining targets we plan to use other facilities such as CHEOPS (Benz et al. 2021) and Twinkle (Edwards et al. 2019), or combined multiple ground-based observations. \nWith this calibration of our ground-based network and TESS, we are in a good position to achieve the most productive use of both resources. We can avoid wasting valuable space telescope time - from facilities like CHEOPS or Twinkle - on following-up targets that can be monitored efficiently from the ground, whilst at the same time we can readily identify the most difficult targets that will definitely require observations from space.', '5.4. TTVs signals': "Our analysis revealed 19 planets with statistically significant signals in the residuals of their linear ephemeris fit (Table 4). Eleven of these planets - namely HD106315c, HD108236b, K2-19b, KOI-94c, KOI-94d, KOI-94e, Kepler-18d, Kepler-396c, TOI-216.01, TOI- \nTable 4. Planets identified with deviations from a linear ephemeris. Long-term refers to variations with periodicities that are close to or longer than the total time span of the data used. In brackets we indicate the peak periodicity in epochs (E), while the 'multiple' label refers to cases were more than one periodicities are significant. \n216.02, TOI-431b - have one or more additional transiting planets in their planetary systems. Hence, it is no surprise that they show TTVs, due to interaction with other planets in the systems. It is beyond the scope of this work to study the dynamics of these systems, but we are flagging them in the ExoClock project, so that observers will continue monitoring them and help with future dynamic analysis. For the remaining eight planets we investigated the different scenarios below. In addition to the periodogramms for the residuals of the linear ephemeris fit, we also applied a quadratic ephemeris fit and studied the periodogramms of these residuals, too (Figure 6). \nHAT-P-7b -An attempt to detect a third body in the system, either an additional planet or companion star, has been made using radial velocity data over a two-year span of observations. Analysis from radial velocity data suggest the presence of a companion star but the results were controversial (Winn et al. 2009a). A possible detection of another Saturn-sized planet in the system was also suggested by Ballard et al. (2011), however, the significance for the detection was low. \nOur results show a significant signal for a long-term periodicity of approximately 2243 epochs, or 4950 days, which is close to, but still lower than, the total time span of the data used. Moreover, we found a significant quadratic term of 6 . 95 ± 0 . 52 × 10 -10 , and after removing it, the long-term periodicity disappeared. The above results suggest that the signal is not periodic yet, but it has a positive curvature at the moment. This means that the planet is not decaying, leaving the possibilities of a third-body or orbital precession still open. \nKOI-12b -Masuda (2017) suggested the presence of a second planet based on the same data (Kepler). \nOur results indicate a few significant short-term periodicities between approximately 20 and 200 epochs. Moreover, we found a non-significant quadratic term of -2 . 49 ± 0 . 97 × 10 -7 , and after removing it, the signals from the short-term periodicities remain strong. From the above we cannot reach a clear conclusion because the multiple short-term periodicities could be caused by stellar activity. More data are required to narrow down the possible scenarios. \nQatar-1b -The first TTVs analysis for the Qatar-1 system was carried out by von Essen et al. (2013). The authors claimed that there are possible TTVs on Qatar1b either due to a weak pertubator in resonance with Qatar-1b or due to a massive body similar to a brown dwarf. The follow-up TTVs studies by Maciejewski et al. (2015) and Collins et al. (2017) did not detect any signal of an additional planet in the system, while Puskullu et al. (2017) found weak evidence of TTVs. It was also reported by Covino et al. (2013) that the orbital period of the planet in the Qatar-1 system is much shorter than the rotation period of the star, so tides produce a decay of the orbit. The most recent analysis, by Su et al. (2021) concluded that no TTV frequencies are identified. \nOur data cover a time span that is double compared to previous studies and our results indicate a statistically significant short-term periodicity at approximately 327 epochs or 465 days. Moreover, we found a nonsignificant quadratic term of 1 . 13 ± 0 . 72 × 10 -10 , and after removing it, the short-term periodicity was not affected. The above suggest that the signal is periodic, and in combination with the low eccentricity of the planet, this means that a pertubator scenario is favoured. \nTrES-3b -So far, studies have concluded that there is no evidence for TTVs for Tres-3b (Kundurthy et al. 2013; Puskullu et al. 2017). Christiansen et al. (2011) mentioned that a long term variability in the light curve of Tres-3b may be due to star spots. Additionally, the lack of periodic TTVs implies that another planetary body \nis absent, according to the study by Mannaday et al. (2020). Finally, precession can be ruled out due to the very low value of eccentricity, whereas the possibility of slow orbital decay cannot (Mannaday et al. 2020). \nOur results show multiple significant short-term periodicities, as well as one prominent long-term periodicity at approximately 4138 epochs or 5400 days. The longterm signal is longer than the total time span of the data. Moreover, we found a significant quadratic term of -1 . 68 ± 0 . 34 × 10 -10 , and after removing it, both the short-term and the long-term periodicities disappeared. The above suggest that the signal is not periodic yet, with a negative curvature at the moment. In combination with the low eccentricity of the planet this means that orbital decay scenario is favoured. \nWASP-4b -From the initial observations of WASP-4b it was assumed that TTVs might be present (Wilson et al. 2008). However, a follow-up study by Petrucci et al. (2013) concluded that the system does not show significant TTV trends. Baluev et al. (2015) proposed that TTVs probably exist in the WASP-4 system with a magnitude of 10-20 seconds and an unknown nature. Additionally, a significant quadratic termm in the OC diagram was reported in the study by Bouma et al. (2019) with the most probable explanation being the planets' orbital decay. Southworth et al. (2019) stated that TTV variations have a smaller magnitude than previously detected, and orbital decay or a third body in the system are both problematic hypotheses. More recently, it was suggested that the line of sight acceleration is the most probable reason for the TTVs (Bouma et al. 2020a). Finally, Baluev et al. (2020) confirmed the existence of quadratic TTVs in the system but without making a new proposal for the origins. \nWefound a significant quadratic term of -1 . 29 ± 0 . 22 × 10 -10 although a long-term periodicity is not shown. In addition, the short-term periodicities disappeared after removing the quadratic term. The above suggest that timing data alone do not provide any indication towards an interpretation. To further investigate this behaviour, other type of data or a longer time span are required. \nWASP-12b -WASP-12b is one of the very first exoplanets with a verified non-linear ephemeris due to orbital decay (Maciejewski et al. 2016a). It is also possible that the planet undergoes apsidal precession as the data indicate that the orbit might be slightly eccentric (Yee et al. 2020). According to (Weinberg et al. 2017) the measured rate of the orbital decay would be reasonable only if WASP-12b was a subgiant that experiences evolutionary changes that cause a rapid orbital decay to the planet. TTVs that were reported later support this \nidea but additional data are needed to confirm this (Maciejewski et al. 2018). More recent data concluded that the orbit is decaying with occultation times occurring about four minutes earlier after 10 years (Yee et al. 2020). WASP-12b is likely to be engulfed by its host star several million years from now (Yee et al. 2020). \nOur results show multiple significant short-term periodicities, as well as one prominent long-term periodicity at approximately 5173 epochs or 5650 days. The longterm signal is longer than the total time span of the data. Moreover, we found a significant quadratic term of -5 . 24 ± 0 . 17 × 10 -10 , and after removing it, both the short-term and the long-term periodicities disappeared. The above suggest that the signal is not periodic yet, with a negative curvature at the moment. In combination with the low eccentricity of the planet this means that orbital decay scenario is favoured. \nWASP-19b -A non-linear ephemeris was reported previously (Mancini et al. 2013a; Espinoza et al. 2019a). Petrucci et al. (2020) conducted the first empirical study of orbital decay by using 74 complete transit light curves covering a 10 yr period. Their results did not show any sign of orbital decay or periodic variations that could indicate the existence of additional bodies. \nOur results show multiple significant short-term periodicities, as well as one prominent long-term periodicity at approximately 4465 epochs or 3520 days. The long-term signal is longer then the total time span of the data. Moreover, we found a significant quadratic term of -0 . 87 ± 0 . 13 × 10 -10 , and after removing it, the majority of the short-term periodicities and the longterm periodicity disappeared. The above suggest that the signal is not periodic yet, with a negative curvature at the moment. In combination with the low eccentricity of the planet this means that orbital decay scenario is favoured. \nWASP-56b -A search for TTVs in the WASP-56 system was carried out recently in a study by (Wang et al. 2021), but statistically significant trends (at levels of 3 σ ) were not found. \nOur results indicate a few significant short-term periodicities between approximately 10 and 100 epochs. Moreover, we found a non-significant quadratic term of -2 . 09 ± 0 . 82 × 10 -8 , and after removing it, the signals from short-term periodicities above 50 epochs became stronger. From the above we cannot reach a clear conclusion as the multiple short-term periodicities could be caused by stellar activity. With more data in the future we will be able to narrow down the possible scenarios.", '6. CONCLUSION': '4 \nFigure 6. Periodogramms for the fitting residuals (linear and quadratic) for the eight planets with TTVs but without transiting companions. The red parts indicate periods with FAP lower than 0.13% and the vertical line indicates the total time span of the data used. \n<!-- image --> \nIn this study, we present a homogeneous analysis for the ephemerides of 450 planets which are currently known candidates for the Ariel mission. The ephemerides resulted from the integration of data from the ExoClock network, mid-time points from the literature, data from the ETD and data from space telescopes (Kepler, K2, and TESS missions). \nThe results showed that the ephemerides produced by the ExoClock project are less biased and hence more reliable for future predictions compared to the ini- \ntial ephemerides reported in the literature, while continuous monitoring is necessary, as 40% of the initial ephemerides for new planets need refinement to achieve the goals of the project. The integrated approach of the project allows us to monitor up to 95% of the Ariel candidates, while identifying missing data and prioritising observations for specific targets. The ExoClock network facilitates effectively the effort of obtaining such highpriority observations, while more difficult targets can be requested to be observed by other space telescopes like CHEOPS and Twinkle. \nThe ExoClock project, after three years of continuous operation, development, and interaction between several communities of academics and non-academics, has became a sustainable platform for providing reliable ephemerides for the Ariel candidate planets. A dynamic evolution of the project is being achieved; new ideas can be implemented with the focus on more specific targets that show special interest (as the ones flagged for TTVs). We plan to continue operating ExoClock within the framework of Open Science with the twofold objective of monitoring the ephemerides and fostering the democratisation of science.', 'SOFTWARE AND DATA': 'Software used: Django, PyLightcurve (Tsiaras et al. 2016), ExoTETHyS (Morello et al. 2020), Astropy (Astropy Collaboration et al. 2013), emcee (ForemanMackey et al. 2013), Matplotlib (Hunter 2007), Numpy (Harris et al. 2020), SciPy (Virtanen et al. 2020). \nAll the data products and their descriptions can found through the OSF repository with DOI: 110.17605/OSF.IO/P298N.', 'ACKNOWLEDGEMENTS': "The ExoClock project has received funding from the UKSA/STFC grants ST/W00254X/1 and ST/W006960/1. \nThis work has made use of data collected with the TESS mission, obtained from the MAST data archive at the Space Telescope Science Institute (STScI). Funding for the TESS mission is provided by the NASA Explorer Program. STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. \nWe would like to acknowledge the support provided by the administrators, designers, and developers of the ETD project and of the Czech Astronomical Society both to the ExoClock project but also to the efforts of the whole amateur community through its 10+ years of operation. \nThis work has made use of observations made by the MicroObservatory which is maintained and oper- \nated as an educational service by the Center for Astrophysics, Harvard & Smithsonian as a project of NASA's Universe of Learning, supported by NASA Award # NNX16AC65A. \nThis work has made use of observations made by the LCOGT network, as part of the LCOGT Global Sky Partners project 'ORBYTS: Refining Exoplanet Ephemerides' (PI B. Edwards). \nASTEP has benefited from the support of the French and Italian polar agencies IPEV and PNRA, and from INSU, the European Space Agency (ESA) through the Science Faculty of the European Space Research and Technology Centre (ESTEC), the University of Birmingham, the European Union's Horizon 2020 research and innovation programme (grants agreements n · 803193/BEBOP), the Science and Technology Facilities Council (STFC; grant n · ST/S00193X/1), the laboratoire Lagrange (CNRS UMR 7293) and the Universit'e Cˆote d'Azur through Idex UCAJEDI (ANR-15-IDEX01). \nMembers from Silesian University of Technology were responsible for (1) observations planning, (2) automation of observatories work, and (3) processing of data from SUTO network. \n- A.A. Belinski is supported by the Ministry of Science and Higher Education of the Russian Federation under the grant 075-15-2020-780 (N13.1902.21.0039).\n- M. Cataneo, E. Russo and M. Cilluffo thank the City Council and Management of Cernusco sul Naviglio for supporting the activity of the Associazione Cernuschese Astrofili and for the construction of the public observatory 'G. Barletta'.\n- B. Edwards is a Laureate of the Paris Region fellowship program supported by the Ile-de-France Region. This project has received funding under the framework program for research and Horizon 2020 innovation under the Marie Sklodowska-Curie grant agreement no. 607 945298.\n- P. Gajdoˇs is supported by the Slovak Research and Development Agency under contract No. APVV-20-0148 and internal grant VVGS-PF-2021-2087 of the Faculty of Science, P. J. ˇ Saf'arik University in Koˇsice.\n- C. Haswell and U. Kolb are supported by STFC under grant ST/T000295/1.\n- M. Maˇsek is supported by MEYS (Czech Republic) under the project MEYS LTT17006.\n- L. Mugnai is funded by ASI grant n. 2021-5-HH.0.", 'REFERENCES': "Addison, B., Wright, D. J., Wittenmyer, R. 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T., et al. 2014, ApJ, 796, 115, doi: 10.1088/0004-637X/796/2/115\n- Zhou, G., Bayliss, D., Penev, K., et al. 2014, AJ, 147, 144, doi: 10.1088/0004-6256/147/6/144\n- Zhou, G., Rodriguez, J. E., Collins, K. A., et al. 2016, AJ, 152, 136, doi: 10.3847/0004-6256/152/5/136\n- Zhou, G., Bakos, G. ' A., Hartman, J. D., et al. 2017, AJ, 153, 211, doi: 10.3847/1538-3881/aa674a\n- Zhou, G., Huang, C. X., Bakos, G. ' A., et al. 2019a, AJ, 158, 141, doi: 10.3847/1538-3881/ab36b5\n- Zhou, G., Bakos, G. ' A., Bayliss, D., et al. 2019b, AJ, 157, 31, doi: 10.3847/1538-3881/aaf1bb", 'A. SUPPLEMENTARY INFORMATION': "Here we append extra information regarding the data sources and results. More specifically, Table 5 includes a list with the amateur private observatories contributing to this work, and is followed by a description of the ASTEP telescope. Table 6 includes a list with the parameters used in the analysis of individual light-curves and the respective references, where the asterisk indicates orbital parameters ( a/R s or i ) that were adjusted based on TESS data to match the observed durations. \nTable 5 . Amateur private observatories contributing to this work. \nFrancois Hurter \nJens Jacobsen \nKevin Johnson \nAziz Ettahar Kaeouach \nBernd Koch \nDidier Laloum \nMassimiliano Mannucci \nJean-Claude Mario \nAntonio Marino \nGiuseppe Marino \nFernando Antonio Mart'ınez-Bravo \nPaolo Arcangelo Matassa \nPhilip Michel \nMike Miller \nDavid Molina \nThomas Mollier \nMario Morales-Aimar \nFabio Mortari \nGabriel Murawski \nJean-Louis Naudin \nRamon Naves \nDavid N'eel \nAlphonso Noschese \nYenal O˘gmen \nOsamu Ohshima \nZlatko Orbanic \nChristian Pantacchini \nNikolaos I. Paschalis \nVal'ere Perroud \nMark Phillips \nJean-Bernard Pioppa \nJean Plazas \nJeff Purcell \nManfred Raetz \nFran¸cois Regembal \nJose Angel Carrion Rodrigo \nLionel Rousselot \nXesco Rubia \nNello Ruocco \nMark Salisbury \nFabio Salvaggio \nJohn Savage \nDanilo Sedita \nAlvaro Fornas Silva \nNick Sioulas \nVojtˇech ˇ Skoln'ık \nMiroslav Smolka \nDimitris Stouraitis \nThiam-Guan Tan \nGeoffrey Thurston \nFernando Pablo Tifner \nAndrea Tomacelli \nAlberto Tomatis \nPierre Valeau \nJean-Pascal Vignes \nAlberto Villa \nAlbireo Observatory, Switzerland \nEgeskov Observatory \nHolbrook Observatory, East Sussex, UK \nHigh Atlas Observatory, Oukaimeden, Morocco \nMPC Code B72 \nObservatoire Priv´e du Mont 40280 Saint-Pierre-du-Mont, \nFrance \nOsservatorio Astronomico Margherita Hack, Firenze, Italy \nObservatoire de la cabergue \nTelescopio Remoto Colacevich c/o Osservatorio Astronomico di Capodimonte di Napoli \nOsservatorio GAC 'Luigi Sturzo', Italy \nChile \nP.M.P.H.R. Deep Sky (MPC K81) Atina (FR) Italy \nVerulamium Private Observatory, St Albans, UK \nGeorgetown Observatory, Georgetown, TX USA \nAnunakiObservatory, Madrid \nTomastro Observatory, France \nObservatorio de Sencelles, Spain \nHypatia Observatory, Italy \nMGAB Observatory \nGatinais French Observatory (GFO) \nMontcabrer MPC-213 \nSadr Observatory, Chile \nElianto Observatory \nGreen Island Observatory IAU B34 \nOhshima Tamashima Observatory \nExplorer Orbanic Observatory, Croatia \nObservatoire de BENAYES ; FRANCE \nNunki Observatory, Skiathos, Greece \nObservatoire de Duines, France \nForthimage Observatory, Edinburgh, Scotland \nLa Roque Esclapon - FRANCE \nRibot Observatory \nOmaha, Nebraska-United States \nPrivat Observatory Herges-Hallenberg, Germany \nHRT Observatory, Spain \nOAO Observatorio Aras de los Olmos \nVierzon Observatory, France \nStupa Observatori, Centelles, Catalonia, Spain \nOsservatorio Astronomico Nastro Verde, Sorrento, Italy \nPOST, UK \nWBRO (K49), Italy \nZ42, Rushay Farm Observatory, Dorset, UK \nOsservatorio Sedita Castrofilippo, Italy \nCentro Astron´omico Alto Turia (CAAT) \nNOAK Observatory L02, Greece \nBroumov NM Observatory, Czech Republic \nMoteˇsice Observatory, SK \nGalileo Observatory, Greece \nPerth Exoplanet Survey Telescope, Australia \nI67, Hartley Wintney, UK \nMPC I32 \nTelescopio Remoto Colacevich UAN c/o Osservatorio Astro- nomico di Capodimonte di Napoli \nAlto-Observatory, Italy \nObservatoire de l'Aiguillon sur Mer, France \nDeep Sky Chile , Chile \nOss Astr G.Galilei-Libbiano Mpc code B33 \nAntoni Vives Sureda Kuldip Vora Martin Vraˇs ˇ t'ak David E. Wright Roberto Zambelli \nAnunnaki observatory Cepheid Observatory, Rawatbhata, India ˇ Zilina-Mojˇs, LSO, Slovakia Yorick Observatory, Hampshire, UK Roberto Zambelli Observatory \nMartin Z'ıbar \nChlumˇcany \nASTEP (Antarctic Search for Transiting ExoPlanets) is a 40 cm telescope installed at the Concordia station, Dome C, Antarctica that operates during the polar winter from March to September (Fressin et al. 2005; Daban et al. 2010; M'ekarnia et al. 2016). The continuous night and excellent atmospheric conditions make it well suited for high precision time series photometry such as exoplanet transit observations. The telescope was installed in 2010 and upgraded in 2022. The project is a collaboration between Laboratoire Lagrange (CNRS UMR 7293), the University of Birmingham, and the European Space Agency. \nTable 6 . Parameters used in the analysis of individual light-curves and the respective references, where the asterisk indicates orbital parameters ( a/R s or i ) that were adjusted based on TESS data to match the observed durations. \nTable 7 . Updated ephemerides and data sources.", 'B. TRANSIT S/N CALCULATION': 'For a light curve with a standard deviation of std , total observing time of T , individual points with exposure time of t e , and overheads of t o , the uncertainty of the relative flux ( σ F ) that can be achieved is: \nσ F = std √ T/ ( t e + t o ) (B1) \nIn the case of a transit (assuming it is square), the transit depth ( d ) is the difference between the out-of-transit relative flux ( F oot ) and the in-transit relative flux ( F int ). Hence the uncertainty on the transit depth ( σ d ) is: \nσ d = √ σ 2 F oot + σ 2 F int = √ std 2 ( t e + t o ) T oot + std 2 ( t e + t o ) T int = std √ ( t e + t o ) ( 1 T oot + 1 T int ) = std √ ( t e + t o )( T oot + T int ) T oot T int (B2) \nHence the square-transit S/N is: \nS/N square -transit = d σ d = d std √ T oot T int ( t e + t o )( T oot + T int ) (B3) \nFinally, due to the fact that in reality the transits are not squares and we are also fitting the light curves for extra parameters, there is an additional x-factor to estimate the final transit S/N. \nS/N transit = xS/N square -transit = x d std √ T oot T int ( t e + t o )( T oot + T int ) (B4) \nFrom simulations, which we verified with current ExoClock and TESS observations, the x-factor is equal to 0.85 for linear or airmass de-trending, and 0.65 for quadratic de-trending. We need to note that for linear and airmass de-trending the x-factor is stable regardless of the length of the out-of-transit observations. However, for quadratic de-trending, in order to maintain the x-factor of 0.65, we need to observe one transit duration before and one after the transit, otherwise the x-factor becomes lower. For example, an observation of a three-hours-long transit with one hour of observations before and after, has an x-factor of 0.5 instead of 0.65.', 'C. TRANSIT S/N PREDICTIONS FOR EXOCLOCK': 'To predict the transit S/N we need to have a prediction for all the values included in Equation B4. The most uncertain one is std , which we predicted from the performance of the current telescopes. Figure 7 (left) shows the std of the current observations made using an R Cousins filter, normalised to one second exposure and to a telescope size of one inch, as a function of the R C magnitude. We have modelled this behaviour as follows: \nstd ExoClock norm = 0 . 135 + 10 -2 . 99+0 . 2 R (C5) \nHence, the predicted std for a light curve obtained by a telescope of diameter D and exposure time of t e will be: \nstd ExoClock = std ExoClock norm √ π ( D/ 2) 2 t e = 0 . 135 + 10 -2 . 99+0 . 2 R √ π ( D/ 2) 2 t e (C6) \nand the predicted transit S/N will be: \nS/N ExoClock transit = x d √ π ( D/ 2) 2 t e 0 . 135 + 10 -2 . 99+0 . 2 R √ T oot T int ( t e + t o )( T oot + T int ) (C7) \nThe ExoClock scheduler calculates the minimum telescope size necessary to observe a transit based on the following assumptions: \n- 1. the targeted S/N transit is 6\n- 2. the de-trending model is expected to be the airmass model, hence x = 0.85\n- 3. the observation includes one hour before and one hour after the transit, hence T oot = 7200 seconds\n- 4. the observation includes the full transit, hence T int = t 14 in seconds\n- 5. the overheads and the exposure time are equal, hencee t e = t o \nHence, the minimum telescope size is: \n6 = 0 . 85 d √ π ( D min / 2) 2 t e 0 . 135 + 10 -2 . 99+0 . 2 R √ 7200 t 14 ( t e + t e )(7200 + t 14 ) D min = 0 . 135 + 10 -2 . 99+0 . 2 R 5 . 1 d √ 7200 + t 14 900 πt 14 (C8) \n<!-- image --> \nFigure 7. Standard deviation of the light-curves as a function of magnitude, for the ExoClock (left, normalised for an one-inch telescope and one-second exposure) and the TESS (right) light-curves, together with the models derived. The red errorbars indicate the median and standard deviation of the data in bins of 0.2 magnitudes. \n<!-- image -->', 'D. TRANSIT S/N PREDICTIONS FOR TESS': 'As far as the TESS observations are concerned, the calculation is less complicated, as many of the parameters are fixed. The std can be predicted from the performance of the telescope. Figure 7 (right) shows the std of the current observation, as a function of the G RP magnitude. For TESS, there is no need to normalise to the telescope size and exposure time, as these are fixed. We have modelled this behaviour as follows: \nstd TESS = (0 . 135 + 10 -2 . 43+0 . 2 G RP +0 . 0039 G 2 RP ) × 10 -3 (D9) \nMoreover, for TESS the de-trending model is the quadratic ( x =0.65), the exposure time is two minutes ( t e = 120) overheads are negligible ( t o = 0), the observations are continuous, so we can select the out-of-transit observations to be equal to one transit duration before and one transit duration after the transit( T oot = 2 t 14 in seconds) and the in-transit observing time is equal to a full transit duration ( T int = t 14 in seconds). Hence, the predicted transit S/N will be: \nS/N TESS transit = 0 . 65 d √ t 14 / 90 0 . 135 + 10 -2 . 43+0 . 2 G RP +0 . 0039 G 2 RP × 10 3 (D10)'} |
2018AJ....156..123A | The Astropy Project supports and fosters the development of opensource and openly developed Python packages that provide commonly needed functionality to the astronomical community. A key element of the Astropy Project is the core package astropy which serves as the foundation for more specialized projects and packages. In this article we provide an overview of the organization of the Astropy project and summarize key features in the core package as of the recent major release version 2.0. We then describe the project infrastructure designed to facilitate and support development for a broader ecosystem of interoperable packages. We conclude with a future outlook of planned new features and directions for the broader Astropy Project. P . | 2018-09-01T00:00:00Z | ['2018AJ....156..123T', '10.48550/arXiv.1801.02634', '10.3847/1538-3881/aabc4f', 'arXiv:1801.02634', '2018AJ....156..123A', '2018arXiv180102634T', '2018arXiv180102634P'] | ['methods: data analysis', 'methods: miscellaneous', 'methods: statistical', 'reference systems', 'Astrophysics - Instrumentation and Methods for Astrophysics'] | The Astropy Project Building an Openscience Project and Status of the v2.0 Core Package | 2,018 | 173 | 0.8 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF'] | 7,617 | https://arxiv.org/pdf/1801.02634.pdf | {'No Header': 'Draft version January 17, 2018 \nTypeset using L A T X modern style in AASTeX61 \nE', 'THE ASTROPY PROJECT: BUILDING AN INCLUSIVE, OPEN-SCIENCE PROJECT AND STATUS OF THE V2.0 CORE PACKAGE': "The Astropy Collaboration, A. M. Price-Whelan, 1 B. M. Sipőcz, H. M. Günther, 2 P. L. Lim, 3 S. M. Crawford, 4 S. Conseil, 5 D. L. Shupe, 6 M. W. Craig, 7 N. Dencheva, 3 A. Ginsburg, 8 J. T. VanderPlas, 9 L. D. Bradley, 3 D. Pérez-Suárez, 10 and M. de Val-Borro 11 \n(primary paper contributors) T. L. Aldcroft, 12 K. L. Cruz, 13, 14, 15 T. P. Robitaille, 16 and E. J. Tollerud 3 (Astropy coordination committee) C. Ardelean, 17 T. Babej, 18 M. Bachetti, 19 A. V. Bakanov, S. P. Bamford, 20 G. Barentsen, 21 P. Barmby, 17 A. Baumbach, 22 K. L. Berry, F. Biscani, 23 M. Boquien, 24 K. A. Bostroem, 25 L. G. Bouma, 1 G. B. Brammer, 3 E. M. Bray, H. Breytenbach, 4, 26 H. Buddelmeijer, 27 D. J. Burke, 12 G. Calderone, 28 J. L. Cano Rodríguez, M. Cara, 3 J. V. M. Cardoso, 29, 21, 30 S. Cheedella, 31 Y. Copin, 32 D. Crichton, 33 D. D'Avella, 3 C. Deil, 34 É. Depagne, 4 J. P. Dietrich, 35, 36 A. Donath, 34 M. Droettboom, 3 N. Earl, 3 T. Erben, 37 S. Fabbro, 38 L. A. Ferreira, 39 T. Finethy, R. T. Fox, L. H. Garrison, 12 S. L. J. Gibbons, 40 D. A. Goldstein, 41, 42 R. Gommers, 43 J. P. Greco, 1 P. Greenfield, 3 A. M. Groener, 44 F. Grollier, A. Hagen, 45, 46 P. Hirst, 47 D. Homeier, 48 A. J. Horton, 49 G. Hosseinzadeh, 50, 51 L. Hu, 52 J. S. Hunkeler, 3 Ž. Ivezić, 53 A. Jain, 54 T. Jenness, 55 G. Kanarek, 3 S. Kendrew, 56 N. S. Kern, 41 W. E. Kerzendorf, 57 A. Khvalko, J. King, 34 D. Kirkby, 58 A. M. Kulkarni, 59 A. Kumar, 60 A. Lee, 61 D. Lenz, 62 S. P. Littlefair, 63 Z. Ma, 64 D. M. Macleod, 65 M. Mastropietro, 66 C. McCully, 50, 51 S. Montagnac, 67 B. M. Morris, 53 M. Mueller, 68 S. J. Mumford, 69 D. Muna, 70 N. A. Murphy, 12 S. Nelson, 7 G. H. Nguyen, 71 J. P. Ninan, 46 M. Nöthe, 72 S. Ogaz, 3 S. Oh, 1 J. K. Parejko, 53 N. Parley, 73 S. Pascual, 74 R. Patil, A. A. Patil, 75 A. L. Plunkett, 76 J. X. Prochaska, 77 T. Rastogi, V. Reddy Janga, 78 J. Sabater, 79 P. Sakurikar, 80 M. Seifert, L. E. Sherbert, 3 H. Sherwood-Taylor, A. Y. Shih, 81 J. Sick, 82 M. T. Silbiger, S. Singanamalla, 83 L. P. Singer, 84, 85 P. H. Sladen, 86 K. A. Sooley, S. Sornarajah, O. Streicher, 87 P. Teuben, 88 S. W. Thomas, 40 G. R. Tremblay, 89 J. E. H. Turner, 47 V. Terrón, 90 M. H. van Kerkwijk, 91 A. de la Vega, 33 L. L. Watkins, 3 B. A. Weaver, 92 J. B. Whitmore, 93 J. Woillez, 57 and V. Zabalza \nCorresponding author: Astropy Coordination Committee \[email protected] \nThe author list has three parts: the authors that made significant contributions to the writing of the paper in order of contribution, the four members of the Astropy Project coordination committee, and contributors to the Astropy Project in alphabetical order. The position in the author list does not correspond to contributions to the Astropy Project as a whole. A more complete list of contributors to the core package can be found in the package repository, and at the Astropy team webpage.", 'The Astropy Collaboration': '(Astropy contributors) \n- 1 Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA \n- 13 Department of Physics and Astronomy, Hunter College, City University of New York, 695 Park Avenue, New York, NY 10065 \n- 15 Department of Astrophysics, American Museum of Natural History, New York, NY, USA \n- 19 INAF-Osservatorio Astronomico di Cagliari, via della Scienza 5, I-09047, Selargius, Italy', 'Astropy Project II': "- 28 Istituto Nazionale di Astrofisica, via Tiepolo 11 Trieste, Italy\n- 29 Universidade Federal de Campina Grande, Campina Grande, PB 58429-900, Brazil\n- 30 Bay Area Environmental Research Institute, Petaluma, CA 94952, USA\n- 31 Department of Physics, Virginia Tech, Blacksburg, VA 24061, USA\n- 32 Université de Lyon, F-69622, Lyon, France; Université de Lyon 1, Villeurbanne; CNRS/IN2P3, Institut de Physique Nucléaire de Lyon\n- 33 Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA\n- 34 Max-Planck-Institut für Kernphysik, PO Box 103980, 69029 Heidelberg, Germany\n- 35 Faculty of Physics, Ludwig-Maximilians-Universität, Scheinerstr. 1, 81679 Munich, Germany\n- 36 Excellence Cluster Universe, Boltzmannstr. 2, 85748 Garching b. München, Germany\n- 37 Argelander-Institut für Astronomie, Auf dem Hügel 71, 53121 Bonn, Germany\n- 38 National Research Council Herzberg Astronomy & Astrophysics, 4071 West Saanich Road, Victoria, BC\n- 39 Instituto de Matemática Estatística e Física - IMEF, Universidade Federal do Rio Grande FURG, Rio Grande, RS 96203-900, Brazil\n- 40 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, UK\n- 41 Department of Astronomy, UC Berkeley, 501 Campbell Hall #3411, Berkeley, CA 94720, USA\n- 42 Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA\n- 43 Scion, Private Bag 3020, Rotorua, New Zealand\n- 44 Drexel University, Physics Department, Philadelphia, PA 19104, USA\n- 45 Vizual.ai, 3600 O'Donnell St, Suite 250, Baltimore, MD 21224\n- 46 Dept of Astronomy and Astrophysics, Pennsylvania State University, University Park, PA 16802\n- 47 Gemini Observatory, 670 N. Aohoku Pl, Hilo, HI 96720, USA\n- 48 Zentrum für Astronomie der Universität Heidelberg, Landessternwarte, Königstuhl 12, 69117 Heidelberg, Germany\n- 49 Australian Astronomical Observatory, 105 Delhi Road, North Ryde NSW 2113, Australia\n- 50 Las Cumbres Observatory, 6740 Cortona Drive, Suite 102, Goleta, CA 93117-5575, USA\n- 51 Department of Physics, University of California, Santa Barbara, CA 93106-9530, USA\n- 52 Imperial College London, Kensington, London SW7 2AZ, United Kingdom\n- 53 Department of Astronomy, University of Washington, Seattle, WA 98155\n- 54 BITS PILANI/Computer Science, Pilani Campus, Rajasthan, India\n- 55 Large Synoptic Survey Telescope, 950 N. Cherry Ave., Tucson, AZ, 85719, USA\n- 56 European Space Agency, Space Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD 21218, USA\n- 57 European Southern Observatory, Karl-Schwarzschild-Straße 2, 85748 Garching bei München, Germany\n- 58 Department of Physics and Astronomy, University of California, Irvine, CA 92697, USA\n- 59 College of Engineering Pune/Department of Computer Engineering and IT, Shivajinagar, Pune 411005, India \n- 61 Department of Physics, University of Berkeley, Califonia, CA94709, USA \n- 62 Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA\n- 63 Department of Physics & Astronomy, University of Sheffield, Sheffield, S3 7RH, UK\n- 64 Department of Physics and Astronomy, University of Missouri, Columbia, Missouri, 65211, USA 65 Cardiff University, Cardiff CF24 3AA, UK\n- 66 Department of Physics and Astronomy, Ghent University, Krijgslaan 281, S9, B-9000 Gent, Belgium", 'ABSTRACT': 'The Astropy project supports and fosters the development of open-source and openly-developed Python packages that provide commonly-needed functionality to the astronomical community. A key element of the Astropy project is the core package astropy , which serves as the foundation for more specialized projects and packages. In this article, we provide an overview of the organization of the Astropy project and summarize key features in the core package as of the recent major release, version 2.0. We then describe the project infrastructure designed to facilitate and support development for a broader ecosystem of inter-operable packages. We conclude with a future outlook of planned new features and directions for the broader Astropy project. \nKeywords: Astrophysics - Instrumentation and Methods for Astrophysics - methods: data analysis - methods: miscellaneous', '1. INTRODUCTION': 'All modern astronomical research makes use of software in some way. Astronomy as a field has thus long supported the development of software tools for astronomical tasks, such as scripts that enable individual scientific research, software packages for small collaborations, and data reduction pipelines for survey operations. Some software packages are, or were, supported by large institutions and are intended for a wide range of users. These packages therefore typically provide some level of documentation and user support or training. Other packages are developed by individual researchers or research groups and are then typically used by smaller groups for more domain-specific purposes. For both packages meant for wider distribution and for scripts specific to particular research projects, a library that addresses common astronomical tasks simplifies the software development process. The users of such a library then also benefit from a community and ecosystem built around a shared foundation. The Astropy project has grown to become this community for Python astronomy software, and the astropy core package is a feature-rich Python library. \nThe development of the astropy core package began as a largely community-driven effort to standardize core functionality for astronomical software in Python . In this way, its genesis differs from, but builds upon, many substantial and former astronomical software development efforts that were commissioned or initiated through large institutional support, such as IRAF (developed at NOAO; Tody 1993), MIDAS (developed at ESO; Banse et al. 1988), or Starlink (originally developed by a consortium of UK institutions and now maintained by the East Asian Observatory; Disney & Wallace 1982; Currie et al. 2014). More recently, community-driven efforts have seen significant success in the astronomical sciences (e.g., Turk et al. 2011). \nPython 1 is an increasingly popular, general-purpose programming language that is available under a permissive open source software license and is free of charge for all major operating systems. The programming language has become especially popular in the quantitative sciences, where researchers must simultaneously produce research, perform data analysis, and develop software. A large part of this success owes itself to the vibrant community of developers and a continuously-growing ecosystem of tools, web services, and stable well-developed packages that enable easier collaboration on software development, easier writing and sharing of software documentation, and continuous testing and validation of software. While dedicated libraries provide support for array representation and arithmetic ( numpy ; Van der Walt et al. 2011), a wide variety of functions for scientific computing ( scipy ; Jones et al. 2001-), and publication-quality plotting ( matplotlib ; Hunter 2007), tens of thousands of other high-quality and easy-to-use packages are available, which can help with tasks that are not specific to astronomy but might be performed in the course of astronomical research, e.g., interfacing with databases, or statistical inferences. More recently, \nthe development and mainstream adoption of package managers such as Anaconda 2 has significantly streamlined the installation process for many libraries, lowering the barriers to entry. \nThe Astropy project aims to provide an open-source and open-development core package ( astropy ) and an ecosystem of affiliated packages that support astronomical functionality in the Python programming language. The astropy core package is now a feature-rich library of sufficiently general tools and classes that supports the development of more specialized code. An example of such functionality is reading and writing FITS files: It would be time consuming and impractical for multiple groups to implement the FITS standard (Pence et al. 2010) and maintain software for such a general-purpose need. Another example of such a common task is in dealing with representations of and transformations between astronomical coordinate systems. \nThe Astropy project aims to develop and provide high-quality code and documentation according to the best practices in software development. The project makes use of different tools and web services to reach those goals without central institutional oversight. The first public release of the astropy package is described in Astropy Collaboration et al. (2013). Since then, the astropy package has been used in hundreds of projects and the scope of the package has grown considerably. At the same time, the scientific community contributing to the project has grown tremendously and an ecosystem of packages supporting or affiliated with the astropy core has developed. In this paper, we describe the current status of the Astropy community and the astropy core package and discuss goals for future development. \nWe start by describing the way the Astropy project functions and is organized in Section 2. We then describe the main software efforts developed by the Astropy project itself: a core package called astropy (Section 3) and several separate packages that help maintain the infrastructure for testing and documentation (Section 4). We end with a short vision for the future of Astropy and astronomical software in general in Section 5. The full paper, including the code to produce the figures, is available in a GitHub repository 3 \nThis article is not intended as an introduction to astropy , nor does it replace the astropy documentation. Instead, it describes the way the Astropy community is organized and the current state of the astropy package.', '2.1. Coordination of Astropy': "From its inception, Astropy has required coordination to ensure the project as a whole and its coding efforts are consistent and reasonably efficient. While many Python projects adopt a 'Benevolent Dictator For Life' (BDFL) model, Astropy has instead opted for a coordination committee . This is in part due to the nature of the \nFigure 1. Left panel : Distribution of number of commits per committer. Right panel : Cumulative number of commits to the astropy core package over time. \n<!-- image --> \nproject as a large-scale collaboration between many contributors with many interests, and in part due to simply the amount of work that needs to get done. For the latter reason, the project has expanded the committee from three to four members starting in 2016. \nFor resolving disagreements about the astropy core package or other Astropymanaged code, the coordination committee primarily acts to work toward consensus, or when consensus is difficult to achieve, generally acts as a 'tie-breaker.' The committee also oversees affiliated package applications to ensure that they are in keeping with Astropy's vision and philosophy, 4 as well as the associated procedures. Additionally, the committee oversees the assignment of roles (primarily driven by already-existing contributions), and increasingly has acted as the 'face' of the Project, providing contact with organizations like NumFOCUS (the body that holds any potential funding in trust for Astropy) and the American Astronomical Society (AAS).", '2.2. Astropy development model': "Code is contributed to the astropy core package or modified through 'pull requests' (via GitHub 5 ) that often contain several git commits. Pull requests may fix bugs, implement new features, or improve or modify the infrastructure that supports the development and maintenance of the package. Individual pull requests are generally limited to a single conceptual addition or modification to make code review tractable. Pull requests that modify or add code to a specific subpackage must be reviewed and approved by one of the subpackage maintainers before they are merged into the core codebase. Bugs and feature requests are reported via the GitHub issue tracker and \nlabeled with a set of possible labels that help classify and organize the issues. The development workflow is detailed in the astropy documentation. 6 \nAs of version 2.0, astropy contains 212244 lines of code 7 contributed by 232 unique contributors over 19270 git commits. Figure 1, left, shows the distribution of total number of commits per contributor as of November 2017. The relative flatness of this distribution (as demonstrated by its log-log slope of -0 . 5 ) shows that the astropy core package has been developed by a broad contributor base. A leading group of 6 developers have each added over 1000 commits to the repository, and ∼ 20 more core contributors have contributed at least 100 commits. However, the distribution of contribution level (number of commits) continues from 100 down to a single commit. In this sense, the development of the core package has been a true community effort and is not dominated by a single individual. It is also important to note that the number of commits is only a rough metric of contribution, as a single commit could be a critical fix in the package or a fix for a typographical error. Figure 1, right, shows the number of commits as a function of time since the genesis of the astropy core package. The package is still healthy: new commits are and have been contributed at a steady rate throughout its existence.", '2.3. APEs - Astropy Proposals for Enhancement': "Central to the success of Astropy is an open environment where anybody can contribute to the project. This model leads to an 'organic' growth, where features are implemented by different people with different programming styles and interfaces. Thus, Astropy has a mechanism to more formally propose significant changes to the core package (e.g., re-writing the coordinates subpackage; Tollerud et al. 2014), to plan out major new features (e.g., a new file format; Aldcroft 2015), or institute new organization-wide policies (e.g., adopting a code of conduct; Cruz et al. 2015). This mechanism is called 'Astropy Proposal for Enhancement' (APE) and is modeled after the 'Python Enhancement Proposals' (PEP) that guide the development of the Python programming language. In an APE, one or more authors describe in detail the proposed changes or additions, including a rationale for the changes, how these changes will be implemented, and in the case of code, what the interface will be (Greenfield 2013). The APEs are discussed and refined by the community before much work is invested into a detailed implementation; anyone is welcome to contribute to these discussions during the open consideration period. APEs are proposed via pull requests on a dedicated GitHub repository 8 ; anyone can therefore read the proposed APEs and leave in-line comments. When a community consensus emerges, the APEs are accepted and become the basis for future work. In cases where consensus cannot be reached, the Astropy coordination committee may decide to close the dis- \ncussion and make an executive decision based on the community input on the APE in question.", '2.4. Concept of affiliated packages': "A major part of the Astropy project is the concept of 'Affiliated Packages.' An affiliated package is an astronomy-related Python package that is not part of the astropy core package, but has requested to be included as part of the Astropy project's community. These packages support the goals and vision of Astropy of improving code re-use, interoperability, and embracing good coding practices such as testing and thorough documentation. \nAffiliated packages contain functionality that is more specialized, have license incompatibilities, or have external dependencies (e.g., GUI libraries) that make these packages more suitable to be separate from the astropy core package. Affiliated packages may also be used to develop substantial new functionality that will eventually be incorporated into the astropy core package (e.g., wcsaxes ). New functionality benefits from having a rapid development and release cycle that is not tied to that of the astropy core (Section 2.5). \nAffiliated packages are listed on the main Astropy website and advertised to the community through Astropy mailing lists; a list of current affiliated packages is included in Table A. Becoming an affiliated package is a good way for new and existing packages to gain exposure while promoting Astropy's high standard for code and documentation quality. This process of listing and promoting affiliated packages is one way in which the Astropy project tries to increase code re-use in the astronomical community. \nPackages can become affiliated to Astropy by applying for this status on a public mailing list. The coordination committee (Section 2.1) reviews such requests and issues recommendations for the improvement of a package, where applicable.", '2.5. Release cycle and Long Term Support': "The astropy package has a regular release schedule consisting of new significant releases every 6 months, with bugfix releases as needed (Tollerud 2013). The major releases contain new features or any significant changes, whereas the bugfix releases only contain fixes to code or documentation but no new features. Some versions are additionally designated as 'Long-term support' (LTS) releases, which continue to receive bug fixes for 2 years following the release with no changes to the API. The LTS versions are ideal for pipelines and other applications in which API stability is essential. The latest LTS release (version 2.0) is also the last one that supports Python 2; it will receive bug fixes until the end of 2019 (Robitaille 2017). \nThe version numbering of the astropy core package reflects this release scheme: the core package version number uses the form x.y.z , where 'x' is advanced for LTS releases, 'y' for non-LTS feature releases, and 'z' for bugfix releases. \nThe released versions of the astropy core package are available from several of the Python distributions for scientific computing (e.g., Anaconda) and from the Python Package Index (PyPI). 9 Effort has been made to make astropy available and easily installable across all platforms; the package is constantly tested on different platforms as part of a suite of continuous integration tests.", '2.6. Support of Astropy': 'The Astropy project, as of the version 2.0 release, does not receive any direct financial support for the development of astropy . Development of the software, all supporting materials, and community support are provided by individuals who work on the Astropy project in their own personal time, by individuals or groups contributing to Astropy as part of a research project, or contributions from institutions that allocate people to work on Astropy. A list of organizations that have contributed to Astropy in this manner can be found in the Acknowledgments. \nDifferent funding models have been proposed for support of Astropy (e.g., Muna et al. 2016), but a long-term plan for sustainability has not yet been established. The Astropy project has the ability to accept financial contributions from institutions or individuals through the NumFOCUS 10 organization. NumFOCUS has, to date, covered the direct costs incurred by the Astropy project.', '3. ASTROPY CORE PACKAGE VERSION 2.0': 'The Astropy project aims to provide Python -based packages for all tasks that are commonly needed in a large subset of the astronomical community. At the foundation is the astropy core package, which provides general functionality (e.g., coordinate transformations, reading and writing astronomical files, and units) or base classes for other packages to utilize for a common interface (e.g., NDData ). In this section, we highlight new features introduced or substantially improved since version 0.2 (previously described in Astropy Collaboration et al. 2013). The astropy package provides a full log of changes 11 over the course of the entire project and more details about individual subpackages are available in the documentation. 12 Beyond what is mentioned below, most subpackages have seen improved performance since the release of the version 0.2 package.', '3.1. Units': "The astropy.units subpackage adds support for representing units and numbers with associated units - 'quantities' - in code. Historically, quantities in code have often been represented simply as numbers, with units implied or noted via comments in the code because of considerations about speed: having units associated \nwith numbers inherently adds overhead to numerical operations. In astropy.units , Quantity objects extend numpy array objects and have been designed with speed in mind. \nAs of astropy version 2.0, units and quantities, prevalent in most of its subpackages, have become a key concept for using the package as a whole. Units are intimately entwined in the definition of astronomical coordinates; thus, nearly all functionality in the astropy.coordinates subpackage (see Section 3.3) depends on them. For most other subpackages, quantities are at least accepted and often expected by default. \nThe motivation and key concepts behind this subpackage were described in detail in the previous paper (Astropy Collaboration et al. 2013). Therefore, we primarily highlight new features and improvements here.", '3.1.1. Interaction with numpy arrays': 'Quantity objects extend numpy.ndarray objects and therefore work well with many of the functions in numpy that support array operations. For example, Quantity objects with angular units can be directly passed in to the trigonometric functions implemented in numpy . The units are internally converted to radians, which is what the numpy trigonometric functions expect, before being passed to numpy .', '3.1.2. Logarithmic units and magnitudes': 'By default, taking the logarithm of a Quantity object with non-dimensionless units intentionally fails. However, some well-known units are actually logarithmic quantities, where the logarithm of the value is taken with respect to some reference value. Examples include astronomical magnitudes, which are logarithmic fluxes, and decibels, which are more generic logarithmic ratios of quantities. Logarithmic, relative units are now supported in astropy.units .', '3.1.3. Defining functions that require quantities': 'When writing code or functions that expect Quantity objects, we often want to enforce that the input units have the correct physical type. For example, we may want to require only length-type Quantity objects. astropy.units provides a tool called quantity\\_input() that can perform this verification automatically to avoid repetitive code.', '3.2. Constants': "The astropy.constants subpackage provides a selection of physical and astronomical constants as Quantity objects (see Section 3.1). A brief description of this package was given in Astropy Collaboration et al. (2013). In version 2.0, the built-in constants have been organized into modules for specific versions of the constant values. For example, physical constants have codata2014 (Mohr et al. 2016) and codata2010 versions. Astronomical constants are organized into iau2015 and iau2012 modules to indicate their sources (resolutions from the International Astronomical Union, IAU). The codata2014 and iau2015 versions are combined into the \ndefault constant value version: astropyconst20 . For compatibility with astropy version 1.3, astropyconst13 is available and provides access to the adopted versions of the constants from earlier versions of astropy . To use previous versions of the constants as units (e.g., solar masses), the values have to be imported directly; with version 2.0, astropy.units uses the astropyconst20 versions. \nAstronomers using astropy.constants should take particular note of the constants provided for Earth, Jupiter, and the Sun. Following IAU 2015 Resolution B3 (Mamajek et al. 2015), nominal values are now given for mass parameters and radii. The nominal values will not change even as 'current best estimates' are updated.", '3.3. Coordinates': 'The astropy.coordinates subpackage is designed to support representing and transforming celestial coordinates and, new in version 2.0, velocities. The framework heavily relies on the astropy.units subpackage, and most inputs to objects in this subpackage are expected to be Quantity objects. Some of the machinery also relies on the Essential Routines of Fundamental Astronomy (ERFA) C library for some of the critical underlying transformation machinery (Tollerud et al. 2017), which is based on the Standards Of Fundamental Astronomy (SOFA) effort (Hohenkerk 2011). \nA key concept behind the design of this subpackage is that coordinate representations and reference systems / frames are independent of one another. For example, a set of coordinates in the International Celestial Reference System (ICRS) reference frame could be represented as spherical (right ascension, declination, and distance from solar system barycenter) or Cartesian coordinates ( x , y , z with the origin at barycenter). They can therefore change representations independent of being transformed to other reference frames (e.g., the Galactic coordinate frame). \nThe classes that handle coordinate representations (the Representation classes) act like three-dimensional vectors and thus support vector arithmetic. The classes that represent reference systems and frames (the Frame classes) internally use Representation objects to store the coordinate data-that is, the Frame classes accept coordinate data, either as a specified Representation object, or using shorthand keyword arguments to specify the components of the coordinates. These preferred representation and short-hand component names differ between various astronomical reference systems. For example, in the ICRS frame, the spherical angles are right ascension ( ra ) and declination ( dec ), whereas in the Galactic frame, the spherical angles are Galactic longitude ( l ) and latitude ( b ). Each Frame class defines its own component names and preferred Representation class. The frame-specific component names map to corresponding components on the underlying Representation object that internally stores the coordinate data. For most frames, the preferred representation is spherical, although this is determined primarily by the common use in the astronomical community. \nFigure 2. The full graph of possible reference frame transformations implemented in astropy.coordinates . Arrows indicate transformations from one frame to another. Arrows that point back to the same frame indicate self-transformations that involve a change of reference frame parameters (e.g., equinox). \n<!-- image --> \nMany of the Frame classes also have attributes specific to the corresponding reference system that allow the user to specify the frame. For example, the Fifth Fundamental Catalogue (FK5) reference system requires specifying an equinox to determine the reference frame. If required, these additional frame attributes must be specified along with the coordinate data when a Frame object is created. Figure 2 shows the network of possible reference frame transformations as currently implemented in astropy.coordinates . Custom user-implemented Frame classes that define transformations to any reference frame in this graph can then be transformed to any of the other connected frames. \nThe typical user does not usually have to interact with the Frame or Representation classes directly. Instead, astropy.coordinates provides a highlevel interface to representing astronomical coordinates through the SkyCoord class, which was designed to provide a single class that accepts a wide range of possible \ninputs. It supports coordinate data in any coordinate frame in any representation by internally using the Frame and Representation classes. \nIn what follows, we briefly highlight key new features in astropy.coordinates .', '3.3.1. Local Earth coordinate frames': 'In addition to representing celestial coordinates, astropy now supports specifying positions on the Earth in a number of different geocentric systems with the EarthLocation class. With this, astropy now supports Earth-location-specific coordinate systems such as the altitude-azimuth ( AltAz ) or horizontal system. Transformations between AltAz and any Barycentric coordinate frame also requires specifying a time using the Time class from astropy.time . With this new functionality, many of the common tasks associated with observation planning can now be completed with astropy or the Astropy-affiliated package astroplan (Morris et al. 2017).', '3.3.2. Proper motion and velocity transformations': 'In addition to positional coordinate data, the Frame classes now also support velocity data. As the default representation for most frames is spherical, most of the Frame classes expect proper motion and radial velocity components to specify the velocity information. The names of the proper motion components all start with pm and adopt the same longitude and latitude names as the positional components. Transforming coordinates with velocity data is also supported, but in some cases the transformed velocity components have limited accuracy because the transformations are done numerically instead of analytically. The low-level interface for specifying and transforming velocity data is currently experimental. As such, in version 2.0, only the Frame classes (and not the SkyCoord class) support handling velocities.', '3.3.3. Solar System Ephemerides': 'Also new is support for computing ephemerides of major solar system bodies and outputting the resulting positions as coordinate objects. These ephemerides can be computed either using analytic approximations from ERFA or from downloaded JPL ephemerides (the latter requires the jplephem 13 optional dependency and an internet connection).', '3.3.4. Accuracy of coordinate transformations': "In order to check the accuracy of the coordinate transformations in astropy.coordinates , we have created a set of benchmarks that we use to compare transformations between a set of coordinate frames for a number of packages 14 . Since no package can be guaranteed to implement all transformations to arbitrary precision and some transformations are sometimes subject to interpretation of standards (in particular in the case of Galactic coordinates), we do not designate any \nof the existing packages as the 'ground truth' but instead compare each tool to all other tools. The benchmarks are thus useful beyond the Astropy project since they allow all of the tools to be compared to all other tools. The tools included in the benchmark at the moment include the astropy core package, Kapteyn (Terlouw & Vogelaar 2015), NOVAS (Barron et al. 2011), PALpy (Jenness & Berry 2013), PyAST (a wrapper for AST, described in Berry et al. 2016), PyTPM 15 , PyEphem (Rhodes 2011), and pySLALIB (a Python wrapper for SLALIB, described in Wallace 1994). \nThe benchmarks are meant to evolve over time and include an increasing variety of cases. At the moment, the benchmarks are set up as follows - we have generated a standard set of 1000 pairs of random longitudes/latitudes that we use in all benchmarks. Each benchmark is then defined using an input and output coordinate frame, using all combinations of FK4, FK5, Galactic, ICRS, and Ecliptic frames. For now, we set the epoch of observation to J2000. We also set the frame to J2000 (for FK5 and Ecliptic) and B1950 (for FK4). In the future, we plan to include a larger variety of epochs and equinoxes, as well as tests of conversion to/from Altitude/Azimuth. For each benchmark, we convert the 1000 longitudes/latitudes from the input/output frame with all tools and quantify the comparison by looking at the median, mean, maximum, and standard deviation of the absolute separation of the output coordinates from each pair of tools. \nFigure 3 visualizes the relative accuracy of the conversion from FK4 to Galactic coordinates for all pairs of tools that implement this transformation. In this figure, the color of the cell indicates the maximum difference (in arcseconds) between the two tools over the 1000 longitude-latitude pairs tested. This figure shows, for example, that astropy , Kapteyn, and PyTPM agree with sub-milliarcsecond differences (light colors, small differences), while PALpy, pySLALIB, and PyAST also agree amongst themselves. However, there is an offset of around 0.2 '' between the two groups. Finally, PyEphem disagrees with all other packages by 0.4-0.8 '' (darker colors, large differences). These values are only meant to be illustrative and will change over time as the benchmarks are refined and the packages updated.", '3.4. Time': 'The astropy.time subpackage focuses on supporting time scales (e.g., UTC, TAI, UT1) and time formats (e.g., Julian date, modified Julian date) that are commonly used in astronomy. This functionality is needed, for example, to calculate barycentric corrections or sidereal times. astropy.time is currently built on the ERFA (Tollerud et al. 2017) C library, which replicates the Standards of Fundamental Astronomy (SOFA; Hohenkerk 2011) but is licensed under a three-clause BSD license. The package was described in detail in Astropy Collaboration et al. (2013) and has stayed stable for the last several versions of astropy . Thus, in what follows, we only highlight significant changes or new features since the previous Astropy paper. \nFigure 3. Comparison matrix of the maximum difference between longitude-latitude values in a set of 1000 random points transformed from FK4 to Galactic with the different packages. Darker colors (larger differences) are more significant disagreements. \n<!-- image -->', '3.4.1. Barycentric and Heliocentric corrections': "Detailed eclipse or transit timing requires accounting for light travel time differences from the source to the observatory because of the Earth's motion. It is therefore common to instead convert times to the Solar System barycenter or heliocenter where the relative timing of photons is standardized. With the location of a source on the sky (i.e., a SkyCoord object), the location of an observatory on Earth (i.e., an EarthLocation object), and time values as Time objects, the time corrections to shift to the solar system barycenter or heliocenter can now be computed with astropy.time using the light\\_travel\\_time method of a Time object.", '3.5.1. nddata': 'The astropy.nddata subpackage provides three types of functionality: an abstract interface for representing generic arbitrary-dimensional datasets intended primarily for subclassing by developers of other packages, concrete classes building on this interface, and utilities for manipulating these kind of datasets. \nThe NDDataBase class provides the abstract interface for gridded data with attributes for accessing metadata, the world coordinate system (WCS), uncertainty arrays matched to the shape of the data, and other traits. Building on this interface, the NDData class provides a minimal working implementation for storing numpy arrays. These classes serve as useful base classes for package authors wishing to develop their own classes for specific use cases and as containers for exchanging gridded data. \nThe classes NDDataRef , NDDataArray , and CCDData extend the base storage functionality with options to do basic arithmetic (addition, subtraction, multiplication, and division), including error propagation in limited cases, and slicing of the dataset based on grid coordinates that appropriately handles masking, errors, and units (if present). Additionally, the CCDData class also provides reading and writing from and to FITS files and uses data structures from astropy , like WCS , to represent the file contents abstractly. \nThe astropy.nddata.utils module provides utilities that can operate on either plain numpy arrays or any of the classes in the astropy.nddata subpackage. It features a class for representing two-dimensional image cutouts, allowing one to easily link pixels in the cutout to those in the original image or vice versa, to convert between world and pixel coordinates in the cutout, and to overlay the cutout on images. Functions to enlarge or reduce an image by doing block replication or reduction are also provided.', '3.5.2. Tables': "The astropy.table subpackage provides functionality for representing and manipulating heterogeneous data. In some respects, this is similar to numpy record arrays (Van der Walt et al. 2011) or pandas dataframes (McKinney 2010) but with modifications for astronomical data. Most notably, tables from astropy.table allow for table or column metadata and can handle vectors or arrays as table entries. The subpackage was described in detail in Astropy Collaboration et al. (2013). Thus, in what follows, we only summarize key new features or updates to astropy.table since the previous Astropy paper. These are support for grouped table operations, table concatenation, and using array-valued astropy objects as table columns. \nA table can contain data that naturally form groups; for example, it may contain multiple observations of a few sources at different points in time and in different bands. Then, we may want to split the table into groups based on the combination of source observed and the band, after which we combine the results for each combination of source and band in some way (e.g., finding the mean or standard deviation of \nthe fluxes or magnitudes over time) or filter the groups based on user-defined criteria. These kinds of grouping and aggregation operations are now fully supported by Table objects. \nTable objects can now be combined in several different ways. If two tables have the same columns, we may want to stack them 'vertically' to create a new table with the same columns but all rows. If two tables are row-matched but have distinct columns, we may want to stack them 'horizontally' to create a new table with the same rows but all columns. For other situations, more generic table concatenation or join are also possible when two tables share some columns. \nThe Table object now allows array-valued Quantity , celestial coordinate ( SkyCoord ), and date/time ( Time ) objects to be used as columns. It also provides a general way for other user-defined array-like objects to be used as columns. This makes it possible, for instance, to easily represent catalogs of sources or time series in Astropy, while having both the benefits of the Table object (e.g., accessing specific rows/columns or groups of them and combining tables) and of, for example, the SkyCoord or the Time classes (e.g., converting the coordinates to a different frame or accessing the date/time in the desired time scale).", '3.6. io': 'The astropy.io subpackage provides support for reading and writing data to a variety of ASCII and binary file formats, such as a wide range of ASCII data table formats, FITS, HDF5, and VOTable. It also provides a unified interface for reading and writing data with these different formats using the astropy.table subpackage. For many common cases, this simplifies the process of file input and output (I/O) and reduces the need to master the separate details of all the I/O packages within astropy . The file interface allows transparent compression of the gzip , bzip2 and lzma ( .xz ) formats; for the latter two if the Python installation was compiled with support the respective libraries.', '3.6.1. ASCII': 'One of the problems when storing a table in an ASCII format is preserving table meta-data such as comments, keywords and column data types, units, and descriptions. The newly defined Enhanced Character Separated Values (ECSV, Aldcroft 2015) format makes it possible to write a table to an ASCII-format file and read it back with no loss of information. The ECSV format has been designed to be both human-readable and compatible with most simple CSV readers. \nThe astropy.io.ascii subpackage now includes a significantly faster Cython/C engine for reading and writing ASCII files. This is available for most of the common formats. It also offers some additional features like parsing of different exponential notation styles, such as commonly produced by Fortran programs. On average, the new engine is about 4 to 5 times faster than the corresponding purePython implementation and is often comparable to the speed of the pandas (McKinney 2010) ASCII \nfile interface. The fast reader has a parallel processing option that allows harnessing multiple cores for input parsing to achieve even greater speed gains. By default, read() and write() will attempt to use the fast Cython/C engine when dealing with compatible formats. Certain features of the full read / write interface are unavailable in the fast version, in which case the reader will by default fall back automatically to the purePython version. \nThe astropy.io.ascii subpackage now provides the capability to read a table within an HTML file or web URL into an astropy Table object. A Table object can now also be written out as an HTML table.', '3.6.2. FITS': 'The astropy.io.fits subpackage started as a direct port of the PyFITS project (Barrett & Bridgman 1999). Therefore, it is pretty stable, with mostly bug fixes but also a few new features and performance improvements. The API remains mostly compatible with PyFITS, which is now deprecated in favor of astropy . \nCommand-line scripts are now available for printing a summary of the HDUs in FITS file(s) ( fitsinfo ) and for printing the header information to the screen in a human-readable format ( fitsheader ). \nFITS files are now loaded lazily by default, i.e., an object representing the list of HDUs is created but the data are not loaded into memory until requested. This approach should provide substantial speed-ups when using the convenience functions (e.g., getheader() or getdata() ) to get an HDU that is near the beginning in a file with many HDUs.', '3.7. Modeling': 'The astropy.modeling subpackage provides a framework for representing analytical models and performing model evaluation and parameter fitting. The main motivation for this functionality was to create a framework that allows arbitrary combination of models to support the Generalized World Coordinate System (GWCS) package. 16 The current FITS WCS specification lacks the flexibility to represent arbitrary distortions and does not meet the needs of many types of current instrumentation. The fact that the astropy modeling framework now supports propagating units also makes it a useful tool for representing and fitting astrophysical models within data analysis tools. \nModels and fitters are independent of each other: a model can be fit with different fitters and new fitters can be added without changing existing models. The framework is designed to be flexible and easily extensible. The goal is to have a rich set of models, but to also facilitate creating new ones, if necessary. \n3.7.1. Single Model Definition and Evaluation \nModels are defined by their parameters and initialized by providing values for them. The names of the parameters are stored in a list, Model.param\\_names . Parameters are complex objects. They store additional information - default value, default unit, and parameter constraints. Parameter values and constraints can be updated by assignment. Supported constraints include fixed , and tied parameters, and bounds on parameter values. The framework also supports models for which the number of parameters and their names are defined by another argument. A typical example is a polynomial model defined by its degree. A model is evaluated by calling it as a function. \nIf an analytical inverse of a model exists it can be accessed by calling Model.inverse . In addition, Model.inverse can be assigned another model which represents a computed inverse. \nAnother useful settable property of models is Model.bounding\\_box . This attribute sets the domain over which the model is defined. This greatly improves the efficiency of evaluation when the input range is much larger than the characteristic width of the model itself.', '3.7.2. Model Sets': 'astropy.modeling provides an efficient way to set up the same type of model with many different sets of parameter values. This creates a model set that can be efficiently evaluated. For example, in PSF (point spread function) photometry, all objects in an image will have a PSF of the same functional form, but with different positions and amplitudes.', '3.7.3. Compound Models': 'Models can be combined using arithmetic expressions. The result is also a model, which can further be combined with other models. Modeling supports arithmetic (+, -, *, /, and **), join ( & ), and composition ( | ) operators. The rules for combining models involve matching their inputs and outputs. For example, the composition operator, | , requires the number of outputs of the left model to be equal to the number of inputs of the right one. For the join operator, the total number of inputs must equal the sum of number of inputs of both the left and the right models. For all arithmetic operators, the left and the right models must have the same number of inputs and outputs. An example of a compound model could be a spectrum with interstellar absorption. The stellar spectrum and the interstellar extinction are represented by separate models, but the observed spectrum is fitted with a compound model that combines both.', '3.7.4. Fitting Models to Data': 'astropy.modeling provides several fitters which are wrappers around some of the numpy and scipy.optimize functions and provide support for specifying parameter constraints. The fitters take a model and data as input and return a copy of \nthe model with the optimized parameter values set. The goal is to make it easy to extend the fitting framework to create new fitters. The optimizers available in astropy version 2.0 are Levenberg-Marquardt ( scipy.optimize.leastsq ), Simplex ( scipy.optimize.fmin ), SLSQP ( scipy.optimize.slsqp ), and LinearLSQFitter ( numpy.linalg.lstsq which provides exact solutions for linear models). \nModeling also supports a plugin system for fitters, which allows using the astropy models with external fitters. An example of this is SABA 17 , which is a bridge between Sherpa (Doe et al. 2007), and astropy.modeling , to bring the Sherpa fitters into astropy .', '3.7.5. Creating New Models': 'If arithmetic combinations of existing models is not sufficient, new model classes can be defined in different ways. The astropy.modeling package provides tools to turn a simple function into a full-featured model, but it also allows extending the built-in model class with arbitrary code.', '3.7.6. Unit Support': 'The astropy.modeling subpackage now supports the representation, evaluation, and fitting of models using Quantity objects, which attach units to scalar values or arrays of values. In practice, this means that one can, for example, fit a model to data with units and get parameters that also have units out, or initialize a model with parameters with units and evaluate it using input values with different but equivalent units. For example, the blackbody model ( BlackBody1D ) can be used to fit observed flux densities in a variety of units and as a function of different units of spectral coordinates (e.g., wavelength or frequency).', '3.8. Convolution': 'The astropy.convolution subpackage implements normalized convolution (e.g., Knutsson & Westin 1993), an image reconstruction technique in which missing data are ignored during the convolution and replaced with values interpolated using the kernel. An example is given in Figure 4. In astropy versions ≤ 1 . 3 , the direct convolution and Fast Fourier Transform (FFT) convolution approaches were inconsistent, with only the latter implementing normalized convolution. As of version 2.0, the two methods now agree and include a suite of consistency checks.', '3.9. Visualization': "The astropy.visualization subpackage provides functionality that can be helpful when visualizing data. This includes a framework (previously the standalone wcsaxes package) for plotting astronomical images with coordinates with matplotlib , functionality related to image normalization (including both scaling and stretching), smart \n(a) Original \n(b) Scipy \n(c) Scipy nan->zero \n<!-- image --> \n(d) Default astropy \n<!-- image --> \nFigure 4. An example showing different modes of convolution available in the Python ecosystem. Each red x signifies a pixel that is set to NaN in the original data (a). If the data are convolved with a Gaussian kernel on a 9 × 9 grid using scipy 's direct convolution (b), any pixel within range of the original NaN pixels is also set to NaN . Panel (c) shows what happens if the NaN s are set to zero first: the originally NaN regions are depressed relative to their surroundings. Finally, panel (d) shows astropy 's convolution behavior, where the missing pixels are replaced with values interpolated from their surroundings using the convolution kernel. \n<!-- image --> \nhistogram plotting, red-green-blue (RGB) color image creation from separate images, and custom plotting styles for matplotlib . \n<!-- image --> \nastropy.visualization provides a framework for transforming values in images (and more generally any arrays), typically for the purpose of visualization. Two main types of transformations are normalization and stretching of image values. \nNormalization transforms the image values to the range [0 , 1] using lower and upper limits ( v min , v max ) , \ny = x -v min v max -v min , (1) \nwhere x represents the values in the original image. \nStretching transforms the image values in the range [0 , 1] again to the range [0 , 1] using a linear or non-linear function, \nz = f ( y ) . (2) \nSeveral classes are provided for automatically determining intervals (e.g., using image percentiles) and for normalizing values in this interval to the [0 , 1] range. \nmatplotlib allows a custom normalization and stretch to be used when displaying data by passing in a normalization object. The astropy.visualization package also provides a normalization class that wraps the interval and stretches objects into a normalization object that matplotlib understands.", '3.9.2. Plotting image data with world coordinates': 'Astronomers dealing with observational imaging commonly need to make figures with images that include the correct coordinates and optionally display a coordinate grid. The challenge, however, is that the conceptual coordinate axes (such as longitude/latitude) need not be lined up with the pixel axes of the image. The astropy.visualization.wcsaxes subpackage implements a generalized way of making figures from an image array and a WCS object that provides the transformation between pixel and world coordinates. \nWorld coordinates can be, for example, right ascension and declination, but can also include, for example, velocity, wavelength, frequency, or time. The main features from this subpackage include the ability to control which axes to show which coordinate on (e.g., showing longitude ticks on the top and bottom axes and latitude on the left and right axes), controlling the spacing of the ticks either by specifying the positions to use or providing a tick spacing or an average number of ticks that should be present on each axis, setting the format for the tick labels to ones commonly used by astronomers, controlling the visibility of the grid/graticule, and overlaying ticks, labels, and/or grid lines from different coordinate systems. In addition, it is possible to pass data with more than two dimensions and slice on-the-fly. Last but not least, it is also able to define non-rectangular frames, such as, for example, Aitoff projections. \nThis subpackage differs from APLpy (Robitaille & Bressert 2012), in that the latter focuses on providing a very high-level interface to plotting that requires very few lines of code to get a good result, whereas wcsaxes defines an interface that is much \nFigure 5. An example of figure made using the astropy.visualization.wcsaxes subpackage, using Spitzer /IRAC 8.0 µ mdata from the Cygnus-X Spitzer Legacy survey (Beerer et al. 2010). \n<!-- image --> \ncloser to that of matplotlib (Hunter 2007). This enables significantly more advanced visualizations. \nAn example of a visualization made with wcsaxes is shown in Figure 5. This example illustrates the ability to overlay multiple coordinate systems and customize which ticks/labels are shown on which axes around the image. This also uses the image stretching functionality from Section 3.9.1 to show the image in a square-root stretch (automatically updating the tick positions in the colorbar).', '3.9.3. Choosing Histogram Bins': "astropy.visualization also provides a histogram function, which is a generalization of matplotlib 's histogram function, to allow for a more flexible specification of histogram bins. The function provides several methods of automatically tuning the histogram bin size. It has a syntax identical to matplotlib 's histogram function, with the exception of the bins parameter, which allows specification of one of four different methods for automatic bin selection: 'blocks', 'knuth', 'scott', or 'freedman'. \nFigure 6. An RGB color image of the region near the Hickson 88 group constructed from SDSS images and the astropy.visualization tools. This example uses astropy.visualization.wcsaxes to display the sky coordinate grid, and the astropy.visualization.make\\_lupton\\_rgb() function to produce the RGB image from three SDSS filter images ( g , r , i ). The left and right panel images show two different parameter choices for the stretch and softening parameters (shown in the titles). \n<!-- image -->", '3.9.4. Creating color RGB images': "Lupton et al. (2004) describe an 'optimal' algorithm for producing RGB composite images from three separate high-dynamic range arrays. The astropy.visualization subpackage provides a convenience function to create such a color image. It also includes an associated set of classes to provide alternate scalings. This functionality was contributed by developers from the Large Synoptic Survey Telescope (LSST) and serves as an example of contribution to Astropy from a more traditional engineering organization (Jenness et al. 2016). \nThe Sloan Digital Sky Survey (SDSS) SkyServer color images were made using a variation on this technique. As an example, in Figure 6, we show an RGB color image of the Hickson 88 group, centered near NGC 6977. 18 This image was generated from SDSS images using the astropy.visualization tools.", '3.10. Cosmology': 'The cosmology subpackage contains classes for representing different cosmologies and functions for calculating commonly used quantities such as look-back time and distance. The subpackage was described in detail in Astropy Collaboration et al. (2013). The default cosmology in astropy version 2.0 is given by the values in Planck Collaboration et al. (2016).', '3.11. Statistics': 'The astropy.stats package provides statistical tools that are useful for astronomy and are either not found in or extend the available functionality of other Python \nstatistics packages, such as scipy (Jones et al. 2001-) or statsmodels (Seabold & Perktold 2010). astropy.stats contains a range of functionality used by many different disciplines in astronomy. It is not a complete set of statistical tools, but rather a still growing collection of useful features.', '3.11.1. Robust Statistical Estimators': 'Robust statistics provide reliable estimates of basic statistics for complex distributions that largely mitigate the effects of outliers. astropy.stats includes several robust statistical functions that are commonly used in astronomy, such as sigma clipping methods for rejecting outliers, median absolute deviation functions, and biweight estimators, which have been used to calculate the velocity dispersion of galaxy clusters (Beers et al. 1990).', '3.11.2. Circular Statistics': 'Astronomers often need to compute statistics of quantities evaluated on a circle, such as sky direction or polarization angle. A set of circular statistical estimators based on Jammalamadaka & Sengupta (2001) are implemented in astropy.stats . These functions provide measurements of the circular mean, variance, and moment. All of these functions work with both numpy.ndarrays (assumed to be in radians) and Quantity objects. In addition, the subpackage includes tests for Rayleigh Test, vtest , and a function to compute the maximum likelihood estimator for the parameters of the von Mises distribution.', '3.11.3. Lomb-Scargle Periodograms': 'Periodic analysis of unevenly-spaced time series is common across many sub-fields of astronomy. The astropy.stats package now includes several efficient implementations of the Lomb-Scargle periodogram (Lomb 1976; Scargle 1982) and several generalizations, including floating mean models (Zechmeister & Kürster 2009), truncated Fourier models (Bretthorst 2003), and appropriate handling of heteroscedastic uncertainties. Importantly, the implementations make use of several fast and scalable computational approaches (e.g., Press & Rybicki 1989; Palmer 2009), and thus can be applied to much larger datasets than Lomb-Scargle algorithms available in, e.g., scipy.stats (Jones et al. 2001-). Much of the Lomb-Scargle code in astropy has been adapted from previously-published open-source code (VanderPlas et al. 2012; VanderPlas & Ivezic 2015).', '3.11.4. Bayesian Blocks and Histogram Binning': "astropy.stats also includes an implementation of Bayesian Blocks (Scargle et al. 2013), an algorithm for analysis of break-points in non-periodic astronomical timeseries. One interesting application of Bayesian Blocks is its use in determining optimal histogram binnings, particularly binnings with unequal bin sizes. This code was adapted, with several improvements, from the astroML package (VanderPlas et al. \nFigure 7. Three approaches to a 1D histogram: left: a standard histogram using matplotlib 's default of 10 bins. center: a histogram with the number of equal-width bins determined automatically using numpy 's bins='auto' . right: a histogram created with astropy , with irregularly-spaced bins computed via the Bayesian Blocks algorithm. Compared to regularly-spaced bins, the irregular bin widths give a more accurate visual representation of features in the dataset at various scales. \n<!-- image --> \n2012). An example of a histogram fit using the Bayesian Blocks algorithm is shown in the right panel of Figure 7.", '4. INFRASTRUCTURE FOR ASTROPY AFFILIATED PACKAGES': 'In addition to astronomy-specific packages and libraries, the Astropy Project also maintains and distributes several general-purpose infrastructure packages that assist the maintenance and upkeep of the astropy core package and other affiliated packages. The following sections describe the most widely-used infrastructure packages developed by the Astropy Project.', '4.1. Package template': 'Astropy provides a package template - as a separate GitHub repository, astropy/package-template 19 -that aims to simplify setting up packaging, testing, and documentation builds for developers of affiliated packages or astropy -dependent packages. Any Python package can make use of this ready-to-go package layout, setup, installation, and Sphinx documentation build infrastructure that was originally developed for the astropy core package and affiliated packages maintained by the Astropy project. The package template also provides a testing framework, template configurations for continuous integration services, and Cython build support.', '4.2. Continuous integration helpers': 'Astropy also provides a set of scripts for setting up and configuring continuous integration (CI) services as a GitHub repository, astropy/ci-helpers . 20 These tools aim to enable package maintainers to control their testing setup and installation process for various CI services through a set of environment variables. While the current development is mostly driven by the needs of the Astropy ecosystem, the \nactual usage of this package is extremely widespread. The current tools support configuration for Travis CI 21 and Appveyor CI 22 .', '4.3. Sphinx extensions': 'The documentation for many Python packages, including all the packages in the Astropy ecosystem, is written using the Sphinx documentation build system. Sphinx supports writing documentation using plain text files that follow a markup language called reStructuredText (RST). These files are then transformed into HTML, PDF, or L A T E Xdocuments during the documentation build process. For the Astropy project, we have developed several Sphinx extensions that facilitate automatically generating API documentation for large projects, like the astropy core package. The main extension we have developed is sphinx-automodapi 23 , which provides a convenient single RST command to generate a set of documentation pages, listing all of the available classes, functions, and attributes in a given Python module.', '5. THE FUTURE OF THE ASTROPY PROJECT': 'Following the release of version 2.0, development on the next major version of the astropy core package (version 3.0) has already begun. On top of planned changes and additions to the core package, we also plan to overhaul the Astropy educational/learning materials and further generalize the infrastructure utilities originally developed for the core package for the benefit of the community.', '5.1. Future versions of the Astropy core and affiliated packages': 'One of the most significant changes coming in this next major release will be removing the support for Python 2 (Robitaille 2017): future versions of astropy will only support Python 3.5 or higher. Removing Python 2 support will allow the use of new Python 3-only features, simplify the code base, and reduce the testing overhead for the package. astropy version 3.0 is currently scheduled for January 2018. \nIn the next major release after version 3.0, scheduled for mid-2018, the focus will be on algorithm optimization and documentation improvement. To prepare for this release, we are subjecting the core package to testing, evaluation, and performance monitoring. As a result, less new functionality may be introduced as a trade-off for better performance. \nBeyond the core code, the Astropy project is also further developing the Astropymanaged affiliated packages. While these may not be integrated into the astropy core package, these projects provide code that is useful to the astronomical community and meet the testing and documentation standards of Astropy. Some of these new efforts include an initiative to develop tools for spectroscopy (Crawford et al. 2017, specutils , specreduc , specviz ), integration of LSST software, and support for HEALPIX projection.', '5.2. Learn Astropy': "The documentation of the astropy core package contains narrative descriptions of the package's functionality, along with detailed usage notes for functions, classes, and modules. While useful as a reference for more experienced Python users, it is not the proper entry-point for other users or learning environments. In the near future, we will launch a new resource for learning to use both the astropy core package and the many packages in the broader Astropy ecosystem, under the name Learn Astropy . \nThe new Learn Astropy site will present several different ways to engage with the Astropy ecosystem: \nDocumentation: The astropy and affiliated package documentation contains the complete description of a package with all requisite details, including usage, dependencies, and examples. The pages will largely remain as-is, but will be focused towards more intermediate users and as a reference resource. \nExamples: These are stand-alone code snippets that live in the astropy documentation that demonstrate a specific functionality within a subpackage. The astropy core package documentation will then gain a new 'index of examples' that links to all of the code or demonstrative examples within any documentation page. \nTutorials: The Astropy tutorials are step-by-step demonstrations of common tasks that incorporate several packages or subpackages. Tutorials are more extended and comprehensive than examples, may contain exercises for the users, and are generally geared towards workshops or teaching. Several tutorials already exist 24 and are being actively expanded. \nGuides: These are long-form narrative, comprehensive, and conceptually-focused documents (roughly one book chapter in length), providing stand-alone introductions to core packages in addition to the underlying astronomical concepts. These are less specific and more conceptual than tutorials. For example, 'using astropy and ccdproc to reduce imaging data.' \nWe encourage any users who wish to see specific material to either contribute or comment on these efforts via the Astropy mailing list or astropy/astropy-tutorials GitHub repository. 25", '6. CONCLUSION': "The development of the astropy package and cultivation of the Astropy ecosystem are still maintaining significant growth while improving in stability, breadth, and reliability. As the astropy core package becomes more mature, several subpackages have reached stability with a rich set of features that help astronomers worldwide to \nperform many daily tasks, such as planning observations, analyzing data or simulation results, and writing publications. The strong emphasis that the Astropy project puts on reliability and high coding standards helps users to trust the calculations performed with astropy and to publish reproducible results. At the same time, the Astropy ecosystem and core package are both growing: new functionality is still being contributed and new affiliated packages are being developed to support more specialized needs. \nThe Astropy project is also spreading awareness of best practices in communitydriven software development. This is important as most practicing astronomers were not explicitly taught computer science and software development, despite the fact that a substantial fraction of many astronomers' workload today is related to software use and development. The astropy package leads by example, showing all interested astronomers how modern tools like git version control or CI testing can increase the quality, accessibility, and discoverability of astronomical software without overly complicating the development cycle. Within Astropy, all submitted code is reviewed by at least one, but typically more, member of the Astropy community, who provide feedback to contributors, which helps to improve their software development skills. As a community, Astropy follows an explicit code of conduct (Cruz et al. 2015) and treats all contributors and users with respect, provides a harassment-free environment, and encourages and welcomes new contributions from all. Thus, while the Astropy project provides and develops software and tools essential to modern astronomical research, it also helps to prepare the current and next generation of researchers with the knowledge to adequately use, develop, and contribute to those tools within a conscientious and welcoming community. \nWe would like to thank the members of the community that have contributed to Astropy, that have opened issues and provided feedback, and have supported the project in a number of different ways. We would like to acknowledge Alex Conley and Neil Crighton for maintaining the cosmology subpackage. \nThe Astropy community is supported by and makes use of a number of organizations and services outside the traditional academic community. We thank Google for financing and organizing the Google Summer of Code (GSoC) program, that has funded severals students per year to work on Astropy related projects over the summer. These students often turn into long-term contributors. We also thank NumFOCUS and the Python Software Foundation for financial support. Within the academic community, we thank institutions that make it possible that astronomers and other developers on their staff can contribute their time to the development of Astropy projects. We would like acknowledge the support of the Space Telescope Science Institute, Harvard-Smithsonian Center for Astrophysics, and the South African Astronomical Observatory. \nThe following individuals would like to recognize support for their personal contributions. HMG was supported by the National Aeronautics and Space Administration through the Smithsonian Astrophysical Observatory contract SV3-73016 to MIT for Support of the Chandra X-Ray Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of the National Aeronautics Space Administration under contract NAS8-03060. JTV was supported by the UW eScience Institute, via grants from the Moore Foundation, the Sloan Foundation, and the Washington Research Foundation. SMC acknowledges the National Research Foundation of South Africa. TLA was supported by NASA contract NAS8-03060. Support for E.J.T. was provided by NASA through Hubble Fellowship grant No. 51316.01 awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS 5-26555, as well as a Giacconi Fellowship. MB was supported by the FONDECYT regular project 1170618 and the MINEDUC-UA projects codes ANT 1655 and ANT 1656. DH was supported through the SFB 881 'The Milky Way System' by the German Research Foundation (DFG). WEK was supported by an ESO Fellowship. C.M. is supported by NSF grant AST-1313484. SP was supported by grant AYA2016-75808-R (FEDER) issued by the Spanish government. JEHT was supported by the Gemini Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., on behalf of the international Gemini partnership of Argentina, Brazil, Canada, Chile, and the United States of America. \nFurthermore, the Astropy packages would not exist in their current form without a number of web services for code hosting, continuous integration, and documentation; in particular, Astropy heavily relies on GitHub, Travis CI, Appveyor, CircleCI, and Read the Docs. \nastropy interfaces with the SIMBAD database, operated at CDS, Strasbourg, France. It also makes use of the ERFA library (Tollerud et al. 2017), which in turn derives from the IAU SOFA Collection 26 developed by the International Astronomical Union Standards of Fundamental Astronomy (Hohenkerk 2011). \nSoftware: astropy (Astropy Collaboration et al. 2013), numpy (Van der Walt et al. 2011), scipy (Jones et al. 2001-), matplotlib (Hunter 2007), Cython (Behnel et al. 2011), SOFA (Hohenkerk 2011), ERFA (Tollerud et al. 2017)", 'REFERENCES': 'Aldcroft, T. 2015, Astropy Proposal for Enhancement 6: Enhanced Character Separated Values table format (APE 6), doi:10.5281/zenodo.1043901. https: //doi.org/10.5281/zenodo.1043901 \nAstropy Collaboration, Robitaille, T. P., \nTollerud, E. J., et al. 2013, A&A, 558, \nStreicher, O., & Weilbacher, P. M. 2012, in Astronomical Society of the Pacific Conference Series, Vol. 461, Astronomical Data Analysis Software and Systems XXI, ed. P. Ballester, D. Egret, & N. P. F. Lorente, 853 Terlouw, J. P., & Vogelaar, M. G. R. 2015, Kapteyn Package, version 2.3, Kapteyn Astronomical Institute, Groningen, available from http://www. astro.rug.nl/software/kapteyn/ Tody, D. 1993, in Astronomical Society of the Pacific Conference Series, Vol. 52, Astronomical Data Analysis Software and Systems II, ed. R. J. Hanisch, R. J. V. Brissenden, & J. Barnes, 173 Tollerud, E. 2013, Astropy Proposal for Enhancement 2: Astropy Release Cycle and Version Numbering (APE 2), doi:10.5281/zenodo.1043888. https: //doi.org/10.5281/zenodo.1043888 Tollerud, E., Pascual, S., Nair, P., et al. 2017, doi:10.5281/zenodo.1021149 Tollerud, E., Price-Whelan, A., Aldcroft, T., & Robitaille, T. 2014, Astropy Proposal for Enhancement 5: Coordinates Subpackage Plan (APE 5), , , doi:10.5281/zenodo.1043897. https: //doi.org/10.5281/zenodo.1043897 Turk, M. J., Smith, B. D., Oishi, J. S., et al. 2011, ApJS, 192, 9', 'APPENDIX A. LIST OF AFFILIATED PACKAGES': 'T able 1 . Registry of affiliate d pac k ages. \ntion \nCita \nMain tainer \nName \nPyPI \nStable \nName \nge \na \nk \nac \nP \nBressert \nEli \nand \nRobitaille \ns \nThoma \ny \nAPLp \nes \nY \ny \nAPLp', 'v an Dokkum 2001': 'McCully \nis \nCurt \ny \nastroscrapp \nes \nY \nAstro-SCRAPPY \nanderplas \nV \ne \nJak \nastroML \nes \nY \nastroML \n2017 \nal. \net \nMorris \nMorris \nBrett \nastroplan \nNo \nastroplan \n2017a \nal. \net \nGinsburg \ncz \no \nip \nS \nBrigitta \nand \nGinsburg \ndam \nA \nquery \nastro \nes \nY \nquery \nastro \nCraig et al. 2015 \nhael \nMic \nand \nCraig, \nMatt \nwford, \nCra \nen \nStev \nc \nccdpro \nes \nY \nc \nccdpro \n2017 \nPrice-Whelan \nPrice-Whelan \ndrian \nA \nastro-gala \nes \nY \ngala \n17 \n20 \nal. \net \nDeil \nDeil \nChristoph \ny \ngammap \nNo \ny \ngammap \n2017 \nal. \net \ne \nhk \nejesc \nLim \ney-Lian \nP \nand \ne \nhk \nJesc \nEric \nginga \nes \nY \nginga \nSeifert \n2016 \nord \nF \nord \nF \nJes \ner-lensing \nclust \nNo \ncluster-lensing \n2015 \nvy \nBo \nvy \nBo \nJo \ny \ngalp \nes \nY \ny \ngalp \n2014 \nal. \net \nt \nBeaumon \nRobitaille \nThomas \nand \nt \nBeaumon \nChris \nglueviz \nes \nY \nGlue \n017 \n2 \nal. \net \na \nhev \nDenc \na \nhev \nDenc \na \nNadi \ncs \ngw \nNo \ncs \ngw \n2017 \nal. \net \nn \nHeari \narin \nHe \nAndrew \nols \nhaloto \nes \nY \nols \nHaloto \nhetti \nBac \nMatteo \nhendrics \nes \nY \nHENDRICS \nh \nc \nBo \nomas \nTh \nand \nDeil \nChristoph \nps \nhi \nNo \nhips \nSosey 2017 \nSosey \nMegan \nimexam \nNo \nimexam \nil \nNe \nand \nejos, \nT \nNicolas \na, \nhask \nc \nPro \nvier \nXa \nJ. \nols \nlineto \nes \nY \nols \nlineto \nton \nCrigh \nT able 1 c ontinue d on next p age', 'T able 1 (c ontinue d)': 'Table 2. \nAstropy Project II Registry of provisionally accepted affiliated packages.'} |
2020A&A...641A..10P | We report on the implications for cosmic inflation of the 2018 release of the Planck cosmic microwave background CMB anisotropy measurements. The results are fully consistent with those reported using the data from the two previous Planck cosmological releases but have smaller uncertainties thanks to improvements in the characterization of polarization at low and high multipoles. Planck temperature polarization and lensing data determine the spectral index of scalar perturbations to be nSUBsSUB 0.9649 0.0042 at 68 CL. We find no evidence for a scale dependence of nSUBsSUB either as a running or as a running of the running. The Universe is found to be consistent with spatial flatness with a precision of 0.4 at 95 CL by combining Planck with a compilation of baryon acoustic oscillation data. The Planck 95 CL upper limit on the tensortoscalar ratio rSUB0.002SUB lt 0.10 is further tightened by combining with the BICEP2Keck Array BK15 data to obtain rSUB0.002SUB lt 0.056. In the framework of standard singlefield inflationary models with Einstein gravity these results imply that a the predictions of slowroll models with a concave potential V lt 0 are increasingly favoured by the data and b based on two different methods for reconstructing the inflaton potential we find no evidence for dynamics beyond slow roll. Three different methods for the nonparametric reconstruction of the primordial power spectrum consistently confirm a pure power law in the range of comoving scales 0.005 MpcSUP1SUP k 0.2 MpcSUP1SUP. A complementary analysis also finds no evidence for theoretically motivated parameterized features in the Planck power spectra. For the case of oscillatory features that are logarithmic or linear in k this result is further strengthened by a new combined analysis including the Planck bispectrum data. The new Planck polarization data provide a stringent test of the adiabaticity of the initial conditions for the cosmological fluctuations. In correlated mixed adiabatic and isocurvature models the nonadiabatic contribution to the observed CMB temperature variance is constrained to 1.3 1.7 and 1.7 at 95 CL for cold dark matter neutrino density and neutrino velocity respectively. Planck power spectra plus lensing set constraints on the amplitude of compensated cold dark matterbaryon isocurvature perturbations that are consistent with current complementary measurements. The polarization data also provide improved constraints on inflationary models that predict a small statistically anisotropic quadupolar modulation of the primordial fluctuations. However the polarization data do not support physical models for a scaledependent dipolar modulation. All these findings support the key predictions of the standard singlefield inflationary models which will be further tested by future cosmological observations. | 2020-09-01T00:00:00Z | ['arXiv:1807.06211', '2018arXiv180706211P', '10.1051/0004-6361/201833887', '2020A&A...641A..10P', '10.48550/arXiv.1807.06211'] | ['inflation', 'cosmic background radiation', 'Astrophysics - Cosmology and Nongalactic Astrophysics'] | Planck 2018 results. X. Constraints on inflation | 2,020 | 173 | 0.71 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF'] | 2,984 | https://arxiv.org/pdf/1807.06211.pdf | {'Planck 2018 results. X. Constraints on inflation': "Planck Collaboration: Y. Akrami 55 ; 57 , F. Arroja 59 , M. Ashdown 65 ; 5 , J. Aumont 94 , C. Baccigalupi 78 , M. Ballardini 21 ; 39 , A. J. Banday 94 ; 8 , R. B. Barreiro 60 , N. Bartolo 28 ; 61 , S. Basak 85 , K. Benabed 53 ; 93 , J.-P. Bernard 94 ; 8 , M. Bersanelli 31 ; 43 , P. Bielewicz 77 ; 8 ; 78 , J. J. Bock 62 ; 10 , J. R. Bond 7 , J. Borrill 12 ; 91 , F. R. Bouchet 53 ; 88 , F. Boulanger 67 ; 52 ; 53 , M. Bucher 2 ; 6 GLYPH<3> , C. Burigana 42 ; 29 ; 45 , R. C. Butler 39 , E. Calabrese 82 , J.-F. Cardoso 53 , J. Carron 23 , A. Challinor 56 ; 65 ; 11 , H. C. Chiang 25 ; 6 , L. P. L. Colombo 31 , C. Combet 69 , D. Contreras 20 , B. P. Crill 62 ; 10 , F. Cuttaia 39 , P. de Bernardis 30 , G. de Zotti 40 ; 78 , J. Delabrouille 2 , J.-M. Delouis 53 ; 93 , E. Di Valentino 63 , J. M. Diego 60 , S. Donzelli 43 ; 31 , O. Dor'e 62 ; 10 , M. Douspis 52 , A. Ducout 53 ; 51 , X. Dupac 34 , S. Dusini 61 , G. Efstathiou 65 ; 56 , F. Elsner 74 , T. A. Enßlin 74 , H. K. Eriksen 57 , Y. Fantaye 3 ; 19 , J. Fergusson 11 , R. Fernandez-Cobos 60 , F. Finelli 39 ; 45 GLYPH<3> , F. Forastieri 29 ; 46 , M. Frailis 41 , E. Franceschi 39 , A. Frolov 87 , S. Galeotta 41 , S. Galli 64 , K. Ganga 2 , C. Gauthier 2 ; 72 , R. T. G'enova-Santos 58 ; 15 , M. Gerbino 92 , T. Ghosh 81 ; 9 , J. Gonz'alez-Nuevo 16 , K. M. G'orski 62 ; 96 , S. Gratton 65 ; 56 , A. Gruppuso 39 ; 45 , J. E. Gudmundsson 92 ; 25 , J. Hamann 86 , W. Handley 65 ; 5 , F. K. Hansen 57 , D. Herranz 60 , E. Hivon 53 ; 93 , D. C. Hooper 54 , Z. Huang 83 , A. H. Ja GLYPH<11> e 51 , W. C. Jones 25 , E. Keihanen 24 , R. Keskitalo 12 , K. Kiiveri 24 ; 38 , J. Kim 74 , T. S. Kisner 71 , N. Krachmalnico GLYPH<11> 78 , M. Kunz 14 ; 52 ; 3 , H. Kurki-Suonio 24 ; 38 , G. Lagache 4 , J.-M. Lamarre 66 , A. Lasenby 5 ; 65 , M. Lattanzi 29 ; 46 , C. R. Lawrence 62 , M. Le Jeune 2 , J. Lesgourgues 54 , F. Levrier 66 , A. Lewis 23 , M. Liguori 28 ; 61 , P. B. Lilje 57 , V. Lindholm 24 ; 38 , M. L'opez-Caniego 34 , P. M. Lubin 26 , Y.-Z. Ma 63 ; 80 ; 76 , J. F. Mac'ıas-P'erez 69 , G. Maggio 41 , D. Maino 31 ; 43 ; 47 , N. Mandolesi 39 ; 29 , A. Mangilli 8 , A. Marcos-Caballero 60 , M. Maris 41 , P. G. Martin 7 , E. Mart'ınez-Gonz'alez 60 , S. Matarrese 28 ; 61 ; 36 , N. Mauri 45 , J. D. McEwen 75 , P. D. Meerburg 65 ; 11 ; 95 , P. R. Meinhold 26 , A. Melchiorri 30 ; 48 , A. Mennella 31 ; 43 , M. Migliaccio 90 ; 49 , S. Mitra 50 ; 62 , M.-A. Miville-Deschˆenes 68 , D. Molinari 29 ; 39 ; 46 , A. Moneti 53 , L. Montier 94 ; 8 , G. Morgante 39 , A. Moss 84 , M. Munchmeyer 53 , P. Natoli 29 ; 90 ; 46 , H. U. Nørgaard-Nielsen 13 , L. Pagano 52 ; 66 , D. Paoletti 39 ; 45 , B. Partridge 37 , G. Patanchon 2 , H. V. Peiris 22 , F. Perrotta 78 , V. Pettorino 1 , F. Piacentini 30 , L. Polastri 29 ; 46 , G. Polenta 90 , J.-L. Puget 52 ; 53 , J. P. Rachen 17 , M. Reinecke 74 , M. Remazeilles 63 , A. Renzi 61 , G. Rocha 62 ; 10 , C. Rosset 2 , G. Roudier 2 ; 66 ; 62 , J. A. Rubi˜no-Mart'ın 58 ; 15 , B. Ruiz-Granados 58 ; 15 , L. Salvati 52 , M. Sandri 39 , M. Savelainen 24 ; 38 ; 73 , D. Scott 20 , E. P. S. Shellard 11 , M. Shiraishi 28 ; 61 ; 18 , C. Sirignano 28 ; 61 , G. Sirri 45 , L. D. Spencer 82 , R. Sunyaev 74 ; 89 , A.-S. Suur-Uski 24 ; 38 , J. A. Tauber 35 , D. Tavagnacco 41 ; 32 , M. Tenti 44 , L. To GLYPH<11> olatti 16 ; 39 , M. Tomasi 31 ; 43 , T. Trombetti 42 ; 46 , J. Valiviita 24 ; 38 , B. Van Tent 70 , P. Vielva 60 , F. Villa 39 , N. Vittorio 33 , B. D. Wandelt 53 ; 93 ; 27 , I. K. Wehus 62 ; 57 , S. D. M. White 74 , A. Zacchei 41 , J. P. Zibin 20 , and A. Zonca 79 \n(A GLYPH<14> liations can be found after the references) \n10 June 2019", 'ABSTRACT': 'We report on the implications for cosmic inflation of the 2018 release of the Planck cosmic microwave background (CMB) anisotropy measurements. The results are fully consistent with those reported using the data from the two previous Planck cosmological releases, but have smaller uncertainties thanks to improvements in the characterization of polarization at low and high multipoles. Planck temperature, polarization, and lensing data determine the spectral index of scalar perturbations to be n s = 0 : 9649 GLYPH<6> 0 : 0042 at 68 % CL. We find no evidence for a scale dependence of n s, either as a running or as a running of the running. The Universe is found to be consistent with spatial flatness with a precision of 0 : 4 % at 95 % CL by combining Planck with a compilation of BAO data. The Planck 95 % CL upper limit on the tensor-to-scalar ratio, r 0 : 002 < 0 : 10, is further tightened by combining with the BICEP2 / Keck Array BK15 data to obtain r 0 : 002 < 0 : 056. In the framework of standard single-field inflationary models with Einstein gravity, these results imply that: (a) the predictions of slow-roll models with a concave potential, V 00 ( GLYPH<30> ) < 0, are increasingly favoured by the data; and (b) based on two di GLYPH<11> erent methods for reconstructing the inflaton potential, we find no evidence for dynamics beyond slow roll. Three di GLYPH<11> erent methods for the non-parametric reconstruction of the primordial power spectrum consistently confirm a pure power law in the range of comoving scales 0 : 005 Mpc GLYPH<0> 1 . k . 0 : 2 Mpc GLYPH<0> 1 . A complementary analysis also finds no evidence for theoretically motivated parameterized features in the Planck power spectra. For the case of oscillatory features that are logarithmic or linear in k ; this result is further strengthened by a new combined analysis including the Planck bispectrum data. The new Planck polarization data provide a stringent test of the adiabaticity of the initial conditions for the cosmological fluctuations. In correlated, mixed adiabatic and isocurvature models, the non-adiabatic contribution to the observed CMB temperature variance is constrained to 1.3 %, 1.7 %, and 1.7 % at 95 % CL for cold dark matter, neutrino density, and neutrino velocity, respectively. Planck power spectra plus lensing set constraints on the amplitude of compensated cold dark matter-baryon isocurvature perturbations that are consistent with current complementary measurements. The polarization data also provide improved constraints on inflationary models that predict a small statistically anisotropic quadupolar modulation of the primordial fluctuations. However, the polarization data do not support physical models for a scale-dependent dipolar modulation. All these findings support the key predictions of the standard single-field inflationary models, which will be further tested by future cosmological observations.', '1. Introduction': "This paper, one of a set associated with the 2018 release of data from the Planck 1 mission, presents the implications for cosmic inflation of the 2018 Planck measurements of the cosmic microwave background (CMB) anisotropies. In terms of data, this paper updates Planck Collaboration XXII (2014) (henceforth PCI13), which was based on the temperature data of the nominal Planck mission ('PR1'), including the first 14 months of observations, and Planck Collaboration XX (2016) (henceforth PCI15), which used temperature data and a first set of polarization data from the full Planck mission ('PR2'), comprising 29 and 52 months of observations for the high frequency instrument (HFI) and low frequency instrument (LFI), respectively. \nThe ideas underlying cosmic inflation were developed during the late 1970s and early 1980s in order to remedy a number of defects of the hot big-bang cosmological model (e.g., the horizon, smoothness, flatness, and monopole problems) (Brout et al. 1978; Starobinsky 1980; Kazanas 1980; Sato 1981; Guth 1981; Linde 1982; Albrecht & Steinhardt 1982; Linde 1983). Subsequently, it was realized that, on account of quantum vacuum fluctuations, cosmic inflation also provides a means to generate the primordial cosmological perturbations (Mukhanov & Chibisov 1981, 1982; Hawking 1982; Guth & Pi 1982; Starobinsky 1982; Bardeen et al. 1983; Mukhanov 1985). The development of cosmic inflation is one of the major success stories of modern cosmology, and in this paper we explore how the latest 2018 release of the Planck data constrains inflationary models. \nPlanck data currently provide the best constraints on the CMB anisotropies, except on very small angular scales beyond the resolution limit of Planck . The Planck data set has enabled a precision characterization of the primordial cosmological perturbations and has allowed cosmological parameters to be constrained at the sub-percent level. One of the main data products, described in more detail in the following section, is the Planck TT , TE , EE , and lensing power spectra, which are shown in Fig. 1, together with the residuals from the six-parameter concordance GLYPH<3> cold dark matter ( GLYPH<3> CDM) model using the best-fit parameter values. \nIn order to provide a quantitative estimate of the improvement achieved by Planck , as well as to show where Planck stands compared to an ultimate cosmic-variance-limited survey, we consider an idealistic estimator for the number of modes [i.e., the e GLYPH<11> ective number of a ' m 's measured (Planck Collaboration I 2016)]: \nN XY modes ( ' ) GLYPH<17> 2 ' X ' 0 = 2 0 B B B B @ C XY ' 0 GLYPH<1> C XY ' 0 1 C C C C A 2 ; (1) \nwhere C XY ' ( GLYPH<1> C XY ' ) is the (error on the) angular power spectrum of the XY channel (Planck Collaboration I 2016; Scott et al. 2016). The number of modes measured by Planck is 1 430 000 and 109 000 for temperature ( XY = TT up to ' = 2500) and polarization ( XY = EE up to ' = 2000), respectively (Planck Collaboration I 2018). The number of modes measured is increased by approximately a factor of 7 (570) for temperature (polarization) with respect to the WMAP 9-year measurement, \nbut there is still a factor of 3 (40) to gain for a cosmic-variancelimited experiment up to ' = 2500 accessing 70 % of the sky. The additional modes measured by Planck play a key role in improving the constraints on the initial conditions for the cosmological perturbations and on models of inflation with respect to previous measurements of CMB anisotropies. \nPlanck data have also greatly improved the constraints on bispectral non-Gaussianity, both for the 'local' pattern, as predicted by many inflationary models, and for other templates such as the equilateral one, as analysed and reported in detail in the dedicated Planck non-Gaussianity papers (Planck Collaboration XXIV 2014; Planck Collaboration XVII 2016; Planck Collaboration IX 2018). The constraints on the non-Gaussianity parameter f NL are limited by a combination of cosmic variance and instrumental noise. An order-of-magnitude estimate for the signal-to-noise ratio for the local pattern (with f loc NL = 1) is given by \nGLYPH<18> S N GLYPH<19> 2 / GLYPH<10> sky ' 2 max ln ' max ' min ! : (2) \nFor the local shape, the logarithm enters because most of the signal derives from detecting the modulation of the smallscale power by the large-scale CMB anisotropy, highlighting the importance of full-sky maps for this kind of analysis. For other shapes such as equilateral, one instead has ( S = N ) 2 GLYPH<24> GLYPH<10> sky ' 2 max . Planck has significantly sharpened the constraints on f NL, largely on account of its measurement of high multipoles with higher signal-to-noise ratio compared to past surveys. Some improvement has also been obtained from including polarization. \nThe Planck measurements have significantly constrained the physics of inflation. The hypothesis of adiabatic Gaussian scalar fluctuations with a power spectrum described by a simple power law, which is the key prediction of the standard single-field slow-roll inflationary models, has been tested to unprecedent accuracy (PCI13; PCI15;Planck Collaboration XXIV 2014; Planck Collaboration XVII 2016). Planck has set tight constraints on the amount of inflationary gravitational waves by exploiting the shape of the CMB temperature spectrum (PCI13). These results have inspired a resurgence of activity in inflationary model building. For more details, see, for example, the following review articles and references therein: Linde (2015); Martin et al. (2014a); Guth et al. (2014); and Burgess et al. (2013). Planck analysis and interpretation have also sparked a debate on the likelihood of initial conditions for some inflationary models (Ijjas et al. 2013; Ijjas & Steinhardt 2016; Linde 2017), which is primarily of theoretical interest and is not addressed in this paper. In combination with more sensitive B -mode ground-based polarization measurements, as from BICEP-Keck Array (BKP), Planck has convincingly ruled out the slow-roll inflationary model with a quadratic potential (PCI15). In terms of physics beyond the simplest slow-roll inflationary models, the prePlanck hints of a running spectral index (Hou et al. 2014) or of large non-Gaussianities (Bennett et al. 2012) have disappeared as a result of the Planck measurements. How to interpret anomalies on the largest angular scales and at high multipoles is a question motivating the search for new nonstandard inflationary models. We discuss how the Planck 2018 release data further test these ideas. \nThis paper is organized as follows. In Sect. 2 we describe the statistical methodology, the Planck likelihoods, and the complementary data sets used in this paper. In Sect. 3 we discuss the updated constraints on the spectral index of the scalar perturbations, on spatial curvature, and on the tensor-to-scalar ratio. \n/lscript \nFig. 1. Planck 2018 CMB angular power spectra, compared with the baseGLYPH<3> CDM best fit to the Planck TT,TE,EE + lowE + lensing data (blue curves). For each panel we also show the residuals with respect to this baseline best fit. Plotted are D ' = ' ( ' + 1) C '= (2 GLYPH<25> ) for TT and TE , C ' for EE , and L 2 ( L + 1) 2 C GLYPH<30>GLYPH<30> L = (2 GLYPH<25> ) for lensing. For TT , TE , and EE , the multipole range 2 GLYPH<20> ' GLYPH<20> 29 shows the power spectra from Commander ( TT ) and SimAll ( TE , EE ), while at ' GLYPH<21> 30 we display the co-added frequency spectra computed from the Plik cross-half-mission likelihood, with foreground and other nuisance parameters fixed to their best-fit values in the baseGLYPH<3> CDM cosmology. For the Planck lensing potential angular power spectrum, we show the conservative (orange dots; used in the likelihood) and aggressive (grey dots) cases. Note some of the di GLYPH<11> erent horizontal and vertical scales on either side of ' = 30 for the temperature and polarization spectra and residuals. \n<!-- image --> \n<!-- image --> \n/lscript \n<!-- image --> \n/lscript \n<!-- image --> \nL \nSection 4 is devoted to constraining slow-roll parameters and to a Bayesian model comparison of inflationary models, taking into account the uncertainties in connecting the inflationary expansion to the subsequent big-bang thermalized era. In Sect. 5 the potential for standard single-field inflation is reconstructed using two di GLYPH<11> erent methodologies. Section 6 describes the primordial power spectrum reconstruction using three different approaches. In Sect. 7, the parametric search for features in the primordial scalar power spectrum is described, including a dedicated study of the axion monodromy model. In Sect. 8, the Planck power spectrum data are combined with information from the Planck bispectrum in a search for oscillations in the primordial spectra. The constraints on isocurvature modes are summarized in Sect. 9. Section 10 updates and extends the constraints on anisotropic inflationary models of inflation. We summarize our conclusions in Sect. 11, highlighting the key results and the legacy of Planck for inflation.", '2. Methodology and data': 'The general theoretical background and analysis methods applied in this paper closely match those of the previous Planck inflation papers (PCI13; PCI15). Consequently, in this section we provide only a brief summary of the methodology and focus on changes in the Planck likelihood relative to previous releases.', '2.1. Cosmological models and inference': "For well over a decade, the baseGLYPH<3> CDM model has been established as the simplest viable cosmological model. Its six free parameters can be divided into primordial and late-time parameters. The former describe the state of perturbations on observable scales (corresponding to a wavenumber range of 10 GLYPH<0> 4 Mpc GLYPH<0> 1 . k . 10 GLYPH<0> 1 Mpc GLYPH<0> 1 today) prior to re-entering the Hubble radius around recombination. In base GLYPH<3> CDM, the initial state of perturbations is assumed to be purely adiabatic and scalar, with the spectrum of curvature perturbations given by the power law \nwhere k GLYPH<3> denotes an arbitrary pivot scale. The late-time parameters, on the other hand, determine the linear evolution of perturbations after re-entering the Hubble radius. By default we use the basis ( ! b ; ! c ; GLYPH<18> MC ; GLYPH<28> ) 2 for the late-time parameters, but occasionally also consider non-minimal late-time cosmologies. Because of the inflationary perspective of this paper, we are mainly interested in exploring modifications of the primordial sector and their interpretation in terms of the physics of the inflationary epoch. \nPerturbations produced by generic single-field slow-roll models of inflation are typically well approximated by the following form of the adiabatic scalar and tensor components: \nln P R ( k ) = ln P 0( k ) + 1 d ln n s ln( k = k GLYPH<3> ) 2 \n2 d ln k + 1 6 d 2 ln n s d ln k 2 ln( k = k GLYPH<3> ) 3 + : : : ; (4) \nln P t( k ) = ln( rA s) + n t ln( k = k GLYPH<3> ) + : : : ; (5) \nwhich allows for a weak scale dependence of the scalar spectral index, n s, modelled by a running, d ln n s = d ln k , or a running of the running, d 2 ln n s = d (ln k ) 2 . 3 The power spectrum parameterization in Eq. 4 can be extended to address wider classes of inflation-related questions, (e.g., the search for isocurvature perturbations, specific primordial features in the spectra, etc.), as described in subsequent sections. We also go beyond simple functions to parameterize the primordial power spectrum. In the spirit of reconstructing the primordial spectrum from the data, we consider some general parameterizations (e.g., taking the power spectrum as an interpolation between knots of freely varied amplitudes at fixed or varying wavenumbers). \nOne could argue that the primordial power spectra are merely intermediate quantities and assess theories directly from more fundamental parameters. By using the slow-roll approximation, or by evolving the mode equations to obtain exact numerical predictions for the spectra without resorting to the slow-roll approximation, we can relate the primordial perturbations to the dynamics of the Hubble parameter during inflation or to the inflaton potential and its derivatives, thus constraining these quantities directly. \nFor any given model, theoretical predictions of CMBrelated and other cosmological observables are calculated using appropriately modified versions of the Boltzmann codes CAMB (Lewis et al. 2000) or CLASS (Blas et al. 2011). As in (PCI13; PCI15), we compare models M 1 and M 2 by the di GLYPH<11> erence in the logarithm of the likelihood of their best fits, or e GLYPH<11> ective GLYPH<1> GLYPH<31> 2 GLYPH<17> 2 [ln L max( M 1) GLYPH<0> ln L max( M 2)]. We apply Bayesian statistical methods to infer the posterior probability distributions of the model parameters and select between competing models (Trotta 2008), using either the Metropolis-Hastings Markov-chain Monte Carlo (MCMC) sampling algorithm, as implemented in CosmoMC (Lewis & Bridle 2002) and Monte Python (Audren et al. 2013), or software based on nested sampling (Skilling 2004), such as MultiNest (Feroz et al. 2009, 2013) or PolyChord (Handley et al. 2015a,b). The latter can simultaneously estimate the Bayesian evidence E i of a model M i , allowing the comparison between di GLYPH<11> erent models via the Bayes factor, B = E 2 = E 1, where j ln B j > 5 is commonly considered \n'strong' evidence for or against the respective model (Je GLYPH<11> reys 1998; Trotta 2007a).", '2.2.1. Planck data': 'The Planck data processing has improved in a number of key aspects with respect to the previous 2015 cosmological release. We briefly summarize the main points here, referring the interested reader to Planck Collaboration II (2018) and Planck Collaboration III (2018) for details. \nThe flagging procedure in the LFI 2018 pipeline has been made more aggressive, in particular for the first 200 operational days. However, the most important improvement in the LFI pipeline is in the calibration approach. Whereas in the 2015 release, the main calibration source for LFI was the Planck orbital dipole (i.e., the amplitude modulation of the CMB dipole induced by the satellite orbit) of each single radiometer model, the 2018 procedure also includes the Galactic emission along with the orbital dipole in the calibration model and becomes iterative (Planck Collaboration II 2018). \nThe HFI data for the 2018 release have also been made more conservative, cutting approximately 22 days of observations under non-stationary conditions with respect to 2015. The main change in the HFI data processing is the use of a new map making and calibration algorithm called SRoll , whose first version was introduced in Planck Collaboration Int. XLVI (2016) for the initial analysis of HFI polarization on large angular scales. This algorithm employs a generalized polarization destriper which uses the redundancy in the data to extract several instrumental systematic-e GLYPH<11> ect parameters directly from the data (Planck Collaboration III 2018). \nThese improvements have a minor impact on Planck temperature maps, but are much more important for polarization, particularly on large angular scales, allowing, for instance, the removal of the high-pass filtering adopted in the 2015 study of isotropy and statistics Planck Collaboration XVI (2016). \nIn the following, we summarize the essentials of the Planck inputs used in this paper (i.e., the Planck likelihood approach to the information contained in the 2-point statistics of the temperature and polarization maps and the Planck CMB lensing likelihood). As for previous cosmological releases, the Planck likelihood approach is hybridized between low- and high-multipole regions, which therefore are summarized separately below. We refer the interested reader to the relevant papers Planck Collaboration V (2018) (henceforth PPL18) and Planck Collaboration VIII (2018) (henceforth PPLe18) for a more complete description of these inputs.', "Planck low-' likelihood": "As in the Planck 2015 release, several options are available for evaluating the temperature likelihood on large angular scales, each with its own computational complexity and approximations. One option is based on the Commander framework and implements full Bayesian sampling of an explicit parametric model that includes both the cosmological CMB signal and non-cosmological astrophysical signals, such as thermal dust, CO, and low-frequency foregrounds. This framework is described in earlier papers [see Planck Collaboration XI (2016) and Planck Collaboration Int. XLVI (2016) and references therein for details]. The only changes since the 2015 implementation concern the data and model selection. As described \nTable 1. Baseline and optional late-time parameters, primordial power spectrum parameters, and slow-roll parameters. All primordial quantities are evaluated at a pivot scale of k GLYPH<3> = 0 : 05 Mpc GLYPH<0> 1 , unless otherwise stated. \nin Planck Collaboration IV (2018), we only use the Planck 2018 data in the current data release, whereas the previous 2015 version additionally included WMAP (Bennett et al. 2013) and Haslam (Haslam et al. 1982) observations. With fewer frequencies available, this requires a simpler model, and in particular we now fit for only a single low-frequency foreground component, rather than individual synchrotron, free-free, and spinning dust emission components, and we only fit a single CO component, rather than for individual CO line components at 100, 217, and 353 GHz. On the one hand, this results in fewer internal foreground degeneracies compared to the 2015 version, and a likelihood that only depends on Planck data, but at the same time the simpler foreground modelling also requires a slightly larger Galactic mask. Overall, the two versions are very compatible in terms of the recovered CMB power spectra, as discussed in PPL18. For additional details on the Commander temperature analysis, see Planck Collaboration IV (2018). \nThe HFI low-' polarization likelihood is based on the fullmission HFI 100-GHz and 143-GHz Q and U low-resolution maps cleaned through a template-fitting procedure with LFI 30 GHz and HFI 353 GHz information 4 used as tracers of polarized synchrotron and thermal dust, respectively (see PPL18 for details about the cleaning procedure). The likelihood method, called SimAll , represents a follow-up of the SimBaL algorithm presented in Planck Collaboration Int. XLVI (2016) and uses the FFP10 simulations to construct empirically the probability for the EE and BB spectra. The method is based on the quadratic maximum likelihood estimation of the cross-spectrum between 100 and 143 GHz, and its multipole range spans from ' = 2 to ' = 29. We only built the likelihood for EE and BB and not \nfor TE , due to the poor statistical consistency of the TE spectrum for ' > 10, and due to the di GLYPH<14> culty of describing accurately the correlation with TT and EE , given the limited number of simulations available; see discussions in section 2.2.6 of PPL18. Further details about the method and consistency tests are presented in PPL18. When combined with the low-' temperature likelihood (based on the Commander CMB solution), the low-' polarization likelihood implies GLYPH<28> = 0 : 051 GLYPH<6> 0 : 009 and r 0 : 002 < 0 : 41 at 95 % CL. \nAs an alternative to the Commander and SimAll low-' likelihood, an update of the joint temperature and polarization pixelbased low-' LFI likelihood used in 2015 is part of this Planck data release. Its methodology (see PPL18 for details) is similar to that of 2015 (Planck Collaboration XI 2016), i.e., a pixel-based approach in TQU at N side = 16, and employs the Commander solution in temperature along with the LFI 70-GHz linear polarization maps, foreground cleaned using the Planck 30-GHz and 353-GHz channels as tracer templates for synchrotron and dust, respectively. This 2018 version allows for a larger sky fraction in polarization (66.4 %, compared to the previous 46 %) and retains the sky surveys 2 and 4 that were excluded in 2015. By performing a two-parameter estimate for A s and GLYPH<28> restricted to ' < 30, we find using this likelihood that GLYPH<28> = 0 : 063 GLYPH<6> 0 : 020 and ln(10 10 As) = 2 : 975 GLYPH<6> 0 : 056 at 68 % CL. The latter value has been derived by varying the TT , EE , and TE CMB spectra.", "Planck high-' likelihood": "The 2018 baseline high-' likelihood ( Plik ) is an update of the 2015 baseline version. The CamSpec likelihood is also used to explore alternative data cuts and modelling of the data and is described below. Both approaches implement a Gaussian likelihood approximation using cross-spectra between the 100-, 143-, and 217-GHz maps. Plik covers the multipoles 30 GLYPH<20> ' GLYPH<20> 2509 in temperature and 30 GLYPH<20> ' GLYPH<20> 1997 in polarization (i.e., for TE \nand EE ). In order to avoid noise bias, the high-' likelihood relies only on half-mission map cross-spectra, which have been demonstrated to be largely free of correlated noise. The spectra are computed on masked maps in order to reduce the anisotropic Galactic contamination (dominated by dust emission), and in the case of TT also strong point sources and CO emission. The Plik masks, identical to the 2015 masks, are tailored to each frequency channel and di GLYPH<11> er in temperature and polarization to take into account di GLYPH<11> ering foreground behaviour and channel beams. The Plik intensity (polarization) masks e GLYPH<11> ectively retain 66, 57, and 47 % (70, 50, and 41 %) of the sky after apodization for the 100, 143, and 217 GHz channels, respectively. Unlike in 2015, when the map beams were computed for an average sky fraction, they are now computed for the exact sky fraction used at each frequency. The data vector used in the likelihood approximation discards multipoles that are highly contaminated by foregrounds or have low signal-to-noise ratios. \nThe Plik power spectra are binned using the same scheme as in 2015. Unbinned likelihoods are also available. When forming the data vector, individual cross-frequency spectra are not co-added. This allows for independent exploration of the calibration, nuisance, and foreground parameter space for each crossspectrum using dedicated templates in the theory vector. \nThe Plik (and CamSpec ) covariance matrices are computed for a fixed fiducial CMB including the latest estimate of the foreground and systematics, which are all assumed to be Gaussian. As verified in 2015, after the masks have been applied this is a reasonable assumption. The covariance matrix computation uses an approximation to account for mask-induced correlations. Plik uses only the large Galactic mask in the analytic computation and then takes extra correlations due to the point-source mask into account using a Monte Carlo estimate of the extra variance induced. Missing pixels are ignored in the covariance. In 2015 its was shown that this approach (i.e., Gaussian approximation and approximate covariance) induced only a less than 0 : 1 GLYPH<27> bias on n s (from the 30 < ' < 100 modes). \nThe Plik noise model has been re-estimated on the latest HFI maps using the same methodology as in 2015, based on a comparison between noise-biased auto-spectra and crossspectra. This procedure avoids correlated glitch residuals, which had biased previous noise estimates (Planck Collaboration XI 2016), particularly in polarization at ' . 500. \nThe 2018 HFI data processing pipeline has refined the maps used in the likelihood relative to 2015. For example, an improved destriping procedure reduced the residual scatter in the polarization maps, in particular at 143 GHz (yielding about 12 % lower noise on the half-mission cross-spectrum). More stringent selection cuts resulted in the discarding of the last 1000 rings of data, increasing the noise in the temperature half-mission spectrum by about 3 %. Also, a higher threshold was imposed on the conditioning of the TQU intra-pixel noise covariance matrix for a pixel to be considered well-measured, resulting in more missing pixels relative to 2015. \nThe data modelling has also significantly improved, in particular for polarization, making cosmological constraints from polarization more reliable. In 2015, the polarization spectra ( TE and EE ) displayed relatively large inter-frequency disagreements. A plausible explanation (at least for TE ) was the temperature-to-polarization leakage induced by beam and calibration di GLYPH<11> erences (so-called 'beam leakage'). The beam-leakage modelling has improved substantially in 2018 (Hivon et al. 2017) so that we can now propagate the beams, gain di GLYPH<11> erences, polarization angles, etc. to compute a reliable template for the beam leakage and thus remove these leakage \ne GLYPH<11> ects. These improvements substantially reduce the TE interfrequency disagreements. \nWe also reassessed the estimates of the polarization e GLYPH<14> -ciency for the polarized channels. Comparing di GLYPH<11> erent databased estimates demonstrates that the ground-based polarization e GLYPH<14> ciency uncertainty estimates (of the order of a fraction of a percent) were too optimistic by a factor of 5 to 10. Correcting for the observed polarization e GLYPH<14> ciency errors (at the percent level) very significantly reduces the EE inter-frequency disagreements. This calibration correction relies on cosmological priors (using the TT best-fit cosmology). Calibrating using either the TE or the EE spectra yields generally consistent results, except at 143 Ghz where there is disagreement at more than 2 GLYPH<27> . At this level, this discrepancy can be caused either by a statistical fluctuation, or by an unknown residual. \nThe Plik baseline likelihood implements a map-based calibration. The TE calibration is deduced from the TT and EE calibrations, including at 143 GHz. Other improvements over the 2015 version are the following. The dust model has been improved in temperature and polarization, using also the latest version of the 353-GHz maps. The level of synchrotron contamination in the 100-GHz and 143-GHz maps has been estimated and shown to be negligible. Sub-pixel noise has been included in TT and EE (and demonstrated to have a negligible e GLYPH<11> ect on the cosmological parameters). Finally, a correlated component of the noise has been observed in the end-to-end HFI simulation, a GLYPH<11> ecting the large scales and very small scales of the EE auto-frequency spectra. The large-scale contribution a GLYPH<11> ects the dust correction and the n s constraints. We constructed an empirical model of this correlated noise from our simulations, which is included in the Plik likelihood. \nCamSpec was the baseline for the 2013 release and was described in detail in Planck Collaboration XV (2014), and used cross-spectra formed from detector-set temperature maps using data from the nominal mission period. It was extended for the 2015 release to include both polarization and temperaturepolarization cross-spectra and to use the data from the full mission period. Similarly to Plik , CamSpec switched from detectorset cross-spectra to cross-spectra formed from frequency maps constructed from separate halves of the full mission data in order to mitigate the e GLYPH<11> ects of noise correlated between detectors. In 2015, the foreground modelling was also modified and the sky fraction retained at each frequency was increased, using common masks with Plik in temperature. CamSpec used a more conservative mask in polarization than Plik . \nDi GLYPH<11> erently from Plik , CamSpec corrects each TE and EE cross-frequency spectrum with a fixed dust and temperature-topolarization leakage template before co-adding them to form the EE and TE components of its data vector and bases its noise estimate on di GLYPH<11> erences between maps constructed using alternating pointing periods. Note also that CamSpec uses an individualspectrum-based calibration scheme, where the TE calibrations are not fixed to be those inferred from the TT and EE ones. \nIn the 2018 release further improvements in the CamSpec foreground modelling have been implemented. The dust model in temperature has been updated in a way similar to Plik . CamSpec now uses a richer model of the cosmic infrared background, allowing for the exploration of any impact on the cosmological parameters. As explained above, the noise modelling was also modified. Further modifications of the masking have been made for polarization, still using the same masks for each frequency channel, but di GLYPH<11> erent from the Plik mask. As we \ndiscussed above, beam-leakage and polarization-e GLYPH<14> ciency corrections are applied to the individual polarization spectra before their addition for inclusion in the likelihood. More details on the Plik and CamSpec likelihoods can be found in PPL18. \nAs in 2015, the high-' Plik and CamSpec likelihoods are in excellent agreement for temperature. The di GLYPH<11> erent assumptions for the polarization-e GLYPH<14> ciency parameters and the masks (Planck Collaboration XI 2016) propagate into di GLYPH<11> erences in cosmological parameter estimates. For the baseline GLYPH<3> CDM model, the di GLYPH<11> erence in cosmological parameters between the Plik likelihood and the CamSpec likelihood (using joint TT,TE,EE in combination with Commander , SimAll , and lensing) is at most 0 : 5 GLYPH<27> (for GLYPH<10> bh 2 ) (Planck Collaboration XIII 2016, henceforth PCP15). Similar di GLYPH<11> erences in cosmological parameters occur in extended cosmological models. The di GLYPH<11> erences between the Plik and CamSpec parameters are dominated by calibration-model di GLYPH<11> erences for the joint TT,TE,EE and TEonly cases, and by mask di GLYPH<11> erences for the EE-only case. To a large extent, the CamSpec results can be reproduced within the Plik framework simply by changing in Plik the calibration model (for TE ) and the polarization mask (for EE ). Below we use Plik as the Planck baseline high-' likelihood. CamSpec results are used to assess the residual uncertainty from modelling and mask choices. We quote values obtained with CamSpec only for a few cases. \nWe use the following conventions for naming the Planck likelihoods: (i) Planck TT + lowE denotes the combination of the high-' TT likelihood at multipoles ' GLYPH<21> 30 and the low-' temperature-only Commander likelihood, plus the SimAll low-' EE-only likelihood in the range 2 GLYPH<20> ' GLYPH<20> 29; (ii) Planck TE and Planck EE denote the TE and EE likelihood at ' GLYPH<21> 30, respectively; (iii) Planck TT,TE,EE + lowE denotes the combination of the combined likelihood using TT , TE , and EE spectra at ' GLYPH<21> 30, the low-' temperature Commander likelihood, and the low-' SimAll EE likelihood; and (iv) Planck TT,TE,EE + lowP denotes the combination of the likelihood using TT , TE , and EE spectra at ' GLYPH<21> 30 and the alternative joint temperaturepolarization likelihood at 2 GLYPH<20> ' GLYPH<20> 29, based on the temperature Commander map and the 70-GHz polarization map. Unless otherwise stated, high-' results are based on the Plik likelihood and low-' polarization information is based on SimAll .", 'Planck CMB lensing likelihood': "The Planck 2018 lensing likelihood, presented in PPLe18, uses the lensing trispectrum to estimate the power spectrum of the lensing potential C GLYPH<30>GLYPH<30> L . This signal is extracted using a minimum-variance combination of a full set of temperature- and polarization-based quadratic lensing estimators (Okamoto & Hu 2003) applied to the SMICA CMB map over approximately 70 % of the sky using CMB multipoles 100 GLYPH<20> ' GLYPH<20> 2048, as described in PPLe18. We use the lensing bandpower likelihood, with bins spanning lensing multipoles 8 GLYPH<20> L GLYPH<20> 400 ; which has been validated with numerous consistency tests. Because its multipole range has been extended down to L = 8 (compared to L = 40 for the Planck 2015 analysis), the statistical power of the lensing likelihood used here is slightly greater.", '2.2.2. NonPlanck data': 'While the data derived exclusively from Planck observations are by themselves already extremely powerful at constraining cosmology, external data sets can provide helpful additional information. The question of consistency between Planck and external data sets is discussed in detail in Planck Collaboration VI (2018) (henceforth PCP18). Here we focus on two data sets that are particularly useful for breaking degeneracies and whose errors can be assessed reliably. We consider the measurement of the CMB B -mode polarization angular power spectrum by the BICEP2 / Keck Array collaboration and measurements of the baryon acoustic oscillation (BAO) scale. The supplementary B -mode data provide independent constraints on the tensor sector, which are better than those that can be derived from the Planck data alone (based on the shape of the scalar power spectrum). The BAO data, on the other hand, do not directly constrain the primordial perturbations. These data, however, provide invaluable low-redshift information that better constrain the late-time cosmology, especially in extensions of GLYPH<3> CDM, and thus allow degeneracies to be broken.', 'BICEP2/Keck Array 2015 B -mode polarization data': 'Although Planck measured the CMB polarization over the full sky, its polarization sensitivity in the cosmological frequency channels is not su GLYPH<14> cient to compete with current suborbital experiments surveying small, particularly low-foreground patches of the sky very deeply using many detectors. In PCI15, constraints on r using the joint BICEP2 / Keck Array and Planck (BKP) analysis (BICEP2 / Keck Array and Planck Collaborations 2015) were reported. Here we make use of the most recent B -mode polarization data available from the analysis of the BICEP2 / Keck field (Ade et al. 2018, henceforth BK15), unless otherwise stated. The BK15 likelihood draws on data from the new Keck array at 220 GHz in addition to those already in use for the BK14 (Ade et al. 2016) likelihood, i.e., the 95 and 150-GHz channel, as well as from Planck and WMAP to remove foreground contamination. The BK15 observations measure B -mode polarization using 12 auto- and 56 cross-spectra between the BICEP2 / Keck maps at 95, 150, and 220 GHz, the WMAP maps at 23 and 33 GHz, and the Planck maps at 30, 44, 70, 100, 143, 217, and 353 GHz, using nine bins in multipole number. By using B -mode information only within the BK15 likelihood, a 95 % upper limit of r < 0 : 07 is found (BK15), which improves on the corresponding 95 % CL r < 0 : 09 (BK14) based on the BK14 likelihood.', 'Baryon acoustic oscillations': 'Acoustic oscillations of the baryon-photon fluid prior to recombination are responsible for the acoustic peak structure of the CMBangular power spectra. The counterpart to the CMB acoustic peaks in the baryon distribution are the BAOs, which remain imprinted into the matter distribution to this day. In the position-space picture, the BAOs of the power spectrum correspond to a peak in the correlation function, defining a characteristic, cosmology-dependent length scale that serves as a standard ruler and can be extracted (e.g., from galaxy redshift surveys). The transverse information of a survey constrains the ratio of the comoving angular diameter distance and the sound horizon at the drag epoch (i.e., when the baryon evolution becomes una GLYPH<11> ected by coupling to the photons), DM = r d, whereas the line-of-sight \ninformation yields a measurement of H ( z ) r d. Sometimes, these two observables are combined to form the direction-averaged quantity DV = r d GLYPH<17> h czD 2 M ( z ) H GLYPH<0> 1 ( z ) i 1 = 3 = r d. \nFor our BAO data compilation, we use the measurements of DV = r d from the 6dF survey at an e GLYPH<11> ective redshift z e GLYPH<11> = 0 : 106 (Beutler et al. 2011), and the SDSS Main Galaxy Sample at z e GLYPH<11> = 0 : 15 (Ross et al. 2015), plus the final interpretation of the SDSS III DR12 data (Alam et al. 2016), with separate constraints on H ( z ) r d and DM = r d in three correlated redshift bins at z e GLYPH<11> = 0 : 38, 0 : 51, and 0 : 61. In Addison et al. (2018), the same set of BAO data combined with either nonPlanck CMB data or measurements of the primordial deuterium fraction was shown to favour a cosmology fully consistent with, but independent of, Planck data.', '3. Planck 2018 results for the main inflationary observables': 'As in PCI13 and PCI15, we start by describing Planck measurements of the key inflationary parameters. Some of the results reported in this section can be found in the Planck Legacy Archive. 5', '3.1. Results for the scalar spectral index': "Planck temperature data in combination with the EE measurement at low multipoles determine the scalar spectral tilt in the GLYPH<3> CDMmodel as \nn s = 0 : 9626 GLYPH<6> 0 : 0057 (68 % CL, Planck TT + lowE) : (6) \nThis result for n s is compatible with the Planck 2015 68 % CL value n s = 0 : 9655 GLYPH<6> 0 : 0062 for Planck TT + lowP (PCP15). The slightly lower value for n s is mainly driven by a corresponding shift in the average optical depth GLYPH<28> , now determined as \nGLYPH<28> = 0 : 052 GLYPH<6> 0 : 008 (68 % CL, Planck TT + lowE) ; (7) \nwhich is to be compared with the Planck 2015 value GLYPH<28> = 0 : 078 GLYPH<6> 0 : 022 (PCP15). This more precise determination of GLYPH<28> is due to better noise sensitivity of the HFI 100- and 143-GHz channels employed in the low-' SimAll polarization likelihood, compared to the joint temperature-polarization likelihood based on the LFI 70-GHz channel in 2015. Because of the degeneracy between the average optical depth and the amplitude of the primordial power spectrum, A s and GLYPH<27> 8 are also lower than in the Planck 2015 release. These shifts from the Planck 2015 values for the cosmological parameters have been anticipated with the first results from the HFI largeangular polarization pattern (Planck Collaboration Int. XLVI 2016; Planck Collaboration Int. XLVII 2016). \nThe trend toward smaller values for ( n s, GLYPH<28> ) with respect to the Planck 2015 release also occurs for di GLYPH<11> erent choices for the low-' likelihood. By substituting Commander and SimAll with the updated joint temperature-polarization pixel likelihood coming from the LFI 70-GHz channel, we obtain in combination with high-' temperature data: \nn s = 0 : 9650 GLYPH<6> 0 : 0061 (68 % CL, Planck TT + lowP) ; (8) GLYPH<28> = 0 : 072 GLYPH<6> 0 : 016 (68 % CL, Planck TT + lowP) : (9) \nAlthough with larger errors, these latter results are consistent with the shifts induced by a determination of a lower optical \ndepth than in the Planck 2015 release, 6 as found in Eqs. (6) and (7). Given this broad agreement and the consistency in the values of GLYPH<28> derived with SimAll separately from the three cross-spectra 70 GLYPH<2> 100, 70 GLYPH<2> 143, and 100 GLYPH<2> 143 (PPL18), we will mainly use the baseline low-' likelihood in the rest of the paper. \nAs anticipated in 2015, the information in the high-' polarization Planck data is powerful for breaking degeneracies in the parameters and to further decrease parameter uncertainties compared to temperature data alone. The addition of high-' polarization leads to a tighter constraint on n s: \nn s = 0 : 9649 GLYPH<6> 0 : 0044 (68 % CL, Planck TT,TE,EE + lowE) : (10) \nThis is in good agreement with the Planck 2015 TT,TE,EE + lowP 68% CL result, n s = 0 : 9645 GLYPH<6> 0 : 0049. In this 2018 release the mean value of n s is approximately 0 : 5 GLYPH<27> larger than the temperature result in Eq. (6). This pull is mainly due to a higher value for the scalar tilt preferred by Planck 2018 polarization and temperature-polarization cross-correlation data only: \nn s = 0 : 969 GLYPH<6> 0 : 009 (68 % CL, Planck TE,EE + lowE) : (11) \nThis pull is then mitigated in combination with temperature due to the larger uncertainty in the determination by TE,EE only. Similar considerations hold for the alternative CamSpec high-' likelihood, which leads to a 68 % CL result n s = 0 : 9658 GLYPH<6> 0 : 0045, consistent with the baseline Plik reported in Eq. (10). Overall, the cosmological parameters from Planck baseline temperature, polarization, and temperature-polarization cross-correlation separately and combined are very consistent, as can be seen from Table 2 and Fig. 2 for the GLYPH<3> CDMmodel. \nAfter combining with Planck lensing, we obtain \nn s = 0 : 9634 GLYPH<6> 0 : 0048 (68 % CL, Planck TT + lowE + lensing) ; (12) n s = 0 : 9649 GLYPH<6> 0 : 0042 \n(68 % CL, Planck TT,TE,EE + lowE + lensing) : (13) \nThe shift in n s (and, more generally, in the cosmological parameters of the baseGLYPH<3> CDMmodel) obtained when Planck lensing is combined with TT,TE,EE + lowE is smaller than in 2015 because of the improved polarization likelihoods. The combination with lensing is, however, powerful for breaking parameter degeneracies in extended cosmological models, and, therefore, for this 2018 release we will consider the full information contained in temperature, polarization, and lensing, i.e., TT,TE,EE + lowE + lensing, as the baseline Planck data set. Figure 3 shows a comparison of the Planck 2018 baseline results with those from alternative likelihoods and from the 2015 baseline for the GLYPH<3> CDMcosmological parameters. \nAs in 2013 and 2015, BAO measurements from galaxy surveys are consistent with Planck . When BAO data are combined, we obtain for the baseGLYPH<3> CDMcosmology: \nn s = 0 : 9665 GLYPH<6> 0 : 0038 (14) (68 % CL, Planck TT,TE,EE + lowE + lensing + BAO) : \nThe combination with BAO data decreases (increases) the marginalized value of GLYPH<10> c h 2 ( GLYPH<10> b h 2 ) obtained by Planck , and this e GLYPH<11> ect is compensated for by a shift in n s towards slightly larger values. \nTable 2. Confidence limits for the cosmological parameters in the baseGLYPH<3> CDM model from Planck temperature, polarization, and temperature-polarization cross-correlation separately and combined, in combination with the EE measurement at low multipoles.", '3.2. Ruling out n s = 1': 'One of the main findings drawn from previous Planck releases was that the scale-independent Harrison-Zeldovich (HZ) spectrum (Harrison 1970; Zeldovich 1972; Peebles & Yu 1970) is decisively ruled out. This conclusion is reinforced in this release: in standard GLYPH<3> CDM late-time cosmology, the scalar spectral index from Table 2 lies 6 : 6, 8 : 0, and 8 : 4 GLYPH<27> away from n s = 1, for Planck TT + lowE, Planck TT,TE,EE + lowE, and Planck TT,TE,EE + lowE + lensing, respectively. The corresponding effective GLYPH<1> GLYPH<31> 2 between the power-law spectrum and the best-fit HZ model are GLYPH<1> GLYPH<31> 2 = 43 : 9, 66 : 9, and 72 : 4. \nSimple one-parameter modifications of the cosmological model are not su GLYPH<14> cient to reconcile a scale-invariant power spectrum with Planck data. For instance, when the e GLYPH<11> ective number of neutrino species N e GLYPH<11> is allowed to float for a cosmology with a scale-invariant spectrum, the e GLYPH<11> ective GLYPH<1> GLYPH<31> 2 with respect to the power-law spectrum are GLYPH<1> GLYPH<31> 2 = 12 : 9, 27 : 5, and 30 : 2, respectively. \nWhen instead the assumption of flat spatial sections is relaxed, 7 we obtain e GLYPH<11> ective GLYPH<1> GLYPH<31> 2 values of GLYPH<1> GLYPH<31> 2 = 11 : 8, 28 : 8, and 40 : 9, respectively, for the same data sets. Therefore, the corresponding closed cosmological models fitting Planck TT + lowE ( GLYPH<10> K = GLYPH<0> 0 : 122 + 0 : 039 GLYPH<0> 0 : 029 , H 0 = 44 : 2 + 3 : 1 GLYPH<0> 4 : 3 kms GLYPH<0> 1 Mpc GLYPH<0> 1 at 68 % CL), Planck TT,TE,EE + lowE ( GLYPH<10> K = GLYPH<0> 0 : 095 + 0 : 029 GLYPH<0> 0 : 019 , H 0 = 47 : 1 GLYPH<6> 3 : 2 km s GLYPH<0> 1 Mpc GLYPH<0> 1 at 68 % CL), and Planck TT,TE,EE + lowE + lensing ( GLYPH<10> K = GLYPH<0> 0 : 032 + 0 : 006 GLYPH<0> 0 : 007 , H 0 = 58 : 9 GLYPH<6> 2 : 0 km s GLYPH<0> 1 Mpc GLYPH<0> 1 at 68 % CL) provide a worse fit compared to the tilted flat GLYPH<3> CDMmodel. 8', '3.3. Constraints on the scale dependence of the scalar spectral index': 'The Planck 2018 data are consistent with a vanishing running of the scalar spectral index. Using Planck \n8 This is not a new result based on the Planck 2018 release, but just an update of a similar conclusion also reached with the Planck 2015 data. Compared to the flat GLYPH<3> CDM tilted model, we obtain GLYPH<1> GLYPH<31> 2 = 12 : 3, 34 : 8, and 45 with Planck 2015 TT + lowP, Planck 2015 TT,TE,EE + lowP, and Planck 2015 TT,TE,EE + lowP + lensing, respectively. Therefore, even with Planck 2015 data, a closed model with n s = 1 provides a worse fit than tilted GLYPH<3> CDM and is not compelling as claimed in Ooba et al. (2017).', 'TT,TE,EE + lowE + lensing we obtain': "dn s d ln k = GLYPH<0> 0 : 0045 GLYPH<6> 0 : 0067 (68 % CL) : (15) \nThese results are consistent with, and improve on, the Planck 2015 result, dn s = d ln k = GLYPH<0> 0 : 008 GLYPH<6> 0 : 008 (PCP15). \nAs discussed in PCI13 and PCI15, a better fit to the temperature low-' deficit was found in 2015, thanks to a combination of non-negative values for the running and the running of the running. The Planck 2018 release has significantly reduced the parameter volume of this extension of the baseGLYPH<3> CDM model. The Planck 2018 TT(TT,TE,EE) + lowE + lensing constraints for the model including running of running are \nn s = 0 : 9587 GLYPH<6> 0 : 0056 (0 : 9625 GLYPH<6> 0 : 0048) ; (16) \ndn s = d ln k = 0 : 013 GLYPH<6> 0 : 012 (0 : 002 GLYPH<6> 0 : 010) ; (17) \nd 2 n s = d ln k 2 = 0 : 022 GLYPH<6> 0 : 012 (0 : 010 GLYPH<6> 0 : 013) ; (18) \nall at 68 % CL. It is interesting to note that the high-' temperature data still allow a sizable value for the running of the running, although slightly decreased with respect to the Planck 2015 results (PCI15). However, when high-' Planck 2018 polarization data are also included, dn s = d ln k and d 2 n s = d ln k 2 are tightly constrained. \nThe model including a scale-dependent running can produce a better fit to the low-' deficit at the cost of an increase of power at small scales; this latter e GLYPH<11> ect is constrained in this release. As an example of a model with suppression only on large scales, we also reconsider the phenomenological model with an exponential cuto GLYPH<11> : \nP R ( k ) = P 0( k ) 8 > > < > > : 1 GLYPH<0> exp 2 6 6 6 6 4 GLYPH<0> k k c ! GLYPH<21> c 3 7 7 7 7 5 9 > > = > > ; ; (19) \nwhich can be motivated by a short stage of inflation (Contaldi et al. 2003; Cline et al. 2003) (see also Kuhnel & Schwarz (2010), Hazra et al. (2014a), and Gruppuso et al. (2016) for other types of large-scale suppressions). We do not find any statistically significant detection of k c using either logarithmic or linear priors and for di GLYPH<11> erent values of GLYPH<21> c, with any combination of Planck baseline likelihoods. We have also checked that these results depend only weakly on the exclusion of the EE quadrupole in SimAll and are stable to the substitution of Commander and SimAll with the joint temperature-polarization likelihood based on the 70-GHz channel. \nFig. 2. Marginalized joint 68 % and 95 % CL regions for the cosmological parameters in GLYPH<3> CDMwith Planck TT, EE, TE, and joint TT,TE,EE, all in combination with the EE likelihood at low multipoles. \n<!-- image -->", '3.4. Constraints on spatial curvature': "Since the vast majority of inflation models predict that the Universe has been driven towards spatial flatness, constraints on the spatial curvature provide an important test of the standard scenario. Therefore in this subsection we extend the baseGLYPH<3> CDM model with the addition of the spatial curvature parameter, GLYPH<10> K . For the case of Planck TT,TE,EE + lowE + lensing, we find a constraint of \nGLYPH<10> K = GLYPH<0> 0 : 011 + 0 : 013 GLYPH<0> 0 : 012 (95 % CL) : (20) \nThe inclusion of Planck lensing information only weakly breaks the geometrical degeneracy (Efstathiou & Bond 1999) which results in the same primary fluctuations while varying the total \nmatter density parameter, GLYPH<3> , and H 0. The degeneracy can be effectively broken with the addition of BAO data, in which case Planck TT,TE,EE + lowE + lensing + BAO gives \nGLYPH<10> K = 0 : 0007 GLYPH<6> 0 : 0037 (95 % CL) : (21) \nAlthough GLYPH<10> K is one of the cosmological parameters exhibiting some di GLYPH<11> erences between Plik and Camspec , the constraints in Eqs. (20) and (21) are quite robust due to the inclusion of lensing (and BAO) information. \nA constraint on the curvature parameter can be translated into a constraint on the radius of curvature, RK , via \nRK = GLYPH<16> a 0 H 0 p j GLYPH<10> K j GLYPH<17> GLYPH<0> 1 ; (22) \nFig. 3. Comparison of the marginalized probability density of the primary parameters and ( GLYPH<27> 8 ; H 0) for the baseline cosmological model from Planck TT + lowE + lensing (black curves), TT,TE,EE + lowE + lensing (red curves), and the alternative likelihood Camspec . For comparison we also display the Planck 2018 TT + lowP + lensing (blue curves) and the corresponding Planck 2015 TT + lowP + lensing (green curves) results. \n<!-- image --> \nTable 3. Constraints on the tensor-to-scalar ratio r and scalar tilt n s for the GLYPH<3> CDM + r model and some important extensions and di GLYPH<11> erent data sets. For each model we quote 68 % confidence limits on measured parameters and 95 % upper bounds on other parameters. \nin units such that c = 1. For the case of Planck TT,TE,EE + lowE + lensing + BAO we find \nRK > 67 Gpc (open) ; (23) \nRK > 81 Gpc (closed) ; (24) \nboth at 95 % confidence. These lengths are considerably greater than our current (post-inflation) particle horizon, at 13 : 9 Gpc. \nOur tightest constraint, Eq. (21), tells us that our observations are consistent with spatial flatness, with a precision of \nabout 0.4 %. However, even if inflation has driven the background curvature extremely close to zero, the presence of fluctuations implies a fundamental 'cosmic variance' for measurements of curvature confined to our observable volume. In particular, the known amplitude of fluctuations implies a standard deviation for GLYPH<10> K of roughly 2 GLYPH<2> 10 GLYPH<0> 5 (Waterhouse & Zibin 2008). Therefore our best constraint is still a factor of roughly 10 2 above the cosmic variance limit for a flat universe. A future measurement of negative curvature above the cosmic variance floor would point to open inflation (Gott 1982; Gott & Statler 1984; Bucher et al. 1995; Yamamoto et al. 1995; Ratra & Peebles 1995; Lyth & Stewart 1990), while a measurement of positive curvature could pose a problem for the inflationary paradigm due to the di GLYPH<14> culty of producing closed inflationary models (Kleban & Schillo 2012). \nAlternatively, excess spatial curvature might be evidence for the intriguing possibility that there was 'just enough' inflation to produce structure on the largest observable scales. Indeed an upper limit on spatial curvature implies a lower limit on the total number of e -folds of inflation (see, e.g., Komatsu et al. 2009). We can relate these limits to the number of e -folds of inflation, N GLYPH<3> = N ( k GLYPH<3> ), after scale k GLYPH<3> left the Hubble radius during inflation, to be given explicitly in Eq. (47). We define the (constant) curvature scale, kK , as the inverse of the comoving radius of curvature, i.e., \nkK GLYPH<17> aH p j GLYPH<10> K j : (25) \nIn the absence of special initial conditions, inflation will begin with a curvature parameter of order unity. Equation (25) then implies that kK GLYPH<24> aH at the start of inflation, i.e., the curvature scale is 'exiting the horizon' at that time. Then the lower limit on the number of e -folds of inflation will simply be NK GLYPH<17> N ( kK ), i.e., the number of e -folds after scale kK left the Hubble radius during inflation. With Eq. (47) this gives 9 \nNK = ln k GLYPH<3> a 0 H 0 GLYPH<0> 1 2 ln j GLYPH<10> K j + N GLYPH<3> : (26) \nWith the pivot scale of k GLYPH<3> = 0 : 002 Mpc GLYPH<0> 1 (for comparison with the values in Sect. 4.2) and our tightest upper limit on GLYPH<10> K from Eq. (21), this becomes \nNK & 4 : 9 + N GLYPH<3> : (27) \nThat is, our constraint on spatial curvature implies that inflation must have lasted at least about 5 e -folds longer than required to produce the pivot scale k GLYPH<3> . Equation (27) provides a modelindependent comparison between the e -folds required to solve the flatness problem (to current precision) and to produce largescale fluctuations (at scale k GLYPH<3> ). We stress that this limit assumes a unity curvature parameter at the start of inflation (although the dependence on this assumption, being logarithmic, is weak). \nFor comparison with the result of Komatsu et al. (2009), we can simplify to the case of instantaneous thermalization and constant energy density during inflation. Then we find \nNK & 34 : 2 + ln T end 1 TeV ; (28) \nwhere T end is the reheating temperature.", '3.5. Constraints on the tensor-to-scalar ratio': "This subsection updates constraints on the tensor-to-scalar ratio r assuming that the tensor tilt satisfies the consistency relation, n t = GLYPH<0> r = 8, which is the case for slow-roll inflation driven by a single scalar field with a canonical kinetic term. \nBy combining Planck temperature, low-' polarization, and lensing we obtain \nr 0 : 002 < 0 : 10 (95 % CL, Planck TT + lowE + lensing) : (29) \nThis constraint slightly improves on the corresponding Planck 2015 95 % CL bound, i.e., r 0 : 002 < 0 : 11 (PCI15), and is unchanged when high-' polarization data are also combined. Note that by using CAMspec instead of Plik as the high-' joint temperature-polarization likelihood, we obtain a slightly looser bound, i.e., r 0 : 002 < 0 : 14 at 95 % CL. By including the Planck B -mode information at 2 < ' < 30 in the low-' polarization likelihood, the 95 % CL constraint is essentially unchanged. \nSince inflationary gravitational waves contribute to CMB temperature anisotropies mostly at ' . 100, the low-' temperature deficit contributes in a nontrivial way to the Planck bound on r . By excising the 2 GLYPH<20> ' GLYPH<20> 29 temperature data, the constraint on r with Planck TT,TE,EE + lensing + lowEB relaxes to \nr 0 : 002 < 0 : 16 (95 % CL) : (30) \nThis result improves on the 2015 95 % CL result, i.e., r . 0 : 24 (PCI15), because of the inclusion of high-' polarization and of the improved determination of GLYPH<28> . \nSince this Planck constraint on r relies on temperature and E -mode polarization, the Planck -only limit depends somewhat on the underlying cosmological model. Table 3 shows the constraints on n s and r for a few important extensions of GLYPH<3> CDMplus tensors, which include a non-zero running, a non-zero spatial curvature, and a non-minimal neutrino sector. We observe that the bound on r is relaxed by at most 30 % when the scale dependence of the scalar tilt is allowed to vary. In all the other extensions the Planck r bound is modified at most by 10 %, demonstrating the constraining power of the Planck 2018 release in reducing the degeneracy of the tensor-to-scalar ratio with other cosmological parameters. As far as the scalar tilt is concerned, we find the largest shift (by roughly 1 GLYPH<27> higher) when the assumption of spatial flatness is relaxed. \nA B -mode polarization measurement can further tighten the constraint on r and help in reducing its degeneracies with other cosmological parameters that may appear when using only temperature and E -mode polarization data. After the release of the first BICEP-Keck ArrayPlanck (BKP) joint cross-correlation, constraints on r from B -mode polarization data alone have become tighter than those based on Planck data alone, thanks to the inclusion of the 95-GHz channel (BK14) and of the 220GHz channel (BK15). By combining the Planck 2018 and BK15 data we obtain \nr 0 : 002 < 0 : 056 (95 % CL, Planck TT,TE,EE + lowE + lensing + BK15) : (31) \nThis bound improves on the corresponding one obtained in combination with BK14, i.e., r 0 : 002 < 0 : 064 at 95 % CL. Note that by using CAMspec instead of Plik as high-' TT,TE,EE likelihood, we obtain a slightly looser bound, i.e., r 0 : 002 < 0 : 069 at 95 % CL. The e GLYPH<11> ectiveness of the combination with the BK15 likelihood in constraining r is also remarkable in the extensions of GLYPH<3> CDM plus tensors, as can be seen from Table 3. By further combining with BAO the limits for r are only slightly modified. \nThe Planck 2018 baseline plus BK15 constraint on r is equivalent to an upper bound on the energy scale of inflation when the pivot scale exits the Hubble radius of \nV GLYPH<3> = 3 GLYPH<25> 2 A s 2 r M 4 Pl < (1 : 6 GLYPH<2> 10 16 GeV) 4 (95 % CL) : (32) \nEquivalently, this last result implies an upper bound on the Hubble parameter during inflation of \nH GLYPH<3> M Pl < 2 : 5 GLYPH<2> 10 GLYPH<0> 5 (95 % CL) : (33)", '3.6. Beyond the tensor-to-scalar ratio consistency condition': "The increasing constraining power of B -mode polarization data allows us to set upper bounds on r without imposing the consistency condition for the tensor tilt, n t = GLYPH<0> r = 8, which is motivated by standard slow-roll single-scalar-field inflation. Deviations can occur in multifield inflation (Bartolo et al. 2001; Wands et al. 2002; Byrnes & Wands 2006), in the models with generalized Lagrangians (Garriga & Mukhanov 1999; Kobayashi et al. 2010), in gauge inflation (Maleknejad et al. 2013), or in a more radical way in alternative models to inflation (Gasperini & Veneziano 1993; Boyle et al. 2004; Brandenberger et al. 2007). \nAs the current data do not lead to a detection of a non-zero tensor amplitude, virtually any value of n t would give a good fit as long as r is close enough to zero. Therefore, as in PCI15, we characterize the tensor perturbations by two well-constrained parameters that we choose to be r at two di GLYPH<11> erent scales, ( rk 1 ; rk 2 ), with k 1 = 0 : 002 Mpc GLYPH<0> 1 and k 2 = 0 : 02 Mpc GLYPH<0> 1 , and assume a power-law power spectrum. We call this two-parameter extension of the GLYPH<3> CDM model the ' GLYPH<3> CDM + r 0 : 002 + r 0 : 02' model. We also quote our results in terms of ( r ˜ k ; n t), calculated from the primary parameters as n t = [ln( rk 2 = rk 1 ) = ln( k 2 = k 1)] + n s GLYPH<0> 1 and r ˜ k = rk 1 ( ˜ k = k 1) n t GLYPH<0> n s + 1 . For ˜ k we choose 0 : 01 Mpc GLYPH<0> 1 , which corresponds roughly to the decorrelation scale of r and n t when using the Planck and BK15 data. \nThe one-dimensional posteriors are displayed in Fig. 4 (which also shows an additional data set, 'LIGO&Virgo2016,' discussed at the end of this subsection). We obtain for the GLYPH<3> CDM + r 0 : 002 + r 0 : 02 model: \nr 0 : 002 < 0 : 044 r 0 : 02 < 0 : 184 ) (95 % CL, Planck TT,TE,EE + lowE + lensing + BK15). (34) \nThe constraints on the derived tensor parameters are r 0 : 01 < 0 : 076 and GLYPH<0> 0 : 55 < n t < 2 : 54 at 95 % CL. \nThe left and right panels of Fig. 5 show the two-dimensional contours for the primary parameters ( r 0 : 002 ; r 0 : 02) and the derived ones ( r 0 : 01 ; n t), respectively. The consistency condition, n t = GLYPH<0> r = 8, denoted by the dashed lines, is fully compatible with Planck + BK15 data. However, a very blue tensor tilt with n t ' 2 and r 0 : 01 ' 0 : 05 is still within the 68 % CL region. Indeed, despite the larger amplitude of the primordial tensor power spectrum at small scales for blue n t, the tensor modes are suppressed when re-entering the Hubble radius, which leads to damping of the observational signal at high k . This explains why the 68 % CL CMB constraint on r 0 : 02 is about a factor of four weaker than the one on r 0 : 002. Fig. 5 also shows a slight improvement of constraints by BK15 compared to the older BK14 data. \nA stochastic background of gravitational waves (GWs) with a blue tensor tilt could be further constrained at much shorter \nFig. 4. Posterior probability density of the tensor-to-scalar ratio at two di GLYPH<11> erent scales in the GLYPH<3> CDM + r 0 : 002 + r 0 : 02 model, i.e., when the inflationary consistency relation is relaxed (top panels). The solid contours show the results when r 0 : 002 and r 0 : 02 are used as sampling parameters with uniform priors, which leads to non-uniform priors for the derived parameters r 0 : 01 and n t (bottom panels). The dotted contours indicate the results after weighting the posterior by the Jacobian J = r 0 : 01 = [ r 0 : 002 r 0 : 02 ln(0 : 02 = 0 : 002)] of the transformation ( r 0 : 002 ; r 0 : 02) ! ( r 0 : 01 ; n t), giving the result we would have obtained had we assigned uniform priors on r 0 : 01 and n t. \n<!-- image --> \nwavelengths, such as those probed by ground-based interferometers dedicated to the direct detection of GWs. For example, assuming a scale-invariant tensor spectrum and using the frequency range (20-85.8) Hz, which corresponds to the wavenumbers k = 2 GLYPH<25> f = (1 : 3-5 : 5) GLYPH<2> 10 16 Mpc GLYPH<0> 1 , LIGO and Virgo set an upper bound on the GW density parameter of GLYPH<10> GW( f ) GLYPH<20> 1 : 7 GLYPH<2> 10 GLYPH<0> 7 at 95 % CL (Abbott et al. 2017). While these scales are likely to be dominated by astrophysical sources, such as GWs from binary mergers, we next examine what constraints the LIGO&Virgo upper bound sets on primordial tensor perturbations, if we assume that they had a power-law spectrum all the way from CMB scales to ultra-short scales. We refer the interested reader to Meerburg et al. (2015) and Cabass et al. (2016) for the use of alternative data on short scales or of additional constraints on the e GLYPH<11> ective energy-momentum tensor of the stochastic background of GWs averaged over wavelengths. \nWe obtain a conservative upper limit on the primordial contribution by demanding that the GW density from our scaledependent primordial tensor perturbations (Meerburg et al. 2015; Abbott et al. 2017; Cabass et al. 2016), \nGLYPH<10> GW( k ) = k GLYPH<26> critical d GLYPH<26> GW dk = A t( k ) 24 z eq = A t1( k = k 1) n t 24 z eq ; (35) \nstays below the above-quoted limit at least at k = 1 : 3 GLYPH<2> 10 16 Mpc GLYPH<0> 1 ( f = 20 Hz). The posterior probability densities when this constraint is included in the analysis as a half-Gaussian prior are compared with those obtained by Planck + BK15 alone in Figs. 4 and 5. LIGO&Virgo sets a very high upper bound 10 on r at ultrahigh k , separated from CMB scales by a factor of 10 18 in k . Due \nr \nFig. 5. 68 % and 95 % CL constraints on tensor perturbations in the GLYPH<3> CDM + r 0 : 002 + r 0 : 02 model, i.e., when the inflationary consistency relation is relaxed. Filled contours in the left panel show the results for our independent primary parameters r 0 : 002 and r 0 : 02, which have uniform priors, and in the right panel for the derived parameters n t and r 0 : 01, which have non-uniform priors. The dotted lines assume uniform priors on r 0 : 01 and n t, calculated as in Fig. 4. The scale k = 0 : 01 Mpc GLYPH<0> 1 is near the decorrelation scale of ( n t, r ) for the Planck + BK15 data. In both panels the dashed black line indicates the inflationary consistency condition, n t = GLYPH<0> r 0 : 01 = 8. (The grey contours follow if we use the older BK14 data instead of the BK15 data.) \n<!-- image --> \n<!-- image --> \nr \nto the long arm length, this e GLYPH<11> ectively provides a cuto GLYPH<11> for n t and excludes the bluest spectra that were allowed by the CMB alone, leading to \nr 0 : 002 < 0 : 064 r 0 : 02 < 0 : 081 ) (95 % CL, Planck TT,TE,EE + lowE + lensing + BK15 + LIGO&Virgo2016), (36) \nor r 0 : 01 < 0 : 066 and GLYPH<0> 0 : 76 < n t < 0 : 52. The consistency condition n t = GLYPH<0> r = 8 is also compatible with these tighter constraints, as can be seen by comparing the red contours and dashed black lines in Fig. 5. As LIGO&Virgo pushes r 0 : 02 down (and we assume a power-law tensor spectrum), the upper bound on r 0 : 002 becomes weaker than with the CMB alone. This is not surprising, since the system is analogous to a see-saw with a pivot point at k GLYPH<24> 0 : 01 Mpc GLYPH<0> 1 , where the data are the most sensitive to the tensor perturbations (taking into account also the transfer function from primordial tensor perturbations to the observable B -mode signal). Once one end of the see-saw is pushed down the other end can go up without disturbing the spectrum too much at the middle scales. We will observe analogous behaviour with isocurvature perturbations, for which we also assume a powerlaw spectrum and have only an upper bound (not a detection); see Sect. 9.3. \nGLYPH<20> 24 z eq GLYPH<2> 1 : 7 GLYPH<2> 10 GLYPH<0> 7 = 1 : 4 GLYPH<2> 10 GLYPH<0> 2 , where we used z eq ' 3400. Assuming further for the scalar perturbations that n s = 0 : 9659 and ln(10 10 A s) = 3 : 044 at k = 0 : 05 Mpc GLYPH<0> 1 , this can be converted into an upper bound r GLYPH<20> 2 : 6 GLYPH<2> 10 7 at k = 1 : 3 GLYPH<2> 10 16 Mpc GLYPH<0> 1 .", '4. Implications for single-field slow-roll inflationary models': 'In this section we discuss the implications of the Planck 2018 likelihood for standard single-field slow-roll inflation. We first update the results for the Hubble flow functions (HFFs) GLYPH<15> i and the potential slow-roll parameters obtained by the analytic perturbative expansion in terms of the HFFs for the primordial spectra of fluctuations. For definitions of the HFF hierarchy and the potential slow-roll parameters see Table 1. We then present a Bayesian comparison for a representative selection of standard slow-roll inflationary models.', '4.1. Constraints on slow-roll parameters': "Exploiting the approximate analytic expressions for the primordial power spectrum of scalar and tensor fluctuations obtained by the Green's function method (Stewart & Lyth 1993; Gong & Stewart 2001; Leach et al. 2002), we can construct constraints on the slow-roll parameters. \nWhen restricting to parameters first order in the HFFs, we obtain with Planck TT,TE,EE + lowE + lensing( + BK15) \nGLYPH<15> 1 < 0 : 0063 (0 : 0039) (95 % CL) ; (37) GLYPH<15> 2 = 0 : 030 + 0 : 007 GLYPH<0> 0 : 005 (0 : 031 GLYPH<6> 0 : 005) (68 % CL) : (38) \nThe Planck TT,TE,EE + lowE + lensing( + BK15) constraints on the slow-roll potential parameters GLYPH<15> V and GLYPH<17> V can be obtained by an exact remapping of the constraints on the HFF parameters (Leach et al. 2002; Finelli et al. 2010) given above: \nGLYPH<15> V < 0 : 0063 (0 : 0039) (95 % CL) ; (39) GLYPH<17> V = GLYPH<0> 0 : 010 + 0 : 004 GLYPH<0> 0 : 008 GLYPH<16> GLYPH<0> 0 : 012 + 0 : 004 GLYPH<0> 0 : 005 GLYPH<17> (68 % CL) : (40) \nFig. 6. Marginalized joint two-dimensional 68 % and 95 % CL regions for ( GLYPH<15> 1 ; GLYPH<15> 2) (top panel) and ( GLYPH<15> V ; GLYPH<17> V ) (bottom panel) for Planck TT,TE,EE + lowE + lensing (red contours), compared with Planck TT,TE,EE + lowE + lensing + BK15 (blue contours). The dashed lines divide between convex and concave potentials. \n<!-- image --> \nAs can be seen from Fig. 6, the 95 % CL allowed contours are in the region of concave potentials when BK15 is combined with Planck 2018 data. \nWhen contributions to the primordial power spectra that are second-order in the HFFs are included, for Planck TT,TE,EE + lowE + lensing( + BK15) we obtain the following constraints on the slow-roll HFFs: \nand on the slow-roll potential parameters we obtain: \nThe marginalized 68 % and 95 % CLs for the slow-roll HFF and potential parameters, allowing GLYPH<15> 3 , 0, with Planck data alone or in combination with BK15, are displayed in Fig. 7.", '4.2. Implications for selected slow-roll inflationary models': "The predictions for ( n s ; r ) to first order in the slow-roll approximation for a few inflationary models are shown in Fig. 8, which updates figure 12 of PCI15 and figure 1 of PCI13 with the same notation. These predictions are calculated for scale \nTable 4. Priors for cosmological parameters used in the Bayesian comparison of inflationary models. \nk = 0 : 002 Mpc GLYPH<0> 1 and include an uncertainty in the number of e -folds of 50 < N GLYPH<3> < 60. \nIn the following we discuss the implications of the Planck 2018 data release by taking into account the uncertainties in the entropy generation stage for a selection of representative standard single-field slow-roll inflationary models, updating the analysis reported in PCI13 and PCI15. As in PCI15, we use the primordial power spectra of cosmological fluctuations generated during slow-roll inflation parameterized by the HFFs, GLYPH<15> i , to second order, which can be expressed in terms of the parameters of the inflationary model and the number of e -folds to the end of inflation, N GLYPH<3> (Liddle & Leach 2003; Martin & Ringeval 2010), given by (PCI13) \nN GLYPH<3> ' 67 GLYPH<0> ln k GLYPH<3> a 0 H 0 ! + 1 4 ln 0 B B B B B @ V 2 GLYPH<3> M 4 pl GLYPH<26> end 1 C C C C C A + 1 GLYPH<0> 3 w int 12(1 + w int) ln GLYPH<26> th GLYPH<26> end ! GLYPH<0> 1 12 ln( g th) ; (47) \nwhere GLYPH<26> end is the energy density at the end of inflation, a 0 H 0 is the present Hubble scale, V GLYPH<3> is the potential energy when k GLYPH<3> left the Hubble radius during inflation, w int characterizes the effective equation of state between the end of inflation and the thermalization energy scale GLYPH<26> th, and g th is the number of e GLYPH<11> ective bosonic degrees of freedom at the energy scale GLYPH<26> th. We fix g th = 10 3 and GLYPH<15> end = 1, and we use modified routines of the public code ASPIC 11 (Martin et al. 2014b). In order to make contact with Fig. 8, we consider the pivot scale k GLYPH<3> = 0 : 002 Mpc GLYPH<0> 1 in this subsection. We assume the uniform priors for the cosmological parameters listed in Table 4, and logarithmic priors on 10 10 A s (over the interval [ e 2 : 5 ; e 3 : 7 ]) and GLYPH<26> th (over the interval [(1 TeV) 4 ; GLYPH<26> end]). Prior ranges for additional parameters in the inflationary models considered are listed in Table 5. In this paper we consider the implications of the Planck 2018 data for the selection of representative models studied in PCI15 by restricting ourselves to w int = ( p GLYPH<0> 2) = ( p + 2), when the potential can be approximated as V ( GLYPH<30> ) / GLYPH<30> p during the coherent oscillation regime after inflation, or simply w int = 0 when the potential considered describes only the inflationary stage. 12 For", '11 http://cp3.irmp.ucl.ac.be/ ~ ringeval/aspic.html': "12 Note that some inflationary potentials in this selection are a valid model for all stages, from the slow-roll phase all the way to coherent oscillations around the minimum during reheating, while others are 'incomplete' in the sense that they only describe the slow-roll regime. The hilltop, D-brane, potential with exponential tails, and spontaneously broken SUSY models fall into the latter category and rely on additional terms, denoted by the ellipses, to complete the potential at the end of inflation. With the increasing precision of CMB data and accompanying accuracy requirements for theoretical predictions, the precise form of the additional terms may a GLYPH<11> ect the scientific interpretation of some incomplete models, as pointed out for the case of quadratic hilltop and double-well inflationary models in PCI15. \nFig. 7. Marginalized joint two-dimensional 68 % and 95 % CL regions for combinations of ( GLYPH<15> 1 ; GLYPH<15> 2 ; GLYPH<15> 3) (upper panels) and ( GLYPH<15> V ; GLYPH<17> V ; GLYPH<24> 2 V ) (lower panels) for Planck TT,TE,EE + lowE + lensing (red contours), compared with Planck TT,TE,EE + lowE + lensing + BK15 (blue contours). \n<!-- image --> \nFig. 8. Marginalized joint 68 % and 95 % CL regions for n s and r at k = 0 : 002 Mpc GLYPH<0> 1 from Planck alone and in combination with BK15 or BK15 + BAO data, compared to the theoretical predictions of selected inflationary models. Note that the marginalized joint 68 % and 95 % CL regions assume dn s = d ln k = 0. \n<!-- image --> \ndata we use the full constraining power of Planck , i.e., Planck TT,TE,EE + lowE + lensing, in combination with BK15. \nThe GLYPH<1> GLYPH<31> 2 and the Bayesian evidence values for a selection of inflationary models with respect to the R 2 model \nTable 5. Bayesian comparison for a selection of slow-roll inflationary models with w int fixed (see text for more details). We quote 0.3 as the error on the Bayes factor. Models are strongly disfavoured when ln B < GLYPH<0> 5. \n(Starobinsky 1980; Mukhanov & Chibisov 1981; Starobinsky 1983) are shown in Table 5. Figure 9 shows the resulting marginalized probability densities of n s and r at k = 0 : 002 Mpc GLYPH<0> 1 for a few inflationary models with the above specified priors, compared to the corresponding 68 % and 95 % CL limits obtained from a GLYPH<3> CDM-plus-tensor fit. We refer the interested reader to PCI15 for a concise description of the inflationary models studied here and we limit ourselves here to a summary of the main results of this analysis. \n- -The inflationary predictions (Mukhanov & Chibisov 1981; Starobinsky 1983) originally computed for the R 2 model (Starobinsky 1980) to lowest order, \nn s GLYPH<0> 1 ' GLYPH<0> 2 N ; r ' 12 N 2 ; (48) \nare in good agreement with Planck 2018 data, confirming the previous 2013 and 2015 results. The 95 % CL allowed range 49 < N GLYPH<3> < 59 is compatible with the R 2 basic predictions N GLYPH<3> = 54, corresponding to T reh GLYPH<24> 10 9 GeV (Bezrukov & Gorbunov 2012). A higher reheating temperature T reh GLYPH<24> 10 13 GeV, as predicted in Higgs inflation (Bezrukov & Shaposhnikov 2008), is also compatible with the Planck data. \n- -Monomial potentials (Linde 1983) V ( GLYPH<30> ) = GLYPH<21> M 4 Pl ( GLYPH<30>= M Pl) p with p GLYPH<21> 2 are strongly disfavoured with respect to the R 2 model. For these values the Bayesian evidence is worse than in 2015 because of the smaller level of tensor modes allowed by BK15. Models with p = 1 or p = 2 = 3 \n(Silverstein & Westphal 2008; McAllister et al. 2010, 2014) are more compatible with the data. \n- -There are several mechanisms which could lower the predictions for the tensor-to-scalar ratio for a given potential V ( GLYPH<30> ) in single-field inflationary models. Important examples are a subluminal inflaton speed of sound due to a nonstandard kinetic term (Garriga & Mukhanov 1999), a nonminimal coupling to gravity (Spokoiny 1984; Lucchin et al. 1986; Salopek et al. 1989; Fakir & Unruh 1990), or an additional damping term for the inflaton due to dissipation in other degrees of freedom, as in warm inflation (Berera 1995; Bastero-Gil et al. 2016). In the following we report on the constraints for a non-minimal coupling to gravity of the type F ( GLYPH<30> ) R , with F ( GLYPH<30> ) = M 2 Pl + GLYPH<24> GLYPH<30> 2 , and a quartic potential. For this model we compute the theoretical predictions in terms of HFFs and number of e -folds to the end of inflation in the Einstein frame as for the R 2 model above, but we omit the technical details for the sake of brevity. 13 Our results show that a quartic potential, which would be excluded at high statistical significance for a minimally-coupled scalar inflaton as seen from Table 5, can be reconciled with the Planck and BK15 data for GLYPH<24> > 0: we obtain a 95 % CL lower limit log 10 GLYPH<24> > GLYPH<0> 1 : 5 with ln B = GLYPH<0> 2 : 4. \nFig. 9. Marginalized probability densities of the scalar tilt n s (top panel) and r (bottom panel) at k = 0 : 002 Mpc GLYPH<0> 1 for natural, R 2 , hilltop quartic, and V ( GLYPH<30> ) / GLYPH<30> 2 = 3 inflation, obtained by marginalizing over the uncertainties in the entropy generation stage, compared to the corresponding 68 % and 95 % CL limits obtained from a GLYPH<3> CDM-plus-tensor fit. \n<!-- image --> \n- -Natural inflation (Freese et al. 1990; Adams et al. 1993) is strongly disfavoured by the Planck 2018 plus BK15 data with a Bayes factor ln B = GLYPH<0> 6 : 6.\n- -Within the class of hilltop inflationary models (Boubekeur & Lyth 2005) we find that a quartic potential provides a better fit than a quadratic one. In the quartic case we find the 95 % CL lower limit log 10 ( GLYPH<22> 2 = M Pl) > 1 : 0.\n- -D-brane inflationary models (Kachru et al. 2003; Dvali et al. 2001; Garc'ıa-Bellido et al. 2002) provide a good fit to Planck and BK15 data for a large portion of their parameter space.\n- -For the simple class of inflationary potentials with exponential tails (Goncharov & Linde 1984; Stewart 1995; Dvali & Tye 1999; Burgess et al. 2002; Cicoli et al. 2009) we find ln B = GLYPH<0> 1 : 0.\n- -Planck 2018 and BK15 data strongly disfavour the hybrid model driven by logarithmic quantum corrections in spontaneously broken supersymmetric (SB SUSY) theories (Dvali et al. 1994), with ln B = GLYPH<0> 6 : 8.\n- -Planck and BK15 data set tight constraints on GLYPH<11> attractors (Kallosh et al. 2013; Ferrara et al. 2013). We obtain log 10 GLYPH<11> E 1 < 1 : 3 and log 10 GLYPH<11> E 2 < 1 : 1 at 95 % CL for the Emodel. We obtain slightly tighter 95 % CL bounds for the T-model, i.e., log 10 GLYPH<11> T 1 < 1 : 0 and log 10 GLYPH<11> T 2 < 1 : 0. Given the relation j RK j = 2 = (3 GLYPH<11> ) between the curvature of the Kahler geometry RK and GLYPH<11> in some of the T-models motivated by supergravity, Planck and BK15 data imply a lower bound on j RK j , which is still in the low-curvature regime. The discrete set of values GLYPH<11> = i = 3 with an integer i in the range [1 ; 7] motivated by maximal supersymmetry (Ferrara & Kallosh 2016; Kallosh et al. 2017) is compatible with the current data.", '5.1. Taylor expansion of V ( GLYPH<30> ) in the observable region': "In this section, as in section 6 of PCI13 and section 7.1 of PCI15, we try to reconstruct the inflaton potential only in its observable window, making no assumptions about the end of inflation. The motivation for being so conservative is that what happens after the inflaton rolls down beyond this range might not be captured by the simplest descriptions. More elaborate treatments would be required, for instance, in the case of a non-trivial potential shape before the end of inflation, a waterfall transition involving extra scalar fields, or several short inflationary stages between the time at which CMB scales exit the Hubble radius and the nucleosynthesis epoch. The analysis of this section relies, however, on the assumption that the potential is smooth enough inside the observable window to be described by a Taylor expansion up to order four. Note that this assumption is much weaker than assuming that a Taylor expansion is valid up to the end of inflation. However, it excludes from the analysis potentials with sharp features in the observable window, such as those studied in the next sections. \nWe perform the Taylor expansion around the value GLYPH<30> GLYPH<3> of the inflaton field evaluated precisely at the time t GLYPH<3> when the pivot scale k GLYPH<3> = 0 : 05 Mpc GLYPH<0> 1 fulfills the relation k GLYPH<3> = a ( t GLYPH<3> ) H ( t GLYPH<3> ). We separately study the cases where the expansion is performed at order n = 2, n = 3, or n = 4. We compute the primordial spectrum with a full integration of the Fourier mode evolution, using the inflationary module of the CLASS code. Although this method assumes no slow-roll approximation at any point, we speed up the convergence of the Markov Chain by taking flat priors not directly on the five Taylor coe GLYPH<14> cients f V ; V GLYPH<30> ; : : : ; V GLYPH<30>GLYPH<30>GLYPH<30>GLYPH<30> g , but on combinations of them matching the definitions of the potential slow-roll parameters f GLYPH<15> V ; GLYPH<17> V ; GLYPH<24> 2 V ; $ 3 V g presented in table 2 of PCI13. Even beyond the slow-roll approximation, these combinations provide nearly linear contributions to the tilt, running, running of the running, etc., of the scalar and tensor spectrum. Hence, they are directly related to observable quantities and well constrained by the data. Instead, if we ran with flat priors on \nf V ; V GLYPH<30> ; : : : ; V GLYPH<30>GLYPH<30>GLYPH<30>GLYPH<30> g , the convergence would be plagued by complicated parameter degeneracies. \nThe results of this analysis are shown in the panels of Fig. 10 and Table 6 for n = 2, 3, and 4, using two data sets for each: Planck TT,TE,EE + lowE alone; or Planck TT,TE,EE + lowE + lensing + BK15. The plot in Fig. 10 deliberately has a lot of white space because, for the sake of comparison, we plotted it over the same parameter ranges as the same plot in PCI15. We notice some significant improvement. Comparing Planck TT,TE,EE + lowE results from 2015 and 2018, we find that error bars on individual parameters have typically been reduced by 30 % thanks to improved polarization data. Including BK data provides further constraining power. Comparing Planck 2015 TT + lowP + BAO and Planck 2018 TT,TE,EE + lowE + lensing + BK15, we find that the error bars on f GLYPH<15> V ; GLYPH<17> V ; GLYPH<24> 2 V ; $ 3 V g shrink by factors of 2 to 4. The new data tend to resolve degeneracies which previously appeared in the n = 4 case and could be understood as a compensation mechanism between potentials with large running of the tilt, running of the running, tensor contribution, etc. The parameters GLYPH<24> 2 V and $ 3 V are perfectly compatible with zero (see Fig. 10 and Table 6), and so are V GLYPH<30>GLYPH<30>GLYPH<30> and V GLYPH<30>GLYPH<30>GLYPH<30>GLYPH<30> (see the contours on the parameters f V ; V GLYPH<30> ; : : : ; V GLYPH<30>GLYPH<30>GLYPH<30>GLYPH<30> g in Fig. 11). This is consistent with the fact that the new data set brings no evidence for running or running of the running. It also explains why the results of this section are close to those of Sect. 4.1, obtained under the slow-roll approximation. Similar to 2015, the best-fit value of running for n = 3 is negative, but has moved down from GLYPH<0> 0 : 013 to GLYPH<0> 0 : 007, and remains compatible with zero at the 1.0 GLYPH<27> level. For n = 4, the trend observed in 2015 to fit the data slightly better with a nonzero tensor contribution has disappeared. The decrease of the minimum e GLYPH<11> ective GLYPH<31> 2 when moving from n = 2 to n = 3 is insignificant and even smaller than in 2015, showing that the data do not require anything more complicated than an approximately parabolic shape for the inflaton potential within the observable window. \nThis can be checked by considering the random sample of well-fitting potentials presented in Fig. 12. Actually, for n = 4, a few of the plotted potentials have a non-parabolic 'spoon-like' shape (with a kink and a plateau), because non-negligible values of j V GLYPH<30>GLYPH<30>GLYPH<30>GLYPH<30> j are still allowed. However, this sub-class of models is by no means preferred over simpler parabolic-like potentials with a negligible j V GLYPH<30>GLYPH<30>GLYPH<30>GLYPH<30> j ; otherwise, we would have obtained a better GLYPH<31> 2 e GLYPH<11> for n = 4. Hence one should not take from Fig. 12 the message that special potentials with a kink and a plateau are favoured by the Planck data. Comparing this plot to figure 15 of PCI15, we see that the models with the largest V ( GLYPH<30> ) amplitude are excluded by stronger bounds on the tensor modes. \nFinally, it is interesting to notice that the predictions for the parameters of the minimal GLYPH<3> CDMmodel, such as n s or GLYPH<28> , remain extremely stable when increasing the freedom in the inflaton potential.", '5.2. Taylor expansion of H ( GLYPH<30> ) in the observable region': "To assess the robustness of our method, in this section we repeat the analysis with a Taylor expansion of the Hubble function H ( GLYPH<30> ) in the observable window, as we did in 2015. We refer the reader to section 7.2 of PCI15 for a precise description of this analysis, and we recall that the di GLYPH<11> erence with respect to the V ( GLYPH<30> ) reconstruction is more than a mere change of priors. For each value of n , the new parameterization covers a slightly di GLYPH<11> erent range of potentials, and, more importantly, it naturally includes \nTable 6. Numerical reconstruction of the potential slowroll parameters beyond any slow-roll approximation, when the potential is Taylor-expanded to n th order, using Planck TT,TE,EE + lowE + lensing + BK15. We also show the corresponding bounds on some related parameters (here n s, dn s = d ln k , and r 0 : 002 are derived from the numerically computed primordial spectra). All error bars are at the 68 % CL and all upper bounds at the 95 % CL. The e GLYPH<11> ective GLYPH<31> 2 value of model n is given relative to model n GLYPH<0> 1. \nFig. 10. Taylor expansion of V ( GLYPH<30> ) at order n = 2, 3, and 4 in the observable region, making no assumption about the end of inflation. The parameters are combinations of Taylor coe GLYPH<14> cients with flat priors. Dashed contours are Planck TT,TE,EE + lowE, while solid contours are Planck TT,TE,EE + lowE + lensing + BK15. The scales are the same as in PCI15. \n<!-- image --> \nV \na marginalization over the uncertainty in the initial value of the derivative ˙ GLYPH<30> when the inflaton enters the observable window. Instead, in the previous analysis, ˙ GLYPH<30> was assumed to have reached the inflaton attractor solution, i.e., there was an implicit assumption that inflation started well before that time. In the analysis based on H ( GLYPH<30> ), inflation models with a minimal duration are not excluded by the priors. \nThe improvement with respect to the 2015 results is even more impressive in this case. Bounds on the n = 4 parameters are typically 3 to 4 times stronger compared with 2015, as can \nFig. 11. Taylor expansion of V ( GLYPH<30> ) at order n = 2, 3, and 4 in the observable region, making no assumption about the end of inflation. In natural units (where p 8 GLYPH<25> M Pl = 1). The parameters are the Taylor coe GLYPH<14> cients, obtained here as derived parameters with non-flat priors. Dashed contours are Planck TT,TE,EE + lowE, while solid contours are Planck TT,TE,EE + lowE + lensing + BK15. The scales are the same as in PCI15. \n<!-- image --> \nFig. 13. Taylor expansion of H ( GLYPH<30> ) at order n = 2, 3, and 4 in the observable region, making no assumption about the end of inflation. The parameters are combinations of Taylor coe GLYPH<14> cients with flat priors. Dashed contours are Planck TT,TE,EE + lowE, while solid contours are Planck TT,TE,EE + lowE + lensing + BK15. The scales are the same as in PCI15. \n<!-- image --> \n- \n∗ \nFig. 12. Representative sample of the observable region of inflaton potentials allowed at the 95 % CL, when the potential is Taylor-expanded at order n = 2, 3, and 4 in the observable region, making no assumption about the end of inflation, and using Planck TT,TE,EE + lowE + lensing + BK15. In natural units (where p 8 GLYPH<25> M Pl = 1). We use the same scales as in PCI15. Note that there is another branch of solutions that is symmetric under ( GLYPH<30> GLYPH<0> GLYPH<30> GLYPH<3> ) !GLYPH<0> ( GLYPH<30> GLYPH<0> GLYPH<30> GLYPH<3> ). \nbe checked from Table 7 and Fig. 13. We found that a factor of 2 improvement comes from switching to the new set of low-' likelihoods, and another factor of 2 from adding the BK likelihood. On the other hand, the use of more recent high-' and lensing likelihoods has a modest impact. \nA consequence of these improved constraints can be seen in Fig. 14, when we compare it to its counterpart from 2015 (figure 20 in PCI15). Again, for a better comparison Fig. 14 uses the same scale as figure 20 of PCI15. For n = 4, the previously best- \nTable 7. Numerical reconstruction of the Hubble slow-roll parameters beyond any slow-roll approximation, using Planck TT,TE,EE + lowE + lensing + BK15. We also show the corresponding bounds on some related parameters (here n s, dn s = d ln k , and r 0 : 002 are derived from the numerically computed primordial spectra). All error bars are at the 68 % CL and all upper bounds at the 95 % CL. The e GLYPH<11> ective GLYPH<31> 2 value of model n is given relative to model n GLYPH<0> 1. \nfitting models included many scenarios starting with a fast-roll stage, producing a tail with large V ( GLYPH<30> ) before pivot-scale crossing. These models are now excluded by better polarization data and tensor constraints. \nGoing beyond the parabolic approximation for H ( GLYPH<30> ) does not improve the goodness-of-fit: as in the potential-based analysis of Sect. 5.1, the GLYPH<1> GLYPH<31> 2 s between n = 2, n = 3, and n = 4 are negligible, and the parameters GLYPH<24> 2 H and $ 3 H related to H GLYPH<30>GLYPH<30>GLYPH<30> and H GLYPH<30>GLYPH<30>GLYPH<30>GLYPH<30> are compatible with zero. \n<!-- image --> \nH \n- \nFig. 14. Representative sample of the observable region of inflaton potentials allowed at the 95 % CL, inferred from H ( GLYPH<30> ) when that function is Taylor-expanded at order n = 2, 3, and 4 in the observable region, making no assumption about the end of inflation, and using Planck TT,TE,EE + lowE + lensing + BK15. In natural units (where p 8 GLYPH<25> M Pl = 1). The scales are the same as in PCI15. Note that there is another branch of solutions symmetric under ( GLYPH<30> GLYPH<0> GLYPH<30> GLYPH<3> ) !GLYPH<0> ( GLYPH<30> GLYPH<0> GLYPH<30> GLYPH<3> ). \n<!-- image --> \n∗", '5.3. Taylor expansion of full V ( GLYPH<30> )': "We now present a new analysis with less conservative assumptions than in the previous subsections. We switch to the hypothesis that the inflaton potential is very smooth not only within its observable window, but also until the end of inflation, such that its whole shape can be captured by a Taylor expansion. We further assume that inflation ends when the first slow-roll condition is violated ( GLYPH<15> V = 1), without invoking any other field. Finally, we fix the number of e -folds between Hubble crossing of the pivot scale and the end of inflation to N GLYPH<3> = 55, which implicitly relies on the hypothesis that no further inflationary stage took place at a later epoch. \nTechnically, the analysis pipeline for this case is similar to that of Sect. 5.1, except for an extra step in which the CLASS inflationary module integrates the background equations until the end of inflation, goes backwards in time by 55 e -folds, and imposes that the Hubble crossing for the pivot scale k GLYPH<3> = 0 : 05 Mpc GLYPH<0> 1 matches that time. \nThese models are much more constrained than those of Sects. 5.1 and 5.2, since the e -fold condition is imposed in addition to having a potential with a good shape within the observable window. The constraining power is then su GLYPH<14> cient for running the MCMC chains directly with flat priors on f V ; V GLYPH<30> ; : : : ; V GLYPH<30>GLYPH<30>GLYPH<30>GLYPH<30> g . \nOur results are presented in Figs. 15 and 16 and in Table 8. For models with a purely quadratic potential, the numerically computed tilt and tensor-to-scalar ratio depend almost exclusively on N GLYPH<3> , thus they remain fixed to ns = 0 : 963 and r 0 : 002 = 0 : 136. Such a large r is in tension with the Planck data, and even more so with the Planck + BK data. Thus the e GLYPH<11> ective GLYPH<31> 2 is poor in the n = 2 case and improves considerably when adding some freedom in going to n = 3. Indeed, the presence of an additional cubic term allows us to reach smaller values of the tensor-toscalar ratio for roughly the same scalar tilt, and lowers GLYPH<31> 2 e GLYPH<11> by more than 13. Instead, when also adding a quartic term, we find no significant improvement in the goodness of fit, and the coefficient of the GLYPH<30> 4 term is consistent with zero. \nThese findings are consistent with the global picture that Planck data prefer potentials which are concave in the observable window. The blue and green curves in the lower left panel of Fig. 16 illustrate the preference of the Planck + lensing + BK15 data for potentials with an inflection point, appearing qualitatively similar to scalar field potentials associated with spontaneous symmetry breaking models, hilltop models, new inflation, natural inflation, etc. \nIn these runs, the value of the scalar tilt running is always very precisely constrained around a value of dn s = d ln k ' GLYPH<0> 6 GLYPH<2> 10 GLYPH<0> 4 . This does not come as a surprise if we keep in mind that these bounds are not imposed directly by the data, but rather by the class of inflationary potentials considered here, with potential parameters fixed by observational bounds on the amplitude and tilt of the scalar and tensor spectra. In other words, the running is not directly measured, but rather predicted as a function of the scalar / tensor amplitudes and scalar tilt. Interestingly, if combinations of future CMB and large-scale structure data with a wide lever arm in wavenumber space could become directly sensitive to such tiny values (which would require a factor of around 10 improvement in sensitivity compared to current CMB + BAO data), a very large class of currently successful inflationary models could be either confirmed or ruled out. \nFig. 15. Taylor expansion of the full V ( GLYPH<30> ) at order n = 3 and 4, trusted until the end of inflation, in natural units (where p 8 GLYPH<25> M Pl = 1). The parameters are the Taylor coe GLYPH<14> cients with flat priors. Dashed contours are Planck TT,TE,EE + lowE, while solid contours are Planck TT,TE,EE + lowE + lensing + BK15. \n<!-- image -->", '5.4. Free-form potential reconstruction': 'As a complementary analysis to the previous three subsections, we next perform a free-form reconstruction of the inflationary potential with cubic splines, in a manner akin to the reconstructions of PCI15 and Sect. 6.2.1. Further plots and theoretical detail can be found in Handley et al. (2019). \nAfree-form reconstruction usually proceeds by parameterizing the function of interest via a spline and taking the locations of the interpolation knots as free parameters in a posterior distribution. These are then varied along with any other model parameters, and then marginalized out to yield a model-independent \nV \nV \n1.2 \n1.0 \n0.8 \n0.6 \n0.4 \n0.2 \n0.0 \n- \n1.2 \n1.0 \n0.8 \n0.6 \n0.4 \n0.2 \n0.0 \n- \n1.2 \n1.0 \n0.8 \n0.6 \n0.4 \n0.2 \n0.0 \n- \n× \n10 \n3.6 \n- \n11 \n- \n- \n- \n- \n- \n- \nn=4 \nn=3 \nn=2 \n- \n- \n2.8 \nφ \nFig. 16. Representative sample of the inflaton potentials allowed at the 95 % CL, when the potential is Taylor-expanded at order n = 2, 3, and 4 and trusted until the end of inflation, and under the assumption of N GLYPH<3> = 55 e -folds of inflation between Hubbleradius crossing for the pivot scale and the end of inflation. In natural units (where p 8 GLYPH<25> M Pl = 1). Left panels: Full potential from the beginning of the observable window till the end of inflation. Right: Zoom on the observable window directly constrained by inflation. Top: Planck TT,TE,EE + lowE. Bottom: Planck TT,TE,EE + lowE + lensing + BK15. Note that there is another branch of solutions that is symmetric under GLYPH<30> !GLYPH<0> GLYPH<30> . \n<!-- image --> \n- \n11 \n- \n- \n3.0 \n11 \n- \n- \n- \n- \n- \n- \n- \n- \n- \n- \n- \nφ \nφ \nreconstruction of the function of interest. The analysis is run for di GLYPH<11> ering numbers of knots, N , and the Bayesian evidence is computed to allow for model comparison to determine how many knots are appropriate from the perspective of the data. \nTo reconstruct the inflationary potential V ( GLYPH<30> ), one cannot take a linear interpolating spline (as in Sect. 6.2.1), since the equations of motion in general depend on first (and sometimes second) derivatives of V . We therefore choose to parameterize the second derivative of the log-potential as a linear spline. The log-potential is computed by integrating this function twice, yielding a function with two additional free parametersa global o GLYPH<11> set and a gradient. Our reconstruction function is \ntherefore \n3.0 \n2.2 \n3.4 \n× \n10 \n3.5 \n× \n10 \n3.5 \n2.5 \n2.5 \n2.0 \n2.0 \n1.5 \n1.5 \n1.0 \n1.0 \nn=4 \nn=3 \nn=2 \n0.5 \n0.0 \nn=4 \nn=3 \nn=2 \n0.5 \n0.0 \nV \n3.2 \n3.0 \n2.6 \n2.4 \nln V = ln V GLYPH<3> + ( GLYPH<30> GLYPH<0> GLYPH<30> GLYPH<3> ) d ln V GLYPH<3> d GLYPH<30> \n+ Z GLYPH<30> GLYPH<30> GLYPH<3> d GLYPH<30> 0 Z GLYPH<30> 0 GLYPH<30> GLYPH<3> d GLYPH<30> 00 LS( GLYPH<30> 00 ; GLYPH<18> ) ; (49) \nGLYPH<18> = GLYPH<30> 1 ; : : : ; GLYPH<30> N ; d 2 ln V 1 d GLYPH<30> 2 ; : : : ; d 2 ln VN d GLYPH<30> 2 ! ; (50) \nLS( GLYPH<30> ; GLYPH<18> ) = ( d 2 ln Vi d GLYPH<30> 2 GLYPH<30> GLYPH<0> GLYPH<30> i + 1 GLYPH<30> i GLYPH<0> GLYPH<30> i + 1 + d 2 ln Vi + 1 d GLYPH<30> 2 GLYPH<30> GLYPH<0> GLYPH<30> i GLYPH<30> i + 1 GLYPH<0> GLYPH<30> i : GLYPH<30> i < GLYPH<30> < GLYPH<30> i + 1 : (51) \nHere LS( GLYPH<30> ; GLYPH<18> ) is a standard linear spline dependent on N knots, ln V GLYPH<3> is the potential at the pivot scale, and d ln V GLYPH<3> = d GLYPH<30> is the gradient of the log-potential at the pivot scale. \nIn general, any reconstruction of the potential will be sensitive only to the observable window of inflation in GLYPH<30> 2 [ GLYPH<30> min ; GLYPH<30> max ], where GLYPH<30> min and GLYPH<30> max are defined as the field values when the largest and smallest observable scales k min and k max exit the Hubble radius during inflation. Regions of the potential outside these GLYPH<30> values are unconstrained by current CMB data. In our analysis, we take k min = 10 GLYPH<0> 4 Mpc GLYPH<0> 1 and k max = 10 GLYPH<0> 0 : 3 Mpc GLYPH<0> 1 , which encompasses the multipole range \n2.0 \nTable 8. Numerical reconstruction of the potential parameters beyond any slow-roll approximation, when the potential is Taylor-expanded to n th order, trusted until the end of inflation, and using Planck high-\' TT,TE,EE + lowE + lensing + BK15. We also show the corresponding bounds on some related parameters (here n s, dn s = d ln k , and r 0 : 002 are derived from the numerically computed primordial spectra). All error bars are at the 68 % CL and all upper bounds at the 95 % CL. The e GLYPH<11> ective GLYPH<31> 2 value of model n is given relative to model n GLYPH<0> 1. \nTable 9. Parameters of the free-form potential reconstruction analysis and details of the priors. There is a further prior constraint in that we require that the inflaton should evolve in an inflating phase throughout the observable window, that the inflaton should be rolling downhill from negative to positive GLYPH<30> throughout, and that any primordial power spectra generated sit in the range 2 < ln 10 10 P R ( k ) < 4. \nconstrained by Planck (see Sect. 6.2.1). The locations GLYPH<30> 1 ; : : : ; GLYPH<30> N of the reconstruction knots should be distributed throughout this observable window. Whilst the locations GLYPH<30> 1 ; : : : GLYPH<30> N and heights d 2 ln V 1 = d GLYPH<30> 2 ; : : : ; d 2 ln VN = d GLYPH<30> 2 themselves influence the size of the observable window, a reasonable approach is to first estimate it using the unperturbed potential (i.e., setting N = 0). This gives an alternative range GLYPH<30> 2 [ ˜ GLYPH<30> min ; ˜ GLYPH<30> max ]. The priors on all our variables are indicated in Table 9. \nOur results are detailed in Fig. 17. The Bayesian evidence shows that the reconstruction preferred by the data is that using N = 1, corresponding to a constant non-zero d 2 ln V = d GLYPH<30> 2 . This indicates that the Planck data do not significantly constrain the inflationary potential within the window any further than up to a quadratic term in a Taylor expansion. It is illuminating, however, to consider adding further structure to the potential, and Fig. 17 shows reconstructions for N = 8. \nConsidering the predictive posterior of the primordial power spectrum, we see that our parameterization is su GLYPH<14> cient to exhibit the deficit at \' \' 30, cosmic variance at low \' , and the loss of resolution at high \' , as seen in Sect. 6.2.1. Consistent with the \nrest of the analyses, " V is unconstrained, whilst the Planck data provide relatively powerful constraints on GLYPH<17> V within the observable window of inflation.', '6. Primordial power spectrum reconstruction': "This section reports results for the non-parametric reconstruction of the primordial scalar power spectrum using the new Planck 2018 likelihoods, as well as comparisons with the previously reported results for the Planck 2013 and 2015 releases. The objective here is to search for deviations from a simple power-law primordial power spectrum (i.e., P R ( k ) = A s( k = k GLYPH<3> ) n s GLYPH<0> 1 ) in a manner that does not presuppose any particular theoretical model giving rise to such deviations. This work is complementary to the searches considered in Sect. 7, where particular functional forms for such deviations motivated by theory are investigated. \nHere we apply three distinct nonparametric methods. In 2013 only the first method was used to reconstruct the primordial power spectrum, the so-called 'penalized likelihood' method, for which the 2018 results are presented in Sect. 6.1. In 2015 two additional methods were also used: a linear spline method (discussed in Sect. 6.2) for which both the number of knots and their positions were allowed to vary, and ideas from Bayesian model selection were applied to determine the appropriate number of knots; and a method using cubic splines (discussed in Sect. 6.3). Although the discussion below includes some description of each method in order to make the paper self-contained, for details the reader is referred to the 2013 and 2015 papers. Here we specify only those details specific to the 2018 analysis or di GLYPH<11> erent from the choices in the 2013 and 2015 analyses. See references in PCI13 and Hunt & Sarkar (2014), Hazra et al. (2014b), and Hunt & Sarkar (2015) for other approaches to nonparametric reconstruction of the primordial power spectrum.", '6.1. Penalized likelihood': 'The underlying idea of the penalized likelihood approach is to add a term to the log-likelihood that penalizes deviations from a perfect power-law spectrum. We parameterize the power spectrum as \nP R ( k ) = P 0( k ) exp GLYPH<2> f ( k ) GLYPH<3> ; (52) \nwhere P 0( k ) = A s( k = k GLYPH<3> ) n s GLYPH<0> 1 , and add the following term to GLYPH<0> 2 ln L : \nf T R ( GLYPH<21>; GLYPH<11> ) f GLYPH<17> GLYPH<21> Z GLYPH<20> max GLYPH<20> min d GLYPH<20> @ 2 f ( GLYPH<20> ) @GLYPH<20> 2 ! 2 + GLYPH<11> Z GLYPH<20> min GLYPH<0>1 f 2 ( GLYPH<20> ) + GLYPH<11> Z + 1 GLYPH<20> max f 2 ( GLYPH<20> ) ; (53) \nwhere GLYPH<20> = ln k : The interval [ GLYPH<20> min ; GLYPH<20> max] is chosen to cover the range over which the likelihood is able to constrain the data. The two GLYPH<11> terms serve to pin the reconstruction to the simple power law where the data have almost no constraining power. One may imagine that GLYPH<11> > 0 should be infinite, but for numerical reasons a large but finite value is used to simplify the numerics. Numerically, for each GLYPH<21> the dimension of f is chosen to be so large that the continuum version of the penalty given in Eq. (53) has been accurately approximated. For more details see Gauthier & Bucher (2012) and the extensive references therein to prior literature, as well as PCI13 and PCI15. \nIn Fig. 18 we show the results using Planck TT + lowE and in Fig. 19 we show the results for Planck TT,TE,EE + lowE. In both cases we have assumed the usual baseGLYPH<3> CDM model specified \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFig. 17. Free-form potential reconstructions using Planck TT + TE + EE + lowl + lowE + lensing (Sect. 5.4). The top-right panel shows Bayes factors for the free-form potential reconstruction. The preferred reconstruction has N = 1, corresponding to a constant nonzero d 2 ln V 1 = d GLYPH<30> 2 . The remaining panels show reconstructions for the N = 8 knot case, focusing on the scalar primordial power spectrum, and the inflationary slow-roll parameters " V and GLYPH<17> V . Red lines indicate sample trajectories from the prior, whilst black lines are from the posterior. Technically the slow-roll parameters are defined as functions of GLYPH<30> , but we instead substitute this for the Hubble-radius-exit value to make for clearer comparison between posterior samples. In all plots, the approximate link between \' and k is via the Limber approximation, \' \' k = D A, where DA = r GLYPH<3> =GLYPH<18> GLYPH<3> is the comoving angular distance to recombination, which is at comoving distance r GLYPH<3> . \n<!-- image --> \nin PCP18, except that the power spectrum is now parameterized by a set of spline points. In addition to these spline points, we also maximize the likelihood with respect to the dimensionless Hubble parameter, h , and the baryon, GLYPH<10> b h 2 , and CDM, GLYPH<10> c h 2 , densities. All other cosmological and nuisance parameters are the same as those quoted in PCP18. \nFor the TT-only case, the maximum deviations are 1 : 55 GLYPH<27> , 2 : 10 GLYPH<27> , 1 : 80 GLYPH<27> , and 1 : 65 GLYPH<27> for GLYPH<21> = 10 3 , 10 4 , 10 5 , and 10 6 , respectively, for which the probabilities to exceed are 13 %, 28 %, 31 %, and 23 % (where we have taken into account the lookelsewhere e GLYPH<11> ect). Similarly, for the TT,TE,EE case, the maximumdeviations are 2 : 07 GLYPH<27> , 1 : 77 GLYPH<27> , 1 : 77 GLYPH<27> , and 1 : 08 GLYPH<27> for GLYPH<21> = 10 3 , \nZ \n10 4 , 10 5 , and 10 6 , respectively, for which the probabilities to exceed are 29 %, 23 %, 32 %, and 25 %. We consequently find no statistically significant evidence for a deviation from the simple power-law hypothesis. This result is consistent with the results previously reported for the Planck 2013 and 2015 releases using essentially the same method. It is likewise consistent with the results below in Sects. 6.2 and 6.3, which use di GLYPH<11> erent methods.', '6.2. Bayesian reconstruction': 'To reconstruct the primordial power spectrum of curvature perturbations, we follow the methodology of section 8.2 of PCI15, using an N -point interpolating logarithmic spline with the positions of the knots considered as free parameters in the full posterior distribution. The positions of the points in the ( k ; P ) plane are treated as likelihood parameters with loguniform priors. Further, the k -positions are sorted a priori such that k 1 < k 2 < GLYPH<1> GLYPH<1> GLYPH<1> < kN , with k 1 and kN fixed. We compute posteriors and evidence values (conditioned on N ) using PolyChord (Handley et al. 2015a,b), also varying all cosmological and nuisance parameters. We then use evidence values for each model to correctly marginalize out the number of knots N . \nTo plot our reconstructions of P ( k ), we compute the marginalized posterior distribution of ln P conditioned on k . The iso-probability confidence intervals are then plotted in the ( k ; P ) plane (see, e.g., Fig. 20), using code recorded in Handley (2018). To quantify the constraining power of a given experiment, we use the conditional Kullback-Leibler (KL) divergence as exemplified by Hee et al. (2016). For two distributions P ( GLYPH<18> ) and Q ( GLYPH<18> ), the KL divergence is defined as \nDKL ( P j Q ) = Z ln " P ( GLYPH<18> ) Q ( GLYPH<18> ) # P ( GLYPH<18> ) d GLYPH<18>; (54) \nand may be interpreted as the information gain in moving from a prior Q to a posterior P (Raveri et al. 2016). For our reconstructions, we compute the KL divergence of each distribution conditioned on k and N , and then marginalize over N using evidence values to produce a k -dependent number which quantifies the compression or information that each data set provides at each value of k . Further plots and theoretical detail can be found in Handley et al. (2019).', '6.2.1. Update on Planck 2015': "In PCI15, our analysis focussed predominantly on the TT + lowTEB data set. Here we present results for TT,TE,EE + lowE + lensing. First, in updating to the lowE likelihood, we find that there is a marked tightening in the constraint on the amplitude of the reconstructed spectrum at all values of k . The improvement in the constraint can be seen directly in the predictive posterior plots (Fig. 20, top-left panel, and Fig. 21), and is quantified in Fig. 20 (bottom-right) via the KL divergence. The reason for the high-' constraint provided by a low-' likelihood change is due to the reduced uncertainty on GLYPH<28> that SimAll EE provides. This can be seen by examining the shifts in the underlying cosmological parameters in Fig. 2. \nUpon adding TE and EE data, we find that the hint of a feature at ' ' 30 is still present, in spite of the additional constraining power provided by polarization. Using polarization data, the N = 3 case is now the most strongly favoured model by the evidence criterion. This indicates that there is some scope for models which account for low-' cosmic variance to be preferred \nTable 10. Priors for the search for sharp features in the primordial power spectrum. Units for k and d are Mpc GLYPH<0> 1 . \nin a Bayesian sense. The other underlying cosmological parameters are una GLYPH<11> ected by the additional degrees of freedom in the primordial power spectrum provided by the reconstruction. \nIn order to combine Planck polarization data with BK15, we also allow the tensor-to-scalar ratio r to vary, and fix the tensor tilt n t via the inflationary consistency condition. As can be seen in the bottom-left panel of Fig. 20, upon adding BK15, the e GLYPH<11> ect of the low-' deficit is softened, but with otherwise little change to the reconstruction. We repeated our analysis with CamSpec in place of Plik and found our results to be qualitatively and quantitatively unchanged.", '6.2.2. Free-form search for features': 'Next we examine the e GLYPH<11> ect that sharp features in the primordial power spectrum can have on cosmological parameters. We model sharp features in the spectrum as a variable number of tophat functions with varying widths, heights, and locations. On top of the traditional A s ; n s parameterization of the power spectrum, we place N sharp top-hat features into the spectrum at locations ki with widths di and heights hi ( i = 1 ; : : : ; N ). That is, we set \nln P R ( k ) = ln A s + ( n s GLYPH<0> 1) ln k k GLYPH<3> ! + N X i = 1 hi " j k GLYPH<0> ki j < di 2 # ; (55) \nwhere the square brackets in the summation denote a logical truth function as introduced by Graham et al. (1994). For values of N = 0 ; : : : ; 8, we treat the variables in parameterization (55) as parameters in a posterior distribution along with the traditional cosmological and Planck nuisance parameters, with priors as detailed in Table 10. We run with both linear and logarithmic priors on the k -locations of the features, as this alters the sensitivity to the type of features uncovered. We sample the posteriors using PolyChord (Handley et al. 2015a,b). \nFigures 22 and 23 show our results. With the linear priors case, there are statistically insignificant features corresponding to the peaks of the TT spectrum, which arise due to the enhanced cosmic variance at these locations. With the logarithmic priors case, a stronger but still statistically insignificant feature is detected at \' \' 30, with a small deficit and surrounding enhancement of power. This case reproduces the results found in Sect. 6.2.1. In both cases, the Bayesian evidence shows preference for a no-features spectrum, and steadily declines as more features are added. The cosmological parameters remain unperturbed despite the introduction of features. \n) \nk \n( \nf \nh \n4 \n3 \n10 \n10 \n- \n4 \n10 \nλ \nλ \nλ \n3 \n5 \n10 \n] \nFig. 18. Planck TT + lowE penalized likelihood primordial power spectrum reconstruction. Top four panels : The deviation f ( k ) for four di GLYPH<11> erent roughness penalties. The red curves indicate the best-fit deviation, while the vertical extents of the dark and light green error bars indicate the GLYPH<6> 1 GLYPH<27> and GLYPH<6> 2 GLYPH<27> errors, respectively. The width of the error bars indicates the minimum reconstructible width (the minimum width for a Gaussian feature such that the mean square deviation of the reconstruction is less than 10 %). The grey regions display where the minimum reconstructible width is undefined, meaning that the reconstruction in these regions is untrustworthy. The hatched region in the GLYPH<21> = 10 6 plot indicates where the fixing penalty has been applied. Lower three panels: GLYPH<6> 1 GLYPH<27> error bars for the three non-primordial-specctrum cosmological parameters included in the reconstruction. The respective best-fit fiducial model values are indicated by the dashed lines. \n<!-- image --> \n0.10 \n0.05 \n0.00 \n0.05 \n- \n0.10 \n- \n0.15 \n- \n10 \n- \n0.6900 \n0.6725 \n0.6550 \n0.6375 \n0.6200 \nλ \n= 10 \nk \n[Mpc \n6 \n10 \n10 \n- \n5 \n2 \n- \n1 \n0.3 \n0.2 \n0.1 \n0.0 \n0.1 \n- \n0.2 \n- \n0.3 \n- \n10 \n- \n0.04 \n0.02 \n0.00 \n0.02 \n- \n0.04 \n- \n0.06 \n- \n10 \n10 \n- \n5 \n4 \n4 \n10 \n2 \nh \nc \nΩ \n10 \n- \n0.135 \n0.130 \n0.125 \n0.120 \n0.115 \n1 \n3 \n10 \n4 \n10 \n) \nk \n( \nf \n) \nk \n( \nf \n6 \n10 \n- \n10 \n- \n3 \n3 \n2 \nh \nb \nΩ \n× \n100 \nλ \n= 10 \nk \n[Mpc \nλ \n= 10 \nk \n[Mpc \n2.300 \n2.225 \n2.150 \n2.075 \n2.000 \n10 \n- \n4 \n2 \n10 \n- \n- \n1 \n] \n6 \n2 \n- \n1 \n3 \n10 \n] \n4 \n10 \n10 \n- \n10 \n- \n5 \n10 \n1 \n1 \n6 \n10 \nFig. 19. Penalized likelihood reconstruction, as Fig. 18 but for Planck TT,TE,EE + lowE. \n<!-- image --> \nZ \nFig. 20. Free-form Bayesian reconstruction of the primordial power spectrum (Sect. 6.2.1) using Planck TT,TE,EE + lowE + lensing. Top-right: Evidence values for each N -knot reconstruction. The evidence is maximal for the N = 2 and N = 3 knot cases, and semicompetitive for the remaining higher knots. Marginalizing over the number of knots produces a predictive posterior plot, shown in the top-left panel. Here we see generic features, with the limit of resolution of Planck at \' \' 2400 and cosmic variance at low \' . Bottom-left: Same as top-left, but using the additional BK15 data and allowing r to vary. Bottom-right: Kullback-Leibler divergence conditional on k , marginalized over the number of knots, showing the increase in compression of the primordial power spectrum over several past CMB missions. The di GLYPH<11> erence in constraining power between Planck 2013 and 2015 is driven entirely by the shift in the GLYPH<28> constraint. \n<!-- image --> \nFig. 21. Free-form Bayesian reconstruction of the primordial power spectrum for varying numbers of knots (Sect. 6.2.1) using TT,TE,EE + lowE + lensing. The amplitude and tilt are consistent with the rest of the results with the same combination of likelihoods. As more knots are added, the \' \' 30 feature in the C \' temperature spectrum is visible as a dip to lower power. \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFig. 22. Free-form Bayesian search for features (Sect. 6.2.2) with Planck TT,TE,EE + lowE + lensing. The upper panels show runs with linear priors on the k -locations. The lower panels use logarithmic priors on the k -features. Left panels show the reconstruction for N = 8 features, while the right panels show the reconstruction marginalized over N = 0 ; : : : ; 8 features. \n<!-- image --> \nFig. 23. The e GLYPH<11> ect on the underlying cosmological parameters of the free-form Bayesian search for features (Sect. 6.2.2), for N = 0 ; : : : ; 3 features with linear k -priors. The parameters remain stable up to N = 8 features, and when changing to logarithmic k -priors. \n<!-- image -->', '6.3. Cubic spline reconstruction': "In this subsection we update the third method of reconstruction used in PCI15, in which ln P R (ln( k )) was expanded in cubic splines localized in ln( k ) about uniformly spaced 'knots,' f ln( kb ) ; b = 1 ; : : : ; N g , whose range was chosen to cover all relevant cosmological scales, from 10 GLYPH<0> 4 Mpc GLYPH<0> 1 to O (1) Mpc GLYPH<0> 1 : We single out the standard scalar power spectrum pivot scale as a 'pivot knot' b = p , with kp = k GLYPH<3> = 0 : 05 Mpc GLYPH<0> 1 : Its associated power ln A s = ln P R ( k GLYPH<3> ) is assigned a uniformly distributed prior. A tilted primordial power spectrum P R ; fid GLYPH<17> A s( k = k GLYPH<3> ) n s ; fid GLYPH<0> 1 , with fixed spectral index n s ; fid is used as the fiducial baseline from which deviations are measured, expressed in terms of N GLYPH<0> 1 relative spectral shape parameters: qb = ln GLYPH<0> P R ( kb ) = P R ; fid( kb ) GLYPH<1> for b , p . For the results presented here, n s ; fid = 0 : 967 was chosen. As in PCI15 we continue to use cubic splines for the k -space modes we expand in, with natural boundary conditions (i.e., vanishing second derivatives at the first and last knots). The treatment here is therefore quite analogous to that in Sect. 5.4, where the inflaton potential rather than the curvature power spectrum is expanded in cubic splines. Knot numbers up to 18 were reported in PCI15, and it was shown that 12 were su GLYPH<14> cient to capture the variations desired by the Planck CMBdata. The mode functions were also varied. For example, linear interpolation leads to similar reconstructions as long as enough knots are used. A weak uniform prior ( GLYPH<0> 1 GLYPH<20> qb GLYPH<20> 1) was imposed on qb . Outside of the spline coverage region [ k 1 ; kN ] we set ln GLYPH<0> P R ( k ) = P R ; fid( k ) GLYPH<1> to be q 1 for k < k 1 and qN for k > kN . The prior on qb and boundary condition choices have little impact on the reconstructions over most of the k -range. \nThe current Planck TT,TE,EE + lowE + lensing + BK15 data give only upper limits to the allowed values of the tensor amplitude, r < 0 : 06. Consequently, adding shape degrees of freedom to the tensor power spectrum would yield a completely prior- \ndriven result. Instead we adopt the standard model power-law parameterization for tensors, P t( k ) = rA s( k = k GLYPH<3> ) n t , with the tensor spectral index constrained by the consistency relation n t = GLYPH<0> r = 8. Without B -mode constraints and with enough knots one could deform the primordial scalar spectrum to mimic a tensor contribution to the CMB power. However, this near-degeneracy is broken with direct B -mode observations, e GLYPH<11> ectively so even if there are only upper limits as for the BK15 data. Our reconstructions here focus on letting r float over a prior range 0 GLYPH<20> r GLYPH<20> 1, but the posterior is strongly constrained by the BK15 data. \nThe joint probability distributions of f qb ; b , p g , ln A s, and the other cosmic and nuisance parameters are determined by CosmoMC modified to incorporate the N -knot parameterization for fixed knot number N . Figure 24 shows the reconstruction. Apart from the mean and 1 GLYPH<27> and 2 GLYPH<27> limits on the ensemble of trajectories allowed by the posterior probability, we also show a set of individual trajectories with parameters taken from 1 GLYPH<27> samples to illustrate the knot-to-knot coherence (dashed curves). The tensor trajectories are straight lines, as required by the adopted tensor power model. \nIn spite of the extra scalar shape freedom in the k -space region over which the tensor modes a GLYPH<11> ect the CMB, the 12 knot reconstruction still leads to a strong constraint of r < 0 : 069, rather close to the r < 0 : 06 limit obtained if the only shape parameter is n s. In fact we find that the current limits on r are such that the scalar-power reconstructions are not sensitive to the details of the r distribution. To illustrate this, the lower panel of Fig. 24 shows the spectrum when r is fixed at the tiny value of r = 0 : 001. One could regard this as a theoretical prior for lowenergy inflation models or a forecast for a future in which r is measured or tightly constrained by B -mode experiments. \nIn PCI15, the main cubic spline reconstruction included nonCMBdata to help pin down cosmological parameters such as H 0, GLYPH<28> , and the late-time expansion history. The improvements in the \nFig. 24. Reconstructed primordial scalar power spectrum derived using Planck TT,TE,EE + lowE + lensing + BK15data and 12 knots for the cubic spline interpolation (with positions marked as GLYPH<1> at the bottom of each panel). Mean (ensemble-averaged) spectra are heavy lines, allowed GLYPH<6> 1 GLYPH<27> and GLYPH<6> 2 GLYPH<27> regions for trajectories are the shaded regions, and the dashed lines denote selected trajectories with parameters sampled within the GLYPH<6> 1 GLYPH<27> posterior. Below the scalar power is the tensor power reconstruction. The addition of the BAO likelihood shown in the middle panel makes almost no visual di GLYPH<11> erence to the reconstructions. In the bottom panel, fixing the tensor-to-scalar ratio to r = 0 : 001 also produces only small di GLYPH<11> erences in reconstruction. Knot positions in k roughly translate to multipoles through kD rec, where D rec is the comoving distance to recombination. \n<!-- image --> \nFig. 25. Reconstructed 12-knot power spectra. The robustness of the reconstruction is apparent when sub-selections of the Planck data are used: Planck TT + lowE + lensing + BK15 (top); Planck TE + lowE + lensing + BK15 (middle); and Planck EE + lowE + lensing + BK15 (bottom). \n<!-- image --> \ndata from 2015 to 2018, especially the decreased errors on GLYPH<28> , result in no non-CMB data being needed. Although GLYPH<28> and ln A s are about 90 % correlated, as they are in the standard power-law model, neither are very correlated with the qb : The strongest is about 40 % for q 3 at k ' 0 : 0006 Mpc GLYPH<0> 1 , corresponding to ' GLYPH<24> 10 where reionization is kicking in. The second strongest is about 30 % for k ' 0 : 02 Mpc GLYPH<0> 1 , similar to the correlation of GLYPH<28> and n s in the standard power-law model. (The correlations among the qb are also relatively small, except for the high k bands b = 10 and 11, where the data are not constraining.) \nThe middle panel in Fig. 24 shows the e GLYPH<11> ect of adding the BAO constraint. Although apparently visually identical, there are slight di GLYPH<11> erences. For example, the 1 GLYPH<27> error on qb at k ' 0 : 02 Mpc GLYPH<0> 1 decreases by about 7 %, from 0 : 0090 to 0 : 0084, while at k ' 0 : 1 Mpc GLYPH<0> 1 the decrease is about 5 %. The restriction to r = 0 : 001 does not change the error bars over the floating r case. At intermediate k for modes 5 to 8 the errors on qb are so close to zero that the reconstruction is quite compatible with a simple power law, corresponding to a straight line in Fig. 24. This was also a main result of the 2015 Planck reconstructions. \nThe errors on the qb grow above GLYPH<6> 0 : 1 for b = 4 as a consequence of increased cosmic variance, giving more freedom in the allowed trajectories. Unfortunately this is also the region of relevance to the TT power spectrum deficit in the ' ' 2030 range. The most significant deviation from zero occurs for q 4 at k ' 0 : 0014 Mpc GLYPH<0> 1 : GLYPH<0> 0 : 254 GLYPH<6> 0 : 127, GLYPH<0> 0 : 255 GLYPH<6> 0 : 125, and GLYPH<0> 0 : 235 GLYPH<6> 0 : 128 for the three cases. Thus the anomaly in terms of deviation from the power law of the standard model hovers at around the 2 GLYPH<27> level. More precisely, the 95 % upper confidence limits on q 4 are GLYPH<0> 0 : 011, GLYPH<0> 0 : 018, and + 0 : 017, for the respective cases. This 2 GLYPH<27> level of the anomaly was also the conclusion of the 2015 Planck reconstructions. Therefore, even though the lowk deficit is robust against the various choices for the reconstruction, we conclude that it is not statistically significant. The associated TT , TE , EE , and BB power spectra responses to the allowed primordial power variations are derived from the mode expansion, and match the D XY ' data well, in particular following the dip in TT at ' ' 20-30. \nIn Fig. 25 we show the reconstructed power spectra using only the TT, TE, and EE data in conjunction with BK15. The fixed r = 0 : 001 cases look very similar. Except at high k , the polarization data using either EE or TE also enforce a nearly uniform n s( k ) over a broad range in k , with values in excellent agreement with those obtained from TT alone, from TE and EE used in combination, and from the combined TT,TE,EE results. For example, the GLYPH<6> 0 : 0087 and GLYPH<6> 0 : 0060 1 GLYPH<27> errors at k ' 0 : 02 Mpc GLYPH<0> 1 and k ' 0 : 1 Mpc GLYPH<0> 1 , respectively, increase only slightly for TT only, to GLYPH<6> 0 : 012 and GLYPH<6> 0 : 0068, but, more significantly, to GLYPH<6> 0 : 017 and GLYPH<6> 0 : 069 with EE alone. The deficit region remains about the same, with the TT,TE,EE result for q 4 of GLYPH<0> 0 : 255 GLYPH<6> 0 : 125 quoted above changing slightly for TT alone, to GLYPH<0> 0 : 252 GLYPH<6> 0 : 130, but with no hint of an anomaly for EE alone, at GLYPH<0> 0 : 126 GLYPH<6> 0 : 460. If just the TE cross data are used, the values are closer to the TT case, namely, GLYPH<0> 0 : 232 GLYPH<6> 0 : 163, now under a 2 GLYPH<27> excursion from the tilted fiducial model. \nAs in PCI15, we can use the P R ( k ) / H 2 =GLYPH<15> and P t ( k ) = P R ( k ) ' 16 GLYPH<15> reconstructions to get an idea of the history of the acceleration of the Universe as a function of time over the significant number of e -folds of the cosmic expansion that the CMB data probe, codified by the dynamical slow-roll parameter GLYPH<15> ( k ) = GLYPH<0> ˙ H = H 2 j k = aH , considered as a function of aH , the value of the wavenumber at Hubble crossing. Results with floating r and r fixed to 0 : 001 are shown in Fig. 26. For the dynamical time variable we use k = aH for the horizontal axis for ease in \nFig. 26. Acceleration history GLYPH<15> ( k ) for reconstructed trajectories using 12 knots (marked as GLYPH<1> at the bottom of the figure), with cubic-spline interpolation and the Planck TT,TE,EE + lowE + lensing + BK15 + BAO data for the two cases of floating r and r fixed at 0.001. Sample 1 GLYPH<27> trajectories for the floating r case allow wide variability, which is naturally greatly diminished if r is fixed to r = 0 : 001. \n<!-- image --> \ncomparing with the P R ( k ) curves of Fig. 24. The wide spread in the GLYPH<15> trajectories for the floating r case is a consequence of being able to fit the n s( k ) shape by a combination of GLYPH<15> ( k ) and d ln GLYPH<15> ( k ) = d ln k . When r is small, n s( k ) is almost entirely determined by d ln GLYPH<15> = d ln k , and the GLYPH<15> ( k ) values cluster near r = 16. \nThe Hamilton-Jacobi energy constraint equation relates the potential to GLYPH<15> and H via V = 3 M 2 P H 2 (1 GLYPH<0> GLYPH<15> = 3). Figure 27 shows the reconstructed inflationary potential shapes in the region over which the allowed inflationary potentials are constrained by the data for the floating r and fixed r cases. Instead of using k for the horizontal axis, we translate into inflaton-field GLYPH<30> -space using the relation between GLYPH<30> and p GLYPH<15> , referenced to the pivot position GLYPH<30> pivot. For the vertical axis we plot ln V = V pivot, with the overall normalization V pivot removed. Its value is set by r , hence there is a distribution of constant V pivot amplitudes to superimpose if we want the total V . The radically di GLYPH<11> erent visual appearance for the floating r and fixed r cases is due to the observable k range being compressed through the smallness of GLYPH<15> into a small precisely determined field range, whereas this range has a distribution in the floating r case. One can monitor whether the shapes of the individual realizations of the potential trajectories bend upwards \n- \nFig. 27. Top: Reconstructed shape of the single-field inflaton potential from the cubic-spline power spectra mode-expansion using 12 knots and the Planck TT,TE,EE + lowE + lensing + BK15 + BAO data. Bottom: Result when r is fixed at 0.001. Instead of plotting as a function of wavenumber k we plot ln V ( GLYPH<30> ) = V pivot about a pivot field value GLYPH<30> pivot. Note that the range on the GLYPH<30> axis is quite di GLYPH<11> erent for the small r case than the floating case. The probability of local convexity evaluated at GLYPH<30> pivot is denoted as p (convex). \n<!-- image --> \nor downwards or do both, an indication of convexity. The sample trajectories shown are not exclusively convex or concave, and a measure of the probability that they are convex can be made from the ensemble. As indicated in Fig. 27 for the 12 knot case, the ensemble-averaged potentials are roughly exponential, with individual trajectories bending away from the mean, but with no strong tendency for convexity or concavity. (The roughly 50 % probability changes somewhat depending upon the combination of data used, whether TT,TE,EE or the individual data sets.) \nThe standard cosmological parameter determinations are highly robust to the addition of these spline shape degrees of freedom. The mean values change little and the error bars grow slightly, by around 10 % for ln A s, GLYPH<28> , and H 0. The largest error increase is for GLYPH<27> 8 ; with GLYPH<27> 8 = 0 : 812 GLYPH<6> 0 : 0058 becoming 0 : 814 GLYPH<6> 0 : 0096. The main conclusions of this section on GLYPH<15> and V , and P R ( k ), remain as in PCI15, but the results have been noticeably sharpened by the improvements in the Planck 2018 data sets.", '7. Search for primordial features in the Planck power spectrum': "The 'bottom-up' power spectrum reconstruction methods of the previous section are an excellent way to search for coarse features in the spectrum, but lack the resolution to detect the higher-frequency features generically predicted by various physical mechanisms [see, e.g., Chluba et al. (2015) for a review]. It is therefore useful to complement power spectrum reconstruction with a 'top-down' approach by fitting specific feature models to the data. In this section we will analyse a representative range of power spectrum templates which parameterize features in terms of a handful of new parameters. \nWith Planck 's temperature and polarization data, we have two essentially independent probes of features at our disposal and will pay particular attention to examining the consistency between the two (Miranda et al. 2015).", '7.1.1. Global oscillation models': "Periodic or quasi-periodic modulations of the power spectrum which extend over the entire observable range of wavenumbers can occur in a variety of models [cf., e.g., Danielsson (2002); Martin & Brandenberger (2003); Bozza et al. (2003); Chen (2012); Jackson & Shiu (2013)]. A general parameterization of models with a sinusoidal modulation of the primordial power spectrum reads \nP X R ( k ) = P 0 R ( k ) GLYPH<2> 1 + A X cos ( ! X GLYPH<4> X ( k ) + ' X ) GLYPH<3> ; (56) \nwhere X 2 f log ; lin ; rf g . Defining GLYPH<20> GLYPH<17> k = k GLYPH<3> , 14 we consider the logarithmic oscillation model, given by GLYPH<4> log GLYPH<17> ln GLYPH<20> , and the linear oscillation model, GLYPH<4> lin GLYPH<17> GLYPH<20> . In addition, we investigate a logarithmic model with running frequency, GLYPH<4> rf GLYPH<17> ln GLYPH<20> (1 + GLYPH<11> rf ln GLYPH<20> ). For 0 GLYPH<20> GLYPH<11> rf < GLYPH<24> 0 : 01, this is a good approximation for the scalar power spectrum in the axion monodromy model (Flauger et al. 2017a), which will be analysed in more detail below, but here we allow for a wider range of the running parameter GLYPH<11> rf , including negative values (i.e., decreasing frequency with increasing k ).", '7.1.2. Localized oscillatory features: inflation with a step': 'A sudden transient event in the evolution of the inflation field, triggered by a sharp feature in the inflaton potential, or a sharp turn in field space, generically leads to a localized oscillatory feature in the power spectrum (Adams et al. 2001; Chen et al. 2007; Ach\'ucarro et al. 2011; Miranda et al. 2012; Bartolo et al. 2013). As an example of this class of feature models, we consider here the case of a tanh-step in an otherwise smooth inflaton potential (Adams et al. 2001), whose power spectrum can be parameterized as (Miranda & Hu 2014) \nln P s R ( k ) = ln P 0 R ( k ) + I 0( k ) + ln GLYPH<16> 1 + I 2 1 ( k ) GLYPH<17> ; (57) \nwhere the first- and second-order terms are given by \nI 0 = A s W 0( k = k s) D k = k s x s ! ; (58) \nI 1 = 1 p 2 " GLYPH<25> 2 (1 GLYPH<0> n s) + A s W 1( k = k s) D k = k s x s !# ; (59) \nTable 11. Prior ranges for the parameters of the feature model templates of Sect. 7.1. \nwith window functions \nW 0( x ) = 1 2 x 4 hGLYPH<16> 18 x GLYPH<0> 6 x 3 GLYPH<17> cos 2 x + GLYPH<16> 15 x 2 GLYPH<0> 9 GLYPH<17> sin 2 x i ; (60) W 1( x ) = GLYPH<0> 3 x 4 ( x cos x GLYPH<0> sin x ) h 3 x cos x + GLYPH<16> 2 x 2 GLYPH<0> 3 GLYPH<17> sin x i ; (61) \nand damping function \nD ( x ) = x sinh x : (62) \nIn this model, the parameter A s determines the amplitude of the oscillatory feature, the step scale k s sets the position of the step in k -space, and the damping parameter x s determines the width of the envelope function.', '7.1.3. Models with suppressed power at large scales': 'The apparent lack of power at the largest scales in the temperature power spectrum with respect to the expectation of base GLYPH<3> CDM serves as a motivation for models with a suppression of primordial perturbations below a cuto GLYPH<11> scale k c . Physically, this e GLYPH<11> ect may be due to fluctuations at the largest observable scales being generated at the onset of the inflationary phase after a prior era of, e.g., kinetic or radiation domination (Vilenkin & Ford 1982; Contaldi et al. 2003), or due to an isolated event such as a kink in the inflaton potential (Starobinsky 1992). \nIn these scenarios, the primordial spectrum can generally be analytically approximated by an expression of the form \nln P Y R ( k ) = ln P 0 R ( k ) + ln GLYPH<7> Y ( k = k c Y ) ; (63) \nwith Y 2 f kin ; rad ; kink g , where GLYPH<7> Y is a function with ln GLYPH<7> Y ! 0 in the limit k GLYPH<29> k c Y that describes the shape of the cuto GLYPH<11> and the transition to a power-law spectrum at smaller scales.', 'Initial kinetic domination': 'If inflation is preceded by an era dominated by the kinetic energy of the inflaton field (i.e., fast roll), we have \nGLYPH<7> kin( y ) = GLYPH<25> 16 y j C c( y ) GLYPH<0> D c( y ) j 2 ; (64) \nwith \nwhere H (2) n denotes the Hankel function of the second kind (Contaldi et al. 2003).', 'Initial radiation domination': 'If inflation begins immediately after a radiation-dominated phase, the cuto GLYPH<11> function reads (Vilenkin & Ford 1982) \nGLYPH<7> rad( y ) = 1 4 y 4 GLYPH<12> GLYPH<12> GLYPH<12> e GLYPH<0> 2 i y (1 + 2 iy ) GLYPH<0> 1 GLYPH<0> 2 y 2 GLYPH<12> GLYPH<12> GLYPH<12> 2 : (67) \nKink in the inflaton potential (Starobinsky model) \nA kink in the inflaton potential, first discussed by Starobinsky (1992), leads to a spectrum approximately given by \nGLYPH<7> kink( y ) = 1 GLYPH<0> 3( R c GLYPH<0> 1) 1 y " 1 GLYPH<0> 1 y 2 ! sin 2 y + 2 y cos 2 y # (68) + 9 2 ( R c GLYPH<0> 1) 2 1 y 2 1 + 1 y 2 ! " 1 + 1 y 2 + 1 GLYPH<0> 1 y 2 ! cos 2 y GLYPH<0> 2 y sin 2 y # ; \nwith the parameter R c expressing the ratio of the slopes of the inflaton potential before and after the kink (Sinha & Souradeep 2006).', '7.2. Data analysis': 'We employ a modified version of CAMB with suitably increased numerical precision settings to calculate the CMB angular power spectra for the feature models. Since variations of the primordial spectrum may be degenerate with late-time cosmology parameters (Obied et al. 2017), we explore a parameter space consisting of the baseGLYPH<3> CDM parameters and the respective additional free parameters of the feature models (see Table 11 for the prior ranges). Note that we take primordial tensor perturbations to be absent in our analysis. In the results presented in Sect. 7.3, nuisance parameters are assumed to be uncorrelated with the feature parameters and kept fixed to their baseGLYPH<3> CDMbest-fit values. \nIn order to maximize sensitivity to narrow features, we use only the unbinned versions of the Planck high-\' likelihoods in the following combinations: (i) temperature data, Planck TT(unbinned) + lowE; (ii) E -polarization data only, Planck EE(unbinned) + lowE; and (iii) temperature plus polarization data, Planck TT,TE,EE(unbinned) + lowE. \nFor all combinations of feature models and data, the parameter space is sampled with the nested sampling algorithm as implemented in MultiNest . The improvement in the fit due \nC c( y ) = e GLYPH<0> i y " H (2) 0 GLYPH<18> y 2 GLYPH<19> GLYPH<0> 1 y + i ! H (2) 1 GLYPH<18> y 2 GLYPH<19> # ; (65) \nD c( y ) = e i y " H (2) 0 GLYPH<18> y 2 GLYPH<19> GLYPH<0> 1 y GLYPH<0> i ! H (2) 1 GLYPH<18> y 2 GLYPH<19> # ; (66) \nto the introduction of a feature is quantified by the e GLYPH<11> ective GLYPH<1> GLYPH<31> 2 GLYPH<17> GLYPH<0> 2(ln L best fit GLYPH<3> CDM GLYPH<0> ln L best fit feature ). Being more complex than a power-law spectrum, feature models will in general have a negative GLYPH<1> GLYPH<31> 2 . However, determining whether the improvement in fit is due to overfitting scatter in the data or due to an actual feature is not straightforward and requires model-dependent simulations (PCI15) or analytic estimates (Fergusson et al. 2015) to determine the expected GLYPH<1> GLYPH<31> 2 under the null-hypothesis of an underlying power-law spectrum. In the Bayesian approach, a feature model\'s general performance relative to base GLYPH<3> CDMcan be expressed in terms of the Bayesian evidence E (Trotta 2007a), which is also evaluated by MultiNest .', '7.3. Feature candidates and their evidence': "We list the best-fit e GLYPH<11> ective GLYPH<1> GLYPH<31> 2 and Bayes factors with respect to a power-law spectrum in Tables 12 and 13. Examining the e GLYPH<11> ective GLYPH<1> GLYPH<31> 2 for the feature models previously considered in PCI15 reveals only minor di GLYPH<11> erences, with a general trend towards smaller improvements due to features. The GLYPH<1> GLYPH<31> 2 of the oscillation and step models fall well within the expected range of GLYPH<1> GLYPH<31> 2 GLYPH<24> 10 found in PCI15. Of note are the relatively high values of the radiation and kink cuto GLYPH<11> models for polarization-only data, partially driven by the high quadrupole of the EE data. However, the best-fit parameters and spectra (see Fig. 30) do not match their counterparts in the temperature data at all, which strongly suggests that this is not a physical e GLYPH<11> ect. The same observation can also be made for the step model: the best fit to the EE data is clearly out of phase with the temperature best fit. \nWe find a similar conclusion for the three oscillation models. As can be seen from the profile likelihood of the frequency parameters in Fig. 31, the likelihood peaks in the modulation frequencies do not match up between the TT and EE data sets. Furthermore, the preferred modulation amplitude for the EE data is in all cases much larger than that for the TT or TT,TE,EE data-given that the polarization data are noisier, this behaviour would be expected for a procedure that is overfitting the data. \nConsequently, the Bayesian evidence for all combinations of models and data lies between barely worth mentioning and substantial evidence against the feature model on the Je GLYPH<11> reys scale. This implies that, currently, the Planck data do not show a preference for the feature models considered here. \nConversely, within the frequency ranges given by our priors, the relative modulation of the power spectrum is constrained to not exceed roughly 3 %, as shown in Fig. 28 for the logarithmic and linear oscillation models. \n<!-- image --> \nFig. 28. Marginalized joint 68 %, 95 %, and 99 % CL regions of the modulation amplitude versus frequency parameter using the TT,TE,EE data set for the logarithmic ( left ) and linear ( right ) oscillation models. \n<!-- image --> \n<!-- image --> \nFig. 29. Marginalized joint 68 % and 95 % CL regions for the lensing parameter A L and the modulation amplitude parameter A lin using the TT data set. Left: Linear oscillation model with log 10 ! lin = 1 : 158 and ' lin = GLYPH<25> . Right: Modified linear oscillation model with a Gaussian envelope function (see text) and log 10 ! lin = 1 : 158, ' lin = GLYPH<25> , GLYPH<22> env = 0 : 2 Mpc GLYPH<0> 1 , and GLYPH<27> env = 0 : 057 Mpc GLYPH<0> 1 : \n<!-- image --> \nIt may also be worth pointing out that in models with oscillations linear in k , the wavelength of the corresponding modulation of the angular power spectra matches that of the CMB's acoustic oscillations, GLYPH<1> ' ' 300, if log 10 ! lin ' 1 : 158. One might therefore suspect that features with frequencies around this value and carefully tuned amplitudes and phases could in principle mimic the (unphysical) e GLYPH<11> ect of a lensing parameter, A L , 1. However, for a model with a modulation at the BAO frequency and a k -independent modulation amplitude A lin, it can be seen in the left panel of Fig. 29 that we find no correlation between A L and A lin. This is due to a different ' -dependence of the respective GLYPH<1> D ' 's. Explaining the lensing discrepancy would thus require a model with a carefully arranged scale-dependent linear modulation of the primordial spectrum. We demonstrate this possibility for a shaped modulation with a Gaussian envelope of the form P R ( k ) = P 0 R ( k ) h 1 + A lin exp( GLYPH<0> ( k GLYPH<0> GLYPH<22> env) 2 = 2 GLYPH<27> 2 env ) cos ( ! lin k = k GLYPH<3> + ' lin) i in the right panel of Fig. 29, but it should be noted that this particular example is of course highly tuned to produce the desired e GLYPH<11> ect. \nAdditionally, while the phenomenology of A L and linear modulation models is similar for temperature and polarization spectra individually, the two scenarios are in principle distinguishable by a combination of temperature and polarization data. This is due to the phase di GLYPH<11> erence of the acoustic peaks in TT , TE , and EE , which leads to similar phase di GLYPH<11> erences for the residuals when varying A L-unlike modifications of the primordial spectrum which do not shift phase in the same way. However, for features with an amplitude chosen to resemble the apparent lensing excess in the Planck TT data, the Planck TE and EE data are not sensitive enough to make this distinction.", '7.4. Axion monodromy': "As in section 10.3 of PCI15, we next derive constraints on the underlying parameters in axion monodromy inflation (Silverstein & Westphal 2008; McAllister et al. 2010; Kaloper et al. 2011; Flauger et al. 2017a), which within string theory motivates a broad class of inflationary potentials of the form \nV ( GLYPH<30> ) = GLYPH<22> 4 GLYPH<0> p GLYPH<30> p + GLYPH<3> 4 0 e GLYPH<0> C 0( GLYPH<30>=GLYPH<30> 0) p GLYPH<3> cos 0 B B B B B @ GLYPH<13> 0 + GLYPH<30> 0 f GLYPH<30> GLYPH<30> 0 ! pf + 1 1 C C C C C A ; (69) \nTable 12. Best-fit e GLYPH<11> ective GLYPH<1> GLYPH<31> 2 and logarithm of the Bayes factors with respect to a featureless power spectrum, as well as best-fit feature parameters, for the step and cuto GLYPH<11> models. Negative values of ln B indicate a preference for a power-law spectrum, while positive ones prefer the feature model. Wavenumbers are in units of Mpc GLYPH<0> 1 . \nTable 13. Same as Table 12, but for the oscillatory feature models. \nwhere GLYPH<22>; GLYPH<3> 0 ; f ; and GLYPH<30> 0 are constants which have dimensions of mass, while C 0 ; p ; p GLYPH<3> ; pf , and GLYPH<13> 0 are dimensionless. In the literature, one can find theoretically motivated models with p = 3 ; 2 ; 4 = 3 ; 1 ; and 2 = 3 (Silverstein & Westphal 2008; McAllister et al. 2010, 2014). In the following, we neglect a possible amplitude drift in the modulation amplitude by fixing C 0 = p GLYPH<3> = 0, focussing instead on a possible frequency drift pf , as was done in previous analyses (Peiris et al. 2013; Easther & Flauger 2014; Jackson et al. 2014; Meerburg & Pajer 2013; Meerburg et al. 2014c,a,b). \nDue to its oscillating nature, a numerical study of this model is restrictive (Peiris et al. 2013). As such, we employ the semianalytic template (Flauger et al. 2017a) used in previous analyses, namely \nP R ( k ) = P R ( k GLYPH<3> ) k k GLYPH<3> ! n s GLYPH<0> 1 8 > > < > > : 1 + GLYPH<14> n s cos 2 6 6 6 6 6 4 GLYPH<30> 0 f GLYPH<30> k GLYPH<30> 0 ! pf + 1 + GLYPH<1> GLYPH<30> 3 7 7 7 7 7 5 9 > > = > > ; : (70) \nWe neglect the e GLYPH<11> ect of small oscillations in the tensor primordial spectrum, and approximate it as a power law with a very small spectral index n t (fixed by the single-field slow-roll self-consistency condition). The most well studied case to date is for p = 4 = 3, but given the high tensor-to-scalar ratio predicted by this model and the current upper bounds on r given in Sect. 3.5, we extend our study to the cases of p = 1 and p = 2 = 3. Furthermore, to completely specify this template, we assume instantaneous reheating, which, for a pivot scale of k GLYPH<3> = 0 : 05 Mpc GLYPH<0> 1 , corresponds to N GLYPH<3> GLYPH<25> 57 : 5, and GLYPH<30> 0 = 12 : 38 M Pl with GLYPH<30> end = 0 : 59 M Pl. This leads to definite predictions for ( r ; n s); namely, (0 : 0922 ; 0 : 971) for p = 4 = 3, (0 : 0692 ; 0 : 974) for p = 1, and (0 : 0462 ; 0 : 977) for p = 2 = 3. \nTo constrain this model, we carry out a Bayesian analysis using a modified version of CLASS (Lesgourgues 2011; Blas et al. \n2011), which has been adapted to allow for a full parameter exploration, using the aforementioned template. As part of these modifications, special care needs to be taken to ensure that a correct sampling GLYPH<1> k in wavenumber space is chosen, at two different levels in the Boltzmann code: when computing an interpolation table for the primordial spectrum of scalars and tensors; and when performing the integral over the squared photon transfer functions multiplied by the primordial spectra to get the multipoles C ' . This sampling needs to be fine enough to guarantee that no features are smoothed out or lost in this convolution, and we checked carefully that this is the case in our runs. The grid of ' values at which the C ' 's are actually computed and not just interpolated also needs to be refined. \nWe fit to the data the five cosmological parameters f ! b ; ! c ; GLYPH<18>; A s ; GLYPH<28> g plus the frequency f of the underlying axion decay constant, the frequency drift pf , and the oscillation amplitude GLYPH<14> n s. We adopt the same priors used in previous analyses: GLYPH<0> 4 GLYPH<20> log 10 ( f = M Pl) GLYPH<20> GLYPH<0> 1 for the frequency; GLYPH<0> 0 : 75 < pf < 1 for the frequency drift; and an upper bound on the amplitude of GLYPH<14> n s < 0 : 5. Furthermore, for the phase parameter GLYPH<1> GLYPH<30> we take a uniform prior of GLYPH<0> GLYPH<25> < GLYPH<1> GLYPH<30> < GLYPH<25> . \nIn Figs. 32, 33, and 34 we show the joint posterior constraints on pairs of primordial parameters for the semi-analytic template, for p = 4 = 3, p = 1, and p = 2 = 3, respectively. \nIn all three cases, we find two expected asymptotic behaviours. First, when the frequency is very high (which means that f is small in our parameterization), the oscillations in the primordial spectrum are smoothed out in the angular power spectrum, and the oscillation amplitude parameter GLYPH<14> n s becomes irrelevant and unconstrained. Second, in the limit of a very small amplitude parameter GLYPH<14> n s, the oscillations become undetectable and the parameter f is also unconstrained. In all cases, no preferred frequency drift is found, which is compatible with previous analyses. \nFig. 31. Profile likelihood of the frequency parameter in the three oscillatory feature models for TT (red curves), EE (green), and TT,TE,EE data (blue). The dotted grey line in the bottom panels marks the frequency for which the linear oscillation model leads to a modulation of the angular power spectra whose wavelength roughly matches that of the CMB's acoustic oscillations. Note the lack of alignment between the temperature and polarization likelihood peaks in the vicinity of this frequency. \n<!-- image --> \n-1 \nFig. 30. Best-fit and central 95 % CL regions for the primordial power spectrum in the three cuto GLYPH<11> and the step models for TT data (red curves), EE data (green), and TT,TE,EE data (blue). Note that for the combination of kink cuto GLYPH<11> model and TT data, the best-fit value for the cuto GLYPH<11> scale k c kink lies close to the prior boundary, and therefore the best-fit spectrum does not fall within the central 95 %-credible band. \n2 \n<!-- image --> \n10 \nX \nFig. 32. Joint 68 % and 95% CL constraints on the axion monodromy parameters using Planck (unbinned) TT,TE,EE + lowE + BK14, for the case of p = 4 = 3. All smoothing has been turned down in the pf GLYPH<0> log 10 ( f = M Pl) posterior to avoid smoothing the features highlighted in red. \n<!-- image --> \nFig. 33. Same as Fig. 32, but for the case of p = 1. \n<!-- image --> \nWe recover the complex structures (highlighted in red in Figs. 32, 33, and 34) found in previous analyses in the frequencyfrequency drift parameter space, which, as was discussed in PCI15, arise due to underlying modulations in the data and the model (Easther et al. 2005). These structures become more apparent as we reduce the index p . \nWe perform a GLYPH<31> 2 comparison with the minimal 6-parameter GLYPH<3> CDM model, and find GLYPH<1> GLYPH<31> 2 ( p = 4 = 3) = GLYPH<3> CDM = 0 : 4, GLYPH<1> GLYPH<31> 2 ( p = 1) = GLYPH<3> CDM = 0 : 6 and GLYPH<1> GLYPH<31> 2 ( p = 2 = 3) = GLYPH<3> CDM = 1 : 2. The reason for higher GLYPH<31> 2 in the axion monodromy models, despite the addition of extra parameters, is that the predicted r values are in tension with the CMB data. This shows that, overall, axion monodromy models are disfavoured due to their high tensor-mode amplitude. \nIn order to check specifically whether the data give any hint of oscillatory patterns in the primordial spectrum matching the axion monodromy template, as well as to compare with the results discussed in the previous subsection, we fitted the data with \n1.0 \nFig. 34. Same as Fig. 32, but for the case of p = 2 = 3. \n<!-- image --> \nGLYPH<3> CDM + r models in which r and n s were fixed to the same values as in the axion monodromy model with p = 2 = 3, 1, and 4 = 3. In each case, the comparison between axion monodromy and GLYPH<3> CDM + r with the same ( r , n s) gives GLYPH<1> GLYPH<31> 2 ( p = 4 = 3) = GLYPH<3> CDM + r = GLYPH<0> 7 : 8, GLYPH<1> GLYPH<31> 2 ( p = 1) = GLYPH<3> CDM + r = GLYPH<0> 7 : 6 and GLYPH<1> GLYPH<31> 2 ( p = 2 = 3) = GLYPH<3> CDM + r = GLYPH<0> 8. That is, in all cases we find GLYPH<1> GLYPH<31> 2 GLYPH<24> 10, which is compatible with the general results shown in Table 13. With three more free parameters, these improvements are statistically insignificant, and we conclude that the data show no preference for axion monodromy models.", '8.1. Approach': "This section establishes constraints on oscillatory models using the power spectrum and the bispectrum simultaneously. Oscillatory features can appear in multiple correlation functions (Chen et al. 2008; Meerburg et al. 2009; Flauger et al. 2010; Flauger & Pajer 2011; Ach'ucarro et al. 2011; Adshead et al. 2012; Ach'ucarro et al. 2014; Flauger et al. 2017a,b) (see, e.g., Chluba et al. (2015) for a recent review). More powerful constraints result when spectra of various orders are combined (Palma 2015; Mooij et al. 2016; Gong & Yamaguchi 2017). Past work has suggested that the statistical weight of the oscillations in the bispectrum (or higher-order correlation functions) is less than that in the power spectrum (Behbahani et al. 2012); however, counterexamples exist as well (see, e.g., Behbahani & Green 2012). The analysis in Sect. 7 used the Planck data to establish stringent constraints on the presence of features in the power spectrum. The 2015 Planck data were analyzed to constrain non-Gaussianties containing features (Planck Collaboration XVII 2016), where, as in the power spectrum analysis, several candidate features were identified at low statistical significance. The analysis here focuses on the location, or frequency, of the feature. Joint analyses of the power spectrum and bispectrum were discussed in several studies (Fergusson et al. 2015; Fergusson et al. 2015; Meerburg et al. 2016). We apply some of the tools developed there to the Planck temperature and polarization data. \nFig. 35. Typical best-fit improvement in units of GLYPH<1> GLYPH<31> 2 in 100 simulations compared to the real data (red dashed lines) for the log feature (left) and the linear feature (right) models. \n<!-- image --> \nThe analysis here is incomplete and limited in several respects. First, we analyse the bispectrum keeping all cosmological parameters fixed. Second, the parameters varied in the bispectrum are not varied in the Bayesian sense. The bispectrum is analyzed using a best-fit analysis based on how well a template shape fits the data. Third, the data suggest the primary bispectrum is close to zero and its covariance dominated by the scalar contributions in the power spectrum. We find that an ideal Bayesian analysis is not computationally feasible (see, e.g., Verde et al. 2013). \nThe output from the bispectrum analysis for features provides us with a map that specifies the significance of a feature in units of GLYPH<27> , given the location (frequency) and the phase of the feature. We can turn this map into a likelihood, which we can simply add to that of the power spectrum; in other words, we take \nln L tot = ln L PS( c ; f ; ! P ; A P ; GLYPH<30> P j dat) + ln L BS( A B ; ! B ; GLYPH<30> B j dat) ; (71) \nwhere c represents the standard cosmological parameters, f the foregrounds, ! P ; B the frequency, A P ; B the amplitude, and GLYPH<30> P ; B the phase of the modulation in, respectively, the power spectrum and the bispectrum. We assume vanishing covariance between the power spectrum and the bispectrum, which has been shown to be a good approximation (Fergusson et al. 2015; Fergusson et al. 2015; Meerburg et al. 2016). Furthermore, the likelihood ln L BS is not normalized (more precisely ln L tot is not normalized in a universe with a non-zero bispectrum). \nStrictly speaking, we do not have a likelihood that measures A B with a certain probability. Furthermore, several studies have shown that the frequency parameter, in combination with the amplitude and the phase, does not obey a GLYPH<31> 2 fitting to the data (Hamann et al. 2010; Meerburg et al. 2014b,c,a; Meerburg 2014; Easther & Flauger 2014). Removing the frequency from the search results in a GLYPH<31> 2 distribution with two degrees of freedom. Since ln L tot is rather large (of order 10 4 when combining all data), we can change the equation above by limiting ourselves \nonly to improvements that are driven by ! B GLYPH<24> ! P GLYPH<17> ! , that is, \nln L tot = ln L PS( c ; f ; !; A P ; GLYPH<30> P j dat) + GLYPH<1> ln L BS( A B ; !; GLYPH<30> B j dat) : (72) \nAssuming that GLYPH<30> B and A B are well described by a twoparameter GLYPH<31> 2 distribution, we can now convert our GLYPH<27> map into a GLYPH<31> 2 improvement via \nGLYPH<31> 2 = GLYPH<0> 2 log h Erf GLYPH<16> GLYPH<27>= p 2 GLYPH<17> + 1 i ; (73) \nor, in terms of the likelihoods, \nGLYPH<0> 2 ln L tot = GLYPH<0> 2 ln L PS ( c ; f ; !; A P ; GLYPH<30> P j dat) + 2 log n Erf h GLYPH<27> ( !; A B ; GLYPH<30> B) = p 2 i + 1 o : (74) \nWe will use the above expression to derive the posterior of the joint fit.", '8.2. Models': 'We will focus on two models: the local or linear feature model and the log feature model. For the log model we set f A P ; !; GLYPH<30> P ; A B ; GLYPH<30> B g = f A log ; ! log ; GLYPH<30> log ; B log ; ˜ GLYPH<30> log g and use for the power spectrum \nP log( k ) = P 0( k ) " 1 + A log cos ! log log k ˜ k 0 + GLYPH<30> log !# ; (75) \nwith P 0( k ) = A s( k = k GLYPH<3> ) n s GLYPH<0> 1 . For the bispectrum we use (Chen 2010) \nB log( k 1 ; k 2 ; k 3) = B log A s k 2 1 k 2 2 k 2 3 cos ! log log X ki ˜ k 0 + ˜ GLYPH<30> log ! : (76) \nThe above parameterized spectra are examples that could be generated in axion monodromy inflation (Flauger et al. 2010; Flauger & Pajer 2011), but generally are expected to appear in models where there exists an oscillatory potential. \nFor the linear model we follow (Fergusson et al. 2015) with f A P ; ! P ; GLYPH<30> P ; A B ; GLYPH<30> B g = f A lin ; ! lin ; GLYPH<30> lin ; B lin ; ˜ GLYPH<30> lin g and write \nP lin( k ) = P 0( k ) " 1 + A lin sin 2 ! lin k ˜ k 0 + GLYPH<30> lin !# (77) \nand (Chen et al. 2007) \nB lin( k 1 ; k 2 ; k 3) = B lin A s k 2 1 k 2 2 k 2 3 cos " ! lin X ki ˜ k 0 ! + ˜ GLYPH<30> lin # : (78) \nIn both models we choose ˜ k 0 = 1 Mpc GLYPH<0> 1 , which is di GLYPH<11> erent from the choice in Sect. 7 for the linear model . As a result the linear frequencies can be related using ! lin ; Sect8 = 10 ! lin ; Sect7. The pivot scale is set to the usual value k GLYPH<3> = 0 : 05 Mpc GLYPH<0> 1 . The above parameterization is a proxy for models that contain sharp oscillatory features (Chen 2010; Hu 2011; Adshead & Hu 2014), although typically such e GLYPH<11> ects would generate decaying features, which will not be considered here. \nFig. 36. Left: Number of aligned peaks in the power spectrum and the bispectrum for the log feature model. Right: Mean improvement of those same peaks in 100 simulated bispectra combined with the unbinned high-\' likelihood. \n<!-- image --> \n<!-- image --> \nN', '8.3.1. Power spectrum': "Our analysis uses a modified version of CAMB (Lewis et al. 2000), which is capable of adaptively changing the sampling in both k and ' depending on the frequency of the feature, allowing us to scan a wide range of frequencies. As in the previous section, we use the unbinned versions of the Planck high-' likelihoods for temperature plus polarization, in combination with lensing and large-scale temperature and polarization (i.e., lowE). We compared the power spectrum results for the limited frequency range considered in Sect. 7 for the log and linear model, and find excellent agreement (sampled with Multinest ). We developed a bispectrum likelihood module based on Eq. (74) using the 2015 data analysis (Planck Collaboration XVII 2016) for both the log and linear feature models, obtained using optimal estimators following Munchmeyer et al. (2014, 2015) and Meerburg & Munchmeyer (2015). For the log model, the frequency range is set to 10 GLYPH<20> ! log GLYPH<20> 1000, while for the linear model we consider the frequency range 10 GLYPH<20> ! lin GLYPH<20> 3000. This joint analysis excludes the very low frequencies known to (weakly) correlate with cosmological parameters. The cosmological parameters are held fixed in the bispectrum analysis. We consider amplitudes 0 GLYPH<20> A log ; lin GLYPH<20> 0 : 9; the highest amplitudes will only be allowed for high frequencies where projection suppresses the power of the oscillating part in the power spectrum significantly. The phase is varied and marginalized over in the joint analysis. We use the PolyChord sampler (Handley et al. 2015a,b), which is powerful enough to include foregrounds (with n live = 512).", '8.3.2. Bispectrum': 'The bispectrum likelihood is derived from the posterior distributions generated in Planck Collaboration IX (2018). Although the linear bispectrum of Eq. (78) can easily be factorized, the log bispectrum of Eq. (76) is not of the factorized form. Using modal techniques developed by Fergusson & Shellard (2009) \nand Fergusson et al. (2010, 2012), any shape can be factorized, with a close-to-optimal estimator. The modal method converts the angular-average bispectrum into a set of factorizable orthogonal mode functions. These functions can be directly constrained using foreground-cleaned CMB maps. From these measurements, a large number of bispectra can be reconstructed and constrained by appropriately weighting the mode functions. The convergence of this method, in terms of how many mode functions are required to accurately reconstruct the shape of interest, depends on the choice of the mode functions. In the 2015 analysis, two di GLYPH<11> erent mode functions were used: a polynomial-based reconstruction; and a trigonometric-based reconstruction. The latter was developed by Munchmeyer et al. (2014, 2015) and relies on expanding around linear oscillations. The polynomialbased reconstruction is extremely powerful for most bispectra, but is non-optimal for oscillatory bispectra, which require a large number of modes (e.g., more than 2000 for ! log = 50). Trigonometric modes allow for faster convergence and provide good reconstruction for much higher frequencies, both for linear- and log-type modulations. For low frequencies, both methods can be compared and results show excellent agreement (Planck Collaboration XVII 2016). In addition, both methods were developed independently, which provides further confidence in the results. In the analysis presented here, we use the results obtained using the trigonometric mode functions. Further details can be found in Planck Collaboration IX (2018).', '8.4. Estimating significance': "Next we will estimate the significance of the improvements driven by the joint analysis. For this purpose we generate 100 mock spectra as in Meerburg et al. (2016) without features and perform an analysis jointly with true CMB power spectrum data, i.e., we use the same power spectrum likelihood (real data) in combination with the simulated bispectrum likelihoods (mock data). We will do this for both the linear and the log models, with 100 simulations in total. Each analysis requires a similar amount of time as does the real data analysis, using about 12 000 CPU hours for the linear feature and about 40 000 CPU hours for the log feature per simulation . More details on the simulated spectra can be found in (Planck Collaboration IX 2018). \nThese simulations help us assess the statistical significance of our results. Improvement in fit is given in units of GLYPH<31> 2 compared to a no-feature model as defined the previous section [i.e., GLYPH<1> GLYPH<31> 2 ' GLYPH<0> 2(ln L best fit GLYPH<3> CDM GLYPH<0> ln L best fit feature )]. The left panel of Fig. 35 shows the typical best-fit improvement from a set of simulations for the log feature model. This first analysis shows that the best fit in the data is perfectly consistent with a standard GLYPH<3> CDM universe, without features, with P ( GLYPH<1> GLYPH<31> 2 GLYPH<21> GLYPH<1> GLYPH<31> 2 data ) = 28 %. This outcome is not unexpected, given earlier analyses for the power spectrum (see, e.g., Meerburg et al. 2014c; Easther & Flauger 2014; Benetti 2013; Miranda & Hu 2014; Fergusson et al. 2015; Planck Collaboration XX 2016; Hazra et al. 2016) and the significance of features in the bispectrum alone (Planck Collaboration XIII 2016). The lookelsewhere e GLYPH<11> ect lowers the significance of features and by jointly constraining features in the power spectrum and the bispectrum it is possible to alleviate some of this suppression. To quantify this, we consider the following two questions: 1) considering the various frequencies with GLYPH<1> GLYPH<31> 2 improvement over no features in the joint analysis, how many of those were present in the power spectrum analysis only; and 2) what is the mean improvement, in units of GLYPH<1> GLYPH<31> 2 , of these fits? We compare the results \nof the simulations, which do not contain any real features, to the data. \nBefore we answer these questions we need a criterion to decide if two frequencies will be considered the same or not, i.e., we need a frequency correlation measure. We will consider a simple ansatz, which will have an analytical solution and will serve to estimate the correlation between frequencies, by defining \nf log[ ! 1 ; ! 2 ; GLYPH<30> ] ' Z dx cos[ ! 1 log x + GLYPH<30> ] cos[ ! 2 log x + GLYPH<30> ] : (79) \nNext we marginalize over phase, defining \ng log[ ! 1 ; ! 2] = Z d GLYPH<30> f log[ ! 1 ; ! 2 ; GLYPH<30> ] = GLYPH<25> 1 + GLYPH<1> ! 2 12 GLYPH<8> x max cos GLYPH<2> GLYPH<1> ! 12 log ( x max) GLYPH<3> GLYPH<0> cos GLYPH<2> GLYPH<1> ! 12 log ( x min) GLYPH<3> x min + GLYPH<1> ! 12 GLYPH<2> sin GLYPH<2> GLYPH<1> ! 12 log ( x max) GLYPH<3> x max GLYPH<0> sin GLYPH<2> GLYPH<1> ! 12 log ( x min) GLYPH<3> x min GLYPH<3>GLYPH<9> ; (80) \nwhere GLYPH<1> ! 12 = ! 1 GLYPH<0> ! 2. For linear oscillations we can derive a similar measure, with \ng lin[ ! 1 ; ! 2] = GLYPH<25> 2 GLYPH<1> ! 12 [sin (2 GLYPH<1> ! 12 x max) GLYPH<0> sin (2 GLYPH<1> ! 12 x min)] : (81) \nThe correlation is given by \nCor( ! 1 ; ! 2) = g [ ! 1 ; ! 2] p g [ ! 1 ; ! 1] g [ ! 2 ; ! 2] : (82) \nThe parameters x min and x max play a role in determining the correlation length. Although strictly speaking they correspond to the minimum and maximum scales observable in the CMB, they can be used to model the correlator to allow for shifts in the frequency coming from a non-optimal analysis. We argue that this is reasonable given the low number of peaks in the analysis. We tested the above on various nearby peaks in the data and found that demanding Cor( ! 1 ; ! 2) GLYPH<20> 0 : 1 is generally su GLYPH<14> cient to e GLYPH<11> ectively identify independent peaks. We explored the sensitivity of the results to the correlation criterion of 0.1. First we increased it to 0.3 and found that in this case many peaks were missed when counting the number of aligned peaks (with little e GLYPH<11> ect on determining the peaks). When we lowered the criterion to 0.01, we obtained many aligned peaks that should not be aligned. Small changes in the correlation criterion have minimal e GLYPH<11> ect on the results presented here. Ideally, more simulations should be generated, which would help to establish the best choice for the correlation criterion. We found that the choice of x min does not a GLYPH<11> ect the correlator as long as x min GLYPH<28> 1. We set x max = 0 : 05 for linear oscillations, which roughly correlates peaks with GLYPH<1> ! lin GLYPH<24> 10, which is within the tails of the observed widths of the peaks in the power and bispectrum analysis. For log oscillations we choose x max = 1, which has the advantage that the correlator has no zero-crossings near the peak (but hardly e GLYPH<11> ects the correlation length). We find GLYPH<1> ! log GLYPH<24> 1 at ! log = 100, which seems reasonable in light of the power and bispectrum peak widths (see Fig. 31). \nIn Fig. 36 we show the number of peaks in the joint analysis that have improved (left panel) as well as their mean improvement (right) over a no-feature analysis. We find P (#peaks GLYPH<21> \nFig. 37. Left: Number of aligned peaks in the power spectrum and the bispectrum for the linear feature model. Right: Mean improvement of those same peaks in 100 simulated bispectra combined with the unbinned high-' likelihood. \n<!-- image --> \n0 \n5 \n10 \n15 \n20 \n25 \n30 \n2 \nΔχ \n#peaks data ) = 16 % and those peaks do not lead to significant improvements in the joint GLYPH<31> 2 , with P ( GLYPH<1> GLYPH<31> 2 peaks GLYPH<21> GLYPH<1> GLYPH<31> 2 peaks ; data ) = 83 %. Assuming that these 100 simulations provide a fair sample of the noisy data, we conclude that there are no significant features present in both the power spectrum and the bispectrum for the models considered within the chosen range of feature parameters. \nWe carry out the same analysis for linear features and show the results in Fig. 35 (right panel), deriving a typical best fit from 100 simulated noisy spectra. The true best fit, as derived from the joint analysis of the 2018 bispectrum and the 2018 power spectrum, shows a relatively small improvement with P ( GLYPH<1> GLYPH<31> 2 GLYPH<21> GLYPH<1> GLYPH<31> 2 data ) = 85 %. Further correlated-features searches find 2 features within the frequency window which may be considered aligned, with a mean GLYPH<1> GLYPH<31> 2 of 12.3, as illustrated in Fig. 37. Compared to 100 simulated noisy spectra, we obtain P (#peaks GLYPH<21> #peaks data ) = 42 % and P ( GLYPH<1> GLYPH<31> 2 peaks GLYPH<21> GLYPH<1> GLYPH<31> 2 peaks ; data ) = 26 %. Since the overall improvement from fitting these aligned peaks does not exceed the 3 GLYPH<27> threshold, we conclude that there is no statistically significant evidence for any of these features. \nWe conclude that the simple parameterization considered in this analysis does not provide any evidence for features. The two models analyzed are representive of a broad class and have well-studied phenomenological spectra; however, other classes of models exist. Features, for example, can have scale dependence [i.e., an 'envelope' (Chen 2010; Achucarro et al. 2014; Torrado et al. 2017)]. Likewise, more realistic modelling of axion models shows that the frequency could depend on scale [i.e., 'running' (Flauger et al. 2017a,b)]. Both these possibilities could substantially change the spectra and likely the joint analysis and significance. \nHere we have not imposed an explicit relation between the amplitude of the bispectrum and the frequency of the power spectrum. In the simplest form of axion monodromy, one has f NL = A log ! 2 log = 8. On account of the quadratic scaling with frequency and the fact that the constraint on the amplitude tends to become poorer as the frequency increases due to \nN \n25 \n20 \n15 \n10 \n5 \n0 \n2018 Analysis \nprojection (see, e.g., Fig. 32), it was already pointed out in Planck Collaboration XX (2016) that there is no evidence for this relation in the data, and with the current data set this situation remains unchanged.", '9.1. Background and modeling': "Single-field inflation with a canonical kinetic term gives rise to primordial super-Hubble comoving curvature perturbations, R . In this case the relative number densities of the various particle species are spatially constant, i.e., the perturbations are adiabatic. Typically photons are chosen as a reference species. Then adiabaticity implies that for every particle species with number density ni the quantity GLYPH<14> ( ni = n GLYPH<13> ) vanishes. However, in addition to R , multi-field inflation can stimulate isocurvature modes, I i , where at primordial times ni = n GLYPH<13> varies spatially (Linde 1985; Polarski & Starobinsky 1994; Linde & Mukhanov 1997; Garc'ıa-Bellido & Wands 1996). In this section we consider all possible non-decaying modes of this type (Bucher et al. 2000): cold dark matter density isocurvature (CDI); baryon density isocurvature (BDI); and neutrino density isocurvature (NDI) modes. For completeness, we also constrain the fourth nondecaying mode, neutrino velocity isocurvature (NVI), although there are no known mechanisms to excite it. Finally, we consider compensated isocurvature perturbations (CIP) between baryons and CDM (Grin et al. 2011a,b). In this case, opposite BDI and CDI perturbations cancel in such a way that the total matter isocurvature perturbation vanishes and there is no first-order isocurvature signal in the CMB. However, we utilize a higherorder lensing-like e GLYPH<11> ect from this mode to obtain constraints on CIP from Planck temperature and polarization power spectra. We find the most powerful power-spectra-based constraints on this mode by exploiting the cosmological information in the lowL lensing potential reconstruction in Sect. 9.5, but leave the use of Planck trispectra in constraining CIP for future work. \nAs the positions of the peaks and dips of the CMB angular power spectra in the density isocurvature models are roughly in opposite phase compared to the pure adiabatic (ADI) spectrum, the primordial CDI, BDI, and NDI modes leave a very distinctive observational imprint on the CMB, whereas the imprint of the NVI mode more closely resembles the pure ADI mode; see, e.g., figure 43 in PCI15. Prior to the detection of CMB anisotropies, studies such as Peebles & Yu (1970) and Efstathiou & Bond (1986, 1987) discussed the possibility that isocurvature perturbations were the sole source of cosmological fluctuations. However, at least after the detection of the first acoustic peak in TT , it became clear that the density isocurvature mode(s) had to be subdominant (Enqvist et al. 2000, 2002), while the adiabatic mode led to a good agreement with observations. Several prePlanck isocurvature constraints were obtained (Stompor et al. 1996; Pierpaoli et al. 1999; Langlois & Riazuelo 2000; Amendola et al. 2002; Peiris et al. 2003; Valiviita & Muhonen 2003; Bucher et al. 2004; Moodley et al. 2004; Beltran et al. 2004; Kurki-Suonio et al. 2005; Dunkley et al. 2005; Bean et al. 2006; Trotta 2007b; Keskitalo et al. 2007; Komatsu et al. 2009; Valiviita & Giannantonio 2009). \nThe mixture of curvature and isocurvature perturbations can be uncorrelated, but typically an arbitrary amount of correlation arises between them if the trajectory in field space is curved between Hubble radius exit and the end of multi-field inflation (Gordon et al. 2001). In extreme cases, such as the simplest curvaton models, there is full correlation or full anticorrelation be- \ntween R and I . In the following subsections, we start with the generic case of generally correlated adiabatic and CDI, NDI, or NVI perturbations. Then we deal with various special CDI (or BDI) cases with no correlation or full (anti)correlation. \nWe parameterize the primordial perturbations as in PCI15, following the notation described there. The primary perturbation parameters scanned by MultiNest (in addition to the four standard GLYPH<3> CDM background cosmological parameters and the Planck nuisance parameters) are the primordial abiabatic perturbation power and isocurvature perturbation power at two scales, corresponding to k 1 = k low = 0 : 002 Mpc GLYPH<0> 1 and k 2 = k high = 0 : 1 Mpc GLYPH<0> 1 , namely, P (1) RR , P (2) RR , P (1) II , P (2) II , and the correlation power between R and I at k 1, i.e., P (1) RI . We assume a powerlaw form for the adiabatic and isocurvature power spectra and denote the spectral indices that can be calculated from the primary parameters by n RR and n II . The correlation spectrum is also assumed to obey a power law, with spectral index n RI = ( n RR + n II ) = 2. Thus P (2) RI is not an independent parameter. This ensures that the correlation fraction cos GLYPH<1> = P RI = ( P RR P II ) 1 = 2 stays inside the interval ( GLYPH<0> 1 ; 1) at every k , as long as we reject any P (1) RI which does not obey this requirement. While the correlation fraction is k -independent in our modelling, the primordial isocurvature fraction GLYPH<12> iso( k ) = P II ( k ) = [ P RR ( k ) + P II ( k )] depends on k , unless n II = n RR . We also report GLYPH<12> iso at an intermediate scale, k mid = 0 : 05 Mpc GLYPH<0> 1 . \nWe do not separately quote constraints on BDI or the total matter density isocurvature (MDI), since these modes are observationally indistinguishable from the CDI case. 15 \nNumerical results for various isocurvature models and selected derived parameters are reported in Table 14, utilizing various data combinations. The table is divided into three main sections: generally correlated models (discussed in Sect. 9.2); oneisocurvature-parameter CDI models (discussed in Sects. 9.4.1 and 9.4.2); and, finally, two-isocurvature-parameter CDI models (discussed in Sects. 9.4.3, 9.4.4, and 9.4.5). For generally correlated CDI we study the stability of constraints (see Sect. 9.3) by using several di GLYPH<11> erent subsets of the Planck data: (1) only high-' TT; (2) high-' TT + lensing; (3) TT,TE,EE; and (4) TT,TE,EE + lensing. For comparison, some Planck 2015 and WMAP results are also cited. Table 14 also includes comparisons to the pure adiabatic model in terms of the di GLYPH<11> erence in the best-fit GLYPH<31> 2 and the natural logarithm of the Bayesian evidence ('model probability') ratios ln B , negative ln B being evidence against the mixed models. 16 \nI MDI = GLYPH<10> c GLYPH<10> m I CDI + GLYPH<10> b GLYPH<10> m I BDI : (83) \nAs we will see, the posteriors for GLYPH<10> c h 2 and GLYPH<10> b h 2 are insensitive to the assumed initial conditions. Thus it is a good approximation to use the mean values obtained in the generally correlated mixed adiabatic and CDI model with TT,TE,EE + lowE + lensing data, namely GLYPH<10> c = GLYPH<10> m ' 0 : 842, GLYPH<10> b = GLYPH<10> m ' 0 : 158, GLYPH<10> c = GLYPH<10> b ' 5 : 33, and ( GLYPH<10> c = GLYPH<10> b) 2 ' 28 : 4. For example, to convert our CDI upper bound on PII to a BDI bound, we should multiply the constraint by ( GLYPH<10> c = GLYPH<10> b) 2 = 28 : 4, and to convert the CDI PRI to BDI, we should multiply the constraint by GLYPH<10> c = GLYPH<10> b ' 5 : 33. If GLYPH<12> iso GLYPH<28> 1, then this also can be converted to a BDI constraint by multiplying the CDI constraint by 28 : 4. The constraint on cos GLYPH<1> will be the same for the CDI and BDI cases, since the conversion factor cancels out. \n16 The values of ln B depend on the priors. We adopt uniform priors in the range (15 ; 40) GLYPH<2> 10 GLYPH<0> 10 for the adiabatic, (0 ; 100) GLYPH<2> 10 GLYPH<0> 10 for \nTable 14. Constraints on mixed adiabatic and isocurvature models. We report 95 % CL intervals or upper bounds on the isocurvature fraction GLYPH<12> iso at three scales ( k low = 0 : 002 Mpc GLYPH<0> 1 , k mid = 0 : 050 Mpc GLYPH<0> 1 ; and k high = 0 : 100 Mpc GLYPH<0> 1 ), the scale-independent correlation fraction, cos GLYPH<1> , and the non-adiabatic contribution to the CMB temperature variance, GLYPH<11> non-adi. Here GLYPH<1> GLYPH<31> 2 is the di GLYPH<11> erence between the GLYPH<31> 2 of the best-fit mixed and pure adiabatic models. In the last column we give the di GLYPH<11> erence between the log of Bayesian evidences. (A negative ln B means that Bayesian model comparison disfavours the mixed model.) The number of extra parameters compared with GLYPH<3> CDM is denoted by GLYPH<1> n in the first column. Note that the uniform priors on the primordial powers at two scales lead to non-uniform priors on the parameters reported in this table. This is particularly significant for GLYPH<12> iso( k mid), where the prior peaks at a non-zero value. The baseline Planck 2018 TT,TE,EE + lowE + lensing results are highlighted in bold. \nP \nFig. 38. Constraints on the primordial perturbation power in generally correlated ADI + CDI (a), ADI + NDI (b), and ADI + NVI (c) models at two scales, k 1 = 0 : 002 Mpc GLYPH<0> 1 (1) and k 2 = 0 : 100 Mpc GLYPH<0> 1 (2). Note that in our modelling P (2) RI is not an independent parameter. \n<!-- image --> \nRR \nII \nII \nP \nP \n- \n<!-- image --> \nII \nRI \nFig. 39. Constraints on the primordial isocurvature fraction, GLYPH<12> iso, at k low = 0 : 002 Mpc GLYPH<0> 1 and k high = 0 : 100 Mpc GLYPH<0> 1 ; the primordial correlation fraction, cos GLYPH<1> ; the isocurvature spectral index, n II ; and the correlation spectral index, n RI = ( n RR + n II ) = 2, for the generally correlated mixed ADI + CDI model (a), for the ADI + NDI model (b), and for the ADI + NVI model (c). All these parameters are derived, and the distributions shown here result from a uniform prior on the primary parameters shown in Fig. 38. However, the e GLYPH<11> ect of the non-flat derived-parameter priors is negligible for all parameters except for n II (and n RI ) where the prior biases the distribution toward unity. Note that these spectral indices are not well constrained, since we do not have a detection of non-zero isocurvature or correlation amplitude. With a su GLYPH<14> ciently small isocurvature or correlation amplitude, an arbitrarily small or large spectral index leads to a very good fit to the data, since the model is then practically adiabatic over the range covered by the Planck data.Fig. 40. Posterior probability density of the observable nonadiabatic fraction of the CMB temperature variance, assuming a generally correlated mixed adiabatic and isocurvature model. These results used Planck TT + lowE + lensing data (dashed lines) and TT,TE,EE + lowE + lensing data (solid lines). \n<!-- image -->", '9.2. Results for generally correlated adiabatic and isocurvature modes': "This subsection explores mixed adiabatic and isocurvature models where only one isocurvature mode at a time is considered. We consider the CDI, NDI, and NVI modes using the Planck 2018 TT(,TE,EE) + lowE( + lensing) data. All five primordial perturbation power amplitudes (of which three describe the isocurvature perturbations) are free parameters. It follows that n II and n RR are independent and cos GLYPH<1> varies between GLYPH<0> 1 and + 1. The constraints for the primary perturbation parameters and the derived parameter P (2) RI are shown in Fig. 38. \nIn all three cases the Planck TT + lowE + lensing results are very similar to the previous results from the Planck 2015 TT + lowP + lensing likelihood. As expected, the lower value of GLYPH<28> preferred by the 2018 (lowE) data is reflected in the adiabatic amplitudes P (1) RR and P (2) RR . For CDI and NDI there is no significant shift in the constraints on isocurvature parameters, but we find slightly tighter constraints than in 2015. For NVI, a minor shift towards more negative correlations is observed (see the last two panels of Fig. 38c). As in 2015, adding the high-' TE,EE data significantly tightens the constraints in all three cases. \nWhen fitting the generally correlated three-isocurvatureparameter models, the Planck data are consistent with null detection, i.e., with the pure adiabatic model, GLYPH<16> P (1) II ; P (2) II GLYPH<17> = (0 ; 0) and P (1) RI = 0 (and P (2) RI = 0). The natural logarithm of the ratio of \nFigure 39 updates the 2015 Planck constraints on the derived primordial fractions and spectral indices. At large scales we find with Planck TT,TE,EE + lowE + lensing that GLYPH<12> iso( k low) < 2 : 5 % for the CDI, 7 : 4 % for the NDI, and 6 : 8 % for the NVI model, all at 95 % CL. Figure 40 shows the non-adiabatic fraction in the observed CMB temperature variance, defined as \n<!-- image --> \nRR \nFig. 41. Comparison of posterior probability density for the standard cosmological parameters in mixed adiabatic and isocurvature models (solid lines) to those in pure adiabatic GLYPH<3> CDM model (ADI, dashed green lines), using Planck TT,TE,EE + lowE + lensing data. \n<!-- image --> \nRR \n- \nFig. 42. Comparison of posterior probability density for selected cosmological parameters with Planck data and prePlanck (i.e., WMAP 9-year) data. Black lines indicate the results obtained for the generally correlated mixed CDI + ADI model, and green lines for the pure adiabatic model. \nmodel probabilities [i.e., the Bayes factor ln B = ln( P ISO = P ADI)] is below GLYPH<0> 10 : 9, corresponding to odds of less than 1 : 54 000 for all three (CDI, NDI, NVI) models. If there were an undetected subdominant isocurvature contribution to the primordial perturbations, a negative correlation between R and I would be favoured, in particular for NDI and NVI (see the last two panels of Figs. 38a,b,c). With our sign convention, this leads to a negative contribution to the Sachs-Wolfe e GLYPH<11> ect and hence reduces the amplitude of the temperature angular power spectrum at low multipoles. \nGLYPH<11> non GLYPH<0> adi = 1 GLYPH<0> ( GLYPH<1> T ) 2 RR ( ' = 2 ; 2500) ( GLYPH<1> T ) 2 tot ( ' = 2 ; 2500) ; (84) \nwhere \n( GLYPH<1> T ) 2 X ( ' = 2 ; 2500) = 2500 X ' = 2 (2 ' + 1) C TT X ;' : (85) \nThe non-adiabatic fraction j GLYPH<11> non GLYPH<0> adi j is below 1.7 % with Planck TT,TE,EE + lowE + lensing data for all three cases at 95 % CL. \nSince the Planck data do not allow a significant isocurvature contribution, the determination of standard cosmological parameters depends only very weakly on the assumed initial conditions, as seen in Fig. 41. We place this result in historical perspective in Fig. 42 (and Table 14) where the parameter determinations of the mixed CDI model and the pure adiabatic model are compared to the prePlanck constraints set by the WMAP 9year data. 17 Planck has dramatically tightened the constraint on the adiabatic spectral index. Its value is now 8 : 4 GLYPH<27> below unity \n(scale-invariance) in the pure ADI case. Allowing for generally correlated CDI reduces the significance of this detection only slightly, to 7 GLYPH<27> , whereas the WMAP 9-year data were consistent with a blue tilt as large as n RR = 1 : 06 at 95 % CL. The non-adiabatic contribution to the CMB temperature variance is constrained (about zero) 5 times more tightly than by WMAP. Finally, the allowed range for the sound horizon angle has shrunk by a factor of 10 in the CDI case, thanks to the Planck data covering more acoustic peaks beyond the first three peaks detected by WMAP.", '9.3. Role of lensing parameter A L and likelihood choices': "The small-scale primordial CDI amplitude is extremely sensitive to the details of the high-' temperature and polarization power spectra and to choices made in constructing the likelihoods. Therefore the general CDI model serves as a robustness test of the Planck data and likelihoods. We now discuss a few curious aspects related to CMB lensing and likelihoods. \nLensing smooths the peaks of the CMB power spectra. This e GLYPH<11> ect is taken into account in our theoretical predictions for the mixed adiabatic and isocurvature models by first calculating the total unlensed CMB spectra as a sum of adiabatic, isocurvature, and correlation C ' 's, and performing a similar summation for the lensing potential power spectrum (Seljak 1996; Lewis & Challinor 2006). The total lensing potential is then used to lens the total CMB spectra. Starting with the WMAP data, accounting for CMB lensing became necessary for calculating constraints on isocurvature models, as Valiviita et al. (2012) showed that there is a strong degeneracy between the lensing e GLYPH<11> ect and the CDI contribution in the generally correlated mixed models. Fixing n II = 1 or n II = n RR (as is done in the next subsection) makes this degeneracy disappear. This is because in these models the CDI contribution modifies only the low-' part of the angular power spectra. The transfer function mapping the primordial CDI mode to the TT (and EE ) angular power is suppressed by a factor ( k = k eq) GLYPH<0> 2 GLYPH<24> ( '=' eq) GLYPH<0> 2 relative to the adiabatic mode. Therefore, to be observable at high ' , the CDI mode must be blue tilted ( n II > 1). A blue-tilted CDI mode a GLYPH<11> ects the total angular power spectra in a manner somewhat similar to lensing. Since the acoustic peaks of the CDI mode have the opposite phase compared to the adiabatic mode, a CDI admixture can 'smooth' the peaks and dips of adiabatic acoustic oscillations. The NDI mode does not have precisely the opposite phase and is not damped relative to the adiabatic mode (see figure 43 in PCI15). Thus we expect a weaker impact of lensing on the primordial NDI amplitude than in the CDI case. Therefore in this subsection we explore the general CDI model as an example. \nStarting with the Planck 2013 release, the consistency of the smoothing e GLYPH<11> ect with the adiabatic GLYPH<3> CDMmodel has been routinely tested by multiplying the lensing power spectrum by a phenomenological lensing consistency parameter, A L, prior to lensing the unlensed CMB spectra (Calabrese et al. 2008). The expectation is that A L = 1. However, the Planck temperature and polarization data prefer a higher level of lensing-like smoothing ( A L > 1) than expected in the adiabatic GLYPH<3> CDM model. In the 2018 release (PCP18) we have \nA L = 1 : 243 GLYPH<6> 0 : 096 (68 % CL, Planck TT + lowE) ; (86) \nA L = 1 : 180 GLYPH<6> 0 : 065 (68 % CL, Planck TT,TE,EE + lowE). (87) \nAdding the Planck CMB lensing likelihood pulls these constraints towards A L = 1 (see also Table 15). The measurement of A L when TT,TE,EE data are included depends on the cali- \nbration of the polarization channels. This procedure and the details of the sky masks di GLYPH<11> er between the Planck baseline Plik TT,TE,EE and the alternative Planck CamSpec TT,TE,EE likelihood, as discussed in PPL18 and PCP18. The Planck CamSpec TT,TE,EE likelihood prefers a smaller value of A L than Plik , but still lying about 2 GLYPH<27> above unity. \nGiven the above motivation, we check the response of the generally correlated CDI model to the various possible choices of likelihoods available in the Planck 2018 release, and, on the other hand, we gauge how the baseline Plik likelihood reacts when allowing A L to vary. For clarity, in Fig. 43 we restrict the analysis to high-' TT,TE,EE and low-' TT,EE(,BB) data without the lensing reconstruction data, but in Table 14 we also report TT + lowE and TT + lowP results, and include Planck lensing in some cases. \nWe notice a considerable variation in the constraints on the isocurvature power at high k , P (2) II , which corresponds to the high-' region in the observed power spectra. In Fig. 43 we can compare the red dashed reference contours (obtained with the Planck baseline Plik TT,TE,EE + lowE likelihood) for the CDI model (where A L = 1) with the solid black contours (obtained with the same data) for the CDI + A L model, where A L is allowed to vary. In many cases, adding an extra free parameter is expected to weaken the constraints on the other parameters, but in this case adding A L tightens the 95 % CL constraint on P (2) II by a factor of 2 : 5 from 28 : 6 GLYPH<2> 10 GLYPH<0> 10 to 11 : 4 GLYPH<2> 10 GLYPH<0> 10 . This is reflected in the derived primordial isocurvature fraction GLYPH<12> iso( k high), whose upper bound changes from 0 : 58 to 0 : 36, according to Table 14. Therefore, we conclude that, when A L = 1, the CDI mode partially accounts for the extra lensing-like smoothing effect required by the Planck TT(,TE,EE) data. Once we allow the lensing amplitude to vary, there is not much need for the CDI contribution at high ' , which should be kept in mind when interpreting the results. In Table 14 we report four cases where A L is allowed to vary. In all the other cases we have fixed A L = 1. In these cases the constraints at high k are 'conservative', i.e., weaker than the Planck data were expected to be capable of (Finelli et al. 2018), due to CDI (or NDI) partially fitting the lensing anomaly. Furthermore, again comparing the red dashed and black solid contours, we observe a slight weakening of the constraint for P (1) II in the CDI + A L model. This is due to the rigidity of the assumed power-law spectrum. When the high-' data allow much less CDI, the low-' (lowk ) CDI amplitude can be larger without much a GLYPH<11> ecting the middle-' range of the CMB power spectra between the first and third acoustic peaks, which is the most sensitive region to departures from adiabaticity. (This is the same see-saw e GLYPH<11> ect discussed in the end of Sect. 3.6 in the case of tensor perturbations.) Finally, in the GLYPH<16> P (1) RR ; P (2) RR GLYPH<17> panel we see a minor shift toward smaller amplitudes, which is an indication of the well-known degeneracy between A L and the overall primordial perturbation amplitude. \nFrom Table 14 it is obvious that adding the lensing data reduces the di GLYPH<11> erences discussed above. This is again as expected, since the lensing data favour values of A L only mildly above unity. For example, with Planck TT,TE,EE + lowE + lensing we obtain GLYPH<12> iso( k high) < 0 : 49 for CDI + A L and 0 : 47 for the CDI model. \nWenowproceed to a comparison of the likelihoods. The grey shaded contours in Fig. 43 indicate the results for the same CDI model as the red dashed contours, but changing the low-' likelihood from the combination Commander TT + SimAll EE to the LFI 70 GHz pixel-based low T,E,B, which is by its methodology and construction very similar to the 2015 baseline low-' \nP \nFig. 43. Comparison of the e GLYPH<11> ect of Planck 2018 likelihood choices and phenomenological lensing amplitude, A L, on the constraints on the generally correlated mixed adiabatic and CDI model. The reference case, indicated by the red dashed curves, is for the 2018 baseline Plik high-' likelihood supplemented by the low-' Commander TT likelihood and the low-' SimAll EE likelihood. This combination is the same as in Fig. 39a (red curves), except now without the lensing likelihood, which to some extent hides the di GLYPH<11> erences between other likelihoods and the e GLYPH<11> ect of A L. Black solid contours show the results using the same likelihood as in the reference case, but now for the mixed adiabatic and CDI model when simultaneously allowing A L to vary. The remaining two curves are for the mixed adiabatic and CDI model (with A L = 1), but now changing the low-' likelihood from Commander TT + SimAll EE to LFI 70 GHz T,E,B (grey), or high-' likelihood from Plik to CamSpec (blue). \n<!-- image --> \nRR \nP \nII \nP \nII \nlikelihood (dotted contours). Indeed, this can be seen in the results: most of the isocurvature parameters follow more closely the 2015 results with this likelihood combination than with the 2018 baseline. This implies that when it comes to isocurvature, not much has changed in high-' TT. 2018 lowP favours slightly smaller values of the optical depth GLYPH<28> than the 2015 version, hence the small shift towards smaller values of the adiabatic amplitude in the GLYPH<16> P (1) RR ; P (2) RR GLYPH<17> panel. With respect to the red dashed contours, the grey contours prefer higher adiabatic amplitudes and have a long degeneracy line in the GLYPH<16> P (1) RR ; P (2) RR GLYPH<17> plane. This is due to lowP having a higher central value and larger uncertainty on GLYPH<28> . \nFinally, the blue shaded contours in Fig. 43 represent the results when using the CamSpec likelihood, to be compared to the red dashed contours obtained by the baseline Plik likelihood. All the other parameters shown are relatively stable against the high-' likelihood, but P (2) II stands out. CamSpec leads to an upper bound of 12 : 6 GLYPH<2> 10 GLYPH<0> 10 , whereas the baseline Plik result was 28 : 6 GLYPH<2> 10 GLYPH<0> 10 , or for GLYPH<12> iso( k high) 0 : 38 versus 0 : 58 at 95 % CL, according to Table 14. This di GLYPH<11> erence is not surprising, given the di GLYPH<11> erent responses of these likelihoods to A L in the adiabatic GLYPH<3> CDM + A L model, and keeping in mind the A L-CDI degener- \nacy in the CDI model. However, this di GLYPH<11> erence is not as concerning as it might appear at first sight: all the cases shown in Fig. 43 are fully consistent with zero isocurvature. It is only the upper bound that varies, with the baseline Plik likelihood and CDI model with A L = 1 leading to the most conservative (i.e., weakest or safest) upper bounds. \nFor CDI the 2015 release Planck high-' TT data favoured a negative correlation fraction but the preliminary high-' TT,TE,EE data favoured a slightly positive correlation. This was confirmed using only the high-' Plik likelihood (and a prior on GLYPH<28> ), as shown by the red curves in the top panel of Fig. 44. Including the low-' data (black curves) did not significantly alter this tension between TT and TT,TE,EE results. In the present 2018 Planck release this tension has disappeared. Both high-' TT and TT,TE,EE data lead to a correlation fraction posterior peaking at zero, as demonstrated in the bottom panel of Fig. 44. Including the low-' TT data (black dashed curve) still shifts the posterior slightly towards negative values, due to the low TT power at low multipoles in the data. \nFig. 44. Scale-independent primordial correlation fraction in the mixed adiabatic and CDI model. The black curves are with Planck high-' and low-' data, while the red ones result from using only the high-' Plik likelihood supplemented with a Gaussian prior on the optical depth. For the 2015 data (top panel) this prior was GLYPH<28> = 0 : 078 GLYPH<6> 0 : 019, whereas for the 2018 case (bottom panel) we have adopted GLYPH<28> = 0 : 055 GLYPH<6> 0 : 007 from the Planck 2018 TT,TE,EE + lowE + lensing CDI chain. \n<!-- image -->", '9.4. Specific CDI models': "In this subsection we constrain CDI models with only one or two isocurvature parameters. The two-parameter cases were not studied in the 2013 and 2015 Planck releases. \nFirst we fix n II to unity and assume no correlation between the CDI and adiabatic modes ('axion'), or we fix n II = n RR and assume full (anti)correlation between the CDI and adiabatic modes ('curvaton I / II'). These models are less sensitive to any residual systematic e GLYPH<11> ects in the high-' data (such as the determination of polarization e GLYPH<14> ciencies or foreground modeling) than the generally correlated models, since CDI now modifies the angular power spectra insignificantly at ' & 200 (see figure 43 in PCI15). As seen in the middle section of Table 14, the Bayesian evidence values for the one-parameter extensions of the adiabatic GLYPH<3> CDM model are higher than for the threeparameter extensions, but all Bayes factors fall below GLYPH<0> 5. None of the one-parameter extensions improve GLYPH<31> 2 over the adiabatic GLYPH<3> CDMmodel. The two-parameter extensions in the bottom section of Table 14 are even more strongly disfavoured, except for the uncorrelated case with free n II ('axion II'), which is actually the only model that improves the best-fit GLYPH<31> 2 by slightly more than the number of extra parameters.", "9.4.1. Uncorrelated ADI+CDI ('axion I')": "Particularly insensitive to any ' & 30 data is the 'axion I' case, since the CDI transfer function has a ( k = k eq) GLYPH<0> 2 suppression and there is no correlation component whose amplitude would be higher than that of the isocurvature alone and hence would modify the adiabatic spectrum. The 'axion I' case is achieved in our parameterization by setting P RI = 0 and P (2) II = P (1) II . \nThus the only varied isocurvature parameter is P (1) II . This uncorrelated case with n II = 1 is a good approximation for many multi-field inflationary models where the slow-roll parameter (in the isocurvature field perturbation direction) GLYPH<17> ss is negligible and the background trajectory in field space is straight between Hubble radius exit and the end of inflation. The predictions for the spectral indices (to first order in the slow-roll parameters) are n RR = 1 GLYPH<0> 6 GLYPH<15> + 2 GLYPH<17>GLYPH<27>GLYPH<27> and n II = 1 GLYPH<0> 2 GLYPH<15> + 2 GLYPH<17> ss , where GLYPH<15> GLYPH<21> 0 and GLYPH<17>GLYPH<27>GLYPH<27> is the second slow-roll parameter in the 'adiabatic' direction (i.e., along the trajectory) in the field space. (An exact match with our model would require GLYPH<17> ss = GLYPH<15> .) The axion model [see, e.g., a recent review by Marsh (2016) and references therein], which was originally proposed to solve the strong CP problem and provides a dark matter candidate, can produce this type of isocurvature modes with n II ' 1 under the following assumptions (PCI13; PCI15): the Peccei-Quinn symmetry should be broken before inflation; it should not be restored by quantum fluctuations of the inflaton nor by thermal fluctuations when the Universe reheats; and axions produced through the misalignment angle should form a significant fraction of the dark matter. \nTable 14 indicates a slight tightening of the 'axion I' constraints using TT + lowE + lensing with respect to 2015 TT + lowP + lensing. This is due to the change of the baseline low-' data from the 2015 LFI 70-GHz pixel-based T,E,B to the 2018 combination of Commander TT and SimAll EE, which in the generally correlated cases also gave tighter constraints at low k . As expected, the addition of high-' TE,EE data only marginally improves the constraints, since the standard (non-isocurvature) parameters are better constrained now. For n II = 1 uncorrelated CDI, we obtain \nGLYPH<12> iso( k mid) < 0 : 038 0 GLYPH<20> GLYPH<11> non GLYPH<0> adi < 1 : 55 % ) (95 % CL, Planck TT,TE,EE + lowE + lensing). (88) \nUsing equation (73) of PCI13, we convert the constraint on the primordial isocurvature fraction to a bound on the inflationary energy scale. If all the dark matter is in axions, the above GLYPH<12> iso( k mid) constraint corresponds to the same limit we quoted in 2015, that is, \nH inf < 0 : 86 GLYPH<2> 10 7 GeV fa 10 11 GeV ! 0 : 408 (95 % CL) ; (89) \nwhere H inf is the expansion rate at Hubble radius exit of the scale corresponding to k mid and fa is the Peccei-Quinn symmetrybreaking energy scale.", "9.4.2. Fully (anti)correlated ADI+CDI ('curvatonI/II')": "If n II = n RR , the low-' data are maximally sensitive to the fully correlated isocurvature perturbations. In this case the correlation component is a geometric average of the adiabatic and isocurvature components, and hence much larger than the isocurvature component alone. We achieve this case in our parameterization by setting P (2) II = GLYPH<16> P (2) RR = P (1) RR GLYPH<17> P (1) II and P (1) RI = GLYPH<6> GLYPH<16> P (1) RR P (1) II GLYPH<17> 1 = 2 , i.e., cos GLYPH<1> = GLYPH<6> 1. The only isocurvature parameter to be varied is again P (1) II . Since n II = n RR , the derived isocurvature fraction GLYPH<12> iso is independent of k . A physically motivated example of this type of model is the simplest curvaton model, where a light scalar field GLYPH<31> that is subdominant (and hence irrelevant for the inflationary dynamics) starts to oscillate at the bottom of its potential after the end of inflation, causing \nits average energy density to evolve like non-relativistic matter. Once fully (or almost fully) dominating the energy density of the Universe, this curvaton field decays either to CDM or to other species (Mollerach 1990; Linde & Mukhanov 1997; Enqvist & Sloth 2002; Moroi & Takahashi 2001; Lyth & Wands 2002; Bartolo & Liddle 2002; Lyth et al. 2003). The amount of isocurvature and non-Gaussianity present after curvaton decay depends on the 'curvaton decay fraction,' rD = 3¯ GLYPH<26>GLYPH<31> = (3¯ GLYPH<26>GLYPH<31> + 4¯ GLYPH<26> radiation), evaluated at curvaton decay time. Under a number of (very) restrictive assumptions discussed in PCI15, the curvaton model can lead to fully (anti)correlated CDI (or BDI) and adiabatic perturbations. \nNot surprisingly, both in the fully correlated and anticorrelated cases, the constraint on GLYPH<12> iso is much (about 40 times) stronger than in the uncorrelated case. At 95 % CL, Planck TT,TE,EE + lowE + lensing leads to \nGLYPH<12> iso < 0 : 00095 (Fully correlated, 'curvaton I') ; (90) \nGLYPH<12> iso < 0 : 00107 (Fully anticorrelated, 'curvaton II'), (91) \nboth rounded to 0 : 001 in Table 14. As in 2015, the TT data favour anticorrelation, due to the low power in the low-' temperature compared to the expectation of the adiabatic GLYPH<3> CDM model. But when the TE,EE data (which do not particularly favour negative correlation) are added, a very tight (one part per thousand) constraint on the primordial isocurvature fraction results. \nFully correlated perturbations are obtained, e.g., in case 4 described in Gordon & Lewis (2003). Many models giving anticorrelation produce too large an isocurvature fraction to be consistent with the above limit, but case 9 of Gordon & Lewis (2003) survives. After the curvaton decay, the primordial isocurvature fraction in these models will be GLYPH<12> iso ' 9(1 GLYPH<0> ˜ r ) 2 = [ r 2 D + 9(1 GLYPH<0> ˜ r ) 2 ], where ˜ r = rD for the fully correlated CDI case and ˜ r = rD = R c GLYPH<21> 1 for the fully anticorrelated CDI case, and R c = ¯ GLYPH<26> c = ( ¯ GLYPH<26> c + ¯ GLYPH<26> b) is the CDMfraction of the total non-relativistic matter. \nOn the other hand, the nonlinearity parameter describing non-Gaussianity is (Sasaki et al. 2006) \nf local NL = GLYPH<16> 1 + GLYPH<1> 2 s GLYPH<17> 5 4 rD GLYPH<0> 5 3 GLYPH<0> 5 rD 6 ; (92) \nwhere GLYPH<1> 2 s = h GLYPH<14>GLYPH<31> 2 i s = ¯ GLYPH<31> 2 is the small-scale variance of the curvaton perturbations, or the ratio of the energy density carried by the curvaton particles to the energy density of the curvaton field (if there is significant production of curvaton particles). The parameter f local NL cannot be smaller than GLYPH<0> 5 = 4, which is obtained when rD = 1 and GLYPH<1> 2 s = 0, as implicitly assumed, for example, in Bartolo et al. (2004a,b). The above GLYPH<12> iso limits correspond to the following rD and f local NL constraints (assuming GLYPH<1> 2 s = 0): 18 \n0 : 98982 < rD GLYPH<20> 1 ('curvaton I') )GLYPH<0> 1 : 2500 GLYPH<20> f local NL < GLYPH<0> 1 : 2287 ; (93) 0 : 84347 GLYPH<20> rD < 0 : 85129 ('curvaton II') )GLYPH<0> 0 : 9077 < f local GLYPH<20> GLYPH<0> 0 : 8876 : (94) \nNL \nEven with the maximal allowed isocurvature fraction, the local non-Gaussianity in the curvaton model is well within the observational Planck limits presented in Planck Collaboration IX \n(2018). The residual isocurvature peturbations in the two studied curvaton models set much tighter constraints on the curvaton decay fraction than do constraints on the observed (consistent with zero) non-Gaussianity.", "9.4.3. Uncorrelated ADI+CDI with free n II ('axion II')": "Axion models do not necessarily produce nearly scale-invariant isocurvature perturbations. In particular, even highly bluetilted spectra (in the observable CMB range) are possible. For example, Kasuya & Kawasaki (2009) construct a model with n II = 2-4. This motivates studying a two-isocurvatureparameter model, where adiabatic and isocurvature modes are uncorrelated, but the isocurvature fraction and spectral index are free to vary. In our parameterization this is achieved by setting P (1) RI = 0, and varying P (1) II and P (2) II independently. The results for this model are presented in the first two rows of the third section of Table 14. The low-' temperature data do not favour any extra contribution beyond the (already too high) abiabatic contribution, whereas the fit to the high-' temperature and polarization data can be improved slightly by the 'smoothing' caused by the CDI mode. This leads to a very blue isocurvature spectrum. Planck TT,TE,EE + lowE + lensing gives at 95 % CL 1 : 55 < n II < 3 : 67, consistent with the recent findings of Chung & Upadhye (2017). Even the very large upper bound GLYPH<12> iso( k high) < 77 % corresponds to a contribution of less than order 1% to the observable CMB TT (or EE ) power spectra at ' ' 1400. The uncertainty in the Planck TT spectrum at these high multipoles is GLYPH<1> D TT ' GLYPH<24> 10 GLYPH<22> K 2 and the actual spectrum is D TT ' GLYPH<24> 1000 GLYPH<22> K 2 . Thus the allowed CDI contribution is only of the same 1 % order as the observable uncertainty. Consequently the non-adiabatic contribution to the observed CMB temperature variance, GLYPH<11> non GLYPH<0> adi, is also vanishingly small, between 7 GLYPH<2> 10 GLYPH<0> 4 and 7 GLYPH<2> 10 GLYPH<0> 3 .", "9.4.4. Arbitrarily correlated ADI+CDI with n II = n RR ('curvaton III')": "Apart from the extremes of GLYPH<6> 100 % correlation, some curvaton models predict an arbitrary degree of correlation. The generic feature of most curvaton models is that the isocurvature and adiabatic spectral indices are equal. This is because both perturbations typically arise from the same source. In the nextto-simplest models, the correlation fraction can be written as cos GLYPH<1> = p GLYPH<21>= (1 + GLYPH<21> ), where GLYPH<21> = (8 = 9) r 2 D GLYPH<15> GLYPH<3> ( M Pl = ¯ GLYPH<31> GLYPH<3> ) 2 . Therefore, the model is fully correlated only if GLYPH<21> GLYPH<29> 1, in which case the results of 'curvaton I' apply. If the slow-roll parameter GLYPH<15> GLYPH<3> is very close to zero or the curvaton field value ¯ GLYPH<31> GLYPH<3> is large compared to the Planck mass, this model leads to almost uncorrelated perturbations and the constraints are well approximated by 'axion I.' Any other case leads to an arbitrary degree of positive correlation between the CDI and adiabatic modes. \nModulated reheating with thermal or non-thermal production of gravitinos can lead to positive or negative correlation, respectively (Takahashi et al. 2009). While the correlation could in principle be arbitrarily large, the observational constrains on GLYPH<12> iso favour only small correlations. \nArbitrarily correlated ADI + CDI with n II = n RR is also a good approximation for those two-field (or multi-field) slowroll models [e.g., double quadratic inflation (Langlois 1999; Beltr'an et al. 2005)] where the trajectory in field space is curved between the Hubble radius exit of perturbations during inflation and the end of inflation. The fraction of isocurvature per- \nurbations converted to adiabatic depends on how the trajectory is curved and this part of the adiabatic perturbations will be fully (anti)correlated with the isocurvature modes, whereas the adiabatic perturbations already present at Hubble radius exit are uncorrelated with isocurvature modes to first order in the slow-roll parameters, and only slightly correlated to second order. [See, e.g., Gordon et al. (2001), Amendola et al. (2002), van Tent (2004), and Byrnes & Wands (2006).] The result is a non-zero correlation between isocurvature and total adiabatic perturbations. The spectral indices of both components are typically 1 GLYPH<0> O (slow-roll parameters), which is well approximated by n II = n RR since the data indicate n RR ' 0 : 965. \nAs expected, the Planck data favour negative correlations, since these n II = n RR models modify only the low-' part of the CMB spectra, where TT power is lower than predicted by the adiabatic GLYPH<3> CDM model. With TT,TE,EE + lowE + lensing we find, at 95 % CL, GLYPH<12> iso < 0 : 039 and GLYPH<0> 0 : 41 < cos GLYPH<1> < 0 : 31.", '9.4.5. Fully (anti)correlated ADI+CDI with free n II': "The remaining two-parameter CDI extensions of the adiabatic GLYPH<3> CDM model are those where the perturbations are fully (anti)correlated, as in the simplest curvaton models, but the isocurvature spectral index is not fixed to the adiabatic one. In this case the free isocurvature parameters are P (1) II and P (2) II , while P (1) RI = GLYPH<6> GLYPH<16> P (1) RR P (1) II GLYPH<17> 1 = 2 . These models are somewhat difficult to motivate, since full (anti)correlation typically implies that the curvature and isocurvature perturbations have their origin in (the decay products of) the same field. Then one would expect equal spectral indices, as in the curvaton model. The conversion of isocurvature perturbations to adiabatic ones (e.g., between Hubble radius exit and the end of inflation, or by curvatontype decay, or by reheating / thermalization) should be scale dependent in order to obtain n II , n RR . Slow-roll two-field inflation leads to an exact match, n II = n RR , in the case where cos 2 GLYPH<1> = 1. [See, e.g., Byrnes & Wands (2006)]. Nevertheless, for completeness we report constraints on these phenomenological models in the last four rows of Table 14. Since the low-' TT data favour negative correlation, a larger isocurvature fraction is allowed in the fully anticorrelated case at low k . This leads to scale-invariant isocurvature perturbations being in the favoured region of parameter space, namely GLYPH<0> 0 : 28 < n II < 1 : 86 with TT,TE,EE + lowE + lensing at 95 % CL. In contrast, in the fully correlated case the low-' TT data disfavour any isocurvature contribution, and hence prefer a blue spectrum, with 1 : 37 < n II < 3 : 65.", '9.5. Compensated BDI-CDI mode': "This subsection presents constraints on uncorrelated adiabatic and scale-invariant CIP modes and discusses the strong degeneracy between the phenomenological lensing parameter A L and the CIP amplitude (Valiviita 2017). Assuming that there are no NVI or NDI perturbations, the total matter density isocurvature perturbation I MDI, given by Eq. (83), vanishes if \nI CDI = GLYPH<0> GLYPH<10> b GLYPH<10> c I BDI : (95) \nThis mode, where the anticorrelated CDI and BDI perturbations cancel even though their individual amplitudes can be large, is called a compensated baryon and cold dark matter isocurvature mode. The CIP mode does not leave a linear-order isocurvature \nFig. 45. Illustration of how the large-scale modulation of the baryon density by CIP gets converted into a smallscale 'smoothing' e GLYPH<11> ect of the temperature and polarization anisotropies. \n<!-- image --> \nsignal in the CMB or matter power spectra (Gordon & Lewis 2003), although it modifies the trispectrum (Grin et al. 2011a, 2014). However, at the next order there is a smoothing e GLYPH<11> ect on the high-' TT , TE , and EE spectra. A formal derivation can be found, for example, in Smith et al. (2017). Here we summarize the heuristic arguments of Mu˜noz et al. (2016). \nOn scales larger than the sound horizon, condition (95) is preserved until last scattering and can be written as \nGLYPH<14>GLYPH<26> c( x ) ' GLYPH<0> GLYPH<14>GLYPH<26> b( x ) : (96) \nConsequently, CIP can be described as a large scale modulation of the baryon and CDM density (Mu˜noz et al. 2016; Heinrich et al. 2016; Valiviita 2017), with \nGLYPH<10> b( ˆ n ) = [1 + GLYPH<1> ( ˆ n )] ¯ GLYPH<10> b ; GLYPH<10> c( ˆ n ) = ¯ GLYPH<10> c GLYPH<0> GLYPH<1> ( ˆ n ) ¯ GLYPH<10> b : (97) \nHere the overbar denotes an average over the whole sky and GLYPH<1> ( ˆ n ) a small perturbation about this average in the direction ˆ n , as illustrated in Fig. 45. In patches of sky where the CMB photons originate from baryon-overdense regions, the odd acoustic peaks at high-' are more pronounced relative to the even peaks compared to the patches where the photons originate from baryonunderdense regions. Averaging over the sky leads to a lensinglike smoothing of the high-' peaks. \nA convenient measure of CIP is the variance GLYPH<1> 2 rms GLYPH<17> hj GLYPH<1> (ˆ n ) j 2 i ' P I BDI I BDI . If GLYPH<1> is a Gaussian random variable, the observed angular power of TT , TE , or EE will be \nC obs ' ( ¯ GLYPH<10> b ; ¯ GLYPH<10> c ; GLYPH<28>; H 0 ; n s ; A s) (98) = 1 p 2 GLYPH<25> GLYPH<1> 2 rms Z C ' GLYPH<0> GLYPH<10> b( GLYPH<1> ) ; GLYPH<10> c( GLYPH<1> ) ; GLYPH<28>; H 0 ; n s ; A s GLYPH<1> e GLYPH<0> GLYPH<1> 2 = (2 GLYPH<1> 2 rms ) d GLYPH<1> ; \nwhere GLYPH<10> b( GLYPH<1> ) = (1 + GLYPH<1> ) ¯ GLYPH<10> b and GLYPH<10> c( GLYPH<1> ) = ¯ GLYPH<10> c GLYPH<0> ¯ GLYPH<10> b GLYPH<1> . For brevity, we will denote the power spectrum in the integrand by C ' j GLYPH<1>= GLYPH<14> . For each GLYPH<14> it can be calculated by assuming adiabatic initial conditions. Approximating the integrand by the first three terms of its Taylor series about GLYPH<1> = 0, we end up with \nC obs ' ' C ' j GLYPH<1>= 0 + 1 2 d 2 C ' d GLYPH<1> 2 GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<1>= 0 GLYPH<1> 2 rms : (99) \nIn the following we describe parameter scans where we vary the six standard (adiabatic) GLYPH<3> CDM parameters, the Planck nuisance parameters, and the CIP variance GLYPH<1> 2 rms , calling this oneparameter extension of the GLYPH<3> CDM model the ' GLYPH<3> CDM + CIP' \nFig. 46. Conservative Planck 2015 lensing data (red points), aggressive Planck 2015 lensing data (blue points with error bars), and conservative Planck 2018 lensing data (black squares in grey boxes), along with the best-fit models to the Planck data: the best-fit adiabatic GLYPH<3> CDM model to 2018 TT + lowE (green dashed line); the best-fit GLYPH<3> CDM + A L model to 2018 TT + lowE (magenta solid line, A L = 1 : 26); and the best-fit GLYPH<3> CDM + CIP model to 2018 TT,TE,EE + lowE and conservative lensing data (black solid line, GLYPH<1> 2 rms = 0 : 0036) and to 2015 TT,TE,EE + lowP and conservative lensing data (black dotted line, GLYPH<1> 2 rms = 0 : 0071). As CIP modifies only the very lowL part of the lensing power spectrum, the conservative 2015 lensing data (40 GLYPH<20> L GLYPH<20> 400) are insensitive to CIP even when GLYPH<1> 2 rms = 0 : 0071. On the other hand, the first two data points of the 2015 aggressive lensing data disfavour the large CIP amplitude (Smith et al. 2017), which gives a very good fit to all the other data. Planck 2018 conservative lensing data cover the range 8 GLYPH<20> L GLYPH<20> 400 and consequently disfavour CIP variances GLYPH<1> 2 rms & 0 : 004. \n<!-- image --> \nmodel. We evaluate the right-hand side of Eq. (99) at each point in parameter space using a finite-di GLYPH<11> erence approximation for the second derivative: \n1 2 d 2 C ' d GLYPH<1> 2 GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<1>= 0 GLYPH<1> 2 rms ' 1 2 C ' j GLYPH<1>= GLYPH<14> GLYPH<0> C ' j GLYPH<1>= 0 + 1 2 C ' j GLYPH<1>= GLYPH<0> GLYPH<14> GLYPH<14> 2 ; (100) \nwhere GLYPH<14> should be 'su GLYPH<14> ciently small.' In practice, good numerical accuracy is achieved if GLYPH<14> is of order p GLYPH<1> 2 rms . So at each point in our MultiNest scan we set GLYPH<14> = p GLYPH<1> 2 rms for the point currently under evaluation, and thus the result of Eq. (99) simplifies to \nC obs ' ' C ' GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<1>= p GLYPH<1> 2 rms + C ' GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<1>= GLYPH<0> p GLYPH<1> 2 rms 2 : (101) \nWith this method each angular power spectra evaluation takes twice as long as for the pure adiabatic case since the spectra are now an average of two spectra, resulting from di GLYPH<11> erent values of GLYPH<10> b and GLYPH<10> c. 19 \nUnlike the high-' TT , TE , and EE spectra, the highL lensing potential power spectrum is virtually una GLYPH<11> ected by CIP. Instead, CIP modifies the low multipoles of [ L ( L + 1)] 2 C GLYPH<30>GLYPH<30> L = (2 GLYPH<25> ) \nA \nFig. 47. Degeneracy between A L and GLYPH<1> 2 rms in the GLYPH<3> CDM + A L + CIP model (top panel) and constraints on GLYPH<1> 2 rms in the GLYPH<3> CDM + CIP model, where A L = 1 (bottom panel). \n<!-- image --> \nby, approximately, adding a term GLYPH<1> 2 rms GLYPH<2> ( L = 0 : 053) GLYPH<0> 2 . For details, see table II in Smith et al. (2017). As illustrated in Fig. 46, when using the Planck 2015 conservative lensing data (40 GLYPH<20> L GLYPH<20> 400) this term does not a GLYPH<11> ect the results. In contrast, the Planck 2018 conservative lensing data also contain the range 8 GLYPH<20> L < 40 and thus CIP variances GLYPH<1> 2 rms & 0 : 004 fit the first data point of the 2018 lensing power spectrum (8 GLYPH<20> L GLYPH<20> 400) worse than in GLYPH<3> CDM. However, even in this case the joint fit of the GLYPH<3> CDM + CIP model to the TT, TE, EE, and lensing data is better than that of the GLYPH<3> CDMmodel, the improvement being of the same order as for the GLYPH<3> CDM + A L model. \nThe top panel of Fig. 47 shows the A LGLYPH<1> 2 rms degeneracy in the GLYPH<3> CDM + A L + CIP model and how it can be broken by the lensing data. The value A L = 1 provides a good fit to the TT + lowE data, if GLYPH<1> 2 rms ' 0 : 016, and to the TT,TE,EE + lowE data, if GLYPH<1> 2 rms ' 0 : 010. The GLYPH<3> CDM + CIP model (where A L = 1) with GLYPH<1> 2 rms ' 0 : 008 provides a better simultaneous fit to the Planck 2015 TT,TE,EE and conservative lensing data (40 GLYPH<20> L GLYPH<20> 400) than does the GLYPH<3> CDM + A L model. When using the Planck 2018 conservative lensing data (8 GLYPH<20> L GLYPH<20> 400), the best-fit value of GLYPH<1> 2 rms decreases to 0 : 0036. This is due to the extra term / L GLYPH<0> 2 brought by CIP to the lensing power estimator, as discussed above and shown in Fig. 46. \nTable 15. Comparison of GLYPH<3> CDM + CIP, GLYPH<3> CDM + A L, and GLYPH<3> CDM + A L + CIP models with various Planck datasets, when using the baseline Plik likelihood at high ' . The first two columns ('Constraints') are the 68% CL ranges or the 95% CL upper bounds on 1000 GLYPH<1> 2 rms (highlighted in bold for GLYPH<3> CDM + CIP) and A L. The remaining columns give the best-fit pameter values, and the di GLYPH<11> erence of the best-fit GLYPH<31> 2 and the di GLYPH<11> erence of the log of the Bayesian evidence with respect to the pure adiabatic GLYPH<3> CDM model. A negative GLYPH<1> GLYPH<31> 2 means that the quoted model fits the data better than GLYPH<3> CDM,while a positive ln B means that the Bayesian model comparison favours the quoted model, when adopting the uniform priors 0 GLYPH<20> 1000 GLYPH<1> 2 rms < 75 and 0 : 3 GLYPH<20> A L GLYPH<20> 1 : 7. \nSince Planck TT,TE,EE + lowE and lensing data can be fit well by A L = 1 in the CIP model, we show in the bottom panel of Fig. 47 the one-dimensional posterior of GLYPH<1> 2 rms in the GLYPH<3> CDM + CIP model. A non-zero value of GLYPH<1> 2 rms is preferred at the 2 : 9 GLYPH<27> (2 : 5 GLYPH<27> ) level by Planck 2018 (2015) TT + lowE(lowP) data and at the 2 : 6 GLYPH<27> (1 : 8 GLYPH<27> ) level by the TT,TE,EE + lowE(lowP) data. Without lensing the 2018 data thus more strongly favour the non-zero CIP amplitude than the 2015 data, which is as we would expect, since the favoured A L value in the GLYPH<3> CDM + A L model has also increased. When using 2018 TT,TE,EE + lowE and the 2018 conservative lensing data the significance decreases to 2 : 0 GLYPH<27> , while switching to the aggressive lensing data (8 GLYPH<20> L GLYPH<20> 2048) leads to 2 : 1 GLYPH<27> . The 68 % CL ranges of GLYPH<1> 2 rms in the GLYPH<3> CDM + CIP model, obtained with the baseline high-' Plik likelihood in combination of other Planck data, are highlighted in Table 15. Replacing Plik with CamSpec (in particular CamSpec TT,TE,EE) leads to somewhat lower values, \n1000 GLYPH<1> 2 rms = 8 > > > > > > > > < > > > > > > > > : 14 : 1 + 5 : 2 GLYPH<0> 5 : 2 TT + lowE, 6 : 5 + 3 : 0 GLYPH<0> 4 : 2 TT,TE,EE + lowE, 2 : 8 + 1 : 2 GLYPH<0> 2 : 2 TT,TE,EE + lowE + lensing (conserv.), 2 : 2 + 1 : 0 GLYPH<0> 1 : 5 TT,TE,EE + lowE + lensing (aggr.), \n(102) \nand reduced significance above zero: 2 : 7 GLYPH<27> , 1 : 9 GLYPH<27> , 1 : 7 GLYPH<27> , and 1 : 9 GLYPH<27> , respectively. \nIn order to check that the preference for GLYPH<1> 2 rms > 0 or A L > 1 is not just a parameter-space volume e GLYPH<11> ect upon marginalization over other parameters, we also report in Table 15 the di GLYPH<11> erence of GLYPH<31> 2 between the best fit in extended models and the base adiabatic GLYPH<3> CDM model. With all data sets, all three extended models lead to an improvement of GLYPH<31> 2 which clearly exceeds the number of extra parameters of the model (1 for GLYPH<3> CDM + CIP and GLYPH<3> CDM + A L, and 2 for GLYPH<3> CDM + A L + CIP). Although the inclusion of lensing data reduces this improvement of fit, the GLYPH<3> CDM + CIP model gives a rather impressive GLYPH<1> GLYPH<31> 2 = GLYPH<0> 4 with Planck TT,TE,EE + lowE and aggressive lensing data. \nSince we observe a moderate preference for a non-zero CIP amplitude, it might be tempting to 'solve' the Planck lensing anomaly by using CIP. However, this explanation seems quite unlikely, since in our treatment the CIP and adiabatic perturbations should be uncorrelated with each other, whereas CDI and BDI should be fully anticorrelated (and have a few orders of magnitude larger amplitude than the adiabatic modes while keeping the perturbations nearly Gaussian). It is di GLYPH<14> cult to imagine a physical model that could lead to this situation. For example, some variants of curvaton model would naturally lead to anticorrelated CDI and BDI, but in these models there would be a correlation with the adiabatic mode too (Gordon & Lewis 2003; He et al. 2015). The above-studied compensated BDI-CDI mode falls into a similar category to NVI: it is an interesting theoretical setup, but a compelling early-Universe model for stimulating this mode has still to be discovered. \nNevertheless, the baseline Planck Plik TT,TE,EE + lowE plus conservative lensing result, GLYPH<1> 2 rms = 0 : 0037 + 0 : 0016 GLYPH<0> 0 : 0021 , is fully compatible with current complementary observations, in particular, the WMAP 95 % CL trispectrum constraint, GLYPH<1> 2 rms . 0 : 012 (Grin et al. 2014), and the upper bound, GLYPH<1> 2 rms . 0 : 006, following from the direct measurements of the variation of the baryon fraction in galaxy clusters (Holder et al. 2010; Grin et al. 2014). It will be interesting to learn what other future CMB anisotropy (Abazajian et al. 2016; Valiviita 2017; Finelli et al. 2018) and complementary measurements, such as observations of the distribution of neutral hydrogen using 21 cm absorption lines (Gordon & Pritchard 2009), BAO (Soumagnac et al. 2016, 2018), or CMB spectral distortion anisotropies (Haga et al. 2018), will tell us about the possible contribution of CIP to the primordial perturbations.", '10. Constraints on anisotropic models of inflation': 'In this section we will test specific physical models for statistical anisotropy in the primordial fluctuations. More phenomenological multipole- or map-space tests are performed in the companion paper, Planck Collaboration VII (2018). Here we update the results of the 2015 release (PCI15) with polarization and new temperature analyses. Incorporating polarization into these tests is particularly important, due to the mild statistical significance of temperature anomalies such as the dipolar asymmetry. Polarization o GLYPH<11> ers the potential to confirm or refute a physical origin for such anomalies via the measurement of independent fluctuation modes. We perform such a new test with k -space dipolar modulation models. In cases such as quadrupolar asymmetry, where no detection has been claimed with temperature, polarization o GLYPH<11> ers the prospect of tightening existing constraints. \nSome asymmetry models predict a modification to the isotropic power spectra, in addition to a dipolar or quadrupolar asymmetry. In other words, for these models, as well as non-zero o GLYPH<11> -diagonal multipole covariance elements, we ex- \npect departures in the diagonal elements relative to the standard GLYPH<3> CDM prediction. Therefore the isotropic spectra can provide independent tests of such models even using temperature data alone (Contreras et al. 2018). The curvaton dipole modulation model we examine in Sect. 10.1.1 exhibits this property, and can be constrained via its predictions for isotropic isocurvature power. Similarly, some versions of the quadrupolar modulation model we study in Sect. 10.2 modify the isotropic spectra via a monopole term. In both cases these isotropic constraints will be important in narrowing the viable parameter space.', '10.1. Dipolar asymmetry': 'A dipolar temperature power asymmetry has long been observed at the largest scales in the CMB (Eriksen et al. 2004), although its statistical significance is not high and is subject to a posteriori (look-elsewhere) corrections (Bennett et al. 2011; Planck Collaboration XXIII 2014; Planck Collaboration XVI 2016). Nevertheless, its large-scale character suggests potential links with inflationary physics and various models have been proposed to explain it. In this subsection we examine several physical models for a dipolar modulation. Some models where a generic CDM density isocurvature (CDI) or tensor component is dipole modulated have already been ruled out due to their isotropic predictions (Contreras et al. 2018), so we do not consider these further here.', '10.1.1. Curvaton model': 'First we update our 2015 study (PCI15) of a specific inflationary model for the dipolar asymmetry: namely, the modulated curvaton model of Erickcek et al. (2009). In that study we showed that that model could not explain the observed asymmetry. Here, we generalize the curvaton model to allow for a non-scale-invariant uncorrelated CDI component. In addition, we treat the power spectrum (isotropic) constraints in a fully unified way with the asymmetry likelihood. Finally, we incorporate polarization. \nThe modulated curvaton model employs a gradient in a background curvaton field to explain the observed large-scale power asymmetry. The curvaton, via coupling GLYPH<20> , produces nearly scaleinvariant CDI fluctuations, as well as a fraction, GLYPH<24> , of the adiabatic fluctuations. Both of these components will be modulated. Up to a sign, GLYPH<24> is equal to the correlation parameter, and is also a measure of the amplitude of dipolar modulation. The isocurvature fraction can be written in terms of these two parameters as \nGLYPH<12> iso = 9 GLYPH<20> 2 GLYPH<24> 1 + 9 GLYPH<20> 2 GLYPH<24> : (103) \nFull details of this model and our treatment of it can be found in Erickcek et al. (2009) and PCI15. \nUsing the dipolar asymmetry estimator from PCI15 we find the posteriors for the dipolar modulation parameters GLYPH<20> and GLYPH<24> ; the results are presented in Fig. 48 (red contours). We see that a substantial amount of asymmetry (as measured by amplitude GLYPH<24> ) can be captured by the model. This preference for asymmetry simply means that the curvaton model can explain the well-known dipolar asymmetry in temperature. However, isocurvature constraints from the power spectra via Eq. (103), which we refer to as the isotropic constraints, can provide independent information (Contreras et al. 2018). This is also shown in Fig. 48, with the blue contours. Here we see that the asymmetry and isotropic posteriors only weakly overlap, and the independent isotropic data do not support the presence of asymmetry for this \nFig. 48. Posteriors for the curvaton dipolar modulation model parameters GLYPH<20> and GLYPH<24> . Contours enclose 68 % and 95 % of the posteriors. The model can explain the well-known dipolar asymmetry: note the preference for GLYPH<24> > 0 in the asymmetry constraint (red contours and curves). However, the modulation preferred by the asymmetry constraint is reduced substantially when the isotropic constraint (blue) is added (black). The asymmetry constraint here uses SMICA , while the isotropic constraint uses Planck TT,TE,EE + lowE + lensing. Resolution is reduced at very small GLYPH<20> due to the sampling in GLYPH<12> iso. \n<!-- image --> \nmodel. No evidence for asymmetry (i.e., no preference for GLYPH<24> > 0) is present in the joint constraints, which treat the isotropic and asymmetry data as independent. In other words, we have no reason to prefer this model over base GLYPH<3> CDM.', '10.1.2. Adiabatic models': 'In the presence of a su GLYPH<14> ciently large bispectrum it is possible that a long-wavelength mode can induce a dipolar asymmetry in the two-point function across our observable volume, although such scenarios appear to require fine tuning (Byrnes et al. 2016b). Nevertheless, examples have been constructed which satisfy the Planck f NL constraints (Byrnes et al. 2016a). In this subsection we consider adiabatic models of this type, in which the isotropic power spectra agree with standard GLYPH<3> CDM, while a scale-dependent dipolar asymmetry is present in the o GLYPH<11> -diagonal multipole covariance (Contreras et al. 2017, 2018). As proposed in Contreras et al. (2017), we fit the asymmetry model parameters to the temperature data and then use those parameters to predict the asymmetry in polarization. We then compare those predictions with the Planck polarization data as a test for a physical modulation. Importantly, a position- (or k -) space model for the modulation is needed for reliable polarization predictions-it is not enough to restrict considerations to multipole space (Contreras et al. 2017). 20 \nAs discussed in detail in Contreras et al. (2017), we take a portion e R lo ( x ) of the adiabatic primordial fluctuations to be spatially linearly modulated according to \ne R lo ( x ) = R lo ( x ) 0 B B B B @ 1 + A x GLYPH<1> ˆ d r LS 1 C C C C A ; (104) \nwhere R lo ( x ) is statistically isotropic with power spectrum P lo R ( k ), A GLYPH<20> 1 and ˆ d are the amplitude and direction of modulation, respectively, and r LS is the comoving radius to last scattering. This leads, to a good approximation, to the total temperature or polarization multipole covariance \nC \' m \' 0 m 0 GLYPH<17> h a \' ma GLYPH<3> \' 0 m 0 i (105) \n= C \'GLYPH<14>\'\' 0 GLYPH<14> mm 0 + GLYPH<14> C \'\' 0 2 X M GLYPH<1> XM GLYPH<24> M \' m \' 0 m 0 ; (106) \nto first order in A . Here C \' is the usual GLYPH<3> CDManisotropy power spectrum; GLYPH<14> C \'\' 0 GLYPH<17> 2( C lo \' + C lo \' 0 ), where C lo \' is the power spectrum calculated in the usual way from P lo R ( k ); GLYPH<1> XM is the multipole decomposition of A ˆ n GLYPH<1> ˆ d ; and the GLYPH<24> M \' m \' 0 m 0 coe GLYPH<14> cients couple \' to \' GLYPH<6> 1 via \nGLYPH<24> M \' m \' 0 m 0 GLYPH<17> r 4 GLYPH<25> 3 Z Y \' 0 m 0 ( ˆ n ) Y 1 M ( ˆ n ) Y GLYPH<3> \' m ( ˆ n ) d GLYPH<10> : (107) \nIn principle the scale dependence of the asymmetry spectrum P lo R ( k ) is completely free, but here we take three phenomenological forms which are capable of producing a large-scale asymmetry with a small number of parameters. First, we consider a simple power-law modulation, \nP lo R ( k ) = P 0 R GLYPH<16> k lo 0 GLYPH<17> 0 B B B B @ k k lo 0 1 C C C C A n lo s GLYPH<0> 1 ; (108) \nwhere P 0 R ( k ) is the usual GLYPH<3> CDM spectrum, and n lo s and k lo 0 are the tilt and pivot scale of the modulation. We consider only red asymmetry tilts with n lo s GLYPH<20> n s, and choose k lo 0 = 1 : 5 GLYPH<2> 10 GLYPH<0> 4 Mpc GLYPH<0> 1 . We also consider a tanh model, defined according to \nP lo R ( k ) = 1 2 P 0 R ( k ) " 1 GLYPH<0> tanh ln k GLYPH<0> ln k c GLYPH<1> ln k !# : (109) \nThis spectrum approaches that of GLYPH<3> CDM on scales larger than k c, with a width determined by GLYPH<1> ln k . That is, scales well above the cuto GLYPH<11> k c will be modulated with amplitude A , and scales below will be unmodulated. Finally, we consider a model with a linear gradient in the scalar tilt, n s, across our volume. In this case the asymmetry spectrum can be written as \nC lo \' = GLYPH<0> GLYPH<1> n s 2 dC \' dn s ; (110) \nwith modulation amplitude GLYPH<1> n s. There will be an implicit dependence on the pivot scale k GLYPH<3> for this model. \nGiven the multipole covariance, Eq. (106), we can construct a maximum likelihood estimator for the modulation, GLYPH<1> XM . In the noise-free, full-sky case this takes the form (Moss et al. 2011; Planck Collaboration XVI 2016) \nGLYPH<1> ˆ XM = 1 4 GLYPH<27> 2 X X \' m \' 0 m 0 GLYPH<14> C \'\' 0 C \' C \' 0 GLYPH<24> M \' m \' 0 m 0 a GLYPH<3> \' m a \' 0 m 0 ; (111) \nto the same scale in E , due to the di GLYPH<11> erent T and E transfer functions (Contreras et al. 2017). \nwhere the cosmic variance of the estimator is given by \nGLYPH<27> 2 X = 12 0 B B B B B @ X \' ( \' + 1) GLYPH<14> C 2 \'\' + 1 C \' C \' + 1 1 C C C C C A GLYPH<0> 1 : (112) \nThe modifications we use to deal with realistic skies are described in detail in Planck Collaboration XVI (2016) and Contreras et al. (2017). \nTo decide whether the polarization data support the modulation model or not, we consider the quantity ˆ Oj 0, which is the ratio of the maximum likelihood for modulation model j to that of GLYPH<3> CDM (Contreras et al. 2017). In Fig. 49 we plot for the three adiabatic models histograms of ˆ Oj 0 calculated for 300 statistically isotropic polarization simulations (sharing the required TE correlation with the real T data) added to the Planck temperature data (red outlines). This indicates our expectation for ˆ Oj 0 for the scenario that the temperature asymmetry is due to a statistical fluctuation and not to a physical modulation. We also plot in Fig. 49 histograms for 300 polarization simulations modulated with the best-fit parameters from the Planck temperature data (black outlines), to represent the scenario that the asymmetry is due to a physical modulation. In both cases the polarization simulations contain realistic levels of noise for Planck . By comparing the isotropic and modulated histograms, we can see that the quantity ˆ Oj 0 can serve to distinguish the two scenarios, but only relatively weakly for Planck noise (Contreras et al. 2017). The blue lines indicate the values using the actual SMICA polarization data (the results for the other component-separation methods are similar). We see that for these models the data do not help to decide whether we have a physical modulation or not, with p -values of 43 %, 30 %, and 57 % for the power-law, tanh, and n s gradient models, respectively, relative to the isotropic simulations.', '10.2. Quadrupolar asymmetry': 'We will next explore models that predict a quadrupolar direction dependence in the primordial power spectrum. In PCI15 we found no evidence for such a modulation, but several inflationary models have been constructed which predict this e GLYPH<11> ect (Ackerman et al. 2007; Soda 2012; Tsujikawa 2014). Therefore it is important to extend those results with the improved polarization data. We now attempt to reduce the e GLYPH<11> ect of unresolved point sources using the bias-hardened estimator approach of Planck Collaboration XV (2016). In PCI15 we pointed out that some models of quadrupolar asymmetry predict a modification to the angular power spectra as well. Here we will account for such modifications in our analysis, increasing the constraining power of temperature data, in particular for tilted models with non-scale-invariant modulation spectral index. Note that independent searches relaxing our approximation of power-law spectra have also been carried out (Durakovic et al. 2018). \nWe assume a modulation of the primordial comoving curvature power spectrum of the form \nP R ( k ) = P 0 R ( k ) GLYPH<20> 1 + g ( k ) GLYPH<16> ˆ k GLYPH<1> ˆ d GLYPH<17> 2 GLYPH<21> ; (113) \nwhich can be rewritten as \nP R ( k ) = P 0 R ( k ) 2 6 6 6 6 6 4 1 + 1 3 g ( k ) + X m g 2 m ( k ) Y 2 m ( ˆ k ) 3 7 7 7 7 7 5 : (114) \nHere \ng 2 m ( k ) GLYPH<17> 8 GLYPH<25> 15 g ( k ) Y GLYPH<3> 2 m ( ˆ d ) ; (115) \n<!-- image --> \n<!-- image --> \nFig. 49. Histograms of the quantity ˆ Oj 0 for the tanh, power-law, and n s gradient modulation models using Planck temperature data combined with 300 statistically isotropic polarization simulations (red outlines) or 300 polarization simulations modulated according to the best-fit parameters from the temperature data (black). The blue lines indicate the values for the actual SMICA polarization data. A large value relative to the isotropic (red) simulations would indicate that the modulation model is preferred over GLYPH<3> CDM. \n<!-- image --> \nwith g 2 m ( k ) satisfying g 2 ; GLYPH<0> m ( k ) = ( GLYPH<0> 1) m g GLYPH<3> 2 m ( k ). We parameterize the scale dependence of the modulation as g ( k ) = g GLYPH<3> ( k = k GLYPH<3> ) q , with pivot scale k GLYPH<3> = 0 : 05 Mpc GLYPH<0> 1 . For q , 0, in addition to producing a quadrupolar modulation of the anisotropies, this model a GLYPH<11> ects the CMB isotropic power spectra via the term g ( k ) = 3 in Eq. (114). We therefore consider a joint constraint with the isotropic power spectra likelihood to improve constraints over the modulation alone. \nAs in PCI15, we obtain constraints on the modulation parameters by forming quadratic maximum-likelihood estimates, ˆ g 2 m , for the data and simulations. For this we use the componentseparated data and 300 simulations provided by NILC , SEVEM , SMICA , and COMMANDER . For brevity we only show the SMICA results. We can then compute a covariance G and likelihood as \nwhere \nL / j G j GLYPH<0> 1 = 2 exp " GLYPH<0> 1 2 M T G GLYPH<0> 1 M # ; 2 m 2 m \nM GLYPH<17> ˆ g GLYPH<0> g ( g GLYPH<3> ; ˆ d ) : (116) \nWethen evaluate the marginalized (over the angles) posterior for g GLYPH<3> . For the isotropic constraints we simply include the modulation parameters in a CosmoMC run using Planck TT,TE,EE + lowE data, and then evaluate the marginalized (over all other GLYPH<3> CDM parameters) posterior for g GLYPH<3> . \nFor q > 0 the e GLYPH<11> ect of g GLYPH<3> on the isotropic spectra occurs mainly at high \' , and is highly degenerate with n s. This degeneracy leads to slightly less stringent constraints than what one would achieve with a fixed n s. We show the marginalized posteriors for this case in the top panels of Fig. 50, where we see that the isotropic constraints are roughly comparable in strength to (and fully consistent with) the constraints from the asymmetry data. \nFor q < 0 the isotropic constraints are much more constraining than the modulation constraints, as seen in the bottom panels of Fig. 50. This is because for large scales the factor k GLYPH<3> = k can become large and a negative g GLYPH<3> will decrease isotropic power on those scales, which is compensated for by increasing A s and GLYPH<28> . Strongly negative g GLYPH<3> values are disallowed by predicting unphysical negative power spectra at low \' . Note that even the parameter ranges in which the power spectra are reduced to close \nto zero are likely beyond the perturbative regime for the models in question, and so should be approached with caution. The isotropic constraints still prefer a slightly negative g GLYPH<3> , likely due to being able to fit the power deficit at large scales. The joint constraint in this case is then greatly improved by the isotropic data. \nMinimumGLYPH<31> 2 and p values (relative to isotropic simulations) for g GLYPH<3> are presented in Table 16. The addition of polarization does not a GLYPH<11> ect the temperature results greatly. \nFinally, when allowing the completely general form of quadrupolar modulation, i.e., \nP R ( k ) = P 0 R ( k ) 2 6 6 6 6 6 4 1 + X m g 2 m ( k = k GLYPH<3> ) q Y 2 m ( ˆ k ) 3 7 7 7 7 7 5 ; (117) \nwith no restriction on the g 2 m , we present results for the quantity g 2 GLYPH<17> p P m j g 2 m j 2 = 5 in Table 17. In all cases there is no significant detection of quadrupolar modulation, as quantified by the p values. \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFig. 50. Marginalized posteriors for quadrupolar modulation parameter g GLYPH<3> , using SMICA data for the TT + EE asymmetry constraints (orange curves) and Planck TT,TE,EE + lowE for the isotropic constraints (blue curves), which probe the modification to the power spectrum via Eq. (114). Top: constraints for q = 2 and 1 (left and right, respectively). Bottom: constraints for q = GLYPH<0> 1 and GLYPH<0> 2 (left and right, respectively). Strongly negative g GLYPH<3> is suppressed for q < 0, due to the unphysical prediction of negative power. \n<!-- image --> \nTable 16. MinimumGLYPH<31> 2 g GLYPH<3> values for quadrupolar modulation, determined from the SMICA foreground-cleaned maps. Also given are p values, defined as the fraction of isotropic simulations with larger j g GLYPH<3> j than the data. The TT results use \' min = 2 and \' max = 1200, while EE uses \' min = 2 and \' max = 850. These results indicate that the data are consistent with cosmic variance in statistically isotropic skies.', '11. Conclusions': "This paper summarizes the status of cosmic inflation in light of the Planck 2018 release. The main improvements are in the Planck polarization likelihoods. The 2018 release now includes a low-' HFI polarization likelihood based on the 100- and 143GHz channels. This likelihood is now the baseline, whereas the \nPlanck 2015 likelihood was based only on the LFI 70-GHz channel data, which also have been updated in this release. Corrections for beam-leakage e GLYPH<11> ects, which had been flagged in the 2015 release as the main limitation of the TE and EE data at that time, have improved the accuracy of the high-' polarization likelihoods. Our analyses focus on the results obtained using the \nTable 17. As for Table 16, but for the quantity g 2 GLYPH<17> p P m j g 2 m j 2 = 5 for a completely general quadrupolar modulation. \nPlanck baseline likelihoods alone, but results supplemented by the BK15 likelihood (when tensors are included) and a compilation of BAO likelihoods are also given in order to help break cosmological parameter degeneracies. We summarize the main results of this paper in the form of responses to a number of key questions.", '1. What is the value of the scalar tilt?': 'Using a characterization of polarization anisotropy better at all multipoles in this release, we find that n s = 0 : 9649 GLYPH<6> 0 : 0042 at 68 % CL, including the full information provided by Planck (TT,TE,EE + lowE + lensing). The 2018 uncertainty is approximately 2 / 3 of that obtained with the Planck 2015 baseline likelihood. Importantly, this determination rules out perfect scale invariance (i.e., n s = 1) at 8 : 4 GLYPH<27> . From an inflationary perspective, this result is consistent with slow-roll inflation evolving towards a natural exit.', '2. Does n s depend on the wavelength?': 'We investigated the possibility of a running spectral index, as well as a running of the running [i.e., the next two (subleading) terms in a power series expansion of ln( P R ) in ln( k )], corresponding to non-negligible third- and fourthorder derivatives of the inflationary potential. Starting with its first 2013 cosmological release, Planck has removed any hint of a running spectral index, which had been suggested by prePlanck data and would have pointed to inflationary models beyond the slow-roll approximation. Planck 2018 sets dn s = d ln k = GLYPH<0> 0 : 005 GLYPH<6> 0 : 013 as the tightest 95 % CL constraint, when d 2 n s = d ln k 2 = 0. No hints of further extensions, such as running of the running, are found with Planck 2018 data. These results are consistent with the simplest slow-roll dynamics for the inflaton. A detection of running at the level predicted by slow-roll models will require a combination of future ambitious CMB anisotropy experiments and galaxy surveys.', '3. Is the Universe spatially flat?': "Most simple models of inflation predict a spatially flat universe, although inflationary models with a minimum degree of fine tuning producing a hyperbolic universe have been constructed. Planck has been the first experiment to constrain the spatial curvature at the percent level without any external information, thanks to the CMB lensing likelihood. Although negative values of GLYPH<10> K GLYPH<24> GLYPH<0> 0 : 01 provide a non-statistically significant improvement to the fit of Planck temperature and polarization data (compared to the minimal GLYPH<3> CDMmodel), Planck 2018 data including lensing constrain GLYPH<10> K = GLYPH<0> 0 : 011 + 0 : 013 GLYPH<0> 0 : 012 at 95 % CL. Combining with BAO data further tightens the uncertainty, constraining GLYPH<10> K to lie within 0.4 % of a flat spatial geometry (at 95 % CL). \n- 4. Are tensor modes required? \nInflationary models predict that tensor modes were also excited during the nearly exponential expansion, with a power spectrum amplitude proportional to the energy scale of inflation. Using the measurement of CMB temperature and E -mode polarization anisotropies from the quadrupole into the acoustic peak region, Planck has reduced the degeneracy between the tensor-to-scalar ratio r and n s ; establishing the bound r 0 : 002 < 0 : 10 at 95 % CL, assuming n t = GLYPH<0> r = 8 as predicted by the simplest inflationary models. When the Planck likelihood is combined with the B -mode polarization likelihood of the BICEP2-Keck Array experiment, a tight 95 % CL upper limit of r 0 : 002 < 0 : 056 is obtained, corresponding to a 95 % CL bound on the energy scale of inflation of V 1 = 4 GLYPH<3> < 1 : 6 GLYPH<2> 10 16 GeV. Planck 2018 and BK15 data also set tight bounds on gravitational waves generated in the early Universe when r and n t are varied independently, complementary to the results obtained by the direct-detection interferometers LIGO and VIRGO at much higher frequencies. \n- 5. Which inflationary models are best able to account for the data? \nStarting with the 2013 release using only a part of the data, Planck has substantially tightened the constraints on slowroll inflationary models, ruling out hybrid models with n s > 1 and power-law inflation (PCI13). In combination with the BK15 data, Planck 2018 now strongly disfavours monomial models with V ( GLYPH<30> ) / GLYPH<30> p and p > 1, natural inflation, and lowscale SUSY models. Within the representative cases studied in this paper, inflationary models such as R 2 , T and E GLYPH<11> -attractor models, D-brane inflation, and those having a potential with exponential tails provide good fits to Planck and BK15 data. We used two methods to reconstruct the inflaton potential beyond the slow-roll approximation: by Taylor expanding the inflaton potential or Hubble parameter in the observable region; and through a free-form reconstruction of the potential with cubic splines. No statistically significant detection beyond the second derivative of the potential was found, suggesting that the slow-roll approximation is adequate for the Planck 2018 likelihood in combination with the BK15 data. \n- 6. What model-independent constraints can be placed on the primordial power spectrum? \nWe reported on three di GLYPH<11> erent methods for the nonparametric reconstruction of the primordial power spectrum (penalized likelihood, a Bayesian spline reconstruction. and a method based on cubic splines). All three methods give broadly consistent results. In no case is any statistically significant evidence for a deviation from a pure power law found. The constraints on the deviations are at the fewpercent level for wavenumbers in the range 0 : 005 Mpc GLYPH<0> 1 . k . 0 : 2 Mpc GLYPH<0> 1 probed by the CMB, the precise constraint depending on the level of smoothing allowed. \n- 7. Is there evidence for features in the primordial power spectrum? \nWe explored several classes of theoretically motivated parametric models with strong departures from a power law for the primordial power spectra and tested their predictions using combinations of Planck temperature and polarization power spectra. We also carried out an analysis using bispectrum data as well. No statistically significant evidence for features was found. \n- 8. Were the primordial cosmological perturbations solely adiabatic? \nA key question is whether the primordial cosmological fluctuations consisted exclusively of adiabatic growing-mode perturbations or whether isocurvature perturbations, possibly correlated with the adiabatic mode and with each other, were also excited. The new polarization data has helped to sharpen constraints on the allowed isocurvature fraction compared to the Planck 2015 results. In correlated mixed adiabatic and isocurvature models, the 95 % CL upper bound for the non-adiabatic contribution to the observed CMB temperature variance is j GLYPH<11> non-adi j < 1.3 %, 1.7 %, and 1.7 % for CDM, neutrino density, and neutrino velocity isocurvature, respectively. For this release we also report constraints on a scale-invariant compensated baryon-CDM isocurvature mode, which is uncorrelated with the adiabatic mode. This mode would cause an additional lensing-like smoothing at high ' and modify the lensing potential at ' . 40. By using the temperature, polarization, and lensing data, we obtain the constraint GLYPH<1> 2 rms = 0 : 0037 + 0 : 0016 GLYPH<0> 0 : 0021 at 68 % CL for the variance of the baryon isocurvature density perturbation. A detection of isocurvature modes would suggest the need for a theory beyond single-field inflation, which is able to excite only one mode. \n- 9. Were the primordial fluctuations statistically isotropic? The Planck analysis has confirmed evidence at low statistical significance of anomalies in the CMB temperature anisotropies on large angular scales that are not alleviated in models with nontrivial topology or an anisotropic expansion (Planck Collaboration XVIII 2016). This motivates an exploration of inflation-based models giving such violation of statistical isotropy. We have found no statistically significant evidence in favour of a curvaton model for dipolar asymmetry (compared to the baseGLYPH<3> CDM model), nor any evidence for a quadrupolar asymmetry in the temperature or polarization anistropies. Theoretical models producing the observed temperature dipolar asymmetry make a prediction for the polarization dipolar asymmetry. We tested whether the fit to the temperature dipolar asymmetry gives a prediction for the polarization asymmetry consistent with the data. Wefound no statistically significant evidence that the pattern seen in temperature is repeated in polarization. However, the discriminating power of this test is weak, due to the low polarization signal-to-noise ratio on large angular scales. \nThe Planck 2013, 2015, and 2018 releases have substantially improved the constraints on the space of inflationary models, as described above. Future CMB polarization data will be crucial for further constraining those inflationary models that currently provide an adequate fit to Planck and other data. Forthcoming E -mode polarization data will be decisive for determining whether the intriguing features in the temperature power spectrum, such as the deficit at ' ' 20-30, the smaller average amplitude at ' . 40, and other anomalies at higher multipoles require new physics or whether these features are simply the result of sta- \ntistical fluctuations plus instrumental noise. Improved measurements of the B modes promise to constrain inflation even more tightly and it will be interesting to see how the search for B modes evolves. One possibility would be a convincing detection of inflationary gravitational waves, but a tighter upper limit of r . 10 GLYPH<0> 3 is also an achievable outcome. Either case would substantially advance our understanding of inflation and the constraints on the physics of the very early Universe. \nAcknowledgements. We are grateful to Jan Hamann and Jim Zibin for extensive help with the final editing of this manuscript. The Planck Collaboration acknowledges the support of: ESA; CNES and CNRS / INSU-IN2P3-INP (France); ASI, CNR, and INAF (Italy); NASA and DoE (USA); STFC and UKSA (UK); CSIC, MINECO, JA, and RES (Spain); Tekes, AoF, and CSC (Finland); DLR and MPG (Germany); CSA (Canada); DTU Space (Denmark); SER / SSO (Switzerland); RCN(Norway); SFI (Ireland); FCT / MCTES(Portugal); ERC and PRACE (EU). 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AIM, Universit'e Paris Diderot, Sorbonne Paris Cit'e, F91191 Gif sur Yvette, France.\n- 2 APC, AstroParticule et Cosmologie, Universit'e Paris Diderot, CNRS / IN2P3, CEA / lrfu, Observatoire de Paris, Sorbonne Paris Cit'e, 10, rue Alice Domon et L'eonie Duquet, 75205 Paris Cedex 13, France\n- 3 African Institute for Mathematical Sciences, 6-8 Melrose Road, Muizenberg, Cape Town, South Africa\n- 4 Aix Marseille Univ, CNRS, CNES, LAM, Marseille, France\n- 5 Astrophysics Group, Cavendish Laboratory, University of Cambridge, J J Thomson Avenue, Cambridge CB3 0HE, U.K.\n- 6 Astrophysics & Cosmology Research Unit, School of Mathematics, Statistics & Computer Science, University of KwaZulu-Natal, Westville Campus, Private Bag X54001, Durban 4000, South Africa\n- 7 CITA, University of Toronto, 60 St. George St., Toronto, ON M5S 3H8, Canada\n- 8 CNRS, IRAP, 9 Av. colonel Roche, BP 44346, F-31028 Toulouse cedex 4, France\n- 9 Cahill Center for Astronomy and Astrophysics, California Institute of Technology, Pasadena CA, 91125, USA\n- 10 California Institute of Technology, Pasadena, California, U.S.A.\n- 11 Centre for Theoretical Cosmology, DAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, U.K.\n- 12 Computational Cosmology Center, Lawrence Berkeley National Laboratory, Berkeley, California, U.S.A.\n- 13 DTU Space, National Space Institute, Technical University of Denmark, Elektrovej 327, DK-2800 Kgs. Lyngby, Denmark\n- 14 D'epartement de Physique Th'eorique, Universit'e de Gen'eve, 24, Quai E. Ansermet,1211 Gen'eve 4, Switzerland\n- 15 Departamento de Astrof'ısica, Universidad de La Laguna (ULL), E38206 La Laguna, Tenerife, Spain\n- 16 Departamento de F'ısica, Universidad de Oviedo, C / Federico Garc'ıa Lorca, 18 , Oviedo, Spain\n- 17 Department of Astrophysics / IMAPP, Radboud University, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands\n- 18 Department of General Education, National Institute of Technology, Kagawa College, 355 Chokushi-cho, Takamatsu, Kagawa 761-8058, Japan\n- 19 Department of Mathematics, University of Stellenbosch, Stellenbosch 7602, South Africa \n- 20 Department of Physics & Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, British Columbia, Canada\n- 21 Department of Physics & Astronomy, University of the Western Cape, Cape Town 7535, South Africa\n- 22 Department of Physics and Astronomy, University College London, London WC1E 6BT, U.K.\n- 23 Department of Physics and Astronomy, University of Sussex, Brighton BN1 9QH, U.K.\n- 24 Department of Physics, Gustaf Hallstromin katu 2a, University of Helsinki, Helsinki, Finland\n- 25 Department of Physics, Princeton University, Princeton, New Jersey, U.S.A.\n- 26 Department of Physics, University of California, Santa Barbara, California, U.S.A.\n- 27 Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois, U.S.A. \n28 \nDipartimento di Fisica e Astronomia G. Galilei, Universit'a degli \nStudi di Padova, via Marzolo 8, 35131 Padova, Italy \n- 29 Dipartimento di Fisica e Scienze della Terra, Universit'a di Ferrara, Via Saragat 1, 44122 Ferrara, Italy\n- 30 Dipartimento di Fisica, Universit'a La Sapienza, P. le A. Moro 2, Roma, Italy\n- 31 Dipartimento di Fisica, Universit'a degli Studi di Milano, Via Celoria, 16, Milano, Italy\n- 32 Dipartimento di Fisica, Universit'a degli Studi di Trieste, via A. Valerio 2, Trieste, Italy\n- 33 Dipartimento di Fisica, Universit'a di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, Italy\n- 34 European Space Agency, ESAC, Planck Science O GLYPH<14> ce, Camino bajo del Castillo, s / n, Urbanizaci'on Villafranca del Castillo, Villanueva de la Ca˜nada, Madrid, Spain\n- 35 European Space Agency, ESTEC, Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands\n- 36 Gran Sasso Science Institute, INFN, viale F. Crispi 7, 67100 L'Aquila, Italy\n- 37 Haverford College Astronomy Department, 370 Lancaster Avenue, Haverford, Pennsylvania, U.S.A.\n- 38 Helsinki Institute of Physics, Gustaf Hallstromin katu 2, University of Helsinki, Helsinki, Finland\n- 39 INAF - OAS Bologna, Istituto Nazionale di Astrofisica -Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, Area della Ricerca del CNR, Via Gobetti 101, 40129, Bologna, Italy\n- 40 INAF -Osservatorio Astronomico di Padova, Vicolo dell'Osservatorio 5, Padova, Italy\n- 41 INAF - Osservatorio Astronomico di Trieste, Via G.B. Tiepolo 11, Trieste, Italy\n- 42 INAF, Istituto di Radioastronomia, Via Piero Gobetti 101, I-40129 Bologna, Italy\n- 43 INAF / IASF Milano, Via E. Bassini 15, Milano, Italy\n- 44 INFN - CNAF, viale Berti Pichat 6 / 2, 40127 Bologna, Italy\n- 45 INFN, Sezione di Bologna, viale Berti Pichat 6 / 2, 40127 Bologna, Italy\n- 46 INFN, Sezione di Ferrara, Via Saragat 1, 44122 Ferrara, Italy\n- 47 INFN, Sezione di Milano, Via Celoria 16, Milano, Italy\n- 48 INFN, Sezione di Roma 1, Universit'a di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, Italy\n- 49 INFN, Sezione di Roma 2, Universit'a di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, Italy\n- 50 IUCAA, Post Bag 4, Ganeshkhind, Pune University Campus, Pune 411 007, India\n- 51 Imperial College London, Astrophysics group, Blackett Laboratory, Prince Consort Road, London, SW7 2AZ, U.K.\n- 52 Institut d'Astrophysique Spatiale, CNRS, Univ. Paris-Sud, Universit'e Paris-Saclay, Bˆat. 121, 91405 Orsay cedex, France\n- 53 Institut d'Astrophysique de Paris, CNRS (UMR7095), 98 bis Boulevard Arago, F-75014, Paris, France\n- 54 Institut fur Theoretische Teilchenphysik und Kosmologie, RWTH Aachen University, D-52056 Aachen, Germany \n- 55 Institute Lorentz, Leiden University, PO Box 9506, Leiden 2300 RA, The Netherlands\n- 56 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, U.K.\n- 57 Institute of Theoretical Astrophysics, University of Oslo, Blindern, Oslo, Norway\n- 58 Instituto de Astrof'ısica de Canarias, C / V'ıa L'actea s / n, La Laguna, Tenerife, Spain\n- 59 Instituto de Astrof'ısica e Ciˆencias do Espac¸o, Faculdade de Ciˆencias da Universidade de Lisboa, Campo Grande, PT1749-016 Lisboa, Portugal\n- 60 Instituto de F'ısica de Cantabria (CSIC-Universidad de Cantabria), Avda. de los Castros s / n, Santander, Spain\n- 61 Istituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, Italy\n- 62 Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, California, U.S.A.\n- 63 Jodrell Bank Centre for Astrophysics, Alan Turing Building, School of Physics and Astronomy, The University of Manchester, Oxford Road, Manchester, M13 9PL, U.K.\n- 64 Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA\n- 65 Kavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.\n- 66 LERMA, CNRS, Observatoire de Paris, 61 Avenue de l'Observatoire, Paris, France\n- 67 LERMA / LRA, Observatoire de Paris, PSL Research University, CNRS, Ecole Normale Sup'erieure, 75005 Paris, France\n- 68 Laboratoire AIM, CEA - Universit'e Paris-Saclay, 91191 Gif-surYvette, France\n- 69 Laboratoire de Physique Subatomique et Cosmologie, Universit'e Grenoble-Alpes, CNRS / IN2P3, 53, rue des Martyrs, 38026 Grenoble Cedex, France\n- 70 Laboratoire de Physique Th'eorique, Universit'e Paris-Sud 11 & CNRS, Bˆatiment 210, 91405 Orsay, France\n- 71 Lawrence Berkeley National Laboratory, Berkeley, California, U.S.A.\n- 72 Leung Center for Cosmology and Particle Astrophysics, National Taiwan University, Taipei 10617, Taiwan\n- 73 Low Temperature Laboratory, Department of Applied Physics, Aalto University, Espoo, FI-00076 AALTO, Finland\n- 74 Max-Planck-Institut fur Astrophysik, Karl-Schwarzschild-Str. 1, 85741 Garching, Germany\n- 75 Mullard Space Science Laboratory, University College London, Surrey RH5 6NT, U.K.\n- 76 NAOC-UKZN Computational Astrophysics Centre (NUCAC), University of KwaZulu-Natal, Durban 4000, South Africa\n- 77 Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences, Bartycka 18, 00-716 Warsaw, Poland\n- 78 SISSA, Astrophysics Sector, via Bonomea 265, 34136, Trieste, Italy\n- 79 San Diego Supercomputer Center, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093, USA\n- 80 School of Chemistry and Physics, University of KwaZulu-Natal, Westville Campus, Private Bag X54001, Durban, 4000, South Africa\n- 81 School of Physical Sciences, National Institute of Science Education and Research, HBNI, Jatni-752050, Odissa, India\n- 82 School of Physics and Astronomy, Cardi GLYPH<11> University, Queens Buildings, The Parade, Cardi GLYPH<11> , CF24 3AA, U.K.\n- 83 School of Physics and Astronomy, Sun Yat-sen University, 2 Daxue Rd, Tangjia, Zhuhai, China\n- 84 School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, U.K.\n- 85 School of Physics, Indian Institute of Science Education and Research Thiruvananthapuram, Maruthamala PO, Vithura, Thiruvananthapuram 695551, Kerala, India\n- 86 School of Physics, The University of New South Wales, Sydney NSW 2052, Australia\n- 87 Simon Fraser University, Department of Physics, 8888 University Drive, Burnaby BC, Canada\n- 88 Sorbonne Universit'e-UPMC, UMR7095, Institut d'Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, France\n- 89 Space Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84 / 32, Moscow, 117997, Russia\n- 90 Space Science Data Center - Agenzia Spaziale Italiana, Via del Politecnico snc, 00133, Roma, Italy\n- 91 Space Sciences Laboratory, University of California, Berkeley, California, U.S.A.\n- 92 The Oskar Klein Centre for Cosmoparticle Physics, Department of Physics, Stockholm University, AlbaNova, SE-106 91 Stockholm, Sweden\n- 93 UPMCUnivParis 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, France\n- 94 Universit'e de Toulouse, UPS-OMP, IRAP, F-31028 Toulouse cedex 4, France\n- 95 Van Swinderen Institute for Particle Physics and Gravity, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands\n- 96 Warsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland"} |
2024A&A...691A..94D | We present the first 3D kinematic analysis of multiple stellar populations MPs in a representative sample of 16 Galactic globular clusters GCs. For each GC in the sample we studied the MP lineofsight planeofthesky and 3D rotation and velocity distribution anisotropy. The differences between firstpopulation FP and secondpopulation SP kinematic patterns were constrained by means of parameters specifically defined to provide a global measure of the relevant physical quantities and to enable a meaningful comparison among different clusters. Our analysis provides the first observational description of the MP kinematic properties and of the path they follow during their longterm dynamical evolution. In particular we find evidence of differences between the rotation of MPs along all velocity components with the SP preferentially rotating faster than the FP. The difference between the rotation strength of MPs is anticorrelated with the cluster dynamical age. We also observe that FPs are characterized by isotropic velocity distributions at any dynamical age probed by our sample. On the contrary the velocity distribution of SP stars is found to be radially anisotropic in dynamically young clusters and isotropic at later evolutionary stages. The comparison with a set of numerical simulations shows that these observational results are consistent with the longterm evolution of clusters forming with an initially more centrally concentrated and more rapidly rotating SP subsystem. We discuss the possible implications these findings have on our understanding of MP formation and early evolution. | 2024-11-01T00:00:00Z | ['2024arXiv240903827D', '10.1051/0004-6361/202451054', '2024A&A...691A..94D', '10.48550/arXiv.2409.03827', 'arXiv:2409.03827'] | ['techniques: photometric', 'techniques: radial velocities', 'stars: abundances', 'Hertzsprung-Russell and C-M diagrams', 'stars: kinematics and dynamics', 'globular clusters: general', 'Astrophysics - Astrophysics of Galaxies', 'Astrophysics - Solar and Stellar Astrophysics'] | A 3D view of multiple populations kinematics in Galactic globular clusters | 2,024 | 173 | 0.6 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF'] | 3 | https://arxiv.org/pdf/2409.03827.pdf | {'A 3D view of multiple populations kinematics in Galactic globular clusters': 'E. Dalessandro 1 , M. Cadelano 1 , 2 , A. Della Croce 1 , 2 , F. I. Aros 3 , E. B. White 3 , E. Vesperini 3 , C. Fanelli 1 , F. R. 1 2 1 2 1 2 1 \nFerraro , , B. Lanzoni , , S. Leanza , , and L. Origlia \n- 3 I-40129 Bologna, Italy\n- 1 INAF - Astrophysics and Space Science Observatory Bologna, Via Gobetti 93 / e-mail: [email protected]\n- 2 Dipartimento di Fisica e Astronomia, Via Gobetti 93 / 2 I-40129 Bologna, Italy\n- 3 Department of Astronomy, Indiana University, Swain West, 727 E. 3rd Street, IN 47405 Bloomington, USA \nReceived June 10, 2024; accepted September 02, 2024', 'ABSTRACT': 'We present the first 3D kinematic analysis of multiple stellar populations (MPs) in a representative sample of 16 Galactic globular clusters (GCs). For each GC in the sample we studied the MP line-of-sight, plane-of-the-sky and 3D rotation as well as the velocity distribution anisotropy. The di ff erences between first- (FP) and second-population (SP) kinematic patterns were constrained by means of parameters specifically defined to provide a global measure of the relevant physical quantities and to enable a meaningful comparison among di ff erent clusters. Our analysis provides the first observational description of the MP kinematic properties and of the path they follow during the long-term dynamical evolution. In particular, we find evidence of di ff erences between the rotation of MPs along all velocity components with the SP preferentially rotating faster than the FP. The di ff erence between the rotation strength of MPs is anti-correlated with the cluster dynamical age. We observe also that FPs are characterized by isotropic velocity distributions at any dynamical age probed by our sample. On the contrary, the velocity distribution of SP stars is found to be radially anisotropic in dynamically young clusters and isotropic at later evolutionary stages. The comparison with a set of numerical simulations shows that these observational results are consistent with the long-term evolution of clusters forming with an initially more centrally concentrated and more rapidly rotating SP subsystem. We discuss the possible implications these findings have on our understanding of MP formation and early evolution. \nKey words. globular clusters: general - Stars: Hertzsprung-Russell and C-M diagrams - kinematics and dynamics - abundances techniques: photometry - spectroscopy - astrometry', '1. Introduction': "Globular clusters (GCs) exhibit intrinsic star-to-star variations in their light element content. In fact, while some GC stars have the same light-element abundances as field stars with the same metallicity (first population / generation-FP), others show enhanced N and Na along with depleted C and O abundances (second population / generation-SP). The manifestation of such light-element inhomogeneities is referred to as multiple populations (MPs - see Bastian & Lardo 2018; Gratton et al. 2019 for a review of the subject). \nLight-element abundance variations can have an impact on both the stellar structures (as in the case of He for example) and atmospheres (as for Na, O, C and N) and they can therefore produce a broadening or splitting of di ff erent evolutionary sequences in color-magnitude-diagrams (CMDs) when appropriate filter combinations are used (Piotto et al. 2007; Sbordone et al. 2011; Dalessandro et al. 2011; Monelli et al. 2013; Piotto et al. 2015; Niederhofer et al. 2017; Cadelano et al. 2023). \nIt is well established now that the MP phenomenon is (almost) ubiquitous among massive stellar clusters. In fact, it has been shown that nearly all massive ( > 10 4 M ⊙ ; e.g., Dalessandro et al. 2014; Piotto et al. 2015; Milone et al. 2017; Bragaglia et al. 2017) and relatively old ( > 1.5-2 Gyr; Martocchia et al. 2018a; Cadelano et al. 2022) GCs host MPs. In addition, MPs are observed in GC systems in any environment where they have been \nproperly searched for. They are regularly found in the Magellanic Clouds stellar clusters (Mucciarelli et al. 2009; Dalessandro et al. 2016), in GCs in dwarf galaxies such as Fornax (Larsen et al. 2012, 2018) and Sagittarius (e.g., Sills et al. 2019), in the M31 GC system (Schiavon et al. 2013; Nardiello et al. 2018) and there are strong indications (though indirect) that they are ubiquitous in stellar clusters in massive elliptical galaxies (e.g., Chung et al. 2011). \nMPs are believed to form during the very early epochs of GC formation and evolution ( ∼ 10 -100 Myr; see Martocchia et al. 2018b; Nardiello et al. 2015; Saracino et al. 2020 for direct observational constraints) and a number of scenarios have been proposed over the years to describe the sequence of physical events and mechanisms involved in their formation. We can schematically group them in two main categories. One includes multi-epoch formation models, which predict that MPs form during multiple (at least two) events of star formation and they typically invoke self-enrichment processes, in which the SP forms out of the ejecta of relatively massive FP stars (e.g. Decressin et al. 2007; D'Ercole et al. 2008; de Mink et al. 2009; D'Antona et al. 2016). The second category includes models where MPs form simultaneously and SPs are able to accrete gas eventually during their pre-main sequence phases (e.g., Bastian et al. 2013; Gieles et al. 2018). However, independently on the specific differences, all models proposed so far have their own caveats and face serious problems to reproduce some or all the available ob- \nrvations. As a matter of fact, we still lack a self-consistent explanation of the physical processes at the basis of MP formation (e.g., see Bastian & Lardo 2018; Gratton et al. 2019). \nUnderstanding the kinematical and structural properties of MPs can provide new insights into the early epochs of GC formation and evolution. In fact, most formation models suggest that MPs form with di ff erent structural and kinematic properties. Di ff erences between the FP and the SP kinematics can be either imprinted at the time of SP formation (see e.g., Bekki 2010; Lacchin et al. 2022) or emerge during a cluster's evolution as a consequence of the initial di ff erences between the FP and SP spatial distributions (see e.g., Tiongco et al. 2019; Vesperini et al. 2021; Sollima 2021). Although the primordial structural and kinematic di ff erences between FP and SP stars are expected to be gradually erased during GC long-term dynamical evolution (e.g., Vesperini et al. 2013; Hénault-Brunet et al. 2015; Tiongco et al. 2019; Vesperini et al. 2021; Sollima 2021), some clusters are expected to still retain some memory of these initial di ff erences. Indeed, Dalessandro et al. (2019) measured the di ff erence in the spatial distributions of FP and SP stars for a large sample of massive Galactic and Magellanic Clouds clusters homogeneously observed with HST. The authors found that the differences between the FP and SP spatial distributions generally follow the evolutionary sequence expected for the long-term dynamical evolution of clusters forming with an initially more centrally concentrated SP subsystem (see also Leitinger et al. 2023; Onorato et al. 2023; Cadelano et al. 2024). \nSpatial distributions alone can provide only a partial picture of the dynamical properties of MPs and further key constraints on the possible formation and dynamical paths of MPs are expected to be hidden in their kinematic properties. Because of the technical limitations to derive kinematic information for large and significant samples of resolved stars in dense environments, most of the available information so far have been obtained by using HST proper motions (PMs) and ESO / VLT MUSE line-ofsight ( LOS ) velocities sampling relatively small portions of the cluster and focusing typically on the innermost regions. In a few particularly well studied systems, MPs have been found to show di ff erent degrees of orbital anisotropy (e.g., Richer et al. 2013; Bellini et al. 2015; Libralato et al. 2023) and possibly di ff erent rotation amplitudes (e.g., Cordero et al. 2017; Kamann et al. 2020; Cordoni et al. 2020; Dalessandro et al. 2021a; Martens et al. 2023). In other cases however, no significant di ff erences between the MP kinematic properties have been observed (see for example Milone et al. 2018; Cordoni et al. 2020; Libralato et al. 2019; Szigeti et al. 2021; Martens et al. 2023) and as a consequence, the emerging internal kinematic results describe a pretty heterogeneous picture. \nTo move a leap forward in our understanding of MP kinematic properties and their possible implications on GC formation, it is fundamental to perform a systematic and homogeneous study of clusters sampling a wide range of dynamical ages and including the analysis of the clusters' outer regions, which are expected to retain some memory of the primordial structural and kinematic di ff erences for longer timescales. As a first step in this direction, in this paper, we perform for the first time a self-consistent study of the 3D kinematics of MPs in a representative sample of Galactic GCs for which it is possible to sample virtually their entire radial extension. This study has the additional advantage to overcome the typical limitations connected with projection e ff ects, which typically arise when LOS or planeof-the-sky velocities (i.e., PMs) are used independently possibly hampering the detection of the actual di ff erences between \n<!-- image --> \nFig. 1. On the left, the vector-point-diagram obtained by using Gaia DR3 PMs for the GC 47 Tuc is shown along with the distribution of stars along the µ ∗ α and µδ velocity components. The red circle represents the 2 σ selection described in Section 2.3. The panel on the right shows the (U, U-I) and (U, C U , B . I ) CMDs for 47 Tuc obtained by using groundbased photometric catalogs published by Stetson et al. (2019). Likely member stars based on the PM selection shown in the left panel are highlighted in black, while in grey likely field interlopers are shown. \n<!-- image --> \nthe MP kinematic properties (see e.g. the discussion concerning these issues in Tiongco et al. 2019). \nThe paper is structured as follows. In Section 2 the adopted sample and the observational database are presented. In Section 3 we describe the kinematic analysis along the three velocity components, while in Section 4 we present the approach adopted for the study of the morphological properties of MPs. In Sections 5 and 6 we present the main observational results along with a detailed comparison with dynamical simulations, and in Section 7 we compare them with the literature. In Section 8 we summarize our findings and discuss their possible implications in the context of massive clusters formation and early evolution.", '2. Sample definition and observational data-sets': 'The analysis presented in this paper targets 16 Galactic GCs. In detail, the sample includes all clusters analyzed by Ferraro et al. (2018) and Lanzoni et al. (2018a,b) in the context of the ESO / VLT Multi Instrument Kinematic Survey of Galactic GCs (MIKiS) but NGC 362 as it lacks near-UV photometric data needed for the study of MPs (see Section 2.2 for details). We added to the target list NGC 104 (47 Tucanae) as it is a massive, relatively close and well studied GC, which can be useful for comparative analysis. We included also NGC 6362 for which we secured a large kinematic data-set in Dalessandro et al. (2018b, 2021a), NGC 6089 (M 80) and NGC 6205 (M 13) as they have been found to show interesting kinematic properties in previous analysis by Cordero et al. (2017) and Kamann et al. (2020). Table 1 summarizes some useful properties of the targets, such as distances, structural properties, age, relaxation times and kinematic sample sizes. The selected clusters are representative of the overall Galactic GC population as they properly encompass the cluster dynamically-sensitive parameter space, spanning a large range of central densities and concentrations, di ff erent stages of dynamical evolution and di ff erent environmental conditions. They are also more massive than M > 10 4 M ⊙ and relatively close to the Earth (within ∼ 16 kpc), thus providing data with good signal-to-noise ratios (S / Ns) for a large sample of stars. \nTable 1. Properties of the 16 clusters fully analyzed in the present work. \nNotes. Distances are from Baumgardt et al. (2019), structural parameters from Harris (1996) and ages come from the compilation by Dotter et al. (2010) with the exception of NGC 1904 for which we used the age derivation by Dalessandro et al. (2013). N LOS and N PM represent the number of LOS velocities and PMs used for the kinematic analysis.', '2.1. Kinematic database': "The analysis performed in this paper is based on two main kinematic data-sets securing LOS velocities (RVs) and PMs for hundreds (or thousands in a few cases - Table 1) of red giant branch stars (RGBs) in each GC. For 12 out of 16 GCs, most of the adopted LOS RVs were obtained by using ESO / VLT KMOS and FLAMESdataobtained as part of the MIKiS survey. We refer the reader to Ferraro et al. (2018) for details about the overall observational strategy and data-analysis. MIKiS LOS RVs were then complemented with those from the publicly available catalog by Baumgardt et al. (2019) to improve the sampling of the external regions of the target clusters. All LOS RVs used for 47 Tuc come from the Baumgardt et al. (2019) catalog. For NGC 6362, M 80 and M 13 LOS RVs were obtained by using ESO / VLTMUSEand FLAMES data (Cordero et al. 2017; Dalessandro et al. 2018b, 2021a; Kamann et al. 2020). \nFor each of the investigated GCs, astrometric information, namely absolute PMs ( µ ∗ α , µδ ) and relative errors, were retrieved from the ESA / Gaia DR3(Gaia Collaboration et al. 2023) archive out to the clusters' tidal radius. Only stars with ruwe < 1 . 3 1 were then used for the kinematic analysis.", '2.2. Photometric dataset, membership selection and differential reddening correction': 'We used the wide-field catalogs published by Stetson et al. (2019) and including U, B, V, R and I bands to identify MPs in the target GCs (see Section 2.3) While these data are seeinglimited and can su ff er of incompleteness in the crowded central regions, they have similar spatial resolution as the kinematic LOS RVs dataset. In addition, they are typically not a ff ected by saturation problems and therefore they maximize the number of bright stars in common with the kinematic samples. The photometric catalogs were cross-matched with the kinematic ones based on their absolute coordinates ( α , δ ) and by using the cross- \nFig. 2. LOS , RAD and TAN velocity distributions of likely member stars of the GC 47 Tuc as a function of the cluster-centric distance. All velocities are shown with respect to the cluster systemic velocity along the corresponding component. \n<!-- image --> \nrrelation tool CataXcorr 2 . For each cluster in the sample, the final catalog includes all stars in common between Gaia and the photometric catalogs. A fraction (typically larger than ∼ 60% along the RGB) of these stars has also LOS RVs and is therefore suited for a full 3D analysis (see Table 1). \nTo separate likely cluster members from field interlopers, we selected stars whose PMs are within 2 σ from the cluster systemic velocity (adopted from Vasiliev & Baumgardt 2021), where σ is the standard deviation of the observed ( µ ∗ α , µδ ) distribution of RGB stars. We have verified that for the clusters in our sample, reasonable variations of the adopted cluster membership selection criteria do not have a significant impact on the main results of the kinematic analysis. As an example, Figure 1 shows \n<!-- image --> \nFig. 3. Left panel: (U, C U , B , I ) CMD of 47 Tuc likely member stars. RGB stars adopted for the kinematic analysis are highlighted in black. The blue and red curves are the fiducial lines adopted to verticalize the color distribution. Right-hand panels: the top panel displays the verticalized color distribution of RGB stars, while the bottom panel shows the corresponding histograms. The red and blue curves represent the two best-fit Gaussians for the FP and SP, respectively, while the solid black curve is their sum. \n<!-- image --> \nthe PM distribution of 47 Tuc stars along with the (U, U-I) and (U, CU , B , I - where CU , B , I = ( U -B ) -( B -I ); Monelli et al. 2013) CMDs for both selected cluster stars and field interlopers. Figure 2 shows instead the distributions of the velocities along the LOS , the PM radial ( RAD ) and PM tangential ( TAN ) components as a function of the cluster-centric distance for likely members RGB stars of the same cluster. It is worth mentioning here that the kinematic catalogs obtained from the MIKiS survey already rely on the cluster membership selection performed by Ferraro et al. (2018) and Lanzoni et al. (2018a,b), which is based on both the LOS RVand [Fe / H] distributions (we refer to those papers for further details). \nAvailable magnitudes were then corrected for di ff erential reddening by using the approach described in Dalessandro et al. (2018c) (see also Cadelano et al. 2020). In short, di ff erential reddening was estimated by using likely member stars selected in a magnitude range typically going from the RGB-bump level down to about one magnitude below the cluster turn-o ff . By using these stars a mean ridge-line (MRL) was defined in the (B, B-I) CMD. Then for all stars within 3 σ (where σ is the color spread around the MRL) the geometric distance from the MRL ( ∆ X ) was computed. For each star in the catalog, di ff erential reddening was then obtained as the mean of the ∆ X values of the 30 nearest (in space) selected stars. ∆ X was then transformed into di ff erential reddening δ E ( B -V ) by using Eq. 1 from Dalessandro et al. (2018c), properly modified to account for the di ff erent extinction coe ffi cients for the adopted filters. Di ff erential reddening corrections turn out to be relatively small ( < 0 . 1mag) for most GCs in the sample, with the most critical cases being NGC 3201 and NGC 5927 for which we find δ E ( B -V ) values larger than ∼ 0 . 2 mag.', '2.3. MP classification': 'Starting from the sample of likely member stars and di ff erential reddening corrected magnitudes, we identified MPs along the RGB by using their distribution in the ( U , CU , B , I ) CMD (Figure 3). It has been shown that this color combination is very e ff ective to identify MPs along the RGB with di ff erent C and N (and possibly He) abundances (Sbordone et al. 2011; Monelli et al. 2013). RGB stars were verticalized in the ( U , CU , B , I ) CMD with respect to two fiducial lines on the blue and red edges of the RGB, calculated as the 5th and 95th percentile of the color distribution in di ff erent magnitude bins (Figure 3 - see, e.g., Dalessandro et al. 2018a,c; Onorato et al. 2023; Cadelano et al. 2023, for similar implementations of the same technique). In the resulting verticalized color distribution ( ∆ CU , B , I ; right panel in Figure 3), stars on the red (blue) side are expected to be N-poor (-rich), i.e., FP (SP), respectively. We run a two components Gaussian Mixture Modeling 3 (GMM) analysis to the resulting ∆ CU , B , I distribution thus assigning to each star a probability of belonging to the FP and SP sub-populations. Stars with a probability larger than 50% to belong to one of the two sub-populations were then flagged as FPs or SPs. Figure 3 shows the result of the MP identification and separation for the GC 47 Tuc. While this approach may introduce a few uncertainties and over-simplifications in the MP classification as we are not directly deriving light-element chemical abundances, it secures statistically large samples of stars with MP tagging that are hard to obtain by using only spectroscopic data.', '3. Kinematic analysis': "For each cluster in the sample we first analyzed the kinematic properties in terms of velocity dispersion, rotation and anisotropy profiles for the LOS and plane-of-the-sky components separately (Sections 3.1 and 3.2) and then, for the fraction of stars for which all velocity components are available, we performed a full 3D study (Section 3.3). We adopted as clusters' centers those reported by Goldsbury et al. (2010). All velocities were corrected for perspective e ff ects induced by the clusters' systemic motions by using the equations reported in van Leeuwen (2009) and following the approach already adopted in Dalessandro et al. (2021b); Della Croce et al. (2023).", '3.1. 1D velocity dispersion and rotation profiles': 'To characterize the kinematic properties of the clusters in the sample and of their sub-populations, we adopted the Bayesian approach described in Cordero et al. (2017) (see also Dalessandro et al. 2018b), which is based on the use of a discrete fitting technique to compare simple kinematic models (including a radial dependence of the rotational amplitude and velocity dispersion of the cluster) with individual radial velocities. We stress that this is a purely kinematic approach aimed at searching for relative di ff erences among di ff erent clusters and sub-populations and it is not aimed at providing a self-consistent dynamical description of each system. \nThe likelihood function for the radial velocities of individual stars depends on our assumptions about the formal descriptions of the rotation and velocity dispersion radial variations. For the velocity dispersion profile we assumed the functional form of the Plummer model (Plummer 1911), which is simply defined by its central velocity dispersion σ 0 and its scale radius a : \nFig. 4. Bottom panels : observed velocity dispersion profiles along the LOS , radial and tangential velocity components for MPs in 47 Tuc obtained by using a maximum-likelihood approach on binned data. Upper panels: Best-fit velocity dispersion profiles as obtained by using the bayesian analysis on discrete velocities described in Section 3.1. \n<!-- image --> \nσ 2 ( R ) = σ 2 0 p 1 + R 2 / a 2 , (1) \nwhere R is the projected distance from the centre of the cluster. We adopted the same formal description for all velocity components. For the rotation curve, we assumed cylindrical rotation and adopted the functional form expected for stellar systems undergoing violent relaxation during their early phases of evolution (Lynden-Bell 1967): \nV rotsin i ( X PA0) = 2 A rot R peak X PA0 1 + ( X PA0 / R peak) 2 (2) \nµ TAN = 2 V peak R peak R 1 + ( R / R peak) 2 (3) \nfor the LOS (Eq. 2) and TAN (Eq. 3) velocity components, respectively. In Eq. 2 Vrotsin i represents the projection of the rotational amplitude along the LOS velocity component at a projected distance X PA0 from the rotation axis. A rot is the peak rotational amplitude occurring at the projected distance R peak from \nthe cluster center. We defined the rotation axis position angle ( PA ) as increasing anti-clockwise in the plane of the sky from north (PA = 0 · ) towards east (PA = 90 · ). Since the inclination of the rotation axis is unknown, Vrotsin i represents a lower limit to the actual rotational amplitude. As an extreme case, if the rotation axis is aligned with the line of sight, the rotation would be in the plane-of-the-sky. In Eq. 3, Vpeak represents the maximum (in an absolute sense) of the mean motion in the TAN component. \nThe fit of the kinematic quantities was performed by using the emcee 4 (Foreman-Mackey et al. 2013) implementation of the Markov chain Monte Carlo (MCMC) sampler, which provides the posterior probability distribution function (PDFs) for σ 0, a , A rot and PA 0. For each quantity, the 50th-, 16th- and 84thpercentile of the PDF distributions were adopted as the best-fit values and relative errors, respectively. We assumed a Gaussian likelihood and flat priors on each of the investigated parameter within reasonably large range of values. It is important to note that in general, since the analysis is based on the conditional probability of a velocity measurement, given the position of a star, our fitting procedure is not biased by the spatial sampling of the stars in the di ff erent clusters and sub-samples. However, the kinematic properties are better constrained in regions that are \nFig. 5. Same as 4, but now for the rotation profiles along the LOS (left panel) and tangential (right panel) components for MPs in 47 Tuc. \n<!-- image --> \nbetter sampled (i.e. larger number of stars with available kinematic information). \nFor sanity check and comparison, we also derived the velocity dispersion and rotation profiles by splitting the surveyed areas in a set of concentric annuli, whose width was chosen as a compromise between a good radial sampling and a statistically significant number of stars. In this case the analysis was limited radially within a maximum distance from the center of the clusters to guarantee a symmetric coverage of the field of view. The adopted limiting distance varies from one cluster to the other depending on the photometric and kinematic dataset field of view limits (Section 2.3). While this approach requires the splitting of the sample in concentric radial bins, whose number and width is at least partially arbitrary and can potentially have an impact on the final results, it has the advantage of avoiding any assumption on the model description of the velocity dispersion and rotation profiles. \nIn each radial bin the velocity dispersion was computed by following the maximum-likelihood approach described by Pryor & Meylan (1993). The method is based on the assumption that \nthe probability of finding a star with a velocity of vi and error ϵ i at a projected distance from the cluster center Ri can be approximated as: \np ( vi , ϵ i , Ri ) = 1 2 π q σ 2 + ϵ 2 i exp ( vi -v 0) 2 -2( σ 2 + ϵ 2 i ) (4) \nwhere v 0 and σ are the systemic velocity and the intrinsic dispersion profile of the cluster along the three components (i.e., LOS , RAD and TAN ), respectively. \nAs for the rotation along the LOS component, we used the method fully described in Bellazzini et al. (2012) and adopted in the previous papers of our group (e.g., Ferraro et al. 2018; Lanzoni et al. 2018a; Dalessandro et al. 2021a; Leanza et al. 2022). In brief, we considered a line passing through the cluster center with position angle varying from -90 · to 90 · by steps of 10 · . For each value of PA , such a line splits the observed sample in two. If the cluster is rotating along the line of sight, we expect to find a value of PA that maximizes the di ff erence between the median \nRVs of the two sub-samples, since one component is mostly approaching and the other is receding with respect to the observer. Moving PA from this value has the e ff ect of gradually decreasing the di ff erence in the median RV. Hence, the appearance of a coherent sinusoidal behavior as a function of PA is a signature of rotation and its best-fit sine function provides an estimate of the rotation amplitude ( A rot) and the position angle of the cluster rotation axis (PA0). For the plane-of-the-sky rotation we used instead the variation of the mean values within each radial bin of the tangential velocity component with respect to the systematic motion. \nExamples of the results obtained with both the Bayesian and maximum-likelihood analyses are shown in Figures 4 and 5 for the MPs of the GC 47 Tuc. For all clusters in the sample we find a good agreement between the discrete and binned analysis, however in the following we will adopt the best-fit results (and errors) obtained with the Bayesian approach. Table A.1 reports the best-fit values and relative errors for the most relevant quantities along both the LOS and plane-of-the-sky for both the FP and SP.', '3.2. Anisotropy profiles': "We adopted a modified version of the Osipkov-Merrit (Osipkov 1979; Merritt 1985) model to provide a formal description of the velocity anisotropy profile for each cluster and subpopulation (see Aros et al., in prep. for further details). This model is isotropic in the cluster centre and it becomes increasingly anisotropic for R > ra (where ra is the anisotropy radius). Our anisotropy description includes two additional parameters, which are the 'outer' anisotropy value ( β ∞ ) and the truncation radius ( rt ): \nβ ( R ) = β ∞ R 2 ( R 2 + ra ) 2 1 -R rt ! (5) \nβ ∞ defines how radial or tangentially biased the velocity anisotropy profile is for R >> ra , and rt is the radius at which the velocity anisotropy becomes isotropic again. In practice, we adopt rt as the Jacobi radius of the cluster. \nFollowing the same approach adopted for the velocity dispersion and rotation, we performed a MCMC fit of the anisotropy profiles (see results in Table A.1) by using the individual stellar velocities along the radial and tangential components ( µ RAD and µ TAN, respectively) and by assuming a Gaussian likelihood in which the best-fit velocity dispersion along the radial direction σ RAD can be described through a Plummer profile (see Equation 1) and the velocity dispersion along the tangential direction is simply given by σ 2 TAN = [1 -β ( R )] σ 2 RAD . It is important to stress that this assumption is only adopted for the best-fit anisotropy parameters derivation, while we will generally consider as best-fit velocity dispersion profiles those derived through the independent fit of σ RAD and σ TAN discussed in Section 3.1. For the binned analysis, the anisotropy profiles were obtained directly from the RAD and TAN velocity dispersion values obtained in Section 3.1. Figure 6 shows an example of the best-fit anisotropy profile for the MPs of 47 Tuc.", '3.3. Full 3D analysis': 'To perform a full 3D analysis we used the kinematic sample of member stars having both LOS RVs and Gaia PMs after quality selection (see Section 2.1). We also limited the analysis to \nthe same radial extension adopted for the binned maximumlikelihood analysis (Section 3.1). \nWe followed the approach described in Sollima et al. (2019), which has the advantage of constraining a cluster full rotation pattern by estimating the inclination angle of the rotation axis ( i ) with respect to the line of sight, the position angle of the rotation axis ( θ 0) and the rotation velocity amplitude ( A ), by means of a model-independent analysis. We refer the reader to that paper for a full description of the method. Here we only report the main ingredients and assumptions. \nIn a real cluster the angular velocity is expected to be a function of the distance from the rotation axis (see for example Eqs. 1 and 2). To account for such a dependence in a rigorous way, a rotating model should be fitted to the data. However, to perform a model-independent analysis we considered an average projected rotation velocity with amplitude A 3D, which has been assumed to be independent on the distance from the cluster center. While of course, this represents a crude approximation of the rotation patterns expected in GCs and provides a rough average of the actual rotation amplitude, it is important to stress that it does not introduce any bias in the estimate of θ 0 and i . \nA 3D, i and θ 0 were derived by solving the equations describing the rotation projection along the LOS ( V LOS) and those perpendicular ( V ⊥ ) and parallel ( V ∥ ) to the rotation axes (see Eq. 2 in Sollima et al. 2019). While the velocity component perpendicular to the rotation axis has a dependence on stellar positions within the cluster along the line of sight, we neglected it in the present analysis. In fact, we note that the dependence on the stellar distance along the line of sight does not a ff ect the mean trend of the perpendicular velocity component, but it can only introduce an additional spread on its distribution. We assumed that i varies in the range 0 · < i < 90 · with respect to the line of sight and the position angle in the range 0 · < θ 0 < 360 · . θ 0 grows anticlockwise from North to East and A 3D is positive for clockwise rotation in the plane of the sky. Following the approach already adopted for the 1D analysis, we derived the best-fit rotation amplitudes, position and inclination angles and relative errors by maximizing the likelihood function reported in Eq. 3 of Sollima et al. (2019) by using the MCMC algorithm emcee . Best-fit results are reported in Table A.1. Figure 7 shows the result of the best-fit analysis along the three velocity components for the FP and SP sub-populations of 47 Tuc.', '4. Morphological analysis: MP ellipticity': "We inferred the ellipticity of the MPs by constructing the socalled shape tensor (Zemp et al. 2011), which is defined by the following equation \nS ij ≡ P k m k( R k)i( R k)j P k m k , (6) \nwhere R k and m k are the projected distance from the cluster center and the mass of the k -th star, within the i -th and j -th element of the shape tensor grid. For the purpose of this study, we assumed that all stars have the same mass. This assumption is well justified as the adopted sample consists of RGB stars (the same used for the kinematic analysis), whose mass variations for a given cluster is expected to be ∼ 0 . 02 M ⊙ , depending on the metallicity. \nThe shape tensor was computed starting from a spherical grid, whose nodes are the stellar radial distribution's 10th, 50th, and 90th percentiles. We adopted an iterative procedure for each bin during which the shape tensor is initially constructed from \nFig. 6. Same as 4, but now for the anisotropy profile (see Section 3.2). \n<!-- image --> \nspherical distances. After that, being ( w 0; w 1) the eigenvalues (with w 0 > w 1) and ( v 0; v 1) the respective eigenvectors of the shape tensor, it follows (Zemp et al. 2011) that \nϵ = 1 -w 1 w 0 and PA = arctan v 0 , y v 0 , x . (7) \nThe particle coordinates were then rotated by the angle -PA, and distances to the center were defined by means of the circularized distance \nR ell ≡ s x ' 2 + y ' 2 ϵ 2 , (8) \nwhere ( x ' ; y ' ) are the rotated locally-Cartesian coordinates of the stars. \nArticle number, page 8 of 26 \nFinally, stars were binned in the new coordinate system according to the initial grid and the shape tensor has been computed using R ell instead of R . Such procedure was then iterated until a relative precision of 5% on the axis ratio was reached. Best-fit ellipticity values and relative errors have been obtained by means of a bootstrapping analysis. In detail, the shape tensor was computed 1000 times for each cluster by randomly selecting at each time a sub-sample including only 90% of stars. The values corresponding to the 50th percentile of the distribution of all the ϵ values has been then adopted as best-fit, while the errors correspond to the 16th and 84th percentiles. As an example, Figure 8 shows the result of the analysis for the MPs in 47 Tuc, while Table A.1 reports the best-fit values for each GC and subpopulation. \nFig. 7. Distribution of the three velocity components as a function of the position angle for MPs in 47 Tuc. Left and right panels refer to the FP and SP sub-population, respectively. The solid lines show the best-fitting trend in all panels. \n<!-- image --> \nFig. 8. 2D density maps of the stars selected in the GC 47 Tuc for the kinematic analysis. FPs are shown on the left panel, while SPs in the right panel. Overplotted to the density distributions are the best-fit ellipses derived as described in Section 4. \n<!-- image --> \n- \n-", '5. Results: rotation': 'To obtain quantitative and homogeneous estimates of the possible kinematic di ff erences among MPs, to follow their evolution and eventually to compare the results obtained for all GCs in the sample with theoretical models and dynamical simulations, we introduced a few simple parameters described in detail in the following. These parameters are meant to incorporate in a meaningful way all the main relevant physical quantities at play in a single value. The general approach of our analysis is not to focus \non the detection of specific and particularly significant kinematic di ff erences of specific targets, but rather to compare the general kinematic behaviors described by MPs in all targets in the sample in the most e ff ective way. \nA description about the approach adopted to quantify the possible impact of the intrinsically limited statistical kinematic samples and of their incompleteness on the final results is discussed in Appendix B. Here we briefly stress that the main effects are not on the derived best-fit values, but rather on their uncertainties. While the main focus of the following sections is \non the MP kinematics, we analyzed also the entire sample of stars with kinematic information (hereafter labeled as TOT) for comparison with previous works and we present the main results in Appendix C.', '5.1. Observations': "To measure the rotation di ff erences between the SP and FP subpopulations for all clusters in the sample for both the LOS and TAN velocity components, we introduced a parameter, hereafter referred as α , defined as the area subtended by the ratio between the best-fit rotation velocity profile and the best-fit velocity dispersion profile for each sub-population in a cluster (Section 3.1) after rescaling the cluster-centric distance to the value of the peak ( R α ) of such a distribution: \nα X = Z 1 0 V rot( R l) /σ ( R l) dR l (9) \nwhere V rot( R l) and σ ( R l) can either refer to the LOS or the TAN velocity components, R l is the cluster-centric distance normalized to the R α and the index X refers to the di ff erent sub-populations (i.e., FP or SP). \nThis parameter has the advantage of providing a robust measure of the relative strength of the rotation signal over the disordered motion at any radial range without making any assumption about the underlying star or mass distribution. By construction α depends on the considered cluster-centric distance and therefore a meaningful cluster-to-cluster comparison requires that the parameter is measured over equivalent radial portions in every system. As shown in a number of numerical studies (see, e.g., Hénault-Brunet et al. 2015; Tiongco et al. 2019), dynamical evolution is expected to smooth out primordial kinematic and structural di ff erences in the innermost regions first and then in the cluster's outskirts. Therefore, capturing rotation di ff erences between MPs require a compromise between probing a fairly wide radial coverage, thus to trace regions where kinematic di ff erences should be present for a longer time, and sampling distances from the cluster center where rotation is more prominent. With this in mind, we decided to measure α within R α . We verified also that the adoption of di ff erent radial selections does not have a significant impact on the overall relative distribution of α values. Errors on α were obtained by propagating the posterior probability distributions obtained from the MCMC analysis for the best-fit rotation and velocity dispersion profiles derivation (see Section 3.1). Di ff erences between SP and FP kinematic patterns are constrained simply by ( α SP -α FP ). With such a definition, a more rapidly rotating SP yields positive values of ( α SP -α FP ). \nFollowing similar lines, we defined a parameter to describe the 3D rotation: \nω X 3 D = ( A 3D /σ 3D 0 ) / ( R m / R h) (10) \nwhere σ 3D 0 represents the 3D central velocity dispersion and it is defined as the quadratic average of the σ 0 values obtained for the three velocity components (i.e., LOS , TAN and RAD - Section 3.1); R m is the average cluster-centric distance of stars for which we have tri-dimensional velocity measures, and R h is the system half-light radius (from Harris 1996). Here R h is adopted as a meaningful radial normalization factor to secure a direct comparison among di ff erent GCs attaining significantly di ff erent projected radial extension. In the assumption of a pure solid-body rotation, ω 3D would represent the best-fit angular rotation. As \nfor the 1D analysis, the introduction of ω 3D is primarily meant to provide a direct and reliable characterization of the 3D rotation based only on quantities that are directly derived from the observations. Di ff erences in the 3D rotation of SP and FP are given by ( ω SP 3D -ω FP 3D ), which yields positive values for a more rapidly rotating SP. \nSeveral works have shown that the rotation strength observed in GCs is primarily shaped by their dynamical age, with dynamically young systems typically showing the larger degree of rotation (e.g. Fabricius et al. 2014; Bianchini et al. 2018; Kamann et al. 2018; Sollima et al. 2019). We used N h = t age / t rh as a proxy of the clusters' dynamical age. We adopted the the half-mass relaxation time ( t rh) values reported by Harris (1996) and ages derived by Dotter et al. (2010) for all clusters but NGC 1904, for which we used the age inferred by Dalessandro et al. (2013). \nIn Figure C.1 in the Appendix we show the distribution of the α values (along both the LOS and TAN velocity components) and of ω 3D as a function of N h for the TOT population. In general, we find a very good agreement with previous analysis (e.g., Bianchini et al. 2018; Kamann et al. 2018; Baumgardt et al. 2019; Sollima et al. 2019) in terms of correlation between cluster rotation strength and dynamical age, thus further strengthening the idea that the present-day cluster rotation is the relic of that imprinted at the epoch of cluster formation, and that it has been then progressively dissipated via two-body relaxation (Einsel & Spurzem 1999; Hong et al. 2013; Tiongco et al. 2017; Livernois et al. 2022; Kamlah et al. 2022). Interestingly, such an agreement provides also an independent assessment about the reliability of the adopted kinematic parameters. \nIn Figures 9, 10 and 11 we show the distributions of α (for both the LOS and PM components) and of ω 3D as a function of N h for both SP and FP stars (bottom and middle panels). Interestingly, a number of common patterns can be highlighted in the three Figures. First, we note that in a large fraction of clusters (up to ∼ 50%) both FP and SP show evidence of non-negligible rotation. Secondly, both sub-populations show evidence of anticorrelation with N h in all three analyzed velocity components, with dynamically young clusters being characterized by a larger rotation strength. This behavior turns out to be more prominent when the PM and 3D analysis are considered. This is somehow expected as PMs are less a ff ected than the LOS velocities by the smoothing introduced by the superposition of stars located at di ff erent cluster-centric distances and attaining di ff erent rotation velocities. In addition, the 3D analysis accounts for any rotation axis inclination and projection e ff ects. Finally, we observe that for all velocity components, in dynamically young clusters the SP is characterized by larger α and ω 3D values (i.e., more rapid rotation) than that observed for the FP at similar N h, and it shows a more rapid decline than the FP as a function of N h. In fact, a Spearman correlation test gives a probability P spear larger than ∼ 99% of correlations between ( α SP )PM or ω SP 3D and N h, while probabilities are smaller when either the LOS component or the FP is considered. We note here that by using the approach described in (Curran 2015) we have verified that the results of the Spearman rank correlation tests performed in our analysis and reported in the following are robust against possible outliers and errors associated to the adopted kinematic parameters. \nTo better highlight such a di ff erential behavior, the upper panels of Figures 9, 10 and 11 show the di ff erence between the rotation strength of the SP and FP as given by ( α SP -α FP ) and ( ω SP 3D -ω FP 3D ). Admittedly, nor ( α SP -α FP ) or ( ω SP 3D -ω FP 3D ) show striking variations in the dynamical age range sampled by the target clusters (2 < N h < 25). In fact, Spearman rank correlation tests give probabilities of correlation of P spear ∼ 90% ( ∼ 95% in \nFig. 9. Bottom and middle panels show the distribution of the ( α )LOS parameter for the FP and SP as a function of the dynamical age Nh for all clusters in the sample (grey circles). The upper panel shows instead the distribution of the rotation di ff erences ( α SP -α SP )LOS. The dashed-lines represent the linear best-fit to the GC distribution. In all panels, the starry symbols refer to the results obtained for the stacked analysis on the dynamically young (green) and old (orange) samples. The size of the star matches the amplitude of the errorbars. \n<!-- image --> \nthe 3D case). Interestingly however, a negative trend between the rotation strength di ff erences and N h is consistently observed in all velocity components. In fact, both ( α SP -α FP ) and ( ω SP 3D -ω FP 3D ) show positive values for dynamically young GCs and then they progressively approach 0 for dynamical older clusters, meaning that FP and SP rotate at the same velocity. The nice agreement between the results obtained in the three analysis definitely provides support to the fact that there is a real correlation between the SP and FP rotation strength di ff erences and N h, and that in general SP shows a more rapid rotation than the FP at dynamically young ages. These results represent the first observational evidence of the link between MP rotation patterns and clusters' long-term dynamical evolution. \nWe do not find any significant di ff erence between the MP rotation axis orientation both for the LOS and 3D analysis. In fact, the mean di ff erence between the best-fit PA 0 values of the SP and FP is -2 · ± 19 · , and those between θ 0 and i are 4 · ± 24 · and 2 · ± 12 · , respectively. \nAs an additional way to analyze the data and search for possible trends, we divided the clusters in two sub-groups according to their dynamical ages. In particular, we defined a group of clusters with N h < 8 and a complementary one ( N h > 8) including all the remaining GCs. In this way the two sub-groups turn out to be populated by the same number of systems. Within each sub-group and sub-population, we then stacked all the available kinematic information after normalizing the cluster-centric dis- \nFig. 10. Same as in Figure 9, but now for ( α ) PM (Section 5.1). \n<!-- image --> \ntances to the cluster's R h (from Harris 1996), the velocities to the central velocity dispersion in a given velocity component (as obtained by the analysis described in Section 3) and rotating all clusters to have the same PA 0 (Section 3.1). In this way, we could jointly compare the behavior of MPs for multiple clusters at once, thus increasing the number of stars that can be used to study the kinematics of each sub-population and narrowing down the uncertainties on the derived kinematic parameters The kinematics of MPs in the two stacked samples was then analyzed by following the same approach described in Section 3 and previously adopted for single GCs. Figure 12 shows the results of the kinematic analysis for the two stacked samples for both the LOS and TAN components (lower and upper panels, respectively). In both cases, a significant di ff erence between the FP and SP rotation profiles is observed for the dynamically young ( N h < 8) \nstacked sample, while they almost disappear for the dynamically old GCs. We then derived the same kinematic parameters described by Equation 9 for a direct comparison with single GCs. Results are shown in Figures 9 and 10 by the two starry symbols. Both in the LOS and TAN components and for each subpopulations, results are fully consistent with the general trend described by single clusters. Interestingly, the reduced uncertainties strengthen the significance of the observed di ff erences discussed above. In particular, the stacked analysis shows that the observed trends between α FP , α SP and N h along both the LOS and TAN components are significant at a large confidence level ( P stacked > 5 σ ). Also, while the ( α SP -α FP ) di ff erence between dynamically young and old clusters is only marginally significant along the LOS component, it turns out to be significant at a ∼ 6 σ level for the tangential component. \nFig. 11. As in Figures 9 and 10, but now for the ω 3D parameter (Section 5.1). \n<!-- image --> \nUnfortunately, we could not apply the 3D rotation analysis (Section 3.3) to the stacked samples as it is not possible to report all clusters to the same values of θ 0 and i .", '5.2. Ellipticity': 'In general a rotating system is also expected to be flattened in the direction perpendicular to the rotation axis (Chandrasekhar 1969). Under the assumption that GCs can be described by the same dynamical model, such as an isotropic oblate rotator (e.g., Varri & Bertin 2012), stronger rotation would be expected in more flattened systems. However, various e ff ects can dilute a possible correlation, the most important ones being anisotropies, inclination e ff ects or tidal forces from the Milky Way (see van den Bergh 2008, for an estimate of the impact of the latter). Nev- \nertheless, Fabricius et al. (2014) and Kamann et al. (2018) were able to reveal a correlation between cluster rotation and ellipticity in a sample of Galactic GCs (see also Lanzoni et al. 2018a; Dalessandro et al. 2021a; Leanza et al. 2023 for similar analysis on specific clusters). \nFollowing on those results, we searched for any link between MP rotation and ellipticity for all clusters in our sample. The results of such a comparison for population TOT are reported in Section B. Interestingly, we find a clear and significant correlation between these two quantities (Spearman rank correlation test gives probabilities ∼ 99 . 9%), in nice agreement with previous results by Fabricius et al. (2014) and Kamann et al. (2018). \nHere we explore in some detail whether also the MP rotation patterns are possibly linked to their morphological 2D spatial distributions. We compared the ellipticity estimates obtained as \n<!-- image --> \nFig. 12. Best-fit results of the kinematic analysis of the dynamically young ( N h < 8) and old ( N h > 8) stacked samples. FP is in red and SP in blue. The lower panels refer to the LOS rotation, while the upper row shows the results for the TAN velocity component. \n<!-- image --> \ndescribed in Section 4 with the ( α )LOS MP values in Figure 13. Both populations show a positive correlation between ( α )LOS and ϵ , with Spearman rank correlation probabilities P spear ∼ 80% and larger than 99 . 9% for the FP and SP, respectively. In detail, the FP shows a pretty flat distribution of ( α FP )LOS up to ellipticity values ϵ ∼ 0 . 2. Then, ( α FP )LOS starts to increase almost linearly with ϵ . As discussed in Section 5.1, the SP tends to show larger values of rotation than the FP. Likely driven by such a stronger rotation, in the upper panel of Figure 13, we observe that ( α SP )LOS follows a nicely linear correlation with ϵ for the entire range of ellipticity values sampled by the target GCs. \nFollowing the analysis by Fabricius et al. (2014) and Kamann et al. (2018) we computed the di ff erences between the PA values obtained in Section 4 for the 2D stellar spatial distribu- \ntion and the best-fit rotation axis position angles ( PA 0). Interestingly, while the distribution of the di ff erence is pretty scattered we find that the average value for the systems in our sample is ∼ 85 · , thus implying that the stellar density distribution is on average flattened in the direction perpendicular to the rotation axis. This behavior is in general agreement with what is expected for a rotating system and it is qualitatively consistent with what predicted, for example, by the models introduced by Varri & Bertin (2012) and previously found in other observational studies (e.g., Bianchini et al. 2013; Bellini et al. 2017; Lanzoni et al. 2018a; Dalessandro et al. 2021a; Leanza et al. 2022). \nFig. 13. Distributions of the ( α )LOS parameter for the FP and SP (lower and upper panel respectively) as a function of the best-fit ellipticity values (Section 4) for the two sub-populations. \n<!-- image -->', '5.3. Dynamical simulations': "To conclude the discussion about the rotational properties of MPs in our target GCs, we briefly present the results of a set of N -body simulations aimed at exploring the evolution of rotating MP clusters. A full discussion and detailed description of the results of these simulations will be presented in White et al. (in prep.), here we just report the evolutionary path followed by the α parameter introduced in this paper to trace the strength of rotation of FP and SP stars and their di ff erence. In our simulations we focus only on the long-term dynamics driven by the e ff ects of two-body relaxation for star clusters evolving in the external tidal field of their host galaxy. Each system starts with 10 5 stars with masses between 0.1 and 1 M ⊙ distributed according to a Kroupa (2001) stellar initial mass function. Our systems start with an equal number of FP and SP stars; following the general properties emerging from a few studies of the formation of SP stars in rotating clusters (see e.g., Bekki 2010, 2011; Lacchin et al. 2022) the SP is initially more centrally concentrated and more rapidly rotating than the FP. The two populations rotate around a common axis. To explore the interplay between internal dynamics and the e ff ects due to the external tidal field we have explored models with di ff erent angles δ between the internal rotation axis and the rotation axis of the cluster orbital motion around the center of the host galaxy. In particular, we have explored systems with values of δ equal to 0 · , 45 · , 90 · , and 180 · . The simulations were run with the NBODY6++GPU code (Wang et al. 2015). In Fig. 14 we show the time 5 evolution of α FP , α SP and ( α SP -α FP ) for these models using rotational velocity profiles calculated for both the LOS (left panel) and the TAN component (right panel). For these plots we adopt an ideal line of sight perpendicular to the cluster angular momentum or parallel to it. We emphasize \nthat these simulations are not aimed at a detailed comparison with observations but they rather serve as a guide to illustrate the extent of the e ff ects of dynamical processes on the initial di ff erences between the rotational kinematics of the FP and SP populations. We also reiterate that our simulations are focussed on the e ff ects of two-body relaxation and do not include early dynamical phases such as those during which a star cluster responds to the mass loss due to stellar evolution, which can have an e ff ect on the sub-populations' dynamical di ff erences (see e.g. Vesperini et al. 2021; Sollima 2021). \nBy construction, the α values derived for the SP are significantly larger by about a factor of 2-3 than those of the FP. The results of our simulations show that the e ff ects of two-body relaxation lead to a rapid and significant reduction of the initial di ff erence between the FP and the SP rotation in the first 2-3 half-mass relaxation times reaching values of ( α SP -α FP ) similar to those found in our observational analysis. Then at later dynamical ages, ( α SP -α FP ) keeps decreasing at a lower pace and it progressively approaches values close to 0 around 10 relaxation times, when FP and SP stars rotate at the same velocity. We note that, as already discussed in Section 5.1 and in agreement with what was found in the observations, the rotation strength for both the FP and SP along with their di ff erence is stronger when a PMlike projection is considered (Fig. 14). The behavior described by the simulations is in nice general agreement with the observed trends (Figures 9, 10 and 11). Such an agreement strongly suggests that both the rotational di ff erences and the mild trend between the rotation strength and the dynamical age revealed by our observational analysis are consistent with those expected for the long-term dynamical evolution of GCs born with a SP initially rotating more rapidly than the FP.", '6.1. Observations': "Weinvestigated the di ff erences in the mean anisotropy properties of MPs by defining the following parameter: \nαβ = Z 1 0 ( β ( Rl ) X ) dRl (11) \nhere Rl represents the distance from the cluster center normalized to the value of the clusters' Jacobi radius (from Baumgardt et al. 2019) to allow a meaningful comparison among different systems. From a geometrical point of view, Equation 11 represents the area subtended by the best-fit radially normalized anisotropy profile (as defined in Section 3.2). By definition, if a sub-population is radially anisotropic, αβ will have positive values, while it will attain negative values in case of tangential anisotropy. Di ff erences between the anisotropy properties of MPs are given by ( α S P β -α FP β ). As before, errors on αβ were obtained by propagating the posterior probability distributions obtained from the MCMC analysis for the best-fit anisotropy profiles. \nSimilarly to the rotation analysis, Figure 15 shows the variation of αβ as a function of the dynamical age for the FP and SP in the bottom and middle panel, while the distribution of ( α S P β -α FP β ) is shown in the top panel. α FP β shows a flat distribution around ∼ 0 at any N h, thus suggesting that the FP is on average isotropic for all GCs in the sample, independently on their dynamical ages. On the contrary, the α S P β shows a non negligible variation as a function of time. In fact, for dynamically young \n<!-- image --> \nFig. 14. Time evolution of the rotational parameters α FP , α SP , and their di ff erences (see Section 5.1) for the simulations described in Section 5.3. The blue line corresponds to the model with δ = 0 · , the pink, cyan and purple correspond to δ = 45 · , 90 · and 180 · , respectively. \n<!-- image --> \nclusters, α S P β is positive (i.e., SP is radially anisotropic) and it becomes isotropic only for the dynamically older systems. To better highlight such a trend, we can use the two GC sub-groups defined in Section 5.1. At a face value, GCs with N h < 8 (dynamically young) have a mean α S P β value of 0 . 40 ± 0 . 20, while for dynamically older clusters the mean value is -0 . 09 ± 0 . 23 (dashed lines in Figure 15). Such a di ff erence is very nicely described by the comparison between the best-fit results obtained for MPs in the two stacked samples as shown in Figure 16. Indeed, when the anisotropy analysis is performed on the two stacked sub-samples, we find that α S P β = 0 . 23 ± 0 . 03 at dynamically young ages, while for dynamically older clusters we obtain α S P β = 0 . 04 ± 0 . 03, thus implying that the di ff erence in the average SP orbital anisotropy is significant at a ∼ 5 σ level. For comparison, we computed the same quantities for the FP on the stacked clusters. As expected, in this case mean values are virtually indistinguishable within the errors. The di ff erent behavior between the average orbital properties of FP and SP stars is further highlighted by the distribution of ( α S P β -α FP β ). In dynamically young GCs SP is systematically more radial anisotropic than the FP (mean di ff erence being 0 . 44 ± 0 . 31), then they both become compatible with being isotropic at later dynamical stages ( -0 . 08 ± 0 . 37). The analysis on the two stacked sub-samples provides values a ( α S P β -α FP β ) = 0 . 28 ± 0 . 07 and 0 . 09 ± 0 . 07 for the dynamically young and old groups, respectively, thus implying that there is a real orbital velocity distribution di ff erence between FP and SP stars with a significance of ∼ 3 . 5 σ .", '6.2. Dynamical models': "The di ff erence between the anisotropy of the FP and SP populations is consistent with the predictions of a number of theoretical investigations of the dynamics of MP clusters. These studies find that in clusters where the SP forms more centrally concen- \ntrated than the FP, the SP is generally characterized by a more radially anisotropic velocity distribution than the FP (see e.g., Bellini et al. 2015; Hénault-Brunet et al. 2015; Tiongco et al. 2019; Vesperini et al. 2021; Sollima 2021). In order to illustrate the expected strength and evolution of the di ff erences between the anisotropy of the FP and SP populations, as measured by the parameter ( α S P β -α FP β ) introduced in this paper, we show in Figure 17 the time evolution of α FP β , α S P β and ( α S P β -α FP β ) for three Monte Carlo simulations following the dynamical evolution of MP clusters with di ff erent initial conditions. The Monte Carlo simulations were run with the MOCCA code (Hypki & Giersz 2013; Giersz et al. 2013; see also Vesperini et al. 2021; Hypki et al. 2022 for the specific case of GCs with MPs). A full presentation of the kinematics and phase space properties of these systems will be presented in a separate paper (Aros et al. in prep.). The three models for which we present the anisotropy evolution in Fig. 17 all start with a ratio of the SP to total mass equal to 0 . 2. The total number of stars is equal to N = 10 6 (hereafter we use the id. N1M for these model) or 0 . 5 × 10 6 (with id. N05M ) and stars have been extracted from a Kroupa (2001) initial stellar mass function between 0.1 and 100 M ⊙ . The SP is initially more concentrated than the FP: for two models the initial ratio for the FP to SP half-mass radii is equal to 20 ( N1Mr20 and N05Mr20 ) and is equal to 10 for one model ( N1Mr10 ). Both populations are initially characterized by a non-rotating, isotropic velocity distribution. All models include the e ff ects of mass loss due to stellar evolution, two-body relaxation, binary interactions and a tidal truncation (see Vesperini et al. 2021 for further details on the evolutionary processes included in these simulations). As shown in Figure 17, in all models the SP is characterized by a more radially anisotropic velocity distribution than the FP. Even if both populations are initially isotropic, the development of a stronger radial anisotropy for the SP is the consequence of its initial more centrally concentrated spatial distribution (see e.g., Tiongco et al. 2019; Vesperini et al. 2021). The di ff erences in anisotropy \nFig. 15. The bottom and middle panels show the distribution of the anisotropy parameter αβ for the FP and SP sub-populations of GCs in the sample (grey circles) as a function of N h. The top panel, shows the di ff erential trend α SP β -α FP β . The dashed lines represent the mean of the observed αβ values derived for the dynamically young ( N h < 8) and old ( N h > 8) GCs. The starry symbols are the result of the analysis on the two stacked samples and their sizes correspond to the errorbars. \n<!-- image --> \nbetween the two populations gradually decrease as the systems evolves. As shown by the results of our simulations, the extent of the di ff erences between the FP and SP anisotropy depends on the initial relative concentration of the two populations being smaller for the model where the initial ratio of the FP to SP half-mass radii equal to 10 ( N1MR10 ) than that for models where this ratio is initially equal to 20 ( N1MR20 ). While also in this case we point out that these simulations have not been specifically designed to provide a detailed fit to observations but rather to provide guidance in the general interpretation of the observational results, it is interesting to note that the extent of the di ff erence between the anisotropy of the FP and the SP popula- \ntions as measured by the ( α S P β -α FP β ) and its time evolution is generally consistent with the findings of our observational analysis (Figure 15). The general agreement between observations and simulations lends further support to the interpretation of the observed di ff erence between the FP and SP anisotropy and its variation with the clusters' dynamical age as the manifestation of the di ff erent dynamical paths followed by subsystems with di ff erent initial spatial distributions. \nFig. 16. Best-fit results of the anisotropy analysis of MPs in the two stacked samples. The upper panels shows the best-fit anisotropy profiles of the FP (red) and SP (blue) sub-populations as obtained for the stacked sample of dynamically young GCs, while the lower panel refers to dynamically old systems. \n<!-- image -->", '7. Comparison with the literature': "In Appendix C we report on a quantitative comparison between the results obtained in the present work and those available in literature for the TOT population in each cluster. Here we detail on the comparison with previous works focusing on the kinematics of MPs. We stress that the full 3D kinematic analysis presented in this paper is the first ever obtained for MPs. Hence, in the following our comparison will be limited to studies considering a single velocity component. \nOur sample has 6 GCs in common with the recent analysis by Martens et al. (2023) based on MUSE LOS velocities. The authors were able to find MP best-fit solutions for both the velocity dispersion and rotation profiles for three of them (namely 47 Tuc, \nNGC5904 and NGC 6093), while for NGC 3201 and NGC 6254 they report only conservative upper limits for the MP rotation amplitudes, and for NGC 1904 they provide information only for the TOT population. Figure 18 shows the distributions of the di ff erences in terms of σ 0 (which is defined as σ max in Martens et al. 2023), A rot ( v max in Martens et al. 2023) and R peak for the FP and SP sub-populations. A good agreement is observed both for the MP central velocity dispersion values (left panel of Figure 18) and the rotation amplitude (middle panel), with the most discrepant value being that corresponding to the rotation of the SP in 47 Tuc. Within the errors there is also a reasonable match between the values of R peak. We note however that, while the sample is certainly small, the estimates of R peak by Martens et al. \nTable 2. MP kinematic parameters describing the radial anisotropy and rotation along the LOS, PM and 3D components. \n(2023) tend to be slightly smaller than those derived in this work. This might be somehow linked to the smaller radial coverage of the Martens et al. (2023) analysis. In fact, all values derived by Martens et al. (2023) for the clusters in common are located well outside the MUSE field of view and therefore they might be only partially constrained by their analysis. \nThe sample analyzed in this work counts also 4 GCs in common with Cordoni et al. (2020), namely 47 Tuc, NGC 288, NGC 5904 and NGC 6254. Our results are in agreement with \ntheirs in that 47 Tuc and NGC 5904 are the systems showing the larger rotation among the clusters in common. However, at odds with their results, we also find that in both GCs the SP shows a larger rotation (as inferred both by A rot and α values) than the FP. Also, within the uncertainties, we do not find evidence of any significant mis-alignment of the FP and SP rotation curves for these systems, neither in terms of position angles and inclination, as constrained by both the LOS and the full 3D analysis. \nFig. 17. Time evolution of the anisotropy parameters defined in Section 6.1 for the FP (bottom panel), the SP (middle panel) and their di ff erence (top panel) in the three Monte Carlo simulations described in Section 6.2 (N1Mr10 blue line, N1M purple line; N05M pink line). Time is normalized by the half-mass relaxation time. \n<!-- image --> \nFinally, our analysis is in nice agreement with the results presented by Cordero et al. (2017) for the MP rotation patterns of M13. \nAs for the anisotropy, the results obtained in this work are in qualitative agreement with those found for the GCs M 13, NGC 2808 and 47 Tuc by Richer et al. (2013); Bellini et al. (2015); Milone et al. (2018). More in general, they are in nice qualitative agreement with those recently obtained in the innermost regions (R < 100 '' ) for a sample of Galactic GCs by Libralato et al. (2023) by using HST PMs to search for average di ff erences between the anisotropy profiles of MPs.", '8. Summary and conclusions': "Wehave presented the first self-consistent 3D kinematic analysis of MPs for a sample of Galactic GCs. The study targets 16 systems spanning a broad range of dynamical ages (2 < N h < 25) and it is based on a large and mostly homogeneous observational dataset securing several hundreds of accurate LOS velocities and PMs for each cluster and sampling virtually their entire extension. \nOur study is mainly focused on the analysis of the MP rotation along the three velocity components and the anisotropy of their velocity distributions. The adopted approach is aimed at providing new insights into the long term evolution of the kinematic properties of MPs (and their di ff erences) for the entire sample of GCs and for the entire dynamical age covered by our analysis instead of focusing on the kinematic di ff erences in specific clusters. To this aim, starting from the observed velocity distributions we defined a few key quantities, to quantitatively and homogeneously compare the results obtained for all the observed GCs. \nOur analysis provides the first observational determination of the dynamical path followed by MP kinematic properties during \ntheir long-term evolution. The main observational results we find can be schematically summarized as follows. \n- -We observe evidence of di ff erential rotation between MPs with the SP preferentially rotating more rapidly than the FP. This result is consistent (although with di ff erent amplitudes) along both the LOS and TAN velocity components, as well as in the full 3D analysis. In all GCs in our sample we find that the rotation position and inclination angles are consistent within the uncertainties between FP and SP.\n- -The strength of the rotation signal of both FP and SP subpopulations nicely correlate with the ellipticity values derived for the two sub-populations. In addition, we find that the rotation axis position angle is typically perpendicular to the ellipses major axis.\n- -The di ff erence in the rotation strength between MPs is mildly anti-correlated with the cluster dynamical age. In particular, di ff erences are larger for dynamically young clusters and they become progressively indistinguishable as dynamical evolution proceeds.\n- -Stars belonging to di ff erent sub-populations show di ff erent average velocity distribution properties. FPs are characterized by isotropic velocity distributions at any dynamical age probed by our sample. On the contrary, the SP of dynamically young GCs have a radially anisotropic velocity distribution which then becomes isotropic in more advanced evolutionary stages. \nThe combination of these results with the analysis of the MP radial distributions of a representative sample of GCs carried out in our previous paper (Dalessandro et al. 2019), provides a full picture of the present-day kinematic and structural properties of MPs in GCs and of their evolution. The comparison with dynamical models following the long-term evolution of MPs in GCs, suggests that these properties, and their evolution with dynamical age are in general good agreement with those expected in clusters forming with a SP subsystem initially more centrally concentrated and more rapidly rotating than the FP (see e.g., D'Ercole et al. 2008; Bekki 2010; Calura et al. 2019; Lacchin et al. 2022). In turn, this would possibly suggest (see e.g., Hénault-Brunet et al. 2015 and discussion in Martens et al. 2023) that GCs experienced multiple events of star formation during their early phases of evolution, with the rotation properties being the more stringent discriminating factors. In fact, according to multi-epoch formation models, the SP is expected to form a low-mass, more centrally concentrated and more rapidly rotating stellar sub-system than the FP. In such a configuration, dissipative accretion processes of material ejected by FP stars and angular momentum conservation in sub-systems with different initial concentration can produce a larger initial rotation of the SP sub-system (Bekki 2011; Hénault-Brunet et al. 2015; Tiongco et al. 2017). Interestingly, it has been shown (e.g., Bekki 2011) that even if only a very small fraction of the kinetic energy of the FP is in the form of bulk rotation energy, SP stars can acquire a much stronger rotation than what remains in the FP. \nIt is important to note that, as shown in this paper and in a number of previous studies, early and long-term evolution can significantly reduce the initial di ff erences between the FP and SP rotational velocities making them virtually indistinguishable in dynamical old systems. The combination of these physical e ff ects with the observational uncertainties arising from the limited available stellar samples, partial cluster coverage and possible biases introduced by the di ff erent rotation inclination angles, can make extremely di ffi cult to capture present-day kinematic \n<!-- image --> \n<!-- image --> \nFig. 18. Comparison between the best-fit σ 0, A rot and R peak values obtained for MPs in the present-work and by Martens et al. (2023). Blue dots refer to SP and red ones to FP results. \n<!-- image --> \ndi ff erences between MPs, in particular in the LOS velocity component. As a consequence, it is important to use some caution in drawing conclusions about the physical mechanisms at the basis of GC formation and early evolution based on the presentday kinematic and structural properties of individual systems or small samples. \nAn homogeneous combination between data obtained with multi-object spectrographs and Gaia (as presented in this paper) which are particularly sensitive to the intermediate and large cluster-centric distances in GCs, with Integral Field Unit LOS RVs and HST absolute PMs sampling more e ffi ciently the clusters' innermost regions (see for example Martens et al. 2023; Libralato et al. 2023), along with an increase of the cluster sample size, represents a natural and promising next step that could potentially enable the exploration of additional physical ingredients at play. \nThe results of this study clearly demonstrates that significant advances in our understanding of cluster formation and early evolution is only possible through a multi-faceted and multidiagnostic approach and by combining state-of-art observations and simulations. \nAcknowledgements. The authors thank the anonymous referee for the careful reading of the paper and the useful comments that improved the presentation of this work. E.D. acknowledges financial support from the Fulbright Visiting Scholar program 2023. E.D. and A.D.C. are also grateful for the warm hospitality of the Indiana University where part of this work was performed. E.D. and M.C. acknowledge financial support from the project Light-on-Dark granted by MIUR through PRIN2017-2017K7REXT. E.V. acknowledges support from NSF grant AST-2009193. E.V. acknowledges also support from the John and A-Lan Reynolds Faculty Research Fund. The research activities described in this paper have been co-funded by the European Union - NextGenerationEU within PRIN 2022 project n.20229YBSAN - Globular clusters in cosmological simulations and in lensed fields: from their birth to the present epoch.", 'References': "Aarseth, S. J. 2003, Gravitational N-Body Simulations, by Sverre J. Aarseth, pp. 430. ISBN 0521432723. 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Best-fit kinematic parameters for the MPs in the target GCs.', 'Appendix B: Incompleteness effects': "We constrained the possible impact of the kinematic samples' size and of their (radially dependent) incompleteness (mainly caused by the intrinsically limited allocation e ffi ciency of multiobject spectrographs) on the results obtained in this work by using the dynamical simulations described in Sections 5 and 6. \nIn detail, we estimated the completeness ( C ) of the observed kinematic samples and its radial variation as the ratio between the number of RGB stars detected in the photometric catalogs and those with LOS RVs and / or PMs measures within concentric radial annuli at di ff erent cluster-centric distances. While we acknowledge that these estimates represent a lower-limit to the real incompleteness, as photometric catalogs are not fully complete, we note that our targets are RGB stars, which are among the brightest stars in GC CMDs and therefore they are only moderately a ff ected by incompleteness even when ground-based catalogs are adopted. \nWe then extracted randomly from the simulations subsamples of stars with similar sizes as the observed ones for a large number of times. We applied the derived completeness curves to these sub-samples to make them as similar as possible to the observed catalogs. Finally we run the same kinematic analysis described in Section 3. We find that, while the limited sample sizes and incompleteness have an impact on the overall noise of the observed kinematic profiles and as a consequence on the uncertainties associated to the derived parameters, they do not have a significant impact on the final results.", 'Appendix C: Global kinematics and comparison with the literature': 'Figure C.1 shows the distribution of the α values derived for the entire population along both the LOS and TAN velocity components, and of the ω TOT 3 D , as a function of N h. As expected, both α TOT (for both LOS and TAN ) and the ω TOT 3 D distributions show pretty clear anti-correlations with N h. In fact, while dynamically young GCs attain larger values of rotation parameters, with the only significant exception being M 3, the rotation strength progressively decreases as dynamical evolution proceeds. By performing a Spearman rank correlation test we find that such anti-correlations have a significance of ∼ 98% for the LOS and > 99 . 9% for both the TAN and the 3D analysis, with the 3D case being the most significant. In general, these results are in very good agreement with previous analysis (e.g., Kamann et al. 2018; Sollima et al. 2019) and they further strengthen the conclusion that the present-day cluster rotation is the relic of that imprinted at the epoch of cluster formation, which has been then progressively dissipated via two-body relaxation. \nFigure C.2 shows the distribution of α TOT LOS with the cluster ellipticity obtained as described in Section 3. As expected (see discussion and references in Section 5.2), a nice correlation between rotation and ellipticity is observed also for the total population in very good agreement with previous findings by Fabricius et al. (2014); Kamann et al. (2018). \nWhile the focus of this work is on the MP kinematics, nevertheless it is useful to compare the results obtained for the TOT population with those largely available in the literature to have an indication about the general performance of the adopted approach and data-sets. \nDetailed one-to-one comparisons with recent results obtained in the literature (Bellazzini et al. 2012; Ferraro et al. 2018; Lanzoni et al. 2018a,b; Baumgardt et al. 2019; Sollima et al. 2019) for the TOT population are shown in Figures C.3 and C.4. \nFig. C.1. Distribution of the three rotation parameters defined in this work for the LOS , PM and 3D velocity components, as a function of the dynamical age ( N h) for the total population (TOT) of GCs in our sample. \n<!-- image --> \nFig. C.2. Distribution of the ( α TOT )LOS parameter for the total population as a function of the best-fit ellipticity values derived as described in Section 4. \n<!-- image --> \nIn general, a quite good agreement is observed with all the compilations considered here. \nOur sample has 6 GCs in common with Bellazzini et al. (2012). A nice match is observed both in terms of σ 0 and A rot (top row of Figure C.3) with the only exception being NGC 6171 for which Bellazzini et al. (2012) finds a rotation amplitude 4-5 times larger than the one obtained in this work. Given the estimate by Bellazzini et al. (2012), NGC 6171 would be a very fast rotator, with Arot /σ 0 ∼ 0 . 7. However, it is important to note, that the sample of LOS RVs used by Bellazzini et al. (2012) for this cluster includes only 31 stars in total, resulting the smallest sample of LOS RVs in their analysis. Here we sample the kinematic profile of NGC 6171 with 184 LOS RVs (see Table 1). We note also that NGC 6171 results to have a significantly smaller rotation amplitude (1 . 2 km / s) than what found by Bellazzini et al. (2012) in the analysis by Ferraro et al. (2018) and it is classified as non rotator by Sollima et al. (2019) \nAs for the comparison with results by Ferraro et al. (2018), we stress that while the spectroscopic sample is largely similar, the adopted kinematic analysis (both the discrete and continuous \n<!-- image --> \nFig. C.3. Comparison between the best-fit σ 0 and A rot values obtained for the TOT sample for the clusters in common between the present work and Bellazzini et al. (2012) - B12, Ferraro et al. (2018) - F18 and Petralia et al. (2024) - P24. \n<!-- image --> \nones) is significantly di ff erent for the rotation study in particular (see Ferraro et al. 2018 for details). Hence, it is not surprising that while the derived central velocity dispersion values are in excellent agreement for the entire sample (middle row of Figure C.3), the distribution of di ff erences for A rot is more scattered, while still showing a reasonable match within the errors. In this case, the most significant discrepancy is observed for NGC 5927, for which Ferraro et al. (2018) derived A rot = 2 . 3 km / s, while we find Arot = 0 . 76 + 0 . 90 -0 . 54 km / s. For this cluster also Sollima et al. (2019) derived a low probability of rotation. \nIn the bottom row of Figure C.3 we compare the results of this work with those recently obtained by Petralia et al. (2024) by using APOGEE spectra for a sample of Galactic GCs. For A rot we use the semi-amplitude of the A fit values reported in their work. A reasonable overall agreement is observed also in this case for the clusters in common, however 47 Tuc and NGC 5904 result to have larger central velocity dispersion values and peak of rotation than in our work. \nFinally, as shown in Figure C.4 (left panel) a reasonably good match is also found with the σ 0 estimates by Baumgardt \n<!-- image --> \nFig. C.4. Left panel: comparison with the best-fit σ 0 values obtained by Baumgardt et al. (2019) - B19 - for the total population of GCs in common with the present work. Right-panel: one-to-one comparison of the 3D rotation amplitude values A 3D found by Sollima et al. (2019) S19 - and the present analysis. \n<!-- image --> \net al. (2019). Among the clusters in common with Sollima et al. (2019), the only significantly discrepant result is that of 47 Tuc, which results to have a ∼ 25% larger rotation in this work. However, we note that in this case, as for the entire sample, both the i and θ 0 values are in very good agreement. In this respect, it is also interesting to highlight the nice match in terms of both the observed rotation amplitude and angles of the 3D rotation of 47 Tuc obtained in this work and those inferred by means of a detailed comparison between HST PMs and theoretical models of rotating clusters by Bellini et al. (2017).'} |
2024MNRAS.534..695C | As part of our comprehensive ongoing characterization of the lowmass end of the main sequence in the Solar neighbourhood we used the OSIRIS instrument at the 10.4 m Gran Telescopio Canarias to acquire low and midresolution Rinlineformulatexmath idTM0001 notationLaTeXapprox texmathinlineformula300 and Rinlineformulatexmath idTM0002 notationLaTeXapprox texmathinlineformula2500 optical spectroscopy of 53 lateM and L ultracool dwarfs. Most of these objects are known but poorly investigated and lacking complete kinematics. We measured spectral indices determined spectral types six of which are new and inferred effective temperature and surface gravity from BTSettl synthetic spectra fits for all objects. We were able to measure radial velocities via line centre fitting and cross correlation for 46 objects 29 of which lacked previous radial velocity measurements. Using these radial velocities in combination with the latest Gaia DR3 data we also calculated Galactocentric space velocities. From their kinematics we identified two candidates outside of the thin disc and four in young stellar kinematic groups. Two further ultracool dwarfs are apparently young field objects 2MASSW J1246467402715 L4 which has a potential weak lithium absorption line and G 1963B L3 which was already known as young due to its wellstudied primary companion. | 2024-10-01T00:00:00Z | ['2024MNRAS.534..695C', 'arXiv:2409.03706', '10.48550/arXiv.2409.03706', '10.1093/mnras/stae2102', '2024MNRAS.tmp.2121C', '2024arXiv240903706C'] | ['Astrophysics - Solar and Stellar Astrophysics', 'Astrophysics - Earth and Planetary Astrophysics', 'Astrophysics - Astrophysics of Galaxies'] | The Gaia ultracool dwarf sampleIV. GTCOSIRIS optical spectra of Gaia lateM and L dwarfs | 2,024 | 173 | 0.58 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF'] | 0 | https://arxiv.org/pdf/2409.03706.pdf | {'The Gaia Ultracool Dwarf Sample - IV. GTC/OSIRIS optical spectra of Gaia late-M and L dwarfs': '- W.J. Cooper, 1 , 2 ★ H. R. A. Jones, 1 R. L. Smart, 2 S. L. Folkes, 1 J. A. Caballero, 3\n- F. Marocco, 4 M.C. Gálvez Ortiz, 3 A. J. Burgasser, 5 J. D. Kirkpatrick, 4 L. M. Sarro, 6\n- B. Burningham, 1 A. Cabrera-Lavers, 7 P. E. Tremblay, 8 C. Reylé, 9 N. Lodieu, 10 , 11\n- Z. H. Zhang 12 , 13 N. J. Cook, 14 J. F. Faherty, 15 D. García-Álvarez, 7 , 10\n- D. Montes, 16 D. J. Pinfield, 1 A. S. Rajpurohit, 17 J. Shi 18 , 19\n- 1 Centre for Astrophysics Research, University of Hertfordshire, Hatfield, Hertfordshire, AL10 9AB, UK\n- 2 Istituto Nazionale di Astrofisica, Osservatorio Astrofisico di Torino, Strada Osservatorio 20, I-10025 Pino Torinese, IT\n- 3 Centro de Astrobiología (CAB), CSIC-INTA, Camino Bajo del Castillo s/n, Campus ESAC, E-28692 Villanueva de la Cañada, Madrid, ES\n- 4 IPAC, Mail Code 100-22, Caltech, 1200 E. California Boulevard, Pasadena, CA 91125, US \n5 \nCenter for Astrophysics and Space Science, University of California San Diego, La Jolla, CA 92093, US \n- 6 Departamento de Inteligencia Artificial, ETSI Informática, UNED, Juan del Rosal, E-16 28040 Madrid, ES\n- 7 GRANTECAN, Cuesta de San José s/n, E-38712, Breña Baja, La Palma, ES\n- 8 Department of Physics, University of Warwick, Coventry CV4 7AL, UK\n- 9 Institut UTINAM, CNRS UMR6213, Université de Bourgogne Franche-Comté, OSU THETA Franche-Comté-Bourgogne,\n- Observatoire de Besançon, BP 1615, 25010, Besançon Cedex, FR\n- 10 Instituto de Astrofísica de Canarias, E-38205 La Laguna, Tenerife, ES\n- 11 Universidad de La Laguna, Departamento de Astrofísica, E-38206 La Laguna, Tenerife, ES \n12 \nSchool of Astronomy and Space Science, Nanjing University, 163 Xianlin Avenue, Nanjing 210023, CN \n- 13 Key Laboratory of Modern Astronomy and Astrophysics, Nanjing University, Ministry of Education, Nanjing 210023, CN\n- 14 Institute for Research on Exoplanets, Université de Montréal, Département de Physique, C.P. 6128 Succ. Centre-ville, Montréal, QC H3C 3J7, CA\n- 15 Department of Astrophysics, American Museum of Natural History, Central Park West at 79th Street, NY 10024, US\n- 16 Departamento de Física de la Tierra y Astrofísica & IPARCOS-UCM (Instituto de Física de Partículas y del Cosmos de la UCM),\n- Facultad de Ciencias Físicas, Universidad Complutense de Madrid, E-28040 Madrid, ES\n- 17 Astronomy and Astrophysics Division, Physical Research Laboratory, Navrangapura, Ahmedabad, 380009, IN\n- 18 College of Astronomy and Space Sciences, University of Chinese Academy of Sciences, Beijing 100049, CN\n- 19 Key Laboratory of Optical Astronomy, National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, CN \nAccepted 2024 September 5. Received 2024 September 5; in original form 2023 July 7', 'ABSTRACT': 'As part of our comprehensive, ongoing characterisation of the low-mass end of the main sequence in the Solar neighbourhood, we used the OSIRIS instrument at the 10.4 m Gran Telescopio Canarias to acquire low- and mid-resolution (R ≈ 300 and R ≈ 2500) optical spectroscopy of 53 late-M and L ultracool dwarfs. Most of these objects are known but poorly investigated and lacking complete kinematics. We measured spectral indices, determined spectral types (six of which are new) and inferred effective temperature and surface gravity from BT-Settl synthetic spectra fits for all objects. We were able to measure radial velocities via line centre fitting and cross correlation for 46 objects, 29 of which lacked previous radial velocity measurements. Using these radial velocities in combination with the latest Gaia DR3 data, we also calculated Galactocentric space velocities. From their kinematics, we identified two candidates outside of the thin disc and four in young stellar kinematic groups. Two further ultracool dwarfs are apparently young field objects: 2MASSW J1246467 + 402715 (L4 𝛽 ), which has a potential, weak lithium absorption line, and G 196-3B (L3 𝛽 ), which was already known as young due to its well-studied primary companion. \nKey words: stars: brown dwarfs - stars: kinematics and dynamics - stars: late-type', '1 INTRODUCTION': "Ultracool dwarfs (UCDs) are objects with effective temperatures 𝑇 eff ⪅ 2700 K (spectral type ≳ M7V, Kirkpatrick et al. 1999) con- \non from the low-mass tail of the main sequence, that consist of spectral types late-M, L, T and Y dwarfs. These UCDs consist of a combination of low-mass stars and brown dwarfs. Brown dwarfs are sub-stellar objects incapable of hydrogen fusion and are defined by mass, between the deuterium minimum mass burning limit, ∼ 13 Jupiter masses (Saumon et al. 1996; Chabrier et al. 2000) and the hydrogen minimum mass burning limit, ∼ 72 Jupiter masses (Chabrier & Baraffe 1997; Baraffe et al. 1997). The majority of known UCDs are within the Solar neighbourhood (e.g. Smart et al. 2019; Kirkpatrick et al. 2021; Sarro et al. 2023) with typically dim apparent optical magnitudes ( Gaia 𝐺 ⪆ 17 mag). The closest stars to the Sun have been catalogued throughout the history of astronomy. For example, the Catalogue of Nearby Stars (CNS) from Gliese (1957) has been updated with every all-sky photometric and astrometric survey, including the most recent release using Gaia DR3 data (CNS5, Golovin et al. 2023). This Solar neighbourhood has been further described in the 'The Solar Neighborhood' series by the Research Consortium on Nearby Stars (RECONS 1 ) team with publications from Henry et al. (1994) to Vrijmoet et al. (2022). Specifically, M dwarfs within 30 pc were covered in another series of articles from Delfosse et al. (1999) to Crifo et al. (2005). Volume limited samples such as the recent Gaia Collaboration et al. (100 pc, 2021b), Kirkpatrick et al. (20 pc, 2021) and Reylé et al. (10 pc, 2021) works provide important constraints on the initial mass function (Salpeter 1955; Scalo 1986; Kroupa 2001; Chabrier 2003), which underpins all aspects of astrophysics from stars to galaxies to cosmology. \nSpectral features of low mass stars, M, L and T dwarfs, and their definitions were initially described by Tinney & Reid (1998), Kirkpatrick et al. (1999), Martín et al. (1999b), Burgasser et al. (2002), Geballe et al. (2002) and Kirkpatrick (2005). The bulk of the flux emitted by L dwarfs lies in the near infrared (NIR) and continues strongly towards the mid-infrared spectral regions for later spectral type UCDs. However, several features of youth, e.g. a weak sodium doublet, 𝜆𝜆 8183,8195 Å (Schiavon et al. 1997a), are apparent in mid- to high-resolution optical spectra. Additionally, in the optical regime features such as the 𝜆 9850-10200 Å FeH Wing-Ford band (Schiavon et al. 1997b) can be seen, which can be indicative of low or high metallicity. Optical spectra have an advantage in that there are fewer and weaker telluric absorption bands than in ground-based infrared spectra, where water and oxygen bands can dominate (Reiners et al. 2007; Smette et al. 2015). However, only the closest and brightest UCDs can be observed with optical spectroscopy due to the low relative flux; further and fainter UCDs require large aperture telescopes and long exposure times. \nUCDs have typically been selected from photometric criteria using optical and near- to mid-infrared imaging surveys, supported by proper motion analysis. Examples of optical surveys include SuperCOSMOS (Hambly et al. 2001), Gaia (Gaia Collaboration et al. 2016), Pan-STARRS (PS1, Chambers et al. 2016) and the SDSS (York et al. 2000; Abazajian et al. 2009), in which UCDs appear red. Notable infrared surveys and catalogues include 2MASS (Cutri et al. 2003; Skrutskie et al. 2006), DENIS (Epchtein et al. 1997), VISTA's VVV/VIRAC/VHS (Minniti et al. 2010; Smith et al. 2018; McMahon et al. 2021) and UKIDSS (Lawrence et al. 2007). Further infrared is the WISE (Wright et al. 2010) survey, which was expanded upon in the unWISE/catWISE (Schlafly et al. 2019; Marocco et al. 2021; Meisner et al. 2023) catalogues. These NIR surveys are complemented by additional surveys constraining \nUCDs in open clusters such as the Pleiades (Steele & Jameson 1995; Pinfield et al. 2000; Lodieu et al. 2012), or elsewhere (Lucas & Roche 2000; Zapatero Osorio et al. 2000; Burningham et al. 2013). \nThe photometry of UCDs is important because the change in colour across the optical and NIR regime (Leggett et al. 2002) correlates with physical and atmospheric properties. These changing processes, such as dust, condensate cloud formation and subsequent clearing as an atmosphere cools, are well covered in the literature (e.g. Marley et al. 2002; Dahn et al. 2002; Saumon & Marley 2008). Understanding a changing atmosphere for different ages with a range of masses has allowed the computing of 'cooling tracks' (Burrows et al. 1997; Baraffe et al. 2015). Accounting for theoretical atmospheric physics has been used in model grids such as BT-Settl (Allard et al. 2011), or Sonora (Marley et al. 2021; Karalidi et al. 2021), and when interpreting the results of retrieval techniques (e.g. Burningham et al. 2017; Calamari et al. 2022). Being able to constrain the mass and/or age has underpinned modern observational UCD astronomy, but is challenging due to the mass/age degeneracy (Burrows et al. 1997). For example, benchmark systems (e.g. Pinfield et al. 2006; Dupuy et al. 2009) allow us to constrain the age of a brown dwarf with the coeval main sequence primary. The metallicity and surface gravity of an object of a given spectral type are the major variables affecting the photometric colour (Stephens et al. 2009), see references to 'blue' and 'red' L dwarfs (e.g. Faherty et al. 2009; Schmidt et al. 2010). Any works that infer spectral type, surface gravity and effective temperature must take into account the atmospheric physics, as these directly correlate with observable features. \nGaia is a European Space Agency mission, launched in 2013 to make high-precision measurements of positions, parallaxes, and proper motions of well over a billion sources and photometry in three different photometric filters ( 𝐺 BP , 𝐺 , 𝐺 RP ). The third Gaia data release (EDR3 and DR3 - Gaia Collaboration et al. 2021a, 2023, respectively) containing astrometric and photometric measurements, was in December 2021, with the remaining measurements and inferred parameters, including spectra, in June 2022 2 . \nObtaining the full 6D (right ascension, declination, proper motions, parallax, radial velocity: 𝛼, 𝛿, 𝜇 𝛼 cos 𝛿, 𝜇 𝛿 , 𝜛, 𝑣 𝑟 ) positional and kinematic information is fundamental to fully characterise the populations of UCDs within a volume limited sample (e.g. Best et al. 2021). Precise measurements of radial velocities (RVs) are obtained from high signal-to-noise observations taken with high resolution spectrographs with resolving powers of R ∼ 100 000, leading to uncertainties ∼ 1-5 m s -1 . This has only been achievable for the nearest, brightest UCDs (e.g. Zechmeister et al. 2019). Blake et al. (2010) achieved 𝛿𝑣 𝑟 ≈ 50-200 m s -1 with the Keck Near-Infrared Spectrometer (NIRSPEC), which had a resolution of R ≈ 25 000. The 'Brown Dwarf Kinematics Project' has gathered further UCD RVs(Burgasser et al. 2015; Hsu et al. 2021) with both the NIRSPEC and the Magellan Echellette (MagE, R ∼ 4100, 𝛿𝑣 𝑟 ≈ 2-3 km s -1 ) spectrographs. By comparison, the lower-resolution spectroscopy such as those discussed in this work (R ≈ 2500) is only capable of theoretical minimum uncertainties of ≳ 5 km s -1 ; this is still useful when constraining the kinematics of the Solar neighbourhood. Parallaxes and proper motions of UCDs were historically gathered from ground based time-domain campaigns (e.g. PARSEC: \nAndrei et al. 2011; Marocco et al. 2013; Smart et al. 2018) that have been generally superseded by Gaia for the brightest objects, 𝐺 ⪅ 20 mag. In the case of most late-L and T dwarfs, ground-based astrometry is still the predominant source (e.g. Vrba et al. 2004; Dupuy & Liu 2012; Liu et al. 2016; Best et al. 2018). For dimmer objects, beyond mid-L dwarfs, parallaxes and proper motions are gathered by space-based infrared surveys and are analysed in-depth by Kirkpatrick et al. (2021). Young moving groups are constrained using these complete kinematics. See the BANYAN Σ series and references therein for detail on nearby young moving groups and clusters (Gagné et al. 2014, to Gagné & Faherty 2018) or similarly, the LACEwING code (Riedel et al. 2017), designed around young objects in the Solar neighbourhood. Subdwarfs, meanwhile, are characterised by their statistically higher space velocities indicative of the older population (e.g. Lodieu et al. 2005; Burgasser et al. 2007; Lodieu et al. 2017; Zhang et al. 2017). \nThis is the fourth item in the Gaia UltraCool Dwarf Sample series (GUCDS, Smart et al. 2017, 2019; Marocco et al. 2020) and is an ongoing, international, multi-year programme aimed at characterising all of the UCDs visible to Gaia . Gaia DR3 produced astrophysical parameters for ≈ 470 million sources (Creevey et al. 2023), including effective temperatures, 𝑇 eff . The ≈ 94 000 Gaia DR3 𝑇 eff values relating to UCDs by Creevey et al. (2023) were provided under the teff\\_espucd keyword. The full sample of UCDs detected by Gaia with Gaia DR3 𝑇 eff values were documented and analysed by Sarro et al. (2023). In our analysis, we will use the values from these Gaia DR3 derivative works to compare with the equivalent values directly measured by us. There is significant overlap between the Sarro et al. (2023) sample and the GUCDS, although the majority of UCD sources as seen by Gaia are as yet not characterised through spectroscopic follow-up. A subset of this Sarro et al. (2023) sample has public Gaia RP spectra (see the Gaia xp\\_summary table 3 ), which covers the 𝐺 RP passband ( Δ 𝜆 ≈ 6200-10420 Å, Riello et al. 2021). This subset from Sarro et al. (2023) was further analysed for spectroscopic outliers by Cooper et al. (2024). The internally calibrated Gaia RP spectra and processing were discussed thoroughly by Carrasco et al. (2021), De Angeli et al. (2023) and Montegriffo et al. (2023). \nThe aim of this work is to complement the literature population with measurements and inferences from low- and mid-resolution optical spectroscopy. In Section § 2 we explain the target selection ( § 2.1) and observation strategy ( § 2.2). Different reduction techniques with a test case are discussed in Section § 3. Section § 4 explains our techniques for determining spectral types ( § 4.1), astrophysical parameters ( § 4.2), and kinematics ( § 4.3) including membership in moving groups ( § 4.4). Section § 5 follows a discussion of our results for spectral types ( § 5.1), kinematics ( § 5.2) and astrophysical parameters ( § 5.3). We also discuss individual objects ( § 5.3.1) before summarising the overall conclusions in Section § 6.", '2 DATA COLLECTION': 'We obtained optical spectroscopy of 53 unique UCDs using the OSIRIS (Optical System for Imaging and low-intermediate Resolution Integrated Spectroscopy - Cepa 1998) instrument on the 10.4 m Gran Telescopio Canarias (GTC) at El Roque de los Muchachos in the island of La Palma, Spain, under proposal IDs GTC54-15A \nand GTC8-15ITP (PIs Caballero and Marocco, respectively). The objects were observed in semesters 2015A, 2015B and 2016A. \nThe observed data from the GTC were complemented with Gaia DR3. Gaia also carries a radial velocity spectrometer, although this was unsuitable for our purposes as all of our targets were fainter than the Gaia selection limit (Katz et al. 2023, 𝐺 < 14 mag,).', '4 W.J. Cooper et al.': 'Table 1: The 53 targets observed at the GTC with OSIRIS and presented in this work. \nReferences - Positions all at 2016.5 except at the indicated epochs: 1. Lawrence et al. (2007) - 2008, 2. Skrutskie et al. (2006) - 1998-2000, \n- 3. Chambers et al. (2016) - 2012-2013, 4. Best et al. (2020) - 2014-2018, 5. Weinberger et al. (2016) - 2007-2013. \nWe acquired 63 spectra in which we observed 53 unique objects, shown in Table 1. These 63 observations are shown in Table A1, including the air mass and humidity of the observation. Of the 63 spectra, 46 were observed with the R2500I volume phased holographic grating (hereafter VPHG), whilst 17 were observed with the R300R grism. Ten of the 53 objects were observed with both dispersive elements. \nTwenty of the 53 objects already had full 6D positional and kinematic information in the literature. Fifty-one had proper motions, 43 had parallaxes, and two had only 𝛼 and 𝛿 . All values along with their provenance are given in Table 1. In the next sub-sections we discuss the target list selection and observations.', '2.1 Target selection': 'Our targets were drawn from a combination of two samples: benchmark systems (system with a star and a UCD, Pinfield et al. 2006) and known L dwarfs with poor or no available spectroscopy. The targets were selected by Marocco et al. (2017) and Marocco et al. (2020), and here we briefly summarise their selection criteria. Both samples were chosen with the aim of gathering low- and midresolution spectra, mostly to achieve radial velocities and to confirm their status as L dwarfs. Benchmark system selection used the procedure of Marocco et al. (2017, their section 4). To summarise, primary systems consisting of possibly metal-rich or metal-poor stars were selected with metallicity cuts of [Fe/H] < -0 . 3 and [Fe/H] > 0 . 2 dex from a number of catalogues (Marocco et al. 2017, their table 2). If more than one value of [Fe/H] was available, the one with the smallest uncertainty was used; Marocco et al. (2017) did not investigate if there were any systematic offsets between different catalogues, as this was beyond the scope of that work. The companions to these systems were filtered by a series of colour, absolute magnitude and photometric quality cuts from 2MASS, SDSS (the Sloan Digital Sky Survey, York et al. 2000) and ULAS (United Kingdom Infrared Telescope Deep Sky Survey, Large Area Survey, Lawrence et al. 2007) photometry in equation (1). These colour cuts in equation (1) are taken directly from Marocco et al. (2017) as that work created part of the target list used in this work. Magnitudes from 2MASS were converted into UKIRT/WFCAM magnitudes via the equations of Stephens & Leggett (2004). \n```\n𝑌 -𝐽 > 0 . 85; (1) 𝐽 -𝐻 > 0 . 50; 𝑧 -𝐽 > 2 . 1; 𝜎 𝐽 < 0 . 1; [ 2 . 5 × ( 𝑧 -𝐽 ) + 4 ] < 𝑀 𝐽 < [ 5 × ( 𝑧 -𝐽 ) + 1 ] ; 𝑀 𝐽 > 11 . 5; 1 . 6 < 𝑖 -𝑧 < 6 . 0; 11 . 5 < 𝑀 𝑧 < [ 3 . 5714 × ( 𝑖 -𝑧 ) + 9 . 286 ] ; 𝑀 𝑧 ≥ 15; 𝑀 𝑧 ≥ [ 3 . 5714 × ( 𝑖 -𝑧 ) + 6 . 5 ] ; 𝑖 -𝑧 ≤ 2 . 1 .\n``` \nThese companions were determined as being candidate benchmark systems with a maximum matching radius of 3 arcmin, i.e. the maximum separation to the primary object. The remaining targets, known L dwarfs, were already spectroscopically confirmed bright L dwarfs that were predicted to be visible to the astrometry and photometry in (at the time, upcoming) Gaia data releases. These \nknown L dwarfs should be single systems. They would, however, not be bright enough for the Gaia radial velocity spectrometer (Katz et al. 2023), and thus were chosen to determine radial velocities for, as a complement to the 30 pc volume-limited sample. This list was complemented with additional targets too dim for Gaia photometry and astrometry, which were detected in UKIDSS, and by a few wellknown L dwarfs, such as G 196-3B, which could serve as template standards.', '2.1.1 Cross-matching': "All observed targets (Table 1) were cross-matched with Gaia , 2MASS, and AllWISE. These surveys were chosen because they are all-sky and we were aiming for completeness in this process. The targets were also cross-matched with Pan-STARRS (50/53 successful matches), for the additional optical components for those sources within the Pan-STARRS footprint. This sample of 53 objects was then also cross-matched against the astrophysical parameter and xp\\_summary tables from Gaia DR3 4 . Thirty-eight of these objects had a teff\\_espucd value, and 28 had a public RP spectrum. Internally calibrated Gaia RP spectra were then extracted from the Gaia archive with a linearly dispersed grid from 6000 Å to 10500 Å using the gaiaxpy.convert (Ruz-Mieres 2022) and gaiaxpy-batch (Cooper 2022a) codes. We also searched for common proper motion systems within Simbad (Wenger et al. 2000) with the selection criteria given in the GUCDS, specifically equation (1) of Marocco et al. (2020): \n𝜌 < 100 𝜛 ; (2) Δ 𝜛 < max [ 3 𝜎 𝜛 , 1 ] ; Δ 𝜇 < 0 . 1 𝜇 ; Δ 𝜃 < 15 deg . \nIn equation (2), 𝜌 is the separation in arcseconds, 𝜃 is the proper motion position angle in degrees, whilst 𝜛 (milli-arcseconds) and 𝜇 (milli-arcseconds per year) are our target list's Gaia DR3 parallax and proper motion, respectively. Like with the photometric selection, equation (1), the common proper motion selection was taken directly from Marocco et al. (2020). This is because the target list in this work is drawn from the same wider target list used in the GUCDS. In effect, this selection is creating a widest possible physical separation of 100 000 AU (see the discussion on binding energies by Caballero 2009).", '2.2 Observations': 'The OSIRIS instrument used a 2 × 1 mosaic of 2048 × 4096 pixel (photosensitive area) red-optimised CCDs (Marconi MAT-44-82 type) with a 7 . 8 × 7 . 8 arcmin 2 unvignetted field of view. We used the standard operational mode of 2 × 2 binning, which has a physical pixel size of 0.254 arcsec pixel -1 . For our purposes, we used the 7.4 arcmin long slit with a width of 1.2 arcsec. We had variable seeing between 0.6 and 2.5 arcsec, with the vast majority having seeing < 1 . 2-1 . 5. The undersampling of the Full Width at HalfMaximum (FWHM) when the seeing is significantly less than the slit width would cause uncertainty in the wavelength calibration. In the worst cases, this can approach the resolution element. This \nwas then included in the systematic uncertainty estimate on the radial velocities. We used the R300R and R2500I grisms and purely read off CCD 2 due to the instrument calibration module having a strong gradient from CCD 1 to 2 in the flat fields. The R300R grism has a wavelength range of ≈ 4800-10 000 Å with a dispersion of ≈ 7 . 74 Å pix -1 for a resolution of ≈ 350 whilst the R2500I VPHG has a wavelength range of ≈ 7330-10 000 Å with a dispersion of ≈ 1 . 36 Å pix -1 for a resolution of ≈ 2500, as per the online documentation 5 . Both dispersive elements experience an increase in fringing at wavelengths ≳ 9200 Å to ≥ 5 per cent. The R300R grism however, had second order light from 4800 to 4900 Å contaminating the 9600 to 9800 Å region. This is because standards, but not UCDs, have flux in the blue regime, hence affecting the flux calibration in the red regime. As a result, the R300R spectra were conservatively truncated to 9000 Å. Our standards were a selection of white dwarfs plus two well-studied bright main sequence dwarf stars, all with literature flux calibrated spectra and spectral types: Ross 640 (DZA6, Oke 1974; McCleery et al. 2020); Hilt 600 (B1, Hamuy et al. 1992, 1994); GD 153 (DA1, Bohlin et al. 1995, 2014); G191-B2B (DA1, Oke 1990; Bohlin et al. 1995, 2014); GD 248 (DC5, Tremblay et al. 2011; McCleery et al. 2020), GD 140 (DA2, Tremblay et al. 2011; McCleery et al. 2020) and G 158-100 (dG-K, Oke 1990). We took a series of short exposures for the brightest objects to avoid saturation and non-linearity. The majority of observations had a bright moon whilst the sky condition varied from photometric to clear with humidity typically ≲ 50 per cent. All calibration frames were taken at the start and end of each night, the arc lamps being used to solve the wavelength solution were: Hg-Ar, Ne and Xe. The full observing log is given in Table A1.', '3 DATA REDUCTION': 'Weaimedtodetermine spectral types, spectral indices and radial velocities from directly measuring the GTC spectra. Furthermore, we inferred astrophysical parameters (effective temperature, 𝑇 eff [K]; surface gravity, log 𝑔 [dex]; and metallicity, [Fe/H] [dex]) from comparisons with atmospheric models. \nOur adopted PypeIt 6 (Prochaska et al. 2020a; Prochaska et al. 2020b) reduction procedure applied to every object was as follows: master calibration files were created by median stacking the relevant flat, bias and arc frames. Basic image processing was performed including bias subtraction, flat fielding, spatial flexure correction and cosmic ray masking via the L.A. Cosmic Rejection algorithm (van Dokkum 2001). We then manually identified the arc lines using the median stacked master arc. These arc lines were used to manually create a wavelength solution through pypeit\\_identify with typical RMS values of ≈ 0 . 0804 Å for the R2500I VPHG and ≈ 0 . 1394 Å for the R300R grism. The R2500I wavelength calibration solution was a 6 th order polynomial, whilst the R300R solution was only 3 rd . The information inside the object headers (observation date, object sky position, longitude and latitude of the observatory) was used to heliocentric correct the wavelength solution. The PypeIt wavelength solution was defined in vacuum. \nThe standard frames were median stacked before the global sky was subtracted and corrected for spectral flexure (to account for fringing). Both the stacked standard and object were then extracted \nusing both boxcar (5 pixel) and optimal (Horne 1986) extraction methods, with the latter being the presented spectra. \nWe then fitted a function to account for the sensitivity, CCD quantum efficiency and zeropoint. The telluric regions listed by Reiners et al. (2007) and Smette et al. (2015) were masked out. Wedivided each standard by its corresponding flux calibrated spectrum from the literature, as listed above. This sensitivity function was then applied to the reduced standard and object to flux calibrate the extracted spectra. If an observation had more than one science frame, those were co-added after wavelength and flux calibration. \nThe standards observed under the R2500I VPHG were used to create a telluric model from a high resolution atmospheric grid derived at Las Campanas. This grid was interpolated through to find the best match across airmass and precipitable water vapour. The telluric model was applied back to the flux calibrated standard and object. This telluric corrected standard was visually checked to confirm that the telluric model was behaving appropriately. The configuration files used in our reduction procedure are given in Appendix A6. \nIt is important to mention here that we made a comparison between this PypeIt reduction and that of a customised reduction (both the full basic image and spectral reductions) using standard IRAF tasks. This was done with the aim of validating the quality of the PypeIt data against that from a well proven reference source. In Appendices A2 and A3 we describe this procedure in detail for one suitably chosen test object from our selection sample, and which is common to both independent reductions: J1745 -1640. \nA comparison between the PypeIt reduction, and that which used standard IRAF routines, is shown in the normalised spectra of J1745 -1640in Figure 1. We show good agreement in the flux profile up to ∼ 8900 Å. The IRAF reduced spectra is brighter in the broad H 2 O region, due to the differing telluric correction methods. The MagEspectrum was not telluric corrected whilst the IRAF spectrum was telluric corrected using a blackbody, instead of Ross 640 (the corresponding white dwarf standard). This difference does not affect the model fitting of the spectra, as this is done in localised, small, chunks. All spectra then agree at wavelengths ⪆ 9800 Å.', '4 ANALYSIS': 'Here, we discuss the analysis of the reduced spectra, in order to produce spectral types, astrophysical parameters and kinematics. We discuss our measurements of astrophysical parameters first because the cross-correlation technique used to measure RV requires the use of a best-fitting model derived template, obtained from the best fit of astrophysical parameters. The code used for both estimating astrophysical parameters and calculating RV is rvfitter (Cooper 2022b). This program was developed to effectively recreate in python older codes (e.g. IRAF.Fxcorr , IRAF.Splot , IDL.gaussfit ) designed for allowing a user to manually cross-correlate spectra and fit line centres with different profiles. All wavelengths discussed in this Section are in standard air, hence we converted our PypeIt spectra from vacuum to air. This was performed via the specutils package, using the corrections by Edlen (1953).', '4.1 Spectral typing': "We spectral typed both the R300R and R2500I spectra using the classifyTemplate method of the kastredux (Burgasser \nFigure 1. R2500I spectra for J1745 -1640, normalised at 8100-8200 Å, comparing two independent reduction procedures: PypeIt in black and IRAF in orange. In blue, the heliocentric corrected MagE spectra (Burgasser et al. 2015) for the same object is shown (which is not telluric corrected). The Earth symbol indicates the telluric bands present in the spectra. \n<!-- image --> \n2021) package. This compared each spectrum against SDSS standards (Bochanski et al. 2007; Schmidt et al. 2010; Kesseli et al. 2017), from M0-T0, and selected the spectral type with the minimumdifference in scaled fluxes ( Δ 𝐹 : equations (3 - 4)) with equally weighted ( 𝑊 ) points. \nΔ 𝐹 = ∑︁ 𝑊 ( 𝐹 object -𝐾𝐹 standard ) 2 𝜎 2 object (3) \n𝐾 = ∑︁ 𝑊𝐹 object 𝐹 standard 𝜎 2 object , ∑︁ 𝑊𝐹 standard 𝐹 standard 𝜎 2 object (4) \nThe spectra had all been smoothed in wavelength with a Gaussian 5 𝜎 kernel, and we only compared the regions from 8000 to 8500 Å for R2500I and 7000 to 8000 Å for R300R. This was decided through experimentation, which deliberately excluded regions with telluric features, as those features can cause poorer solutions. Each object was also visually checked against known standards (Kirkpatrick et al. 1999), the spectral sub-types by which we refer to as 'by eye'. Any spectra with indicators of youth are given optical gravity classes as defined by Cruz et al. (2009), from 𝛽, 𝛾, 𝛿 in order of decreasing surface gravity. The kastredux spectral types were our adopted spectral types.", '4.1.1 GTC spectral sequence': 'The 46 spectra from the R2500I VPHG, ordered by our adopted spectral type, are shown in Figures 2 and 3. All spectra are heliocentric corrected, such that the relative motion of the Earth has been removed. Each spectrum shown had an outlier masking routine applied such that points within a rolling ≈ 15 Å (ten data points) chunk \nare removed if they had a difference greater than the standard deviation from the median. Additionally, some objects had problematic O 2 A-band tellurics. In those cases, we interpolated over the region 7540-7630 Å from the maximum of the first ≈ 7 . 5 Å to minimum of the last ≈ 7 . 5 Å. Where appropriate, spectra were co-added. All spectra appear noisy in the primary H 2 O band of ≈ 9200-9600 Å. The 17 heliocentric corrected, reduced spectra from the R300R grism are shown in Figure 4. The R300R spectra were trimmed from 6500 < 𝜆 < 9000 Å due to (a) the lack of signal in the blue regime and (b) to constrain to purely the first order light. Unlike the R2500I spectra, the R300R spectra were not telluric corrected.', '4.2 Fundamental astrophysical parameters': "Weused the rvfitter.crosscorrelate code on our R300R and R2500I spectra with BT-Settl CIFIST model grids from 1200 ≤ 𝑇 eff ≤ 4000 K and 4 . 5 ≤ log 𝑔 ≤ 5 . 5 dex (Allard et al. 2011). Lower surface gravity grids were available but not routinely used as the focus was on RV measurement with an a priori expectation of field surface gravity, ≈ 5 dex. These models assume a solar metallicity with no variation and are linearly dispersed in steps of 100 K and 0 . 5 dex. This code allowed us to visually select the best fitting model from the array of model grids and for each spectral line from Table 2. \nWe used these chosen lines rather than correlating against the entire model because the models do not exactly match the flux profile of ground based spectra. It was also known that the BT-Settl grids were generated using a different line list to our selected alkali lines, taken from the NIST database (Kramida et al. 2021). For efficiency purposes, each model when being loaded into the code, was interpolated onto the wavelength array of the object being compared against. The models could optionally be Gaussian smoothed, \nWavelength[ ˚ A] \n<!-- image --> \nFigure 2. The first 24 of the R2500I VPHG spectra with a linear offset applied, sorted by spectral sub-type. We show the short names and the spectral sub-types from this work, attached to each spectrum. At the top of the figure are grey lines denoting a selection of spectral features typical to L dwarfs, plus the two main telluric bands. \nWavelength[ ˚ A] \n<!-- image --> \nFigure 3. Same as Figure 2 but for the second half of the R2500I VPHG sample. \noffset \n+ \n] \nλ \nF \n[ \nFlux \nNormalised \n<!-- image --> \nWavelength[ ˚ A] \nFigure 4. Same as Figure 2 but for the R300R grism spectra. Instead of the spectral features visible in Figures 2 and 3, we only show where any lithium detection would be.Table 2. The list of atomic alkali metal lines used when estimating astrophysical parameters and calculating radial velocities. Wavelengths are as measured by Kramida et al. (2021) and are defined in standard air. \nwhich was helpful for fitting any 'messy' regions of models (e.g. \ntelluric bands in models with 𝑇 eff ⪆ 2000 K). We normalised the model and data by their respective medians in a given variably sized 'chunk' around each spectral line. We noted that around certain lines, particular models appeared almost identical to each other, e.g. around 7000-8000 Å, the 1900 and 2000 K models are not visually distinct. This means there is a higher uncertainty for effective temperatures within the 1900-2000 K region. Not every spectral line was used for each object as some have poorly resolved features or low signal-to-noise. Our selected 𝑇 eff was the mean 𝑇 eff from each line measurement, as was log 𝑔 . To determine the error on each 𝑇 eff and log 𝑔 final value, we chose to use the standard deviation from each independent line fit divided by square root of the number of lines used. This error was added in quadrature with half of the separation between each grid, i.e. 50 K for 𝑇 eff and 0 . 25 dex for log 𝑔 . \nAdditionally, we created an 'expected' effective temperature, c 𝑇 eff , using the Filippazzo, sixth order field 𝑇 eff relation (Filippazzo et al. 2015) and our adopted spectral types. The errors on c 𝑇 eff correspond with the mean difference in 𝑇 eff across ± 0 . 5spectral subtypes (our spectral sub-type uncertainty), plus the quoted relation RMS of 113 K.", '4.3 Calculating the radial velocities': 'Only our R2500I spectra were used to determine RVs as the features in R300R spectra are mostly blended/unresolved. We used two methods by which to measure an adopted RV: line centre fitting and cross correlation. We note that our seeing (Table A1, corrected for airmass) was almost always smaller than the slit width, which affects the RV offset as the slit is not fully illuminated. The full width at half-maximum was typically 3-4 pixels, corresponding to ≈ 0 . 75-1 arcseconds. Most observations were seeing-limited, whilst a few, taken in poorer conditions, were slit-limited. The following methods were performed only on heliocentric corrected spectra, hence any quoted RV values are heliocentric corrected.', '4.3.1 Line centre fitting': 'Using the same atomic absorption lines listed in Table 2, we applied the rvfitter.linecentering code to interactively fit Gaussian, Lorentzian and Voigt profiles with the minimum possible width. This minimum possible width is equal to the number of free parameters plus one (although this does not guarantee a successful fit). We used these different profiles to obtain the best fit for a particular line given its underlying absorption characteristics and the available signal-to-noise of the spectral region. The fitting technique used was least-mean-square 7 minimisation. For each spectral line, we subtracted a linear continuum from the data. The continuum corresponds to the medians of selected regions to the blueward and redward sides of the spectral line. Each continuum region is chosen to follow the shape of the spectra with a minimum width of ≈ 50 Å within 100-200 Å of the spectral line. Also shown during the fitting routine is a fifth order spline, as a visual aid; the minima of the spline does not necessarily correspond to the line position. A example of this routine is given for J1745 -1640 in Figure 5. The fits were only accepted if they appeared to accurately represent the spectral lines profile upon visual inspection. In general, the most consistently reliable lines were the rubidium lines, sodium doublet and first caesium line. The potassium doublet often was affected by rotational broadening whilst the second caesium line was often affected by neighbouring features. The uncertainty for each line, was the value in the diagonal of the covariance matrix corresponding to centroid position from the least-squares fit, plus the wavelength calibration RMS for that object, Doppler shifted into RV space. \nAfter measuring every line, we then calculated the overall weighted mean ( 𝜇 LC ) and weighted standard deviation ( 𝜎 LC ), the weights were the inverse of the uncertainties of each line used, squared. The uncertainty from the vacuum to air conversion was negligible ( ≪ 0 . 1 km s -1 ) compared to the fitting uncertainties calculated from the eight (or less, if rejected) aforementioned lines. The final line centre RV standard error was the weighted standard deviation divided by the square root of the number of lines fit.', '4.3.2 Cross-correlation': 'In addition to estimating the astrophysical parameters with rvfitter.crosscorrelate in Section § 4.2, we also used the same package to measure RV by manually shifting the best fitting BT-Settl model as a template. No smoothing was applied to the model template to match the spectral resolution of the object spectrum. This was because smoothing could confuse where the centroid of a line was, when looking by-eye. Likewise, there was no continuum subtraction applied to the object spectrum. The RV shift was in steps of 5 , 10 , 100 km s -1 , which in turn defined the RV uncertainty on each line (2 . 5 , 5 , 50 km s -1 , i.e. the margin of error). These RV errors are added to the wavelength calibration RMSfor the given object (Doppler shifted into an RV error). Not all atomic lines were always used, only in the cases where the model appeared to closely match the apparent line profile. The typical technique was to select a broad region ( Δ 𝜆 = 100-200 Å) around each spectral line, find the best fitting template in terms of 𝑇 eff and log 𝑔 , then narrow that region ( Δ 𝜆 ≈ 50 Å) to then find an RV. This was a predominantly by-eye technique, although root-mean-square deviation divided by interquartile range (RMSDIQR) values were computed as a numerical guide when comparing models. We also show a fifth order spline, as with the line centering method, as a visual aid. This initial broad region is shown for J1745 -1640 in Figure 6. \nAs in Section § 4.3.1, the overall cross-correlated weighted mean RV value ( 𝜇 XC ) and weighted standard deviation ( 𝜎 XC ) was calculated using all of the manually selected lines. We used the same method to estimate the uncertainty in final cross-correlation derived RVs as for the line centre results, by finding the standard error of the mean.', '4.3.3 Adopted RV': 'We created an adopted RV by constructing a weighted mean, using the deviation in each method as the weighting. The different RV values for each line, method and the corresponding probability distribution functions (PDFs) are shown in Figure 7, for J1745 -1640. We also note that our final adopted RV for J1745 -1640 obtained from combining the results of the two measurement techniques (32 . 7 ± 6 . 5 km s -1 ) is in agreement with the values obtained from both the customised IRAF reduced data and the value reported by Burgasser et al. (2015), within their respective uncertainties. See Appendix A3 for a full description. \nThe adopted RV was the mean ( 𝜇 RV ) whilst the standard error ( 𝛿 RV ) was equal to the standard deviation ( 𝜎 RV ) divided by √ 2. The mean and standard deviation was calculated through the inverse variance weighting equations (5 and 6). Typically, we found that the cross-correlation technique was more precise (being more controlled by-eye) and robust. The line centre fitting was often more accurate, however, and performed best on the higher quality spectra. \n𝜇 RV = 𝜇 LC 𝜎 2 XC + 𝜇 XC 𝜎 2 LC 𝜎 2 LC + 𝜎 2 XC (5) \n𝜎 RV = v u t 𝜎 2 LC 𝜎 2 XC 𝜎 2 LC + 𝜎 2 XC (6) \n] \nλ \nF \n[ \nFlux \nNormalised \nFigure 5. J1745 -1640 RV calculation via different line profiles (orange: solid - Gaussian; dash-dot - Voigt) against the data (black squares) and fifth order spline fit (blue) in the regime around the eight listed line centres. The flux uncertainty is smaller than the height of each square. The shift from the laboratory line position (vertical dashed grey line) is shown as the vertical solid black line. The horizonal black line (solid or dash-dot, depending on the fitted line profile as above) is the continuum, as is subtracted from the data. A grey band is given, corresponding to the region of data the line profiles are fitted to. The shown region is between the inner edges of the continuum regions. \n<!-- image --> \n] \nλ \nF \n[ \nFlux \nNormalised \nFigure 6. J1745 -1640 RV calculation via the manually shifted BT-Settl model (orange) against the data (black squares) and fifth order spline fit (blue). The flux uncertainty is smaller than the height of each square. The laboratory line position (vertical dashed grey line) has been manually shifted by the RV given on the sub-plot title (vertical solid black line). Effective temperature, gravity and metallicity are also indicated on each features title. \n<!-- image --> \nFigure 7. J1745 -1640 RV values for each given line. In the top panel, orange squares are cross-correlated RVs, blue diamonds are line centre RVs; each spectral feature has been indicated on the 𝑦 axis. In the bottom panel, the orange curve is the cross-correlated PDF; the blue curve is the line centre PDF; and the black curve is the adopted PDF. The dotted vertical lines are the mean RV values as associated with each PDF. \n<!-- image -->', '4.4 Kinematics': 'Galactic UVW velocities were calculated using our adopted RVs plus Gaia astrometric measurements, using the equations from astrolibpy . We corrected for the Local Standard of Rest (LSR) using the values from Coşkunoˇglu et al. (2011) where U, V, W = ( -8 . 50 , + 13 . 38 , + 6 . 49) km s -1 . These equations follow the work by Johnson & Soderblom (1987), except that U is orientated towards the Galactic anti-centre. We also used BANYAN Σ (Gagné et al. 2015a, 2018), which provided moving group classification with associated probability. When using BANYAN Σ , we checked the resultant probabilities both with and without RV. This was because RV has by far the lowest precision, thus could reduce a likely membership candidate into a field object in error. Our final values are the ones which include RV. Notably, when using velocities in the Galactic reference frame, one can select a Galactic component with 𝑉 total (where 𝑉 total is the total space velocity, 𝑉 total = √ 𝑈 2 + 𝑉 2 + 𝑊 2 ). We followed the work by Nissen & Schuster (2010) and define thick disc and halo objects as having 𝑉 total > 70 km s -1 and 𝑉 total > 180 km s -1 respectively. This definition, especially for separating the thin and thick disc, is indicative of metallicity; see the Besançon Galaxy models (Czekaj et al. 2014; Lagarde et al. 2021).', '5 RESULTS': 'In this Section, we present the spectral types, radial velocities and astrophysical parameters. In Table A2, we provide photometry from the Gaia , 2MASS and ALLWISE catalogues. We discuss individually interesting objects and objects where our measured results differ significantly from published values.', '5.1 Spectral types': "In Table 3 we list published spectral types based on optical spectra, near-infrared spectra and the 'by eye' and kastredux methods discussed in Section § 4.1. This work has produced the first spectral type estimates for six of the 53 objects. \nThe 47 objects with known spectral types have a standard \nFigure 8. Comparison between this works spectral types and the literature spectral types. Blue squares are spectral types from our adopted, kastredux method whilst orange circles are from the manual 'by eye' method. Grey lines connect these two methods and we show a one-to-one dashed grey line with associated ± 2 spectral sub-types confidence bands. \n<!-- image --> \ndeviation of 0.5 sub-types between the published values and the 'by eye'/ kastredux results, which we adopt as the error on the new spectral sub-types. When the literature values for a given object differ we adopted the optical spectral type. Our spectral types across the two methods are displayed against the adopted literature spectral types in Figure 8. \nAll objects except J1004 -1318 have sub-type differences between the spectral type derived in this work and the adopted literature spectral type of less than two sub-types. J1004 -1318, has an optical (Opt) spectral sub-type of L0 (Martín et al. 2010) whilst Marocco et al. (2013) found a sub-type of L1 using nearinfrared (NIR) spectra; we find a sub-type of L3. However, a more recent study, Robert et al. (2016), found a sub-type of L4 (NIR), which is more consistent with our result. The fit statistic from kastredux is about twice larger for L1 than for L3. In Figure 2, J1004 -1318 does not seem dissimilar to the neighbouring objects, whereas the L0/L1 spectra appear different (e.g. weaker alkali lines). The different spectral typing of J1004 -1318 may be due to lower signal-tonoise (S/N) ratios of some observations. For example, Martín et al. (2010) exposed for 2400 s at the 2.56 m Nordic Optical Telescope, while we exposed for 1500 s, and with moderately good seeing and low airmass, with a telescope with an aperture over 16 times larger.", '5.2 Radial velocity analysis': "We have derived RVs for 46 of the observed 53 objects, the seven objects that we did not measure RVs were only observed with the R300R grism. For 20 of the 53 objects, there are published RVs and for 17 of these we have measured RVs. The objects J1004 + 5022, J1441 -0945 and J1617 + 7733B are candidate members of benchmark systems (Section § 2.1), and we adopt the RVs of their primary stars as a comparison with our measured values for the secondary, for a total of 20 comparison RVs. In Figure 9, we plot histograms of the 20 published and the 46 measured values. We also show \nTable 3. Our spectral types compared with the literature optical and near-infrared types for each object. \nLiterature Spectral Types: 1. Reid et al. (2008), 2. Bardalez Gagliuffi et al. (2014), 3. Hawley et al. (2002), 4. Kendall et al. (2003), 5. Cruz et al. (2003), 6. Schmidt et al. (2010), 7. Cruz et al. (2007), 8. Marocco et al. (2017), 9. West et al. (2011), 10. Kirkpatrick et al. (1999), 11. Kirkpatrick et al. (2008), 12. Allers & Liu (2013), 13. Martín et al. (2010), 14. Marocco et al. (2013), 15. Martín et al. (1999b), 16. Knapp et al. (2004), 17. Schneider et al. (2014), 18. Gagné et al. (2015a), 19. Kirkpatrick et al. (2000), 20. Kirkpatrick et al. (2010), 21. Kendall et al. (2004), 22. Kendall et al. (2007), 23. Phan-Bao et al. (2011), 24. Phan-Bao et al. (2008). \nThe ':' after a spectral type indicates uncertainty of ± 1 whilst 'p' indicates peculiarity. The surface gravity flag 𝛽 is given when appropriate, and is discussed in Section § 5.3.1. The adopted spectral type is the kastredux method, only overwritten where there are gravity flags in the 'by eye' method. In addition, J1246 + 4027 has been typed as having a potential Li /i.pc detection ( 𝜆 6708 Å), which can only be seen in the R300R spectra. \nthe difference between the published and measured values of the 20 overlapping objects. If there is more than one literature value, we take the weighted mean RV and standard error on the mean, to compare against the adopted RV from this work. We show literature measurements with respect to their resolutions and define these as: low, 𝑅 < 2 500; mid, 2 500 ≤ 𝑅 ≤ 25 000; high, 𝑅 > 25 000. The error used to define 𝜎 are the quadrature summed errors from the literature and our adopted RV. \nOur 46 RVs in the heliocentric reference frame are presented in Table 4. This reference frame has been experimented with, in that the heliocentric/barycentric correction via pypeit has been compared with a manual barycentric correction using barycorrpy (Kanodia & Wright 2018). Resultant RV differences from the manual barycentric correction to the pipeline barycentric correction differ by ≈ 0 . 1 km s -1 . The difference between heliocentric and barycentric correction is 0 . 5 km s -1 in the case of J1745 -1640. \nThe median difference between our adopted RVs and the literature RVs was 7.8 km s -1 . This 7.8 km s -1 was then added in quadrature to our adopted RV error. We used this value to account for systematic uncertainties such as night-to-night instrumental drift and any FWHM undersampling. A S/N ratio of 20-30 also correlates with an RV uncertainty of ≈ 8 km s -1 , which was the typical S/N ratio seen around our alkali lines. Some lines, such as the potassium doublet, had lower S/N ratios and lower local resolutions due to a combination of wider features and lower flux values. All objects \nexcept J0940 + 2946 and J1221 + 0257 have an adopted and literature RV difference less than twice the sum of the respective errors in quadrature. J0940 + 2946 was 2 . 69 𝜎 from the weighted mean literature value. Of the two literature values constructing this weighted mean, our value is < 2 𝜎 from the value from Kiman et al. (2019), which is notably larger than the value from Schmidt et al. (2010). J1221 + 0257 was 2 . 08 𝜎 from the weighted mean literature value. OurRVvaluewasclosest to the value from Kiman et al. (2019), with less agreement shown with the value from Schmidt et al. (2010), which itself was most similar to the values from Burgasser et al. (2015) and Hsu et al. (2021). We note for both of these objects that the RV values from Schmidt et al. (2010) utilised considerably lower resolution spectra, hence a worse agreement being shown. Any objects in Table 4 which have known primary stars with literature RVs are discussed below: \nJ1004 + 5022 : G 196-3B is the binary companion to G 1963A (Kirkpatrick et al. 2008). G 196-33A has a mean RV of -1 . 6 ± 0 . 4 km s -1 (Shkolnik et al. 2012; Schlieder et al. 2012b; Binks & Jeffries 2016; Gaia Collaboration et al. 2018a). This mean RV of the primary is 0 . 1 𝜎 away from the RV of the secondary companion from this work. \nJ1441 -0945 : DENISJ144137.2 -094558 is the binary companion to G 124-62 (Bouy et al. 2003; Seifahrt et al. 2005). G 12462 has an RV of -41 . 65 ± 5 . 91 km s -1 (Gaia Collaboration et al. \n<!-- image --> \nFigure 9. [Left Panel]: Histograms of the RVs calculated in this work (orange) and from the literature (blue) to show the relevant population densities. The dashed vertical lines indicate the means of the associated distributions. [Right Panel]: The RV values from the literature on the 𝑥 axis with our adopted RV values, on the 𝑦 . We show a one-to-one relation, over which our 20 comparison RVs are plotted. Squares are from low-resolution literature measurements, whereas circles and diamonds are mid- and high-resolution literature measurements respectively. Orange points are like-for-like comparisons and blue points are for the three benchmark systems, i.e., comparisons between our measured secondary RV against the literature RV of the primary. \n<!-- image --> \n2018a), which is within 1 . 4 𝜎 of the companion (which had large uncertainties). \nJ1617 + 7733B : TYC4571-1414-1B is the binary companion of TYC4571-1414-1A (Alonso-Floriano et al. 2015). TYC4571-14141Ahas an RV of -19 ± 0 . 8 km s -1 (Gaia Collaboration et al. 2018a), this RV is 0 . 1 𝜎 from the companion RV.", '5.2.1 Moving groups': "Our results for UVW Galactic kinematic components are presented in Table 5 with each object's moving group classification and associated probability from BANYAN Σ . When accounting for RV in BANYAN Σ , the resultant probability was often lower than the calculation without RV. This was due to the Bayesian probabilities being designed for a higher recovery rate (moving from 82 per cent to 90 per cent) when accounting for the RV (see the BANYAN Σ cautionary note 8 , Gagné et al. 2018). In addition, the RV uncertainties from this work are much higher than proper motion or parallax uncertainties from Gaia . \nWe find four objects are members of the following young moving groups and clusters: Argus (30-50 Myr, Makarov & Urban 2000); 𝛽 Pictoris (Zuckerman et al. 2001), 20-26 Myr (Mamajek & Bell 2014; Couture et al. 2023, and references therein); Carina-Near ( ∼ 200 Myr, Zuckerman et al. 2006); and the Hyades cluster (600-800 Myr, Perryman et al. 1998; Martín et al. 2018; Lodieu et al. 2018). These objects (J1058 -1548, J0453 -1751, J1213 -0432, and J0502 + 1442 - respectively) are discussed below in Section § 5.3.1.", '5.2.2 Galactic components': 'Thin disc objects were differentiated from thick disc and halo objects using the LSR corrected UVW Galactic velocities; the thick disc and halo objects were those with 𝑉 total > 70 km s -1 and 𝑉 total > 180 km s -1 respectively (Nissen & Schuster 2010). 𝑉 total is the total space velocity. We calculated upper and lower bounds for UVW Galactic velocities using the propagated parallax, proper motion, and RV errors; these UVW velocities with associated uncertainties are shown in Figure 10. The objects J1109 -1606 ( 𝑉 total = 103 ± 5 km s -1 ) and J1539 -0520 ( 𝑉 total = 69 ± 4 km s -1 ) are found using the above criteria to be most likely thick disc objects, and are highlighted in Figure 10. J1539 -0520, is a borderline thick disc object, within 1 𝜎 of the thick disc cut-off. Considering that a nearby object is most likely within the thin disc (Holmberg et al. 2009), J1539 -0520 is a reasonable thick disc candidate, hence the inclusion here. It was also assigned a 64.6 per cent probability of being in the thick disc by Cooper et al. (2024), although it did not pass the conservative subdwarf candidate selection criteria in that work. Without metallicity information, an object being in the thick disc is not a direct inference on age. These objects are worth visiting with higher resolution spectroscopy to gain metallicity information, to confirm any potential subdwarf candidacy. This future work would also involved gathering NIR spectra, as in work by Zhang (2018); Zhang et al. (2018, and references therein).', '5.3 Astrophysical parameters': "We present the 𝑇 eff and log 𝑔 values from the model fitting (Section § 4.2) in Table 6 along with c 𝑇 eff , assuming our adopted spectral type and equation (4) by Filippazzo et al. (2015) and teff\\_espucd values from Gaia DR3. In Figure 11, we plot the difference be- \nTable 4. RVs measured in this work and compared to the literature. \nLiterature Radial Velocities: 1. Kiman et al. (2019), 2. Schmidt et al. (2010), 3. Binks & Jeffries (2016), 4. Gaia Collaboration et al. (2018a), 5. Shkolnik et al. (2012), 6. Schlieder et al. (2012b), 7. Burgasser et al. (2015), 8. Hsu et al. (2021), 9. Blake et al. (2010). \nIndices: 1 if line from Table 2 used, 0 otherwise. \nQuoted RVs are already heliocentric corrected. A ' † ' symbol next to an RV means the RV is that of the primary star in the common proper motion system a given object is part of. \ntween our value and the expected value. In the cases of objects with both R2500I and R300R spectra available, we default to the higher resolution result. \nAlthough the best-fitting surface gravity values can be indicative of youth, they are quite degenerate and without corresponding metallicity values, therefore they are not relied upon in our discussion below. The best fitting spectral sub-types and BT-Settl models are shown in a spectral sequence for R2500I VPH spectra in Figures A1 and A2. \nFigure 12 shows a set of colour-absolute magnitude diagrams \n(CAMD), 2MASS 𝐽 -𝐾 𝑠 , 2MASS 𝑀 𝐽 , AllWISE 𝑊 1 -𝑊 2 and AllWISE 𝑀 𝑊 1 . Parallaxes from Gaia were used to generate the absolute magnitudes. Highlighted here are the objects with spectral features that are indicative of youth. These are compared to known young UCDs from Faherty et al. (2016) and Liu et al. (2016, 'VLG' or 'Young'), as well as the full sample from the GUCDS. These young objects tend to be over-bright, although the effect varies across filters and is further complicated by intrinsic scatter plus variability. \nTable 5. The UVW velocities and BANYAN Σ classification (with associated probability) from this work. \nLiterature astrometry used to generate UVWs: 1. Smith et al. (2014), 2. Burgasser et al. (2015), 3. Best et al. (2020), 4. Weinberger et al. (2016), 5. Blake et al. (2010). \nU is in the direction of the Galactic anti-centre. Derived using this work's adopted radial velocity in combination with Gaia DR3 kinematics unless otherwise indicated. We also show the predicted Galaxy component, taken from the UVW velocities and 𝑉 total cuts in Nissen & Schuster (2010). † : J1213 -0432 had an additional probability (26 per cent) of being a member of Argus, for a total non-field probability of 98 per cent.", '5.3.1 Individual objects': 'We further discuss here objects we have indicated as being nontypical, with interesting features or results. We check for any age classifications, based on the moving group membership from BANYAN Σ and location on the CAMD in Figure 12. There are additional objects which exist in the same colour space as our highlighted objects in Figure 12 which are not discussed below. This is because there can be large implicit colour scatter due to unresolved binarity, metallicity and dust. Hence, only objects which are interesting either spectrally or kinematically are discussed. The following four objects were found to be members of the moving groups listed above, in Section § 5.2.1. \nJ0453 -1751 : This L3 object, 2MASS J04532647 -1751543, is \na probable member of 𝛽 Pictoris with a 99 per cent confidence, this is an increase on the 55 per cent categorisation by Ujjwal et al. (2020, using Gaia DR2 data). Gagné et al. (2015b) by comparison found this object as a member (96 per cent) of the similarly aged Columba association (20-40 Myr, Torres et al. 2008). We have used Gaia DR3 kinematics, which are consistent with the values from Gaia DR2 but with reduced uncertainties. In Gaia DR2 (Gaia Collaboration et al. 2018a), this was 𝜛 = 33 . 2 ± 0 . 6 mas, 𝜇 𝛼 cos 𝛿 = + 44 . 6 ± 0 . 7 mas yr -1 and 𝜇 𝛿 = -20 . 8 ± 0 . 8 mas yr -1 . In Gaia DR3 (Gaia Collaboration et al. 2023), 𝜛 = 33 . 1 ± 0 . 5 mas, 𝜇 𝛼 cos 𝛿 = + 44 . 4 ± 0 . 4 mas yr -1 and 𝜇 𝛿 = -20 . 6 ± 0 . 4 mas yr -1 . The work by Best et al. (2020) is in broad agreement, with larger uncertainties, 𝜋 = 37 . 4 ± 5 . 7 mas, 𝜇 𝛼 cos 𝛿 = + 34 . 7 ± 4 . 9 mas yr -1 and 𝜇 𝛿 = -24 . 0 ± 3 . 9 mas yr -1 . \nTable 6. Effective temperatures and surface gravities from this work. \nThese 𝑇 eff values are generated using fits to preferentially R2500I spectra if available, else R300R. Model fits assume solar metallicities. c 𝑇 eff represents the expected effective temperature, based on an object\'s spectral type. Gaia 𝑇 eff are the teff\\_espucd effective temperatures from Gaia DR3. \nFigure 10. Toomre diagram, as done by Bensby et al. (2005), using Gaia DR3 astrometry in combination with our calculated RVs. V is on the 𝑥 axis, against the velocity dispersion ( √ U 2 + W 2 ) on the 𝑦 axis. Black circles are UVW velocities calculated with the RVs from this work, with associated error-bars given. We show the respective thick disc and halo selection lines at 𝑉 total > 70 km s -1 and 𝑉 total > 180 km s -1 respectively. \n<!-- image --> \nThe change of from Gaia DR2 to Gaia DR3 in isolation did not alter the confidence (99.2 per cent), whereas the inclusion of our adopted RV value dropped this to 98.9 per cent. Our adopted RV was 15 . 0 ± 8 . 3 km s -1 , which is within 1 𝜎 of the \'optimal\' RV from BANYAN Σ , 21 . 5 ± 1 . 5 km s -1 . From Figure 12, we see J0453 -1751 (a) is photometrically similar to known young objects. Its 𝑇 eff of 1850 ± 70 K is in good agreement with c 𝑇 eff and \nteff\\_espucd , although is cooler than the 2100 K from Gagné et al. (2015a). We can conclude that this object is an L3 within 𝛽 Pictoris. \nJ0502 + 1442 : 2MASS J05021345 + 1442367, an L0, we find as a member of the Hyades cluster with a 99 per cent probability. This improves the membership confidence by Gagné & Faherty (2018, 75 per cent) and concurs with the classifications by Gaia Collaboration et al. (100 per cent confidence, 2018b); Cantat-Gaudin et al. (100 per cent confidence, 2020, using the Melotte 25 name). Works by Oh & Evans (2020) and Spina et al. (2021) also placed this object in Melotte 25 with 96 per cent and 99 per cent confidences, respectively. It also agrees with the classification by Lodieu et al. (2019), which had a \' c parameter\' of 5.88, well within their Hyades membership limit, c < 25 . 9. Figure 12, places J0502 + 1442 (b) also as photometrically similar to known young objects, being somewhat over-bright, although there is considerable overlap with standard M-L sequence. With a 𝑇 eff of 2212 ± 126 K, J0502 + 1442 is an L0 object in the Hyades cluster. \nJ1058 -1548 : Another L3 object, SIPS J1058 -1548, is classified with 93 per cent confidence as a member of Argus. Gagné et al. (2015b) had the same classification with a much lower probability (35 per cent). Gaia DR2 astrometry in isolation gave a confidence of 96.3 per cent, whilst Gaia DR3 reduced this to 94.8 per cent, the inclusion of our adopted RV value further dropped this to 93.1 per cent. Our adopted RV was -0 . 9 ± 11 . 1 km s -1 , which is within 1 𝜎 of the \'optimal\' RV from BANYAN Σ , 8 . 5 ± 1 . 4 km s -1 . Specifically, in Gaia DR2, this was 𝜛 = 54 . 6 ± 0 . 5 mas, 𝜇 𝛼 cos 𝛿 = -258 . 1 ± 0 . 8 mas yr -1 and 𝜇 𝛿 = + 31 . 1 ± 0 . 7 mas yr -1 . In Gaia DR3, 𝜛 = 55 . 1 ± 0 . 3 mas, 𝜇 𝛼 cos 𝛿 = -258 . 6 ± 0 . 3 mas yr -1 and 𝜇 𝛿 = + 30 . 8 ± 0 . 3 mas yr -1 . These values are in broad agreement with nonGaia works, where 𝜋 ranges from 49.2 mas-66.5 mas, \nFigure 11. The expected c 𝑇 eff (calculated via spectral type through a Filippazzo relation, Filippazzo et al. 2015) on the 𝑥 axis and the best-fitting BT-Settl model mean 𝑇 eff on the 𝑦 axis. Blue crosses are for objects with a fit to the R300R spectra whilst black crosses are objects with a fit to the R2500I spectra. \n<!-- image --> \n𝜇 𝛼 cos 𝛿 from -60 mas yr -1 ( ± 160 mas yr -1 ) to -276 mas yr -1 and 𝜇 𝛿 from + 14 mas yr -1 to + 210 mas yr -1 ( ± 150 mas yr -1 ); c.f. Dahn et al. (2002); Caballero (2007); Deacon & Hambly (2007); Schmidt et al. (2007); Faherty et al. (2009, 2012); Weinberger et al. (2016); Dahn et al. (2017); Smart et al. (2018). J1058 -1548 has a 𝑇 eff = 1900 ± 102 K (in exact agreement with Gagné et al. 2015a), but is not as convincingly over-bright as neighbouring known young objects, see (c) in Figure 12. Sanghi et al. (2023) conclude that for J1058 -1548, \'it is probable that the YMG assignment [Argus] is incorrect", because their spectrum well matched L-dwarf FLD-G standards, although the log 𝑔 value of 4 . 27 dex was an outlier and more typical of a VL-G object (their figure (21)). The log 𝑔 value in this work was 5 . 0 ± 0 . 3 dex, although this less robust than that from Sanghi et al. (2023), who also had a much lower 𝑇 eff = 1570 K, which itself is more akin to a cooler object, ≈ L5. We would argue that this a probable L3 member of Argus but more high resolution spectra and modelling is required to ascertain youth. \nJ1213 -0432 : 2MASS J12130336 -0432437 (L4) we classify as a member of Carina-Near or Argus (98 per cent), which is an update on the 75 per cent classification of being in Carina-Near by Gagné & Faherty (2018). Just using Gaia DR2 astrometry gave a confidence of 68.5 per cent (with a 30.6 per cent likelihood of being in Argus), whilst Gaia DR3 increased this to 74.3 per cent (24.7 per cent for Argus), the inclusion of our adopted RV value (with large uncertainty) updated this to 72.0 per cent, with a 26.0 per cent likelihood of being in Argus. Our adopted RV was -25 . 3 ± 22 . 4 km s -1 , which is within 1 . 5 𝜎 of the \'optimal\' RV from BANYAN Σ , 2 . 4 ± 0 . 8 km s -1 . In Gaia DR2, it was 𝜛 = 59 . 5 ± 1 . 0 mas, 𝜇 𝛼 cos 𝛿 = -368 . 1 ± 2 . 2 mas yr -1 and 𝜇 𝛿 = -34 . 6 ± 1 . 4 mas yr -1 . In Gaia DR3, 𝜛 = 59 . 1 ± 0 . 6 mas, 𝜇 𝛼 cos 𝛿 = -367 . 9 ± 0 . 7 mas yr -1 and 𝜇 𝛿 = -34 . 0 ± 0 . 5 mas yr -1 . The work by Best et al. (2020) is also in good agreement, 𝜋 = 56 . 3 ± 3 . 8 mas, 𝜇 𝛼 cos 𝛿 = -380 . 9 ± 2 . 7 mas yr -1 and 𝜇 𝛿 = -33 . 4 ± 2 . 4 mas yr -1 . Figure 12 (d) shows this object as being under-bright compared \nwith known young objects, with a 𝑇 eff of 1783 ± 143 K. Being the age of Carina-Near could explain this relative under-brightness, as it should be tending towards field-like behaviour. This object can be classified then as an L4 member of Carina-Near. \nThere are two further field objects that we have highlighted as interesting due to their spectral features: \nJ1246 + 4027 : The L4 dwarf, 2MASSW J1246467 + 402715, observed at the two resolutions, is of interest due to the potential Li /i.pc detection at ≈ 6708 Å. As this feature is only in the wavelength regime of the R300R spectra, this is not definitive enough a detection to confirmlithium (see discussion by Martín et al. 2018, using the equation from Cayrel (1988)). Higher resolution (R ⪆ 2000) spectra would be required for confirmation (Gálvez-Ortiz et al. 2014). Assuming a true detection, employing the lithium test (Rebolo et al. 1992) alongside our fitted effective temperature of 𝑇 eff = 1750 ± 91 K would identify this object as being substellar. This 𝑇 eff is in good agreement with the expected temperature of c 𝑇 eff = 1717 ± 116 K and the Gaia DR3 𝑇 eff of 1780 ± 162 K. This substellar argument is in line with discussion by Basri (1998), Martín et al. (1999a) and Kirkpatrick et al. (1999), because our 𝑇 eff is in the range 2670 > 𝑇 eff > 1400 K. Figure 12 suggests J1246 + 4027 (e) neighbours some known young objects. The best fitting model had a surface gravity of log 𝑔 = 4 . 6 ± 0 . 3 dex, although we have no complementary metallicity information. BANYAN Σ finds no correlation with any known young moving groups. J1246 + 4027 could be classed as an L4 𝛽 field object. \nJ1004 + 5022 : G 196-3B is known to be a low gravity brown dwarf (Rebolo et al. 1998; Kirkpatrick et al. 2008; Allers & Liu 2013), to which we concur, with a spectral sub-type of L3 𝛽 . Our log 𝑔 value is 4 . 5 ± 0 . 2 dex ( 𝑇 eff = 1740 ± 113 K), as would be expected from the already known young nature. This object sits extremely red and over-bright in Figure 12 (f), even more extremely than most known young objects. It is a companion to the well known G 196-3A M3 star, to which we compared our kinematics in Section § 5.2, finding a 0 . 1 𝜎 difference. There is much deeper discussion on this benchmark system by Zapatero Osorio et al. (2010), which measures an angular separation of 𝜌 = 15 . 99 ± 0 . 06 \'\' . Combined with a Gaia DR3 parallax of 𝜛 = 46 . 1952 ± 0 . 5452 mas (in agreement with the 49 . 0 ± 2 . 3 mas and 41 . 0 ± 4 . 1 mas measurements by Liu et al. 2016; Zapatero Osorio et al. 2014, respectively), this implies a projected separation of 𝑠 = 739 ± 1 AU. This is slightly more than the projected physical separation range calculated by Zapatero Osorio et al. (2010), 285-640 AU. We found a probability of the secondary being a field object of 99.9 per cent, which is an increase on the 32 per cent probability of being a member of AB Doradus by Gagné et al. (2014). Liu et al. (2016) kinematically confirmed that G 196-3B is a young field object. This is also in agreement with the 50 per cent classification of the primary being a member of AB Doradus by Schlieder et al. (2012a), which was later downgraded to 0 per cent by Binks & Jeffries (2016); however, the primary was also classified as being a member of the controvertible Castor moving group (Barrado y Navascues 1998) with 75 per cent confidence (Klutsch et al. 2014). The Castor moving group was not included in BANYAN Σ , hence not being included in our analysis. We classify this object as an L3 𝛽 object.', '6 SUMMARY AND CONCLUSIONS': 'Wehave presented the low and mid resolution optical GTC/OSIRIS spectra of 53 objects observed between 2015 and 2016. Our data \nFigure 12. CAMDs of 2MASS and AllWISE photometry, focused on the majority of this sample (an inset of the full sequence is shown in the upper right). The 2MASS 𝐽 -𝐾 𝑠 colour is on the 𝑥 axis for the first column, with the AllWISE 𝑊 1 -𝑊 2 colour on the 𝑥 axis on the second column. Absolute 2MASS 𝐽 magnitude is on the 𝑦 axis for the first row whilst AllWISE 𝑀 𝑊 1 is the 𝑦 axis of the second row. Underlying the plots as grey points is the full UCD sequence from the GUCDS. Known young objects from Faherty et al. (2016) and Liu et al. (2016) are displayed as black diamonds. Each object is coloured by our adopted spectral type, with absolute magnitude error shown. Coloured diamonds are the young candidates discussed in Section § 5.3.1. Key: a-J0453 -1751, b-J0502 + 1442, c-J1058 -1548, d-J1213 -0432, e-J1246 + 4027, f-J1004 + 5022. \n<!-- image --> \n- \n- \nreduction was non-standard, using a pipeline package, PypeIt ; this reduction was validated with an independent IRAF spectral extraction and calibration for one of the objects. We used kastredux to create 53 automated spectral types, six of which are for objects not yet spectrally typed, alongside the established technique of comparing against spectral standard template spectra. We found that \nour chosen spectral reduction package, PypeIt , introduced some non-optimal artefacts during reduction. One example is a spike appearing near the O 2 A band from the telluric correction procedure, which required interpolating over for visualisation purposes (it does not affect wavelength solutions). \nIn addition to using new data reduction software, we also \nused novel analysis software, rvfitter , that we developed to perform manual line centering and cross-correlation (against BT-Settl CIFIST models). The rvfitter code also used an uncertaintyweighted mean to create an adopted RV. This produced 46 RVs, 29 of which are new, which we have validated against standard IRAF and IDL software techniques. There were 17 RVs which were compared against literature values, showing good agreement with a median difference of 7.8 km s -1 , adopted as our systematic uncertainty. Our median RV uncertainty was 11.2 km s -1 , indicating that further high-resolution spectroscopy would be necessary to validate our RV values and conclusions. The cross-correlation also produced mean 𝑇 eff and log 𝑔 values for all 53 objects. \nIn this work, we performed further analysis on our spectral types, RVs and 𝑇 eff values by making comparisons to the literature where appropriate and ensuring all results were within two spectral sub-types, Δ RV < 2 𝜎 and Δ 𝑇 eff < 2 𝜎 (against c 𝑇 eff and Gaia DR3 teff\\_espucd ). We then discussed any measurements which did not conform with these standards, such as J0940 + 2946, which had a Δ RV = 2 . 69 𝜎 . There were four objects that we classified through BANYAN Σ as being a member of a young moving group: SIPS J1058-1548 (J1058 -1548), 2MASS J04532647-1751543 (J0453 -1751), 2MASS J12130336-0432437 (J1213 -0432), and 2MASS J05021345+1442367 (J0502 + 1442). There were two objects we placed as members of the thick disc: SIPS J1109-1606 (J1109 -1606) and 2MASS J15394189-0520428 (J1539 -0520). \nFinally, by relating to gravity sensitive alkali lines and the aforementioned young moving group members, we discuss the interesting young candidates J1246 + 4027 and J1004 + 5022. 2MASSW J1246467 + 402715 (J1246 + 4027) has a potential lithium indication and is otherwise an L4 𝛽 field object. G 196-3B (J1004 + 5022) is confirmed as a young object, as was known from its primary companion. \nIn conclusion, this work was part of the GUCDS series of papers. A search of the GUCDS yields 145 known L dwarfs with measured RVs, excluding those from the SDSS. The 29 new L dwarf RVs presented in this work are therefore an ≈ 20 per cent increase to the number of 6-D complete L dwarfs. A number of interesting objects were identified or confirmed, either into young moving groups or young field objects. We used novel open-source techniques at all stages of our procedure, which we make available to the astronomical community. These techniques have been compared with established and accepted techniques in order to generate a baseline of trust. The observation campaign to complete the 30 pc sample is ongoing, with predominantly NIR spectrographs. This campaign will continue to produce work discussing, expanding and exploring this 30 pc sample.', 'Data availability': 'The data underlying this article will be available in CDS VizieR 9 , the GUCDS Data Browser 10 , and the SIMPLE Database 11 . The code used to generate the reduced spectra and analysis is available either through open-source repositories (see Cooper 2022b, and the acknowledgements) or upon any reasonable request. \n10 \nhttps://gucds.inaf.it', 'ACKNOWLEDGEMENTS': "Wewould like to thank the anonymous referees for their very useful and much appreciated feedback, which has much improved this manuscript. Based on observations made with the Gran Telescopio Canarias (GTC), installed in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias, on the island of La Palma. This work is based on data obtained with the instrument OSIRIS, built by a Consortium led by the Instituto de Astrofísica de Canarias in collaboration with the Instituto de Astronomía of the Universidad Autónoma de México. OSIRIS was funded by GRANTECAN and the National Plan of Astronomy and Astrophysics of the Spanish Government. \nThis research has made use of the SIMBAD database, operated at CDS, Strasbourg, France. This work presents results from the European Space Agency (ESA) space mission Gaia. Gaia's data are processed by the Gaia Data Processing and Analysis Consortium (DPAC). Funding for the DPAC is provided by national institutions, in particular the institutions participating in the Gaia Multi Lateral Agreement (MLA). This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. This publication makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, and NEOWISE, which is a project of the Jet Propulsion Laboratory/California Institute of Technology. WISE and NEOWISE are funded by the National Aeronautics and Space Administration. \nIRAF is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. We have made use of the on-line resources available from the IDL Astronomy Library hosted by the NASA Goddard Space Flight Center, in particular the DeFringeFlat.pro routine. We acknowledge the relevant open source packages used in our python (van Rossum & de Boer 1991) codes: astropy (Astropy Collaboration et al. 2013, 2018), barycorrpy (Kanodia & Wright 2018), kastredux (Burgasser 2021), matplotlib (Hunter 2007), numpy (Harris et al. 2020), pandas (Wes McKinney 2010; pandas development team 2020), scipy (Virtanen et al. 2020), specutils (Earl et al. 2021), splat (Burgasser & Splat Development Team 2017) and tqdm (da Costa-Luis et al. 2021). This research also made use of PypeIt (v.1.4.0), 12 a Python package for semi-automated reduction of astronomical slit-based spectroscopy (Prochaska et al. 2020a; Prochaska et al. 2020b). \nWJC is funded by a University of Hertfordshire studentship. WJC, HRAJ, SF, BB and DJP recognise the computing infrastructure provided via STFC grant ST/R000905/1 at the University of Hertfordshire. RLS has been supported by a STSM grant from COST Action CA18104: MW-Gaia. Funded in part by Chinese Academy of Sciences President's International Fellowship Initiative, Grant No. 2020VMA0033. DM, JAC, MCGO and NL acknowledge financial support from the Spanish Agencia Estatal de Investigación of the Ministerio de Ciencia e Innovación (AEI/10.13039/501100011033) and the ERDF 'A way of making \nEurope' through projects PID2019-109522GB-C51, -53 and -54 and PID2022-137241NB-C41, -42 and -44.", 'A1 Supplementary Tables': 'Table A1: Additional information for all observations carried out as part of the two programmes presented here. Note, multiple objects were observed multiple times, with either the same grism or the other. Seeing is given as a range corresponding to reverse wavelength, and is corrected for airmass.', 'A2 Comparison with standard routines': 'In the reduction we use two procedures based on IRAF and Python packages with a comparison target (J1745 -1640, DENIS J174534.6 -164053, Phan-Bao et al. 2008) as a sanity check. A full image and spectral reduction was carried out using standard tasks within the IRAF package on one of our target objects (J1745-1640) plus complimentary flux standard (Ross 640). This was done to assess both the quality of the data and to ascertain the necessary required reduction steps to maximise data quality. The results from this bespoke reduction method served as a reliable reference by which to measure the performance of a python pipeline (with support for the GTC/OSIRIS instrument recently added), which was later applied to all objects within our sample.', 'A2.1 Bespoke IRAF Reduction': "Our IRAF reduction was applied to the science and calibration frames of J1745 -1640 (L1-1.5) and Ross 640 (DZA6) as appropriate using the following tasks, beginning with basic image reduction: \nCCDPROC : Pre-scan bias level and bias structure removal; flatfielding; illumination correction; data section trimming. \nRESPONSE : Spectroscopic flat-field lamp colour removal (normalisation). \n- Illumination and CCDPROC : Correction for spatial axis illumination gradients, made from the extensive sky lines of a well exposed object frame. \nIDENTIFY , FITCOORDS and TRANSFORM : Correction for geometric image distortion (curvature) along the spatial axis sky background. \nFor the spectral reduction: \nAPALL : Trace and extraction using both optimal and fixed-width aperture summing using image distortion corrected arc frames. \nIDENTIFY and DISPCOR : Wavelength calibration to a linear wavelength dispersion using image distortion corrected arc frames. \nSTANDARD , SENSFUNC and CALIBRATE : Flux calibration from the flux standard Ross 640 taken on same night as the target. \nIn addition to the IRAF tasks mentioned above, two extra reduction software tools were utilised during the reduction process: \nDeFringFlat : An IDL routine aquired from the NASA IDL Astronomy library (Landsman 1993) was used to provide capabilities in de-fringing the flat field frames ( DeFringFlat.pro; Rojo & Harrington 2006). \nSKYCALC : ESO Sky Model Calculator provides additional telluric correction during flux calibration. A telluric sky model was queried using meteorological (e.g. moon phase, precipitable water vapour) and astrometric parameters (e.g. altitude, angular separation) appropriate for the object in question. \nDuring the bias subtraction we discovered that the pre-scan region of the second CCD containing the spectrum displayed a gradient across it in ADU. A carefully chosen restricted section of the pre-scan region was used ( ∼ 3 pixels wide), which was found to be reliable for row-by-row bias level subtraction, before the 2D image bias structure was removed. \nTo correct for illumination gradients evident along the spatial axis of the 2D image introduced by the slit illumination function, we utilised the extensive sky lines of the well exposed object frames as a pseudo twilight sky flat (no sky flats were available). The IRAF Illumination task provided this functionality for correction, and \nwe estimate that, after the correction was applied, the error introduced by the slit illumination gradient was reduced to a maximum of ∼ 1 . 5 per cent in the flat-field frames. \nThe latter, longer wavelength half of the flat-field frames showed evidence of fringing between wavelengths of approximately 8500 Å to 10 , 000 Å, coincident with the area of the CCD containing the spectra of interest. We used the IDL routine DeFringFlat as mentioned above to attempt to remove as much of the fringing as possible and found the best fit using the Morlet 'wavemother' model, and near default parameters. We estimate from measuring the cleaned flat-fielded image that the amplitude of the fringing was reduced from an original level of approximately 7 per cent, to a maximum of about 1 . 7 per cent. \nA combined arc frame was made from the three arcs available from the night of observation to cover the entire wavelength region of the spectrum. An initial wavelength solution was created and applied as part of the geometric image distortion correction, which resulted in a wavelength solution with an RMS error of 0 . 016 Å. A second wavelength calibration was subsequently made after additional reduction steps to ensure no systematic errors had been introduced, resulting in a more reasonable final RMS to the fitted wavelength solution of 0 . 025 Å. The final wavelength corrected spectrum had a linear dispersion 1 . 396 Å pixel -1 over the entire extracted range of 7339 Å -10 , 155 Å. \nTwo separate flux calibrations were then made: one used a blackbody to represent the DZ white dwarf flux standard with an effective temperature 8070 K (Blouin et al. 2018) and with an 𝐼 -band magnitude of 13.66 mag (Bergeron et al. 2001); the second used the low resolution calibrated flux standard spectrum of Ross 640 contained in the IRAF database. In both cases, the sensitivity functions were created by interpolating over the affected telluric regions, and regions of intrinsic absorption features. Both of these sensitivity functions provided flux calibrations with almost identical results. A correction for atmospheric extinction and telluric features to the target was included during the flux calibration. An initial extinction correction was made from using a file containing tabulated extinction magnitudes as a function of wavelength applicable to the observatory site, that was provided on the GTC instrument website. However, an improved extinction correction was obtained from the much higher spectral resolution telluric sky model mentioned above (via the ESO Sky Model Calculator). The improvement is particularly evident over the wavelength regions containing the potassium K/i.pc 𝜆𝜆 7665,7699 Å doublet and the H 2 O band at about 9500 Å.", 'A3 Radial velocity method validation': 'In keeping with our strategy outlined in Section A2 we again invoked an independent check, this time to validate our methods by helping to identify any problems with our RV measurements relating to the PypeIt reduced data set. The techniques used to measure RVs via the centres of atomic neutral alkali lines and through crosscorrelation of spectra were employed by Burgasser et al. (2015), and we adopt a similar twin measurement approach to derive our final RVs. We achieved this through the use of both IRAF and custom prepared routines within IDL to measure the RV via the Fourier cross-correlation and the line centre fitting methods. This analysis was conducted on the bespoke IRAF reduced data of our test object J1745 -1640. We then used our validated RVs to classify any objects into young moving groups and stellar associations.', '28': 'Table A2. Cross-matched absolute photometry from Gaia , 2MASS & WISE, using Gaia parallaxes.', 'A3.1 Line centres': 'Twointeractive methods were employed here: the first using routines in IDL to measure the 1D centroids of fitted Gaussian profiles to the atomic lines of J1745 -1640, while the second used the IRAF task Splot to again measure the same lines but via fitting Voigt profiles. \nIn the first case, sub-sections of the spectrum surrounding the line features to be measured were extracted and interpolated onto a ten times finer wavelength grid, to facilitate the manual fitting of Gaussian profiles with a different number of terms via the Gaussfit.pro routine. Best fitting model profiles to spectral features were initially determined by eye, and determined by how closely the profile matched the feature with more emphasis being given around the line centre region. The reported RMS error and FWHMof fitted profiles were also taken into account for when the different Gaussian profiles produced similar results, such that the number of terms which fitted with the least error and narrowest FWHM were chosen. The measured wavelength shifts from labo- \nratory rest-frame line centres (in standard air: Kramida et al. 2021) were then converted to Doppler RVs. \nSecondly, and by using Splot , Voigt profiles were fitted to the same line features of appropriately pseudo-continuum subtracted sub-sections of the spectrum, and Doppler RVs were then found in the same manner as previously from the reported line centres. We obtained results for all eight line features from both measurement sets. However, it was apparent that four of the measurements gave the least error and particularly consistent results between both sets, these being Rb /i.pc-a, Rb /i.pc-b, Na /i.pc-a, Cs /i.pc-a with mean values for RV found from these four selected for each measurement set. The RV derived from the Gaussian fitted profiles ( IDL ) was found to be 35.1 km s -1 , and via Voigt profiles ( Splot ) 29.0 km s -1 (all test results are Heliocentric: barycentric correction calculated using baryvel.pro ). Typically, we found that Gaussian profiles were more reliable to fit but Voigt profiles were best for lines which could be successfully fit. From the spread among the individually \nmeasured line shifts we place more confidence in the latter result, and assign uncertainties based on the 1𝜎 standard deviation of the respective RV measurements of 4.3 km s -1 and 3.8 km s -1 . \nThe RV as measured by our line centering method using the PypeIt reduced data for J1745 -1640 is 36 . 2 ± 4 . 4 km s -1 (see Table 4) which is in broad agreement with those from this independent measurement test. The RV measured via line centre fitting as reported by Burgasser et al. (2015) is 28 ± 9 km s -1 . Thus, we have confidence in our RV results derived from our chosen method, which contribute to the final adopted values.', 'A3.2 Cross-correlation': "To validate this second technique of measuring RVs as part of our adopted method, and to ascertain the best way forward in its application, we used the Fourier cross-correlation task Fxcor within IRAF to conduct tests. Our choice of RV rest-frame models were a BT-Settl model spectrum and custom-made synthetic atomic absorption spectra. Our object was again the bespoke IRAF reduced J1745 -1640 spectrum. \nThe BT-Settl spectrum used was the best fitting model with the physical parameters of 𝑇 eff = 2000 K, log 𝑔 = 5 dex and Fe/H = 0 dex, corresponding to ≃ L1 in spectral type. We smooth the spectrum using a Gaussian kernel to match the dispersion and resolution of the J1745 -1640, and appropriate FITS header keywords added for the Fxcor task to recognise the template spectrum as rest-frame. \nTo help highlight any potential systematic wavelength shifts introduced by the use of the BT-Settl model, and therefore to help assess its suitability as an RV template, we measured the line centre locations of the most reliable Rb /i.pc-b and Cs /i.pc-a lines by fitting Voigt profiles in Splot . BT-Settl is known to generate models using a different line list to those selected in this work, where we used the NIST database. A maximum difference compared to laboratory rest-frame line centres of 0 . 13 Å was found, corresponding to 4.5 km s -1 . This shift is similar to the uncertainty found earlier from the fitted line profiles suggesting that the BT-Settl model is reliable for use as a template, however, we add this uncertainty in velocity units in quadrature to the subsequent Fxcor individual RV region measurements. \nTo facilitate the most accurate RV measurements we extracted sections of both object and template spectra into discrete spectral regions, each covering the main atomic absorption features as well as the FeH Wing-Ford band at ∼ 9900 Å, then each region was appropriately pseudo-continuum subtracted and normalised. \nDuring the RV measurements, we interactively adjusted the sample test wavelength range around the features of interest to reduce noise in Fourier space domain. Next, the width of the crosscorrelation function (CCF) fit was changed to facilitate a best-fit (Gaussian fit to the CCF was used). The results of these changes to the CCF height, the goodness-of-fit 'R-value' and fit error were noted, until the best RV estimate was obtained. The shape of the CCF profile was also informative to this end, it tended to be broad, with no apparent double peaks seen. No Fourier filtering was applied as it was not found to be beneficial. \nFor this test, three regions gave consistent results covering both of the rubidium lines, the first caesium line ( ≈ 8500 Å) and the FeH Wing-Ford band. The average of these individual results gave an RV of 21 . 2 ± 5 . 2 km s -1 . \nFor our second test, we created a noise-free synthetic absorption spectrum of unity continuum with line widths and depths as measured by Voigt profiles of the neutral atomic lines in of \nJ1745 -1640, with no attempt to include the FeH band. The line centres were fixed to the laboratory rest-frame wavelength values. Results from all four regions were averaged which covered both of the rubidium lines, the sodium doublet and both caesium lines. Including the potassium doublet gave a similar result for that region but gave a very large increase in uncertainty, so was not included. We find a resulting RV of 24 . 6 ± 1 . 7 km s -1 . \nOur final test was conducted to ascertain the intrinsic level of uncertainty in RV from the application of this method through the use of Fxcor on a representation of our spectral data. This involved making a cross-correlation between two noise-free synthetic absorption spectra: the same RV rest-frame template as used above in the second test, and with the object being a wavelength shifted version of the same synthetic spectrum, with the FITS header updated accordingly. The shift in wavelength was set at a value corresponding to the adopted RV presented in Burgasser et al. (2015), of 26 . 2 ± 2 . 3 km s -1 . We found the average combined RV of the four measured regions used to be 26 . 7 ± 1 . 2 km s -1 , indicating that 1.2 km s -1 is our base level uncertainty in using this method. This is, however, in addition to any uncertainty introduced from a real object spectrum (i.e. J1745 -1640). \nBoth of these cross-correlation RV test results for J1745 -1640 are in agreement with the equivalent value presented in Burgasser et al. (2015), within their respective uncertainties. The measured RVfor J1745 -1640 using the cross-correlation package we adopted and apply to our data set (see Section § 4.3) has a value of 28 . 8 ± 4 . 7 km s -1 . Again, the results of this cross-correlation test validate our method and provide us with confidence in the separately derived RVs as well as in our final adopted values combined from both methods (see Section § 4.3.3).", 'A4 Radial velocity measurement confidence': 'We demonstrate here a worked example for our test object, J1745 -1640, including measurement uncertainties and our confidence metric. J1745 -1640 had a wavelength calibration RMS of 0 . 077 Å. The wavelength shifts and uncertainties excluding this wavelength calibration RMS, i.e. the uncertainty corresponding to the fitted profile centre from the square root of the diagonal of the covariance matrix, are: K /i.pc-a 0 . 767 ± 0 . 397 Å; K /i.pc-b 0 . 713 ± 0 . 190 Å; Rb /i.pc-a 0 . 916 ± 0 . 112 Å; Rb /i.pc-b 0 . 542 ± 0 . 168 Å; Na /i.pca 0 . 537 ± 0 . 114 Å; Na /i.pc-b 1 . 237 ± 0 . 088 Å; Cs /i.pc-a 1 . 363 ± 0 . 051 Å; Cs/i.pc-b 0 . 330 ± 0 . 264 Å. We had experimented with several different metrics such as 𝜒 2 but found that the root mean square deviation divided by the interquartile range (RMSDIQR) gave the most robust metric, especially when comparing across the two distinct techniques; those values were logged as follows. J1745 -1640, Line Centering: \nK/i.pc-a - Gaussian Profile with 17 . 4 Å 𝜎 ; 30 . 0 ± 18 . 5 km s -1 ; RMSDIQR = 0.74. K /i.pc-b - Gaussian Profile with 12 . 2 Å 𝜎 ; 27 . 8 ± 10 . 4 km s -1 ; RMSDIQR = 0.16. Rb /i.pc-a - Gaussian Profile with 2 . 1 Å 𝜎 ; 35 . 2 ± 7 . 2 km s -1 ; RMSDIQR = 0.09. Rb /i.pc-b - Gaussian Profile with 2 . 2 Å 𝜎 ; 20 . 4 ± 9 . 2 km s -1 ; RMSDIQR = 0.16. Na /i.pc-a Voigt Profile with 2 . 4 Å 𝜎 ; 19 . 7 ± 7 . 0 km s -1 ; RMSDIQR = 0.08. Na/i.pc-b -Voigt Profile with 2 . 8 Å 𝜎 ; 45 . 2 ± 6 . 0 km s -1 ; RMSDIQR = 0.06. Cs /i.pc-a - Voigt Profile with 2 . 3 Å 𝜎 ; 47 . 9 ± 4 . 5 km s -1 ; RMSDIQR = 0.04. Cs/i.pc-b - Gaussian Profile with 2 . 0 Å 𝜎 ; 11 . 1 ± 11 . 4 km s -1 ; RMSDIQR = 0.25. RV Line Centre = 36 . 2 ± 4 . 4 km s -1 . J1745 -1640, Cross Correlation: \nK/i.pc-a -2200 K, log 𝑔 = 5 . 0 dex ; 30 . 0 ± 5 . 0 km s -1 ; RMSDIQR = 0.48. K/i.pc-b - 2200K, log 𝑔 = 5 . 0 dex ; 20 . 0 ± 5 . 0 km s -1 ; RMSDIQR = 0.20. Rb /i.pc-a - 2200 K, log 𝑔 = 5 . 0 dex', 'A5 Spectral sequence': "We compare here in Figures A1 and A2 the sequence of R2500I spectra, as in Figures 2 and 3, to their appropriate standards and bestfitting BT-Settl models. All spectra are normalised by the median flux from 8100-8200 Å. The standards and BT-Settl models have been interpolated onto the wavelength grid of the spectra from this work. BT-Settl models have been additionally smoothed by a 2 𝜎 Gaussian kernel, so as to not 'dominate' the plot. These models are only plotted within ± 100 Å of each spectral line listed in Table 2.", 'A6 PypeIt Configuration Files': 'A6.1 Reduction \nWavelength[ ˚ A] \n<!-- image --> \nFigure A1. Same as Figure 2 with additional comparison spectra. Light blue shows the corresponding standard optical spectra whilst light orange is the best-fitting BT-Settl model around the relevant spectral lines. \nFigure A2. Same as Figure A1 but for the second half of the R2500I VPHG spectral sample. \n<!-- image --> \nWavelength[ ˚ A] \n```\n[ s e n s f u n c ] a l g o r i t h m = IR mask\\_abs\\_lines = True polyorder = 5 s a m p \\_ f a c t = 1.0 e x t r a p \\_ b l u = 0.5 e x t r a p \\_ r e d = 0.5 [ [ IR ] ] objmodel = poly polyorder = 3 d e l t a \\_ r e d s h i f t = 0. fit\\_wv\\_min\\_max = [7350 , 7550 , 7750 , 8000 , 8350 , 8900 , 9850 , 10150] A6.3 Flux Calibration [ f l u x c a l i b ] e x t i n c t \\_ c o r r e c t = False f l u x r e a d . . / Science / < spec1d -s t a n d a r d . f i t s > s e n s f u n c . f i t s . . / Science / < spec1d -object . f i t s > s e n s f u n c . f i t s f l u x end A6.4 Coadding [ coadd1d ] c o a d d f i l e = . . / Science / < s t a n d a r d . f i t s > coadd1d read . . / Science / < spec1d -s t a n d a r d . f i t s > SPAT0240 -SLIT0457 -DET02 coadd1d end [ coadd1d ] c o a d d f i l e = . . / Science / < o b j e c t . f i t s > coadd1d read . . / Science / < spec1d -object . f i t s > SPAT0240 -SLIT0457 -DET02 coadd1d end A6.5 Telluric Correction [ t e l l u r i c ] objmodel = poly polyorder = 5 fit\\_wv\\_min\\_max = 7350 , 7550 , 7750 , 8000 , 8350 , 8900 , 9850 , 10150 maxiter = 1 popsize = 300 pi x \\_ s h i f t \\_ b o u n d s = -10. , 10.\n``` \nThis paper has been typeset from a T E X/L A T E X file prepared by the author.'} |
2024ApJ...973..111J | JWST is uncovering the properties of everincreasing numbers of galaxies at z gt 6 during the epoch of reionization. Connecting these observed populations to the process of reionization requires understanding how efficiently they produce Lyman continuum LyC photons and what fraction f SUBescSUB of these photons escape into the intergalactic medium. By applying the Cox proportional hazards model a survival analysis technique to the Lowredshift Lyman Continuum Survey LzLCS we develop new empirical multivariate predictions for f SUBescSUB. The models developed from the LzLCS reproduce the observed f SUBescSUB for z 3 samples which suggests that LyC emitters may share similar properties at low and high redshift. Our bestperforming models for the z 3 galaxies include information about dust attenuation ionization andor morphology. We then apply these models to z 6 galaxies. For large photometric samples we find a median predicted f SUBescSUB 0.0470.14. For smaller spectroscopic samples which may include stronger emissionline galaxies we find that 33 of the galaxies have f SUBescSUB gt 0.2 and we identify several candidate extreme leakers with f SUBescSUB 0.5. The current samples show no strong trend between predicted f SUBescSUB and UV magnitude but limited spectroscopic information makes this result uncertain. Multivariate predictions can give significantly different results from singlevariable predictions and the predicted f SUBescSUB for highredshift galaxies can differ significantly depending on whether star formation rate surface density or radius is used as a measure of galaxy morphology. We provide all parameters necessary to predict f SUBescSUB for additional samples of highredshift galaxies using these models. SUPSUP Based on observations made with the NASAESA Hubble Space Telescope obtained at the Space Telescope Science Institute which is operated by the Association of Universities for Research in Astronomy Inc. under NASA contract NAS 526555. These observations are associated with programs GO15626 GO13744 GO14635 GO15341 and GO15639. | 2024-10-01T00:00:00Z | ['10.48550/arXiv.2406.10179', '2024arXiv240610179J', 'arXiv:2406.10179', '2024ApJ...973..111J', '10.3847/1538-4357/ad5557'] | ['Astrostatistics', 'Reionization', 'High-redshift galaxies', 'Starburst galaxies', 'Interstellar medium', 'Ultraviolet astronomy', 'Radiative transfer', '1882', '1383', '734', '1570', '847', '1736', '1335', 'Astrophysics - Astrophysics of Galaxies'] | Multivariate Predictors of Lyman Continuum Escape. II. Predicting Lyman Continuum Escape Fractions for Highredshift Galaxies | 2,024 | 173 | 0.6 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF'] | 12 | https://arxiv.org/pdf/2406.10179.pdf | {'Multivariate Predictors of LyC Escape II: Predicting LyC Escape Fractions for High-Redshift Galaxies ∗': "Anne E. Jaskot, 1 Anneliese C. Silveyra, 1, 2 Anna Plantinga, 3 Sophia R. Flury, 4 Matthew Hayes, 5 John Chisholm, 6 Timothy Heckman, 7 Laura Pentericci, 8 Daniel Schaerer, 9 Maxime Trebitsch, 10 Anne Verhamme, 9, 11 Cody Carr, 12, 13 Henry C. Ferguson, 14 Zhiyuan Ji, 15 Mauro Giavalisco, 4 Alaina Henry, 14 Rui Marques-Chaves, 9 Goran Ostlin, 5 Alberto Saldana-Lopez, 5 Claudia Scarlata, 16 G'abor Worseck, 17 and Xinfeng Xu 18 \n1 Department of Astronomy, Williams College, Williamstown, MA 01267, USA \n2 Department of Physics, University of Nevada, Reno, NV 89557, USA \n3 Department of Mathematics & Statistics, Williams College, Williamstown, MA 01267, USA \n4 Department of Astronomy, University of Massachusetts Amherst, Amherst, MA 01002, USA \n5 Department of Astronomy, Oskar Klein Centre, Stockholm University, AlbaNova, SE-10691 Stockholm, Sweden \n6 Department of Astronomy, University of Texas at Austin, Austin, TX 78712, USA \n7 Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA \n8 INAF - Osservatorio Astronomico di Roma, via Frascati 33, 00078, Monteporzio Catone, Italy \n9 Observatoire de Gen'eve, Universit'e de Gen'eve, Chemin Pegasi 51, 1290 Versoix, Switzerland \n10 Astronomy, Kapteyn Astronomical Institute, Landleven 12, 9747 AD Groningen, The Netherlands \n11 Univ. Lyon, Univ. Lyon 1, ENS de Lyon, CNRS, Centre de Recherche Astrophysique de Lyon UMR5574, 69230 Saint-Genis-Laval, France \n12 Center for Cosmology and Computational Astrophysics, Institute for Advanced Study in Physics, Zhejiang University, Hangzhou 310058, China \n13 Institute of Astronomy, School of Physics, Zhejiang University, Hangzhou 310058, China \n14 Space Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD 21218, USA \n15 Steward Observatory, University of Arizona, Tucson, AZ 85721, USA \n16 Minnesota Institute for Astrophysics, School of Physics and Astronomy, University of Minnesota, 316 Church St. SE, Minneapolis, MN 55455, USA \n17 VDI/VDE Innovation+Technik, Berlin, Germany \n18 Center for Interdisciplinary Exploration and Research in Astrophysics, Northwestern University, Evanston, IL 60201, USA", 'ABSTRACT': 'JWST is uncovering the properties of ever increasing numbers of galaxies at z > 6, during the epoch of reionization. Connecting these observed populations to the process of reionization requires understanding how efficiently they produce Lyman continuum (LyC) photons and what fraction ( f esc ) of these photons escape into the intergalactic medium. By applying the Cox proportional hazards model, a survival analysis technique, to the Low-redshift Lyman Continuum Survey (LzLCS), we develop new, empirical, multivariate predictions for f esc . The models developed from the LzLCS reproduce the observed f esc for z ∼ 3 samples, which suggests that LyC emitters may share similar properties at low and high redshift. Our best-performing models for the z ∼ 3 galaxies include information about dust attenuation, ionization, and/or morphology. We then apply these models to z ≳ 6 galaxies. For large photometric samples, we find a median predicted f esc =0.047-0.14. For smaller spectroscopic samples, which may include stronger emission line galaxies, we find that ≥ 33% of the galaxies have f esc > 0 . 2, and we identify several candidate extreme leakers with f esc ≥ 0 . 5. The current samples show no strong trend between predicted f esc and UV magnitude, but limited spectroscopic information makes this result uncertain. Multivariate predictions can give significantly different results from single variable predictions, and the predicted f esc for high-redshift galaxies can differ significantly depending on whether star formation rate surface density or radius is used as a measure of galaxy morphology. We provide all parameters necessary to predict f esc for additional samples of high-redshift galaxies using these models. \n- ∗ Based on observations made with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. These observations are associated with programs GO-15626, GO-13744, GO-14635, GO-15341, and GO-15639.', '1. INTRODUCTION': "Reionization represents a fundamental transformation of the universe's hydrogen gas and dramatically illustrates the effect galaxies can have on their surroundings. The presence of Ly α absorption in quasar spectra suggests that reionization ended sometime near z ∼ 6 (e.g., Fan et al. 2006; McGreer et al. 2015; Robertson 2022). The electron scattering optical depth of the cosmic microwave background is consistent with this general picture, constraining the midpoint of reionization to z ∼ 7 . 8 (Planck Collaboration et al. 2020). However, the exact timeline of reionization remains uncertain, in part because of the inhomogeneous nature of the reionization process (e.g., Eilers et al. 2018; Jung et al. 2020; Becker et al. 2021). \nThe recently launched James Webb Space Telescope (JWST) is rapidly expanding our knowledge of galaxy properties during the epoch of reionization. Surveys are detecting more bright galaxies than anticipated, challenging our understanding of early galaxy evolution (e.g., Castellano et al. 2022; Naidu et al. 2022; Finkelstein et al. 2023; Donnan et al. 2023; Harikane et al. 2023; McLeod et al. 2024). JWST images suggest that galaxies in the reionization era are morphologically compact (e.g., Robertson et al. 2023; Ormerod et al. 2024; Morishita et al. 2024), and spectroscopic observations are tracing the evolution of nebular ionization and metallicity to redshifts well above 7 (e.g., Schaerer et al. 2022; Sanders et al. 2023; Tang et al. 2023; Fujimoto et al. 2023; Curti et al. 2023; Backhaus et al. 2024). Measurements of nebular emission lines in early galaxies are also constraining their ionizing, Lyman continuum (LyC) photon production rate, ξ ion , a key quantity needed to understand reionization. Such observations suggest that faint galaxies with bursty star formation histories may have elevated ξ ion (e.g., Atek et al. 2024; Simmonds et al. 2024; Saxena et al. 2024). \nSeveral complementary approaches can give insights as to which galaxy populations dominated the reionization process. A galaxy's contribution to reionization depends on its ξ ion and the fraction f esc of these LyC photons that escape into the intergalactic medium (IGM). Using constraints on galaxy number densities at z > 6, some studies explore different distributions of ξ ion and f esc among the galaxy population and seek to match the observed constraints on the reionization timeline. Depending on the model assumptions, either moderately bright or very faint galaxies may provide the majority of the ionizing photons (e.g., Finkelstein et al. 2019; Naidu et al. 2020). \nCosmological hydrodynamical simulations can also model the progress of reionization over time and can ex- \nplore how and why LyC photons escape simulated galaxies. Simulations consistently find that feedback plays an essential role in clearing out obscuring material (e.g., Wise & Cen 2009; Cen & Kimm 2015; Paardekooper et al. 2015; Trebitsch et al. 2017). LyC input may peak shortly following a burst of star formation after supernovae explode (e.g., Ma et al. 2015; Trebitsch et al. 2017), although other factors such as binary star evolution or an extended star formation history may affect this timeline (e.g., Ma et al. 2015; Barrow et al. 2020; Katz et al. 2023). The geometry and mechanism of LyC escape may also vary across the galaxy population (e.g., Katz et al. 2023; Bremer & Dayal 2023). Despite general agreement on the importance of feedback, however, simulation predictions have yet to be confirmed observationally and the galaxies that are the primary drivers of reionization have not yet been conclusively identified. \nDirectly detecting LyC and constraining f esc observationally becomes difficult above z > 4 (e.g., Inoue et al. 2014) due to high IGM attenuation. Fortunately, LyC observations at lower redshifts can help test simulation predictions regarding which galaxy properties regulate LyC escape. Over the past decade, the number of LyC detections has grown rapidly, with dozens of LyC Emitters (LCEs) now known at z ∼ 2 -3 (e.g., Mostardi et al. 2015; Shapley et al. 2016; Vanzella et al. 2016, 2018; Bassett et al. 2019; Fletcher et al. 2019; RiveraThorsen et al. 2019; Ji et al. 2020; Saxena et al. 2022), multiple detections of LyC emission in stacked samples at z ∼ 2 -4 (e.g., Marchi et al. 2018; Steidel et al. 2018; Bian & Fan 2020; Nakajima et al. 2020), and more than 50 known LCEs at z < 0 . 5 (e.g., Leitet et al. 2011, 2013; Borthakur et al. 2014; Izotov et al. 2016b, 2018b, 2021; Leitherer et al. 2016; Wang et al. 2019; Flury et al. 2022a). At both low and intermediate redshift, LCEs appear deficient in absorbing gas and dust (e.g., Gazagnes et al. 2018; Chisholm et al. 2018; SaldanaLopez et al. 2022; Steidel et al. 2018; Ji et al. 2020), morphologically compact (e.g., Borthakur et al. 2014; Izotov et al. 2018b; Flury et al. 2022b; Vanzella et al. 2016; Marchi et al. 2018; Rivera-Thorsen et al. 2019), and bright in higher ionization emission lines such as [O iii ] λ 5007 (e.g., Izotov et al. 2018b; Flury et al. 2022b; Vanzella et al. 2016; Fletcher et al. 2019; Nakajima et al. 2020). \nBuilding on these observational and theoretical efforts, several studies have proposed diagnostics to predict f esc at z > 6 based on observable properties. Using simulated galaxies from the SPHINX cosmological radiation hydrodynamics simulation, Choustikov et al. (2024) develop a method to predict f esc from a linear combination of observables, including the UV slope, E(B-V), \nH β luminosity, UV magnitude, and nebular line ratios. Most other studies have taken an empirical approach, constructing f esc diagnostics based on the large observational sample of LCEs at z ∼ 0 . 3 (e.g., Verhamme et al. 2017; Wang et al. 2021; Flury et al. 2022b; Xu et al. 2023). The largest of these z ∼ 0 . 3 samples is the Low-redshift Lyman Continuum Survey (LzLCS; Flury et al. 2022a), a set of 66 galaxies with LyC measurements from the Hubble Space Telescope (HST) and ancillary ultraviolet and optical data from HST and the Sloan Digital Sky Survey (SDSS; Blanton et al. 2017). Combined with archival datasets (Izotov et al. 2016a,b, 2018a,b, 2021; Wang et al. 2019), this z ∼ 0 . 3 sample, hereafter the LzLCS+, consists of 89 galaxies with measured LyC or stringent upper limits. Based on an analysis of the LzLCS+, Chisholm et al. (2022) find that the UV slope β 1550 shows one of the strongest correlations with f esc and propose that this single variable can serve as a predictor of f esc at high redshift. Other recent studies of the LzLCS+ consider f esc diagnostics that incorporate information from multiple variables. SaldanaLopez et al. (2022) generate an equation to predict f esc from E(B-V) and low-ionization UV absorption lines. By tracing the gas and dust that destroy LyC photons, these parameters closely track f esc . Employing this method at high redshift may be a challenge, however, as measuring weak absorption lines requires high signal-to-noise observations of galaxy continua (but see Saldana-Lopez et al. 2023). Emission lines and photometry offer a simpler, if less direct, means of predicting f esc . Mascia et al. (2023) propose a new f esc diagnostic using β 1550 , [O iii ] λ 5007/[O ii ] λ 3727=O32, and halflight radius, three of the variables that correlate strongly with f esc in the LzLCS+. By applying this diagnostic at high redshift using JWST observations from the GLASS and CEERS surveys, they find predicted f esc of ∼ 0 . 1 for galaxies at z > 6 (Mascia et al. 2023, 2024). With a similar combination of variables ( β 1550 , O32, and UV magnitude), Lin et al. (2024) develop a regression model for the probability of LyC escape. Their model suggests that f esc may be high ( ∼ 0 . 2) in brighter galaxies in the epoch of reionization (Lin et al. 2024). These studies all agree that plausible LCE candidates exist at high redshift, although they differ in their f esc prediction methods. \nIn Jaskot et al. (2024), hereafter Paper I, we use the LzLCS+ to develop new empirical multivariate models for predicting f esc . Because the LzLCS+ contains both LyC detections and upper limits, we adopt the statistical techniques of survival analysis, which are suitable for such censored data. Specifically, we employ the semiparametric Cox proportional hazards model to generate \nf esc predictions based on a desired set of input observables. We show that a model limited to observables accessible at z > 6 can reproduce the observed f esc in the LzLCS+ with a root-mean-square scatter of 0.46 dex. Of these observables, three variables (O32, β 1550 , and the star formation rate surface density) are statistically significant in the fit, and a model limited to these three input observables predicts f esc as well as the full model. The Cox model technique can be customized to include any combination of variables that are available for most of the LzLCS+ galaxies and hence offers a flexible tool for predicting f esc at high redshift. \nIn this paper, we apply Cox models for f esc to samples of high-redshift galaxies. Following the techniques in Paper I, which we summarize in § 2, we generate new models for the sets of variables in published high-redshift samples. In § 3, we first test the models' performance at z ∼ 3 using samples with published LyC measurements. With published samples at z ≳ 6, we then generate f esc predictions for galaxies in the epoch of reionization in § 4. In § 5, we compare our models with alternative proposed f esc prediction methods from the literature, and we discuss the implications of our results for studies of reionization in § 6. We summarize our conclusions in § 7. In the Appendix, we provide parameters for all models as well as examples on how to apply these models to future samples. We adopt a cosmology of H 0 = 70 km s -1 Mpc -1 , Ω m = 0 . 3, and Ω Λ = 0 . 7.", '2.1. Sample: The Low-redshift Lyman Continuum Survey': "We derive our empirical predictions of f esc from the LzLCS+, a combined, homogeneously processed dataset consisting of the Low-redshift Lyman Continuum Survey (Flury et al. 2022a) and archival samples with HST Cosmic Origins Spectrograph (COS) LyC observations (Izotov et al. 2016a,b, 2018a,b, 2021; Wang et al. 2019). A full description of the data processing and UV and optical measurements appears in Flury et al. (2022a), Saldana-Lopez et al. (2022), and Paper I, but we summarize key points here. \nThe LzLCS+ contains 89 galaxies at z ∼ 0 . 3, a distance where the LyC redshifts into a sensitive wavelength range for the COS detector. The LzLCS targets were selected on properties proposed to indicate LyC escape: high O32 ratios (O32 ≥ 3), high star formation rate surface densities (Σ SFR > 0 . 1 M ⊙ yr -1 kpc -2 ), and/or blue UV slopes ( β < -2). The LzLCS+ covers a wide range of parameter space, spanning ∼ 2 dex in O32, ∼ 2 dex in Σ SFR , and an observed (not dust-corrected) UV absolute magnitude range of M 1500 = -18 . 3 to - \n21.5. In Paper I and this study, we exclude one galaxy from the LzLCS+ sample: J1333+6246 (Izotov et al. 2016b). This galaxy has visibly truncated emission lines in its SDSS spectrum and unphysical Balmer line ratios, which together suggest that its nebular line flux measurements may be inaccurate. For this work, our total sample therefore includes 88 z ∼ 0 . 3 galaxies, 49 of which have detected LyC. \nTo measure the LyC, we use COS G140L observations, processed using the calcos pipeline (v3.3.9) and the FaintCOS software routines (Worseck et al. 2016; Makan et al. 2021). We measure the LyC in a 20 ˚ A-wide wavelength bin near rest-frame 900 ˚ A, while excluding any wavelengths above 1180 ˚ A in the observed frame because of telluric emission. We follow the definitions of LyC detections and upper limits from Flury et al. (2022a), where detections are observations with a probability < 0 . 02275 of originating from background counts and the upper limit for non-detections represents the the 84th percentile of the background count distribution. To correct for Milky Way attenuation, we adopt the Green et al. (2018) dust maps and Fitzpatrick (1999) attenuation law. \nFlury et al. (2022a) investigate several different measures of f esc . Here, as in Paper I, we adopt the absolute f esc , with Starburst99 (Leitherer et al. 2011, 2014) spectral energy distribution (SED) fits to the UV continuum providing the estimate of the initial intrinsic LyC (Chisholm et al. 2019; Saldana-Lopez et al. 2022). Alternative estimates of the intrinsic LyC from H β neglect LyC photons absorbed by dust, assume an isotropic geometry, and require an assumed simplistic star formation history (e.g., Flury et al. 2022a). Another alternative measure of LyC escape is the F λ LyC /F λ 1100 flux ratio. As a ratio of two observed fluxes, this method requires no model assumptions. However, we found in Paper I that this quantity was more difficult to predict from easily accessible observables with the survival analysis models. Hence, for the f esc models in this paper, we proceed with the absolute f esc estimates from the UV SED fits for the LzLCS+. \nOptical and UV observations supply information about numerous other properties of the LzLCS+ galaxies (see Flury et al. 2022a and Saldana-Lopez et al. (2022) for details). We obtain stellar mass ( M ∗ ) estimates from Prospector (Leja et al. 2017; Johnson et al. 2019) fits to the SDSS and Galaxy Evolution Explorer ( GALEX ; Martin et al. 2003) photometry (Flury et al. 2022a, Ji et al. in prep.). With multi-Gaussian fits to nebular lines, we measure nebular line fluxes and equivalent widths (EWs), and we estimate the nebular dust attenuation, E(B-V) neb , from Balmer line ratios. \nWe calculate the oxygen abundance using the direct method and the pyneb package (Luridiana et al. 2015), adopting n e = 100 cm -3 and the estimated [O iii ] λ 4363 flux from the Pilyugin et al. (2006) 'ff-relation' in cases where the [S ii ] λλ 6716,6731 doublet or [O iii ] λ 4363 auroral line are undetected. We estimate star formation rates (SFRs) from the Kennicutt & Evans (2012) SFR calibration using the dust-corrected H β luminosities and Case B H α /H β ratio (Storey & Hummer 1995). The COS near-UV acquisition images allow us to measure the UV half-light radius r 50 , NUV , which we then use to calculate Σ SFR as \nΣ SFR = SFR 2 πr 2 50 , NUV . (1) \nIn addition to f esc , we derive several other parameters from the HST COS UV spectra. We obtain estimates of the dust attenuation E(B-V) from the UV spectrum Starburst99 SED fits; we label this parameter as E(BV) UV to distinguish it from the nebular dust attenuation E(B-V) neb derived from the Balmer lines. Although the G140L spectra do not extend to rest-frame 1500 ˚ A, we can estimate the 'observed' (non-extinction corrected) absolute magnitude at 1500 ˚ A ( M 1500 ) and the power law index slope at 1550 ˚ A ( β 1550 ) by extrapolating the SED fits to longer wavelengths (Saldana-Lopez et al. 2022). The inferred β 1550 values do match the observed values for the few galaxies with existing longer wavelength UV spectra (Chisholm et al. 2022). We also measure Ly α properties from the COS spectra. We derive Ly α fluxes and EWs by linearly fitting the continuum within 100 ˚ A of Ly α , excluding regions affected by nebular or stellar features, and integrating all emission above the continuum. We then derive the Ly α escape fraction ( f esc , Ly α ) using the dust-corrected H β flux and intrinsic Case B Ly α /H β ratio (Storey & Hummer 1995) appropriate for the galaxies' measured electron temperatures and densities. We note that the Ly α measurements represent the net sum of both underlying absorption along the line of sight and scattered Ly α emission within the aperture. Nine galaxies have detectable Ly α absorption troughs that overlap with the Si iii λ 1206 absorption feature. We increase our uncertainties to account for the change in flux from omitting wavelengths within 500 km s -1 of the Si iii line. The inclusion or exclusion of this region has only minor effects, changing the Ly α EW by < 3 ˚ A and with typical changes to f esc , Ly α of only 0.001.", '2.2. Multivariate Survival Analysis: The Cox Proportional Hazards Model': "In order to generate multivariate diagnostics for f esc , we need to incorporate information from both the f esc \ndetections and the upper limits. Within the field of statistics, survival analysis techniques are appropriate for censored datasets that contain limits. One such survival analysis method is the Cox proportional hazards model (Cox 1972; see Clark et al. 2003 and Bradburn et al. 2003 for reviews and Feigelson & Nelson 1985 and Isobe et al. 1986 for examples in astronomy). Here we describe the basic form of the Cox model and its assumptions. We refer the reader to Paper I for a more thorough discussion of this model and its application to the LzLCS+. To implement the Cox model, we use the CoxPHFitter routine in the lifelines python package (Davidson-Pilon 2019). \nThe Cox proportional hazards model predicts the probability of a particular f esc value given a set of input variables. Like many other survival analysis techniques, implementations of the Cox model typically assume the dataset contains measurements and lower limits (so-called 'right-censored' data). In contrast, our dependent variable data consist of f esc measurements and associated upper limits. Consequently, we transform our f esc values to the absorbed fraction of LyC, f abs = 1 f esc , for use in the Cox model. However, for ease of interpretation, we put the f abs results back into the form of f esc in all figures and in the tabulated results in the Appendix. \nAs applied to the LzLCS+, the Cox proportional hazards regression model fits for the probability of a LyC detection in an infinitesimally small increment of f abs , for a given set of independent variables and assuming no detection at a lower value of f abs (higher value of f esc ). The Cox model assumes a particular functional form for this f esc probability, which is known as the 'hazard function': \nh ( f abs | x ) = h 0 ( f abs ) exp[ n ∑ i =1 b i ( x i -¯ x i )] . (2) \nIn this equation, b i are the best-fit coefficients for each input variable x i , and ¯ x i is the mean value of each input variable in the reference LzLCS+ dataset. The term h 0 ( f abs ) is the baseline hazard function, the probability of having f abs in the case where all input variables match their average values within the LZLCS+. The Cox model is semi-parametric, in that the dependence on the input variables has a fixed exponential functional form, but the baseline hazard function, h 0 , is estimated non-parametrically. Despite its fixed functional form for the input variable dependence, the Cox model allows the input variables to take any form. For instance, we could define an input variable x i as a measured value j , the logarithm of that measurement log 10 ( j ), the square of that measurement j 2 , or any other functional form of our \nchoosing. In our case, we opt to use logarithmic forms of our input variables where possible (e.g, log 10 (O32), M 1500 , log 10 ( M ∗ )). In this logarithmic form, most variables have a similar order of magnitude and scale in the same manner, and an order of magnitude increase in a particular input variable simply translates into increasing the probability h ( f abs | x ) by a factor of e b i . In this paper, the only variables that we do not use in a logarithmic fashion are EW(Ly α ) and f esc , Ly α , because they range from negative (Ly α absorption) to positive (Ly α emission) within the LzLCS+. \nAlthough Equation 2 gives the probability of f abs , we would like to predict the expected values of f abs and f esc and their associated uncertainty. We adopt the median of the probability distribution as this expected value (e.g., Bradburn et al. 2003; Davidson-Pilon 2019); the model predicts that f esc will be above this value 50% of the time and below it 50% of the time. To determine this median f abs mathematically, we first calculate the survival function, S ( f abs ), the probability that we do not detect LyC at f abs , detect < f abs (i.e., at f esc , detect > f esc ). In the Cox model, the survival function is \nS ( f abs ) = exp[ -HF 0 ( f abs ) · ph( x )] , (3) \nwhere HF 0 is the baseline cumulative hazard function \nHF 0 ( f abs ) = ∫ f abs 0 h 0 ( f ) df (4) \nand ph( x ) is the partial hazards function, which describes how the probability scales with the set of input variables x : \nph( x ) = exp[ n ∑ i =1 b i ( x i -¯ x i )] . (5) \n(e.g., Cox 1972; Bradburn et al. 2003; Davidson-Pilon 2019; McLernon et al. 2023). The median f abs , which corresponds to our predicted f esc value, is the f abs value where S ( f abs )= 0 . 5. In some cases, S ( f abs ) > 0 . 5, even for our largest tabulated value of f abs , which implies that f abs is closer to 1 than we can determine and f esc ∼ 0. In this circumstance, we infer that f esc is arbitrarily small and report a predicted f esc = 0. \nThe lifelines CoxPhFitter returns the best-fit coefficients b i and the cumulative baseline hazard HF 0 ( f abs ) for each value of f abs corresponding to a LyC detection in the LzLCS+ (see Paper I for more details about the lifelines methodology). In the Appendix, we provide these best-fit parameters for the models in Paper I and the models in this paper, and we give examples of how to use these models to calculate the predicted f esc for a set of observed input variables. The coefficients and \nhazards can be used to predict the expected f esc for any galaxy, as long as it has estimated values for each of the independent variables used in the model.", '2.2.1. Uncertainty in the Predicted f esc': 'Because the survival function represents a probability distribution, we can also use it to calculate the expected uncertainty in our predicted f esc estimate. The f abs values where S ( f abs ) = 0.159 and S ( f abs ) = 0.841 represent the range in the predicted f esc corresponding to the Normal theory 1σ uncertainty. This uncertainty range reflects the inherent scatter of the relationships between f esc and the input variables, where this scatter can come both from measurement uncertainties and from genuine variation among the galaxy population. In Paper I, we tested the effect of measurement uncertainty by performing a Monte Carlo (MC) resampling of each independent and dependent variable according to its observational uncertainty. We found that the distribution of predicted f esc from the resampled inputs was nearly always smaller than the 1σ bounds inferred from the survival function. In other words, the inherent scatter in the correlations, not the measurement error, is the dominant source of uncertainty in the predicted f esc , and the survival function bounds serve as a reasonable estimate of the uncertainty in the f esc predictions.', '2.2.2. Goodness-of-Fit Metrics': "As in Paper I, we evaluate the Cox models' performance using several complementary metrics. The concordance index, C , is particularly useful, as it is appropriate for censored data. The concordance index assesses whether the model f esc predictions correctly sort the dataset in the order of its observed f esc . To evaluate this sorting, the concordance index calculation compares each possible pair of data points. Concordant pairs are those where the galaxy with higher observed f esc also has higher predicted f esc , discordant pairs are the opposite, and tied pairs have identical predicted f esc . Some pairs with upper limits in f esc lead to ambiguous rankings and do not appear in the concordance index calculation. With all pairs evaluated, the concordance index is calculated as \nC = n c +0 . 5 n t n c + n d + n t , (6) \nwhere n c , n d , and n t are the number of concordant, discordant, and tied pairs, respectively. A value of C = 1 . 0 indicates a perfect rank ordering, 0.5 is perfectly random ordering, and 0 is perfect disagreement. The major advantage of C is that it includes both LyC detections and non-detections. However, it only assesses the relative order of the predicted f esc values, not their accuracy. To \nmeasure the latter, we turn to alternative quantities. One such metric is the R 2 statistic \nR 2 = 1 -∑ i ( y i -f i ) 2 ∑ i ( y i -¯ y ) 2 , (7) \nwhere y i are the observed values of log 10 ( f esc ), ¯ y is their mean value, and f i are the predicted f esc values from the Cox model. As in Paper I and Maji et al. (2022), we also calculate a variant of R 2 , the adjusted R 2 , which accounts for the number of free parameters p in the model and number of data points n : \nR 2 adj = 1 -(1 -R 2 ) n -1 n -p -1 . (8) \nWhen increasing the number of variables, R 2 adj increases only if a variable improves the predictions more than expected by chance. Finally, we report the root-meansquare (RMS) dispersion \nRMS = √ ∑ i ( y i -f i ) 2 n . (9) \nAs explained in Paper I, we evaluate these three quantities ( R 2 , R 2 adj , and RMS) using log 10 ( f esc ). The scatter in the log 10 ( f esc ) predictions is relatively consistent across the full range of observed f esc , whereas the scatter in the linear f esc changes systematically across this range. We can only calculate the R 2 , R 2 adj , and RMS metrics for the LzLCS+ galaxies with both detected LyC and non-zero predicted f esc , since these metrics require observed values of log 10 ( f esc ) and finite predicted log 10 ( f esc ). Hence, these metrics only indicate the accuracy of the model predictions for LCEs. However, the most successful models according to the R 2 , R 2 adj , and/or RMS metrics also tend to have high C and vice versa (see Paper I).", '2.2.3. Choice of Input Variables': 'In generating Cox models from the LzLCS+, we consider a limited set of input variables. Although the dependent variable in the Cox model can contain measurement limits, the independent variables cannot. We therefore only use independent variables that have measurements for nearly all the LzLCS+ galaxies, which requires us to exclude fainter emission line measurements (e.g., [O i ] λ 6300 and [S ii ] λλ 6716,6731) or measurements not widely available for the full sample (e.g., Ly α velocity peak separation). In this paper, all 88 galaxies in the LzLCS+ have measurements for our chosen input variables with one exception. The non-leaker J1046+5827 does not have a reported Σ SFR or r 50 , NUV , and we do not include it in deriving Cox models that use these variables. The Cox model may also fail to \nconverge if we include highly collinear variables, which trace nearly identical properties. Given this limitation, we choose to use only one measure of UV dust attenuation (E(B-V) UV or β 1550 ) per model, and we do not include both Σ SFR and r 50 , NUV in a single model. \nWe explored a variety of variable combinations in Paper I, and our best model attained R 2 = 0 . 69, R 2 adj = 0 . 60, RMS = 0 . 31 dex, and C = 0 . 91. We provide the parameters for this model in the Appendix. However, this model included the average EW of Lyman-series absorption lines (EW(H i ,abs)) and f esc , Ly α , both of which will be affected by the IGM at z > 6. Models in Paper I without UV absorption lines or Ly α showed higher scatter but an overall ability to identify LCEs, with R 2 = 0 . 29 -0 . 40, R 2 adj = 0 . 14 -0 . 35, RMS = 0 . 44 -0 . 47 dex, and C = 0 . 83 (Paper I; see Appendix for these model parameters). In this paper, we are specifically concerned with variables that are easily observable at z > 6 or measured in large samples at z ∼ 3. Hence, we likewise omit UV absorption line measurements, and we avoid Ly α measurements for most of the models tailored to z > 6 galaxies.', '3. TESTING THE COX MODELS AT Z ∼ 3': 'We apply the Cox models developed on the LzLCS+ sample to galaxies at z ∼ 3 and z ≳ 6. We use the z ∼ 3 galaxies to test whether the models based on the lowredshift LzLCS+ galaxies can successfully predict f esc for galaxies at high redshift. We then apply the Cox models to z ≳ 6 samples to predict f esc for galaxies in the epoch of reionization.', '3.1. High-redshift Datasets and Models': "First, we compile samples of z ∼ 3 galaxies with reported global absolute LyC f esc and at least three of the input variables from Paper I, which include stellar mass M ∗ , M 1500 , nebular EWs, metallicity, optical nebular line ratios, Σ SFR , r 50 , NUV , E(B-V) UV , β 1550 , and Ly α measurements. The z ∼ 3 LyC measurements include individual detections, reported LyC upper limits, and stacked samples that average over variations in IGM attenuation (Hainline et al. 2009; Vasei et al. 2016; James et al. 2018; de Barros et al. 2016; Vanzella et al. 2016; Steidel et al. 2018; Pahl et al. 2021; Bassett et al. 2019; Fletcher et al. 2019; Nakajima et al. 2020; Bian & Fan 2020; Marques-Chaves et al. 2021, 2022; Liu et al. 2023; Kerutt et al. 2024). We exclude one AGN from the Fletcher et al. (2019) sample and one shock-dominated galaxy from the Bassett et al. (2019) sample. Some wellknown z ∼ 3 LCEs do not appear in our sample, because they lack published global absolute f esc measurements (e.g., Ion3 and the Sunburst Arc). We also note that \nsome of the high-redshift samples fall outside of the parameter space probed by the LzLCS+, such that our model predictions will extrapolate for these galaxies. The z ∼ 3 data has limitations as well; the compiled z ∼ 3 samples differ in the methods used for f esc and input variable measurements, and individual z ∼ 3 f esc measurements have high uncertainty due to unknown variations in IGM attenuation. We discuss the limitations of the z ∼ 3 comparison further in the following sections. \nIn Paper I, we introduced a JWST Cox Model, which could be applied to an ideal z > 6 sample, with eight relevant variables from the LzLCS+ included. We also found that a 'limited JWST Model', fit using only the three top-ranked variables ( β 1550 , log 10 (O32), and log 10 (Σ SFR )) performed equally well for predicting f esc in the LzLCS+ sample. Unfortunately, many of the required variables for both the full and limited JWST models have not been measured for large samples of z ∼ 3 LCEs or z > 6 galaxies. Consequently, we run new models limited to the variables available for the z ∼ 3 and for z ≳ 6 samples. In Table 1 and Table 2, we list these models, the variables they include, and the samples to which they apply. \nFor the z ≳ 6 samples, we prioritize models that can apply to large samples (e.g., Endsley et al. 2021, 2023; Morishita et al. 2024), models that can apply to faint galaxies (Atek et al. 2024), and models that have at least two of the most statistically significant variables in Paper I (a measure of dust attenuation plus a measurement of ionization or morphology). The 'TopThree' model includes only the three top-ranked variables from Paper I: dust, O32, and Σ SFR . In the TopThree model, we use E(B-V) UV rather than β 1550 , since it enables us to compare with a larger z ∼ 3 sample. As noted in Chisholm et al. (2022), the β 1550 and E(B-V) UV measurements for the LzLCS+ track each other almost perfectly and provide equivalent information. Most models in Table 1 use E(B-V) UV ; models with ' β ' in their name use β 1550 instead, and one model, LAE-O32-nodust, has no dust attenuation measurement. All models except the TopThree model include M 1500 , and most models include M ∗ as well. The only models without M ∗ are the LAE, LAE-O32-nodust, and ELG-O32β models. \nWe provide the best-fit coefficients and cumulative baseline hazards for each model in the Appendix, which can be used to derive the median predicted f esc for a given galaxy and the uncertainty from the 16-84th percentiles of the f esc probability distribution. As with the LzLCS+ galaxies, we use an MC method to sample the variable uncertainties for the z ∼ 3 and z ≳ 6 observations and re-generate the predicted f esc . We again \nfind that the uncertainty estimated from the Cox model survival function dominates over the uncertainty from sampling the input variables in nearly all cases. \nTable 1. Cox Models for High-Redshift Predictions", 'T able 2 . Co x Mo del High-Redshift Samples': 'Note -(a) F rom de Barros et al. ( 2016 ); V anzella et al. ( 2016 , 2020 ). (b) F rom Hainline et al. ( 2009 ); V asei et al. ( 2016 ); James et al. ( 2018 ). W e adopt the a v erage dust-corrected O32 ratio o f the t w o similar-flux regio ns in Hainline et al. ( 2009 ). The E(B-V) UV and Σ SFR v alues are flux-w eigh ted a v erages of the regions in Jame s et al. ( 2018 ). (c) F rom Marques-Cha v es et al. ( 2022 ). (d) F rom Fletc her et al. ( 2019 ); Nak a jima et al. ( 2020 ). (e) W e adopt the f esc determined from the SED-fitting metho d in Liu et al. ( 2023 ). (f ) F rom Marques-Cha v es et al. ( 2021 ). (g) F rom Steidel et al. ( 2018 ); P ahl et al. ( 2021 ). (h) W e adopt M ∗ calculated using the burst y non-parametric star formation history , but include the alternativ e v alues in the unce rtain ties. W e use the direct metho d meta llicities for the t w o ga lax i es with secure [O i i i ] λ 4363 detections and the mo deled metallicities for the others. (i) W e correct M ∗ for magnification, usin g the v alues in Sc haerer et al. ( 2022 ). F or the source without a direct metho d metallicit y , w e adopt the strong -line metho d 12+log 10 (O/H). (j) W e exclude so urces flagged as ha ving significan t residuals in the morp hological fits. (k) Metallicities calculated using strong-line metho ds. \nWe first test the Cox model predictions on z ∼ 3 samples to see whether the high-redshift LCEs behave similarly to the LzLCS+ sample. We derive Cox models for the variable sets in Table 1 using only the LzLCS+ data; the high-redshift samples are not included in the fitting process. We then use the models to predict f esc for each set of z ∼ 3 measurements, and we calculate goodness-of-fit metrics for the LzLCS+ sample alone, the high-redshift sample alone, and for the combined sample of low- and high-redshift galaxies. We list these goodness-of-fit metrics in Table 3.', '3.2. Model Performance': 'Figures 1 and 2 show the predicted vs. observed f esc for the LzLCS+ and z ∼ 3 samples. As seen from the plots and the goodness-of-fit statistics in Table 3, these models do not perform as well as the fiducial model from Paper I, but several (TopThree, LAE-O32, ELG-O32, ELG-O32β , ELG-O32β -Ly α , R50β ) are comparable to or better than the Paper I JWST model (see Tables 11-13 for the list of input variables used in the Paper I models). The metrics for the LzLCS+ sample are R 2 = 0 . 02 -0 . 42 (vs. 0.59 for the fiducial model and 0.29 for JWST), R 2 adj = -0 . 06 to 0.36 (vs. 0.48 for the fiducial model and 0.14 for JWST), RMS= 0 . 44 -0 . 59 (vs. 0.37 for the fiducial model and 0.47 for JWST), and C = 0 . 77 to 0.84 (vs. 0.88 for the fiducial model and 0.83 for JWST). \nThe models developed on the LzLCS+ sample generally reproduce the observed f esc values of the z ∼ 3 LCEs. In fact, the high-redshift samples often have a lower RMS scatter than the predictions for the LzLCS+, and in several models, the R 2 and C values for the combined high- and low-redshift sample are comparable to or higher than the R 2 and C values for the LzLCS+ sample alone. Notably, predictions for the two strongest high-redshift LCEs, Ion2 and J1316+2614, match the observations nearly perfectly in most cases (Figures 1 and 2), demonstrating the success of the model in identifying extreme LCEs. \nThe goodness-of-fit metrics give clues as to which variables are most important in predicting f esc . As seen in Table 3, the models with the highest R 2 and lowest RMS for the LzLCS+ sample are the TopThree, LAE-O32, ELG-O32 and variants, and the R50β models. These same models are the only ones that include both a measure of UV dust attenuation and either O32 or radius as variables. Turning to the C metric, which includes nondetections, the best-performing models for the LzLCS+ sample are the R50β , TopThree, LAE-O32, ELG-O32- \nβ -Ly α , and LAE models. To properly estimate f esc for weak and non-LCEs, UV dust attenuation again appears important, but each of the best-performing models also includes either morphological information or Ly α measurements. These additional variables may help distinguish non-LCEs from LCEs. \nBy all metrics, the LAE-O32-nodust and β -Metals models perform the worst for the LzLCS+ galaxies. The LAE-O32-nodust model is the only one that lacks E(BV) UV or β 1550 , which highlights the crucial role of dust extinction in LyC escape. Conversely, the β -Metals model relies almost exclusively on dust attenuation to infer f esc and lacks information on O32, Ly α , or morphology, which may better constrain the LyC absorption due to H i .', '3.2.1. Model Performance for z ∼ 3 Samples': "Comparing the models' performance for the highredshift and combined low- and high-redshift samples is difficult, because each high-redshift model applies to a different set of galaxies. For example, the ELG-O32β model has the lowest RMS for the high-redshift galaxies, but this RMS is based on a single galaxy: Ion2. Ion2 is the only galaxy that is included in most of the models (all but ELG-O32β -Ly α , where it has an input limit). For Ion2, we adopt an observed absolute f esc =0.75, in the middle of the 0.5-1 bounds reported by Vanzella et al. (2016). Using this f esc , Ion2's LyC escape is predicted most accurately by the ELG-O32β (RMS=0.02 dex) and ELG-O32 (RMS=0.04 dex) models. For Ion2, dust attenuation, luminosity, and ionization appear to be key factors in predicting its f esc , although most other models are still consistent with its reported range of f esc (RMS=0.09-0.15 dex for the next six models). The β -Metals model, which lacks information about O32 and morphology, is the only inaccurate model (RMS=0.43 dex). More surprisingly, however, the TopThree model gives the lowest predicted f esc =0.53 (RMS=0.15 dex), which highlights the role of UV magnitude in the f esc predictions, a point we discuss further in § 4. \nNine galaxies, including four LCEs, appear in each of the LAE, LAE-O32, LAE-O32-nodust, and ELG-O32 model test samples. Although the LAE-O32-nodust model performed worst for the LzLCS+, for this common subsample of nine high-redshift galaxies, the LAEO32-nodust model has the lowest RMS (0.09 dex) and highest C (0.76), compared to RMS values of 0.27-0.32 dex and C values of 0.65-0.71 for these galaxies in the other three models. Given the generally successful performance of all four of these models, models with O32 and/or EW(Ly α ) appear able to predict f esc for highredshift galaxies equally well. Hence, while models with \nTable 3. Metrics for High-Redshift Cox Modelsz ∼ 3 Sample", 'Combined Sample': 'Note -N Gal is the number of galaxies assessed in the fit and does not include any galaxies with limits for input variables. N Detect is the number of galaxies with LyC detections, finite f esc predictions, and no upper or lower limits for input variables; the R 2 and RMS metrics use only these galaxies. The R 2 statistic measures how well the predictive model explains the observed variance in the log 10 ( f esc ) data, with higher R 2 values corresponding to more accurate models. The R 2 adj metric accounts for the number of parameters used in the model and increases only if a variable improves the fit more than expected by chance. RMS is the rootmean-square dispersion of the predicted vs. observed log 10 ( f esc ) for the LyC detections. Higher C values indicate that the model more accurately sorts the observations in the correct order of increasing f esc . C includes both detections and galaxies with f esc upper limits. See § 2.2.2 for a full description of these statistics. \nFigure 1. The f esc predictions from the TopThree (a), LAE (b), LAE-O32 (c), and LAE-O32-nodust (d) models. See Table 1 for model descriptions. Red circles represent LzLCS+ LyC detections, and blue triangles represent upper limits. We plot z ∼ 3 galaxies in green, with stars denoting LyC detections and triangles denoting upper limits. We identify the two strongest highredshift LCEs, Ion2 and J1316+2614, by a teal and a green diamond, respectively. Light green symbols indicate high-redshift galaxies that have a limit for one or more input variables. We draw circles around data points representing high-redshift galaxy stacks, which are less subject to uncertainty in IGM attenuation. The error bar in the upper left corner indicates the median size of the uncertainties in the observed and predicted f esc for the combined sample of low- and high-redshift galaxies. The dashed line shows a one-to-one correspondence. Several of the Cox models predict f esc for both low- and high-redshift galaxies with comparable accuracy. \n<!-- image --> \nELG-EW Model \n<!-- image --> \nELG-O32- Model \n<!-- image --> \nR50- Model \n2 \n10 \nObserved \nf \nesc(LyC) \n(e) \n<!-- image --> \nELG-O32 Model \n<!-- image --> \nFigure 2. The f esc predictions from the ELG-EW (a), ELG-O32 (b), ELG-O32β (c), ELG-O32β -Ly α (d), R50β (e), and β -Metals (f) models. See Table 1 for model descriptions. Ion2 is shown by an open symbol in panel d, because its prediction is based on a limit in f esc , Ly α . Other symbols are the same as in Figure 1. \n<!-- image --> \n10 \n) \nC \ny \nL \n( \nc \ns \ne \nf \nd \ne \nt \nc \ni \nd \ne \nr \nP \n10 \n10 \n10 \n10 \nesiduals \nR \n0 \n1 \n2 \n3 \n1 \n0 \n1 \n10 \n3 \n1 \n10 \n0 \nE(B-V) UV and O32 or morphology work best for the LzLCS+ sample, we cannot rule out the possibility that EW(Ly α ) may also be important in predicting f esc at z ∼ 3. \nNevertheless, using EW(Ly α ) as the sole nebular measurement is not sufficient to accurately predict f esc . The LAE model (Figure 1b), which applies to the largest set of z ∼ 3 LyC measurements, is one of the least successful models at predicting f esc in the high-redshift galaxies. As seen in Table 3, this model has the largest RMS scatter for the z ∼ 3 sample (0.9 dex), lowest C (0.49), and the second worst R 2 (-4.2). The other model with comparably poor R 2 and C for z ∼ 3 galaxies is the ELG-EW model, which is the only other model that exclusively relies on an emission line EW for its input nebular information. Both models also have some of the lowest R 2 values for the LzLCS+ sample, which reflects the fact that these models substantially under-predict the f esc of the strongest LCEs in the LzLCS+. The LAE model shows this same tendency towards underprediction for the z ∼ 3 LCEs as well; 13 z ∼ 3 galaxies with observed f esc ≥ 0 . 2 are predicted to have f esc of only ≤ 0 . 03 by the model. \nThe poor performance of the LAE model may result from limitations in both the model itself and in the z ∼ 3 data. The fact that both the LAE and ELG-EW models fare poorly for the LzLCS+ and for the high-redshift galaxies, especially for strong LCEs, suggests that nebular EW is a flawed tracer of f esc . Indeed, [O iii ] EW and Ly α EW should be low both for the weakest and for the strongest LCEs. In non-leakers, a high optical depth and corresponding lack of Ly α escape should lead to a low Ly α EW. Low [O iii ] and Ly α EWs could also indicate a weak current burst of star formation, without significant feedback or LyC production, which might result in undetected LyC. At lower optical depths and moderate f esc , Ly α EW may increase due to enhanced escape, and [O iii ] EWs may likewise increase if these lower optical depths are preferentially associated with stronger starbursts. However, the [O iii ] and Ly α EWs will decrease again in the strongest LCEs, as a general lack of nebular gas results in the limited production of nebular emission lines (e.g., Zackrisson et al. 2013; Nakajima & Ouchi 2014). Thus, EW may be an ambiguous indicator of LyC escape. \nThe z ∼ 3 data also have limitations. The strong LCEs that are most severely under-predicted by the LAE model all come from the Liu et al. (2023) and Kerutt et al. (2024) samples of individual LAEs with LyC detections. Both papers note that their LyC detections represent the extreme of the population and may not be representative of the average galaxy population \nwith these parameters, whose f esc may be much lower. In Paper I, we found a similar result for the strongest LCEs in the LzLCS+, whose only distinguishing feature compared to more moderate LCEs is their low line-ofsight H i content, suggestive of a favorable orientation. Another limitation of the high-redshift data is the uncertainty in the IGM attenuation. At z ∼ 3, this attenuation is significant and varies along different lines of sight (e.g., Rudie et al. 2013; Inoue et al. 2014; Vanzella et al. 2016; Steidel et al. 2018). This unknown sightline dependence leads to additional uncertainty in the reported f esc for individual galaxies and could cause some genuine LCEs to appear as non-detections or to have overestimated f esc . \nThe stacked samples (Steidel et al. 2018; Nakajima et al. 2020; Bian & Fan 2020) shown circled in Figure 1, average over these variations and are less subject to this uncertainty. In the LAE model in Figure 1b, we see that the stacked samples do indeed show less scatter than the individual high-redshift detections. However, the model does not perfectly predict f esc for the stacked samples, and their scatter is still comparable to the scatter in the individual LzLCS+ galaxies. In addition to averaging over IGM attenuation variations, the stacks also average over any galaxy-to-galaxy variation in physical properties. Hence, the input variables may not represent the true set of properties of an individual system, leading to some uncertainty in the predicted f esc . \nUnfortunately, the small sample sizes in most models and the lack of a common high-redshift sample across the models make it difficult to discern which model parameters are most important for predicting f esc at high redshift. Our results suggest that if high-redshift galaxies behave like their lower redshift counterparts in the LzLCS+ sample, E(B-V) UV or β 1550 , O32, and Σ SFR or half-light radius are essential variables to include. For the existing z ∼ 3 samples, models with at least two of these parameters do successfully predict the f esc of the available high-redshift galaxies to within ∼ 0.3 dex. \nThis performance also shows that the LzLCS+ galaxies may indeed be reasonable analogs for high-redshift galaxies and that the same observable and physical properties may correlate with LyC escape at both low and high redshift (e.g., Saldana-Lopez et al. 2023; Schaerer et al. 2022; Mascia et al. 2023). This agreement is not trivial, as in principle, the z ∼ 0 . 3 and z ∼ 3 galaxies could have different star formation histories, dust properties, morphologies, or other properties, any of which could affect f esc . The LzLCS+ parameter space does cover the properties of the z ∼ 3 samples, with some exceptions. Seventeen z ∼ 3 targets (8 LCEs and 9 non-LCEs) are brighter than the LzLCS+ UV magni- \nge by 0.1-3.18 magnitudes, including the strong LCEs Ion2 and J1316+2614. An additional two LCEs in the LAE model are fainter than the LzLCS+ by 0.50.7 magnitudes. The predictions for most models, all those with M 1500 as a variable, therefore extrapolate to an unobserved part of parameter space, yet still perform well. \nThe data in Figures 1 and 2 and metrics in Table 3 include z ∼ 3 galaxies that have measured values for all of the input independent variables in the models 1 . However, additional galaxies in these samples have upper or lower limits for some independent variables, which can still provide potentially useful constraints on f esc . For these galaxies, we adopt the limit as the value of the independent variable and predict their f esc using the LzLCS+ models. We plot these approximate estimates in Figures 1 and 2 as light green symbols and an open symbol for Ion2, but we do not include these galaxies in the goodness-of-fit metrics. \nThe galaxies with limits have one or more of the following: upper limits in M ∗ , lower limits in Σ SFR , lower limits in EW([O iii ]+H β ), lower limits in O32, lower limits in EW(Ly α ), lower limits in M 1500 , and lower limits in f esc , Ly α . One galaxy has an upper limit in EW([O iii ]+H β ) rather than a lower limit. Given the coefficients for these variables in the models, the mass and magnitude limits would cause the model to overestimate f esc , while the lower limits in the nebular emission lines should generally cause the model to underestimate f esc . For example, a galaxy with a lower limit of EW(Ly α ) > 100 ˚ Ais an even stronger Ly α emitter than we assume, and its predicted f esc will be an underestimate. \nMost of the z ∼ 3 LCEs with input limits in the LAEO32 and LAE-O32-nodust models only have lower limits in O32, such that their predicted f esc should be an underestimate. For galaxies with lower limits in both a nebular property and M 1500 , we cannot easily interpret the predicted f esc as a lower or upper limit. However, the best-fit coefficients in the Appendix show that the models have a steeper dependence on the nebular lines than they do on M 1500 . Figures 1 and 2 show that for most of the galaxies with input limits, the measured limits still constrain f esc as accurately as for the rest of the sample. However, a few galaxies (90675, 101846, 105937 from Fletcher et al. 2019) have strongly underpredicted f esc , which suggests that their nebular lines may be much stronger than the reported limit. \nFinally, we note that our model predictions do not account for systematic uncertainties, including differences in methodology. We adopt the published values for all data. However, each paper makes different assumptions, which could affect the tabulated values. For instance, papers differ in their adopted IGM transmission models, the models used in SED fits, and adopted dust attenuation laws. These assumptions could lead to systematic differences in properties like f esc , M ∗ , and E(B-V) UV among the different samples. In addition, LyC measurements may be photometric or spectroscopic and may cover different wavelength ranges. Despite these systematics, the models work well in predicting f esc . Adopting consistent methodologies could potentially result in better predictions; however, as shown by the scatter in the LzLCS+ sample, which has a consistent methodology, the inherent uncertainty in the model itself also limits the possible accuracy of predictions.', '3.2.2. Summary of z ∼ 3 Results': 'In conclusion, Cox models derived using the z ∼ 0 . 3 LzLCS+ sample can successfully predict f esc in z ∼ 3 LCEs. This agreement suggests that LCEs may have similar physical properties at both low and high redshift (e.g. Saxena et al. 2023). We find that the most accurate models include E(B-V) UV and O32 or morphology measurements as variables. However, larger high-redshift samples with a full suite of measurements are required to test this result. Future observations of z ∼ 3 LCEs with JWST will further clarify whether the relationship between f esc and physical properties evolves with redshift or remains constant. Preliminary JWST observations suggest that z ≳ 6 galaxies may indeed share numerous physical properties with low-redshift analog samples like the LzLCS (e.g., Schaerer et al. 2022; Endsley et al. 2023; Mascia et al. 2023; Lin et al. 2024). The models developed on the LzLCS+ sample and tested on z ∼ 3 LCEs may therefore apply equally well at the reionization epoch.', '4.1. Model Predictions for z ≳ 6 Samples': "Given the success of our models in predicting f esc at z ∼ 3, we now use the Cox models to estimate f esc for galaxies at z ≳ 6, where LyC is not detectable because of the IGM opacity. We consider several models: the 'ELG' models, R50β , and β -Metals. The β -Metals model is the most limited, as it only includes measurements of luminosity, mass, UV slope, and metallicity and has no information on nebular line strength or morphology. Based on the LzLCS+ galaxies, these models have an RMS scatter of 0.44-0.55 dex in f esc (see Sec- \nand Table 3), with the ELG-O32 or R50β models giving the most accurate predictions, depending on the metric considered. \nWe can apply these models to several samples of reionization-era galaxies, which span z ∼ 6 -14: photometric samples from Endsley et al. (2021), Endsley et al. (2023), Bouwens et al. (2023), and Morishita et al. (2024) and spectroscopic samples from Williams et al. (2023), Schaerer et al. (2022), Tang et al. (2023), Fujimoto et al. (2023), Saxena et al. (2023, 2024), Mascia et al. (2023), and Atek et al. (2024). The Endsley et al. (2021) sample is based on Spitzer , HST , and groundbased observations; all other samples incorporate JWST NIRCam photometry and several (Williams et al. 2023; Schaerer et al. 2022; Tang et al. 2023; Fujimoto et al. 2023; Saxena et al. 2023, 2024; Mascia et al. 2023; Atek et al. 2024) use JWST NIRSpec spectroscopy as well. We select all galaxies at z > 5 . 9 from these samples to investigate f esc in the epoch of reionization.", '4.1.1. Photometric Samples': 'We list the predicted f esc values and their associated uncertainties for the z ≳ 6 galaxies in Tables 4-8. The ELG-EW and R50β models (Tables 4 and 5) apply to the largest compilations of z ≳ 6 galaxies, with 183 measurements corresponding to 180 unique galaxies for the ELG-EW samples and 278 measurements at z > 5 . 9 for the R50β samples, although most of the galaxies for both models only have photometric redshifts. Most of the galaxies in the ELG-EW samples are from Endsley et al. (2021, 2023), and most of the R50β sample galaxies come from Morishita et al. (2024). \nWeshow histograms of the ELG-EW and R50β model f esc predictions in Figure 3. The median of the f esc predictions for the ELG-EW model samples is fairly low, f esc = 0 . 047, which is lower than the average f esc value of 0.1-0.2 required to reionize the universe assuming a canonical ionizing photon production efficiency log 10 ( ξ ion ) ∼ 25 . 3(e.g., Finkelstein et al. 2015, 2019; Robertson et al. 2015; Naidu et al. 2020). Of these z ≳ 6 galaxies, 27% have f esc ≥ 0 . 1 and only 6% have f esc ≥ 0 . 2. However, as shown in Figure 2a, the ELGEW model is one of the less accurate models, with a greater tendency to underpredict the true f esc of the LzLCS+ sample. This underprediction implies that the ELG-EW variable set, which includes EW([O iii ]+H β ) but not O32, does not distinguish strong LCEs from weaker LCEs. The more accurate R50β model (see Table 3 and Figure 2) implies a higher fraction of LCEs, with a median value of f esc = 0 . 14, and 56% and 39% of the galaxies with f esc > 0 . 1 and f esc > 0 . 2, respectively. Both models find a substantial fraction of weak \nor non-leakers, and 25% of the galaxies in both models have f esc < 0 . 03 -0 . 04. Taken at face value, our preliminary results would suggest that f esc values > 0 . 1 are common but not ubiquitous among moderate to bright galaxies ( M 1500 < -18) in the epoch of reionization (see also Mascia et al. 2023, 2024). \nWe caution that some of the z ≳ 6 galaxies in the ELG-EW sample and many in the R50β sample fall outside of the parameter space probed by the LzLCS+, although a similar extrapolation did not seem to adversely affect the f esc predictions for the z ∼ 3 galaxies (Section 3). Of the 183 measurements in the ELG-EW model sample, 14 galaxies have brighter UV magnitudes than the LzLCS+ galaxies, one is fainter, and 17 have higher EW([O iii ]+H β ). Likewise, the R50β sample contains 8 galaxies brighter than and 30 galaxies fainter than the LzLCS+ sample. More concerningly, 114 of the 278 galaxies in the R50β compilation are more compact than the LzLCS+ galaxies, which could mean that their high inferred f esc values result from an incorrect extrapolation into this compact regime.', '4.1.2. Spectroscopic Samples': 'The ELG-O32 and ELG-O32β model predictions (Tables 6 and 7) apply to fewer z ≳ 6 galaxies, only 17 and 27 galaxies, respectively, but all of these galaxies have spectroscopic redshifts. Half the galaxies in the ELGO32 model have predicted f esc < 0 . 05, although most of these are only lower limits in f esc , and 25% of the ELGO32β sample galaxies have f esc ≤ 0 . 01. Like the R50β model predictions, low f esc is common, but the overall distribution also extends to very high f esc . Both models include some galaxies with lower limits in O32, whose predicted f esc values therefore also correspond to lower limits. Depending on how high the true f esc for these galaxies are, 35-82% of the ELG-O32 model galaxies and 33-41% of the ELG-O32β galaxies have f esc ≥ 0 . 2. \nBecause these galaxies are all spectroscopically confirmed, these samples could be biased toward galaxies with stronger emission lines and hence higher f esc ; the median EW([O iii ]+H β ) of the spectroscopic ELG-O32 sample is 1790 ˚ A (Fujimoto et al. 2023; Tang et al. 2023; Saxena et al. 2023), compared with 690 ˚ A for the photometric samples in the ELG-EW model, for example (Endsley et al. 2021, 2023; Bouwens et al. 2023). In addition, these models may not underpredict f esc for strong LCEs to the same extent as the ELG-EW model (see Figure 2 and Table 3). For the six galaxies with pre- \nTable 4. Predictions for z ≳ 6 Galaxies from the ELG-EW Model \nNote -Predicted f esc values from the ELG-EW model for the z ≳ 6 galaxies from Endsley et al. (2021, 2023), Bouwens et al. (2023), Tang et al. (2023), Fujimoto et al. (2023), and Saxena et al. (2023). z phot and z spec are the photometric and spectroscopic redshifts. f esc , min and f esc , max represent the 15.9 and 84.1 percentiles of the model f esc predictions. For galaxies with upper limits on EW([O iii ]+H β ), the f esc predictions are also upper limits and are marked accordingly. The Reference column lists the publication used for the model input variables. The full, machine-readable version of this table is available online. We show the first five rows as an example here. \nTable 5. Predictions for z ≳ 6 Galaxies from the R50β Model \nNote -Predicted f esc values from the R50β model. z phot and z spec are the photometric and spectroscopic redshifts. f esc , min and f esc , max represent the 15.9 and 84.1 percentiles of the model f esc predictions. The Reference column lists the publication used for the model input variables. The full, machine-readable version of this table is available online. We show the first five rows as an example here. \ndicted f esc > 0 . 2 in the ELG-O32 model, the ELG-EW predictions are indeed lower by ∆ f esc =0.26 on average, which suggests that model differences also account for some of the higher f esc values compared to the ELG-EW distribution. \nWhile we have generated initial predictions for z ≳ 6 galaxies using our models, predictions for the largest samples are limited by a lack of spectroscopic information. We do find evidence of high f esc > 0 . 2 among smaller spectroscopic samples at z ≳ 6, but these samples may be biased toward stronger emission-line galaxies. To obtain more accurate predictions, high-redshift \nstudies should prioritize observations of nebular line ratios and galaxy sizes for larger samples. As JWST surveys continue, larger, more representative spectroscopic samples will improve estimates of the average f esc at z > 6.', '4.2. Trends with UV Magnitude': "We examine the predicted f esc values as a function of magnitude in Figure 4. We plot predictions from the three models that apply to the largest sample sizes (ELG-EW, ELG-O32β , and R50β ) and from the model that applies to the faintest galaxies ( β -Metals; Table 8). We see no strong trend with magnitude, although the few brightest sources with M 1500 ≲ -21 . 5 do appear to have higher f esc on average, with a median f esc = 0 . 18 vs. 0.07 for fainter sources. Aside from these \nTable 6. Predictions for z ≳ 6 Galaxies from the ELG-O32 Model \nNote -Predicted f esc values from the ELG-O32 model. z spec is the spectroscopic redshift. f esc , min and f esc , max represent the 15.9 and 84.1 percentiles of the model f esc predictions. For galaxies with lower limits on O32, the f esc predictions are also lower limits and are marked accordingly. The Reference column lists the publication used for the model input variables. \nrare sources, the ELG-EW, ELG-O32β , and β -Metals samples tend to show f esc within ∼ 0 . 02 -0 . 3 across a wide range of magnitudes, from -16 to -22. The R50β model (Figure 4b) also shows a flat distribution of f esc with magnitude, but it predicts many more galaxies at higher f esc . However, as previously noted, 41% of the galaxies in this model require extrapolating predictions to galaxies with extremely compact morphologies, which makes these predictions uncertain. \nAny trends with magnitude or lack thereof in the other models should also be taken with caution. All the models have a tendency to under-predict f esc , but this tendency is most pronounced in the ELG-EW and β -Metals models (Figure 2), which trace the extremes of the magnitude range in Figure 4. The more reliable ELG-O32β model, which better reproduces the LzLCS+ f esc values, suggests that some of the galaxies in the M 1500 = -19 to -21 magnitude range may have significantly high f esc . The ELG-O32β sample does not include enough faint \ngalaxies to reveal whether or not a trend between f esc and magnitude exists, however. \nAt the faintest magnitudes, the Atek et al. (2024) sample, included in the β -Metals model includes several extremely low-luminosity lensed galaxies observed as part of the JWST UNCOVERsurvey (Bezanson et al. 2022; Weaver et al. 2024). For these galaxies, the β -Metals model therefore extrapolates the LzLCS+ sample to an unobserved parameter space. The only statistically significant predictor variable in the β -Metals model is β 1550 , and Figure 4 illustrates that these galaxies' blue slopes suggest at least moderate f esc (0.02-0.18; Table 8). Crucially, even these moderate f esc values, which may be under-predicted, are still greater than what is required for such faint galaxies to dominate reionization given their high LyC photon production rates (Atek et al. 2024). \nWhile moderate or high f esc for faint, blue galaxies matches past predictions (e.g., Chisholm et al. 2022), the high f esc in the brighter sources in Figure 4 may seem surprising at first glance. In all four models, the fit- \nTable 7. Predictions for z ≳ 6 Galaxies from the ELG-O32β Model \nNote -Predicted f esc values from the ELG-O32β model. z spec is the spectroscopic redshift. f esc , min and f esc , max represent the 15.9 and 84.1 percentiles of the model f esc predictions. For galaxies with lower limits on O32, the f esc predictions are also lower limits and are marked accordingly. The Reference column lists the publication used for the model input variables. \nted coefficients (see the Appendix) show that a brighter M 1500 results in a higher f esc at fixed values of the other inputs (fixed E(B-V), fixed O32, etc). This effect does not come from the association of higher observed UV luminosity with lower dust content. The data do not show such a correlation, and, because the models already include dust measurements, the model implies that brighter UV luminosities increase f esc at fixed dust attenuation. In addition, we see a similar relationship with M ∗ , where higher M ∗ is also associated with higher f esc at fixed values of the other parameters. Lin et al. (2024) find a similar result using a different technique. Using the LzLCS+ data, they fit a logistic regression \nmodel for the probability of having LyC escape as a function of M 1500 , β 1550 , and O32 and likewise find a brighter M 1500 increases the probability of LyC escape for fixed values of the other variables. This scaling does not imply that higher UV luminosities correlate with f esc in the LzLCS+ sample; if anything, the LzLCS+ indicates a slight trend in the opposite direction (Flury et al. 2022b). Instead, this scaling implies that a bright galaxy with LCE-like properties, such as high O32 and blue UV slope, is a more extreme object than a faint galaxy with identical properties. For instance, at fixed O32, the LzLCS+ galaxies with higher luminosities tend to have higher f esc , but high O32 values are also uncom- \nTable 8. Predictions for z ≳ 6 Galaxies from the β -Metals Model \nNote -Predicted f esc values from the β -Metals model. z spec is the spectroscopic redshift. f esc , min and f esc , max represent the 15.9 and 84.1 percentiles of the model f esc predictions. The Reference column lists the publication used for the model input variables. \nmon among the bright galaxies as a whole, leading to an overall trend of decreasing f esc with luminosity (e.g., Flury et al. 2022b). \nThe enhanced f esc for bright LCE candidates affects the predictions in Figure 4 by boosting the f esc of a bright, dust-poor emission-line galaxy or compact galaxy relative to an otherwise similar faint galaxy. For instance, for the ELG-EW model, the brightest galaxies should have higher predicted f esc only if they have equally low E(B-V) and equally high EW as the fainter galaxies. Indeed, the ELG-EW samples, mostly galaxies from Endsley et al. (2021, 2023), show no trend between E(B-V) UV and M 1500 or between EW([O iii ]+H β ) and M 1500 ; with comparable dust and emission line strengths, the brighter galaxies therefore end up with higher predicted f esc . Endsley et al. (2023) note that lower metallicities or higher f esc among the fainter galaxies could suppress their EWs and account for the lack of an observed trend with M 1500 . Hence, other properties, such as O32, might be necessary to distinguish faint galaxies with high f esc . The EWs in the Endsley et al. (2021, 2023) samples also come from photometry, which may be more uncertain than spectroscopic measurements (e.g., Duan et al. 2024). \nThe ELG-O32β samples have O32 measurements, but they similarly show no strong trend between β 1550 and M 1500 or between O32 and M 1500 . Likewise, the Morishita et al. (2024) sample, which constitutes the bulk of the R50β model sample, exhibits a relatively flat trend between radius and M 1500 . Only the β -Metals samples, \nmostly from Atek et al. (2024), show any trend between physical properties and M 1500 , with the lowest metallicities and bluest slopes appearing among the faintest galaxies. \nThe limitations of the models and the available z ≳ 6 datasets make it difficult to discern any trends between M 1500 and f esc . Limited measurements for galaxies at the brightest and faintest magnitudes restrict us to some of the less accurate Cox models. The models also require extrapolating to make predictions for galaxies outside the magnitude or radius range of the LzLCS+ sample. Better constraints on f esc within the z > 6 population will require larger spectroscopic samples across a wide range of galaxy magnitudes as well as better observational constraints on f esc in a more diverse set of galaxies at lower redshifts.", '4.3. Notes on Individual Sources': "4.3.1. Strong LCES \nSeveral galaxies in the ELG-O32 and ELG-O32β models appear to be extremely strong LCEs (Tables 6 and 7). One unusual galaxy, CEERS-1019, has a predicted f esc = 1. Such a high f esc would seemingly conflict with the presence of strong emission lines, since it should correspond to a complete absence of absorbing gas. However, given the uncertainty in the model predictions, the f esc of CEERS-1019 could be as low as 0.6. Two other galaxies in the ELG-O32 model sample, CEERS-698 and CEERS-1027, have f esc ≥ 0 . 5, and f esc as high as 1 is consistent with their uncertainties. In- \nFigure 3. The distribution of predicted f esc from the ELG-EW model (a) for z ≳ 6 galaxies from Endsley et al. (2021, 2023), Bouwens et al. (2023), Tang et al. (2023), Fujimoto et al. (2023), and Saxena et al. (2023) and from the R50β model (b) for z ≳ 6 galaxies from Morishita et al. (2024) and Mascia et al. (2023). The vertical dotted lines show the median and quartiles of the distributions ( f esc = 0 . 027 , 0 . 047 , 0 . 105 for ELG-EW and f esc = 0 . 039 , 0 . 141 , 0 . 467 for R50β ). Both models have some limitations but suggest that many, but not all, high-redshift galaxies may have f esc > 0 . 1. \n<!-- image --> \nFigure 4. Predicted f esc as a function of M 1500 for z ≳ 6 galaxies. Panel a shows the predicted f esc from the ELG-EW model (circles) for galaxies with photometric redshifts (gray) and spectroscopic redshifts (black), from the ELG-O32β model (blue stars), and from the β -Metals model (green diamonds). Panel b shows the predicted f esc from the R50β model (squares) with photometric (gray) and spectroscopic (black) redshifts. The crosses in the upper corners show the median uncertainties for each panel. Dark gray dotted lines connect the predictions for the same galaxy in different models. We see no strong trend between f esc and M 1500 , but many of the predictions suffer from a limited set of variables (ELG-EW and β -Metals models) or require extrapolation outside the LzLCS+ parameter space (R50β model). \n<!-- image --> \nPredicted \nf \nesc vs. \nM \n1500 for \nz \n6 Galaxies \n18 \n19 \nM \n(b) \n20 \n1500 \nR50- Model \n21 \n22 \nc \ns \ne \nf \nd \ne \nt \nc \ni \nd \ne \nr \nP \n10 \n10 \n10 \n10 \n0 \n1 \n2 \n3 \n17 \n23 \nterestingly, all three of these strong LCE candidates are strong Ly α emitters, which suggests that they reside within ionized bubbles (Tang et al. 2023). Similarly, two galaxies in the ELG-O32β samples have predicted f esc > 0 . 5: 18846 and 9422 from JADES (Saxena et al. 2024). With O32= 70 . 6, the ELG-O32β model infers f esc = 0 . 85 for 9422, much higher than the f esc = 0 . 01 value derived by Saxena et al. (2024) using the multivariate prediction method of Choustikov et al. (2024); we discuss the differences between our predictions and those of other models in § 5. \nThe true f esc of some of these extreme galaxies may not be quite as high as the models predict, however. Because two of these strong LCEs, CEERS-1019 ( f esc = 1) and CEERS-698 ( f esc ∼ 0 . 7), have brighter UV magnitudes than the LzLCS+ sample by 0.4-0.9, their high predicted f esc could be unreliable. However, these same models successfully predicted the high f esc of the z ∼ 3 galaxies Ion2 and J1316+2614 (Figure 2), which are as bright as these z > 6 galaxies. The LzLCS+ sample itself also contains only three LCEs with f esc > 0 . 5, which limits the models' ability to accurately determine predictive relationships in the high f esc regime (Mascia et al. 2023). CEERS-1019 may differ from the LzLCS+ galaxies for additional reasons; it has unusually strong nitrogen emission and is a possible candidate for supermassive star formation (Marques-Chaves et al. 2024) or an AGN (Larson et al. 2023). The physical conditions occurring in extreme galaxies like CEERS-1019 likely are not present within the LzLCS+ sample. Empirical predictions based on z = 0 . 3 galaxies would not be suitable for such objects. \nThe predictions for the other strong LCE candidates are likely more reliable, provided their star-formation and nebular conditions are not significantly different from the LCEs at z ∼ 3. Galaxy 9422 has a higher O32 ratio than the LzLCS+, but the other strong leaker candidates fall within the LzLCS+ parameter space, so that the model is not extrapolating to an unobserved regime. Hence, our models suggest that several strong emission-line galaxies identified in the epoch of reionization may be strong LCEs with f esc well above 0.2.", '4.3.2. JADES-GS-z7-LA': "Saxena et al. (2023) report the detection of a z = 7 . 3 galaxy with strong Ly α emission, JADES-GS-z7-LA, which is presumably located within an ionized bubble. With f esc , Ly α = 0 . 96, IGM absorption seems to have had little effect on the galaxy's Ly α emission. We therefore choose to incorporate the measured EW(Ly α ) and f esc , Ly α in our model predictions. We can apply most of the models in Table 1 to JADES-GS-z7-LA, omitting \nonly the TopThree, R50β , and β -Metals models due to a lack of reported Σ SFR , radius, and 12+log 10 (O/H). We compare the different f esc predictions in Table 9. JADES-GS-z7-LA does fall outside of the LzLCS+ parameter space, which may limit the accuracy of the f esc predictions; it is fainter than the LzLCS+ galaxies by 1.6 mag, and it has a higher EW(Ly α ) by 144.1 ˚ A. \nThe different Cox models disagree regarding the f esc of JADES-GS-z7-LA, with predicted f esc ranging from 0 to 0.40. The only nonzero predictions come from models that have information about UV dust attenuation and Ly α , but even the extreme Ly α properties of JADES-GS-z7-LA do not guarantee f esc > 0 . 1. The LAE-O32 and ELG-O32β -Ly α models include O32 and stellar mass as well as Ly α and find f esc = 0 . 017 -0 . 085. These two models are also among the best-performing models for the LzLCS+ sample (Table 3). JADES-GSz7-LA is not devoid of dust, with E(B-V) UV = 0 . 10, and its UV slope β = -2 . 1 is not extreme. These properties are consistent with the properties of z ∼ 0 . 3 galaxies with moderate f esc ; all the LzLCS+ galaxies with higher f esc > 0 . 1 have lower E(B-V) UV < 0 . 1, and a slope of β 1550 = -2 . 1 matches the median value for the moderate LCEs with f esc = 0 . 05 -0 . 1. \nFurthermore, as discussed above, given JADES-GSz7-LA's low mass (log 10 ( M ∗ / M ⊙ )=7 . 15), its O32 value of 8.8 may not be extreme. Even its strong Ly α emission does not necessarily imply a low optical depth along the line of sight, since Ly α photons can scatter. For instance, the non-leaking z ∼ 0 . 3 galaxy J1248+4259 (Izotov et al. 2018b) has f esc ≤ 0 . 013 and EW(Ly α )= 256 ˚ A. One piece of evidence that favors a high f esc for JADESGS-z7-LA is the small offset between the Ly α velocity and the systemic redshift (Saxena et al. 2023). With the low resolution of most of the LzLCS+ UV spectra, we cannot currently include this parameter in our models, and it may boost the predicted f esc for JADES-GS-z7LA. However, this low velocity offset could still be consistent with little to no LyC escape; its offset of 120 ± 80 kms -1 also resembles the Ly α profile of the non-leaker J1248+4259, whose red peak is offset from the systemic velocity by < 200 kms -1 . \nIn agreement with the assessment of Saxena et al. (2023), we thus find that the evidence for LyC escape in JADES-GS-z7-LA is ambiguous. The multivariate models that incorporate the most information suggest that it likely has a moderate f esc ∼ 0 . 017 -0 . 085 and may therefore not be solely responsible for producing its ionized bubble in the IGM (e.g., Witstok et al. 2024). \nThe case of JADES-GS-z7-LA highlights the fact that the Cox multivariate predictions can differ from singlevariable estimates (e.g., based on Ly α alone). Galaxies with high O32 provide another example of this result. For instance, Williams et al. (2023) present JWST observations of a lensed low-mass (log 10 ( M ∗ / M ⊙ ) = 7.7) galaxy, 11027, at z = 9 . 51 and hypothesize that the galaxy has a high f esc > 0 . 1 based on its high O32 ratio of 12 ([O iii ] λλ 5007+4959/[O ii ]=16). However Lin et al. (2024) suggest that 11027 is not likely to be a strong LCE considering the combination of O32, UV magnitude, and UV slope. Our ELG-O32 and ELGO32β Cox models (Tables 6-7) likewise predict f esc of only 0.012-0.021. O32 alone is not sufficient to constrain f esc , and high O32 values are common among the lowest mass galaxies in the LzLCS+ sample (Flury et al. 2022b), including weak LyC emitters. In contrast, the ELG-O32β model predicts that several galaxies with lower O32 than galaxy 11027 (e.g., 6355, 10000, 10612, 12637 with O32=6.3-10.6, Schaerer et al. 2022; Mascia et al. 2023; Saxena et al. 2024) are actually more likely to be LCEs because of their higher luminosities and/or bluer UV slopes (Table 7). The Cox predictions emphasize the fact that not all galaxies with high O32 have high f esc (e.g., Izotov et al. 2018b; Flury et al. 2022b) and additional properties such as mass, UV luminosity, and dust extinction are important to consider. \nMultivariate models are an important tool for predicting f esc in the epoch of reionization. Our results highlight the fact that multivariate models can give substantially different predictions from single variable estimates, and f esc predictions should therefore incorporate as much information as possible.", '5.1. Single-Variable Predictions': "Among the observable properties measured for the LzLCS+ sample, the UV slope β 1550 shows one of the strongest correlations with f esc . Consequently, Chisholm et al. (2022) propose that β 1550 could predict f esc for high-redshift galaxies and derive a relationship between β 1550 and f esc based on the LzLCS+ dataset. In Figure 5, we compare the predicted f esc from this method with the observed f esc for the LzLCS+ and for z ∼ 3 galaxies (Vanzella et al. 2016; Bassett et al. 2019; Marques-Chaves et al. 2022). We list the goodness-offit statistics for the LzLCS+ with the Chisholm et al. (2022) model in Table 10. The Chisholm et al. (2022) model's high R 2 = 0 . 45 and low RMS= 0 . 43 are comparable to the best-performing Cox models in Tables 3 \nand 10, which demonstrates that β 1550 alone can indeed predict f esc for LCEs reasonably well. Consistent with this result, Paper I finds that, of the variables accessible at high redshift, β 1550 is the most important variable to include in the Cox models. \nDespite its success for LCEs, however, the Chisholm et al. (2022) model has more difficulty in constraining f esc in the LzLCS+ non-detections, as indicated by its lower concordance C = 0 . 76. This concordance is lower than all the Cox models in Tables 3 and 10, which have C = 0 . 77 -0 . 83. The difference between the predictions for detections and non-detections is apparent in Figure 5. For LzLCS+ galaxies with f esc detections or upper limits between 0 . 01 -0 . 1, the one-to-one relation between the Chisholm et al. (2022) model predictions and the observations runs right between the LzLCS+ detections. However, nearly all the non-detections are above this line, indicating that the model is systematically over-predicting their f esc . In contrast, for one of the better performing Cox models, such as the ELG-O32 model in Figure 2b, both detections and non-detections in this same f esc range fall on either side of the one-toone relation, indicating that predictions for detections and non-detections are comparable. \nFigure 5 also applies the Chisholm et al. (2022) model to two z ∼ 3 LCEs: Ion2 and J1316+2614. The model correctly identifies both galaxies as LCEs but underpredicts their f esc , although Ion2's observed f esc is consistent with the stated uncertainties in the Chisholm et al. (2022) predictions. Comparing the Ion2 predictions with those from the Cox models discussed in § 3, we find that the Chisholm et al. (2022) single-variable β 1550 model is one of the least accurate at predicting Ion2's f esc , with a predicted f esc = 0 . 29, corresponding to RMS=0 . 40 dex. As discussed in Chisholm et al. (2022), β 1550 only tracks the loss of LyC photons due to dust. Scatter below and above the Chisholm et al. (2022) β 1550 -f esc relation results from variations in the absorbing H i column. Other properties, such as O32, Σ SFR , and UV magnitude, may better track the H i component of LyC absorption for galaxies like Ion2. \nFigure 6 illustrates how including these other variables affects predictions at z > 4. We plot the difference between the observed LzLCS+ f esc and the Chisholm et al. (2022) predictions as a function of β 1550 . Galaxies are color-coded by their O32 ratio (Figure 6a) or Σ SFR (Figure 6b). For the reddest β 1550 , all the model predictions and observations agree that galaxies have little to no LyC escape. However, at blue UV slopes, the disagreement increases. The Chisholm et al. (2022) model assigns all galaxies a single value of f esc , whereas the LzLCS+ observations show a range of f esc and deviate \nTable 9. f esc Predictions for JADES-GS-z7-LA \nNote -Predicted f esc values for JADES-GS-z7LA from different models. f esc , min and f esc , max represent the 15.9 and 84.1 percentiles of the model f esc predictions. \nfrom the predictions by as much as ∆ f esc = 0 . 2 lower or ∆ f esc = 0 . 6 higher at the bluest slopes. For the LzLCS+ galaxies, this difference is associated with other differences in the galaxies' properties; galaxies with higher O32 or higher Σ SFR tend to be offset to higher f esc , and galaxies with lower f esc than the Chisholm et al. (2022) predictions tend to have lower O32 or lower Σ SFR . \nThe multivariate model predictions reflect these trends with other properties and assign galaxies a higher or lower f esc at fixed β 1550 accordingly. In Figure 6, we also plot the difference between the predicted f esc for z > 4 galaxies from multivariate models vs. the singlevariable Chisholm et al. (2022) model. All these multivariate models use β 1550 in addition to two or more other variables. The Mascia et al. (2023) model (squares) predicts f esc using a linear fit to β 1550 , log 10 (O32) and log 10 ( r 50 , NUV ) for their sample of z > 4 JWST-GLASS galaxies. We show predictions from two Cox models in crosses and stars: a Cox model using the three topranked, high-redshift accessible variables from Paper I ( β 1550 , O32, and Σ SFR ) for the Mascia et al. (2023) sample and the ELG-O32β Model ( §§ 3-4), which uses β 1550 , O32, and M 1500 as inputs to predict f esc for z > 4 galaxies from Williams et al. (2023), Schaerer et al. (2022), Mascia et al. (2023), and Saxena et al. (2024). Like the LzLCS+ observations, the multivariate z > 4 predictions can significantly deviate from the Chisholm et al. (2022) predictions, especially at blue UV slopes. Highredshift surveys relying only on β 1550 could therefore miss some of the physical differences among blue galaxies that might indicate variations in f esc . Properties such as O32 and Σ SFR are sensitive to feedback in galaxies and may complement β 1550 measurements by tracing a galaxy's ability to carve low column density H i channels.", '5.2. Multivariate Predictions': "Recently, Choustikov et al. (2024) and Mascia et al. (2023) have developed multivariate linear regression models to predict f esc from a set of observable variables. Choustikov et al. (2024) derive their model for f esc from synthetic spectra of z > 4 galaxies from the SPHINX cosmological radiation hydrodynamics simulation. Mascia et al. (2023) base their predictions on the LzLCS+ sample but adopt the upper limit in f esc as the observed value for non-detections. Here, we use galaxies at low and high redshift to compare the predictions of the Cox models with the predictions from these literature models. \nIn Figure 7a, we show the results of applying the Choustikov et al. (2024) model to the LzLCS+ sample, and we list the corresponding goodness-of-fit statistics in Table 10. The Choustikov et al. (2024) model fails to predict the observed f esc in the LzLCS+ sample. This result does not depend on our method of calculating f esc ; we find a similarly poor fit if we substitute the f esc derived from H β , as used in the Choustikov et al. (2024) simulations. In Figure 7b, we show f esc predictions from a Cox model run using the same set of variables ( β 1550 , E(B-V) neb , L (H β ), M 1500 , R23=([O iii ] λλ 5007,4959+[O ii ] λ 3727)/H β , and O32) as in the Choustikov et al. (2024) linear regression mode, and we list the goodness-of-fit metrics in Table 10. \nThis model performs comparably to the other Cox models in Table 3, which shows that the reason for the Choustikov et al. (2024) model's difficulty is not its set of variables. Rather, in the Choustikov et al. (2024) model, f esc anti-correlates with O32, whereas the LzLCS+ data \nFigure 5. The f esc predictions for the LzLCS+ and z ∼ 3 galaxies from the Chisholm et al. (2022) model, which uses a single predictor variable: β 1550 . Symbols are the same as in Figure 1. The model reproduces the f esc of LCEs but over-predicts the f esc for the LzLCS+ non-detections and under-predicts f esc for the strongest LCEs. \n<!-- image --> \nTable 10. Goodness-of-Fit Statistics for Literature vs. Cox Models \nNote -A comparison of the goodness-of-fit statistics for f esc predictions for the LzLCS+ sample from the Chisholm et al. (2022), Choustikov et al. (2024), and Mascia et al. (2023) literature models and various Cox models. We list statistics for the JWST Model from Paper I and a model limited to its three top-ranked variables. We also list statistics for Cox models run using the same sets of variables as in the Choustikov et al. (2024) and Mascia et al. (2023) models. \n<!-- image --> \nFigure 6. Circles (detections) and triangles (upper limits) show the difference between LzLCS+ observations and Chisholm et al. (2022) predicted f esc as a function of observed β 1550 . Other symbols show the difference between multivariate model predictions for z > 4 samples and the Chisholm et al. (2022) single-variable predictions as a function of β 1550 . Squares compare the f esc predictions from Mascia et al. (2023), using β 1550 , O32, and r 50 , NUV . Crosses show f esc predictions for the same z > 4 Mascia et al. (2023) sample for a Cox model using β 1550 , O32, and Σ SFR . Stars show the ELG-O32β Cox Model predictions for z > 4 galaxies from Williams et al. (2023), Schaerer et al. (2022), Mascia et al. (2023), and Saxena et al. (2024), where the input variables are β 1550 , O32, and M 1500 . We color-code galaxies by their observed O32 ratio in panel (a) and by Σ SFR in panel (b). At blue UV slopes, the LzLCS+ observations and high-redshift model predictions can differ significantly from the Chisholm et al. (2022) model predictions, depending on the galaxies' other properties, such as O32 and Σ SFR . \n<!-- image --> \nand the Cox model using the Choustikov et al. (2024) variable set indicate a correlation between f esc and O32. The compact, lower-mass LCEs in the LzLCS do not have counterparts in the SPHINX simulations; these compact LCEs often have high O32 ratios ≳ 10 (Flury et al. 2022b), whereas the SPHINX galaxies with O32 > 10 typically have lower values of f esc . Hence, while we find that high O32 increases f esc in our multivariate predictions, Choustikov et al. (2024) find the opposite, that at fixed β and E(B-V), galaxies with older ages and lower O32 have higher f esc . Radiative feedback with a turbulent gas structure may allow f esc from high O32 galaxies at earlier ages than predicted by cosmological simulations (e.g., Kakiichi & Gronke 2021; Kimm et al. 2019; Choustikov et al. 2024). However, further studies of z ∼ 3 galaxies are also necessary to test whether the LzLCS population and the LzLCS-derived Cox models correctly describe the high-redshift population. \nLike our Cox model predictions in § 4, Choustikov et al. (2024) predict a low f esc = 0 . 03 for JADES-GS-z7-LA, due to its moderate UV slope and dust content. However, because of the different dependence of f esc on O32, the Choustikov et al. (2024) model predicts a dramatically lower f esc for CEERS-44, CEERS-698, CEERS1019, and CEERS-1027 ( f esc = 0 . 006 -0 . 1) and for JADES 9422 ( f esc = 0 . 01; Saxena et al. 2024) than our model predictions ( f esc > 0 . 4 to 1). Choustikov et al. (2024) point out that their predicted f esc values are consistent with f esc , Ly α > f esc as observed locally (e.g., Flury et al. 2022b), but IGM effects may complicate any comparison with f esc , Ly α . As noted above ( § 4), the Cox model predictions for some of these galaxies require extrapolating outside the LzLCS+ parameter space, and their f esc may not truly be as extreme as this model predicts. Still, these galaxies remain plausible candidates for high f esc given the observed trends seen at z ∼ 0 . 3 and z ∼ 3 ( § 3). LyC observations of galaxies with similar properties are necessary to better constrain their f esc . \nIn contrast to the results for the Choustikov et al. (2024) model, the Mascia et al. (2023) model (Figure 8a and Table 10) reproduces f esc for the LzLCS+ galaxies reasonably well. This finding is unsurprising, since the Mascia et al. (2023) model is in fact derived from the LzLCS+ dataset. A Cox model run using the same variables ( β 1550 , r 50 , NUV , and O32) performs comparably to the Mascia et al. (2023) model (Figure 8b and Table 10). The largest difference occurs for weak LCEs ( f esc < 0 . 05) and non-detections. In these cases, the Mascia et al. (2023) model tends to systematically overpredict f esc , whereas by incorporating information from non-detections, Cox model predictions are more evenly \ndistributed above and below the observed values. However, we note that the over-prediction of f esc in the Mascia et al. (2023) model is small, ∆ f esc of only a few percent. \nDespite its derivation from the same dataset, the Cox and Mascia et al. (2023) models give different predictions at high redshift. The Mascia et al. (2023) model predicts f esc = 0 . 19 for Ion2, the only z ∼ 3 LCE with the required measurements, whereas the Cox models more accurately match its observed f esc > 0 . 5 ( § 3). This difference suggests that galaxy luminosity, incorporated in the Cox models in either Σ SFR or in M 1500 may be important in reproducing f esc for the strongest LCEs. \nIn Figure 9, we compare the model predictions for GLASS-JWST galaxies (Treu et al. 2022) at z = 4 -8 from Mascia et al. (2023) with predictions from Cox models using similar combinations of variables. For a Cox model using the same variables as in Mascia et al. (2023), the predictions from Mascia et al. (2023) and the corresponding Cox model track each other (Figure 9a), aside from the aforementioned tendency of the Mascia et al. (2023) model to give higher predictions at low f esc . However, the Mascia et al. (2023) and Cox models disagree more strongly when Σ SFR is used as a variable in the Cox model instead of r 50 , NUV (Figure 9b). The inclusion of Σ SFR shifts some galaxies with compact sizes but weak star formation to much lower f esc . We conclude that including some measure of luminosity or SFR may be important to accurately identify non-leakers and extreme LCEs like Ion2. \nMascia et al. (2024) also apply their multivariate model to galaxies from CEERS and find significantly different results than the predictions from our ELG-O32 Cox model for these galaxies ( § 4, Table 6). We compare these predictions in Figure 9c. Again, the disagreement primarily arises from the inclusion of luminosity in the ELG-O32 Cox model but not the Mascia et al. (2023) model. As with the GLASS sample, when we run a Cox model with the same input variables ( β 1550 , r 50 , NUV , and O32), our predictions agree closely with the Mascia et al. (2024) predictions (Figure 9a). Adding one additional variable, M 1500 , as a measure of luminosity begins to bring the predicted f esc into agreement with the ELGO32 predictions (Figure 9d). The remaining disagreement between the Mascia et al. (2024) and ELG-O32 Cox model predictions is due to the use of radius as a variable in the Mascia et al. (2023) model and different estimates of dust content and ionization from different publications (Mascia et al. 2024; Fujimoto et al. 2023; Tang et al. 2023). \nObservational and theoretical studies agree that f esc depends on multiple physical parameters. However, \n<!-- image --> \nFigure 7. (a) The f esc predictions from the Choustikov et al. (2024) literature model compared with the LzLCS+ observations. (b) The f esc predictions from a Cox model run using the same variables as the Choustikov et al. (2024) model: β 1550 , E(B-V) neb , L (H β ), M 1500 , R23, and O32. Symbols are the same as in Figure 1. The Choustikov et al. (2024) model does not reproduce the f esc observations from the LzLCS+, but a Cox model using the same input variables does recover the observed f esc . \n<!-- image --> \n<!-- image --> \nFigure 8. (a) The f esc predictions from the Mascia et al. (2023) literature model compared with the LzLCS+ observations. (b) The f esc predictions from a Cox model run using the same variables as the Mascia et al. (2023) model: β 1550 , r 50 , NUV , and O32. Symbols are the same as in Figure 1. Both models perform similarly, but the Cox model does slightly better at predicting f esc in weak LCEs and non-detections. \n<!-- image --> \nstudies have not yet reached a consensus as to which parameters matter and how. Predictions for f esc in z ≳ 6 galaxies from this work, Choustikov et al. (2024), Mascia et al. (2023) and Mascia et al. (2024) disagree because of different adopted scalings with O32 and with luminosity. Observationally testing and distinguishing between these predictions with larger samples at z < 6 \nwill be critical to reliably predict f esc in the epoch of reionization.", '6. IMPLICATIONS FOR REIONIZATION': 'As demonstrated in § 3, the empirical Cox models derived from the LzLCS+ show promise as diagnostics of f esc at high redshift. However, with the limited measurements at z > 6 available so far, the models do', 'Cox Model Variables: O32, r50, 1550': 'Figure 9. A comparison of f esc predictions for high-redshift galaxies from Cox models and the Mascia et al. (2023) model. (a) A Cox model using β 1550 , r 50 , NUV , and O32 gives similar predicted f esc as the Mascia et al. (2023) model for z > 4 galaxies from the GLASS-JWST (black) and CEERS (blue) surveys (Mascia et al. 2023, 2024). (b) If Σ SFR is substituted for r 50 , NUV in the Cox model, the Cox model f esc predictions for the GLASS galaxies deviate from those of the Mascia et al. (2023) model. (c) For the CEERS galaxies, the f esc predictions from the ELG-O32 Cox model disagree with the Mascia et al. (2024) predictions. (d) A Cox model using M 1500 in addition to the Mascia et al. (2023) variables begins to agree more closely with the ELG-O32 Cox model predictions. Adding a measurement of galaxy luminosity in the form of Σ SFR or M 1500 can significantly affect the f esc predictions. \n<!-- image --> \n(a) \nComparison of \nf \nesc Predictions for \nz \n>6 CEERS Galaxies \n<!-- image -->', 'Cox Model Variables: O32, SFR, 1550': "3 \nPredicted \nf \nesc from Mascia et al. (2023) \nComparison of \nf \nesc Predictions for \nz \n>6 CEERS Galaxies \n10 \n3 \nPredicted \nf \nesc from 1550, \nr \n50, O32, \nM \n1500 Cox Model \n(d) \nnot decisively show which galaxy populations dominate reionization ( § 4.2). Because a combination of properties regulates a galaxy's optical depth, we need estimates of many factors to accurately predict f esc . Dust attenuation, galaxy morphology, ionization, and UV luminosity all play a role in promoting LyC escape. Based primarily on dust attenuation, we would expect faint galaxies to be strong LCEs (e.g., Chisholm et al. 2022; Atek et al. 2024). Nevertheless, at a fixed value of β 1550 , f esc still \nshows considerable spread in the LzLCS+ and in multivariate predictions (Figure 6), spanning a range of up to ∼ 0 . 7 in f esc . Without estimates of other properties (e.g., Σ SFR , O32), we therefore cannot determine whether faint galaxies have f esc above or below the average LzLCS+ value for the same UV color. \nAt the same time, we cannot yet rule out the contribution of more luminous galaxies. Several galaxies with M 1500 ≤ -20 (e.g., CEERS 498, 1027, 698, 1019; \n10 \n2 \n10 \n1 \n10 \n0 \n(b) \n10 \n2 \n10 \n1 \nl \ne \nd \no \nM \nx \no \nC \nm \no \nr \nf \nc \ns \ne \nf \nd \ne \nt \nc \ni \nd \ne \nr \nP \nl \ne \nd \no \nM \nx \no \nC \n2 \n3 \nO \n- \nG \nL \nE \nm \no \nr \nf \nc \ns \ne \nf \nd \ne \nt \nc \ni \nd \ne \nr \nP \n10 \n10 \n10 \n10 \n10 \n10 \n10 \n10 \n0 \n1 \n2 \n3 \n0 \n1 \n2 \n3 \n10 \n10 \n0 \nJADES 18846; GLASS-JWST 100003, 10021) have the low dust content and high ionization suggestive of extreme f esc ≥ 0 . 5. The LzLCS+ data seem to suggest that for the same O32 ratio and β 1500 , a more luminous galaxy will have higher f esc (see also Lin et al. 2024). This effect could possibly be an observational bias, as the LyC flux should be easier to detect for more luminous objects. However, we note that at fixed O32, brighter LzLCS+ galaxies show higher ratios of 900 ˚ A to 1100 ˚ A flux, which suggests that their high f esc is genuine. Moreover, the trend of higher f esc with luminosity only appears in galaxies that have properties associated with LyC escape, such as high O32. Overall, the brighter galaxies in the LzLCS+ have fewer LyC detections (Flury et al. 2022b), which suggests that the detections are not biased toward the brightest galaxies. Furthermore, including UV luminosity in the multivariate models appears necessary to reproduce the f esc of Ion2 ( § 3 and § 5), which suggests that the increase of f esc with luminosity may be a real phenomenon. \nIn the multivariate models, M 1500 might help to break the degeneracy between ionization parameter and optical depth in galaxies with high O32. For instance, lowmetallicity galaxies tend to have both low luminosities and inherently high ionization parameters (e.g., Nagao et al. 2006; Dopita et al. 2006), such that they may be more likely to have high O32 even at high optical depth. Alternatively, LCEs with higher UV luminosities could be able to ionize gas over a wider opening angle, so that we are statistically more likely to observe a high f esc line of sight. \nOf course, to evaluate a galaxy population's influence on reionization, we must also consider how many ionizing photons are produced and the total LyC input into the IGM, not just f esc . Unfortunately, the Cox models do not do as well at predicting the ionizing to nonionizing UV flux ratio or the total LyC luminosity (Paper I). Galaxies with high O32 and high nebular EWs may possess elevated ξ ion values (e.g., Schaerer et al. 2016; Tang et al. 2019; Maseda et al. 2020; Naidu et al. 2022), such that at fixed f esc , these galaxies will provide more LyC photons to the IGM. These same properties scale with f esc in the multivariate Cox models, and our identified candidate strong LCEs at z > 6 generally have O32 > 10 and/or EW([O iii ]+H β ) > 1500 ˚ A. High f esc may therefore be coupled with high ξ ion (e.g., Schaerer et al. 2016; Naidu et al. 2022). \nHowever, ionizing photon production may vary with other parameters as well, such as galaxy luminosity (e.g., Bouwens et al. 2016; Finkelstein et al. 2019). Fujimoto et al. (2023) estimate ξ ion for spectroscopically confirmed galaxies at z ∼ 8 -9 and find that \nξ ion is two times higher at M UV ∼ -19 . 5 compared to M UV ∼ -21 . 5. Although this enhanced efficiency is not enough to compensate for the six times fainter UV luminosity at M UV ∼ -19 . 5 vs. -21.5, these fainter galaxies are also approximately 40 times more numerous (Bouwens et al. 2015). Atek et al. (2024) infer similarly high ξ ion for even fainter ( M UV ∼ -17 to -15) and even more numerous galaxies, which only need f esc < 0 . 05 to drive reionization. For the faintest galaxies we consider ( M 1500 > -17 . 5), the median of the Cox model predicted f esc values is near this threshold, with median f esc ∼ 0 . 04. However, these f esc estimates require refinement with additional parameters (e.g., O32, Σ SFR ). If f esc does not vary strongly with luminosity, as suggested by our preliminary, limited models, the fainter population would dominate reionization due to their higher ξ ion and greater numbers. \nAssessing the main contributors to reionization requires progress on several fronts. At z > 6, we need estimates of nebular and morphological parameters in addition to β 1550 and M 1500 for galaxy samples spanning a wide range of luminosities. Parameters such as O32 and Σ SFR have some of the greatest effects on the f esc prediction accuracy for both the LzLCS+ (Paper I) and galaxies at z ∼ 3 ( § 3). The limited magnitude range of the LzLCS+ reference sample ( M 1500 ∼ -18 . 5 to ∼ -21 . 5) also introduces uncertainty. Our f esc estimates for spectroscopic samples within this magnitude range are likely reasonable and suggest moderately high median f esc ∼ 0 . 04 -0 . 05, with some galaxies reaching f esc as high as 0.6-0.7 (see Figure 4 and predictions in Table 6 for the ELG-O32 model and Table 7 for the ELG-O32β model). However, our f esc estimates for fainter and brighter galaxy populations rely on extrapolation and are less trustworthy. To confirm these estimates, we need to explore f esc and its dependence on galaxy properties across a wider magnitude range at z < 6. Lastly, observations of the IGM and galaxy population at z > 6 will provide a further test of f esc predictions. An accurate model of f esc should reproduce both the timing and topology of reionization with the z > 6 galaxy population. While much progress is needed to both improve and confirm f esc estimates, our current f esc predictions imply that plausible contributors to reionization appear at all magnitude ranges, and star-forming galaxies with the required levels of LyC escape do indeed exist at z > 6.", '7. SUMMARY': "Quantifying the LyC escape fraction ( f esc ) of galaxies is critical to understand the sources of cosmic reionization. We have developed a flexible tool for predicting \nf esc empirically using combinations of observable variables available in the z ∼ 0 . 3 LzLCS+ reference sample (Paper I). We generate multivariate diagnostics for f esc with the Cox proportional hazards model (Cox 1972), a survival analysis technique that appropriately treats data with upper limits. We test Cox models developed from the z ∼ 0 . 3 galaxies on f esc observations at z ∼ 3, and we apply the models to several samples of z ≳ 6 galaxies to predict f esc for galaxies in the epoch of reionization. The Appendix provides all the information necessary to use the multivariate Cox models to predict f esc for other samples of galaxies, including samples at z ∼ 3 and samples at z ≳ 6, where the LyC is unobservable. 2 \nWe summarize our main findings below. \n- 1. The models successfully reproduce the observed f esc values for the high-redshift z ∼ 3 galaxies and often have a lower RMS for the z ∼ 3 samples than for the z ∼ 0 . 3 galaxies. The success of these models suggests that low-redshift and highredshift LCEs may share similar properties. ( § 3)\n- 2. The best-performing models for the z ∼ 3 galaxies include the dust attenuation inferred from the UV SED (E(B-V) UV ) or the UV slope β 1550 , plus either O32 or a morphological measurement ( r 50 , NUV or Σ SFR ). However, this result is tentative; each model is tested on a different subset of high-redshift galaxies, which makes comparing the models difficult. To determine which variable combinations best predict f esc at high redshift, we require larger z ∼ 3 samples of LyC-emitters with the full suite of input variables. ( § 3)\n- 3. We generate new Cox models based on the LzLCS+ observations, which incorporate variables measured for z ≳ 6 samples. One model, using M 1500 , M ∗ , E(B-V) UV and EW([O iii ]+H β ) as input variables, applies to 180 galaxies from Endsley et al. (2021, 2023); Bouwens et al. (2023); Tang et al. (2023); Fujimoto et al. (2023); Saxena et al. (2023) and predicts a median f esc = 0 . 047 for this combined sample. Of these 180 galaxies, 27% have f esc > 0 . 1 and only 6% have f esc > 0 . 2. However, this set of variables results in predictions that tend to underestimate f esc , and the galaxy samples mostly consist of photometric measurements. A second model, using M 1500 , M ∗ , β 1550 , and \nr 50 , NUV as input variables, applies to 278 galaxies, mostly with photometric redshifts, from Morishita et al. (2024) and Mascia et al. (2023). This model predicts a higher median f esc = 0 . 14, with 56% and 39% of the galaxies having f esc > 0 . 1 and f esc > 0 . 2, respectively. Many of these galaxies are more compact than the LzLCS+ sample, and this model therefore requires extrapolating beyond the LzLCS+ parameter space. ( § 4.1) \n- 4. Smaller samples of spectroscopically-confirmed z ≳ 6 galaxies have higher predicted f esc , likely because they tend to include stronger emissionline galaxies. We use a model with β 1550 , M 1500 , and O32 to predict f esc for 27 spectroscopicallyconfirmed galaxies (Williams et al. 2023; Schaerer et al. 2022; Mascia et al. 2023; Saxena et al. 2023, 2024) and find that 33-41% of these galaxies have f esc > 0 . 2. For a smaller sample of 17 galaxies (Williams et al. 2023; Tang et al. 2023; Fujimoto et al. 2023; Saxena et al. 2023), using M ∗ as an additional input variable and E(B-V) UV instead of β 1550 , we find that 46-85% of the galaxies have f esc > 0 . 2. These models identify five galaxies at z > 5 . 9 that may have a line-of-sight f esc as high as 0.5-1: CEERS-698, CEERS-1019, CEERS-1027 from Tang et al. (2023) and JADES 18846 and JADES 9422 from Saxena et al. (2024). ( §§ 4.1, 4.3).\n- 5. Currently, the predicted f esc values for different galaxy samples show no strong trend with UV magnitude across the range of M 1500 = -16 to -22. However, many of these models are limited by a lack of spectroscopic information. The models suggest that low-luminosity galaxies ( M 1500 > -18) have at least moderate f esc ∼ 0 . 05. If this f esc is coupled with a high ionizing photon production efficiency, such faint galaxies could substantially contribute to reionization (e.g., Atek et al. 2024; Simmonds et al. 2024). Additional measurements of variables such as O32 and Σ SFR for more z ≳ 6 galaxies could enable more accurate f esc predictions for both bright and faint galaxies and might increase the numbers of suspected strong LCEs. ( § 4.2)\n- 6. The multivariate predictions for z ≳ 6 galaxies can differ strongly from f esc predictions based on a single variable. We predict f esc for the strong Ly α Emitter, JADES-GS-z7-LA, discovered at z = 7 . 278 (Saxena et al. 2023). Despite its exceedingly high f esc , Ly α , which would seem to imply a high f esc , the two Cox models that include \nthe most available information (dust, Ly α , O32, and luminosity) predict that JADES-GS-z7-LA is a moderate LCE with f esc = 0 . 017 -0 . 085. Another example is the galaxy 11027 from Williams et al. (2023), which has O32 = 12, similar to the z ≳ 6 galaxies with predicted f esc ≥ 0 . 6. Yet, models incorporating other information about the galaxy, such as its mass (log 10 ( M ∗ / M ⊙ )=7.7), UV magnitude ( M 1500 = -17 . 4), and dust attenuation ( β 1550 = -2) predict f esc ≤ 0 . 02. However, we caution that some z ≳ 6 galaxies, including these ones, fall outside the parameter space covered by the z ∼ 0 . 3 galaxy sample, which make make our f esc predictions less reliable. ( § 4.3) \n- 7. We compare our model predictions to single variable predictions (Chisholm et al. 2022) and to other multivariate f esc predictions derived from simulations (Choustikov et al. 2024) or observations (Mascia et al. 2023, 2024). The Cox models more accurately predict f esc in non-detections than the β 1550 model from Chisholm et al. (2022), and the inclusion of feedback-sensitive variables such as O32 and Σ SFR may better trace the effect of absorption from H i in LCEs with low dust content. Turning to multivariate models, the LzLCS+ sample implies a correlation between O32 and f esc , rather than the anti-correlation adopted in the Choustikov et al. (2024) model. As a result, we find different predicted f esc for high-redshift galaxies, and the Choustikov et al. (2024) model fails to reproduce the f esc of the LzLCS+ galaxies. In contrast, our predictions agree with the Mascia et al. (2023) model, when we use their same set of input variables. However, our high-redshift f esc predic- \nons differ strongly when we add UV luminosity or Σ SFR as input variables. \nLyC-emitting galaxies show similar physical characteristics at both low and high redshift, which suggests that the same processes may govern LyC escape across cosmic time. Because a variety of factors can influence LyC escape, accurately predicting f esc requires information about galaxy luminosities, dust, nebular properties, and ionization. Direct measurements of LyC at low redshift, combined with multivariate statistical models for f esc , can give insight into the possible f esc of galaxies in the epoch of reionization. Nevertheless, additional observations are necessary before these techniques can reach their full potential. At low redshift, LyC observations that push to new parameter space, especially fainter and brighter UV magnitudes, will aid the application of low-redshift results to high-redshift samples. Measurements of relevant galaxy properties, especially in the rest-frame optical, for larger samples of z ∼ 3 LCEs will better test the relative performance of these models at high redshift. Based on our current results, models using O32 and Σ SFR appear promising but are not yet applicable to large samples at z > 6. By measuring these properties for larger, representative galaxy samples at z > 6, high-redshift surveys can connect galaxies in the epoch of reionization with their LyCemitting counterparts at low redshift. \nWe thank Aayush Saxena for providing information about the dust attenuation in JADES-GS-z7-LA, and we thank the anonymous referee for feedback that improved the paper's readability. AEJ and SRF acknowledge support from NASA through grant HST-GO-15626. STScI is operated by AURA under NASA contract NAS-526555. ASL acknowledges support from Knut and Alice Wallenberg Foundation. MT acknowledges support from the NWO grant 0.16.VIDI.189.162 ('ODIN').", 'APPENDIX': "In Tables 11-23, we provide the information necessary to predict f esc using the models discussed in this paper, and we give an example of how to use these models below. We have also developed a python script, available on GitHub 3 , that allows the user to generate and apply new Cox proportional hazards models to predict f esc using a desired set of observed variables (version 0.1.0 of the code is archived in Zenodo; Flury et al. 2024). \nHere, we provide the parameters for the main models presented in Paper I and in this work. The data for the fiducial model from Paper I appears in Table 11. The best-performing model for the LzLCS+ sample is the fiducial model modified to use EW(H i ,abs), and these model parameters appear in Table 12. In Table 13, we list the parameters for \nthe Paper I JWST model, which only uses variables accessible at z > 6. Two alternative JWST models use only the top-ranked accessible variables: β 1550 , Σ SFR , and either O32 or [Ne iii ]/[O ii ]. We provide the data for these models in Tables 14 and 15. Finally, Tables 16-25 present the parameters for the models used for the z ∼ 3 and z ≳ 6 f esc predictions discussed in Sections §§ 3-4. \nEach table first lists the goodness-of-fit statistics for the LzLCS+ galaxies as a measure of the model performance. It then provides the fitted coefficients ( b i ) for each included variable and the reference values ¯ x i , which are the mean of the LzLCS+ x i values, where x i is the input variable ( β 1550 , log 10 (O32), etc). Finally, the table lists the baseline cumulative hazard function, HF 0 , calculated by the lifelines Cox Model fitting routine for each of the observed f esc values for LCEs in the LzLCS+. Together, these parameters predict f esc for a given galaxy with a set of input variables x as follows (see Section § 2.2 for more detail). \nTo predict f esc for a galaxy, we first calculate the partial hazards function ph( x ) for its set of variables: \nph( x ) = exp[ n ∑ i =1 b i ( x i -¯ x i )] . (1) \nWe use the coefficients b i given in the model table, the galaxy's observed values x i for each variable, and the mean values ¯ x i also given in the model table. For example, for the galaxy Ion2, we have observed values of E(B-V) UV = 0, log 10 (O32)=0.851, and log 10 (Σ SFR /( M ⊙ yr -1 kpc -2 ))=2.03. Using b i (-10.083, 1.240, 1.369) and ¯ x i (0.158, 0.521, 0.705) from Table 16, we calculate \nph( x ) = exp[ -10 . 083 ∗ (0 -0 . 158) + 1 . 240 ∗ (0 . 851 -0 . 521) + 1 . 369 ∗ (2 . 03 -0 . 705)] = 45 . 43 . (2) \nWe then use this value of ph( x ) to scale the baseline cumulative hazard function HF 0 ( f esc ) and calculate the Survival Function S ( f esc ), which represents the probability that the true escape fraction, f esc , true , is lower than each tabulated f esc : \nS ( f esc ) = exp[ -HF 0 ( f esc ) ∗ ph( x )] . (3) \nFrom Table 16, for f esc = 0 . 8890, HF 0 is 0.003949 and S (0.8890) is therefore exp[ -0 . 003949 ∗ 45 . 43] = 0 . 8358. For f esc = 0 . 6247, HF 0 = 0 . 008253 and S (0 . 6247) = exp[ -0 . 008253 ∗ 45 . 43] = 0 . 6873, and so on, for each value of f esc . Thus, according to the 'TopThree model, there is a 68.73% probability that Ion2 has a measured f esc < 0 . 6247 and an 83.58% probability that Ion2's f esc < 0 . 8890. To find the median predicted f esc , we calculate S for each value of f esc in the table and find the value where S reaches 0.5; f esc is predicted to be above this value 50% of the time and below this value 50% of the time. For Ion2, we see that S (0 . 5838) = 0 . 5629 and S (0 . 4911) = 0 . 4577. We linearly interpolate to find where S ( f esc )= 0 . 5, finding that S (0 . 53) ∼ 0 . 5. The predicted f esc of Ion2 is then f esc = 0 . 53. Similarly, to determine the uncertainties in the predicted f esc , we find where S ( f esc ) reaches 0.159 and 0.841. According to the model, f esc will be between these f esc values 68% of the time. For some galaxies, S ( f esc ) is always > 0 . 5 for the tabulated f esc values, indicating a > 50% probability that f esc is smaller than the smallest tabulated f esc . This situation corresponds to an arbitrarily small predicted f esc , f esc ∼ 0. Conversely, if S ( f esc ) is always < 0 . 5, f esc is arbitrarily large and f esc ∼ 1. \nTable 11. Fiducial Model (Paper I)Baseline Cumulative Hazard \nTable 12. Fiducial Model with EW(H i ,abs) (Paper I)Baseline Cumulative Hazard \nTable 13. Full JWST Model (Paper I)Baseline Cumulative Hazard \nStatistics \nR \n2 \n0.34 \n0.29 \n0.46 \n0.83 \nVariable \nb \nlog \nlog \nβ \n10 \n10 \n([O \n(Σ \n1550 \n] \nλ \n3727) \n0.996 \n0.521 \n- \n2 \nii \nkpc \n) \n1.404 \n0.705 \n-2.274 \n-1.810", 'Baseline Cumulative Hazard': "Table 18. LAE-O32 Model \na Positive EW(Ly α ) denotes net emission. \nClark, T. G., Bradburn, M. J., Love, S. B., & Altman, D. G. 2003, British Journal of Cancer, 89, 232, doi: 10.1038/sj.bjc.6601118 \nCox, D. R. 1972, Journal of the Royal Statistical Society: Series B (Methodological), 34, 187, \ndoi: https://doi.org/10.1111/j.2517-6161.1972.tb00899.x \nCurti, M., D'Eugenio, F., Carniani, S., et al. 2023, MNRAS, 518, 425, doi: 10.1093/mnras/stac2737 \nDavidson-Pilon, C. 2019, Journal of Open Source Software, 4, 1317, doi: 10.21105/joss.01317 \nde Barros, S., Vanzella, E., Amor'ın, R., et al. 2016, A&A, 585, A51, doi: 10.1051/0004-6361/201527046 \nModel Parameters \ni \n-0.935 \n-19.84 \n¯ x \ni \n⊙ \nR \n2 \nadj \nStatistics \nRMS \nC \n) \n0.961 \n8.920 \n-15.987 \n0.157 \nii \nTable 19. LAE-O32-nodust Model \nDonnan, C. T., McLeod, D. J., Dunlop, J. S., et al. 2023, \nMNRAS, 518, 6011, doi: 10.1093/mnras/stac3472 Dopita, M. A., Fischera, J., Sutherland, R. S., et al. 2006, ApJS, 167, 177, doi: 10.1086/508261 Duan, Q., Conselice, C. J., Li, Q., et al. 2024, MNRAS, 529, 4728, doi: 10.1093/mnras/stae872 \nEilers, A.-C., Davies, F. B., & Hennawi, J. 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S. 2019, Prospector: Stellar population inference from spectra and SEDs, Astrophysics Source Code Library, record ascl:1905.025. http://ascl.net/1905.025\n- Jung, I., Finkelstein, S. L., Dickinson, M., et al. 2020, ApJ, 904, 144, doi: 10.3847/1538-4357/abbd44 \nKakiichi, K., & Gronke, M. 2021, ApJ, 908, 30, doi: 10.3847/1538-4357/abc2d9 Katz, H., Saxena, A., Rosdahl, J., et al. 2023, MNRAS, 518, 270, doi: 10.1093/mnras/stac3019 Kennicutt, R. C., & Evans, N. J. 2012, ARA&A, 50, 531, doi: 10.1146/annurev-astro-081811-125610 Kerutt, J., Oesch, P. A., Wisotzki, L., et al. 2024, A&A, 684, A42, doi: 10.1051/0004-6361/202346656 \n- Kimm, T., Blaizot, J., Garel, T., et al. 2019, MNRAS, 486, 2215, doi: 10.1093/mnras/stz989 \nTable 23. ELG-O32β -Ly α ModelBaseline Cumulative Hazard \nLarson, R. L., Finkelstein, S. L., Kocevski, D. D., et al. \n2023, ApJL, 953, L29, doi: 10.3847/2041-8213/ace619 Leitet, E., Bergvall, N., Hayes, M., Linn'e, S., & Zackrisson, E. 2013, A&A, 553, A106, doi: 10.1051/0004-6361/201118370 Leitet, E., Bergvall, N., Piskunov, N., & Andersson, B. G. 2011, A&A, 532, A107, doi: 10.1051/0004-6361/201015654 \nLeitherer, C., Ekstrom, S., Meynet, G., et al. 2014, ApJS, 212, 14, doi: 10.1088/0067-0049/212/1/14 \nLeitherer, C., Hernandez, S., Lee, J. C., & Oey, M. S. 2016, ApJ, 823, 64, doi: 10.3847/0004-637X/823/1/64 \nLeitherer, C., Schaerer, D., Goldader, J., et al. 2011, Starburst99: Synthesis Models for Galaxies with Active Star Formation, Astrophysics Source Code Library, record ascl:1104.003. http://ascl.net/1104.003 \nTable 24. R50β ModelBaseline Cumulative Hazard \nLeja, J., Johnson, B. D., Conroy, C., van Dokkum, P. G., & Byler, N. 2017, ApJ, 837, 170, \ndoi: 10.3847/1538-4357/aa5ffe \nLin, Y.-H., Scarlata, C., Williams, H., et al. 2024, MNRAS, 527, 4173, doi: 10.1093/mnras/stad3483 \nLiu, Y., Jiang, L., Windhorst, R. A., Guo, Y., & Zheng, Z.-Y. 2023, ApJ, 958, 22, doi: 10.3847/1538-4357/acf9fa Luridiana, V., Morisset, C., & Shaw, R. A. 2015, A&A, 573, A42, doi: 10.1051/0004-6361/201323152 \nMa, X., Kasen, D., Hopkins, P. F., et al. 2015, MNRAS, 453, 960, doi: 10.1093/mnras/stv1679 \nMaji, M., Verhamme, A., Rosdahl, J., et al. 2022, A&A, 663, A66, doi: 10.1051/0004-6361/202142740 \nMakan, K., Worseck, G., Davies, F. B., et al. 2021, ApJ, 912, 38, doi: 10.3847/1538-4357/abee17 \nMarchi, F., Pentericci, L., Guaita, L., et al. 2018, A&A, 614, A11, doi: 10.1051/0004-6361/201732133 \nTable 25. β -Metals ModelBaseline Cumulative Hazard \nAlvarez-M'arquez, J., \nMarques-Chaves, R., Schaerer, D., ' \net al. 2021, MNRAS, 507, 524, doi: 10.1093/mnras/stab2187 \n- -. 2022, MNRAS, 517, 2972, doi: 10.1093/mnras/stac2893 \nMarques-Chaves, R., Schaerer, D., Kuruvanthodi, A., et al. 2024, A&A, 681, A30, doi: 10.1051/0004-6361/202347411 \nMartin, C., Barlow, T., Barnhart, W., et al. 2003, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 4854, Future EUV/UV and Visible Space Astrophysics Missions and Instrumentation., ed. J. C. Blades & O. H. W. Siegmund, 336-350, doi: 10.1117/12.460034 \nMascia, S., Pentericci, L., Calabr'o, A., et al. 2023, A&A, 672, A155, doi: 10.1051/0004-6361/202345866 \n- -. 2024, A&A, 685, A3, doi: 10.1051/0004-6361/202347884 \nWitstok, J., Smit, R., Saxena, A., et al. 2024, A&A, 682, A40, doi: 10.1051/0004-6361/202347176 \nWorseck, G., Prochaska, J. X., Hennawi, J. F., & McQuinn, \nM. 2016, ApJ, 825, 144, \ndoi: 10.3847/0004-637X/825/2/144 \nXu, X., Henry, A., Heckman, T., et al. 2023, ApJ, 943, 94, doi: 10.3847/1538-4357/aca89a \nZackrisson, E., Inoue, A. K., & Jensen, H. 2013, ApJ, 777, 39, doi: 10.1088/0004-637X/777/1/39", 'Model Parameters': 'i \n¯ \nx \ni \nR \n2 \nadj \nRMS \nC \n⊙ \nStatistics \nR \n2 \n0.40 \n0.35 \n0.44 \n0.83 \nVariable \nb \nlog \nlog \nβ \n10 \n10 \n([Ne \n(Σ \n1550 \n] \nλ \n3727) \n1.315 \n-0.532 \n- \n2 \n] \nλ \n3869/[O \niii \nSFR \n/( M \n- \n1 \nyr \nii \nkpc \n) \n1.343 \n0.705 \n-2.192 \n-1.810 \nBaseline Cumulative Hazard \nTable 15. 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2024arXiv240800064S | It is commonly believed that our own Milky Way is on a collision course with the neighbouring Andromeda galaxy. As a result of their merger predicted in around five billion years the two large spiral galaxies that define the present Local Group would form a new elliptical galaxy. Here we consider the latest and most accurate observations by the Gaia and Hubble space telescopes along with recent consensus mass estimates to derive possible future scenarios and identify the major sources of uncertainty in the evolution of the Local Group over the next 10 billion years. We find that the next most massive Local Group member galaxies namely M33 and the Large Magellanic Cloud distinctly and radically affect the Milky Way Andromeda orbit. While including M33 increases the merger probability the orbit of the Large Magellanic Cloud runs perpendicular to the Milky Way Andromeda orbit and makes their merger less likely. In the full system we find that uncertainties in the present positions motions and masses of all galaxies leave room for drastically different outcomes and a probability of close to 50 that there is no Milky Way Andromeda merger during the next 10 billion years. | 2024-07-01T00:00:00Z | ['10.48550/arXiv.2408.00064', '2024arXiv240800064S', 'arXiv:2408.00064'] | ['Astrophysics - Astrophysics of Galaxies'] | Apocalypse When No Certainty of a Milky Way Andromeda Collision | 2,024 | 173 | 0.59 | ['EPRINT_HTML', 'EPRINT_PDF'] | 2 | https://arxiv.org/pdf/2408.00064.pdf | {'Apocalypse When? No Certainty of a Milky Way Andromeda Collision': "Till Sawala 1,2* , Jehanne Delhomelle 1,3 , Alis J. Deason 2 , Carlos S. Frenk 2 , Peter H. Johansson 1 , Atte Keitaanranta 1 , Alexander Rawlings 1 , Ruby Wright 1 \n1 Department of Physics, University of Helsinki, Gustaf Hallstromin katu 2, Helsinki, FI-00014, Finland. 2 Institute for Computational Cosmology, Durham University, South Road, Durham, DH1 3LE, United Kingdom. Universit'e Toulouse III - Paul Sabatier, 118 Route de Narbonne, \n3 Toulouse, F-31062, France. \n*Corresponding author(s). E-mail(s): [email protected];", 'Abstract': "It is commonly believed that our own Milky Way is on a collision course with the neighbouring Andromeda galaxy. As a result of their merger, predicted in around five billion years, the two large spiral galaxies that define the present Local Group would form a new elliptical galaxy. Here we consider the latest and most accurate observations by the Gaia and Hubble space telescopes, along with recent consensus mass estimates to derive possible future scenarios and identify the major sources of uncertainty in the evolution of the Local Group over the next 10 billion years. We find that the next most massive Local Group member galaxies - namely, M33 and the Large Magellanic Cloud - distinctly and radically affect the Milky Way - Andromeda orbit. While including M33 increases the merger probability, the orbit of the Large Magellanic Cloud runs perpendicular to the Milky Way - Andromeda orbit and makes their merger less likely. In the full system, we find that uncertainties in the present positions, motions, and masses of all galaxies leave room for drastically different outcomes, and a probability of close to 50% that there is no Milky Way - Andromeda merger during the next 10 billion years. \nThe Local Group (LG) contains two large spiral galaxies, our own Milky Way (MW) and the Andromeda galaxy (hereafter M31), along with approximately 100 known smaller galaxies [1]. In addition, it likely hosts additional galaxies yet to be discovered [2] and, according to the standard cosmological model, a vast number of completely dark substructures [3]. \nThe negative radial velocity of M31 towards the MW has been known for over a century [4], even before its distance was first accurately measured [5]. However, while indirect methods had since been used to constrain the transverse components of M31's velocity vector [6, 7, 8], direct measurements of the minute proper motions were only achieved much more recently, with the Hubble Space Telescope (HST) [9]. \nThe first numerical studies [10] of a possible MW-M31 merger predate even the early estimates of the transverse velocity. The finding that the MW-M31 motion is close to radial immediately led to the prediction of a likely future collision and merger [11, 12, 13]. This scenario has since become the prevalent narrative [14, 15] and textbook knowledge [16, 17].", 'Predicting the future of the Local Group': 'The MW and M31 both contain remnants of past mergers and interactions with other galaxies [18, 19, 20, 21]. Predicting future mergers requires knowledge about the present coordinates, velocities, and masses of the systems partaking in the interaction. In addition to the gravitational force between galaxies, dynamical friction is the dominant process in the lead-up to galactic mergers. It describes a transfer of orbital kinetic energy to internal energy of the objects involved, and consequently leads to the decay of galactic orbits. \nIn this study, we parameterise the density profile of each halo as isotropic NavarroFrenk-White profiles [22], use standard assumptions about their concentrations and velocity dispersion profiles [23] and calculate dynamical friction using an analytic formalism (see Methods for a detailed description). While non-gravitational processes, such as gas drag and star formation leading to increased central densities, etc., also shape the final stages of a merger, the phase of the orbit that defines the occurrence of mergers is largely determined by gravity, which in turn, is dominated by the dark matter component in the standard cosmological model [24]. \nThe simplest model for the MW-M31 orbit, as considered by [11], contains only the two main galaxies. Due to the planar symmetry, only five parameters are required: the two masses, along with the initial separation and the two velocity components. In general, an orbit with N > 2 galaxies requires specifying 3 × ( N -1) coordinates and 3 × ( N -1) velocity components, along with the N masses. More recently, [12] and [13] have considered three-body orbits including M33, the third most massive LG galaxy; and [13] also considered the Large Magellanic Cloud (LMC), the fourth most massive LG galaxy. They still conclude that a merger is certain. We will consider two-body, three-body, and four-body systems to study the future evolution of the LG and reveal the distinct effects of the M33 and LMC on the MW-M31 orbit.', 'A fiducial model based on the most accurate values available': 'In the context of the Local Group, it is important to note that apart from the sky positions, all parameters including distance moduli, line-of-sight velocities, proper motions, masses, and concentrations carry non-negligible uncertainties. We will use Monte Carlo (NC) sampling of all values to investigate how these observational errors propagate to uncertainties on the future evolution, and in particular, the probability of a merger between the MW and M31. \nOur fiducial LG model contains the four most massive LG members: the MW, M31, M33, and the LMC, based on the latest and most accurate available data. The mass of the MW has recently been extensively studied using Gaia data, with a consensus emerging of a total mass close to 10 12 M /circledot . We adopt a value of M 200 = 1 ± 0 . 2 × 10 12 M /circledot (excluding the mass of the LMC, which we treat separately). For all other galaxies, there is significantly more uncertainty. For M31, we adopt M 200 = 1 . 3 ± 0 . 4 × 10 12 M /circledot . For M33, we assume M 200 = 3 ± 1 × 10 11 M /circledot and for the LMC we assume M 200 = 1 . 5 ± 0 . 5 × 10 11 M /circledot . See Methods for a review of mass estimates. \nThe line-of-sight velocities are well-known and we adopt the values given by [1]. For the distance moduli, µ , we choose the most accurate and precise recent values in the literature: [25] for the LMC and [26] for M33. For M31, we use recent HST Cepheid results [27]. \nFor the proper motions, where we use the notation, µ δ for the proper motion in declination and µ ∗ α = µ α cos δ for the proper motion in right ascension, we use the Hubble Space Telescope (HST) proper motions of [28] for the LMC, and the combined HST and Gaia DR2 proper motions of [13] for M33. Our fiducial M31 proper motions are the most precise values in the literature based on Gaia DR3 astrometry [29]. However, we find very similar results using the combined Gaia DR2 and HST proper motions of [13], as shown in Extended Data Figure 1. All assumed values are listed in Extended Table 1. We assume Galactocentric coordinates of RA GC = 266 . 41 · , Dec GC = -28 . 94 · [30], d GC = 8 . 122 kpc [31], and a velocity of (12 . 9 , 245 . 6 , 7 . 78) kms -1 with respect to the Sun [30, 31, 32]. \nTo account for the fact that the true probability distributions may not be Gaussian and to exclude possible effects caused by unrealistic or even unphysical outliers, we truncate all distributions at ± 2 σ . This truncation increases the probability of a MWM31 merger by only ∼ 10% (see Extended Data Figure 2). For the fiducial model, we use 50000 MC samples while for all other variants, we use 2500 samples, ensuring that all statistical errors on the merger rate are below 1%.', 'The evolution of the MW-M31-M33-LMC system': "The Monte Carlo initial conditions are integrated numerically using a symplectic direct scheme (see Methods for details). Figure 1 shows 100 realisations each of the MW-M31 orbit, in the two-body MW-M31, and the four-body MW-M31-M33-LMC systems. Figure 2 shows the evolution of the distances between the MW and M31 for the same sets of orbits, and additionally, for the MW-M31-M33 and MW-M31-LMC three-body systems. \nThe MW-M31 two-body orbit evolves in a plane and leads to a merger in slightly less than half of cases, the majority of which occur during the second pericentre. The addition of M33 increases the merger probability to ∼ 2 / 3, with a similar median merger time. However, the addition of the LMC has the opposite effect: the pure MWM31-LMC system experiences a merger in only slightly more than 1 / 3 of cases and the merger probability of the full M31-MW-M33-LMC system is just over 50%. \nFor each system, we also show the single orbit obtained by adopting the most likely value of each observable, either in the fiducial model or assuming HST+ Gaia DR2 proper motions for M31 [13]. In the case of the MW-M31-M33 system, which is the one considered by [12] and [13], we reproduce similar orbits and times of first pericentre and merger (the differences are likely due to differences in the other model parameters). However, despite using newer and more precise measurements, when we perform a Monte Carlo analysis, we find considerable uncertainty in the outcome that was not previously reported. In particular, in the full MW-M31-M33-LMC system, a merger between the MW and M31 occurs within the next 10 Gyr in only approximately half of the cases. This is in stark contrast to all previous results that had only considered the most likely values without accounting for the numerous and significant uncertainties. \nIn Figure 3, we show the probability distributions of the merger time and of the minimum distance between the MW and M31, as well as the 'survival' rate of the MW over time, i.e. the probability that no merger with M31 has occurred. In the fiducial model, we consider a merger to occur when the distance between any two galaxies is below 20 kpc, but we find that our results are not sensitive to this particular choice (see Methods for details). Due to the effect of dynamical friction, we find that there are two distinct possibilities for the eventual fate of the MW and M31: orbits that come within less than ∼ 200 kpc eventually merge, which would likely lead to the formation of an intermediate-mass elliptical galaxy [10, 33, 34]. For systems that merge, we find a median time of 7 . 6 Gyr in the fiducial model, or 8 . 0 Gyr adopting a 10 kpc threshold. By contrast, orbits with larger pericentres do not decay due to dynamical friction. In this case, the MW and M31 continue to evolve in isolation. Based on the best current data, both outcomes are almost equally likely.", 'The roles of M33 and the LMC': 'The distinct effects of each satellite on the MW-M31 orbit are illustrated in Figures 4 and 5, where we compare the trajectories of the MW and M31 in two-body systems and in systems that also include either M33 or the LMC. Both satellites provide some additional acceleration in the radial direction of the MW-M31 orbit. However, importantly, both satellites also affect the motion of their respective hosts. Including M33 in the calculation decreases the transverse velocity of M31 with respect to the MW. By contrast, as already pointed out in [35], at its current orbital phase, the recoil due to the LMC results in a lower transverse velocity measured between the MW and M31. During its short orbital period of ∼ 1 . 5 Gyr, the LMC will accelerate the MW to a higher transverse velocity. In addition, the inclusion of M33 largely provides momentum in the original MW-M31 plane, while the inclusion of the LMC also provides significant momentum perpendicular to the MW-M31 plane. In our \nanalysis, the LMC is certain to merge with the MW, and M33 is highly likely ( ∼ 86%) to merge with M31 before any possible MW-M31 merger. The net effect of adding M33 to the two-body system is to increase the merger probability, while the net effect of adding the LMC is to decrease it.', 'Sources of uncertainty': 'In Figure 6, we show how the merger probability depends on the different observables. In each panel, we show the dependence on two variables in the ± 2 σ ranges, with the remaining variables Monte Carlo sampled. The effects of the concentration parameters are shown in Extended Data Figure 3. We find that all masses, the proper motions of M31 and M33, and the distance moduli of M31 and M33, significantly impact the probability of a merger. The merger probability is positively correlated with the masses of the MW, M31, and M33 and negatively correlated with the mass of the LMC. The impact of the satellite masses is more pronounced for lower combined masses of the MWand M31. The ± 2 σ ranges of the M31 proper motions include values that imply a merger probability above 90%, but also values that imply a merger probability close to zero. The most likely proper motions (assuming no errors) only lead to a merger in ∼ 2 / 3 of cases. \nFuture more precise proper motion measurements may significantly change the expected outcome, although they could make the merger either more or less likely. However, if they fall within ± 1 σ of the current most likely values, even precise M31 proper motion measurements alone will not suffice to determine the outcome. \nEven the comparatively high precision of the line-of-sight velocity and distance moduli for M33 and M31 contribute significant uncertainty, with the probability of a merger varying between 40% and over 60% for different combinations in the ± 2 σ ranges around the most likely values. \nGiven the considerable measurement errors, it is worth noting that cosmological simulations result in a present-day MW-M31 transverse velocity prior of v t = 75 +65 -40 kms -1 [36]. There is thus no reason to assume that the transverse velocity measured using Gaia DR3 (most likely value of v t = 76 kms -1 ) is overestimated or to expect that more precise measurements will result in a lower value. It is also worth noting that a perfectly radial present-day M31-MW motion ( µ ∗ α = 38 . 03 mas yr -1 , µ δ = -21 . 37 mas yr -1 ) is neither compatible with current observations nor does it result in the highest merger probability in the four-body system.', 'Summary': 'Even using the latest and most precise available observational data, the future evolution of the Local Group is uncertain. Intriguingly, we find an almost equal probability for the widely publicised merger scenario (albeit with a later median time to merger) and one where the MW and M31 survive unscathed. We reach this conclusion by including the LMC and importantly, for the first time, considering the relevant uncertainties in the observables. \nOur results are not sensitive to the necessary choices of gravitational softening (see Extended Data Figure 4), merger threshold (Extended Data Figure 5), or dynamical friction scheme (Extended Data Figure 6)). \nWhile we have shown that considerable uncertainty results from the proper motion measurements of M31, we also find that a more accurate prediction requires more precise measurements of the positions, motions, and masses of all participating galaxies. The dependence of the evolution of the MW-M31 system on the treatment of other LG galaxies points to further uncertainties. Cosmological simulations suggest that ∼ 25% of the bound mass of the LG is outside the two main haloes [37]. The next most massive individual Local Group galaxies that could impact the MW-M31 orbit are M32 (an M31 satellite and possible merger remnant [38]) and the Small Magellanic Cloud (SMC), a satellite of the MW. Both are at least a factor of five less massive than the LMC, and we find that including the SMC, for which proper motion measurements are available, does not significantly change the MW-M31 merger probability (see Extended Data Figure 7). However, the unaccounted cumulative effects of additional substructures as well as that of the cosmic environment introduce further uncertainty, particularly towards the far future. An accurate prediction of the evolution even from perfectly precise observations may require cosmological constrained simulations [39, 40, 41] that can account for these effects. 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On the top and bottom panels, respectively, trajectories are projected in the orbital plane and perpendicular to the orbital plane defined by the initial positions and velocities of the MW and M31. White markers denote MW-M31 mergers. In the left column, we show the MW-M31 two-body system, while in the right, we show the four-body system that includes the MW, M31, M33, and the LMC. \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFig. 2 Distance between the MW and M31. On each panel, we show 100 realisations of our fiducial MC model and also state the probability for a MW-M31 merger within 10 Gyr. In the top row, we show the MW-M31 two-body orbits (top-left) and MW-M31-M33-LMC four-body orbits (top-right) shown in Figure 1. In the bottom row, we show the MW-M31-M33 (bottom-left) and MW-M31-LMC (bottom-right) three-body orbits. White markers denote MW-M31 mergers; percentages indicate the fraction of orbits that merge within 10 Gyr. Only slightly more than half of the four-body orbits lead to a merger within 10 Gyr. The inclusion of M33 increases the likelihood of a merger, whereas the inclusion of the LMC decreases it. White lines show the individual orbits using the most likely values of every variable, either assuming the Gaia DR3 proper motions [29] of the fiducial model (solid) or HST + Gaia DR2 proper motions [13] (dashed). \n<!-- image --> \n50% \n20% \n10% \n5% \n1% \n<5 \ndm \n=20 kpc \ndm \n=10 kpc \n5 \n6 \n7 \n8 \n9 \n10 >10 \ntm \n( \nM \n31 \nMW \n) [Gyr] \n<15 15 25 \n50 100 200 400>400 \nmin( \ndM \n31 \nMW \n)[kpc] \nFig. 3 Distributions of the merger time, t m , the minimum distance, min( d M 31 -MW ), and the 'survival' rate of the MW in the MW-M31-M33-LMC system. Blue and red colours show results using a distance threshold of 20 kpc (our default) or 10 kpc for a merger, respectively. Mergers happen on average later when the distance threshold is lower, but the fraction of systems that merge within 10 Gyr is similar. The median time to merger is 7.6 Gyr for systems that merge with a 20 kpc threshold and 8.0 Gyr for systems that merge with a 10 kpc threshold. The distributions of minimum distance show a clear bimodality, independently of the threshold: about half of the systems reach the merger threshold within 10 Gyr while the vast majority of the remaining systems do not approach closer than 200 kpc. The 'survival' rate of the MW drops sharply between ∼ 6 -9 Gyr and levels off afterwards. \n<!-- image --> \n50% \n20% \n10% \n5% \n1% \ndm \n=20 kpc \ndm \n=10 kpc \nFig. 4 Effects of M33 and the LMC on the trajectory of the MW. As in Figures 1 and 5, panels in the top row are projected in the orbital plane defined by the initial MW and M31 positions and velocities, while those in the bottom row are projected perpendicular to this orbital plane. Arrows point towards the initial position of M31. On the left, we compare MW trajectories in the two-body MW-M31 system to those in the three-body MW-M31-M33 system. On the right, we show the orbit of the LMC, and that of the MW before and after the merger with the LMC in the MW-M31-LMC system. Compared to the two-body orbit, the inclusion of M33 reduces the transverse velocity of the MW relative to M31 and introduces only a small velocity perpendicular to the MW-M31 orbital plane. By contrast, the LMC increases the MW-M31 transverse velocity and causes a larger velocity perpendicular to the MW-M31 orbital plane. \n<!-- image --> \nFig. 5 Effects of M33 and the LMC on the trajectory of M31. As in Figures 1 and 4, panels in the top row are projected in the orbital plane defined by the initial MW and M31 positions and velocities, while those in the bottom row are projected perpendicular to this orbital plane. Arrows point towards the initial position of the MW. On the right, we compare M31 trajectories in the two-body MW-M31 system to those in the three-body MW-M31-LMC system. On the left, we show the orbit of M33, and that of M31 before and after a possible merger with M33 in the MW-M31-M33 system. Compared to the 2-body orbit, the inclusion of the LMC increases the transverse velocity of M31 and introduces motion perpendicular to the initial orbital plane. On the other hand, the inclusion of M33 reduces the transverse velocity of M33 with respect to the MW. \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFig. 6 Dependence of the merger probability on observables of the MW-M31-M33-LMC system. In each panel, we show the probability of a MW-M31 merger within 10 Gyr as a function of two variables, with all other observables sampling the probability distributions of the fiducial model. White lines on the colour bars indicate the minimum and maximum merger probabilities for the range of values shown on each panel, indicating the sensitivity of the merger probability on the two corresponding variables. In the top row, we show the dependence on different masses, the second row shows the dependence on proper motions, and the third row shows the dependence on distance moduli and line of sight velocities. The axes extend to ± 2 σ of the fiducial model. The merger probability is positively correlated with the mass of the MW, M31, and M33, and negatively correlated with the mass of the LMC. The merger probability strongly depends on µ δ ( M 31) and µ ∗ α ( M 31), but also varies significantly with µ δ ( M 33) and µ ∗ α ( M 33). The uncertainties in the distance moduli for M31 and M33 also contribute to the uncertainty of the outcome, while the effect of the line-of-sight velocities is small. \n<!-- image -->", 'Methods': 'Our results are based on numerically integrated initial conditions, which are in turn based on Monte Carlo samples of the observational data. We describe here the generation of the Monte Carlo samples, the numerical integration, and our treatment of dynamical friction and galaxy mergers. We also demonstrate the robustness of our findings to the particular choices made and show that our fiducial model and the results are conservative in predicting an uncertain future and relatively low merger probability. We also present results when the SMC is included in the analysis, in addition to the four main galaxies. To facilitate the reproduction of our results and allow future work to easily incorporate new observational data, our analysis code is flexible and public.', 'Monte Carlo Samples': 'Our fiducial model consists of the Milky Way and three additional galaxies, M31, M33, and the LMC. We assume that the sky coordinates (RA & Dec) for the centres of M31, M33, and the LMC are known. We furthermore assume that the position of the Galactic centre with respect to the sun is fixed at ( Ra,Dec ) = (266.4051, 28.936175) [30], d = 8 . 122 kpc [31], and that the galactocentric velocity of the Sun is (12 . 9 , 245 . 6 , 7 . 78) km -1 [32, 31, 30]. \nWe create Monte Carlo samples for the remaining 20 variables, the four halo masses, M MW , M M 31 , M M 33 , M LMC , the four halo concentration parameters, the three distance moduli, µ , the three sets of proper motions, µ δ and µ ∗ α , and three lineof-sight velocities, v los . Sampling directly in the space of the observables rather than sampling in a space of derived variables such as Cartesian coordinates or velocities minimises the effect of possible correlations. \nTo allow reproducibility and identification of individual orbits across our figures when changing parameters of the model such as the set of galaxies included and numerical parameters such as the gravitational softening length and merger threshold, we re-initialise the pseudo-random number generator with the same values for every new set of Monte Carlo samples. While we generally only show the first 100 orbits on each plot, all quoted probabilities are computed from at least 2500 samples each, so that statistical errors are less than 1%. \nThe most likely values of each variable together with the ± 1 σ uncertainty are either taken directly from single sources in the literature or in the case of masses and concentrations (see below), estimated by us based on multiple sources. In creating our Monte Carlo samples, we assume that each variable follows a Gaussian probability distribution. However, in our fiducial model, we truncate all distributions at ± 2 σ , corresponding to the central ∼ 95% of values. While Gaussian distributions are a natural assumption for measurement errors, this is not always explicitly stated, and it may not reflect the true probability distribution, especially at some distance from the most likely values. Indeed, for some variables, untruncated distributions extend to unphysical values with finite probabilities. A truncation at 2 σ also ensures that all variables in the samples are physical. \nGiven that our central claim is uncertainty in the future evolution of the LG, truncating the probability distributions of the observables is a conservative assumption that leads to lower uncertainty about the outcome. However, as shown in Extended Data Figure 2, our results are not sensitive to the truncation at ± 2 σ , and the merger probability is only slightly lower when the distributions are not truncated. On the other hand, even a truncation at only ± 1 σ leaves approximately 1 / 4 of orbits that do not merge within 10 Gyr.', 'Numerical Integration': 'The orbits are integrated using a symplectic leapfrog algorithm in the centre-of-mass frame, with a step size of 1 Myr, approximately the time it takes for a galaxy moving at 100 kms -1 to travel 0.1 kpc - 1/200 th of our merger threshold (see below). Our results are not affected by the finite time step. \nTo account for the fact that the haloes are extended objects, the gravitational force between the haloes is softened with a constant softening length of 20 kpc, similar to the scale radius of an NFW halo in the mass range we consider here. We also consider different choices for the softening and show in Extended Data Figure 4 results with softening lengths of 10, 20, or 30 kpc, respectively. A softening length that is too small can lead to unphysical hard scattering events during close encounters while a softening length that is too large artificially reduces the gravitational force. In the context of the Local Group, both of these effects could reduce the merger probability. However, we find no strong dependence of the merger probability on the softening length, with our adopted fiducial value of 20 kpc resulting in the highest merger probability.', 'Dynamical Friction': "To estimate the effect of dynamical friction, we use a modified Chandrasekhar formula, similar to that used in [24]. The classic Chandrasekhar formula assumes a point mass orbiting in the potential of a much more massive host halo, which is in turn composed of much less massive particles. This approach has been expanded to account for extended satellites [46], and we use the following expression to calculate the acceleration of a satellite due to dynamical friction [16] once inside r 200 of the host halo: \nd v d t = -4 πG 2 Mρ ln Λ v 2 [ erf( X ) -2 X √ π e -X 2 ] v v , (1) \nwhere G is the gravitational constant, M is the mass of the satellite, v is the velocity of the satellite relative to the host, ρ is the density of the host at the position of the satellite, X = v/ (2 σ ) is the ratio between the orbital speed of the satellite and the 1D-velocity dispersion, σ , of the host at the location of the satellite, and Λ is the Coulomb factor expressed as r//epsilon1 , with /epsilon1 a scale length that depends on the density of the satellite. To determine /epsilon1 , we adopt an empirical expression derived from N-body simulations [47]: \n/epsilon1 = { 2 . 2 r s -14 kpc if r s ≥ 8kpc 0 . 45 r s if r s < 8kpc (2) \nwhere r s is the scale radius of the satellite's NFW halo. Finally, to approximate the velocity dispersion of the host at the location of the satellite, we use the expression derived in [23] for NFW haloes. \nStandard dynamical friction schemes assume a clear hierarchy between the (much more massive) host and the (much less massive) satellite. According to the assumptions underlying Equation 1, the satellite and host enter the calculation in clearly defined and distinct roles, with the dynamical friction force applied only on the satellite, while the host remains unaffected. \nHowever, in the Local Group context where galaxies and haloes of similar mass are interacting, this introduces an inconsistency. In particular, in the (relatively likely) scenario that M31 has a similar mass to the MW, the roles of the satellite and host are unclear, but their assignment changes the result of the calculation. For example, if the MW is considered the satellite, it would be accelerated by its interaction with M31 while M31 would remain unaffected, and only the motion of the MW with respect to the other galaxies would be affected while that of M31 would remain unchanged. If M31 is considered the satellite, the roles would be reversed. Accelerating only one galaxy also violates momentum conservation. \nTo make the calculation more symmetrical, in our dynamical friction calculation we distribute the dynamical friction force proportionally between the satellite ( s ) and host ( h ), conserving the total momentum: \nd v s d t = a DF M h M s + M h , (3) \nd v h d t = -a DF M s M s + M h , (4) \nwhere v s and M s are the velocity and mass of the satellite, v h and M h are the mass and velocity of the host, and a DF is the acceleration computed using Equation 1. In the limit that the satellite is much less massive than the host, the standard hierarchical scheme is recovered and only the satellite is accelerated, while in the limit that both galaxies have equal mass, both receive equal and opposite accelerations. \nA small inconsistency remains in that even when the differences in mass are small, we still assign the more massive galaxy as the host and the less massive galaxy as the satellite when calculating the magnitude of the dynamical friction, where the velocity dispersion of the host, σ , but not that of the satellite, and through the Coulomb factor, the scale radius of the satellite, but not that of the host, are considered. In our spherical halo models, both the velocity dispersion and Coulomb factor depend only on the assumed masses and concentrations, and as we discuss below, the concentration of the satellite has a greater impact on the dynamical friction force. For two halos of significantly different masses, the assignment of host and satellite is clear. For an individual case of two haloes of nearly identical masses but different (randomly \nassigned) concentrations, the choice of calculating the dynamical friction force by treating either halo as the satellite seems arbitrary. However, for a large number of samples of nearly equal-mass interactions with randomly assigned concentrations, the dynamical friction calculations are not biased. We also assume identical distributions of concentration parameters for all galaxies. \nIn Extended Data Figure 6, we compare the orbits of the fiducial system using no dynamical friction, 'hierarchical' dynamical friction (only from the more massive host to the less massive satellite), and our default 'proportional' dynamical friction. It is clear that without dynamical friction a MW-M31 merger is highly unlikely. In fact, the finite merger rate without dynamical friction depends strongly on our default impact parameter threshold of 20 kpc, with a lower threshold, the merger rate can become arbitrarily small. On the other hand, when dynamical friction is included, the evolution of each orbit is quite similar in the 'hierarchical' and 'proportional' schemes, and the merger rate is not significantly affected by the exact choice of scheme. \nIt is worth noting that our semi-analytical approach to dynamical friction is still quite simplistic, and while the average behaviour of N-body simulations has been used to calibrate parameters, numerical simulations also show that individual systems with non-zero internal angular momenta and substructures can have different merger times than predicted by these simple equations [48]. A precise prediction of the MW-M31 orbit will likely require full N-body simulations. On the other hand, we show that even a simple dynamical model that assumes no spin, no triaxiality, and no substructure results in considerable uncertainty in the future evolution of the MW-M31 system.", 'Mergers': 'Below a certain distance, the interactions of the gas and stellar components become significant, and our simple approach is no longer appropriate for predicting the remaining orbital evolution. Our aim here is not to predict the precise time of the merger (in fact, we argue that such a prediction is futile based on the current data), so we simply assume that any system that passes below a threshold distance will eventually merge, and identify this time as a lower limit for the time of the merger. \nIn our fiducial model, we have adopted a value of 20 kpc as the merger threshold for all galaxy interactions. When such a merger occurs between two galaxies, we combine the masses and momenta of the two galaxies at their common centre of mass and continue the integration. The concentration parameter is set to that of the most massive galaxy. \nIn Extended Data Figure 5, we compare the sensitivity of our results to adopting merger thresholds of 10, 20, and 30 kpc. With a threshold of 10 kpc, we find a slightly reduced MW-M31 merger probability. However, raising the threshold from 20 to 30 kpc has no significant impact on our results. This confirms our assertion in the main section: due to the effect of dynamical friction, orbits either inspiral and eventually merge, or do not come close enough for dynamical friction to become effective and hence do not merge. \nThe small fraction of MW-M31 orbits that merge with a threshold of 20 kpc, but do not when the threshold is set to 10 kpc, approach with a small impact parameter and high velocity. This reduces the effect of dynamical friction and also allows them \nto escape to a large apocentre. While this may be a possible scenario for the Local Group, our methods are not adequate for studying such close interactions between galaxies. To be conservative in our prediction of a low merger probability, by setting the merger threshold at 20 kpc we assume that these orbits also merge, and with an even larger threshold of 30 kpc, we would still predict a similar merger rate. \nFor comparison, the best-studied merger of two galaxies with similar stellar masses to the MW and M31 are the Antenna galaxies. Their past evolution is reproduced with an orbit with pericentre ∼ 10 kpc [49], predicted to lead to coalescence ∼ 1 . 3 Gyr later [50].', 'Fate of the LMC and M33': 'While our main focus here is on the future evolution of the MW-M31 orbit, we naturally also make predictions for the evolution of the LMC and M33. \nIn the fiducial model, we find that the LMC is certain to merge with the MW before any eventual MW-M31 merger. With a merger threshold of 20 kpc, we find a median time of the LMC-MW merger of 1.3 Gyr, while with a threshold of 10 kpc, we find a median time of 1.9 Gyr. \nFor M33, with a merger threshold of 20 kpc, we find an ∼ 86% chance of a merger with M31 and a median time of 3.3 Gyr, while with a merger threshold of 10 kpc, we find an ∼ 83% probability of a merger with M31 and a median time of 3.9 Gyr. In both cases, we also find a small probability of ∼ 1 -2% for a merger of M33 with the MW-M31 remnant after a MW-M31 merger within the next 10 Gyr. \nIt must be noted that our simulations are not designed to study these mergers in detail and ignore, for example, the impact of the disk of the Milky Way. Nevertheless, they broadly agree with the results of [24], who have previously studied the LMC-MW encounter in the presence of M31.', 'Additional galaxies': 'The next most massive LG member galaxy for which proper motion data [28] is available is the Small Magellanic Cloud (SMC), a satellite galaxy of the MW that has likely been accreted together with the LMC. We repeat our analysis including the SMC as a fifth system, and show results in Extended Data Figure 7. Adding the SMC, whose mass is approximately 10% of that of the LMC, has no significant effect on the merger rate. The SMC properties used are listed along with those of the other galaxies in Extended Data Table 1.', 'DR2 + HST proper motions': 'Due to the fact that the M31 proper motions have the largest impact on the probability of a MW-M31 orbit, and for easier comparison with earlier works, particularly [13], we also repeat our analysis adopting the HST+ Gaia DR2 M31 proper motions of [13]. In Extended Data Figure 1, we show the corresponding evolution of the MWM31 distance in the same two-body MW-M31, three-body MW-M31-M33 and MWM31-LMC, and four-body MW-M31-M33-LMC systems. The results can be directly compared to Figure 2 which shows the same quantities in our fiducial model that used \nGaia DR3 proper motions. In each case, the merger probability is slightly lower using the HST+ Gaia DR2 proper motions: 40% instead of 44% for the MW-M31 system, 56% instead of 63% for the MW-M31-M33 system, 34% instead of 37% for the MWM31-LMC system, and 48% instead of 54% for the full MW-M31-M33-LMC system. However, both sources of proper motions predict similar distributions of outcomes and a similar uncertainty about the MW-M31 merger. This difference can be attributed to the slightly lower precision of the HST+ Gaia DR2 proper motions.', 'Sources for the masses': 'As discussed in Figure 6, the assumed masses of all four galaxies and their associated uncertainties have a strong impact on the likely evolution of the MW-M31 orbit in our fiducial model. Here, we review the mass measurements and show the implications for some alternative scenarios. \nMW The total mass of the MW has been extensively studied with different methods and tracers, and the accurate astrometry of the Gaia space telescope has brought a flurry of recent measurements. Estimates for the total MW mass based on Gaia DR2 or DR3 satellite dynamics include 1 . 17 +0 . 21 -0 . 15 × 10 12 M /circledot [51], 1 . 51 0 . 45 -0 . 40 × 10 12 M /circledot [52], 1 . 23 +0 . 21 -0 . 18 × 10 12 M /circledot [53], and 1 . 1 +0 . 1 -0 . 1 × 10 12 M /circledot [54] (where the latter two works also use a simulation based prior). \nEstimates using rotation curves include 1 . 08 +0 . 20 -0 . 14 × 10 12 M /circledot based on Gaia DR2 [55], 0 . 89 +0 . 1 -0 . 08 × 10 12 M /circledot using stars in the galkin catalogue [56], 0 . 822 ± 0 . 052 × 10 12 M /circledot using classical Cepheids [57], and 1 . 08 +0 . 12 -0 . 11 × 10 12 M /circledot from the H3 survey and Gaia DR3 [58]. Other recent measurements include 1 . 54 +0 . 75 -0 . 44 × 10 12 M /circledot using combined Gaia DR2 and HST kinematics of globular clusters [59], 1 . 16 ± 0 . 24 × 10 12 M /circledot (including the mass of the LMC) using the kinematics of halo stars, 1 . 26 +0 . 40 -0 . 22 × 10 12 M /circledot from high velocity RR-Lyrae stars [60], and 1 . 19 +0 . 49 -0 . 32 × 10 12 M /circledot from a Bayesian estimate using dwarf galaxy kinematics from multiple sources [61]. \n[36] contains an overview of recent measurements and [62] includes a comprehensive review of earlier results. \nAnalysis from several different tracers, and particularly from the latest studies using the latest Gaia observations consistently point towards a Milky Way total mass that is close to 10 12 M /circledot . Both the simple mean and median of the above measurements are 1 . 16 × 10 12 M /circledot , but it is worth noting that most methods of measuring the total mass of the MW include that of the LMC (at a galactocentric distance of ∼ 50 kpc, well within r vir or r 200 ), which is not always made explicit. When the LMC is excluded, the mass of the MW is reduced by ∼ 0 . 15 × 10 12 M /circledot (see our discussion of the LMC mass below). We adopt M MW = 1 . 0 ± 0 . 2 × 10 12 M /circledot excluding the LMC in our fiducial model, reflecting the consensus of recent observations. \nM31 While M31 does not enjoy the benefit of accurate Gaia proper motions, there nevertheless exist a number of studies estimating its mass. Most recently, [63] measured a total mass of 1 . 14 +0 . 51 -0 . 35 × 10 12 M /circledot using rotation curves based on LAMOST data release 9 and DESI. [64] measured 1 . 4 ± 0 . 4 × 10 12 M /circledot derived from satellite kinematics, [65] derived 1 . 2 +0 . 9 -0 . 7 × 10 12 M /circledot from kinematics of M31 dwarf spheroidals, and [66] measured a total mass of 1 . 05 +0 . 15 -0 . 15 × 10 12 M /circledot from SED fitting together with \nthe rotation curve and the kinematics of outer globular clusters and satellite galaxies. [67] measured 1 . 0 × 10 12 M /circledot from the HI rotation curve, [68] measured 2 . 0 +0 . 4 -0 . 3 × 10 12 M /circledot from kinematics of the Giant southern stream. [69] measured 1 . 35 +0 . 15 -0 . 15 × 10 12 M /circledot and [70] found 1 . 4 +0 . 2 -0 . 2 × 10 12 M /circledot both using outer halo globular clusters, while [71] found 0 . 8 ± 0 . 1 × 10 12 M /circledot from high-velocity planetary nebulae, and [72] found 1 . 39 ± 0 . 26 × 10 12 for the total mass combining disk rotation velocities and radial velocities of satellite galaxies and globular clusters. Both the simple mean and median of the above values are 1 . 27 × 10 12 M /circledot . Our adopted mass of M M 31 = 1 . 3 ± 0 . 4 × 10 12 M /circledot in the fiducial model reflects the broad consensus of M31 mass estimates using different methods, but also the considerable remaining uncertainty. \nM33 Mass estimates of M33 are much more sparse. More than twenty years ago, [73, 74] measured a dark matter mass of 5 × 10 10 M /circledot , extrapolated out to a virial mass of 5 × 10 11 M /circledot from the measured HI rotation curve, but noted that this results in a very low baryon fraction. More recently, [75] obtained a similar result using the Hα rotation curve, but noted that because the measurements only extend to a few percent of the virial radius, there are no strong constraints on the total dark matter halo. On the other hand, abundance matching based on the observed stellar mass results in a significantly lower total mass of 1 . 7 ± 0 . 55 × 10 11 M /circledot [76, 77]. Citing both the direct measurements and abundance matching, [77] adopt a mass range of 0 . 8 -3 . 2 × 10 11 M /circledot in their dynamical models of the M33-M31 interaction, while [78] and [13] assume a total mass of 2 . 5 × 10 11 M /circledot . \nWe adopt M M 33 = 3 ± 1 × 10 11 M /circledot in the fiducial model, marginally compatible with both the extrapolated masses from rotation curve measurements and the results of abundance matching and in line with previous studies. However, the results from the two methods are certainly in tension. \nLMC For the mass of the LMC, recent mass estimates based on its effect on Galactic stellar streams include 1 . 38 +0 . 37 -0 . 24 × 10 11 M /circledot [79], 1 . 30 ± 0 . 3 × 10 11 M /circledot [80], 1 . 88 +0 . 4 -0 . 35 × 10 11 M /circledot [81] and 1 . 29 +0 . 28 -0 . 23 × 10 11 M /circledot . [82] obtain good agreement between the perturbations of Milky Way halo stars with an LMC mass of 1 . 5 × 10 11 M /circledot . Using the abundance of likely satellites of the LMC, [83] obtain a lower limit of 1 . 24 × 10 11 M /circledot , and using kinematics of satellites associated with the LMC, [84] find 1 . 65 +0 . 47 -0 . 49 × 10 11 M /circledot . Most recently, [85] used 30 LMC globular clusters to infer a total mass of 1 . 80 +1 . 05 -0 . 54 × 10 11 M /circledot while [35] find an even higher value of 2 . 5 +0 . 9 -0 . 8 × 10 11 M /circledot using a timing argument and to be most consistent with the Hubble Flow around the Local Group. The simple mean of the above values is 1 . 6 × 10 11 M /circledot while the median is 1 . 5 × 10 11 M /circledot . A comprehensive recent review on the effect of the LMC on the Milky Way, including a discussion of mass measurements, is given by [86], who conclude that the LMC mass is likely in the range 1 -2 × 10 11 M /circledot , which matches the choice of M LMC = 1 . 5 ± 0 . 5 × 10 11 M /circledot in our fiducial model.', 'Halo Concentrations': "As explained above, we assume that for the purpose of integrating their orbits, all galaxies are represented by NFW haloes [22] with the total masses M 200 defined above. We assume concentration parameters of 10 ± 2 for all galaxies, consistent with results \nof cosmological simulations [87, 88, 89, 90, 91] in this mass range. While cosmological simulations show the average concentration parameter to be mass-dependent, the halo-to-halo scatter is significantly larger than the change in the mean concentration over this narrow mass range [87]. \nThe concentration parameter does not affect the orbital calculation, except for the dynamical friction, where (for a given mass), it sets the scale radius, r s = r 200 /c , of the 'satellite' that enters in Equation 2. \nIn Extended Data Figure 3, we show the dependence of the merger probability on the concentration. Except for a very low M31 mass, there is only a weak dependence of the merger probability on the concentration of M31, which is more likely to be the more massive 'host' galaxy in the MW-M31 encounter. For the concentration of the MW, which, in the MW-M31 interaction is more likely to be the satellite, we find that the merger probability is reduced if the concentration is below ∼ 8, i.e. below -1 σ of our fiducial value. We find no significant dependence on the merger probability on the concentration assumed for M33 or the LMC. \nAcknowledgments. We thank Marius Cautun for his generous help, and Joonas Nattila for helpful suggestions. TS and JD are supported by the Research Council of Finland grant 354905, and TS and PHJ are also supported by the Research Council of Finland grant 339127. JD is supported by an Erasmus+ grant. TS and CSF are supported by the European Research Council (ERC) Advanced Investigator grant DMIDAS (GA 786910) and the STFC Consolidated Grant ST/T000244/1. PHJ, AK, AR and RW also acknowledge support from the European Research Council (ERC) Consolidator Grant KETJU (no. 818930). This work used facilities hosted by the CSC-IT Centre for Science, Finland, and the DiRAC@Durham facility, managed by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk) and funded by BEIS capital funding via STFC capital grants ST/K00042X/1, ST/P002293/1, ST/R002371/1 and ST/S002502/1, Durham University and STFC operations grant ST/R000832/1. DiRAC is part of the UK National e-Infrastructure. We thank the authors of the open source software listed below. \nAuthors' contributions. TS planned the project and performed the analysis together with JD. TS, AJD, CSF and PHJ planned the paper, and TS wrote the analysis code, created the figures and wrote the first draft. AK and AR contributed to the dynamical friction calculations. TS, JD, AJD, CSF, PHJ, AK, AR and RW jointly discussed and edited the manuscript.", 'Data availability': 'All data used in this work is publicly available and provided as part of the analysis code listed below.', 'Code availability': "The analysis in this paper was performed using Python 3.10, and makes extensive use of the following open-source libraries: Astropy 6.0.1 [42], Matplotlib 3.8.3, NumPy 1.26.4 [43], SciPy 1.13.0 [44] and Colossus 1.3.5 [45]. A documented Juypter notebook containing the code to produce all figures in this paper is available at: https://github. com/TillSawala/MW-M31. \nConflict of interest. The authors declare that they have no conflict of interest. \nExtended Data Table 1 Parameters of the Fiducial model \n<!-- image --> \n<!-- image --> \n<!-- image --> \nExtended Data Figure 1 Distance between the MW and M31, using HST+ Gaia DR2 proper motions for M31 [13], analogous to Figure 2. As in Figure 2, solid and dashed white lines denote the orbits using the most likely values using either Gaia DR3 proper motions [29] or HST+ Gaia DR2 proper motions. White markers denote MW-M31 mergers, percentages indicate the fraction of orbits that merge within 10 Gyr. The merger rate is slightly lower in all cases when compared to the fiducual model that uses the more precise Gaia DR3 proper motions, but the results are qualitatively similar. \n<!-- image --> \nExtended Data Figure 2 Effect of truncating the assumed probability distributions of observables. The distance between the MW and M31 (analogous to Figure 2) in the four-body MW-M31-M33LMC system for different truncations of the observables. From left to right, we show results where the probability distribution for each observable in the fiducial model is truncated to ± 1 σ , ± 2 σ (our default model, same as in Figure 2), or left untruncated. The fraction of systems that merge is only slightly increased when the distributions are clipped at ± 2 σ or above. \n<!-- image --> \n<!-- image --> \n<!-- image --> \nc \n<!-- image --> \nc \nc \nExtended Data Figure 3 Effect of the concentration parameters on the merger probability in the four-body MW-M31-M33-LMC system, similar to Figure 6. From left to right, in the top row, we show the dependencies on mass and concentration of the MW and M31, while in the bottom row, we show those of M33 and the LMC, respectively. The concentration parameter affects the merger rate only for the lower mass system in the MW-M31 encounter, which is the MW in most cases. A concentration parameter of the MW below ∼ 7 ( -1 . 5 σ ), particularly in combination with a low MW mass results in a significantly lower merger rate. Unlike their masses, the concentration parameters for M33 and the LMC have no discernible effects on the MW-M31 merger probability. \n<!-- image --> \n600 \n400 \n200 \n0 \n200 \n400 \n600 \n<!-- image --> \nx \nxMW \n[kpc] \n600 \n400 \n200 \n0 \n200 \n400 \n600 \n<!-- image --> \n<!-- image --> \n1000 \n800 \n600 \n400 \n200 \n0 \n<!-- image --> \n<!-- image --> \n<!-- image --> \nx \nxMW \n[kpc] \n600 \n400 \n200 \n0 \n200 \n400 \n600 \n<!-- image --> \nx \nxMW \n[kpc] \n600 \n400 \n200 \n0 \n200 \n400 \n600 \n<!-- image --> \n1000 \n800 \n600 \n400 \n200 \n0 \nExtended Data Figure 4 Effect of the gravitational softening. MW-M31 orbits (analogous to Figure 1) and distance between the MW and M31 (analogous to Figure 2) in the four-body MW-M31M33-LMC system for different softening lengths. From left to right, we show results with a softening length of 10 kpc, 20 kpc (our default value), and 30 kpc. A softening length that is too small can lead to some unrealistically strong kicks in close encounters, while a softening length that is too large weakens the overall gravitational attraction. However, the merger fraction is not significantly affected by the choice of softening length within this range. \n<!-- image --> \nExtended Data Figure 5 Effect of the merger threshold on the MW-M31 distance evolution and merger rate. The distance between the MW and M31 (analogous to Figure 2) in the four-body MWM31-M33-LMC system for different merger thresholds. From left to right, we show results with a threshold of 10 kpc, 20 kpc or 30 kpc for all mergers. The MW-M31 merger probability is not very sensitive to the assumed merger threshold, and we obtain almost the same merger probability even with a (very generous) threshold of 30 kpc. \n<!-- image --> \n600 \n400 \n200 \n0 \n200 \n400 \n600 \n<!-- image --> \n<!-- image --> \n600 \n400 \n200 \n0 \n200 \n400 \n600 \n<!-- image --> \n<!-- image --> \n1000 \n800 \n600 \n400 \n200 \n0 \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n600 \n400 \n200 \n0 \n200 \n400 \n600 \n600 \n400 \n200 \n0 \n200 \n400 \n600 \n1000 \n800 \n600 \n400 \n200 \n0 \nExtended Data Figure 6 Effect of different schemes of dynamical friction. MW-M31 orbits (topt two rows, analogous to Figure 1) and distance between the MW and M31 (bottom row, analogous to Figure 2) in the four-body MW-M31-M33-LMC system for different softening lengths. The left column assumes no dynamical friction, the middle column uses our default 'proportional' scheme where the dynamical friction force is divided such that equal and opposite dynamical forces are applied to both host and satellite, the right column uses the 'hierarchical' scheme where dynamical friction is only applied to the less massive object. Dynamical friction is essential for orbits to decay and for the MWM31 merger to occur, but the probability of an MW-M31 merger is not sensitive to the scheme used. However, due to its momentum-conserving property, mergers in the proportional scheme are more likely to occur close to the original orbital plane compared to the hierarchical scheme. \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nExtended Data Figure 7 The effect of including the SMC on the MW-M31 distance and merger rate. In the top row, we show the MW-M31 two-body system, and the MW-M31-M33-LMC fourbody system in the fiducial model, corresponding to the top row of Figure 2. In the bottom row, we add the SMC to both systems, i.e. we show the MW-M31-SMC three-body system and the MWM31-M33-LMC-SMC five-body system. The inclusion of the SMC has only a small effect on most MW-M31 orbits and does not significantly affect the total merger rate. \n<!-- image -->", 'Methods References': "- [46] White, S. D. M. A note on the minimum impact parameter for dynamical friction involving spherical clusters. MNRAS 174 , 467-470 (1976).\n- [47] Jethwa, P., Erkal, D. & Belokurov, V. A Magellanic origin of the DES dwarfs. MNRAS 461 , 2212-2233 (2016). 1603.04420.\n- [48] Boylan-Kolchin, M., Ma, C.-P. & Quataert, E. Dynamical friction and galaxy merging time-scales. MNRAS 383 , 93-101 (2008). 0707.2960.\n- [49] Karl, S. J. et al. One Moment in Time-Modeling Star Formation in the Antennae. 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2024arXiv240712867R | We present results from a search for Xraygammaray counterparts of gravitationalwave GW candidates from the third observing run O3 of the LIGOVirgoKAGRA LVK network using the Swift Burst Alert Telescope SwiftBAT. The search includes 636 GW candidates received in low latency 86 of which have been confirmed by the offline analysis and included in the third cumulative GravitationalWave Transient Catalogs GWTC3. Targeted searches were carried out on the entire GW sample using the maximumlikelihood NITRATES pipeline on the BAT data made available via the GUANO infrastructure. We do not detect any significant electromagnetic emission that is temporally and spatially coincident with any of the GW candidates. We report flux upper limits in the 15350 keV band as a function of sky position for all the catalog candidates. For GW candidates where the SwiftBAT false alarm rate is less than 103 Hz we compute the GWBAT joint false alarm rate. Finally the derived SwiftBAT upper limits are used to infer constraints on the putative electromagnetic emission associated with binary black hole mergers. | 2024-07-01T00:00:00Z | ['arXiv:2407.12867', '2024arXiv240712867R', '10.48550/arXiv.2407.12867'] | ['Astrophysics - High Energy Astrophysical Phenomena', 'General Relativity and Quantum Cosmology'] | SwiftBAT GUANO followup of gravitationalwave triggers in the third LIGOVirgoKAGRA observing run | 2,024 | 173 | 0.57 | ['EPRINT_HTML', 'EPRINT_PDF'] | 2 | https://arxiv.org/pdf/2407.12867.pdf | {'The Swift-BAT/GUANO Team': "Gayathri Raman, 1 Samuele Ronchini, 1 James Delaunay, 2, 1 Aaron Tohuvavohu, 3, 4 Jamie A. Kennea, 1 Tyler Parsotan, 5 \nThe Swift Collaboration \nElena Ambrosi, 6 Maria Grazia Bernardini, 7 Sergio Campana, 7 Giancarlo Cusumano, 6 Antonino D'A'ı, 6 Paolo D'Avanzo, 7 Valerio D'Elia, 8, 9 Massimiliano De Pasquale, 10 Simone Dichiara, 1 Phil Evans, 11 Dieter Hartmann, 12 Paul Kuin, 13 Andrea Melandri, 14 Paul O'Brien, 11 Julian P. Osborne, 11 Kim Page, 11 David M. Palmer, 15 Boris Sbarufatti, 16 Gianpiero Tagliaferri, 7 Eleonora Troja, 17 \nThe LIGO Scientific Collaboration, the Virgo Collaboration, and the KAGRA Collaboration \nA. G. Abac, 18 R. Abbott, 19 H. Abe, 20 I. Abouelfettouh, 21 F. Acernese, 22, 23 K. Ackley, 24 C. Adamcewicz, 25 S. Adhicary, 26 N. Adhikari, 27 R. X. Adhikari, 19 V. K. Adkins, 28 V. B. Adya, 29 C. Affeldt, 30, 31 D. Agarwal, 32 M. Agathos, 33 O. D. Aguiar, 34 I. Aguilar, 35 L. Aiello, 36 A. Ain, 37 T. Akutsu, 38, 39 S. Albanesi, 40, 41 R. A. Alfaidi, 42 A. Al-Jodah, 43 C. All'en'e, 44 A. Allocca, 45, 23 S. Al-Shammari, 36 P. A. Altin, 29 S. Alvarez-Lopez, 46 A. Amato, 47, 48 L. Amez-Droz, 49 A. Amorosi, 49 C. Amra, 50 S. Anand, 19 A. Ananyeva, 19 S. B. Anderson, 19 W. G. Anderson, 19 M. Andia, 51 M. Ando, 52 T. Andrade, 53 N. Andres, 44 M. Andr'es-Carcasona, 54 T. Andri'c, 18, 55 J. Anglin, 56 S. Ansoldi, 57, 58 J. M. Antelis, 59 S. Antier, 60 M. Aoumi, 61 E. Z. Appavuravther, 62, 63 S. Appert, 19 S. K. Apple, 64 K. Arai, 19 A. Araya, 65 M. C. Araya, 19 J. S. Areeda, 66 N. Aritomi, 21 F. Armato, 67 N. Arnaud, 51, 68 M. Arogeti, 69 S. M. Aronson, 28 G. Ashton, 70 Y. Aso, 38, 71 M. Assiduo, 72, 73 S. Assis de Souza Melo, 68 S. M. Aston, 74 P. Astone, 75 F. Aubin, 76 K. AultONeal, 59 G. Avallone, 77 S. Babak, 78 F. Badaracco, 67 C. Badger, 79 S. Bae, 80 S. Bagnasco, 41 E. Bagui, 81 Y. Bai, 19 J. G. Baier, 82 R. Bajpai, 38 T. Baka, 83 M. Ball, 84 G. Ballardin, 68 S. W. Ballmer, 85 S. Banagiri, 86 B. Banerjee, 55 D. Bankar, 32 P. Baral, 27 J. C. Barayoga, 19 B. C. Barish, 19 D. Barker, 21 P. Barneo, 53, 87 F. Barone, 88, 23 B. Barr, 42 L. Barsotti, 46 M. Barsuglia, 78 D. Barta, 89 S. D. Barthelmy, 90 M. A. Barton, 42 I. Bartos, 56 S. Basak, 91 A. Basalaev, 92 R. Bassiri, 35 A. Basti, 93, 37 M. Bawaj, 94, 62 P. Baxi, 95 J. C. Bayley, 42 A. C. Baylor, 27 M. Bazzan, 96, 97 B. B'ecsy, 98 V. M. Bedakihale, 99 F. Beirnaert, 100 M. Bejger, 101 D. Belardinelli, 102 A. S. Bell, 42 V. Benedetto, 103 D. Beniwal, 104 W. Benoit, 105 J. D. Bentley, 92 M. Ben Yaala, 106 S. Bera, 107 M. Berbel, 108 F. Bergamin, 30, 31 B. K. Berger, 35 S. Bernuzzi, 109 M. Beroiz, 19 C. P. L. Berry, 42 D. Bersanetti, 67 A. Bertolini, 48 J. Betzwieser, 74 D. Beveridge, 43 N. Bevins, 110 R. Bhandare, 111 U. Bhardwaj, 112, 48 R. Bhatt, 19 D. Bhattacharjee, 82, 113 S. Bhaumik, 56 S. Bhowmick, 114 A. Bianchi, 48, 115 I. A. Bilenko, 116 G. Billingsley, 19 A. Binetti, 117 S. Bini, 118, 119 O. Birnholtz, 120 S. Biscoveanu, 86, 46 A. Bisht, 31 M. Bitossi, 68, 37 M.-A. Bizouard, 60 J. K. Blackburn, 19 C. D. Blair, 43, 74 D. G. 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Chaty, 78 A. Chen, 161 A. H.-Y. Chen, 162 D. Chen, 163 H. Chen, 158 H. Y. Chen, 164 K. H. Chen, 159 X. Chen, 43 Yi-Ru Chen, 158 Yanbei Chen, 165 Yitian Chen, 166 H. P. Cheng, 56 P. Chessa, 93, 37 H. T. Cheung, 95 H. Y. Chia, 56 F. Chiadini, 167, 121 C. Chiang, 159 G. Chiarini, 97 A. Chiba, 168 R. Chiba, 169 R. Chierici, 170 A. Chincarini, 67 M. L. Chiofalo, 93, 37 A. Chiummo, 23, 68 C. Chou, 162 S. Choudhary, 43 N. Christensen, 60 S. S. Y. Chua, 29 K. W. Chung, 79 G. Ciani, 96, 97 P. Ciecielag, 101 M. Cie'slar, 101 M. Cifaldi, 102 A. A. Ciobanu, 104 \nR. Ciolfi, 171, 97 F. Clara, 21 J. A. Clark, 19, 69 T. A. Clarke, 25 P. Clearwater, 172 S. Clesse, 81 F. Cleva, 60 E. Coccia, 55, 129, 54 E. Codazzo, 55 P.-F. Cohadon, 130 M. Colleoni, 107 C. G. Collette, 49 J. Collins, 74 S. Colloms, 42 A. Colombo, 135, 136, 173 M. Colpi, 135, 136 C. M. Compton, 21 L. Conti, 97 S. J. Cooper, 126 T. R. Corbitt, 28 I. Cordero-Carri'on, 174 S. Corezzi, 94, 62 N. J. Cornish, 98 A. Corsi, 175 S. Cortese, 68 C. A. Costa, 34 R. Cottingham, 74 M. W. Coughlin, 105 A. Couineaux, 75 J.-P. Coulon, 60 S. T. Countryman, 176 J.-F. Coupechoux, 170 B. Cousins, 26 P. Couvares, 19, 69 D. M. Coward, 43 M. J. Cowart, 74 D. C. Coyne, 19 R. Coyne, 177 K. Craig, 106 R. Creed, 36 J. D. E. Creighton, 27 T. D. Creighton, 178 P. Cremonese, 107 A. W. Criswell, 105 J. C. G. Crockett-Gray, 28 M. Croquette, 130 R. Crouch, 21 S. G. Crowder, 179 J. R. Cudell, 127 T. J. Cullen, 19 A. Cumming, 42 E. Cuoco, 68, 180, 37 M. Cusinato, 152 P. Dabadie, 139 T. Dal Canton, 51 S. Dall'Osso, 75 G. D'alya, 100 B. D'Angelo, 67 S. Danilishin, 47, 48 S. D'Antonio, 102 K. Danzmann, 31, 30, 31 K. E. Darroch, 132 L. P. Dartez, 21 A. Dasgupta, 99 S. Datta, 181 V. Dattilo, 68 A. Daumas, 78 N. Davari, 182, 147 I. Dave, 111 A. Davenport, 114 M. Davier, 51 T. F. Davies, 43 D. Davis, 19 L. Davis, 43 M. C. Davis, 110 E. J. Daw, 183 M. Dax, 18 J. De Bolle, 100 M. Deenadayalan, 32 J. Degallaix, 184 M. De Laurentis, 45, 23 S. Del'eglise, 130 V. Del Favero, 90 F. De Lillo, 131 D. Dell'Aquila, 182, 147 W. Del Pozzo, 93, 37 F. De Marco, 75, 123 F. De Matteis, 149, 102 V. D'Emilio, 36 N. Demos, 46 T. Dent, 140 A. Depasse, 131 N. DePergola, 110 R. De Pietri, 185, 186 R. De Rosa, 45, 23 C. De Rossi, 68 R. De Simone, 167 A. Dhani, 18 S. Dhurandhar, 32 R. Diab, 56 M. C. D'ıaz, 178 M. Di Cesare, 45 G. Dideron, 187 N. A. Didio, 85 T. Dietrich, 18 L. Di Fiore, 23 C. Di Fronzo, 49 F. Di Giovanni, 152 M. Di Giovanni, 123, 75 T. Di Girolamo, 45, 23 D. Diksha, 48, 47 A. Di Michele, 94 J. Ding, 78, 188 S. Di Pace, 123, 75 I. Di Palma, 123, 75 F. Di Renzo, 170 Divyajyoti, 189 A. Dmitriev, 126 Z. Doctor, 86 E. Dohmen, 21 P. P. Doleva, 132 L. Donahue, 190 L. D'Onofrio, 75 F. Donovan, 46 K. L. Dooley, 36 T. Dooney, 83 S. Doravari, 32 O. Dorosh, 191 M. Drago, 123, 75 J. C. Driggers, 21 Y. Drori, 19 J.-G. Ducoin, 192, 78 L. Dunn, 146 U. Dupletsa, 55 D. D'Urso, 182, 147 H. Duval, 193 P.-A. Duverne, 51 S. E. Dwyer, 21 C. Eassa, 21 M. Ebersold, 194, 44 T. Eckhardt, 92 G. Eddolls, 42 B. Edelman, 84 T. B. Edo, 19 O. Edy, 138 A. Effler, 74 J. Eichholz, 29 H. Einsle, 60 M. Eisenmann, 38 R. A. Eisenstein, 46 A. Ejlli, 36 M. Emma, 70 E. Engelby, 66 A. J. Engl, 35 L. Errico, 45, 23 R. C. Essick, 195 H. Estell'es, 18 D. Estevez, 76 T. Etzel, 19 M. Evans, 46 T. Evstafyeva, 33 B. E. Ewing, 26 J. M. Ezquiaga, 141 F. Fabrizi, 72, 73 F. Faedi, 73, 72 V. Fafone, 149, 102 S. Fairhurst, 36 P. C. Fan, 190 A. M. Farah, 141 B. Farr, 84 W. M. Farr, 196, 197 G. Favaro, 96 M. Favata, 198 M. Fays, 127 M. Fazio, 106 J. Feicht, 19 M. M. Fejer, 35 E. Fenyvesi, 89, 199 D. L. Ferguson, 164 I. Ferrante, 93, 37 T. A. Ferreira, 28 F. Fidecaro, 93, 37 A. Fiori, 37, 93 I. Fiori, 68 M. Fishbach, 195 R. P. Fisher, 132 R. Fittipaldi, 200, 121 V. Fiumara, 201, 121 R. Flaminio, 44 S. M. Fleischer, 202 L. S. Fleming, 203 E. Floden, 105 E. M. Foley, 105 H. Fong, 155 J. A. Font, 152, 153 B. Fornal, 204 P. W. F. Forsyth, 29 K. Franceschetti, 185 N. Franchini, 78 S. Frasca, 123, 75 F. Frasconi, 37 A. Frattale Mascioli, 123, 75 Z. Frei, 205 A. Freise, 48, 115 O. Freitas, 206, 152 R. Frey, 84 W. Frischhertz, 74 P. Fritschel, 46 V. V. Frolov, 74 G. G. Fronz'e, 41 M. Fuentes-Garcia, 19 S. Fujii, 169 I. Fukunaga, 207 P. Fulda, 56 M. Fyffe, 74 W. E. Gabella, 208 B. Gadre, 83 J. R. Gair, 18 S. Galaudage, 25, 209 S. Gallardo, 210 B. Gallego, 210 R. Gamba, 109 A. Gamboa, 18 D. Ganapathy, 46 A. Ganguly, 32 S. G. Gaonkar, 32 B. Garaventa, 67, 142 J. Garcia-Bellido, 128 C. Garc'ıa-N'u˜nez, 203 C. Garc'ıa-Quir'os, 194 J. W. Gardner, 29 K. A. Gardner, 155 J. Gargiulo, 68 A. Garron, 107 F. Garufi, 45, 23 C. Gasbarra, 149, 102 B. Gateley, 21 V. Gayathri, 27 G. Gemme, 67 A. Gennai, 37 J. George, 111 R. George, 164 O. Gerberding, 92 L. Gergely, 211 N. Ghadiri, 66 Archisman Ghosh, 100 Shaon Ghosh, 198 Shrobana Ghosh, 30, 31 Suprovo Ghosh, 32 Tathagata Ghosh, 32 L. Giacoppo, 123, 75 J. A. Giaime, 28, 74 K. D. Giardina, 74 D. R. 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Guo, 48, 47 Anchal Gupta, 19 Anuradha Gupta, 216 Ish Gupta, 26 N. C. Gupta, 99 P. Gupta, 48, 83 S. K. Gupta, 56 T. Gupta, 98 N. Gupte, 18 R. Gurav, 143 J. Gurs, 92 N. Gutierrez, 184 F. Guzman, 145 D. Haba, 20 M. Haberland, 18 L. Haegel, 78 G. Hain, 132 S. Haino, 217 E. D. Hall, 46 E. Z. Hamilton, 194 G. Hammond, 42 W.-B. Han, 218 M. Haney, 194, 48 J. Hanks, 21 C. Hanna, 26 M. D. Hannam, 36 O. A. Hannuksela, 154 A. G. Hanselman, 141 H. Hansen, 21 J. Hanson, 74 R. Harada, 52 T. Harder, 60 K. Haris, 48, 83 T. Harmark, 148 J. Harms, 55, 129 G. M. Harry, 219 I. W. Harry, 138 B. Haskell, 101 C.-J. Haster, 220 J. S. Hathaway, 221 K. Haughian, 42 H. Hayakawa, 61 K. Hayama, 222 J. Healy, 221 A. Heffernan, 107 A. Heidmann, 130 M. C. Heintze, 74 J. Heinze, 126 J. Heinzel, 46 H. Heitmann, 60 F. Hellman, 223 P. Hello, 51 A. F. Helmling-Cornell, 84 G. Hemming, 68 M. Hendry, 42 I. S. Heng, 42 E. Hennes, 48 J.-S. Hennig, 47, 48 M. Hennig, 47, 48 C. Henshaw, 69 A. Hernandez, 198 T. Hertog, 117 M. Heurs, 30, 31 A. L. Hewitt, 33, 224 S. Higginbotham, 36 S. Hild, 47, 48 P. Hill, 106 S. Hill, 42 Y. Himemoto, 225 A. S. Hines, 145 N. Hirata, 38 C. Hirose, 226 J. Ho, 159 S. Hoang, 51 S. Hochheim, 30, 31 D. Hofman, 184 N. A. Holland, 48, 115 K. Holley-Bockelmann, 208 I. J. Hollows, 183 Z. J. Holmes, 104 D. E. Holz, 141 C. Hong, 35 J. Hornung, 84 S. Hoshino, 226 J. Hough, 42 S. Hourihane, 19 E. J. Howell, 43 C. G. Hoy, 138 D. Hoyland, 126 C. A. Hrishikesh, 149 H.-F. Hsieh, 158 C. Hsiung, 227 H. C. Hsu, 159 S.-C. Hsu, 64, 158 W.-F. Hsu, 117 P. Hu, 208 Q. Hu, 42 H. Y. Huang, 159 Y.-J. Huang, 26 Y. Huang, 46 Y. T. Huang, 64 A. D. Huddart, 228 B. Hughey, 59 D. C. Y. Hui, 229 V. Hui, 44 R. Hur, 84 S. Husa, 107 R. Huxford, 26 T. Huynh-Dinh, 74 A. Iakovlev, 230 G. A. Iandolo, 47 A. Iess, 180, 37 \nK. Inayoshi, 231 Y. Inoue, 159 G. Iorio, 96 J. Irwin, 42 M. Isi, 196, 197 M. A. Ismail, 159 Y. Itoh, 207, 232 M. Iwaya, 169 B. R. Iyer, 91 V. JaberianHamedan, 43 P.-E. Jacquet, 130 S. J. Jadhav, 233 S. P. Jadhav, 172 T. Jain, 33 A. L. James, 36 P. A. James, 132 R. Jamshidi, 49 A. Z. Jan, 164 K. Jani, 208 L. Janiurek, 42 J. Janquart, 83, 48 K. Janssens, 122, 60 N. N. Janthalur, 233 S. Jaraba, 128 P. Jaranowski, 234 P. Jasal, 53 R. Jaume, 107 W. Javed, 36 A. Jennings, 21 W. Jia, 46 J. Jiang, 56 H.-B. Jin, 235, 236 K. Johansmeyer, 198 G. R. Johns, 132 N. A. Johnson, 56 R. Johnston, 42 N. Johny, 30, 31 D. H. Jones, 29 D. I. Jones, 237 R. Jones, 42 S. Jose, 189 P. Joshi, 26 L. Ju, 43 K. Jung, 238 J. Junker, 30, 31 V. Juste, 76 T. Kajita, 239 C. Kalaghatgi, 83, 48, 240 V. Kalogera, 86 M. Kamiizumi, 61 N. Kanda, 232, 207 S. Kandhasamy, 32 G. Kang, 241 J. B. Kanner, 19 S. J. Kapadia, 32 D. P. Kapasi, 29 S. Karat, 19 C. Karathanasis, 54 S. Karki, 113 R. Kashyap, 26 M. Kasprzack, 19 W. Kastaun, 30, 31 J. Kato, 168 T. Kato, 169 S. Katsanevas, 68, ∗ E. Katsavounidis, 46 W. Katzman, 74 T. Kaur, 43 R. Kaushik, 111 K. Kawabe, 21 D. Keitel, 107 J. Kelley-Derzon, 56 J. Kennington, 26 R. Kesharwani, 32 J. S. Key, 242 S. Khadka, 35 F. Y. Khalili, 116 F. Khan, 30, 31 I. Khan, 243, 50 T. Khanam, 175 E. A. Khazanov, 230 M. Khursheed, 111 W. Kiendrebeogo, 60, 244 N. Kijbunchoo, 104 C. Kim, 245 J. C. Kim, 246 K. Kim, 247 M. H. Kim, 248 S. Kim, 229 W. S. Kim, 249 Y.-M. Kim, 247 C. Kimball, 86 N. Kimura, 61 M. Kinley-Hanlon, 42 M. Kinnear, 36 J. S. Kissel, 21 T. Kiyota, 207 S. Klimenko, 56 T. Klinger, 36 A. M. Knee, 155 N. Knust, 30, 31 P. Koch, 30, 31 S. M. Koehlenbeck, 35 G. Koekoek, 48, 47 K. Kohri, 250 K. Kokeyama, 36 S. Koley, 55 P. Kolitsidou, 126 M. Kolstein, 54 K. Komori, 52 A. K. H. Kong, 158 A. Kontos, 251 M. Korobko, 92 R. V. Kossak, 30, 31 X. Kou, 105 A. Koushik, 122 N. Kouvatsos, 79 M. Kovalam, 43 N. Koyama, 226 D. B. Kozak, 19 S. L. Kranzhoff, 47, 48 V. Kringel, 30, 31 N. V. Krishnendu, 91 A. Kr'olak, 252, 191 G. Kuehn, 30, 31 P. Kuijer, 48 S. Kulkarni, 216 A. Kulur Ramamohan, 29 A. Kumar, 233 Praveen Kumar, 140 Prayush Kumar, 91 Rahul Kumar, 21 Rakesh Kumar, 99 J. Kume, 52 K. Kuns, 46 S. Kuroyanagi, 128, 253 S. Kuwahara, 52 K. Kwak, 238 K. Kwan, 29 G. Lacaille, 42 P. Lagabbe, 44 D. Laghi, 137 S. Lai, 162 A. H. Laity, 177 M. H. Lakkis, 49 E. Lalande, 254 M. Lalleman, 122 M. Landry, 21 B. B. Lane, 46 R. N. Lang, 46 J. Lange, 164 B. Lantz, 35 A. La Rana, 75 I. La Rosa, 107, 123, 44 A. Lartaux-Vollard, 51 P. D. Lasky, 25 J. Lawrence, 175 M. Laxen, 74 A. Lazzarini, 19 C. Lazzaro, 96, 97 P. Leaci, 123, 75 S. LeBohec, 204 Y. K. Lecoeuche, 155 H. M. Lee, 246 H. W. Lee, 255 K. Lee, 248 R.-K. Lee, 158 R. Lee, 46 S. Lee, 247 Y. Lee, 159 I. N. Legred, 19 J. Lehmann, 30, 31 L. Lehner, 187 A. Lemaˆıtre, 256 M. Lenti, 73, 257 M. Leonardi, 258, 38 E. Leonova, 112 M. Lequime, 50 N. Leroy, 51 M. Lesovsky, 19 N. Letendre, 44 M. Lethuillier, 170 C. Levesque, 254 Y. Levin, 25 K. Leyde, 78 A. K. Y. Li, 19 K. L. Li, 157 T. G. F. Li, 154, 117 X. Li, 165 Chien-Yu Lin, 159, 158 Chun-Yu Lin, 259 E. T. Lin, 158 F. Lin, 159 H. Lin, 159 L. C.-C. Lin, 157 F. Linde, 240, 48 S. D. Linker, 150, 210 T. B. Littenberg, 260 A. Liu, 154 G. C. Liu, 227 Jian Liu, 43 F. Llamas, 178 J. Llobera-Querol, 107 R. K. L. Lo, 19 J.-P. Locquet, 117 L. London, 112 A. Longo, 72, 73 D. Lopez, 194 M. Lopez Portilla, 83 M. Lorenzini, 149, 102 V. Loriette, 51 M. Lormand, 74 G. Losurdo, 37 T. P. Lott IV, 69 J. D. Lough, 30, 31 H. A. Loughlin, 46 C. O. Lousto, 221 M. J. Lowry, 132 H. Luck, 31, 30, 31 D. Lumaca, 102 A. P. Lundgren, 138 A. W. Lussier, 254 L.-T. Ma, 158 S. Ma, 165 M. Ma'arif, 159 R. Macas, 138 M. MacInnis, 46 R. R. Maciy, 30, 31 D. M. Macleod, 36 I. A. O. MacMillan, 19 A. Macquet, 54 D. Macri, 46 K. Maeda, 168 S. Maenaut, 117 I. Maga˜na Hernandez, 27 S. S. Magare, 32 C. Magazz'u, 37 R. M. Magee, 19 E. Maggio, 18 R. Maggiore, 48, 115 M. Magnozzi, 67, 142 M. Mahesh, 92 S. Mahesh, 261 M. Maini, 177 S. Majhi, 32 E. Majorana, 123, 75 C. N. Makarem, 19 J. A. Malaquias-Reis, 34 S. Maliakal, 19 A. Malik, 111 N. Man, 60 V. Mandic, 105 V. Mangano, 75, 123 B. Mannix, 84 G. L. Mansell, 85, 46 M. Manske, 27 M. Mantovani, 68 M. Mapelli, 96, 97 F. Marchesoni, 63, 62, 262 D. Mar'ın Pina, 53, 87, 263 F. Marion, 44 S. M'arka, 176 Z. M'arka, 176 C. Markakis, 161 A. S. Markosyan, 35 A. Markowitz, 19 E. Maros, 19 A. Marquina, 174 S. Marsat, 137 F. Martelli, 72, 73 I. W. Martin, 42 R. M. Martin, 198 B. B. Martinez, 145 M. Martinez, 54, 264 V. Martinez, 139 A. Martini, 118 K. Martinovic, 79 J. C. Martins, 34 D. V. Martynov, 126 E. J. Marx, 46 L. Massaro, 47, 48 A. Masserot, 44 M. Masso-Reid, 42 M. Mastrodicasa, 75 S. Mastrogiovanni, 75 M. Mateu-Lucena, 107 M. Matiushechkina, 30, 31 M. Matsuyama, 207 N. Mavalvala, 46 N. Maxwell, 21 G. McCarrol, 74 R. McCarthy, 21 D. E. McClelland, 29 S. McCormick, 74 L. McCuller, 19 G. I. McGhee, 42 K. B. M. McGowan, 208 M. Mchedlidze, 198 C. McIsaac, 138 J. McIver, 155 K. McKinney, 179 A. McLeod, 43 T. McRae, 29 S. T. McWilliams, 261 D. Meacher, 27 A. K. Mehta, 18 Q. Meijer, 83 A. Melatos, 146 S. Mellaerts, 117 A. Menendez-Vazquez, 54 C. S. Menoni, 114 R. A. Mercer, 27 L. Mereni, 184 K. Merfeld, 84 E. L. Merilh, 74 J. R. M'erou, 107 J. D. Merritt, 84 M. Merzougui, 60 C. Messenger, 42 C. Messick, 27 M. Meyer-Conde, 207 F. Meylahn, 30, 31 A. Mhaske, 32 A. Miani, 118, 119 H. Miao, 265 I. Michaloliakos, 56 C. Michel, 184 Y. Michimura, 19, 52 H. Middleton, 126 A. L. Miller, 48 S. Miller, 19 M. Millhouse, 69 E. Milotti, 266, 58 Y. Minenkov, 102 N. Mio, 267 Ll. M. Mir, 54 L. Mirasola, 268, 75 M. Miravet-Ten'es, 152 C.-A. Miritescu, 54 A. K. Mishra, 91 A. Mishra, 32 C. Mishra, 189 T. Mishra, 56 A. L. Mitchell, 48, 115 J. G. Mitchell, 59 S. Mitra, 32 V. P. Mitrofanov, 116 G. Mitselmakher, 56 R. Mittleman, 46 O. Miyakawa, 61 S. Miyamoto, 169 S. Miyoki, 61 G. Mo, 46 L. Mobilia, 72, 73 L. M. Modafferi, 107 S. R. P. Mohapatra, 19 S. R. Mohite, 27 M. Molina-Ruiz, 223 C. Mondal, 214 M. Mondin, 210 M. Montani, 72, 73 C. J. Moore, 126 M. Morales, 66 D. Moraru, 21 F. Morawski, 101 A. More, 32 S. More, 32 C. Moreno, 59 G. Moreno, 21 S. Morisaki, 52, 169 Y. Moriwaki, 168 G. Morras, 128 A. Moscatello, 96 P. Mourier, 107 B. Mours, 76 C. M. Mow-Lowry, 48, 115 S. Mozzon, 138 F. Muciaccia, 123, 75 D. Mukherjee, 260 Samanwaya Mukherjee, 32 Soma Mukherjee, 178 Subroto Mukherjee, 99 Suvodip Mukherjee, 269, 187, 112 N. Mukund, 46 A. Mullavey, 74 J. Munch, 104 C. L. Mungioli, 43 M. Munn, 21 W. R. Munn Oberg, 270 M. Murakoshi, 271 P. G. Murray, 42 S. Muusse, 29 S. L. Nadji, 30, 31 A. Nagar, 41, 272 N. Nagarajan, 42 K. N. Nagler, 59 K. Nakamura, 38 H. Nakano, 273 M. Nakano, 19 D. Nandi, 28 V. Napolano, 68 P. Narayan, 216 I. Nardecchia, 149, 102 H. Narola, 83 L. Naticchioni, 75 R. K. Nayak, 274 B. F. Neil, 43 J. Neilson, 103, 121 A. Nelson, 145 T. J. N. Nelson, 74 M. Nery, 30, 31 A. Neunzert, 21 S. Ng, 66 C. Nguyen, 78 P. Nguyen, 84 L. Nguyen Quynh, 275 S. A. Nichols, 28 A. B. Nielsen, 276 G. Nieradka, 101 A. Niko, 159 Y. Nishino, 38, 277 \nA. Nishizawa, 52 S. Nissanke, 112, 48 E. Nitoglia, 170 W. Niu, 26 F. Nocera, 68 M. Norman, 36 C. North, 36 J. Novak, 124, 278, 279, 280 J. F. Nu˜no Siles, 128 G. Nurbek, 178 L. K. Nuttall, 138 K. Obayashi, 271 J. Oberling, 21 J. O'Dell, 228 M. Oertel, 124, 278, 279, 281, 280 A. Offermans, 117 G. Oganesyan, 55, 129 J. J. Oh, 249 K. Oh, 229 S. H. Oh, 249 T. O'Hanlon, 74 M. Ohashi, 61 M. Ohkawa, 226 F. Ohme, 30, 31 H. Ohta, 52 A. S. Oliveira, 176 R. Oliveri, 124, 278, 279 V. Oloworaran, 43 B. O'Neal, 132 K. Oohara, 282, 283 B. O'Reilly, 74 N. D. Ormsby, 132 M. Orselli, 62, 94 R. O'Shaughnessy, 221 Y. Oshima, 284 S. Oshino, 61 S. Ossokine, 18 C. Osthelder, 19 D. J. Ottaway, 104 A. Ouzriat, 170 H. Overmier, 74 B. J. Owen, 175 A. E. Pace, 26 R. Pagano, 28 M. A. Page, 38 A. Pai, 156 S. A. Pai, 111 A. Pal, 285 S. Pal, 274 M. A. Palaia, 37, 93 O. Palashov, 230 M. P'alfi, 205 P. P. Palma, 149, 102 C. Palomba, 75 K. C. Pan, 158 P. K. Panda, 233 L. Panebianco, 72, 73 P. T. H. Pang, 48, 83 F. Pannarale, 123, 75 B. C. Pant, 111 F. H. Panther, 43 C. D. Panzer, 105 F. Paoletti, 37 A. Paoli, 68 A. Paolone, 75, 286 E. E. Papalexakis, 143 L. Papalini, 37, 93 G. Papigkiotis, 287 A. Parisi, 48, 112 J. Park, 247 W. Parker, 74 G. Pascale, 30, 31 D. Pascucci, 100 A. Pasqualetti, 68 R. Passaquieti, 93, 37 D. Passuello, 37 O. Patane, 21 M. Patel, 132 D. Pathak, 32 M. Pathak, 104 A. Patra, 36 B. Patricelli, 93, 37 A. S. Patron, 28 S. Paul, 84 E. Payne, 19 T. Pearce, 36 M. Pedraza, 19 R. Pegna, 37 A. Pele, 19 F. E. Pe˜na Arellano, 61 S. Penn, 270 M. D. Penuliar, 66 A. Perego, 118, 119 A. Pereira, 139 J. J. Perez, 56 C. P'erigois, 171, 97, 96 C. C. Perkins, 56 G. Perna, 96 A. Perreca, 118, 119 J. Perret, 78 S. Perri'es, 170 J. W. Perry, 48, 115 D. Pesios, 287 C. Petrillo, 94 H. P. Pfeiffer, 18 H. Pham, 74 K. A. Pham, 105 K. S. Phukon, 126, 48, 240 H. Phurailatpam, 154 O. J. Piccinni, 54 M. Pichot, 60 M. Piendibene, 93, 37 F. Piergiovanni, 72, 73 L. Pierini, 75 G. Pierra, 170 V. Pierro, 103, 121 M. Pietrzak, 101 M. Pillas, 51 F. Pilo, 37 L. Pinard, 184 C. Pineda-Bosque, 210 I. M. Pinto, 103, 121, 288, 45 M. Pinto, 68 B. J. Piotrzkowski, 27 M. Pirello, 21 M. D. Pitkin, 33, 224 A. Placidi, 62, 94 E. Placidi, 123, 75 M. L. Planas, 107 W. Plastino, 289, 290 R. Poggiani, 93, 37 E. Polini, 44 L. Pompili, 18 J. Poon, 154 E. Porcelli, 48 J. Portell, 53, 87, 263 E. K. Porter, 78 C. Posnansky, 26 R. Poulton, 68 J. Powell, 172 M. Pracchia, 44 B. K. Pradhan, 32 T. Pradier, 76 A. K. Prajapati, 99 K. Prasai, 35 R. Prasanna, 233 P. Prasia, 32 G. Pratten, 126 M. Principe, 150, 103, 288, 121 G. A. Prodi, 291, 119 L. Prokhorov, 126 P. Prosposito, 149, 102 L. Prudenzi, 18 A. Puecher, 48, 83 J. Pullin, 28 M. Punturo, 62 F. Puosi, 37, 93 P. Puppo, 75 M. Purrer, 177 H. Qi, 161 J. Qin, 29 G. Qu'em'ener, 215, 124, 214 V. Quetschke, 178 C. Quigley, 36 P. J. Quinonez, 59 R. Quitzow-James, 113 F. J. Raab, 21 G. Raaijmakers, 112, 48 N. Radulesco, 60 P. Raffai, 205 S. X. Rail, 254 S. Raja, 111 C. Rajan, 111 B. Rajbhandari, 221, 175 D. S. Ramirez, 59 K. E. Ramirez, 74 F. A. Ramis Vidal, 107 A. Ramos-Buades, 18 D. Rana, 32 E. Randel, 114 S. Ranjan, 69 P. Rapagnani, 123, 75 B. Ratto, 59 S. Rawat, 105 A. Ray, 27 V. Raymond, 36 M. Razzano, 93, 37 J. Read, 66 M. Recaman Payo, 117 T. Regimbau, 44 L. Rei, 67 S. Reid, 106 S. W. Reid, 132 D. H. Reitze, 19 P. Relton, 36 A. Renzini, 19 P. Rettegno, 41 B. Revenu, 78, 292 A. Reza, 48 M. Rezac, 66 A. S. Rezaei, 75, 123 F. Ricci, 123, 75 M. Ricci, 75 D. Richards, 228 C. J. Richardson, 59 J. W. Richardson, 143 A. Rijal, 59 K. Riles, 95 H. K. Riley, 36 S. Rinaldi, 93, 37 J. Rittmeyer, 92 C. Robertson, 228 F. Robinet, 51 M. Robinson, 21 A. Rocchi, 102 L. Rolland, 44 J. G. Rollins, 19 M. Romanelli, 125 A. E. Romano, 293 R. Romano, 22, 23 A. Romero, 193 I. M. Romero-Shaw, 33 J. H. Romie, 74 T. J. Roocke, 104 L. Rosa, 23, 45 T. J. Rosauer, 143 C. A. Rose, 27 D. Rosi'nska, 133 M. P. Ross, 64 M. Rossello, 107 S. Rowan, 42 S. K. Roy, 196, 197 S. Roy, 83 D. Rozza, 182, 147 P. Ruggi, 68 E. Ruiz Morales, 294, 128 K. Ruiz-Rocha, 208 S. Sachdev, 69 T. Sadecki, 21 J. Sadiq, 140 P. Saffarieh, 48, 115 M. R. Sah, 269 S. S. Saha, 158 T. Sainrat, 76 S. Sajith Menon, 213, 123, 75 K. Sakai, 295 M. Sakellariadou, 79 T. Sako, 168 S. Sakon, 26 O. S. Salafia, 173, 136, 135 F. Salces-Carcoba, 19 L. Salconi, 68 M. Saleem, 105 F. Salemi, 123, 75 M. Sall'e, 48 S. Salvador, 215, 214, 124 A. Sanchez, 21 E. J. Sanchez, 19 J. H. Sanchez, 86 L. E. Sanchez, 19 N. Sanchis-Gual, 296, 152 J. R. Sanders, 297 E. M. Sanger, 18 T. R. Saravanan, 32 N. Sarin, 25 A. Sasli, 287 P. Sassi, 62, 94 B. Sassolas, 184 H. Satari, 43 R. Sato, 226 S. Sato, 168 Y. Sato, 168 O. Sauter, 56 R. L. Savage, 21 T. Sawada, 61 H. L. Sawant, 32 S. Sayah, 44 D. Schaetzl, 19 M. Scheel, 165 J. Scheuer, 86 M. G. Schiworski, 104 P. Schmidt, 126 S. Schmidt, 83 R. Schnabel, 92 M. Schneewind, 30, 31 R. M. S. Schofield, 84 K. Schouteden, 117 H. Schuler, 26 B. W. Schulte, 30, 31 B. F. Schutz, 36, 30, 31 E. Schwartz, 36 J. Scott, 42 S. M. Scott, 29 T. C. Seetharamu, 42 M. Seglar-Arroyo, 54 Y. Sekiguchi, 298 D. Sellers, 74 A. S. Sengupta, 299 D. Sentenac, 68 E. G. Seo, 42 J. W. Seo, 117 V. Sequino, 45, 23 A. Sergeev, 230 M. Serra, 75 G. Servignat, 278 Y. Setyawati, 83 T. Shaffer, 21 U. S. Shah, 69 M. S. Shahriar, 86 M. A. Shaikh, 246 B. Shams, 204 L. Shao, 231 A. K. Sharma, 91 P. Sharma, 111 S. Sharma-Chaudhary, 113 P. Shawhan, 134 N. S. Shcheblanov, 300, 256 B. Shen, 134 Y. Shikano, 301, 302 M. Shikauchi, 52 K. Shimode, 61 H. Shinkai, 303 J. Shiota, 271 D. H. Shoemaker, 46 D. M. Shoemaker, 164 R. W. Short, 21 S. ShyamSundar, 111 A. Sider, 49 H. Siegel, 176, 196, 197 M. Sieniawska, 131 D. Sigg, 21 L. Silenzi, 62, 63 M. Simmonds, 104 L. P. Singer, 90 A. Singh, 216 D. Singh, 26 M. K. Singh, 91 A. Singha, 47, 48 A. M. Sintes, 107 V. Sipala, 182, 147 V. Skliris, 36 B. J. J. Slagmolen, 29 T. J. Slaven-Blair, 43 J. Smetana, 126 J. R. Smith, 66 L. Smith, 42 R. J. E. Smith, 25 W. J. Smith, 208 J. Soldateschi, 257, 304, 73 S. N. Somala, 305 K. Somiya, 20 K. Soni, 32 S. Soni, 46 V. Sordini, 170 F. Sorrentino, 67 N. Sorrentino, 93, 37 R. Soulard, 60 T. Souradeep, 32, 306 A. Southgate, 36 E. Sowell, 175 V. Spagnuolo, 47, 48 A. P. Spencer, 42 M. Spera, 96, 97 P. Spinicelli, 68 A. K. Srivastava, 99 F. Stachurski, 42 D. A. Steer, 78 J. Steinlechner, 47, 48 S. Steinlechner, 47, 48 N. Stergioulas, 287 P. Stevens, 51 M. StPierre, 177 L. C. Strang, 146 G. Stratta, 307, 308, 75, 309 M. D. Strong, 28 A. Strunk, 21 R. Sturani, 310 A. L. Stuver, 110 M. Suchenek, 101 S. Sudhagar, 32, 101 N. Sueltmann, 92 A. G. Sullivan, 176 K. D. Sullivan, 28 L. Sun, 29 S. Sunil, 99 A. Sur, 101 J. Suresh, 52, 131 P. J. Sutton, 36 Takamasa Suzuki, 226 Takanori Suzuki, 20 B. L. Swinkels, 48 A. Syx, 76 M. J. Szczepa'nczyk, 56 P. Szewczyk, 133 M. Tacca, 48 H. Tagoshi, 169 S. C. Tait, 42 H. Takahashi, 311 R. Takahashi, 38 A. Takamori, 65 K. Takatani, 207 H. Takeda, 312 M. Takeda, 207 C. J. Talbot, 106 C. Talbot, 46 M. Tamaki, 169 N. Tamanini, 137 D. Tanabe, 159 K. Tanaka, 313 S. J. Tanaka, 271 T. Tanaka, 312 A. J. Tanasijczuk, 131 D. Tang, 43 S. Tanioka, 85 D. B. Tanner, 56 L. Tao, 56 R. D. Tapia, 26 E. N. Tapia San Mart'ın, 48 R. Tarafder, 19 C. Taranto, 149, 102 A. Taruya, 314 J. D. Tasson, 190 M. Teloi, 49 \nR. Tenorio, 107 H. Themann, 210 A. Theodoropoulos, 152 M. P. Thirugnanasambandam, 32 L. M. Thomas, 126 M. Thomas, 74 P. Thomas, 21 J. E. Thompson, 165 S. R. Thondapu, 111 K. A. Thorne, 74 E. Thrane, 25 J. Tissino, 55 A. Tiwari, 32 Shubhanshu Tiwari, 194 Srishti Tiwari, 32 V. Tiwari, 126 M. R. Todd, 85 A. M. Toivonen, 105 K. Toland, 42 A. E. Tolley, 138 T. Tomaru, 38 K. Tomita, 207 T. Tomura, 61 C. Tong-Yu, 159 A. Toriyama, 271 N. Toropov, 126 A. Torres-Forn'e, 152, 153 C. I. Torrie, 19 M. Toscani, 137 I. Tosta e Melo, 315 E. Tournefier, 44 A. A. Trani, 52 A. Trapananti, 63, 62 F. Travasso, 63, 62 G. Traylor, 74 J. Trenado, 53 M. Trevor, 134 M. C. Tringali, 68 A. Tripathee, 95 L. Troiano, 316, 121 A. Trovato, 58, 266 L. Trozzo, 23 R. J. Trudeau, 19 T. T. L. Tsang, 36 R. Tso, 165, † S. Tsuchida, 317 L. Tsukada, 26 T. Tsutsui, 52 K. Turbang, 193, 122 M. Turconi, 60 C. Turski, 100 H. Ubach, 53, 87 A. S. Ubhi, 126 N. Uchikata, 169 T. Uchiyama, 61 R. P. Udall, 19 T. Uehara, 318 K. Ueno, 52 C. S. Unnikrishnan, 269 T. Ushiba, 61 A. Utina, 47, 48 M. Vacatello, 37, 93 H. Vahlbruch, 30, 31 N. Vaidya, 19 G. Vajente, 19 A. Vajpeyi, 25 G. Valdes, 145 J. Valencia, 107 M. Valentini, 115, 48 S. A. Vallejo-Pe˜na, 293 S. Vallero, 41 V. Valsan, 27 N. van Bakel, 48 M. van Beuzekom, 48 M. van Dael, 48, 319 J. F. J. van den Brand, 47, 115, 48 C. Van Den Broeck, 83, 48 D. C. Vander-Hyde, 85 M. van der Sluys, 48, 83 A. Van de Walle, 51 J. van Dongen, 48, 115 K. Vandra, 110 H. van Haevermaet, 122 J. V. van Heijningen, 131 J. Vanosky, 19 M. H. P. M. van Putten, 320 Z. van Ranst, 47, 48 N. van Remortel, 122 M. Vardaro, 47, 48 A. F. Vargas, 146 V. Varma, 18 M. Vas'uth, 89 A. Vecchio, 126 G. Vedovato, 97 J. Veitch, 42 P. J. Veitch, 104 S. Venikoudis, 131 J. Venneberg, 30, 31 P. Verdier, 170 D. Verkindt, 44 B. Verma, 151 P. Verma, 191 Y. Verma, 111 S. M. Vermeulen, 19 D. Veske, 176 F. Vetrano, 72 A. Veutro, 75 A. M. Vibhute, 21 A. Vicer'e, 72, 73 S. Vidyant, 85 A. D. Viets, 321 A. Vijaykumar, 91 A. Vilkha, 221 V. Villa-Ortega, 140 E. T. Vincent, 69 J.-Y. Vinet, 60 S. Viret, 170 A. Virtuoso, 266, 58 S. Vitale, 46 H. Vocca, 94, 62 D. Voigt, 92 E. R. G. von Reis, 21 J. S. A. von Wrangel, 30, 31 S. P. Vyatchanin, 116 L. E. Wade, 82 M. Wade, 82 K. J. Wagner, 221 R. C. Walet, 48 M. Walker, 132 G. S. Wallace, 106 L. Wallace, 19 H. Wang, 284 J. Z. Wang, 95 W. H. Wang, 178 Z. Wang, 159 G. Waratkar, 156 R. L. Ward, 29 J. Warner, 21 M. Was, 44 T. Washimi, 38 N. Y. Washington, 19 D. Watarai, 52 K. E. Wayt, 82 B. Weaver, 21 C. R. Weaving, 138 S. A. Webster, 42 M. Weinert, 30, 31 A. J. Weinstein, 19 R. Weiss, 46 C. M. Weller, 64 R. A. Weller, 208 F. Wellmann, 30, 31 L. Wen, 43 P. Weßels, 30, 31 K. Wette, 29 J. T. Whelan, 221 D. D. White, 66 B. F. Whiting, 56 C. Whittle, 46 J. B. Wildberger, 18 O. S. Wilk, 82 D. Wilken, 30, 31, 31 K. Willetts, 36 D. Williams, 42 M. J. Williams, 42 N. S. Williams, 126 J. L. Willis, 19 B. Willke, 31, 30, 31 M. Wils, 117 C. C. Wipf, 19 G. Woan, 42 J. Woehler, 47, 48 J. K. Wofford, 221 N. E. Wolfe, 46 D. Wong, 155 H. T. Wong, 159 H. W. Y. Wong, 154 I. C. F. Wong, 154 J. L. Wright, 29 M. Wright, 42 C. Wu, 158 D. S. Wu, 30, 31 H. Wu, 158 D. M. Wysocki, 27 L. Xiao, 19 V. A. Xu, 46 Y. Xu, 194 N. Yadav, 101 H. Yamamoto, 19 K. Yamamoto, 168 M. Yamamoto, 168 T. S. Yamamoto, 253 T. Yamamoto, 61 S. Yamamura, 169 R. Yamazaki, 271 S. Yan, 35 T. Yan, 126 F. W. Yang, 204 F. Yang, 176 K. Z. Yang, 105 L.-C. Yang, 162 Y. Yang, 162 Z. Yarbrough, 28 S.-W. Yeh, 158 A. B. Yelikar, 221 S. M. C. Yeung, 27 X. Yin, 46 J. Yokoyama, 52 T. Yokozawa, 61 J. Yoo, 166 H. Yu, 165 H. Yuzurihara, 61 A. Zadro˙zny, 191 A. J. Zannelli, 132 M. Zanolin, 59 M. Zeeshan, 221 T. Zelenova, 68 J.-P. Zendri, 97 M. Zeoli, 127, 131 M. Zerrad, 50, 243 M. Zevin, 86 A. C. Zhang, 176 J. Zhang, 29 L. Zhang, 19 R. Zhang, 56 T. Zhang, 126 Y. Zhang, 29 C. Zhao, 43 Yue Zhao, 204 Yuhang Zhao, 169, 38, 78 Y. Zheng, 113 H. Zhong, 105 S. Zhong, 43 R. Zhou, 223 Z.-H. Zhu, 144, 322 A. B. Zimmerman, 164 M. E. Zucker, 46, 19 J. Zweizig, 19 \n- 1 Department of Astronomy and Astrophysics, The Pennsylvania State University, 525 Davey Lab, University Park, PA 16802, USA\n- 2 Department of Physics and Astronomy, University of Alabama, Tuscaloosa, AL 35487, USA 3 Department of Astronomy & Astrophysics, University of Toronto, Toronto, ON M5S 3H4 4 Dunlap Institute for Astronomy & Astrophysics, University of Toronto, Toronto, ON M5S 3H4 5 Astrophysics Science Division, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA 6 INAF - IASF-Palermo, via Ugo La Malfa 153, 90146 Palermo PA, Italy 7 INAF-Osservatorio Astronomico di Brera, Via E. Bianchi 46, 23807 Merate, LC, Italy 8 Space Science Data Center (SSDC) - Agenzia Spaziale Italiana (ASI), 00133 Roma, Italy 9 INAF - Osservatorio Astronomico di Roma, Via Frascati 33, 00040 Monte Porzio Catone, Italy 10 MIFT Department, Polo Papardo, University of Messina, Viale Ferdinando Stagno d'Alcontres, 31, 98166 Messina, Italy 11 School of Physics and Astronomy, University of Leicester, University Road, Leicester LE1 7RH, UK 12 Department of Physics & Astronomy, Clemson University, Kinard Lab of Physics, Clemson, SC 29634, USA 13 Mullard Space Science Laboratory, University College London, Holmbury St. Mary, Dorking, Surrey RH5 6NT, UK 14 INAF-Osservatorio Astronomico di Roma, Via di Frascati 33, 00040 Monte Porzio Catone, RM, Italy 15 Los Alamos National Laboratory, PO Box 1663, Los Alamos New Mexico 87545 16 INAF - Osservatorio Astronomico di Brera, Via E. Bianchi 46, 23807 Merate, Italy 17 University of Rome Tor Vergata, via Cracovia 50, 00100 Roma, Italy 18 Max Planck Institute for Gravitational Physics (Albert Einstein Institute), D-14476 Potsdam, Germany 19 LIGO Laboratory, California Institute of Technology, Pasadena, CA 91125, USA 20 Graduate School of Science, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan 21 LIGO Hanford Observatory, Richland, WA 99352, USA \n22 Dipartimento di Farmacia, Universit'a di Salerno, I-84084 Fisciano, Salerno, Italy 23 \nINFN, Sezione di Napoli, I-80126 Napoli, Italy \n54 \n61 \n53 \n24 University of Warwick, Coventry CV4 7AL, United Kingdom \n25 OzGrav, School of Physics & Astronomy, Monash University, Clayton 3800, Victoria, Australia \n26 The Pennsylvania State University, University Park, PA 16802, USA \n27 University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA \n28 Louisiana State University, Baton Rouge, LA 70803, USA \n29 OzGrav, Australian National University, Canberra, Australian Capital Territory 0200, Australia \nMax Planck Institute for Gravitational Physics (Albert Einstein Institute), D-30167 Hannover, Germany \n31 Leibniz Universitat Hannover, D-30167 Hannover, Germany \n32 \nInter-University Centre for Astronomy and Astrophysics, Pune 411007, India \n33 University of Cambridge, Cambridge CB2 1TN, United Kingdom \n34 Instituto Nacional de Pesquisas Espaciais, 12227-010 S˜ao Jos'e dos Campos, S˜ao Paulo, Brazil \n35 Stanford University, Stanford, CA 94305, USA \n36 Cardiff University, Cardiff CF24 3AA, United Kingdom \n37 INFN, Sezione di Pisa, I-56127 Pisa, Italy \n38 Gravitational Wave Science Project, National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka City, Tokyo 181-8588, Japan \n39 Advanced Technology Center, National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka City, Tokyo 181-8588, Japan \n40 Dipartimento di Fisica, Universit'a degli Studi di Torino, I-10125 Torino, Italy \n41 INFN Sezione di Torino, I-10125 Torino, Italy \n42 SUPA, University of Glasgow, Glasgow G12 8QQ, United Kingdom \n43 OzGrav, University of Western Australia, Crawley, Western Australia 6009, Australia \n44 Univ. Savoie Mont Blanc, CNRS, Laboratoire d'Annecy de Physique des Particules - IN2P3, F-74000 Annecy, France \n45 Universit'a di Napoli 'Federico II', I-80126 Napoli, Italy \n46 LIGO Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139, USA \n47 Maastricht University, 6200 MD Maastricht, Netherlands \n48 Nikhef, 1098 XG Amsterdam, Netherlands \n49 Universit'e Libre de Bruxelles, Brussels 1050, Belgium \n50 Institut Fresnel, Aix Marseille Universit'e, CNRS, Centrale Marseille, F-13013 Marseille, France \n51 Universit'e Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France \n52 University of Tokyo, Tokyo, 113-0033, Japan. \nInstitut de Ci'encies del Cosmos (ICCUB), Universitat de Barcelona (UB), c. Mart'ı i Franqu'es, 1, 08028 Barcelona, Spain \nInstitut de F'ısica d'Altes Energies (IFAE), The Barcelona Institute of Science and Technology, Campus UAB, E-08193 Bellaterra \n(Barcelona), Spain \n55 Gran Sasso Science Institute (GSSI), I-67100 L'Aquila, Italy \n56 University of Florida, Gainesville, FL 32611, USA \n57 Dipartimento di Scienze Matematiche, Informatiche e Fisiche, Universit'a di Udine, I-33100 Udine, Italy \n58 INFN, Sezione di Trieste, I-34127 Trieste, Italy \n59 Embry-Riddle Aeronautical University, Prescott, AZ 86301, USA \n60 Universit'e Cˆote d'Azur, Observatoire de la Cˆote d'Azur, CNRS, Artemis, F-06304 Nice, France \nInstitute for Cosmic Ray Research, KAGRA Observatory, The University of Tokyo, 238 Higashi-Mozumi, Kamioka-cho, Hida City, \nGifu 506-1205, Japan \n62 \nINFN, Sezione di Perugia, I-06123 Perugia, Italy \n63 Universit'a di Camerino, I-62032 Camerino, Italy \n64 University of Washington, Seattle, WA 98195, USA \n65 Earthquake Research Institute, The University of Tokyo, 1-1-1 Yayoi, Bunkyo-ku, Tokyo 113-0032, Japan \n66 \nCalifornia State University Fullerton, Fullerton, CA 92831, USA \n67 INFN, Sezione di Genova, I-16146 Genova, Italy \n68 European Gravitational Observatory (EGO), I-56021 Cascina, Pisa, Italy \n69 Georgia Institute of Technology, Atlanta, GA 30332, USA \n70 Royal Holloway, University of London, London TW20 0EX, United Kingdom \n71 The Graduate University for Advanced Studies (SOKENDAI), 2-21-1 Osawa, Mitaka City, Tokyo 181-8588, Japan \n72 Universit'a degli Studi di Urbino 'Carlo Bo', I-61029 Urbino, Italy \n73 INFN, Sezione di Firenze, I-50019 Sesto Fiorentino, Firenze, Italy \n74 LIGO Livingston Observatory, Livingston, LA 70754, USA \n75 INFN, Sezione di Roma, I-00185 Roma, Italy \n76 Universit'e de Strasbourg, CNRS, IPHC UMR 7178, F-67000 Strasbourg, France \n77 Dipartimento di Fisica 'E.R. Caianiello', Universit'a di Salerno, I-84084 Fisciano, Salerno, Italy \n78 Universit'e Paris Cit'e, CNRS, Astroparticule et Cosmologie, F-75013 Paris, France \n30 \n115 \n80 \n119 \n79 King's College London, University of London, London WC2R 2LS, United Kingdom \nKorea Institute of Science and Technology Information, Daejeon 34141, Republic of Korea \n81 Universit'e libre de Bruxelles, 1050 Bruxelles, Belgium \n82 Kenyon College, Gambier, OH 43022, USA \n83 Institute for Gravitational and Subatomic Physics (GRASP), Utrecht University, 3584 CC Utrecht, Netherlands \n84 University of Oregon, Eugene, OR 97403, USA \n85 Syracuse University, Syracuse, NY 13244, USA \n86 Northwestern University, Evanston, IL 60208, USA \n- 87 Departament de F'ısica Qu'antica i Astrof'ısica (FQA), Universitat de Barcelona (UB), c. Mart'ı i Franqu'es, 1, 08028 Barcelona, Spain \n88 Dipartimento di Medicina, Chirurgia e Odontoiatria 'Scuola Medica Salernitana', Universit'a di Salerno, I-84081 Baronissi, Salerno, \nItaly \n89 Wigner RCP, RMKI, H-1121 Budapest, Hungary \n90 NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA \n91 International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India \n92 Universitat Hamburg, D-22761 Hamburg, Germany \n93 Universit'a di Pisa, I-56127 Pisa, Italy \n94 Universit'a di Perugia, I-06123 Perugia, Italy \n95 University of Michigan, Ann Arbor, MI 48109, USA \nUniversit'a di Padova, Dipartimento di Fisica e Astronomia, I-35131 Padova, Italy \n97 INFN, Sezione di Padova, I-35131 Padova, Italy \n98 Montana State University, Bozeman, MT 59717, USA \n99 Institute for Plasma Research, Bhat, Gandhinagar 382428, India \n100 Universiteit Gent, B-9000 Gent, Belgium \n101 Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences, 00-716, Warsaw, Poland \n102 INFN, Sezione di Roma Tor Vergata, I-00133 Roma, Italy \n103 Dipartimento di Ingegneria, Universit'a del Sannio, I-82100 Benevento, Italy \n104 OzGrav, University of Adelaide, Adelaide, South Australia 5005, Australia \n105 University of Minnesota, Minneapolis, MN 55455, USA \n106 SUPA, University of Strathclyde, Glasgow G1 1XQ, United Kingdom \n107 IAC3-IEEC, Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain \n108 Departamento de Matem'aticas, Universitat Aut'onoma de Barcelona, 08193 Bellaterra (Barcelona), Spain \n109 Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universitat Jena, D-07743 Jena, Germany \n110 Villanova University, Villanova, PA 19085, USA \n111 RRCAT, Indore, Madhya Pradesh 452013, India \n112 GRAPPA, Anton Pannekoek Institute for Astronomy and Institute for High-Energy Physics, University of Amsterdam, 1098 XH Amsterdam, Netherlands \n113 Missouri University of Science and Technology, Rolla, MO 65409, USA \n114 Colorado State University, Fort Collins, CO 80523, USA \nDepartment of Physics and Astronomy, Vrije Universiteit Amsterdam, 1081 HV Amsterdam, Netherlands \n116 Lomonosov Moscow State University, Moscow 119991, Russia \n117 Katholieke Universiteit Leuven, Oude Markt 13, 3000 Leuven, Belgium \n118 Universit'a di Trento, Dipartimento di Fisica, I-38123 Povo, Trento, Italy \nINFN, Trento Institute for Fundamental Physics and Applications, I-38123 Povo, Trento, Italy \n120 Bar-Ilan University, Ramat Gan, 5290002, Israel \n121 \nINFN, Sezione di Napoli, Gruppo Collegato di Salerno, I-80126 Napoli, Italy \n122 Universiteit Antwerpen, 2000 Antwerpen, Belgium \n123 Universit'a di Roma 'La Sapienza', I-00185 Roma, Italy \n124 Centre national de la recherche scientifique, 75016 Paris, France \n125 Univ Rennes, CNRS, Institut FOTON - UMR 6082, F-35000 Rennes, France \n126 University of Birmingham, Birmingham B15 2TT, United Kingdom \n127 Universit'e de Li'ege, B-4000 Li'ege, Belgium \n128 Instituto de Fisica Teorica UAM-CSIC, Universidad Autonoma de Madrid, 28049 Madrid, Spain \n129 INFN, Laboratori Nazionali del Gran Sasso, I-67100 Assergi, Italy \n130 Laboratoire Kastler Brossel, Sorbonne Universit'e, CNRS, ENS-Universit'e PSL, Coll'ege de France, F-75005 Paris, France \n131 \nUniversit'e catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium \n132 Christopher Newport University, Newport News, VA 23606, USA \n133 Astronomical Observatory Warsaw University, 00-478 Warsaw, Poland \n96 \n184 \n137 \n144 \n150 \n134 University of Maryland, College Park, MD 20742, USA \n135 Universit'a degli Studi di Milano-Bicocca, I-20126 Milano, Italy \n136 INFN, Sezione di Milano-Bicocca, I-20126 Milano, Italy \nL2IT, Laboratoire des 2 Infinis - Toulouse, Universit'e de Toulouse, CNRS/IN2P3, UPS, F-31062 Toulouse Cedex 9, France \n138 University of Portsmouth, Portsmouth, PO1 3FX, United Kingdom \n139 \nUniversit'e de Lyon, Universit'e Claude Bernard Lyon 1, CNRS, Institut Lumi'ere Mati'ere, F-69622 Villeurbanne, France \n140 IGFAE, Universidade de Santiago de Compostela, 15782 Spain \n141 University of Chicago, Chicago, IL 60637, USA \n142 Dipartimento di Fisica, Universit'a degli Studi di Genova, I-16146 Genova, Italy \n143 University of California, Riverside, Riverside, CA 92521, USA \nDepartment of Astronomy, Beijing Normal University, Xinjiekouwai Street 19, Haidian District, Beijing 100875, China \n145 Texas A&M University, College Station, TX 77843, USA \n146 OzGrav, University of Melbourne, Parkville, Victoria 3010, Australia \n147 INFN, Laboratori Nazionali del Sud, I-95125 Catania, Italy \n148 Niels Bohr Institute, Copenhagen University, 2100 København, Denmark \n149 Universit'a di Roma Tor Vergata, I-00133 Roma, Italy \nUniversity of Sannio at Benevento, I-82100 Benevento, Italy and INFN, Sezione di Napoli, I-80100 Napoli, Italy \n151 University of Massachusetts Dartmouth, North Dartmouth, MA 02747, USA \n152 Departamento de Astronom'ıa y Astrof'ısica, Universitat de Val'encia, E-46100 Burjassot, Val'encia, Spain \n153 Observatori Astron'omic, Universitat de Val'encia, E-46980 Paterna, Val'encia, Spain \n154 The Chinese University of Hong Kong, Shatin, NT, Hong Kong \n155 University of British Columbia, Vancouver, BC V6T 1Z4, Canada \n156 Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India \n157 Department of Physics, National Cheng Kung University, No.1, University Road, Tainan City 701, Taiwan \n158 National Tsing Hua University, Hsinchu City 30013, Taiwan \n159 National Central University, Taoyuan City 320317, Taiwan \n160 OzGrav, Charles Sturt University, Wagga Wagga, New South Wales 2678, Australia \n161 Queen Mary University of London, London E1 4NS, United Kingdom \n162 Department of Electrophysics, National Yang Ming Chiao Tung University, 101 Univ. Street, Hsinchu, Taiwan \n163 Kamioka Branch, National Astronomical Observatory of Japan, 238 Higashi-Mozumi, Kamioka-cho, Hida City, Gifu 506-1205, Japan \n164 \nUniversity of Texas, Austin, TX 78712, USA \n165 CaRT, California Institute of Technology, Pasadena, CA 91125, USA \n166 Cornell University, Ithaca, NY 14850, USA \n167 Dipartimento di Ingegneria Industriale (DIIN), Universit'a di Salerno, I-84084 Fisciano, Salerno, Italy \n168 Faculty of Science, University of Toyama, 3190 Gofuku, Toyama City, Toyama 930-8555, Japan \n169 Institute for Cosmic Ray Research, KAGRA Observatory, The University of Tokyo, 5-1-5 Kashiwa-no-Ha, Kashiwa City, Chiba 277-8582, Japan \n170 Universit'e Lyon, Universit'e Claude Bernard Lyon 1, CNRS, IP2I Lyon / IN2P3, UMR 5822, F-69622 Villeurbanne, France \n171 INAF, Osservatorio Astronomico di Padova, I-35122 Padova, Italy \n172 OzGrav, Swinburne University of Technology, Hawthorn VIC 3122, Australia \n173 INAF, Osservatorio Astronomico di Brera sede di Merate, I-23807 Merate, Lecco, Italy \n174 Departamento de Matem'aticas, Universitat de Val'encia, E-46100 Burjassot, Val'encia, Spain \n175 Texas Tech University, Lubbock, TX 79409, USA \n176 Columbia University, New York, NY 10027, USA \n177 University of Rhode Island, Kingston, RI 02881, USA \n178 The University of Texas Rio Grande Valley, Brownsville, TX 78520, USA \n179 Bellevue College, Bellevue, WA 98007, USA \n180 Scuola Normale Superiore, I-56126 Pisa, Italy \n181 Chennai Mathematical Institute, Chennai 603103, India \n182 Universit'a degli Studi di Sassari, I-07100 Sassari, Italy \n183 The University of Sheffield, Sheffield S10 2TN, United Kingdom \nUniversit'e Lyon, Universit'e Claude Bernard Lyon 1, CNRS, Laboratoire des Mat'eriaux Avanc'es (LMA), IP2I Lyon / IN2P3, UMR \n5822, F-69622 Villeurbanne, France \n185 Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Universit'a di Parma, I-43124 Parma, Italy \n186 \nINFN, Sezione di Milano Bicocca, Gruppo Collegato di Parma, I-43124 Parma, Italy \n187 \nPerimeter Institute, Waterloo, ON N2L 2Y5, Canada \n188 Corps des Mines, Mines Paris, Universit'e PSL, 60 Bd Saint-Michel, 75272 Paris, France \n235 \n189 Indian Institute of Technology Madras, Chennai 600036, India \n190 Carleton College, Northfield, MN 55057, USA \n191 National Center for Nuclear Research, 05-400 ' Swierk-Otwock, Poland \n192 Institut d'Astrophysique de Paris, Sorbonne Universit'e, CNRS, UMR 7095, 75014 Paris, France \n193 Vrije Universiteit Brussel, 1050 Brussel, Belgium \n194 University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland \nCanadian Institute for Theoretical Astrophysics, University of Toronto, Toronto, ON M5S 3H8, Canada \n196 Stony Brook University, Stony Brook, NY 11794, USA \n197 Center for Computational Astrophysics, Flatiron Institute, New York, NY 10010, USA \n198 Montclair State University, Montclair, NJ 07043, USA \n199 Institute for Nuclear Research, H-4026 Debrecen, Hungary \n200 CNR-SPIN, I-84084 Fisciano, Salerno, Italy \n201 Scuola di Ingegneria, Universit'a della Basilicata, I-85100 Potenza, Italy \n202 Western Washington University, Bellingham, WA 98225, USA \n203 SUPA, University of the West of Scotland, Paisley PA1 2BE, United Kingdom \n204 The University of Utah, Salt Lake City, UT 84112, USA \n205 Eotvos University, Budapest 1117, Hungary \n206 Centro de F'ısica das Universidades do Minho e do Porto, Universidade do Minho, PT-4710-057 Braga, Portugal \n207 Department of Physics, Graduate School of Science, Osaka Metropolitan University, 3-3-138 Sugimoto-cho, Sumiyoshi-ku, Osaka City, Osaka 558-8585, Japan \n208 Vanderbilt University, Nashville, TN 37235, USA \n209 Universit'e Cˆote d'Azur, Observatoire de la Cˆote d'Azur, CNRS, Lagrange, F-06304 Nice, France \n210 California State University, Los Angeles, Los Angeles, CA 90032, USA \n211 University of Szeged, D'om t'er 9, Szeged 6720, Hungary \n212 INAF, Osservatorio Astronomico di Capodimonte, I-80131 Napoli, Italy \n213 \nAriel University, Ramat HaGolan St 65, Ari'el, Israel \n214 Universit'e de Normandie, ENSICAEN, UNICAEN, CNRS/IN2P3, LPC Caen, F-14000 Caen, France \n215 \nLaboratoire de Physique Corpusculaire Caen, 6 boulevard du mar'echal Juin, F-14050 Caen, France \n216 The University of Mississippi, University, MS 38677, USA \n217 Institute of Physics, Academia Sinica, 128 Sec. 2, Academia Rd., Nankang, Taipei 11529, Taiwan \nShanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030, China \n219 American University, Washington, DC 20016, USA \n220 University of Nevada, Las Vegas, Las Vegas, NV 89154, USA \n221 Rochester Institute of Technology, Rochester, NY 14623, USA \n222 Department of Applied Physics, Fukuoka University, 8-19-1 Nanakuma, Jonan, Fukuoka City, Fukuoka 814-0180, Japan 223 University of California, Berkeley, CA 94720, USA \n224 University of Lancaster, Lancaster LA1 4YW, United Kingdom \n225 College of Industrial Technology, Nihon University, 1-2-1 Izumi, Narashino City, Chiba 275-8575, Japan 226 Faculty of Engineering, Niigata University, 8050 Ikarashi-2-no-cho, Nishi-ku, Niigata City, Niigata 950-2181, Japan 227 Department of Physics, Tamkang University, No. 151, Yingzhuan Rd., Danshui Dist., New Taipei City 25137, Taiwan 228 Rutherford Appleton Laboratory, Didcot OX11 0DE, United Kingdom \n229 Department of Astronomy and Space Science, Chungnam National University, 9 Daehak-ro, Yuseong-gu, Daejeon 34134, Republic of Korea \n230 Institute of Applied Physics, Nizhny Novgorod, 603950, Russia \n231 \n232 \nKavli Institute for Astronomy and Astrophysics, Peking University, Yiheyuan Road 5, Haidian District, Beijing 100871, China \nNambu Yoichiro Institute of Theoretical and Experimental Physics (NITEP), Osaka Metropolitan University, 3-3-138 Sugimoto-cho, \nSumiyoshi-ku, Osaka City, Osaka 558-8585, Japan \n233 Directorate of Construction, Services & Estate Management, Mumbai 400094, India \n234 University of Biaglyph[suppress]lystok, 15-424 Biaglyph[suppress]lystok, Poland \nNational Astronomical Observatories, Chinese Academic of Sciences, 20A Datun Road, Chaoyang District, Beijing, China \n- 236 School of Astronomy and Space Science, University of Chinese Academy of Sciences, 20A Datun Road, Chaoyang District, Beijing, China \n237 University of Southampton, Southampton SO17 1BJ, United Kingdom \n- 238 Department of Physics, Ulsan National Institute of Science and Technology (UNIST), 50 UNIST-gil, Ulju-gun, Ulsan 44919, Republic of Korea \n239 Institute for Cosmic Ray Research, The University of Tokyo, 5-1-5 Kashiwa-no-Ha, Kashiwa City, Chiba 277-8582, Japan 240 Institute for High-Energy Physics, University of Amsterdam, 1098 XH Amsterdam, Netherlands \n218 \n195 \n244 \n259 \n264 \n267 \n271 \n282 \n241 Chung-Ang University, Seoul 06974, Republic of Korea \n242 University of Washington Bothell, Bothell, WA 98011, USA \n243 Aix Marseille Universit'e, Jardin du Pharo, 58 Boulevard Charles Livon, 13007 Marseille, France \nLaboratoire de Physique et de Chimie de l'Environnement, Universit'e Joseph KI-ZERBO, 9GH2+3V5, Ouagadougou, Burkina Faso \n245 Ewha Womans University, Seoul 03760, Republic of Korea \n246 Seoul National University, Seoul 08826, Republic of Korea \n247 Korea Astronomy and Space Science Institute, Daejeon 34055, Republic of Korea \n248 Sungkyunkwan University, Seoul 03063, Republic of Korea \n249 National Institute for Mathematical Sciences, Daejeon 34047, Republic of Korea \n250 Institute of Particle and Nuclear Studies (IPNS), High Energy Accelerator Research Organization (KEK), 1-1 Oho, Tsukuba City, Ibaraki 305-0801, Japan \n251 Bard College, Annandale-On-Hudson, NY 12504, USA \n252 Institute of Mathematics, Polish Academy of Sciences, 00656 Warsaw, Poland \n253 Department of Physics, Nagoya University, ES building, Furocho, Chikusa-ku, Nagoya, Aichi 464-8602, Japan \n254 Universit'e de Montr'eal/Polytechnique, Montreal, Quebec H3T 1J4, Canada \n255 Inje University Gimhae, South Gyeongsang 50834, Republic of Korea \n256 \nNAVIER, \n' \nEcole des Ponts, Univ Gustave Eiffel, CNRS, Marne-la-Vall'ee, France \n257 Universit'a di Firenze, Sesto Fiorentino I-50019, Italy \n258 Department of Physics, University of Trento, via Sommarive 14, Povo, 38123 TN, Italy \nNational Center for High-performance computing, National Applied Research Laboratories, No. 7, R&D 6th Rd., Hsinchu Science \nPark, Hsinchu City 30076, Taiwan \n260 NASA Marshall Space Flight Center, Huntsville, AL 35811, USA \n261 West Virginia University, Morgantown, WV 26506, USA \n262 School of Physics Science and Engineering, Tongji University, Shanghai 200092, China \n263 Institut d'Estudis Espacials de Catalunya, c. Gran Capit'a, 2-4, 08034 Barcelona, Spain \nInstitucio Catalana de Recerca i Estudis Avan¸cats (ICREA), Passeig de Llu'ıs Companys, 23, 08010 Barcelona, Spain \n265 Tsinghua University, Beijing 100084, China \n266 Dipartimento di Fisica, Universit'a di Trieste, I-34127 Trieste, Italy \nInstitute for Photon Science and Technology, The University of Tokyo, 2-11-16 Yayoi, Bunkyo-ku, Tokyo 113-8656, Japan \n268 \nINFN Cagliari, Physics Department, Universit'a degli Studi di Cagliari, Cagliari 09042, Italy \n269 Tata Institute of Fundamental Research, Mumbai 400005, India \n270 Hobart and William Smith Colleges, Geneva, NY 14456, USA \nDepartment of Physical Sciences, Aoyama Gakuin University, 5-10-1 Fuchinobe, Sagamihara City, Kanagawa 252-5258, Japan \n272 \nInstitut des Hautes Etudes Scientifiques, F-91440 Bures-sur-Yvette, France \nFaculty of Law, Ryukoku University, 67 Fukakusa Tsukamoto-cho, Fushimi-ku, Kyoto City, Kyoto 612-8577, Japan \n274 \nIndian Institute of Science Education and Research, Kolkata, Mohanpur, West Bengal 741252, India \n275 Department of Physics and Astronomy, University of Notre Dame, 225 Nieuwland Science Hall, Notre Dame, IN 46556, USA 276 University of Stavanger, 4021 Stavanger, Norway \n277 \nDepartment of Astronomy, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan \n278 \nLaboratoire Univers et Th'eories, Observatoire de Paris, 92190 Meudon, France \n279 Observatoire de Paris, 75014 Paris, France \n280 Universit'e PSL, 75006 Paris, France \n281 Universit'e de Paris Cit'e, 75006 Paris, France \nGraduate School of Science and Technology, Niigata University, 8050 Ikarashi-2-no-cho, Nishi-ku, Niigata City, Niigata 950-2181, \nJapan \n283 Niigata Study Center, The Open University of Japan, 754 Ichibancho, Asahimachi-dori, Chuo-ku, Niigata City, Niigata 951-8122, Japan \n284 Department of Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan \n285 \nCSIR-Central Glass and Ceramic Research Institute, Kolkata, West Bengal 700032, India \n286 \nConsiglio Nazionale delle Ricerche - Istituto dei Sistemi Complessi, I-00185 Roma, Italy \n287 Department of Physics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece \n288 \nMuseo Storico della Fisica e Centro Studi e Ricerche 'Enrico Fermi', I-00184 Roma, Italy \n289 Dipartimento di Ingegneria Industriale, Elettronica e Meccanica, Universit'a degli Studi Roma Tre, I-00146 Roma, Italy \n290 INFN, Sezione di Roma Tre, I-00146 Roma, Italy \n291 Universit'a di Trento, Dipartimento di Matematica, I-38123 Povo, Trento, Italy \nSubatech, CNRS/IN2P3 - IMT Atlantique - Nantes Universit'e, 4 rue Alfred Kastler BP 20722 44307 Nantes C \n293 Universidad de Antioquia, Medell'ın, Colombia \n292 \n273 \n' \nEDEX 03, France \n303 \n307 \n294 Departamento de F'ısica - ETSIDI, Universidad Polit'ecnica de Madrid, 28012 Madrid, Spain \n295 Department of Electronic Control Engineering, National Institute of Technology, Nagaoka College, 888 Nishikatakai, Nagaoka City, Niigata 940-8532, Japan \n- 296 Departamento de Matem'atica da Universidade de Aveiro and Centre for Research and Development in Mathematics and Applications, 3810-183 Aveiro, Portugal \n297 Marquette University, Milwaukee, WI 53233, USA \nFaculty of Science, Toho University, 2-2-1 Miyama, Funabashi City, Chiba 274-8510, Japan \nIndian Institute of Technology, Palaj, Gandhinagar, Gujarat 382355, India \n298 299 \n300 Laboratoire MSME, Cit'e Descartes, 5 Boulevard Descartes, Champs-sur-Marne, 77454 Marne-la-Vall'ee Cedex 2, France \n301 \nGraduate School of Science and Technology, Gunma University, 4-2 Aramaki, Maebashi, Gunma 371-8510, Japan \n302 Institute for Quantum Studies, Chapman University, 1 University Dr., Orange, CA 92866, USA \nFaculty of Information Science and Technology, Osaka Institute of Technology, 1-79-1 Kitayama, Hirakata City, Osaka 573-0196, \nJapan \n304 INAF, Osservatorio Astrofisico di Arcetri, I-50125 Firenze, Italy \n305 Indian Institute of Technology Hyderabad, Sangareddy, Khandi, Telangana 502285, India \n306 Indian Institute of Science Education and Research, Pune, Maharashtra 411008, India \nInstitut fur Theoretische Physik, Johann Wolfgang Goethe-Universitat, Max-von-Laue-Str. 1, 60438 Frankfurt am Main, Germany \n308 \nIstituto di Astrofisica e Planetologia Spaziali di Roma, 00133 Roma, Italy \n309 INAF, Osservatorio di Astrofisica e Scienza dello Spazio, I-40129 Bologna, Italy \n310 Universidade Estadual Paulista, 01140-070 Campinas, S˜ao Paulo, Brazil \n- 311 Research Center for Space Science, Advanced Research Laboratories, Tokyo City University, 8-15-1 Todoroki, Setagaya, Tokyo 158-0082, Japan\n- 312 Department of Physics, Kyoto University, Kita-Shirakawa Oiwake-cho, Sakyou-ku, Kyoto City, Kyoto 606-8502, Japan\n- 313 Institute for Cosmic Ray Research, Research Center for Cosmic Neutrinos, The University of Tokyo, 5-1-5 Kashiwa-no-Ha, Kashiwa City, Chiba 277-8582, Japan\n- 314 Yukawa Institute for Theoretical Physics (YITP), Kyoto University, Kita-Shirakawa Oiwake-cho, Sakyou-ku, Kyoto City, Kyoto 606-8502, Japan \n315 University of Catania, Department of Physics and Astronomy, Via S. Sofia, 64, 95123 Catania CT, Italy \n- 316 Dipartimento di Scienze Aziendali - Management and Innovation Systems (DISA-MIS), Universit'a di Salerno, I-84084 Fisciano, Salerno, Italy \n317 National Institute of Technology, Fukui College, Geshi-cho, Sabae-shi, Fukui 916-8507, Japan \n- 318 Department of Communications Engineering, National Defense Academy of Japan, 1-10-20 Hashirimizu, Yokosuka City, Kanagawa 239-8686, Japan \n319 Eindhoven University of Technology, 5600 MB Eindhoven, Netherlands \n- 320 Department of Physics and Astronomy, Sejong University, 209 Neungdong-ro, Gwangjin-gu, Seoul 143-747, Republic of Korea 321 Concordia University Wisconsin, Mequon, WI 53097, USA \n322 School of Physics and Technology, Wuhan University, Bayi Road 299, Wuchang District, Wuhan, Hubei, 430072, China", 'ABSTRACT': 'We present results from a search for X-ray/gamma-ray counterparts of gravitational-wave (GW) candidates from the third observing run (O3) of the LIGO-Virgo-KAGRA (LVK) network using the Swift Burst Alert Telescope ( Swift -BAT). The search includes 636 GW candidates received in low latency, 86 of which have been confirmed by the offline analysis and included in the third cumulative Gravitational-Wave Transient Catalogs (GWTC-3). Targeted searches were carried out on the entire GW sample using the maximum-likelihood NITRATES pipeline on the BAT data made available via the GUANO infrastructure. We do not detect any significant electromagnetic emission that is temporally and spatially coincident with any of the GW candidates. We report flux upper limits in the 15-350 keV band as a function of sky position for all the catalog candidates. For GW candidates where the Swift -BAT false alarm rate is less than 10 -3 Hz, we compute the GW-BAT joint false alarm rate. Finally, the derived Swift -BAT upper limits are used to infer constraints on the putative electromagnetic emission associated with binary black hole mergers.', '1. INTRODUCTION': 'The discovery of gravitational waves (GWs) from coalescing binary black holes (BBH) by the Laser Interferometer Gravitational-Wave Observatory (LIGO) opened \na new window to the Universe (Abbott et al. 2016a). In addition to GWs, compact binary mergers with at least one neutron star (NS) component are likely to generate electromagnetic (EM) radiation (e.g., Nakar 2020a; Kyutoku et al. 2021). Coincident detection of EM emission from compact binary mergers provides a complete picture of the merger process and can have huge implications for our understanding of the Universe. Such coincidences play a crucial role in tracing the properties of the source host galaxy (Troja et al. 2017; Alexander et al. 2017), mitigating degeneracies in GW parameter estimation (Abbott et al. 2017a; Hughes & Holz 2003; Wang & Giannios 2021), placing constraints on the NS equation of state (Bauswein et al. 2017; Radice et al. 2018), and investigating the expansion rate of the Universe, thereby testing cosmological models (Abbott et al. 2017a,b; Hotokezaka et al. 2019; Nissanke et al. 2013; Schutz 1986). Additionally, they allow for the measurement of arrival time differences between photons and gravitons, providing limits to the mass of a graviton, exploring potential violations to the equivalence principle and Lorentz invariance (Abbott et al. 2017e). \nThe joint detection of the first GW event consistent with a binary NS (BNS) coalescence GW170817 (Abbott et al. 2017a), and a coincident short gamma-ray burst GRB 170817A (Goldstein et al. 2017a; Savchenko et al. 2017), accompanied by the optical/infrared kilonova counterpart AT 2017gfo (Arcavi et al. 2017; Coulter et al. 2017; Tanvir et al. 2017; Evans et al. 2017; Pian et al. 2017; Smartt et al. 2017; Drout et al. 2017; Cowperthwaite et al. 2017) and the GRB afterglow (in the X-rays: Troja et al. 2017; Margutti et al. 2017 and radio: Hallinan et al. 2017), together ushered in a new era in the field of multi-messenger astrophysics and forever impacted our comprehension of compact binary coalescences (CBC) involving an EM counterpart. Massive coordinated EM follow-up efforts were dedicated to deeply monitor the error regions derived from the joint sky localization of GW detectors and high-energy satellites, helping to reduce the initial three detector network sky localization from 28 deg 2 to within a few arcseconds of the host galaxy NGC 4993 (Abbott et al. 2017d,a). The spectacular spectral and light curve evolution of this transient (Abbott et al. 2017d; Villar et al. 2017) suggested that this explosive event was an active site for r-process nucleosynthesis (Pian et al. 2017; Smartt et al. 2017; Coulter et al. 2017; Drout et al. 2017) (for a detailed review of the multimessenger observations of GW170817, see, e.g., Nakar 2020b; Margutti & Chornock 2021). \nThe expected EM counterpart emission from BNS or neutron star-black hole (NSBH) mergers can poten- \ntially be weak due to various factors such as considerable source distances, an off-axis viewing angle, or limited amount of ejected mass. For the specific case of GW170817, despite a coincident GRB detection, it took nearly half a day to localize the host galaxy and begin observations of the kilonova (Abbott et al. 2017a). Prompt targeted searches around the GW trigger times, leveraging facilities with enhanced localization capabilities, can refine search strategies and assist optical or infrared (IR) facilities in correctly identifying and pursuing transient candidates for subsequent follow-up studies. In addition to prompt searches, Fermi -GBM analysis of triggers from the first and second LIGO-Virgo observing runs showed that targeted offline searches are capable of recovering additional candidate joint events that may be of astrophysical relevance (Hamburg et al. 2020; Pillas et al. 2023). Temporal and spatial coincidence information can be used to derive the joint false alarm rate (FAR). These estimates have the potential to elevate subthreshold triggers in either the GW or GRB domains to the status of an above-threshold candidate detection (Nitz et al. 2019). \nUnlike Fermi , Swift has been for a long time incapable of relaying a continuous stream of event mode data to the ground in real time. Such a capability was enabled through GUANO (Gamma-ray Urgent Archiver for Novel Opportunities, described in Section 2) (Tohuvavohu et al. 2020), which recovers event data from the Swift Burst Alert Telescope (BAT, Barthelmy et al. 2005), that then get processed by the Non-Imaging Transient Reconstruction And TEmporal Search (NITRATES, DeLaunay & Tohuvavohu 2022) pipeline (see Section 4) to search for subthreshold transient candidates. 1 In addition to other astronomical transients, such as GRBs, fast radio bursts (FRBs), and highenergy neutrinos, the GUANO-NITRATES infrastructure performs targeted searches on GW events communicated by the LVK Collaboration, to detect possible GRBs associated with CBCs. \nThe impact and potential of Swift -BAT subthreshold searches are crucial for multi-messenger related goals. Indeed, deeper targeted searches increase the joint detection horizon, thus enhancing the probability of finding weak EM counterparts of CBCs in the hard X-ray domain. Moreover, thanks to the high spatial accuracy enabled by the BAT coded mask, subthreshold searches open the possibility of recovering the position of the \ncandidate EM event at the precision level of a few arcminutes, fundamental to drive the subsequent follow-up with ground and space-based EM facilities. \nCurrently, the targeted search analysis carried out thanks to GUANO, has enabled the discovery of more than 35 GRBs with arcminute localization. A total of 7 of the detected GRBs have a duration < 2 s (e.g., DeLaunay et al. 2020; Tohuvavohu et al. 2022a), hence they are potentially associated with CBCs containing at least one NS. GUANO data have also been used for the localization of 29 long GRBs through imaging (e.g., DeLaunay et al. 2021a) and non-imaging analysis techniques (e.g., DeLaunay et al. 2021b; Tohuvavohu et al. 2022b). GRB 220107A, detected during BAT slew and localized with arcminute precision, enabled the first optical redshift measured using GUANO data (DeLaunay et al. 2022a). The arcminute localization of GRB 211106A enabled prompt multiband follow up and led to the discovery of the first afterglow in the millimeter band from a short GRB (Tohuvavohu et al. 2021a). With GUANO, one can additionally recover coarse localization information on GRB-like transients that originate from outside the BAT field of view (FOV; e.g., DeLaunay et al. 2023). \nIn addition to the application to real-time analysis, the availability of BAT data enables us to perform a systematic, deeper targeted search focused on archival LVK triggers. The goal of this study is to perform such an analysis on all the LVK triggers received during the third LIGO-Virgo observing run, during which the GUANO pipeline started to be fully operational. The run duration was comprised of two segments: O3a, which operated from April 1, 2019, 15:00 UTC to October 1, 2019, 15:00, and O3b which operated from November 1, 2019, 15:00 UTC, to March 27, 2020, 17:00 UTC. The alerts distributed during O3 were reporting the following parameters: FAR, the signal classification (CBC or unmodeled Burst), and the associated astrophysical probabilities. The results of O3 are summarized in Gravitational-Wave Transient Catalog data releases GWTC-2 (Abbott et al. 2021), GWTC-2.1 (Abbott et al. 2024), and GWTC-3 (Abbott et al. 2023). \nIn this work, we use Swift -BAT observations to carry out offline targeted subthreshold searches for EM counterparts of the GW triggers obtained during O3. The rest of the paper is organized as follows: In Section 2, we describe the Swift -BAT instrument and its new capabilities, and in Section 3, we provide details about the GW trigger sample used for the analysis. In Section 4 we summarize the targeted search method adopted for the analysis. We present the results from our targeted search analysis on the various subcategories of triggers \nin Section 5, and discuss the scientific interpretation in Section 6.', '2. Swift -BAT': 'The Neil Gehrels Swift Observatory (henceforth, Swift ) is a GRB-focused mission launched in 2004, with three onboard payloads - the Burst Alert Telescope (BAT), the X-ray Telescope (XRT), and the UltraViolet/Optical Telescope (UVOT) - covering the EM spectrum from hard X-rays and gamma-rays all the way to the optical (Gehrels et al. 2004). The BAT instrument (Barthelmy et al. 2005) is a hard X-ray coded mask imager with a wide FOV that operates in the broad energy band of 15-350 keV. It is the primary instrument that detects GRBs and performs an onboard imaging analysis via a cross-correlation between the spatial pattern of the counts across the detector array and the pattern of the coded mask. The sensitivity of BAT is capable of providing arcminute localizations of GRBs triggered onboard (Gehrels et al. 2004). Due to the lack of continuous downlinking of timing, spatial, and energy information for each detector count (event mode data), carrying out targeted searches offline has not been possible in the past. A new infrastructure, called GUANO, was incorporated into the Swift -BAT operations in 2019. Details of the GUANO operations can be found in Tohuvavohu et al. (2020). From the outset, GUANO has demonstrated that recovering event mode data from astrophysically compelling time windows can enhance the overall transient detection rate and sensitivity of BAT (Tohuvavohu et al. 2020).', '3. GRAVITATIONAL-WAVE TRIGGER SAMPLE': "This paper focuses on the Swift -BAT subthreshold analysis of a sample of GW triggers with a FAR < 2 per day, distributed by the LVK Collaboration during O3. The subthreshold GW alerts were received by the EM follow-up groups that were part of a Memorandum of Understanding with the LVK Collaboration. For candidates found with CBC search pipelines (Dal Canton et al. 2021; Sachdev et al. 2019; Messick et al. 2017; Aubin et al. 2021; Nitz et al. 2018; Hooper et al. 2012) and Burst search pipelines (Klimenko et al. 2005, 2016), the alerts contain basic information about the GW FAR, the probability of the candidate being astrophysical ( p astro ) and trigger time. In the case of CBC candidates, the alerts received in low latency report the p astro split in the four CBC classes: BBH, BNS, NSBH, and Mass Gap. The Mass Gap category includes CBC candidates in which at least one component has a mass in the range [3-5] M ⊙ . Using the GW trigger information received in low latency, we can further perform a search for associations in BAT data. \nFrom the list of 1552 alerts that were communicated via low latency channels, we obtained successful GUANO data dumps for 636 triggers. The GW information of the candidates received in low latency are reported in Table 1. The FAR and the p astro classification reported here correspond to the preferred event, namely the one with the highest SNR. Swift -BAT event mode data coincident with the GW trigger time were made available for these triggers in real time via the GUANO data dumps. Post-processing on the data was carried out on this sample from O3 using the NITRATES analysis pipeline (see Section 4). \nOut of the 636 low-latency alerts, a total of 86 GW candidates have been confirmed by the offline analysis and included in the GWTC-2.1 and GWTC-3 data releases (Abbott et al. 2024; Abbott et al. 2023). Among the 86 confirmed candidates, 14 triggers have p astro > 0 . 5 and 72 triggers have p astro < 0 . 5. We indicate the details of the confirmed candidates with p astro > 0 . 5 and p astro < 0 . 5 in Table 2 and Table 3, respectively. The values of FAR, p astro and CBC Class given in Table 2 and Table 3 are derived from offline analyses, as reported in GWTC-2.1 and GWTC-3 data releases, hence are considered more reliable than the values obtained in low latency, reported in Table 1. The FAR and the p astro classification reported in Tables 2 and 3 come from the pipeline that gives the highest value of p astro . For high significance events, if multiple pipelines derive a p astro ≃ 1, we select the one with highest SNR. According to these rules, in the case of S200225q, reported in Table 2, the selected pipeline is cWB, but the event is classified as CBC. The CBC class is determined by the highest among p BBH , p NSBH and p BNS . In Tables 2 and 3 we report the value of p Class defined as max[ p BBH , p NSBH , p BNS ]. It is possible that some candidates marked as 'Burst' in Table 1 are then re-classified as 'CBC' by the offline analysis. Therefore, for each catalog event the most updated group is the one reported in Tables 2 and 3. \nIn Figure 1 we show the histograms of the p astro probabilities for all the low-latency CBC candidates processed by GUANO, for a total of 424 candidates, divided into 67 BBH, 130 BNS, 148 NSBH, and 79 Mass Gap events (Fig. 2, left panel). In the offline postprocessing of GW candidates, the Mass Gap classification was removed, classifying an object with M > 3 M ⊙ as a black hole. This implies that all the CBC events with at least one component mass in the range [3 -5] M ⊙ are re-distributed into the BBH and NSBH classes. The distribution of the updated p astro classification released by LVK is over-plotted in Figure 1, and the classes are \ndivided in 39 BBH, 22 BNS and 17 NSBH candidates (Fig. 2, right panel).", '3.1. GW sky localization': "For the selection of the GW sky localization for each candidate we adopt the following scheme: \n- 1. Above-threshold GW candidate: The GW candidate is contained in the list of high-probability ( p astro > 0 . 5) candidates reported in Table 2 of GWTC-2.1 (Abbott et al. 2024) or Table 1 of GWTC-3 (Abbott et al. 2023). The GW sky localizations are downloaded from the parameter estimation data releases of GWTC-2.1 and GWTC3. 2 We use the results derived from a combination of IMRPhenomXPHM (Pratten et al. 2021) and SEOBNRv4PHM (Ossokine et al. 2020) waveforms (labeled as 'Mixed' in the release).\n- 2. Subthreshold GW candidate: The GW candidate is classified as low-probability ( p astro < 0 . 5) in the offline analysis. The GW sky localization is produced by BAYESTAR (Singer & Price 2016; Singer et al. 2016) and is taken from the GWTC-3 release, which contains both O3b events and updated O3a events. If multiple events are present for a single GW candidate, the pipeline with the highest p astro is considered for the selection of the sky localization.\n- 3. Non-confirmed low-latency GW candidate: The GWcandidate has an associated low-latency alert, but the event has not been confirmed by the offline analysis. The GW sky localization is downloaded from GraceDB, selecting the preferred event.", '4. TARGETED SEARCHES USING GUANO-NITRATES': 'Targeted searches are carried out in real time for all types of transients such as GRBs, FRBs, neutrino events as well as GW triggers, on the event mode data obtained using GUANO. The targeted search pipeline that is currently operational is NITRATES. This is a maximum-likelihood framework that forward models signals through the entire instrument response (DeLaunay & Tohuvavohu 2022). The BAT responses are created by simulating the photon paths through all detector segments using Geant4, which is a particle-interaction simulator software toolkit (Allison et al. 2016). Unlike the standard BAT responses, the NITRATES responses \n40 \n35 \nReceived in low-latency \nConfirmed by offline analysis \n<!-- image --> \n<!-- image --> \n30 \n1 \n. \n5 \nlog10[P(BBH)] \n<!-- image --> \n<!-- image --> \nFigure 1. Distribution of the p astro values for the CBC triggers detected during O3, with available GUANO data dumps. We distinguish with different colors the triggers received in low latency and the ones confirmed by offline analysis. \n<!-- image --> \nFigure 2. Left: distribution of the CBC triggers from O3 received in low latency, which had successful GUANO data dumps, divided in the BBH, NSBH, BNS, and Mass Gap classes. Right: analogous distribution for the CBC candidates confirmed by the offline analysis. In the post-processing, the Mass Gap classification has been subsumed into BBH and NSBH. \n<!-- image --> \n- \n1 \n. \n0 \n<!-- image --> \naccount for all the detectors on the focal plane, regardless of their coding by the mask. The responses also encode details on the gamma-ray interactions inside the instrument, which carry additional information. This approach enables substantial sensitivity gain compared to the conventional technique of cross-correlation imaging, which translates to a 50% increase in the detection horizon distance for a GRB 170817A-like burst compared to the onboard imaging. Details of the NITRATES response generation and the analysis pipeline are provided in DeLaunay & Tohuvavohu (2022). \nA GRB-like transient signal is described using the sky localization and parameters that are specific to the assumed spectral model (the peak energy and spectral slope). This framework then computes the significance of each signal using a test statistic (TS) by maximizing the log-likelihood (LLH) as a function of signal parameters. The likelihood ratio test statistic Λ is used to compare the source signal+background model (described using a set of parameters, Θ sig that maximizes the LLH) to a background-only model (described by the set of parameters Θ off bkg , that maximizes the LLH in the off-time window) and is defined as follows (DeLaunay & Tohuvavohu 2022): \nΛ = -2[LLH(Θ off bkg | N on ) -LLH(Θ sig , Θ off bkg | N on )] , (1) \nwhere N on is on-time data. \nThe search pipeline workflow can be summarized as follows: \n- 1. The event mode data is cleaned and filtered to discard potential glitches and artifacts from cosmic rays, and to flag poorly behaving detectors. Good time intervals (GTIs) are determined, where there is quality data for the analysis.\n- 2. A time window of 50 s (from the pre- and posttrigger intervals) is identified as the background interval. It is then utilized to model contributions to the background from known bright sources and diffuse sources.\n- 3. To narrow down the search parameter space, a set of simple analyses are performed to select a list of interesting start times and durations (hereafter referred to as time seeds) as well as portions of the BAT FOV (position seeds).\n- 4. Finally, the log-likelihoods are computed for all parameters corresponding to the shortlisted time and position seeds. \nIn essence, the set of signal parameters that maximizes the log-likelihood is the most preferred set of parameters. \nThe NITRATES likelihood analysis outperforms the onboard mask-weighted imaging analysis by delivering superior sensitivity, given the increased effective area (see Fig. 2 in DeLaunay & Tohuvavohu 2022). At the cost of a significantly increased computational time, this method is capable of delivering arcminute scale localization for events that fall inside the BAT FOV, even when the transient event does not trigger Swift -BAT onboard (Tohuvavohu et al. 2021b; Tohuvavohu 2023; DeLaunay et al. 2022b). The NITRATES pipeline has the ability to distinguish between bursts that come from in and outside the BAT FOV. NITRATES has also accurately localized sufficiently bright bursts outside the FOV (DeLaunay & Tohuvavohu 2022). \nSwift -BAT GUANO was operating during the O3 and was successfully procuring event mode information in response to GW subthreshold triggers (Tohuvavohu et al. 2020). We describe the targeted search analysis that has been carried out using the NITRATES version 0.0.1 which was available in early 2022. 3 The targeted search analysis that was operational in O3 corresponded to a preliminary version of the NITRATES code, that has since undergone several stages of development. The most updated version is publicly available on GitHub. 4 \nDuring O3, for a total of 636 GW triggers, GUANO dumped either 200 s or 90 s of event mode data, for public triggers and for privately communicated triggers, respectively. The choice of the width of the temporal window is made to avoid an overload of downlink data in the process of GUANO data dump. The targeted search pipeline was run in a time window of ± 20 s centered around the trigger time. The search was carried out on 8 time bins (0.128 s, 0.256 s, 0.512 s, 1.024 s, 2.048 s, 4.096 s, 8.192 s and 16.384 s) and 9 energy bins (between 15-350 keV). The results from the search are reported using the following set of parameters: 1) the maximum √ TS describes the statistical significance of a potential detection (see Section 5); 2) ∆LLH out indicates the preference of the search to a location inside or outside the BAT FOV, and 3) ∆LLH peak indicates the confidence of the search in localizing the source to arcminute scales. \nThe NITRATES search was performed on the ROAR supercomputing cluster on a set of 200 virtual cores for a total of ∼ 600 × 800 CPU hours for the entire GW sample.', '5. RESULTS FROM NITRATES': 'The targeted search analysis provides a list of top candidates whose spatial, temporal, and spectral parameters maximize the log-likelihood. In order for a candidate to be qualified as a confident detection, we require that the resulting detection significance parameter √ TS must exceed the threshold value of 8, corresponding to a FAR ∼ 4 × 10 -5 Hz. Being a targeted search, the NITRATES analysis can give a false positive with √ TS > 8 with a probability which follows a Poissionian distribution: \nP ( N det ≥ 1) = 1 -P ( N det = 0) = 1 -e -FAR × ∆ t , (2) \nwith ∆ t = 40 s being the width of the search window. This leads to a pre-trial p-value of 1 . 6 × 10 -3 . Since the NITRATES analysis is performed on all GW triggers with a FAR < 2 day -1 , and considering that there are N GW -search = 5 independent GW pipelines, the rate of expected false positive candidates with √ TS > 8 is ∼ 5 × 2 × 1 . 6 × 10 -3 /day ∼ 1/(60 day). 5 \nFor the entire sample of 636 low-latency triggers processed using NITRATES, we have no candidates that qualify as detection of a signal of astrophysical origin. None of the top candidates within the ± 20 s search window are coincident with the GW triggers. A temporal coincidence with a GW trigger is claimed if the NITRATES search finds a candidate with √ TS > 8 and | t 0 -t start | < 20 s, where t 0 is the GW trigger time and t start is the starting time of the temporal bin with highest ranking statistics. A detailed list of all the NITRATES results for the entire sample analyzed during O3 is provided in Table 1. We discuss specific false positive candidates in Section 5.1. \nIf the GW trigger time is included in the time window corresponding to slew mode of BAT, the analysis cannot be performed using NITRATES since the targeted search requires stable attitude information to compute the background. Similarly, some triggers have insufficient exposure time, preventing the NITRATES analysis. In this case neither TS results nor flux upper limits can be computed. As a cut to narrow down the parameter space, the targeted search selects time seeds as described in Section 4. If there are no time seeds that pass the preliminary cuts then there will be no final likelihood computations. Results for these types of triggers are indicated as NFL (No Final Likelihood) in Table 1. In the case of NFL triggers, though, the flux upper limit can be computed, since a full likelihood analysis is not required.', '5.1. False positives': 'We did not find any candidate associations from any of the triggers with BAT. However, the targeted search pipeline did result in the detection of six candidates with a significance above the NITRATES detection threshold of √ TS = 8. These candidates were examined to understand our false positive population. S200327j ( √ TS ∼ 22), S200324ax ( √ TS ∼ 11), and S200225af ( √ TS ∼ 10 . 5) are triggers that occurred during the passage of Swift in the proximity of the South Atlantic Anomaly (SAA). The background characterization becomes unreliable when the spacecraft is either entering or leaving the SAA on account of increased background contamination. In S190919au, a peculiar dip ( ∼ 20 s) in the background may have contributed to a false elevation in the signal detection statistic, by causing an under representation of the background rate. For S190919u, we obtain a √ TS ∼ 8 . 0, which corresponds to a FAR of ∼ 4 × 10 -5 Hz. The presence of such a detection, in a total sample of 636 triggers that were analyzed, is compatible with the expected number of false positives, which corresponds to (4 × 10 -5 Hz) × (40 s) × 636 ∼ 1. \nS200130ai corresponds to a subthreshold GW trigger at T 0 = 2020-01-30T09:59:58 that was identified by the CBC search as a NSBH candidate with a p astro = 0 . 008 and a GW FAR ∼ 1 . 8 × 10 -5 Hz. It was detected using NITRATES at a significance of √ TS ∼ 16 . 3 with a ∆LLH out = -19 . 68 and ∆LLH peak = 2 . 14, consistent with a sky localization outside the BAT-FOV. The highest log-likelihood candidate, was identified to arise 1.5 s prior to the GW trigger time. Due to the low value of ∆LLH peak , we do not have an arcminute-level precision on the sky localization. The candidate was associated with a Fermi trigger 602071201 (GCN 26944, Fermi GBM Team 2020) and was classified as a long GRB. The Fermi localization is RA = 137 . 5 deg, Dec = -51 . 3 deg, with a statistical uncertainty of 3.5 degrees. The Interplanetary Gamma-Ray Burst Timing Network (IPN) further localized the event in a 3-sigma error box with an area of 1487 arcmin 2 and centered at an RA = 134 . 742 deg and Dec = -49 . 627 deg (Hurley et al. 2020). Although this event presents a temporal coincidence with the GW trigger, on account of the lack of spatial coincidence with the GW location, this event is discarded from being associated with the GW subthreshold trigger. Additionally, this low-latency GW candidate has not been confirmed by offline analyses.', '5.2. Computation of flux upper limits': 'Since each GW trigger processed in this analysis resulted in a non-detection in Swift -BAT, we estimate the flux upper limits in the following manner. The \nFigure 3. Flux upper limit maps are shown for all the O3 catalog events with a p astro > 0 . 5 that were processed successfully using NITRATES. The color bar indicates the upper limit in the 15-350 keV Swift -BAT band as a function of the sky position. The part of the sky in white corresponds to the area covered by the Earth. The solid and dashed contours are the GW 90% and 50% credible levels, respectively. \n<!-- image --> \nNITRATES analysis generates rates curves in the 15350 keV energy band from the GTIs of the filtered event list. The number of active detectors corresponding to each trigger is read out from its respective detector mask file. A linear fit is carried out to the background window of duration 50 s. We then estimate the 5 σ count rate and the corresponding uncertainty over the full signal window which has a ± 20 s duration. This is computed for all the 8 time bins (see Section 4). We further convert the 5 σ count rates to flux upper limits, as a function of sky position, in the 15-350 keV band, by convolving different spectral models with the NITRATES responses for each time bin iteration. We select 929 grid points on the sky and interpolate upper limit values for locations in between. We assume the following different spectral templates: \n- 1. Band function (Band et al. 1993) with a soft template ( E peak = 70 keV, α = -1 . 9, β = -3 . 7)\n- 2. Band function with a normal template ( E peak = 230 keV, α = -1.0, β = -2.3)\n- 3. Cutoff power law function with a hard template ( E peak = 1500 keV, α = 1.5)\n- 4. Cutoff power law function that has been used to describe GRB 170817A ( E peak = 185 keV, α = 0.62) (Goldstein et al. 2017b) \nThe parameters α and β correspond to the low-energy and high-energy photon indices of the spectrum, respectively. The first three spectral templates are identical to the ones that are routinely adopted by Fermi -GBM (Goldstein et al. 2016a). In the rest of the paper, all the \nFigure 3. (continued) \n<!-- image --> \nresults are reported assuming a Normal spectral template. \nCalling Ω = (RA , Dec) the coordinates variable, for each temporal bin and spectral template we convert the upper limit map ϕ UL (Ω) into a unique marginalized upper limit value: \nϕ UL = ∫ Ω / ∈ Ω ⊕ ϕ UL (Ω) P GW (Ω)dΩ , (3) \nwhere P GW (Ω) is the posterior probability distribution of the GW sky position. The notation Ω / ∈ Ω ⊕ means that the integral is limited to the region of the sky not occulted by the Earth. We report in Table 1 the marginalized flux upper limits for a 1 s time bin, assuming the normal spectral template. In Fig. 3, we provide the sky maps reporting both the flux upper limits as a function of sky position and the GW contours (50% and 90% credible levels) for the GW candidates with p astro > 0 . 5. \nAs additional information, we also report the quantity ε in BAT , which quantifies the probability that the GW source is inside the BAT coded FOV and corresponds to \nε in BAT = ∫ Ω ∈ Ω in P GW (Ω)dΩ , (4) \nwhere the integral is limited to the solid angle Ω in , namely the portion of the sky where the BAT partial coding fraction is larger than 0.01. The location of Ω in , i.e., the BAT FOV, is identified by the yellow region in the sky maps of Figure 3. The higher the ε in BAT , the better the BAT covering of the GW error region, and the more constraining the derived upper limit. The flux upper limits as a function of ε in BAT for all GW trigger candidates is shown in Fig. 4. We also indicate with different markers the sample of low-latency triggers, the confirmed list of subthreshold candidates ( p astro < 0 . 5), and the above-threshold candidates ( p astro > 0 . 5). In Table 1 we also report the probability that the GW \n⊕ \n0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 ε in BAT Figure 4. The 15-350 keV flux upper limit ϕ UL derived with NITRATES as a function of ε in BAT , namely the probability that the GW candidate is contained inside the BAT FOV. The plot includes all the GW candidates received during O3, including both Burst and CBC events. With different symbols we distinguish the GW candidates received in low latency and the ones confirmed and included in the O3 catalog, separated in p astro > 0 . 5 and p astro < 0 . 5. The dashed and solid red lines are the 50% and 90% containment regions of the scatter plot, respectively. The one-dimensional histograms of ϕ UL and ε in BAT are reported on the sides. The color bar indicates the value of ε ⊕ , the probability that the GW candidate is occulted by the Earth. \n<!-- image --> \nsource is occulted by the Earth, defined as: \nε ⊕ = ∫ Ω ∈ Ω ⊕ P GW (Ω)dΩ , (5) \nwhere Ω ⊕ is the solid angle subtended by the Earth from the Swift reference system.', '5.3. Computation of luminosity upper limits': 'We further convert the flux upper limits into luminosity upper limits for all the GW triggers with available information about the distance posterior distribution, namely only triggers identified by CBC searches. The luminosity upper limit in the rest frame band 1 keV10 MeV is estimated as \nL UL = ⟨ 4 πD 2 L kϕ UL ⟩ , (6) \nwhere D L is extracted from the posterior probability P ( D L ) reported in the GW sky localization files, while k is the k-correction and corresponds to \nk = I [1 keV / (1 + z ) , 10 MeV / (1 + z )] I [15 keV , 350 keV] , (7) \nwhere \nI [ a, b ] = ∫ b a E d N d E ( E )d E, (8) \nand d N /d E is the assumed photon spectrum. The band 1 keV-10 MeV is chosen since it is usually adopted to report the bolometric luminosity of GRBs. \nThe luminosity upper limits as a function of the mean value of the luminosity distance is reported in Figure 5. Similar to what was shown previously, we demarcate the various samples. Candidates with a low latency classification of Mass Gap were later re-distributed to other categories as part of post-processing, which is evident from the Mass Gap panel in Figure 5. As expected and as evident already from Figure 4, we see a clear correlation between the luminosity upper limit and ε in BAT in Figure 5, indicating that the inferred constraints on the EM counterpart are more stringent when the GW probability integrated inside the BAT FOV is higher. Since ϕ UL is an upper limit and not a measure coming from a detection, Eq. (6) is an approximated method to convert ϕ UL in a luminosity upper limit, averaging over the P ( D L ) distribution provided by the GW analysis. In Appendix A we show a more accurate way to estimate the luminosity upper limit, but we find no relevant differences with respect to the method reported in this section. The Eq. (6) is used only to produce the plots of Fig. 5, but this approximation is not used \nin Section 6 to perform inference about the EM model parameters. Instead, in Section 6 a reverse process is followed, namely the EM model is used to predict the luminosity, which is then convolved with P ( D L ) to obtain a probability distribution of the flux in the BAT energy band.', '5.4. Computation of joint FAR': 'To calculate the joint GW-BAT FARs for each GW trigger, we elaborate on the methods used to compute the individual FARs and subsequently combine them. To derive the sensitivity of the NITRATES search, timetagged event data assembled from intervals corresponding to calibration runs and data from before and after known GRB signal times (total exposure time of ∼ 51 ks) were analyzed. The behavior of the background population and its associated FAR were then identified (see Section 7.6 and Fig. 16 in DeLaunay & Tohuvavohu 2022). We further compute the joint Swift -BAT-GW FAR by combining the BAT FAR (calculated using the method described above) with the GW FAR. A targeted joint FAR threshold routine is constructed as part of the Rapid, on-source VOEvent Coincident Monitor (RAVEN; Abbott et al. 2017c; Urban 2016), which combines the FARs obtained from GWs along with those from BAT and computes the joint temporal as well as the joint spatial FAR. We also specify details of the search pipeline used in the process, Burst or CBC. The joint FAR prescription as reported in the RAVEN documentation, 6 is computed as \nFAR GRB+GW = Z I Ω [ 1 + ln( Z max Z ) ] (9) \nwhere Z is the joint ranking statistic given by, \nZ = FAR GW FAR GRB ∆ t, (10) \nZ max = FAR GW , max FAR GRB , max ∆ t, (11) \nand we adopt ∆ t = 30 s, FAR GW , max = 2 day -1 and FAR GRB , max = 10 -3 Hz. I Ω is an integral that quantifies the spatial overlap between the GW localization and the GRB localization (Ashton et al. 2018). Even if the search of subthreshold candidates in NITRATES is done in a temporal window [ t 0 -20 s , t 0 + 20 s] around the trigger time t 0 , for the RAVEN joint alert the adopted temporal window is [ t 0 -10 s , t 0 + 20 s]. Since none of the BAT candidates analyzed in this work has a confident estimation of the sky localization, we adopt a uniform posterior probability on the full sky \nfor the EM candidate. Hence, by definition, we set I Ω = 1. The candidate triggers a RAVEN alert when the FAR GRB+GW × N t < FAR max , with N t being the trials factor of the joint search and FAR max = (1/30) day -1 for CBC events and FAR max = 1 yr -1 for Burst events. The trials factor corresponds to N t = S GW ( S GW -1), where S GW is the number of search GW pipelines, 4 for CBC events and 3 for Burst events. Since the GW pipelines are not fully independent, a realistic value of the trials factor is smaller than the one adopted here, therefore the RAVEN threshold can be considered as a conservative estimate for the significance of a joint detection. \nWe quote the derived joint FARs along with other trigger-specific details only for those triggers with a FAR GRB , max < 10 -3 Hz in Table 4. We find that, after rejecting false positives, 2 CBC events pass the joint FAR detection threshold to trigger a RAVEN alert. Specifically, S191110x ( √ TS = 7 . 2) and S200108p ( √ TS = 7 . 4) have a joint FAR of 3 . 02 × 10 -4 yr -1 and 21 . 3 yr -1 , respectively. These values are obtained considering the GW FAR received with the low-latency alert. In the offline analysis of the GW candidates, neither S191110x or S200108p have been confirmed. We therefore conclude that, considering the offline joint analysis of GW and Swift -BAT data, none of the candidates is eligible to claim a significant joint detection. \nIn Fig. 6 we report the location in the GW FAR -√ TS plane of all the candidates that pass the condition FAR GW , max < 2 day -1 and FAR GRB , max < 10 -3 Hz (i.e., √ TS ≳ 7), to be considered for a potential joint alert. The astrophysical origin of all the candidates with √ TS > 8 has been rejected as discussed in Section 5.1, and therefore they are not reported in Fig. 6. The dashed black and red lines mark the separation line for the event to pass the RAVEN alert threshold, for CBC and Burst candidates, respectively. Candidates below those lines would have triggered a RAVEN alert.', '6. SCIENCE DISCUSSION': 'In this section, we describe how the upper limits derived from the joint subthreshold search can be used to infer constraints about possible EM emission from the GWcandidates. Starting from a model of the EM emission, the luminosity in the BAT band can be estimated, whose value will depend on some internal parameters of the model ( λ 1 , ..., λ k ). The goal is to explore the model parameter space and test if the estimated flux is in agreement with the upper limit constraints derived in this paper. \nFor this purpose, a knowledge of the distance of the GWcandidate is needed, and the GW sky localization is \n) \n1 \n- \ns \n(erg \nUL \nL \n10 \n10 \n10 \n10 \n2 \nFigure 5. Upper limits on the luminosity computed in the rest frame 1 keV-10 MeV energy band, as a function of the mean luminosity distance extracted from the sky localization map of each GW candidate. The color bar indicates the quantity ε in BAT , namely the probability that the GW candidate is located inside the BAT coded FOV. With different symbols, we distinguish the GW candidates received in low latency and the ones confirmed and included in the O3 catalog, separated in p astro > 0 . 5 and p astro < 0 . 5. \n<!-- image --> \n<!-- image --> \nMass Gap \n- Low-latency events\n- Catalog events with p astro < 0 . 5\n- Catalog events with p astro > 0 . 5 \nDL \n(Mpc) \n<!-- image --> \n10 \n<!-- image --> \nBNS \nused to extract the posterior distribution P ( D L ). Since only CBC events have such information, Burst events are not considered in this discussion. For the CBC events, we consider a phenomenological model which describes the probability distribution of the luminosity L (in the 15-350 keV rest-frame) \nP ( L ) = (1 -f ) δ ( L = 0) + f Π( L ) . (12) \nHere, the f parameter is a proxy for the EM-bright nature of the event, i.e., given a CBC source described by a set of GW parameters ⃗ θ GW , f ( ⃗ θ GW ) corresponds to the probability that the EM luminosity of the source is nonzero. On the other hand, Π( L ) is the intrinsic luminosity function of the EM transient associated with the specific CBC class. In the case of BNS and NSBH candidates, \nthe assumption on Π( L ) should be informed by our prior knowledge of the luminosity function of merger-driven GRBs. A detailed study of the impact of this work on our knowledge of merger-driven GRBs luminosity function will be reported in a follow-up paper. In this section, instead, we focus only on the BBH class, for which no strong prior exists for Π( L ). For simplicity and in order to show the constraining power of our joint subthreshold search, we assume that the EM process associated with BBH, if present, produces a universal, viewing angle-independent luminosity L 0 . Therefore, in the scenario specified above, we have ( λ 1 , ..., λ k ) = ( f, L 0 ) and \nP ( L ) = (1 -f ) δ ( L = 0) + fδ ( L -L 0 ) = P ( L ; f, L 0 ) . (13) \n49 \n48 \n47 \n3 \n0 \n. \n6 \n0 \n. \n4 \n0 \n. \n2 \n0 \n. \n0 \nT \nBA \nin \nε \nFigure 6. Distribution in the GW FAR-√ TS plane of all the triggers which passed the threshold FAR GRB < 10 -3 Hz. Triggers in the regions below the black and red dashed lines (marked with a green cross) would have triggered the RAVEN alert system, for CBC and Burst events, respectively. The plot does not include all the triggers that have a √ TS > 8 which turned out to be spurious artifacts. \n<!-- image --> \nOnce the model for the EM emission is specified, the probability distribution of the predicted flux is \nP ( ϕ ) = (1 -f ) δ ( ϕ = 0) + fP EM ( ϕ ) , (14) \nwhere P EM ( ϕ ) = P ( L 0 / 4 πkD 2 L ) is the flux probability distribution in the assumption that the source is EM bright. Hence, for the i -th candidate, the probability that the predicted flux is below the estimated upper limit ϕ 0 ,i corresponds to \nP i ( ϕ < ϕ 0 ,i ) = (1 -f ) + f ∫ ϕ 0 ,i 0 P i ( ϕ )d ϕ, (15) \nvalid in the limit in which the GW candidate is assumed to be real. Therefore, given a candidate GW with a probability of being astrophysical p astro ,i = π i , there are only three possibilities to have a non-detection in BAT: \n- 1. The source is not astrophysical, with a probability 1 -π i ;\n- 2. The source is astrophysical, but it is occulted by the Earth, with a probability π i ε ⊕ ;\n- 3. The source is astrophysical, it is not occulted by the Earth and the predicted flux by the EM model is below the BAT upper limit, with a probability π i (1 -ε ⊕ ) P i ( ϕ < ϕ 0 ,i ). \nThis allows us to define a non-detection likelihood corresponding to \nL i = (1 -π i ) + π i [ ε ⊕ +(1 -ε ⊕ ) P i ( ϕ < ϕ 0 ,i )] . (16) \nIn the case of L 0 → 0, P i ( ϕ < ϕ 0 ,i ) → 1, so L i → 1. For very large values of L 0 , instead, P i ( ϕ < ϕ 0 ,i ) → 0 and therefore L i → (1 -π i ) + π i ε ⊕ . This last result shows how, even if the luminosity predicted by the model is exceedingly large, a non-detection can occur if the GW source is not real (1 -π i ), or if it is real but occulted by the Earth ( π i ε ⊕ ). Since the analysis is focused only on BBHevents, we consider only those candidates that have p BBH > p NSBH , p BNS . By definition, p BBH + p NSBH + p BNS = p astro and typically for the candidates classified as BBH we have that p BBH ≫ p NSBH , p BNS . The last condition allows us to consider Eq. (16) still valid if we replace π i with p BBH ,i , since the contribution of p NSBH ,i and p BNS ,i to the non-detection probability is negligible. \nGiven the definition of Eq. (16), L i indicates the probability, given a set of ( λ 1 , ..., λ k ) EM parameters, that the BAT upper limit is not violated, taking into account the possible non-astrophysical origin of the candidate and also the probability that, even if astrophysical, the source is occulted by the Earth and therefore not detectable by Swift . Having a collection of E 1 , ..., E N GW candidates, the posterior distribution of the model parameters can be obtained following the Bayes theorem \nP ( L 0 , f | E 1 , ..., E N ) = N ∏ i =1 L i π ( L 0 ) π ( f ) / ∫ N ∏ i =1 L i π ( L 0 ) π ( f )d L 0 d f, \n(17) \nwhere π ( L 0 ) and π ( f ) are the prior distributions of L 0 and f . We assume a log-uniform prior for both L 0 and f in the respective intervals 46 < log 10 [ L 0 (erg s -1 )] < 53 and -3 < log 10 ( f ) < 0. The choice of the prior boundaries are poorly informed by theoretical expectations, which are still affected by large uncertainties. Instead, the priors are chosen on the basis of the typical range of upper limit luminosity derived in this work for BBH events and the total number of candidates considered in this analysis. The constraints reported in the following may strongly depend on the choice of the prior boundaries. Therefore, the final goal of this simulation, more than deriving strong limits on the putative EM model, is to show the predictive power of the present analysis in the context of model inference and how this analysis can improve with the addition of more GW events in the future. \nIn the specific case of our simulation, we consider all the GW candidates released in GWTC-3 (Abbott et al. 2023), including both the above threshold ( p astro > 0 . 5) \nL \nL \nFigure 7. Constraints on the two parameters L 0 and f of the model for the putative EM counterpart of BBH mergers. The color map reports the likelihood L , for the full analysis including all the O3 catalog events with p astro > 0 . 5. L 0 is in units of erg s -1 . The thick blue solid and dashed contours indicate the exclusion regions in the [ L 0 , f ] plane at 90% and 50% credibility levels, respectively. The magenta solid and dashed lines report the same contours, but for an analysis that includes all O3 catalog events, with no cut in p astro . \n<!-- image --> \nand the subthreshold ( p astro < 0 . 5) candidates. For the latter, we emphasize that the classification of the CBC candidate as BBH merger is valid under the condition that the subthreshold GW event is of astrophysical origin. The simulation described in this section is set up in such a way that this assumption is taken into account for the final constraints of the physical parameters. All the low-latency candidates not confirmed by the offline analysis are not included in the simulation. The considered BBH sample with full NITRATES results and available flux upper limits consists of 32 events, 12 of which with p astro > 0 . 5. \nIn order to compute numerically the functional behavior of the likelihood, we set up a simulation to evaluate P ( L 0 , f | E 1 , ..., E N ) in the full [ L 0 , f ] plane defined by the prior boundaries. The details of the simulation setup are reported in Appendix B. The results of the simulation for the sample of above threshold BBH candidates are reported in Fig. 7, where the color map indicates the value of L = ∏ L i , normalized by the maximum max( L ) over the full domain. The contour levels defining the 50% and 90% exclusion regions are reported as well. For comparison, in Fig. 7 we include also the same contour levels obtained with an analysis that considers all the BBH candidates without imposing any cut on the p astro . The level of constraining power of the analysis can be quantified by defining the fraction of the \nFigure 8. Posterior distribution of log 10 ( f ), including the 50% and 90% upper limits with dot-dashed and dashed lines, respectively. The function is derived from Fig. 7, marginalizing over L 0 . \n<!-- image --> \nFigure 9. Posterior distribution of log 10 ( L 0 ), including the 50% and 90% upper limits with the dot-dashed and dashed lines, respectively. The function is derived from Fig. 7, marginalizing over f . \n<!-- image --> \nfull parameter space excluded with a credibility level η , corresponding to: \nR η = I η /I tot , (18) \nwhere I tot = ∫ d f d L 0 , being the integral extended to the full parameters domain, and with \nI η = ∫ S d f d L 0 , (19) \ncorresponding to the dimension of the region S of the parameter space excluded with a credibility level η . The \nanalysis performed using only BBH with p astro > 0 . 5 gives a R 90% = 22%, while the analysis performed with the inclusion of subthreshold BBH candidates gives R 90% = 10%. This result indicates that with the inclusion of GW subthreshold events the analysis allows us to exclude a smaller portion of the parameter space, with respect to an analysis carried out using only events with high p astro . Therefore, even adding further information with the inclusion of more GW events, if these last are likely originated by noise, the full Bayesian analysis is affected by more uncertainty, resulting in an overall less constraining power of the analysis. \nFigs. 8 and 9 report the posterior distribution of P ( f ) and P ( L 0 ), respectively, obtained as: \nP ( f ) = ∫ P ( L 0 , f | E 1 , ..., E N )d L 0 (20) \nand \nP ( L 0 ) = ∫ P ( L 0 , f | E 1 , ..., E N )d f. (21) \nBoth P ( L 0 ) and P ( f ) are normalized such that max[ P ( L 0 )] = max[ P ( f )] = 1. The posterior is reported in magenta and blue for both samples, with and without cut in p astro , respectively. The 50% and 90% upper limits are reported as well. From the shape of the posteriors and the values of the 50% and 90% upper limits, it is evident that both the L 0 and f posteriors only slightly differ from the flat priors, if no cut in p astro is applied. On the other hand, for the above threshold sample, the analysis is more informative and more stringent constraints on the parameters can be obtained, especially for L 0 . For the sample with p astro > 0 . 5, the 50% and 90% upper limits for f are log 10 ( f 50% ) = -1 . 67 and log 10 ( f 90% ) = -0 . 42, while for L 0 are log 10 [ L 0 , 50% (erg s -1 )] = 47 . 8 and log 10 [ L 0 , 90% (erg s -1 )] = 49 . 2, respectively. \nIn the limit of a collection of triggers which correspond only to non-astrophysical events, i.e., all with π i = 0, the likelihood is constant in the full parameter space, not allowing to infer any constraints on the EM model parameters. On the other hand, if we increase the fraction of confident GW events and we keep fixed the total number N , their distance distribution P ( D L ) and the derived upper limits, then we obtain that L decreases accordingly. This implies that increasing the number of events with π i close to 1, the overall exclusion region in the ( λ 1 , ..., λ k ) parameter space increases as well. This demonstrates that with the collection of more data, in the limit of a GW detector horizon constant in time, this method allows us to improve incrementally our constraints on the EM models of CBC events. Although, realistically the GW detection horizon will increase with time (Abbott et al. 2020), implying an overall increase of \nthe median values of D L of the candidate events. Such an effect increases in turn the values of the luminosity upper limits, increasing as well the values of P i ( ϕ < ϕ 0 ) and hence of L . This effect tends to decrease the dimension of the exclusion region of the ( λ 1 , ..., λ k ) parameter space. Overall, the final outcome of the inclusion of additional GW data, in terms of the constraining power of this analysis, will depend on the simultaneous combined effect of increasing the number of confident events and of increasing the detection horizon. \nIn order to show how the inclusion of more significant GW candidates can improve the constraining power of the present analysis, we carried out the following simulation. We repeated the same procedure adopted to produce the exclusion regions of Fig. 7, but replacing the real π i with π i = 1 for all the confirmed BBH candidates, hence imposing that they are all significant events. All the values of ϕ UL , ε ⊕ and P ( D L ) of each candidate are left unchanged. The resulting 50% and 90% exclusion regions are reported in Fig. 10, with black dashed and solid lines, respectively. The fraction of the 90% excluded region increases to a value of R 90% = 17%, clearly demonstrating that, even if the BAT flux upper limit are the same, the increase of confidence about the astrophysical nature of the GW improves our final constraints on the model parameter space. Furthermore, Fig. 10 reports also the 50% and 90% exclusion regions (with red dashed and solid lines, respectively), obtained as before, but imposing both π i = 1 and ε ⊕ = 0 for each candidate. This combination corresponds to simulate all real BBH candidates, whose sky localization does not overlap with the sky region covered by the Earth. In this case R 90% = 35%, showing that also the fraction of GW sky posterior occulted by Earth has a significant impact on our final results.', '7. CONCLUSIONS': 'In this work we report the systematic search of signals jointly detected by the LIGO-Virgo interferometers and the Swift -BAT telescope, during the third LVK observing run. Thanks to the prompt availability of BAT data using GUANO and the sensitive targeted search capabilities of the NITRATES pipeline, we conducted deep follow-up searches for EM signals on a sample of 636 GW triggers. The search results did not yield any confident joint detection, allowing us to derive upper limits in the 15-350 keV band. We provide comprehensive details on all analyzed GW triggers along with their NITRATES search statistics. This information can be valuable for calibrating and comparing with other offline targeted search pipelines that are currently operational or may be developed in the future. \nL \nL \nFigure 10. Same as Fig. 7, but simulating all the O3 catalog candidates with an associated p astro = π i = 1. The black dashed and solid lines identify the 50% and 90% exclusion regions, respectively. The red dashed and solid lines have the same meaning, but derived imposing both π i = 1 and ε ⊕ = 0. \n<!-- image --> \nIn the specific case of the BBH class, the BAT flux upper limits have been used to perform a stacking analysis and to derive constraints on the possible nature of an associated EM emission. As illustrated in Section 6, the presence of several BBH candidates with large values of p astro in our sample, enhances our ability to better constrain the parameter space for EM emission from BH mergers, with minimal assumptions on our prior knowledge of the nature of the emission. The prospect of detecting EM emission from BBH mergers has been debated and discussed in detail in recent times. Particularly, the GBM trigger that accompanied the first BBH merger event, GW150914, has served as a case study to test possible association and potential implications (Abbott et al. 2016b; Connaughton et al. 2016; Goldstein et al. 2016b). Though not likely, there are a number of physical models that have been proposed that could give rise to detectable emission in the gamma-ray band. A summary of the various different models has been discussed in Fletcher et al. (2023) and Veres et al. (2019). The models involve parameters pertaining to potential remnant accretion effects, magnetic field strength, black hole charge and spin, among others (e.g., Loeb 2016; Dai et al. 2017; Woosley 2016; Zhang 2016). The method described in Section 6 can be easily extended to any of these models, provided that the luminosity function Π( L ) of the putative BBH EM emission is known. Additionally, effects possibly related to the viewing angle dependency of the EM emission can be easily included in this approach. Regarding CBCs containing at least one \nNS (BNS and NSBH), it was not possible to conduct a similar stacking analysis as the one described for BBH in Section 6, due to the paucity of such events with a large enough value of p astro . Further observations, including the fourth LVK observing run (O4), could lead to the collection of a larger number of BNS and NSBH candidates with moderate values of p astro , giving the possibility to repeat the analysis performed in this paper and to derive informative constraints on the EM emission of these classes and the properties of the associated GRB populations. Data products associated with the present analysis are reported in a separate data release. 7 \nO4 commenced on the 24th of May, 2023. The number of significant detections is expected to increase by several times during the entire duration of O4 (Abbott et al. 2018; Petrov et al. 2022). Targeted search results using the GUANO-NITRATES infrastructure are publicly available in real-time 8 . In the case of non-detection of an EM counterpart, the GUANO team reports the 15-350 keV flux upper limit for all the GW triggers classified as significant , via GCN Circulars. Additional enhancements to the likelihood search code have reduced the search latency by a factor of 2, with respect to O3. \nThanks to its sensitivity in the hard X-ray band and the possibility to localize EM transients down to a precision of an arcminute, Swift represents one of the main discovery machines for the detection of EM counterparts of GW transients. This paper shows how the GUANO infrastructure has a deep impact on the multi-messenger science case, in particular for optimally exploiting the sensitivity of the Swift -BAT instrument for the detection of EM counterparts of CBCs detected by the LVK Collaboration. The deep subthreshold search enabled by the NITRATES pipeline sensibly increases the detection horizon of Swift , giving the chance to detect transients also outside the BAT FOV and allowing us to possibly detect faint X-ray/gamma-ray transients associated to relativistic jets observed off-axis, as in the case of GW170817. In the case of a confident joint Swift -GW detection, the GUANO team will promptly disseminate all the information about the EM candidate via GCN Circulars, providing an estimate of the sky localization when available. Moreover, also in the case of non-detection, this paper shows how the upper limits derived from the NITRATES analysis can be combined to have the most sensitive constraints on the EM emission from all the CBC classes. The cumulative collection of \nnon-detection will gradually improve our knowledge of the EM nature of CBCs.', 'ACKNOWLEDGMENTS': "Gayathri Raman, Samuele Ronchini, and Jamie Kennea acknowledge the support of NASA grants 80NSSC19K0408 and 80NSSC22K1498 awarded as part of the NASA Neil Gehrels Swift Observatory Guest Investigator program. Jamie Kennea and James Delaunay acknowledge the support of NASA contract NAS5-0136. \nThis material is based upon work supported by NSF's LIGO Laboratory which is a major facility fully funded by the National Science Foundation. The authors also gratefully acknowledge the support of the Science and Technology Facilities Council (STFC) of the United Kingdom, the Max-Planck-Society (MPS), and the State of Niedersachsen/Germany for support of the construction of Advanced LIGO and construction and operation of the GEO 600 detector. Additional support for Advanced LIGO was provided by the Australian Research Council. The authors gratefully acknowledge the Italian Istituto Nazionale di Fisica Nucleare (INFN), the French Centre National de la Recherche Scientifique (CNRS) and the Netherlands Organization for Scientific Research (NWO), for the construction and operation of the Virgo detector and the creation and support of the EGO consortium. The authors also gratefully acknowledge research support from these agencies as well as by the Council of Scientific and Industrial Research of India, the Department of Science and Technology, India, the Science & Engineering Research Board (SERB), India, the Ministry of Human Resource Development, India, the Spanish Agencia Estatal de Investigaci'on (AEI), the Spanish Ministerio de Ciencia, Innovaci'on y Universidades, the European Union NextGenerationEU/PRTR (PRTR-C17.I1), the ICSC - CentroNazionale di Ricerca in High Performance Computing, Big Data and Quantum Computing, funded by the European Union NextGenerationEU, the Comunitat Auton'oma de les Illes Balears through the Direcci'o General de Recerca, Innovaci'o i Transformaci'o Digital with funds from the Tourist Stay Tax Law ITS 2017-006, the Conselleria d'Economia, Hisenda i Innovaci'o, the FEDER Operational Program 2021-2027 of the Balearic Islands, the Conselleria d'Innovaci'o, Universitats, Ci'encia i Societat Digital de la Generalitat Valenciana and the CERCA Programme Generalitat de Catalunya, Spain, the National Science Centre of Poland and the European Union - European Regional Development Fund; Foundation for Polish Science (FNP), the Polish Ministry of Science and Higher Education, the Swiss National Science Foundation (SNSF), the Russian Science Foun- \nthe European Commission, the European Social Funds (ESF), the European Regional Development Funds (ERDF), the Royal Society, the Scottish Funding Council, the Scottish Universities Physics Alliance, the Hungarian Scientific Research Fund (OTKA), the French Lyon Institute of Origins (LIO), the Belgian Fonds de la Recherche Scientifique (FRS-FNRS), Actions de Recherche Concert'ees (ARC) and Fonds Wetenschappelijk Onderzoek - Vlaanderen (FWO), Belgium, the Paris ˆ Ile-de-France Region, the National Research, Development and Innovation Office Hungary (NKFIH), the National Research Foundation of Korea, the Natural Science and Engineering Research Council Canada, Canadian Foundation for Innovation (CFI), the Brazilian Ministry of Science, Technology, and Innovations, the International Center for Theoretical Physics South American Institute for Fundamental Research (ICTPSAIFR), the Research Grants Council of Hong Kong, the National Natural Science Foundation of China (NSFC), the Leverhulme Trust, the Research Corporation, the National Science and Technology Council (NSTC), Taiwan, the United States Department of Energy, and the Kavli Foundation. The authors gratefully acknowledge the support of the NSF, STFC, INFN and CNRS for provision of computational resources. \nThis work was supported by MEXT, JSPS Leadingedge Research Infrastructure Program, JSPS Grant-inAid for Specially Promoted Research 26000005, JSPS Grant-in-Aid for Scientific Research on Innovative Areas 2905: JP17H06358, JP17H06361 and JP17H06364, JSPS Core-to-Core Program A. Advanced Research Networks, JSPS Grant-in-Aid for Scientific Research (S) 17H06133 and 20H05639 , JSPS Grant-in-Aid for Transformative Research Areas (A) 20A203: JP20H05854, the joint research program of the Institute for Cosmic Ray Research, University of Tokyo, National Research Foundation (NRF), Computing Infrastructure Project of Global Science experimental Data hub Center (GSDC) at KISTI, Korea Astronomy and Space Science Institute (KASI), and Ministry of Science and ICT (MSIT) in Korea, Academia Sinica (AS), AS Grid Center (ASGC) and the National Science and Technology Council (NSTC) in Taiwan under grants including the Rising Star Program and Science Vanguard Research Program, Advanced Technology Center (ATC) of NAOJ, and Mechanical Engineering Center of KEK. \nAdditional acknowledgements for support of individual authors may be found in the following document: https://dcc.ligo.org/LIGO-M2300033/public . For the purpose of open access, the authors have applied a Creative Commons Attribution (CC BY) license to any Author Accepted Manuscript version arising. We \nrequest that citations to this article use 'A. G. Abac et al. (LIGO-Virgo-KAGRA Collaboration), ...' or similar phrasing, depending on journal convention. \nMatplotlib (Hunter 2007), SEABORN (Waskom 2021), NumPy (Harris et al. 2020) and SciPy (Virtanen et al. 2020) were used in the preparation of the manuscript.", 'REFERENCES': 'where λ is the credibility level of the contour. \n- 4. The contour is drawn imposing L = L ( ˆ L 0 , ˆ f ).', 'A. LUMINOSITY UPPER LIMIT': 'A more accurate method to derive the luminosity upper limit should be based on the knowledge of P ( D L ) and P ( ϕ ), where ϕ is the flux measured in the BAT energy band. Having only an upper limit, P ( ϕ ) can be approximated as \nP ( ϕ ) ∝ Π( ϕ ) , ϕ < ϕ UL 0 , ϕ > ϕ UL , (A1) \nwhere Π( ϕ ) is our prior distribution for the flux. Using the conversion from flux to luminosity L = 4 πD 2 L ϕ , the probability distribution of the luminosity can be computed as \nP ( L ) = P (4 πD 2 L ϕ ) ∝ ∫ 1 ϕ P ϕ ( ϕ ) P D 2 L ( L 4 πϕ ) d ϕ, (A2) \nwhere P ϕ is the flux probability distribution and P D 2 L is the probability distribution of D 2 L . In the conversion from flux to luminosity, the k-correction has been omitted, since it introduces a mild dependence on the redshift, which is not relevant for the purposes of this section. The 5 σ luminosity upper limit L UL can be found imposing that \n∫ L UL 0 P ( L )d L = 1 -ε 5 σ , (A3) \nwith ε 5 σ = 3 × 10 -7 . The value of L UL has been computed adopting two different assumptions for the flux prior, corresponding to Π( ϕ ) ∝ const. and Π( ϕ ) ∝ ϕ -3 / 2 , with the latter being inspired by the usual trend followed by GRBs (e.g., Salafia et al. 2023). In both cases, we find that L UL ∼ 3 × 4 π ⟨ D 2 L ⟩ ϕ UL .', 'B. SIMULATION SETUP': 'In this section we specify the details of the simulation used to compute numerically the L ( L 0 , f | E 1 , ..., E N ) function. For each simulated GW candidate, the single L i is computed for each pairs of values ( L 0 ,n , f m ). The flux predicted by the EM model is predicted injecting 1000 sources whose luminosity distance is distributed according to P ( D L ), derived from the GW localization. The probability P i ( ϕ < ϕ 0 ,i ) is derived computing the fraction of cases that have a flux below the sky-averaged BAT upper limit, defined by Eq. (3). The computation of L is performed on a 100 × 100 grid of ( L 0 ,n , f m ). Once the previous steps are performed for all the GW candidates, the final combined likelihood is computed as \nL ( L 0 , f | E 1 , ..., E N ) = ∏ i L i ( L 0 , f ) . (B4) \nIn order to produce the credibility contours in the [ L 0 ,n , f m ] plane, we adopt the following steps: \n- 1. L is normalized such that \n∑ n,m L ( L 0 ,n , f m ) = 1 . (B5) \n- 2. A one-dimensional array L [ x n ] is created flattening the two-dimensional grid L ( L 0 ,n , f m ), then L [ x n ] is sorted in ascending order.\n- 3. We find the element [ ˆ L 0 , ˆ f ] = [ x n ∗ ] such that \nn ∗ ∑ n =0 L [ x n ] = λ, (B6)', 'C. FLUX UPPER LIMIT DERIVATION': 'In this appendix we show an alternative method to compute the non-detection likelihood presented in Section 6. The NITRATES analysis allows us to derive a flux upper limit at a given confidence level for each pixel of the GW sky localization, corresponding to the function ϕ UL (RA , Dec) defined in Eq. (3). Then combined probability of being located in the pixel x i and to have a non-detectable EM emission is \nP (non-det , x i ) ∝ P GW ( x i ) P [ ϕ < ϕ UL ( x i )]∆Ω i , (C7) \nwhere \nP [ ϕ < ϕ UL ( x i )] = (1 -f ) + f ∫ ϕ UL ( x i ) 0 P ( ϕ | x i )d ϕ, (C8) \nand ∆Ω i is the area of the pixel. Here we express P ( ϕ | x i ) as the conditional flux probability distribution, namely the flux probability distribution assuming that the GW source is contained in the pixel x i . To compute the P ( ϕ | x i ), for a fixed luminosity L , the luminosity distance is extracted from the the conditional probability distribution P ( D L | x i ), which is derived from the GW sky localization. Finally the overall non-detection probability is obtained integrating Eq. (C7) over the full sky: \nP (non-det | f, L 0 ) = ∑ x i P GW ( x i ) P [ ϕ < ϕ UL ( x i )]∆Ω i = (1 -f )+ f [ ε ⊕ + ∑ x i / ∈ Ω ⊕ P GW ( x i )∆Ω i ∫ ϕ UL ( x i ) 0 P ( ϕ | x i )d ϕ ] , (C9) \nwhere we have used that \n∫ ϕ UL ( x i ) 0 P ( ϕ | x i )d ϕ = 1 if x i ∈ ⊕ , and ∑ x i ∈ Ω ⊕ P GW ( x i )∆Ω i = ε ⊕ . (C10) \nThe resulting probability of non-detecting any EM emission in correspondence to a GW trigger with a given p astro = π i is \nP (non-det | f, L 0 , π i ) = (1 -π i ) + π i P (non-det | f, L 0 ) . (C11) \nEq. (C9) has to be compared with the method used in Section 6, where instead we used the approximation: \nP (non-det | f, L 0 ) = (1 -f ) + f ∫ ϕ UL 0 P ( ϕ )d ϕ, (C12) \nϕ UL = ∫ Ω / ∈ Ω ⊕ ϕ UL (Ω) P GW (Ω)dΩ , (C13) \nwith \nand P ( ϕ ) is obtained extracting D L from the full sky marginalized distribution P ( D L ), corresponding to \nP ( D L ) = ∑ x i P GW ( x i ) P ( D L | x i )∆Ω i . (C14) \nThe two methods give comparable results in the assumption that the following approximation is valid: \n∫ ϕ UL 0 P ( ϕ )d ϕ ≈ ε ⊕ + ∑ x i / ∈ Ω ⊕ P GW ( x i )∆Ω i ∫ ϕ UL ( x i ) 0 P ( ϕ | x i )d ϕ. (C15) \nFor completeness, we clarify here the main differences in the two methods.', 'METHOD 1:': 'This is the method used in Section 6 and is based on the following steps: \n- 1. The marginalized upper limit ϕ UL is computed over the full sky, weighting by the GW sky localization.\n- 2. Once L 0 is fixed, the flux probability distribution P ( ϕ ) is computed extracting randomly D L from the P ( D L ), the latter corresponding to the posterior distribution of the GW luminosity distance, marginalized over the full sky (excluding the part occulted by the Earth).\n- 3. The integral ∫ ϕ UL 0 P ( ϕ )d ϕ which appears in Eq. (C12) is evaluated counting the fraction of simulated events that have a predicted flux below the sky-averaged upper limit ϕ UL .', 'METHOD 2:': 'This is the method presented in this appendix and summarized by Eqs. (C9) and (C11), consisting in the following procedure: \n- 1. A set of sources is injected in space and the distribution follows the volumetric probability distribution of the GW candidate. First the coordinates of the injected source are extracted from the sky localization P GW (RA , Dec), then for each position the distance is extracted according to the conditional probability P ( D L | RA , Dec).\n- 2. For each injected source, once the luminosity L 0 is fixed, the predicted flux is compared with the coordinatesdependent BAT upper limit map ϕ UL (RA , Dec).\n- 3. We define ρ / ∈⊕ the fraction of all the sources injected which are not occulted by the Earth and also have a predicted flux below ϕ UL (RA , Dec). Given this definition, we have: \n∑ x i / ∈ Ω ⊕ P GW ( x i )∆Ω i ∫ ϕ UL ( x i ) 0 P ( ϕ | x i )d ϕ = (1 -ε ⊕ ) ρ / ∈⊕ . (C16) \nThe last equality can be justified considering that, if for each pixel i we inject N tot ,i sources, we can define ρ i = N ND ,i /N tot , i , where N ND ,i is the fraction of injected sources that are not detected, i.e., with a predicted flux below ϕ UL ( x i ). Therefore: \n∫ ϕ UL ( x i ) 0 P ( ϕ | x i )d ϕ = ρ i . (C17) \nLet us call N tot the total number of sources injected on the full sky. Then we have \nN tot , i = N tot P GW ( x i )∆Ω i , (C18) \nand therefore \n∑ x i / ∈ Ω ⊕ P GW ( x i )∆Ω i ∫ ϕ UL ( x i ) 0 P ( ϕ | x i )d ϕ = ∑ x i / ∈ Ω ⊕ N tot ,i N tot ρ i = 1 N tot ∑ x i / ∈ Ω ⊕ N ND ,i . (C19) \nSince the total number of injected sources not occulted by Earth are N tot , / ∈⊕ = (1 -ε ⊕ ) N tot , and using that \nρ / ∈⊕ = 1 N tot , / ∈⊕ ∑ x i / ∈ Ω ⊕ N ND ,i , (C20) \nwe finally recover Eq. (C16). \nIn order to quantify the difference between the two methods, the following test is performed. Having fixed the two parameters f and L 0 , we compute the likelihood L for the two methods and we derive the quantity \nε L = 2 abs( L 1 -L 2 ) L 1 + L 2 . (C21) \nHere we use the subscripts 1 and 2 for the respective methods. Both likelihoods are computed considering only BBH candidates with p astro > 0 . 5. In both cases, the total number of injected sources for each BBH candidate is N tot = 1000. The distribution of ε L is evaluated sampling randomly f and L 0 , for a total of 100 sampled pairs ( f, L 0 ). We obtain that the median value of ε L is 0.04 and that in ∼ 80% of the sampled cases ε L < 0 . 2. Since the difference between the two methods is limited and since the Method 2 is more computationally expensive, all the results are used adopting Method 1. \n√ \nTable 1 . List of 636 low latency GW triggers analyzed using NITRATES is shown along with their respective p astro values and 15-350 keV band flux upper limits. The maximum √ TS is indicated for all the triggers with successful NITRATES results. Observations corresponding to triggers with insufficient exposure time during the BAT pointing mode do not have valid NITRATES results or flux upper limits. For those triggers that do have NITRATES results but fail to meet the criterion for a full likelihood analysis, the max √ TS is indicated as NFL (No Final Likelihood). The GW triggers from the Burst pipeline do not have associated p astro values and are therefore left blank. The fraction of the GW sky posterior distribution inside the BAT coded FOV and the fraction of the GW posterior occulted by the Earth are denoted by ε in BAT and ε ⊕ , respectively. \nTable 2. Details of the O3 candidates confirmed by the offline analysis and with a p astro > 0 . 5, for which GUANO data dumps are available. The reported p astro and FAR are relative to the pipeline with the highest p astro . If two pipelines have equal p astro , we select the one with the highest SNR. The GW FAR, p astro and Class details are quoted from Abbott et al. (2024) and Abbott et al. (2023). \nTable 3 . List of the O3 candidates confirmed by the offline analysis with p astro < 0 . 5, for which GUANO data dumps were available. GW FAR, p astro , Class and p Class are reported from Abbott et al. (2023). CBC or Burst group categories are quoted as per the offline analysis and not from the low-latency information. \n√ \nTable 4. Details of the joint FAR computed according to the procedure detailed in Section 5.4 for all the triggers with FAR GRB , max < 10 -3 Hz. The RAVEN alert is, by definition, evaluated considering only information received in low latency. The events marked with a (*) are GW candidates with p astro > 0 . 5.'} |
2024ApJ...975..177B | Spitzer hidden observations of the background are used to construct a catalog of 4090 spectra and examine the signature of polycyclic aromatic hydrocarbon PAH molecules and their connection to extinction by dust. A strong positive correlation is recovered between WISE12 EB V and the 11.2 m PAH band. For 0.06 EB V 5.0 correlations of the 6.2 11.2 and 12.7 m PAH band are positive with EB V. Three dust temperature regimes are revealed. Correlations with WISE12 are well constrained and that with 12.711.2 is flat. Decomposition with the NASA Ames PAH IR Spectroscopic Database reveals a tentative positive correlation between the 6.211.2 and the PAH ionization fraction while that with 12.711.2 is slightly negative suggesting PAH structural changes. The relation with PAH size and 6.211.2 is negative while that with 12.711.2 is positive. Averaging spectra into five EB V and three T SUBdustSUB bins shows an evolution in PAH emission and variations in 12.711.2. Databasefits show an increase in f SUBiSUB and the PAH ionization parameter but a more stable large PAH fraction. While the largest s are associated with the highest T SUBdustSUB there is no onetoone correlation. The analysis is hampered by lowquality data at short wavelengths. There are indications that PAHs in the morediffuse backgrounds behave differently from those in the general interstellar medium. However they are often still associated with larger scale filamentary cloudlike structures. The spectra and auxiliary data have been made available through the Ames Background Interstellar Medium Spectral Catalog and may guide JWST programs. | 2024-11-01T00:00:00Z | ['10.3847/1538-4357/ad7d08', '10.48550/arXiv.2409.12324', '2024arXiv240912324B', '2024ApJ...975..177B', 'arXiv:2409.12324'] | ['Interstellar medium', 'Interstellar dust', 'Dust continuum emission', 'Infrared spectroscopy', 'Astronomy databases', 'Polycyclic aromatic hydrocarbons', '847', '836', '412', '2285', '83', '1280', 'Astrophysics - Astrophysics of Galaxies', 'Astrophysics - Solar and Stellar Astrophysics'] | The Background Interstellar Medium as Observed from Offorder Lowresolution SpitzerIRS Spectra | 2,024 | 173 | 0.49 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF'] | 1 | https://arxiv.org/pdf/2409.12324.pdf | {'The Background Interstellar Medium as Observed from Off-Order Low-Resolution Spitzer-IRS Spectra': 'C. Boersma , 1 J.D. Bregman , 1 L.J. Allamandola , 1 P. Temi , 1 and A. Maragkoudakis 1 \n1 NASA Ames Research Center, MS 245-6, Moffett Field, CA 94035-1000, USA \n(Received July 22, 2024; Revised September 11, 2024; Accepted September 19, 2024) \nSubmitted to ApJ', 'ABSTRACT': "Spitzer 'hidden' observations of the background are used to construct a catalog of 4,090 spectra and examine the signature of polycyclic aromatic hydrocarbon (PAH) molecules and their connection to extinction by dust. \nA strong positive correlation is recovered between WISE12, E(B-V), and the 11.2 µ m PAH band. For 0.06 ≤ E(B-V) ≤ 5.0, correlations of the 6.2, 11.2, and 12.7 µ m PAH band are positive with E(B-V). Three dust temperature regimes are revealed. Correlations with WISE12 are well-constrained and that with 12.7/11.2 is flat. \nDecomposition with the NASA Ames PAH IR Spectroscopic Database reveals a tentative positive correlation between the 6.2/11.2 and the PAH ionization fraction, while that with 12.7/11.2 is slightly negative, suggesting PAH structural changes. The relation with PAH size and 6.2/11.2 is negative, while that with 12.7/11.2 is positive. \nAveraging spectra into five E(B-V) and three T dust bins shows an evolution in PAH emission and variations in 12.7/11.2. Database-fits show an increase in f i and the PAH ionization parameter γ , but a more stable large PAH fraction. While the largest γ s are associated with the highest T dust , there is no one-to-one correlation. The analysis is hampered by low-quality data at short wavelengths. \nThere are indications that PAHs in the more-diffuse backgrounds behave differently from those in the general interstellar medium. However, they are often still associated with larger scale filamentary cloud-like structures. \nThe spectra and auxiliary data have been made available through the Ames Background Interstellar Medium Spectral Catalog and may guide JWST programs. \nKeywords: Interstellar Medium (847) - Interstellar Dust (386) - Dust Continuum Emission (412) - Infrared Spectroscopy (2285) - Astronomy Databases (83) - Polycyclic Aromatic Hydrocarbons (1280)", '1. INTRODUCTION': 'The life cycle of interstellar polycyclic aromatic hydrocarbon (PAH) molecules starts with their formation in the ejecta from carbon-rich AGB stars followed by a residency in the interstellar medium (ISM), where they subsequently evolve via processing by ultraviolet (UV) photons. After some 10 7 yr, PAHs are incorporated into dark clouds where they are thought to freeze out onto dust grains and are processed further, along with new PAH formation, now via ice grain chemistry. Once new stars form, PAHs are exposed to UV radiation and processed yet again. This PAH-evolution is observed as spectral changes from one environment to another and within individual objects, e.g., in reflection nebulae (RNe) as a function of distance from the illuminating star (see e.g., Bregman & Temi 2005; Boersma et al. 2016). PAHs have been well-studied in planetary nebulae (PNe), H IIregions, and RNe, but little work has been done on PAHs in the background, diffuse ISM. \nThe direct association of dust with PAHs has been inferred from the correlation of IRAS 100 µ m measurements with that of the 3.3 µ m PAH band strength (Tanaka et al. 1996) and with 4.5-11.7 µ m spectra (Onaka et al. 1996) obtained by the Infrared Telescope in Space (IRTS). While the 3.3 µ m data show a correlation with both IRAS 12 µ m (PAHs) and the 100 µ m (classical dust) data in the intensity range corresponding to the diffuse background ISM, almost all of the mid-IR data sample directions towards dense clouds rather than the background diffuse ISM. Broadband WISE 12 µ m band 3 data was selected to sample PAH emission based on the IRTS results. Consequently, WISE band 3 data is commonly used as a proxy for PAH emission (see e.g., Meisner & Finkbeiner 2014; Lan et al. 2015). \nUtilizing data obtained by the InfraRed Spectrograph (IRS; Houck et al. 2004) onboard the \nSpitzer Space Telescope (Spitzer; Werner et al. 2004) during its cryogenic mission that ended on May 15, 2009, this work generates a catalog of 4,090 low-resolution spectra of the background ISM by extracting them from the off-order positions. In turn, this catalog is used to study the PAH emission from the background ISM and its connection to other dust components. The catalog is made available as The Ames Background ISM Spectral Catalog and can be accessed through a comprehensive website. \nThis paper is organized as follows. Section 2 describes the observations and data reduction, Section 3 sets out the analysis, Section 4 discusses the results, Section 5 draws astronomical implications, Section 6 describes The Ames Background ISM Spectral Catalog, and Section 7 concludes the paper with a summary and its main takeaways.', '2.1. Spitzer': 'Candidate Spitzer observations were taken from The Nominal Science Operations - Schedule of Executed Science and Calibration Observations, which was obtained from IRSA 1 . The observation log lists 58,575 entries up to the end of the cryogenic mission, of which 18,018 are labeled as AOT=irsstare . The left panel of Fig. 1 shows the positions of these observations on the sky overlain on an all-sky GAIA color image (2 nd data release 2 ). To avoid the crowded Milky Way, a one-degree exclusion zone around the galactic plane is imposed, i.e., half a degree above and below it, based on the reported target positions in the observation log. This brings the number of IRS staring observations down to 17,600. Next, meta data associated with each observation were retrieved using the \nAPI 3 exposed by the Spitzer Heritage Archive (SHA) using the AOR key associated with each observation. Subsequently, these were used to check for Short-Low (SL) observations and ensure both SL1 (7 . 5 ≲ λ ≲ 14 . 5 µ m) and SL2 (5 . 2 ≲ λ ≲ 14 . 5 µ m) orders are present. This brings the number of observations to consider down to 10,190. Lastly, using the same meta data from the SHA the observations were further limited to those having either a ramp time of 60 or 240 s to ensure enough exposure to detect the weak emission associated with the background ISM. This reduced the number of observations to 3,294; with 2,813 and 481 having 60 and 240 s ramp times, respectively. The galactic positions of these observations are shown in the right panel of Fig.1.', '2.2. Data Reduction': "The raw data associated with each of the 3,294 observations were retrieved from the SHA 4 . The CUPID software tool at version 2.0 5 was used to generate Basic Calibrated Data (BCD), taking into account different pointings--traced by the CLNUMPOS FITS header keyword--in a single observation. BCDs were created both with and without using the pipeline's default dark subtraction. BCDs for a total of 4,090 positions were generated, where those having multiple pointings and those only containing peak-up data were taken into account. It is noted that for a few pointings the slit information was missing ( PA SLIT FITS header keyword). Table A1 in the Appendix lists the five observations that could not be processed. \nNext, spectra were extracted from the BCDs using the The CUbe Builder for IRS Spectra \n- 3 sha.ipac.caltech.edu/applications/Spitzer/SHA/help/ doc/api.html\n- 4 sha.ipac.caltech.edu/applications/Spitzer/SHA/\n- 5 irsa.ipac.caltech.edu/data/SPITZER/docs/ dataanalysistools/tools/cupid/ \nMaps (CUBISM 6 ; Smith et al. 2007) tool. CUBISM was modified to allow automation of the process. Spectra were extracted using a 24x2 pixel window (see Fig. 2 for details) in the offorder position to obtain a background spectrum and saved to disk using the IPAC table format 7 for both the dark and non-dark corrected BCDs. The associated headers track relevant information, e.g., remapped center position, target name, program name, pointing, reduction history, etc. The IPAC-formatted tables for the non-dark corrected spectra have been made publicly available (see Sect. 6). Figure 3 showcases and compares typical dark and non-dark extracted spectra. Also shown is an estimate for the zodiacal light spectrum as determined by the Zodiacal Light Model available through IPAC 8 . The figure shows that most of the non-dark emission can be attributed to zodiacal light. \nLastly, on- and off-order cross-dispersion profiles are constructed for emission between 1111.6 µ m. First, for each target and pointing the Data Collection Event (DCE) files for a single nod position are combined for the non-dark subtracted BCDs, FUNCs, and BMASKs. Second, the resulting combined images are collapsed in the cross-dispersion direction after straightening the source trace to achieve a sub-pixel sampling of the profile. Third, wavelength and order are associated with each data point using available calibration files ( irs sl wavesamp-[omask,wave] v5.fits ). \nFinally, the cross-dispersion profile between 11-11.6 µ m is constructed for each exposure and fitted with a Gaussian profile plus an off- \n- 6 irsa.ipac.caltech.edu/data/SPITZER/docs/ dataanalysistools/tools/cubism/ \n- 8 irsa.ipac.caltech.edu/data/SPITZER/docs/ dataanalysistools/tools/contributed/general/ zodiacallight/ \nFigure 1. Positions of the Spitzer-IRS staring observations considered in this work overlain on a GAIA all-sky color image using an isotropic Aitoff projection. Some well-known constellations are indicated. Left: All 18,018 irsstare observations. Right: The 3,294 irsstare observations with 60 or 240 s ramp time and outside one-degree of the Galactic plane. See Sect. 2 for details. \n<!-- image --> \nset. The IDL MPFITPEAK -procedure by Craig Markwardt is used for the fitting, forcing a positive peak and a minimum FWHM of 2'.355, the extent of a point-source. Prior to fitting, the data are sigma clipped over 4 surrounding elements. Figure 4 provides an example of the extracted and fitted cross-dispersion profile for the same target and pointing as shown in Fig. 3 after shifting its center to 0' and subtracting the offset. The figure reveals slightly extended PAH emission originating from the target with a FWHM of about 2'.5, while the signal from the background is spatially unresolved, thus fully extended. It is noted that for all background cross-dispersion profiles the PAH signal is spatially unresolved.", '3. ANALYSIS': "To isolate any PAH emission bands in the background spectra, a broad continuum is subtracted. This continuum primarily consists of low surface brightness zodiacal light (cf. Fig. 3), but at SL2 wavelengths is dominated by the IRS' detector response and appears as a broad discrete feature centered around 6 µ m. Removing this continuum is achieved in three steps. First an average zodiacal light spectrum is con- \ntructed from the non-dark subtracted spectra that fall below b = -60 · using a weighted mean. The resulting zodiacal light spectrum is the average of 108 and 15 spectra with 60- and 240 s ramp times, respectively. Second, this spectrum is scaled and added to a 1 st -order polynomial to match the emission at parts of the SL1 spectrum not affected by PAH emission. For the SL2 spectrum the average zodiacal spectrum is only scaled. Wavelength elements between 9-11 and beyond 13.5 µ m are used to set the continuum level for SL1, while for SL2 wavelength elements between 5.7 and 5.87 µ m are selected. Third, the continuum constructed this way is subtracted. Figure 5 demonstrates this approach for the spectrum in Fig. 3. Next, the resulting difference spectrum, e.g., that shown in the bottom panel of Fig. 5, is integrated over intervals associated with known PAH emission. For the 12.7, 11.2 and 6.2 µ m PAH bands integration ranges are set as 12.2-13.1, 10.5-11.7 and 5.9-6.4 µ m, respectively. Because the 7.7 µ m PAH emission complex is split across SL1 and SL2 it is not considered here.", '4. DISCUSSION': "Figure 2. Slit layout. The top four narrow rectangles resemble the long slit of the low-resolution (SL) Infrared Spectrograph (IRS) module onboard Spitzer projected onto the sky. Each slit depicts the position of the target (yellow star) in the slit when employing a nodding strategy. When the target is, in either nod position, on the left half of the slit its light is diffracted by the grating in first order (SL1). Subsequently, when the source is on the right half the light is diffracted in second order (SL2). As a bonus, the SL2 configuration also produces a spectral segment covering 7.3 ≲ λ ≲ 8.7 µ m (SL3). Note that the 22' part of the slit shaded black is not used. Simultaneously, when the target is observed in one order the background is observed in the other. The on-sky geometry shown at the bottom reveals a spatial separation of 158' between the SL1 and SL2 observations of the background and a 3'.6 × 38' (2 × 21 pixels) overlapping aperture. NB the observations considered in this work all make use of the depicted nodding strategy. \n<!-- image -->", '4.1. PAHs and Dust Extinction': "The determined PAH band strengths are compared to extinction measurements, where the latter are retrieved from the Galactic Dust Reddening and Extinction service at IPAC 9 (IRSA 2022). The returned information has a spatial resolution of five arcminutes and includes E(BV), 100 µ m emission, and the dust temperature. Here, the values at the reference pixel ( RefPixel ) are used. As an alternative mea- \nure, the WISE 12 micron full-sky dust map 10 is also considered. The 430, 8,000x8,000-pixel 'cleaned' tiles covering the entire sky at a spatial resolution of 6'.5 were obtained and the WISE12 µ m flux and standard error were computed over a 3-pixel circular aperture. \nFigure 6 shows the strength of the WISE12 band versus E(B-V) for background positions with a WISE12 band of at least 1 × 10 -4 MJy/sr and an associated uncertainty in the 11.2 µ m PAH band strength of less than 3 × 10 -21 W cm -2 . The 11.2 µ m PAH band strengths are \nFigure 7 presents six correlation plots that involve either WISE12 or E(B-V) and the PAH band strengths for data with a signal-to-noise ratio (SNR) of at least 3 and 0.06 ≤ E(BV) ≤ 5.0. In each panel the data points have been color-coded according their associated dust temperature. Trend lines determined \n<!-- image --> \n30 \nFigure 3. Extracted spectra. The SL1, 2 and 3 orders for the dark corrected spectrum have been colored light red, red and dark red, respectively. Those for the non-dark corrected spectrum light green, green and dark green, respectively. The modeled zodiacal light is shown as the grey line. See Sect. 2.2 for details. \n<!-- image --> \nFigure 4. Determined and fitted cross-dispersion profiles for the 11.2 µ m PAH band. The two colors represent the two different nod positions. Left: Science target. Right: Background. \n<!-- image --> \nshown as the (filled) contours. The contours have been constructed from a 21x21 pixel image of the average 11.2 µ m PAH band strength of the points falling in a single pixel. The figure establishes the known correlation between WISE12 and E(B-V) and shows that the 11.2 µ m PAH band strength correlates as well. \nFigure 5. Top: Constructed broad band continuum (greens) matched to the non-dark corrected spectrum (reds) from Fig. 6. Bottom: Resulting background spectrum from subtracting the broad band continuum from the non-dark corrected spectrum. See Sect. 3 for details. \n<!-- image --> \nby fitting a first-order polynomial, not taking into account uncertainties, have been overlain. For each correlation the linear correlation coefficient R 2 , taking uncertainties into account, has been provided. \nThe figure shows that the 6.2 µ mPAHband is significantly hampered by poor SNR as can be inferred from Fig. 5 and the top-left and middleright panels of Fig. 7, where only 83 data points have a SNR ≥ 3. In contrast, the correlations involving the other two PAH band intensities count almost ten times as many points. The correlations of the 11.2 and 12.7 µ m PAH band intensities with the WISE12 band are far more significant with a R 2 of 0.79 and 0.84, respectively. In addition, the correlations are quite \ntight and show, overall, an increase in dust temperature with an increase in both the PAH band intensity and WISE12 band. \nTurning to the correlations of the PAH band intensities with E(B-V), while the trend lines hint at tentative correlations, none are supported by their R 2 values. Though, there appears to be a stratification with an increase in PAH band strengths with dust temperature. That is, lower temperatures are associated with lower PAH band strength intensities. This is further explored in Fig. 8, which plots the correlations with E(B-V) using a linear scale. \nWhen plotting the PAH band strength correlations with E(B-V) on a linear scale (Fig. 8), stratification with dust temperature becomes \nFigure 9 presents the 12.7/11.2 µ m PAH band strength ratio versus E(B-V). Since both bands are obtained from the same spectral segment (SL1), their ratios are relatively well constrained. Traditionally the 12.7/11.2 µ m PAH band strength ratio has been considered a measure for PAH edge structure, notably the ratio of (duo+trio)/solo-hydrogens. However, the ratio typically also shows a strong correlation with the 6.2/11.2 µ m PAH band strength ratio, which is considered a tracer for PAH charge (e.g., Hony et al. 2001; Boersma et al. 2014a). The figure reveals a ratio hovering around 0.7 for all the backgrounds, with R 2 indicating no correlation, and no obvious stratification with dust temperature is discerned. This lack of stratification of the 12.7 µ m band is more fully explored in the next section. \n<!-- image --> \n10 \nFigure 6. WISE12 band versus E(B-V) for target positions with a WISE12 band of at least 1 × 10 -4 MJy/sr and an associated uncertainty in the 11.2 µ m PAH band strength of less than 3 × 10 -21 W cm -2 . The 11.2 µ m PAH band strengths are shown using (filled) contours (logarithmically scaled). \nquite apparent and hints at three discrete temperature regimes. To emphasize this, trend lines have been added by separately fitting the data for T dust < 18, 18 ≤ T dust ≤ 20, and T dust > 20 K. For all three PAH band strengths, points falling in the lowest temperature bin have the weakest correlation per R 2 and, except for 6.2 µ m, those falling in the highest bin have the strongest. The only meaningful correlation with the 6.2 µ m PAH band strength is for 18 ≤ T dust ≤ 20. Overall, the trend line for temperatures > 20 K visually matches best the correlation of the 11.2 µ m PAH band strength with E(B-V), which is reflected by having the best R 2 (0.84). \nA likely explanation for the observed bifurcation is that those background positions with a higher dust temperature, and subsequently stronger PAH band emission, are influenced by an additional radiation source rather than the \ninterstellar radiation field alone. This could either be because they are still receiving some portion of the radiation from the on-target source or the off-target background position happens to fall near a not too distant radiation source in, for example, a crowded star-forming region like the Orion Molecular Cloud.", '4.2. Emission Lines': 'For many astronomical objects, the 12.7 µ m PAH emission band blends with the 12.8 µ m Ne II and the 12.3 µ m H 2 S(2) lines. These contributions to the 12.7 µ m PAH band are removed using the approach from Shannon et al. (2015). Here, an emission-line-free 12.7 µ m PAH band is used as a template to fit the feature. The template used is that from Boersma et al. (2018), which is extracted from SpitzerIRS observations of the RN NGC 7023. The fitting region is selected such that it excludes resolution elements affected by the two emission lines. Figure 10 demonstrates the approach for the spectrum shown in Figs. 3 and 5. The figure shows that some of the 12.7 µ m emission can be attributed to Ne II, while any contribution from \n2 \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 7. Correlation plots involving either WISE12 or E(B-V) and the 6.2, 11.2 and 12.7 µ m PAH band strengths. Each data point is color-coded according to its associated dust temperature. The data have a SNR ≥ 3 and 0.06 ≤ E(B-V) ≤ 5.0. Trend lines determined by fitting the data have been overlain. For each correlation the linear correlation coefficient R 2 , taking uncertainties into account, has been provided. See Sect. 4.1 for details. \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 8. Correlation of the 6.2 (top-left), 11.2 (top-right), and 12.7 µ m (bottom-left) PAH band strength with E(B-V) presented on a linear scale counter to the logarithmic scale used in Fig. 7. Each data point is color-coded according to its dust temperature. The data have a SNR ≥ 3 and 0.06 ≤ E(B-V) ≤ 5.0. Trend lines determined by fitting the data, taking uncertainties into account, for three separate dust temperature intervals have been overlain. For each temperature interval the linear correlation coefficient R 2 , that takes uncertainties into account, has been provided. See Sect. 4.1 for details. \n<!-- image --> \nFigure 9. The 12.7/11.2 µ m PAH band strength ratio versus E(B-V). Each data point is color-coded according to its associated dust temperature. The data have a SNR ≥ 3 and 0.06 ≤ E(B V) ≤ 5.0. A trend line determined by fitting the data has been overlain. Provided is, taking uncertainties into account, the linear correlation coefficient R 2 . See Sect. 4.1 for details. \n<!-- image --> \nFigure 10. Disentangling Ne II, H 2 , and 12.7 µ m PAH band emission by establishing a straight-line continuum (grey-dashed line) and fitting a scaled generic 12.7 µ m PAH profile (blue- and green-solid line) from the RN NGC 7023. Shown is the fitted profile, where any excess between the observations (red) and the blue and green is attributed to H 2 S(2) and Ne II emission, respectively. See Sect. 4.2 for details. \n<!-- image --> \nthe H 2 S(2) line could not be reliably identified (i.e., I H 2 < 0 W cm -2 ). \nFigure 11 reproduces the correlations presented in Fig. 7 that involve the 12.7 µ m PAH band, but now uses the strength determined from the approach described above (PAH 12 . 7 ). The figure shows that there is a drastic decrease in the number of points that pass the SNR threshold (3), with that of the PAH 12 . 7 versus WISE12 and E(B-V) going from 399 to 212 and 636 to 244, respectively. This would suggest that for many, if not most, background observations the emission at 12.7 µ m is affected strongly by line emission. \nIn terms of WISE12, above a surface brightness of ∼ 3 MJy sr -1 the correlation with PAH 12 . 7 seen in Fig. 7 is maintained. However, below this value PAH 12 . 7 flattens out around 0.05 × 10 -19 W cm -2 , albeit with a significant intrinsic spread. Though, overall the correlation remains strong with R 2 dropping only from 0.84 to 0.71. A rather unlikely explanation for the flattening out would be that the correlation seen in Fig. 7 is initially driven solely by line emission and that any background PAH 12 . 7 emission is absent-Ne II emission is typically associated with distinct astronomical objects like H II-regions and RNe. A far more plausible explanation is that the already low SNR for these low-intensity points is not enough to confidently identify and remove any line emission. On top of that, the 12.7 µ mPAHbandis not universal and the emission template taken from NGC 7023 likely not characteristic for all background spectra. \nRegarding E(B-V), the visually spurious correlation with I 12 . 7 present in Fig. 7 is now entirely absent, flattened out, and hovering around 0.02 × 10 -19 Wcm -2 in the PAH 12 . 7 case. Though, there is two orders ( ∼ 0.001-0.1) of intrinsic scatter, R 2 seemingly improves from 0.02 to 0.47, and the obvious stratification with dust temperature is now gone. \nFigure 12 correlates E(B-V) with the PAH 12 . 7 /11.2 µ mPAH band strength as a coun- \n<!-- image --> \nFigure 11. The correlations from Fig. 7 involving the 12.7 µ m PAH band strength presented now using PAH 12 . 7 instead of I 12 . 7 . Trend lines determined by fitting the data have been overlain. For each correlation the linear correlation coefficient R 2 , that takes uncertainties into account, has been provided. See Sect. 4.2 for details. \n<!-- image --> \nerpart to Fig 9 that uses I 12 . 7 instead. Again, the number of viable data points drops significantly, going from 484 down to 160. The visually apparent decrease in the ratio with E(B-V) is perhaps somewhat more prominent here, with the flattening out occurring around E(B-V) ≃ 1.0 and hovering at a PAH 12 . 7 /11.2 µ m ratio of 0.2, down from ∼ 0.7 in Fig. 9. Here, again R 2 seems to suggest a somewhat stronger correlation, albeit still marginal.', '4.3. Spectroscopic Database Fitting': 'Turning to the library of computed spectra at version 3.20 of the NASA Ames PAH IR Spectroscopic Database (PAHdb hereafter 11 , Bauschlicher et al. 2010; Boersma et al. 2014b; Bauschlicher et al. 2018; Mattioda et al. 2020), PAH emission spectra are synthesized for an excitation energy of 7 eV and used to perform a fit to each background spectrum using software tools also provided by PAHdb. The propagated observational uncertainties are taken into account and, to obtain errors for the derived ionization- ( f i ≡ n cation / ( n cation + n neutral )) and large-PAH fraction ( f large ; N carbon > 50), a \nFigure 12. PAH 12 . 7 /11.2 µ m PAH band strength ratio. Each data point is color-coded according to its associated dust temperature. The data have a SNR ≥ 3 and 0.06 ≤ E(B V) ≤ 5.0. A trend line determined by fitting the data has been overlain. Provided is, taking uncertainties into account, the linear correlation coefficient R 2 . See Section 4.2 for details. \n<!-- image --> \nMonte Carlo technique is employed in which the spectra are permuted uniformly within the observational uncertainties and re-fitted 1024 times. The error ( σ SL1 ) is computed as the area of the absolute value of the residual over the area of the astronomical spectra. The integration is done in frequency-space (cm -1 ) as it is \nFigure 13. PAHdb-fit (Sl1=red, Sl2=green) to the background spectrum (in grey) from Fig. 5. Indicated are the single run error ( σ SL1 ), cation- ( f i ) and large-PAH ( f large ) fractions and, in parenthesis, the corresponding values with their associated uncertainties from a Monte Carlo technique. See Section 4.3 for details. \n<!-- image --> \nlinear in energy. See Boersma et al. (2018) for a description of the employed modeling and a discussion of potential caveats. The mean and standard deviation of f i and f large are subsequently determined for each spectrum. Figure 13 presents the results following this approach for the spectrum shown in Fig. 5 and indicates both the single-run and Monte Carlo derived values for σ SL1 , f i , and f large . Note that any potential emission lines were not removed (see Sect. 4.2). The figure shows a reasonably good fit in the SL1 region with σ SL1 of 0.17 for the non-perturbed case, which matches general expectations (e.g., Maragkoudakis et al. 2022). The permuted average error is larger at 0.23 ± 0.08 and reflects the considerable uncertainty on the data. However, f i and f large are more consistent at 0.43 and 0.43 ± 0.08 and 0.69 and 0.60 ± 0.08, respectively. Assessment of the fit to the SL2 segment of the spectrum is significantly hampered for this particular background spectrum due to its poor quality, which is unfortunately also the case for most of the other background spectra. \nThe top two panels in Fig. 14 present the 6.2/11.2 (left panel) and 12.7/11.2 µ m (right panel) PAH band strength ratios plotted against the Monte Carlo-derived f i , while the two middle panels do the same for f large . The two bottom panels of Fig. 14 take the PAH 12 . 7 PAH band strength instead of I 12 . 7 . Trend lines have been added, constructed from straight-line fits to the data, and R 2 values that take uncertainties into account have been provided. Because of the 6.2 µ m PAH band falling in the poorquality SL2 part of the spectrum, the number of points in the correlations involving it are sparse (64). \nNotwithstanding the low number of 6.2/11.2 µ m data points, an interesting trend emerges from Fig. 14 for the correlations involving the 12.7 µ m PAH band (as reflected by their R 2 values). While none of the correlations are particularly strong, the 6.2/11.2 µ m PAH band strength versus ionization fraction, f i , shows an overall positive trend while that of the 12.7/11.2 µ m PAH band strength versus f i hints at a negative trend. This is somewhat surprising as a positive correlation between the 6.2/11.2 versus 12.7/11.2 µ m PAH band strength ratio has been well-established for ISM sources (e.g., Hony et al. 2001; Boersma et al. 2014a). As shown in the lower two frames of the figure, this tentative negative trend also holds when considering PAH 12 . 7 instead of I 12 . 7 , where R 2 improves somewhat. Turning to the large PAH fraction, f large , the negative trend with the 6.2/11.2 µ m PAH band strength ratio is again surprising as the correlation between f i and f large is generally shown to be positive for ISM sources (e.g., Boersma et al. 2015). The same holds for the 12.7/11.2 µ m PAH band strength ratio. This unexpected behavior strongly suggests that PAH edge structure is an important variable parameter that should be considered when analyzing (diffuse) ISM PAH background spectra, particularly when using ratios of indi- \nband strengths as proxies for single PAH properties such as charge and size. As done here, individual bands are often normalized to the strong, well-defined 11.2 µ m band. However, the 11.2, 12.2, 12.7, 13.5, and 14.2 µ m PAH bands are produced by CH out-of-plane bending motions (CH oop ), associated with solo, duo, trio, quartet, and quintet adjacent hydrogen atoms per edge ring, respectively (e.g., Hony et al. 2001). The number of these different hydrogen adjacency types depend on PAH structure and size, with small and irregularly shaped PAHs generally carrying substantially more duo through quintet hydrogens than solo hydrogens, while very large, compact structures with straight edges are dominated by solo hydrogens. The relative intensities of these bands vary with PAH size and structure. Thus, one could argue here that, when considering the 12.7/11.2 µ m PAH band strength ratio as a pure measure for PAH edge structure, as PAH size increases the relative number of solo hydrogens increases and the 11.2 µ m increases, causing the 12.7/11.2 µ m PAH band strength ratio to decrease. \nOf course, the R 2 values are not particularly convincing, though some of that could be driven by the outliers. In addition, the results are likely susceptible to the systematic effect of the poor quality of much of the SL2 data. Nonetheless, as illustrated by Fig. 13, the fit to the 6.2 µ m band is somewhat constrained by the error bars. This points to the importance of having a complete ∼ 5-15 µ m spectrum when fitting (see e.g., Boersma et al. 2015).', '4.4. PAH Spectra Averaged by E(B-V)': 'To increase spectral fidelity, Fig. 15 takes the background SL1 and SL2 spectra (in grey) that have an uncertainty associated with the 11.2 µ m PAH band strength of less than 3 × 10 -21 W cm -2 and averages their broad band continuum subtracted spectra, normalized to the emission at 10 µ m, across five E(B-V) and three T dust \nbins. The figure shows a steady increase in the overall PAH emission when moving both towards higher E(B-V) values and dust temperatures. This is accompanied by changes in the relative strengths of the 11.2 and 12.7 µ m PAH bands. Especially noticeable is the increased fidelity of the SL2 segment when moving towards higher E(B-V) and T dust values. \nThe average PAH spectra are, as in Sect. 4.3, fitted using PAHdb, and show generally good matches in the SL1 region of the spectra. However, the poor SNR at the shorter wavelengths covered by the SL2 segment are poorly constrained for all but the bins with the higher E(B-V) and T dust values. Note that the SL3 bonus order has been discarded. The figure indicates both the single-run total error ( σ ) and that for the SL1 segment alone σ SL1 , as well as f i and f large with their associated uncertainties derived from the fit using a Monte Carlo technique. σ and σ SL1 systematically drop as E(BV) increases. f i generally increases when moving up an E(B-V) bin or crossing a T dust boundary. Compared to f i , f large shows less variance, typically staying between 0.4-0.5. \nThe almost systematic variations observed in f i , and perhaps f large less so, imply altered PAH populations driven by changing astrophysical environments. This can be quantified by turning to the PAH ionization parameter γ , which relates the ionization state of the PAH population to the the strength of the radiation field (G 0 ), the electron density ( n e ), and the temperature of the gas (T gas ) as γ ∝ G 0 T 1 / 2 gas /n e (see Tielens 2005). Figure 15 provides the PAH ionization parameter γ inferred from the fit as 2.66 · f i / f 0 [ × 10 4 K 1 / 2 cm 4 ], where f 0 is the neutral PAH fraction (e.g., Boersma et al. 2018). These are listed in Table 1. \nIn general, but not across the board, the table shows the highest values of γ for the warmest dust temperatures. The lower values of γ are \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 14. Top: The 6.2/11.2 (left) and 12.7/11.2 µ m (right) PAH band strength ratio versus the PAHdbderived ionization fraction ( f i ). Middle: The 6.2/11.2 (left) and 12.7/11.2 µ m (right) PAH band strength ratio versus the PAHdb-derived large PAH fraction ( f large ). Bottom: The PAH 12 . 7 /I 11 . 2 µ m PAH band strength ratio versus f i (left) and f large (right). The data have a SNR ≥ 3 and 0.06 ≤ E(B V) ≤ 5.0. Trend lines determined by fitting the data have been overlain. For each correlation the linear correlation coefficient R 2 , that takes uncertainties into account, has been provided. NB Some extraneous data have been clipped for presentation purposes. See Sect. 4.3 for details. \n<!-- image --> \nFigure 15. Average background PAH spectra across five E(B-V) and three T dust bins. Only those background spectra with an uncertainty of less than 3 × 10 -21 Wcm -2 for their 11.2 µ m PAH band strength and 0.06 ≤ E(B-V) ≤ 5 have been considered. The number of spectra in each bin is indicated by n. The average spectra (SL1+SL2 in grey) have been fitted (red=SL1, green=SL2) with PAH emission spectra synthesized using PAHdb. Indicated are the error ( σ and σ SL1 ), ionization- ( f i ) and large PAH-fraction ( f large ), and the PAH ionization parameter γ determined from the fits through a Monte Carlo technique with the derived uncertainty given in parenthesis. See Sect. 4.4 for details. \n<!-- image --> \nTable 1. PAHdb-derived PAH Ionization Parameter γ for different E(B-V) and T dust Bins. \nconsistent with the warm neutral medium ( γ ≃ 10 4 K 1 / 2 cm 3 ), with the others pushing into the photo-dissociation region (PDR) domain ( γ ≃ few × 10 4 -10 5 K 1 / 2 cm 3 ; Tielens 2005). \nThis shows that many of the background positions are not entirely isolated and are, to one degree or other, influenced by an additional radiation source rather than the interstellar radiation field alone. Obviously, the PDR-like backgrounds are far from isolated. By happenstance the off-target background position could have fallen on a nearby radiation source or it was simply not sufficiently separated from the extended on-source target. This could indeed be easily the case for a close-by star-forming region like the Orion Molecular Cloud.', '5. ASTRONOMICAL IMPLICATIONS': "A sizable fraction (18%; SNR 11 . 2 > 3) of the background spectra show detectable PAH emission, with the bulk (85%) in directions where 0.06 ≤ E(B-V) ≤ 5.0. \nWhile the correlation between PAHs and classical dust shown in Fig. 7 appears at first glance to be poor, with a wide range in PAH emission strengths at all values of E(B-V), the correlations of the 11.2 and 12.7 µ m PAH bands with WISE 12 Band 3 observations point to a much stronger connection. Figure 8 shows these bands plotted linearly against E(B-V). As discussed in Sect. 4.1, these plots show tempera- \nure stratification and a linear upper bound relationship between the PAHs and classical dust. \nAll of the points along the upper boundary have high dust temperatures while points with lower temperature dust fills in below the upper bound. This implies that PAH abundances and dust densities are well-correlated and that more intense radiation fields produce more PAH emission and warmer dust. This suggests that the PAH ionization parameter γ , which is connected to the intensity of the radiation field through G 0 , would correlate with dust temperature when assuming the electron density n e and gas temperature T gas are largely invariant. Though, while Table 1 indeed has the largest γ 's associated with the highest T dust bins, a clear one-to-one correlation is lacking. Inspection of Fig. 15 shows signs of Ne II 12.8 µ m emission in some of the averaged spectra (See also Sect. 4.2). This indicates possible contributions from ionized regions along those lines of sight and hence those values of γ may not be indicative of the isolated background ISM. \nEach line of sight is a composite of all the material along each direction and consists of regions with differing densities and illumination. Those lines of sight with the lowest E(B-V) values can only be composed of low extinction regions, while those with high E(B-V) values could include a multitude of low, moderate, and high extinction regions. The lowest extinction lines of sight have low γ s, indicative of the warm neutral phase of the ISM, while the other lines of sight are consistent with PAH emission from PDRs. In this case a PDR includes any cloud or filament surface with somewhat higher density than the warm neutral phase that is excited by non-ionizing UV photons. Those directions that show obvious Ne II line emission must have a component that is illuminated by ionizing UV photons from luminous O and B stars. \nOne last thing to consider is the separation of the background from the intended science \ntarget and, on a larger scale, that from associated cloud structures. The former seems to be sufficiently scrutinized by the cross-dispersion profiles of the 11.2 µ m PAH emission, which are spatially completely unresolved for all backgrounds (Fig. 4). \nConcerning the latter, the presence of Ne II line emission in at least some of the background spectra does seem to indicate, not unexpectedly, that some of the backgrounds are associated with luminous O-B stars likely connected to larger extended structures. Appendix. B examines this in more detail for the more-diffuse backgrounds (0.06 ≤ E(B-V) ≤ 5.0) by first spatially clustering by position and then constructing images of the 128 cluster regions from GAIA and WISE 12 full-sky dust maps. Indeed, many regions show the background positions to be associated with filamentry cloud-like structures.", '6. THE AMES BACKGROUND INTERSTELLAR MEDIUM SPECTRAL CATALOG': "The Ames Background Interstellar Medium Spectral Catalog makes available the non-dark off-module extracted background spectra as well as a separate downloadable complementary table containing all derived measurement from this work. The catalog can be accessed at www.astrochemistry.org/bism. Spectra can be retrieved using a known AOR-key, by specifying coordinates, or by providing a target name that will be resolved using SIMBAD services. In the latter two cases the catalog is scanned at 1 · increments from the resolved position until there is at least a single hit. Figure 16 shows the search interface as presented at the website. \nQuery results are organized per AOR-key and the number of available spectra are indicated. For each result a spectrum is shown, with the orders color-coded separately and associated error bars. Also provided are the statistical representations for E(B-V), IRAS100 µ m emission, the dust temperature as retrieved from the Galac- \nFigure 16. The Ames Background Interstellar Medium Spectral Catalog's landing page showing the search interface. \n<!-- image --> \nFigure 17. The Ames Background Interstellar Medium Spectral Catalog's results page showing one of the non-dark subtracted spectra found when searching for 'Orion Bar'. \n<!-- image --> \nic Dust Reddening and Extinction service at IPAC, and WISE12 measurements determined as described in this work. The spectra can be downloaded in IPAC format by clicking the 'download data'-link. The file size is indicated for convenience. Figure 17 shows an example when querying and resolving for 'Orion Bar'. \nThe link to download the complementary data can be found near the bottom of the page, as is shown in Fig. 18. \nFigure 18. Ames Background Interstellar Medium Spectral Catalog website showing the link to the complementary data. \nThe website is written in PHP8 with SQLite3 as the database back-end and runs on Apache2 under Ubuntu Server 22.04 LTS. The figures and downloadable data have all been preproduced.", '7. SUMMARY AND CONCLUSIONS': "Many data sets obtained by the Spitzer-IRS contain 'hidden' observations of the IR background ISM. These are often used to subtract line-of-sight contamination, or are simply ignored. Here, these background observations are considered in their own right. A catalog of 4,090 spectra is constructed to examine the PAH spectral signature in the background ISM and its connection to extinction by (classical) dust. \nTo isolate any background PAH emission, carefully selected high-galactic latitude spectra, dominated by zodiacal light, were averaged and subtracted from the background data. Subsequently, the 6.2, 11.2, and 12.7 µ m PAH band strengths were determined. A strong, positive correlation is recovered between the 11.2 µ m PAH band strength with extinction (E(B-V)) and WISE12 observations . \nFocusing on the more-diffuse emission with 0.06 ≤ E(B-V) ≤ 5.0, correlations of the 6.2, 11.2, and 12.7 µ m PAH band strength is positive with E(B-V), albeit with considerable scatter. In addition, the correlations reveal a clear separation into three distinct dust temperature regimes when presented on a linear scale. The correlations with WISE12 data are far better constrained. \nDecomposition of the PAH emission in terms of charge and size using the data and tools made available through PAHdb reveals a tentative positive correlation between the 6.2/11.2 µ m PAH band strength ratio and f i , while that with the 12.7/11.2 µ m PAH band strength ratio, surprisingly, hints to a slight negative trend. The 12.7/11.2 µ mPAHband strength ratio normally tracks with PAH ionization along with the 6.2/11.2 µ mPAHband strength ratio. Since the 11.2 and 12.7 µ mbands are also tracers for PAH edge structure and size, this behavior suggests PAH structures are changing along these lines of sight. \nThe relation between f large and the 6.2/11.2 µ m PAH band strength ratio is negative, with that with the 12.7/11.2 µ m PAH band strength ratio hinting at being positive. \nIncreasing the SNR by averaging the background spectra into five E(B-V) and three T dust bins shows a clear evolution in the strength of the PAH emission and variations in the relative strength of the 11.2 and 12.7 µ m PAH bands. Database-fits show, overall, an increase in f i and γ but a somewhat more stable f large . While the largest found γ s are associated with the highest T dust bins, a clear one-to-one correlation is lacking. However, much of the analysis remains, in many cases, limited by the low SNR at shorter wavelengths ( λ ≲ 7.5 µ m). \nTaking everything together, there are some hints that the PAH population in the morediffuse background behaves differently from that of the general ISM. However, in most cases the backgrounds are still associated with larger scale filamentary cloud-like structures or, in a few cases, PDR-like environments located somewhere along the line of sight. \nThe spectra and auxiliary data have been made publicly available for download through the Ames Background Interstellar Medium Spectral Catalog and, as they may guide JWST \nprograms, focused on explicitly studying the (more-diffuse) background ISM. \n- 10\n- 11\n- 12 \nC.B. is grateful for an appointment at NASA Ames Research Center through the San Jos'e State University Research Foundation (80NSSC22M0107). C.B., J.D.B., L.J.A. and A.M. acknowledge support from the Internal Scientist Funding Model (ISFM) Laboratory Astrophysics Directed Work Package at NASA Ames (22-A22ISFM-0009). L.J.A., J.D.B. and A.M. are thankful for an appointment at NASA Ames Research Center through the Bay Area Environmental Research Institute (80NSSC19M0193). 1 2 3 4 5 6 7 8 9 \nFacilities: Spitzer \nSoftware: astropy (Astropy Collaboration et al. 2013, 2018), amespahdbidlsuite (Bauschlicher et al. 2018)", 'REFERENCES': 'Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, aa, 558, A33, doi: 10.1051/0004-6361/201322068 Astropy Collaboration, Price-Whelan, A. M., Sip"ocz, B. M., et al. 2018, aj, 156, 123, doi: 10.3847/1538-3881/aabc4f Bauschlicher, C. W., Boersma, C., Ricca, A., et al. 2010, ApJS, 189, 341, doi: 10.1088/0067-0049/189/2/341 Bauschlicher, Jr., C. W., Ricca, A., Boersma, C., & Allamandola, L. J. 2018, apjss, 234, 32, doi: 10.3847/1538-4365/aaa019 Boersma, C., Bregman, J., & Allamandola, L. J. 2014a, ApJ, 795, 110, doi: 10.1088/0004-637X/795/2/110 -. 2015, ApJ, 806, 121, doi: 10.1088/0004-637X/806/1/121 -. 2016, apj, 832, 51, doi: 10.3847/0004-637X/832/1/51 -. 2018, apj, 858, 67, doi: 10.3847/1538-4357/aabcbe Boersma, C., Bauschlicher, C. W., Ricca, A., et al. 2014b, ApJS, 211, 8, doi: 10.1088/0067-0049/211/1/8 Bregman, J., & Temi, P. 2005, aa, 621, 831, doi: 10.1086/427738 Hony, S., Van Kerckhoven, C., Peeters, E., et al. 2001, A&A, 370, 1030, doi: 10.1051/0004-6361:20010242', 'The Spitzer Background ISM': 'Figure 20. Zoom-in on the 128 cluster regions from Fig. 20 using the 40kx20k GAIA all-sky image. Scale bars indicate 2 · . In white nearby SIMBAD IR sources are shown and in blue those with known sizes. Parent objects are indicated in purple, clipped to the extent of the field-of-view. Acknowledgment: Gaia Data Processing and Analysis Consortium (DPAC); A. Moitinho / A. F. Silva / M. Barros / C. Barata, University of Lisbon, Portugal; H. Savietto, Fork Research, Portugal. \n<!-- image -->', 'A. OBSERVATIONS WITH COMPLICATIONS': 'Table A1 lists and provides a brief description of those observations for which analysis complications were encountered.', 'B. SPATIAL CLUSTERING': "Figure 19 shows the background positions where the uncertainty associated with the 11.2 µ m PAH band strength is less than 3 × 10 -21 W cm -2 and 0.06 ≤ E(B-V) ≤ 5.0. The positions have been grouped into regions using hierarchical clustering based on complete linkage and a maximum link distance of 6 · . Figure 20 zooms in on each of the regions using the 40kx20k-pixels all-sky GAIA color image (2 nd data release) and indicates each background position by its cluster number and a unique color. Utilizing SIMBAD's TAP service 12 , IR sources within a 117' radius of each position were identified as well as any hierarchical links. This information is displayed in Fig. 21, where IR sources are shown as white dots or as blue boxes when size information is available. Parent objects are indicated in purple. Many regions show the background positions associated with dark filamentary-like cloud structures. \nFigure 21 does the same as Fig. 20, but now uses the 430, 8,000x8,000-pixel WISE 12 micron full-sky dust map. For each region the appropriate sub-images were extracted from the contributing 'clean' WISE-tiles and combined into a single image. The images are displayed using a logarithmic scaling. \nFigure 19. Position of the 687 backgrounds with an associated uncertainty of less than 3 × 10 -21 W cm -2 for the 11.2 µ m PAH band strength and 0.06 ≤ E(B-V) ≤ 5.0. Each background position is indicated by its region number established through hierarchical clustering and uniquely color-coded. \n<!-- image -->", 'Boersma et al.': "<!-- image --> \nFigure 21. Zoom-in on the 128 cluster regions from Fig. 19 using the 430, 8,000x8,000-pixel 'cleaned' tiles from the WISE 12 micron full-sky dust map. Scale bars indicate 2 · . The background positions are shown as the white points. Each image is displayed using a logarithmic scaling."} |
2024A&A...686A..42H | Context. The census of open clusters has exploded in size thanks to data from the Gaia satellite. However it is likely that many of these reported clusters are not gravitationally bound making the open cluster census impractical for many scientific applications. BR Aims We aim to test different physically motivated methods for distinguishing between bound and unbound clusters using them to create a cleaned star cluster catalogue. BR Methods We derived completenesscorrected photometric masses for 6956 clusters from our earlier work. Then we used these masses to compute the size of the Roche surface of these clusters their Jacobi radius and distinguish between bound and unbound clusters. BR Results We find that only 5647 79 of the clusters from our previous catalogue are compatible with bound open clusters dropping to just 11 of clusters within 250 pc. Our catalogue contains 3530 open clusters in a more strongly cut highquality sample of objects. The moving groups in our sample show different trends in their size as a function of age and mass suggesting that they are unbound and undergoing different dynamical processes. Our cluster mass measurements constitute the largest catalogue of Milky Way cluster masses to date which we also use for further science. Firstly we inferred the massdependent completeness limit of the open cluster census showing that the census is complete within 1.8 kpc only for objects heavier than 230 MSUBSUB. Next we derived a completenesscorrected age and mass function for our open cluster catalogue including estimating that the Milky Way contains a total of 1.3 10SUP5SUP open clusters only 4 of which are currently known. Finally we show that most open clusters have mass functions compatible with the Kroupa initial mass function. BR Conclusions We demonstrate Jacobi radii for distinguishing between bound and unbound star clusters and publish an updated star cluster catalogue with masses and improved cluster classifications. P Full Tables 1 and 2 are available at the CDS via anonymous ftp to A hrefhttpscdsarc.cds.unistra.frcdsarc.cds.unistra.frA ftp130.79.128.5 or via A hrefhttpscdsarc.cds.unistra.frvizbincatJAA686A42httpscdsarc.cds.unistra.frvizbincatJAA686A42A | 2024-06-01T00:00:00Z | ['10.1051/0004-6361/202348662', '2024A&A...686A..42H', '10.48550/arXiv.2403.05143', 'arXiv:2403.05143', '2024arXiv240305143H'] | ['methods: data analysis', 'catalogs', 'astrometry', 'open clusters and associations: general', 'Astrophysics - Astrophysics of Galaxies', 'Astrophysics - Solar and Stellar Astrophysics'] | Improving the open cluster census. III. Using cluster masses radii and dynamics to create a cleaned open cluster catalogue | 2,024 | 173 | 0.7 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF'] | 25 | https://arxiv.org/pdf/2403.05143.pdf | {'III. Using cluster masses, radii, and dynamics to create a cleaned open cluster catalogue ⋆': 'Emily L. Hunt 1 and Sabine Re ff ert 1 \nLandessternwarte, Zentrum für Astronomie der Universität Heidelberg, Königstuhl 12, 69117 Heidelberg, Germany e-mail: [email protected] \nReceived 17 November 2023; accepted 5 March 2024', 'ABSTRACT': 'Context. The census of open clusters has exploded in size thanks to data from the Gaia satellite. However, it is likely that many of these reported clusters are not gravitationally bound, making the open cluster census impractical for many scientific applications. Aims. We aim to test di ff erent physically motivated methods for distinguishing between bound and unbound clusters, using them to create a cleaned star cluster catalogue. \nMethods. We derived completeness-corrected photometric masses for 6956 clusters from our earlier work. Then, we used these masses to compute the size of the Roche surface of these clusters (their Jacobi radius) and distinguish between bound and unbound clusters. \nResults. We find that only 5647 (79%) of the clusters from our previous catalogue are compatible with bound open clusters, dropping to just 11% of clusters within 250 pc. Our catalogue contains 3530 open clusters in a more strongly cut high-quality sample of objects. The moving groups in our sample show di ff erent trends in their size as a function of age and mass, suggesting that they are unbound and undergoing di ff erent dynamical processes. Our cluster mass measurements constitute the largest catalogue of Milky Way cluster masses to date, which we also use for further science. Firstly, we inferred the mass-dependent completeness limit of the open cluster census, showing that the census is complete within 1.8 kpc only for objects heavier than 230 M ⊙ . Next, we derived a completenesscorrected age and mass function for our open cluster catalogue, including estimating that the Milky Way contains a total of 1 . 3 × 10 5 open clusters, only ∼ 4% of which are currently known. Finally, we show that most open clusters have mass functions compatible with the Kroupa initial mass function. \nConclusions. We demonstrate Jacobi radii for distinguishing between bound and unbound star clusters, and publish an updated star cluster catalogue with masses and improved cluster classifications. \nKey words. open clusters and associations: general - Methods: data analysis - Catalogs - Astrometry', '1. Introduction': "Data releases from the Gaia satellite have completely revolutionised the census of open clusters (OCs) (Cantat-Gaudin 2022). Since the first full Gaia data release (Brown et al. 2018), strides have been made in many aspects of the census, including ruling out many OCs reported before Gaia as asterisms (CantatGaudin & Anders 2020; Piatti et al. 2023; Hunt & Re ff ert 2021, 2023), detecting many thousands of new objects thanks to Gaia's high-precision astrometry (e.g. Liu & Pang 2019; Castro-Ginard et al. 2020), and determining cluster parameters to higher levels of accuracy than previously possible (e.g. Bossini et al. 2019; Cantat-Gaudin et al. 2020). Nevertheless, the OC census still has room for improvement, with a major issue being that current observational definitions of OCs do not seem to be robust enough to distinguish them from unbound moving groups (MGs) (Hunt &Re ff ert 2023). \nCantat-Gaudin & Anders (2020) improve on the first major catalogue of OCs in the Gaia era, (Cantat-Gaudin et al. 2018), searching for additional OCs in Gaia data and using \na set of observational criteria to distinguish between plausible OCs and asterisms. Their criteria are as follows: firstly, a candidate OC should be a clear overdensity, including that it has at least roughly ten member stars; secondly, it must have a colourmagnitude diagram (CMD) that follows a clear isochrone, indicating that a given cluster is a co-evolutionary population of stars with the same age and chemical composition; and finally, a candidate OC must pass criteria that distinguish asterisms that cannot physically be gravitationally bound from potential bound OCs: namely, a median radius r 50 less than 15 pc, and a proper motion dispersion that corresponds to an internal velocity dispersion smaller than 5 km s -1 (or 1 mas yr -1 for distant clusters where Gaia measurement uncertainties are dominant.) \nThe criteria on the density (or number of stars) and CMD quality of an OC candidate are common practice in the literature. For instance, Froebrich et al. (2007), Cantat-Gaudin et al. (2019), and Hunt & Re ff ert (2021) (hereafter Paper I) require a candidate new cluster to be a clear overdensity, while works such as Platais et al. (1998), Castro-Ginard et al. (2018), and Liu & Pang (2019) are examples of works that use cluster CMDs to validate candidate new objects. The Cantat-Gaudin & Anders (2020) criteria on the possibility of a cluster being bound have also been adopted in the literature, with works such as Hunt & \nRe ff ert (2021) and Castro-Ginard et al. (2022) using them to validate new OC candidates. \nIn Hunt & Re ff ert (2023) (hereafter Paper II), we used Gaia DR3 data (Gaia Collaboration et al. 2023) to construct a large, homogeneous catalogue of star clusters. However, despite our methodology being originally intended to only detect OCs, many of the clusters we detected appear to be MGs, often having sparse or 'flat' stellar distributions - unlike the clustered appearance of canonically bound OCs such as the Pleiades. Many of the suspected MGs we detected are consistent with being single populations of co-evolutionary stars based on our CMD classifier in Paper II, and most of the MGs we detected still pass the observational criteria on the boundness of an OC proposed in CantatGaudin & Anders (2020). In Paper II, we suggested that these criteria are too permissive to accurately classify the many sparse star clusters we are able to detect near to the Sun. The inability to distinguish precisely between bound and unbound clusters limits the scientific usability of catalogues such as the one in Paper II, with a significant proportion of the catalogue's content being likely to be MGs - particularly within around 1 kpc from the Sun. \nIn this work, we aim to create a new way to distinguish between bound and unbound clusters, utilising a relationship between the mass and Jacobi radius of a self-gravitating cluster. With this method, we aim to classify all objects in Paper II into bound and unbound objects, in addition to demonstrating the applicability of this method to future studies of the Milky Way's clusters, such as with upcoming data releases like Gaia DR4. Firstly, we provide an overview of some background theory in Sect 2. In Sect. 3, we outline how we calculate cluster masses and radii, including how we correct for selection e ff ects and unresolved binary stars. Section 4 outlines the results of this work, including how many clusters in Paper II are bound OCs and how they are distributed. We explore the results of this work further in Sect. 5, including using our cluster masses to estimate the massdependent completeness of the Gaia DR3 OC census, the age and mass functions of the OC census, and the compatibility of clusters with a Kroupa IMF (Kroupa 2001). Section 6 concludes this work.", '2. Theoretical relations on the boundness of a cluster': 'In this section, we review some theory on how the boundness of a star cluster could be measured. We aim to find relations that can be straightforwardly applied to Gaia data of star clusters.', '2.1. The virial theorem': "One of the most common and widely used relations in astrophysics is the virial theorem, which states that a system under the influence of gravitation and in equilibrium should have twice as much kinetic energy T as it has potential energy U , 2 T = | U | . For a star cluster with a distribution function following a Plummer (1911) model, this implies that a star cluster in virial equilibrium should have a one-dimensional velocity dispersion σ 1D equal to an ideal virial velocity dispersion σ vir given by (Portegies Zwart et al. 2010): \nσ vir = s GM η r hm ≈ σ 1D for a bound cluster , (1) \nArticle number, page 2 of 20 \nwhere G is the gravitational constant, M is the cluster's mass, rhm is the cluster's half-mass radius, and η is a constant equal to ∼ 10 for a typical cluster - although it can be as low as ∼ 5 or as high as ∼ 30 depending on the cluster's spatial distribution (Portegies Zwart et al. 2010). This relation is used to analyse the dynamics of a small subset of nearby OCs in works including Bravi et al. (2018), Kuhn et al. (2019), and Pang et al. (2021). \nEquation 1 may o ff er some explanation on why the individual empirical cuts presented in Cantat-Gaudin & Anders (2020) seem to be inadequate to distinguish between OCs and MGs from Paper II. As an example, consider a small cluster with a radius of r 50 = 1 pc and a velocity dispersion of around 2 kms -1 . Equation 1 predicts that this cluster would need a mass of ∼ 10 4 M ⊙ to be virialised - a value far higher than almost all OCs in the Milky Way, and clearly unrealistic for typical small MGs that we detect in Paper II. Instead of adopting individual radius and velocity dispersion cuts, it appears that the expected radius and velocity dispersion of OCs must be 'calibrated' individually based on a cluster's mass. \nHowever, during the preparation of this work, we found this relation impossible to apply successfully to all clusters in Paper II. Velocity dispersions are easily contaminated by Gaia measurement uncertainties, binary stars, unbound stars (including stars in cluster tidal tails), perspective expansion, and interloping field stars. For instance, we found that binary stars (resolved or unresolved) often contribute 500 m s -1 or more to cluster velocity dispersions derived using proper motions, and incorrect removal of tidal tails can contribute as much as 2 km s -1 in the worst cases. In addition, Gaia measurement uncertainties become dominant in the proper motion dispersion of most clusters above a few kpc, making it di ffi cult to make a meaningful measurement of σ 1D for many clusters. Accounting for all of these e ff ects for all clusters in Paper II and arriving at accurate measurements of σ 1D was not possible. In addition, it is predicted theoretically that star clusters are often supervirial, such as during phases of expansion for young clusters (Banerjee & Kroupa 2017; Krause et al. 2020) - making it di ffi cult to make a scientifically motivated cut on σ vir in many cases, as the velocity dispersion of many bound clusters can be expected to be supervirial, with measurements in multiple studies supporting this hypothesis (e.g. Bravi et al. 2018; Kuhn et al. 2019; Pang et al. 2021).", '2.2. Jacobi radii': "Using currently available data, we found it was much more successful to only rely on cluster masses and radii alone to distinguish between OCs and MGs. An instantaneously bound cluster should have a Roche surface, within which its potential is stronger than that of its host galaxy. In principle, a cluster with no radius at which its potential is stronger than that of the Milky Way will have no Roche surface, and is hence not selfgravitating. The Roche surface of a given cluster can be measured by considering its Jacobi radius, rJ , which is the distance from the centre of a cluster to its L 1 Lagrange point. This is given by (Portegies Zwart et al. 2010; Ernst et al. 2011): \nrJ = GLYPH<18> GM 4 Ω 2 -k 2 GLYPH<19> 1 3 , (2) \nwhich relates rJ to the mass M of a cluster, weighted by the circular frequency Ω and epicyclic frequency k of the cluster's orbit around its host galaxy, assuming that the orbit is circular. Outside of rJ , the host galaxy's potential is dominant - such as \n- \n/circledot \nFig. 1. CMDofmember stars of NGC 2451A shaded by their calculated stellar mass. 100 sampled isochrones for the cluster from Paper II are shown in black. (Adapted from Hunt 2023) \n<!-- image --> \nfor stars in the tidal tails of a cluster, which are no longer bound to their parent cluster (Meingast et al. 2021). Assuming that a cluster fills its Roche surface, rJ ≈ rt from a King (1962) model fit (Binney & Tremaine 1987), with this relationship being used by works such as Piskunov et al. (2008) to derive the masses of OCs based on their size. \nDespite the fact that some open clusters, such as the Hyades, have been shown to have higher than expected stellar velocity dispersions and are supervirial and dissolving, these dissolving clusters are nevertheless dense enough to be currently selfgravitating (Oh & Evans 2020; Meingast et al. 2021). The same applies for young clusters that have recently been observed to be in possible supervirial expansion phases not long after their initial formation (Kuhn et al. 2019). On the other hand, MGs of all kinds (including sparse OB associations) are not bound, and are actively dissolving into the disk, meaning that they should have no radius at which they have a Roche sphere, which should be possible to measure with Eqn. 2. For the remainder of this work, we aim to apply this equation to distinguish between bound and unbound star clusters.", '3. Mass and radius calculations': 'In this section, we describe how we calculated photometric masses and Jacobi radii for all clusters within 15 kpc from Paper II. Much of this method was originally described in Hunt (2023), but is outlined again here to aid in the reading of this work. This method contains five steps that we discuss in the following subsections. Firstly, we derived the photometric masses of member stars in every cluster. Next, we corrected for selection e ff ects. We then applied a correction for unresolved binary stars. Mass functions were then fitted and integrated to calculate total cluster mass. Finally, this process was repeated at di ff erent radii to find the Jacobi radius of each cluster.', '3.1. Calculation of stellar masses': 'Following a similar method to that used by works including Meingast et al. (2021) and Cordoni et al. (2023), we began by using PARSEC (Bressan et al. 2012) isochrone fits from Paper II to \nestimate the masses of member stars of each cluster. To calculate stellar masses, we used the predicted mass of stars as a function of G -band magnitude from our fitted Paper II isochrones, m ( G ), which was accurate for most cluster members. However, the oldest clusters in our sample often contain evolved giant stars whose G -band magnitude is less than the tip of the main sequence in the cluster - meaning that m ( G ) is not a one-to-one mapping from magnitude to mass for some cluster members. Therefore, in regions where our fitted PARSEC isochrones do not have a oneto-one mapping from magnitude to mass, we also used BP -RP colour indices to decide on the best stellar mass for a given cluster member. We elected not to use the BP -RP index of most stars as BP and RP magnitudes are frequently underestimated for very red or blue stars with magnitudes G ≳ 19 (Riello et al. 2021). The regions where we do use BP -RP colour indices were not within ranges where BP and RP are underestimated, however, due to the blue colour of these regions and due to our studied clusters being within 15 kpc. \nTo incorporate the uncertainty on our isochrone fits from Paper II, we repeated this process 100 times for 100 sampled isochrones from our variational inference neural network in Paper II. This incorporated uncertainties on the age, extinction, and distance to stars into our mass estimates for them. Figure 1 illustrates this process for NGC 2451A, showing 100 sampled isochrones from Paper II and our estimated stellar masses for each star with the shading of points. \nIt is worth discussing further the limitations and assumptions of this method. Firstly, since our Paper II photometric parameters do not include metallicities, our masses are biased for clusters with particularly low or high metallicities. To quantify this systematic, we repeated our entire pipeline on 143 randomly selected clusters but assuming metallicities of [Fe / H] = + 0 . 5 and -0 . 5 dex, testing how masses change given [Fe / H] values at the upper and lower limit of those observed in OCs (Kharchenko et al. 2013; Bossini et al. 2019). Assuming a high metallicity increases masses by an average of + 7%, while a low metallicity decreases masses by an average of -12%. The mean metallicity of OCs is approximately solar (Kharchenko et al. 2013), and so these values are edge-case limits that will mostly impact clusters at particularly high or low galactocentric radii which are most likely to have non-solar metallicities (Spina et al. 2022). In the future, it will be important to include spectroscopic metallicity estimates in machine learning OC parameter inference to improve the accuracy of OC masses further. \nNext of note is that the e ff ects of binary stars were not included in our interpolation scheme. Our stellar mass estimates are hence only estimates of the mass of the primary star in any binary system. To mitigate this e ff ect, we applied a correction to the overall cluster mass function for unresolved binaries in Sect. 3.3. \nFinally, our use of PARSEC isochrones also influences our derived stellar masses, and our mass estimates may di ff er to those derived using other stellar evolution models. We investigated how using MIST isochrones (Choi et al. 2016) would change our mass estimates, using limited comparisons between our fitted PARSEC isochrones and MIST isochrones at the same age. m ( G ) for PARSEC and MIST isochrones is generally very similar for m ≳ 0 . 8 M ⊙ at all ages, and hence m ( G ) for clusters at distances greater than 1 kpc (where almost all observed stars are greater than this mass) will be similar. We estimate that MIST-derived total cluster masses would still be lower than PARSEC ones for such clusters, although no more than ∼ 5% lower. However, m ( G ) is appreciably lower in MIST compared to PARSEC for stars with masses below ∼ 0 . 8 M ⊙ at all ages, \nFig. 2. Computed cluster selection functions for Blanco 1 (top row) , Ruprecht 134 (middle row) , and Berkeley 72 (bottom row) . The left panel in each row shows our adopted Gaia (blue), subsample (orange), and algorithm (HDBSCAN, red) selection functions as a function of magnitude for each cluster, in addition to the multiplicative total selection function (purple). The CMD of each cluster is shown for reference on the right panels. (Adapted from Hunt 2023) \n<!-- image --> \nmeaning that low mass stars would be assigned lower masses by MIST isochrones given the same brightness. We estimate that this would result in total cluster masses that are at most ∼ 15% lower for clusters within 300 pc (which have mass functions most strongly impacted by the low-mass stars where MIST and PARSEC isochrones have the weakest agreement.)', '3.2. Correction for selection effects': "Although our Paper II cluster membership lists aimed to be as complete as possible, with membership lists including stars as faint as G ∼ 20, there remain a number of selection e ff ects that limit the completeness of our membership lists that must be accounted for to derive accurate cluster masses. From inspection of the CMDs of clusters that are challenging to recover, such as those in regions of high crowding where Gaia data becomes incomplete (Gaia Collaboration et al. 2021) or distant clusters where our adopted clustering technique can miss member stars (Paper II), there are clearly selection e ff ects that would otherwise influence our derived mass functions. In this subsection, we describe how we model the selection e ff ects that impact each of our cluster membership lists. We refer readers to Hunt (2023) for more detail on our method. \nWe consider three di ff erent e ff ects that could result in a real star to be missing from our membership lists. Firstly, there is the probability that a given star with parameters q appears in the Gaia DR3 catalogue of 1.8 billion sources, S Gaia( q ). Then, there is the conditional probability that a source in Gaia DR3 was included in the subset of Gaia data that we used for clustering analysis, which is all 729 million stars with a full astromet- \nric solution, BP and RP photometry, and a Rybizki et al. (2022) v1 quality flag of greater than 0.5, S subsample( q | q in Gaia). Finally, there is the additional probability that our adopted clustering algorithm in Paper II, HDBSCAN (Campello et al. 2013; McInnes et al. 2017), assigns this star as a member of the cluster, S algorithm( q | q in subsample) - which is decreasingly likely depending on how clearly separated a cluster is from the surrounding field. These e ff ects are multiplicative (Rix et al. 2021; Castro-Ginard et al. 2023), hence giving an overall probability S cluster( q ) that a star with parameters q appears in our adopted cluster membership list: \nS cluster( q ) = S Gaia( q ) · S subsample( q | q in Gaia) · S algorithm( q | q in subsample) . (3) \nThe first two terms, S Gaia( q ) and S subsample( q | q in Gaia), are calculated directly from the works of Cantat-Gaudin et al. (2023) and Castro-Ginard et al. (2023). In the first work, Cantat-Gaudin et al. (2023) derive an empirical probability that a source appears in Gaia DR3 by comparing the Gaia dataset to photometric surveys deeper than Gaia itself. They describe the probability that a source is included in Gaia based on its position, which is a good tracer for the extent of crowding in a given region, in addition to its G -band magnitude, which is a strong predictor of how well it would be processed by the Gaia telescope and data processing pipeline. Values of S Gaia( q ) as a function of position and magnitude were queried directly from the gaiaunlimited Python package 1 (Cantat-Gaudin et al. 2023). \nNext, Castro-Ginard et al. (2023) outline a method to determine the probability that a source appears in a given subsample of the Gaia dataset S subsample( q | q in Gaia), using a method from Rix et al. (2021). We implemented the empirical CastroGinard et al. (2023) method as a function of position and G -band magnitude alone, which we found to be good predictors of the probability of a source being in our adopted subsample of the Gaia dataset. The subsample of Gaia data we used in Paper II was largely to restrict our analysis to only sources with a good-quality astrometric solution, which is strongly influenced by the position and brightness (S / N) of a source. Since the method in Castro-Ginard et al. (2023) bins sources to calculate S subsample( q | q in Gaia), we selected all stars in the on-sky region a given cluster covers and binned them by G -band magnitude in bins of size 0.2 mag. To prevent under-sampling of bins for bright sources, bins were merged until every bin contained at least ten stars. \nFinally, to model the impact of incompleteness due to the clustering algorithm we used in Paper II, we developed a stochastic technique to simulate the probability of a true member star of a given cluster being assigned as a member by the algorithm. The chance of a star being correctly assigned as a cluster member depends strongly on its astrometric precision (Paper II). For a star with less astrometric precision, its position in the five dimensional Gaia astrometry that we performed clustering for will be further away from the centre of a cluster, meaning that it is more likely to be missed by our clustering algorithm. This is particularly the case for distant clusters, where Gaia uncertainties are often larger than the true underlying parallax or proper motion dispersion of a cluster. Since the clusters in this work are generally no smaller than 0.1 · in angular extent, astrometric errors on star position in Gaia DR3 are negligible compared to the size of clusters, and so this e ff ect is only dependent on the proper motion and parallax precision of cluster members. \nThis e ff ect was modelled by performing simulations of whether or not stars with simulated astrometry would appear within a cluster. For every cluster, we simulated 100 000 stars with magnitudes uniformly distributed in the range 2 < G < 21. Astrometric errors were assigned to each star by randomly selecting stars with a similar magnitude in the vicinity of the real cluster and using their errors directly. Then, ten random samplings of the proper motions and parallaxes of each star were performed. To estimate whether or not each simulated star would be assigned as a member of a given cluster, we fitted a 3D ellipsoid to the proper motions and parallaxes of our Paper II cluster members for each cluster, and then calculated how often each simulated star appeared within each fitted ellipsoid to calculate the probability that a given star was included in our membership list. \nThe estimated selection functions and the CMDs of three OCs are shown in Fig. 2, showing how di ff erent OCs have CMDs dominated by di ff erent e ff ects. In the first case, Blanco 1 is a high galactic latitude, nearby ( d = 240 pc) cluster that is easy to detect and is clearly separated from the field. It has a CMD that is visually well-populated to magnitudes fainter than even G ∼ 20. This is reflected in its estimated selection function, which is largely complete for G < 19. On the other hand, Ruprecht 134 is a cluster in an extremely crowded field near the galactic centre ( d = 2 . 3 kpc), being one of the most incomplete clusters in our Paper II catalogue. It is strongly impacted by the selection e ff ect of our subsample, which removes a large number of sources with anomalous astrometry due to crowding - in addition to the selection function of Gaia DR3, which reduces sharply at G ∼ 20 in this region. Finally, Berkeley 72 is a more distant cluster ( d = 5 . 1 kpc). Owing to its distance and relative sparsity, its selection function is mostly dominated by the selection function of our clustering algorithm, although the subsample selection function also makes a contribution due to the cluster's location in a somewhat crowded region of the galactic disk. From these three examples alone, it is clear that all three selection e ff ects impact every cluster in di ff erent ways that must all be considered. \nIn addition, it is worth noting that no cluster is estimated to be 100% complete at any magnitude. We suggest that many of these potentially missing stars are likely to be multiple stars. During the processing of Gaia DR3, all stars were assumed to be single; however, binaries with large deviations from ideal singlestar astrometry will have higher errors in their astrometric fits (Lindegren et al. 2021), and are less likely to appear in the subsample of stars with good-quality astrometry that our Paper II catalogue was constructed from.", '3.3. Correction for unresolved binaries': "The next step in our method corrected for unresolved binary stars. Since our inferred stellar masses in Sect. 3.2 assumed that stars are single, an additional correction for unresolved binaries is important to avoid our final cluster masses being biased to low values. \nIdeally, it would be possible to directly detect all binaries in a given cluster and measure the mass ratio q of each binary system, using this as a correction to each star's estimated mass. However, such direct measurements are not possible based on Gaia DR3 data alone, and particularly not for all 7167 clusters in Paper II. Some works have recently studied the binary star fraction in a subset of reliable OCs, including Cordoni et al. (2023) who measure it for 78 OCs and Donada et al. (2023) who measure it for 202 OCs within 1.5 kpc. Nevertheless, both works are only able to measure the binary fraction for mass ratios q ≳ 0 . 6, \ndue to the di ffi culty of distinguishing between low-mass ratio binary stars and single stars on the main sequence, especially in the presence of di ff erential reddening. Particularly since the binary star fraction in the solar neighbourhood has been tentatively shown to peak at q ∼ 0 . 3 for most stars below 2 to 5 M ⊙ (Moe & Di Stefano 2017), meaning that many stars are most likely to have binaries with q below values that can be measured with Gaia DR3, we conclude that robust direct measurements of the mass ratio of most binaries in OCs are not possible, and instead used an approximate correction to our adopted stellar masses that accounts for binaries at all q values. \nWe derived corrections to apply to our final cluster mass functions by using the selection e ff ect corrected multiplicity fraction, companion star frequency, and mass ratio distribution of field stars from Moe & Di Stefano (2017). Binary stars for each cluster were simulated based on these distributions. To simulate whether a binary is resolved, which is frequently the case for nearby clusters (Donada et al. 2023), the period and eccentricity distributions from Moe & Di Stefano (2017) were used to simulate the mean separation of each simulated binary, which was then compared to the angular resolution of Gaia DR3 (Gaia Collaboration et al. 2021). Depending on the distance to a cluster and the mass range of the mass function bin to be corrected, this binary star correction increases mass bins by 10% to 30%, while also inflating our quoted uncertainties on cluster masses significantly, owing to the approximate nature of this method. We estimate that in the worst cases, errors due to our assumed field-like binary star population could contribute additional systematics of up to ∼ 20%onour final cluster masses. In the future, it will be important to improve methods to determine which stars are binaries to improve the accuracy of OC masses further.", '3.4. Mass function fits': "The final step in our cluster mass measurement pipeline was to fit a mass function to each cluster and integrate it to derive a total cluster mass, including stars too faint to be observed. There are a number of di ff erent functional forms for mass functions that can be adopted (Krause et al. 2020), with a common form being a broken power law with a break point at a value like 0.5 M ⊙ (Kroupa 2001). Some works, such as Cordoni et al. (2023), derive OC masses while fitting bespoke broken power law mass functions to every cluster. However, since most clusters in our sample are more distant than 1 kpc, or have AV ≳ 2, they contain few or no stars below the typical mass function break point of ∼ 0 . 5 M ⊙ , making it impossible to fit two-part mass functions to them. Extrapolating single power law mass functions measured for high mass stars down to lower stellar masses would cause severe over-estimates of our cluster masses in these cases, since it has been robustly measured that clusters form with mass functions that are significantly less steep at masses of around 0.5 M ⊙ and below (Krause et al. 2020). Instead, we adopted a 'safer' approach, and fit only a Kroupa (2001) IMF (hereafter Kroupa IMF) to every cluster. \nTo fit cluster mass functions, we used the imf Python package 2 and performed least squares fitting of the amplitude of each cluster mass function after correcting for selection e ff ects and unresolved binary stars. In general, we found that the majority of clusters had mass functions well approximated by a Kroupa IMF, albeit only after correcting masses for selection e ff ects and binary stars. Figure 3 shows mass functions for the three clusters from Fig. 2, all of which have slopes well approximated by \n/circledot \nFig. 3. Mass functions for the three OCs from Fig. 2. Original binned stellar masses are shown by the blue squares, while binned masses corrected for selection e ff ects and unresolved binary stars are shown by the orange squares. The dashed black line shows our fitted Kroupa IMF, with our calculated total cluster mass and corresponding uncertainty in the top right. (Adapted from Hunt 2023) \n<!-- image --> \nKroupa IMFs after incorporating corrections. Section 5.3 compares our cluster mass functions to the Kroupa IMF further. \nFinally, to convert our fitted IMF into a total cluster mass, each fitted IMF was integrated from a lower limit of 0.03 M ⊙ to the highest observed stellar mass in the cluster. This lower limit is slightly lower than the 0.08 M ⊙ lower limit used in some other works (e.g. Meingast et al. 2021) which corresponds to the minimum mass at which nuclear fusion still occurs. Our lower limit of 0.03 M ⊙ intentionally also includes brown dwarfs, which are also observed in OCs (Moraux et al. 2003) - but stops short of integrating from 0 M ⊙ , as companion objects around stars with masses below 0.03 M ⊙ are often considered planets and have a poorly constrained IMF (Akeson et al. 2013), and the quantity of these objects that are free-floating is also poorly constrained. Nevertheless, the choice of lower limit makes a negligible di ff erence on the final cluster mass on the order of ∼ 1%.", '3.5. Jacobi radius inference': "The last step of our method was to calculate the mass of each cluster at all radii, which was then compared against the theoretically predicted Jacobi radius of a cluster of that mass and radius. This produced a probability that a given cluster has some radius rJ at which its gravitational potential is stronger than that of the Milky Way, hence measuring whether a given cluster is self-gravitating and (currently) bound. \nFirstly, we repeated our mass measurement pipeline at all cluster radii, deriving cluster mass as a function of cluster radius \nM obs( r ). We did not consider cluster radii where a cluster had fewer than ten member stars, as OCs are usually defined to contain at least ten member stars to di ff erentiate them from multiple star systems (Cantat-Gaudin & Anders 2020; Portegies Zwart et al. 2010). In addition, we calculated the total cluster mass including all assigned member stars (such as tidal tails) MA . \nNext, we used the method of Meingast et al. (2021) to calculate the theoretical Jacobi mass as a function of radius for each cluster, MJ ( r ), by inverting Eqn. 2. One must assume a model of the galactic potential to calculate Ω and k within Eqn. 2, for which we used the galpy MWPotential2014 model of the Milky Way's potential (Bovy 2015). This potential is smooth, and does not include spiral arms or giant molecular clouds (GMCs), but was fit to a wide variety of data and should be accurate enough for our circumstances. In practice, rJ depends relatively weakly on the assumed potential model, due to its cube root dependence on the quantity 3 √ 4 Ω 2 -k 2 calculated from the potential. Within our adopted potential model, this quantity is only a factor of four larger between clusters at the lowest galactocentric radii in this study ( ∼ 2 kpc) and the highest ( ∼ 20 kpc). In addition, we are most interested in this work in distinguishing between MGs and OCs in the solar neighbourhood, for which we expect the galactic potential to be most well determined by this model. These frequencies could of course be less accurate at higher distances from the Sun, for which the galactic potential is not as well constrained. \nNevertheless, the smoothness of our adopted potential model could be a source of bias. Due to their increased gas density relative to the rest of the galaxy, GMCs and spiral arms will have a stronger potential, with collisions with GMCs and spiral arms being major contributors to mass loss and destruction of OCs (Krause et al. 2020). Given the weak dependence of Eqn. 2 on our assumed potential model, this e ff ect is likely to be small for spiral arms, which have been measured to have around a 50% increase in gas mass (Colombo et al. 2022) - which results in only a small change in local potential, as the local potential acting on an OC in a spiral arm will still be dominated by the Milky Way's (assumed) smooth dark matter halo (Bovy 2015; Cautun et al. 2020). However, due to their significantly higher gas density, the potential in a typical GMC will be significantly higher (Krause et al. 2020). Since we are most interested in classifying suspicious candidate new OCs in the solar neighbourhood, it is somewhat fortunate that the Sun is within a bubble that contains little gas (Zucker et al. 2023). As an additional check, we matched our catalogue against the catalogue of nearby molecular clouds in (Cahlon et al. 2024). Only one cluster (HSC 598) is within 25 pc of a molecular cloud, at a separation of 8 pc. However, this cluster is already classified as an MG by our pipeline below, and so the influence of its neighbouring molecular cloud on our final results is negligible. Future work using deeper datasets (such as Gaia DR4 or DR5) will presumably work with a deeper catalogue that includes more clusters out to larger distances, and hence the influence of GMCs and spiral arms on the local potential surrounding clusters may have to be considered. \nFinally, with theoretical values of MJ ( r ) calculated for each cluster, MJ ( r ) and M obs( r ) were compared to identify a possible Jacobi radius for each cluster. If a cluster has some radius r at which the enclosed mass within this radius M obs( r ) = MJ ( r ), then r is taken as the cluster's Jacobi radius rJ , with corresponding enclosed mass MJ . In cases where M obs( r ) < MJ ( r ) at all radii, the cluster is considered to have no valid Jacobi radius and is an MG. In addition, some clusters have M obs( r ) > MJ ( r ) at all radii, meaning that the observed cluster is smaller than the \n5 \nFig. 4. Jacobi radius calculation method shown for the three OCs from Fig. 2, with each row corresponding to each OC. Within each row, the left panel shows the cluster mass as a function of radius from the centre of the cluster, where the blue line is our calculated total cluster mass with a shaded uncertainty region, and the orange line is the theoretical Jacobi mass for a cluster of that size given Eqn. 2. The intersection of these lines is the cluster's rJ . The right panel in each row shows the cluster in an arbitrary coordinate frame centred on the cluster centre. Member stars within rJ are shown in orange, with member stars outside of rJ shown in blue. rJ is indicated by the dashed red line. The dotted purple line denotes our approximate calculated King (1962) tidal radius for each cluster from Paper II. (Adapted from Hunt 2023) \n<!-- image --> \nJacobi radius of the cluster given its observed mass. This is the case for some distant or di ffi cult to detect clusters, for which it is likely that we only observe the innermost parts of the cluster, and do not detect stars out to the true cluster rJ . In these cases, we used the theoretical rJ of all observed stars in the cluster as the cluster's rJ , although such values probably underestimate the true cluster rJ . \nM obs( r ) and MJ ( r ) are shown in Fig. 4 for Blanco 1, Ruprecht 134, and Berkeley 72. As would be expected for these reliable clusters, all of them clearly have radii at which their enclosed mass is higher than the theoretical Jacobi mass, implying that they are self-gravitating bound clusters. As previously discussed, Ruprecht 134 is a di ffi cult to detect cluster, for which we find M obs( r ) > MJ ( r ) at all radii, implying that additional member stars in the outskirts of this cluster are yet to be detected. \nHowever, since we are interested in using Jacobi radii to distinguish between bound and unbound clusters, the performance of this method on suspect clusters is most relevant. In the penultimate section of Paper II, we highlighted three example candidate new OCs - two of which appeared particularly suspicious due to their on-sky distributions not showing a clear 'clumpy' cluster core as would be expected for an OC (King 1966). Figure 5 shows the mass as a function of radius of these three clusters. The two clusters that we highlighted as looking unlike OCs, HSC 1131 and HSC 2376, have M obs( r ) < MJ ( r ) at all radii, strongly suggesting that they are indeed unbound MGs. On the other hand, HSC 1186, a cluster that does have a small central clump, appears compatible with being a small bound OC with a bound mass of M ∼ 65 M ⊙ , as was also expected in Paper II. \nIn total, it took less than 10% of the total CPU wall time to perform mass and radius calculations as our original cluster- \nFig. 5. Same as Fig. 4, but for three candidate new OCs from Paper II: HSC 1131 (top row) , HSC 2376 (middle row) , and HSC 1185 (bottom row) . Although HSC 1131 and HSC 2376 do not appear to have a Jacobi radius, their most likely Jacobi radius is still shown in the plots in the right column. (Adapted from Hunt 2023) \n<!-- image --> \nFig. 6. Distribution of P ( rJ ) for all clusters in this work, which is the probability that a cluster has a valid Jacobi radius. (Adapted from Hunt 2023) \n<!-- image --> \ning analysis in Paper II took. Not only does this method appear viable for distinguishing between OCs and MGs, it is also not especially computationally challenging, meaning that it could feasibly be incorporated into any future cluster blind searches.", '4. Results': "We derived Jacobi radii and masses for 6956 clusters that are not globular clusters and that are closer than 15 kpc. Since clusters further than 15 kpc away were not included in the training data for our Paper II neural network, their age and extinction estimates were too unreliable to be used. An attempt was made to fit isochrones to cluster photometry using other methods, although most of these distant clusters had poor quality CMDs, making it impossible to derive accurate estimates of their parameters using Gaia data alone. These distant clusters should be investigated separately in a di ff erent work, particularly since our adopted model of the Milky Way's potential used to calculate Ω \nTable 1. Catalogue of star clusters with masses, object classifications, and Jacobi radii. \nNotes. Shown for a random selection of ten OCs and ten MGs in the high-quality sample within 250 pc. Errors on masses are in the brackets. The full table is available in the online material, and includes all parameters derived in Paper II. \nand k may be less accurate at distances greater than 15 kpc. In the next section, we present these overall results and compare our cluster masses with literature values.", '4.1. Updated definitions for clusters from Paper II': "Thanks to our Jacobi radius inference method, we are now able to provide updated definitions for the clusters in our Paper II catalogue. In the following section, we discuss how the incorporation of this method changes our catalogue. \nWe calculated the probability that a cluster has a valid Jacobi radius, P ( rJ ), the distribution of which is shown in Fig. 6. 827 clusters are strongly incompatible with having a bound component, with P ( rJ ) < 0 . 05. On the other hand, around 5733 clusters are strongly compatible with having a valid Jacobi radius ( P ( rJ ) > 0 . 95), with 397 clusters having values between these two limits. Masses and Jacobi radii appear to be a successful method for di ff erentiating between bound and unbound objects. Unlike previous attempts to use the virial theorem to discriminate between OCs and MGs during the preparation of this work (see Sect. 2), the probability that a given cluster has a valid Jacobi radius is more successful at distinguishing between bound and unbound clusters. \nNevertheless, our current method still appears to have limitations at the low-mass end. Some MGs from Paper II, such as the densest region of the β Tucanae MG, are measured as having small Jacobi masses - typically less than 40 M ⊙ , but often lower than 20 M ⊙ . While this suggests that these clusters have compact low-mass bound regions that lie somewhere between the definitions of a multiple star system or a star cluster, there are also multiple reasons why these low Jacobi radii may be errors. Firstly, by assuming a Kroupa IMF, our mass estimates will be biased towards conservative, higher values for dynamically evolved stellar groups that have lost low mass stars in long-term two-body \ninteractions. In these cases, a dense group of a dozen high mass stars will have an overestimated total mass using our method, which will be especially the case for MGs that are mass segregated. Secondly, an implicit assumption in our use of Eqn. 2 is that a star cluster is spherically symmetric (Binney & Tremaine 1987). This assumption may break down for small groups of a dozen stars in the densest region of an MG. Some examples of low-mass components of MGs that appear to have a valid Jacobi radius clearly violate this assumption, and may hence be erroneously measured as having a valid Jacobi radius. \nConsequently, we also recommend using an additional minimum MJ of 40 M ⊙ when deciding between OCs and MGs. This lower limit is higher than the MJ of even the smallest widely accepted OCs, such as Melotte 111 (Coma Ber) or Platais 9, but excludes edge cases that appear to be dense regions of MGs where our method breaks down, or cases that may be better classified as a resolved multiple star system. Clusters below this mass limit that have a measured high value of P ( rJ ) would still be interesting objects for a follow-up study on why some MGs appear to have dense cores. Such dense cores could, for instance, be the remnant of a dissolved OC. \nIn total, our Paper II catalogue contains 5647 clusters (82%) with P ( rJ ) > 0 . 5, a minimum mass of 40 M ⊙ , and at least ten observed stars within rJ . In the solar neighbourhood, most clusters from Paper II are classified as MGs, with 11% (26 of 234) clusters within 250 pc being compatible with our OC definition. Within 100 pc, there are only two OCs: Melotte 25 (the Hyades) and Melotte 111 (Coma Ber). Of the new clusters reported in Paper II, 1441 of 2387 are compatible with being OCs, or 487 of the 739 high quality new clusters from Paper II. This is in line with our belief in Paper II that a significant fraction of our newly reported clusters did not appear to be OCs. Surprisingly, seven new clusters reported in Paper II within 250 pc are compatible \nTable 2. Member stars of OCs and MGs within 15 kpc with individual stellar masses. \nNotes. Shown for ten member stars of Blanco 1. The full table is available in the online material, and includes all parameters listed in the Paper II table of member stars from Gaia DR3. \nTable 3. Total counts of cluster types. \nNotes. ( a ) Count of how many clusters of a given type are also in the high-quality sample of clusters from Paper II, which are those with a median CMD class greater than 0.5 and an astrometric S / N (CST) greater than 5 σ . ( b ) Clusters defined as GCs in Vasiliev & Baumgardt (2021), Kharchenko et al. (2013), or Gran et al. (2022). ( c ) Clusters later matched to galaxies or dwarf galaxies, or removed due to being obvious clustering algorithm errors (see Sect. 4.1). \nwith being OCs, although all but one have low masses of 66 M ⊙ or lower. \nAnupdated version of the Paper II catalogue including object classifications and masses is given in Table 1. In addition, an updated version of the Paper II stellar membership lists for each cluster (including individual stellar masses) is given in Table 2. Table 3 shows overall statistics on the total number of clusters by object type and sample. \nIn addition, some naming updates were incorporated into the catalogue. Firstly, eleven additional clusters were updated to be labelled as GCs: firstly, HSC 134 and HSC 2890, which are in fact Gran 3 and Gran 4 and were already reported in Gran et al. (2022). In addition, Palomar 2, 6, 8, 10, 11, and 12, IC 1276, 1636-283 (whose name was changed to the more widely used ESO 452-11), and Pismis 26 were updated to be labelled as GCs as in Kharchenko et al. (2013) and Perren et al. (2023). \nNext, 17 clusters clearly compatible with galaxies, dwarf galaxies, or errors in our clustering algorithm are flagged in the catalogue (Type = r ). These clusters were highlighted by members of the community in the months since the publication of Paper II (Großschedl, private communication; Alessi, private com- \nunication), and should not be used in studies of galactic star clusters. \nFinally, we incorporated a handful of naming corrections from Perren et al. (2023), Teutsch (private communication), and Röser and Schilbach (private communication). These corrections are listed in Table A.1, and include corrections to typos, changes to certain names for consistency with other clusters with the same designation, changes to clusters from Liu & Pang (2019) to have the more common designation 'LP' instead of 'FoF', and incorporation of a paper missed from crossmatching in Paper II and by other input OC catalogues (Kronberger et al. 2006). Clusters retain the same id number between this work and Paper II, and previous names from Paper II are listed in the full online catalogue.", '4.2. Overall catalogue distributions': "In this subsection, we compare di ff erences between the spatial and parameter distributions of OCs and MGs. Figure 7 shows the distribution in Cartesian heliocentric coordinates of the highquality samples of OCs and MGs. In Paper II, we remarked that our catalogue had a nonphysical peak in density near to the Sun, with many hundreds of additional clusters compared to catalogues such as that of Cantat-Gaudin & Anders (2020), which we suggested were MGs. Figure 7 confirms this hypothesis, showing that objects now classified as MGs were responsible for the density peak near to the Sun, as MGs are all at much lower distances. As suspected from Paper II, MGs dominate the distribution of clusters in the catalogue near to the Sun. \nOCs and MGs in this work have di ff erent parameter distributions, with one particularly strong di ff erence being in their masses. The distributions of OC and MG masses in Fig. 8 show that MGs in this work are generally much less massive, with a modal total mass of ∼ 75 M ⊙ . OCs are generally much heavier, with a modal mass of around ∼ 400 M ⊙ . \nOC and MG radii also show a number of interesting di ff erences and correlations that can be compared against theoretical predictions. Figure 9 shows OC radii and concentrations against mass and age: namely, the radius containing 50% of members within rJ ( r 50 , J), rJ itself, and the ratio between these two radii (analogous to the concentration of the cluster) r 50 , J / rJ . These are only shown for OCs in the high-quality sample of objects that are within 2 kpc; namely, those for which radii, masses and ages are most robustly measured. As a function of mass, r 50 , J is lightly correlated, with higher mass clusters generally having slightly larger cores. r J is strongly correlated with mass although this is to be expected, since r J is calculated directly from cluster mass with Eqn. 2. Cluster concentrations are also strongly correlated with cluster mass, with the lowest mass clusters being the least centrally concentrated, strongly suggesting that OCs are less centrally concentrated as a function of mass, likely due to dynamical processes within them (Portegies Zwart et al. 2010; Krause et al. 2020). However, as a function of age, cluster radii and concentrations are generally uncorrelated, although the youngest clusters (log t < 7) may be slightly smaller and more concentrated, which is in line with existing theory that OCs undergo a phase of expansion (Krause et al. 2020). Since the minimum age that the neural network from Paper II can measure is log t = 6 . 4, young cluster ages may not be well measured enough to adequately sample this range of cluster formation. \nAlthough our methodology was not originally intended to detect MGs (Paper I), the MGs in our catalogue are still an interesting point of comparison against our detected OCs. Figure 10 shows cluster median radii r 50, total radii including all member \n/circledot \n/circledot \nFig. 7. Comparison of the spatial distribution of OCs and MGs. Left column: distribution in Cartesian heliocentric coordinates of 3530 OCs in the high quality sample of OCs from Table 3. The Sun is at X = Y = 0 pc, the galactic centre is to the right, and the Z axis denotes height above or below the plane. OCs are shaded by the mass of the entire detected cluster, including tidal tails. Right column: identical plot, but for 539 MGs in the high quality sample. \n<!-- image --> \nFig. 8. Histogram of total cluster masses MA for all clusters divided into di ff erent samples. This is shown for all clusters (black dotted line), those with P ( rJ ) > 0 . 5 (blue dotted line), and those with P ( rJ ) < 0 . 5 (orange dotted line). The dashed and solid variants of these lines show the mass distribution for these clusters but restricted to only those in the high quality object sample. (Adapted from Hunt 2023) \n<!-- image --> \nstars and any tidal tails r total, and the ratio between cluster median radius over total radius r 50 / r total for the high-quality OCs and MGs in our sample within 2 kpc. The size of detected MGs strongly correlates with their mass - although this may be a selection e ff ect, as it could be easier to detect member stars of MGs out to higher radii on-sky if they are also higher mass. MGs clearly occupy a di ff erent region of radius-mass parame- \nter space, generally being much larger than OCs at a given mass. MG concentration does not appear to change as a function of mass, which is di ff erent to OCs whose structural evolution is driven by their internal (bound) dynamics and gradual dissolution due to the Milky Way's potential (Krause et al. 2020). \nMGs and OCs di ff er strongly as a function of age. OCs and MGs have similar sizes at young ages for log t < 7, suggesting a similar origin. However, whereas OCs only undergo a small phase of expansion, MGs expand much more strongly, eventually being significantly larger than OCs at all older ages (particularly for r 50.) This increase in observed size is consistent with the MGs in our catalogue being unbound groups of coeval stars that expand over time. Nevertheless, many MGs are older than the expected time it would take for them to disperse (Zucker et al. 2022). If these MGs are real, co-evolutionary groups of stars, then they could be unbound remnants of bound OCs. These objects (and their dynamical evolution, such as whether or not their member stars are expanding from a common origin) should be investigated further in a future work.", '4.3. Comparison of masses with literature results': "Finally, an important step in validating our results is to compare our derived cluster masses against results from the literature, although cluster masses are generally not frequently measured in the literature and di ff erent methodologies can produce highly di ff erent results. Cluster masses from the literature are compared against masses derived in this work in Fig. 11. We compare our masses to masses derived without Gaia data and using profile fitting techniques in Piskunov et al. (2008) and Just et al. (2023); using Gaia DR2 photometry in Meingast et al. (2021); and us- \n/circledot \nFig. 9. Jacobi radii and concentrations of high-quality OCs within 2 kpc, shown for r 50 , J ( upper row ), r J ( middle row ), and cluster concentrations r 50 , J / r J ( bottom row ) against cluster Jacobi mass ( left column ) and cluster age ( right column ). In each panel, a trend line of binned medians is shown in blue, with error bars showing standard error. \n<!-- image --> \ning Gaia DR3 photometry in Cordoni et al. (2023) and Almeida et al. (2023). \nWe find that our masses are most similar to the small sample of ten nearby clusters studied in Meingast et al. (2021), who applied a similar methodology of assuming a Kroupa IMF and fitting it to a cluster's mass function. Our mass estimates are higher than theirs, with this discrepancy being largest for Platais 9 a cluster that Meingast et al. (2021) estimate to have a mass of only 13.1 M ⊙ , compared to our measurement of 62 . 1 ± 7 . 9 M ⊙ . In fact, our mass estimates are generally higher than the estimates of all Gaia -based works. This is likely due to our incorporation of corrections for selection e ff ects and unresolved binaries, both of which will cause our mass estimates to be higher than existing Gaia works that do not correct for both e ff ects. Even amongst Gaia -based works, there is currently little general agreement on the masses of most clusters. \nWe have limited similarity in mass measurements for some clusters to the OC mass catalogue of Cordoni et al. (2023), who fitted bespoke mass functions to clusters in their sample but without correcting for incompleteness. Some clusters in their work have significantly higher cluster masses than in this work, which is likely due to the bespoke cluster mass functions that they use. The cluster with the largest discrepancy is Ha ff ner 26, which we measure to have a mass of 868 ± 86 M ⊙ , compared to their mass of 14563 M ⊙ . For Ha ff ner 26, Cordoni et al. (2023)'s fitted mass \nFig. 10. Radii and concentrations of high-quality OCs (blue) and highquality MGs (orange) within 2 kpc, shown for r 50 ( upper row ), r total ( middle row ), and cluster concentrations r 50 / r total ( bottom row ) against total cluster mass ( left column ) and cluster age ( right column ). Trend lines are plotted with the same formatting as in Fig. 9. \n<!-- image --> \n/circledot \nfunction has power law indices of 3.37 and 4.78 above and below a break point at 1 M ⊙ . This mass function is much steeper than the Kroupa IMF used in this work, which has indices of 2.3 and 1.3 above and below a 0.5 M ⊙ break point. However, after correcting for selection e ff ects, our mass function for Ha ff ner 26 is highly compatible with a Kroupa IMF, and is strongly incompatible with the strong power law indices fitted in Cordoni et al. (2023). In addition, since Ha ff ner 26 is at a distance of around 3 kpc from the Sun, few of its low-mass stars are resolved by Gaia . Our approach of assuming a Kroupa IMF may be less accurate for some nearby clusters for which their mass function can be clearly resolved, but it is at least a safe and consistent approach for clusters at all distances. Extrapolation of a steep mass function of 4.78 below 1 M ⊙ in Cordoni et al. (2023) likely contributes most of the mass towards this cluster in their measurement, even though few stars below that mass are actually observed by Gaia for Ha ff ner 26. \nFinally, Almeida et al. (2023) publish a catalogue of cluster masses based on Gaia DR3 data, created by extracting estimated stellar masses (including accounting for binaries) through comparison with simulated clusters, then fitting bespoke mass functions to each cluster. The overall trend of our results matches theirs, although our mass estimates are once again generally higher, which is likely due to our additional corrections for Gaia incompleteness. Similar to Cordoni et al. (2023), some of our \nFig. 11. Cluster masses in this work compared against those in the literature. The x -axes show literature mass values while the y -axes show cluster Jacobi masses derived in this work. The dashed y = x line shows where mass measurements that are in perfect agreement would be. (Adapted from Hunt 2023) \n<!-- image --> \n/circledot \n/circledot \n/circledot \nmass estimates for clusters are significantly lower than theirs which once again may be due to di ff erences from extrapolating mass functions derived for high mass stars across the entire (unobserved) mass range of a cluster, or due to di ff erences in cluster membership list. \nOur results show poor agreement with masses derived in preGaia works. Piskunov et al. (2008) presented the largest catalogue of cluster masses that was available before the release of Gaia . Their masses are calculated in two ways: firstly, by fitting a King (1962) profile to clusters and assuming that the King tidal radius rt = rJ , then inverting Eqn. 2 to derive a cluster mass given its radius; and secondly, by fitting only semi-major axes to clusters and deriving a mass in the same way. However, these methods are extremely sensitive to the derived cluster membership list and cluster radius, since MJ ∝ r 3 J in Eqn. 2. Particularly as Piskunov et al. (2008) relied on cluster membership lists that do not use Gaia astrometric data and are hence much harder to clean of field stars, in addition to being less complete (Cantat-Gaudin 2022), di ff erences in cluster membership alone can explain why our mass measurements have poor agreement with theirs. Cluster membership lists in Kharchenko et al. (2013) contain four times fewer member stars than in our Paper II catalogue, and even though our approximate King tidal radii in Paper II were only ≈ 1 . 5 × larger than the tidal radii derived in Piskunov et al. (2008), this already corresponds to a ∼ 3 × increase in cluster mass based on Eqn. 2. It is hence likely that data and methodological di ff erences alone can explain the large inconsistencies between Gaia and preGaia cluster masses. Just \net al. (2023) also use a similar method relying on preGaia OC membership lists, for whom we also have poor agreement with their results. \nIn summary, some of our mass results are in good agreement with literature catalogues, although the majority are not. This can be explained by di ff erences in methodology, in particular di ff erences in accounting for selection e ff ects meaning our mass estimates are generally higher, in addition to di ff erences in adopted mass functions and cluster membership lists. OC masses derived with preGaia cluster membership lists are generally in poor agreement with masses using Gaia data.", '5. Discussion': "To the best of the authors' knowledge, this work represents the largest catalogue of Milky Way star cluster masses ever derived, in addition to being the first to classify clusters robustly into bound and unbound objects. In this section, we discuss a number of interesting scientific use cases of this work, beginning with deriving a mass-dependent completeness estimate of our catalogue.", '5.1. Completeness of the Gaia DR3 open cluster census': "The completeness of the OC census is an important but difficult to measure quantity. For instance, although Kharchenko et al. (2013) derived that their OC catalogue was complete within 1.8 kpc, this claim has since been disproven by many studies that \nFig. 12. Kernel density estimates of the two-dimensional distance from the Sun R = √ X 2 + Y 2 distribution of clusters in di ff erent mass ranges. All curves are normalised to have a peak of one for easier comparison between curves. (Adapted from Hunt 2023) \n<!-- image --> \nFig. 13. Kernel density estimate of the R -mass distribution of clusters in this work. To enhance the clarity of the peak of this distribution, the density estimate at each mass is normalised to have a peak of one. The best-fit log-linear completeness model (see Sect. 5.1) is shown by the red dotted line. (Adapted from Hunt 2023) \n<!-- image --> \n/circledot \nreport new OCs within this distance using Gaia data (e.g. CastroGinard et al. 2018, 2019, 2020, 2022; Liu & Pang 2019; Sim et al. 2019; Hunt & Re ff ert 2021, 2023). Any investigation of the OC census must be conducted carefully. In the Gaia era, Anders et al. (2021) derive a completeness estimate of the OC census, although this was performed without cluster masses, with it being unknown how masses may a ff ect the completeness of an OC census. In this section, using our catalogue of cluster masses, we will derive an approximate mass-dependent completeness estimate for our catalogue, demonstrating the importance of cluster masses in deriving the completeness of the OC census. \nIt is helpful to first consider what the distribution of OCs as a function of radius from the Sun should be. Since the scale height of OCs in the disk is small ( ∼ 100 pc) compared to the kpcscales out to which OCs are observed, one can approximate the \nexpected OC distribution in two dimensions X and Y looking at the galaxy top-down. Given a uniform top-down surface density of clusters per square parsec n , the expected number of clusters N within some radius R = √ X 2 + Y 2 is hence given by: \nN ( R ) = n π R 2 , (4) \nwith a derivative of: \nd N ( R ) d R = 2 n π R , (5) \nimplying that the radius distribution of the OC census should increase linearly, assuming that n is constant and that R does not exceed the distance to the edge of the Milky Way's disk. \nIn practice, the actual observed distribution of OCs is unlikely to follow this simple model exactly. Estimation of the true completeness of the OC census is challenging, as the distribution of OCs depends on some distribution function of OCs in the Milky Way, and cannot be assumed to be uniform (Anders et al. 2021). For instance, the distribution of young OCs is known to be correlated with the Milky Way's spiral arms (Castro-Ginard et al. 2021). Deriving such a model for OCs is beyond the scope of this work; however, we can produce a rough estimate of the OCcompleteness distribution as a function of mass as a proof of concept. \nAs an initial test, the R distribution of clusters when divided into separate mass bins in Fig. 12 shows clear signs of incompleteness depending on cluster mass, with lower mass clusters being more likely to occur at low distances in our catalogue. In the lowest mass bin (50 ≤ MJ < 100 M ⊙ ), the cluster distribution peaks at ∼ 1 kpc, while it peaks at ∼ 2 . 7 kpc for the two highest mass bins (800 ≤ MJ < 1600 M ⊙ and 1600 ≤ MJ < 3200 M ⊙ ). In the four lowest mass bins, P ( R )normed appears roughly linear up to a peak, after which the distribution falls o ff exponentially. This is roughly the expected model of the OC distribution implied by Eqns. 4 and 5, given some limiting 100% completeness radius R 100% at each given mass. On the other hand, the highest mass bins do not appear linear up to their peak radius. This may be because high-mass clusters seem more likely to be found in the direction of the galactic centre (see Fig. 7), and that assuming their n is uniform is a poor assumption. \nTo investigate the distribution of clusters without mass binning, Fig. 13 shows the complete massR distribution of OCs smoothed with kernel density estimation. Kernel density estimates were normalised based on mass, in e ff ect meaning that every vertical strip in the figure has a peak at one, helping to make clear where the distribution peaks at a given cluster mass. The trend in peaks shows a log-linear relation up to a mass of ∼ 1000 M ⊙ , after which the distribution does not rise further. This suggests that ∼ 2800 pc is the approximate upper limit of Gaia 's 100% completeness. This could be due to multiple limitations of current Gaia DR3 data, such as its magnitude limit, astrometric accuracy, and extinction. \nTo quantify this relationship, we fitted a log-linear model with a break point after which the model is flat to the peaks of this distribution from 40 ≤ MJ ≤ 10 4 M ⊙ . This gives the approximate 100% completeness limit of our OC census R 100%, with the model taking the form: \nR 100% = ( α log( MJ [ M ⊙ ]) + β R 100% < R break R break R 100% ≥ R break (6) \nArticle number, page 13 of 20 \nFig. 14. Completeness-corrected age function of OCs in this work (black points) compared against various age functions in the literature. Dashed lines show broken power law fits while dotted lines show Schechter function fits. Literature age functions are normalised to ages below 0.2 Gyr for easier visualisation of di ff erences in shape of the upper end of the distributions. The blue lines show fits from Krumholz et al. (2019), the orange lines from Anders et al. (2021), and the red lines are fits from this work. Poisson uncertainties on the data are indicated by the error bars. Schechter function fits from Krumholz et al. (2019) and Anders et al. (2021) have characteristic ages scaled by a factor log 10 e to correct for an error in their Schechter function fitting codes. \n<!-- image --> \nwhere the constraint 0 ≥ R 100% < R break was also applied during fitting. Our best fit had values α = 633 . 1 ± 7 . 3 pc, β = -1582 . 6 ± 39 . 5 pc, and R break = 2792 . 9 ± 8 . 2 pc. \nThis model is at clear odds with the claim of Kharchenko et al. (2013), who claimed that the OC census is complete within 1.8 kpc. Within 1.8 kpc, our all-sky OC census is only complete for clusters heavier than ∼ 230 M ⊙ - a cluster mass similar to that of Melotte 25 (the Hyades). Our catalogue is only complete within 1 kpc for clusters heavier than 100 M ⊙ .", '5.2. Estimating the age and mass functions of OCs in the Milky Way': "Using the approximate completeness estimate in Sect. 5.1, it is also possible to estimate the age and mass functions of OCs in the Milky Way from the number density of OCs as a function of age or mass, in addition to the total number of OCs in the Milky Way. To do so, the number of OCs at distances below R 100% at their given mass was counted into bins, and then divided by the total 2D area of a circle of radius R 100% at the central mass of each bin. \nOur completeness-corrected age function for OCs in the Milky Way is plotted in Fig. 14. Unlike Krumholz et al. (2019) and Anders et al. (2021), who find that the cluster age function is well approximated by a broken power law or a Schechter function, we find that our cluster age function is only fitted well by a broken power law, with a much sharper 'knee' in our cluster age function than that of Anders et al. (2021) or Krumholz et al. (2019). This may be due to our di ff erent definition of an OC in terms of its gravitational potential, which may mean our catalogue is more strongly cleaned of older, unbound star clusters. Nevertheless, we confirm the results of Anders et al. \nFig. 15. Mass function of OCs in this work. Top: Completenesscorrected mass function of OCs in this work (black points) compared against a κ = -2 power law (Krumholz et al. 2019, blue dashed line) fit to clusters with masses greater than 400 M ⊙ . Bottom: completenesscorrected mass functions for clusters, separated into age ranges and including power-law fits to each age range. \n<!-- image --> \n/circledot \n(2021), who found that the number of old clusters in Gaia is much lower than previous preGaia results such as Piskunov et al. (2018). Based on our results in Paper II, it is likely that the reduced number of old OCs in Gaia -derived results (including this work) is due to many old OCs reported before Gaia being unlikely to be real. Our broken power law fit to our data gives α 1 = -0 . 594 ± 0 . 038, α 2 = -2 . 321 ± 0 . 127, and with a break point at log t break = 8 . 33 ± 0 . 04 log yr. The slopes of this distribution are compatible within uncertainty to the results of Anders et al. (2021), although our log t break is slightly lower than their value of log t break = 8 . 49 + 0 . 21 -0 21 log yr. \n. The upper plot of Fig. 15 shows our completeness-corrected mass function for OCs in the Milky Way. Above a mass of around 400 M ⊙ , this mass function is well approximated by a power law with index κ = -2, which is identical to the cluster initial mass function found in numerous other galaxies that is well approximated by a power law with slope κ = -2 for clusters with masses below ≈ 10 4 M ⊙ (Portegies Zwart et al. 2010; Krumholz et al. 2019), implying a log-uniform rate of cluster formation as a function of mass. However, for clusters below a mass of 400 M ⊙ , we find a slightly less steep power law function is a better fit to the data. This appears to be a trend based on age. The lower plot of Fig. 15 shows the age-binned mass \nFig. 16. Slope of power-law fits in Fig. 15 as a function of age. A fit to this slope with κ 0 fixed to -2 is shown in blue, with a fit with κ 0 free shown in orange. \n<!-- image --> \nFigure 16 shows the slopes of power law fits to the agebinned mass function of clusters as a function of age. We fit o ff -set power laws of the form κ = At ν + κ 0 to this distribution; firstly, for κ 0 fixed to -2, as predicted for zero-age clusters (Krumholz et al. 2019); and secondly, with all parameters free. In the first ( κ 0 fixed) case, we find ν = 0 . 423 ± 0 . 078; in the second case, we find κ 0 = -1 . 832 ± 0 . 024 and ν = 0 . 853 ± 0 . 111. These observations should be compared to large-scale N-body simulations of cluster dissolution in the future. \n<!-- image --> \n/circledot \nFig. 17. Completeness-corrected estimated total number of OCs in the Milky Way as a function of mass ( N ( m ), blue) including Poisson uncertainties on bins, compared against the distribution of OC masses in this work (orange). \nfunction of the same clusters, including fits by unbroken power laws. For the youngest clusters, their mass function is close to a κ = -2 power law, which is the prediction for young clusters in Krumholz et al. (2019). With increasing age, the cluster mass function appears to flatten, in addition to decreasing at all masses. This flattening of the mass function slopes with age may suggest accelerated cluster dissolution for low-mass clusters as a function of age compared to high-mass clusters, and should be investigated with theoretical studies. \nWe also used our derived cluster number density to calculate an estimate of the total number of OCs in the Milky Way at a given mass, N ( M ), assuming a flat disk distribution of OCs with a radius of 12.5 kpc, which corresponds to the approximate limit out to which OCs are observed in the galactic disk (see \ne.g. Fig. 7). Figure 17 shows the total number of OCs in this work compared against a completeness-corrected estimated total number of OCs in the Milky Way. Summing this distribution, we estimate that the Milky Way contains a total of ∼ 1 . 3 × 10 5 OCs with masses in the range 40 < M J < 10 4 M ⊙ , which is comparable to the ∼ 10 5 OCs in the Milky Way estimated to exist by Dias et al. (2002). This estimated total number implies that only around ∼ 4% of the Milky Way's total number of OCs are known at this time, with this incompleteness being strongest for low-mass clusters. \nSumming our predicted N ( M ) distribution, we estimate that the Milky Way contains ∼ 4 . 8 × 10 7 M ⊙ of stars that are currently bound to OCs. Cautun et al. (2020) used Gaia DR2 to estimate that the Milky Way contains 5 . 04 + 0 . 43 -0 . 52 × 10 10 M ⊙ of stellar mass; compared to our prediction, this suggests that around 0.1% of the Milky Way's stars are currently in an OC. This is similar to the ratio between our total number of input stars in Paper II and the final number of stars that we find to be currently bound to an OC. In Paper II, we used an input list of 729 million stars from Gaia DR3 to construct our catalogue. In this work, we find that 614 358 of those stars are currently within the Jacobi radius of an OC, which is around 0.1% of the stars considered in our Paper II clustering analysis.", '5.3. Comparison between cluster mass functions and the Kroupa IMF': "Throughout this work, we relied on the Kroupa IMF to calculate total cluster masses. After extensive correction for cluster membership selection e ff ects in Sect. 3.2, we find that cluster mass functions were widely compatible with the Kroupa IMF, and across a wide range of cluster ages. Figure 18 shows data points from all mass functions in this work and plotted as a 2D histogram. \nThere are some notable outliers in this figure that are worth discussing initially. Firstly, the highest mass points (with masses greater than ∼ 20 M ⊙ ) appear to be over-counted. This is because the cluster ages from Paper II have a lower limit of log t = 6 . 4, meaning that high mass stars in star clusters younger than this age cannot have masses higher than this limit assigned to them, and the highest mass bins in young clusters are hence overestimated due to contamination from even higher mass and shortlived O stars. This is visible in the 6 ≤ log t < 7 subplot of the figure - only young clusters have these erroneously high measurements. \nSecondly, some mass bins in the range 2 ≤ M < 10 M ⊙ for some clusters contain around an order of magnitude fewer stars than would be expected for these clusters. These low-count bins have correspondingly high Poisson uncertainties, and do not dramatically change our overall total cluster mass measurements, but are nevertheless still worth discussion. There are likely to be multiple reasons for the missing high-mass stars in these clusters, including poor-quality CMDs, small number statistics, poor isochrone fits, and unaccounted for selection e ff ects. 902 of the 6956 (12.9%) of clusters with mass measurements in this work have at least one mass bin more than 5 × below the expected value from a Kroupa IMF, and hence have points within the previously identified region. 200 of these clusters have low-quality CMDs (Paper II CMD score below 0.5) which may mean they are not a real single population of stars or that they are a poor detection of a real cluster, which hence may have gaps for nonphysical reasons. \nOf the remaining 702 clusters with good-quality CMDs, 572 have fewer than 100 member stars, which could plausibly have \nFig. 18. Comparison between mass function points of clusters in this work and the Kroupa IMF. Left: 2D histograms of all points from all cluster mass functions in this work for 1235 OCs within 2 kpc in the high-quality sample of clusters and with at least 50 member stars compared against the Kroupa IMF (dashed red line). Individual cluster mass functions are normalised before combining. The colour of histogram bins denotes how many mass function points went into each individual bin, corresponding to the colour bar in the upper-right. Right: same as left panel, except clusters are divided into four separate age ranges. \n<!-- image --> \n/circledot \n/circledot \ngaps simply due to missing stars due to small number statistics, or due to a small number of stars making it di ffi cult to constrain an accurate isochrone fit, as our Paper II parameter inference accuracy is strongly correlated with number of member stars. Almost all erroneously low mass function points are at the tip of the main sequence within clusters - a region within a cluster CMDthat is generally sparsely populated but that covers a wide range in stellar masses, particularly for young clusters - meaning that a small error in an isochrone fit or an isochrone itself can correspond to a large error in derived stellar mass, hence faking the appearance of a gap in a measured cluster mass function. \nNevertheless, 54 clusters with good quality CMDs and at least 200 member stars still have mass function gaps. Most of these clusters are nearby ( d < 1 kpc), young (log t < 8), and have well-inferred photometric parameters. All but one of the gaps in these clusters are brighter than G = 12, are at or near the tip of the main sequence, and occur in well-studied clusters such as Blanco 1 (9 < G < 9 . 5, see Fig. 2). In the case of Blanco 1, this gap appears robust between di ff erent works, appearing in other Gaia -based works (e.g. Zhang et al. 2020; Cantat-Gaudin et al. 2020). The gaps being at brighter magnitudes may be significant for three reasons. Firstly, Gaia 's CCDs become saturated above G = 12, and sources above this limit undergo di ff erent photometric processing (Riello et al. 2021). Bright sources in Gaia often have much higher astrometric errors, which could mean that they are missed from a cluster membership list due to poor astrometry, that they may only have a two-parameter astrometric solution and were hence not included in our clustering analysis, or that they are more likely to be tagged as a false positive by the Rybizki et al. (2022) method we used to clean the Gaia DR3 dataset in Paper II. Secondly, at high magnitudes, there are sig- \nnificantly fewer stars, meaning that Gaia 's selection function is much more di ffi cult to accurately characterise empirically. This could impact the Gaia or subsample selection functions applied in Sect. 3.2. With just a handful of stars in a given wide magnitude range to use to empirically determine a selection function, uncertainty on a selection function in a given range is higher. It is notable that in Fig. 2, Blanco 1's subsample selection function is lower in the range where it has a gap, suggesting that one reason for missing stars in this range could be an underestimated selection e ff ect. In the future, it may be necessary to improve subsample selection functions further to be more accurate at bright magnitudes where the presence of few bright stars in most fields makes it di ffi cult to empirically determine subsample selection functions for bright stars accurately. Finally, since stars in the mass range 2 ≤ M < 10 M ⊙ are usually binaries (Moe & Di Stefano 2017), it may also be that binary star-induced astrometric errors cause some stars to be missing from our cluster membership lists - particularly if they are on ∼ 1 year orbits with motion similar to that of parallax (Lindegren et al. 2021). Improved binary star astrometry and classifications in Gaia DR4 will help to reduce the number of missed binary stars in future works (Gaia Collaboration et al. 2023). \nAside from these outliers, the bulk of points in cluster mass functions are a good fit to a Kroupa IMF. Some deviation from a Kroupa IMF is visible for the oldest clusters in Fig. 18, where mass functions appear flatter, with a possible physical cause being preferential mass loss of low-mass stars in the oldest clusters. Fundamentally, however, we are unable to reproduce the results of works including Cordoni et al. (2023), who find that cluster mass functions are compatible with power law slopes that deviate significantly from a Kroupa IMF. To investigate cluster mass \nfunctions further in the future, and with a higher accuracy than was possible in this work, cluster mass functions incorporating more accurate cluster-by-cluster binary star corrections should be conducted. This is likely to be possible with future surveys such as Gaia DR4, which will provide epoch astrometry for better identification of binaries (Gaia Collaboration et al. 2023), or using stellar spectra in upcoming large spectral surveys such as 4MOST (de Jong et al. 2012) to identify spectroscopic binaries. Currently, photometric identification of binaries with Gaia data is only able to detect the highest mass ratio binary stars ( q ≳ 0 . 6) in the most reliable clusters (e.g. Cordoni et al. 2023; Donada et al. 2023).", '6. Conclusion': "In this work, we investigated methods to classify star clusters in the Milky Way as being bound or unbound. By measuring cluster masses and Jacobi radii, we were able to classify 6956 clusters from our catalogue of star clusters in Paper II as being bound OCs or unbound MGs. This classification method provides a new, more precise way to distinguish between OCs and MGs in Gaia data compared to simply using individual cuts on parameters. \nAs a component of this work, we release a catalogue of star cluster masses and radii, which is the largest catalogue of Milky Way cluster masses to date, being around seven times larger than the largest catalogue of OC masses made using Gaia data so far (Almeida et al. 2023). Our cluster masses were precisely calculated by considering three CMD selection e ff ects and the impact of unresolved binaries. We compare our mass estimates against those in the literature, finding that our masses are typically higher than previous literature results. We suggest that this is due to our inclusion of selection e ff ect corrections. \nWe use our cluster masses to estimate the fraction of clusters from Paper II that are compatible with bound (instantaneously self-gravitating) objects, publishing an updated star cluster catalogue with improved cluster classifications. Within 15 kpc (the maximum distance that we provide mass measurements for), we find that only 79% of the clusters from Paper II are compatible with being bound. Nearby to the Sun, within 250 pc, our catalogue is dominated by MGs, with just 11% of clusters being compatible with bound objects. Our final catalogue contains 5647 OCs, 3530 of which are in a high-quality sample with higher astrometric S / N and good-quality CMDs. The catalogue contains 1309 MGs, 539 of which are of high quality by the same definition. \nComparisons between OCs and MGs in our catalogue show interesting di ff erences between these objects. The structural concentration of OCs is a strong function of their mass, and not their age. On the other hand, older MGs are significantly larger than young ones, which is compatible with them being unbound, expanding objects. Young MGs and OCs in our catalogue appear to form at similar initial sizes, but with MGs expanding significantly more with age. Our detection of so many MGs in our cluster search in Paper II was an accident, as we only intended to detect OCs; however, given that both MGs and OCs are remnants of coeval star formation, and some unbound MGs may even be remnants of bound OCs, it would make sense in future blind cluster searches to continue searching for both classes of object and conducting comparisons between them. \nWe also used these results to derive approximate estimates of the completeness, age function, and mass function of the OC census in Gaia DR3. The completeness of our catalogue is well described by a logarithmic function of only cluster mass up to \n∼ 2800 pc, beyond which the 100% completeness limit does not increase further. Kharchenko et al. (2013) stated that the OC census is complete within 1.8 kpc, a claim that has since been disproven by numerous Gaia -based works (Castro-Ginard et al. 2018, 2019, 2020, 2022; Liu & Pang 2019; Sim et al. 2019; Hunt & Re ff ert 2021, 2023); in this work, we find that our OC census is only approximately 100% complete at 1.8 kpc for clusters heavier than 230 M ⊙ , suggesting that many more low-mass OCs are still yet to be discovered within this distance range. \nUsing this completeness estimate, we confirm the results of Anders et al. (2021) that the Gaia census of OCs is significantly younger than preGaia works. We also derive a completenesscorrected mass function of our OC catalogue, finding that OCs above around 400 M ⊙ are compatible with a power law with slope equal to -2, which is compatible with observations of the cluster mass function of numerous other galaxies (Krumholz et al. 2019). However, below this mass, we find that there are fewer clusters than expected. Separated into age bins, our cluster mass function appears to flatten with increasing age, suggesting an accelerated rate of cluster dissolution for low-mass clusters. Our cluster mass function implies that the Milky Way should contain a total of around 1 . 3 × 10 5 OCs, ∼ 4% of which are currently known. Finally, in investigation of the mass functions of individual clusters, we find that most OCs are broadly compatible with a Kroupa IMF for ages below 1 Gyr - but only after extensive correction of their mass functions for selection e ff ects. \nSince the release of Gaia DR2 (Brown et al. 2018), there has been an explosion in studies reporting detections of new OCs (e.g. Sim et al. 2019; Castro-Ginard et al. 2020, 2022; Liu & Pang 2019; He et al. 2021, 2022; Hao et al. 2022). Works generally agree that an OC should be an overdensity in Gaia data, with at least ∼ 10 member stars, and a CMD compatible with a single population of stars. However, until now, there has been no way to further observationally define detected star clusters into bound and unbound objects. The e ff ectiveness of measuring cluster Jacobi radii for this purpose will improve the accuracy and clarity of both the current OC census and future OC censuses based on upcoming data releases. \nAcknowledgements. We thank the anonymous referee for their comments that improved the quality of this paper, as well as Siegfried Röser and Elena Schilbach for further helpful comments. E.L.H. and S.R. gratefully acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Project-ID 138713538 - SFB 881 ('The Milky Way System', subproject B5). We thank Josefa Großschedl, Bruno Alessi, Philipp Teutsch, Siegfried Röser, and Elena Schilbach for providing feedback on unreliable clusters or primary names from our Paper II work. This work has made use of data from the European Space Agency (ESA) mission Gaia ( https://www. cosmos.esa.int/gaia ), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/ consortium ). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. This research has made use of NASA's Astrophysics Data System Bibliographic Services. This research also made use of the SIMBAD database, operated at CDS, Strasbourg, France (Wenger, M. et al. 2000). In addition to those cited in the main body of the text, this work made use of the open source Python packages NumPy (Harris et al. 2020), SciPy (Virtanen et al. 2020), IPython (Pérez & Granger 2007), Jupyter (Kluyver et al. 2016), Matplotlib (Hunter 2007), pandas (McKinney 2010; The pandas development team 2020), and Astropy (Robitaille et al. 2013; Astropy Collaboration et al. 2018). 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2024ApJ...977....8K | Extrememassratio inspirals EMRIs of stellarmass black holes BHs are among the main targets for upcoming lowfrequency gravitational wave GW detectors such as the Laser Interferometer Space Antenna. In the classical scenario EMRIs are formed when BHs scatter off each other and are driven onto highly eccentric orbits that gradually inspiral due to GW emission. If the cluster is in a state of strong mass segregation the BHs are expected to reside in a steep cusp around the central massive black hole MBH which would facilitate more efficient EMRI formation. However strong mass segregation may also lead to an increased rate of ejections due to close encounters between the BHs. Here we test the relevance of such ejections for EMRI formation by numerically solving a twodimensional FokkerPlanck equation. Our formalism includes the effects of twobody relaxation GW dissipation and ejections. We find that the EMRI formation rate can be suppressed due to ejections by more than an order of magnitude for strongly segregated BH cusps with density index 2.25 around central MBHs of mass M SUBSUB 10SUP6SUP M SUBSUB. The EMRI formation rate levels off up to a maximum value of 200 GyrSUP1SUP due to ejections which is roughly an order of magnitude lower than the usual scenarios ignoring ejections for steep BH cusps around lowmass MBHs. Our analysis reveals the significance of strong scatterings for EMRI formation in galactic nuclei. | 2024-12-01T00:00:00Z | ['2024arXiv240910618K', '2024ApJ...977....8K', 'arXiv:2409.10618', '10.48550/arXiv.2409.10618', '10.3847/1538-4357/ad89bd'] | ['Galaxy nuclei', 'Galactic center', 'Stellar dynamics', 'Supermassive black holes', 'Stellar mass black holes', 'Gravitational wave sources', '609', '565', '1596', '1663', '1611', '677', 'Astrophysics - High Energy Astrophysical Phenomena'] | Extrememassratio Inspirals in the Strong Segregation Regimeto Inspiral or to Get Ejected | 2,024 | 173 | 0.49 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF'] | 0 | https://arxiv.org/pdf/2409.10618.pdf | {'Extreme-mass ratio inspirals in strong segregation regime - to inspiral or to get ejected?': '<!-- image --> \n1 Technion - Israel Institute of Technology, Haifa, 3200002, Israel \n(Received September 18, 2024)', 'ABSTRACT': 'Extreme-mass ratio inspirals (EMRIs) of stellar-mass black holes (BHs) are among the main targets for upcoming low-frequency gravitational wave (GW) detectors such as the Laser Interferometer Space Antenna (LISA). In the classical scenario, EMRIs are formed when BHs scatter off each other and are driven onto highly eccentric orbits that gradually inspiral due to GW emission. If the cluster is in a state of strong mass segregation, the BHs are expected to reside in a steep cusp around the central massive black hole (MBH), which would facilitate more efficient EMRI formation. However, strong mass segregation may also lead to an increased rate of ejections due to close encounters between the BHs. Here, we test the relevance of such ejections for EMRI formation by numerically solving a two-dimensional Fokker-Planck equation. Our formalism includes the effects of two-body relaxation, GW dissipation, and ejections. We find that the EMRI formation rate can be suppressed due to ejections by more than an order of magnitude for strongly segregated BH cusps with density index γ ≳ 2 . 25 around central MBHs of mass M · ≲ 10 6 M ⊙ . The EMRI formation rate levels off up to a maximum value of ≃ 200 Gyr -1 due to ejections, which is roughly an order of magnitude lower than the usual scenarios ignoring ejections for steep BH cusps around low mass MBHs. Our analysis brings forth the significance of strong scatterings for EMRI formation in galactic nuclei. \nKeywords: Galaxy nuclei (609), Galactic center (565), Stellar dynamics (1596), Supermassive black holes (1663), Stellar mass black holes (1611), Gravitational wave sources (677)', '1. INTRODUCTION': "In the near future, space-based gravitational wave observatories like LISA and TianQin will usher in a new era of mHz gravitational wave astronomy, providing a window into the galactic nuclei harboring massive black holes (MBHs) (eLISA Consortium et al. 2013; Mei et al. 2021; Kormendy & Ho 2013). Many of these MBHs reside within dense nuclear star clusters (NSCs), dynamic environments where a multitude of astrophysical phenomena can occur, including the formation of extreme-mass ratio inspirals (EMRIs) (Alexander 2017a). EMRIs, consisting of stellar-mass black holes (BHs) captured in close orbits around MBHs, are key targets for this low-frequency gravitational wave (GW) detectors (Amaro-Seoane et al. 2007). \nThe classical EMRI formation channel posits that twobody (2B) scatterings drive BHs into highly eccentric orbits, \nCorresponding author: Karamveer Kaur \[email protected] \nwhich subsequently undergo inspiral due to GW emission (Hils & Bender 1995; Sigurdsson & Rees 1997; Freitag 2001; Hopman & Alexander 2005) 1 . However, theoretical predictions of EMRI rates remain uncertain, varying by up to two orders of magnitude (Hopman & Alexander 2005; AmaroSeoane & Preto 2011; Merritt 2015; Bar-Or & Alexander 2016; Aharon & Perets 2016; Raveh & Perets 2021; Kaur et al. 2024; Rom et al. 2024). This uncertainty stems from our incomplete knowledge of the internal structure of NSCs, in particular, the distribution and number density of stellar BHs, both of which significantly influence relaxation timescales and the rate of BH influx into the GW loss cone (AmaroSeoane 2018). \nFokker-Planck investigations dealing with scatteringdriven energy relaxation have sought to elucidate the structure of NSCs theoretically. Bahcall & Wolf (1977) identi- \nfied a zero-flux solution characterized by a density profile index γ =7/4 for the dominant stellar population. However, Alexander & Hopman (2009) proposed an alternative solution branch with non-zero net flux, leading to steeper density profiles with γ ≃ 2 -11 / 4 , known as the strong segregation regime. This regime is particularly relevant when BHs constitute a minority population compared to stars. In this scenario, the more numerous stars establish a shallower profile consistent with the zero-flux solution, while the BHs, experiencing dynamical friction against the stellar background, form a steeper cusp (Alexander & Hopman 2009; see Linial & Sari 2022 for a different perspective). The appeal of the strong segregation regime lies in its ability to produce high EMRI rates, on the order of hundreds per Gyr for a Milky Way-like galaxy, even with physically plausible BH number fractions as low as 10 -3 (Amaro-Seoane & Preto 2011). These high rates are facilitated by the steep BH density profiles, which yield high BH densities in the inner regions ( ≈ 0 . 01 r h ; r h being the radius of influence of central MBH) where EMRIs predominantly form (Hopman & Alexander 2005), and also shorten the relaxation timescales associated with EMRI formation. \nHowever, the increased BH densities in strongly segregated cusps raise the possibility of strong encounters that could eject stars and compact objects. Such ejection effects have been previously explored in the context of globular clusters (H'enon 1960a; Lin & Tremaine 1980; Goodman 1983), where they are important for cluster evaporation. In NSCs, such strong encounters can eject stars before they achieve sufficiently eccentric orbits to interact with the MBH strongly. In particular, Teboul et al. (2024) showed that the ejections of stars on high eccentricity ( e ) orbits can significantly quench the rates of tidal disruption events (TDEs) in strongly segregated NSCs. This brings out the significance of strong scatterings for high e orbits, particularly relevant for general loss cone dynamics around an MBH (Weissbein & Sari 2017; Teboul et al. 2024). Indeed, strong scatterings can induce substantial changes in BH orbital parameters (Agekyan 1959; H'enon 1960a,b; Ashurov 2004; H'enon 2011; Teboul et al. 2024), potentially ejecting BHs from the highly eccentric orbits necessary for EMRI formation, especially in the strongly segregated NSCs which have enhanced BH densities in their inner regions. \nIn this study, we examine the impact of strong scatteringdriven ejections on EMRI formation within strongly segregated BH cusps. We evaluate a numerical solution of a twodimensional Fokker-Planck equation that simultaneously accounts for 2B relaxation, GW-induced orbital decay, and ejections triggered by strong scatterings. For the first time in consideration of strong scatterings, we take into account the depletion of the inner cusp of (BH) scatterers due to GW loss cone. This prevents a significant overestimation of ejection \nrates due to artificially high densities in the inner regions of the cluster. We employ the Fokker-Planck solution to calculate time-dependent EMRI rates and assess the level of suppression induced by ejections. It turns out that strong scatterings can lower the EMRI rates up to 1-2 orders of magnitude in strongly segregated cusps with steepest profiles ( γ ≳ 2 . 25 ) around MBHs with low mass M · ≈ 10 5 -6 M ⊙ (table 1, figure 3). This is especially interesting given the maximum LISA sensitivity for EMRI detections around these low-mass MBHs (Babak et al. 2017; Rom et al. 2024). \nThis paper is structured as follows. In Section 2, we delineate the relevant physical processes and present our FokkerPlanck formalism. Section 3 discusses the numerical solution of the equation and presents the resulting EMRI rates. Finally, in Section 4 we summarize our findings and discuss their astrophysical implications.", '2. PHYSICAL SET-UP': 'We consider a single BH population of mass m forming a compact power-law density profile, with index γ , well inside the radius r h of influence of the central MBH of mass M · . The number profile N ( a ) = N f ( a/a f ) 3 -γ represents the BHs with semi-major axes ≤ a , such that N f = f bh ( M · /m ⋆ ) is the total number of BHs within the reference radius a f ≃ 0 . 1 r h (Amaro-Seoane & Preto 2011). Here, stars of mass m ⋆ form the most abundant stellar population within r h , and f bh is the number fraction for BHs relative to stars. In spite of the presence of more abundant lighter components (for e.g. stars), the high number density of BHs in these inner regions makes them the dominant scatterers, for both weak scatterings responsible for angular momentum ( L ) relaxation, and strong scatterings leading to ejections. \nIn the classical picture of EMRI formation, many random weak or 2B scatterings may ultimately lead a BH onto a highly eccentric orbit, such that it loses its orbital energy and gradually shrinks in semi-major axis ( a ) due to GW emission upon pericentric interactions with MBH. However, as BHs attain these high eccentricity e orbits, they become susceptible to ejections owing to a possible close encounter with another BH near its pericenter. This can lead to suppression in the formation rates of EMRIs. We discuss these physical phenomena below and implement them in a Fokker-Planck (FP) framework to study the evolution of BH distribution.', '2.1. Two-body scatterings': 'Due to numerous 2B scatterings among each other, BHs undergo a random walk in two-dimensional space of energy ε = GM · / (2 a ) and angular momentum L = √ 2 GM · r p over 2B relaxation time T 2b ; r p being the pericentric distance from central MBH. While T 2b represents a reference time for near-circular orbits, relaxation in r p for highly eccentric orbits, relevant for EMRI formation, occurs over much shorter \ntimescale T L 2b = (2 r p /r ) T 2b compared to relaxation in a (Binney & Tremaine 1987). Hence, we neglect the relaxation in a and consider a fixed spatial profile N ( a ) of scatterers at all times. We employ the following form of 2B timescale (Merritt et al. 2011; Merritt 2013; Bortolas & Mapelli 2019): \nT 2b ( a ) = 3 √ 2 π 2 32 C T Kep ( a ) ln Λ ( M · m ) 2 1 N ( a ) (1) \nwith dynamical timescale T Kep = √ a 3 / ( GM · ) , and Coulomb logarithm ln Λ ≃ log ( M · /m ) . The factor C has a weak dependence on γ (see for e.g. appendix A of Bortolas & Mapelli 2019), and we use C = 1 . 35 corresponding to a γ = 7 / 4 Bahcall-Wolf (BW) cusp (Kaur et al. 2024). Similar to previous studies (Merritt 2013), the inclusion of 2B scatterings in FP equation 7 considers that: (1) background scatterers (BHs themselves in our scenario) follow a thermal distribution, and (2) drift and diffusion coefficients arising from L -relaxation are combined using fluctuation-dissipation theorem to get an effective diffusion term in r p (Lightman & Shapiro 1977). Further, we neglect resonant relaxation (Rauch & Tremaine 1996), as it has been found to be ineffective in influencing overall EMRI rates (Alexander 2017b).', '2.2. GW-induced dissipation': 'As a fraction of BHs attain high e , they can come sufficiently close to MBH during pericentric passages leading to energy and angular momentum losses by GW emission. The resulting orbital shrinkage or inspiral occurs over a GW timescale T gw given explicitly as (Peters 1964; Kaur et al. 2024): \nT gw ( r, r p ) ≡ a | ˙ a | = 96 √ 2 85 R s c M · m ( r p R s ) 4 √ a r p , (2) \nas suited for high e orbits. Here R s = 2 GM · /c 2 is the Schwarzschild radius. We take into account this orbital shrinkage due to energy and angular momentum losses in the FP equation 7. \nFor high e orbits with T gw ≤ T L 2b , orbital inspiral dominates over the stochastic effect of 2B scatterings, and orbit eventually becomes an EMRI. This condition effectively presents a GW loss cone, for r p ≤ r p, gw , with the critical periapsis r p, gw given explicitly as: \nr p, gw ( a ) = R s [ ξ ln Λ m ⋆ m 1 f bh ] 2 / 5 ( a f a ) 2(3 -γ ) / 5 (3) \nwhere ξ ≃ 1 . 7 (Kaur et al. 2024). \nThe critical semi-major axis a c roughly demarcates the outer boundary within which most EMRIs occur (Hopman & Alexander 2005), because GW loss cone for a c coincides \nwith the capture radius ≃ 4 R s 2 . Using r p, gw ( a c ) = 4 R s , we get: \na c = a f [ ξ 32 ln Λ m ⋆ m 1 f bh ] 1 / (3 -γ ) . (4) \nThis implies a c /r h ≃ 0 . 03 -0 . 06 for γ = 7 / 4 -5 / 2 using a f = 0 . 1 r h , ln Λ = 10 , m = 10 m ⋆ , f bh = 10 -3 . Outside a c , most highly eccentric orbits lead to direct plunges into the MBH. However, for a central intermediate-mass black hole ( M · ≲ 10 4 M ⊙ ), this boundary becomes somewhat blurred and a fraction of EMRIs may arise from a ≫ a c (Qunbar & Stone 2023). \nFurther, for highly bound orbits a ≤ a gw , GW-induced orbital inspiral begins to dominate even for circular orbits. Using a gw = r p, gw ( a gw ) , we have: \na gw = R s [ ξ ln Λ m ⋆ m 1 f bh ( a f R s ) (3 -γ ) ] 1 / (11 / 2 -γ ) (5) \nwhich gives a gw ≃ 23 -170 R s for γ = 7 / 4 -5 / 2 3 , and M · = 4 × 10 6 M ⊙ and r h = 2 pc corresponding to our Galactic center (Ghez et al. 2008; Gillessen et al. 2009). This roughly marks the inner edge till where BH cusp defined by power-law density index γ may sustain, while inside this region the cusp is depleted due to GW losses.', '2.3. Strong scatterings': "For highly eccentric orbits, ejection from the system is the most probable outcome of strong scatterings. In fact, it might not even require strong scatterings (with impact parameter b 90 = Gm/v 2 such that ∆ v ∼ v ), since the test BH is moving near escape speed v esc ≃ √ 2 GM · /r for most phases of its orbit satisfying r ≪ a . Only a small velocity impulse is needed for its ejection ∆ v ej = v esc -v = v esc r/ (4 a ) ≪ v ∼ v esc . This corresponds to a large ejection impact parameter b ej = Gm/ ( v ∆ v ej ) ≃ 2 am/M · ≫ b 90 , making ejections likely for high e orbits as they navigate through high-density regions of inner BH cusp during the pericentric passage. \nH'enon (1960a); Henon (1969) calculated the local probabilistic rate of ejection ˙ P ej of a test star for general stellar clusters, and more recently Teboul et al. (2024) evaluated it for a Keplerian potential. This gives the ejection probability ˙ P ej ≈ πb 2 ej vN ( r ) / (4 πr 3 ) of a test star with Keplerian orbital parameters { a, r p } near phase r of its orbit; equation A3 gives an exact form of ˙ P ej . The actual time spent by a star at the phase r is proportional to ( r/a ) 3 / 2 . This implies an orbitaveraged ejection rate of ⟨ ˙ P ej ⟩ ≈ ( T Kep ) -1 ∫ d r ˙ P ej /v ≈ \n˙ P ej ( r/a ) 3 / 2 , which gives: \n⟨ ˙ P ej ⟩≃ 3 √ 2 π 32 C 1 ln Λ T 2b ( a f ) Γ( γ +1) Γ( γ +1 / 2) √ a a f r 0 a f ( a f r 0 -a f 2 a ) γ (6) \nsee appendix A for the details of derivation. The dominant contribution to ejections comes from the phase r 0 = max[ r p , a gw ] . This is because ejections due to strong scatterings dominate for small r phases of an orbit near its pericenter, which is in contrast to the case of L -relaxation owing to weak scatterings (Binney & Tremaine 1987). \nDepletion of inner cusp : Here we account for the depletion of inner BH cusp by GW loss cone, because dense parts of BH cusp cease to exist within ≈ a gw , marking the inner edge of the depleted cusp. This depletion radius corresponds to a few 100 R s for most realistic galactic nuclei with M · ≃ 10 4 -7 M ⊙ . Hence, it is important to account for this depletion by choosing the appropriate phase r 0 that contributes maximally to ejections, as we consider above. This effect of depletion of inner cusp by GW loss cone has not been accounted in previous studies, and might have led to overestimation of ejection rates. \nFurther, rather than using the above approximate expression for ⟨ ˙ P ej ⟩ , we evaluate numerically the integral in equation A8 for an exact expression of ⟨ ˙ P ej ⟩ , which is then employed in the Fokker-Planck framework described below. The exact evaluation of ⟨ ˙ P ej ⟩ , as detailed in the appendix A, takes into account the chosen (physically motivated) form of orbital configuration (equation A5) of depleted region a ≲ a gw . But, it is in close agreement with the above approximate form of ⟨ ˙ P ej ⟩ that just depends on the radius of depletion a gw 4 .", '2.4. Fokker-Planck framework': 'We track the evolution of 2D density N of BH distribution in { a, R } -plane with R = 1 -e 2 = 1 -(1 -r p /a ) 2 , because these coordinates provide a suitable rectangular grid for the numerical solution (Merritt 2013). We include the influence of all three processes described above into the following Fokker-Planck (FP) equation: \n∂ N ∂t = 1 a γ -3 / 2 ∂ ∂ R ( R ∂ N ∂ R ) + ∂ ∂a ( | ˙ a | N ) -∂ ∂ R ( | ˙ R | N ) -F ej ( a, R ) N . (7) \nThe first diffusion term on the right side of the above partial differential equation (PDE), refers to the relaxation in R due \nto weak two-body scatterings (Merritt 2013). The second and third advection terms account for losses due to GWs, with GW advection speeds | ˙ a | and | ˙ R | respectively in a and R directions (Peters 1964) given as: \n| ˙ a | = 96 × 2 7 / 2 425 a 3 R 7 / 2 A ( 1 + 73 24 (1 -R ) + 37 96 (1 -R ) 2 ) | ˙ R | = 304 × 2 7 / 2 (1 -R ) 425 a 4 R 5 / 2 A ( 1 + 121 304 (1 -R ) ) . (8) \nHere the constant factor A = T gw ( a f ) /T 2b ( a f ) is the ratio of GW to 2B timescales (equations 1 and 2) for circular orbits of radius a f . The fourth sink term depicts ejections due to strong scatterings, with the ejection factor 5 : \nF ej ≡ ⟨ ˙ P ej ⟩ T 2b ( a f ) ≃ 3 √ 2 π 32 C ln Λ Γ( γ +1) Γ( γ +1 / 2) √ ar 0 ( 1 r 0 -1 2 a ) γ (9) \nThe above approximate form of F ej employs the simplified analytical expression for ⟨ ˙ P ej ⟩ in equation 6. However, for implementation in our code, we generate tables for F ej (using equations A9 and A12) by numerically solving the integrals in the exact expression of ⟨ ˙ P ej ⟩ in equation A8. Here, we have described all lengths in the unit of a f , and time t in a unit of T 2b ( a f ) . \nWe solve the above PDE in the domain a ∈ [ a gw /a f , 2 a c /a f ] for an isotropic initial distribution (with N independent of R ), respecting the initial power-law profile N ( a ) ∝ a 3 -γ . This initial condition follows an empty (capture) loss-cone 6 , such that N = 0 for R ≤ R cap ( a ) = 8 R s / ( aa f ) . Here, R cap ( a ) defines a capture boundary corresponding to pericenter r p = 4 R s . The evolution of BH distribution N ( a, R ) is followed till time T 2b ( a c ) , which is 2B relaxation time at the critical semi-major axis a c (equation 4) and is the timescale over which BHs channelling most of the EMRIs are expected to get depleted. \nThe upper boundary for circular orbits R = 1 has a zero in-flow of BHs, with ∂ N /∂ R | R =1 = 0 . For outer-most grid column a = 2 a c , we consider zero GW fluxes because this boundary lies well outside GW loss cone defined by equation 3. For grid cells just above the capture boundary \nR cap ( a ) , the GW flux in R is not considered to avoid unphysical flow out of the empty loss cone, and only advective flux in a maintains the GW-induced orbital inspiral. \nWe consider a 50 × 50 rectangular and logarithmically uniform grid in a ∈ [ a gw /a f , 2 a c /a f ] and R ∈ [ R cap (2 a c ) / 2 , 1] 7 . Our numerical scheme is based on mid-point discretization for diffusion term in R , and upwind scheme for GW advection terms so that physical direction of flow is maintained. We employ a forward Euler scheme for the time evolution, that respects Courant condition for numerical stability (Press et al. 1992). We discuss the numerical solution and implications for resulting EMRI rates in the coming section.', '3. RESULTS': 'In this section, we discuss the results obtained from the numerical solution, while reporting the EMRI rates for a wide range of physically interesting parameters for NSCs. This includes MBH masses M · = 10 4 -7 M ⊙ relevant for the low-frequency GW detectors (Babak et al. 2017), and density slopes γ = 1 . 75 -2 . 5 of BH cluster to check the impact of strong segregation (Alexander & Hopman 2009). Further, we consider the cluster to host a single BH population with mass m = 10 M ⊙ , and choose a BH number fraction of f bh = 10 -3 motivated by realistic stellar initial mass functions (IMFs; discussed below in more detail). This also ensures that the strong segregation regime is valid inside the influence radius r h of MBH (Alexander & Hopman 2009). We employ the form of influence radius r h = 2pc √ M · / (4 × 10 6 M ⊙ ) , motivated by M · -σ relation (Kormendy & Ho 2013). So, the BH cusp has a total of N f = 10 -3 M · /M ⊙ BHs within a f = 0 . 1 r h , considering most stars as solar-type with mass m ⋆ = 1 M ⊙ . Here, we present the time-dependent EMRI rates, Γ (accounting for ejections) and Γ 0 (without the sink term induced by ejections) resulting from the numerical solution of FP equation 7. \nMost numerical works on strong segregation lead to relaxed BH cusps with γ ∼ 2 (Preto & Amaro-Seoane 2010). Further, one can theoretically estimate γ for various realistic stellar IMFs (Salpeter 1955; Miller & Scalo 1979; Kroupa 2001), by computing the associated relaxation coupling constant, ∆ = 4 f bh ( m/m ⋆ ) 2 / (3 + m/m ⋆ ) which separates the strong-segregation regime ( ∆ ≪ 1 ) from that of weak segregation ( ∆ > 1 ) (Alexander & Hopman 2009). For a typical m/m ⋆ ≃ 10 , these IMFs correspond to f bh ≃ 4 × 10 -4 -10 -3 , leading to ∆ ≃ 0 . 01 -0 . 03 8 that implies γ ≃ 2 -2 . 2 (see figure 4 of Alexander & Hopman \n2009). For γ ≥ 2 . 5 , unrealistically low values of ∆ ≲ 10 -3 are needed. Hence, the moderate slopes with γ ≈ 2 -2 . 25 can be deemed as the more realistic scenarios for the strong segregation regime. \nFirst, we identify a critical semi-major axis a c0 from the numerical solution, within which EMRIs are expected to dominate over plunges. Its analytical counterpart a c is given in equation 4. Technically, a c0 corresponds to the initial semi-major axis which is connected to an EMRI (progenitor) orbit with final r p = 4 R s (at a = a gw ) through a stream-line defined by the direction of the net-flux (see figure 5). For this, we employ the final state reached (without the inclusion of ejections) at time T 2b ( a c ) marking the end of simulations. So, only the fluxtubes originating at a ≤ a c0 can channel BHs onto the inspiral orbits near MBH. Table 1 gives both these lengthscales, a c0 and a c , for some representative values of M · and γ . Evidently, a c0 ∼ a few × 0 . 1 -10 mpc, is always smaller than a c ∼ a few × 10 -100 mpc. This trend stands out especially for steeper cusps with high γ , because associated higher BH densities imply shorter 2B timescales, leading to a narrower GW loss cone defined by r p ≤ r p, gw (see equation 3). While smaller physical values of a c0 are associated with smaller MBH masses and steeper BH cusps, the ratio a c0 /r h turns out to be almost independent of M · throughout the full range of parameters, with a c0 /r h ≃ { 0 . 018 , 0 . 0094 , 0 . 0032 , 0 . 0007 } for typical values γ = { 1 . 75 , 2 , 2 . 25 , 2 . 5 } . Further, the values of critical semi-major axis a c0 recovered from our FP approach is compatible with those quoted in earlier Monte-Carlo approaches (Raveh & Perets 2021). \nWe evaluate the instantaneous EMRI rates Γ (and Γ 0 for simulations without considering ejections) from the numerical solution as the net advective flow in a -direction at the inner boundary a = a gw . This presents a suitable method especially for the scenario including ejections, because the cusp is effectively depleted and ejections are not important inside a gw . Figure 1 showcases the evolving EMRI rates for various M · and γ , where we also present the well-studied case of the Milky Way (MW)-type NSC (with M · = M · mw = 4 × 10 6 M ⊙ ) in the left panel, enabling an easier comparison with earlier studies (Hopman & Alexander 2005; AmaroSeoane & Preto 2011; Raveh & Perets 2021) 9 .The impact of ejections is quite evident with lower EMRI rates Γ , in comparison to Γ 0 for the case without ejections. This rate suppression is strongest for the smallest M · and highest γ , \n<!-- image --> \nFigure 1. Evolution of EMRI rates, Γ (including ejections, in blue) and Γ 0 (without ejections, in magenta), shown as function of time in units of T 2b ( a c ) for various density slopes γ of BH cusp. [Left panel] Rates for various BH density index γ = 2 -2 . 5 are presented while fixing MBH mass M · = 4 × 10 6 M ⊙ . [Right panel] Rates for γ = 2 while varying M · = 10 4 -6 M ⊙ . Evidently, the suppression is strongest for steepest cluster densities with higher γ and/or lower MBH mass M · . Further, the level of rate suppression due to ejections driven by strong scatterings, is time-dependent. The comparison at time T 0 = T 2b ( a c0 ) (vertical gray lines) is apparently the most relevant in astrophysical settings that may sustain a steady state (see the text for more details). \n<!-- image --> \nTable 1. EMRI rates, Γ 0 (usual scenario without ejections) and Γ (with ejections) from the numerical solution are quoted for various cluster density slopes γ and MBH mass M · . These are the instantaneous rates at time T 0 = T 2b ( a c0 ) . Here a c0 is the numerically deduced maximum semi-major axis within which most EMRIs occur; while a c is its analytical counterpart as detailed in the text. \nFigure 2. EMRIrate ratio Γ / Γ 0 as a function of density slopes γ for a MW type NSC with M · = 4 × 10 6 M ⊙ at various times in color. The black curve corresponds to the suitable reference timescale T 0 for comparison with a steady-state undepleted BH cusp. The rates (at T 0 ) are suppressed by a factor of ≃ 3 for γ = 2 , while becoming an order of magnitude lower only for very steep cusps with γ ≳ 2 . 4 . \n<!-- image --> \nthough it builds up over time due to loss of BHs owing to ejections. The increasing magnitude of rate suppression with time is evident from figure 2, which gives the ratio Γ / Γ 0 for M · = M · mw at different values of time. \nFigure 1 shows the time evolution of EMRI rates Γ and Γ 0 , which is quite significant for steeper cusps. For γ ≤ 2 . 25 with dominant diffusive fluxes N/T 2b ∝ a 2(2 . 25 -γ ) at larger a , there is only a moderate decay of rates over time T 0 = T 2b ( a c0 ) (gray vertical lines in the figure); see table 1 for the typical values of T 0 for various M · and γ . This is because of the evolution of the initially isotropic system (uniform in R ) towards the Cohn & Kulsrud (1978) logarithmic profile in the diffusion regime ( r p ≳ r p, gw ) 10 . Afterward, rates, both Γ and Γ 0 , decay steeply due to the depletion of BHs (see figures 6 and 7) owing to losses due to the capture by MBH and ejections driven by strong scatterings 11 . So, T 0 can serve as a reference time over which BHs need to be replenished to channel a steady flux of EMRIs, and we quote the instantaneous values of Γ and Γ 0 at this time (black points on the evolution curves) for a better comparison with steady rates reported in literature. From table 1, T 0 ∼ 10 7 -9 yr for MW- \nlower MBH masses this timescale can be shorter by upto 2 orders of magnitude. The replenishment of BHs (possibly leading to a steady-state) in real NSCs can occur as a result of dynamical friction arising from energy relaxation, that we do not consider here. Further, galactic nuclei may undergo episodes of star formation (Levin & Beloborodov 2003; Lu et al. 2013; Aharon & Perets 2015, 2016; Schodel et al. 2020), that can also rejuvenate the supply of BHs. From figure 2 for the case of a MW-type NSC, rates Γ (at the relevant time T 0 ) are suppressed by more than a factor of 3 for γ ≳ 2 , while becoming an order of magnitude lower than Γ 0 for steepest cusps with γ ≳ 2 . 4 . \nWe report the EMRI rates over a wide range of M · and γ in figure 3 and table 1 at the time T 0 . The EMRI rates Γ 0 , without ejections, are higher for steeper density profiles and lower MBH masses following roughly Γ 0 ∝ M -1 / 4 · (for γ ≤ 2 . 25 ), as obtained previously in the works without consideration of strong scatterings 12 (Hopman & Alexander 2005; Aharon & Perets 2016; Broggi et al. 2022). For realistic values of γ ≃ 2 -2 . 25 , Γ 0 ≃ 100 -1500 Gyr -1 throughout the explored span of M · , and the suppressed rates Γ due to ejections, are lower by a factor of ≈ 3 -30 depending on M · and γ . As star clusters with high γ and low M · are expected to have higher BH number densities ∝ M · /r h 3 ∝ M -1 / 2 · , both the probability of ejections and EMRI formation tend to be higher for these systems. This effect levels off the suppressed rates to Γ ≃ 30 -50 Gyr -1 for all M · and realistic segregated profiles with γ ≃ 2 -2 . 25 . The extremely steep cusps with γ = 2 . 5 retain Γ ≃ 30 -200 Gyr -1 , although these are suppressed by a factor ≈ 10 -300 for all M · . However, for unsegregated cusps with BW profile γ = 1 . 75 , the rate suppression occurs upto a factor of ≃ 2 for all M · . It is notable that usual EMRI rates Γ 0 span a large range of values with a factor ≈ 30 -60 for various γ ∈ [1 . 75 , 2 . 5] for a given M · . This expanse of values attained is more restricted for suppressed rates Γ , and is limited to a factor of ≈ 2 -7 for M · = 10 5 -7 M ⊙ . Further, Γ becomes an increasing function of M · for steeper profiles γ ≳ 2 . 25 . \nFigure 3 (right panel) summarizes the level of suppression for all parameters. For a given M · , the rate suppression due to ejections becomes increasingly important for steeper profiles with larger γ . This behavior is similar to that found by Teboul et al. (2024) for TDE rates. For non-segregated BW cusps with γ = 1 . 75 , ejections suppress the EMRI rates by a factor of ≈ 2 . For segregated cusp with γ = 2 , the suppression level slightly increases upto a factor of ≈ 3 . For the realistic and moderately steep profiles with γ = 2 . 25 , the rates are suppressed by more than an order of magni- \n<!-- image --> \nFigure 3. EMRI rates, Γ (including ejections, solid curves) and Γ 0 (without ejections, dashed curves), in left panel, and their ratio Γ / Γ 0 in the right panel, are presented as a function of M · for various γ (in color) at time T 0 . Both lower MBH masses (low M · ) and steeper density slopes of clusters (high γ ) support a higher number density of BHs in the inner regions, which can potentially increase the rates of both EMRIs and ejections. From the left panel, Γ 0 is higher for higher γ and lower M · , and roughly follows the classical relation Γ 0 ∝ M -1 / 4 · found in many previous works without the inclusion of ejections. However, impact of ejections: (1) destroys these simplistic trends for EMRI rates Γ that become an increasing function of MBH mass M · for steep profiles with γ ≳ 2 . 25 , (2) caps the EMRI rates at a maximum Γ ≃ 200 Gyr -1 for the overall range of parameters explored here, while EMRI rates without ejections can attain high values of Γ 0 ∼ a few × 1000 Gyr -1 . From the right panel, suppression is stronger for higher γ and lower M · . For γ = 2 , the rates are suppressed by a factor ≈ 3 for all M · , while the suppression becomes important by an order of magnitude for (a) M · ≲ 10 6 M ⊙ for γ = 2 . 25 , and (b) M · ≲ 10 7 M ⊙ for γ = 2 . 5 . \n<!-- image --> \n· ≲ 10 6 M ⊙ . Highly steep cusps with γ = 2 . 5 have rates suppressed upto a factor ≳ 10 for all M · , while suppression gets important upto 2 orders of magnitude for M · ≲ 10 5 M ⊙ . These lower MBH masses otherwise have the highest unsuppressed rates Γ 0 and can channel EMRIs at LISA frequencies with the highest sensitivity (Babak et al. 2017), and are thus expected to dominate the future detection rates in the standard relaxation-driven scenario of EMRI formation. Thus, ejections driven by strong scatterings in strongly segregated cusps can have significant consequences for upcoming EMRI detections.', '4. SUMMARY AND DISCUSSION': 'Strong scatterings can significantly alter the loss cone dynamics around massive black holes in galactic nuclei (Teboul et al. 2024). In the current study, we hereby test the relevance of strong scatterings for EMRI formation in a strongly segregated BH cusp lying in the inner regions of an NSC. We numerically solve a time-dependent two-dimensional FokkerPlanck equation in an effective energy-angular momentum space to track the evolution of BH distribution. This takes into account angular momentum relaxation driven by 2B scatterings, GW induced losses, and ejections triggered by strong scatterings. We also improve upon the previous treatment of orbit-averaged ejection rates, by considering depletion of inner cusp of strong scatterers due to GW loss cone \n(see equations A8-A12 for exact expression, and equation 6 for a good approximation). \nEmploying the FP solution, we evaluate the timedependent EMRI rates, that decrease with time due to depleting BHs (figure 1), as a result of the capture by central MBH and ejections. The resulting EMRI rates Γ are lower than those Γ 0 without consideration of ejections, however the magnitude of rate suppression depends sensitively on the central MBH mass M · and steepness of BH cusp defined by density profile index γ . We explore a wide and physically interesting range of γ = 1 . 75 -2 . 5 and M · = 10 4 -7 M ⊙ for a single BH population of mass 10 M ⊙ with a fixed BH number fraction of f bh = 10 -3 within a ≤ 0 . 1 r h . The choice of a compact BH distribution with a low f bh is motivated by the previous studies on EMRI formation in strongly segregated cusps (Amaro-Seoane & Preto 2011; Raveh & Perets 2021). \nWe find that the impact of strong scatterings limits the per-galaxy rate of EMRI formation to the highest value of Γ ≃ 200 Gyr -1 for all M · and γ (Figure 3 left panel), which otherwise can reach upto Γ 0 ∼ a few × 10 3 Gyr -1 for steep BH cusps around low-mass MBHs. Contrary to the usual trend, suppressed rates Γ become an increasing function of M · for steep cusps with γ ≳ 2 . 25 . The level of suppression is highest for steepest cusps and lowest mass MBHs (Figure 3 right panel). Weakly segregated cusps with BahcallWolf γ = 7 / 4 profile, display only moderate rate suppres- \nr of 2. However, for strongly segregated BH cusps with γ ≳ 2 . 25 , the rate suppression becomes significant upto an order of magnitude for M · ≲ 10 6 M ⊙ . For steepest cusps with γ = 2 . 5 , rates are suppressed upto 2 orders of magnitude for small MBH masses of M · ≲ 10 5 M ⊙ . Thus, the phenomenon of strong scatterings, which is not traditionally accounted for EMRI formation, can significantly suppress the EMRI rates in galactic nuclei. The high levels of rate suppression for steep BH cusps around low-mass MBHs, brings forth the significance of strong scatterings for future EMRI detections by LISA, which is expected to be maximally sensitive for these low-mass MBHs (Babak et al. 2017). \nWe note that our framework captures the time-dependent effects only partially due to the assumption of a non-evolving background field of scatterers (see for eg., Vasiliev 2017; Pan & Yang 2021; Broggi et al. 2022 for a consistent timedependent approach). This fixes in time both the: (1) the diffusion coefficient for L -relaxation owing to weak 2B scatterings, and (2) ejection probability rate due to strong scatterings, which would otherwise decay with time because of depletion of BHs. However, this effect of expected depletion might be naturally countered by replenishment of BHs due to energy relaxation (that is not considered here) leading to dynamical friction driven in-flow of BHs into these \ninner regions of NSC. Further, the growth of NSC owing to in-situ star formation episodes and/or mergers with star clusters sinking-in from larger distances, can contribute towards the replenishment. This may result into a quasi steady state of scatterers, similar to our assumption. Further, consideration of multiple BH populations may also impact the EMRI rates (Aharon & Perets 2016), by altering the rate of ejections (Henon 1969). We defer these more detailed investigations to a future study. \nOur work brings out the significance of ejections driven by strong scatterings for EMRI formation in strongly segregated NSCs. However, the actual level of suppression implied for the future EMRI detection rates will depend sensitively on the structure and evolution of BH cusps in real NSCs and the lower end of the MBH mass function. The upcoming observational facilities, LISA and TianQin, necessitate more thorough future studies to evaluate realistic detection rates of these gravitational wave transients. \n- We would like to thank Odelia Teboul and Kartick C. Sarkar 1\n- for helpful comments and discussions. KK gratefully ac2\n- knowledges the support to access HPC at Technion, and com3\n- puting facilities at Indian Institute of Technology, Kanpur. 4', 'A. EJECTION RATE DRIVEN BY STRONG SCATTERINGS': "We consider a subject BH of mass m moving with velocity ⃗v in a uniform-density sea of similar BHs with an isotropic distribution function F . The probability that the BH undergoes a velocity kick ∆ ⃗v per unit d 3 ∆ ⃗v per unit time (Agekyan 1959; H'enon 1960a): \nR ( ⃗v, ∆ ⃗v ) = 8 πG 2 m 2 (∆ v ) 5 ∫ ∞ v '' d v ' v ' F ( v ' ) (A1) \nwhere v '' = ∆ v | 1 + ⃗v glyph[squaresmallsolid] ∆ ⃗v/ (∆ v ) 2 | for equal-mass encounters considered here. \nThe BH can get ejected from the system if its final speed exceeds that escape speed, i.e. the condition, C : | ⃗v +∆ ⃗v | ≥ v esc , is satisfied. This implies a probability of ejection per unit time, ˙ P ej = ∫ C R ( ⃗v, ∆ ⃗v ) d 3 ∆ ⃗v , given explicitly as (H'enon 1960b; Goodman 1983): \n˙ P ej = 32 π 2 G 2 m 2 3 v ( v 2 esc -v 2 ) 2 ∫ v esc √ v 2 esc -v 2 d v ' v ' F ( v ' ) ( v ' 2 + v 2 -v 2 esc ) 3 / 2 (A2) \nThe above integral is analytical for a near-Keplerian system and can be explicitly written as (Teboul et al. 2024): \n˙ P ej = P 0 a 2 v 1+2 γ ; with P 0 = Γ( γ +1) Γ( γ +1 / 2) ( m M · ) 2 N f a 3 f ( a f 2 GM · ) γ (A3) \nfor a power-law number profile N ( r ) = N f ( r/a f ) 3 -γ of BH cusp, with reference BH number N f = f bh M · /m ⋆ within a f , as considered earlier in Section 2. This corresponds to a distribution function F ∝ ε γ -3 / 2 . \nHence, the orbit-averaged ejection rate ⟨ ˙ P ej ⟩ for the BH on an orbit with elements { a, r p } can be evaluated as ⟨ ˙ P ej ⟩ = ( πT Kep ( a )) -1 ∫ d r ˙ P ej v -1 r , with v r as the radial speed along the Keplerian orbit at given phase r . We approximate this integral by considering only the dominant contribution at the phase r 0 = max[ r p , a gw ] , which gives the following simplified \nexpression: \n⟨ ˙ P ej ⟩ ≃ P 0 a 2 πT Kep ( a ) r 0 ( 2 GM · r 0 -GM · a ) γ (A4) \nUsing equation 1, the above expression can be posed in an alternative form presented in equation 6. \nThe above expression for ⟨ ˙ P ej ⟩ is accurate to a few tens of per cent to the actual integral in the expression for ⟨ ˙ P ej ⟩ , that is derived below. But, first we need to model the distribution of BHs in the depleted cusp within a gw . \nA.1. Depleted cusp \nWe assume that initial BH number profile N ( a ) = N f ( a/a f ) 3 -γ continues till a ≥ a gw , while orbital evolution for a ≲ a gw is dominated by GW emission even for circular orbits. Hence, depletion due to capture of stellar BHs by central MBH becomes important inside a gw , and the initial density profile with power-law index γ can not continue inside this critical distance. \nSince the orbits with a ≤ a gw correspond to EMRI progenitors, these are sourced by diffusive fluxes in the outer regions with semi-major axis a gw < a ' < a c . We denote the depleted profile of scatterers N d ( a ) for a ≤ a gw , which can be estimated in the steady state limit (Sari & Fragione 2019; Kaur et al. 2024) as N d ( a ) /T gw ( a ) = N ( a ' ) / (ln Λ 0 T 2b ( a ' )) . Here ln Λ 0 ≃ log ( a ' /r p, gw ( a ' )) ≃ 10 is the logarithmic size of GW loss cone 13 , and a ' > a gw is the initial semi-major axis that channels the BHs at a = r p, gw ( a ' ) < a gw , where the loss cone boundary r p, gw is given by equation 3. The inherent assumption in this approach is that BH orbits predominantly diffuse in r p due to 2B scatterings for r p > r p, gw , while for r p < r p, gw advection in a occurs due to GW emission. This gives a power-law form of depleted cusp: \nN d ( a ) = N fd ( a a gw ) 3 -γ d where N fd = N ( a gw ) ln Λ 0 = N f ln Λ 0 ( a gw a f ) 3 -γ and γ d = 33 -16 γ 4(3 -γ ) . (A5) \nThe depleted profile is much shallower with a power-law index γ d which is always a decreasing function of γ . For γ ∈ [1 . 75 , 2 . 5] , γ d ∈ [ -3 . 5 , 1] where the change in sign occurs at γ ≃ 2 . 06 . \nWe assign a function N ' ( a ) = N ' f ( a/a ' f ) 3 -γ ' to the overall BH number profile, with piecewise functions: \n{ N ' f , a ' f , γ ' } = { N f , a f , γ } , a ≥ a gw { N fd , a gw , γ d } , a < a gw (A6) \nA.2. Orbit-averaged ejection rate \nHere we formulate an exact way to evaluate orbit-averaged ejection rate ⟨ ˙ P ej ⟩ of a BH with orbital elements { a, r p } . For this, we also need a general local ejection rate ˙ P ej , valid for the entire test BH orbit as it traces both undepleted and depleted portions of the cusp, which is defined as: \n˙ P ej = P ' 0 a 2 v 1+2 γ ' with piece-wise function: \nP ' 0 = P 0 = Γ( γ +1) Γ( γ +1 / 2) ( m M · ) 2 N f a 3 f ( a f 2 GM · ) γ for a ≥ a gw P 0d = ( m M · ) 2 N fd a 3 gw ( a gw 2 GM · ) γ d for a < a gw (A7) \nas implied by equation A3 and A6. \nWe can then orbit-average this local ejection rate over true anomaly f as: \n⟨ ˙ P ej ⟩ = 1 2 πT Kep ( a ) √ GM · a (1 -e 2 ) ∫ 2 π 0 d f r 2 ˙ P ej = P ' 0 a 3 ( GM · /a ) γ ' 2 πT Kep ( a ) √ 1 -e 2 ∫ 2 π 0 d f r ' 2 v ' 1+2 γ ' where r ' = 1 -e 2 1 + e cos f , v ' = √ 2 r ' -1 = √ 1 + e 2 +2 e cos f 1 -e 2 . (A8) \nThis leads to the final ejection term F ej = ⟨ ˙ P ej ⟩ T 2b ( a f ) (defined earlier in equation 7) for the two cases, (a) r p ≥ a gw where whole orbit remains in undepleted cusp, and (b) r p < a gw where a portion of the orbit penetrates into depleted cusp inside a gw .", 'Case a. r p ≥ a gw': 'Since the orbit remains only in the undepleted cusp defined by density index γ , we have the following form of ejection term: \nF ej = Γ( γ +1) Γ( γ +1 / 2) 3 √ 2 π 2 γ +5 C ln Λ ( a f a ) γ -3 / 2 I ( e, γ, 0 , 2 π ) (1 -e 2 ) γ -1 (A9) \nwhere the function I is defined as: \nI ( e, γ, f 1 , f 2 ) = ∫ f 2 f 1 d f (1 + e 2 +2 e cos f ) γ +1 / 2 (1 + e cos f ) 2 (A10)', 'Case b. r p < a gw': 'The BH orbit is inside the depleted cusp for f ∈ [ -f gw , f gw ] , while it remains in the undepleted cusp with r > a gw for the remaining orbital phases, where \nf gw = cos -1 ( a (1 -e 2 ) a gw e -1 e ) . (A11) \nHence, for this case the ejection term F ej = F (u) ej + F (d) ej , where F (u) ej and F (d) ej arise due to the time spent in undepleted and depleted cusp respectively, and are given as: \nF (u) ej = Γ( γ +1) Γ( γ +1 / 2) 3 √ 2 π 2 γ +4 C ln Λ ( a f a ) γ -3 / 2 I ( e, γ, f gw , π ) (1 -e 2 ) γ -1 , F (d) ej = 3 √ 2 π 2 γ d +4 C ln Λ ln Λ 0 ( a f a gw ) γ -γ d ( a a f ) 3 / 2 -γ d I ( e, γ d , 0 , f gw ) (1 -e 2 ) γ d -1 . (A12)', 'B. ADDITIONAL FIGURES': '<!-- image --> \nFigure 4. Evolution of EMRI rates Γ (including strong scatterings; in blue) and ejection rates ˙ N ej (in pink) is shown as a function of time for [Left panel] for various γ = 2 -2 . 5 and fixed M · = 4 × 10 6 M ⊙ , and [Right panel] different M · = 10 4 -6 M ⊙ and fixed γ = 2 . The ejection rate ˙ N ej is computed for BHs with a ≤ a c0 , that are relevant for EMRI formation. As earlier, we depict the rates at time T 0 = T 2b ( a c0 ) (vertical gray lines) as black points. \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 5. Evaluation of the critical semi-major axis a c0 . Here, we show the streamlines (in blue) in { a, R } -plane indicating the net-flow of BHs at the end of simulations ( t = T 2b ( a c ) ) without strong scatterings. Different panels correspond to γ = { 2 , 2 . 25 , 2 . 5 } while fixing M · = 10 6 M ⊙ . Evidently, the analytical estimate a c (dotted magenta line) of the critical semi-major axis (equation 4) is not completely accurate and stream-lines initiating at a ∈ ( a c0 , a c ) end up into plunging orbits inside the capture radius (solid gray line). Hence, a c0 ≪ a c is a better estimate of the critical semi-major axis, as most stream-lines inside with initial a ≤ a c0 lead to inspiral orbits that eventually evolve into EMRIs. As γ increases towards right, a c0 /a f becomes smaller (and its departure from a c /a f is higher) due to narrowing GW loss cone (region under the dashed gray line defined by r p ≤ r p, gw ; equation 3). However, a c0 /a f is almost independent of M · , with a c0 /a f = { 0 . 094 , 0 . 032 , 0 . 007 } for γ = { 2 , 2 . 25 , 2 . 5 } . \n<!-- image --> \n<!-- image --> \nFigure 6. Evolution of BH number profiles N ( a ) , defined as number of BHs with semi-major axis < a , with time (in color). The solid (dashed) curves indicate the solution with (without) strong scatterings. The two panels correspond to γ = 2 , 2 . 25 for fixed M · = 10 6 M ⊙ . The black curves depict the solutions at the reference time T 0 ; while the initial power-law profile is depicted in yellow. The depletion of BHs due to ejections triggered by strong scatterings is evidently higher in the inner regions of the cusp (smaller a ), and for clusters with higher γ . The kink in these curves approximately corresponds to the BH depletion due to the capture by central MBH, satisfying log [1 / R cap ( a )] T 2b ( a ) ≈ t . The solid gray line corresponds to the critical semi-major axis a c0 , while dashed gray line to its analytical counterpart a c . \n<!-- image --> \n<!-- image --> \nFigure 7. Evolution of 2D density N of BHs, as function of R at fixed a = a c0 , with time in color. As earlier, solid (dashed) curves represent the solution with (without) strong scatterings, and black curves indicate the solution at time T 0 . The panels are for γ = 2 , 2 . 25 and fixed M · = 10 6 M ⊙ . Evidently, the solutions without strong scatterings follow the expected logarithmic profile in R , while solutions, including this effect, is more depleted for low R corresponding to higher eccentricity orbits, that are more prone to ejections. The impact of strong scatterings is higher for the cluster with higher γ (right panel). \n<!-- image -->', 'EMRI RATES FOR STRONGLY-SEGREGATED CUSPS': 'Qunbar, I., & Stone, N. C. 2023, arXiv e-prints, arXiv:2304.13062, doi: 10.48550/arXiv.2304.13062 \nRauch, K. P., & Tremaine, S. 1996, NewA, 1, 149, \ndoi: 10.1016/S1384-1076(96)00012-7 \nRaveh, Y., & Perets, H. B. 2021, MNRAS, 501, 5012, \ndoi: 10.1093/mnras/staa4001 \nRom, B., Linial, I., Kaur, K., & Sari, R. 2024, arXiv e-prints, \narXiv:2406.19443, doi: 10.48550/arXiv.2406.19443 \nSalpeter, E. E. 1955, ApJ, 121, 161, doi: 10.1086/145971 \nSari, R., & Fragione, G. 2019, ApJ, 885, 24, doi: 10.3847/1538-4357/ab43df Sch¨odel, R., Nogueras-Lara, F., Gallego-Cano, E., et al. 2020, A&A, 641, A102, doi: 10.1051/0004-6361/201936688 Sigurdsson, S., & Rees, M. J. 1997, MNRAS, 284, 318, doi: 10.1093/mnras/284.2.318 Teboul, O., Stone, N. C., & Ostriker, J. 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2024RAA....24j5008C | We used the Fivehundredmeter Aperture Spherical radio Telescope FAST to search for the molecular emissions in the Lband between 1.0 and 1.5 GHz toward four comets C2020 F3 NEOWISE C2020 R4 ATLAS C2021 A1 Leonard and 67PChuryumovGerasimenko during or after their perihelion passages. Thousands of molecular transition lines fall in this lowfrequency range many attributed to complex organic or prebiotic molecules. We conducted a blind search for the possible molecular lines in this frequency range in those comets and could not identify clear signals of molecular emissions in the data. Although several molecules have been detected at high frequencies of greater than 100 GHz in comets our results confirm that it is challenging to detect molecular transitions in the Lband frequency ranges. The nondetection of Lband molecular lines in the cometary environment could rule out the possibility of unusually strong lines which could be caused by the masers or nonLTE effects. Although the line strengths are predicted to be weak for FAST using the ultrawide bandwidth receiver and improving the radio frequency interference environments would enhance the detectability of those molecular transitions at low frequencies in the future. | 2024-10-01T00:00:00Z | ['10.1088/1674-4527/ad7823', '2024RAA....24j5008C', '10.48550/arXiv.2409.06227', 'arXiv:2409.06227', '2024arXiv240906227C'] | ['astrochemistry', 'ISM: molecules', 'comets: general', 'line: identification', 'Astrophysics - Earth and Planetary Astrophysics'] | FAST Observations of Four Comets to Search for the Molecular Line Emissions between 1.0 and 1.5 GHz Frequencies | 2,024 | 173 | 0.52 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML'] | 0 | https://arxiv.org/pdf/2409.06227.pdf | {'No Header': 'R esearch in \nA \nstronomy and \nA \nstrophysics', 'FAST Observations of Four Comets to Search for the Molecular Line Emissions between 1.0 and 1.5 GHz Frequencies': "Long-Fei Chen 1 , 2 , 3 , Chao-Wei Tsai 4 , 5 , 6 , Jian-Yang Li 7 , Bin Yang 8 , Di Li 4 , 3 , 9 , Yan Duan 10 , Chih-Hao Hsia 11 , Zhichen Pan 4 , Lei Qian 4 , Donghui Quan 3 , Xue-Jian Jiang 3 , Xiaohu Li 12 , Ruining Zhao 13 , 14 and Pei Zuo 4 \n- 1 School of Physics and Electronic Science, Guizhou Normal University, Guiyang 550025, China\n- 2 Guizhou Provincial Key Laboratory of Radio Astronomy and Data Processing, Guiyang 550025, China\n- 3 Research Center for Astronomical Computing, Zhejiang Laboratory, Hangzhou 311100, China\n- 4 National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, China; [email protected]\n- 5 Institute for Frontiers in Astronomy and Astrophysics, Beijing Normal University, Beijing 102206, China\n- 6 Key Laboratory of Radio Astronomy and Technology, Chinese Academy of Sciences, A20 Datun Road, Chaoyang District, Beijing, 100101, China\n- 7 School of Atmospheric Sciences, Sun Yat-sen University, Zhuhai, Guangdong, China\n- 8 N'ucleo de Astronom'ıa, Facultad de Ingenier'ıay Ciencias, Universidad Diego Portales, Chile\n- 9 Department of Astronomy, College of Physics and Electronic Engineering, Qilu Normal University, 2 Wenbo Road, Zhangqiu District, Jinan 250200, China\n- 10 Department of Electronic and Optical Engineering, Space Engineering University, Beijing, China\n- 11 Laboratory for Space Research, Faculty of Science, The University of Hong Kong, Hong Kong (SAR), China\n- 12 Xinjiang Astronomical Observatory, Chinese Academy of Sciences, No. 150 Science 1-Street, Urumqi 830011, China\n- 13 CAS Key Laboratory of Optical Astronomy, National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, China\n- 14 School of Astronomy and Space Sciences, University of Chinese Academy of Sciences, Beijing 100049, China \nReceived 20xx month day; accepted 20xx month day \nAbstract We used the Five-hundred-meter Aperture Spherical radio Telescope (FAST) to search for the molecular emissions in the L-band between 1.0 and 1.5 GHz toward four comets, C/2020 F3 (NEOWISE), C/2020 R4 (ATLAS), C/2021 A1 (Leonard), and 67P/Churyumov-Gerasimenko during or after their perihelion passages. Thousands of molecular transition lines fall in this low-frequency range, many attributed to complex organic or prebiotic molecules. We conducted a blind search for the possible molecular lines in this frequency range in those comets and could not identify clear signals of molecular emissions in the data. Although several molecules have been detected at high frequencies of great than 100 GHz in comets, our results confirm that it is challenging to detect molecular transitions in the L-band frequency ranges. The non-detection of L-band molecular lines in the cometary environment could rule out the possibility of unusually strong lines, which could be caused by the masers or non-LTE effects. Although the line strengths are predicted to be weak, for FAST, using the ultra-wide bandwidth receiver and \nimproving the radio frequency interference environments would enhance the detectability of those molecular transitions at low frequencies in the future. \nKey words: astrochemistry - ISM: molecules - comets: general - line: identification", '1 INTRODUCTION': "Comets are considered 'fossils' of our Solar System according to the planet formation theories (Shu 1977; Brasser & Morbidelli 2013). Following the formation of the Solar Nebula from molecular cloud to protostar and protoplanetary disk, planetesimals formed and finally evolved to the present day Solar System (Ehrenfreund & Charnley 2000; Chambers 2023). Thus, the composition of comets may reflect the initial environments of our Solar System (van Dishoeck et al. 2013; Willacy et al. 2022). For example, the various molecular species detected in the cometary coma may have inherited from the protosolar nebula (Bergner & Ciesla 2021), where they are frozen onto and preserved in the ice mantles of the dust grains until the comets travel to inner Solar System to the Sun to release the species into the gas phase under solar heating. \nThe composition of comets offers a useful chemical way to study the molecular inheritance from the Solar Nebula's evolutionary history. More than 70 molecules have been identified from various comets by remote or in-situ measurements (Rubin et al. 2019), with 67P/Churyumov-Gerasimenko having the highest number of discovered molecules (Biver & Bockel'ee-Morvan 2019) from measurements by the Rosetta mission, including both simple species such as hydroxyl (Smith et al. 2021), and complex species such as glycine (Altwegg et al. 2016). \nBesides the identification of molecules by in situ mass spectrometer measurements and by electron excitation in ultraviolet and visible wave ranges, the rotationally excited transitions of the gas-phase molecules located in the (sub-)millimeter frequency range provide the best way to detect them remotely by radio observations to reveal the kinetics and excitation conditions of the environment where these molecules are located. For example, using Atacama Large Millimeter/Submillimeter Array (ALMA), Roth et al. (2021) detected the emissions from methanol, formaldehyde, and other species from comet C/2015 ER61 (PanSTARRS) in the radio frequency around ∼ 350 GHz, revealing the low coma kinetic temperature and its asymmetric expansion velocity. Bergman et al. (2022) detected the methanol and hydrogen cyanide in several comets using the Onsala 20-m telescope at a frequency near 90 GHz, revealing short-time activity variations of those comets. \nAlthough many observations of molecular transitions have been reported in the 10 GHz and higher frequency range, the lower frequency range is under-explored, and the molecules detected in these bands only represent a small fraction of the molecules detected in comets. Detection of molecular transitions in the low-frequency ranges is challenging. Salter et al. (2008) conducted a line search within a bandwidth between 1.1 and 10 GHz using the Arecibo Telescope toward the starburst galaxy, Arp 220. Several absorption lines were reported, with their rest frequencies all above 4 GHz. Only one absorption feature around 1.638 GHz was found, which was attributed to 18 OH or formic acid (HCOOH) due to line confusion. Tan et al. (2020) performed surveys toward star-forming regions to search for molecular lines in the frequency range 6.0-7.4 GHz using the same telescope, resulting in the detection of only three molecules, CH 3 OH, H 2 CS, and OH. In the low frequencies down to 1 GHz of the radio L-band, only the OH 18 cm Λ -doublet transitions have been detected for many comets using the Arecibo Telescope, Nanc¸ay Radio Telescope, and Green Bank Telescope (Lovell et al. 2002; Tseng et al. 2007; Smith et al. 2021; Drozdovskaya et al. 2023). At even lower frequencies around 0.1 GHz, tentative detection of NO and its isotopologues, t-DCOOH, 17 OO, as well as SH were reported toward Galactic Centre and Orion using the Murchison Widefield Array (Tremblay et al. 2017, 2018, 2020). \nThe Five-hundred-meter Aperture Spherical radio Telescope (FAST) is the world's largest singledish radio telescope with three times the sensitivity of the Arecibo Telescope (Nan et al. 2011; Li et al. 2018; Qian et al. 2020). Thus, it provides a great opportunity to search for molecules in the frequency range from 1.0 to 1.5 GHz with its 19-beam receiver. In this paper, we report the first attempt to search \nfor molecular lines in four comets observed by FAST in 2020 and 2021. The observational setups are presented in Section 2. Section 3 and 4 discuss the results and the interpretations, respectively. We reach our conclusions in Section 5.", '2 OBSERVATIONS': "Four comets were observed from August 2020 to December 2021, including three long-period comets, C/2020 F3 (NEOWISE) (hereafter NEOWISE, (Bauer et al. 2020; Biver et al. 2022)), C/2020 R4 (ATLAS) (Manzini et al. 2021), and C/2021 A1 (Leonard) (Leonard et al. 2021; Zhang et al. 2021), and one Jupiter-family comet, 67P/Churyumov-Gerasimenko (hereafter 67P/C-G) (Biver et al. 2023). Table 1 summarizes the observation log. The Ephemeris of each comet was obtained from the HORIZONS system of Jet Propulsion Laboratory (Giorgini et al. 1996). All comets were moving at less than 15 ' per minute during the tracking, well within the 30 ' per minute tracking limit of FAST (Jiang et al. 2020). The observations of comet NEOWISE were conducted during the gap between two maintenance periods of FAST when the comet was near its closest approach to Earth, and the observations of other comets were Target of Opportunity (ToO) observations. Table 2 lists the FAST programs to observe these comets. \nFor comet NEOWISE, the target was observed with 12 observation blocks, each with an on-source integration time of 10 minutes in a tracking observation mode centered in the central beam (M01) of the FAST 19-beam receiver. To obtain off-source measurement for baseline subtraction, every 4 on-source observation schedules were accompanied by an observation schedule at ∼ 3 degrees from the target. Pulsar and spectral data were simultaneously recorded in the FAST backend during the observations. The pulsar data were used to search for pulsar candidates with no positive detections (Pan et al. 2021; Qian & Pan 2021). The spectral data were used in this work to search for molecular lines from the comet. It should be noted that for the tracking mode used for comet NEOWISE, the comet was actually not tracked. The telescope pointing was targeted at a fixed RA and Dec coordinate that was consistent with the position of the comet NEOWISE during each observation block. However, due to the moving speed is faster for comet than for background stars, the comet would move out of the beam after a period of time. Thus, the tracking time for comet NEOWISE was limited to 10 minutes at each observation block to ensure the comet remained within the 2 . ' 9 (at 1.4 GHz) beam size of M01. The left panel of Figure 2 shows the optical image of comet NEOWISE taken at the site of FAST during the observations. \nFor the other three comets that we observed, a custom observation mode was developed in order to improve the observational efficiency for moving targets. Compared with the tracking mode used for comet NEOWISE, these three comets were tracked at their non-sidereal rates in this custom mode, i.e. their RA and Dec coordinates were update in real time. In this mode, the target comet was placed at the central beam M01 and the side beam M11 of the 19-beam receiver alternatively with a 5-minute integration in each beam. This new mode would allow one beam to acquire the source signal while the other simultaneously acquires the off-source sky background. When alternating between two beams, the source can be continuously observed, although the sky subtraction has to be performed for each beam individually because different beams have different gains. For our observations, the separation of 10 . ' 22 between M01 and M11 required 30 seconds to switch between them to avoid exceeding the acceleration limit of 90 '' per second for the FAST receiver maneuver. Figure 1 shows the tracking positions and the orbital coordinates of comet C/2020 R4 (ATLAS) on 30th April 2021. The right panel of Figure 2 shows its optical image centered at the M01 beam and overlaid by the FAST 19-beam receiver. \nTable 2 summarizes the observation setups. The spectral backend with 1,048,576 channels was configured to cover 1.0 to 1.5 GHz at a frequency resolution of 476.8 Hz, corresponding to a velocity resolution of 0.1 km/s at 1.420 GHz. The sampling time was 1 second for the observations on comet NEOWISE and 0.1 second for other comets. The flux calibration utilized the noise diode with an injection to the optical path in the low-intensity mode (1 K) for NEOWISE and the high-intensity mode (10 K) for other comets. The noise injection period varied for different comets, as listed in Table 2. Assuming a system temperature of 24 K (Jiang et al. 2020) and an hour of integration, the rms noise level could reach 13 mK. \nTable 1: Observing conditions of the four comets observed by FAST. \nNotes: a Line-of-sight velocity of the target with respect to the observer. A positive value means the target center is moving away from the observer, negative indicates movement toward the observer. Data were taken from JPL/Horizons ephemeris. \nTable 2: Receiver configurations for our observations of the four comets. \nNotes: The dates of observations for these comets are in Table 1. The PIs of these observations are Zhichen Pan for comet NEOWISE, Chao-Wei Tsai for program ID PT2020 0166, and Zhong-Yi Lin for program ID PT2021 0045, respectively. \nFig. 1: The telescope tracking (blue) and the actual RA and Dec coordinates (orange) for comet C/2020 R4 (ATLAS) on 30 April 2021. \n<!-- image --> \n<!-- image --> \nFig. 2: (a) The image of comet NEOWISE stacked with 20 pictures, each with an exposure of 30 seconds. It was taken with a 127 mm refraction telescope at the FAST site on 30 July 2020. (b) The 25 ' × 23 ' optical image of comet C/2020 R4 (ATLAS) from Lulin observatory taken on 30 April 2021 overlaid with the beam positions of the FAST 19-beam receiver. \n<!-- image -->", '3 DATA PROCESSING AND ANALYSIS': 'The receiver recorded the full polarization of the signal in the backend during the observations. We first averaged the total power in the two perpendicular polarization directions, and converted the total power to antenna temperature using Equation 1, \nT a = T noise × Power cal -off Power cal -on -Power cal -off , (1) \nwhere T noise is the temperature of noise injection, and Power cal -on and Power cal -off are the total power when the noise injection was switched on and off, respectively. From the spectra for CH 3 OH (Figure 3), we can see that the rms of the final spectrum is consistent with the estimated noise level.', '3.1 Molecular line searching': "To search for the molecular transition lines in the L-band, we collected a list of lines from the Splatalogue 1 database in the frequency range from 1.0 to 1.5 GHz. We carried out a systematic blind \nsearch based on the prospective molecular lines in selected frequency ranges without strong radio frequency interference (RFI). Table 3 shows examples of the prospective molecular transition lines for CH 3 OH and 17 OH in the L-band. \nMethanol is one of the common ice components and has been detected in a number of comets (Rubin et al. 2019). Figures 3 and 4 show our spectra that cover the expected CH 3 OH lines for comets 67P/C-G and C/2021 A1 (Leonard), respectively. However, no clear methanol emissions are discernible. A suspicious signal of CH 3 OH at 1.120 GHz for comet C/2021 A1 (Leonard) is visible in Figure 4. However, it is present in both the on-source and off-source observations for both beam M01 and beam M11 and therefore is most likely RFI. Assuming a rotation temperature of 36 K and a line width of 0.7 km/s for the methanol emission as indicated by the IRAM 30-m observations (Biver et al. 2023), the estimated 3 σ upper limit production rate of CH 3 OH at 1.443 GHz for 67P/C-G is 2 × 10 32 molecules s -1 for two hours of integration time using the equation from Drahus et al. (2010). Compared with the derived production rate of CH 3 OH(around 10 26 molecules s -1 ) from Biver et al. (2023), this estimated upper limit is too high to give a reasonable constrains. Due to the non-detection of molecular lines, we therefore constrained the integrated intensity of CH 3 OH at 1.443 GHz based on its reported production rate and described the results in the later section. \nThe 18 cm OH line has been detected in many comets in the L-band by the Arecibo Telescope, Nanc¸ay Radio Telescope, and Green Bank Telescope (Lovell et al. 2002; Tseng et al. 2007; Smith et al. 2021; Drozdovskaya et al. 2023). Unfortunately, the OH transitions at 1.665 and 1.667 GHz are out of the frequency range of the FAST L-band receiver. However, several tens of 17 OH transitions and six 18 OH transitions are located within the FAST L-band receiver. Figure 5 shows those spectra for comet C/2021 A1 (Leonard). Again, no 17 OH and 18 OH transitions are visible in our data. However, there is a suspicious absorption feature near the expected transitions of 1073.214 MHz. Further inspection of the data suggests that this 'absorption' resulted from the subtraction of the off-source spectrum from the on-source spectrum, which means that it is an emission both in the on-source and off-source spectrum. Therefore, we conclude that this is an artifact due to RFI. \nNo observations were reported for the detection of 17 OHin comets in the literature. To estimate the upper limit column density of 17 OH, we assumed a line width of 2.5 km/s based on the OH line profile for comet NEOWISE observed by Arecibo (Smith et al. 2021). Since the excitation temperature of 17 OH in comet comae is unknown, we simply adopted a range of values from 10-100 K and calculated the 3 σ upper limit column density of 17 OHat 1.302 GHz. It should be noted that the excitation temperature used here is not physically appropriate since the maser effects due to the absorption of solar radiation are very important for the OH excitation (Schleicher & A'Hearn 1988). Therefore, the above constrains on the 17 OH column density are very crude. We found that the upper limit column density for 17 OH is in the range of 7-10 × 10 12 cm -2 . The estimated value is about 2 times lower than the 16 OH column density previously reported (Smith et al. 2021). \nIn the cometary environments, the production of OH is dominated by the photodissociation of H 2 O. The molecule 17 OH would be produced by the photodissociation of the isotopologue of water, H 17 2 O. The oxygen isotopic abundance ratio of 17 O/ 16 O as derived from H 17 2 O/H 16 2 O in comet 67P/CG is ∼ 4 × 10 -4 (Altwegg et al. 2015; Muller et al. 2022). If we assume the same abundance ratio of 17 OH/ 16 OH for comet NEOWISE and 67P/C-G, then that would indicate that our upper limit column density of 17 OH is highly overestimated. This suggests that the integrated intensity of 17 OH used in the calculations, i.e., the rms noise level with the current integration time is too high to constrain the signal strength of 17 OH. This also indicated that the non-detection of 17 OH is consistent with the expected 17 O/ 16 O elemental ratio. \nInstead of constraining the molecular column density or production rate, we can use the molecular production rate reported in the literature to estimate the expected integrated intensity of L-band molecular emissions. The molecular production rate can be expressed by the following Equation 2, assuming optically thin and LTE (Drahus et al. 2010), \nQ = 2 √ πln 2 k B h b ∆ v exp DI ( T ) ν ( e hν/k B T -1) ∫ T mb dv, (2) \nwhere Q is the production rate in molecules per second, k B and h are Boltzmann constant and Planck constant, respectively, b = 1.2 is the dimensionless factor of the full width at half maximum (FWHM) of the telescope's beam, D is the antenna's aperture, ∆ is the geocentric distance of the comet, v exp is the expansion velocity of gas, ν and I ( T ) are the rest frequency of the molecule and line intensity at temperature T , respectively, and ∫ T mb dv is the integrated intensity. Using the production rates of molecules reported for comets in the literature, we can derive the intensity of molecular emissions. Table 4 shows the estimated integrated intensity for CH 3 OH at 1.443 GHz and 17 OH at 1.302 GHz for comets NEOWISE, 67P/C-G, and C/2021 A1 (Leonard), which have the production rates for CH 3 OH and OH previously reported. It should also be noted here that these estimations are based on the LTE assumption, so the results are rough estimate. The maser effects or non-LTE assumption should be considered if one wants to obtain a more accurate estimate. The integrated intensity for OH at 1.667 GHz is also estimated for comparison. The estimated integrated intensity for CH 3 OH at 1.443 GHz and 17 OHat 1.302 GHz are orders of magnitude lower than that for OH at 1.667 GHz. Therefore, we do not expect the detection of those molecular lines in the L-band in our data, at least for CH 3 OH and 17 OH. The possible reasons are discussed in the following section. \nExcept for CH 3 OH and 17 OH, there are various other molecules whose transition lines fall in the L-band. Table 5 lists a subset of molecules of interest. None of these molecules were identified, and their detection thresholds are far below the limit of current constrains. Several reasons can lead to the nondetection. Apart from CH 3 OHor H 2 CO, whose abundances are a few to ten percent relative to water, for other molecules, such as C 2 H 5 OH shown in the list, their abundances relative to water could be lower than one percent. On the other hand, large molecules tend to have more transition lines. However, due to the energy distribution in these lines, the intensity for transition lines at L-band is weaker than the high frequency lines. And also the non-LTE effects for the molecule excitation in the cometary coma environment may also complicate the situation (Bockel'ee-Morvan et al. 2004). \nFinally, we examined the potential line features of CH 3 OH, 17 OH, and other molecules for the comets C/2020 R4 (ATLAS) and C/2020 F3 (NEOWISE). However, the RFI was more severe in the data obtained before November 2021 because the electronic environment of the FAST feed cabin was not improved until after that time. Therefore, fewer RFI-free bands in the 1.0-1.5 GHz were available for these two comets. In summary, we did not identify convincing molecular emission signals in the observations for these comets, either.", '3.2 The neutral hydrogen line': 'Since our primary scientific goal is to search the molecular lines, we did not specifically consider the background contamination of comets when scheduling our observation plans, which is important if we want to identify the HI emission from the comet itself. We have visually checked the background contamination by stacking the comets position and the HI4PI survey (HI4PI Collaboration et al. 2016). We found that, for comet NEOWISE and the last two scheduled observations of comet C/2020 R4 (ATLAS), the background was clean. While for other comets during their observations, they were either coincide with a background cloud or on the edge of a background cloud, thus unsuitable for constraining the HI emission from those comets. As a result, the HI spectrum from comet 67P/C-G, C/2021 A1 (Leonard), and C/2020 R4 (ALTAS) have a complex profile with multiple components, likely due to the background contamination. Most importantly, the off-source positions were also characterised by similar HI spectrum, thus we were also unable to perform a good baseline subtraction. \nFor comet NEOWISE, although it was visually located in a clean background during its observations, there were still HI emissions around 1420.4 MHz both from the on-source position and offposition. The HI emission profile was less complicated than in the other scheduled observations of comets. Figure 6 shows the 20 minutes of integration of the HI emission for the on-source position and off-position without Doppler correction from comet NEOWISE. Due to the different total on-source and off-source times for comet NEOWISE, we were unable to give a baseline subtraction from the off-source. It was also noted from Figure 6 that off-source HI emission was slightly stronger than the on-source HI emission by less than 1 K. The calculated HI column density is 1.5 × 10 20 cm -2 and 1.9 × \nTable 3: Molecular transition lines and their spectral parameters for CH 3 OH and 17 OH in the L-band.Table 4: The estimated integrated intensity for molecular lines of CH 3 OH (at 1.443 GHz), 17 OH (at 1.302 GHz), and OH (at 1.667 GHz) for comets C/2020 F3 (NEOWISE), 67P/C-G, and C/2021 A1 (Leonard). \nTable 5: A subset of molecules of interest used for searching in this study.', 'CHBOH 1120.37 MHZ': "Fig. 3: The spectra of comet 67P/C-G obtained on 2021-10-18 and 2021-11-02. The red vertical dashed lines mark the rest frequencies of CH 3 OH. The upper and lower panels are for CH 3 OH at 1.120 and 1.443 GHz, respectively. \n<!-- image --> \n10 20 cm -2 for the on-source and off-source positions, respectively. This difference could be due to the spatial variation of background HI gas. On the other hand, due to the large FAST beam size (174 '' ) and small comet nucleus size (0.009 '' ) at the observed distance, the obstruction of background emission byFig. 4: The spectra of comet C/2021 A1 (Leonard) obtained on 2021-12-07. The red vertical dashed lines mark the rest frequencies of CH 3 OH. The upper and lower panels for CH 3 OH at 1.120 and 1.443 GHz, respectively. \n<!-- image --> \nFig. 5: The 17 OH spectra of comet C/2021 A1 (Leonard) obtained on 2021-12-07. The red vertical dashed lines mark the rest frequencies of 17 OH. \n<!-- image --> \nthe comet could be ignored. Therefore, from these observations we could not constrain whether comet NEOWISE has an HI emission or not. \nTo our best knowledge, there are almost no report of HI in the radio wavelength range for comets to-date. Recently, Pal & Manna (2024) reported an HI absorption detection from comet NEOWISE using the Giant Metrewave Radio Telescope (GMRT), and derived an HI column density on the order of 10 21 cm -2 . If we simply assume that HI and OH are mainly produced by the photodissociation of H 2 O, then the production rate and column density for HI and OH should be similar. Keller & Lillie (1974) reported such a similar production rate for HI and OH with no less than 2 times difference for the comet Bennett (1970 II). For comet NEOWISE, compared with the derived OH column density of 1.1 × 10 13 cm -2 from Smith et al. (2021), the derived HI column density from Pal & Manna (2024) was highly overestimated. From our observations, although the dip around 1420.4 MHz may be interpreted as an absorption feature from the on-source spectrum, we could not rule out the possibility of the multicomponents of emission feature when compared with the off-source profile. Moreover, if we assume that the production rate for HI is comparable with that of OH, the integrated line intensity for HI would be below our detection limit.", '4 DISCUSSION': "Molecules in the coma originate from the outgassing of the nucleus, sublimation of ice species in the coma, or formed directly in the coma by photochemical processes (Cordiner et al. 2023). The chemical composition of comets is essential to study the chemical inheritance from the protosolar nebula to the planetary system. Both in-situ measurements and remote observations are used to identify the molecular species in comets. Numerous molecules have been discovered in Jupiter-family comet 67P/C-G, mainly due to the Rosetta mission of the European Space Agency. On the other hand, molecules in many longperiod Oort Cloud comets have been observed during their perihelion passages by remote observations in the optical, infrared, or radio wavelengths. Table 6 summarizes the detected molecules reported in the literature for the four comets we observed as derived from observations in various wavelengths. \nThe detectability of molecules depends on the activity of the comet. Molecules in highly active comets are relatively easy to detect (Biver et al. 2015; Protopapa et al. 2021; Faggi et al. 2023). Volatile species can be released during the sublimation of H 2 O and CO 2 ices (Rubin et al. 2023), which is a \nFig. 6: The 20 minutes of integration of HI emission both for the on-source and off-source position during the observations of comet NEOWISE. \n<!-- image --> \nconsequence of the outburst due to the thermodynamic evolution of the cometary nucleus surface and subsurface layers (Wesołowski 2022). Astrochemical models also show that abundant organic molecules can be present in the coma due to the outgassing of the nucleus and gas-phase chemistry in the coma (Cordiner & Charnley 2021; Ahmed & Acharyya 2022). For the four comets that we observed, C/2020 R4 (ATLAS) was observed by only optical facilities. Lin et al. (2021) found at least three outbursts in their observations and did not see any new jet features and fragments based on its coma morphology. For comet NEOWISE and 67P/C-G, although multiple molecules have been observed in the frequency range higher than 100 GHz by IRAM 30-m radio telescope and NOEMA interferometry array, OH is the only molecule detected in the low-frequency L-band. Comet C/2021 A1 (Leonard) was observed by FAST about one month before its perihelion, although many volatile species were detected in the near-infrared range two weeks before the perihelion (Faggi et al. 2023). \nIn the previous section, the LTE condition was assumed to calculate the molecular production rates. However, in the cometary coma environment, the LTE condition may not be valid due to the processes of various other excitation mechanisms (Bockel'ee-Morvan et al. 2004). For example, although OH could be excited by the 2.7 K background radiation, the dominated mechanism to excite OH is the resonance fluorescence by the absorption of solar radiation (Schleicher & A'Hearn 1988). Therefore, the nondetection of radio wavelength of L-band molecular lines in the cometary environment could rule out the possibility of anomalously strong lines, which would be caused by pumping of ultraviolet/infrared radiation or other non-LTE effects. \nIn addition to the production rates of molecules, the detectability of molecules in the radio wavelengths depends on their intrinsic transitional characteristics and environmental excitation conditions, such as the Einstein A-coefficient and upper/lower energy at a specific level of excitation, as well as the gas temperature and number density of molecules in the coma. For the detection of more than 300 molecules in the ISM 2 , the observed frequencies are in the range of tens to hundreds of GHz, and most of them are in star-forming regions (McGuire 2022), suggesting that the excitation conditions of molecules are related to their environment. Although many molecular transitions fall in the L-band, including the organic and prebiotic molecules, the high upper-level energy (E up ) and/or low value of Einstein A-coefficient (A ij ) of the molecular excitation conditions do not favor their detection in the L-band in the cometary environment. For example, methanol, one of the molecules commonly detected in comets with high-frequency transitions, has E up and A ij generally several of tens Kelvin and higher \nTable 6: Summary of the molecules detected by ground-based observations in the optical, infrared, and radio wavelengths for comets C/2020 F3 (NEOWISE), C/2020 R4 (ATLAS), 67P/C-G, and C/2021 A1 (Leonard). \nNotes: 1. Cambianica et al. (2021a), 2. Cambianica et al. (2021b), 3. Faggi et al. (2020), 4. Munaretto et al. (2023), 5. Faggi et al. (2021), 6. Smith et al. (2021), 7. Biver et al. (2022), 8. Drozdovskaya et al. (2023), 9. Manzini et al. (2021), 10. Biver et al. (2023), 11. Mugrauer (2021), 12. Faggi et al. (2023), 13. Skirmante & Jasmonts (2022), 14. Biver et al. (2024). \nthan 10 -6 s -1 , respectively. However, the highest A-coefficient for molecules in the L-band is less than 10 -9 s -1 , and the E up can be as high as hundreds of Kelvin, making the detection much harder than in the high-frequency bands. \nFinally, in the L-band, RFI is usually more severe than at high frequencies and has to be considered in data analysis. The wide frequency range of RFI reduces the usable frequency bandwidth and introduces ambiguities in the identifications of molecular spectral lines. The ultra-wide bandwidth receiver (Zhang et al. 2023), which will cover the 0.5-3.3 GHz frequency range, once installed on FAST, could greatly benefit the detection of molecules in comets.", '5 SUMMARY': "We observed four comets, C/2020 F3 (NEOWISE), C/2020 R4 (ATLAS), C/2021 A1 (Leonard), and 67P/Churyumov-Gerasimenko, to detect their molecular emission or absorption features in the radio Lband from 1.0 to 1.5 GHz using FAST from August 2020 to December 2021. We searched for thousands of line transitions associated with hundreds of molecular species in the RFI-free frequency channels in the data. No clear evidence of the emission lines was present, resulting in a null detection of those molecules in these four comets. Under the LTE conditions, we estimated the integrated intensity for CH 3 OH at 1.443 GHz and 17 OH at 1.302 GHz using their production rates reported in the literature, and found that the expected intensity for the searched molecular lines is too weak to be detected in our observations. Therefore, it is not surprising for the non-detection of molecular lines in the L-band from 1.0 to 1.5 GHz. This non-detection of molecular lines in the cometary environment could also rule out the possibility of unusually strong lines. Observing highly active comets and the implementation of the ultra-wide bandwidth receiver on FAST expected in the near future will improve the detectability of molecular lines in comets. \nAcknowledgements We thank the reviewer for the suggestions that helped us greatly improve the manuscript. The authors thank Qingliang Yang, Chun Sun at FAST Operation Center for assisting in the development of the special observation mode for moving target observations. The authors thank ZhongYi Lin and Wing-Huen Ip for their discussions on this project. This work is supported by a grant from the \nNational Natural Science Foundation of China (NSFC) No. 11988101. L.-F.C. acknowledges the support from the China Postdoctoral Science Foundation grant No. 2023M733271 and the Foundation of Education Bureau of Guizhou Province, China (Grant No. KY (2020) 003). Z.P. and L.Q. acknowledge the support from the National Key R&D Program of China grant No. 2022YFC2205202 and 2020SKA0120100, and by the NSFC grant No. 11703047, 11773041, U2031119, 12173052, 12173053, 12373032, and 11963002. Z.P. and L.Q. are supported by the CAS 'Light of West China' Program and the Youth Innovation Promotion Association of the Chinese Academy of Sciences (ID No. 2023064, 2018075, and Y2022027). D.L. is a New Cornerstone Investigator. 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2024arXiv240903941B | This chapter provides an indepth overview of white dwarfs the evolutionary terminus of the vast majority of stars. It discusses their discovery their nature as degenerate objects their connections to earlier phases of stellar evolution their subsequent evolution as they gradually cool down the varied physical conditions from their dense cores to their tenuous atmospheres some key statistics about the properties of the ever expanding population of known white dwarfs the diversity of their spectra the accretion of planetary material and the presence of magnetic fields. The chapter also highlights the instrumental role of white dwarfs in other areas of astronomy. | 2024-09-01T00:00:00Z | ['arXiv:2409.03941', '10.48550/arXiv.2409.03941', '2024arXiv240903941B'] | ['Astrophysics - Solar and Stellar Astrophysics'] | White dwarf fundamentals | 2,024 | 173 | 0.56 | ['EPRINT_HTML', 'EPRINT_PDF'] | 0 | https://arxiv.org/pdf/2409.03941.pdf | {'Simon Blouin 1 ,a': 'a University of Victoria, Department of Physics and Astronomy, Victoria, BC V8W 2Y2, Canada \n© 20xx Elsevier Ltd. All rights reserved.', 'Abstract': 'This chapter provides an in-depth overview of white dwarfs, the evolutionary terminus of the vast majority of stars. It discusses their discovery, their nature as degenerate objects, their connections to earlier phases of stellar evolution, their subsequent evolution as they gradually cool down, the varied physical conditions from their dense cores to their tenuous atmospheres, some key statistics about the properties of the ever expanding population of known white dwarfs, the diversity of their spectra, the accretion of planetary material, and the presence of magnetic fields. The chapter also highlights the instrumental role of white dwarfs in other areas of astronomy. \nKeywords: Chandrasekhar limit, Compact objects, Degenerate matter, Stellar atmospheres, Stellar evolution, White dwarf stars', 'Glossary': "Chandrasekhar limit Maximum mass that a stable white dwarf star can have before collapsing \nCooling track Theoretical model describing how a white dwarf of a particular mass and composition cools over time \nCore crystallization Liquid-solid phase transition undergone by the dense plasma in white dwarf cores \nDegenerate matter State of matter in which quantum mechanical e ff ects dominate over thermal e ff ects due to Pauli's exclusion principle \nEquation of state Model describing the relationship between properties such as pressure, volume, and temperature for a given substance \nGravitational settling Process where heavier elements sink while lighter elements rise to the surface \nLuminosity function Distribution showing the number of white dwarfs at di ff erent luminosities \nMass-radius relation Relation describing how higher-mass white dwarfs have smaller radii \nSpectral evolution Changes in the surface composition of white dwarfs as they cool \nWhite dwarf cosmochronology The study of cooling white dwarfs to determine the age of stellar populations", 'Nomenclature': 'DA White dwarf displaying hydrogen absorption lines \nDB White dwarf displaying neutral helium absorption lines \nDC White dwarf with a featureless spectrum \nDO White dwarf displaying ionized helium absorption lines \nDQ White dwarf displaying carbon absorption features \nDZ White dwarfs displaying absorption lines from metals other than carbon', 'Learning objectives': '- · White dwarfs were first recognized as a fundamentally di ff erent class of stars in the early 1900s due to their small radii.\n- · White dwarfs are supported by electron degeneracy pressure and obey a mass-radius relationship.\n- · White dwarfs are the final stage in the evolution of the vast majority of stars.\n- · The composition profiles of white dwarfs bear the imprint of previous evolutionary phases.\n- · The evolution of white dwarfs is characterized by a long cooling process.\n- · White dwarf masses are narrowly distributed around ≃ 0 . 6 M ⊙ .\n- · The faint end of the white dwarf cooling sequence provides a valuable tool for measuring the ages of stellar populations.\n- · White dwarf atmospheres vary widely in temperature and composition, resulting in a broad spectrum of spectral types.\n- · The pollution of white dwarf atmospheres by planetary debris o ff ers insights into exoplanetary systems.\n- · White dwarfs can have strong magnetic fields, the origin of which remains debated.', '1 Introduction': "Around 98% of stars in our Galaxy will eventually become white dwarfs at the end of their long lives, 6% of stars in the solar neighbourhood are currently white dwarfs, and over 350,000 white dwarfs have now been identified with high confidence. Yet, unless they deliberately sought them out, chances are the reader of this chapter has never seen a white dwarf in the night sky. If white dwarfs are so inconspicuous despite being so prevalent, it is because they are intrinsically faint. Only about the size of the Earth, they shine orders of magnitude less brightly than more voluminous main-sequence and giant stars with the same temperature, such that none is visible to the naked eye. White dwarfs are small, but their masses are comparable to that of the Sun, making them extremely dense objects, 100,000 times denser than the center of the Earth. \nThese stellar remnants have historically proven to be exceptionally fertile grounds for advances in physics and astronomy. In the 1920s and 1930s, understanding the very nature of white dwarfs as degenerate stars constituted one of the first applications of quantum mechanics to astrophysics (this groundbreaking work was later recognized in the 1983 Nobel Prize in Physics). Since the 1980s, the oldest white dwarfs are used to place tight constraints on the ages on various stellar populations within the Milky Way. Type Ia supernovae, powerful thermonuclear detonations that occur when a white dwarf accretes too much mass from a companion, led to the paradigm-shifting discovery of the accelerating expansion of the Universe in 1998 (also later recognized with a Nobel Prize in 2011). Today, the James Webb Space Telescope is searching for biosignatures on planets orbiting white dwarfs, the European Space Agency's Gaia mission is enabling us to glimpse at the exotic physics of crystallization in the dense plasmas found in the cores of white dwarfs, and in the not-so-distant future LISA will detect gravitational waves emanating from white dwarf pairs. But white dwarfs are more than useful testbeds for the physics of extreme densities, tracers of cosmology, or laboratories for astrobiology. They are also fascinating objects in their own rights, and in large part, this is what this chapter will attempt to demonstrate. \nThe chapter is organized as follows. Section 2 presents a brief historical account of the discovery of white dwarfs. The conundrum that arose from these early observations will naturally lead us to Section 3, which details the nature of white dwarfs as degenerate stars and introduces the famous Chandrasekhar limit. We will then look at the origins of white dwarfs in Section 4 by detailing the stellar evolution phases that shape the composition profiles of white dwarfs. In particular, this section will expand on the numerous connections between the helium-burning phases of stellar evolution and white dwarfs. We will then explore in Section 5 how the star evolves once it has become a white dwarf. This is a vast topic in itself, and we will cover only the basic notions required for the rest of this chapter. This section will also provide an opportunity to more closely examine the physical conditions within white dwarfs, from their dense cores to their sparse atmospheres. The subsequent section, Section 6, delves into the analysis of white dwarfs as a population, focusing on their characteristically narrow mass distribution, as well as the importance of the luminosity function in age-dating applications. Section 7 will then discuss important notions pertaining to the atmospheres of white dwarfs, including spectral classification, external pollution by planetary material, and magnetic fields. Finally, a brief conclusion is given in Section 8.", '2 The discovery of white dwarfs': "The earliest observation of a white dwarf is attributed to William Herschel (1738-1822), who in 1783 discovered two faint companions to the much brighter 40 Eridani A (abbreviated 40 Eri A). 1 We now know that one of these companions, 40 Eri B, is a white dwarf. Although it was discovered first, 40 Eri B is not the brightest white dwarf in the night sky. This distinction goes to Sirius B. However, Sirius B is much harder to observe as it is swamped by the intense brightness of its companion Sirius A, the brightest star visible at night. In fact, the existence of Sirius B was hypothesized some two decades before it could be discovered. In 1844, Wilhelm Friedrich Bessel (1784-1846) suggested that yet unseen binary companions were responsible for irregularities in the motions of Sirius and Procyon across the sky. Sirius B was finally discovered in 1862 and found to be 10,000 times fainter than Sirius A. \nSince the components of a given star system are at a very similar distance from the Earth, large di ff erences in apparent brightness imply large di ff erences in intrinsic luminosity. The luminosity L of a star is given by \nL = 4 π R 2 σ T 4 e ff , (1) \nwhere R is the star's radius, σ is the Stefan-Boltzmann constant and T e ff is the e ff ective temperature. At the time, the large di ff erence in L between Sirius A and Sirius B (and between 40 Eri A and 40 Eri B) could a priori be attributed to di ff erences in temperatures, because their temperatures were yet unknown. With this convenient explanation, there was apparently nothing unusual left to explain. It is only decades later that the truly peculiar nature of 40 Eri B and Sirius B would become clear. \nIn 1910, Henry Norris Russell, Edward Charles Pickering and Williamina Fleming discovered that 40 Eri B was of spectral class A, meaning that it is hot enough to show hydrogen spectral lines. This was totally unexpected, because it meant that it had to be hotter than 40 Eri A, a K-type star. 40 Eri B being ∼ 100 times fainter than 40 Eri A, this realization implied that 40 Eri B was remarkably small (Equation 1). Similar conclusions were reached for Sirius B and van Maanen's Star (an isolated star, also known as van Maanen 2) during the following decade, clearly establishing a new and puzzling class of stars. This newest addition to the stellar zoo was first coined 'white dwarfs' by Luyten (1922) to highlight their small size and white color (indicative of hot temperatures). In addition, because 40 Eri B and Sirius B belong to triple and binary star systems, their masses could be determined from their orbital motions. Their substantial \nmasses ( M ∼ 1 M ⊙ ) and small radii ( R ∼ 0 . 01 R ⊙ ) implied very high densities ( ρ ∼ 10 6 g cm -3 ) that seemed aberrant at the time, before the development of quantum mechanics. In the words of Eddington (1927): \nThe message of the Companion of Sirius when it was decoded ran: 'I am composed of material 3,000 times denser than anything you have come across; a ton of my material would be a little nugget that you could put in a matchbox.' What reply can one make to such a message? The reply which most of us made in 1914 was-'Shut up. Don't talk nonsense.' \nIn the next section, we will make sense of this 'nonsense' by invoking the quantum mechanical concepts that were not yet available to the discoverers of white dwarfs at the beginning of the previous century.", '3.1 Electron degeneracy pressure': "Given their high temperatures and extreme densities, white dwarfs interiors are completely ionized plasmas, where atoms are stripped bare of their electrons. Moreover, particles are so closely packed together that the average interparticle distance is smaller than the electrons' thermal de Broglie wavelength, \nλ th = s 2 π ℏ 2 mekBT , (2) \nwhere ℏ is the reduced Planck constant, me is the electron mass, where kB is the Boltzmann's constant, and T is the temperature. This is key to understanding the nature of white dwarfs, as quantum e ff ects arise when the de Broglie wavelengths overlap. In particular, the Pauli exclusion principle significantly alters the behavior of electrons in white dwarfs. This principle states that identical fermions (e.g., electrons) cannot simultaneously occupy the same quantum state. In the extremely dense plasmas of white dwarfs, this forces electrons to occupy higher-energy levels, leading to the formation of 'degenerate matter'. Crucially, trying to reduce the volume occupied by these degenerate electrons would force them to occupy yet higher-energy states. This would require the application of a compression force against this 'degeneracy pressure'. \nFor a completely degenerate electron gas (a good approximation for the extreme densities that characterize most of a white dwarf's interior) and assuming that the density is not too high (see below), this degeneracy pressure P is given by \nP = GLYPH<16> 3 π 2 GLYPH<17> 2 / 3 5 ℏ 2 me n 5 / 3 e , (3) \nwhere ne is the electron number density. This P ∝ ρ 5 / 3 equation of state was derived by Fowler (1926), who was investigating the nature of matter at the mean white dwarf densities established during the previous decade from astronomical observations. This was one of the first applications of quantum mechanics to astrophysics, and it contained the key insight required to understand the structure of white dwarfs. More specifically, it is this degeneracy pressure that opposes the white dwarf's strong gravity and prevents it from collapsing under its own weight. This is to be contrasted with lower-density main sequence stars, where thermal pressure (as in the ideal gas law, P = nkBT ) plays this role.", '3.2 The mass-radius relationship and the Chandrasekhar limit': "By combining this equation of state (Equation 3) with two of the basic equations of stellar structure, the hydrostatic equilibrium and the mass conservation equations, the famous white dwarf mass-radius relationship can be derived. In the simple case where Equation (3) applies, it takes the form M ∝ R -3 . This implies that higher-mass white dwarfs have smaller radii. The higher the mass, the stronger the gravity and the higher ne must be to provide the degeneracy pressure required to avoid collapse. There is however a limit to this relation. Equation (3) relied on the assumption that the electrons are not relativistic. However, with increasing densities, electrons are forced into higher and higher energy states and eventually exceed the rest energy of an electron, thereby becoming relativistic particles. As the electronic velocities increase, the equation of state 'softens' and eventually becomes P ∝ ρ 4 / 3 in the extreme relativistic limit: \nP = GLYPH<16> 3 π 2 GLYPH<17> 1 / 3 4 ℏ cn 4 / 3 e , (4) \nwhere c is the speed of light. Note that we have again assumed that the electrons are completely degenerate. For simplicity, equations for the general case of partial degeneracy and arbitrary relativity are not provided here. The softening of the equation of state from P ∝ ρ 5 / 3 to P ∝ ρ 4 / 3 indicates that past a certain point, increasing the star's density provides a diminishing increase in the degeneracy pressure. Ultimately, this sets an upper limit of ≃ 1 . 4 M ⊙ on the mass of a stable white dwarf. This is the famous Chandrasekhar (1931) limit. A white dwarf with a mass exceeding this critical threshold would collapse into a neutron star or black hole. \nFigure 1 illustrates modern theoretical mass-radius relationships. Today, detailed numerical models are used instead of the analytical relations described above. These models take into consideration the detailed composition of the white dwarf's degenerate interior, the size of its non-degenerate outer layers, and the small but non-negligible e ff ects of finite temperature. Note how increasing the temperature of \nFig. 1 White dwarf mass-radius relationships. The curves correspond to predictions from a modern stellar evolution code (B'edard et al., 2022) assuming different effective temperatures and hydrogen layer thicknesses (see legend). The masses and radii of three classical white dwarfs are also shown (Bond et al., 2015, 2017a,b). \n<!-- image --> \nthe star inflates its radius for a given mass as a result of the e ff ect of thermal pressure. Figure 1 also illustrates the properties of three classical white dwarfs, whose masses and radii were independently determined due to their belonging to binary or triple star systems (in general the mass and radius of a white dwarf cannot be independently measured). These measurements match the theoretical mass-radius relations. The same conclusion has been reached for many other stars using other observational techniques that can yield independent mass and radius measurements for specific stellar systems, such as astrometric microlensing and the characterization of eclipsing binaries. In fact, the mass-radius relationship is now well established to the extent that determining one, the mass or the radius, is generally considered to directly provide an accurate determination of the other, and vice-versa. Similarly, a measurement of the surface gravity g = GM / R 2 , a convenient quantity in the analysis of stellar spectra, can be used to uniquely determine the mass and radius of a white dwarf.", '4 White dwarfs as the fossil records of stellar evolution': 'Having explored the basic properties and nature of white dwarfs, it is now time we turn to their origins. White dwarfs represent the end point in the evolution of stars with initial masses lower than approximately 10 M ⊙ . This encompasses 98% of the stars in our Milky Way, with the remainder collapsing into neutron stars or black holes. Today, around 6% of stars in our Galaxy are white dwarfs. The path to this final state involves a sequence of nuclear fusion processes that are summarized below. We first focus on the formation of white dwarf cores, where most of the mass is contained, in Sections 4.1 and 4.2 before discussing the outer layers in Section 4.3.', '4.1 The formation of white dwarf cores': "The first step in the creation of a white dwarf core is when hydrogen undergoes fusion into helium, a process that begins in the main sequence stage and continues into the subgiant and red giant branch (RGB) phases. Stars with a mass inferior to about 0 . 5 M ⊙ do not continue further down the periodic table to ignite helium fusion. These stars generally remain on the main sequence for durations exceeding the age of the universe and therefore cannot a priori contribute to the current white dwarf population. Nonetheless, helium-core white dwarfs, stars that did not undergo further nuclear burning, do exist. The formation of these rare objects is generally attributed to evolution within binary systems that experience mass-transfer episodes. Indeed, the majority of these helium-core white dwarfs are found in binary systems. These stars have masses below ≃ 0 . 45 M ⊙ , and the most extreme specimens, with masses inferior to ≃ 0 . 25 M ⊙ , are known as extremely low-mass white dwarfs (ELMs). \nExcluding the progenitors of helium-core white dwarfs, the second step in the creation of a white dwarf core starts when a star progresses to ignite helium burning. During the horizontal branch phase, helium is burned in a convective core surrounded by a stable helium envelope. The helium-fusing core produces carbon-12 through the tripleα reaction and oxygen-16 via the α -capture reaction 12 C( α, γ ) 16 O, ultimately leading to the formation of a carbon-oxygen core. This region is homogeneous (i.e., the carbon-to-oxygen ratio is the same everywhere) because it was produced in a convective environment where fluid motions e ffi ciently mix the di ff erent chemical species. After the horizontal branch, the star will continue burning helium during the asymptotic giant branch (AGB) phase. AGB stars are characterized by a sandwichlike structure, with a carbon-oxygen core (the remnant of the core helium burning phase) surrounded by helium-burning shell, a hydrogen burning shell, and an inert intershell in between. Evolved AGB stars undergo periodic instabilities known as 'thermal pulses'. This \nFig. 2 Representative chemical composition profile of a typical white dwarf. The abundances of each species are given as mass fractions. The center of the star is at the left, and the surface is towards the right. m ( r ) is the mass enclosed within a sphere of radius r ; hence, q corresponds to the mass fraction outside of radius r . For example, 90% of the white dwarf's mass is located to the left of log q = -1 . It is useful to think of a white dwarf as being divided into three parts: a core (in most cases made of carbon and oxygen), atop of which sits a helium-hydrogen envelope, and an atmosphere (the very thin layer at the surface that is observable, log q ≲ -14 ). Note how the envelope makes up only 1% of the mass but accounts for 15% of the radius given its relatively low density. \n<!-- image --> \nphenomenon is due to the mismatch between the helium production rate by the hydrogen-burning shell and the rate at which helium is fused in the helium-burning shell underneath. Helium thus piles up and is compressed to become moderately degenerate. This catastrophically increases the helium burning rate as the helium-burning shell can now heat up without significantly expanding (remember that thermal pressure is negligible in a degenerate gas). This is also known as a helium shell flash. \nThe helium-burning phases detailed in the previous paragraph ultimately result in the formation of a carbon-oxygen core that comprises approximately 99% of the mass of the soon-to-be-born white dwarf. As indicated in Figure 2, this core bears the imprints of the di ff erent helium-burning phases. A large portion of the mass is contained in a central homogeneous region formed during the core helium burning phase, and above that we find a more complex composition profile resulting from the AGB evolution. As we will see in Section 4.2, the core composition profile remains uncertain, reflecting unresolved questions related to the helium-fusing phases of stellar evolution. \nThe carbon-oxygen core also harbor a small trace of neon-22, representing 1-2% of the white dwarf's mass. 2 This isotope is formed during the helium-burning phases, through the capture of alpha particles by nitrogen-14, which itself is a byproduct of the hydrogen-burning CNO cycle. Although neon-22 is only a trace species in carbon-oxygen white dwarfs, its presence is noteworthy due to its neutron-rich composition, featuring 12 neutrons and 10 protons, unlike carbon-12 and oxygen-16 which have equal numbers of neutrons and protons. This excess of neutrons means that more mass is necessary to achieve the same pressure, which as we have seen is primarily sustained by electron degeneracy pressure. Neon-22 requires 2.2 nucleons per electron, compared to 2.0 nucleons per electron for carbon-12 and oxygen-16. This has interesting implications for the subsequent evolution of the white dwarf (Section 5.2), which are the subject of active research and have yet to be fully understood. \nFinally, stars with an initial mass close to the 10 M ⊙ upper limit for white dwarf progenitors can go one step further in their nuclear burning. These objects can reach the super asymptotic giant branch (SAGB), where the temperature is high enough to ignite carbon and ultimately produce an oxygen-neon core. This core is dominated by oxygen-16 and neon-20, but also contains significant traces of carbon12, neon-22, sodium-23, and magnesium-24. Oxygen-neon white dwarfs are rare. A white dwarf needs to have a mass ≳ 1 . 05 M ⊙ (these stars are known as ultramassive white dwarfs) to have evolved through the SAGB phase, and only ∼ 2% of white dwarfs reach that mass. In fact, the fraction of white dwarfs hosting oxygen-neon cores might be even smaller, as there is now convincing evidence that some ultramassive white dwarfs actually harbor carbon-oxygen core, contrary to what standard (single-star) stellar evolution theory predicts (Cheng et al., 2019).", '4.2 The uncertain core composition profile': 'The white dwarf core composition profile shown in Figure 2 left of log q = -2 aligns with the predictions of current stellar evolution theory, but these predictions remain uncertain. First, there is the question of the 12 C( α, γ ) 16 O reaction rate. As we have seen, this reaction is responsible for the production of oxygen-16 during the helium-burning phases, and as such, it largely controls the carbon-to-oxygen ratio of white dwarf cores. However, the cross section of 12 C( α, γ ) 16 O at the temperatures relevant for helium burning in stars is very small, making its empirical determination challenging. Decades of experimental work have significantly narrowed down the uncertainties on the 12 C( α, γ ) 16 O reaction rate, but these reduced uncertainties still imply a relatively wide range of possible oxygen abundances. Varying the reaction rate within the 2 σ experimental confidence interval results in changes of the order of ± 0 . 1 in the oxygen mass fraction at the center of white dwarfs (Chidester et al., 2022). This ≃ 15% uncertainty has important consequences for the subsequent evolution of the white dwarf. \nSecond, we still do not know how large the central oxygen-rich region really is. The mass extent of this homogeneous region corresponds to the mass extent of the former helium-burning convective core, where species were e ffi ciently mixed by convective eddies. Standard stellar evolution provides a simple prescription, the Schwarzschild criterion, to define the location of a convective boundary. According to this criterion, the star is convective in regions that satisfy the condition ∇ rad > ∇ ad, where ∇ rad is the so-called radiative gradient, \n∇ rad = 3 16 π acG κ LP mT 4 , (5) \nwith a the radiation constant, κ the opacity of the gas, and m the mass enclosed within the radius at which ∇ rad is calculated. ∇ ad is the adiabatic gradient, the temperature variation in a fluid element undergoing a change in pressure at constant entropy, \n∇ ad = ∂ ln T ∂ ln P ! s . (6) \nThe value of ∇ ad depends on the equation of state of the fluid (e.g., ∇ ad = 0 . 4 for an ideal gas without molecules). Despite the pervasiveness of the Schwarzschild criterion in stellar evolution models, it is well established that mixing due to convection can occur passed the Schwarzschild boundary, in regions where ∇ rad < ∇ ad. In fact, establishing just how far this convective boundary mixing extends is an area of active study for all sorts of stellar environments, from the hydrogen-burning cores of main-sequence cores to the convective envelopes of white dwarfs. This problem turns out to be particularly challenging in the case of convective helium-burning cores. A complication arises due to the production of carbon and oxygen in the convective core, which are much more opaque than the helium located above. This tends to create a discontinuity in the κ profile, which makes the convective boundary unstable. Indeed, any amount of extra mixing beyond the convective boundary would push high-opacity material in the helium-rich layer above, thereby increasing its ∇ rad and making it convective. But to due the particular interactions between the di ff erent thermodynamic quantities at play, this outward extension eventually splits the convective region in two. This outcome is generally considered to be unphysical and many numerical schemes have been devised to circumvent this problem. These di ff erent prescriptions ultimately lead to di ff erent white dwarf composition profiles, and it remains unclear which of the multiple approaches implemented in various stellar evolution codes should be preferred. Note that uncertainties on core boundary mixing do not only a ff ect the size of the oxygen-rich region, but also its carbon-to-oxygen ratio. \nThird, the region formed during the thermally-pulsing AGB phase ( -2 ≲ log q ≲ -1 . 5 in Figure 2) is also fraught with uncertainty. Successive thermal pulses convert helium into carbon and oxygen, and homogenize the shell due to the development of a convection zone. This is thought to ultimately result in the formation of a complex double-layered structure at the bottom of the helium envelope. However, the size and composition of this region are very uncertain. They are a ff ected by the number of thermal pulses experienced by the star, which itself depend on the rate at which the star loses mass during the thermally-pulsing AGB phase and on the uncertain extent of convective boundary mixing in the flash-driven convection zone. \nThese di ff erent sources of uncertainties are inconvenient in that they currently limit our ability to precisely model white dwarfs. Yet, the fact that white dwarfs are sensitive to these earlier processes also represents a unique opportunity. In particular, asteroseismology, the study of pulsations in stars, is emerging as a promising tool to semi-empirically infer the composition profiles of the interiors of white dwarfs (Giammichele et al., 2018). The hope is that white dwarfs can then play the role of stellar fossils that, when correctly examined, shed light on earlier evolutionary phases to constrain the 12 C( α, γ ) 16 O reaction rate (Chidester et al., 2022) or the physics of convective boundary mixing.', '4.3 The envelope': "We have discussed at length the formation of the carbon-oxygen core, but what about the helium and hydrogen layers above (Figure 2)? These layers represent only ∼ 1% of the total mass, but they play a fundamental role in our understanding of white dwarfs. Not only is the only observable region of the star (the atmosphere, see Section 7) located there, but also the hydrogen / helium envelope largely controls the subsequent evolution of the star by mediating the flow of energy between the core and the exterior. \nStandard stellar evolution predicts an helium envelope making up ∼ 1% of the star's total mass ( M He ∼ 10 -2 M ⋆ ) and a much thinner superficial hydrogen layer, M H ∼ 10 -4 M ⋆ , where the exact numbers depend on the metallicity and stellar mass (thinner hydrogen and helium layers with increasing mass). Asteroseismological studies largely confirm that these predictions are correct for typical white dwarfs (Giammichele et al., 2022). The size of the hydrogen layer is determined by the maximum extent of the hydrogen-burning region. If an hydrogen envelope is too thick, the conditions for nuclear burning are met at its base, and the hydrogen layer then shrinks at the expense of the helium envelope. This residual burning is negligible for most white dwarfs, but not for low-metallicity white dwarfs (Miller Bertolami et \nFig. 3 Theoretical white dwarf cooling tracks for carbon-oxygen white dwarfs with thick hydrogen envelopes and masses of 0.6 and 1.0 M ⊙ (top and bottom, respectively). Small black circles with adjacent labels indicate cooling times in Gyr. The portions of the cooling tracks colored in light blue correspond to white dwarfs in the process of crystallizing, while the dark blue portions mark complete core crystallization. Some well-known white dwarfs are shown in grey. \n<!-- image --> \nal., 2013). That being said, we also know from observations (see Section 5.2) that ∼ 25% of white dwarfs must have a hydrogen layer that is much thinner than this canonical value of M H ∼ 10 -4 M ⋆ . The main evolutionary pathway associated with the formation of these so-called hydrogen-deficient white dwarfs is the born-again scenario (Werner and Herwig, 2006). According to this scenario, a post-AGB star or a white dwarf experiences a late helium-shell flash after having left the AGB. The envelope then rapidly expands and becomes convective. This convection zone engulfs the residual hydrogen and transports it to deeper (and hotter) regions where it can be burnt. When this final burning episode is completed, the star contracts into a white dwarf, now with almost no hydrogen left (as little as M H ∼ 10 -12 M ⋆ in some cases).", '5 The evolution of white dwarfs': 'Having discussed in some detail the evolutionary history leading to the formation of a white dwarf, we now turn our attention to their subsequent evolution upon reaching the white dwarf stage. This has been a subject of active research for decades. Given the extensive nature of this topic, it is covered in greater depth elsewhere in this Encyclopedia. However, a concise overview is essential here to provide context and framework for the discussions that follow. This section aims to o ff er a brief summary of the main themes and processes involved in white dwarf cooling. We will also take a closer look at the physical conditions that characterize white dwarfs in Section 5.3.', '5.1 White dwarfs as cooling embers': "The vast majority of white dwarfs do not experience any nuclear burning. 3 Deprived of this energy source, white dwarfs behave as stellar embers that can only cool down. This makes white dwarf evolution relatively simple compared to other types of stars. As shown in Figure 3, white dwarfs simply follow a straight trajectory in a luminosity-e ff ective temperature diagram (Hertzsprung-Russell diagram), becoming ever cooler and fainter. This evolutionary pathway is often called a cooling track. \nThe fact that the cooling tracks in Figure 3 are nearly perfectly linear in this log-log diagram implies that the stars' radii remain virtually constant as they cool down (see Equation 1). This is of course a direct consequence of the degenerate nature of white dwarfs, which largely prevents further gravitational contraction (see also the small di ff erence in radius between white dwarfs with identical masses but di ff erent temperatures in Figure 1). This is an important property, as it implies that white dwarfs cannot tap into gravitational contraction to maintain their luminosity. Very young and hot white dwarfs do experience a non-negligible decrease ( ∼ 50%) of their radii in their first few Myr of evolution. However, this contraction mostly takes place in their non-degenerate outer envelopes. As we have seen (Figure 2), the envelope represents a very small fraction of the white dwarf's total mass, meaning that the gravitational energy released by this process is only a minor source of energy. \nMost of a white dwarf's thermal energy is stored by the ions in the core, where ≃ 99% of the mass is. The electrons contribute little to \nthe total thermal reservoir, since as degenerate particles, they already lie in the lowest energy states available to them. White dwarf cooling is then mostly a matter of emptying that ionic thermal energy reservoir outside the star. For most of the white dwarf's evolution, this is ultimately done by photons escaping from the surface, but in the early stages of the cooling process, neutrinos produced and escaping from the core are the main energy sink. \nThere are two main questions when it comes to white dwarf cooling: how large is the thermal reservoir that is slowly leaking, and how fast is this leaking taking place. The first question is closely connected to the discussion of Section 4.2. If we assume for simplicity that the core is made of non interacting ions, classical thermodynamics tells us that the heat capacity of each ion in the core is given by 3 2 kB . Since a core of a given mass will contain a di ff erent number of ions depending on its composition, it follows that the heat capacity and stored thermal energy is dependent on the core composition. For example, a more oxygen-rich core will have fewer ions than a more carbon-rich core, and all else being equal, cool down more quickly as a result. As we have seen, uncertainties in pre-white dwarf evolution imply that the core composition remains only loosely constrained. \nThe second question is a complicated one that depends on a wide range of interesting physical processes, including thermal conduction in dense plasmas, convection, radiative opacities in the atmosphere, and the physics of phase transitions. This last point is particularly noteworthy and merits further discussion, even in this brief overview of white dwarf cooling. As the ions in the core gradually lose their thermal energy, they become more and more influenced by the Coulomb interactions with their neighbours. The ratio of their electrostatic interaction energy to their kinetic energy gradually increases. This is measured by the Coulomb coupling parameter, \nΓ = Z 2 e 2 aikBT , (7) \nwhere Z is the ionic charge and ai is the average interionic distance. The plasma freezes into a solid state when Γ reaches a critical threshold ( Γ = 175 for the simple case of a plasma with just one ionic species). This liquid-solid phase transition is known as core crystallization. 4 Core crystallization unfolds over Gyr timescales (see the light blue highlighting in Figure 3) as a crystallization front slowly moves outward starting from the center of the star. This inside-out progression is the result of a weak temperature gradient (Section 5.3) and strong density gradient in the core. The center being more dense, the Coulomb interactions are stronger there ( ai is smaller in Equation 7), which favors crystallization. Importantly for white dwarf evolution, this transition is accompanied by the release of latent heat, which, as a new energy source, slows down the cooling process. Ultimately, this transformation into a solid state accelerates the cooling of the coolest white dwarfs by dramatically decreasing the heat capacity of the core. This is known as Debye cooling.", '5.2 Chemical evolution': "As a consequence of being compact objects, white dwarfs have an intense surface gravity of ∼ 10 8 cms -2 (log g = 8, compared to log g = 4 . 4 for the Sun). This implies that gravitational settling is particularly e ffi cient, with heavier elements sinking down and lighter ones floating up. However, this simple picture must be incomplete, since as we will see in Section 7, elements heavier than hydrogen are often detected at the surface of white dwarfs. Additional physical mechanisms compete with gravitational settling so that the very thin observable layer at the surface can take di ff erent compositions. Interestingly, these transport mechanisms change in importance during the evolution of white dwarfs, such that the surface composition of a given white dwarf changes over the course of its evolution. This is often referred to as spectral evolution. \nThe general picture that has emerged from decades of empirical and theoretical studies is as follows. As we have seen in Section 4.3, standard stellar evolution theory predicts that white dwarfs have a hydrogen content corresponding to M H ∼ 10 -4 M ⋆ . This is generally referred to as a 'thick' hydrogen layer, and with that amount of hydrogen no internal transport mechanism can alter the composition of the surface layer. Hence, these white dwarfs, which represent approximately 75% of the white dwarf population maintain a pure-hydrogen atmosphere throughout their evolution (assuming that no external accretion contaminates their surface). The other 25% are more interesting. They enter the white dwarf cooling track with much less hydrogen, probably following a late helium shell flash (Section 4.3). Initially, at very high temperatures (down to ≃ 75 , 000 K), their atmospheres are rich in carbon and oxygen. These heavy elements can remain at the surface despite the incessant pull of gravitational settling because of competition from stellar winds. Eventually, the winds fade and gravitational settling can operate more freely. The small amount of residual hydrogen contained in these stars then rises to the surface, and most display a hydrogen-dominated atmosphere by the time they reach ≃ 30 , 000 K. The amount of hydrogen required to complete this transformation into a hydrogen-atmosphere white dwarfs is surprisingly small, with M H ∼ 10 -12 M ⋆ thought to be su ffi cient. As the cooling process continues, these stars eventually recover a hydrogen-deficient atmosphere. For stars with very little hydrogen, this is done through the process of convective dilution in the 30 , 000 K ≳ T e ff ≳ 15 , 000 K range. The thin superficial hydrogen layer is eroded from beneath by convective motions in the much thicker helium layer. For stars with more hydrogen, convection instead develops in the hydrogen layer, thereby mixing it with the thicker helium mantle underneath provided that the hydrogen layer is not too thick ( M H ≲ 10 -6 M ⋆ ). This takes place below 15 , 000 K, with more massive hydrogen layers necessitating cooler temperatures (the convection zone deepens as the white dwarf cools). Spectral evolution contains many more complexities and unsolved riddles than what has been presented in this brief outline: the interested reader should consult B'edard (2024) for a more complete up-to-date overview of this subfield. \nThe surface layers are not the only ones to undergo compositional changes. First, gravitational settling also operates in the core. For \nexample, in a carbon-oxygen white dwarf neon-22 slowly di ff uses downward as a result of its higher mass-to-charge ratio, a process often referred to as neon-22 sedimentation. This is noteworthy because the transport of heavier species toward the center releases gravitational energy that can materially slow down the cooling process by providing a new energy source to the star. By the time neon-22 di ff usion stops because of the core being completely crystallized, this process can induce a cooling delay of a few hundreds of Myr (meaning that the white dwarf would have reached a given temperature / luminosity hundreds of Myr earlier if it were not for neon-22 sedimentation). A more spectacular chemical transformation takes place during the crystallization process itself. The newly formed solid phase generally does not have the same composition as the coexisting liquid. This fractionation process is similar to the formation of sea ice, where salt is largely expelled from the solid phase. In white dwarfs, this chemical separation rearranges the core composition profile in a way that can release copious amounts of gravitational energy. This is enough to delay the cooling process by many hundreds of Myr for the vast majority of white dwarfs, and for some specific core compositions this can extend to several Gyr (Blouin et al., 2021).", '5.3 A detailed look at the stratification': 'In this section, we focus on the specific physical conditions within white dwarfs, examining their internal structure from the dense core all the way to the tenuous atmosphere at their surface. Figure 4 shows density-temperature profiles for a standard 0 . 6 M ⊙ carbon-oxygen core white dwarf with a thick hydrogen layer ( M H / M ⋆ = 10 -4 ) at di ff erent e ff ective temperatures. The thickness of the lines indicate the dominant atomic constituent, with carbon and oxygen in the dense core on the right (thickest lines), followed by helium, and finally hydrogen on the left towards the surface (thinnest lines). Note that this representation greatly expands the outer layers of the star: in Figure 2 we saw that the carbon-oxygen core represents 85% of the radial extent and 99% of the mass of the star. \nThe densities and temperatures encountered in white dwarfs span many orders of magnitude, and as such, matter is found in di ff erent states. At high temperatures and / or high densities, we have a completely ionized plasma. This corresponds to the region with no color shading in Figure 4. If the plasma is both very dense and not too hot, we have seen in Section 5.1 that the plasma freezes into a solid state. This liquid-solid boundary is indicated by triangular symbols in Figure 4. At the other extreme, at very low temperatures and densities close to the surface, we find a neutral hydrogen gas (region shaded in dark blue). For the coolest white dwarfs, this neutral hydrogen gas is dominated by molecular hydrogen. Note that Figure 4 only considers the case of a thick hydrogen layer: white dwarfs with much less hydrogen can instead have a neutral helium gas close to the surface. Finally, there is a region of partial ionization (region shaded in light blue) in between the completely ionized plasma in the interior and the neutral hydrogen gas close to the surface. This region of partial ionization is associated with a convection zone. This is due to the strong opacity increase that makes convection the most e ffi cient way of transporting to the surface heat from the thermal reservoir in the core. This particular region of the star is also responsible for the driving of pulsations in pulsating white dwarfs, which is discussed in more details elsewhere in this Encyclopedia. Also shown in Figure 4 is the boundary of the degenerate interior (diamond symbols), defined here as the point where the temperature is inferior to the local Fermi temperature, the temperature at which thermal e ff ects are comparable to quantum e ff ects due to the Pauli exclusion principle. We can see that both the helium layer and the carbon-oxygen core are in this electron degenerate state. \nWementioned convection as an energy transport mechanism in the previous paragraph, but this is just one out of four ways of transporting energy out of the core in white dwarfs. First, we already saw earlier how neutrino cooling be an important energy sink for very hot white dwarfs. Second, radiative transport is the dominant energy transport mechanism for the outer layers of the star above the convection zone (if there is one). Radiative transport is negligible in the interior because radiative opacities are too high for this to be an e ffi cient transport mechanism. Third, electron thermal conduction dominates energy transport for the bulk of the star. In fact, it is so e ffi cient in dense degenerate plasmas that the cores of white dwarfs are nearly isothermal (notice how shallow the slope of the density-temperature profiles becomes at very high densities in Figure 4). The region where the electron thermal conductivity drops (in the vicinity of the degenerate interior boundary) acts as a sort of bottleneck for white dwarf cooling.', '6 White dwarfs demographics': 'We are today far from the early years of white dwarf science where only a handful of objects were known. Over 350,000 white dwarf candidates have been identified with high confidence based on their distances and magnitudes (Gentile Fusillo et al., 2021), and tens of thousands of stars are confirmed to be white dwarfs given the appearance of their optical spectra (Dufour et al., 2017). These large numbers enable statistical studies of the properties of white dwarfs. However, interpreting these statistics can be challenging due to biases in sample selection (e.g., brighter stars are easier to detect than fainter ones), a common issue in astronomical studies. To mitigate this, we can focus on a volume-complete sample, where all white dwarfs within a given radius from the Sun are analyzed. Such a sample, with a very high degree of completeness, is now available up to 40 pc. We take a closer look at this sample below.', '6.1 The mass and temperature distributions': "Figure 5 shows the distribution in the mass-e ff ective temperature plane of all known white dwarfs within 40 pc of the Sun. 5 On a diagram like this one, white dwarfs evolve from left to right (decreasing temperature) at constant mass. The most striking aspect of Figure 5 is the clustering of stars around 0 . 6 M ⊙ , indicating that most white dwarfs have masses close to 0 . 6 M ⊙ . This fact is more explicitly illustrated in \nFig. 4 White dwarf density-temperature profiles. These profiles were calculated for a 0 . 6 M ⊙ white dwarf with a thick hydrogen layer ( M H / M ⋆ = 10 -4 ) for effective temperatures ranging from 30,000 K (top profile) to 3000 K (bottom profile). The center of the star is on the right; the surface is on the left. The thickness of the lines indicates the dominant atomic constituent (hydrogen, helium, or carbon/oxygen). The white region corresponds to a fully ionized plasma, while the different shadings indicate a partially ionized plasma (light blue) or a neutral hydrogen gas (darker blue). The regions highlighted in orange are convective. A small black circle on each profile marks the location of the photosphere, a diamond indicates the outer boundary of the degenerate interior, and a triangle delimits the extent of the solid core for objects that are cool enough to have undergone crystallization. \n<!-- image --> \nFig. 5 Distribution of white dwarfs in the mass-effective temperature plane. Every dot represents a star from the 40 pc volume-complete sample of O'Brien et al. (2024). A dashed gray line marks 0 . 6 M ⊙ to guide the eye. \n<!-- image --> \nFigure 6, where the white dwarf mass distribution is shown. The distribution shows a sharp peak centered at 0 . 61 M ⊙ . There is also a broad shoulder at larger masses (whose exact origin remains an open question), but overall the mass distribution is surprisingly narrow given the wide range of stellar masses spanned by stars that eventually become white dwarfs. This question is explored in greater details elsewhere in this Encyclopedia (see the chapter on the initial-final mass relation), but in short, more massive stars lose more mass than less massive ones during the late stages of stellar evolution. This implies that white dwarf progenitors with very di ff erent initial masses converge to comparable masses by the time they reach the white dwarf state. For example, a 1 M ⊙ star will evolve to become a ≃ 0 . 57 M ⊙ white dwarf, while a star twice that mass (2 M ⊙ ) yields a barely more massive white dwarf ( ≃ 0 . 65 M ⊙ ). \nAnother remarkable aspect of Figure 5 concerns the distribution of white dwarfs along the e ff ective temperature axis. In particular, notice how the number of white dwarf per 1000 K interval increases with decreasing temperature (at least down to ≃ 5000 K). In an unbiased white \nFig. 6 The mass distribution of white dwarfs. The histogram is normalized so that it integrates to one. The same sample as in Figure 5 was used to generate this figure. \n<!-- image --> \ndwarf sample, hot white dwarfs are much more rare than cool white dwarfs. The fundamental reason for this distribution is shown in Figure 3: the cooling rate of a white dwarf generally decreases with time. A 0 . 6 M ⊙ white dwarf needs more time to cool down from T e ff = 100 , 000 K all the way to 6000 K, than for cooling down from 6000 K to 5000 K. Part of this behavior is due to specific physical processes that slow down the cooling process (e.g., crystallization), but a simple, general explanation is that the cooler the surface becomes, the more slowly the star can radiate away its energy (Equation 1).", '6.2 The white dwarf luminosity function': 'Interestingly, this trend of increasing numbers of white dwarfs with decreasing temperature eventually comes to an end. Very few white dwarfs cooler than 4000 K are known, even beyond the 40 pc sample. Of course, such cool white dwarfs are harder to identify given their intrinsic faintness, but this fact alone cannot explain the clear decrease in the number of white dwarfs below ≃ 5000 K in the almost complete 40 pc sample of Figure 5. The solution to this conundrum is simply that the disk of our Galaxy is still too young to have produced older white dwarfs. A 0 . 6 M ⊙ white dwarf had to go through ≃ 4 Gyr of stellar evolution before even reaching the white dwarf phase, and it then needs another ≃ 6 Gyr to cool down to 5000 K. Given that the galactic disk is estimated to be ≃ 10 Gyr old, this naturally explains the paucity of cooler stars in Figure 5. \nThis property of the temperature distribution is not merely a curiosity. It implies that white dwarfs can be used as cosmic clocks to constrain the age of the galactic disk and other stellar populations. The oldest white dwarf of any given stellar population yields a firm lower limit on the age of that population. The analysis of white dwarfs with the aim of constraining ages is known as white dwarf cosmochronology, and it represents one of the key motivations underpinning current-day studies of white dwarfs. A common way to characterize this downturn in the white dwarf population at old ages is through the luminosity function (Figure 7). It consists of a volumenormalized luminosity (or bolometric magnitude) distribution of white dwarfs. In other words, it shows for a cubic parsec of space how many white dwarfs there are within a given luminosity bin. Figure 7 shows an increasing number of white dwarfs with decreasing luminosity, which reflects the decreasing cooling rate. This is followed by an abrupt cut-o ff around log L / L ⊙ = -4 . 5 that marks the finite age of the galactic disk (Winget et al., 1987). Beyond age-dating applications, the luminosity function is also a useful tool to test and constrain white dwarf cooling models as its detailed shape is a ff ected by the various physical ingredients (e.g., crystallization physics, opacities) that control the cooling rate.', '7 White dwarf atmospheres': "Being only ∼ 100 m thick, the atmosphere of a white dwarf represents a tiny fraction of the star's total radius and mass. Nonetheless, the importance of this minuscule layer is hard to overstate, as it is the only observable portion of the star. \nThe light emerging from a white dwarf mostly comes from a region known as the photosphere (marked by circles in Figure 4), which is defined as the layer where 50% of the photons escape the atmosphere without further absorption or scattering. While the photosphere represents the typical depth from which photons that we observe come from, it is important to note that when a white dwarf is observed, we are e ff ectively integrating along a line of sight and many other layers contribute to shaping the observed spectrum. In particular, the \nL \n/ \nL \nFig. 7 The white dwarf luminosity function. The curve shows the number of white dwarfs per luminosity bin (or, similarly, per bolometric magnitude bin, M bol = 4 . 74 -2 . 5 log 10 ( L / L ⊙ ) ) and normalized by volume. The gray error bars show the 1 σ uncertainties based on Poisson statistics in each bin. The same sample as in Figure 5 was used to generate this figure. \n<!-- image --> \natmospheric gas is not uniformly transparent to radiation across all wavelengths. If the opacity is larger for a given wavelength (e.g., in the core of a spectral absorption line), then the average photon detected at that wavelength originates from higher up in the atmosphere than the average photon at other wavelengths where the opacity is lower. \nTo a first approximation, the flux emerging at the top of the atmosphere is that of a blackbody spectrum with temperature T e ff . But in practice, many absorption and scattering processes contribute to shaping a much richer spectrum. Among the main opacity sources there are spectral lines from atomic bound-bound transitions (e.g., hydrogen Balmer lines), bound-free absorption (photoionization), free-free absorption, Thomson scattering from electrons, Rayleigh scattering from atoms and molecules, and molecular opacities (e.g., molecular absorption bands, collision-induced absorption). Precisely characterizing these opacity sources under the conditions found in white dwarf atmospheres remains an active area of research. Of particular interest is the theory of spectral line broadening. In white dwarfs, the width of spectral absorption lines is mostly controlled by collisional broadening (i.e., interactions between the radiating atom and neighbouring particles). Describing this accurately is a complex challenge, but the e ff ort is worthwhile as it enables the extraction of valuable information about the physical conditions in the atmosphere from the spectral line shapes (Bergeron et al., 1992).", '7.1 Spectral classification': "The wide range of atmospheric temperatures and compositions found in white dwarfs leads to a diversity of spectral types (Figure 8). The spectral classification scheme currently used is that of Sion et al. (1983). In its simplest form, the spectral type of a white dwarf is given by the combination of the letter 'D' (for degenerate) and one of six letters that identify the primary spectral types. The majority of spectroscopically observed white dwarfs belong to the DA spectral class, an example of which is shown at the top of Figure 8. DA white dwarfs are those that display hydrogen Balmer in their spectra. As we have seen, most white dwarfs have a thick hydrogen layer (Section 4.3), and therefore exhibit a pure hydrogen atmosphere, which most of the time produces detectable Balmer lines. Next are the DB white dwarfs, which are characterized by neutral helium spectral lines. DB white dwarfs are much more rare than DAs. This is not only due to the fact that most white dwarfs have hydrogen-dominated atmospheres, but also that helium lines cannot be detected below T e ff ≃ 11 , 000 K since there is then too little thermal energy to excite the appropriate electronic transitions. This brings us to the third spectral type, the DC white dwarfs. These are stars that display a featureless, continuous spectrum. Most DC white dwarfs have helium- or hydrogen-dominated atmospheres that are too cool to produce optical absorption lines (the threshold is around 5000 K for hydrogen). After that, there are the DO white dwarfs, which show ionized helium lines. They are even more rare than DB white dwarfs, as the presence of ionized helium lines requires very hot temperatures where white dwarfs spend very little time due to the initially rapid cooling (Figure 3). \nIn apparent contradiction to earlier discussions where the outer layers have been described as being composed of hydrogen and / or helium, DQ and DZ white dwarfs are characterized by absorption lines from elements heavier than hydrogen and helium. In the case of DQ white dwarf, that element is carbon, and it can manifests itself either as atomic absorption lines (above ≃ 10 , 000 K) or absorption bands from the C2 molecule (below ≃ 10 , 000 K, such as in the example shown in Figure 8). The presence of carbon in the atmospheres of most DQ \nFig. 8 Spectra of white dwarfs showcasing diverse spectral types. The spectral type, name of the star and effective temperature are given next to each spectrum. To enhance clarity, linear trends have been subtracted from the original spectra, which are also normalized and vertically offset. The wavelengths of the main atomic absorption lines (or molecular band heads in the case of C 2 ) are indicated above the figure. The spectra shown here are taken from a variety of sources and are freely available on the Montreal White Dwarf Database 7 (Dufour et al., 2017). \n<!-- image --> \nwhite dwarfs is understood as the result of the convective dredge-up of carbon from the deep interior. This leads to the presence of relatively small traces of carbon in the atmosphere (of the order of one part per million) that are readily detectable thanks to the strong opacity of carbon features. While this dredge-up process successfully explains the presence of carbon in most white dwarfs, it cannot account for the carbon abundances measured in all DQ white dwarfs. Finally, DZ white dwarfs are those that display absorption lines of atomic species other than hydrogen, helium, or carbon. We return to this special case in Section 7.2. \nWhite dwarf spectral types can contain more than just two letters. First, additional letters can be appended to the primary spectral class to signal the presence of secondary absorption features. For example, Figure 8 shows the example of a DAZ white dwarf, which has both hydrogen and calcium absorption lines. Second, there are other letters that can be used to signal various peculiarities, such as 'H' to signal the detection of a magnetic field (more on this in Section 7.3), 'V' for a variable white dwarf, and 'e' for emission lines. Before closing this section, it is important to highlight a frequent source of confusion regarding the atmospheric composition and spectral type of a white dwarf. The spectral type is only a description of the appearance of the star's spectrum, not a direct indicator of its atmospheric composition. For example, some DA white dwarfs actually contain more helium than hydrogen in their atmospheres, and most DQs and DZs have helium-dominated atmospheres.", '7.2 Metal pollution': 'What is responsible for the presence of metals in the atmospheres of DZ white dwarfs (and other similar variants)? We know that the metals frequently observed in the atmospheres of these stars (e.g., Ca, Mg, Fe) should sink out of view below the photosphere in timescales that are very short (days to millions of years) compared to cooling timescales (billions of years). So how do we explain that 25-50% of white dwarfs show signs of metal pollution (Koester et al., 2014)? One way to explain this would be the presence of a mechanism that counterbalances gravitational settling. Such a mechanism exists in the form of radiative levitation, but it only operates in very hot stars and can be ruled out for the vast majority of metal-polluted white dwarfs. The alternative explanation, and the correct one, is that these metals arrived in the atmosphere only recently and therefore did not yet have the chance to completely sink below the photosphere. More specifically, metal pollution in white dwarf atmospheres is the result of the accretion of planetary material, an idea that is now very well supported by observations. In fact, some metal-polluted white dwarfs have detectable debris disks (Jura, 2003) or disintegrating planetesimals (Vanderburg et al., 2015), and the X-ray signature of the accretion process has now been observed (Cunningham et al., 2022). \nThe spectral analysis of metal-polluted white dwarfs provides a direct window into the chemical composition of the accreted material, o ff ering insights into the makeup of rocky bodies within exoplanetary systems. This is to be contrasted with other methods that infer \nplanetary composition indirectly from bulk density measurement or are limited to surface or atmospheric characteristics. To first order, most metal-polluted white dwarfs show that the accreted material closely resembles the composition of bulk Earth, but there is much more to learn from this unique approach. For instance, variations in the ratios of volatile to refractory elements can provide clues about the conditions under which the accreted bodies were initially formed. Moreover, measuring the composition of bodies accreted by white dwarfs of di ff erent ages allows to probe changes in the chemical makeup of rocky bodies throughout the evolution of our galaxy. Needless to say, this field represents an area of very active research, with potential to significantly advance our understanding of the architecture and evolution of planetary systems.', '7.3 Magnetic fields': "In unbiased samples, around 20% of white dwarfs are found to host a detectable magnetic field (Bagnulo and Landstreet, 2021). The strength of this field spans orders of magnitude, from a few kilogauss for the weakly magnetic objects all the way to gigagauss levels at the other extreme (for comparison, sunspots have magnetic fields of a few kilogauss). Most of the time, magnetic fields are detected thanks to Zeeman splitting, where the presence of a magnetic field causes the spectral lines of atoms to split into multiple components at di ff erent wavelengths. A white dwarf with Zeeman-split Balmer lines is of the DAH spectral type. The magnitude of the Zeeman splitting depends on the strength of the field, thereby allowing a spectroscopic measurement of the field strength. High-resolution spectroscopy may be required to detect very weak fields. Of course, no Zeeman splitting can be observed in featureless DC white dwarfs even if some are strongly magnetic. For these objects, the only option is to use spectropolarimetry, as polarization of the continuum indicates the presence of a magnetic field. \nThere are striking demographic trends within the magnetic white dwarf population that suggest di ff erent origins for magnetic fields in white dwarfs. Notably, more massive white dwarfs exhibit a higher incidence of magnetism compared to their less massive counterparts and have stronger fields. Since a larger fraction of more massive white dwarfs are believed to be the products of stellar mergers, it is possible that the merger process can generate strong magnetic fields. Among normal-mass white dwarfs, it is found that magnetic fields are rarely detected for objects younger than 2-3 Gyr. However, magnetic white dwarfs become gradually more common at lower temperatures. This progression could be due to the outward di ff usion to the surface of a magnetic field generated prior to the white dwarf phase and initially buried in the core. In addition, it is possible that the white dwarf generates a field of its own. This could potentially be achieved during crystallization. When the solid core is formed and has a di ff erent composition than the surrounding liquid (Section 5.2), instabilities due to the redistribution of ionic species within the core trigger fluid motions that could generate an internal dynamo (Isern et al., 2017). This elegant idea is similar to the mechanism that powers Earth's magnetic field, and testing its validity is an area of active research. \nFinally, it is important to note that magnetic fields have minimal impact on the structure and cooling of white dwarfs. The magnetic pressure remains much smaller than the electron degeneracy pressure in the core unless the internal field strength surpasses ∼ 10 13 G, which is orders of magnitude stronger than anything observed and hence very unlikely. Therefore, magnetic fields do not a ff ect the mass-radius relation of any known white dwarf. Likewise, magnetic fields smaller than 10 9 G are too weak to impact energy transfer at the interface of the degenerate interior (where cooling rates are regulated, see Section 5.3), meaning that these fields cannot significantly alter the cooling process.", '8 Conclusion': "This chapter has endeavored to provide a foundational understanding of white dwarfs. From their discovery and physical nature to the details of their evolutionary history and the diversity of their spectra, we have covered a wide range of topics. However, the scope of this discussion has only scratched the surface of the field. Many exciting topics, some touched upon briefly in this chapter and others barely mentioned, are discussed in more depth elsewhere in this Encyclopedia. This includes the initial-final mass relation that prescribes the masses of white dwarfs, the burgeoning field of white dwarf asteroseismology, and white dwarf exoplanetary systems. Most egregiously, we have not discussed the vast and rich topic of white dwarfs in binary systems, which encompasses a broad spectrum of phenomena from Type Ia supernovae to cataclysmic variables and gravitational waves. It is the author's hope that this chapter has furnished the reader with the essential background required to delve deeper into the many fascinating aspects of white dwarfs.", 'Acknowledgments': 'The author thanks the Canadian Institute for Theoretical Astrophysics (CITA) National Fellowship program for financial support.', 'References': "Bagnulo S and Landstreet JD (2021), Nov. New insight into the magnetism of degenerate stars from the analysis of a volume-limited sample of white dwarfs. MNRAS 507 (4): 5902-5951. \nB'edard A (2024). The spectral evolution of white dwarfs: where do we stand? in press . \nB'edard A, Brassard P, Bergeron P and Blouin S (2022). On the Spectral Evolution of Hot White Dwarf Stars. II. Time-dependent Simulations of Element Transport in Evolving White Dwarfs with STELUM. ApJ 927 (1), 128. \nBergeron P, Saffer RA and Liebert J (1992), Jul. A Spectroscopic Determination of the Mass Distribution of DA White Dwarfs. ApJ 394: 228. 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A Cooling Anomaly of High-mass White Dwarfs. ApJ 886 (2), 100. \nChidester MT, Farag E and Timmes FX (2022). On Trapped Modes in Variable White Dwarfs as Probes of the 12 C( α , γ ) 16 O Reaction Rate. ApJ 935 (1), 21. \nCunningham T, Wheatley PJ, Tremblay PE, Gansicke BT, King GW, Toloza O and Veras D (2022), Feb. A white dwarf accreting planetary material determined from X-ray observations. Nature 602 (7896): 219-222. \nDufour P, Blouin S, Coutu S, Fortin-Archambault M, Thibeault C, Bergeron P and Fontaine G (2017), The Montreal White Dwarf Database: A Tool for the Community, Tremblay PE, Gaensicke B and Marsh T, (Eds.), 20th European White Dwarf Workshop, Astronomical Society of the Pacific Conference Series, 509, pp. 3. \nEddington AS (1927). Stars and Atoms. \nFontaine G, Brassard P and Bergeron P (2001). The Potential of White Dwarf Cosmochronology. \nPASP \n113 (782): 409-435. \nFowler RH (1926). 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ApJL 315: L77.", 'Further information': '- · For a classic (although now somewhat dated) review article on white dwarfs: Fontaine et al. (2001);\n- · For an accessible historical account of the development of the field of white dwarf studies: Van Horn (2015);\n- · For a recent review focusing on the physics of white dwarfs: Saumon et al. (2022).'} |
2024arXiv240804938S | On the largest scales the universe appears to be almost perfectly homogeneous and isotropic adhering to the cosmological principle. On smaller scales inhomogeneities and anisotropies become increasingly prominent reflecting the origin emergence and formation of structure in the Universe and its cosmological impact. Also a range of tensions between various cosmological observations may suggest it to be necessary to explore the consequences of such deviations from the ideal uniform universe. In this study we restrict this to an investigation of anisotropies on the nature of the Universe. This motivates a more thorough understanding of the manifestation of anisotropy in cosmological applications. When letting go of the assumption of isotropy spacetime metrics become homogeneous and completely anisotropic. As such the Lie algebras of Killing vector fields will be 3D and fit into the socalled 9part Bianchi classification. This work strives to given a suitable 3D Lie algebra of vector fields xiaa123 reconstruct the basis for a metric on which this Lie algebra is Killing. Through finding a determining equation for the frame invariant under xia and using the method of characteristics to solve it expressions for said invariant frame in terms of the xia are obtained. This leads to general equations for the invariant frame in terms of the xia organized by Bianchi class. Some examples demonstrating this method are worked out. | 2024-08-01T00:00:00Z | ['arXiv:2408.04938', '2024arXiv240804938S', '10.48550/arXiv.2408.04938'] | ['General Relativity and Quantum Cosmology', 'Astrophysics - Cosmology and Nongalactic Astrophysics', 'Mathematical Physics'] | Cosmic Anisotropy and Bianchi Characterization Killing vector fields and the implied finding of their metric frame | 2,024 | 173 | 0.31 | ['EPRINT_HTML', 'EPRINT_PDF'] | 0 | https://arxiv.org/pdf/2408.04938.pdf | {'Cosmic Anisotropy and Bianchi Characterization: Killing vector fields and the implied finding of their metric frame': 'Robbert W. Scholtens, 1, 2, ∗ Marcello Seri, 2, † Holger Waalkens, 2, ‡ and Rien van de Weygaert 1, § \n1 \nKapteyn Astronomical Institute, University of Groningen, Groningen, The Netherlands 2 Bernoulli Institute for Mathematics and Computer Science, University of Groningen, Groningen, The Netherlands (Dated: September 13, 2024) \nOn the largest scales the universe appears to be almost perfectly homogeneous and isotropic, adhering to the cosmological principle. On smaller scales inhomogeneities and anisotropies become increasingly prominent, reflecting the origin, emergence and formation of structure in the Universe and its cosmological impact. Also, a range of tensions between various cosmological observations may suggest it to be necessary to explore the consequences of such deviations from the ideal uniform universe. In this study, we restrict this to an investigation of anisotropies on the nature of the Universe. This motivates a more thorough understanding of the manifestation of anisotropy in cosmological applications. When letting go of the assumption of isotropy, spacetime metrics become homogeneous and completely anisotropic. As such, the Lie algebras of Killing vector fields will be 3D, and fit into the so-called 9-part Bianchi classification. This work strives to, given a suitable 3D Lie algebra of vector fields { ξ a } a =1 , 2 , 3 , reconstruct the basis for a metric on which this Lie algebra is Killing. Through finding a determining equation for the frame invariant under { ξ a } and using the method of characteristics to solve it, expressions for said invariant frame in terms of the { ξ a } are obtained. This leads to general equations for the invariant frame in terms of the { ξ a } , organized by Bianchi class. Some examples demonstrating this method are worked out.', 'I. INTRODUCTION': "The present study centers on an investigation of the mathematical structure and of anisotropic homogeneous \nBianchi metrics for the geometry of the universe on a global scale. \nThe current cosmological worldview is based on the central tenet of the cosmological principle . According to the cosmological principle the universe on the largest scales is spatially homogeneous and isotropic . In other words, it is uniform, the same everywhere and looking the same in every direction. This constraint on the nature of our universe mathematically implies a strong constraint on the geometry. The curvature is restricted to only three possible options, specified by the RobertsonWalker (RW) metric, \nds 2 = -dt 2 + a ( t ) 2 d x 2 , (1) \nwhere d x 2 is the line element either of Euclidean (flat) space, or unit curvature hyperbolic or spherical space, and a ( t ) the scale factor, normalized to a ( t = now) = 1 . \nThe fundamental importance of the highly symmetric RW geometry is its ability to simplify the complex, nonlinear system of Einstein's field equations of General Relativity. Instead of a nonlinear system of 10 equations, cosmologists can work with just two independent equations of motion: the Friedmann-Lemaˆıtre-RobertsonWalker (FLRW) equations [see e.g. 1-8]. These equations, which describe the universe's evolution in terms of the scale factor a , have been instrumental in probing the depths and history of the universe from its primordial beginnings 13.8 Gyrs ago up to the present epoch. Only with the help of the RW metric and FLRW equations the incessantly growing body of cosmological observations can be turned into understandable physical insights and interpretations. \nWhile an overwhelming body of observational evidence has shown that the cosmological principle represents an \namazingly accurate description of the global universe, at least asymptotically, we still need to establish how close the universe fulfils these conditions and what the scale of homogeneity actually is. First and foremost evidence is that of the amazing isotropy of the Cosmic Microwave Background, at a level of ≈ 10 -5 [9-11]. Strong indications also exist for the universe to be close to homogeneous on large scales. Most notably this is indicated by the scaling of the angular clustering of galaxies as a function of depth of the galaxy survey: the angular two-point correlation function of galaxies is seen to scale almost perfectly with the survey depth as expected for a spatial galaxy distribution that assumes uniformity on large scales [12, 13]. Nonetheless, it has not yet been unequivocally established beyond what scale the universe may be considered to be homogeneous and isotropic. General contention is that the universe may be considered homogeneous and isotropic beyond a scale of a few to several hundreds of Megaparsec [see e.g. 14]. \nAn increasing number of observations indicate that although cosmological principle may be a very accurate description of physical reality, it may also be in need of either a more precise formulation or, at worst, a reconsideration. Even within the context of the observed universe, it is more accurate to consider the cosmological principle as a statistical principle. Indeed, since the COBE CMB maps, we know that even on the largest scales the mass distribution in the universe is marked by a stochastic spatial pattern of density variations (be it of a minute amplitude). In fact, we know that structure in the universe arose through gravitational growth of these primordial perturbations, and that these were imprinted on all scales. Since COBE revealed the map of the large scale angular structure of the cosmic microwave background temperature, we know that at the time of recombination there were structures much larger than the particle horizon at the time. By implication, this is still true at the current cosmic epoch, so that the visible Universe presents us with a minute deviation from the global cosmological average. Hence, it is more precise to state that the statistical properties of the corresponding inhomogeneous mass distribution obey the cosmological principle and do not vary with direction and location, but that the particular realization within our visible universe may not be perfectly uniform [see e.g. 2]. \nAt a more fundamental level, various studies forwarded observational indications for deviations from pure isotropy or homogeneity, and hence may have serious implications for the validity of FLRW-based cosmologies. An intriguing indication for possible intrinsic anomalous anisotropies is the finding by Copi et al. [15, 16] that the low multipole components of the cosmic microwave background anisotropies are mutually aligned, and aligned with our motion through the universe (i.e. the dipole motion). The existence of this Axis of Evil would seriously question the validity of the isotropy of our universe, and hence the cosmological principle. At an even more critical level, Sarkar and collaborators have highlighted some \nserious problems in the regular interpretation of cosmological observations within the context of the standard FLRW cosmology. One aspect concerns the fact that the dipole in the CMB is fully ascribed to the gravitational attraction by local inhomogeneities, and hence can be interpreted as a dipole effect due to our Galaxy moving within the restframe blackbody cavity of the cosmic microwave background radiation. However, recently Secrest et al. [17] found that the dipole with respect to the high redshift sample of quasars does not coincide with the CMB dipole, which it would be expected to do when our Galaxy's motion is solely a local effect. It may imply the existence of an anomalously large dipole that is not of local origin and directly calls into question the validity of the assumed isotropy of the universe. This finding is augmented by a potentially even more disruptive result that questions the reality of dark energy and instead may indicate a bulk flow on a cosmological scale. Colin et al. [18] inferred from a sample of 740 Supernovae Ia that the cosmological acceleration parameter is direction dependent, which may imply that we are embedded in a deep bulk flow that conflicts with the assumption of an isotropic universe. \nFollowing the indications for the reality of possible deviations from the cosmological principle, the incentive for the present study is to explore cosmologies that do not entail uniformity. With the ultimate goal to extend this to the geometric structure of fully inhomogeneous and anisotropic cosmologies, we here first explore the implications for cosmologies that are anisotropic, yet homogeneous. No longer involving the symmetry of isotropy, there is a substantial extension of possible metrics with respect to the 3 types of highly symmetric RW metrics. This consideration gives rise to the class of so-called Bianchi models , derived from the classification scheme for three-dimensional Lie algebras [19]. These models, first introduced by Bianchi in 1898 [20], provide a framework for studying universes that are perfectly homogeneous yet have varying degrees of anisotropy. \nIn this work, we investigate the mathematical basis of the Bianchi models from the perspective of the infinitesimal isometries, or Killing vector fields (KVFs), belonging to the metrics involved. Namely, requirements on homogeneity and isotropy of a metric manifest themselves mathematically as such infinitesimal isometries. Denoting by g the relevant metric, a KVF ξ is characterized by the following equation: \nL ξ g = 0 , or equivalently, ξ µ ; ν + ξ ν ; µ = 0 , (2) \nwhere L indicates Lie derivative and semicolon notation is used for covariant derivative. For instance, if g is taken as a RW metric (1), one will find exactly six solutions for ξ to this equation: three characterizing homogeneity, and three characterizing full isotropy. Moreover, due to the property of Lie derivatives that L [ ξ,η ] = [ L ξ , L η ]-so that the commutator of KVFs is again a KVF-we see that KVFs form a Lie algebra of vector fields (Killing Lie \nalgebra; KLA). It is known that a metric in dimension n may allow for up to n ( n +1) / 2 KVFs [21, § 17.1], and certainly does not need to reach this bound. \nRetracting the requirement of isotropy, and replacing it by that of anisotropy while retaining full homogeneity, substantially extends the pool of potential metrics with respect to the three RW type metrics. Instead, we obtain the so-called Bianchi models, named after the classification scheme for 3D Lie algebras [19]. This is because the KVFs effecting homogeneity will be three in number, and hence the KLA will be three-dimensional. An example metric of the so-called Bianchi I type is given by \nds 2 = -dt 2 + a ( t ) 2 dx 2 + b ( t ) 2 dy 2 + c ( t ) 2 dz 2 , (3) \nfor a ( t ), b ( t ), and c ( t ) mutually different-a generalization of the flat RW metric, allowing for different length scales in the different directions. This metric is anisotropic, as the 'rotation generating' vector fields, e.g. x∂ y -y∂ x , fail to be Killing for all t . (One could make the KVFs dependent on time, but since we wish for our symmetries to be purely spatial, this is outside of the scope of this work.) \nBianchi models, with their inherent anisotropy, have been investigated extensively in the literature-see [22] for a review. The hopes are that, due to their inherent anisotropy, they could (partially) explain some of the anisotropies that are currently observed, such as that of the CMB. Particular focus has been placed on Bianchi types I, V, VII, and IX, as these provide 'natural' anisotropic extensions of the flat, hyperbolic, and spherical FLRW models, respectively, and so would allow for the FLRW model of cosmology to remain somewhat standing. A calculation detailing contributions of Bianchi-type perturbations to the CMB was performed in [23]. A Bianchi type VII contribution was tested against the Planck data, though it was ruled out [24]. \nSince a Bianchi model is defined by the structure of its metric's KLA, our approach shall be to build the models starting from this structure. That is to say, given a suitable 'candidate' 3D Lie algebra of vector fields, we wish to find a (frame basis for a) metric on which those vector fields will be Killing. This stands in contrast with the usual approach, where a priori a metric is chosen whose KLA is of a certain Bianchi type. Such metrics are tabulated, e.g. [25, § 6.4] and [26, § 8.2]. In some sense, we aim to reverse the traditional treatment: instead of being given a metric and finding its KVFs through the Killing equation (2), we declare that we want a given Lie algebra to be Killing, and then find a metric on which it is so. \nBy linking the KLA structure with the metric in this manner, we establish a more robust methodology for studying Bianchi metrics. This approach ideally eliminates the need for any specific choice of KLA within a given Bianchi type, thereby elminating the frame dependence of the metric. \nThe key observation is to consider vector fields that commute with each proposed KVF-these are known as invariant . Namely, it can be shown that the Lie deriva- \ntive w.r.t. a candidate KVF of the dual of an invariant vector field, vanishes. As such, if three invariant vector fields which form a frame are found (invariant frame; IF), then the dual to this IF can be used as a basis for constructing a metric; the candidate KLA will then be a bona fide KLA for that metric. \nOur approach in this article is as follows. We first recall mathematical details pertaining to homogeneity and isotropy. We then introduce the Bianchi classification of Lie algebras, and define what we mean by a Bianchi model. In the subsequent section we implement the approach as outlined above, by explicitly calculating the invariant vector fields belonging to a candidate KLA. The result is a tabulation of invariant frames in terms of the candidate KLAs, ordered by Bianchi type, and dependent only on the particular KLA chosen. We then conclude the article by exemplifying some of the constructions, and discussing further potential research questions.", 'A. General symmetry considerations': "Definition II.1 (Isometry) . Let ( M , g ) a (pseudo)Riemannian manifold. We say that a diffeomorphism ϕ from M to itself is an isometry if the pullback of the metric by ϕ leaves it invariant, that is, ϕ ∗ g = g . We denote the resulting collection of isometries of a given space with known metric as Isom( M ). \nThe set of all isometries forms a group, acting on the manifold. In fact, by the Myers-Steenrod theorem, this group is a Lie group. As such, we may also consider the infinitesimal version of the isometries, i.e. the corresponding elements in the Lie algebra. These are captured by vector fields, which are known as Killing vector fields (KVFs). A KVF ξ has the (defining) property that \nL ξ g = 0 , (4) \nwhere L denotes the Lie derivative. Since all KVFs belonging to a certain metric form a Lie algebra under commutation of vector fields, we may speak of a metric's Killing Lie algebra (KLA). \nIn a local coordinate system { x i } , we can write an explicit equation to solve for a KVF's components. In said coordinate system, expand the KVF as ξ = ξ i ∂ i , where ∂ i indicates the derivative in the x i -direction, and then apply this to the equation \nL ξ [ g ( ∂ i , ∂ j ) ] = ( L ξ g ) ( ∂ i , ∂ j ) + g ( L ξ ∂ i , ∂ j ) + g ( ∂ i , L ξ ∂ j ) , (5) \nfrom which we can deduce that \nξ k g ij,k = -ξ k ,i g kj -ξ k ,j g ik , (6) \nwhere comma notation was used to indicate partial differentiation ( ,a ≡ ∂ a ). Given the metric, this defines \na collection of PDEs to solve for the ξ i . If instead covariant differentiation is used, the lhs vanishes and we are left with the famous equation ξ i ; j + ξ j ; i = 0 defining KVFs. \nIn general a metric may allow for multiple KVFs; the maximum allowed is n ( n +1) / 2, where n is the dimension of the manifold [21, § 17.1]. This also means that the dimension r of the isometry group is bounded by r ≤ n ( n +1) / 2. While on a general manifold it is possible that no KVF is allowed for a metric, this is never the case in the class of spacetime manifolds considered in this work: we wish to investigate homogeneous spacetimes, and as we shall see below, this means they have KVFs. \nOnce we have our infinitesimal descriptions of isometries, we can quantify the orbit of a starting point under these infinitesimal isometries. To this end, we need the following definition. \nDefinition II.2 (Transitivity) . A (sub)group of isometries G ≤ Isom( M ) is said to act transitively on some submanifold S if G ( s ) = S [27, § 5] for all s ∈ S . In this case S is the orbit of the isometries, and also referred to as surfaces of transitivity. The dimension of the transitivity subgroup dim S we denote by s . \nEffectively, through transitivity we infer an equivalence relation on M of which points can be reached via an isometry. However, there may be 'redundancy' in the surface of transitivity; we may reach individual points via more than one isometry. We say that G acts simply transitively on the surface of transitivity if there is precisely one isometry that connects pairs of points. Or, equivalently, when the infinitesimal generators of isometries are linearly independent as vector fields [25, § 6.2]. Otherwise, we call G multiply transitive . \nWe also need to touch on the concept of isotropy . The isotropy subgroup (of isometries) at some p ∈ M consists of those isometries which leave the point p unchanged: \nStab( p ) := { ϕ ∈ Isom( M ) : ϕ ( p ) = p } . (7) \nThe symbol Stab comes from the alternative name for the isotropy subgroup, being the stabilizer . Directly, this implies that the infinitesimal generators of those isometries, i.e. their associated KVFs, vanish at p . \nLemma II.3. The dimension of the isotropy subgroup dimStab ≡ q is constant over a surface of transitivity. \nProof. Suppose a and b lie in the same surface of transitivity, and let ϕ be an isometry so that ϕ ( a ) = b . We notice that we can send an isotropy transformation at a to one at b by noticing \nψ ∈ Stab( b ) = ⇒ ϕ -1 · ψ · ϕ ∈ Stab( a ) = ⇒ ψ · ϕ ∈ ϕ · Stab( a ) = ⇒ Stab( b ) · ϕ ⊆ ϕ · Stab( a ) (8) \nBy interchanging the roles of a and b , the reverse inclusion can also be established, so that we have equality. Hence, Stab( a ) ∼ = Stab( b ), and so they have the same dimension. \nOne also says that the surface of transitivity is completely anisotropic if the isotropy subgroup is trivial (and so generated by the zero vector field). \nFollowing the above Lemma, we recognize that dimStab( S ) is a well-defined entity for a surface of transitivity S . We can then make an important conclusion about the structure of symmetries, by noticing the following: we have that \ndimIsom( S ) = dim Stab( S ) + dim S (9) \n[27, § 5.1]; or, r = q + s . Here, Isom( S ) indicates the group of isometries defined on the surface of transitivity. That this is true can be seen heuristically by considering that KVFs (which generate isometries) either vanish at some point on S or they do not. If no KVF vanishes anywhere on S , then evidently dim Isom( S ) = dim S ; if not, then dim Stab( S ) is the maximum number of KVFs that simultaneously vanish at some p ∈ S . Then S must be of dimension dim Isom( S ) -dimStab( S ), as that is how many KVFs at said point still generate the surface of transitivity.", 'B. Bianchi models': "We now narrow our symmetry considerations down to the case of interest, namely that of a 4-dimensional pseudo-Riemannian manifold ( M , g ). \nDefinition II.4 (Spatial homogeneity) . We say that if ( M , g ) permits a transitive KLA of dimension 3, then ( M , g ) is spatially homogeneous . We may also shorten this to homogeneous if it is clear from context that spatial homogeneity is meant. \nA surface of transitivity generated by such a KLA is called a spatial section . \nIt is then clear that we should consider the spatial sections as 'space,' with the remaining dimension fulfilling the role of 'time.' This gives rise to the split M = R × S , a consequence of the assumption of global hyperbolicity that is often present in mathematical treatises of general relativity (e.g. [28, Thm. 5.44]), and implicitly made in the study of cosmology. Using the symmetry language built up in the previous section, we can then articulate precisely what is meant with a Bianchi model. \nDefinition II.5 (Bianchi model) . A Bianchi model is a spatially homogeneous spacetime, with a KLA of dimension precisely 3. \nRemark II.6 (Differing nomenclature) . One may also adhere to the definition that a Bianchi model is a spacetime which is spatially homogenous, without the dimensionality restriction. Under this definition, locally rotationally symmetric (LRS) Bianchi models (which have a KLA of dimension 4) and the FLRW models (dimension 6) will be special cases of Bianchi models, instead of being completely separated. Since for our purposes we \nwish to restrict to three-dimensional KLAs, we make the additional dimensionality requirement. \nIn view of equation (9), it follows immediately that a Bianchi model is completely anisotropic. Furthermore, the vector fields which compose the KLA form a frame for the tangent space T p S for all p ∈ S : indeed, if somewhere they did not form a frame, then locally the spatial section would be of a dimension smaller than 3, which is not possible due to (9). \nRemark II.7 (Different choices q , s ) . Effectively in the definition of Bianchi models we prescribed dimensions for the isotropy subgroup and surfaces of transitivity. Since mathematically there is nothing special about these choices, others can be considered as well. For instance, [27, § 5.2] presents a table outlining the valid choices of dimensions, and the names of the resulting models. \nNow that we have defined what a Bianchi model is, we may introduce a further level of detail by invoking the Lie algebraic structure of the KLA. Since for Bianchi models the dimension of the KLA is 3, a Bianchi model is effectively characterized by a 3-dimensional Lie algebrawe shall write its basis as { ξ a } a =1 , 2 , 3 . Note that here the subscripts do not indicate coordinate dependence, but are simply the labels of the vector fields. \nThere are infinitely many Lie algebras of dimension 3, but if we construct equivalence classes through GL(3 , R ) invariance of the Lie algebra's basis, it turns out that there are only nine distinct classes of Lie algebras. One could classify these in various ways that one sees fit, and depending on the intended use. In the astrophysics literature it is common to consider the Bianchi(-Behr) classification of Lie algebras. This is given in Table I, with additional mathematical information [22, 29]. The structure constants are determined from the table's entries as \n [ ξ 1 , ξ 2 ] = n 3 ξ 3 -aξ 2 [ ξ 2 , ξ 3 ] = n 1 ξ 1 [ ξ 3 , ξ 1 ] = n 2 ξ 2 + aξ 3 . (10) \nThat this indeed yields a valid classification (i.e. every Lie algebra belongs to precisely one of these types by a suitable GL(3) transformation of its basis) is shown in [19]. The classes are thus composed of Lie algebras which, when their basis is suitably chosen, have the prototypical structure constants. \nWe are then able to articulate what we mean when we consider metrics of Bianchi type n . \nDefinition II.8 (Bianchi type n metric) . A spacetime model ( M , g ) (and in particular its metric) is of Bianchi type n if it is \n- 1. spatially homogeneous,\n- 2. its KLA has dimension precisely 3, and\n- 3. the KLA is of Bianchi type n , i.e. it permits a basis with structure constants as found through Table I and equation (10), for Bianchi type n . \nTABLE I: Table of Bianchi types, together with further mathematical detail. Here, i refers to the 1-dimensional Lie algebra, and l (2) to the irreducible 2-dimensional Lie algebra. \nWe call the metric g of such a Bianchi model a Bianchi type n metric. \nFor further details, see [21, § 2.7.3 & § 17.1.3] and [27, § 5].", 'III. FROM PRE-KLA TO METRIC': "In the preceding section, we have assumed that there was a spacetime model featuring a metric, for which we would then find the appropriate Killing vector fields (KVFs), for instance through use of the Killing equation (4). Now, we wish to turn this scenario around: for some given 3D Lie algebra of smooth, non-vanishing, transitive vector fields { ξ a } a =1 , 2 , 3 on a 3D manifold Σ 3 , which we interpret as a spatial section, we desire to find a metric so that these vector fields are Killing with respect to it. That is to say, knowing the isometries that we want, we wish to find a metric on which they are indeed realized as isometries. Let us call a Lie algebra with the above characterisation a pre-KLA. \nIn order to cook up a suitable metric, we need i) a collection of vector fields, { X i } i =1 , 2 , 3 , and ii) a collection of one-forms { e i } i =1 , 2 , 3 , both on Σ 3 , which satisfy the following relations: \n[ ξ a , X i ] = 0 and e i ( X j ) = δ i j . (11) \nIn this case, the collection { X i } is known as an invariant frame (IF), and { e i } is its dual (or the dual frame , DF). In the mathematical literature, what we have termed an IF is known as a left-invariant frame [30, § 8]. This terminology stems from the frame { X i } being invariant under the left action of the ξ a , emblemized by their commutator vanishing as in (11). \nWe note that the first condition implies that the invariant frame is also a Lie algebra. Assume there are functions D k ij on Σ 3 such that [ X i , X j ] = D k ij X k . Then, we can calculate \nL ξ a ( D k ij X k ) = ( ξ a D k ij ) X k (12a) \nand \nL ξ a [ X i , X j ] = [ ξ a , [ X i , X j ]] = 0 , (12b) \ndue to the Jacobi identity. Hence, we conclude that ξ a D k ij = 0 for all a , everywhere, so that the functions D k ij must in fact be constants. It follows that the IF is a Lie algebra. \nThe two properties (11) are sufficient to construct a tensor product which vanishes under Lie differentiation by a pre-KLA, which is shown in the following proposition. \nProposition III.1. Let { ξ a } a pre-KLA, and { X i } and { e i } an IF and DF, respectively. Furthermore, let γ = γ ij e i ⊗ e j for γ ij arbitrary constants. Then, L ξ a γ = 0 . \nProof. Let us first show that L ξ a e i = 0. To see this, observe that \n0 = L ξ a ( e i ( X j )) = ( L ξ a e i ) ( X j ) + e i ( L ξ a X j ) = ( L ξ a e i ) ( X j ) , (13) \nwhere we used that [ ξ a , X j ] = 0. So we conclude that L ξ a e i = 0 for all i and a . We can then calculate the Lie derivative of γ : \nL ξ a γ = ( ξ a γ ij ) e i ⊗ e j + γ ij ( L ξ a e i ) ⊗ e j + γ ij e i ⊗ ( L ξ a e j ) , (14) \nand see that it vanishes: the latter two terms due to our previous observation, and the first since γ ij is constant. Finally, since ξ a was chosen arbitrarily, this holds for all a . \nAlthough the tensor product γ introduced above is certainly not a metric, it could be made into one by symmetrizing the tensor product and demanding that the constants γ ij form a positive, symmetric matrix. This then results in a constant metric on the spatial slices Σ 3 . However, for our purposes, we wish to obtain a metric for spacetime, and so we somehow need to involve time as well. \nThis is done by 'adding a time axis,' and creating a spacetime manifold R × Σ 3 (cf. the global hyperbolicity assumption). We furthermore introduce the vector field X 0 along the added dimension, and its corresponding dual e 0 . \nThe requirements we place on e 0 is that { e µ } µ =0 , 1 , 2 , 3 and its dual { X µ } µ =0 , 1 , 2 , 3 satisfy i) the properties as outlined in (11), but with the Roman indices replaced by Greek ones: \n[ ξ a , X µ ] = 0 and e µ ( X ν ) = δ µ ν ; (15) \nand ii) the relation [ X 0 , X i ] = 0 for all i . From this latter condition, we see that essentially X 0 = ∂ 0 is the only choice up to a coefficient, which we could absorb into the definition of t . \nAs such, we can construct a spacetime metric on which a pre-KLA is indeed Killing: set g = g µν ( t ) e µ e ν for some appropriate functions g µν ( t ). The { ξ a } do not see the dependence on t in the metric components, and hence the reasoning for the proof of Proposition III.1 is valid to show L ξ a g = 0 for all a . \nSince commutativity of the Killing and IF vector fields persists if we multiply the latter with functions of time, a linear combination of IF vector fields with functions in time-say Y µ ( t ) := y ν µ ( t ) X ν , for functions y ν µ ( t )-will still form an IF. This freedom corresponds to the freedom to 'line up' the spatial slices along the t -axis in the product M = R × Σ 3 [25, § 6.4]. \nRemark III.2 (Orthonormal synchronous form) . For any metric g = g µν ( t ) e µ e ν , we can also consider a redefinition of the dual basis into ˜ e µ ( t ) := b µ ν ( t ) e ν , for b having the property that η µν b µ α b ν β = g αβ , where η µν = diag( -1 , 1 , 1 , 1) is the Minkowski metric. In that case, g = η µν ˜ e µ ˜ e ν , and so the metric becomes the Minkowski one. This form is called the orthonormal synchronous form of a Bianchi model [21]. \nAs many equations greatly simplify/trivialize with constant metric components, this form may be useful to utilize if one is willing to make the tradeoff for more complicated (time evolutions of) basis vectors. \nSo, let us summarize what we have found up to now. Given some pre-KLA { ξ a } , if we find \n1. some vector fields { X µ } satisfying [ ξ a , X µ ] = [ X 0 , X µ ] = 0, and \n- 2. the covector fields { e µ } dual to these, \nthen we can find a spacetime metric, depending only on time t , so that the { ξ a } are Killing on it. It rests us to construct these vector and covector fields, i.e. to detail the steps 1 and 2 above. \nIn fact, it is known that the second step can be made in a unique way, and its computation is straightforward. Since the first step is nontrivial, however, that is what we shall detail in the following subsection.", 'A. To frame a Killing': "To simplify matters, let us make the choice X 0 ≡ ∂ 0 , so that the problem reduces to needing to find only { X i } for i = 1 , 2 , 3. \nLet us note that since a pre-KLA { ξ a } forms a basis for the tangent space at each point in Σ 3 (by spatial homogeneity), we may decompose any vector field in its basis. Utilizing this for our proposed invariant basis { X i } , and \ntaking inspiration from [31], we make the calculation \n0 ! = [ ξ a , X i ] = [ ξ a , X b i ξ b ] = ( ξ a X b i ) ξ b + X b i [ ξ a , ξ b ] = ( ξ a X b i + X c i C b ac ) ξ b , (16) \nwhere the C b ac are the structure constants of the Lie algebra spanned by { ξ a } . Thus, the X i satisfy the linear differential equations, \nξ a X b i = -C b ac X c i . (17) \nfor all a and b . We emphasize that the above equations are actually three sets of equations, one for each i . That is to say, one for each of three vector fields { X i } which together form an IF. In order to make headway in solving these differential equations, we employ the method of characteristics [32, § 3.2]. Assume that we parametrize a path in (a local coordinate) ( x 1 , x 2 , x 3 )-space with parameter s , so that the tangent to this path is given by F ( x ) = F j ( x ) ∂ j : \ndx j ds = F j , (18) \nwhere x = x ( s ). That is, we follow the flow of F from a given starting point. Then, along this flow, we can examine how the components of X i evolve: \ndX b i ds = ∂X b i ∂x j dx j ds = F j ∂ j X b i ⋆ = F a ξ a X b i = -C b ac F a X c i , (19) \nwhere in ⋆ we used that F j ∂ j = F a ξ a for suitable components F a ( x ). \nObserve now that if we supply an initial condition, then along this flow from this initial condition, our problem will boil down to an initial value problem (IVP), and standard results in the theory of ODEs will give us existence and uniqueness of the evolution of (the components of) X i along said flow. \nThe initial condition we pick is that at some p ∈ Σ 3 , we have X i | p := δ a i ξ a | p . The reason for this is that, if this is the case, we have the following useful Proposition at our disposal. \nProposition III.3. Let { ξ a } a pre-KLA on Σ 3 , and { X i } its IF. Let C c ab denote the structure constants of the pre-KLA, and D k ij those of the IF. Assume moreover that there exists a point p ∈ Σ 3 such that X i | p = δ a i ξ a | p . Then we have D k ij = -C k ij . \nConsequently, the Lie algebra spanned by { X i } is of the same Bianchi type as that spanned by { ξ a } . \nProof. We adopt the proof from [25, § 6.3]. Since we assumed homogeneity, we have \nD k ij X k = [ X i , X j ] = [ X a i ξ a , X b j ξ b ] = ✘ ✘ ✘ ✘ ✘ ✘ X a i ( ξ a X b j ) ξ b -X b j ( ξ b X a i ) ξ a + ✘ ✘ ✘ ✘ ✘✘ X a i X b j C c ab ξ c = X a j X c i C b ac ξ b , (20) \nwhere the cancellation follows upon applying (17). As such, we may derive the equality \nD k ij X b k = X a j X c i C b ac . (21) \nThe premise implies that at p , X a i | p = δ a i . If we consider the equality at point p , we find the relation D k ij = C k ji = -C k ij . This completes the first part of the proof. \nTo see the second point, simply consider the isomorphism x ↦→ -x for x in the Lie algebra spanned by { X i } . Under this isomorphism, the structure constants are sent to their negative, and so will be the same as those for the Lie algebra spanned by { ξ a } . Hence, they must have the same Bianchi type. \nWithout loss of generality, we can choose p , the point of coincidence in the above Proposition, to be the origin in a local coordinate space. We may then formulate an IVP for what we shall name the IF: \n- 1. along the flow of F , parametrized by s , evolve X b i as: \ndX b i ds = -C b ac F a X c i ; (22a) \n2. initial condition at p ∈ Σ 3 : \nX i | p = δ a i ξ a | p . (22b) \nIn the especially simple case F a = δ a b , for b ∈ { 1 , 2 , 3 } , we get the flow along the corresponding KVF-this flow is known as the characteristic . Hence, we can also reformulate our IVP (22) as being along the characteristic curves: \n- 1'. the characteristic of ξ a , parametrized by s : \ndx j ds = ξ j a ; (23a) \n2'. evolve the components of X i along this characteristic as: \ndX b i ds = -C b ac X c i ; (23b) \n3'. initial condition at p ∈ Σ 3 : \nX i | p = δ a i ξ a | p . (23c) \nNote that since we assumed that our vector fields were transitive, we know that we can reach any point p ∗ ∈ Σ 3 from p by following certain characteristics for a certain amount of distance. If the number of characteristics required is more than one, we see that our path from p to p ∗ will be only piecewise smooth. \nThe IVP (23), for each i and a , can be regarded as a matrix differential equation: if we define ⃗ X i := ( X 1 i X 2 i X 3 i ) ⊺ , then \nd ds ⃗ X i = -C a ⃗ X i = ⇒ ⃗ X i ( s ) = exp( -C a s ) ⃗ X i, 0 , (24) \nfor the matrix C a with entries ( C a ) b c = C b ac . Note that due to the skew symmetry of the structure constants, we will have that the a th column in C a will be the zero column, so that the matrix exponential will look like, e.g. for a = 1, \nC 1 = 0 ∗ ∗ 0 ∗ ∗ 0 ∗ ∗ = ⇒ exp( -C 1 s ) = 1 ∗ ∗ 0 ∗ ∗ 0 ∗ ∗ . (25) \nRemark III.4. The matrices C a are also the images of the adjoint representation of the pre-KLA spanned by { ξ a } , when expressed in the { ξ a } -basis. \nConsidering the relative simplicity of exponentiating constant matrices, the formulation (23) suggests a good way to find an IF when presented with a pre-KLA { ξ a } . Namely, if we want to know the IF at some point p ∗ ∈ Σ 3 , p ∗ = p , we make the following steps: \n̸ \n- 1. Find a path consisting of at most three segments, where each segment\n- (a) follows a characteristic, say ξ a ,\n- (b) for a suitable length s a , \nso that the end point of the full path is p ∗ . \n- 2. For each segment, calculate the matrix exp( -C a s a ) (no sum).\n- 3. Multiply ⃗ X i | p with the first, and then second and third previously calculated matrices (if applicable), corresponding to the order of the matrices. \nThe result is, for each p ∗ , the 'vector of vectors' ⃗ X i | p ∗ , which when read in the { ξ a } -basis, gives vector fields of the IF. \nRemark III.5. We may also consider a different flavor of the IVP (23). Since the pre-KLA { ξ a } is homogeneous, there should exist an inverse tensor A = A ( x ) so that A a e ξ f a = δ f e for ξ a expressed in a local coordinate neighborhood around p as ξ a = ξ e a ∂ e . Here { ∂ e } are the coordinate vector fields. If we then contract equation (17) with A a e , we obtain \n∂ e X b i = -A a e C b ac X c i . (26) \nWith the identical justifcation as the one leading up to (22), we can write down what the evolution of (the components of) X i is along some curve parametrized by s whose tangent curve is F : \n- 1. along the flow of F , parametrized by s , evolve X b i as: \ndX b i ds = -C b ac F e A a e X c i ; (27a) \n- 2. initial condition at p ∈ Σ 3 : \nX i | p = δ a i ξ a | p (27b) \n(note that F = F e ∂ e ). This formulation makes it very simple to find a path connecting the starting point p to some other (nearby) p ∗ ∈ Σ 3 : simply follow the flows of ∂ 1 , ∂ 2 , and ∂ 3 for the required length. The tradeoff is in the complexity of the integrand: this now depends on the (generally nontrivial) functions A a e , so that one has to resort to complex integration methods.", 'B. The question of uniqueness': "So far, then, given a pre-KLA { ξ a } , we can find the components of the IF (that is, an invariant frame { X i } with a point p ∈ Σ 3 so that X i | p = δ a i ξ a | p ) in the basis of the pre-KLA at any given point within our spatial section (recall that the spatial section is by definition all those points that can be reached via the isometries generated by the KVFs): we have existence for the IF. Now then rises the question of whether the so found IF is unique to this pre-KLA. \nThis question is not related to the unique solvability of the IVP along the characteristics: standard ODE results tell us that uniqueness is guaranteed in this case. The issue arises when one chooses to use a path not strictly composed of characteristics in order to reach a point at which the IF is desired. \nThe following theorem tells us that uniqueness is indeed the case. \nTheorem III.6. For a given pre-KLA { ξ a } , the corresponding IF { X i } with X i | p = δ a i ξ a | p at some p ∈ Σ 3 , is unique. \nProof. Suppose there are two such IFs, { A i } and { B i } , with p ∈ Σ 3 such that A i | p = B i | p = δ a i ξ a | p . Then their difference, D i := A i -B i , satisfies \n[ ξ a , D i ] = 0 = ⇒ ξ a ( D b i ) = -C b ac D c i , (28) \nby the same reasoning as we used above to write equation (17) (note D i is decomposed in the { ξ a } -basis). So, we can create an IVP for the D i along the same lines we did above: let F = F a ξ a a the tangent to some curve parametrized by s , so then \n- 1. along the flow of F , parametrized by s , evolve D b i as: \ndD b i ds = -C b ac F a D c i ; (29a) \n2. initial condition at p : \nD i | p = 0 . (29b) \nRewrite the evolution equation into a matrix system: ⃗ D i := ( D 1 i D 2 i D 3 i ) ⊺ , so that we can write \nd ds ⃗ D i = -F a C a ⃗ D i = ⇒ ⃗ D i ( s ) = P exp ( -∫ s 0 F a C a ds ' ) ⃗ D i, 0 , (30) \nwith C a defined as in (24) and P exp denoting the pathordered exponential. This latter is necessary, as the integrand depends on position (namely, F a = F a ( x ( s )) generally). \nObserve that if we choose to start from the point p , then ⃗ D i, 0 = ⃗ D i | p = 0, so that ⃗ D i ( s ) = 0 for all s -more importantly, this conclusion holds for any vector field F . Hence, the vector field D i is well-defined as a function of the spatial section Σ 3 , and moreover this function returns the zero vector everywhere: D i | q = 0 for all q ∈ Σ 3 . Therefore A i = B i everywhere, completing the proof. \nCorollary III.7. Let { X i } be the IF for the pre-KLA { ξ a } with X i | p = ξ i | p , p ∈ Σ 3 . Suppose that { Y i } is a collection of vector fields on Σ 3 such that [ ξ a , Y i ] = 0 for all a and i , and Y i | p = y a i ξ a | p for constants y a i forming a GL(3) matrix. Then Y i = y j i X j . \nProof. Define ˜ Y i := ( y -1 ) j i Y j , and observe that \n[ ξ a , ˜ Y i ] = 0 and ˜ Y i | p = δ a i ξ a | p . (31) \nBy the above Theorem, the only collection of vector fields which has this property is the IF { X i } , so ˜ Y i = X i .", 'C. Change of the invariant frame': "In the previous two subsections we have established that, given a pre-KLA { ξ a } on Σ 3 , we have existence and uniqueness of the accompanying IF. That is to say, we have that there exist vector fields { X i } so that [ ξ a , X i ] = 0 for all a and i , that X i | p = δ a i ξ a | p for some p ∈ Σ 3 , and this collection of vector fields is unique. \nWe would now like to understand how the IF changes if we apply some manner of transformation to the original { ξ a } . After all, it would hardly be efficient if for each possible pre-KLA { ξ a } we would need to calculate the IF all over again, even if it is clear that there exist a relation to some other pre-KLA for which the IF was known already. Ideally, we want to export a relation between the pre-KLAs to one between their corresponding IFs. \nTo this, non-exhaustive effect, we have the following Propositions. \nProposition III.8. Let { ξ a } a pre-KLA on M with IF { X i } , and { ξ ' a } one on N . Suppose that ϕ : ξ a ↦→ ξ ' a for a = 1 , 2 , 3 . If ϕ = Φ ∗ for some diffeomorphism Φ : M→ N , then the IF belonging to { ξ ' a } is given by { ϕ ( X i ) } . \nProof. It is known that for a diffeomorphism Φ : M→ N and vector fields A,B ∈ X ( M ), we have that [Φ ∗ A, Φ ∗ B ] = Φ ∗ [ A,B ]-see for instance [30, Ch. 8]. Applying this to our case, observe \n[ ξ ' a , ϕ ( X i )] = [Φ ∗ ξ a , Φ ∗ X i ] = Φ ∗ [ ξ a , X i ] = 0 , (32) \nestablishing commutativity. Now let p ∈ M be a point so that X i | p = δ a i ξ a | p . Then, we have \nΦ ∗ X i | Φ( p ) f = X i | p ( f · Φ) = δ a i ξ a | p ( f · Φ) = δ a i Φ ∗ ξ a | Φ( p ) f, (33) \nfor any function f ∈ C ∞ ( N ). Conclude therefore that ϕ ( X i ) | Φ( p ) = δ a i ξ ' a | Φ( p ) , and so that { ϕ ( X i ) } constitutes the IF for { ξ a } . \nProposition III.9. Under a GL(3) transformation ϕ of the pre-KLA { ξ a } , ξ a ↦→ ξ ' s = ϕ a s ξ a , we have that the components of the corresponding IF change as X a i ↦→ X ' t s = ϕ i s X a i ( ϕ -1 ) t a . \nProof. Since the KVFs and the IF commute with each other, so does any constant linear combination of them. As such, we see that for the GL(3) transformation ϕ , we have that \n[ ϕ a s ξ a , ϕ i s X i ] = 0 and ϕ i s X i | p = ϕ a s ξ a | p , (34) \nfor p ∈ Σ 3 such that X i | p = δ a i ξ a | p . Hence, { ϕ i s X i } is the IF for the transformed pre-KLA. The components of the IF vector fields in the new basis are given by \nϕ i s X i = ϕ i s X a i ξ a = ϕ i s X a i ( ϕ -1 ) t a ϕ b t ξ b = ϕ i s X a i ( ϕ -1 ) t a ϕ t . (35) \nIf we write X ' s = ϕ i s , then from the above equation we conclude that X ' t s = ϕ i s X a i ( ϕ -1 ) t a , which completes the proof. \nFrom this second Proposition, we make an important observation. Namely, GL(3) actions acting on a pre-KLA conserve Bianchi type of said pre-KLA. As such, once we know what the Bianchi class of our pre-KLA is, we also know that there will exist a certain GL(3) to bring it into its tabulated, standard form. In finding IFs for pre-KLAs, we can thus assume that the pre-KLA is in standard form; if a given pre-KLA were not, we would perform a suitable GL(3) transformation to make it so, find the IF, and then 'untransform' utilizing Proposition III.9. \nThough Proposition III.9 tells us that any pre-KLA can be transformed into standard Bianchi form, this does not imply that any two pre-KLAs of the same Bianchi type can be transformed into each other. To illustrate, consider the following example of two Bianchi II preKLAs (even with identical structure constants C 1 23 = 1 and rest zero), in some local coordinate frame: \n ξ 1 = ∂ 2 ξ 2 = ∂ 3 ξ 3 = ∂ 1 + x 3 ∂ 2 and ξ 1 = ∂ 1 ξ 2 = ∂ 2 -1 2 x 3 ∂ 1 ξ 3 = ∂ 3 + 1 2 x 2 ∂ 1 . (36) \n(The former is type II in Ryan & Shepley [25, § 6.4], and the latter is the Heisenberg algebra.) Evidently one cannot perform a GL(3) transformation from one to the other, and it can be verified that they are not diffeomorphic. Hence, these two cannot be related to each other by the previous two Propositions. We shall discuss this case further in the next section.", 'D. List of invariant frames': 'From subsection III C we learned that if we know the IF to some pre-KLA, we can also calculate the IF belonging to a GL(3)-transformed pre-KLA, and from III B that any found IF-with a given initial condition-is unique to the pre-KLA. Combining these two observations, we see that it has value to find expressions for the IFs belonging to pre-KLAs in the various Bianchi classes. The methodology from subsection III A allows us to do this, given only the structure constants of a pre-KLA. \nThis has been done here, with the Table II as its result. For each class, one considers a pre-KLA { ξ a } such that they possess the structure constants as indicated (with any unmentioned structure constants equal to zero). 1 The general form of the IF is then given, with relations to be satisfied by the functions p ( x ), q ( x ), and r ( x ). Initial conditions for these functions are such that, at some reference point (e.g. a coordinate origin x = 0), X i = δ a i ξ a . \nRemark III.10. Since the relation [ ξ a , X i ] is symmetric, we could also reverse the roles that ξ a and X i are playing: take the ξ a as the invariant frame, and X i as the Killing vectors. In this way, the table allows us to find KVFs belonging to a metric without having to employ the Killing equation. If a given metric can be written in the form ds 2 = -dt 2 + γ ij ( t ) e i e j , and constants C i jk found so that the 1-forms e i satisfy \nde i = -1 2 C i jk e j ∧ e k , (37) \nthen inputting ξ i as the dual of e i in the table will yield the X i as the non-vanishing KVFs.', 'IV. EXAMPLES': 'In the preceding section we developed a methodology for calculating the IF belonging to a given pre-KLA. The result was the listing in § III D, which gives the IF in terms of the pre-KLA that was given, with some functions that need to be determined and that are pre-KLA specific. This was done so that a general structure may be observed, without reference to any particular choice of pre-KLA, say, in terms of coordinates. \nHowever, in order to gain some intuition into Bianchi models, the symmetries that may arise, and practical application of this result, it is evidently useful to consider some specific examples. This is what we shall do in this section.', 'A. Twice Bianchi V': "Let us work out two examples of the Bianchi V type. This is the anisotropic 'generalization' of the open FLRW model, and so one may already expect that somehow a hyperbolic nature to these models will arise. \nModel 1: Consider the pre-KLA \nξ 1 = ∂ 2 , ξ 2 = ∂ 3 , ξ 3 = ∂ 1 + x 2 ∂ 2 + x 3 ∂ 3 (38) \n(this is type V in [25, § 6.4]). Its structure constants are C 1 13 = C 2 23 = 1, and the rest is zero. This is precisely type V in the above list, and so no additional GL(3) transformation is required. The solution for the functions p , q , and r are found to be: \np ( x ) = x 2 , q ( x ) = x 3 , r ( x ) = x 1 , (39) \nso that the IF, and corresponding dual frame, is \n X 1 = e x 1 ∂ 2 X 2 = e x 1 ∂ 3 X 3 = ∂ 1 = ⇒ e 1 = e -x 1 dx 2 e 2 = e -x 1 dx 3 e 3 = dx 1 . (40) \nThese are thus the basis components for our spacetime metric. Upon setting e 0 ≡ dt , we can then find the general spacetime metric to be \nds 2 = g µν ( t ) e µ e ν , (41) \nwhere the coefficients g µν ( t ) are functions of time t and satisfy the usual assumptions. Making the particular choice of diagonal spacetime metric, setting g 00 ≡ -1, and writing out the coordinate components, we obtain the form \nds 2 = -dt 2 + g 11 ( t ) 2 ( dx 1 ) 2 + e -2 x 1 ( g 22 ( t ) 2 ( dx 2 ) 2 + g 33 ( t ) 2 ( dx 3 ) 2 ) . (42) \nThe functions g ii ( t ), i = 1 , 2 , 3 (no sum) are essentially arbitrary for the purposes of ensuring this metric is Bianchi V (indeed, has the KLA defined by (38)). \nModel 2: We now pick as pre-KLA \nξ 1 = e x 1 ∂ 2 , ξ 2 = e x 1 ∂ 3 , ξ 3 = ∂ 1 . (43) \n(This is exactly the IF in the above construction!) Its structure constants are C 1 13 = C 2 23 = -1; this does not occur as a standard one in the above list, and so we need a GL(3) transformation. The relevant transformation is ξ a ↦→ ξ ' a = -ξ a , for which then C ' 1 13 = C ' 2 23 = 1, and this is again exactly type V in the above list. We can then solve for the functions p , q , and r (remembering that we are doing this for ξ ' a ), \np ( x ) = -x 2 e -x 1 , q ( x ) = -x 3 e -x 1 , r ( x ) = -x 1 , (44) \n̸ \nTABLE II: General forms of the IFs { X i } belonging to pre-KLAs { ξ a } of the various Bianchi types. The functions p = p ( x ), q = q ( x ), and r = r ( x ) must be solved for any particular choice of pre-KLA, to satisfy the indicated relations. In Bianchi VII h , ∆ := 1 2 √ 4 -h 2 . \nand so construct the IF and corresponding dual frame: \n X ' 1 = ∂ 2 X ' 2 = ∂ 3 X ' 3 = ∂ 1 + x 2 ∂ 2 + x 3 ∂ 3 (45) \n⋆ = ⇒ X 1 = ∂ 2 X 2 = ∂ 3 X 3 = ∂ 1 + x 2 ∂ 2 + x 3 ∂ 3 (46) \n= ⇒ e 1 = -x 2 dx 1 + dx 2 e 2 = -x 3 dx 1 + dx 3 e 3 = dx 1 . (47) \nThe step ⋆ boiled down to a relabeling, as a scaling does not affect the components of the IF (cf. Proposition III.9). Now that we have the dual basis, we can construct a spacetime metric along the same lines as above. Once more assuming a diagonal spacetime metric (in the basis { e µ } ), g 00 ≡ -1, and writing the spacetime metric in coordinates, we find \nds 2 = -dt 2 + g 33 ( t ) 2 ( dx 1 ) 2 + g 11 ( t ) 2 ( dx 2 -x 2 dx 1 ) 2 + g 22 ( t ) 2 ( dx 3 -x 3 dx 1 ) 2 (48) \nHow do we interpret the metrics resulting from both these models? Ignoring the temporal components, the spatial metrics of these models are \ndℓ 2 1 = a 2 1 ( dx 1 ) 2 +( b 1 e -x 1 ) 2 ( dx 2 ) 2 +( c 1 e -x 1 ) 2 ( dx 3 ) 2 (49a) \nand \ndℓ 2 2 = a 2 2 ( dx 1 ) 2 + b 2 2 ( dx 2 -x 2 dx 1 ) 2 + c 2 2 ( dx 3 -x 3 dx 1 ) 2 , (49b) \nfor constants a 1 etc. The spheres of radius R around a point x ∗ , S R ( x ∗ ), for the former case are ellipsoids, whilst for the latter the shape is more difficult to categorize. Note that these S R ( x ∗ ) are also the shapes of the lightcones in the relevant spacetime: as such, they are also the shape of a 'flash of light' emitted from x ∗ . Some representative S R ( x ∗ ) for both cases are shown in Figures 1 and 2. \nIt must also be noted that these shape seem to have some rotational symmetry still, which would beleaguer the assumption of anisotropy. However, we recall that these metrics only hold for a specific time, namely such a time at which the metric coefficients become the constants a 1 etc. Evolving the full spacetime metric in time (with the metric coefficients following Einstein's equation), these S R ( x ∗ ) will start to evolve as well. The only way the apparent rotational symmetry might persist is if, in general, the KVFs were dependent on time, which we demanded they not be.", 'B. Twice Bianchi II': 'In the above subsection § III C we touched upon two forms for Bianchi II pre-KLAs: \n ξ RS 1 = ∂ 2 ξ RS 2 = ∂ 3 ξ RS 3 = ∂ 1 + x 3 ∂ 2 & ξ H 1 = ∂ 1 ξ H 2 = ∂ 2 -1 2 x 3 ∂ 1 ξ H 3 = ∂ 3 + 1 2 x 2 ∂ 1 , (50) \nwith RS and H denoting Ryan & Shepley and Heisenberg, respectively. Both have the tabulated structure constants C 1 23 = 1 belonging to type II. As both algebras are written in the { ∂ i } -basis we can find a linear transformation mapping one to the other: ϕ : ξ RS a ↦→ ξ H a . In the { ∂ i } -basis, it is expressed as \nϕ = 1 2 x 2 -x 3 1 -1 2 x 3 0 0 1 1 0 0 . (51) \nThis is not a Jacobian matrix, and so we cannot use Proposition III.8 to our advantage; the methodology will be calculate the IFs separately, using the Table II. So we do this: \n X RS 1 = ∂ 2 X RS 2 = x 1 ∂ 2 + ∂ 3 X RS 3 = ∂ 1 & X H 1 = ∂ 1 X H 2 = ∂ 2 + 1 2 x 3 ∂ 1 X H 3 = ∂ 3 -1 2 x 2 ∂ 1 . (52) \n(In fact, if ϕ were the Jacobian matrix of a diffeomorphism, then due to Proposition III.8 it would have to map ϕ : X RS a ↦→ X H a ; we can verify by direct calculation that this does not occur.) The dual frames follow: \n e RS 1 = dx 2 -x 1 dx 3 e RS 2 = dx 3 e RS 3 = dx 1 (53a) \nand \n e H 1 = dx 1 + 1 2 ( x 2 dx 3 -x 3 dx 2 ) e H 2 = dx 2 e H 3 = dx 3 . (53b) \nAssuming a diagonal metric (in the { e µ } -bases), the spatial parts of the metrics when written out in coordinates become \ndℓ 2 RS = a 2 RS ( dx 1 ) 2 + b 2 RS ( dx 3 ) 2 + c 2 RS ( dx 2 -x 1 dx 3 ) 2 (54a) \nand \ndℓ 2 H = a 2 H ( dx 1 + 1 2 ( x 2 dx 3 -x 3 dx 2 )) 2 + b 2 H ( dx 2 ) 2 + c 2 H ( dx 3 ) 2 , (54b) \nfor constants a RS etc. Examples of various S R ( x ∗ ) are presented in Figures 3 and 4 for both metrics. \nFIG. 1: The sphere of radius R about x ∗ , S R ( x ∗ ) at various x ∗ , for the spatial metric (49a); R = 5, a 1 = b 1 = c 1 = 1. Since S R ( x ∗ ) does not depend on x 2 nor x 3 , only x 1 is fixed in the subfigures. In this particular case, they are all ellipsoids. Note also the different scales used for the different subfigures. \n<!-- image --> \nFIG. 2: The sphere of radius R about x ∗ , S R ( x ∗ ) at various x ∗ , for the spatial metric (49b); R = 5, a 2 = b 2 = c 2 = 1. Since S R ( x ∗ ) does not depend on x 1 , only x 2 and x 3 are fixed in the subfigures. \n<!-- image --> \nFIG. 3: The sphere of radius R about x ∗ , S R ( x ∗ ), at various x ∗ , for the spatial metric (54a); R = 7, a RS = b RS = c RS = 1. Since S R ( x ∗ ) does not depend on x 2 nor x 3 , only x 1 is fixed in the subfigures. \n<!-- image --> \nFIG. 4: The sphere of radius R about x ∗ , S R ( x ∗ ), at various x ∗ , for the spatial metric (54b); R = 7, a H = b H = c H = 1. Since S R ( x ∗ ) does not depend on x 1 , only x 2 and x 3 are fixed in the subfigures. \n<!-- image -->', 'C. Separated form of homogeneous metrics': 'The existence of the IF when given a pre-KLA not only allows us to formulate metrics on which the pre-KLA is realized as a KLA, but also to rewrite given homogeneous metrics into a standard form, wherein the components depend only on time. This is evidently useful for GR-related purposes, as then the Einstein equation will reduce to coupled ODEs, instead of PDEs, simplifying the solving procedure. \nTheorem IV.1. Let g be a spatially homogeneous metric. Then there exists a frame of 1-forms { e µ } and coefficients g µν ( t ) such that \ng = g µν ( t ) e µ e ν . (55) \nProof. Let { ξ i } i =1 , 2 , 3 be those KVFs of g which are purely spatial and form a frame everywhere, which exist due to spatial homogeneity. Since the KLA spanned by { ξ i } is trivially a pre-KLA, we can find the IF { X i } by the results of Section III. Set X 0 ≡ ∂ 0 = ∂ t , and then observe that we have \nξ i [ g ( X µ , X ν )] = ( L ξ i g )( X µ , X ν ) + g ([ ξ i , X µ ] , X ν ) + g ( X µ , [ ξ i , X ν ]) = 0 , (56) \n( µ, ν = 0 , 1 , 2 , 3), where the first term vanishes due to ξ i being a KVF, and the second and third term due to the IF commuting with KVFs. (That [ ξ i , X 0 ] = 0 follows from ξ i being purely spatial.) Since the KVFs form a frame, this means that the functions g ( X µ , X ν ) are constant in space, and hence are dependent only on time: g ( X µ , X ν ) = g ( X µ , X ν )( t ). \nGiven the IF { X i } , we can uniquely find its dual frame of 1-forms { e i } . Moreover, with the added vector field X 0 = ∂ t ↔ e 0 = dt , we still maintain the desired dual frame relations e µ ( X ν ) = δ µ ν . When we then write the metric in said basis, \ng = g ( X µ , X ν ) e µ e ν , (57) \nwe see that we have exactly recovered the form of the metric we wanted to show exists. \nWe emphasize that the proof is constructive: due to Table II, once the KVFs are determined, the IF follows after having solved for the functions p , q , and r from that table. Furthermore, the proof shows that any spatially homogeneous metric can be written into a form in which the temporal and spatial dependencies-generally e µ = e µ ( x )-are separated. As such, we dub this to be the separated form of the spatially homogeneous metric.', 'V. SUMMARY AND CONCLUSIONS': "According to the standard cosmological worldview, based on the Friedman-Lemaˆıtre-RobertsonWalker (FLRW) equations as its foundation, the universe on the largest scales adheres to the cosmological \nprinciple. It means that the universe is perfectly uniform, isotropic and homogeneous, on scales in the order of the Hubble radius. Yet, we do know that in physical reality the universe contains a rich infrastructure, reflected in the existence of density and velocity deviations whose amplitude becomes more prominent towards smaller scales. It is these deviations which are essential for the formation and evolution of all structure and objects in our cosmos. A range of recent observational studies have started to question the validity of the assumption that globally the universe is very close to perfectly isotropic and homogeneous, while the deviations represent a minor perturbation to the overall FLRW universe. \nOf fundamental significance for assessing the consequences of a universe that does not obey the cosmological principle is to establish its geometric foundation. In this paper we illustrate the implications for the spacetime metric once we abandon the assumption of isotropy. This entails a study of the metric structure implied by the specific symmetry properties of the cosmological model considered. In the present investigation this involves cosmologies with a perfect homogeneity, yet with a distinctly anisotropic angular structure. For this configuration, the corresponding homogeneous and fully anisotropic spacetime metric fits into the so-called 9-part Bianchi classification [see 21]. \nThe intention of the current study is to investigate the mathematical basis of the Bianchi models from the perspective of the infinitesimal isometries, or Killing vector fields (KVFs), belonging to the metrics involved. Requirements on homogeneity and isotropy of a metric manifest themselves mathematically as such infinitesimal isometries. The resulting homogeneous, completely anisotropic metric has precisely three KVFs, which together form a Lie algebra. As such, the resulting Lie algebra of KVFs (Killing Lie algebra; KLA) belongs to one of Bianchi's classes I-IX. \nThe approach of the current study is that of a reversal of the process. Given a suitable 3D Lie algebra of vector fields { ξ a } a =1 , 2 , 3 (termed a pre-KLA ), the program is to construct a metric on which those vector fields would be Killing. We proceeded by showing that the basis of the metric should be the frame dual to the invariant frame (IF), i.e. that frame { X i } i =1 , 2 , 3 so that [ ξ a , X i ] = 0 for all a and i . Expanding the X i in the ξ -basis, and substituting this into the commutator relation above, we infer a differential equation that is solved by the metric of characteristics. This yields an immediate expression for the X i in terms of ξ a , albeit involving some functions that need to be determined. \nSeveral results are obtained and proven that expand the applicability of our method. Specifically, this concerns the situation in which an initial condition is provided and for which the IF is unique to the pre-KLA. We demonstrate that when given a pre-KLA that is not in canonical Bianchi form, the following procedure should be followed: perform a GL(3) operation to bring it to a \ncanonical form, construct the IF, and then operate with the inverse GL(3) to get the IF belonging to the original pre-KLA. To complete our inventory, we calculate and tabulate for the different Bianchi classes the above relations between pre-KLA and IF. The results are tabulated in Table II. \nThe relevance of the presented formalism and results lies in the generalization of the implications of Bianchi type metrics. It aims to establish a notation rooted solely in the Lie algebra structure of the KVFs, independent of any specific coordinate formulation. This approach opens the road for further research, particularly in constructing tensors of cosmological interest-such as the Ricci and shear tensors-for the various Bianchi type cosmologies. 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2024arXiv240903522E | To constrain models beyond LambdaCDM the development of the Euclid analysis pipeline requires simulations that capture the nonlinear phenomenology of such models. We present an overview of numerical methods and Nbody simulation codes developed to study the nonlinear regime of structure formation in alternative dark energy and modified gravity theories. We review a variety of numerical techniques and approximations employed in cosmological Nbody simulations to model the complex phenomenology of scenarios beyond LambdaCDM. This includes discussions on solving nonlinear field equations accounting for fifth forces and implementing screening mechanisms. Furthermore we conduct a code comparison exercise to assess the reliability and convergence of different simulation codes across a range of models. Our analysis demonstrates a high degree of agreement among the outputs of different simulation codes providing confidence in current numerical methods for modelling cosmic structure formation beyond LambdaCDM. We highlight recent advances made in simulating the nonlinear scales of structure formation which are essential for leveraging the full scientific potential of the forthcoming observational data from the Euclid mission. | 2024-09-01T00:00:00Z | ['arXiv:2409.03522', '10.48550/arXiv.2409.03522', '2024arXiv240903522E'] | ['Astrophysics - Cosmology and Nongalactic Astrophysics'] | Euclid preparation. Simulations and nonlinearities beyond LambdaCDM. 1. Numerical methods and validation | 2,024 | 173 | 0.52 | ['EPRINT_HTML', 'EPRINT_PDF'] | 2 | https://arxiv.org/pdf/2409.03522.pdf | {'ABSTRACT': 'To constrain models beyond Λ CDM, the development of the Euclid analysis pipeline requires simulations that capture the nonlinear phenomenology of such models. We present an overview of numerical methods and N -body simulation codes developed to study the nonlinear regime of structure formation in alternative dark energy and modified gravity theories. We review a variety of numerical techniques and approximations employed in cosmological N -body simulations to model the complex phenomenology of scenarios beyond Λ CDM. This includes discussions on solving nonlinear field equations, accounting for fifth forces, and implementing screening mechanisms. Furthermore, we conduct a code comparison exercise to assess the reliability and convergence of di ff erent simulation codes across a range of models. Our analysis demonstrates a high degree of agreement among the outputs of di ff erent simulation codes, providing confidence in current numerical methods for modelling cosmic structure formation beyond Λ CDM. We highlight recent advances made in simulating the nonlinear scales of structure formation, which are essential for leveraging the full scientific potential of the forthcoming observational data from the Euclid mission. \nKey words. cosmological N -body simulations - dark matter - dark energy', '1. Introduction': "Significant progress in cosmological observations is expected in the upcoming years, in particular from the Euclid survey (Laureijs et al. 2011; Euclid Collaboration: Mellier et al. 2024; Euclid Collaboration: Scaramella et al. 2022; Nesseris et al. 2022; Martinelli et al. 2021; Euclid Collaboration: Castro et al. 2023), Vera Rubin Observatory's Legacy Survey of Space and Time (LSST, Ivezi'c et al. 2019), the Roman Space Telescope (Spergel et al. 2015) and the Dark Energy Spectroscopic Instrument (DESI, DESI Collaboration: Aghamousa et al. 2016). These surveys will o ff er precision observations to high redshifts, allowing us to study the evolution of the Universe with unprecedented accuracy and potentially uncover the nature of dark matter and dark energy (DE). Gaining a deeper understanding of the nature of DE and addressing the long-standing question of whether the cosmological constant ( Λ ) is responsible for the late-time accelerated expansion of the Universe is indeed one of the primary goals of the Euclid survey (Amendola et al. 2018). \nThe Euclid space telescope was launched on July 1, 2023, and is going to observe billions of galaxies out to redshift z ≈ 2, covering more than a third of the sky in optical and near-infrared wavelengths. Euclid will deliver precise measurements of the shapes and redshifts of galaxies (Euclid Collaboration: Bretonnière et al. 2022, 2023; Euclid Collaboration: Merlin et al. 2023; Euclid Collaboration: Desprez et al. 2020; Euclid Collaboration: Ilbert et al. 2021), from which we will measure weak gravitational lensing (Euclid Collaboration: Ajani et al. 2023) and galaxy clustering (Euclid Collaboration: Adam et al. 2019). These primary probes can be used to rigorously investigate different cosmological scenarios, in particular those related to DE that go beyond the Λ -Cold-Dark-Matter ( Λ CDM) concordance model. \nAlthough the Λ CDM model is generally very successful in matching observations, the true identities of CDM and the cosmological constant Λ remain unknown. Additionally, some tensions have persisted in recent years, most notably the Hubble tension (see Di Valentino et al. 2021a, for a summary and references) where local measurements of the Hubble parameter today, H 0, appear to disagree with those inferred from high-redshift observations by around 5 σ . Further examples are the S 8 tension (see Di Valentino et al. 2021b, for a summary and references) and some anomalies found in measurements of the cosmic microwave background (Abdalla et al. 2022). The presence of these tensions may hint at a breakdown of the Λ CDM model and further motivates the exploration of alternative scenarios. \nOver the past few years, cosmologists have explored di ff erent possibilities to account for the late-time accelerating expansion of the Universe (Tsujikawa 2010; Clifton et al. 2012; Joyce et al. 2016) either by introducing a new field, referred to as the DE field or by proposing a modified theory of gravity (MG). A wide range of MG or DE models is equivalent to adding a new light scalar degree of freedom to the theory of General Relativity (GR). \nIn these theories, the scalar degree of freedom exhibits time evolution, sometimes accompanied by spatial fluctuations within the cosmic horizon. Even in the absence of such fluctuations, the background evolution may be di ff erent from Λ CDM, leading to modifications in structure formation. Significant spatial fluctuations in these models may arise due to various factors, including a low characteristic speed of sound in the theory (Gleyzes et al. 2014; Hassani et al. 2019), or as a result of the non-minimal coupling of the scalar field to matter or gravity (see Amendola 2004, for an example). MG and DE theories featuring a coupling of the scalar field to matter can further a ff ect perturbations at subhorizon scales by mediating a fifth force. If the coupling is universal and includes baryons, a screening mechanism is essential to evade the precise constraints of local experiments (Will 2014). Screening mechanisms are typically achieved through nonlinear phenomena in such theories. If, on the other hand, the coupling to matter is non-universal and is confined entirely to the dark sector, local experiments have no constraining power, and cosmological observations provide the main constraints. \nGiven the diversity of possible DE or MG scenarios, a large information gain is expected from nonlinear scales in the cosmological large-scale structure. These scales must be studied using N -body simulations that capture the essential aspects of the DE or MG models under consideration. This usually means that at least one additional equation needs to be solved for the extra degree of freedom. In many cases, this leads to a di ffi cult nonlinear problem that could require special techniques or approximations that need to be developed. This makes N -body simulations for models of DE or MG a challenging task. \nIn this paper, we first review the main features of the di ff erent classes of DE and MG models that have been proposed over the past years (see also Amendola et al. 2018, and Frusciante et al. in prep. for a more comprehensive and detailed overview). For each of them, we then discuss the numerical methods implemented within a selection of existing N -body codes (summarised in Table 1). Focusing on MG models with a universal coupling, we then compare the results of di ff erent N -body implementations for two well-studied theories, namely the Hu-Sawicki f ( R ) gravity (Hu & Sawicki 2007) and the 'normal branch' of the DvaliGabadadze-Porrati braneworld model (nDGP, Dvali et al. 2000; \nSchmidt 2009a). We choose simulation parameters following the code comparison paper by Winther et al. (2015) [W15 hereafter], allowing us to validate a number of new codes against existing results. \nThis article is part of a series that collectively explores simulations and nonlinearities beyond the Λ CDMmodel: \n- 1. Numerical methods and validation (this work).\n- 2. Results from non-standard simulations (Rácz et al. in prep.).\n- 3. Cosmological constraints on non-standard cosmologies from simulated Euclid probes (D'Amico et al. in prep.).\n- 4. Constraints on f ( R ) models from the photometric primary probes (Koyama et al. in prep.). \nFor further details, see our companion papers. The purpose of this first article in the series is to serve as a reference for models beyond Λ CDMand their existing implementations in various codes. This paper is structured as follows. In Sect. 2 we give a broad overview of di ff erent numerical approaches to treat the additional physics of models beyond Λ CDM. In Sect. 3 we discuss a number of di ff erent codes that implement those approaches and carry out a validation exercise, comparing several recently developed codes with the existing state of the art. We conclude in Sect. 4. In an Appendix, we discuss some performance considerations.", '2.1. Non-standard background evolution': "A wide range of models beyond the simplest cosmological constant scenario are based on an additional scalar degree of freedom - e.g. a classical scalar field ϕ - that evolves dynamically in the expanding Universe and whose background energy density ρϕ provides the source for the observed DE abundance. To induce cosmic acceleration and to match existing constraints on the background expansion history, the equation-of-state parameter w of such an additional field must be su ffi ciently negative at recent epochs but is poorly constrained at earlier times, which allows for models where w also evolves dynamically as long as it converges to values close to w ≈ -1 in the late Universe. For these models, the DE component modifies the background expansion history of the Universe, which is encoded by the general expression of the Hubble function, \nH 2 ( a ) H 2 0 = Ω m a -3 + Ω r a -4 + Ω k a -2 + Ω DEe -3 R a 1 1 + w ( a ' ) a ' d a ' , (1) \nwhere the equation-of-state parameter of DE can be obtained by solving the background field equations - including the evolution of the additional scalar degree of freedom ϕ - or can be parameterised. A common parameterisation suggested by Chevallier, Polarski & Linder (CPL, Chevallier & Polarski 2001; Linder 2003) is based on the desired evolution of w at low redshifts, \nw ( a ) = w 0 + wa (1 -a ) . (2) \nAlternatively, one can set the desired relative abundance of DE at late ( Ω DE = 1 -Ω m) and early ( Ω EDE) epochs as in the Early Dark Energy (EDE, Wetterich 2004) parameterisation, \nw ( a ) = w 0 1 + b ln(1 / a ) , b = 3 w 0 ln 1 -Ω EDE Ω EDE + ln 1 -Ω m Ω m . (3) \nThe modified expansion history expressed by Eq. (1) will indirectly a ff ect the evolution of matter density fluctuations and modify the formation process of collapsed structures by changing the Hubble friction term in the equation for linear matter perturbations, which in Newtonian gauge and in Fourier space for sub-horizon scales reads: \n¨ δ m + 2 H ˙ δ m = 4 π G ( ρ m δ m + ρ DE δ DE) , (4) \nwhere δ m and δ DE are the density contrasts of matter and DE perturbations, respectively, G is Newton's constant, and a dot represents a derivative with respect to cosmic time. \nBesides the richer background dynamics that is endowed by an evolving field, whenever DE is promoted from a cosmological constant to a dynamical degree of freedom, the model also acquires an additional layer of complexity: the presence and evolution of DE fluctuations around the mean-field configuration. This corresponds to the situation where δ DE in Eq. (4) is nonnegligible, whereas in Λ CDM it would vanish identically at all scales. Like any other density perturbations, inhomogeneities in the DE would then contribute to the peculiar gravitational potential that governs the evolution of matter perturbations and thus the formation of cosmic structures as shown by Eq. (4). \nHowever, in many of the simplest scalar-field scenarios, such perturbations are negligible at sub-horizon scales because the speed of sound c s of the scalar field is naturally close to the speed of light. Ignoring them for the purpose of numerical simulations, the only modification of N -body algorithms required to simulate these DE models is given by an appropriate calculation of the cosmic expansion rate. The most common approach amounts to tabulating the specific expansion rate of the universe for the model to be simulated according to Eq. (1) and replacing the standard analytical calculation of the Hubble function within the N -body algorithm with an interpolated value from the tabulated solution that is provided to the code as an input. This approach is implemented by most of the simulation codes employed within the Euclid Collaboration to perform cosmological simulations in homogeneous DE models beyond Λ CDM.", '2.2. Linearised DE perturbations': 'Although a wide range of DE models are characterised by negligible DE fluctuations as discussed above, some specific scenarios may not fulfil such a condition at all scales and / or at all times, either because they feature a lower value of the DE speed of sound, allowing DE perturbations to grow on scales above the associated Jeans length that then falls inside the cosmological horizon, or because additional interactions - besides gravity - can induce the growth of such perturbations. The former case corresponds to the class of clustering DE models, while the latter is known as coupled DE.', '2.2.1. Clustering DE': 'The clustering DE models are characterised by two timedependent variables: the speed of sound c s and the equation of state parameter w . In these theories, the DE component clusters on scales larger than the associated sound-horizon / Jeans scale, λ s = H / c s and DE perturbations decay quickly below the soundhorizon scale. For a su ffi ciently small speed of sound, we may even expect nonlinear DE structures to form. At a fundamental level, clustering DE models are analogous to the k-essence type of theories, so that the action reads (Armendariz-Picon et al.', '2001)': 'interacting species in the so-called Einstein frame, \nS = Z d 4 x √ -g " c 4 R 16 π G + P ( X , ϕ ) + L m # , (5) \nwhere P is a general function of the kinetic term X ≡ -1 2 ∇ µϕ ∇ µ ϕ and the scalar field ϕ , and L m is the matter Lagrangian. For a given P ( X , ϕ ), the speed of sound and the equation of state are given by (Armendariz-Picon et al. 2000) \nw = P P -2 XP , X , c 2 s = P , X 2 XP , XX + P , X , (6) \nwhere the subscript \' , X \' denotes the partial derivative with respect to X . We therefore need to specify the function P ( X , ϕ ) to derive the equations of motion for the k-essence scalar field. However, since there are many possible choices, we can instead employ the e ff ective field theory (EFT) approach to model the dynamics of k-essence DE. The EFT framework, although not a fundamental theory, o ff ers several advantages (Gleyzes et al. 2014; Cheung et al. 2008), such as being a description of a wide range of theories within some scales. The EFT is a perturbative approach based on the assumption that the scalar field perturbations remain small over the scales of interest. It is worth noting that the regime of nonlinear matter clustering is accessible to the EFT framework as long as the scalar field perturbations remain small. The k-essence theories or clustering DE models are implemented in several N -body and Einstein-Boltzmann codes. In CLASS (Lesgourgues 2011) and CAMB (Lewis et al. 2000), these theories are implemented using the fluid picture. In hi\\_class (Zumalacárregui et al. 2017), the EFT equations are implemented and can be controlled using the EFT parameter α K ≡ 3(1 + w ) c -2 s within the code. On the other hand, in k-evolution (Hassani et al. 2019, 2020), which is an N -body code based on gevolution (Adamek et al. 2016), nonlinear equations for clustering DE are implemented as an independent component, and the k-essence field for small c s can form nonlinear structures. In some N -body codes, for example in gevolution (Adamek et al. 2016), clustering DE is implemented through a linear solution from an Einstein-Boltzmann solver. This is a good assumption for large speeds of sound, but for small ones, this method does not allow for the response of DE to the nonlinear matter structures.', '2.2.2. Coupled quintessence': 'Moving to the case of coupled DE models, the interaction can be formulated at a fundamental level by introducing a direct coupling between the scalar field and the spatial curvature R in the so-called Jordan frame (see e.g. Pettorino & Baccigalupi 2008), so that the action reads \nS = Z d 4 x √ -g " c 4 f ( ϕ, R ) 16 π G -1 2 Z ( ϕ ) ∇ µ ϕ ∇ µϕ -V ( ϕ ) + L m # , (7) \nwhere f ( ϕ, R ) is a function that couples the scalar field to the curvature, Z ( ϕ ) is a function that allows for non-standard kinetic terms, V ( ϕ ) is the scalar field self-interaction potential, and the matter Lagrangian contains at least one cold species characterised by some rest mass m 0. \nAlternatively, the interaction can be formulated by including source terms in the covariant conservation equations of the \nArticle number, page 4 of 20 \n∇ µ T (c) µν = -β (c)( ϕ ) M Pl T (c) ∇ νϕ , (8) \n∇ µ T (b) µν = -β (b)( ϕ ) M Pl T (b) ∇ νϕ , (9) \n∇ µ T ( ϕ ) µν = 1 M Pl h β (c)( ϕ ) T (c) + β (b)( ϕ ) T (b) i ∇ νϕ , (10) \nwhere T (Y) µν is the stress-energy tensor of a given species Y, T (Y) is its trace, β (Y)( ϕ ) is the coupling function of species Y, the labels c , b , ϕ refer to the dark matter, baryon, and scalar field species, respectively, and M Pl ≡ ( ℏ c ) 1 / 2 (8 π G ) -1 / 2 is the reduced Planck mass. \nIn the case of a universal coupling (that is, if β (b) = β (c)), the two approaches can be related to one another through a Weyl transformation of the metric (see again Pettorino & Baccigalupi 2008), and are therefore equivalent. On the other hand, the possibility to leave the baryonic component of the Universe only minimally coupled evades Solar System constraints (see e.g. Will 2014) on the deviations from standard gravity thereby avoiding the need for screening mechanisms. This is the case of Coupled Quintessence models (Wetterich 1995; Amendola 2000), where the direct coupling between the scalar field and massive (nonbaryonic) particles can support stable perturbations of the DE field at sub-horizon scales (see e.g. Amendola 2004). In general, such perturbations may even become nonlinear in the presence of a su ffi ciently strong coupling (as in the case of Growing Neutrino Quintessence models, see e.g. Amendola et al. 2008; Mota et al. 2008; Baldi 2012b; Ayaita et al. 2016). Nonetheless, a large class of widely studied coupled DE models is known to feature scalar perturbations of the order of the standard Newtonian gravitational potential ( δϕ ∼ Φ N, see again Amendola 2004, for an extended derivation), thereby remaining in the linear regime at all times and scales of cosmological interest. This allows us to linearise the corresponding field equations and derive modified equations of motions for massive particles, including the contribution of the additional force arising from the direct coupling with the scalar field (see e.g. Baldi 2011). \nWhile in the former case the interaction will be universal (i.e. involving all matter species with the same strength), which goes under the name of Extended Quintessence, the latter approach allows for non-universal couplings that may selectively involve individual species, for example by separately choosing the coupling functions for baryons and dark matter. \nIn fact, a general feature of coupled DE models is the existence of a \'fifth force\' mediated by the scalar field. The new force can be expressed as an additional acceleration experienced by a massive coupled particle, which in comoving coordinates will be given by \na Y , 5 th = -β (Y)( ϕ ) ∇ δϕ, (11) \nwhere Y identifies a coupled matter species, and δϕ is the scalar field fluctuation. This extra acceleration term is added to the standard Newtonian acceleration acting on all massive particles, \na N = -H v -∇ Φ N , (12) \nwhere v is the peculiar particle velocity in comoving coordinates, and Φ N is the peculiar Newtonian potential obeying the standard Poisson equation \n∇ 2 Φ N = 4 π Ga 2 X Y ρ Y δ Y , (13) \nwhere the sum runs over all clustering species in the Universe. \nTherefore, solving for the dynamical evolution of massive coupled particles requires solving for the scalar field perturbation δϕ entering in Eq. (11), which in the most general case follows a nonlinear elliptic equation, \n∇ 2 δϕ = F ( δϕ ) + X Y 8 π Ga 2 β (Y)( ϕ ) δ Y , (14) \nwith F a function of the scalar field fluctuation δϕ , and where the sum runs over all the coupled matter species with their respective couplings β (Y)( ϕ ). \nFor the particular case of a coupled DE model with a nonuniversal interaction (Damour et al. 1990) involving only dark matter and leaving baryons uncoupled (i.e. β (b) = 0) the function F ( δϕ ) in Eq. (14) is negligible compared to the term associated with matter density perturbations (see Amendola 2004, for a derivation) and can be safely discarded. As a result, the scalarfield equation reduces to \n∇ 2 δϕ ≈ 8 π Ga 2 β (c)( ϕ ) δ c = 2 β (c)( ϕ ) ∇ 2 Φ c , (15) \nwhere Φ c is the Newtonian potential generated by the distribution of the coupled dark matter particles, that is \n∇ 2 Φ c = 4 π Ga 2 ρ c δ c . (16) \nTherefore, the solution for the scalar field perturbations will be directly proportional to the potential Φ c according to the relation \nδϕ ≈ 2 β (c) Φ c . (17) \nThe acceleration equation for a coupled particle can then be rewritten as \na c = -H v c -∇ Φ b -∇ GLYPH<16> 1 + 2 β 2 (c) ( ϕ ) GLYPH<17> Φ c , (18) \nassuming here for simplicity that other clustering species (such as massive neutrinos) give a negligible contribution to the total Newtonian potential such that Φ N = Φ c + Φ b. This modified acceleration equation introduces a further modification to be implemented in N -body simulation codes for Coupled Quintessence cosmologies besides the specific expansion history of each particular model. This often requires substantial modifications in the gravity solvers of conventional N -body codes, as the algorithms need to evolve coupled and uncoupled massive particles (typically dark matter and baryons, respectively) with di ff erent equations and should therefore treat these components separately. Even under the approximation of a purely collisionless treatment (i.e., ignoring the hydrodynamical and astrophysical processes that a ff ect standard baryonic matter leading to the formation of stars and galaxies) that is often employed for largevolume simulations targeted at galaxy surveys such as Euclid , both coupled and uncoupled matter species must be included in the simulation to provide a consistent representation of the dynamics at all scales: as baryons and dark matter evolve di ff erently, assuming that all matter is dark would lead to an overestimation of the e ff ects of the coupling and, thus, biased results. This approach is implemented in the C-Gadget code (Baldi et al. 2010; Baldi 2012a) which is employed for Coupled DE simulations performed within the Euclid Collaboration. \nDistinguishing between coupled and uncoupled particle types in simulations of Coupled Quintessence is also crucial for proper treatment of two other e ff ects that characterise these cosmological models beyond the fifth force described by Eq. (11). The first is the mass variation of coupled particles due to the \nexchange of rest-frame energy with the DE scalar field, which arises as a direct consequence of the modified continuity equations (8) and of the assumption of particle number conservation. More specifically, the mass of coupled particles evolves as a result of the evolution of the background scalar field according to \nm Y( a ) = m 0 exp -Z ϕ ( a ) ϕ 0 β (Y)( ϕ ) d ϕ M Pl ! . (19) \nSuch a mass variation, which involves only particle species with a non-vanishing coupling to the scalar field, must be taken into account in N -body algorithms by changing the mass of individual simulation particles at every time step. This is normally done by tabulating the mass as a function of scale factor a by numerically integrating Eq. (19) along with the background dynamics of the scalar field ϕ , and interpolating from that table as the simulation progresses. \nThe second e ff ect is an additional force (on top of the fifth force) acting on coupled particles as a consequence of momentum conservation due to the particles\' mass variation described by Eq. (19), which takes the form of a velocity-dependent extra acceleration behaving either as a friction or as a drag, depending on the relative signs of the coupling function β ( ϕ ) and of the scalar field velocity ˙ ϕ (see e.g. Baldi et al. 2010), \na Y , v = -β (Y)( ϕ ) M Pl ˙ ϕ v . (20) \nSuch a velocity-dependent acceleration is responsible for a very rich phenomenology characterising Coupled Quintessence models, especially on highly nonlinear scales (see e.g. Baldi et al. 2010; Baldi 2012a; Li & Barrow 2011; Baldi 2023), and must be included in N -body simulations as well for a fully consistent treatment of these scenarios. This is done by adding the extra acceleration described in Eq. (20) to the total acceleration (i.e. Newtonian plus fifth force) of all coupled particles in each time step, \na Y = a N + a Y , 5 th + a Y , v . (21) \nThe relevant quantities β ( ϕ ) and ˙ ϕ can again be interpolated from a table obtained by integrating the background dynamics of the system. This is the approach implemented in the C-Gadget code that has been used to run Coupled Quintessence simulations within the Euclid Collaboration.', '2.2.3. Momentum exchange and dark scattering': "A further example of interacting DE cosmologies characterised by scalar-field perturbations that always remain linear is given by models of pure momentum exchange (see e.g. Pourtsidou et al. 2013; Skordis et al. 2015) between the DE field and massive particles like dark matter or baryons. A limiting case is given by the Dark Scattering scenario (Simpson 2010) where the momentum transfer between the two components is modelled as the elastic scattering of massive particles moving through a homogeneous DE fluid with equation of state w . This results in an extra force acting on the moving massive particles which is proportional to their comoving velocity, similar to the velocity-dependent force described by Eq. (20) for Coupled Quintessence models. However, the origin of this force is completely di ff erent in this case, as it does not originate from the mass variation of particles but rather from the momentum transfer with the DE field. As a result, the scattering acceleration can be expressed as \na s = -(1 + w ) 3 H 2 Ω DE 8 π G ξ v , (22) \nwhere the parameter ξ is defined as \nξ ≡ σ m , (23) \nwith σ denoting the scattering cross section and m the typical mass of the scattering particle species. \nAlthough Dark Scattering represents a limiting case of the more general class of pure momentum-exchange models between matter and DE (also known as 'Type 3' models in the classification of Skordis et al. 2015), for which further modifications to the standard particle dynamics are expected besides the drag force of Eq. (22), recent works (Palma & Candlish 2023) have shown that such additional modifications are generally subleading with respect to the drag force so that their e ff ect on structure formation can be neglected. This ensures that the current implementation of Dark Scattering within the simulations used in the Euclid Collaboration can be considered representative of the general class of momentum-exchange cosmologies. \nThis type of interaction can be implemented in N -body algorithms (see e.g. Baldi & Simpson 2015, 2017) in a very similar way as the velocity-dependent acceleration in Coupled Quintessence scenarios, as the factors entering Eq. (22) are all either constants or background quantities that can be interpolated at every timestep from tabulated data. This is the approach implemented in the C-Gadget code that has been used to run the DAKAR and DAKAR2 simulations (Baldi & Simpson 2017, Rácz et al. in prep.).", '2.3. Nonlinear scalar field perturbations': "In models where a scalar field couples to matter universally, or at least to baryons in a relevant way, some mechanism to suppress the coupling is required to satisfy the stringent local tests of gravity. This is commonly referred to as 'screening'. Screening mechanisms are achieved by nonlinearity in the scalar-field equation coupled to matter. The equation determining the evolution of the scalar field is typically a wave equation of the form \n□ ϕ = S ( ϕ, ∇ µϕ, ∇ µ ∇ νϕ, ρ m) . (24) \nSeveral approximations are often used to solve these nonlinear equations. The most common one is the quasi-static approximation. The scalar field can be split into a background part, ¯ ϕ , and a perturbation, δϕ , as ϕ = ¯ ϕ + δϕ . The quasi-static approximation amounts to ignoring the time dependence of the scalar field perturbation, i.e. assuming ˙ ϕ ≃ ˙ ¯ ϕ . The partial differential equation (PDE) of the field perturbation, which in its original form may have been of the hyperbolic or parabolic type, is therefore cast into an elliptic form so that the scalar field solution at any given time depends solely on the matter configuration at that time. This is a good approximation whenever the speed of sound of the scalar field is small (Sawicki & Bellini 2015), which is the case for the MG models considered here. Non-quasistatic cosmological simulations have been conducted for several MG models using di ff erent techniques, such as the explicit leap-frog method and the implicit Newton-Gauss-Seidel method (Llinares & Mota 2013; Bose et al. 2015; Winther & Ferreira 2015b). \nHere, □ ≡ ∇ µ ∇ µ represents the d'Alembertian operator and S is a nonlinear function that depends on the matter density, the scalar field, and its derivatives. Various methods have been developed to solve this nonlinear scalar field equation in N -body simulations, where the nonlinear density ρ m is modelled by collisionless particles (see W15, for more details). \nThe scalar-field solution is required for the computation of the total gravitational potential Φ that acts on the matter particles, \n∇ 2 Φ = 4 π Ga 2 GLYPH<16> δρ m + δρ e ff ( ϕ ) GLYPH<17> , (25) \nwhere the e ff ective density depends on the scalar field. There are two common ways of solving for this total gravitational force. The first option is to solve first for ϕ and then use this solution to compute the source term in Eq. (25) and solve for Φ using a standard Poisson solver to get the total force ∇ Φ . The other option is to apply the total force ∇ Φ N + ∇ ϕ to the particles. \nUnder the quasi-static approximation, the scalar-field equation assumes the same form as the usual Poisson equation, \n∇ 2 ϕ = S ( ϕ, ∇ i ϕ, ∇ i ∇ j ϕ, δρ m) . (26) \nThe main di ff erence is that the scalar-field equation is generally nonlinear. This nonlinear behaviour implies that conventional techniques, such as using Fourier analysis, cannot be used to solve the equation. Numerous approaches have been developed to address this challenge, and we refer the reader to W15 for details. For computational methods that aim to accurately solve a nonlinear equation on refined grids, the approach typically involves discretising the equation in a suitable way and employing an iterative algorithm, such as the Newton-Raphson method, to successively refine solutions based on an initial guess. To speed up convergence, many of these methods incorporate socalled 'multigrid' acceleration techniques which we quickly review here.", '2.3.1. Nonlinear multigrid algorithm': 'A generic way to solve nonlinear elliptic PDEs is to couple the multigrid algorithm to the Newton-Raphson method, \nu new = u old -L ( u old ) ∂ L /∂ u old , (27) \nwhere u is the discretised field, L is the di ff erential operator (which is a Laplacian for Newtonian gravity) and the superscripts refer to the new or old estimate of the solution in one Newton-Raphson iteration. The Newton-Raphson method produces linear equations for the correction terms, which are solved by the Full-Approximation-Storage Multigrid algorithm (Brandt 1977; Wesseling 2004; Guillet & Teyssier 2011). For a review of these methods applied to MG simulations, see e.g. Li (2018), Llinares (2018) and W15. \nA simple sketch of the algorithm goes as follows. One starts with a guess for the solution on a grid; this could be anything from a constant value across the grid to using the solution from the previous timestep in the simulation. One then loops a few times over all cells in the grid, updating the solution using Eq. (27). This solution is then restricted to a grid with half the resolution, the solution is updated again, and this process is repeated recursively up to the coarsest grid (one with only 2 3 cells). The solution is then interpolated to the finer grid, updated once more, and this is done recursively until one reaches the finest grid we started with. One such cycle is called a V-cycle, and one repeats such V-cycles until convergence is achieved. The advantage of having this stack of coarser grids is that it helps to accelerate the convergence of the largest modes in the solution.', '2.3.2. Screening with nonlinearity in potentials': 'In models where screening is achieved by nonlinearity in a potential or coupling function, the equation for the scalar field becomes \n∇ 2 ϕ = 4 π Ga 2 β ( ϕ ) δρ m + V ( ϕ ) , (28) \nwhere β ( ϕ ) and V ( ϕ ) are nonlinear functions of ϕ . A typical example is f ( R ) gravity. In this class of models, the value of the scalar field changes by orders of magnitude. To enhance numerical stability, a common technique involves redefining the scalar field in terms of a new variable. The redefinition to choose depends on the specific model under consideration. It is typically chosen to prevent the occurrence of unphysical values of the scalar field during Newton-Raphson iterations. For example, for f ( R ) models the scalar field ϕ = f , R will be driven towards zero in high-density regions, but at the same time f , R cannot cross zero, as the potential becomes singular in this scenario. To avoid this issue, a commonly used field redefinition is u ≡ ln[ f , R / ¯ f , R ( a )] (Oyaizu 2008). However, this transformation introduces additional nonlinearity and in some models, such as Hu-Sawicki f ( R ) models, this transformation is not necessary and might even lead to considerable performance losses in a simulation. \nIn some cases, this can be avoided. For example, Bose et al. (2017) noticed that for Hu-Sawicki f ( R ) gravity with n = 1 (Hu & Sawicki 2007), when making the change of variable u = p -f , R , the field equation could be recast as a depressed cubic equation, \nu 3 + pu + q = 0 , (29) \nwhich possesses analytical solutions (Ruan et al. 2022). Although the Gauss-Seidel smoothing procedure is still needed (because p depends on the values of the field u in neighbouring cells), this removes the Newton-Raphson part and expensive exponential / logarithmic operations from the method of Oyaizu (2008), therefore leading to significant performance gains. Ruan et al. (2022) also generalised this improved relaxation approach to the cases of n = 0 (which strictly speaking is not a variation of the Hu-Sawicki model) and n = 2. \nFor other models like the symmetron, which has a Higgslike potential, the scalar field is free to cross zero, and no field redefinition is needed (apart from a simple rescaling).', '2.3.3. Screening with nonlinearity in kinetic terms': 'In another class of models, nonlinearity emerges within the kinetic term. For example, in models with the Vainshtein screening mechanism (Vainshtein 1972), the equation exhibits nonlinearity in the second derivatives of the scalar field, \n∇ 2 ϕ = 4 π Ga 2 β ( a ) δρ m + g ( ∇ i ϕ, ∇ i ∇ j ϕ ) , (30) \nwhere β ( a ) is a time-dependent coupling function. The simplest example here is the DGP model which was first simulated by Schmidt (2009b). In such cases, the operator-splitting trick (Chan & Scoccimarro 2009) can be employed. This approach can simplify the equations, avoiding potential issues associated with imaginary square roots, and improving code performance. This trick is particularly useful for the DGP braneworld models and other Vainshtein screening models, such as Cubic and Quartic Galileons (see Sect. 4.2.2 of W15, for more information).', '2.3.4. Approximate treatments of screening': "Some models allow linearisation of the nonlinear equation using some approximation. One approach (Khoury & Wyman 2009; Winther & Ferreira 2015a; see also the appendix of Schmidt 2009a) is to introduce the screening factor for the matter density perturbation \n∇ 2 ϕ = c 2 m 2 a 2 ℏ 2 ϕ + 4 π Ga 2 δρ m ϵ screen( Φ N , |∇ Φ N | , ∇ 2 Φ N) , (31) \nwhere the screening function depends on the Newtonian potential Φ N. This type of parameterised modified gravity is referred to as 'type 1' in Table 1. One specific method, developed in Brando et al. (2023), starts from linearising Klein-Gordon's equation. In this formalism, one solves the Poisson equation in Fourier space, \n-k 2 Φ = 4 π G e ff ( a , k ) a 2 δρ m , , (32) \nwhere the function G e ff ( a , k ) approximates the e ff ective Newton's constant introduced by the screening e ff ect of the scalar field on small scales. This function is given by \nG e ff ( a , k ) = G + ∆ G e ff ( a , k ) = G + GLYPH<16> G lin e ff ( a ) -G GLYPH<17> M ( a , k ) , (33) \nwith G lin e ff ( a ) being the asymptotic linear e ff ective Newton's constant that depends only on time, and M ( a , k ) is a function that approximately captures the nonlinear corrections introduced by the scalar field on small scales. This function allows Eq. (33) to transition from G e ff ( a , k ) → G lin e ff ( a ) on large scales to G e ff ( a , k ) → G on small scales. This type of parameterised modified gravity is referred to as 'type 2' in Table 1. A procedure to fix M ( a , k ) is described in Brando et al. (2023), which has the advantage of avoiding additional parameters to tune the screening e ffi ciency. \nOne can also choose to parameterise the nonlinear contribution using an e ff ective Newton's constant at both small and large scales. If the modifications of gravity are encoded in a scaledependent function ∆ G e ff ( a , k ) as in Eq. (33), then we can propose a similar equation in real space, \n˜ G e ff ( a , r ) = G + ∆ ˜ G e ff ( a , r ) , (34) \nwhere ∆ ˜ G e ff ( a , r ) is the Fourier transform of ∆ G e ff ( a , k ). In practice, an additional approximation is made, namely ∆ ˜ G e ff ( a , r ) ≈ ∆ G e ff ( a , k → 1 / r ). \nThis approach allows the encoding of nonlinear contributions over the whole range of scales modelled by N -body algorithms through real-space equations, for instance, the Tree Particle-Mesh (TreePM) method implemented in codes like Gadget4 . Provided the parameterisation is e ff ective, this is expected to increase the accuracy of the estimation of the nonlinear e ff ects. \nSeveral parameterisations have been proposed for this kind of approach, either with additional tuning parameters, such as in Lombriser (2016), or based on local small-scale environmental properties to avoid the need for any extra parameters, such as in Winther & Ferreira (2015a).", '3. Additional code validation': 'A code comparison for simulations that solve nonlinear scalar field perturbations is presented in W15. Since then, various new codes have been developed. In this section, we show a comparison of the predictions for the power spectrum and the halo mass \nTable 1: Summary table of the N -body codes implementing various extensions to the standard Λ CDM cosmology that have been used to produce simulations employed in Euclid pre-launch analysis, validation, and forecasting. \nfunction using the simulations of W15 as a reference and starting from the same initial conditions. These were generated using second-order Lagrangian perturbation theory in a Λ CDM cosmology with Ω m = 0 . 269, ΩΛ = 0 . 731, h = 0 . 704, n s = 0 . 966 and σ 8 = 0 . 801. The simulations have N p = 512 3 particles of mass M p ≃ 8 . 756 × 10 9 h -1 M ⊙ in a box of size B = 250 h -1 Mpc and start at redshift z = 49. As in W15, we compare simulations for f ( R ) and nDGP models. In these models (Schmidt 2009a), the background expansion history is closely approximated by that of Λ CDM. Furthermore, the e ff ect of modified gravity can be ignored at z = 49, thus it is justified to use the initial conditions of Λ CDM. The measurements of the power spectrum and mass function are performed by the pipeline developed in the second article of this series, Rácz et al. (in prep.). Based on two models only, our comparison does not encompass the full diversity of numerical methods discussed in the previous section. In many cases some validation of the various implementations can be found in the corresponding references.', '3.1. Summary of codes used in the validation': 'Table 1 shows an overview of the simulation codes considered in this section and provides a quick reference of their capabilities \nand limitations. For each of them, a short summary is presented here. In the Appendix we comment on the trade-o ff between accuracy and computational cost of the implementations.', '3.1.1. MG-Arepo': "First presented in Arnold et al. (2019), this code is based on the moving-mesh N -body and hydrodynamical simulation code Arepo (Springel 2010; Weinberger et al. 2020), which uses a TreePM algorithm to calculate gravitational forces. The additional modified gravity force (fifth force) is calculated with a relaxation solver (Bose et al. 2017) that is accelerated by the multigrid method and uses adaptive mesh refinement (AMR). It currently also supports simulations for the nDGP model (Hernández-Aguayo et al. 2021), as well as massive neutrinos implemented using the δ f method (Elbers et al. 2021). \nTo solve the modified gravity equations, the density field is projected onto the AMR grid, constructed in such a way that each cell on the highest refinement level contains at most one particle (except if a pre-set maximum refinement level is reached; the cell size at this level is of the order of the smoothing length of the standard gravity solver). Once the field equation is solved to \nobtain the scalar field configuration, the modified gravity force can be computed from its gradient using finite di ff erencing. \nSince MG-Arepo computes the forces on all particles simultaneously, and the modified gravity field equations are generally highly nonlinear (with a poor convergence rate of the relaxation algorithm), this is computationally expensive compared to Arepo 's Newtonian gravity solver. However, the maximum acceleration of the modified gravity force is smaller than that of Newtonian gravity, mainly because the latter occurs in regions with high density where screening occurs. This allows the modified gravity solver to run using larger time steps (Arnold et al. 2019), resulting in significantly reduced computational cost. Together with Arepo 's e ffi cient MPI parallelisation and lean memory footprint, these have made it possible to run the large number of f ( R ) simulations used in various recent works, such as the FORGE -BRIDGE (Arnold et al. 2022; Harnois-Déraps et al. 2023; Ruan et al. 2024) simulation suite of 200 f ( R ) and nDGP models. This has allowed accurate emulators of various physical quantities or observables to be constructed. \nAnother highlight of MG-Arepo is its capability to run realistic galaxy formation simulations in a cosmological box (Arnold et al. 2019; Hernández-Aguayo et al. 2021), thanks to the use of the Illustris-TNG subgrid physics model (Weinberger et al. 2017; Pillepich et al. 2018). More recently, it has been used for larger-box hydrodynamical simulations with a realistic recalibrated Illustris-TNG model (Mitchell et al. 2022), enabling the study of galaxy clusters in modified gravity. \nThe MG-Arepo simulations used in this paper were run using a residual criterion of ϵ = 10 -2 and a maximum refinement level ( MaxAMRLevel ) of 10 for the nDGP simulations and 18 for the f ( R ) models with a gravitational softening of 0 . 01 h -1 Mpc.", '3.1.2. MG-Gadget': 'MG-Gadget (Puchwein et al. 2013) is a modified version of the TreePM N -body code Gadget-3 (which in turn is based on the public code Gadget-2 , see Springel 2005) implementing an AMR solver for the scalar degree of freedom f , R characterising the widely-studied Hu-Sawicki f ( R ) gravity model. In MG-Gadget , the same tree structure that is employed to solve for standard Newtonian gravity is also used as an adaptive grid to solve for the scalar field configuration through an iterative Newton-Gauss-Seidel (NGS) relaxation scheme (see Sect. 2.3) with the Full-Approximation-Storage Multigrid method (see Sect. 2.3.1) and with the field redefinition u ≡ ln[ f , R / ¯ f , R ( a )] (see Sect. 2.3.2). MG-Gadget also allows MG simulations to be run with massive neutrinos (see e.g. Baldi et al. 2014; Giocoli et al. 2018) using the neutrino particle method (see Adamek et al. 2023, for a review on numerical methods for massive neutrino simulations). \nFor the simulations presented here, the relative tree opening criterion was used with an acceleration relative error threshold of 0.0025, and a uniform grid with 512 3 cells was employed to compute long-range Newtonian forces. Concerning the MG field solver, a residual tolerance of ϵ = 10 -2 was set for the V-cycle iteration and a maximum refinement level of 18 was used for the AMR grid, corresponding to a spatial resolution of 1 h -1 kpc at the finest grid level, compared to a gravitational softening of 18 h -1 kpc, following the setup adopted in W15.', '3.1.3. ECOSMOG': 'ECOSMOG (Li et al. 2012) is a generic modified gravity simulation code based on the publicly-available N -body and hydrodynamical simulation code RAMSES (Teyssier 2002). Originally developed for f ( R ) gravity, this code takes advantage of the adaptive mesh refinement of RAMSES to achieve the high resolution needed to solve the scalar field and hence the fifth force in highdensity regions. The nonlinear f ( R ) field equation is solved with the standard Gauss-Seidel approach as first applied by Oyaizu (2008), but it was later replaced by the more e ffi cient algorithm of Bose et al. (2017). The code has since been extended for simulations for the generalised chameleon (Brax et al. 2013), symmetron and dilaton (Brax et al. 2012), nDGP (Li et al. 2013b), cubic Galileon (Barreira et al. 2013, 2015), quartic Galileon (Li et al. 2013a), vector Galileon (Becker et al. 2020) and nonlocal gravity (Barreira et al. 2014). \nFor the ECOSMOG simulations used in this paper, we have used a domain grid (the uniform mesh that covers the whole simulation domain) with 2 9 = 512 cells per dimension, and the cells are hierarchically refined if they contain 8 or more e ff ective 1 particles. The highest refined levels have e ff ectively 2 16 cells, leading to a force resolution of about 0 . 0075 h -1 Mpc.', '3.1.4. ISIS': 'The ISIS code (Llinares et al. 2014), like the ECOSMOG code above, is based on RAMSES (Teyssier 2002). It contains a scalar field solver that can be used to simulate generic MG models with nonlinear equation of motion and has been used to simulate models such as f ( R ) gravity, the symmetron model, nDGP and disformal coupled models (Gronke et al. 2014; Winther & Ferreira 2015a; Winther et al. 2015; Hagala et al. 2016; Llinares et al. 2020). It also allows for hydrodynamical simulations that have been used to study the interplay between baryonic physics and modified gravity (Hammami et al. 2015). Furthermore, the code has the capability to go beyond the quasistatic limit and study the full time dependence of the scalar field (Llinares & Mota 2013; Llinares & Mota 2014; Hagala et al. 2017). The scalar field solver used in the code is a Gauss-Seidel relaxation method with multigrid acceleration, very similar to the one in ECOSMOG described above. \nFor the ISIS simulations presented in this paper, we have used a domain grid (the uniform mesh that covers the whole simulation domain) with 2 9 = 512 cells per dimension, and the cells are hierarchically refined if they contain 8 or more e ff ective particles.', '3.1.5. PySCo': 'PySCo 2 is a particle-mesh (PM) code written in Python and accelerated with Numba which currently supports Newtonian and f ( R ) gravity (parameterised as in Hu & Sawicki 2007, with n = 1 or n = 2). While multiple flavours of solvers based on Fast Fourier Transforms (FFT) are available, for the present paper, we use a multigrid solver to propose something di ff erent from other codes in this comparison project (other codes that also use multigrid are AMR-based). PySCo uses a triangular-shaped \ncloud mass assignment scheme and solves the linear Poisson equation using multigrid V-cycles with a tolerance threshold of the residual of 10 -3 , and two F-cycles (such cycles go through the mesh levels more often than V-cycles, resulting in a higher convergence rate of the residuals at the cost of increased runtime, for details on multigrid cycles see also Ruan et al. 2022) to solve the additional field in f ( R ) gravity with the nonlinear multigrid method described in Sect. 2.3.1 and Eq. (29). Our convergence threshold is very conservative since we do not intend to conduct a convergence study in this paper (a less conservative threshold could still give reasonable results at much lower computational cost). Furthermore, to resolve the small scales we use a coarse grid with 2048 3 cells, resulting in roughly 500 time steps to complete the simulations.', '3.1.6. MG-COLA': 'MG-COLA simulations were performed using the FourierMultigrid Library FML . 3 These simulations are based on the COmoving Lagrangian Acceleration (COLA) method (Tassev et al. 2013), which combines Lagrangian perturbation theory with the PM method to reduce the number of time steps that are required to recover clustering on large scales. The MG-COLA N -body solver in the FML library contains implementations of various DE and MG models like the DGP model, the symmetron model, f ( R ) gravity and the Jordan-Brans-Dicke model (see Winther et al. 2017, for more details). For the PM part, we used N 1 / 3 mesh = 5 N 1 / 3 p , i.e. a mesh discretisation five times smaller than the mean particle separation, and the total of 150 time-steps linearly spaced along the scale factor to achieve a good agreement of the mass function with AMR simulations. We also ran lowresolution simulations with N 1 / 3 mesh = 3 N 1 / 3 p with 100 time steps, and checked that the nonlinear enhancement of the power spectrum from these low-resolution simulations agrees well with the one from the high-resolution ones. Screening is included using approximate treatments described in Sect. 2.3.4. For f ( R ) gravity models, the FML code has one parameter to tune the strength of chameleon screening called screening efficiency . For the model with ¯ f , R = 10 -6 , we used the default value screening efficiency=1 while we used screening efficiency=2 for ¯ f , R = 10 -5 . For nDGP, we used Gaussian smoothing with a smoothing radius of 1 h -1 Mpc to compute the density field for screening and did not use an option to enforce the linear solution at small wavenumbers k . We also ran simulations based on the screening approximation using G e ff ( a , k ) given by Eq. (33) for nDGP, and in these simulations, we have used the same COLA settings as in the other screening approximation implementation. The biggest advantage of this screening implementation is that it does not require any additional tuning parameter related to screening, that is, the function G e ff ( a , k ) is completely defined by the theoretical model one wants to simulate.', '3.1.7. PANDA': "PANDA is an extension of the TreePM code Gadget4 . It introduces modifications at large scales in the PM part with an effective Newton's constant according to Eq. (33), while the force at small scales is modified in the tree part with Eq. (34). In this respect, PANDA implements an approximate solver for the extra force induced by di ff erent possible MG theories. On the other hand, di ff erently from other approximate methods, the dynamics of matter particles under the e ff ect of the (dominant) standard \ngravity force is treated with the full TreePM solver of Gadget4 of which it retains the accuracy in modeling the nonlinear density field. For nDGP, the functional form of ˜ G e ff ( a , r ) is described in Lombriser (2016), while for f ( R ) it is introduced as in Winther & Ferreira (2015a). In the latter case, no additional parameters are needed to describe the screening. For the first nDGP case instead, in addition to the theoretically defined parameters N 0 = 1, B = 1 / [3 β ( a )], b = 2 and a = 3, we consider the screening scale k th = 0 . 4 h Mpc -1 defined by the parameterisation \nr V GLYPH<27> 2 3 3 Ω m H 2 r 2 c β 2 ! 1 / 3 r th . (35) \nwhere r V is the Vainshtein radius of the model for a spherically symmetric density perturbation (see Lombriser 2016, for more details).", '3.2. Comparison of the power spectrum': "Before discussing the e ff ect of extra degrees of freedom, we compare the di ff erent codes within a Λ CDM cosmology. For reference and only for the case of Λ CDM, we include results from the code PKDGRAV3 that was also used for the Flagship simulations in Euclid (Potter et al. 2017; Euclid Collaboration: Castander et al. 2024). Figure 1 shows the matter power spectra at two di ff erent redshifts, z = 1 (left panels) and z = 0 . 667 (right panels). The lower panels show the relative di ff erence to the simulation carried out with MG-Arepo that we use as a reference throughout. Since ISIS and ECOSMOG have both been developed from the RAMSES code, the agreement between these two codes is better than 1%. These AMR codes tend to underestimate the power on small scales when compared to tree-based codes, mainly due to the mesh refinement criterion used in our simulations. Fine-tuning some precision parameters, we expect that a better agreement can be achieved, cf. Schneider et al. (2016). As we can see, the codes that perform closely to our benchmark at small scales are the ones that also share the same tree-PM gravity solver, PANDA and MG-Gadget . PKDGRAV3 , which uses a tree structure and the fast multipole method to compute gravitational forces, is in excellent agreement with MG-Gadget up to k ∼ 1 h Mpc -1 , yielding slightly more power at higher k . \nFor modified gravity models, we compare the ratio between the power spectrum computed for that model and the one found in the Λ CDM reference cosmology. Taking such ratios largely cancels out the di ff erences seen in Λ CDM between the codes, making the di ff erences due to the treatments of the extra scalar field more apparent. Following W15, we consider two MG models that exhibit two di ff erent types of growth dependence and \nAlso shown in the figure are the results from COLA , a pure PM code that uses a fixed grid to solve the Poisson equation. The fixed resolution of the PM grid explains why COLA consistently su ff ers from low force resolution at k ≳ 1 h Mpc -1 . All our simulations use the exact same initial particle data, such that our comparisons should not be contaminated by cosmic variance. However, for COLA simulations, we also reconstructed the initial density field from the initial particle distribution to estimate the displacement fields required for the 2LPT calculations in those simulations. While PySCo incorporates a full N -body solver, its small-scale accuracy is constrained by resolution limitations that arise from the absence of AMR. Moreover, within the scope of this code comparison project, PySCo employs a multigrid algorithm, resulting in reduced small-scale clustering compared to FFT (on a regular grid), thereby explaining the observed deficiency in power at wavenumbers k ≳ 1 h Mpc -1 . \n∆ \nFig. 1: Matter power spectra from simulations carried out with di ff erent codes (di ff erent line styles) for a Λ CDM cosmology. The left panels show the spectra at redshift z = 1 whereas the right panels show the spectra at redshift z = 0 . 667. The bottom panels show the relative di ff erence with respect to the simulation carried out with MG-Arepo . \n<!-- image --> \nscreening mechanisms. The first one is the so-called Hu-Sawicki f ( R ) which exhibits a scale-dependent growth factor at linear order, and a screening mechanism realised by nonlinearities in the potential, see Sect. 2.3.2. The strength of the modification of gravity is characterised by the background value of the scalar degree of freedom, and we study the cases ¯ f , R = 10 -5 and ¯ f , R = 10 -6 , labelled 'F5' and 'F6', respectively, the latter being closer to GR. The second one is nDGP, a braneworld theory defined in a five-dimensional spacetime. The growth factor in this theory is scale independent, and screening is realised through nonlinearities in the kinetic terms, as discussed in Sect. 2.3.3. Here, the strength of the modification of gravity is characterised by the value of the cross-over scale, and we study the cases r c = 1 . 2 H -1 0 and r c = 5 . 6 H -1 0 , labelled 'N1.2' and 'N5.6', respectively, the latter being closer to GR. \nFigure 2 shows the relative change (with respect to Λ CDM) of the matter power spectrum due to modifications of gravity, sometimes called 'boost', for the two f ( R ) scenarios at redshifts z = 1 (left panels) and z = 0 . 667 (right panels). Since f ( R ) already exhibits a scale-dependent growth factor at linear order, we can see that on scales of k ≈ 0 . 1 h Mpc -1 this e ff ect is already present, and gets enhanced at large wavenumbers due to nonlinearities of the density field. As we can see the codes MG-Arepo , MG-Gadget , ECOSMOG , ISIS , and PySCo all roughly agree to better than one percent down to scales of k ≈ 10 h Mpc -1 , as expected. These minor discrepancies can be caused by the refinement criterion used in the latter code or by slight variations in the redshift of the particle snapshot output. The present results are largely in agreement with a similar analysis performed in W15. \nIn the same plots, we can also see how approximation schemes to introduce screening from nonlinearities in the potential perform in contrast to the exact solutions. The results that use these methods are showcased by the examples of COLA and PANDA , where each uses a di ff erent scheme to approximate the e ff ects of the dynamics of the scalar field in dense environments. \nAs expected, the two codes do not exhibit the same level of agreement down to scales of k ≈ 10 h Mpc -1 as their counterparts using full solvers. However, in both cases, we can see that the deviations from the MG-Arepo reference results are limited to 2% even in the most extreme regime of departure from GR. Through closer inspection, we can see small di ff erences in the agreement between the approximate schemes and full solvers at di ff erent values of k . These deviations are caused by di ff erent implementations of the approximate MG solvers in the two codes. In fact, while COLA is an approximate method that uses a PM algorithm to solve the Poisson equation on a fixed grid with the use of the screening approximation to linearise the Poisson equation, PANDA is a new implementation that exploits the TreePM structure of the baseline Gadget-4 code to solve for the small-scale particle dynamics by incorporating the MG e ff ects (including the screening) through a scale-dependent Newton's constant in both real and Fourier space. \nFigure 3 shows results for the matter power spectra in the two nDGP scenarios we consider, characterised by the two values of the cross-over scale, r c = 1 . 2 H -1 0 (N1.2) and r c = 5 . 6 H -1 0 (N5.6). As before, we compare simulation results at two different redshifts, z = 1 (left panels) and z = 0 . 667 (right panels). Since the linear growth function has a scale-independent enhancement compared to Λ CDM, we see a constant amplification of power at small values of k . The Vainshtein mechanism suppresses the deviation from Λ CDM on small scales, leading to a diminishing boost at large k . The agreement between different codes at k < 1 h Mpc -1 is better than 1% for all codes, including the approximate simulations with COLA and PANDA . At k > 1 h Mpc -1 , the AMR-based codes, ECOSMOG and ISIS , show larger deviations at the level of 2%, where the suppression of the deviation from Λ CDM is slightly underestimated compared with MG-Arepo . This could be caused by the di ff erence in the Λ CDMpower spectrum shown in Fig. 1, even though this di ff erence in the baseline code is largely cancelled out in the boost. On the other hand, despite their approximations in the MG solvers, \nFig. 2: Amplification factor of the matter power spectra from simulations of two f ( R ) scenarios, ¯ f , R = 10 -5 (F5, top row ) and ¯ f , R = 10 -6 (F6, bottom row ), relative to the reference Λ CDM cosmology. The amplification factor BP = P / P Λ CDM is measured for di ff erent codes (di ff erent line styles) at two di ff erent redshifts, z = 1 ( left panels ) and z = 0 . 667 ( right panels ). The bottom panel of each plot shows the relative agreement of the individual measurements, using MG-Arepo as a common reference. \n<!-- image --> \nCOLA and PANDA agree with MG-Arepo at the level of 1% even on these scales.", '3.3. Comparison of the halo mass function': 'To gain further insight into the nonlinear dynamics of the di ff erent models and their respective implementations we also compare the cumulative halo mass functions measured in our simulations. For this purpose, halo catalogues are obtained with the Rockstar halo finder by running a pipeline described in the second paper of this series (Rácz et al. in prep.). To establish the baseline for the comparison, in the spirit of the previous section, Fig. 4 shows a comparison of the cumulative halo mass function for the di ff erent codes in a Λ CDM cosmology. Since MG-Arepo and MG-Gadget are based on the same TreePM gravity solver, their agreement is excellent. They also agree very well with results from PKDGRAV3 which, as we like to remind the reader, are only available for the case of Λ CDM and are shown for reference here. On the other hand, ECOSMOG and ISIS are based on the AMR method. They agree with each other, but these simulations underestimate the abundance of low-mass ha- \nbelow 10 13 h -1 M ⊙ . COLA uses a fixed-grid PM method, thus it also underestimates the abundance of low-mass halos. With relatively high resolution in these simulations ( N 1 / 3 mesh = 5 N 1 / 3 p ), COLA agrees well with ECOSMOG and ISIS . We note that PySCo exhibits a similar behaviour to COLA , ECOSMOG and ISIS , albeit with a lower amplitude. This discrepancy can be attributed to the fact that PySCo has the least small-scale clustering (see Fig. 1), resulting in a smaller number of halos within the simulation. \nThe chameleon screening in f ( R ) depends on the halo mass such that high-mass halos are typically screened. The critical mass above which screening is e ff ective is determined by the parameter ¯ fR . As we can see from Fig. 5, the ratio of the halo mass function between f ( R ) and Λ CDM is enhanced for lowmass halos but approaches unity at the high-mass end where all halos are e ff ectively screened. The agreement between full simulations ( ECOSMOG and ISIS , MG-Arepo , MG-Gadget ) is around 4% for all halo masses. COLA uses an approximation for screening based on the thin-shell condition for a spherically symmetric object. Although this captures an overall e ff ect of screening, it fails for low-mass halos in F6 and high-mass halos in F5, leading to larger deviations. \nFig. 3: Amplification factor of the matter power spectra from simulations of two nDGP scenarios, r c = 1 . 2 H -1 0 (N1.2, top row ) and r c = 5 . 6 H -1 0 (N5.6, bottom row ), relative to the reference Λ CDM cosmology. The amplification factor BP = P / P Λ CDM is measured for di ff erent codes (di ff erent line styles) at two di ff erent redshifts, z = 1 ( left panels ) and z = 0 . 667 ( right panels ). The bottom panel of each plot shows the relative agreement of the individual measurements, using MG-Arepo as a common reference. \n<!-- image --> \nIn the case of the Vainshtein mechanism, there is no halomass dependence on the screening. Despite screening being effective inside dark matter halos, these halos still feel enhanced gravitational attraction. This increases the merger rate and ultimately leads to larger enhancements of the halo mass function for halos of larger masses. In nDGP, shown in Fig. 6, all codes agree within 4%, with a sub-per cent agreement seen in the intermediate mass range 10 12 h -1 M ⊙ < M < 10 13 h -1 M ⊙ . ECOSMOG shows a large deviation for M ≲ 5 × 10 11 h -1 M ⊙ . For these masses, there are less than 50 dark matter particles assigned to each halo, which indicates that the results could be a ff ected by the refinement criteria of the simulations. The deviations are larger for the most massive halos, but the number of these halos is low and the halo mass function therefore becomes very noisy in this regime.', '4. Conclusions': "In this work, we have presented a comprehensive review of numerical methods for cosmological N -body simulations in scenarios extending beyond the standard Λ CDM model. Our ex- \nloration spanned a variety of alternative DE and MG theories, highlighting the critical role of N -body simulations in connecting theoretical models with observational data. Through the detailed examination of numerical solvers and approximations tailored to these extended theories, we have showcased the state of the art of modelling the nonlinear scales of cosmic structure formation under a wide range of cosmological scenarios. Our code comparison exercise, based on the simulations from W15 and extended by incorporating new codes and approximation techniques, has demonstrated a fair consensus among di ff erent numerical implementations. This validation is particularly important for the Euclid mission, as the forthcoming observational data will require precise nonlinear modelling to constrain cosmological parameters e ff ectively. \nThis article is part of a series that explores simulations and nonlinearities in models beyond Λ CDM. The simulation codes that we have considered in this article are used to generate simulation products in the companion paper by Rácz et al. (in preparation) and are crucial for generating the simulations needed to create the nonlinear modelling in the companion paper by Koyama et al. (in preparation), which forecasts constraints for \nFig. 4: Cumulative halo mass function from simulations carried out with di ff erent codes (di ff erent line styles) for a Λ CDMcosmology. The left panels show the halo mass function at redshift z = 1 whereas the right panels show the halo mass function at redshift z = 0 . 667. The bottom panels show the relative di ff erence with respect to the simulation carried out with MG-Arepo . \n<!-- image --> \n/circledot \n/circledot \nf ( R ) gravity from photometric primary probes of the Euclid mission. The validation checks performed in this paper are therefore critical for being able to trust these results. \nThe main outcomes of this work can be summarised as follows: \n- ⋆ N -body simulation codes have been developed for a wide range of extended cosmological scenarios, ranging from simple DE models to more complex interacting scalar field models and MG theories, to non-standard dark matter and initial conditions; the availability of such codes will be a crucial asset for the future developments of large galaxy surveys such as Euclid ; we have provided a concise yet comprehensive overview of several such codes, their main features, implementation methods, assumptions and approximations.\n- ⋆ As a result of our validation e ff ort, we found agreement in the power spectrum boost at ≲ 1% up to k ≲ 1 h Mpc -1 and at ≲ 3% up to k ≲ 10 h Mpc -1 among all the codes implementing full field solvers ( MG-Arepo, MG-Gadget, ECOSMOG, ISIS, PySCo ), while approximate methods ( PANDA, MG-COLA ) display slightly larger deviations not exceeding 3% up to k ≲ 7 h Mpc -1 .\n- ⋆ As a matter of fact, among these simulation codes the ones involving algorithms for the solution of nonlinear di ff erential equations of some additional degrees of freedom, as e.g. for the case of MG theories, are the most challenging in terms of implementation and numerical convergence; we have therefore performed a thorough validation e ff ort of these methods through a code comparison study, extending the approach adopted in W15 to more recent and diverse algorithms.\n- ⋆ The halo mass function shows larger deviations among the codes, also due to larger di ff erences in the outcomes for the baseline Λ CDM simulations; nonetheless, all codes agree within less than 5% on the relative change of the halo mass function except for the very largest and the smallest mass ranges where poor statistics and insu ffi cient resolution, respectively, may impact the results. \n- ⋆ Wealso compared the computational requirements of the different codes by measuring the CPU time needed to complete a reference MG simulation starting from identical initial conditions; we found that while full field solvers generally imply a substantial increase - up to a factor of ten - of the total CPU time relative to a Λ CDMsimulation, approximate solvers are not significantly more demanding for MG simulations compared to standard Λ CDMruns. \nLooking forward, the continued evolution of simulation techniques will be paramount in leveraging the full potential of upcoming large-scale structure surveys such as Euclid . N -body simulations therefore continue to set a solid foundation for the next generation of cosmological inquiries. By persistently pushing the boundaries of computational astrophysics, we are poised to uncover the underlying physics driving the accelerated expansion of the Universe, thereby opening new windows onto the fundamental nature of DE, dark matter, and gravity itself. \nAcknowledgements. The lead authors thank A. Schneider and R.E. Smith for their diligent work as internal referees and P. Schneider for carefully proofreading the manuscript. The work of JA is supported by the Swiss National Science Foundation. During part of this work, AMCLB was supported by a fellowship of PSL University hosted by the Paris Observatory. This project was provided with computer and storage resources by GENCI at CINES thanks to the grant 2023A0150402287 on the supercomputer Adastra's GENOA partition. GR's research was supported by an appointment to the NASA Postdoctoral Program administered by Oak Ridge Associated Universities under contract with NASA. GR was supported by JPL, which is run under contract by the California Institute of Technology for NASA (80NM0018D0004). The Euclid Consortium acknowledges the European Space Agency and a number of agencies and institutes that have supported the development of Euclid , in particular the Agenzia Spaziale Italiana, the Austrian Forschungsförderungsgesellschaft, funded through BMK, the Belgian Science Policy, the Canadian Euclid Consortium, the Deutsches Zentrum für Luft- und Raumfahrt, the DTU Space and the Niels Bohr Institute in Denmark, the French Centre National d'Etudes Spatiales, the Fundação para a Ciência e a Tecnologia, the Hungarian Academy of Sciences, the Ministerio de Ciencia, Innovación y Universidades, the National Aeronautics and Space Administration, the National Astronomical Observatory of Japan, the Netherlandse Onderzoekschool Voor Astronomie, the Norwegian Space Agency, the Research Council of Finland, the Romanian Space Agency, the State Secretariat for Education, Research, and Innovation (SERI) at the Swiss Space O ffi ce (SSO), and \nFig. 5: Increase of the cumulative halo mass function from simulations for the two f ( R ) scenarios, ¯ f , R = 10 -5 (F5, top row ) and ¯ f , R = 10 -6 (F6, bottom row ), relative to the reference Λ CDM cosmology. The enhancement factor Bn = n / n Λ CDM is measured for di ff erent codes (di ff erent line styles) at two di ff erent redshifts, z = 1 ( left panels ) and z = 0 . 667 ( right panels ). The bottom panel of each plot shows the relative agreement of the individual measurements, using MG-Arepo as a common reference. \n<!-- image --> \n/circledot \n/circledot \nthe United Kingdom Space Agency. A complete and detailed list is available on the Euclid web site ( http://www.euclid-ec.org ).", 'References': 'Abdalla, E., Abellán, G. F., Aboubrahim, A., et al. 2022, Journal of High Energy Astrophysics, 34, 49 \nAdamek, J., Angulo, R. E., Arnold, C., et al. 2023, JCAP, 06, 035 \nAdamek, J., Daverio, D., Durrer, R., & Kunz, M. 2016, Nature Phys., 12, 346 \nAdamek, J., Daverio, D., Durrer, R., & Kunz, M. 2016, JCAP, 07, 053 \nAmendola, L. 2000, Phys. Rev. D, 62, 043511 \nAmendola, L. 2004, Phys. Rev. D, 69, 103524 \n- Amendola, L., Appleby, S., Avgoustidis, A., et al. 2018, Living Rev. 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G. 2017, JCAP, 08, 019\n- 1 Department of Astrophysics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland\n- 3 Dipartimento di Fisica e Astronomia, Università di Bologna, Via Gobetti 93 / 2, 40129 Bologna, Italy\n- 2 Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth PO1 3FX, UK\n- 4 INAF-Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, Via Piero Gobetti 93 / 3, 40129 Bologna, Italy\n- 6 Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Muhlenberg 1, D-14476 Potsdam-Golm, Germany\n- 5 INFN-Sezione di Bologna, Viale Berti Pichat 6 / 2, 40127 Bologna, Italy\n- 7 Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans, s / n, 08193 Barcelona, Spain\n- 8 Institut de Ciencies de l\'Espai (IEEC-CSIC), Campus UAB, Carrer de Can Magrans, s / n Cerdanyola del Vallés, 08193 Barcelona, Spain\n- 10 Institute of Theoretical Astrophysics, University of Oslo, P.O. Box 1029 Blindern, 0315 Oslo, Norway\n- 9 Laboratoire Univers et Théorie, Observatoire de Paris, Université PSL, Université Paris Cité, CNRS, 92190 Meudon, France\n- 11 Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA, 91109, USA\n- 13 Department of Physics, Institute for Computational Cosmology, Durham University, South Road, DH1 3LE, UK\n- 12 Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 85748 Garching, Germany\n- 14 Jodrell Bank Centre for Astrophysics, Department of Physics and Astronomy, University of Manchester, Oxford Road, Manchester M13 9PL, UK\n- 16 Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, Via Irnerio 46, 40126 Bologna, Italy\n- 15 INAF-IASF Milano, Via Alfonso Corti 12, 20133 Milano, Italy\n- 17 Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK\n- 19 Institut für Theoretische Physik, University of Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany\n- 18 Higgs Centre for Theoretical Physics, School of Physics and Astronomy, The University of Edinburgh, Edinburgh EH9 3FD, UK\n- 20 Institut de Recherche en Astrophysique et Planétologie (IRAP), Université de Toulouse, CNRS, UPS, CNES, 14 Av. Edouard Belin, 31400 Toulouse, France\n- 22 Institut de Physique Théorique, CEA, CNRS, Université ParisSaclay 91191 Gif-sur-Yvette Cedex, France\n- 21 Université St Joseph; Faculty of Sciences, Beirut, Lebanon\n- 23 School of Mathematics and Physics, University of Surrey, Guildford, Surrey, GU2 7XH, UK\n- 25 IFPU, Institute for Fundamental Physics of the Universe, via Beirut 2, 34151 Trieste, Italy\n- 24 INAF-Osservatorio Astronomico di Brera, Via Brera 28, 20122 Milano, Italy\n- 26 INAF-Osservatorio Astronomico di Trieste, Via G. B. Tiepolo 11, 34143 Trieste, Italy\n- 28 SISSA, International School for Advanced Studies, Via Bonomea 265, 34136 Trieste TS, Italy\n- 27 INFN, Sezione di Trieste, Via Valerio 2, 34127 Trieste TS, Italy\n- 29 INAF-Osservatorio Astrofisico di Torino, Via Osservatorio 20, 10025 Pino Torinese (TO), Italy\n- 31 INFN-Sezione di Genova, Via Dodecaneso 33, 16146, Genova, Italy\n- 30 Dipartimento di Fisica, Università di Genova, Via Dodecaneso 33, 16146, Genova, Italy\n- 32 Department of Physics "E. Pancini", University Federico II, Via Cinthia 6, 80126, Napoli, Italy\n- 34 INFN section of Naples, Via Cinthia 6, 80126, Napoli, Italy\n- 33 INAF-Osservatorio Astronomico di Capodimonte, Via Moiariello 16, 80131 Napoli, Italy\n- 35 Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP, Rua das Estrelas, PT4150-762 Porto, Portugal\n- 37 Aix-Marseille Université, CNRS, CNES, LAM, Marseille, France\n- 36 Faculdade de Ciências da Universidade do Porto, Rua do Campo de Alegre, 4150-007 Porto, Portugal\n- 38 Dipartimento di Fisica, Università degli Studi di Torino, Via P. Giuria 1, 10125 Torino, Italy\n- 40 INAF-Osservatorio Astronomico di Roma, Via Frascati 33, 00078 Monteporzio Catone, Italy\n- 39 INFN-Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy\n- 41 INFN-Sezione di Roma, Piazzale Aldo Moro, 2 - c / o Dipartimento di Fisica, Edificio G. Marconi, 00185 Roma, Italy\n- 43 Port d\'Informació Científica, Campus UAB, C. Albareda s / n, 08193 Bellaterra (Barcelona), Spain\n- 42 Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas (CIEMAT), Avenida Complutense 40, 28040 Madrid, Spain \n- 44 Institute for Theoretical Particle Physics and Cosmology (TTK), RWTH Aachen University, 52056 Aachen, Germany\n- 45 Institut d\'Estudis Espacials de Catalunya (IEEC), Edifici RDIT, Campus UPC, 08860 Castelldefels, Barcelona, Spain\n- 46 Dipartimento di Fisica e Astronomia "Augusto Righi" - Alma Mater Studiorum Università di Bologna, Viale Berti Pichat 6 / 2, 40127 Bologna, Italy\n- 47 Instituto de Astrofísica de Canarias, Calle Vía Láctea s / n, 38204, San Cristóbal de La Laguna, Tenerife, Spain\n- 48 European Space Agency / ESRIN, Largo Galileo Galilei 1, 00044 Frascati, Roma, Italy\n- 49 ESAC / ESA, Camino Bajo del Castillo, s / n., Urb. Villafranca del Castillo, 28692 Villanueva de la Cañada, Madrid, Spain\n- 50 Université Claude Bernard Lyon 1, CNRS / IN2P3, IP2I Lyon, UMR 5822, Villeurbanne, F-69100, France\n- 51 Institute of Physics, Laboratory of Astrophysics, Ecole Polytechnique Fédérale de Lausanne (EPFL), Observatoire de Sauverny, 1290 Versoix, Switzerland\n- 52 Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona (IEEC-UB), Martí i Franquès 1, 08028 Barcelona, Spain\n- 53 Institució Catalana de Recerca i Estudis Avançats (ICREA), Passeig de Lluís Companys 23, 08010 Barcelona, Spain\n- 54 UCB Lyon 1, CNRS / IN2P3, IUF, IP2I Lyon, 4 rue Enrico Fermi, 69622 Villeurbanne, France\n- 55 Departamento de Física, Faculdade de Ciências, Universidade de Lisboa, Edifício C8, Campo Grande, PT1749-016 Lisboa, Portugal\n- 56 Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal\n- 57 Department of Astronomy, University of Geneva, ch. d\'Ecogia 16, 1290 Versoix, Switzerland\n- 58 Université Paris-Saclay, CNRS, Institut d\'astrophysique spatiale, 91405, Orsay, France\n- 59 INFN-Padova, Via Marzolo 8, 35131 Padova, Italy\n- 60 INAF-Istituto di Astrofisica e Planetologia Spaziali, via del Fosso del Cavaliere, 100, 00100 Roma, Italy\n- 61 Université Paris-Saclay, Université Paris Cité, CEA, CNRS, AIM, 91191, Gif-sur-Yvette, France\n- 62 FRACTAL S.L.N.E., calle Tulipán 2, Portal 13 1A, 28231, Las Rozas de Madrid, Spain\n- 63 INAF-Osservatorio Astronomico di Padova, Via dell\'Osservatorio 5, 35122 Padova, Italy\n- 64 Max Planck Institute for Extraterrestrial Physics, Giessenbachstr. 1, 85748 Garching, Germany\n- 65 Universitäts-Sternwarte München, Fakultät für Physik, LudwigMaximilians-Universität München, Scheinerstrasse 1, 81679 München, Germany\n- 66 Dipartimento di Fisica "Aldo Pontremoli", Università degli Studi di Milano, Via Celoria 16, 20133 Milano, Italy\n- 67 Felix Hormuth Engineering, Goethestr. 17, 69181 Leimen, Germany\n- 68 Technical University of Denmark, Elektrovej 327, 2800 Kgs. Lyngby, Denmark\n- 69 Cosmic Dawn Center (DAWN), Denmark\n- 70 Université Paris-Saclay, CNRS / IN2P3, IJCLab, 91405 Orsay, France\n- 71 Max-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany\n- 72 NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA\n- 73 Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK\n- 74 Department of Physics and Helsinki Institute of Physics, Gustaf Hällströmin katu 2, 00014 University of Helsinki, Finland\n- 75 Aix-Marseille Université, CNRS / IN2P3, CPPM, Marseille, France\n- 76 Université de Genève, Département de Physique Théorique and Centre for Astroparticle Physics, 24 quai Ernest-Ansermet, CH1211 Genève 4, Switzerland\n- 77 Department of Physics, P.O. Box 64, 00014 University of Helsinki, Finland \n- 78 Helsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, Finland\n- 80 Centre de Calcul de l\'IN2P3 / CNRS, 21 avenue Pierre de Coubertin 69627 Villeurbanne Cedex, France\n- 79 NOVA optical infrared instrumentation group at ASTRON, Oude Hoogeveensedijk 4, 7991PD, Dwingeloo, The Netherlands\n- 81 Universität Bonn, Argelander-Institut für Astronomie, Auf dem Hügel 71, 53121 Bonn, Germany\n- 83 Université Paris Cité, CNRS, Astroparticule et Cosmologie, 75013 Paris, France\n- 82 Dipartimento di Fisica e Astronomia "Augusto Righi" - Alma Mater Studiorum Università di Bologna, via Piero Gobetti 93 / 2, 40129 Bologna, Italy\n- 84 University of Applied Sciences and Arts of Northwestern Switzerland, School of Engineering, 5210 Windisch, Switzerland\n- 86 Institut d\'Astrophysique de Paris, UMR 7095, CNRS, and Sorbonne Université, 98 bis boulevard Arago, 75014 Paris, France\n- 85 Institut d\'Astrophysique de Paris, 98bis Boulevard Arago, 75014, Paris, France\n- 87 Institut de Física d\'Altes Energies (IFAE), The Barcelona Institute of Science and Technology, Campus UAB, 08193 Bellaterra (Barcelona), Spain\n- 89 DARK, Niels Bohr Institute, University of Copenhagen, Jagtvej 155, 2200 Copenhagen, Denmark\n- 88 European Space Agency / ESTEC, Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands\n- 90 Waterloo Centre for Astrophysics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada\n- 92 Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada\n- 91 Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada\n- 93 Space Science Data Center, Italian Space Agency, via del Politecnico snc, 00133 Roma, Italy\n- 95 Institute of Space Science, Str. Atomistilor, nr. 409 M˘agurele, Ilfov, 077125, Romania\n- 94 Centre National d\'Etudes Spatiales - Centre spatial de Toulouse, 18 avenue Edouard Belin, 31401 Toulouse Cedex 9, France\n- 96 Dipartimento di Fisica e Astronomia "G. Galilei", Università di Padova, Via Marzolo 8, 35131 Padova, Italy\n- 98 Universität Innsbruck, Institut für Astro- und Teilchenphysik, Technikerstr. 25 / 8, 6020 Innsbruck, Austria\n- 97 Departamento de Física, FCFM, Universidad de Chile, Blanco Encalada 2008, Santiago, Chile\n- 99 Satlantis, University Science Park, Sede Bld 48940, Leioa-Bilbao, Spain\n- 101 Universidad Politécnica de Cartagena, Departamento de Electrónica y Tecnología de Computadoras, Plaza del Hospital 1, 30202 Cartagena, Spain\n- 100 Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências, Universidade de Lisboa, Tapada da Ajuda, 1349-018 Lisboa, Portugal\n- 102 Kapteyn Astronomical Institute, University of Groningen, PO Box 800, 9700 AV Groningen, The Netherlands\n- 104 Dipartimento di Fisica, Università degli studi di Genova, and INFN-Sezione di Genova, via Dodecaneso 33, 16146, Genova, Italy\n- 103 INFN-Bologna, Via Irnerio 46, 40126 Bologna, Italy\n- 105 Infrared Processing and Analysis Center, California Institute of Technology, Pasadena, CA 91125, USA\n- 107 Astronomical Observatory of the Autonomous Region of the Aosta Valley (OAVdA), Loc. Lignan 39, I-11020, Nus (Aosta Valley), Italy\n- 106 INAF, Istituto di Radioastronomia, Via Piero Gobetti 101, 40129 Bologna, Italy\n- 108 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK\n- 110 Junia, EPA department, 41 Bd Vauban, 59800 Lille, France\n- 109 School of Physics and Astronomy, Cardi ff University, The Parade, Cardi ff , CF24 3AA, UK \n- 111 ICSC - Centro Nazionale di Ricerca in High Performance Computing, Big Data e Quantum Computing, Via Magnanelli 2, Bologna, Italy\n- 113 CERCA / ISO, Department of Physics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106, USA\n- 112 Instituto de Física Teórica UAM-CSIC, Campus de Cantoblanco, 28049 Madrid, Spain\n- 114 INFN-Sezione di Milano, Via Celoria 16, 20133 Milano, Italy\n- 116 Departamento de Astrofísica, Universidad de La Laguna, 38206, La Laguna, Tenerife, Spain\n- 115 Departamento de Física Fundamental. Universidad de Salamanca. Plaza de la Merced s / n. 37008 Salamanca, Spain\n- 117 Dipartimento di Fisica e Scienze della Terra, Università degli Studi di Ferrara, Via Giuseppe Saragat 1, 44122 Ferrara, Italy\n- 119 Center for Data-Driven Discovery, Kavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan\n- 118 Istituto Nazionale di Fisica Nucleare, Sezione di Ferrara, Via Giuseppe Saragat 1, 44122 Ferrara, Italy\n- 120 Ludwig-Maximilians-University, Schellingstrasse 4, 80799 Munich, Germany\n- 122 Dipartimento di Fisica - Sezione di Astronomia, Università di Trieste, Via Tiepolo 11, 34131 Trieste, Italy\n- 121 Max-Planck-Institut für Physik, Boltzmannstr. 8, 85748 Garching, Germany\n- 123 Minnesota Institute for Astrophysics, University of Minnesota, 116 Church St SE, Minneapolis, MN 55455, USA\n- 125 Université Côte d\'Azur, Observatoire de la Côte d\'Azur, CNRS, Laboratoire Lagrange, Bd de l\'Observatoire, CS 34229, 06304 Nice cedex 4, France\n- 124 Institute Lorentz, Leiden University, Niels Bohrweg 2, 2333 CA Leiden, The Netherlands\n- 126 Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu, HI 96822, USA\n- 128 Department of Astronomy & Physics and Institute for Computational Astrophysics, Saint Mary\'s University, 923 Robie Street, Halifax, Nova Scotia, B3H 3C3, Canada\n- 127 Department of Physics & Astronomy, University of California Irvine, Irvine CA 92697, USA\n- 129 Departamento Física Aplicada, Universidad Politécnica de Cartagena, Campus Muralla del Mar, 30202 Cartagena, Murcia, Spain\n- 131 Department of Physics, Oxford University, Keble Road, Oxford OX1 3RH, UK\n- 130 Instituto de Astrofísica de Canarias (IAC); Departamento de Astrofísica, Universidad de La Laguna (ULL), 38200, La Laguna, Tenerife, Spain\n- 132 CEA Saclay, DFR / IRFU, Service d\'Astrophysique, Bat. 709, 91191 Gif-sur-Yvette, France\n- 134 Instituto de Astrofísica de Canarias, c / Via Lactea s / n, La Laguna E-38200, Spain. Departamento de Astrofísica de la Universidad de La Laguna, Avda. Francisco Sanchez, La Laguna, E-38200, Spain\n- 133 Department of Computer Science, Aalto University, PO Box 15400, Espoo, FI-00 076, Finland\n- 135 Ruhr University Bochum, Faculty of Physics and Astronomy, Astronomical Institute (AIRUB), German Centre for Cosmological Lensing (GCCL), 44780 Bochum, Germany\n- 137 Department of Physics and Astronomy, Vesilinnantie 5, 20014 University of Turku, Finland\n- 136 Univ. Grenoble Alpes, CNRS, Grenoble INP, LPSC-IN2P3, 53, Avenue des Martyrs, 38000, Grenoble, France\n- 138 Serco for European Space Agency (ESA), Camino bajo del Castillo, s / n, Urbanizacion Villafranca del Castillo, Villanueva de la Cañada, 28692 Madrid, Spain\n- 140 Centre for Astrophysics & Supercomputing, Swinburne University of Technology, Hawthorn, Victoria 3122, Australia\n- 139 ARC Centre of Excellence for Dark Matter Particle Physics, Melbourne, Australia\n- 141 School of Physics and Astronomy, Queen Mary University of London, Mile End Road, London E1 4NS, UK\n- 142 Department of Physics and Astronomy, University of the Western Cape, Bellville, Cape Town, 7535, South Africa\n- 143 ICTP South American Institute for Fundamental Research, Instituto de Física Teórica, Universidade Estadual Paulista, São Paulo, Brazil\n- 145 Oskar Klein Centre for Cosmoparticle Physics, Department of Physics, Stockholm University, Stockholm, SE-106 91, Sweden\n- 144 IRFU, CEA, Université Paris-Saclay 91191 Gif-sur-Yvette Cedex, France\n- 146 Astrophysics Group, Blackett Laboratory, Imperial College London, London SW7 2AZ, UK\n- 148 Dipartimento di Fisica, Sapienza Università di Roma, Piazzale Aldo Moro 2, 00185 Roma, Italy\n- 147 INAF-Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, 50125, Firenze, Italy\n- 149 Centro de Astrofísica da Universidade do Porto, Rua das Estrelas, 4150-762 Porto, Portugal\n- 151 INFN, Sezione di Roma 2, Via della Ricerca Scientifica 1, Roma, Italy\n- 150 Dipartimento di Fisica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, Roma, Italy\n- 152 HE Space for European Space Agency (ESA), Camino bajo del Castillo, s / n, Urbanizacion Villafranca del Castillo, Villanueva de la Cañada, 28692 Madrid, Spain\n- 154 INAF-Osservatorio Astronomico di Brera, Via Brera 28, 20122 Milano, Italy, and INFN-Sezione di Genova, Via Dodecaneso 33, 16146, Genova, Italy\n- 153 Aurora Technology for European Space Agency (ESA), Camino bajo del Castillo, s / n, Urbanizacion Villafranca del Castillo, Villanueva de la Cañada, 28692 Madrid, Spain\n- 155 Theoretical astrophysics, Department of Physics and Astronomy, Uppsala University, Box 515, 751 20 Uppsala, Sweden\n- 157 Mullard Space Science Laboratory, University College London, Holmbury St Mary, Dorking, Surrey RH5 6NT, UK\n- 156 Department of Physics, Royal Holloway, University of London, TW20 0EX, UK\n- 158 Department of Astrophysical Sciences, Peyton Hall, Princeton University, Princeton, NJ 08544, USA\n- 160 Niels Bohr Institute, University of Copenhagen, Jagtvej 128, 2200 Copenhagen, Denmark\n- 159 Cosmic Dawn Center (DAWN)\n- 161 Center for Cosmology and Particle Physics, Department of Physics, New York University, New York, NY 10003, USA\n- 162 Center for Computational Astrophysics, Flatiron Institute, 162 5th Avenue, 10010, New York, NY, USA \nFig. A.1: Comparison of the computational costs of simulations run under di ff erent gravity models (labels on the x -axis) and with di ff erent codes (markers and colours as described in the legend). \n<!-- image -->', 'Appendix A: Computational cost': 'One of the obstacles in producing accurate predictions in the nonlinear regime of structure formation is presented by the computational cost of running large N -body simulations. This issue becomes even more pronounced in the case of MG simulations, due to the additional computational cost associated with the solution of the Klein-Gordon equation for the scalar field describing the additional degree of freedom in these theories. To provide some insights into the trade-o ff s between accuracy and time-tosolution, we attempt here a comparative analysis of the computational cost of the simulations run for this paper. Given that these simulations were run on di ff erent machines and with di ff erent parallelisation settings, we cannot conduct a precise assessment of their computational cost. Instead, we limit this analysis to an order-of-magnitude comparison of the simulations run with the various codes and models discussed in this paper. \nFor this exercise, we use the information on the wall-clock time, T real, and the number of cores, N cores, as recorded in the log files of the simulation runs. This information was available for all simulations except for the ones run with ISIS . We estimate the computational cost of the simulations, C , as the product of the wall-clock time and the number of cores: \nC ≡ N cores T real . (A.1) \nThe performance of multigrid codes depends on the convergence criterion chosen. For PySCo , we chose an extremely con- \nThe estimates of the computational cost for the simulations are compared in Fig. A.1. We can see that the cost of Λ CDM simulations of Tree-PM and AMR codes is C ∼ 10 3 CPUh, while for MG-COLA and PySCo the cost is about one order of magnitude lower at C ∼ 10 2 CPUh. For f ( R ) gravity models instead, the computational cost increases significantly (approximately by a factor of ten) for the codes that solve the full Klein-Gordon equation of the scalar field, namely MG-Arepo , MG-Gadget , ECOSMOG , and PySCo , while the overhead is smaller for the codes that adopt screening approximations, namely PANDA and MG-COLA . Finally, in nDGP gravity, only ECOSMOG has a significant overhead compared to Λ CDM, while MG-Arepo and the approximate codes, PANDA and MG-COLA , have just a small overhead. \nservative approach (see Sect. 3.1.5) with a very low tolerance threshold, resulting in more V-cycles and almost double the CPU time needed to solve the linear Poisson equation compared to a more standard setup. Regarding the nonlinear solver, we use two F-cycles instead of a single one (which should in principle be enough, but it is not the goal of the present paper to provide a convergence study), therefore roughly doubling the CPU time needed for the f ( R ) gravity models. \nWe stress that a thorough assessment of the e ffi ciency of the codes is beyond the scope of this paper and would have required a much more methodical e ff ort including (but not limited to) \n- -running the simulations in a controlled environment,\n- -testing the scaling performance of each code.\n- -conducting convergence tests for the various hyperparameters, \nIn fact, when focusing only on predictions of the amplification factors, it is possible to achieve a similar level of accuracy with lower force, mass or time resolution, since resolution effects mostly cancel out when taking ratios of quantities a ff ected by the same inaccuracies (Brando et al. 2022). This has been shown to be the case for MG-COLA simulations in Fiorini et al. (2023), where the use of a lower resolution allowed accurate predictions of power spectrum boosts in nDGP gravity with a theoretical gain of ∼ 300 with respect to the computational cost of the COLA simulations presented here. \nSuch large speed-ups have paved the way for creating emulators for the nonlinear amplification of the power spectrum in models beyond Λ CDM in a cost-e ff ective way, i.e. without the need for supercomputers. This has already been done for some of the models we consider in this paper (see e.g. Ramachandra et al. 2021; Mauland et al. 2024; Fiorini et al. 2023). For instance, Fiorini et al. (2023) found that an emulator for the nDGP model can be constructed with as little as a few thousand CPUh worth of computational time. Likewise, Mauland et al. (2024) who presented a generic pipeline for using COLA to create such emulators, used f ( R ) gravity as an example and found similar numbers for the required computational time. Gordon et al. (2024) described a simulation setup that can also be used for emulating the full power spectrum (up to a reasonable high wavenumber k ∼ 1 h Mpc -1 ), requiring around ∼ 100 CPUh per simulation on a modern CPU. Emulators have also been constructed for beyondΛ CDM models using high-resolution direct simulations in the same way as has been done for Λ CDM. This approach generally gives more accurate emulators than those created with approximate methods, but this comes at a much higher cost. For example, both Sáez-Casares et al. (2023) and Arnold et al. (2022) have each presented a high-fidelity emulator for the f ( R ) model considered in this paper, but at a higher cost of about 3.5-4 million CPUh.'} |
2024arXiv240803920P | We present the design and observations of low resolution JWSTNIRSpec PRISM spectroscopy from the Ultradeep NIRSpec and NIRCam ObserVations before the Epoch of Reionization UNCOVER Cycle 1 JWST Treasury program. Targets are selected using JWSTNIRCam photometry from UNCOVER and other programs and cover a wide range of categories and redshifts to ensure the legacy value of the survey. These categories include the first galaxies at zgtrsim10 faint galaxies during the Epoch of Reionization zsim68 high redshift AGN zgtrsim6 Population III star candidates distant quiescent and dusty galaxies 1lesssim z lesssim 6 and filler galaxies sampling redshiftcolormagnitude space from zsim 0.113. Seven NIRSpec MSA masks across the extended Abell 2744 cluster were observed along with NIRCam parallel imaging in 8 filters F090W F115W F150W F200W F277W F356W F410M F444W F480M over a total area of 26 arcmin2 overlapping existing HST coverage from programs including the Hubble Frontier Fields and BUFFALO. We successfully observed 553 objects down to mmathrmF444Wsim30mathrmAB and by leveraging mask overlaps we reach total ontarget exposure times ranging from 2.416.7h. We demonstrate the success rate and distribution of confirmed redshifts and also highlight the rich information revealed by these ultradeep spectra for a subset of our targets. An updated lens model of Abell 2744 is also presented including 14 additional spectroscopic redshifts and finding a total cluster mass of MmathrmSL2.1pm0.3times1015mathrmModot. We publicly release reduced 1D and 2D spectra for all objects observed in Summer 2023 along with a spectroscopic redshift catalog and the updated lens model of the cluster httpsjwstuncover.github.ioDR4.html. | 2024-08-01T00:00:00Z | ['2024arXiv240803920P', '10.48550/arXiv.2408.03920', 'arXiv:2408.03920'] | ['Astrophysics - Astrophysics of Galaxies'] | The UNCOVER Survey First Release of Ultradeep JWSTNIRSpec PRISM spectra for 700 galaxies from z0.313 in Abell 2744 | 2,024 | 173 | 0.65 | ['EPRINT_HTML', 'EPRINT_PDF'] | 21 | https://arxiv.org/pdf/2408.03920.pdf | {'The UNCOVER Survey: First Release of Ultradeep JWST/NIRSpec PRISM spectra for ∼ 700 galaxies from z ∼ 0 . 3 -13 in Abell 2744': "Sedona H. Price, 1 Rachel Bezanson, 1 Ivo Labbe, 2 Lukas J. Furtak, 3 Anna de Graaff, 4 Jenny E. Greene, 5 Vasily Kokorev, 6 David J. Setton, 5, ∗ Katherine A. Suess, 7, † Gabriel Brammer, 8 Sam E. Cutler, 9 Joel Leja, 10, 11, 12 Richard Pan, 13 Bingjie Wang ( 王 冰 洁 ), 10, 11, 12 John R. Weaver, 9 Katherine E. Whitaker, 9, 14 Hakim Atek, 15 Adam J. Burgasser, 16 Iryna Chemerynska, 15 Pratika Dayal, 17 Robert Feldmann, 18 Natascha M. Forster Schreiber, 19 Yoshinobu Fudamoto, 20 Seiji Fujimoto, 6, † Karl Glazebrook, 21 Andy D. Goulding, 5 Gourav Khullar, 1 Mariska Kriek, 22 Danilo Marchesini, 13 Michael V. Maseda, 23 Tim B. Miller, 24 Adam Muzzin, 25 Themiya Nanayakkara, 21 Erica Nelson, 26 Pascal A. Oesch, 27, 8 Heath Shipley, 28 Renske Smit, 29 Edward N. Taylor, 2 Pieter van Dokkum, 30 Christina C. Williams, 31 and Adi Zitrin 3 \n1 Department of Physics and Astronomy and PITT PACC, University of Pittsburgh, Pittsburgh, PA 15260, USA 2 Centre for Astrophysics and Supercomputing, Swinburne University of Technology, Melbourne, VIC 3122, Australia 3 Physics Department, Ben-Gurion University of the Negev, P.O. Box 653, Be'er-Sheva 84105, Israel 4 Max-Planck-Institut fur Astronomie, Konigstuhl 17, D-69117, Heidelberg, Germany 5 Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08544, USA 6 Department of Astronomy, The University of Texas at Austin, Austin, TX 78712, USA 7 Kavli Institute for Particle Astrophysics and Cosmology and Department of Physics, Stanford University, Stanford, CA 94305, USA 8 Cosmic Dawn Center (DAWN), Niels Bohr Institute, University of Copenhagen, Jagtvej 128, København N, DK-2200, Denmark 9 Department of Astronomy, University of Massachusetts, Amherst, MA 01003, USA 10 Department of Astronomy & Astrophysics, The Pennsylvania State University, University Park, PA 16802, USA 11 Institute for Computational & Data Sciences, The Pennsylvania State University, University Park, PA 16802, USA 12 Institute for Gravitation and the Cosmos, The Pennsylvania State University, University Park, PA 16802, USA 13 Department of Physics and Astronomy, Tufts University, 574 Boston Ave., Medford, MA 02155, USA 14 Cosmic Dawn Center (DAWN), Denmark 15 Institut d'Astrophysique de Paris, CNRS, Sorbonne Universit'e, 98bis Boulevard Arago, 75014, Paris, France 16 Department of Astronomy & Astrophysics, UC San Diego, La Jolla, CA, USA 17 Kapteyn Astronomical Institute, University of Groningen, 9700 AV Groningen, The Netherlands 18 Department of Astrophysics, University of Zurich, CH-8057, Switzerland 19 Max-Planck-Institut fur extraterrestrische Physik, Giessenbachstrasse 1, 85748 Garching, Germany 20 Center for Frontier Science, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522, Japan 21 Centre for Astrophysics and Supercomputing, Swinburne University of Technology, PO Box 218, Hawthorn, VIC 3122, Australia 22 Leiden Observatory, Leiden University, P.O.Box 9513, NL-2300 AA Leiden, The Netherlands 23 Department of Astronomy, University of Wisconsin-Madison, 475 N. Charter St., Madison, WI 53706, USA 24 Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA), Northwestern University,1800 Sherman Ave, Evanston, IL 60201, USA 25 Department of Physics and Astronomy, York University, 4700 Keele Street, Toronto, Ontario, ON MJ3 1P3, Canada 26 Department for Astrophysical and Planetary Science, University of Colorado, Boulder, CO 80309, USA 27 Department of Astronomy, University of Geneva, Chemin Pegasi 51, 1290 Versoix, Switzerland 28 Department of Physics, Texas State University, San Marcos, TX 78666, USA 29 Astrophysics Research Institute, Liverpool John Moores University, 146 Brownlow Hill, Liverpool L3 5RF, UK 30 Astronomy Department, Yale University, 219 Prospect St, New Haven, CT 06511, USA 31 NSF's National Optical-Infrared Astronomy Research Laboratory, 950 North Cherry Avenue, Tucson, AZ 85719, USA", 'ABSTRACT': 'We present the design and observations of low resolution JWST/NIRSpec PRISM spectroscopy from the Ultradeep NIRSpec and NIRCam ObserVations before the Epoch of Reionization (UNCOVER) Cycle 1 JWST Treasury program. Targets are selected using JWST/NIRCam photometry from UNCOVER and other programs, and cover a wide range of categories and redshifts to ensure the legacy value of the survey. These categories include the first galaxies at z ≳ 10, faint galaxies during the Epoch of Reionization ( z ∼ 6 -8), high redshift AGN ( z ≳ 6), Population III star candidates, distant quiescent and dusty galaxies (1 ≲ z ≲ 6), and filler galaxies sampling redshift-color-magnitude space \nfrom z ∼ 0 . 1 -13. Seven NIRSpec MSA masks across the extended Abell 2744 cluster were observed, along with NIRCam parallel imaging in 8 filters (F090W, F115W, F150W, F200W, F277W, F356W, F410M, F444W, F480M) over a total area of ∼ 26 arcmin 2 , overlapping existing HST coverage from programs including the Hubble Frontier Fields and BUFFALO. We successfully observed 553 objects down to m F444W ∼ 30AB, and by leveraging mask overlaps, we reach total on-target exposure times ranging from 2 . 4 -16 . 7h. We demonstrate the success rate and distribution of confirmed redshifts, and also highlight the rich information revealed by these ultradeep spectra for a subset of our targets. An updated lens model of Abell 2744 is also presented, including 14 additional spectroscopic redshifts and finding a total cluster mass of M SL = (2 . 1 ± 0 . 3) × 10 15 M ⊙ . We publicly release reduced 1D and 2D spectra for all objects observed in Summer 2023 along with a spectroscopic redshift catalog and the updated lens model of the cluster (https://jwst-uncover.github.io/DR4.html). \nKeywords: Galaxy evolution (594) - Galaxy formation (595) - High-redshift galaxies (734)', '1. INTRODUCTION': "Deep JWST imaging from early programs has already begun to revolutionize our understanding of the faint, distant universe. The observatory has met or exceeded nearly every pre-flight expectation (Rieke et al. 2023), and early data has enabled us to find and begin characterizing many galaxy populations that were previously inaccessible: from the first generation of galaxies at Cosmic Dawn (e.g., Naidu et al. 2022, Atek et al. 2023, Finkelstein et al. 2023, Robertson et al. 2023, 2024, Casey et al. 2024), to the faint galaxies driving the reionization of the universe at z ∼ 6 -9 (e.g., P'erez-Gonz'alez et al. 2023), to early quiescent galaxies at z ∼ 3 -5 (e.g., Carnall et al. 2023a, Valentino et al. 2023). JWST imaging also provides new insights into galaxies' detailed structures (at z ≲ 6; e.g., Ferreira et al. 2022, 2023, Kartaltepe et al. 2023, Martorano et al. 2023, Nelson et al. 2023, van der Wel et al. 2024, among many others), including reaching low stellar masses approaching those of the dwarf galaxy population ( M ∗ ∼ 10 6 M ⊙ , e.g., Suess et al. 2023) and revealing the structures of heavily dustobscured galaxies which were previously observable only in the sub-millimeter (e.g., Kokorev et al. 2023, Price et al. 2023, Wu et al. 2023). Ultradeep JWST imaging has additionally enabled detections of possible globular clusters as early as z ∼ 1 . 4 (e.g., Mowla et al. 2022, Claeyssens et al. 2023, Forbes & Romanowsky 2023), as well as more detailed studies of globular clusters within galaxies out to at least z ∼ 0 . 3 (e.g., Harris & ReinaCampos 2023, 2024). Early JWST imaging has also yielded surprises, including larger than anticipated numbers of very luminous early galaxies (e.g., Naidu et al. 2022, Atek et al. 2023, Austin et al. 2023, Bradley et al. \n2023, Finkelstein et al. 2023, Adams et al. 2024, Casey et al. 2024, Chemerynska et al. 2024a, Robertson et al. 2024) and an unexpected, relatively numerous population of obscured active galactic nuclei (AGN) candidates at high redshift (e.g., Labbe et al. 2023, Furtak et al. 2023a, Barro et al. 2024, Kokorev et al. 2024, Williams et al. 2024). \nTaking the next step in exploring these newly uncovered parameter spaces requires leveraging JWST's spectroscopic capabilities to both confirm galaxies' redshifts and to probe their internal physical properties in detail. Even with the high sensitivity of JWST/NIRSpec (Boker et al. 2023), pushing to the most distant and faint regimes is best accomplished with very deep observations in cluster fields, where the strong gravitational lensing boost reaches intrinsically fainter populations by 1 -2 magnitudes relative to blank fields. Complementing the aforementioned imaging results, spectra from early JWST programs have already revealed new discoveries and unprecedented measurements. Results from this early spectroscopy include confirming the redshifts and properties of galaxies at z ≳ 9 (e.g., Arrabal Haro et al. 2023a,b, Curtis-Lake et al. 2023, Roberts-Borsani et al. 2023), and confirming and characterizing high-redshift obscured AGN (e.g., Harikane et al. 2023, Maiolino et al. 2023, Matthee et al. 2024) as well as quiescent galaxies at z ≳ 3 (e.g., Carnall et al. 2023b, de Graaff et al. 2024a, Glazebrook et al. 2024, Carnall et al. 2024). \nThe Ultradeep NIRSpec and NIRCam ObserVations before the Epoch of Reionization (UNCOVER) Cycle 1 Treasury survey (Bezanson et al. 2022) was designed to collect these deep spectra early in the JWST mission. UNCOVER was designed to obtain ultradeep, multiband NIRCam imaging, photometrically detect and characterize galaxies down to mag F444W ∼ 30 AB (Weaver et al. 2024), and then select targets from these newly-observable populations for follow up ultra- \nep NIRSpec/PRISM multi-object spectroscopy (Ferruit et al. 2022). The low resolution PRISM mode provides both high sensitivity and wide spectral coverage (i.e., Boker et al. 2023), enabling us to constrain continuum breaks down to ∼ 29AB and measure rest-frame ultraviolet (UV) to near infrared (NIR) emission and absorption features raging from galaxies within the cluster itself at z ∼ 0 . 3 out to the earliest epochs at z ≳ 10. Early UNCOVER spectroscopic results already address many of these aims, including finding objects among the first generation of galaxies (e.g., Wang et al. 2023a), characterizing distant obscured AGN (e.g., Greene et al. 2024), and uncovering early quiescent galaxy formation (e.g., Setton et al. 2024). \nIn this paper we present an overview of the UNCOVER NIRSpec/PRISM spectroscopic observations of 668 targets in the Abell 2744 strong lensing cluster field, as well as our coordinated parallel NIRCam imaging which overlaps with existing HST observations from the Hubble Frontier Fields (HFF; Lotz et al. 2017) and BUFFALO (Steinhardt et al. 2020) programs. We detail the target selection and mask design and the observations (Sec. 2), and the spectroscopic reduction and redshift measurements (Sec. 3). We also present the redshift success rate and distribution of measured redshifts, and discuss example cases of spectra addressing the scientific objectives of the UNCOVER survey (Sec. 4). This paper accompanies the public release of early reduced NIRSpec/PRISM spectra, spectroscopic redshifts, and the NIRCam parallel imaging. All magnitudes given are in the AB system (Oke 1974).", '2.1. Target Selection': "Targets are primarily selected from photometric catalogs constructed from all publicly available HST and JWST imaging over Abell 2744 as of June 2023. The JWST/NIRCam observations are: UNCOVER (PIs Labbe & Bezanson, JWST-GO-2561; Bezanson et al. 2022), the Early Release Science program GLASS (PI: Treu, JWST-ERS-1324; Treu et al. 2022), and a Director's Discretionary program (PI: Chen, JWST-DD2756), providing a total of 8 filters: F090W, F115W, F150W, F200W, F277W, F356W, F410M, and F444W. The archival HST data consists of HST-GO-11689 (PI: Dupke), HST-GO-13386 (PI: Rodney), HST-DD-13495 (PI: Lotz; Lotz et al. 2017), and HST-GO-15117 (PI: Steinhardt; Steinhardt et al. 2020), providing coverage in 7 filters: F435W, F606W, F814W, F105W, F125W, F140W, and F160W. The majority of the targets are selected from the UNCOVER NIRCam-selected catalog (as presented in Weaver et al. 2024), using inter- \nnal version v2.2.0. This version, containing ∼ 50 , 000 objects down to a combined long-wavelength (LW; F277W+F356W+F444W) depth of ∼ 30 . 5AB in the deepest regions, included improved treatment of PSFhomogenization and estimates of total magnitudes compared to the initial public DR1 (January 2023). 1 While selecting targets, UNCOVER stellar population modeling including Prospectorβ and eazy were considered (as in Weaver et al. 2024, Wang et al. 2024). However, the default UNCOVER catalogs excluded a small number of interesting sources, e.g., highly lensed, multiply imaged and/or shredded objects. In these cases, targets were added by hand (with target IDs > 60000). Furthermore, a subset of the targets were selected based on information from other wavelengths, including ALMA sub-mm/mm (DUALZ, PI: Fujimoto, Fujimoto et al. 2023a; ALCS, PI: Kohno, Fujimoto et al. 2023b; ALMA Frontier Fields, PI: Bauer; Mu˜noz Arancibia et al. 2023) and Chandra X-ray (e.g., Bogd'an et al. 2024) observations. \nFor target selection, the updated version of the Furtak et al. (2023b) analytic lens model of Abell 2744 was used ( v1.1 ). 2 This version includes one additional multiple image system in the northern sub-structure (system 82), and more importantly, an additional spectroscopic redshift in the north-western sub-structure from new VLT/MUSE observations of the cluster (system 68 at z = 2 . 584, Bergamini et al. 2023a; see also Appendix B.2). The v1.1 lens model achieved a lens plane average image reproduction root-mean-square (RMS) of ∆ RMS = 0 . 51 '' . \nAs the UNCOVER science goals cover a wide range of topics, including potentially risky unknown-unknowns, the final spectroscopic targeting is complex. The prioritization scheme for assigning targets to masks is as follows. Categories corresponding to the originally proposed science cases (see Bezanson et al. 2022) are roughly prioritized corresponding to rarity and scientific value: 1) any z> 12 candidates, 2) z> 9 galaxies prioritized by brightness, 3) Pop III candidate sources, \nFigure 1. UNCOVER NIRSpec MSA mask footprints within the Abell 2744 cluster field. Shaded regions denote the regions of magnification µ > 2 , 10 , 100 (grayscale, light to dark) from the updated UNCOVER lensing maps ( v2.0 ) for a source at redshift z s = 8, and existing NIRCam coverage (from Cycle 1 imaging) is shown with the black outline. The masks, shown with colored outlines, span most of the imaging footprint over a range of low- and high-magnification regions. The electrical short-impacted Mask 1 is marked with a dotted outline. (Note Masks 5-7 have near-complete overlap.) \n<!-- image --> \nRA \n4) faint highly magnified 6 <z< 7 galaxies, 3 5) z> 4 quiescent galaxies, 6) z> 6 AGN, 7) z> 4 dusty galaxies, and other galaxies with ALMA detections (e.g., Fujimoto et al. 2023a), 8) low mass quiescent galaxies at 1 <z< 6, 9) any unusual or unexpected sources, 10) extreme emission line galaxies, and finally 11) mass-selected 'filler' galaxies sampled in bins of redshift, mass, and F150W -LW color (using the LW noise equalized-F277W+F356W+F444W image, and eazy -derived mass and redshifts). For these filler targets, the numerical priority class n was set to be proportional to the log 2 inverse of the cumulative surface density in each property. As the mask design software eMPT (Bonaventura et al. 2023) maps priority class n to weight according to a 1 / 2 n weighting scheme, this approximately equates to an importance sampling scheme that is flat in color, magnitude, and redshift (i.e., sparsely sam- \nTable 1. NIRSpec MSA Masks \nNote -The sample includes 668 unique targets, with some targets on multiple masks. \npling regions of parameter space with many objects, and densely sampling where objects are less common).", '2.2. Mask Designs & Observations': 'The NIRSpec/PRISM observations are split into 7 microshutter array (MSA) mask configurations, with permask exposure times of 2.6-4.4h (see Table 1). As shown in Figure 1, these masks cover the UNCOVER NIRCam primary footprint, with overlaps allowing for repeated observations of faint, high-priority targets. The masks were designed iteratively using eMPT (Bonaventura et al. 2023), designing each mask in sequence according to target priority, then modifying the priorities to ensure targets requiring deeper integrations are placed on additional masks until the required exposure time is met. This procedure was repeated using hand-specified mask positions until an optimal design (in terms of both number of highest priority targets and total number of targets) was reached. In total, 668 unique targets are assigned to masks, with total planned exposure times ranging from 2.6 to 17.4 hours. \nThe NIRSpec observations were taken on 31 July - 2 August 2023, with a 2-POINT-WITH-NIRCam-SIZE2 dither pattern and a 3 shutter slitlet nod pattern. The NIRSpec NRSIRS2RAPID and NRSIRS2 readout patterns were adopted for Masks 1-3 and 4-7, respectively. Coordinated parallel NIRCam imaging was also taken (as described in Appendix A). The observations were taken with a V3PA angle ∼ 266 or NIRSpec MSA aperture PA ∼ 44 . 56 (see exact values in Table 1), to ensure \nefficient MSA coverage over the UNCOVER NIRCam footprint and to overlap the parallel NIRCam imaging with existing HST/ACS and WFC3 observations from the HFF (Lotz et al. 2017) and BUFFALO (Steinhardt et al. 2020) programs. \nAn electrical short early in Visit 1 severely impacted both detectors, with complete loss for most sources and severely reduced data quality in a minority of objects; repeat observations of a slightly modified Mask 1 (due to small differences in PA) were approved, and were observed on 30-31 July 2024. Additionally, a solid state recorder (SSR) drive exception (relating to drive space) impacted the Visit 3 observations, leading to a loss of 7% of the NIRSpec integration time in Mask 3 (1 frame each for both detectors; yielding a total exposure of 2.4h) as well as 66% of the NIRCam parallel imaging (all of F150W, F200W, F356W, F444W). Repeat observations of the NIRCam parallel for Visit 3 (in all 6 filters, given a probable observing PA change) were also approved, and observed on 31 July 2024. All repeat observations will be included in a future release. Given these setbacks, and a small percentage of failed reduction/extractions or other data quality issues, here we present robust spectra for 553 objects, with exposure times of 2.4-16.7 hours.', '3.1. Spectroscopic reduction & 1D extraction': "The PRISM spectra are reduced using msaexp (v0.8.5; Brammer 2023a), grizli (v1.11.9; Brammer 2023b), and the JWST jwst pipeline (v1.14.0; Bushouse et al. 2024) using the jwst 1241.pmap reference files. Level 1 products are downloaded from MAST 4 , and then msaexp (using grizli ) runs the jwst stage 1 pipeline, inserting the snowblind 5 (Davies 2024) improved 'snowball' identification and correction procedure after the Jump step. msaexp next applies a 1 /f correction, and, finally, a median pedestal bias offset of the science data ( sci extension) and multiplicative scaling factor to the read noise array ( rnoise extension) are calculated from empty parts of each exposure that should not have any contribution from sky or source photons. Further steps of the jwst stage 2 pipeline are then run to assign the world coordinate system (WCS), flag open microshutters, identify and extract 2D slits, apply slit-level flat-fielding, correct for vignetting of the MSA bars, and apply the photometric calibration. \nFor this first spectroscopic data release, local background subtraction is performed by taking differences of the 2D spectrum arrays at the different telescope nod positions. 6 This local background subtraction is performed on the original 2D slitlet cutouts before performing drizzle resampling. msaexp then rectifies the 2D spectra from each exposure and resamples them into a final stack with an algorithm analogous to drizzle (Fruchter & Hook 2002), adopting a pixel fraction and wavelength sampling of 1.0. In contrast to the STScI jwst drizzle resampling algorithm, the spectra here are only rectified along the columns of the cross-dispersion axis and all wavelength bins are kept fully independent, which eliminates the correlated noise in the dispersion direction that results from a full 2D drizzle resampling. \nThe final 1D spectra are then extracted from the local background-subtracted 2D spectra using an optimal extraction (Horne 1986) scheme, modified to account for the variable spatial resolution across the full PRISM wavelength range. A 2D Gaussian profile for this optimal extraction is fit to the curved traces of the original spectral cutouts with parameters for both the profile width and a spatial offset relative to the position expected from the mask and input catalog metadata. The 2D profile is rebinned and rectified in the same way as the science data, and the optimally-weighted extraction is performed in the rectified frame. Path-loss corrections computed by msaexp are included in the final spectra using the predicted intra-shutter position and assuming an axisymmetric Gaussian shape with the width determined from the fit to the cross-dispersion profile described above. \nWe note that some objects are observed on multiple masks. In the current reduction, all frames of a target are directly combined during the reduction, implicitly assuming that the slitlets of different masks cover the same spatial region of that source. We also note that we have not attempted to apply any aperture corrections (beyond the path-loss correction described above); in some cases, it may be beneficial for users to derive a wavelength-dependent aperture correction when jointly modeling photometry and spectroscopy. Finally, we note that for this first release, spectra from the shortimpacted Mask 1 are not reduced. The spectra from the repeat observation of Mask 1, and the spectra taken along with the repeat of the Visit 3 NIRCam parallel imaging, will be included in future spectroscopic releases. \nFigure 2. Total magnification map of our new v2.0 SL model of Abell 2744 for a source at redshift z s = 10. The black contours represent magnification thresholds of µ = 2 and µ = 4. \n<!-- image --> \n| \n|", '3.2. Spectroscopic redshifts and line fluxes': 'The spectroscopic redshifts for this data release are determined from the reduced, full-depth 1D spectra using msaexp . First, redshift fits are performed using the eazy (Brammer et al. 2008) corr sfhz 13 galaxy template set with a wide allowed redshift range ( z = [0 . 05 , 14]). A second redshift fit is then performed with a library of spectral lines and cubic splines for a flexible continuum model, restricted within ± 0 . 03(1 + z ) of the template best-fit redshift (or within the range z = [0 . 05 , 14] if the template fit failed). Models for the emission lines are generated in msaexp as pixel-integrated Gaussians with widths taken from the wavelength-dependent spectral resolution curve provided by STScI and used by the JWST exposure time calculator ( R ∼ 50 at 1.5 µ m, R ∼ 300 at 5 µ m; 7 jwst nirspec prism disp.fits). The prism disperser does not spectrally resolve typical galaxy emission lines, though extremely broad emission (e.g., due to broad-line AGN or outflows) can be resolved. \nThe spectroscopic redshift for each object is determined as follows: (1) from the template fit, for objects with only continuum features based on visual inspection (i.e., only breaks or stellar bumps and no emission lines); or else (2) from the lines+splines fit, if at least one emission line is detected with signal-to-noise S / N ≥ 3 in that fit (and the target was not flagged as only having \ncontinuum features in visual inspection); or finally (3) from the template fit, from the template fit, if no line is detected. The redshift uncertainties for all targets are taken from the 16, 84 th percentiles of the full redshift range template fit (or from the lines+splines fit, if the template fit failed). \nThe redshift fits are examined by multiple (minimum 3) team members, and flagged based on the number and robustness of the detected spectral features, as described in Table 2. The redshift quality flag, flag zspec qual , denotes secure redshifts (= 3; from two or more secure spectral features, e.g., two robustly-detected emission lines, one clear break and one robust emission line, two robustly-detected absorption features), solid redshifts (= 2; from one broad continuum feature, either a break or stellar bump, or from two less robust features, e.g., two marginally-detected emission lines or one marginally detected emission line and a break), tentative but unreliable redshifts (= 1), and no redshift solution (= 0). A flag flag successful spectrum is also included, indicating whether the target spectrum was successfully observed and reduced (= 1) or not (= 0; due to data quality issues or missing spectra). \nIn select cases identified during the visual fit inspection (14 objects; 2.5%), the redshifts are manually refit with alternative settings (i.e., multiple robust emission lines where the initial template fits yielded inaccurate redshift estimates; noise misidentified as lines when the redshifts are more robustly measured from template fits to continuum breaks) or are fixed (the 3 brown dwarfs at z spec = 0; see Sec. 4.2). The redshift quality flag is updated based on these modified redshift solutions. \nIn addition to spectroscopic redshifts, we also determine line fluxes from the msaexp fits for each object. We adopt the values from the same fit as the best-fit redshift (described above). The reported line fluxes are not corrected for lensing magnification. \nAccompanying this paper, we publicly release reduced spectra and spectroscopic redshifts from the UNCOVER NIRSpec/MSA observations taken in Summer 2023. 8 This data release (UNCOVER DR4) includes the 1D optimally extracted spectra and the 2D spectra with local background subtraction, for all successfully reduced spectra. The redshift catalog for this release (described in Table 2 and the downloadable machine readable format version) includes the measured redshifts (if any), redshift and spectra quality flags, the total exposure time, and the assigned masks for the full set of targeted \nFigure 3. Redshift distribution of spectroscopically confirmed galaxies with robust redshifts ( flag zspec qual ≥ 2). Panel a: Redshift histogram, split by redshift quality flag. Panel b: Spectroscopic versus photometric redshifts, using Prospectorβ -derived z phot from the internal v2.2.0 catalog (used during MSA target selection, including only HST and JWST/NIRCam broad-band filters), with the uncertainties showing the 16, 84 th percentiles. Catastrophic outliers (with | ∆ z | = ( z phot -z spec ) / (1+ z spec ) > 0 . 15; boundary denoted with dotted lines) are colored red. \n<!-- image --> \nobjects. Subsequent spectroscopic releases will include the observations from the repeated Visits 1 & 3 and both global- and local-background subtracted spectra (optimized for extended and point sources, respectively) and an updated redshift catalog. \nReduced mosaics of the NIRCam parallel observations are also available, constructed following the same procedures as the cluster NIRCam observations (except that modeling and subtraction of bright cluster galaxy and intracluster light is not performed). Full details about the parallel mosaics are presented in the UNCOVER survey paper (Bezanson et al. 2022).', '3.3. Updating the UNCOVER lens model': "We also use the UNCOVER spectroscopy to update the lens model of Abell 2744 presented in Furtak et al. (2023b) and include the new v2.0 model in the data release. As described in detail in Appendix B, this model incorporates the UNCOVER DR4 spectroscopic redshifts of multiple images, and all currently available JWST imaging for cluster member selection. In total, we added 14 spectroscopic redshifts compared to our initial v1.0 model. The model is constructed with an updated version of the analytic lens modeling method by Zitrin et al. (2015). We refer the reader to Appendix B and Furtak et al. (2023b) for details of the parameterization for our lensing model of Abell 2744. \nWith these constraints, the model achieves an average image reproduction error of ∆ RMS = 0 . 60 '' , which \nis slightly better than our v1.0 model (∆ RMS = 0 . 66 '' Furtak et al. 2023b). The critical lines and multiple image positions are shown in Figure 9 in Appendix B and we show an updated magnification map at source redshift z s = 10 in Figure 2. We find the cluster to have a total critical area of A crit = 0 . 63 arcmin 2 for a source at z s = 2. This translates to an effective Einstein radius of θ E = 26 . 9 '' ± 2 . 7 '' enclosing a mass of M ( < θ E ) = (1 . 0 ± 0 . 2) × 10 14 M ⊙ . These also agree well with our measurements from our v1.0 model (Furtak et al. 2023b). Summing the surface mass density over the entire field (see Figure 9), we obtain a total cluster mass of M SL = (2 . 1 ± 0 . 3) × 10 15 M ⊙ . This is comparable to an M 200 mass and thus places Abell 2744 well within the mass range of typical clusters with the same Einstein radius (e.g. Fox et al. 2022). \nThe v2.0 lens model is included in the UNCOVER DR4. The public lensing products include deflection α , convergence κ , shear γ , magnification µ and potential ψ maps, normalized to D ds /D s = 1, as well as catalogs of the cluster member galaxies and multiple images used. The JWST cluster member selection and spectroscopic redshifts of multiple images are further detailed in Appendices B.1 and B.2 respectively. We also updated the UNCOVER photometric and spectroscopic catalogs with magnification and shear parameters from the v2.0 model. Individual models of each of the three sub-structures separately are also available on re- \nFigure 4a shows that successfully-observed spectroscopic targets with and without robust redshifts ( flag zspec qual ≥ and < 2, purple filled and gray open circles, respectively) have similar F277W-F444W colors. On average, targets without robust redshifts are fainter in F444W than spectroscopically-confirmed objects, though both have overlapping distributions down \n<!-- image --> \n<!-- image --> \na \nm \ng \na \nm \nFigure 4. Distribution of the spectroscopic sample relative to the full UNCOVER photometric catalog ( left ) and the redshift measurement success rate ( right ) over total F444W magnitude versus F277W-F444W color. All values are taken from the internal v2.2.0 catalog (used for designing masks). Panel a: Points indicate the spectroscopically-confirmed objects ( flag zspec qual ≥ 2; filled purple circles), targets without robust redshifts ( flag zspec qual < 2; dark gray open circles), and targets with data quality issues (e.g., those on MSA1; gray crosses). Contours denote the parent photometric sample distribution (with use phot = 1; see Weaver et al. 2024; 1,2,3 σ levels). Side panels show histograms over mag F444W and F277W-F444W (line colors the same as points in the main panel). Though the sample selection incorporates multiple disparate categories, overall the targets follow the distribution of the parent sample down to mag F444W ∼ 29 AB. Successfully-observed targets without measured z spec do not have systematically redder/bluer colors compared to the spectroscopically-confirmed ones. The unconfirmed targets are fainter on average than the confirmed objects, though their distribution does overlap down to the very faintest magnitudes (mag F444W ≳ 30 AB) Panel b: 2D and 1D histograms of the redshift measurement success fraction over mag F444W and F277WF444W, defined as the fraction of objects with robust redshifts over the total number of successfully-observed targets. The success fraction is very high over most of this space, though drops to ∼ 30 -50% at mag F444W ∼ 29 AB. \nquest, each achieving local image reproduction errors of ∆ RMS ≃ 0 . 2 '' .", '4.1. Success rate and redshift distribution for spectroscopically-confirmed objects': 'The UNCOVER spectroscopic redshift catalog includes a 74% success rate, with robust redshifts (i.e., defined as flag zspec qual ≥ 2; see Table 2 and Sec. 3.2) for 409 of the 553 targets with successfully observed and reduced spectra. A histogram of the redshift distribution of spectroscopically confirmed targets, split by flag zspec qual , is shown in Figure 3a. We measure secure redshifts (based on two or more secure spectral features; flag zspec qual = 3) for 327 objects, spanning from z ∼ 0 . 3 to z ∼ 10. The 82 galaxies with solid redshifts (based on one broad continuum feature or 2 less robust features; flag zspec qual = 2) also span a wide redshift range ( z ∼ 0 . 2 -13). This latter category includes most of the targeted galaxies in the Abell \n2744 cluster itself, as most have very red spectra with no emission lines and only a broad stellar bump (resulting in lower redshift precision). \nWe compare the spectroscopic and photometric redshifts for our sample of spectroscopically-confirmed galaxies in Figure 3b. We find the majority of the Prospectorβ -derived z phot (from the internal v2.2.0 catalog, the most up-to-date catalog used during MSA design in early Summer 2023) are in good agreement with the measured z spec , with a low normalized median absolute deviation σ NMAD = 0 . 060. However, there is a relatively high fraction of catastrophic photometric redshift outliers (with | ∆ z | = ( z phot -z spec ) / (1 + z spec ) > 0 . 15; 17.6%, red circles). \nto the very faintest magnitudes (mag F444W ≳ 30 AB). This is quantified in Figure 4b, as the spectroscopic success fraction is very high for bright targets, but drops only to ∼ 30 -50% at mag F444W ∼ 29 AB (excepting a few extremely faint targets at mag F444W ≳ 30 AB). The distributions of the targets with and without robustlymeasured redshifts suggest that while low S/N does contribute to failed spectroscopic confirmations, low S/N is not entirely responsible for the failed spectroscopic confirmations. Color likewise appears to not drive failed z spec measurements. \nWe similarly find catastrophic photometric redshift failures within of the spectroscopically-confirmed sample ( | ∆ z | > 0 . 15; shown with red points in Figure 3) are not primarily driven by low S/N or color, as these objects exhibit a wide range of magnitudes and F277WF444W colors similar to the complete spectroscopically confirmed sample. Preliminary visual inspection suggests some of these outliers are due to confusion of the Lyman and Balmer breaks in the Prospectorβ redshift fits, while others may be explained by emission line boosting of the broad-band photometry. We note that for the few very red targets (F277W -F444W ≳ 1), nearly all have catastrophic photometric redshift failures, suggesting additional spectral templates for photometric redshift fitting may be needed to capture the extreme colors of these objects. A more detailed discussion of photometric redshift outliers relative to the measured z spec will be presented in a forthcoming paper (see also Suess et al. 2024).', '4.2. Scientific objectives addressed by the UNCOVER spectroscopic sample': "With a high redshift success rate (74% of targets with robust redshifts) and very deep spectra (up to 16.7h for five individual targets, and up to 38h with multiply lensed images of one object), these NIRSpec observations provide a treasure trove for studies ranging from galaxies within the Abell 2744 cluster itself out to the first galaxies at Cosmic Dawn. We demonstrate the wide range of spectral features seen in the full UNCOVER sample of 409 galaxies with robust z spec ( flag zspec qual ≥ 2) ranging from z ∼ 0 . 3 to z ∼ 13 in Figure 5. An incredible diversity of features can be seen in these ultradeep spectra: Paschen lines, HeI -10833 ˚ A, and the polycyclic aromatic hydrocarbon (PAH) 3.3 µ m feature are seen in galaxies at the lowest redshifts, and [SIII] -9068 , 9531 ˚ A ˚ A are seen out to z ∼ 5. H α is detected out to z ∼ 7, and H β and [OIII] -4959 , 5007 ˚ A ˚ A are seen in all galaxies except those at the very highest redshifts. The Balmer break is \nseen in galaxies at z ≳ 1, while the Lyman break (and Ly α ) are seen at z ≳ 4 . 5. \nAn overview of the science cases addressed by the UNCOVER spectra are highlighted in Figure 6, showing a subset of objects from our sample. We have spectroscopically confirmed and begun characterizing the rest-frame UV spectra of 10 early galaxies at z ≥ 8 . 5 (see e.g., Fujimoto et al. 2023c), including two among the first generation of galaxies at z spec = 13 . 03 and z spec = 12 . 39 (as presented in Wang et al. 2023a). Our sample also features a number of early AGN at z > 6, including an X-ray luminous AGN at z spec = 10 . 1 (Bogd'an et al. 2024, Goulding et al. 2023) and a broad-line AGN at z spec = 8 . 5 (Kokorev et al. 2023). Other targets include a number of dust-reddened, high-redshift objects described as 'little red dots' (e.g., Labbe et al. 2023, Furtak et al. 2024a, Greene et al. 2024). The PRISM spectra also yield the first spectroscopic constraints on low-mass, low-luminosity galaxies during the Epoch of Reionization ( z ∼ 6 -8), including direct constraints on the ionizing photon production efficiency that yield evidence that these faint galaxies are the primary drivers of the reionization of the Universe (Atek et al. 2024, Dayal et al. 2024), and extending the mass-metallicity relation to the low-mass end (Chemerynska et al. 2024b). \nOur spectroscopic sample additionally includes a range of dusty galaxies out to z ∼ 4, both with (e.g., some of the objects presented in Kokorev et al. 2023 and Price et al. 2023) and without ALMA continuum detections (from e.g., DUALZ, Fujimoto et al. 2023a). Two targeted galaxies at low redshift ( z ≲ 0 . 5) reveal detections of the 3.3 µ m PAH emission feature and ice absorption features. Also targeted are a number of quiescent galaxies extending from low redshift to z ≳ 3. This includes a massive, dusty quiescent galaxy confirmed at z spec = 3 . 97 (Setton et al. 2024), with the deep PRISM spectra revealing its detailed star formation history that indicates the early formation of its dense stellar core. Finally, we obtained spectra for three brown dwarfs located within our own Milky Way (Langeroodi & Hjorth 2023, Burgasser et al. 2024): one explicitly targeted, and two that were selected based on the photometric criteria for dust-reddened 'little red dots' and AGN at high redshift. These deep spectra reveal the spectral classifications, temperatures, and metallicities, as well as characterizing molecular features within the brown dwarf atmospheres. \nz \nFigure 5. NIRSpec/PRISM spectra for all UNCOVER targets with secure ( flag zspec qual = 3, N = 327; top ) and solid ( flag zspec qual = 2, N = 82; bottom ) redshifts, shifted to the restframe and ordered by increasing redshift. The wavelength axis is split, with linear and log scaling below and above 1 . 2 µ m, respectively. The locations of notable emission and absorption/break features are annotated above and below the spectra. \n<!-- image --> \nf \nRest-frame Wavelength [ Å ] \n<!-- image --> \nFigure 6. Overview of 1D spectra for a subset of our sample, highlighting key science themes addressed by the UNCOVER survey and mask design strategy. All spectra are shown in the restframe in f λ units (with arbitrary normalization and shifting), with the shaded contour denoting the uncertainty. The redshift and MSA ID of each object are annotated next to the spectra. Vertical lines mark the wavelengths of selected emission features. \nTable 2. Redshift catalog from UNCOVER NIRSpec/PRISM spectra \nNote -The full table is available in machine readable format from https://jwst-uncover.github.io/DR4.html. \nColumns: \n- (1) MSA ID (corresponding to internal v2.2.0).\n- (2) Targeted Right Ascension and Declination (internal v2.2.0 catalog; J2000, decimal degrees).\n- (3) Spectroscopic redshift.\n- (4)-(6) 16/50/84th percentile from redshift fit p ( z ) distribution.\n- (7) Redshift quality flag: 3 = secure, based on two or more secure spectral features (e.g., two robustly-detected emission lines, one clear break and one robust emission line, two robustly-detected absorption features); 2 = solid, based on one broad continuum feature or two less robust features (e.g., a break or stellar bump, or two marginally-detected emission lines, or one marginally-detected emission lines and a break); 1 = tentative but unreliable redshift; 0 = no redshift. For analysis, using redshifts with quality flag = 3 or = 2 is recommended.\n- (8) Spectrum flag: 1 = successfully observed and reduced spectrum; 0 = no spectrum/data quality issue.\n- (9) Feature flag, for spectra containing two or more emission lines (1=yes, 0=no).\n- (10) Feature flag, for spectra containing a break + an emission line (1=yes, 0=no).\n- (11) Feature flag, for spectra containing only a break and strong absorption features (1=yes, 0=no).\n- (12) Feature flag, for spectra containing only a break (1=yes, 0=no).\n- (13) Feature flag, for spectra containing only a stellar bump (1=yes, 0=no).\n- (14) Fit method for best-fit redshift.\n- (15) Fit method for redshift uncertainties/percentiles.\n- (16) Total exposure time (hours).\n- (17) List of masks on which each object was included (comma separated string).\n- (18) Closest match DR3 ID (Weaver et al. 2024, Wang et al. 2024).\n- (19) Separation of DR3 & MSA RA/Dec, in arcsec.", '5. FINAL REMARKS': 'The ultradeep PRISM spectra from the UNCOVER program add immense value to the already rich - and still growing - treasure trove of public observations in the Abell 2744 lensing cluster field. This first data release of the 1D and 2D spectra (with local background subtraction), along with derived catalogs with quality flags, is publicly available on the survey website (https://jwst-uncover.github.io/DR4.html). Future spectroscopic releases will include the repeat observations of MSA1 and spectra accompanying the Visit 3 repeat of the NIRCam parallel imaging (observed 30-31 July 2024). Additional improvements in the reduction and released products will include global background subtraction and more sophisticated modeling of emission lines. \nThis work is based in part on observations made with the NASA/ESA/CSA James Webb Space Telescope . The data were obtained from the Mikulski Archive for Space Telescopes at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-03127 for JWST. These observations are associated with JWST-GO-2561. Support for program JWST-GO- \n2561 was provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Associations of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555. The specific observations analyzed can be accessed via 10.17909/8k5c-xr27. Cloud-based data processing and file storage for this work is provided by the AWS Cloud Credits for Research program. The Cosmic Dawn Center is funded by the Danish National Research Foundation (DNRF) under grant #140. The BGU lensing group acknowledges support by grant No. 2020750 from the United States-Israel Binational Science Foundation (BSF) and grant No. 2109066 from the United States National Science Foundation (NSF), by the Israel Science Foundation Grant No. 864/23, and by the Ministry of Science & Technology, Israel. \nFacilities: JWST(NIRSpec, NIRCam) \nSoftware: astropy (Astropy Collaboration et al. 2013, 2018, 2022), eMPT (Bonaventura et al. 2023), jwst pipeline (v1.14.0; Bushouse et al. 2024), msaexp (v0.8.5; Brammer 2023a), grizli (v1.11.9; Brammer 2023b), eazy (Brammer et al. 2008), matplotlib (Hunter 2007), numpy (Harris et al. 2020), scipy (Virtanen et al. 2020), seaborn (Waskom et al. 2017), snowblind (Davies 2024)', 'A. PARALLEL NIRCAM IMAGING': 'Coordinated parallel NIRCam imaging was taken simultaneously with the primary NIRSpec/PRISM multiobject spectroscopy. Altogether imaging was taken in 7 broadband and 2 medium band filters (see Table 3), using the MEDIUM8 readout pattern for all exposures. This parallel imaging overlaps existing HST/ACS and WFC3 observations (HFF, Lotz et al. 2017; BUFFALO, Steinhardt et al. 2020; see Figure 7). The cumulative exposure time per filter over the parallel footprint ranges from 0.9 to 5.9 hours, with total areas ranging from 9.2 to 26.9 sq. arcmin. This includes imaging in six of the broadband filters that covers the full parallel area (excepting observation issues), and imaging in F090W and the two medium bands F410M and F480M that were only taken in parallel with Mask 5 (see Table 3).', 'B. UPDATES TO THE UNCOVER STRONG LENSING MODEL OF ABELL 2744': "We use the UNCOVER spectroscopy, described in this work, as well new JWST/NIRCam imaging (Suess et al. \n2024) and grism spectroscopy (R. Naidu & J. Matthee, et al., in prep.) of the Abell 2744 field to update the UNCOVER strong lensing (SL) model of the cluster, as presented in Section 3.3. \nThe parametric lens model of Abell 2744 is constructed with an updated version of the Zitrin et al. (2015) analytical method. It comprises five smooth cluster-scale dark matter halos centered on each of the sub-clusters' BCG, modeled as pseudo-isothermal elliptical mass distributions (PIEMDs; Kassiola & Kovner 1993), and 552 cluster member galaxies (see Appendix B.1), modeled as dual pseudo-isothermal ellipsoids (dPIEs; El'ıasd'ottir et al. 2007). We refer the reader to Furtak et al. (2023b) for more details on the implementation and setup of our Abell 2744 model. \nWhile the currently available v1.1 SL model presented in Furtak et al. (2023b) is based on HST-selected cluster members and mostly photometric multiple image systems in the northern and north-western extended cluster sub-structures, the v2.0 model presented here adds additional cluster member galaxies selected with JWST (Appendix B.1) and new spectroscopic redshifts \nFigure 7. The NIRCam parallel footprints, plotted over the existing NIRCam and HST/ACS+WFC3 coverage footprints and the lensing contours as shown in Figure 1. Coverage of F090W, F410M, and F480M is restricted to Visit 5 (taken in parallel to Mask 5) and is shown in purple, and all other filters with full parallel pointing coverage (F115W, F150W, F200W, F277W, F356W, F444W) are shown in blue. \n<!-- image --> \nFigure 8. JWST/NIRCam color-magnitude diagram of objects detected in Abell 2744, showing the cluster's red sequence. Known spectroscopic and photometric cluster members from Bergamini et al. (2023b) are shown as red and orange dots and our red sequence selection is shown as the blue shaded area. \n<!-- image --> \nof multiple image systems as constraints (section B.2). The new SL model (Appendix 3.3) maps are also made public on the UNCOVER website in the framework of DR4 (https://jwst-uncover.github.io/DR4.html; see Appendix 3.3). \nTable 3. NIRCam Parallel Imaging \nNote -Depths are calculated within 0. '' 16 and 0. '' 32 diameter apertures in the short and long wavelength bands, respectively, using noise properties derived from the weight maps and corrected to total assuming a point source geometry. As the footprint is inhomogenous, these estimates correspond to a 0.7 arcmin 2 box centered at (3 . 6012969 , -30 . 4908199). Columns: (1) NIRCam filter. (2) Filter exposure time across footprint (hours). (3) Total filter footprint area (sq. arcmin). (4) Imaging 5 σ depth. (5) Mask(s) with which the filter was observed in parallel.", 'B.1. JWST cluster member selection': "Thanks to the JWST Medium Bands, Mega Science program ( MegaScience ; Suess et al. 2024), we now have NIRCam F070W and F090W imaging data covering the entire UNCOVER field at our disposal. These two filters straddle the 4000 ˚ A break at the cluster's redshift z d = 0 . 308 and are therefore ideally suited for photometric selection of cluster members from the red sequence (e.g. Repp & Ebeling 2018). We use SExtractor (Bertin &Arnouts 1996) in dual-imaging mode to detect sources in the F090W mosaic and measure their photometry in F070W and F090W. Following our approach in Furtak et al. (2023b) and Furtak et al. (2024b), we then use the colors of the known spectroscopic cluster members from Bergamini et al. (2023b) to calibrate the cluster's red sequence in the color-magnitude diagram (see Figure 8). Cluster members are then selected in a color-window of width 0.1 around the red sequence and brighter than 23 magnitudes in the F090W band. The resulting sample is cross-matched with the known spectroscopic and HST-selected cluster members (Furtak et al. 2023b) to make sure no galaxy is counted doubly. \nAs a result, we complement our previous cluster member sample from Furtak et al. (2023b) with 132 new NIRCam selected cluster members. This bring the total \nTable 4. New spectroscopic redshifts of multiply-imaged sources included in our v2.0 SL model of Abell 2744. \nNote -A full table of multiple images used in the v2.0 model is included in the public SL model release at https://jwst-uncover.github. io/DR4.html. \nColumns: (1) ID number of the multiple image system. (2) ID number of the MSA-slit on one of the images. (3) Spectroscopic redshift. (4) Reference to the spectroscopic redshift measurement. \nnumber of cluster members included in the SL model to 552, now spanning the entire 45 arcmin 2 of the UNCOVER field. The new, NIRCam-selected sample in particular adds cluster members in the north-east to north-west of the cluster, areas which were not covered with HST.", 'B.2. New spectroscopic redshifts of multiple images': "The unprecedented depth and areal coverage of the UNCOVER survey's imaging (Bezanson et al. 2022) enabled us to detect new multiple image systems in northwestern and northern extensions of Abell 2744 which were previously not know to be dense enough to produce strong lensing (Furtak et al. 2023b). These new systems \nwere however not constrained with spectroscopic redshifts in the first UNCOVER model ( v1.0 ) due to lack of spectroscopic coverage in those areas. Multiple images without precise redshift information are known to significantly bias SL models of galaxy clusters (e.g. Johnson & Sharon 2016) which is why spectroscopic redshifts are paramount for accurate SL modeling and magnification estimates. \nAfter the publication of our v1.0 model (Furtak et al. 2023b), new VLT/MUSE observations found system 68 to lie at z spec = 2 . 584 (Bergamini et al. 2023a). We included that new redshift in our v1.1 model release in June 2023, but the model remained mostly constrained with photometric systems in the north-west and the north. With the UNCOVER JWST/NIRSpec observations presented in this work, we are now able to spectroscopically confirm numerous multiple image systems in the whole UNCOVER field. In total, we obtained 10 new spectroscopic redshifts. These in particular include the triply-imaged high-redshift AGN A2744-QSO1 at z spec = 7 . 045 (system 53; Furtak et al. 2024a), a low-mass heavily star-forming object at z spec = 6 . 875 (system 86; Atek et al. 2024), and a massive quiescent galaxy at z spec = 2 . 322 stretched into an arc (system 67; Siegel et al. in prep.). In addition, the JWST Cycle 2 program All the Little Things (ALT; Program-IS: 3516 PIs J. Matthee & R. Naidu) observed the Abell 2744 field with JWST/NIRCam grism spectroscopy in the F356W filter, which enabled the discovery of two new multiple image systems at z spec = 6 . 873 (system 84) and z spec = 4 . 753 (system 85) respectively (R. Naidu & J. Matthee, et al., in prep.), which we also included in the model. Note that system 84 also has an UNCOVER NIRSpec redshift which agrees with the ALT redshifts. \nWe list all new multiple image redshifts in Table 4 and show them in Figure 9. In total, our new v2.0 SL model is constrained by 187 multiple images belonging to 66 individual sources. Of these, 60 sources now have spectroscopic redshifts, leaving only 6 multiply-imaged sources with free redshifts in the model. For additional constraining power, we now also use parity information of 4 very close knot systems, systems 65.3, 67.3, 78.3, and 80.2, as constraints in the model (see equations 8 and 9 in Furtak et al. 2023b). A full list of multiple images, including coordinates and redshifts, is included in the public v2.0 SL model release.", 'REFERENCES': "Adams, N. J., Conselice, C. J., Austin, D., et al. 2024, ApJ, \n965, 169, doi: 10.3847/1538-4357/ad2a7b \nArrabal Haro, P., Dickinson, M., Finkelstein, S. L., et al. \n2023a, ApJL, 951, L22, doi: 10.3847/2041-8213/acdd54 \nFigure 9. A 4 . 3 ' × 4 . 8 ' cutout of an UNCOVER and MegaScience NIRCam composite-color image of Abell 2744 including all broad and medium bands. Overlaid we show the critical curves of our SL model for source redshifts z s = 1 . 6881 (corresponding to system 1) and z s = 10 in blue and purple respectively. Multiple images from Bergamini et al. (2023b), used with spectroscopic redshifts in our v1.0 model, are shown in yellow and multiple images with new spectroscopic redshifts in our v2.0 model are shown in green. Photometric multiple images are shown in red. The area between the main cluster and the north-western sub-structure in particular has high magnifications of order µ ≳ 4 for sources at z s = 10 (see Fig. 2). Note, a vectorized full 0.04 '' /pix resolution version of this figure is included in the public v2.0 SL model release. \n<!-- image --> \nWang, B., Leja, J., Bezanson, R., et al. 2023b, ApJL, 944, L58, doi: 10.3847/2041-8213/acba99 \nWang, B., Leja, J., Labb'e, I., et al. 2024, ApJS, 270, 12, doi: 10.3847/1538-4365/ad0846 \nWaskom, M., Botvinnik, O., O'Kane, D., et al. 2017, Mwaskom/Seaborn: V0.8.1 (September 2017), v0.8.1, Zenodo, Zenodo, doi: 10.5281/zenodo.883859 \nWeaver, J. R., Cutler, S. E., Pan, R., et al. 2024, ApJS, \n270, 7, doi: 10.3847/1538-4365/ad07e0 Williams, C. C., Alberts, S., Ji, Z., et al. 2024, ApJ, 968, 34, doi: 10.3847/1538-4357/ad3f17 Wu, Y., Cai, Z., Sun, F., et al. 2023, ApJL, 942, L1, doi: 10.3847/2041-8213/aca652 \nZitrin, A., Fabris, A., Merten, J., et al. 2015, ApJ, 801, 44, doi: 10.1088/0004-637X/801/1/44"} |
2024arXiv240913324G | The vacuum and electrovacuum Einstein equations for spacetimes with two commuting Killing vectors can be solved by indirect methods of integrable systems. But if in addition the spacetime admits an irreducible Killing tensor and the corresponding KleinGordon equation is separable they can be integrated directly by separation of variables as shown by Carter in 1968. We generalize this approach to supergravity and derive a metric ansatz that ensures the above properties for Petrovtype I. Our derivation is based on the BenentiFrancavilla ansatz for metrics admitting two commuting Killing vectors and an irreducible Killing tensor. We find additional constraints that guarantee the existence of two shearfree null geodesic congruences and the separability of the KleinGordon equation. The resulting class of metrics belongs to a certain sector of Petrov type I called IB whose algebraically special subsector contains only type D. For this class a direct integration of the supergravity equations seems possible. We also show that these spacetimes admit a general description of the generalized photon and massive particle surfaces recently introduced in connection with black hole shadows. | 2024-09-01T00:00:00Z | ['arXiv:2409.13324', '2024arXiv240913324G', '10.48550/arXiv.2409.13324'] | ['General Relativity and Quantum Cosmology', 'High Energy Physics - Theory', 'Mathematical Physics'] | Petrov types separability and generalized photon surfaces of supergravity black holes | 2,024 | 173 | 0.3 | ['EPRINT_HTML', 'EPRINT_PDF'] | 1 | https://arxiv.org/pdf/2409.13324.pdf | {'Petrov types, separability and generalized photon surfaces of supergravity black holes': "Dmitri Gal'tsov ∗ and Aleksandr Kulitskii † \nFaculty of Physics, Moscow State University, 119899, Moscow, Russia", 'Abstract': 'The vacuum and electrovacuum Einstein equations for spacetimes with two commuting Killing vectors can be solved by indirect methods of integrable systems. But if, in addition, the spacetime admits an irreducible Killing tensor and the corresponding Klein-Gordon equation is separable, they can be integrated directly by separation of variables, as shown by Carter in 1968. We generalize this approach to supergravity and derive a metric ansatz that ensures the above properties for Petrov-type \n- I . Our derivation is based on the Benenti-Francavilla ansatz for metrics admitting two commuting Killing vectors and an irreducible Killing tensor. We find additional constraints that guarantee the \nexistence of two shear-free null geodesic congruences and the separability of the Klein-Gordon equation. \nThe resulting class of metrics belongs to a certain sector of Petrov type I , called I B , whose algebraically special subsector contains only type D . For this class, a direct integration of the supergravity equations seems possible. We also show that these spacetimes admit a general description of the generalized photon and massive particle surfaces recently introduced in connection with black hole shadows.', 'I. INTRODUCTION': 'Hidden symmetries of spacetime, exhibited by second-rank Killing tensors, play no less an important role than isometries; for a review, see [1-3]. Over the past decade, significant progress has been made in the analysis of geometries admitting Killing-Yano (KY) tensors [2-4], which are the strongest symmetries in the Killing hierarchy for D -type spacetimes. \nHidden symmetries of spacetimes beyond type D , which do not allow for KY structures, have also been examined, but their analysis is not yet so complete. Here we would like to investigate hidden symmetries of black holes in supergravities N = 4 , 8, which include scalar fields and belong to general Petrov type I . The exact solutions for stationary black holes in these theories are fairly well known, largely due to hidden symmetries of a different kind that arise when these theories are dimensionally reduced in stationary spacetimes. As is well known, the dimensional reduction of supergravities to three dimensions leads to sigma models on homogeneous target spaces [5-7]. The target space isometries include Harrison transformations, which allow generating charged supergravity black holes from the Kerr metric. Apart from static solutions, which are easily obtained directly from the Einstein equations, most of the known stationary solutions were found just in this way, see e.g. [8-11]. A few exceptions are the supersymmetric solutions obtained using the Bogomolny equations [12, 13] or solving the null geodesic equations in the target space [14] (also known as the nilpotent orbit method [15]). \nMeanwhile, in vacuum and electrovacuum gravity, Carter showed that direct integration of stationary axisymmetric Einstein equations is possible for spacetimes admitting both HamiltonJacobi and Klein-Gordon separability [16]. By explicit integration he found several classes of solutions, among which were Kerr and Kerr-Newman black holes of the Petrov type D (for some details of the electromagnetic dressing see also [17]). \nBlack holes in four-dimensional extended supergravities N = 4 , 8 generically belong to Petrov type I , though they still admit the second rank Killing tensor like their type D cousins in vacuum and electrovacuum gravity. We use the Benenti-Francaviglia (BF) [18] parameterization of metrics admitting a Killing tensor independently of their Petrov type (Section 2). We then require for them the fulfillment of another important property of stationary axisymmetric spacetimes which holds in type D : the existence of two null geodesic shearfree congruences (Section 3). This is accomplished by imposing two additional conditions on the BF ansatz. Using the Newman-Penrose formalism, we find that the property of allowing null geodesic shearfree congruences extends beyond type D also to a certain class of metrics of type I . \nWe then impose a third constraint, ensuring the separability of the Klein-Gordon equation (Section 4). The resulting class of metrics, which we call I B , turns out to be similar to that used by Carter to perform direct integration of Einstein equations in vacuum and electrovacuum gravity [16]. But our ansatz is now valid for type I B and is not constrained by the assumptions about the sources of matter used by Carter. This opens the way to a direct integration of the supergravity equations for this class. Leaving an explicit integration for a separate publication, here we verify that all explicitly known solutions for black holes do admit a simple polynomial description in terms of the constrained BF ansatz. \nSection 5 is devoted to deriving conditions for the BF metrics to belong to type D. In Section 6 we show that for our class of metrics it is possible to have a unified description of the photon and massive particle surfaces, recently introduced [19, 20] as a tool for analyzing black hole shadows [21, 22]. This follows from a special property of the BF-Killing tensor, which was called slice-reducibility. Our treatment is valid for all basic black holes in extended supergravities. Finally, in Section 7 we present an explicit description of the known black hole solutions in terms of BF functions and give explicit MPS equations for them.', 'II. GENERAL BENENTI-FRANCAVIGLIA ANSATZ': "Our starting point is the class of four-dimensional metrics admitting a pair of commuting Killing vectors. More specifically, we will be interested in the stationary axisymmetric orthogonally transitive spacetimes (SAS) that can be parameterized by a block-diagonal metric, one of whose blocks is spanned by Killing vectors. For parametrization of the SAS spacetime that guarantees the separability of the Hamilton-Jacobi equation we use an ansatz given by Benenti and Francaviglia [18], based on Benenti's theorems [23, 24] proving that in any dimension a necessary and sufficient condition for this to happen is the existence of a commuting system of Schouten-Nijenhuis brackets [25, 26] (for a later discussion, see [27]) for Killing vectors and Killing tensors with a total number equal to the dimension of the spacetime. In this algebra, certain conditions on the eigenvectors of the Killing tensors must also be satisfied.In the fourdimensional case of SAS metrics, to which we restrict ourselves here, one (trivial) Killing tensor is the metric itself, so one irreducible Killing tensor is needed for separability. \nRecently, SAS spacetimes with prescribed hidden symmetry have attracted attention in the search for viable alternatives to the Kerr metric for astrophysical modeling [28-31]. In these applications, it was found, in particular, that separation of variables in the Klein-Gordon \nequation for general BF metrics is not guaranteed. \nA general BF parametrization consists of ten arbitrary functions, each depending on one variable. The metrics are off-shell in the sense that Einstein's equations are not imposed. The SAS metric is written in the coordinates x µ = ( x a , x i ), where x a = t, ϕ correspond to the subspace spanned by the Killing vectors K ( t ) = ∂ t and K ( ϕ ) = ∂ ϕ and x i = r, y , belong to orthogonal two-dimensional space whose metric without loss of generality can be assumed diagonal. BF ansatz looks somewhat simpler in terms of the contravariant metric tensor g µν = ( g ab , g ij ) as follows: \nwhere two sets of arbitrary functions are introduced A k ( r ) , B k ( y ) , k = 1 .. 5 depending each on one variable, r and y respectively. In order to ensure existence of an exact Killing tensor (EKT), the conformal factor Σ = Σ( r, y ) must be of the special form \ng ab = Σ -1 A 3 -B 3 A 4 -B 4 A 4 -B 4 A 5 -B 5 , g ij = -Σ -1 A 2 0 0 B 2 (1) \nΣ = A 1 + B 1 . (2) \nThen the Killing tensor, satisfying the equation \n∇ ( α K µν ) = 0 , (3) \nwhere symmetrization over indices is understood, also has a block diagonal form K µν = ( K ab , K ij ) , where \nThe inverse metric (1) and the Killing tensor (4) have the following automorphism: \nΣ K ab = A 1 B 3 + A 3 B 1 A 1 B 4 + A 4 B 1 A 1 B 4 + A 4 B 1 A 1 B 5 + A 5 B 1 , Σ K ij = -A 2 B 1 0 0 A 1 B 2 (4) \nA 1 ↔-B 1 , A i ↔ B i , ( i = 2 .. 4) , g rr ↔-g yy , K rr ↔-K yy . (5) \nFor an arbitrary conformal factor Σ only a conformal Killing tensor may exist, we will come back to this later. We keep separate notation for the conformal factor for further convenience. \nTwo blocks for the covariant metric tensor g µν = ( g ab , g ij ): read \ng ab = Σ P A 5 -B 5 -A 4 + B 4 -A 4 + B 4 A 3 -B 3 , g ij = -Σ A -1 2 0 0 B -1 2 (6) \nwhere \nP = ( A 3 -B 3 )( A 5 -B 5 ) -( A 4 -B 4 ) 2 . (7) \nWe shall assume that in a significant region of space-time (e.g. beyond the horizon or the ergosphere) all the BF coefficient functions are positive, and we shall not consider analytic continuation into the negative region, which of course can be done in the usual way. Therefore we shall often use square roots of the BF coefficients, assuming that they are real. \nOther sign conditions follow from the metric signature with the same reservations: \nA 3 -B 3 > 0 , A 5 -B 5 < 0 , P < 0 . (8) \nWe also assume that the conformal factor Σ is positive in the case where it is not assumed to be given by the formula (2). \nThe static limit corresponds to \nA 4 ≡ 0 , B 4 ≡ 0 . (9) \nOne may wonder what is the gauge freedom inside the BF ansatz. The form of the metric suggests possibility of two coordinate transformations \nr → ˜ r ( r ) , y → ˜ y ( y ) , (10) \ncontaining two arbitrary functions of independent variables. These can be used to impose additional conditions on A 2 , B 2 , e.g., A 2 = 1 = B 2 (conformally flat metric in the r, y block). Other useful conditions, which turn out to be satisfied by all known supergravity black holes read as follows: \nA 2 A 5 = a 2 = const , B 2 B 5 = b 2 = const . (11) \nIn what follows we will assume the validity of this gauge.", 'A. Null tetrad': 'We proceed by choosing some natural Newman-Penrose (NP) null tetrad [32] for the inverse metric tensor(1): \ng µν = l µ n ν + n µ l ν -m µ ¯ m ν -¯ m µ m ν . (12) \nWe choose real vectors of the tetrad symmetrically: \nl = 1 / √ 2Σ ( √ A 3 ∂ t + C ∂ ϕ -√ A 2 ∂ r ) , m = 1 / √ 2Σ ( √ B 3 ∂ t + D∂ ϕ -i √ B 2 ∂ y ) , (13a) \nwhere \nn = 1 / √ 2Σ ( √ A 3 ∂ t + C ∂ ϕ + √ A 2 ∂ r ) , ¯ m = 1 / √ 2Σ ( √ B 3 ∂ t + D∂ ϕ + i √ B 2 ∂ y ) . (13b) \n( A 3 -B 3 ) C = √ A 3 ( A 4 -B 4 )+ √ B 3 √ -P , ( A 3 -B 3 ) D = √ B 3 ( A 4 -B 4 )+ √ A 3 √ -P . (14) \nRecall that so far the number of arbitrary Benenti functions are ten.', 'III. NULL SHEAR-FREE GEODESIC CONGRUENCES': 'Starting from the general BF class, we would like to select a subclass with additional properties noticed for some supergravity black holes [33]. Vacuum solutions of type D have the property, according to the Goldberg-Sachs theorem [34], of admitting two null geodesic congruences without shear. Electrovacuum black holes arising in pure N = 2 supergravity without matter multiplets are also of type D and also have similar congruences. Recall that stationary charged black holes in N = 2 supergravity can be obtained by Harrison transformations in the three-dimensional sigma model description [5] from the Kerr vacuum metric, so it is likely (although no actual proof has been given) that Harrison transformations preserve this property as well as geodesic integrability. Black hole solutions in extended supergravities with scalar moduli N = 4 , 8, can also be obtained by Harrison transformations from Kerr metric [8, 3537], so we can expect them to share both of the above properties (as mentioned e.g. in [33] for the two-charge STU solution). But Harrison transformations certainly do not preserve the Petrov type of the metric, since supergravity black holes with scalar fields generically belong to type I . \nThus, first of all we look for a reduced BF ansatz ensuring existence of two null geodesic shear-free congruences.', 'A. First constraints': 'Using algebraic computing and hints from the known supergravity black hole solutions, one is led to consider the following constraints excluding two of ten BF functions: \nA 4 = √ A 3 A 5 , B 4 = √ B 3 B 5 . (15) \nWith these conditions the functions entering (14) simplify to \n√ -P = √ A 3 B 5 -√ A 5 B 3 , ⇒ C = √ A 5 , D = √ B 5 . (16) \nThis reduced BF ansatz leads to significant simplification of the null tetrad: \nl = 1 / √ 2Σ ( √ A 3 ∂ t + √ A 5 ∂ ϕ -√ A 2 ∂ r ) , m = 1 / √ 2Σ ( √ B 3 ∂ t + √ B 5 ∂ ϕ -i √ B 2 ∂ y ) , n = 1 / √ 2Σ ( √ A 3 ∂ t + √ A 5 ∂ ϕ + √ A 2 ∂ r ) , ¯ m = 1 / √ 2Σ ( √ B 3 ∂ t + √ B 5 ∂ ϕ + i √ B 2 ∂ y ) . (17) \nIn the static limit our relations (15) degenerate, so we have to be careful in making correct assignments for the remaining BF coefficients. Looking at the constrained tetrad 17, we realize that a correct choice in the static limit will be \nA 5 ≡ 0 , B 3 ≡ 0 , A 3 = 0 , B 5 = 0 , (18) \n/negationslash \n/negationslash \nimplying a = 0 , b = 0 . \n/negationslash', 'B. Newman-Penrose analysis': "To understand which restrictions on the nature of spacetime is put by the first constraints, let's continue our NP analysis. Recall the definitions of the NP projections of the covariant derivatives \nD = l µ ∇ µ , ∆ = n µ ∇ µ , δ = m µ ∇ µ , ¯ δ = ¯ m µ ∇ µ , (19) \nand the action of D, ∆ on the vectors l µ , n µ \nDl µ = ( /epsilon1 +¯ /epsilon1 ) l µ -¯ κm µ -κ ¯ m µ (20) \n∆ n µ = -( γ + ¯ γ ) n µ + νm µ + ¯ ν ¯ m µ . (21) \nConsider null congruences aligned with l µ , n µ . If κ = 0 = ν they are geodesic , with /epsilon1, γ being measure of non-affinity. Another important quantity of null congruences is shear , which is defined for them as \nσ = -m µ δ l µ , ¯ λ = m µ δ n µ (22) \nrespectively. Calculating the spin coefficients (see Appendix A) for our tetrad, we find: \nκ = ν = 0 , σ = λ = 0 , (23) \nwhich means that both congruences are geodesic and shearfree . Other spin coefficients are generically non-zero and pairwise equal: \nµ = ρ, τ = π, /epsilon1 = γ, α = β. (24) \nSuch properties are typical for Petrov type D , once two congruences are principal null directions of the Weyl tensor (which is also the case, as we shall see shortly). To establish the Petrov type in our case, we calculate the NP projections of the Weyl tensor: \nΨ 0 = -C αβγδ l α m β l γ m δ , Ψ 1 = -C αβγδ l α n β l γ m δ , Ψ 2 = -C αβγδ ( l α n β l γ n δ -l α n β m γ ¯ m δ ) / 2 , (25) Ψ 3 = -C αβγδ n α l β n γ m δ , Ψ 4 = -C αβγδ n α ¯ m β n γ ¯ m δ , \nFrom the computer assisted calculations one finds that two of them are zero, \nΨ 0 = 0 = Ψ 4 , (26) \nwhile the others are rather cumbersome in terms of BF coefficients. Still one can extract the following relation between the other two: \nΨ 1 = Ψ 3 , (27) \nreflecting obvious symmetry of the tetrad with respect to interchange A ↔ B . Vanishing of Ψ 0 and Ψ 4 means that the real vectors l µ , n µ are two distinct principal null directions of the Weyl tensor for a constrained BF metric. At the same time, this means that our tetrad is not canonical for determination of the Petrov type. We therefore proceed by computing the values of the quadratic and cubic curvature invariants of the Weyl tensor \nI = Ψ 0 Ψ 4 -4Ψ 1 Ψ 3 +3Ψ 2 2 , J = ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ Ψ 4 Ψ 3 Ψ 2 Ψ 3 Ψ 2 Ψ 1 Ψ 2 Ψ 1 Ψ 0 ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ . (28) \n∣ ∣ As is known, in order for a metric to be algebraically special, the following relationship between these invariants must be satisfied: \nI 3 = 27 J 2 . (29) \nIt is easy to see that (27) implies that the constrained BF metrics are algebraically special if in addition to (27) the following conditions hold \nΨ 2 1 = k Ψ 2 2 , with k = 9 / 16 , or 0 . (30) \n/negationslash \nIf (30) does not hold, the metric is of the Petrov type I. If (30) if satisfied with k = 0 and Ψ 2 = 0 then the metric type is D . Other algebraically special types are not possible within the constraint (27), for example if one supposes type II, for which Ψ 0 = 0 = Ψ 1 , one immediately finds that the Weyl tensor completely vanishes, i.e. the metric is of type O . \nBy Goldberg-Sachs theorem, the vacuum spacetime is algebraically special if it contains a null geodesic shear-free congruence. If there are two such congruences, the Petrov type is D . Therefore in the case of type I spacetime, admitting a null geodesic shear-free congruence, the Ricci tensor should be non-zero. This is the case for supergravity black holes. Thus our class I B consists of non-vacuum metrics, possessing a Killing tensor and a pair of null geodesic shear-free congruences. These properties are close to properties of D type, they will ensure separability of Hamilton-Jacobi equation and Klein-Gordon equation if some further conditions on the Ricci tensor are fulfilled (see below). However a Killing-Yano tensor exists only in the algebraically special case D , so the Dirac equation generally is non-separable. \nNote that in the null tetrad (17) the vectors l µ , n µ are not affinely parameterized (the spin coefficients /epsilon1, γ being non-zero). One can pass to an affinely parameterized congruence l µ (like in the case of the Kinnersley tetrad for Kerr metric) by rescaling the tetrad vectors as \nl µ → f -1 ( r, y ) l µ , n µ → f ( r, y ) n µ , f = √ Σ /A 2 . (31) \nThis leaves unchanged the coefficients (23) and the Weyl projections (27), so our conclusions will be the same. \nOne can also use the classification scheme based on expansion of the Weyl tensor in terms of bivectors. This leads to the traceless symmetric 3 × 3 matrix Q , related to Weyl projections via \nQ = Ψ 2 -1 2 (Ψ 0 +Ψ 4 ) 1 2 i (Ψ 4 -Ψ 0 ) Ψ 1 -Ψ 3 1 2 i (Ψ 4 -Ψ 0 ) Ψ 2 + 1 2 (Ψ 0 +Ψ 4 ) i (Ψ 1 +Ψ 3 ) Ψ 1 -Ψ 3 i (Ψ 1 +Ψ 2 ) -2Ψ 2 . (32) \nWith our conditions (27) this matrix becomes \nQ = Ψ 2 0 0 0 Ψ 2 2 i Ψ 1 0 2 i Ψ 1 -2Ψ 2 . (33) \nThe case of three different eigenvalues corresponds to type I , one degenerate eigenvalue (in the case Ψ 1 = 0) lead to type D . \nThus, we have proved that the BF metrics (1) with the additional conditions (15) defines a special Petrov type I B class, which shares with the type D two important properties: 1) it admits two independent null shearfree geodesic congruences, 2) it has an irreducible Killing tensor of the second rank. In terms of the constrained tetrad (17) the Killing tensor has only two non-vanishing Newman-Penrose projections exactly as in the case of the type D metrics: \nK ln = B 1 , K m ¯ m = A 1 . (34) \nThis gives some geometric interpretation of two Benenti coefficients entering the conformal factor (2). \nWe further show that the known solutions for black holes in N = 2 , 4 , 8 extended supergravity theories belong to this class I B , which ensures the separability of the Hamilton-Jacobi equation. Moreover, they satisfy the third constraint, which we are going to establish, ensuring separability of the Klein-Gordon equation.", 'C. Separability of geodesic equations': "Separability of the Hamilton-Jacobi equation \n∂S ∂x µ ∂S ∂x ν g µν = µ 2 (35) \ncan be easily demonstrated. Assuming \nS = -Et + Lϕ + S r ( r ) + S y ( y ) , (36) \ndenoting p µ = ∂S/∂x µ , and taking into account that all A k depend only on r , while all B k depend only on y , we obtain \nA 2 p 2 r + U r = 0 , B 2 p 2 y + U y = 0 , (37) \nU r = C ε 2 a + A 1 µ 2 , U y = -C + ε 2 b + B 1 µ 2 , (38) \nε a = E √ A 3 -L √ A 5 , ε b = E √ B 3 -L √ B 5 , (39) \nwhere C is the Carter separation constant related to the Killing tensor as \nC = p µ p ν K µν = Σ -1 [ A 1 ( ε 2 b + B 2 p 2 y ) + B 1 ( ε 2 a -A 2 p 2 r )] . (40) \nNote that the above definition of potentials is such that Carter's constant enters additively. This has certain advantages when discussing spherical and conical orbits. The radii of spherical orbits, e.g., correspond to solutions of the equations \nU r = 0 = U ' r . (41) \nThe derivative equation does not contain the Carter integral, and thus define radii of spherical orbits as functions of E, L . Then substitution of this radius into the equation U = 0 will show, that the Carter integral for spherical orbits is a function of Killing vector integrals.", 'D. Variable mass': 'In conclusion to this section we explore in which case separability of the Hamilton-Jacobi equation can be extended to the case of variable coordinate-depending mass. This may be of interest in the study of shadows of black holes surrounded by plasma [38-41]. A photon in electron plasma acquires an effective mass determined by the refractive index, which depends on the coordinates through the electron density. Since the latter in the vicinity of black holes depends on the coordinates, the propagation of photons in the geometrical optics approximation is described by the Hamilton-Jacobi equations with mass depending on the coordinates: \nµ 2 ⇒M 2 ( x ) . \nSeparation of variables in the Hamilton-Jacobi equation with variable mass is possible only if \nM 2 ( x ) = M 2 r ( r ) + M 2 y ( y ) A 1 + B 1 . (42) \nThis is consistent with what was found for the Kerr metric in [41].', 'IV. SEPARABILITY OF KLEIN-GORDON EQUATION': "Generic type I B class of metrics still does not quarantee separability of the wave equations. Consider the Klein-Gordon equation for a real scalar field φ : \n/square φ = 1 √ -g ∂ µ ( √ -gg µν ∂ ν φ ) = -µ 2 φ. (43) \nThe metric determinant is crucial for separability. It is easier to calculate it in BF form using contravariant components (1). One obtains \n√ -g = Σ 2 √ A 2 B 2 √ -P (44) \nWith the first two constraints (15), one can use (7) to obtain P , leading to \n√ -g = Σ 2 √ A 2 B 2 ( √ A 3 B 5 -√ A 5 B 3 ) . (45) \nConsidering the inverse metric (1), it becomes clear that the condition for separability is \n√ -g = Σ . \nThis leads to the third constraint on the Benenti coefficient functions \nΣ = √ A 2 B 2 ( √ A 3 B 5 -√ B 3 A 5 ) , (46) \nwhich can also be rewritten as \nA 1 + B 1 = bA 23 -aB 23 , (47) \nwhere we introduced \nA 23 = √ A 2 A 3 , B 23 = √ B 2 B 3 , (48) \nand used the gauge (11). Differentiating this with respect to the relevant arguments, one can find useful differential relations following from the third constraint: \nA ' 1 = bA ' 23 , B ' 1 = aB ' 23 . (49) \n(Note that primes cannot be omitted in these ratios!) With the help of (44) it is easy to establish the separability of equation (43). Substituting the product \nφ ( x µ ) = e -iωt + imϕ R ( r ) Y ( y ) , (50) \nand dividing the Eq.(43) by φ we obtain: \n(( A 2 ) ' R ) ' R + (( B 2 ) ' Y ) ' Y + U ( r ) -V ( y ) = 0 , (51) \nwhere the primes denote derivatives with respect to the corresponding arguments r, y , and the potential terms are equal to \nU ( r ) = ( ω √ A 3 -m √ A 5 ) 2 -µ 2 A 1 , (52) \nV ( y ) = ( ω √ B 3 -m √ B 5 ) 2 + µ 2 B 1 . (53) \nThe separability of Eq. (51) is obvious. The third constraint effectively reduces the number of arbitrary functions to seven, two of which A 2 , B 2 can still be fixed using the gauge freedom, so the number of essentially independent functions is five. \nThe separability of the Klein-Gordon equation can also be investigated using the secondorder Carter differential operator associated with the Killing tensor: \nˆ K = ∇ µ K µν ∇ ν , (54) \nwhich must commute with D'Alembert operator [33]. The commutator was elaborated in [42]: \n[ /square , ˆ K ] φ = 4 3 ∇ α ( K σ [ α R β ] σ ) ∇ β φ. (55) \nProjecting the tensor on the right side onto the NP tetrad, we obtain \nK σ [ α R β ] σ = 2( K ln + K m ¯ m )( n β ( ¯ m α Φ 01 + m α Φ 10 ) -n α ( ¯ m β Φ 01 + m β Φ 10 )+ +( l β ¯ m α -l α ¯ m β )Φ 12 +( l β m α -l α m β )Φ 21 ) . (56) \nThus, a sufficient condition for commutativity is that two Ricci scalars vanish: \nΦ 01 = R µν l µ m ν / 2 = Φ 10 , Φ 12 = R µν n µ m ν / 2 = Φ 21 . (57) \nTheir expression, taking into account the third constraint (46) in terms of Benenti quantities, is given by formula (B2) in Appendix B. Equating them to zero, we obtain: \naA '' 23 -bB '' 23 = 0 , (58) \nwhere the primes denote derivatives with respect to the corresponding arguments. Given (49), this can also be rewritten as \na 2 A '' 1 -b 2 B '' 1 = 0 . (59) \nSince one term is a function of r and the other is a function of y , each must be a constant. In other words, A 1 and B 1 must be at most quadratic polynomials of the corresponding arguments. \nLet us give an NP-description of the operators /square and ˆ K in terms of derivatives (19) and the spin coefficients: \n/square = ( D + ε + ¯ ε -ρ -¯ ρ )∆ + (∆ + µ + ¯ µ -γ -¯ γ ) D --( δ -τ -¯ α + β + ¯ π ) ¯ δ -( ¯ δ -α -¯ τ + π + ¯ β ) δ, (60) \nˆ K = ( D + ε + ¯ ε -ρ -¯ ρ ) K ln ∆+(∆+ µ + ¯ µ -γ -¯ γ ) K ln D + +( δ -τ -¯ α + β + ¯ π ) K m ¯ m ¯ δ +( ¯ δ -α -¯ τ + π + ¯ β ) K m ¯ m δ, (61) \nIt is useful to introduce the following set of operators: \nD ± n = √ A 3 /A 2 ∂ t ± ∂ r + √ A 5 /A 2 ∂ ϕ ± n∂ r (ln A 2 ) , (62a) \nwhere n and s are integers. These operators satisfy two simple identities: \nL ± s = √ B 3 /B 2 ∂ t ± i ∂ y + √ B 5 /B 2 ∂ ϕ ± is ∂ y (ln B 2 ) , (62b) \nD ± n A 2 k = A 2 k D ± n + k , L ± s B 2 l = B 2 l L ± s + l . (63) \nIn terms of these operators, the directional derivatives acting on a scalar function will have the form: \nD = √ A 2 / 2Σ D -0 , ∆ = √ A 2 / 2Σ D + 0 , δ = √ B 2 / 2Σ L -0 , ¯ δ = √ B 2 / 2Σ L + 0 . (64) \nAlso, taking into account the explicit form of the spin coefficients (A1-A4), we can obtain the following representations for the combinations included in the d'Alembertian: Also, taking into account the explicit form of the spin coefficients (A1-A4), we can obtain the following representations for the combinations included in the d'Alembertian: \nε + ¯ ε -ρ -¯ ρ = √ A 2 / 8Σ 3 ( 2Σ ∂ r ln [ √ A 3 B 5 -√ A 5 B 3 ] -3 ∂ r Σ ) , (65a) \nIt is easy to see that in the general case, after substitution into the Klein-Gordon equation, the latter does not allow separation of variables unless the second constraint (46) is imposed. In this case, our expressions are simplified to \n¯ π -τ + β -¯ α = -i √ B 2 / 8Σ 3 ( 2Σ ∂ y ln [ √ A 3 B 5 -√ A 5 B 3 ] -3 ∂ y Σ ) . (65b) \nε + ¯ ε -ρ -¯ ρ = √ A 2 ∂ r (2Σ) -1 / 2 -(2Σ) -1 / 2 ∂ r √ A 2 , (66a) \nUsing Eqs. (62-66), the d'Alembert operator (60) can be cast into the form \n¯ π -τ + β -¯ α = i √ B 2 ∂ y (2Σ) -1 / 2 -i (2Σ) -1 / 2 ∂ r √ B 2 . (66b) \n/square = 1 2Σ [ A 2 ( D -1 D + 0 + D + 1 D -0 ) -B 2 ( L -1 L + 0 + L + 1 L -0 )] , (67) \nwhile the Carter operator can be presented as \nˆ K = 1 2Σ [ A 2 B 1 ( D -1 D + 0 + D + 1 D -0 ) + B 2 A 1 ( L -1 L + 0 + L + 1 L -0 )] = B 2 2 ( L -1 L + 0 + L + 1 L -0 ) + B 1 /square . (68) \nBy direct substitution one can verify that it commutes with the d'Alembertian (67), taking into account the fact that the second term in the solution space /square φ = 0 vanishes.", 'A. Separable supergravity backgroounds': "By imposing three constraints on the BF coefficients, we arrive at the following metric paremeterization: \nds 2 = A 2 B 2 Σ ( √ B 5 dt -√ B 3 dϕ ) 2 -A 2 B 2 Σ ( √ A 5 dt -√ A 3 dϕ ) 2 -Σ A 2 dr 2 -Σ B 2 dy 2 , (69) \nΣ = √ A 2 B 2 ( √ A 3 B 5 -√ A 5 B 3 ) . (70) \nwhere \nThis is precisely the Carter metric ansatz [16] for which Einstein's vacuum and electrovacuum equations were solved directly. Several families of solutions were obtained among which Kerr and Kerr-Newman black holes belonging to the Petrov type D (for further applications see also [17]) \nConsider now the generic 4D supergravity bosonic action which is a special scalar-vectortensor theory with multiple scalar and vector fields [5, 6]. It includes n s scalar moduli Ψ A , A = 1 , . . . , n s (dilatons and axions) and n v abelian vector fields F I = dA I , I = 1 , . . . , n v : \nS = ∫ d 4 x [( R -1 2 f AB ∂ µ Ψ A ∂ µ Ψ B -1 2 K IJ F I µν F Jµν ) √ -g -1 2 H IJ F I µν F J λτ /epsilon1 µνλτ ] . (71) \nThe scalar moduli parametrize a four-dimensional coset (e.g. U (8) /E 7(7) for N = 8 supergravity) with an associated target metric f AB . Vector fields transform under the same global symmetry implemented by real symmetric matrices K IJ , H IJ depending on scalar fields Ψ A (summation over the repeated indices I, J is understood). The corresponding Einstein equations read: \nR µν = 1 2 f AB Ψ A ,µ Ψ B ,ν -K IJ ( F I µλ F Jλ ν + 1 4 g µν F I αβ F Jαβ ) . (72) \nThe scalar fields depend only on r, y , so they contribute directly only in transverse part of the Ricci tensor. The Maxwell sector at the right hand side of this equations has the same structure as in pure Einstein-Maxwell system. All this ensure (the details will be given elsewhere) that supergravity Einstein equations will separate for our metric ansatz similarly to Einstein-Maxwell case [16]. Note, that the Smarr mass formulas for black hole solutions of the theory (71) was shown recently to repeat the case of the Einstein-Maxwell theory [43]. \nTo our knowledge, no direct integration of the Einstein supergravity equations by separation of variables has been performed so far, although the leading solutions for black holes have been obtained using indirect methods. In addition to Harrison transformations, one can mention the guesswork of obtaining the BPS solution [9] or the integration of the null geodesic equations for \nthe target space of the sigma model [14] (also known as the nilpotent orbit method [15]). The situation is different in the electrovacuum case, where large classes of solutions were obtained by direct integration of the Ernst equations.", 'V. KILLING-YANO AND TYPE D': 'Here we find conditions on the BF functions that guarantee that the solution is algebraically special, which for our class I B means type D . In this case, the Killing-Yano tensor that guarantees the separability of the Dirac equation also exists. We will use this as a tool to find conditions for type D .', 'A. Killing-Yano': "The Killing-Yano tensor Y µν = -Y νµ satisfying the equation \n∇ ( α Y µ ) ν = 0 , (73) \ncan be regarded as a 'square root' of the Killing tensor: \nY µ α Y αν = K µν . (74) \nSince we know the Killing tensor (4) independently of Petrov type of the metric, we may consider Eqs.(73) and (74) as independent conditions which prescribe the metric to be of type D , and define the KY tensor itself. In NP description, our Killing tensor has only two non-zero components (34): K ln and K m ¯ m . So projecting (74) on the NP tetrad, one obtains: \nY 2 ln = K ln , Y 2 m ¯ m = -K m ¯ m , (75) \nthe other NP components of the KJ tensor being zero. Together with Eq. (34) this gives \nY ln = p √ B 1 , Y m ¯ m = iq √ A 1 , p = ± 1 , q = ± 1 , (76) \nwhere we introduced sign factors to be fixed later.", 'B. Consistency conditions for type D': "Now we have to satisfy the KY equation (73) for consistency, in other words, we have to find new constraint equations on the BF coefficient functions which ensure that solution belongs to \ntype D . Projecting KJ equation (73) onto the NP tetrad we get sixteen equations, from which, with account for (34) and pairwise equality of the spin coefficients (24), only the following six are relevant: \n( τ + ¯ π ) Y ln -( τ -¯ π ) Y m ¯ m = 0 , ( ρ + ¯ ρ ) Y ln -( ρ -¯ ρ ) Y m ¯ m = 0 , (77a) \nDY m ¯ m -ρ ( Y ln -Y m ¯ m ) = 0 , δY ln -¯ π ( Y ln + Y m ¯ m ) = 0 , (77b) \n∆ Y m ¯ m + µ ( Y ln -Y m ¯ m ) = 0 , δY ln + τ ( Y ln -Y m ¯ m ) = 0 . (77c) \nFor the spin coefficients involved, from (A1,A2) with account for the constraint (46) one obtains: \nµ = ρ = b 2Σ √ A 2 / 2Σ( A ' 23 -i B ' 23 ) , τ = π = a 2Σ √ B 2 / 2Σ( A ' 23 -i B ' 23 ) , (78) \nThen from the first pair of equations of the system we get the following relation: \np A ' 23 √ B 1 = q B ' 23 √ A 1 , (79) \nwhich separates into pair of one-variable equations: \npA ' 23 = 2 √ A 1 , qB ' 23 = 2 √ B 1 (80) \nwhere we introduced the coefficient two for further convenience. Now we have to satisfy the second and third pairs of equations (77b, 77c). Taking into account definition of NP derivatives (19), we obtain \n( √ A 1 ) ' = b/p, ( √ B 1 ) ' = -a/q (81) \nFrom (2) and (46) follows that \nA 1 + B 1 = bA 23 -aB 23 . \nTo satisfy this, one has to choose p = 1 and q = -1, Thus, we have found a system of restrictions on the BF coefficients that guarantee the existence of the KY tensors, i.e. the belonging of the metric to the D type: \nA ' 1 = 2 b √ A 1 , B ' 1 = 2 a √ B 1 ; (82a) \nIntergration of this system provides generic form for some of the metric coefficients for type D BF sector: \nA ' 23 = 2 √ A 1 , B ' 23 = -2 √ B 1 . (82b) \nA 1 = ( br + c 1 ) 2 , B 1 = ( ay + d 1 ) 2 , (83a) \nA 23 = br 2 +2 c 1 r + c 2 , B 23 = -( ay 2 +2 d 1 y + d 2 ) , (83b) \nwhere the constants c 1 , d 1 , c 2 and d 2 according to equations (2, 46) are subject to the following condition: \nb c 2 + ad 2 = c 1 2 + d 1 2 . (84) \nOnly when all these conditions are satisfied does the Killing-Yano tensor become legitimate and finally given by the NP components \nY ln = √ B 1 , Y m ¯ m = -i √ A 1 . (85) \nIt remains to check that with these conditions the Weyl tensor projections Ψ 1 , 3 vanish. Taking into account the conditions (11 15, 46), Ψ 1 , 3 can be written in the form \n8Σ 3 Ψ 1 , 3 = -√ A 2 B 2 { Σ( aA '' 23 -bB '' 23 ) -ab ( A ' 23 2 + B ' 23 2 ) } , (86) \nSubstituting here (83a 83b 84), one finds that Ψ 1 , 3 = 0 indeed. Also note that in the static case, when A 5 = 0 = B 3 and hence a = 0 , B 23 = 0, we have Ψ 1 , 3 = 0 without imposing a condition of type D . \nThe Ricci scalars and the Weyl scalar Ψ 2 for generic type I B are given in the Appendix B.", 'C. Conformal Killing tensor': 'The conformal Killing tensor (CKT) must satisfy the equations \n∇ ( α K µν ) = Ω ( α g µν ) , 6 Ω α = (2 ∇ σ K α σ + ∇ α K ) , K = g µν K µν . (87) \nFor Ω α = 0, these equations in our I B class of metrics have a solution with only two NP projections K ln , K m ¯ m , so it is natural to look for CKT of similar structure. Projecting Eq. (87) onto the chosen tetrad (17) one gets the following system of four equations \n( D + ρ + ¯ ρ )( K ln + K m ¯ m ) = 0 , ( δ + τ -¯ π )( K ln + K m ¯ m ) = 0 , (∆ -µ -¯ µ )( K ln + K m ¯ m ) = 0 , ( ¯ δ + ¯ τ -π )( K ln + K m ¯ m ) = 0 . (88) \nAfter substitution of an explicit expression (A1, A2) for the spin coefficients, this can be easily integrated. Thus, up to an arbitrary function S ( r, y ), we can write down the NP projections \nK ln = Σ -S ( r, y ) , K m ¯ m = S ( r, y ) . (89) \nThe conformal factor Σ = Σ( r, y ) here may be non-separable. The conformal tensor transforms into an exact one for S = A 1 and the condition (2).', 'A. Black hole shadows and characteristic surfaces': 'Recent interest to black hole shadows [21, 22] gave rise to new theoretical approaches. Strong gravitational lensing, quasi-normal modes of black holes and the formation of black hole shadows are determined by the motion of massless particles near photon surfaces [44-46], on which photons winds before scatter to infinity. Such surfaces exist in static metrics, while in rotating case similar role is played by surfaces where non-planar spherical orbits are located [47]. In the first case, the corresponding hypersurfaces in space-time are umbilic [48] (the tensor of external curvature is proportional to the induced metric), in the second case they are partially umbilic, which means that the latter property is satisfied not for tensors as a whole, but by their convolutions with a part of the vectors of the tangent space. \nIn a similar way, one can consider the characteristic surfaces of massive particles, as well as particles of variable mass, for example, photons in plasma [49, 50]. It was noted [51-53] that the existence of hypersurfaces with the above properties correlates with the existence of the Killing tensor. Eventually, it was shown [19, 20] that Killing tensors, which reduce to trivial (products of Killing vectors) on hypersurfaces that can be used to stratify the entire spacetime, ensure that these hypersurfaces contain generalized photon surfaces, including those associated with the motion of massive particles. \nIt is remarkable that the BF Killing tensor (4) possesses the slice-reducibility property. As a result, the BF ansatz can be used to give unified description of particle surfaces and shadows of supergravity black holes [19, 20, 49, 50].', 'B. Slice-reducibility': 'To see that Benenti Killing tensor (4) is slice-reducible [20], it is enough write it in the form \nK µν = -A 1 g µν -A 2 δ µ r δ ν r + ˜ K µν r , (90) \nwhere \n˜ K µν r = A 3 δ µ t δ ν t +2 A 4 δ ( µ t δ ν ) ϕ + A 5 δ µ ϕ δ ν ϕ . (91) \nThe first term in (90) is trivial Killing tensor on S r , since A 1 = const there. The second term is orthogonal to S r and thus irrelevant, while the third term ˜ K µν r is a reducible Killing tensor \non this hypersurface, being presented as linear combination of the tensor products of Killing vectors projected onto it. \nSimilarly the BF Killing tensor can be presented in terms of S y foliation: \nK µν = B 1 g µν + B 2 δ µ y δ ν y + ˜ K µν y . (92) \nwith the slice projection \n˜ K µν y = B 3 δ µ t δ ν t +2 B 4 δ ( µ t δ ν ) ϕ + B 5 δ µ ϕ δ ν ϕ . (93) \nIf we omit terms with the metric tensor in Eqs. (90) and (92), we get simple expressions for conformal Killing tensors, which depend only on one coordinate. \nThus, the Benenti ansatz ensures the Killing tensor is slice-reducible with respect to both foliations of spacetime. In particular, automorphism (5) naturally arises as a symmetry of slices discussed in [20].', 'C. Photon and massive particle surfaces': "The black hole horizon r h is the largest root of the equation \nA 2 ( r h ) = 0 , A 2 > 0 for r > r h . (94) \nIn spacetime this is a null hypersurface. Consider the timelike three-dimensional hypersurface S r for r = const > r h with the unit outward normal spacelike covector n µ = -√ Σ /A 2 δ r µ . The induced metric and the extrincsic curvature of S r in the bulk coordinates read \nh µν = g µν + n µ n ν , χ µν = h α µ h β ν ∇ α n β = h α µ h β ν ( n β,α -Γ λ αβ n λ ) . (95) \nχ µν dx µ dx ν = n r 2 g rr ( g ab,r dx a dx b + g yy,r dy 2 ) (96) \nFrom here one finds: \nThe hypersurface S r in the static spacetime contains a photon surface, like r = 3 M in Schwarzschild metric, which is the loci of confined photon orbits. Its radius is determined by the umbilicity condition \nχ µν = χ α α 3 h µν , µ, ν = t, ϕ, y (97) \n(ln g tt ) ,r = (ln | g ϕϕ | ) ,r = (ln | g yy | ) ,r (98) \nwhich reduces to \nSubstituting BF coefficients one finds that the last equality is trivially satisfied (showing that we are dealing with geometrical sphere), while the first one defines the radius of the photon sphere. It may be written as \n(ln A 2 ) ' = 2(ln A 23 ) ' , or simply A ' 3 = 0 , (99) \nwhere prime denotes the derivative with respect to r . Let's test this equation for the ReissnerNordstrom metric, in which case A 2 = r 2 -2 Mr + Q 2 , A 23 = r 2 . From Eq. (99) one obtains \nr 2 -3 Mr +2 Q 2 = 0 ⇒ r = 3 2 ( M ± √ M 2 -8 Q 2 / 9 ) , (100) \nwhich is indeed a correct expression for the photon spheres in Reissber-Nordstrom spacetime. One can also look for timelike hypersurfaces such that a particle of mass µ moving along an initially tangent worldline to S r with tangential vector p µ (normalized as p µ p ν = µ 2 ) remains there forever. Such a surface was named massive particle surface (MPS). The surface radius r will depend on the particle energy and angular momentum integrals E, L , and the family of MPS will foliate certain four-dimensional subspace in space-time. The (partial) umbilicity condition in this case must be satisfied only in those directions of the tangent space of S r that are orthogonal to the Killing vectors. \nIn the general stationary case, the induced metric and the second quadratic form can be written as \nh µν dx µ dx ν = A 2 Σ ( bdt -B 23 dϕ ) 2 -B 2 Σ ( adt -A 23 dϕ ) 2 -Σ B 2 dy 2 , (101) \nχ µν dx µ dx ν = 1 2 √ A 2 Σ ∂ r h µν dx µ dx ν . (102) \nIn this case the MPS radius is a solution of the following equation of partial umbilicity [49]: \n1 2 h yy √ | g rr | ∂ r p 2 = χ yy ( p 2 -µ 2 ) , p 2 = g ab p a p b , p a = ( -E,L ) , (103) \nwhere E, L are constants of motion defined in (36). Substition of BF coefficient functions leads to \n( √ A 3 E -√ A 5 L ) ' ( √ A 3 E -√ A 5 L ) = 1 2 µ 2 A ' 1 (104) \nwhere primes denotes derivatives over r . This equation is nothing but U ' r = 0 for the radial potential defined in (38). As was noted in Sec. 3, for spherical orbits the Carter constant is determined by two other integrals of motion. This explains why the equation for MPS radius defines it as function of two parameters only. \nDetailed description of spherical orbits of massive particles in Kerr spacetime was given by Teo, [47], where the suitable domains of C are determined. More general setting, including the case of charged particles and presence of the electromagnetic field can be found in [49]. In the massless case, spherical orbits are determined by the ratio L/E , called the impact parameter, due to the scale invariance of the geodesic equations. \nOur Eq. (104) is a general equation defining the MPS radius in terms of the motion integrals E, L for the entire class of I B -type metrics. The regions of these parameters is restricted by consistency conditions, this is discussed in detail in [47, 49]. The particular cases are described in the next section.", 'VII. SUPERGRAVITY BLACK HOLES': 'Black hole solutions in N = 2 , 4 , 8 supergravity models have been constructed using the Harrison transformations applied to the Kerr metric. The most general results were obtained in this way by Chow and Compere [37]. Here we show that the known solutions indeed belong to our doubly restricted (15,46) BF class I B . This class ensures the separability of not only the Hamilton-Jacobi and Klein-Gordon equations, but also the Einstein equations [16]. All metrics also satisfy the gauge condition (11) in Boyer-Lindquist coordinates with the gauge constant a equal to the rotation parameter. Therefore, in this section we will use the a symbol in both senses.', 'A. Kerr-Newman': 'Kerr-Newman metric may be viewed as type D solution of N = 2 pure supergravity which does not involve scalar fields. In Boyer-Lindquist coordinates, the interval reads: \nds 2 = ∆ r / Σ ( dt -a sin 2 θ dϕ ) 2 -1 / Σ sin 2 θ ( adt -( r 2 + a 2 ) dϕ ) 2 -Σ / ∆ r dr 2 -Σ dθ 2 , ∆ r = r 2 -2 Mr + a 2 + Q 2 , Σ = r 2 + a 2 cos θ, \nThe corresponding BF metric coefficients A k ( r ) and B k ( y ) read: \nA 1 = r 2 , B 1 = a 2 y 2 A 3 = ( r 2 + a 2 ) 2 / ∆ r , B 3 = a 2 (1 -y 2 ) , A 5 = a 2 / A 2 = a 2 / ∆ r , B 5 = 1 / B 2 = 1 / (1 -y 2 ) , (105) \nwhere y = cos θ . It is easy to verify that conditions (15,46) are fulfilled: \nA 2 4 = A 3 A 5 , B 2 4 = B 3 B 5 , b = 1 , A 1 + B 1 = √ A 2 B 2 ( √ A 3 B 5 -√ A 5 B 3 ) . (106) \nThe BF formulas for type D (83a,83b,84) also hold. The non vanishing Weyl and Ricci scalars for this metric are \nΨ 2 = Q 2 -M ( r -i ay ) ( r -i ay ) 3 ( r + i ay ) , Φ 11 = Q 2 2( r 2 + a 2 ) 2 . (107) \nThe MPS equation is a 5-th order polynomial \n( E ( r 2 + a 2 ) -aL ) [ Er (∆ r -Mr + Q 2 ) + aL ( r -m ) + a 2 EM ] -µ 2 r ∆ r 2 = 0 , (108) \nwhich can be subjected to numerical analysis.', "B. Gal'tsov-Kechkin solution": "The seven-parametric family of rotating dilaton-axion-NUT dyons in truncated N = 4 supergravity obtained in [8] reads: \nds 2 = ∆ r -a 2 sin 2 θ Σ ( dt -wdϕ ) 2 -Σ ( dr 2 ∆ r + dθ 2 + ∆ r sin 2 θ ∆ r -a 2 sin 2 θ dϕ 2 ) , ∆ r = ( r -r -)( r -2 M ) + a 2 -( N -N -) 2 , Σ = r ( r -r -) + ( a cos θ + N ) 2 -N 2 -, w = 2 a 2 sin 2 θ -∆ r [ N ∆ r cos θ + a sin 2 θ ( M ( r -r -) + N ( N -N -)) ] , r -= M | Q -iP | 2 M + iN 2 , N -= N | Q -iP | 2 2 M + iN 2 . \n| | | | \nThe physical parameters are mass M , NUT-charge N , electric and magnetic charges Q, P , a rotation parameter a , the asymptotic values of dilaton and axion a set zero (for P = 0 = N this metric transforms to the Sen metric [54] via the coordinate shift r → r + r -). In the BF form (1) with cos θ = y we have: \nA 1 = r ( r -r -) , B 1 = ( ay + N ) 2 -N 2 -, A 3 = ( r ( r -r -) + a 2 + N 2 -N 2 -) 2 / ∆ r , B 3 = [ a (1 -y 2 ) -2 Ny ] 2 / 1 -y 2 , A 5 = a 2 / A 2 = a 2 / ∆ r , B 5 = 1 / B 2 = 1 / 1 -y 2 , (110) A 2 4 = A 3 A 5 , B 2 4 = B 3 B 5 , a = a, b = 1 , A 1 + B 1 = √ A 2 B 2 ( √ A 3 B 5 -√ A 5 B 3 ) , \n(109) \nthe last relation being the separability condition (46) for the Klein-Gordon equation. The metric is a non-vacuum Petrov type I B solution, with the following set of non-zero Weyl and Ricci scalars: \nΨ 1 = Ψ 3 = a (4 N 2 -+ r 2 -) sin θ √ ∆ r 8Σ 3 , \n12Σ 3 Ψ 2 = -12( M -iN )( r + i ( a cos θ + N )) 3 +6 NN -( 2( r + i ( N + a cos θ )) -r -) 2 + + r 3 -(8 M -r ) + 8 NN 3 --4 N 4 --6 ar -cos θ (5 M -3 iN )( a cos θ +2 N -2 ir )+ +2 r -( M ( 15( r + iN ) 2 +7 N 2 -) +18 N 2 r +9 iN 3 +3 iN ( N 2 --3 r 2 ) -2 N 2 -r ) + +4 N 2 -( 2 a 2 -a cos θ (3 i ( M + iN ) + a cos θ ) -5 Mr +2 N 2 -3 iN ( M + r ) ) + +4 N 2 -r 2 + r 2 -( 2 a 2 -24 iMN -7 N 2 +2 NN --N 2 -) --r 2 -( a cos θ (6 i (4 M -iN ) + a cos θ ) + 26 Mr -6 iNr -r 2 ) , \n16Σ 3 Φ 11 = a 2 cos 2 ( θ ) ( 8 Mr -+16 NN --4 N 2 --r 2 -) +16 MN 2 -r --24 MN 2 -r + +8 aN cos θ ( 2 Mr -+4 NN --4 N 2 --r 2 -) + N 2 ( 8 Mr --28 N 2 --7 r 2 -) + +8 Mr -r 2 -14 Mr 2 -r +16 N 3 N -+2 NN -( 4 N 2 -+3 r 2 -+8 r 2 -8 r -r ) + + N 2 -r 2 --4 N 2 -r 2 +4 N 2 -r -r +4 N 4 --r 2 -r 2 + r 3 -r +6 Mr 3 -, Λ = R 24 = -( 4 N 2 -+ r 2 -) ( a 2 sin 2 θ +∆ r ) 48Σ 3 . (111) \nMatter fields ensure that the Carter and d'Alembert operators commute, since the NPprojectors of the Ricci tensor 57 are zero. The MPS equation reads: \nE ( r ( r -r -) + N 2 -N 2 -) [ (2 r -r -)( r 2 -2 rr -+ a 2 + N 2 -2 NN -+3 N 2 -) --2 M (3 r 2 -rr -+2 r 2 -+ N 2 -N 2 -) ] -4 a 2 ELM + a 4 E 2 (2 r +2 M -r -)+ + a 2 [ 2 E 2 (2(2 r -r -)( N 2 --NN -) -2 M (( r -r -) 2 ) -N 2 + N 2 -) -L 2 (2 r -r --2 M ) ] + +4 aEL [ M ( r -r -) -N 2 + N 2 -+ N -( N -N -)(2 r -r -) ] -µ 2 ∆ 2 (2 r -r -) = 0 .", 'C. SWIP solutions': "Another family of supersymmetric extremal stationary solutions of N = 4 , D = 4 supergravity containing a set of electric and magnetic charges satisfying Bogomol'nyi (BPS) bound \nwas obtained in [9]. The metric has the same form (109) with \n2 \nw = ∆ r -a 2 sin 2 θ ( N ∆ r cos θ + a sin 2 θ [ m ( r -( m + | Υ | )) + 1 2 ( |M| 2 -| Υ | 2 ) ]) , ∆ r = r [ r -2( m + | Υ | )] + a 2 +( m + | Υ | ) 2 , Σ = r ( r -2 | Υ | ) + ( a cos θ + N ) 2 . (112) \nThe solution depends on complex mass M = m + iN , axion-dilaton charge Υ = -2 M ∑ n [ ¯ Γ ( n ) ] 2 , and electromagnetic charges Γ ( n ) = 1 2 ( Q ( n ) + iP ( n ) ) , n = 1 , 2. The BPS identity reads: \n|M| 2 + | Υ | 2 -4 ∑ n | Γ ( n ) | 2 = 0 . (113) \nIn this case \nA 1 = r ( r -2 | Υ | ) , B 1 = N + ay, A 3 = ( r ( r -2 | Υ | ) + N 2 + a 2 ) 2 / ∆ r , B 3 = ( a (1 -y 2 ) -2 Ny ) 2 / 1 -y 2 , A 5 = a 2 / A 2 = a 2 / ∆ r , B 5 = 1 / B 2 = 1 / (1 -y 2 ) . (114) \nThe non-zero NP quantities are: \n2 √ \nΨ 1 = Ψ 3 = a sin θ | Υ | ∆ r 2Σ 3 , 3Σ 3 Ψ 2 = 3( m -iN )( m -r -ia cos θ )( r + i ( N + a cos θ )) 2 + +3( m -iN )(3( r + ia cos θ ) -2 m + iN )( r + i ( N + a cos θ )) | Υ |--2 | Υ | 2 (3 N 2 -2( m 2 + r 2 ) + 7 mr +6 iN ( m -r ) + a 2 cos 2 θ ) --2 | Υ | 3 (4( r -m ) + 3 i ( N + a cos θ )) + 3 ia | Υ | 2 cos θ (2( m -iN ) -r ) + 4 | Υ | 4 , 4Σ 3 Φ 11 = 3 | Υ | 2 ( a 2 cos 2 θ +( m -r + | Υ | ) 2 )+ +2Σ( m 2 -a cos θ (2 N + a cos θ ) -( r -| Υ | ) 2 +Σ) , Φ 00 = Φ 22 = | Υ | 2 ∆ r 2Σ 3 , Φ 02 = -a 2 | Υ | 2 sin 2 θ 2Σ 3 , Λ = -| Υ | ( a 2 (1 + sin 2 θ ) + ( m -r + | Υ | ) 2 ) 12Σ 3 . (115) \nSince Φ 01 = Φ 12 = 0, the separability condition for the Klein-Gordon equation aldo holds. \nThe MPS equation reads: \nE ( r ( r -2 | Υ | ) + N 2 + a 2 ) [ { E (∆ r + | Υ | 2 -M 2 -N 2 ) + aL } ( r -M -| Υ | ) + 2 a 2 EM ] --µ 2 ∆ 2 r ( r -| Υ | ) = 0 . (116)", 'D. STU black holes': 'The STU solution is a general asymptotically flat, stationary black hole in supergravity N = 8 [37], parameterized by mass M , rotation parameter a , and four electric charges s I , I = 1 , .., 4. The metric has the same form (109) with \n∆ r = r 2 -2 Mr + a 2 , w = -2 Maω sin 2 θ ∆ r -a 2 sin 2 θ , ω = ((Π c -Π s ) r +2 M Π s ) , Σ 2 = 4 ∏ I =0 ( r +2 Ms 2 I ) + a 4 cos 4 θ + \n+2 a 2 cos 2 θ ( r 2 + Mr 3 ∑ I =0 s 2 I +4 M 2 (Π c -Π s )Π s -2 M 2 3 ∑ I<J<K s 2 I s 2 J s 2 K ) , \nand the products of charges are defined as follows \nΠ s = 4 ∏ i =1 s I = 4 ∏ I =1 sinh δ I , Π c = 4 ∏ I =1 √ 1 + s 2 I = 4 ∏ I =1 cosh δ I , s 2 I = sinh 2 δ I . \nIn the case of pairwise equality of charges s 1 = s 3 = S 1 , s 2 = s 4 = S 2 we are dealing with the so-called two-charge solution. If S 2 = 0, then the metric reduces to the Kerr-Sen solution [54]. For a more general classification, see [35]. It is possible to transform the metric to the BF form (1) only in the case of a two-charge solution, for which \nΣ 2 ch = r 2 + a 2 cos 2 θ +2 Mr ( S 2 1 + S 2 2 ) + 4 M 2 S 2 1 S 2 2 = r 2 -2 Mr + a 2 cos 2 θ +2 Mw 2 ch (117) \nThen for the functions A i ( r ), B i ( x ) we have: \n2 2 2 \nA 1 = r -2 Mr +2 Mw 2 ch , B 1 = a y , A 3 = ( r 2 -2 Mr + a 2 +2 Mw 2 ch ) 2 / ∆ r , B 3 = a 2 (1 -y 2 ) , A 5 = a 2 / A 2 = a 2 / ∆ r , B 5 = 1 / B 2 = 1 / (1 -y 2 ) . (118) \nThe Weyl and Ricci scalars are \n4Σ \nΨ 1 = Ψ 3 = aM 2 sin θ ( S 2 1 -S 2 2 ) 2 √ ∆ r 2Σ 3 2 ch , 3Σ 3 2 ch M Ψ 2 = -3( r + ia cos θ ) 3 -3( r + ia cos θ ) 2 ( M + r + ia cos θ )( S 2 1 + S 2 2 ) --2 M ( r 2 +2 Mr -a 2 +3 iar cos θ + a 2 y 2 )( S 4 1 + S 4 2 )+ +12 M 2 S 2 1 S 2 2 ( M -r -ia cos θ )( S 2 1 + S 2 2 )+ +4 M S 2 1 S 2 2 (5 a 2 cos 2 θ -a 2 +4 Mr -5 r 2 +3 iaM cos θ -9 iar cos θ ) , Φ 00 = Φ 22 = M 2 ∆ r ( S 2 1 -S 2 2 ) 2 2Σ 3 2 ch , Φ 02 = Φ 20 = -a 2 M 2 sin 2 θ ( S 2 1 -S 2 2 ) 2 2Σ 3 2 ch , 3 2 ch M 2 Φ 11 = 4( r 2 + a 2 cos 2 θ )( S 2 1 + S 2 2 ) + (2 Mr +3 r 2 +3 a 2 cos 2 θ )( S 4 1 + S 4 2 )+ +16 M ( r + M ) S 2 1 S 2 2 ( S 2 1 + S 2 2 ) + 32 M 2 S 4 1 S 4 2 +2 S 2 1 S 2 2 ( r 2 +14 Mr + a 2 cos 2 θ ) , Λ = 1 24 R = -M 2 ( S 2 1 -S 2 2 ) 2 ( ∆ r + a 2 sin 2 θ ) 12Σ 3 2 ch . (119) \nSimilar to the above cases of N = 4, the quantum separability condition (55) is also satisfied for the two-charge STU solution. \nThe MPS equation reads \nE 2 [ ∆ 2 ( r -M +2 M Σ s 2 ) -4 M 2 r ( Mr -a 2 )Σ 2 s 2 -16 M 4 ( r -M )Π 2 s 2 -8 M 3 ( r 2 -a 2 )Π s 2 Σ s 2 ] + +2 aELM [ 4 M ( r -M )Π s 2 -( r 2 -a 2 )Σ s 2 ] -a 2 L ( r -M ) -µ 2 ∆ 2 ( r -M + M Σ s 2 ) = 0 , (120) \nwhere Σ s 2 = S 2 1 + S 2 2 , Π s 2 = S 2 1 S 2 2 . \nFrom the above analysis of the various supergravity solutions for black holes, their similarity in the form of BF is obvious. For all of them, the coefficient functions A 1 , B 1 , A 23 , B 23 are just simple quadratic polynomials, which make us to believe that these solutions can be obtained by directly integrating the general supergravity equations.', 'VIII. CONCLUSIONS': "We have shown that the suitably refined Benenti-Francaviglia ansatz for stationary axisymmetric spacetimes, admitting a non-trivial Killing tensor of rank two, defines a class of non-algebraically special metrics that admits two null geodesic congruences without shear. If we additionally impose the separability condition of the Klein-Gordon equation, the resulting ansatz coincides with Carter's parametrization, which led him to perform a direct integration of \nEinstein's vacuum and electrovacuum equations, reproducing, in particular, the Kerr-Newman metric and its extensions of type D. Our class of algebraically non-special metrics opens the way to a generalization of this approach to the general supergravity bosonic action (71) containing a set of vector and scalar fields. It was demonstrated explicitly that all known N = 2 , 4 , 8 supergravitational black holes, with the exception of the so-called four-charge solution, belong to our class indeed. \nAn algebraically special subclass of our metrics contains only the type D . For this case, we have obtained an explicit parametrization of the BF coefficient functions integrating the Killing-Yano equations. \nOur class of metrics leads to a general description of the confining surfaces of photons and massive particles that have recently been introduced in the theory of black hole shadows. The Killing tensor given by the BF formula has a crucial property of slice-reducibility [20] with respect to spacetime bundles by three-dimensional timelike hypersurfaces with constant r or y . This opens a way to universal description of supergravity black hole shadows. Here we have given a general equation for the confining surfaces of massive particles and its particular form for all examples of supergravity solutions. \nTo summarize, the restricted Benenti-Francavilla metric gives a unified description of type I supergravity black holes, revealing many of their common properties. This was established by combining the Newman-Penrose analysis of the BF class of metrics with a specific form of known solutions. We hope that it will be possible to directly solve the supergravity equations of motion in the obtained parameterization, thus extending Carter's program to solve the Einstein equations for metrics admitting second-rank Killing tensors.", 'Acknowledgements': "The authors thanks Kirill Kobialko, Igor Bogush and G'erard Cl'ement for useful suggestions and discussions. This work was supported by Russian Science Foundation under Contract No. 23-22-00424.", 'Appendix A: Spin coefficients': 'Here we give NP spin coefficients calculated for the tetrad (17) without constraint (46) on the conformal factor Σ: \nµ = ρ = -√ A 2 ∂ ∂ r 1 √ 2Σ -iB 5 √ B 2 2 √ 2 Σ ∂ y √ B 3 /B 5 √ A 3 B 5 -√ A 5 B 3 , (A1) \n/epsilon1 = γ = 1 2 ρ + √ A 2 2 √ 2Σ 3 / 2 ( Σ ∂ r ln [ √ A 3 B 5 -√ A 5 B 3 ] -∂ r Σ ) , (A3) \nτ = π = -i √ B 2 ∂ ∂ y 1 √ 2Σ + A 5 √ A 2 2 √ 2 Σ ∂ r √ A 3 /A 5 √ A 3 B 5 -√ A 5 B 3 , (A2) \nα = β = 1 2 τ + i √ B 2 2 √ 2Σ 3 / 2 ( Σ ∂ y ln [ √ A 3 B 5 -√ A 5 B 3 ] -∂ y Σ ) . (A4)', 'Appendix B: Ricci and Weyl scalars': "With account for the second constraint the NP projections of te Ricci and Weyl scalars are \nΦ 00 = Φ 22 = bA 2 8Σ 3 ( bA ' 23 2 + bB ' 23 2 -2Σ A '' 23 ) , (B1) \nΦ 01 = Φ 10 = Φ 12 = Φ 21 = -√ A 2 B 2 8Σ 3 ( aA '' 23 -b B '' 23 ) , (B2) \nΦ 02 = Φ 20 = -aB 2 8Σ 3 ( aA ' 23 2 + aB ' 23 2 +2Σ B '' 23 ) , (B3) \n64Σ 3 Φ 11 = Σ 2 { A 2 [( ln A 5 ) ,r 2 -4 ( ln A 5 ) ,rr ] -B 2 [( ln B 5 ) ,r 2 -4 ( ln B 5 ) ,rr ] } + +3 { 4 b 2 A 2 B ' 23 2 -4 a 2 B 2 A ' 23 2 + A 2 [ Σ(ln A 5 ) ,r +2 bA ' 23 ] 2 -B 2 [ Σ(ln B 5 ) ,y -2 aB ' 23 ] 2 } + +4Σ { A 2 ((ln A 5 ) ,r ) [ Σ(ln A 5 ) ,r +2 bA ' 23 ] -A 2 [ Σ(ln A 5 ) ,r +2 bA ' 23 ] ,r } --4Σ { B 2 ((ln B 5 ) ,y ) [ Σ(ln B 5 ) ,y -2 aB ' 23 ] -B 2 [ Σ(ln B 5 ) ,y -2 aB ' 23 ] ,y } , (B4) \n192 Σ 3 Λ = 8 Σ 3 R = 3Σ 2 { A 2 [ 3(ln A 5 ) ,r 2 -4(ln A 5 ) ,rr ] + B 2 [ 3(ln B 5 ) ,y 2 -4(ln B 5 ) ,yy ]} + +4 A 2 [ Σ(ln A 5 ) ,r +2 bA ' 23 ] ,r +4 B 2 [ Σ(ln B 5 ) ,y -2 aB ' 23 ] ,y -4 a 2 B 2 A ' 23 2 --A 2 [ Σ(ln A 5 ) ,r +2 bA ' 23 ] 2 -B 2 [ Σ(ln B 5 ) ,y -2 aB ' 23 ] 2 -4 b 2 A 2 B ' 23 2 . 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2024ApJ...975..234L | We investigate the radial dependence of the scaling relations of dust attenuation in nearby galaxies using integral field spectroscopy data from MaNGA. We identify ionized gas regions of kiloparsec size from MaNGA galaxies and for each region we estimate both the stellar attenuation EB VSUBstarSUB and gas attenuation EB VSUBgasSUB. We then quantify the correlations of 15 regionalglobal properties with EB VSUBgasSUB and EB VSUBstarSUB using both the feature importance obtained with the Random Forest regression technique and the Spearman correlation coefficients. The importance of stellar mass metallicity and nebular velocity dispersion found previously from studies based on the Sloan Digital Sky Survey can be reproduced if our analysis is limited to the central region of galaxies. The scaling relations of both EB VSUBgasSUB and EB VSUBstarSUB are found to vary strongly as one goes from the galactic center to outer regions and from Hbright regions to Hfaint regions. For EB VSUBgasSUB N IIS II is topranked with a much higher correlation coefficient than any other property at 0 lt R R SUB e SUB while O IIIO II outperforms N IIS II as the leading property in the outermost region. For EB VSUBstarSUB stellar age shows the strongest correlation with noweak dependence on radial distance although SUBH SUB and specific star formation rate present similarly strong correlations with EB VSUBstarSUB in the galactic center. We find Hbright regions to generally show stronger correlations with EB VSUBgasSUB while Hfaint regions are more strongly correlated with EB VSUBstarSUB although this depends on individual properties and radial distance. The implications of our results for studies of highz galaxies are discussed. | 2024-11-01T00:00:00Z | ['10.48550/arXiv.2409.13340', '2024ApJ...975..234L', 'arXiv:2409.13340', '10.3847/1538-4357/ad7dea', '2024arXiv240913340L'] | ['Interstellar dust', 'Interstellar dust extinction', 'Scaling relations', 'Random Forests', '836', '837', '2031', '1935', 'Astrophysics - Astrophysics of Galaxies'] | Estimating Dust Attenuation from Galactic Spectra. III. Radial Variations of Dust Attenuation Scaling Relations in MaNGA Galaxies | 2,024 | 174 | 0.5 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF'] | 1 | https://arxiv.org/pdf/2409.13340.pdf | {'Estimating Dust Attenuation From Galactic Spectra. III. Radial variations of dust attenuation scaling relations in MaNGA galaxies': '<!-- image --> \nNiu Li \n1 \nand Cheng Li \n1 \n1 Department of Astronomy, Tsinghua University, Beijing 100084, China', 'ABSTRACT': 'We investigate the radial dependence of the scaling relations of dust attenuation in nearby galaxies using integral field spectroscopy (IFS) data from MaNGA. We identify ionized gas regions of kpc sizes from MaNGA galaxies, and for each region we estimate both the stellar attenuation E ( B -V ) star and gas attenuation E ( B -V ) gas . We then quantify the correlations of 15 regional/global properties with E ( B -V ) gas and E ( B -V ) star , using both the feature importance obtained with the Random Forest regression technique and the Spearman correlation coefficients. The importance of stellar mass, metallicity and nebular velocity dispersion found previously from SDSS-based studies can be reproduced if our analysis is limited to the central region of galaxies. The scaling relations of both E ( B -V ) gas and E ( B -V ) star are found to strongly vary as one goes from the galactic center to outer regions, and from H α -bright regions to H α -faint regions. For E ( B -V ) gas , [N ii ]/[S ii ] is top ranked with a much higher correlation coefficient than any other property at 0 < R ≲ R e , while [O iii ]/[O ii ] outperforms [N ii ]/[S ii ] as the leading property in the outermost region. For E ( B -V ) star , stellar age shows the strongest correlation with no/weak dependence on radial distance, although Σ H α and sSFR present similarly strong correlations with E ( B -V ) star in the galactic center. We find H α -bright regions to generally show stronger correlations with E ( B -V ) gas , while H α -faint regions are more strongly correlated with E ( B -V ) star , although depending on individual properties and radial distance. The implications of our results on studies of highz galaxies are discussed. \nKeywords: Dust attenuation - Integral field spectroscopy', '1. INTRODUCTION': "Ubiquitously distributed in interstellar medium (ISM) and accounting for only ∼ 1% of the mass of a typical galaxy (e.g. R'emy-Ruyer et al. 2014; Driver et al. 2018), dust forms through the ejecta of asymptotic giant branch (AGB) stars and supernovae (e.g. Dwek 1998; Popping et al. 2017), grows by accreting gas-phase metals (Dominik & Tielens 1997; Dwek 1998; Zhukovska 2014), and can be destroyed through supernova shocks, thermal sputtering and grain collisions, or incorporated into newly formed stars (e.g., Dwek 1998; Bianchi & Ferrara 2005; Nozawa et al. 2007). Interstellar dust is composed of grains of different sizes ranging from 5 to 250 nm typically (Weingartner & Draine 2001), and made of various chemical elements (Draine 2003). The \nCorresponding author: Niu Li & Cheng Li \[email protected] \[email protected] \nsize, mass and chemical composition of dust grains may evolve through various mechanisms (Asano et al. 2013). \nDust plays important roles in the thermodynamics and chemistry of the ISM as well as star formation and radiative transfer in the host galaxy (e.g. Conroy 2013). Dust can facilitate star formation in galaxies by acting as a catalyst in the formation of molecular Hydrogen (Draine 2003) and cooling down the ISM through thermal emission in infrared (Montier & Giard 2004) and depletion of metals onto dust grains (Schneider et al. 2006). Dust grains may absorb or scatter the light emitted by stars, thus reshaping the spectral energy distribution (SED) of the galaxy, an effect known as dust attenuation or dust extinction (Galliano et al. 2018; Salim & Narayanan 2020). Dust attenuation laws can be generally characterized by three distinct properties: the dust attenuation curve (attenuation as a function of wavelength) which is approximately a power law in optical and UV, the UV bump which accounts for the excess attenuation at around 2175 ˚ A, and the relation between the stellar and nebular attenuation which is usually quanti- \nied by comparing the optical color excess E ( B -V ) star and E ( B -V ) gas as measured from the stellar continuum and the Balmer emission lines. In a recent review article, Salim & Narayanan (2020) summarized previous studies of the attenuation curve and UV bump in galaxies of different redshifts, which have been mostly limited to global measurements of the galaxies. Spatially-resolved dust attenuation curves and the UV bump strength have recently been derived for a sample of nearby galaxies based on the Swift /UVOT data (e.g. Molina et al. 2020; Belles et al. 2023; Duffy et al. 2023; Zhou et al. 2023). \nThe correlation between E ( B -V ) gas and E ( B -V ) star was originally studied by Fanelli et al. (1988) who found E ( B -V ) gas to be significantly higher than E ( B -V ) star . Calzetti et al. (1994, 2000) analyzed a sample of local starburst galaxies and found a typical ratio of the stellarto-gas attenuation of f ≡ E ( B -V ) star / E ( B -V ) gas ≈ 0 . 44. Extensive studies in the past two decades in different types of galaxies and at different redshifts have confirmed the larger attenuation in gas than in stellar populations, but finding f to vary over a wide range, from 0.44 to ∼ 1 (e.g., Calzetti et al. 2000; Riffel et al. 2008; Reddy et al. 2010; Wuyts et al. 2011; Kashino et al. 2013; Kreckel et al. 2013; Wuyts et al. 2013; Price et al. 2014; Pannella et al. 2015; Valentino et al. 2015; Puglisi et al. 2016; Zahid et al. 2017; Buat et al. 2018; Koyama et al. 2019; Qin et al. 2019). The discrepancy between E ( B -V ) gas and E ( B -V ) star may be generally explained by a two-component dust model in which the dust disk of a galaxy is formed of two components: a diffuse and optically-thin component distributed throughout the ISM, plus dense and optically-thick clouds (birth clouds) where young stars are born (e.g., Charlot & Fall 2000; Wild et al. 2011; Chevallard et al. 2013). In this model, the emission lines produced in star-forming regions are attenuated by both the dust in the birth clouds and that in the ambient ISM, while the continuum radiation of older stars is attenuated only by the diffuse dust in ISM. Consequently, if spatially-resolved observations are available, one would expect a stronger correlation between E ( B -V ) gas and E ( B -V ) star in star-forming regions than in regions dominated by diffuse ionized gas (DIG), as well as a strong dependence of the value of f on the average age of stars in the ionized regions. In fact, recent studies of the spatially-resolved E ( B -V ) gas and E ( B -V ) star in lowz galaxies based on integral field spectroscopy data from the Mapping Nearby Galaxies at Apache Point Observatory (MaNGA; Bundy et al. 2015) survey have produced consistent results with this model (e.g., Greener et al. 2020; Lin & Kong 2020; Li et al. 2021; Riffel et al. 2021). \nThere have been many studies focused on the gas attenuation only, which can be easily measured from the Balmer decrement (H α /H β ) for galaxies at different redshifts, aiming to understand how E ( B -V ) gas depends on the physical properties of galaxies. These studies have established that, among many properties and fol- \nwed by star formation rate and gas-phase metallicity, the stellar mass appear to be the most important property in driving E ( B -V ) gas for both low-redshift galaxies (e.g., Asari et al. 2007; Garn & Best 2010; Maheson et al. 2024) and high-redshift galaxies (e.g., Pannella et al. 2009; Garn et al. 2010; Whitaker et al. 2017; McLure et al. 2018; Shapley et al. 2022, 2023). As pointed out by Maheson et al. (2024), the strong dependence of dust attenuation on stellar mass is expected from simple analytical relations between dust-to-gas mass, stellar mass and metallicity. However, the previous studies have been largely based on single-fiber or slitless spectroscopy, thus limited to dust attenuation and galaxy properties measured in the central region of lowz galaxies, or the global measurements for galaxies at higher redshifts. The use of IFS data should provide valuable insights into the scaling relations of dust attenuation which essentially rely on how the two dust components are distributed across the galaxy. Based on the MaNGA data, Lin & Kong (2020) and Li et al. (2021, hereafter Paper II) have recently highlighted the significance of gas-phase metallicity and ionization level in relation to E ( B -V ) gas at kpc scales, with the [N ii ]/[S ii ] flux ratio being particularly identified as the most important property. The discrepancy between the IFS-based studies and those based on single-fiber spectroscopy data should be partly (if not fully) attributed to the different spatial resolutions of the data used in different studies. The discrepancy also strongly implies that the scaling relations of dust attenuation vary from region to region within galaxies. \nThis work is the third of a series of papers dedicated to spatially-resolved dust attenuation in MaNGA galaxies. In Li et al. (2020, hereafter Paper I), we developed a new technique to estimate a model-independent attenuation curve from the observed optical spectrum for each spaxel in MaNGA datacubes. In Paper II, we applied this technique to approximately 8000 unique galaxies from MaNGA to investigate the correlation between E ( B -V ) gas and E ( B -V ) star in both DIG and H ii regions, examining how the correlation depends on the physical properties of galaxies. A total number of 16 regional or global properties were considered. In this work, we extend the study of Paper II by further examining how the scaling relations of both E ( B -V ) gas and E ( B -V ) star depend on the radial distance of the ionized regions. We will firstly concentrate on the central region of our galaxies in order to make comparisons with previous studies of single-fiber spectroscopy data. We then consider the regions at all different radial distances. We use Random Forest regression to assess the feature importance of the galactic properties, and also use the Spearman correlation coefficient to quantify the correlation strength between dust attenuation and the properties. \nThe paper is organized as follows. In section 2 we describe the observational data, the quantities measured from the data, and the samples for our analysis. We \nthen present our results in section 3 and discuss on the results in section 4. Finally, we summarize in section 5. Throughout this paper we assume a Λ cold dark matter cosmology model with Ω m = 0 . 3, Ω Λ = 0 . 7, and H 0 = 70 km s -1 Mpc -1 . We assume a Chabrier (2003) initial mass function (IMF).", '2. DATA': 'The data used in this work is the same as in Paper II. In this section we briefly describe the data and measurements, and refer the reader to § 2 of Paper II for a comprehensive description.', '2.1. MaNGA': "As one of the key components of the fourth-generation Sloan Digital Sky Survey project (SDSS-IV; Blanton et al. 2017), the MaNGA survey obtained IFS data for 10,010 nearby galaxies over its six-year operation from July 2014 through August 2020 (Bundy et al. 2015). MaNGA utilize 17 hexagonal science integral field units with a field of views ranging from 12 '' to 32 '' to obtain IFS datacubes covering out to 1.5 or 2.5 effective radii ( R e ) of the target galaxies (Drory et al. 2015). The target galaxies are selected from the SDSS-based catalog, NASA Sloan Atlas (NSA Blanton et al. 2011) so as to cover a stellar mass range of 5 × 10 8 M ⊙ h -2 ≤ M ∗ ≤ 3 × 10 11 M ⊙ h -2 and a redshift range of 0 . 01 < z < 0 . 15 with a median redshift of z ∼ 0 . 03 (Wake et al. 2017). The MaNGA spectra span a wavelength range from 3622 ˚ A to 10354 ˚ A with a spectral resolution R ∼ 2000 and an r -band signal-to-noise (S/N) ratio of 4 -8 per ˚ A per 2 '' fiber at 1 -2 R e of the galaxies. \nMaNGA raw data are reduced with the Data Reduction Pipeline (DRP; Law et al. 2016) which produces a datacube for each target galaxy with a spaxel (spatial pixel) size of 0 '' .5 × 0 '' .5 and an effective spatial resolution of ∼ 2 '' .5. The absolute flux calibration of the MaNGA spectra is < 5% for > 80% of the wavelength range. Spectral flux calibration, survey execution strategy and data quality tests are detailed in Yan et al. (2016a) and Yan et al. (2016b). Additionally, the Data Analysis Pipeline (DAP) performs full spectral fitting to the reduced datacubes, deriving measurements of stellar kinematics, emission lines, and spectral indices (Westfall et al. 2019; Belfiore et al. 2019). The final data from MaNGA including DRP and DAP products of 11,273 datacubes for 10,010 unique galaxies are released as a part of the SDSS Data Release 17 (SDSS DR17; Abdurro'uf et al. 2022). In this work, we make use of an earlier sample, the MaNGA Product Launch 9, which includes 8113 datacubes for 8000 unique galaxies, a random subset of the final sample in SDSS/DR17.", '2.2. Identifying Ionized Gas Regions': "We first identify ionized gas regions in the MaNGA galaxies, and for each region we then quantify the stellar and gas attenuation, along with other relevant prop- \nerties. Following the methodology described in Liang et al. (2020), we apply the publicly available H ii -region finder HII EXPLORER (S'anchez et al. 2012) to the H α flux map to detect the ionized gas regions. Originally intended for identifying H ii regions, HII EXPLORER commonly applies a relatively high threshold density, for instance, H α surface brightness Σ H α > 10 39 . 5 erg s -1 kpc -2 . Instead, we opt for a much lower threshold of Σ H α > 10 37 . 5 erg s -1 kpc -2 to include DIG regions in our analysis. The identified ionized gas regions each consist of a few to tens of spaxels with similar H α surface densities by definition. We stack the spectra within each region to derive an averaged spectrum with a high S/N (Liang et al. 2020), from which we then calculate the stellar and gas attenuation, as well as other pertinent properties required for our study. \nDue to the limitation in resolution, it is not possible to resolve individual H ii regions, which typically range in size from a few to hundreds of parsecs (e.g., Kennicutt 1984; Hunt & Hirashita 2009; Anderson et al. 2019). As a result, each of the ionized gas regions identified from MaNGA, which have a size of approximately 1 kpc, may encompass a few to hundreds of individual H ii regions, or represent a combination of DIG and H ii regions. Zhang et al. (2017) found that Σ H α can be effectively utilized to differentiate DIG-dominated regions from H ii -dominated regions within the MaNGA galaxies, with an empirical dividing value of Σ H α ∼ 10 39 erg s -1 kpc -2 . Accordingly we divide the ionized regions in our sample into two subsets, and for simplicity we will refer to H ii -dominated regions as H α -bright regions and DIG-dominated regions as H α -faint regions in what follows.", '2.3. Measuring gas and stellar attenuations and regional properties': "For each of the ionized regions as identified above, we firstly obtain a model-independent attenuation curve ( A λ -A V ) by applying the technique developed in Paper I to the stacked spectrum of the region. In this method, the observed spectrum is first decomposited into two components, separately including the small-scale features ( S ) and large-scale spectral shape ( L ), by applying a moving-average filter. The spectrum of each single stellar-population model template is also separated into the S and L components in the same way. The intrinsic dust-free model spectrum of the stellar component is then obtained by fitting the observed ratio of the smallto large-scale spectral components ( S/L ) obs to the corresponding ratio of the model spectra ( S/L ) model . Since the small- and large-scale components are affected by dust in the same way, their ratio S/L is expected to be dust-free, as long as the dust attenuation curves are similar for different stellar populations in a galactic region. Finally, A λ -A V is determined by comparing the observed spectrum with the best-fit model spectrum. Sub- \nquently, the value of E ( B -V ) star , defined as A B -A V , is obtained from the attenuation curve. \nNext, we use the attenuation curve to correct the effect of dust attenuation for the observed spectrum. We then perform full spectral fitting to the dust-free spectrum, and measure stellar population parameters from the best-fit spectrum of the stellar component as well as emission line parameters from the starlight-subtracted spectrum. For the spectral fitting, we employ the simple stellar population (SSP) library provided by Bruzual & Charlot (2003, hereafter BC03), which provides spectra for a comprehensive set of 1326 SSPs with 221 different ages ranging from t = 0 Gyr to t = 20 Gyr and six metallicities from Z = 0 . 005 Z ⊙ to Z = 2 . 5 Z ⊙ . The computation of SSPs is based on the Padova evolutionary track (Bertelli et al. 1994) and adopts the stellar initial mass function (IMF) of Chabrier (2003). We select a subset of 150 SSPs, covering 25 ages that are evenly spaced in log 10 t at each of the six metallicities. The effect of stellar velocity and velocity dispersion is taken into account by shifting/broadening the SSP model spectra according to the stellar velocity v ∗ and velocity dispersion σ ∗ measurements provided by the MaNGA DAP. Based on the best-fit stellar spectrum, we then obtain the following parameters to quantify the stellar populations in each region: surface density of stellar mass Σ ∗ , light-weighted stellar age t L , light-weighted stellar metallicity Z L , and the narrow-band version of the 4000 ˚ A break D n 4000 defined by Balogh et al. (1999). \nWe subtract the best-fit stellar spectrum from the stacked spectrum, and we use Gaussian profiles to fit different emission lines, thus measuring the flux, surface density, velocity dispersion, and equivalent width (EW) of the emission lines within each region. The lines considered particularly in this work include [O ii ] λλ 3726 , 3729, H β , [O iii ] λλ 4959 , 5007, [N ii ] λλ 6548 , 6583, H α , and [S ii ] λλ 6717 , 6731. The fluxes have been corrected for attenuation, using the H α -toH β flux ratio from the observed spectrum and assuming the case-B recombination: \nE ( B -V ) gas = 2 . 5 k ( λ H β ) -k ( λ H α ) log 10 [ (H α/ H β ) obs 2 . 86 ] , (1) \nwhere k ( λ H β ) and k ( λ H α ) are the attenuation at the wavelength of H α and H β given by the same attenuation curve as obtained above for the stellar component. Derived from the measurements of emission lines, we have estimated the following parameters to characterize properties related to the gas component: H α surface density Σ H α , H α equivalent width EW H α , H α velocity dispersion σ H α , logarithm of specific star formation rate (SFR) defined by the ratio of SFR to stellar mass sSFR, gas-phase metallicity 12+log 10 (O/H) estimated from the parameter O3N2 ≡ ([O iii ] λ 5007/H β )/([N ii ] λ 6583/H α ) (Pettini & Pagel 2004), logarithm of the flux ratio log 10 ([N ii ]/[O ii ]) \nbetween [N ii ] λ 6583 and [O ii ] λλ 3726 , 3729, logarithm of the flux ratio log 10 ([O iii ]/[O ii ]) between [O iii ] λ 5007 and [O ii ] λλ 3726 , 3729, logarithm of the flux ratio log 10 ([N ii ]/[S ii ]) between [N ii ] λ 6583 and [S ii ] λ 6717. In the subsequent sections, we employ shorthand notations in certain figures to enhance visual clarity: Z gas =12+log 10 (O/H), N2O2=log 10 [N ii ]/[O ii ], O3O2=log 10 [O iii ]/[O ii ], N2S2=log 10 [N ii ]/[S ii ]. \nIn our analysis, we refine our focus to a subset of ionized gas regions exhibiting significantly elevated S/N in both the stellar continuum and the relevant emission lines. Specifically, we stipulate a requirement of S/N > 5 for both the stellar continuum and the H α and H β lines, while a S/N > 3 criterion is imposed for the [O ii ] λλ 3726 , 3729, [O iii ] λ 5007, [N ii ] λ 6583, and [S ii ] λλ 6717 , 6731 lines. Additionally, we exclude regions classified as AGN on the Baldwin-PhillipsTerlevich diagram (BPT; Baldwin et al. 1981). We first exclude all the regions that are classified as Seyferts in the [O iii ]/H β versus [S ii ]/H α diagram. We also exclude the regions that are located within 3 '' from their galactic center and are classified as either low-ionization emission-line regions (LINER) in the [O iii ]/H β versus [S ii ]/H α diagram or as active galactic nuclei (AGN) in the [O iii ]/H β versus [N ii ]/H α diagram. The final sample includes ∼ 2 . 7 × 10 5 ionized gas regions, of which ∼ 40% are H α -faint regions and ∼ 60% are H α -bright regions.", '2.4. Regional and global properties in consideration': "This work is aimed to investigate the dependency of both E ( B -V ) gas and E ( B -V ) star on a variety of physical properties at various radial distances, both at regional and global scales. In alignment with Paper II, we consider 12 regional properties: log 10 Σ H α , log 10 EW H α , sSFR, log 10 Σ ∗ , log 10 t L , Z L , D n 4000, 12+log 10 (O/H), log 10 [N ii ]/[O ii ], log 10 [O iii ]/[O ii ], log 10 [N ii ]/[S ii ], log 10 σ H α . Note that when compared to Paper II, we have omitted t M and Z M due to their similarities with t L and Z L , while additionally including σ H α which was found in Maheson et al. (2024) to be strongly correlated with E ( B -V ) gas . Furthermore, we consider three global properties: total stellar mass ( M ∗ ), morphological type ( T -type), and the r -band minor-to-major axial ratio ( b/a ). The M ∗ and b/a measurements are derived from NSA, and the T -type parameters are taken from Dom'ınguez S'anchez et al. (2018). A negative value of T -type typically corresponds to early-type morphology, whereas a positive T -type value signifies late-type morphology. In total, we consider a total of 15 properties including 12 regional properties and 3 global properties.", '3. RESULTS': "We present our results in this section. In subsection 3.1, we first focus on the gas-phase attenuation in ionized gas regions that are located less than 3 '' from the \n<!-- image --> \nFigure 1. Random Forest regression is employed to analyze the relationship between E ( B -V ) gas and galactic properties in the central region of galaxies. The histograms display the FI of the properties. Note that the maximum FI has been normalized to 1. Panel (a) excludes properties associated with H α flux and [N ii ]/[S ii ], whereas Panel (b) encompasses the entirety of the 15 properties under consideration. In each panel the solid/blue histogram and red histogram show the results respectively for the full sample and the subsample at z ≤ 0 . 03. The errorbar shows standard deviation of feature importance, estimated by bootstrap resampling of all the spaxels in the sample and re-training the Random Forest regression for 100 times. \n<!-- image --> \nFigure 2. Random Forest regression is utilized to analyze the relationship between E ( B -V ) gas and galactic properties across various radial bins, as indicated in the legend, using the unit of R e . The histograms depict the FI of the properties. It is noteworthy that the maximum FI in each radial bins has been normalized to 1. \n<!-- image --> \ncenter of their host galaxy, for comparison with previous SDSS-based studies (e.g., Maheson et al. 2024). In subsection 3.2, we extend the analysis of gas attenuation to the outer regions of our galaxies. Next, we consider both gas and stellar attenuations in subsection 3.3, and compare H α -bright and H α -faint regions in subsection 3.4. We note that, although we focus on gas attenuation in subsection 3.1 and subsection 3.2, the same analysis is also done for stellar attenuation. The results are consis- \nnt with those presented in subsection 3.3 and so are not included for simplicity.", '3.1. Feature importance to gas attenuation at galactic center': "We first focus on the central regions of our galaxies. We utilize the technique of Random Forest regression to analyze the relationship between E ( B -V ) gas and the 15 regional/global properties. In practice, we use the Ran- \ndom Forest regressor from the Python Scikit-learn package (Pedregosa et al. 2011). Figure 1 displays the result of the feature importance (FI) analysis as the solid/blue histograms. In Panel (a), following the approach outlined by Maheson et al. (2024), we exclude Σ H α , sSFR and EW H α from the analysis considering that all the three parameters are based on the H α flux which is corrected for dust attenuation through the Balmer decrement. Additionally, for this panel we omit [N ii ]/[S ii ], a parameter that was not considered in Maheson et al. (2024). As can be seen, σ H α and M ∗ emerge as the most influential properties in predicting E ( B -V ) gas , with subsequent importance attributed to Z L and [O iii ]/[O ii ], both intricately linked to stellar and gas metallicity. The results are in good agreement with Maheson et al. (2024), demonstrating that the scaling relations derived from single-fiber spectroscopy data within the SDSS are replicable when our analysis is confined to the central region of MaNGA galaxies. \nNext, we consider all the 15 properties. As shown in Panel (b) of Figure 1, [N ii ]/[S ii ] emerges as the most important property, with a feature importance that is much higher than that of any other property. The most important property in Panel (a), σ H α is now ranked in the second position, followed by M ∗ and Σ H α which exhibit comparable levels of significance. Previous studies have shown that [N ii ]/[S ii ] is sensitive to both metallicity and ionization parameters. Specifically, this parameter exhibits an increasing trend toward higher metallicity (ionization parameter) when the ionization parameter (metallicity) is fixed (Dopita et al. 2013). As mentioned, [N ii ]/[S ii ] was found to strongly correlate with E ( B -V ) gas at kpc scales in MaNGA galaxies regardless of radial distance (Lin & Kong 2020; Li et al. 2021). \nThe smearing effect duo to the limited spatial resolution of both SDSS and MaNGA may lead to underestimation of the parameters in galactic centers. To test the potential effect of smearing on the FI analysis, we select all the spaxels from those galaxies at z < 0 . 03, where the physical resolution is better than the full sample. The results of the FI analysis for this subsample are shown in Figure 1 as red histograms. For the analysis with the subset of parameters as shown in the left panel, obvious variation is seen between the full sample and the subsample, particularly for M ∗ , Σ ∗ and Z L . At z < 0 . 03 where the smearing effect is less severe, M ∗ is no longer the most important parameter, while regional properties Σ ∗ and Z L become more important. In contrast, for the FI analysis with all the 15 parameters as shown in the right panel, the results of the subsample are consistent with that of the full sample. This test suggests that the FI analysis is robust to the smearing effect as long as the truly relevant parameters are included. It is interesting to note that the FI of σ H α shows little variation between the two samples, a result that is true in both panels. \n3.2. Feature importance to gas attenuation at all radii \nWe then consider all the regions and examine how the feature importance to E ( B -V ) gas may vary as one goes from the galactic center to larger radial distances. In Figure 2, we present the FI analysis depicting the relationship between the 15 galactic properties and E ( B -V ) gas across various radial regions from the center to the outer regions. Different colors correspond to different radial bins as indicated. With an increase in distance from the galactic center, several noticeable results can be read from the figure. First, [N ii ]/[S ii ] consistently maintains a high FI across almost all the radii. Specifically, [N ii ]/[S ii ] is the leading parameter out to ∼ R e , before it gives place to Σ H α at 0 . 9 R e < R < 1 . 2 R e with a FI that is only slightly smaller than that of the new leader. Although its FI further drops at larger radii, [N ii ]/[S ii ] ends up with a relatively high FI, taking the fifth position in the largest radial bin. Second, the most important parameters identified previously in Maheson et al. (2024), the FI of M ∗ , Z L and σ H α decreases with increasing distance. In particular, σ H α experiences an immediate and dramatic decline in FI as one goes beyond the first radial bin '[0, 0]', strongly indicating that this parameter is only important in the galactic center. Third, properties associated with H α flux such as Σ H α , EW H α and sSFR exhibit much higher FI in the outer regions than in the galactic center, and they become the dominate parameters in the largest radial bin. Finally, properties linked to gas metallicity and ionization, [N ii ]/[O ii ] and [O iii ]/[O ii ] manifest significantly greater FI at larger radii. In the outermost regions, [O iii ]/[O ii ] even presents a higher FI than [N ii ]/[S ii ]. \nApparently, our findings indicate a pronounced radial variability in the FI of galactic properties to E ( B -V ) gas . This variability introduces a heightened level of complexity to the gas attenuation scaling relations, when compared to previous studies which have been mostly limited to either the galactic center or the whole galaxy.", '3.3. Radial profiles of dust attenuation scaling relation': 'To enable a direct and more intuitive comparison of the radial variations of the dust attenuation scaling relations, we further utilize the Spearman rank correlation coefficients ( ρ ) to quantify the correlation strength of both E ( B -V ) gas and E ( B -V ) star with all the 15 properties and in all the radial bins. The result of this analysis is shown in Figure 3, where the two panels show the radial profiles of ρ for E ( B -V ) gas and E ( B -V ) star , respectively. In each panel the different lines/colors correspond to different properties, as indicated. Note that, we show the absolute values of ρ , thus comparing the strength of correlations while ignoring whether the correlations are positive or negative. \nFor E ( B -V ) gas , as seen from Panel (a) of Figure 3, both the relative ranks of the correlation coefficient for different properties at given radial distance and the radial variations of the correlation coefficient for given \n<!-- image --> \nFigure 3. Panel (a): Radial profiles of correlations between E ( B -V ) gas and galactic properties. The x-axis represents the radial distance scaled by R e from the galactic center, while the y-axis depicts the Spearman rank correlation coefficients between E ( B -V ) gas and various galactic properties at different radii. Panel (b): The symbols and lines are the same as in Panel (a) but for E ( B -V ) star . \n<!-- image --> \nFigure 4. The three rows describe correlations between E ( B -V ) gas and [O iii ]/[O ii ], [N ii ]/[S ii ], σ H α . The gray-scale background in each panel represents the distributions of all ionized gas regions, with the median relation and 1 σ scatter delineated by black lines. The Spearman rank correlation coefficients ( ρ ) and the range of distance to the galactic center scaled in R e are indicated in the upper-left corner. \n<!-- image --> \nproperty are generally consistent with what we have seen above from the analysis of FI provided by Random Forest. For instance, [N ii ]/[S ii ] presents the highest value of ρ out to R ∼ 1 . 3 R e , ranging from ρ = 0 . 76 at R = 0 to ρ ∼ 0 . 5 at 1 . 3 R e , while σ H α shows the maximum decline, starting with a relatively high value of ρ ∼ 0 . 6 and ending with ρ ∼ 0 . 15 at the largest radius. In fact, a negative gradient is seen in all the eight properties that exhibit relatively strong correlations ( ρ > 0 . 5) with E ( B -V ) gas at the galactic center. Ordered by decreasing ρ at R = 0, these properties are [N ii ]/[S ii ], M ∗ , Z L , Σ ∗ , σ H α , Σ H α , 12+log 10 (O/H), and [N ii ]/[O ii ]. In contrast, all the other 7 properties except [O iii ]/[O ii ] \nshow no/weak radial gradient, thus maintaining relatively weak correlations ( ρ < 0 . 5) at all radial distances. The only exception, [O iii ]/[O ii ] presents significant increase with radius, starting with a low value of ρ ∼ 0 . 3 at R = 0 and reaching the maximum value of ρ ∼ 0 . 5 at R > 1 . 3 R e , even surpassing [N ii ]/[S ii ]. Figure 4 displays the distribution of individual regions on the diagrams formed by E ( B -V ) gas against [O iii ]/[O ii ], [N ii ]/[S ii ] and σ H α . Panels from left to right correspond to different distances from the galactic center, as indicated. The median relation and the 1 σ scatter are plotted as the thick line and the two thin lines in each panel. We see that, [O iii ]/[O ii ] exhibits an increasingly strong and \nFigure 5. The three rows depict correlations between E ( B -V ) star and Σ H α , sSFR, t L . The symbols and lines are the same as in Figure 4. \n<!-- image --> \nnegative correlation with E ( B -V ) gas as the radius increases, while [N ii ]/[S ii ] presents a strong correlation with E ( B -V ) gas in the galactic center and maintains a relatively strong correlation in the outer regions. Regarding σ H α , a strong correlation is found only in the first radial bin, with ρ declining rapidly in the outer regions. \nFor E ( B -V ) star , as seen from Panel (b) of Figure 3, the correlation coefficient declines for all properties except t L as one goes beyond the galactic center. As a result, in the outermost region t L becomes the only dominant parameter while all the other properties are weakly correlated with E ( B -V ) star . In the galactic center, only three properties show relatively strong correlations with ρ > 0 . 5: Σ H α , sSFR, and t L . Although Σ H α and sSFR exhibit a stronger correlation with E ( B -V ) star than t L in the center, their correlations rapidly decrease with increasing radius. The correlation coefficient of t L is roughly constant at ρ ∼ 0 . 55 -0 . 6 at all radii. These results can also be seen from Figure 5 which displays the individual regions on the diagrams of E ( B -V ) star versus Σ H α , sSFR and t L in different radial bins. \nIn conclusion, [N ii ]/[S ii ] is the main driver of E ( B -V ) gas , while t L drives E ( B -V ) star from the center to the outer regions, consistent with our findings in Paper II. Several properties exhibit a relatively strong correlation with E ( B -V ) gas in the center, although these correlations notably weaken as the radius increases. Regarding E ( B -V ) star , only SFR-related properties are comparable to correlation of t L in the central regions.', '3.4. Radial profiles of dust attenuation scaling relation in H α -faint and H α -bright regions': 'Figure 6 displays the radial profiles of the correlation coefficient ρ for E ( B -V ) gas (left panels) and E ( B -V ) star (right panels), and for H α -faint (upper \npanels) and H α -bright (lower panels) regions separately. For E ( B -V ) gas , we see that all of the properties present moderate or weak correlations ( ρ < 0 . 5) in H α -faint regions, while the H α -bright regions show similar results to what we have seen in Panel (a) of the previous figure. This indicates that the scaling relations of E ( B -V ) gas observed for the full sample are predominantly contributed by H α -bright regions. In addition, comparing the H α -bright regions with the full sample, we find EW H α , D n 4000, and Z L to display notably stronger correlations with E ( B -V ) gas , particularly in outer regions with R ≳ R e . Even in the outermost regions, five properties still maintain ρ > 0 . 5, with three of them displaying stronger correlations with E ( B -V ) gas compared to [N ii ]/[S ii ]. This complexity adds challenges to determining the true driving factor among these properties or establishing a scaling relation for estimating E ( B -V ) gas . \nFor E ( B -V ) star , the result appears less complex in the sense that only a few properties show relatively strong correlations with ρ > 0 . 5 in both H α -faint and H α -bright regions and at any given radius. Nonetheless, the two types of regions are different in several aspects. First, if we only consider H α -faint regions, we find the strong correlation of E ( B -V ) star with t L as seen from Panel (b) of the previous figure remains similarly strong or even stronger over the full range of radial distance, with ρ ∼ 0 . 7 here versus ρ ∼ 0 . 6 in Figure 3. In contrast, for H α -bright regions the correlation of E ( B -V ) star with t L is only moderate, with ρ ∼ 0 . 5, though still independent of radial distance. Second, Σ H α in H α -faint regions shows no correlation with E ( B -V ) star at all radii, while Σ H α in H α -bright regions presents a radial profile of ρ similar to that of the full sample. Third, different from both t L and Σ H α , sSFR presents similar \nFigure 6. Panel (a) and Panel (c) are Radial profiles of correlations between E ( B -V ) gas and galactic properties in H α -faint regions with Σ H α ≤ 10 39 erg s -1 kpc -2 and H α -bright regions with Σ H α > 10 39 erg s -1 kpc -2 , respectively. Panel (b) and Panel (d) are Radial profiles of correlations between E ( B -V ) star and galactic properties in H α -faint and H α -bright regions, respectively. The symbols and lines are the same as in Figure 3. \n<!-- image --> \n(a) Σ \nH \nα \n≤ \n10 \n39 \nerg s \n- \n1 \nkpc \n- \n2 \n<!-- image --> \n(c) Σ \nH \nα \n> \n10 \n39 \nerg s \n- \n1 \nkpc \n- \n2 \n<!-- image --> \n<!-- image --> \n(d) Σ \nH \nα \n> \n10 \n39 \nerg s \n- \n1 \nkpc \n- \n2 \nprofiles in H α -faint and H α -bright regions, with a moderate coefficient of ρ ∼ 0 . 5 at R = 0 and a low coefficient of ρ ∼ 0 . 2 -0 . 3 in the outermost region. Finally, we notice that, although Σ ∗ in H α -bright regions roughly follows the radial profile of the full sample, with rather weak correlations ( ρ ≲ 0 . 2) at all radii, this parameter presents almost the same profile as sSFR in H α -faint regions. This result may be understood from the fact that Σ ∗ is largely contributed by old stellar populations which are mostly associated with H α -faint regions. Considering both that H α -faint and H α -bright regions are dominated respectively by Σ ∗ and Σ H α and that sSFR is defined by the ratio of SFR (which can be estimated from Σ H α ) to Σ ∗ , it is natural to see that sSFR behaves similarly to Σ ∗ in H α -faint regions but similarly to Σ H α in H α -bright regions.', '4.1. Gas attenuation': "Our work has clearly established that [N ii ]/[S ii ] outperforms all the other properties as the most important parameter in relation to E ( B -V ) gas . This result was \nfound in Paper II for all the ionized gas regions as a whole. We further found here that the conclusion is true for a wide range of radial distance from the galactic center out to R ≳ R e , and it holds for both H α -faint and H α -bright regions although the overall correlations in H α -faint regions are relatively weak. The physical reason behind the high importance of [N ii ]/[S ii ] to gas attenuation is not immediately clear, however. When [N ii ]/[S ii ] is excluded from the analysis, both stellar mass and metallicity are highly ranked in feature importance as shown in Figure 1, consistent with the previous SDSS-based study by Maheson et al. (2024). As pointed out by Maheson et al. (2024), the dependence of dust attenuation on stellar mass and metallicity is expected from simple analytical equations relating dust mass with gas mass and metallicity, as well as the molecular gas main sequence for resolved regions in ALMaQUEST (MGMS Lin et al. 2019). Since [N ii ]/[S ii ] is commonly employed as a diagnostic for gas-phase metallicity (e.g., P'erez-Montero & Contini 2009; Dopita et al. 2016; Teklu et al. 2020) and given the known stellar mass-metallicity relation (e.g., Tremonti et al. 2004; Koppen et al. 2007; \nFigure 7. Predicted [N ii ]/[S ii ] versus the observed value for ionized gas regions in different radial bins as indicated in each panel. In the upper panels, the prediction is made by using Σ ∗ and 12+log 10 (O/H) as inputs to train a Random Forest regressor. In the lower panels, [O iii ]/[O ii ] is used in addition to Σ ∗ and 12+log 10 (O/H) for the prediction. In each panel, the dashed red line represents the 1:1 relation, while the solid thick and thin lines indicate the median and the 1 σ scatter of the relation. \n<!-- image --> \nYates et al. 2012; Dayal et al. 2013; Ma et al. 2016), one can well expect [N ii ]/[S ii ] to also strongly depend on E ( B -V ) gas . It is likely that [N ii ]/[S ii ] integrates the favorable factors in both stellar mass and metallicity, thus rendering it the most sensitive property to variations in E ( B -V ) gas , and this is the reason why the feature importance of both stellar mass and metallicity is reduced dramatically when [N ii ]/[S ii ] is included in the Random Forest analysis. To test this conjecture, we train a Random Forest regressor using Σ ∗ and 12+log 10 (O/H) as inputs to predict [N ii ]/[S ii ]. In the upper panels of Figure 7, we compare the predicted [N ii ]/[S ii ] with the observed value for ionized gas regions in different radial bins, with the correlation coefficient ρ indicated in each panel. As expected, [N ii ]/[S ii ] can be reasonably well predicted by the two properties, with ρ > 0 . 85 in all cases. This prediction is most accurate in the galactic center ( ρ = 0 . 91), and becomes slightly less accurate as the radial distance increases. \nAnother important parameter identified in our work is [O iii ]/[O ii ] which presents similarly strong or even stronger correlations with E ( B -V ) gas in the outer regions, when compared to [N ii ]/[S ii ]. In fact, [O iii ]/[O ii ] as an indicator of ionization parameter and nitrogen enrichment can also influence [N ii ]/[S ii ]. At a fixed metallicity, [N ii ]/[S ii ] can increase with ionization parameter or primary nitrogen enrichment (Dopita et al. 2013; Blanc et al. 2015). We have repeated the above analysis, using [O iii ]/[O ii ] in addition to Σ ∗ and 12+log 10 (O/H) as inputs to train the Random Forest regressor. The result is shown in the lower panels of Figure 7. We see that the correlation between the predicted and real [N ii ]/[S ii ] becomes slightly tighter at given radial bin, with an average increase of ∼ 5% in the correlation coefficients. On one hand, this result confirms that [N ii ]/[S ii ] and [O iii ]/[O ii ] indeed share common information (e.g. ion- \non parameter) that is closely related to dust attenuation in ionized gas. On the other hand, however, the limited improvement in predicting [N ii ]/[S ii ] by the inclusion of [O iii ]/[O ii ] strongly implies that [O iii ]/[O ii ] represents some important and distinct factor that is not fully encoded in [N ii ]/[S ii ], and this factor is very likely the ionization parameter, which plays a more important role in gas attenuation in the outer regions of galaxies when compared to [N ii ]/[S ii ]. \nThe nebular velocity dispersion as quantified by σ H α was previously found to be important in determining gas attenuation based on SDSS single fiber spectra (Maheson et al. 2024). Those authors suggested that the unexpected importance of σ H α may be understood if the nebular velocity dispersion trances the gravitational potential in the galaxy, which determines how much the galaxy can retain dust and metals against radiation pressure and gas outflows (Chisholm et al. 2015). Our finding that the strong correlation of σ H α with gas attenuation holds only in the galactic centers provides further evidence in support of their conjecture, considering that the gravitational potential well is deeper in the center than the outer regions. Alternatively, σ H α may be tracing non-gravitational motions such as outflows, which may also play important roles by pushing the dust to large radii, thus reducing the content of dust in the central region. In addition, the importance of σ H α for gas attenuation could be related to the contamination of low-ionization nuclear emission regions (LINERs) in galactic centers. As found in Law et al. (2021), regions typically associated with AGN or LINER exhibit much higher σ H α values (reaching 100-200 km/s) when compared to H ii regions located at larger radii. Figure 4 suggests that the strong correlation between σ H α and E ( B -V ) gas in galactic centers is largely attributed to the component with σ H α greater than 100 km/s. More \nwork would be needed in future if one were to fully understand the important role of σ H α for gas attenuation in galactic centers.", '4.2. Stellar attenuation': 'The strong correlation between E ( B -V ) star and stellar age in resolved regions has been found in Paper II for all the ionized gas regions as a whole. The agedependent stellar attenuation has also been considered in some earlier studies (e.g., Panuzzo et al. 2007; Noll et al. 2009; Buat et al. 2012; Lo Faro et al. 2017; Tress et al. 2019). The result in our work further shows that the driving role of stellar age for E ( B -V ) star holds across the whole galaxy, independent of radial distance, and the strong correlation is predominantly contributed by H α -faint regions. This is expected considering that the H α -faint regions are dominated by old stellar populations. Moreover, the importance of Σ H α and sSFR in determining E ( B -V ) star in the galactic center was overlooked by Paper II due to the contamination of outer regions where the importance of Σ H α and sSFR is low. When dividing the ionized regions into H α -faint and H α -bright regions, we further find that the important role of Σ H α and sSFR holds only in H α -bright regions at the galactic center. This may be understood from the fact that the central star formation is usually associated with environments with increased dust content which in turn plays a vital role in the star formation process (e.g., Dwek 1998; Panuzzo et al. 2007; Dwek & Cherchneff 2011; Zhukovska 2014).', '4.3. Implications for highz studies': "The strong dependence of dust attenuation on stellar mass has been found also for galaxies at higher redshifts, and recent studies have consistently reported a lack of significant evolution in the relation between dust attenuation and stellar mass over a wide range of redshift from z ∼ 0 up to z ∼ 6 . 5 (e.g. Shapley et al. 2022, 2023; Maheson et al. 2024). Studies of high -z galaxies are mostly limited to either measurements of global properties due to poor angular resolution of ground-based observations, or small samples of spatially resolved properties from space telescopes with poor spectral resolution. In most cases where ground-based observations are used, the best available seeing ∼ 0 . 6 '' roughly corresponds to ∼ 4 -5 kpc at z > 1. Our work has revealed strong radial variations in the dust attenuation scaling relations in nearby galaxies, which (if hold true also at higher redshifts) could be barely resolved with current observations of highz galaxies. It is unclear to what extent the lack of evolution in the attenuation-mass relation found in previous studies is caused by the poor angular resolution of the data. In addition, the significantly different attenuation relations as found in H α -faint and H α -bright regions need to be considered when interpreting results at high redshifts, because the relative contribution of star-forming regions to the global dust attenuation rela- \nted to be a function of redshift given the evolution of the cosmic star formation rate density (e.g. Madau & Dickinson 2014).", '5. SUMMARY': 'This is the third paper of a series of studies on resolved dust attenuation in nearby galaxies. In the previous two papers, we have developed a new technique to estimate a model-independent attenuation curve from a given optical spectrum of MaNGA galaxies (Paper I), which has enabled us to study the correlations of both stellar and gas attenuations at kpc scales with a large number of regional/global properties of nearby galaxies (Paper II). In this work we have extended our study by further examining the radial variations of the dust attenuation relations, using the same data as in Paper II. We apply the Random Forest regression technique to obtain the feature importance (FI) of all the regional/global properties in relation to E ( B -V ) gas and E ( B -V ) gas . We also calculate the Spearman correlation coefficients ( ρ ) to quantify the strength of the correlations between dust attenuation and galactic properties. We perform these analyses for ionized regions in different radial bins separately, and for H α -bright regions and H α -faint regions separately. \nFirst of all, the dust attenuation scaling relations obtained previously from single-fiber spectroscopy data within the SDSS are reproducible when our analysis is confined to the central region of MaNGA galaxies. \n- · If [N ii ]/[S ii ] is not included in the analysis as in previous studies, nebular velocity dispersion ( σ H α ), global stellar mass ( M ∗ ), luminosityweighted stellar metallicity ( Z L ) and surface stellar mass density (Σ ∗ ) are found to be most important for E ( B -V ) gas . This result is in good agreement with previous SDSS-based studies. If included, [N ii ]/[S ii ] outperforms all other properties as the most important property in the galactic center in relation to E ( B -V ) gas .\n- · For E ( B -V ) gas , following [N ii ]/[S ii ], quite a number of properties including σ H α , M ∗ , Σ ∗ , Z L , H α surface brightness (Σ H α ), gas-phase metallicity (12+log 10 O/H) and the [N ii ]/[O ii ] line ratio (N2O2) exhibit relatively high feature importance and strong correlation coefficient. For E ( B -V ) star , in contrast, only a few properties (Σ H α , sSFR, and t L ) show relatively strong correlations in the galactic center. \nWhen considering regions at all radii, the scaling relations of both E ( B -V ) gas and E ( B -V ) star are found to strongly vary as one goes from the galactic center towards outer regions, and from H α -faint regions and H α -bright regions. \n- · For E ( B -V ) gas , [N ii ]/[S ii ] is top ranked in feature importance and presents a much higher correla- \non coefficient than any other properties and over a wide range of radial distance from R = 0 out to R ∼ R e . In the outermost regions ( R > 1 . 2 R e ), [O iii ]/[O ii ] outperforms [N ii ]/[S ii ] as the leading property in relation to E ( B -V ) gas , although [N ii ]/[S ii ] still remains a comparably high correlation coefficient. \n- · For E ( B -V ) star , stellar age t L shows a strong correlation with ρ ∼ 0 . 55 -0 . 6 only weakly dependent on radial distance, making itself the most important property over all radii except the galactic center where H α surface density (Σ H α ) and specific star formation rate (sSFR) present similarly strong correlations. The correlation strengths of the latter two properties decline rapidly with radial distance.\n- · When dividing the ionized regions into H α -bright regions and H α -faint regions, we find the former to generally show stronger correlations with E ( B -V ) gas and the latter to be more strongly correlated with E ( B -V ) star , although depending on individual properties and radial distance.', 'ACKNOWLEDGEMENTS': "We are grateful to the anonymous referee whose comments have helped improve this paper. This work is supported by the National Key R&D Program of China (grant NO. 2022YFA1602902), and the National Natural Science Foundation of China (grant Nos. 12433003, 11821303, 11973030). \nFunding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. \nDepartment of Energy Office of Science, and the Participating Institutions. SDSS-IV acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. The SDSS web site is www.sdss.org. \nSDSS-IV is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, the Chilean Participation Group, the French Participation Group, HarvardSmithsonian Center for Astrophysics, Instituto de Astrof'ısica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU) / University of Tokyo, Lawrence Berkeley National Laboratory, Leibniz Institut fur Astrophysik Potsdam (AIP), Max-Planck-Institut fur Astronomie (MPIA Heidelberg), Max-Planck-Institut fur Astrophysik (MPA Garching), Max-Planck-Institut fur Extraterrestrische Physik (MPE), National Astronomical Observatories of China, New Mexico State University, New York University, University of Notre Dame, Observat'ario Nacional / MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Aut'onoma de M'exico, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University. \nSoftware: Astropy (Astropy Collaboration et al. 2013, 2018), Scikit-learn (Pedregosa et al. 2011)", 'REFERENCES': 'Abdurro\'uf, Accetta, K., Aerts, C., et al. 2022, ApJS, 259, 35, doi: 10.3847/1538-4365/ac4414 Anderson, L. 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2021ApJS..255....8R | We present a highprecision radial velocity RV survey of 719 FGKM stars which host 164 known exoplanets and 14 newly discovered or revised exoplanets and substellar companions. This catalog updated the orbital parameters of known exoplanets and longperiod candidates some of which have decadeslonger observational baselines than they did upon initial detection. The newly discovered exoplanets range from warm subNeptunes and superEarths to cold gas giants. We present the catalog sample selection criteria as well as over 100000 RV measurements which come from the KeckHIRES APFLevy and LickHamilton spectrographs. We introduce the new RV search pipeline RVSearch httpscaliforniaplanetsearch.github.iorvsearch that we used to generate our planet catalog and we make it available to the public as an opensource Python package. This paper is the first study in a planned series that will measure exoplanet occurrence rates and compare exoplanet populations including studies of giant planet occurrence beyond the water ice line and eccentricity distributions to explore giant planet formation pathways. We have made public all radial velocities and associated data that we use in this catalog. | 2021-07-01T00:00:00Z | ['2021ApJS..255....8R', 'arXiv:2105.11583', '2021arXiv210511583R', '10.48550/arXiv.2105.11583', '10.3847/1538-4365/abe23c'] | ['Exoplanet catalogs', 'Exoplanet astronomy', 'Radial velocity', '488', '486', '1332', 'Astrophysics - Earth and Planetary Astrophysics'] | The California Legacy Survey. I. A Catalog of 178 Planets from Precision Radial Velocity Monitoring of 719 Nearby Stars over Three Decades | 2,021 | 174 | 0.69 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF'] | 199 | https://arxiv.org/pdf/2105.11583.pdf | {'The California Legacy Survey I. A Catalog of 178 Planets from Precision Radial Velocity Monitoring of 719 Nearby Stars over Three Decades': "Lee J. Rosenthal, 1 Benjamin J. Fulton, 1, 2 Lea A. Hirsch, 3 Howard T. Isaacson, 4 Andrew W. Howard, 1 Cayla M. Dedrick, 5, 6 Ilya A. Sherstyuk, 1 Sarah C. Blunt, 1, 7 Erik A. Petigura, 8 Heather A. Knutson, 9 Aida Behmard, 9, 7 Ashley Chontos, 10, 7 Justin R. Crepp, 11 Ian J. M. Crossfield, 12 Paul A. Dalba, 13, 14 Debra A. Fischer, 15 Gregory W. Henry, 16 Stephen R. Kane, 13 Molly Kosiarek, 17, 7 Geoffrey W. Marcy, 1, 7 Ryan A. Rubenzahl, 1, 7 Lauren M. Weiss, 10 and Jason T. Wright 18, 19, 20 \n1 Cahill Center for Astronomy & Astrophysics, California Institute of Technology, Pasadena, CA 91125, USA \n2 \nIPAC-NASA Exoplanet Science Institute, Pasadena, CA 91125, USA \n3 Kavli Institute for Particle Astrophysics and Cosmology, Stanford University, Stanford, CA 94305, USA \n4 Department of Astronomy, University of California Berkeley, Berkeley, CA 94720, USA \n5 Cahill Center for Astronomy & Astrophysics, California Institute of Technology, Pasadena, CA 91125, USA \n6 Department of Astronomy & Astrophysics, The Pennsylvania State University, 525 Davey Lab, University Park, PA 16802, USA \n7 NSF Graduate Research Fellow \n8 Department of Physics & Astronomy, University of California Los Angeles, Los Angeles, CA 90095, USA \n9 Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA 91125, USA \n10 Institute for Astronomy, University of Hawai'i, Honolulu, HI 96822, USA \n11 Department of Physics, University of Notre Dame, Notre Dame, IN, 46556, USA \n12 Department of Physics and Astronomy, University of Kansas, Lawrence, KS, USA \n13 Department of Earth and Planetary Sciences, University of California, Riverside, CA 92521, USA \n14 NSF Astronomy & Astrophysics Postdoctoral Fellow \n15 Department of Astronomy, Yale University, New Haven, CT 06511, USA \n16 Center of Excellence in Information Systems, Tennessee State University, Nashville, TN 37209 USA \n17 Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064, USA \n18 Department of Astronomy and Astrophysics, The Pennsylvania State University, University Park, PA 16802, USA \n- 19 Center for Exoplanets and Habitable Worlds, The Pennsylvania State University, University Park, PA 16802, USA\n- 20 Penn State Extraterrestrial Intelligence Center, The Pennsylvania State University, University Park, PA 16802, USA \n(Received July 3, 2020; Revised November 17, 2020; Accepted February 2, 2021) \nSubmitted to AAS Journals", 'ABSTRACT': 'We present a high-precision radial velocity (RV) survey of 719 FGKM stars, which host 164 known exoplanets and 14 newly discovered or revised exoplanets and substellar companions. This catalog updated the orbital parameters of known exoplanets and long-period candidates, some of which have decades-longer observational baselines than they did upon initial detection. The newly discovered exoplanets range from warm sub-Neptunes and super-Earths to cold gas giants. We present the catalog sample selection criteria, as well as over 100,000 radial velocity measurements, which come from the Keck-HIRES, APF-Levy, and Lick-Hamilton spectrographs. We introduce the new RV search pipeline RVSearch that we used to generate our planet catalog, and we make it available to the public as an opensource Python package. This paper is the first study in a planned series that will measure exoplanet occurrence rates and compare exoplanet populations, including studies of giant planet occurrence beyond the water ice line, and eccentricity distributions to explore giant planet formation pathways. We have made public all radial velocities and associated data that we use in this catalog. \nCorresponding author: Lee J. Rosenthal \[email protected] \nKeywords: exoplanets - catalogs', '1. INTRODUCTION': "Expanding and characterizing the population of known exoplanets with measured masses, orbital periods, and eccentricities is crucial to painting a more complete picture of planet formation and evolution. A census of diverse exoplanets sheds light on worlds radically different than Earth, and can provide insight into how these planets, as well our own Solar System, formed. For instance, the mass, semi-major axis, and eccentricity distributions of giant planets can be used to constrain formation scenarios for these objects. Nielsen et al. (2019) and Bowler et al. (2020) used mass and eccentricity constraints from direct imaging surveys to show that planetary-mass gas giants likely form via core accretion (Pollack et al. 1996), while more massive brown dwarfs and other substellar companions likely form via gravitational instability in protoplanetary disks (Boss 1997). The present-day architectures and orbital properties of planetary systems can also be used to constrain their migration histories. Dawson & Murray-Clay (2013) used a sample of giant planets with minimum masses and orbits constrained by radial velocity (RV) observations to provide evidence that giant planets orbiting metal-rich stars are more likely to be excited to high eccentricities or migrate inward due to planet-planet interactions. Many related questions remain unanswered. What is the mass-period distribution of planets out to 10 AU? How abundant are cold gas giants beyond the water ice line, and what can this abundance tell us about planet formation across protoplanetary disks? How do small, close-in planets arrive at their final masses and system architectures? What is the relationship between these warm small planets and cold gas giants; are their formation processes related? These questions can only be answered with an expansive and rigorously constructed census of exoplanets with measured masses and wellconstrained orbits. \nThe community has made substantial progress on these fronts over the past few decades via targeted RV surveys. For instance, Bryan et al. (2016) surveyed 123 known giant hosts to study outer giant companions; they found that half of all giants have an outer companion, with tentatively declining frequency beyond 3 AU. Similarly, Knutson et al. (2014) found a 50% companion rate for transiting hot Jupiters using a sample of 51 stars. These two results suggest a planet formation process that favors giant multiplicity. On the small-planet front, Bryan et al. (2019) constructed an RV survey of 65 super-Earth hosts and found a giant companion rate of \n39 ± 7%. This suggests that these two populations are related in some way. Some questions have seen conflicting answers, requiring further work with a more expansive RV survey. For instance, Fernandes et al. (2019) studied planet occurrence as a function of orbital period by extracting the planetary minimum masses and periods, as well as completeness contours, from a catalog plot shown in Mayor et al. (2011), which presented a HARPS and CORALIE blind radial velocity survey of 822 stars and 155 planets over 10 yr (corresponding to a 4.6 AU circular orbit around a solar-mass star). The HARPS and CORALIE radial velocities were not published in Mayor et al. (2011), which measured giant planet occurrence as a function of orbital period out to 4000 days, in the range of the water ice line. Fernandes et al. (2019) pushed out to low-completeness regimes and estimated a sharp falloff in occurrence beyond the water ice line. In sharp contrast, Wittenmyer et al. (2020) used their radial velocities from the Anglo-Australian Planet Search to construct a blind survey of 203 stars and 38 giant planets over 18 yr. They found that giant planet occurrence is roughly constant beyond the water ice line, out to almost 10 AU. The discrepancy between these two results needs to be resolved. \nThe California Planet Search team (CPS; Howard et al. 2010a) has conducted many RV surveys over the past three decades, in order to find exoplanets, measure their minimum masses, and characterize their orbits. Many of these surveys were designed explicitly for the purpose of studying planet occurrence. Therefore, they used stellar samples that were constructed without bias toward stars with known planets, or an increased likelihood of hosting planets, such as metal-rich stars (Gonzalez 1997). For instance, the Keck Planet Search (Cumming et al. 2008) used 8 yr of Keck-HIRES data collected from 585 FGKM stars to study the occurrence of gas giants with periods as long as the survey baseline, measured the mass-period distribution of giant planets out to 5 AU, and found an increase in gas giant occurrence near the water ice line. The Eta-Earth Survey (Howard et al. 2010b) used 5 yr of Keck-HIRES data collected from 166 Sun-like stars to measure the occurrence of planets with orbital periods less than 50 days, ranging from super-Earths to gas giants, and found both an abundance of planets within 10 day orbits and a mass function that increases with decreasing mass for close-in planets. The APF-50 Survey combined 5 yr of highcadence Automated Planet Finder data on a sample of 50 bright, nearby stars with 20 yr of Keck-HIRES data \nto constrain the mass function of super-Earths and subNeptunes, and discovered several planets of both varieties (Fulton et al. 2016). \nWe constructed an aggregate survey from these distinct RV surveys, known hereafter as the California Legacy Survey (CLS), in order to measure exoplanet occurrence, particularly for planets with long orbital periods. We selected every star in the CPS catalog that was observed as part of an occurrence survey, added 31 CPS stars that satisfied our stellar selection criteria (described below), and regularly observed these stars using the Keck and UCO-Lick observatories. The California Legacy Survey contains 103,991 RVs, and reaches observational baselines beyond three decades. We wrote an automated planet search pipeline to systematically recover all planets that are detectable in the CLS and to measure the search completeness of each star's RV time series. We can use these completeness contours to calculate exoplanet occurrence rates with respect to planetary and host-star properties (e.g. Cumming et al. 2008; Howard et al. 2010b). \nIn this paper, we present the CLS stellar sample and the 164 known exoplanets orbiting these stars, as well as 14 newly discovered and vetted exoplanets and substellar companions. In Section 2, we describe our methodology for stellar selection. In Section 3, we describe the RVs measured for this survey. In Section 4, we describe our methods for computing the stellar properties of our sample. In Section 5, we describe the methods by which we search for exoplanets in the RVs, confirm their planetary status, and characterize their orbits. In Section 6, we present the catalog of known exoplanets, and describe in detail each of the new exoplanet candidates. In Section 7, we discuss the significance of our catalog, and conclude with plans for future work.", '2. STELLAR SAMPLE SELECTION': "Our goal for this study was to construct a sample of RV-observed FGKM stars and their associated planets, in order to provide a stellar and planetary catalog for occurrence studies. We want a survey that is quantifiably complete in some way, such as being volume- or magnitude-limited, so that we can perform unbiased occurrence measurements. One way to do this would be to observe every HD star within our desired range of stellar parameters, with the same cadence and a thirty year baseline. Given the constraints of finite observing time and instrumental magnitude limits, this is not possible. More importantly, there is no achievable, Platonic ideal of a quantifiably complete survey. However, we can approximate one by selecting CPS-observed stars that were originally chosen without bias toward a higher- or \nlower-than-average likelihood of hosting planets. Multiple CPS surveys, including the Keck Planet Search and Eta-Earth Survey, performed their stellar selection with these criteria. \nWe began with the Keck Planet Search sample, so that we can make direct comparisons to their results. We then supplemented those 585 stars with 135 stars that were not originally included as part of that sample, but they have since been observed by the CPS team and satisfy a set of criteria intended to ensure survey quality and statistical rigor for planet occurrence measurements. We selected these criteria to ensure data quality, both of individual measurements and stellar datasets, and proper stellar selection, without bias toward known or likely planet hosts, which would skew our occurrence measurements. We included CPS-observed stars that have at least 20 total RVs and at least 10 High Resolution Echelle Spectrometer (HIRES) RVs collected after the HIRES CCD upgrade in 2004, to guarantee enough RVs for well-constrained Keplerian fits, and have an observational baseline of at least 8 yr, which is the maximum baseline of the Cumming et al. (2008) sample at the time of publication. All stars in the Keck Planet Search sample pass these criteria, since we have collected more than 10 new HIRES RVs for each of them since 2004. \nIn order to ensure proper stellar selection, we did not include CPS-observed stars that were chosen for surveys that deliberately selected known planet hosts, metal-rich stars, or non-main-sequence stars, since these surveys would bias planet occurrence measurements. We excluded stars that were observed as part of the 'N2K' and 'M2K' surveys, which targeted metal rich stars to search for gas giants (Fischer et al. 2005; Apps et al. 2010). We excluded all massive stars that were observed as part of a search for planets orbiting subgiants (Johnson et al. 2010b), since that survey used a particular observing strategy geared solely toward detecting giant planets. We excluded all young stars that were selected for CPS observing based on photometric IR excess, since such stars were selected for an increased probability of planet occurrence (Hillenbrand et al. 2015). We excluded all stars from the 'Friends of Hot Jupiters' surveys, which targeted known planet hosts (Knutson et al. 2014). For the same reason, we excluded all stars that were observed as part of Kepler , K2, TrES, HAT, WASP, or KELT transiting planet surveys (Bakos et al. 2002; Alonso et al. 2004; Pollacco et al. 2006; Pepper 2007; Borucki 2016). \nThis selection process left us with 719 stars. Figure 1 shows the entire CLS samples as a Venn diagram, illustrating the overlap of the Cumming et al. (2008) sam- \nple with the Eta-Earth (Howard et al. 2010b) and 25 pc northern hemisphere volume-limited (Hirsch et al. 2020) samples. The 25 pc sample includes 255 G and early K dwarfs with apparent V magnitudes ranging from V ≈ 3 to V ≈ 9. These stars have a median temperature of 5360 K and a median mass of 0.86 M glyph[circledot] . The median number and duration of RV observations for this sample was 71 RVs spanning 21 yr, while the minimum number and duration of observations in the sample was 20 RVs spanning 3 yr. The architects of all three of these surveys designed them for planet occurrence studies. Therefore, they did not construct these catalogs by selecting on properties known to correlate or anticorrelate with planet occurrence. There are only 31 stars in the California Legacy Survey that do not belong to any of these three surveys but do still pass of our selection criteria. This survey has no hard constraints on distance, apparent magnitude, or color, as seen in Figure 4. \nFigure 1. Venn diagram showing the overlap between the stars in the Keck Planet Search sample (Cumming et al. 2008), the Eta-Earth sample (Howard et al. 2010a), and a 25 pc northern hemisphere volume-limited survey (Hirsch et al. in prep). 31 stars in the California Legacy Survey do not belong to the union of these three surveys. \n<!-- image -->", '3.1. Keck-HIRES': "HIRES (Vogt et al. 1994) has been in operation on the Keck I Telescope since 1994 and has been used to measure stellar RVs via the Doppler technique since 1996 (Cumming et al. 2008). This technique relies on measuring the Doppler shift of starlight relative to a reference spectrum of molecular iodine, which is at rest in the observatory frame (Butler et al. 1996). We consistently set up HIRES with the same wavelength format on the CCDs for each observation and followed other standard \nprocedures of CPS Howard et al. (2010a). With the iodine technique, starlight passes through a glass cell of iodine gas heated to 50 · C, imprinting thousands of molecular absorption lines onto the stellar spectrum, which act as a wavelength reference. We also collected an iodine-free 'template' spectrum for each star. This spectrum is naturally convolved with the instrumental point spread function (PSF) and is sampled at the resolving power of HIRES ( R = 55,000-86,000, depending on the width of the decker used). These spectra are deconvolved using PSF measurements from spectra of featureless, rapidly rotating B stars with the iodine cell in the light path. The final, deconvolved intrinsic stellar spectra serve as ingredients in a forward-modeling procedure from which we measure relative Doppler shifts of each iodine-in spectrum of a given star (Valenti et al. 1995). We also used this process to compute uncertainties on the Doppler shifts. The uncertainty for each measurement is the standard error on the mean of the RVs for 700 segments of each spectrum (each 2 ˚ Awide) run through the Doppler pipeline. We distinguish between 'pre-upgrade' RVs (1996-2004; ∼ 3 ms -1 uncertainties) and 'post-upgrade' RVs (2004-present; ∼ 1 ms -1 uncertainties). In 2004, HIRES was upgraded with a new CCD and other optical improvements. We account in the time series modeling for different RVs zero points ( γ ) for data from the two different eras. \nThe RVs reported here stem from HIRES observations with a long history. The RVs from 1996 to 2004 are based on HIRES spectra acquired by the California & Carnegie Planet Search (CCPS) collaboration and were reported in Cumming et al. (2008). CCPS continued to observe these stars, but split into two separate collaborations: CPS and the Lick-Carnegie Exoplanet Survey (LCES). This paper principally reports results from 41,804 CPS and CCPS HIRES spectra that were obtained and analyzed by our team during 19962020. In addition, we have included RVs computed by our pipeline for 7530 spectra of CLS stars taken by LCES during 2008-2014. These HIRES spectra were acquired with the same instrumental setup as the CPS spectra and are publicly available in the Keck Observatory Archive. Butler et al. (2017) separately published RVs based on the same HIRES observations from CCPS, CPS, and LCES for the 1996-2014 time span. The LCES and CPS Doppler pipelines diverged in ∼ 2007. Tal-Or et al. (2019) uncovered the 2004 zero-point offset, which we model with two independent offsets. They also claimed two second-order systematics in the LCES 2017 dataset: a long-term drift of order < 1 ms -1 , and a correlation between stellar RVs and time of night with respect to midnight. They estimated the long-term drift \nby averaging the zero points of three RV-quiet stars on each night, where possible. However, by our estimates, even the quietest stars exhibit 1-2 m s -1 jitter in HIRES time series. Averaging the zero points of three such stars will likely yield a scatter of 1 m s -1 across many nights. Additionally, they did not remove planet RV signals from their data before estimating the linear correlation between RV and time of night, and it is unclear how they derived the uncertainty in that correlation.", '3.2. Automated Planet Finder': "The APF-Levy spectrograph is a robotic telescope near the summit of Mt. Hamilton, designed to find and characterize exoplanets with high-cadence Doppler spectroscopy (Vogt et al. 2014; Radovan et al. 2014). The facility consists of a 2.4-m telescope and the Levy Spectrometer, which has been optimized for optical Doppler shift measurements. The Doppler pipeline that was developed for Keck-HIRES also extracts RV measurements from APF spectra. Most of the APF data in the California Legacy Survey was collected as part of the APF-50 Survey (Fulton 2017), the stellar sample of which was drawn entirely from the Eta Earth sample. These two surveys have slightly different selection criteria. While both surveys have a distance cut d < 25 pc and luminosity cut M V < 3, Eta-Earth cuts on apparent magnitude V < 11, whereas APF-50 has V < 7; Eta-Earth cuts on chromospheric activity log R ' HK < -4 . 7, whereas APF-50 has log R ' HK < -4 . 95; and Eta-Earth cuts on declination > -30 · , whereas APF-50 has declination > -10 · . These stricter cuts were made to ensure higher data quality for the high-cadence APF survey.", '3.3. Lick-Hamilton': 'The Hamilton Spectrograph is a high-resolution echelle spectrometer, attached to the 3 m Shane telescope on Mt. Hamilton. Beginning in 1987, and ending in 2011 with a catastrophic iodine cell failure, the Lick Planet Search program (Fischer et al. 2014) monitored 387 bright FGKM dwarfs to search for and characterize giant exoplanets. This was one of the first surveys to produce precise RVs via Doppler spectroscopy with iodine cell calibration, and yielded RVs with precision in the range 3-10 m s -1 . The Lick Planet Search overlaps heavily with the Keck Planet Search and other CPS surveys, since these surveys drew from the same bright-star catalogs.', '3.4. Activity Indices': 'For each HIRES and APF spectrum from which we measure radial velocities, we also measure the strength of emission in the cores of the Ca II H & K lines (Svalues) following the techniques of Isaacson & Fischer \n(2010) and Robertson et al. (2014). There is a small, arbitrary offset between the HIRES and APF activity indices. We adopted uniform S-value uncertainties with values of 0.002 and 0.004 for HIRES and APF respectively. We provide activity indices along with our RV measurements. Missing values are the result of sky contamination and/or low SNR.', '3.5. APT Photometry': "We collected long-term photometric observations of the subset of our sample that were included in the APF50 survey (Fulton 2017), in order to search for evidence of rotation-induced stellar activity. We collected these measurements with Tennessee State University's Automated Photometric Telescopes (APTs) at Fairborn Observatory as part of a long-term program to study stellar magnetic activity cycles (Lockwood et al. 2013). Most stars have photometric datasets spanning 15 - 23 yr. The APTs are equipped with photomultiplier tubes that measure the flux in the Stromgren b and y bands relative to three comparison stars. We combined the differential b and y measurements into a single ( b + y ) / 2 'passband' then converted the differential magnitudes into a relative flux normalized to 1.0. The precision in relative flux is typically between 0.001 and 0.0015. Further details of the observing strategy and data reduction pipeline are available in Henry (1999); Eaton et al. (2003); Henry et al. (2013). We make the photometric data available as a machine-readable table.", '3.6. Observational Statistics': 'We examined the range of observing cadences and observational baselines within the CLS sample, to determine whether stars without known planets were observed with strategies that differed significantly from those for stars with known planets. Figure 2 shows the distribution of number of observations and observational baselines for three groups of stars: the entire sample, the stars around which we detected planets, and the star around which we did not detect planets. Each of these three samples has a median baseline of 21 yr. Stars with detected planet have a median of 74 observations, compared to 35 observations for stars without detected planets and 41 observations for the entire CLS sample. A factor of two in number of observations will have a small but measurable impact on planet detectability of a given data set - and therefore on its search completeness contours.', '4. STELLAR PROPERTIES': 'We derived stellar properties for our sample by applying the SpecMatch (Petigura 2015) and Isoclassify \n<!-- image --> \nAll stars \nStars with detections \nStars without detections \n<!-- image --> \nFigure 2. Distributions of observational baseline versus number of observations. The top panel shows these statistics for all stars in the CLS sample; the center panel shows stars around which we detect planets; the bottom panel shows stars around which we do not detect planets. Median baseline and number of observations for each sample are overplotted as translucent circles. \n<!-- image --> \n(Huber 2017) software packages to the template KeckHIRES spectra of our stars. Specmatch takes an optical stellar spectrum as input, and by interpolating over \na grid of template spectra with known associated stellar properties, returns three spectral properties and uncertainties. For stars hotter than 4700 K, we interpolated over synthetic spectra to derive spectral parameters (Petigura 2015). For stars below this threshold, we interpolated over real spectra of cool stars with wellcharacterized stellar properties, since synthetic spectral models are unreliable below this temperature (Yee et al. 2017). \nSpecmatch produces metallicity, effective temperature, and surface gravity when interpolating over synthetic spectra; it produces metallicity, effective temperature, and radius when interpolating over empirical spectra. Isoclassify takes effective temperature, metallicity, and surface gravity as spectral parameter inputs, and uses isochrone models and multinest Bayesian sampling (Buchner 2016) to produce estimates and uncertainties of physical parameters, in particular stellar mass. For stars cooler than 4700 K, we passed Isoclassify a wide Gaussian input prior on surface gravity, since temperature and metallicity strongly constrain the masses of cool, main-sequence stars (Johnson et al. 2017). \nAlmost all stars in the California Legacy Survey have both Gaia-measured parallaxes (Gaia Collaboration et al. 2016, 2018; Lindegren et al. 2018) and apparent K -band magnitudes. For stars with both of these measurements available, we pass them and their uncertainties into Isoclassify as additional inputs, since taken together, they constrain stellar luminosity and therefore place tighter constraints on stellar mass. Isoclassify also returns more precise estimates of stellar radius when provided with parallax and apparent magnitude. With the inclusion of this luminosity constraint, the median precision of our stellar mass measurements is 3.6%. \nIn Table 2 in Appendix B, we report stellar mass, radius, surface gravity, effective temperature, and metallicity for a subsection of the CLS sample. We make this table available for the entire sample in machine-readable format, with additional columns including V -band magnitude and Gaia parallax. Figure 3 is a visualization of these stellar properties, while Figure 4 shows individual histograms for mass, metallicity, and effective temperature, as well as for the following observational properties: parallax-inverse distance, V , and B -V .', '5.1. Planet Search': "We developed an iterative approach to a search for periodic signals in RV data in order to generate the CLS planet catalog. We outline this algorithm, which we de- \nFigure 3. Stellar property measurements of the California Legacy Survey, in effective temperature, surface gravity, and mass. The sample consists of stars spanning spectral types F, G, K, and M, some of which have evolved off of the main sequence. Most stars have metallicities within 0.4 dex of Solar metallicity, with the exception of a small handful of extremely metal-poor stars, which lie below the main sequence on this plot. \n<!-- image --> \nveloped as the open-source Python package RVSearch and have made public alongside the publication of this paper. Figure 5 is a flowchart that lays out each step of the algorithm, and Figure 6 is a visualization of an example RVSearch output, where the top two panels show the final model, and each successive row shows an iterative search for each signal in the model. First, we provide an initial model, from which the iterative search begins. This initial model contains an RV data set, and a likelihood function. The natural logarithm of the latter is defined as \nln( L ) = -1 2 ∑ i [ ( v i -m ( t i ) -γ D ) 2 σ 2 i +ln(2 πσ 2 i ) ] , (1) \nwhere i is the measurement index, v i is the i th RV measurement, γ D is the offset of the instrumental dataset from which the i th measurement is drawn, and σ 2 i is the quadrature sum of the instrumental error and the stellar jitter term of the i th measurement's instrumental data set. Here, m ( t i ) is the model RV at time t i , defined as \nm ( t ) = ∑ n K( t | K n , P n , e n , ω n , t cn )+˙ γ ( t -t 0 )+¨ γ ( t -t 0 ) 2 , \nwhere n is a given Keplerian orbit in the model, K( t | K,P,e,ω,t c ) is the Keplerian orbit RV signature at time t given RV amplitude K , period P , eccentricity e , argument of periastron ω , and time of inferior conjunction t c , ˙ γ is a linear trend term, ¨ γ is a quadratic trend term, and t 0 is a reference time, which we defined as the median time of observation. \nWe used RadVel (Fulton et al. 2018) to fit Keplerian orbits. The initial likelihood model contains either a one-planet Keplerian model with undefined orbital parameters, or a predefined model including trend/curvature terms and/or Keplerian terms associated with known orbital companions. We defaulted to performing a blind search starting with the undefined single-planet model, and we only supply a predefined model if there is evidence for a highly eccentric companion whose period is misidentified by our search algorithm. Several highly eccentric stellar binaries satisfy this criterion, as do two planets: HD 120066 b (Blunt et al. 2019), and HD 80606 b (Wittenmyer et al. 2007b). \nBefore beginning a blind search, RVSearch determines whether the data merits a trend with curvature, a linear trend, or no trend. It does this by fitting each of these three models to the data, then performing a goodness-of-fit test to decide which model is favored. \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 4. Stellar parameter distributions. The left column shows mass, metallicity, and effective temperature, while the right column shows parallax-inferred distance, V -band magnitude, and B -V color. Black lines are histograms of the stellar parameter median values. For the left column, colored lines are 500 histograms per panel, with parameters redrawn from normal distributions with width equal to their individual measurement uncertainties. We left these redrawn parameter histograms out for the plots in the right column because distance, magnitude, and color have uncertainties that are smaller than the chosen bin size. \n<!-- image --> \nWe measured the Bayesian Information Criterion (BIC) for each of the three models, and computed the ∆BIC between each model. RVSearch selects the linear model if it has ∆BIC = 5 with respect to the flat model, and \nthe quadratic model if it has ∆BIC = 5 with respect to the linear model. We did not perform this test on datasets that contain eccentric companions with orbital periods greater than the data's observational baseline, \nsince such datasets would be better fit with a long-period Keplerian orbit than with linear and parabolic trends. The Bayesian information criterion is defined as \nBIC = k ln( n obs ) -2ln( L ) , (3) \nwhere n obs is the number of observations, k is the number of free model parameters, and ln( L ) is the loglikelihood of the model in question. \nOnce we provide an initial model, RVSearch defines an orbital period grid over which to search, with sampling such that the difference in frequency between adjacent grid points is 1 2 πτ , where τ is the observational baseline. We chose this grid spacing in accordance with Horne & Baliunas (1986), who state that, in frequency space, a Lomb-Scargle periodogram has a minimum peak width of 1 2 πτ . For each dataset, we searched for periodicity between two days and five times the observational baseline. Searching out to five times the baseline only adds a few more points to the period grid, and it allows for the possibility of recovering highly eccentric, ultra-longperiod planet candidates with best-fit orbital period. \nThe search algorithm then computes a goodness-of-fit periodogram by iterating through the period grid and fitting a sinusoid with a fixed period to the data. We measure goodness-of-fit as the ∆BIC at each grid point between the best-fit, n +1-planet model with the given fixed period, and the n -planet fit to the data (this is the zero-planet model for the first planet search). \nAfter constructing a ∆BIC periodogram, the algorithm performs a linear fit to a log-scale histogram of the periodogram power values. The algorithm then extrapolates a ∆BIC detection threshold corresponding to an empirical false-alarm probability of 0.1%, meaning that, according to the power-law fit, only 0.1% of periodogram values are expected to fall beyond this threshold. This process follows the detection methodology outlined in Howard & Fulton (2016). \nIf a periodic signal exceeds this detection threshold, RVSearch refines the fit of the corresponding Keplerian orbit by performing a maximum a posteriori (MAP) fit with all model parameters free, including eccentricity, and records the BIC of that best-fit model. RVSearch includes two hard-bound priors, which constrain K > 0 and 0 < = e < 1. The algorithm then adds another planet to the RV model and conducts another grid search, leaving all parameters of the known Keplerian orbits free so that they might converge to a more optimal solution. In the case of the search for the second planet in a system, the goodness of fit is defined as the difference between the BIC of the best-fit one-planet model and the BIC of the two-planet model at each fixed period in the grid. RVSearch once again sets a detection \nthreshold in the manner described above, and this iterative search continues until it returns a nondetection. \nThis iterative periodogram search is superior to a Lomb-Scargle residual subtraction search in two key ways. First, this process fits for the instrument-specific parameters of each dataset, stellar jitter and RV-offset, as free parameters throughout the search. Second, by leaving the known model parameters free while searching for each successive planet, we allow the solutions for the already discovered planets to reach better maxlikelihood solutions that only become evident with the inclusion of another planet in the model. \nNote that our search and model comparison process is not Bayesian; we do not use priors to inform our model selection, and we do not sample posteriors, beyond a grid search in period space, until we settle upon a final model. We use the BIC as our model comparison metric because it incorporates the number of free parameters as a penalty on more complex models, which, in our case, corresponds to models with additional planets. \nWe make RVSearch publicly available alongside this paper via a GitHub repository. See the RVSearch website for installation and use instructions.", 'RVSearch Flowchart': 'Figure 5. Search algorithm flowchart. \n<!-- image --> \nYear \nFigure 6. Example RVSearch summary plot, for the known two-planet system HIP 109388. Panel a) shows the total model plotted over the radial velocity time series, while panel b) shows the model residuals. Each successive row shows a phase-folded signal discovered by RVSearch on the left, and the associated periodogram on the right. The final row shows the running periodograms of each signal, generated with Lomb-Scargle power, on the left, and the final periodogram on the right. \n<!-- image -->', '5.2. Search Completeness': 'We characterized the search completeness of each individual dataset, and of the entire survey, by running injection-recovery tests. Once RVSearch completed an iterative search of a dataset, it injected synthetic planets into the data and ran one more search iteration to determine whether it recovers these synthetic planets in that particular dataset. We ran 3000 injection tests for each star. We drew the injected planet period and M sin i from log-uniform distributions, and drew eccentricity from the beta distribution described in Kipping (2013), which was fit to a population of RV-observed planets. \nWe used the results of these injection tests to compute search completeness for each individual dataset, and report constant M sin i / a contours of detection probability. Figure 7 shows examples of these contours and the corresponding RVs for three different stars, all early G-type: one with 25 observations, one with 94, and one with 372. We make the 10th, 50th, and 90th percentile completeness contours for each individual star available in machine-readable format.', '5.3. Model posteriors': "Once RVSearch returned max-likelihood estimates of the orbital model parameters for a given dataset, we sampled the model posterior using affineinvariant sampling, implemented via emcee and RadVel (Foreman-Mackey et al. 2013; Fulton et al. 2018). We sampled using the orbital parameter basis { log P K t c √ e sin ω √ e cos ω } . We placed uniform priors on all fitting parameters, with hard bounds such that K > 0 and 0 ≤ e < 1. We fit in log P space to efficiently sample orbits with periods longer than our observational baseline, and in √ e sin ω and √ e cos ω to minimize bias toward higher eccentricities (Lucy & Sweeney 1971). We reported parameter estimates and uncertainties as the median and ± 1 σ intervals. \nIf a dataset is so poorly constrained by a Keplerian model that emcee 's affine-invariant sampler cannot efficiently sample the posterior distribution, we instead used a rejection sampling algorithm to estimate the posterior. In these cases, we used TheJoker (Price-Whelan et al. 2017), a modified MCMC algorithm designed to sample Keplerian orbital fits to sparse radial velocity measurements. We chose a flat prior on log P , with a minimum at the observing baseline and a maximum at twenty times the observing baseline. We drew orbital eccentricity from a beta prior weighted toward zero, as modeled in Kipping (2013), in order to downweight orbits with arbitrarily high eccentricity, which can be vi- \nable fits to sparse or otherwise underconstraining RV data sets.", '5.4. False-positive vetting': 'We performed a series of tests to vet each planet candidate discovered by our search pipeline. The following subsections each detail one test we perform to rule out one way in which a signal might be a false-positive. We also represent this process with a flowchart in Figure 9, and include a table of all false-positive signals recovered by RVSearch in Table 6.', '5.4.1. Stellar activity, magnetic/long-period': "Many main-sequence stars, particularly F- and Gtype, have magnetic activity cycles on timescales of several to tens of years. These fluctuations in activity can cause changes in the core depths of stellar Calcium H & K lines, which manifest as apparent RV shifts (Isaacson & Fischer 2010). To evaluate whether stellar activity may be the cause of a signal recovered by our search pipeline, we measure the linear correlation between the RV signature of that signal and a measured stellar activity metric-in our case, S-values. We computed S-values for both post-upgrade HIRES and APF data by measuring the core flux of Calcium H & K lines. \nIf we found a periodic signal in the S-value data that has a similar period and phase similar to one of the Keplerian terms in our RV model, we searched for correlations between our RV model and S-values. If we found one periodic signal in an RV dataset, we measured its correlation with stellar activity simply as the linear correlations between the RVs of each instrument and their associated S-values. If we found multiple periodic signals, then for each signal, we subtracted the associated RV models of all other signals from the data, and measured the correlations between these residuals and the S-values. A significant linear correlation between a signal's RV residuals and the associated S-values does not necessarily mean that this signal is caused by stellar activity, even when these signals also have the same period and phase, but we took it as sufficient evidence to remove such signals from our catalog of confirmed planets. \nIt is important to note that our approach to vetting our planet candidates is systematic but not exhaustive, particularly with respect to stellar activity. One might use activity metrics beyond S-values and photometry, such as H α line modulation. Furthermore, there are more sophisticated ways to deal with activity than searching for linear correlations with RVs. For instance, one might actively model stellar activity during the search process, using a Gaussian Process (Haywood et al. 2014) or some other correlated noise model. Such techniques might improve the accuracy of our planet \nFigure 7. RVs and completeness contours for three datasets with similar baselines, median measurement errors, and stellar jitter. The left column plots RVs with respect to time, while the right column plots injected signals in the M sin i and a plane, where blue dots are recovered injections and red dots are not. The right column also shows detection probability contours, with 50% plotted as a solid black line. From top to bottom, we show RVs and contours for HD 44420, for which we have 24 RVs; HD 97343, for which we have 94 RVs; and HD 12051, for which we have 372 RVs. \n<!-- image --> \ncandidate parameters and catalog selection, but require case-by-case analysis for each stellar system, as activity modeling is sometimes unwarranted or even counterproductive, e.g., for low-activity stars or confirmed planets \nthat have periods similar to their host star's activity cycle. We chose to perform uniform, after-the-fact vetting for our catalog, and invite others to perform more sophisticated modeling for individual systems of interest.", '5.4.2. Stellar activity, rotation/short-period': 'We only detected planet candidates that are lowamplitude and short-period enough to possibly be stellar rotation false positives in sustained, high-cadence datasets. Almost all CLS datasets that satisfy this criteria were collected as part of the APF-50 survey. We collected APT photometry of all APF-50 stars, which we can use to search for evidence of stellar rotation with moving-average smoothing and periodogram analysis. If we find strong evidence for rotation in APT photometry, or spectral S-value measurements, we discount planet candidates with periods close to the apparent rotation timescale or its harmonics.', '5.4.3. Yearly alias': 'When we find a signal with a period of a year or an integer fraction of a year, we investigate whether it is an alias of long-period power, or a systematic that is correlated with the barycentric velocity at the time of observation or Doppler fitting parameters. We do this by recomputing the associated RVs using a different template observation. When another template observation was unavailable, we were able to take one using KeckHIRES during collaborator observing nights. Templates taken in poor observing conditions or when barycentric velocity with respect to the observed star is high can produce systematic errors in the Doppler code. If a search of this new dataset returns a nondetection, or detection at a significantly different period, we conclude that this signal is an alias. Figure 8 shows the presence of yearly alias power in our survey, seen in a stack of the the final nondetection periodograms of all CLS stars. \nFigure 8. Stack of all final nondetection periodograms in the CLS planet search, linearly interpolated to the same period grid. A broad peak around 1 year is evident, as well as narrow peaks at 1/2-year, 1/3-year, and 1/4-year. \n<!-- image --> \nFigure 9. Candidate vetting flowchart. \n<!-- image -->', '6. PLANET AND STELLAR/SUBSTELLAR COMPANION CATALOG': 'We present orbital solutions for the known planets, substellar companions, and stellar binaries that RVSearch has recovered in the California Legacy Survey. As mentioned in Section 5.1, where appropriate, we modeled long-period companions with linear or parabolic trends. We included in the appendix portions of the tables associated with each class of object: one for \nplanets, one for stellar and substellar companions that are best modeled by Keplerian orbits, and one for stars with linear or parabolic RV trends. We also present 14 newly confirmed or significantly revised exoplanets and substellar companions. We list them and their orbital parameters in Table 1, and include individual notes on each system in Appendix A. Figure 10 shows all recovered planets in our survey, and distinguishes between known planets and new discoveries. \nTable 1 . Discovered or Revised Planets and Substellar Companions', '7. DISCUSSION': "Through the use of high-cadence APF observations and long-baseline HIRES observations, we have expanded the population of known exoplanets along the current mass and semi-major axis boundary of detectability, as seen in Figure 10. We recovered 43 planets with M sin i < 30 M ⊕ , including four new discoveries within 1 AU. In a future paper in the California Legacy Survey series, we will leverage the decades-long-baseline datasets in which these planets were discovered, in order to constrain the probability that a host of a small planet also hosts an outer companion, as explored in Bryan et al. (2019) and Zhu & Wu (2018). We will also directly place a lower limit on the conditional occurrence of inner small planets given the presence of an outer gas giant. \nIn addition to expanding the population of small planets with measured M sin i , we discovered or revised the orbits of ten planets with orbital separations greater than 1 AU, six of them beyond 4 AU. We represent the model posteriors for the coldest of these planets in Figure 11, and show a gallery of some of their orbits in \nFigure 12. These discoveries include two new detections with incomplete orbits, HD 213472 b and HD 26161 b. Details are provided in Appendices A.3 and A.14. Using HIRES to extend the observational baseline of our survey by another decade will tighten our M sin i and orbital parameter constraints for these planets, and may reveal more cold companions beyond 10 AU. \nIn a future paper in the CLS series, we will use our sample of long-period planets and completeness contours to measure the mass-period planet occurrence distribution out to 10 AU, extending beyond the Keck Planet Search's limit of 5 AU (Cumming et al. 2008) and the 9 AU limit of Wittenmyer et al. (2020). This will provide novel constraints on planet occurrence beyond the water ice line, resolve the discrepancy between the results of Fernandes et al. (2019) and those of Wittenmyer et al. (2020), and provide new insight into planet formation across protoplanetary disks. \nFigure 13 is a visualization of the eccentricities of all planets in the California Legacy Survey. In future work, we will quantify the eccentricity distribution of gas giants in our sample and its dependence on planet mass and multiplicity, as well as the eccentricity distributions \nFigure 10. Scatterplot of best-fit M sin i and semi-major axis values for planets in the CLS catalog. Blue dots represent known planets, while green circles represent newly discovered planets and planets with significantly revised orbits. \n<!-- image --> \nFigure 11. Contours (1- and 2σ ) of M sin i and semi-major axes for planets in the CLS sample whose semi-major axis posteriors extend beyond 10 AU. Contours for HD 26161 b have hard cutoffs due to sparsity below 7 M J and 12 AU; these limits come from the data's baseline and RV increase to date. \n<!-- image --> \nof brown dwarfs and other substellar companions, in order to clarify possible formation pathways. We will extend the wide-orbit population comparisons of Bowler et al. (2020) to our sample of planets and brown dwarfs within 20 AU of their hosts. We will also explore the eccentricity distribution of gas giants beyond 7 AU. As Figures 12 and 13 show, all planets recovered beyond 7 AU are eccentric with significance e > 2 σ e . This may \nbe a selection effect, as the median baseline of observations in our sample is 21 yr, which corresponds to a semi-major axis of 7.6 AU for a planet orbiting a solarmass star. It is possible that planets with orbital periods beyond our observational baselines are more easily detectable if they are eccentric. We can use injectionrecovery tests to determine whether there is a detection bias toward eccentric planets beyond observational baselines. If this phenomenon is not a selection effect, it might imply that most giant planets beyond 7 AU have undergone a scattering event or otherwise been excited to high eccentricity. Taken together, these studies will leverage this decades-long observational undertaking to provide new insights into planet formation and evolution. \n] \n1 \ns \nm \n[ \nV \nR \n120 \n100 \n80 \n60 \n40 \n20 \n0 \n20 \n40 \nHD 26161 b \n2000 \n2005 \n2010 \n2015 \n2020 \n<!-- image --> \n2000 \n2005 \n2010 \n2015 \n2020 \nHD 120066 b \n2000 \n2005 \n2010 \n2015 \n2020 \nYear \n2005 \n2010 \n2015 \n2020 \n2000 \n2005 \n2010 \n2015 \n2020 \nYear \nFigure 12. Orbit gallery for six of the coldest companions in our survey. We plot RV data and Keplerian model versus year, and subtract off the model signatures of inner companions and stellar activity. We did not include UMa 47 d, seen in Figure 11, in this plot, because its detection relied on early Lick-Hamilton RVs, and we wanted to showcase HIRES RV measurements from the past twenty-four years.Figure 13. M sin i , a , and eccentricity of the CLS sample. Eccentricity is plotted in medians and 68% confidence intervals, while scatter size is proportional to M sin i posterior mode. \n<!-- image -->", 'ACKNOWLEDGMENTS': "L.J.R. led the construction of this paper, including finalizing the stellar sample, running the Keplerian search, assessing planet candidates and generating the planet catalog, generating most of the figures, and writing this manuscript. The RVsearch pipeline was developed by L.J.R., B.J.F., and L.A.H., with assistance from A.W.H., H.T.I., E.A.P., and C.M.D. B.J.F., A.W.H., L.A.H., H.T.I., E.A.P., and I.A.S. assisted L.J.R. in vetting the planet candidates and insuring the integrity of the RVs and the planet catalog. A.W.H., G.W.M. (though 2015), D.A.F., and J.T.W. provided leadership and funding to CPS and CCPS. L.J.R., B.J.F., L.A.H., H.T.I., A.W.H., S.C.B., E.A.P., A.B., A.C., J.R.C., I.J.M.C., P.A.D., D.A.F., M.K., G.W.M., R.A.R., L.M.W., and J.T.W. contributed significantly to the Doppler observations. H.T.I., A.W.H., B.J.F., and G.W.M. executed and refined the Doppler pipeline that produced the RVs reported here. G.W.H. contributed photometry and analysis that were used to rule out stellar activity signals. I.A.S. provided similar analysis of activity based on a suite of indicators. B.J.F., A.W.H., E.A.P., L.M.W., R.A.R., and H.T.I. created an internal data visualization system ('Jump') that was integral to this project. L.J.R., B.J.F., A.W.H., L.A.H., H.T.I., E.A.P., H.A.K., S.R.K., P.A.D., and L.M.W. contributed to the discussion section and structure of this paper, as well as the strategy of this paper and successors in the CLS series. \nHD 50499 c \n50 \n0 \n50 \n100 \n150 \n60 \n40 \n20 \n0 \n20 \nHD 68988 c \n0 \n10 \n20 \n30 \n40 \n50 \n60 \n30 \n20 \n10 \n0 \n10 \n20 \n30 \nHD 213472 b \nWe thank Jay Anderson, G'asp'ar Bakos, Mike Bottom, John Brewer, Christian Clanton, Jason Curtis, Fei Dai, Steven Giacalone, Sam Grunblatt, Michelle Hill, Lynne Hillenbrand, Rebecca Jensen-Clem, John A. Johnson, Chris McCarthy, Sean Mills, Teo Moˇcnik, Ben Montet, Jack Moriarty, Tim Morton, Phil Muirhead, Sebastian Pineda, Nikolai Piskunov, Eugenio Rivera, Julien Spronck, Jonathan Swift, Guillermo Torres, Jeff Valenti, Sharon Wang, Josh Winn, Judah van Zandt, Ming Zhao, and others who contributed to the observations and analysis reported here. We acknowledge R. P. Butler and S. S. Vogt for many years of contributing to this dataset. This research has made use of the Keck Observatory Archive (KOA), which is operated by the W. M. Keck Observatory and the NASA Exoplanet Science Institute (NExScI), under contract with the National Aeronautics and Space Administration. We acknowledge RVs stemming from HIRES data in KOA with principal investigators from the LCES collaboration (S. S. Vogt, R. P. Butler, and N. Haghighipour). We gratefully acknowledge the efforts and dedication of the Keck Observatory staff for support of HIRES and remote observing. We are grateful to the time assignment committees of the Caltech, the University of California, the University of Hawaii, NASA, and NOAO for their generous allocations of observing time. Without their longterm commitment to radial velocity monitoring, these planets would likely remain unknown. \nWe thank Ken and Gloria Levy, who supported the construction of the Levy Spectrometer on the Automated Planet Finder, which was used heavily for this research. We thank the University of California and Google for supporting Lick Observatory, and the UCO staff as well as UCO director Claire Max for their dedicated work scheduling and operating the telescopes of Lick Observatory. G.W.H. acknowledges long-term support from NASA, NSF, Tennessee State University, and the State of Tennessee through its Centers of Excellence program. A.W.H. acknowledges NSF grant 1753582. H.A.K. acknowledges NSF grant 1555095. P.D. gratefully acknowledges support from a National Science Foundation (NSF) Astronomy & Astrophysics Postdoctoral Fellowship under award AST-1903811. \nThis work has made use of data from the European Space Agency (ESA) mission Gaia (https://www. cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www. cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. \nFinally, we recognize and acknowledge the cultural role and reverence that the summit of Maunakea has within the indigenous Hawaiian community. We are deeply grateful to have the opportunity to conduct observations from this mountain. \nFacilities: Keck:I (HIRES), Automated Planet Finder (Levy), Lick (Hamilton) \nSoftware: All code, plots, tables, and data used in this paper are available at github.com/leerosenthalj/ CLSI. Data and tables, including the full stellar catalog with { M , R , T eff , log g , [Fe/H] } , as well as APT photometry, are also available in the associated .tar.gz file available through ApJ. RVSearch is available at github.com/ California-Planet-Search/rvsearch. This research makes use of GNU Parallel (Tange 2011). We made use of the following publicly available Python modules: pandas (McKinney 2010), numpy/scipy (van der Walt et al. 2011), emcee (Foreman-Mackey et al. 2013), Specmatch (Petigura 2015; Yee et al. 2017), Isoclassify (Huber 2017), TheJoker (Price-Whelan et al. 2017), RadVel (Fulton et al. 2018), RVSearch (this work).", 'APPENDIX': 'In our appendices, we have included individual notes on each planet discovery reported in this paper; complete tables of recovered planets, Keplerian-resolved stellar binaries, and substellar companions in the California Legacy Survey; signals that RVSearch recovered and we determined to be false-positives; linear and parabolic RV trends; and excerpts from the stellar sample and RV dataset, which are available in their entirety in machinereadable format.', 'A.1. HD 3765': 'HD 3765 is a K2 dwarf at a distance of 17.9 pc (Gaia Collaboration et al. 2018). Figure 14 shows the RVSearch results for this star. We recovered a signal with a period of 3.36 yr. Table 1 reports all planet parameters. There is significant periodicity in the S-value time series, but concentrated around a period of 12 yr. Furthermore, we find no correlation between the RVs and S-values. Figure 15 shows a Lomb-Scargle periodogram of the S-value time series. Thus, we label this signal as a confirmed planet, with M sin i =0 . 173 ± 0 . 014 M J and a = 2 . 108 ± 0 . 033 AU. The magnetic activity cycle is too weak for RVSearch to recover, but is evident in the best-fit RV residuals. We used RadVel to model this activity cycle with a squared-exponential Gaussian process, and report MCMC-generated posteriors for both orbital and Gaussian process parameters in Figure 16 and Figure 17. \nYear \nFigure 14. RVSearch summary plot for HD 3765. See Figure 6 for plot description. \n<!-- image --> \nHD 3765 Activity \nFigure 15. Lomb-Scargle periodogram of HIRES S-values for HD 3765. Significant power at and beyond 4,300 days. \n<!-- image --> \n1 \nYear \nFigure 16. RadVel model orbital plot for HD 3765, including a Gaussian process with a squared-exponential kernel. The grey shaded curve represents the 68% interval for the Gaussian process RV signature. \n<!-- image --> \nFigure 17. Orbital and Gaussian process parameter posteriors for HD 3765. η 1 is the GP amplitude, while η 2 is the GP exponential decay timescale. \n<!-- image -->', 'A.2. HD 24040': 'HD 24040 is a G1 dwarf at a distance of 46.7 pc. Figure 18 shows the RVSearch results for this star. It hosts a known gas giant (Wright et al. 2007; Feng et al. 2015) with a semi-major axis that we measured as a = 4 . 72 ± 0 . 18 AU, an orbital period of 9 . 53 ± 10 -4 yr, and a minimum mass M sin i = 4 . 09 ± 0 . 22 M J . We have extended the observational baseline of our HIRES measurements to 21.7 yr, constrained the long-term trend and curvature of the RVs, and discovered a new exoplanet, a sub-Saturn ( M sin i = 0 . 201 ± 0 . 027 M J ) on a 1.4 yr orbit ( a = 1 . 30 ± 0 . 021 AU) that is consistent with circular. The S-values are uncorrelated with the the RVs of both planet signals, after removing the long-term trend. Figure 19 shows a Lomb-Scargle periodogram of the S-value time series. Table 1 reports all planet parameters. \nIn addition to the newly detected sub-Saturn, we further constrained the known linear trend in the RVs and found evidence for a curvature term as well. RVSearch detected a curvature term with model preference ∆BIC > 10 over a purely linear trend. We measured the linear trend to be 0 . 00581 ± 0 . 00044 ms -1 d -1 , and the curvature to be -6 . 6 × 10 -7 ± 1 . 2 × 10 -7 ms -1 d -2 , a 5.5 σ detection. The trend and curvature parameters are slightly correlated in the posterior, but neither is correlated with any of the Keplerian orbital parameters in the model. Therefore, we kept the curvature term that RVSearch selected in our model. This long-term trend is low-amplitude enough that it may be caused by another planet in the system, orbiting beyond 30 AU. Gaia astrometry or another two decades of RVs may provide further constraints on this object. \nFigure 18. RVSearch summary plot for HD 24040. See Figure 6 for plot description. \n<!-- image --> \nHD 24040 Activity \nFigure 19. Lomb-Scargle periodogram of HIRES S-values for HD 24040. No periods show power that is statistically significant. \n<!-- image -->', 'A.3. HD 26161': 'HD 26161 is a G0 dwarf located at a distance of 50.0 pc. Figure 20 shows the RVSearch results for this star. Our RVs are consistent with a long-period, eccentric companion, and RVSearch detected this long-period signal. Due to the sparseness of the data and the fractional orbital coverage, traditional MCMC methods fail to return a well-sampled model posterior. Since the data underconstrains our model, we used TheJoker to sample the posterior, which is consistent with an extremely long-period gas giant with minimum mass M sin i = 13 . 5 +8 . 5 -3 . 7 M J , semi-major axis a = 20 . 4 +7 . 9 -4 . 9 AU, and eccentricity e = 0 . 82 +0 . 06 -0 . 05 . Table 1 reports current estimates of all orbital parameters, and Figure 21 shows their posterior distributions. A Keplerian model is significantly preferred over a quadratic trend, with ∆BIC > 15. \nThe Simbad stellar catalog designates HD 26161 as a stellar multiple. We used Gaia to identify a binary companion with similar parallax and within 60 arcseconds. This companion has an effective temperature identified from Gaia colors of 4053 K, and a projected separation of 562 AU. A stellar companion that is currently separated from its primary by more than 560 AU could not cause a change in RV of 100 m s -1 over 4 yr. This curve is far more likely caused by an inner planetary or substellar companion approaching periastron. \nFigure 22 shows a sample of possible orbits for HD 26161 b, drawn from our rejection sampling posteriors and projected over the next decade. We will continue to monitor HD 26161 with HIRES at moderate cadence, and have begun observing this star with APF. As we gather more data during the approach to periastron, we can tighten our constraints on the minimum mass, eccentricity, and orbital separation of HD 26161 b. \nFigure 20. RVSearch summary plot for HD 26161. See Figure 6 for plot description. \n<!-- image --> \nFigure 21. Rejection sampling posterior for HD 26161. \n<!-- image --> \nFigure 22. Possible orbits for HD 26161 b. RV curves are drawn from the rejection sampling posterior generated with TheJoker . The color of each orbit drawn from the posterior scales with M sin i . \n<!-- image -->', 'A.4. HD 66428': "HD 66428 is a G8 dwarf found at a distance of 53.4 pc. Figure 23 shows the RVSearch results for this star. This system has one well-constrained cold Jupiter (Butler et al. 2006b) and an outer companion candidate first characterized in Bryan et al. (2016) as a linear trend. With four more years of HIRES data, we now see curvature in the RVs and a clear detection in RVSearch , and can place constraints on this outer candidate's orbit with a Keplerian model. The Keplerian orbit for the outer candidate is preferred to a parabolic trend with ∆BIC > 30. A maximum likelihood fit gives an orbital period of P = 36 . 4 yr. However, since we have only observed a partially resolved orbit so far, the orbit posterior in period space is wide and asymmetric. MCMC sampling produces P = 88 +153 -49 yr. Table 1 reports current estimates of all orbital parameters. \nThe model parameters are M sin i = 27 +22 -17 M J , a = 23 . 0 +19 . 0 -7 . 6 AU, and e = 0 . 31 +0 . 13 -0 . 13 . This orbital companion could be a massive gas giant or a low-mass star, if we only consider constraints from RV modeling. However, Bryan et al. (2016) used NIRC2 Adaptive-Optics images to place upper bounds on the mass and semi-major axis of an outer companion, at a time when it only presented as a linear trend in HIRES RVs. They found an upper bound of ≈ 100 M J on mass, not just M sin i , and an upper bound of ≈ 150 AU on a . We will continue to monitor this star with HIRES to further constrain the mass and orbit of HD 66428 c. \nFigure 23. RVSearch summary plot for HD 66428. See Figure 6 for plot description. \n<!-- image -->", 'A.5. HD 68988': "HD 68988 is a G0 dwarf found at a distance of 61 pc. Figure 24 shows the RVSearch results for this star. This system has one well-constrained hot Jupiter (Vogt et al. 2002) and an outer companion candidate that was first characterized in Bryan et al. (2016) as a partially resolved Keplerian orbit. With four more years of HIRES data, we can place tighter constraints on this outer candidate's orbit. A maximum likelihood fit gives an orbital period of 49.2 yr. However, since we have only observed a partially resolved orbit so far, the orbit posterior is wide and asymmetric in period space. MCMC sampling produces P = 61 +28 -20 yr. The model parameters are M sin i = 17 . 6 +2 . 4 -2 . 5 M J , a = 16 . 5 +4 . 8 -3 . 8 AU, and e = 0 . 53 +0 . 13 -0 . 09 . Table 1 reports all companion parameters. \nRVSearch detects a third periodic signal, with P = 1900 days, that has the same period and phase as the peak period in the S-value time series. This signal also has a low RV amplitude, 6 m s -1 . Therefore, we designated this signal as a false-positive corresponding to stellar activity. \n<!-- image --> \nFigure 24. RVSearch summary plot for HD 68988. See Figure 6 for plot description. \n<!-- image -->", 'A.6. HD 95735': "HD95735 (GJ 411) is an M2 dwarf found at a distance of 2.55 pc. Figure 25 shows the RVSearch results for this star. This system has one known short-period superEarth, with M sin i = 3.53 M ⊕ and an orbital period of 12.9 days. Our detection of this planet was driven by high-cadence APF data. This planet was first reported by D'ıaz et al. (2019), who also noted long-period power in their SOPHIE RV data, but they did not have a sufficiently long baseline or the activity metrics necessary to determine the origin of this power. With our HIRES post-upgrade and APF observations, we have an observational baseline of 14 yr, allowing us to confirm this long-period signal as a planet with M sin i = 24 . 7 ± 3 . 6 M ⊕ and an orbital period P = 8 . 46 yr. Table 1 reports all planet parameters. Since GJ 411 is a cool M dwarf, the Lick-Hamilton and HIRES pre-upgrade data are not reliable, because those detectors are not sufficiently high-resolution to capture a cool M dwarf's dense spectral lines (Fischer et al. 2014). \nThere is a long-period trend in the HIRES S-value time series, with significant power at and beyond 25 yr, but no significant power near the orbital period of the outer candidate. Therefore, we included this candidate in our catalog as a new planet candidate, to be verified and constrained with several more years of HIRES observations. \nRVSearch also recovered a highly eccentric, 216 day signal, but this signal correlates with APF systematics. Therefore, we labeled it as a false positive. This systematic remained when we applied RVSearch only to the HIRES post-upgrade and APF data and left out the problematic pre-upgrade and Lick data. \n<!-- image --> \nFigure 25. RVSearch summary plot for HD 95735. See Figure 6 for plot description. \n<!-- image --> \nHD 95735 Activity \nFigure 26. Lomb-Scargle periodogram of HIRES S-values for HD 95735. There is evidence for an activity cycle longer than 10,000 days, but no significant power near the period of our 3,000-day planet candidate. \n<!-- image -->", 'A.7. HD 107148': 'HD 107148 is a G5 dwarf at a distance of 49.5 pc. Figure 27 shows the RVSearch results for this star. Butler et al. (2006b) reported a planet with a period of 44 days. They reported periodicity at 77 days, but determined that this was an alias of the 44 day signal. The 77 day signal is significantly stronger in our likelihood periodogram, as seen in Figure 27, and better fits the data than a 44 day Keplerian by a significant ∆BIC ˙ This constitutes strong evidence that the true period of this planet is 77 days. We report new orbital parameters for this planet in Table 3. \nWe also recovered a signal with a period of 18.3 days. There is significant periodicity in the S-value time series, a periodogram of which is shown in Figure 28. However, it is concentrated around a period of 6 yr, and there is no significant power near 18.3 days. Furthermore, we find no correlation between the RVs and S-values. Thus, we report this signal as a confirmed planet, with M sin i = 19 . 9 ± 3 . 1 M ⊕ and a = 0 . 1406 ± 0 . 0018 AU. \nFigure 27. RVSearch summary plot for HD 107148. See Figure 6 for plot description. \n<!-- image --> \nFigure 28. Lomb-Scargle periodogram of HIRES S-values for HD 107148. Significant power at and beyond 4,300 days. \n<!-- image -->', 'A.8. HD 136925': 'HD 136925 is a G0 dwarf, found at a distance of 47.9 pc. RVSearch detected two periodic signals in this dataset, as seen in Figure 29, at 311 days and 12.4 yr. This dataset is currently sparse, with two gaps of several years in the post-upgrade HIRES data, but there is clear long-period variation in the RVs. Keplerian modeling predicts M sin i = 0.84 M J for the giant planet. \nThe S-value periodogram seen in Figure 30 shows no significant power beyond 1000 days, suggesting that the long-period HD 136925 b is a real planet. There is broad power around 300 days, overlapping with the period of the inner signal. It is unclear whether this periodicity is caused by real stellar variability or is a product of sparse data. Table 1 reports current estimates of all planet parameters. We need more data in order to clarify our model, and determine whether the inner signal is caused by a planet or a product of stellar activity and sparse data. Therefore, we designated HD 136925 b as a planet, and the inner signal as a probable false positive, to be clarified with continued HIRES observing. \n<!-- image --> \nFigure 29. RVSearch summary plot for HD 136925. See Figure 6 for plot description. \n<!-- image --> \nFigure 30. RVSearch summary plot for HD 136925. Periodogram of HIRES S-values for HD 136925. Significant periodicity around 300 days, near the period of the inner signal. \n<!-- image -->', 'A.9. HD 141004': 'HD 141004 is a G0 dwarf found at a distance of 11.8 pc. Figure 31 shows the RVSearch results for this star. Roy et al. (2021, in preparation) discovered a subNeptune at an orbital period of 15.5 days, with M sin i = 13 . 9 ± 1 . 5 M ⊕ , and will report on the analysis of this system in greater detail. Table 1 reports current estimates of all planet parameters. \nFigure 31. RVSearch summary plot for HD 141004. See Figure 6 for plot description. \n<!-- image -->', 'A.10. HD 145675': "HD 145675 (14 Her) is a K0 dwarf found at a distance of 17.9 pc. Figure 32 shows the RVSearch results for this star. This system has one known cold gas giant, with M sin i = 5.10 M J and an orbital period of 4.84 yr, which was first reported in Butler et al. (2003). Wittenmyer et al. (2007a) conducted further analysis with a longer observational baseline of twelve years, and noted a long-period trend. Wright et al. (2007) used additional RV curvature constraints to show that this trend must correspond to a companion with P > 12 yr and M sin i > 5 M J . The observational baseline has since increased from 12 yr to 22, and regular observations with HIRES and APF allow us to place further constraints on this long-period companion. We find M sin i = 5.8 +1 . 4 -1 . 0 M J , P = 68 +64 -25 yr, semi-major axis a = 16 . 4 +9 . 3 -4 . 3 AU, and eccentricity e = 0 . 45 +0 . 17 -0 . 15 . Table 1 reports all planet parameters. \nFigure 33 shows a Lomb-Scargle periodogram of the HIRES S-value time series. There is strong periodicity in the HIRES S-value time series, peaking around 10 yr, but no significant power near the supposed orbital period of the long-period candidate. These S-values strongly correlate with a third Keplerian signal picked up by our search, also with a period of 10 yr, as seen in the Figure 34, therefore we designate this signal as stellar activity. \nThere is a potential complication owed to a stellar binary candidate. Roberts et al. (2011) conducted a direct-imaging survey of known exoplanet hosts and reported a candidate stellar companion to 14 Her, with a differential magnitude of 10 . 9 ± 1 . 0, an angular separation of 4.3', and a minimum orbital separation of 78 AU. This is a single-epoch detection, and therefore could be only a visual binary. Additionally, Rodigas et al. (2011) conducted a deep direct imaging study of 14 Her, to constrain the mass and orbital parameters of 14 Her c, which, at the time, presented only as a parabolic trend in RV data. They used the Clio-2 photometer on the MMT, which has a 9' x 30' field of view; the authors only looked at imaging data within 2', to filter out background stars. Although this deep imaging study did not mention any stellar companion, the candidate reported by Roberts et al. (2011) falls outside of their considered imaging data, which corresponds to a minimum separation of 112.8 AU. Wittrock et al. (2017) also found a null binary detection, using the Differential Speckle Survey Instrument (DSSI) at the Gemini-North Observatory. A 6 Jupiter mass object would not have been detected by the above surveys, as they were designed only to rule out stellar companions and therefore used shorter imaging exposures that would miss planetary-mass companions. \nAdditionally, we used Gaia DR2 to search for bound stellar companions within 10', and found no such companions. We conclude that 14 Her does not have a bound stellar companion. Therefore, we designated 14 Her c as an eccentric, long-period planet. We will continue to monitor this star with Keck/HIRES and APF, to further constrain the orbit of this planet. \nFigure 32. RVSearch summary plot for HD 145675. See Figure 6 for plot description. \n<!-- image --> \nHD 145675 Activity \nFigure 33. Lomb-Scargle periodogram of HIRES S-values for HD 145675 showing significant power at 3,600 days. \n<!-- image --> \n14 Her, j \n14 Her, apf \n<!-- image --> \nFigure 34. Activity vetting plots for HD 145675. For all panels, the horizontal axis shows the S-value activity metric of each observation, while the vertical axis shows corresponding RV residuals for each individual Keplerian orbit. The left-hand panels show HIRES post-upgrade observations, while the right-hand panels show APF observations. Each row shows RVs with the model residuals of one Keplerian model, with the other Keplerian models subtracted from the data. The blue lines show linear correlations between these residuals and the corresponding S-values. In the HIRES and APF data, we measured > 3 σ correlations for the third Keplerian signal. The APF and HIRES linear correlations are within 3 σ of each other, implying that this signal is caused by stellar activity. We find correlations between the residuals and S-values for the second signal as well, but they are significantly different for HIRES and APF. Since the period of this signal is much greater than the APF baseline of this star, we discount this second correlation as caused by the limited baseline of the data with respect to the signal. \n<!-- image -->", 'A.11. HD 156668': 'HD 156668 is a K3 dwarf found at a distance of 24.4 pc. Figure 35 shows the RVSearch results for this star. This system has one known short-period super-Earth, with M sin i = 4.15 M ⊕ and an orbital period of 4.64 days. This planet was first reported by Howard et al. (2011), who also noted a long-period ( P ≈ 2 . 3 yr) signal with insufficient RV observations or additional data for confirmation as a planet. The observational baseline has since increased from five years to fourteen, allowing us to confirm this long-period signal as a planet with M sin i = 0.167 M J and an orbital period P = 2 . 22 yr. \nThere is a strong periodicity in the HIRES S-value time series, peaking around 10 yr, but no significant power near the orbital period of the long-period candidate. If we do not model this activity, a one-year alias signal appears in the periodogram search (Fig. 35). The data do not sufficiently constrain a Keplerian fit with a 10 yr period, but we find that a linear trend models the activity well enough to remove the one-year alias from the search. We opt to include this linear trend, which we treat as a nuisance parameter. \nFigure 35. RVSearch summary plot for HD 156668. See Figure 6 for plot description. \n<!-- image -->', 'A.12. HD 164922': 'HD 164922 is a G9 V dwarf located at a distance of 22.1 pc. Figure 36 shows the RVSearch results for this star. It hosts two known planets: a 0.3 M J planet with an orbital period of 1207 days (Butler et al. 2006b), and a super-Earth with M sin i = 14.3 M ⊕ and an orbital period of 75.8 days. This super-Earth was reported by Fulton et al. (2016), who also reported residual power around 41.7 days but did not find it significant enough to merit candidate status. With approximately two more years of HIRES and APF data, we identified the 41.7 day signal as a strong planet candidate and confirmed the 12.5 day planet reported in Benatti et al. (2020). Both planets are of sub-Neptune mass and have eccentricity posteriors that are consistent with circular orbits. The 41.7 day planet has M sin i = 10 . 7 ± 1 . 0 M ⊕ and a semi-major axis a = 0 . 2294 ± 0 . 0031 AU. The 12.5 day planet has M sin i = 4 . 63 ± 0 . 70 M ⊕ and a semi-major axis a = 0 . 1024 ± 0 . 0014 AU. Table 1 reports all planet parameters. \nTo validate these candidates, we searched for periodicity in both S-value activity metrics and APT photometry. We found no evidence for stellar rotation in Svalues, but estimated a stellar rotation period of 62.1 days from our APT photometry. Figure 37 shows periodograms and a phase-folded curve from this APT analysis, and Figure 38 shows equivalent analysis for HIRES S-values. The 1 yr alias of 62.1 days is 75.8 days, but the 75.8 day planet detection is high-amplitude and clean, without an additional peak near 62 days in any of the RVSearch periodograms. Therefore, within the limits of our activity metrics and vetting process, we ruled out stellar rotation as a cause of the 41.7 day signal. \nBenatti et al. (2020) used multiple HARPS-N spectral activity indicators to estimate a stellar rotation period of 41.6 days, and they note that this rotation period is to be expected from empirical activity-rotation relationships. Therefore, they determined that the strong 42 day signal present in their HARPS RVs is caused by rotation. However, we find no evidence of significant 42 day periodicity in our analysis of spectral activity indicators or APT photometry, as seen in Figures 37 and 38, and both datasets reflect significant periodicity near 60 days. Since our RV detection of this planet candidate is clean and does not conflict with our activity analysis, we chose to include this signal in our catalog as a planet candidate, to be confirmed or refuted by independent analysis. \nFigure 36. RVSearch summary plot for HD 164922. See Figure 6 for plot description. \n<!-- image --> \nFigure 37. Visualization of APT photometry analysis for HD 164922. The top panel shows a Lomb-Scargle periodogram of the photometry, with a moving-average filter to reduce alias issues. The middle panel shows an unfiltered periodogram. \n<!-- image --> \nFigure 38. Visualization of HIRES S-value analysis for HD 164922. The top panel shows a Lomb-Scargle periodogram of the S-values, with a moving-average filter to reduce alias issues. The middle panel shows an unfiltered periodogram. \n<!-- image -->', 'A.13. HD 168009': 'HD 168009 is a G1 dwarf found at a distance of 23.3 pc. Figure 39 shows the RVSearch results for this star. Roy et al. (2021, in preparation) discovered a superEarth candidate at an orbital period of 15.5 days, with M sin i = 10 . 3 ± 1 . 1 M ⊕ , and will report on the analysis of this candidate in greater detail. Table 1 reports current estimates of all planet parameters. \nRVSearch also recovered a highly eccentric 1 yr signal, but this signal correlates with APF systematics. Therefore, we labeled it as a false positive. Roy et al. (2021, in preparation) will model these systematics in greater detail. \nFigure 39. RVSearch summary plot for HD 168009. See Figure 6 for plot description. \n<!-- image -->', 'A.14. HD 213472': "HD 213472 is a G5 dwarf located at a distance of 64.6 pc. Figure 40 shows the RVSearch results for this star. There is an approximately eleven-year gap in RV observations of this star. The first post-upgrade HIRES observation was measured in 2005, shortly after the last pre-upgrade observation, and the second post-upgrade observation was measured in 2016. The 40 ms -1 difference between these two observations prompted the CPS team to begin observing HD 213472 regularly. Together with observations since 2016, and the thirteen pre-upgrade HIRES measurements, the data are consistent with a long-period, eccentric, planetary companion. Our periodogram search detects such a long-period signal. Due to the sparseness of the data, traditional MCMC methods fail to return a well-sampled model posterior. We used the rejection sampling algorithm TheJoker (Price-Whelan et al. 2017) to estimate the posterior, and found it to be unimodal. This model is consistent with a very long-period gas giant, with M sin i = 3 . 48 +1 . 10 -0 . 59 M J orbital period P = 46 +33 -13 yr, semi-major axis a = 13 . 0 +5 . 7 -2 . 6 AU, and eccentricity of e = 0 . 53 +0 . 12 -0 . 09 . Table 1 reports all planet parameters. Figure 41 shows the orbital parameter posteriors generated by TheJoker. \nTo investigate the possibility of a stellar or substellar companion, we compared this Keplerian model to a simple linear trend by computing the ∆BIC between the two max-likelihood models. The Keplerian model is significantly preferred with ∆BIC = 23.7. Additionally, we used Gaia to search for bound companions within 10', and found no such companions. Therefore, we inferred that HD 213472 b is either a planet or low-mass substellar companion, and not a wide-orbit stellar companion. \nFigure 42 shows a sample of possible orbits for HD 213472 b, drawn from our rejection sampling posteriors and projected over the next decade. More HIRES observations will further constrain this object's mass and orbital parameters. \nFigure 40. RVSearch summary plot for HD 213472. See Figure 6 for plot description. \n<!-- image --> \nFigure 41. Rejection sampling posterior for HD 213472 b orbital parameters. ∆ γ is the relative linear offset between different instrumental datasets, in this case pre-upgrade and post-upgrade HIRES. \n<!-- image --> \nFigure 42. Possible orbits for HD 213472 b. RV curves are drawn from the rejection sampling posterior generated with TheJoker. The color of each orbit drawn from the posterior scales with M sin i . \n<!-- image -->", 'B. STELLAR CATALOG': 'We record a subset of the stellar catalog and its associated stellar parameters in Table 2. We make this table of CLS stars available in its entirety in machine-readable format. \nTable 2 . Stellar Catalog', 'C. KNOWN PLANETS': 'We record all planets recovered by RVSearch in Table 3, with M sin i and key orbital parameter medians and uncertainties. We record all fitting parameter values in machine-readable format. \nTable 3 . Planet Catalog', 'Rosenthal et al.': 'Table 3 . Planet Catalog (Continued) \nTable 3 . Planet Catalog (Continued) \nTable 3 . Planet Catalog (Continued) \nTable 3 . Planet Catalog (Continued)', 'D. RESOLVED BINARIES AND SUBSTELLAR COMPANIONS': 'We record all stellar binaries and substellar companions recovered by RVSearch in Table 4. \nTable 4 . Binary and Substellar Catalog \nTable 4 . \nRosenthal et al. Binary and Substellar Catalog (Continued)', 'E. LONG-TERM TRENDS': 'We record all linear and parabolic trends recovered by \nRVSearch in Table 5. \nTable 5 . Long Term Trends', 'California Legacy Survey I: The Catalog': 'Table 5 . Long Term Trends (Continued) \nTable 5 . \nRosenthal et al. Long Term Trends (Continued)F. DATA \nWe include a sample table of RVs in Table 6. \nTable 6 . Sample of RV Data', 'G. FALSE POSITIVES': "We record all RVSearch-detected false positives in Table 7. The 'cause' column denotes why a signal was \nlabeled as a false positive. 'A' refers to a long-period \nmagnetic activity cycle, 'R' refers to stellar rotation, and 'N' refers to an annual and/or instrumental systematic. Long-period instrumental systematics are occasionally caused by offsets between dewars in the Lick data. Sev- \neral of these false positives correspond to reported planets in the literature, or to stars that have been discussed extensively in the literature. We elaborate on each of these cases in the subsections below. \nTable 7 . False Positives \nTable 7 . False Positives (Continued) \nTable 7 . False Positives (Continued)", 'G.1. HD 115617': 'Vogt et al. (2010) reported three planets orbiting this star, with periods of 4.2, 38, and 124 days. RVSearch recovered signals at all three periods. However, the 124 day signal (1/3rd of a year) has a strong harmonic at 1/4th of a year, and there is significant residual power at roughly one year, as seen in panels h and j of Figure 43. We investigated this candidate by computing periodograms for the 12 HIRES PSF parameters computed for each RV measurement, and found periodicity at 1 yr and harmonics of 1 yr for several parameters, as seen in Figure 44. Additionally, several of these PSF parameters correlate strongly with the corresponding RVs, after subtracting the RV models of the two inner planets, as seen in Figure 45. Therefore, we designated the 124 day signal as a yearly systematic.', 'G.2. HD 154345': "Here, we confirm the planetary status of the planet claim for HD 154345. Wright et al. (2008) announced the detection of a true Jupiter analog, with M sin i = 0.95 M J and an orbital period of 9.2 yr, corresponding to an orbital separation of 4.2 AU. This paper also presented strong evidence for a stellar magnetic activity cycle with a periodic timescale of roughly nine years. As the CPS group continued to observe HD 154345 over the next few years, the planet candidate's RV signature and the corresponding S-values appeared to be strongly in phase, and Wright (2016) noted that the candidate may be a false positive. However, in the twelve years since HD 154345 b was initially reported, HIRES RV measurements and activity metrics have drifted from being completely in phase to being completely out of phase, as seen in Figure 46, and therefore are not linearly correlated. This strongly implies that this Jupiter analog \ncandidate cannot be attributed to stellar activity, and that this candidate should be cemented as a confirmed planet. RVSearch detects two signals in our HD 154345 dataset, both close to 9 yr, as seen in Figure 47. We attribute the circular orbit with a greater RV amplitude to HD 154345 b, and the weak, eccentric signal to stellar activity.", 'G.3. HD 26965': 'Ma et al. (2018) reported a 42.4 day super-Earth orbiting the nearby star HD 26965, using datasets taken by multiple spectrographs, including HIRES. We detected significant periodicity at 42 days in the HIRES S-value measurements as seen in Figure 48, and determined that 42 days is the likely stellar rotation period of HD 26965. There is also evidence of a long-period magnetic activity cycle, as seen in the juxtaposition of S-values and RVs in Figure 49.', 'G.4. HD 34445': "Howard et al. (2010a) reported a giant planet orbiting this star at a period of 1049 days. Vogt et al. (2017) reported five small planets, claiming evidence in LCESderived HIRES radial velocities. RVSearch detected the giant planet and three of the five small planet claims, as seen in the summary plot shown in Figure 50. The longest-period candidate among the five, not modeled as a Keplerian here, clearly correlates with HIRES Svalues; we model this signal with a linear trend, for simplicity. Figure 51 juxtaposes the HIRES S-values and corresponding RVs, minus the Keplerian signal of the system's giant planet. As for the three other periodic signals that we detect, two are likely HIRES systematics and one is likely stellar rotation We detected significant periodicity at 52 days in the HIRES S-value measure- \nFigure 43. RVSearch summary plot for HD 115617. See Figure 6 for plot description. Note the nearly equivalent-height peaks at 1/3 and 1/4 year in panel h, corresponding to the 124 day reported planet. Panel j shows that there is residual power at 1 year after subtracting the 122 day signal, suggesting the presence of yearly systematic noise in the data. \n<!-- image --> \nments as seen in Figure 52, and determined that 52 days is the likely stellar rotation period of HD 34445. This places our weak detection of the 49 day claimed planet candidate under suspicion, and we have labeled it as a false positive in our catalog. There is also evidence of semiannual HIRES systematics, as seen in Figure 54, \nwhich shows the correlation between HIRES RVs minus the giant planet signature and PSF parameters, and in Figure 53, which shows periodograms of each PSF parameter time series. Multiple PSF parameters correlate ( | R | > 0.15) with the RV residuals, and multiple parameters show periodicity around one-third and one-fourth \nFigure 44. PSF Lomb-Scargle periodograms for HD 115617. Each panel corresponds to a Doppler code PSF fitting parameter. \n<!-- image --> \nof a year. The two claimed planets at 118 and 215 days are close to one-third and one-half of a year, respectively, and show weak and equal-strength signatures in their RVSearch periodograms, as seen in Figure 50. Therefore, we have labeled these signals as false positives in our catalog. \nFigure 45. PSF correlation plots for the candidate HD 115617 d. Each panel corresponds to a Doppler code PSF fitting parameter, with PSF value on the x-axis and RV without the signatures of the inner two planets on the y-axis. Dashed blue lines are least-squares linear fits. R is the Pearson correlation value; multiple PSF parameters have | R | > 0.15. \n<!-- image --> \nFigure 46. HIRES post-upgrade RV and S-value activity timeseries for HD 154345. Note that the two datasets share minima and appear to be in phase when post-upgrade observations began, but have drifted completely out of phase over the following 23 years. \n<!-- image --> \nFigure 47. RVSearch summary plot for HD 154345; see Figure 6 for description. RVSearch first recovered a strong signal at 9 years, but then recovered additional power at a similar period due to stellar activity. The final orbit fit switched the two models, so that panels e) and d) show the planetary signal, while panels c) and f) show the stellar activity signal. \n<!-- image --> \nFigure 48. Stellar rotation analysis of HIRES S-values for HD 26965. The top panel shows a Lomb-Scargle periodogram of the S-values after we applied a high-pass filter to them, to remove the impact of the long-period magnetic activity cycle. The middle panel shows a periodogram of the raw S-values. The top panel shows significant periodicity near 40 days, with a maximum at 41.6 days. The bottom-left panel shows the filtered S-values, while the bottom-right panel shows the filtered S-values phased to 41.6 days; there appears to be a coherent signal at this period, implying stellar rotation with this period. \n<!-- image --> \nFigure 49. HIRES post-upgrade S-values and RVs for HD 26965. The two datasets both have long-period power and are in phase with each other. \n<!-- image --> \nFigure 50. RVSearch summary plot for HD 34445; see Figure 6 for description. RVSearch first recovered the known giant planet, \n<!-- image --> \nFigure 51. HIRES post-upgrade RV and S-value activity timeseries for HD 34445, with the giant planet RV model subtracted. Note that these two datasets share a negative long-term trend, which we believe accounts for the claimed 5,700-day planet in the system. \n<!-- image --> \nFigure 52. Stellar rotation analysis of HIRES S-values for HD 34445. The top panel shows a Lomb-Scargle periodogram of the S-values after we applied a high-pass filter to them, to remove the impact of the long-period magnetic activity cycle. The middle panel shows a periodogram of the raw S-values. The top panel shows significant periodicity around 52.1 days. The bottom-left panel shows the filtered S-values, while the bottom-right panel shows the filtered S-values phased to 52.1 days; there appears to be a coherent signal at this period, implying stellar rotation with this period. This led us to label the 49 day claimed planet as a false positive, since there is insufficient evidence to distinguish it from stellar rotation. \n<!-- image --> \nFigure 53. PSF Lomb-Scargle periodograms for HD 34445. Each panel corresponds to a Doppler code PSF fitting parameter. \n<!-- image --> \nFigure 54. PSF correlation plots for HD 34445, without the RV signature of the star's giant planet. Each panel corresponds to a Doppler code PSF fitting parameter, with PSF value on the x-axis and RV without the giant planet signature on the y-axis. Dashed blue lines are least-squares linear fits. R is the Pearson correlation value; multiple PSF parameters have | R | > 0.15. \n<!-- image -->"} |
2017Natur.551...85A | On 17 August 2017 the Advanced LIGO and Virgo detectors observed the gravitationalwave event GW170817a strong signal from the merger of a binary neutronstar system. Less than two seconds after the merger a ray burst GRB 170817A was detected within a region of the sky consistent with the LIGOVirgoderived location of the gravitationalwave source. This sky region was subsequently observed by optical astronomy facilities resulting in the identification of an optical transient signal within about ten arcseconds of the galaxy NGC 4993. This detection of GW170817 in both gravitational waves and electromagnetic waves represents the first multimessenger astronomical observation. Such observations enable GW170817 to be used as a standard siren meaning that the absolute distance to the source can be determined directly from the gravitationalwave measurements to measure the Hubble constant. This quantity represents the local expansion rate of the Universe sets the overall scale of the Universe and is of fundamental importance to cosmology. Here we report a measurement of the Hubble constant that combines the distance to the source inferred purely from the gravitationalwave signal with the recession velocity inferred from measurements of the redshift using the electromagnetic data. In contrast to previous measurements ours does not require the use of a cosmic distance ladder the gravitationalwave analysis can be used to estimate the luminosity distance out to cosmological scales directly without the use of intermediate astronomical distance measurements. We determine the Hubble constant to be about 70 kilometres per second per megaparsec. This value is consistent with existing measurements while being completely independent of them. Additional standard siren measurements from future gravitationalwave sources will enable the Hubble constant to be constrained to high precision. | 2017-11-01T00:00:00Z | ['2017arXiv171005835A', 'arXiv:1710.05835', '2017Natur.551...85A', '10.48550/arXiv.1710.05835', '10.1038/nature24471'] | ['Astrophysics - Cosmology and Nongalactic Astrophysics'] | A gravitationalwave standard siren measurement of the Hubble constant | 2,017 | 174 | 0.68 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML'] | 1,028 | https://arxiv.org/pdf/1710.05835.pdf | {'A GRAVITATIONAL-WAVE STANDARD SIREN MEASUREMENT OF THE HUBBLE CONSTANT': 'THE LIGO SCIENTIFIC COLLABORATION AND THE VIRGO COLLABORATION, THE 1M2H COLLABORATION, THE DARK ENERGY CAMERA GW-EM COLLABORATION AND THE DES COLLABORATION, THE DLT40 COLLABORATION, THE LAS CUMBRES OBSERVATORY COLLABORATION, THE VINROUGE COLLABORATION, THE MASTER COLLABORATION, et al.', 'ABSTRACT': "The detection of GW170817 (Abbott et al. 2017a) in both gravitational waves and electromagnetic waves heralds the age of gravitational-wave multi-messenger astronomy. On 17 August 2017 the Advanced Laser Interferometer Gravitational-wave Observatory (LIGO) (LIGO Scientific Collaboration et al. 2015) and Virgo (Acernese et al. 2015) detectors observed GW170817, a strong signal from the merger of a binary neutron-star system. Less than 2 seconds after the merger, a gamma-ray burst (GRB 170817A) was detected within a region of the sky consistent with the LIGO-Virgo-derived location of the gravitational-wave source (Abbott et al. 2017b; Goldstein et al. 2017; Savchenko et al. 2017). This sky region was subsequently observed by optical astronomy facilities (Abbott et al. 2017c), resulting in the identification of an optical transient signal within ∼ 10 arcsec of the galaxy NGC 4993 (Coulter et al. 2017; Soares-Santos et al. 2017; Valenti et al. 2017; Arcavi et al. 2017; Tanvir et al. 2017; Lipunov et al. 2017). These multi-messenger observations allow us to use GW170817 as a standard siren (Schutz 1986; Holz & Hughes 2005; Dalal et al. 2006; Nissanke et al. 2010, 2013), the gravitational-wave analog of an astronomical standard candle, to measure the Hubble constant. This quantity, which represents the local expansion rate of the Universe, sets the overall scale of the Universe and is of fundamental importance to cosmology. Our measurement combines the distance to the source inferred purely from the gravitational-wave signal with the recession velocity inferred from measurements of the redshift using electromagnetic data. This approach does not require any form of cosmic 'distance ladder' (Freedman et al. 2001); the gravitational-wave (GW) analysis can be used to estimate the luminosity distance out to cosmological scales directly, without the use of intermediate astronomical distance measurements. We determine the Hubble constant to be 70 . 0 +12 . 0 -8 . 0 kms -1 Mpc -1 (maximum a posteriori and 68% credible interval). This is consistent with existing measurements (Planck Collaboration et al. 2016; Riess et al. 2016), while being completely independent of them. Additional standard-siren measurements from future gravitational-wave sources will provide precision constraints of this important cosmological parameter. \nThe Hubble constant H 0 measures the mean expansion rate of the Universe. At nearby distances ( d glyph[lessorsimilar] 50 Mpc) it is well approximated by the expression \nv H = H 0 d, (1) \nwhere v H is the local 'Hubble flow' velocity of a source, and d is the distance to the source. At such distances all cosmological distance measures (such as luminosity distance and comoving distance) differ at the order of v H /c where c is the speed of light. As v H /c ∼ 1% for GW170817 we do not distinguish between them. We are similarly insensitive to the values of other cosmological parameters, such as Ω m and Ω Λ . \nTo obtain the Hubble flow velocity at the position of GW170817, we use the optical identification of the host galaxy NGC 4993 (Abbott et al. 2017c). This identification is based solely on the 2-dimensional projected offset and is independent of any assumed value of H 0 . The position and redshift of this galaxy allow us to estimate the appropriate value of the Hubble flow velocity. Because the source is relatively nearby the random relative motions of galaxies, known as peculiar velocities, need to be taken into account. The peculiar velocity is ∼ 10% of the measured recessional velocity (see Methods). \nThe original standard siren proposal (Schutz 1986) did not rely on the unique identification of a host galaxy. By combining information from ∼ 100 independent GW detections, each with a set of potential host galaxies, a ∼ 5% estimate of H 0 can be obtained even without the detection of any transient optical counterparts (Del Pozzo 2012). This is particularly relevant, as gravitational-wave networks will detect many binary black hole mergers over the coming years (Abbott et al. 2016a), and these are not expected to be accompanied by electromagnetic counterparts. Alternatively, if an EM counterpart has been identified but the host galaxy is unknown, the same statistical method can be applied but using only those galaxies in \na narrow beam around the location of the optical counterpart. However, such statistical analyses are sensitive to a number of complicating effects, including the incompleteness of current galaxy catalogs or the need for dedicated follow-up surveys, as well as a range of selection effects (Messenger & Veitch 2013). In what follows we exploit the identification of NGC 4993 as the host galaxy of GW170817 to perform a standard siren measurement of the Hubble constant (Holz & Hughes 2005; Dalal et al. 2006; Nissanke et al. 2010, 2013). \nAnalysis of the GW data associated with GW170817 produces estimates for the parameters of the source, under the assumption that general relativity is the correct model of gravity (Abbott et al. 2017a). We are most interested in the joint posterior distribution on the luminosity distance and binary orbital inclination angle. For the analysis in this paper we fix the location of the GW source on the sky to the identified location of the counterpart (Coulter et al. 2017). See the Methods section for details. \nAn analysis of the GW data alone finds that GW170817occurred at a distance d = 43 . 8 +2 . 9 -6 . 9 Mpc (all values are quoted as the maximum posterior value with the minimal width 68 . 3% credible interval). We note that the distance quoted here differs from that in other studies (Abbott et al. 2017a), since here we assume that the optical counterpart represents the true sky location of the GW source instead of marginalizing over a range of potential sky locations. The ∼ 15% uncertainty is due to a combination of statistical measurement error from the noise in the detectors, instrumental calibration uncertainties (Abbott et al. 2017a), and a geometrical factor dependent upon the correlation of distance with inclination angle. The GW measurement is consistent with the distance to NGC4993 measured using the Tully-Fisher relation, d TF = 41 . 1 ± 5 . 8 Mpc (Sakai et al. 2000; Freedman et al. 2001). \nThe measurement of the GW polarization is crucial for inferring the binary inclination. This inclination, ι , is defined as the angle between the line of sight vector from the source to the detector and the orbital angular momentum vector of the binary system. For electromagnetic (EM) phenomena it is typically not possible to tell whether a system is orbiting clockwise or counter-clockwise (or, equivalently, face-on or face-off), and sources are therefore usually characterized by a viewing angle: min( ι, 180 · -ι ) . By contrast, GW measurements can identify the sense of the rotation, and thus ι ranges from 0 (counter-clockwise) to 180 deg (clockwise). Previous GW detections by LIGO had large uncertainties in luminosity distance and inclination (Abbott et al. 2016a) because the two LIGO detectors that were involved are nearly co-aligned, preventing a precise polarization measurement. In the present case, thanks to Virgo as an additional detector, the cosine of the inclination can be constrained at 68 . 3% ( 1 σ ) confidence to the range [ -1 . 00 , -0 . 81] corresponding to inclination angles between [144 , 180] deg . This implies that the plane of the binary orbit is almost, but not quite, perpendicular to our line of sight to the source ( ι ≈ 180 deg ), which is consistent with the observation of a coincident GRB (LVC, GBM, & INTEGRAL 2017 in prep.; Goldstein et al. 2017, ApJL, submitted; Savchenko et al. 2017, ApJL, submitted). We report inferences on cos ι because our prior for it is flat, so the posterior is proportional to the marginal likelihood for it from the GW observations. \nEM follow-up of the GW sky localization region (Abbott et al. 2017c) discovered an optical transient (Coulter et al. 2017; Soares-Santos et al. 2017; Valenti et al. 2017; Arcavi et al. 2017; Tanvir et al. 2017; Lipunov et al. 2017) in close proximity to the galaxy NGC 4993. The location of the transient was previously observed by the Distance Less Than 40 Mpc (DLT40) survey on 2017 July 27.99 UT and no sources were found (Valenti et al. 2017). We estimate the probability \nFigure 1. GW170817 measurement of H 0 . Marginalized posterior density for H 0 (blue curve). Constraints at 1 - and 2 σ from Planck (Planck Collaboration et al. 2016) and SHoES (Riess et al. 2016) are shown in green and orange. The maximum a posteriori value and minimal 68 . 3% credible interval from this PDF is H 0 = 70 . 0 +12 . 0 -8 . 0 kms -1 Mpc -1 . The 68 . 3% ( 1 σ ) and 95 . 4% ( 2 σ ) minimal credible intervals are indicated by dashed and dotted lines. \n<!-- image --> \nof a random chance association between the optical counterpart and NGC 4993 to be 0 . 004% (see the Methods section for details). In what follows we assume that the optical counterpart is associated with GW170817, and that this source resides in NGC4993. \nTo compute H 0 we need to estimate the background Hubble flow velocity at the position of NGC4993. In the traditional electromagnetic calibration of the cosmic 'distance ladder' (Freedman et al. 2001), this step is commonly carried out using secondary distance indicator information, such as the Tully-Fisher relation (Sakai et al. 2000), which allows one to infer the background Hubble flow velocity in the local Universe scaled back from more distant secondary indicators calibrated in quiet Hubble flow. We do not adopt this approach here, however, in order to preserve more fully the independence of our results from the electromagnetic distance ladder. Instead we estimate the Hubble flow velocity at the position \nFigure 2. Inference on H 0 and inclination. Posterior density of H 0 and cos ι from the joint GW-EM analysis (blue contours). Shading levels are drawn at every 5% credible level, with the 68 . 3% ( 1 σ , solid) and 95 . 4% ( 2 σ , dashed) contours in black. Values of H 0 and 1 - and 2 σ error bands are also displayed from Planck (Planck Collaboration et al. 2016) and SHoES (Riess et al. 2016). As noted in the text, inclination angles near 180 deg ( cos ι = -1 ) indicate that the orbital angular momentum is anti-parallel with the direction from the source to the detector. \n<!-- image --> \nof NGC4993 by correcting for local peculiar motions. \nNGC4993 is part of a collection of galaxies, ESO-508, whose center-of-mass recession velocity relative to the frame of the CMB (Hinshaw et al. 2009) is (Crook et al. 2007) 3327 ± 72 km s -1 . We correct the group velocity by 310 km s -1 due to the coherent bulk flow (Springob et al. 2014; Carrick et al. 2015) towards The Great Attractor (see Methods section for details). The standard error on our estimate of the peculiar velocity is 69 km s -1 , but recognizing that this value may be sensitive to details of the bulk flow motion that have been imperfectly modelled, in our subsequent analysis we adopt a more conservative estimate (Carrick et al. 2015) of 150kms -1 for the uncertainty on the peculiar velocity at the location of NGC 4993, and fold this into our estimate of the uncertainty on v H . From this, we obtain a Hubble velocity v H = 3017 ± 166 km s -1 . \nOnce the distance and Hubble velocity distributions have been determined from the GW and EM data, respectively, we can constrain the value of the Hubble constant. The measurement of the distance is strongly correlated with the measurement of the inclination of the orbital plane of the binary. The analysis of the GW data also depends on other parameters describing the source, such as the masses of the components (Abbott et al. 2016a). Here we treat the uncertainty in these other variables by marginalizing over the posterior distribution on system parameters (Abbott et al. 2017a), with the exception of the position of the system on the sky which is taken to be fixed at the location of the optical counterpart. \nWe carry out a Bayesian analysis to infer a posterior distribution on H 0 and inclination, marginalized over uncertainties in the recessional and peculiar velocities; see the Methods section for details. Figure 1 shows the marginal posterior for H 0 . The maximum a posteriori value with the minimal 68 . 3% credible interval is H 0 = 70 . 0 +12 . 0 -8 . 0 kms -1 Mpc -1 . Our estimate agrees well with state-of-the-art determinations of this quantity, including CMB measurements from Planck (Planck Collaboration et al. 2016) ( 67 . 74 ± 0 . 46 km s -1 Mpc -1 , 'TT,TE,EE+lowP+lensing+ext') and Type Ia supernova measurements from SHoES (Riess et al. 2016) ( 73 . 24 ± 1 . 74 km s -1 Mpc -1 ), as well as baryon acoustic oscillations measurements from SDSS (Aubourg et al. 2015), strong lensing measurements from H0LiCOW (Bonvin et al. 2017), highl CMB measurements from SPT (Henning et al. 2017), and Cepheid measurements from the HST key project (Freedman et al. 2001). Our measurement is a new and independent determination of this quantity. The close agreement indicates that, although each method may be affected by different systematic uncertainties, we see no evidence at present for a systematic difference between GW and established EM-based estimates. As has been much remarked upon, the Planck and SHoES re- \nlts are inconsistent at glyph[greaterorsimilar] 3 σ level. Our measurement does not resolve this tension, and is broadly consistent with both. \nOne of the main sources of uncertainty in our measurement of H 0 is due to the degeneracy between distance and inclination in the GW measurements. A face-on or face-off binary far away has a similar gravitational-wave amplitude to an edgeon binary closer in. This relationship is captured in Figure 2, which shows posterior contours in the H 0 -cos ι parameter space. \nThe posterior in Figure 1 results from the vertical projection of Figure 2, marginalizing out uncertainties in the cosine of inclination to derive constraints on the Hubble constant. Alternatively, it is possible to project horizontally, and thereby marginalize out the Hubble constant to derive constraints on the cosine of inclination. If instead of deriving H 0 independently we take the existing constraints on H 0 (Planck Collaboration et al. 2016; Riess et al. 2016) as priors, we are able to significantly improve our constraints on cos ι as shown in Figure 3. Assuming the Planck value for H 0 , the minimal 68.3% credible interval for the cosine of inclination is [ -1 . 00 , -0 . 92] (corresponding to an inclination angle range [157 , 177] deg ). For the SHoES value of H 0 , it is [ -0 . 97 , -0 . 85] (corresponding to an inclination angle range [148 , 166] deg ). For this latter SHoES result we note that the face-off ι = 180 deg orientation is just outside the 90% confidence range. It will be particularly interesting to compare these constraints to those from modeling of the short GRB, afterglow, and optical counterpart associated with GW170817 (Abbott et al. 2017c). \nWe have presented a standard siren determination of the Hubble constant, using a combination of a GW distance and an EM Hubble velocity estimate. Our measurement does not use a 'distance ladder', and makes no prior assumptions about H 0 . We find H 0 = 70 . 0 +12 . 0 -8 . 0 kms -1 Mpc -1 , which is consistent with existing measurements (Riess et al. 2016; Planck Collaboration et al. 2016). This \n( \ndeg \n) \ncos \nFigure 3. Constraints on the inclination angle of \n<!-- image --> \nGW170817. Posterior density on cos ι , for various assumptions about the prior distribution of H 0 . The analysis of the joint GW and EM data with a 1 /H 0 prior density gives the blue curve; using values of H 0 from Planck (Planck Collaboration et al. 2016) and SHoES (Riess et al. 2016) as a prior on H 0 give the green and red curves, respectively. Choosing a narrow prior on H 0 converts the precise Hubble velocity measurements for the group containing NGC 4993 to a precise distance measurement, breaking the distance inclination degeneracy, and leading to strong constraints on the inclination. Minimal 68 . 3% ( 1 σ ) credible intervals are indicated by dashed lines. Because our prior on inclination is flat on cos ι the densities in this plot are proportional to the marginalised likelihood for cos ι . \nfirst GW-EM multi-messenger event demonstrates the potential for cosmological inference from GW standard sirens. We expect that additional multimessenger binary neutron-star events will be detected in the coming years, and combining subsequent independent measurements of H0 from these future standard sirens will lead to an era of precision gravitational-wave cosmology.", 'PROBABILITY OF OPTICAL COUNTERPART ASSOCIATION WITH NGC4993': "We calculate the probability that an NGC 4993like galaxy (or brighter) is misidentified as the host by asking how often the centre of one or more such galaxies falls by random chance within a given angular radius θ of the counterpart. Assuming Poisson counting statistics this probability is given by P = 1 -exp [ -πθ 2 S ( < m )] where S ( < m ) is the surface density of galaxies with apparent magnitude equal to or brighter than m . From the local galaxy sample distribution in the infrared (Kband) apparent magnitude (Huang et al. 1998) we obtain S ( < K ) = 0 . 68 × 10 (0 . 64( K -10 . 0) -0 . 7) deg -2 . As suggested by (Bloom et al. 2002), we set θ equal to twice the half-light radius of the galaxy, for which we use NGC 4993's diameter of ∼ 1 . 1 arcmin, as measured in the near infrared band (the predominant emission band for early-type galaxies). Using K = 9 . 2 mag taken from the 2MASS survey (Skrutskie et al. 2006) for NGC 4993, we find the probability of random chance association is P = 0 . 004 %.", 'FINDING THE HUBBLE VELOCITY OF NGC4993': "In previous EM determinations of the cosmic 'distance ladder', the Hubble flow velocity of the local calibrating galaxies has generally been estimated using redshift-independent secondary galaxy distance indicators, such as the Tully-Fisher relation or type Ia supernovae, calibrated with more distant samples that can be assumed to sit in quiet Hubble flow (Freedman et al. 2001). We do not adopt this approach for NGC 4993, however, in order that our inference of the Hubble constant is fully independent of the electromagnetic distance scale. Instead we estimate the Hubble flow velocity at the position of NGC 4993 by correcting its measured recessional velocity for local peculiar motions. \nNGC4993 resides in a group of galaxies whose center-of-mass recession velocity relative to the Cosmic Microwave Background (CMB) frame (Hinshaw et al. 2009) is (Crook et al. 2007, 2008) 3327 ± 72 km s -1 . We assume that all of the galaxies in the group are at the same distance and therefore have the same Hubble flow velocity, which we assign to be the Hubble velocity of GW170817. This assumption is accurate to within 1% given that the radius of the group is ∼ 0 . 4 Mpc. To calculate the Hubble flow velocity of the group, we correct its measured recessional velocity by the peculiar velocity caused by the local gravitational field. This is a significant correction (Springob et al. 2014; Carrick et al. 2015); typical peculiar velocities are 300 km s -1 , equivalent to ∼ 10% of the total recessional velocity at a distance of 40 Mpc. \nWe employ the 6dF galaxy redshift survey peculiar velocity map (Springob et al. 2014; Jones et al. 2009), which used more than 8,000 Fundamental Plane galaxies to map the peculiar velocity field in the Southern hemisphere out to redshift z glyph[similarequal] 0 . 055 . We weight the peculiar velocity corrections from this catalog with a Gaussian kernel centered on NGC 4993's sky position and with a width of 8 h -1 Mpc ; the kernel width is independent of H 0 and is equivalent to a width of 800 km s -1 in velocity space, typical of the widths used in the catalog itself. There are 10 galaxies in the 6dF peculiar velocity catalog within one kernel width of NGC 4993. In the CMB frame (Hinshaw et al. 2009), the weighted radial component of the peculiar velocity and associated uncertainty is 〈 v p 〉 = 310 ± 69 km s -1 . \nWe verified the robustness of this peculiar velocity correction by comparing it with the velocity field reconstructed from the 2MASS redshift survey (Carrick et al. 2015; Huchra et al. 2012). This exploits the linear relationship between the peculiar velocity and mass density fields smoothed on scales larger than about 8 h -1 Mpc, and the constant of proportionality can be determined by com- \nparison with radial peculiar velocities of individual galaxies estimated from e.g. Tully-Fisher and Type Ia supernovae distances. Using these reconstructed peculiar velocities, which have a larger associated uncertainty (Carrick et al. 2015) of 150 km s -1 , at the position of NGC 4993 we find a Hubble velocity in the CMB frame of v H = 3047kms -1 - in excellent agreement with the result derived using 6dF. We adopt this larger uncertainty on the peculiar velocity correction in recognition that the peculiar velocity estimated from the 6dF data may represent an imperfect model of the true bulk flow at the location of NGC 4993. For our inference of the Hubble constant we therefore use a Hubble velocity v H = 3017 ± 166 km s -1 with 68.3% uncertainty. \nFinally, while we emphasise again the independence of our Hubble constant inference from the electromagnetic distance scale, we note the consistency of our GW distance estimate to NGC 4993 with the Tully-Fisher distance estimate derived by scaling back the Tully-Fisher relation calibrated with more distant galaxies in quiet Hubble flow (Sakai et al. 2000). This also strongly supports the robustness of our estimate for the Hubble velocity of NGC4993.", 'SUMMARY OF THE MODEL': "Given observed data from a set of GW detectors, x GW , parameter estimation is used to generate a posterior on the parameters that determine the waveform of the GW signal. Parameters are inferred within a Bayesian framework (Veitch et al. 2015) by comparing strain measurements (Abbott et al. 2017a) in the two LIGO detectors and the Virgo detector with the gravitational waveforms expected from the inspiral of two point masses (Hannam et al. 2014) under general relativity. We use algorithms for removing short-lived detector noise artifacts (Abbott et al. 2017a; Cornish & Littenberg 2015) and we employ approximate pointparticle waveform models (Buonanno & Damour 1999; Blanchet 2014; Hannam et al. 2014). We have verified that the systematic changes in the re- \nsults presented here from incorporating non-pointmass (tidal) effects (Hinderer & Flanagan 2008; Vines et al. 2011) and from different data processing methods are much smaller than the statistical uncertainties in the measurement of H 0 and the binary orbital inclination angle. \nFrom this analysis we can obtain the parameter estimation likelihood of the observed GW data, marginalized over all parameters characterizing the GWsignal except d and cos ι , \np ( x GW | d, cos ι ) = ∫ p ( x GW | d, cos ι, glyph[vector] λ ) p ( glyph[vector] λ )d glyph[vector] λ. (2) \nThe other waveform parameters are denoted by glyph[vector] λ , with p ( glyph[vector] λ ) denoting the corresponding prior. \nGiven perfect knowledge of the Hubble flow velocity of the GW source, v H , this posterior distribution can be readily converted into a posterior on cos ι and H 0 = v H /d , \np ( H 0 , cos ι | x GW ) ∝ ( v H /H 2 0 ) p ( x GW | d = v H /H 0 , cos ι ) × p d ( v H /H 0 ) p ι (cos ι ) , (3) \nwhere p d ( d ) and p ι (cos ι ) are the prior distributions on distance and inclination. For the Hubble velocity v H = 3017kms -1 , the maximum a posteriori distance from the GW measurement of 43 . 8 Mpc corresponds to H 0 = 68 . 9 km s -1 Mpc -1 , so this procedure would be expected to generate a posterior on H 0 that peaks close to that value. \nWhile the above analysis is conceptually straightforward, it makes a number of assumptions. In practice, the Hubble-flow velocity cannot be determined exactly and it must be corrected for uncertain peculiar velocities. The above does not explicitly set a prior on H 0 , but instead inherits a 1 /H 4 0 prior from the usual p d ( d ) ∝ d 2 prior used in GW parameter estimation. In addition, the logic in this model is that a redshift has been obtained first and the distance is then measured using GWs. As GW detectors cannot be pointed, \nwe cannot target particular galaxies or redshifts for GW sources. In practice, we wait for a GW event to trigger the analysis and this introduces potential selection effects which we must consider. We will see below that the simple analysis described above does give results that are consistent with a more careful analysis for this first detection. However, the simple analysis cannot be readily extended to include second and subsequent detections, so we now describe a more general framework that does not suffer from these limitations. \nWe suppose that we have observed a GW event, which generated data x GW in our detectors, and that we have also measured a recessional velocity for the host, v r , and the peculiar velocity field, 〈 v p 〉 , in the vicinity of the host. These observations are statistically independent and so the combined likelihood is \np ( x GW , v r , 〈 v p 〉 | d, cos ι, v p , H 0 ) = p ( x GW | d, cos ι ) p ( v r | d, v p , H 0 ) p ( 〈 v p 〉 | v p ) . (4) \nThe quantity p ( v r | d, v p , H 0 ) is the likelihood of the recessional velocity measurement, which we model as \np ( v r | d, v p , H 0 ) = N [ v p + H 0 d, σ 2 v r ] ( v r ) (5) \nwhere N [ µ, σ 2 ] ( x ) is the normal (Gaussian) probability density with mean µ and standard deviation σ evaluated at x . The measured recessional velocity, v r = 3327kms -1 , with uncertainty σ v r = 72 km s -1 , is the mean velocity and standard error for the members of the group hosting NGC 4993 taken from the two micron all sky survey (2MASS) (Crook et al. 2007, 2008), corrected to the CMB frame (Hinshaw et al. 2009). We take a similar Gaussian likelihood for the measured peculiar velocity, 〈 v p 〉 = 310kms -1 , with uncertainty σ v p = 150 km s -1 : \np ( 〈 v p 〉 | v p ) = N [ v p , σ 2 v p ] ( 〈 v p 〉 ) . (6) \nFrom the likelihood (4) we derive the posterior \np ( H 0 , d, cos ι, v p | x GW , v r , 〈 v p 〉 ) ∝ p ( H 0 ) N s ( H 0 ) p ( x GW | d, cos ι ) p ( v r | d, v p , H 0 ) × p ( 〈 v p 〉 | v p ) p ( d ) p ( v p ) p (cos ι ) , (7) \nwhere p ( H 0 ) , p ( d ) , p ( v p ) and p (cos ι ) are the parameter prior probabilities. Our standard analysis assumes a volumetric prior, p ( d ) ∝ d 2 , on the Hubble distance, but we explore sensitivity to this choice below. We take a flat-in-log prior on H 0 , p ( H 0 ) ∝ 1 /H 0 , impose a flat (i.e. isotropic) prior on cos ι , and a flat prior on v p for v p ∈ [ -1000 , 1000] km s -1 . These priors characterise our beliefs about the cosmological population of GW events and their hosts before we make any additional measurements or account for selection biases. The full statistical model is summarized graphically in Extended Data Figure 1. This model with these priors is our canonical analysis. \nIn Eq. (7), the term N s ( H 0 ) encodes selection effects (Loredo 2004; Mandel et al. 2016; Abbott et al. 2016a). These arise because of the finite sensitivity of our detectors. While all events in the Universe generate a response in the detector, we will only be able to identify, and hence use, signals that generate a response of sufficiently high amplitude. The decision about whether to include an event in the analysis is a property of the data only, in this case { x GW , v r , 〈 v p 〉} , but the fact that we condition our analysis on a signal being detected, i.e., the data exceeding these thresholds, means that the likelihood must be renormalized to become the likelihood for detected events. This is the role of \nN s ( H 0 ) = ∫ d glyph[vector] λ d d d v p dcos ι d x GW d v r d 〈 v p 〉 \ndetectable × [ p ( x GW | d, cos ι, glyph[vector] λ ) p ( v r | d, v p , H 0 ) × p ( 〈 v p 〉 | v p ) p ( glyph[vector] λ ) p ( d ) p ( v p ) p (cos ι ) ] , (8) \nwhere the integral is over the full prior ranges of the parameters, { d, v p , cos ι, glyph[vector] λ } , and over data sets \nthat would be selected for inclusion in the analysis, i.e., exceed the specified thresholds. If the integral was over all data sets it would evaluate to 1 , but because the range is restricted there can be a non-trivial dependence on parameters characterizing the population of sources, in this case H 0 . \nIn the current analysis, there are in principle selection effects in both the GW data and the EM data. However, around the time of detection of GW170817, the LIGO-Virgo detector network had a detection horizon of ∼ 190 Mpc for binary neutron star (BNS) events (Abbott et al. 2017a), within which EM measurements are largely complete. For example, the counterpart associated with GW170817 had brightness ∼ 17 mag in the I band at 40 Mpc (Valenti et al. 2017; Arcavi et al. 2017; Tanvir et al. 2017; Lipunov et al. 2017; Coulter et al. 2017); this source would be ∼ 22 mag at 400 Mpc, and thus still detectable by survey telescopes such as DECam well beyond the GW horizon. Even the dimmest theoretical lightcurves for kilonovae are expected to peak at ∼ 22 . 5 mag at the LIGO-Virgo horizon (Metzger & Berger 2012). We therefore expect that we are dominated by GW selection effects at the current time and can ignore EM selection effects. The fact that the fraction of BNS events that will have observed kilonova counterparts is presently unknown does not modify these conclusions, since we can restrict our analysis to GW events with kilonova counterparts only. \nIn the GW data, the decision about whether or not to analyse an event is largely determined by the signal-to-noise ratio (SNR), ρ , of the event. A reasonable model for the selection process is a cut in SNR, i.e., events with ρ > ρ ∗ are analysed (Abbott et al. 2016b). In that model, the integral over x GW in Eq. (8) can be replaced by an integral over SNR from ρ ∗ to ∞ , and p ( x GW | d, cos ι, glyph[vector] λ ) replaced by p ( ρ | d, cos ι, glyph[vector] λ ) in the integrand. This distribution depends on the noise properties of the operating detectors, and on the intrinsic strain amplitude of the source. The former are clearly independent of \nthe population parameters, while the latter scales like a function of the source parameters divided by the luminosity distance. The dependence on source parameters is on redshifted parameters, which introduces an explicit redshift dependence. However, within the ∼ 190 Mpc horizon, redshift corrections are at most glyph[lessorsimilar] 5% , and the Hubble constant measurement is a weak function of these, meaning the overall impact is even smaller. At present, whether or not a particular event in the population ends up being analysed can therefore be regarded as a function of d only. When GW selection effects dominate, only the terms in Eq. (8) arising from the GW measurement matter. As these are a function of d only and we set a prior on d , there is no explicit H 0 dependence in these terms. Hence, N s ( H 0 ) is a constant and can be ignored. This would not be the case if we set a prior on the redshifts of potential sources instead of their distances, since then changes in H 0 would modify the range of detectable redshifts. As the LIGO-Virgo detectors improve in sensitivity the redshift dependence in the GW selection effects will become more important, as will EM selection effects. However, at that point we will also have to consider deviations in the cosmological model from the simple Hubble flow described in Eq. (1) of the main article. \nMarginalising Eq. (7) over d , v p and cos ι then yields \n∫ \np ( H 0 | x GW , v r , 〈 v p 〉 ) ∝ p ( H 0 ) d d d v p dcos ι × p ( x GW | d, cos ι ) p ( v r | d, v p , H 0 ) × p ( 〈 v p 〉 | v p ) p ( d ) p ( v p ) p (cos ι ) . (9) \nThe posterior computed in this way was shown in Figure 1 in the main article and has a maximum a posteriori value and minimal 68 . 3% credible interval of 70 . 0 +12 . 0 -8 . 0 kms -1 Mpc -1 , as quoted in the main article. The posterior mean is 78 km s -1 Mpc -1 and the standard deviation is 15 km s -1 Mpc -1 . Various other summary statistics are given in Extended Data Table 1. \nROBUSTNESS TO PRIOR SPECIFICATION \n<!-- image --> \nExtended Data Figure 1. Graphical model illustrating the statistical relationships between the data and parameters. Open circles indicate parameters which require a prior; filled circles described measured data, which are conditioned on in the analysis. Here we assume we have measurements of the GW data, x GW , a recessional velocity (i.e. redshift), v r , and the mean peculiar velocity in the neighborhood of NGC 4993, 〈 v p 〉 . Arrows flowing into a node indicate that the conditional probability density for the node depends on the source parameters; for example, the conditional distribution for the observed GW data, p ( x GW | d, cos ι ) , discussed in the text, depends on the distance and inclination of the source (and additional parameters, here marginalized out). \nOur canonical analysis uses a uniform volumetric prior on distance, p ( d ) ∝ d 2 . The distribution of galaxies is not completely uniform due to clustering, so we explore sensitivity to this prior choice. We are free to place priors on any two of the three variables { d, H 0 , z } , where z = H 0 d/c is the Hubble flow redshift of NGC 4993. A choice of prior for two of these variables induces a prior on the third which may or may not correspond to a natural choice for that parameter. A prior on z could be obtained from galaxy catalog observations (Dalya et al. 2016), but must be corrected for incompleteness. When setting a prior on H 0 and z , \nthe posterior becomes \np ( H 0 , z, cos ι, v p | x GW , v r , 〈 v p 〉 ) ∝ p ( H 0 ) N s ( H 0 ) p ( x GW | d = cz/H 0 , cos ι ) p ( v r | z, v p ) × p ( 〈 v p 〉 | v p ) p ( z ) p ( v p ) p (cos ι ) , (10) \nbut now \nN s ( H 0 ) = ∫ detectable d z d v p dcos ι d x GW d v r d 〈 v p 〉 × p ( x GW | d = cz/H 0 , cos ι ) p ( v r | z, v p ) × p ( 〈 v p 〉 | v p ) p ( z ) p ( v p ) p (cos ι ) . (11) \nWhen GW selection effects dominate, the integral is effectively \nN s ( H 0 ) = ∫ d z dcos ι d x GW × p ( x GW | d = cz/H 0 , cos ι ) p ( z ) p (cos ι ) = ∫ d d dcos ι d x GW × p ( x GW | d, cos ι ) p ( dH 0 /c ) p (cos ι ) ( H 0 /c ) , (12) \nwhich has an H 0 dependence, unless p ( z ) takes a special, H 0 -dependent form, p ( z ) = f ( z/H 0 ) /H 0 . However, if the redshift prior is volumetric, p ( z ) ∝ z 2 , the selection effect term is ∝ H 3 0 , which cancels a similar correction to the likelihood and gives a posterior on H 0 that is identical to the canonical analysis. \nFor a single event, any choice of prior can be mapped to our canonical analysis with a different prior on H 0 . For any reasonable prior choices on d or z , we would expect to gradually lose sensitivity to the particular prior choice as further observed events are added to the analysis. However, to illustrate the uncertainty that comes from the prior choice for this first event, we compare in Extended Data Figure 2 and Extended Data Table 1 the results from the canonical prior choice p ( d ) ∝ d 2 to those from two other choices: using a flat prior \non z , and assuming a velocity correction due to the peculiar velocity of NGC 4993 that is a Gaussian with width 250 km s -1 . (To do the first of these, the posterior samples from GW parameter estimation have to be re-weighted, since they are generated with the d 2 prior used in the canonical analysis. We first 'undo' the default prior before applying the desired new prior.) \nThe choice of a flat prior on z is motivated by the simple model described above, in which we imagine first making a redshift measurement for the host and then use that as a prior for analysing the GW data. Setting priors on distance and redshift, the simple analysis gives the same result as the canonical analysis, but now we set a prior on redshift and H 0 and obtain a different result. This is to be expected because we are making different assumptions about the underlying population, and it arises for similar reasons as the different biases in peculiar velocity measurements based on redshiftselected or distance-selected samples (Strauss & Willick 1995). As can be seen in Extended Data Table 1, the results change by less than 1 σ , as measured by the statistical error of the canonical analysis. \nBy increasing the uncertainty in the peculiar velocity prior, we test the assumptions in our canonical analysis that (1) NGC 4993 is a member of the nearby group of galaxies, and (2) that this group has a center-of-mass velocity close to the Hubble flow. The results in Extended Data Table 1 summarizes changes in the values of H 0 and in the error bars. \nWe conclude that the impact of a reasonable change to the prior is small relative to the statistical uncertainties for this event.", 'INCORPORATING ADDITIONAL CONSTRAINTS ON H 0': "By including previous measurements of H 0 (Planck Collaboration et al. 2016; Riess et al. 2016) we can constrain the orbital inclination more precisely. We do this by setting the H 0 prior in Eq. (7) to p ( H 0 | µ H 0 , σ 2 H 0 ) = N [ µ H 0 , σ 2 H 0 ] , \n<!-- image --> \nExtended Data Figure 2. Using different assumptions compared to our canonical analysis. The posterior distribution on H 0 discussed in the main text is shown in black, the alternative flat prior on z (discussed in the Methods section) gives the distribution shown in blue, and the increased uncertainty ( 250 kms -1 ) applied to our peculiar velocity measurement (also discussed in the Methods section) is shown in pink. Minimal 68 . 3% ( 1 σ ) credible intervals are shown by dashed lines. \nwhere for ShoES (Riess et al. 2016) µ H 0 = 73 . 24 km s -1 Mpc -1 and σ H 0 = 1 . 74 km s -1 Mpc -1 , while for Planck (Planck Collaboration et al. 2016) µ H 0 = 67 . 74 km s -1 Mpc -1 and σ H 0 = 0 . 46 km s -1 Mpc -1 . The posterior on cos ι is then \np (cos ι | x GW , v r , 〈 v p 〉 , µ H 0 , σ 2 H 0 ) ∝ ∫ d d d v p d H 0 × p ( x GW | d, cos ι ) p ( v r | d, v p , H 0 ) p ( 〈 v p 〉 | v p ) × p ( H 0 | µ H 0 , σ 2 H 0 ) p ( d ) p ( v p ) . (13) \nThis posterior was shown in Figure 3 of the main article. \nThe authors gratefully acknowledge the support of the United States National Science Foundation (NSF) for the construction and operation of the LIGO Laboratory and Advanced LIGO as well as the Science and Technology Facilities Council (STFC) of the United Kingdom, the MaxPlanck-Society (MPS), and the State of Niedersachsen/Germany for support of the construction \nExtended Data Table 1. Summary of constraints on the Hubble constant, binary inclination, and distance \nNOTE-We give both one-sigma (68.3%) and 90% credible intervals for each quantity. 'Symm.' refers to a symmetric interval (e.g. median and 5% to 95% range), while 'MAP' refers to maximum a posteriori intervals (e.g. MAP value and smallest range enclosing 90% of the posterior). Values given for ι are derived from arc-cosine transforming the corresponding values for cos ι , so the 'MAP' values differ from those that would be derived from the posterior on ι . \nof Advanced LIGO and construction and operation of the GEO600 detector. Additional support for Advanced LIGO was provided by the Australian Research Council. The authors gratefully acknowledge the Italian Istituto Nazionale di Fisica Nucleare (INFN), the French Centre National de la Recherche Scientifique (CNRS) and the Foundation for Fundamental Research on Matter supported by the Netherlands Organisation for Scientific Research, for the construction and operation of the Virgo detector and the creation and support of the EGO consortium. The authors also gratefully acknowledge research support from these agencies as well as by the Council of Scientific and Industrial Research of India, the Department of Science and Technology, India, the Science & Engineering Research Board (SERB), India, the Ministry of Human Resource Development, India, the Spanish Agencia Estatal de Investigaci'on, the Vicepresid'encia i Conselleria d'Innovaci'o, Re- \nrca i Turisme and the Conselleria d'Educaci'o i Universitat del Govern de les Illes Balears, the Conselleria d'Educaci'o, Investigaci'o, Cultura i Esport de la Generalitat Valenciana, the National Science Centre of Poland, the Swiss National Science Foundation (SNSF), the Russian Foundation for Basic Research, the Russian Science Foundation, the European Commission, the European Regional Development Funds (ERDF), the Royal Society, the Scottish Funding Council, the Scottish Universities Physics Alliance, the Hungarian Scientific Research Fund (OTKA), the Lyon Institute of Origins (LIO), the National Research, Development and Innovation Office Hungary (NKFI), the National Research Foundation of Korea, Industry Canada and the Province of Ontario through the Ministry of Economic Development and Innovation, the Natural Science and Engineering Research Council Canada, the Canadian Institute for Advanced Research, the Brazilian Ministry of \nScience, Technology, Innovations, and Communications, the International Center for Theoretical Physics South American Institute for Fundamental Research (ICTP-SAIFR), the Research Grants Council of Hong Kong, the National Natural Science Foundation of China (NSFC), the Leverhulme Trust, the Research Corporation, the Ministry of Science and Technology (MOST), Taiwan and the Kavli Foundation. The authors gratefully acknowledge the support of the NSF, STFC, MPS, INFN, CNRS and the State of Niedersachsen/Germany for provision of computational resources. This article has been assigned the document number LIGO-P1700296. \nWe thank the University of Copenhagen, DARK Cosmology Centre, and the Niels Bohr International Academy for hosting D.A.C., R.J.F., A.M.B., E.R., and M.R.S. during the discovery of GW170817/SSS17a. R.J.F., A.M.B., E.R., and D.E.H. were participating in the Kavli Summer Program in Astrophysics, 'Astrophysics with gravitational wave detections.' This program was supported by the the Kavli Foundation, Danish National Research Foundation, the Niels Bohr International Academy, and the DARK Cosmology Centre. \nThe UCSC group is supported in part by NSF grant AST-1518052, the Gordon & Betty Moore Foundation, the Heising-Simons Foundation, generous donations from many individuals through a UCSC Giving Day grant, and from fellowships from the Alfred P. Sloan Foundation (R.J.F), the David and Lucile Packard Foundation (R.J.F. and E.R.) and the Niels Bohr Professorship from the DNRF (E.R.). A.M.B. acknowledges support from a UCMEXUS-CONACYT Doctoral Fellowship. Support for this work was provided by NASA through Hubble Fellowship grants HSTHF-51348.001 and HST-HF-51373.001 awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-26555. \nThe Berger Time-Domain Group at Harvard is supported in part by the NSF through grants AST-1411763 and AST-1714498, and by NASA through grants NNX15AE50G and NNX16AC22G. \nFunding for the DES Projects has been provided by the DOE and NSF (USA), MEC, MICINN, MINECO (Spain), STFC (UK), HEFCE (UK), NCSA(UIUC), KICP (U. Chicago), CCAPP (Ohio State), MIFPA (Texas A&M), CNPQ, FAPERJ, FINEP (Brazil), DFG (Germany) and the Collaborating Institutions in the Dark Energy Survey. The Collaborating Institutions are Argonne Lab, UC Santa Cruz, University of Cambridge, CIEMATMadrid, University of Chicago, University College London, DES-Brazil Consortium, University of Edinburgh, ETH Zurich, Fermilab, University of Illinois, ICE (IEEC-CSIC), IFAE Barcelona, Lawrence Berkeley Lab, LMU Munchen and the associated Excellence Cluster Universe, University of Michigan, NOAO, University of Nottingham, Ohio State University, University of Pennsylvania, University of Portsmouth, SLAC National Lab, Stanford University, University of Sussex, Texas A&M University, and the OzDES Membership Consortium. Based in part on observations at Cerro Tololo Inter-American Observatory, National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. The DES Data Management System is supported by the NSF under Grant Numbers AST-1138766 and AST-1536171. The DES participants from Spanish institutions are partially supported by MINECO under grants AYA201571825, ESP2015-88861, FPA2015-68048, and Centro de Excelencia SEV-2012-0234, SEV-20160597 and MDM-2015-0509. Research leading to these results has received funding from the ERC under the EU's 7 th Framework Programme including grants ERC 240672, 291329 and 306478. We acknowledge support from the Australian Research Council Centre of Excellence for All-sky \nAstrophysics (CAASTRO), through project number CE110001020. This manuscript has been authored by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of Science, Office of High Energy Physics. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paidup, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. \nD.J.S. acknowledges support for the DLT40 program from NSF grant AST-1517649. \nSupport for I.A. was provided by NASA through the Einstein Fellowship Program, grant PF6170148. G.H., D.A.H. and C.M. are supported by NSF grant AST-1313484. D.P. acknowledges support by Israel Science Foundation grant 541/17. \nVINROUGE is an European Southern Observatory Large Survey (id: 0198.D-2010). \nMASTER acknowledges the Lomonosov MSU Development Programm and the Russian Federation Ministry of Education and Science. \nThis research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. \nAll authors contributed to the work presented in this paper. \nThe authors declare that they have no competing financial interests. \nCorrespondence and requests for materials should be addressed to the LVC spokespeople (email: [email protected], [email protected]). \nAvailable public codes can be found at the LIGO Open Science Center ( https://losc.ligo. org ). \nAvailable public data can be found at the LIGO Open Science Center ( https://losc.ligo. org ).", 'All Authors and Affiliations': "THE LIGO SCIENTIFIC COLLABORATION AND THE VIRGO COLLABORATION, THE 1M2H COLLABORATION, THE DARK ENERGY CAMERA GW-EM COLLABORATION AND THE DES COLLABORATION, \nTHE DLT40 COLLABORATION, THE LAS CUMBRES OBSERVATORY COLLABORATION, \nTHE VINROUGE COLLABORATION, THE MASTER COLLABORATION, B. P. ABBOTT, 1 R. ABBOTT, 1 T. D. ABBOTT, 2 F. ACERNESE, 3, 4 K. ACKLEY, 5, 6 C. ADAMS, 7 T. ADAMS, 8 P. ADDESSO, 9 R. X. ADHIKARI, 1 V. B. ADYA, 10 C. AFFELDT, 10 M. AFROUGH, 11 B. AGARWAL, 12 M. AGATHOS, 13 K. AGATSUMA, 14 N. AGGARWAL, 15 O. D. AGUIAR, 16 L. AIELLO, 17, 18 A. AIN, 19 P. AJITH, 20 B. ALLEN, 10, 21, 22 G. ALLEN, 12 A. ALLOCCA, 23, 24 P. A. ALTIN, 25 A. AMATO, 26 A. ANANYEVA, 1 S. B. ANDERSON, 1 W. G. ANDERSON, 21 S. V. ANGELOVA, 27 S. ANTIER, 28 S. APPERT, 1 K. ARAI, 1 M. C. ARAYA, 1 J. S. AREEDA, 29 N. ARNAUD, 28, 30 K. G. ARUN, 31 S. ASCENZI, 32, 33 G. ASHTON, 10 M. AST, 34 S. M. ASTON, 7 P. ASTONE, 35 D. V. ATALLAH, 36 P. AUFMUTH, 22 C. AULBERT, 10 K. AULTONEAL, 37 C. AUSTIN, 2 A. AVILA-ALVAREZ, 29 S. BABAK, 38 P. BACON, 39 M. K. M. BADER, 14 S. BAE, 40 P. T. BAKER, 41 F. BALDACCINI, 42, 43 G. BALLARDIN, 30 S. W. BALLMER, 44 S. BANAGIRI, 45 J. C. BARAYOGA, 1 S. E. BARCLAY, 46 B. C. BARISH, 1 D. BARKER, 47 K. BARKETT, 48 F. BARONE, 3, 4 B. BARR, 46 L. BARSOTTI, 15 M. 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EHRENS, 1 J. EICHHOLZ, 1 S. S. EIKENBERRY, 5 R. A. EISENSTEIN, 15 R. C. ESSICK, 15 D. ESTEVEZ, 8 Z. B. ETIENNE, 41 T. ETZEL, 1 M. EVANS, 15 T. M. EVANS, 7 M. FACTOUROVICH, 50 V. FAFONE, 32, 33, 17 H. FAIR, 44 S. FAIRHURST, 36 X. FAN, 82 S. FARINON, 59 B. FARR, 91 W. M. FARR, 58 E. J. FAUCHON-JONES, 36 M. FAVATA, 111 M. FAYS, 36 C. FEE, 84 H. FEHRMANN, 10 J. FEICHT, 1 M. M. FEJER, 51 A. FERNANDEZ-GALIANA, 15 I. FERRANTE, 23, 24 E. C. FERREIRA, 16 F. FERRINI, 30 F. FIDECARO, 23, 24 D. FINSTAD, 44 I. FIORI, 30 D. FIORUCCI, 39 M. FISHBACH, 91 R. P. FISHER, 44 M. FITZ-AXEN, 45 R. FLAMINIO, 26, 112 M. FLETCHER, 46 H. FONG, 90 J. A. FONT, 85, 113 P. W. F. FORSYTH, 25 S. S. FORSYTH, 76 J.-D. FOURNIER, 66 S. FRASCA, 97, 35 F. FRASCONI, 24 Z. FREI, 55 A. FREISE, 58 R. FREY, 69 V. FREY, 28 E. M. FRIES, 1 P. FRITSCHEL, 15 V. V. FROLOV, 7 P. FULDA, 5 M. FYFFE, 7 H. GABBARD, 46 B. U. GADRE, 19 S. M. GAEBEL, 58 J. R. GAIR, 114 L. GAMMAITONI, 42 M. R. GANIJA, 72 S. G. GAONKAR, 19 C. GARCIA-QUIROS, 102 F. GARUFI, 78, 4 B. GATELEY, 47 S. GAUDIO, 37 G. GAUR, 115 V. GAYATHRI, 116 N. GEHRELS, 79, ∗ G. GEMME, 59 E. GENIN, 30 A. GENNAI, 24 D. GEORGE, 12 J. GEORGE, 60 L. GERGELY, 117 V. GERMAIN, 8 S. GHONGE, 76 ABHIRUP GHOSH, 20 ARCHISMAN GHOSH, 20, 14 S. GHOSH, 65, 14, 21 J. A. GIAIME, 2, 7 K. D. GIARDINA, 7 A. GIAZOTTO, 24 K. GILL, 37 L. GLOVER, 109 E. GOETZ, 118 R. GOETZ, 5 S. GOMES, 36 B. GONCHAROV, 6 G. GONZ'ALEZ, 2 J. M. GONZALEZ CASTRO, 23, 24 A. GOPAKUMAR, 119 M. L. GORODETSKY, 61 S. E. GOSSAN, 1 M. GOSSELIN, 30 R. GOUATY, 8 A. GRADO, 120, 4 C. GRAEF, 46 M. GRANATA, 26 A. GRANT, 46 S. GRAS, 15 C. GRAY, 47 G. GRECO, 121, 122 A. C. GREEN, 58 E. M. GRETARSSON, 37 P. GROOT, 65 H. GROTE, 10 S. GRUNEWALD, 38 P. GRUNING, 28 G. M. GUIDI, 121, 122 X. GUO, 82 A. GUPTA, 63 M. K. GUPTA, 105 K. E. GUSHWA, 1 E. K. GUSTAFSON, 1 R. GUSTAFSON, 118 O. HALIM, 18, 17 B. R. HALL, 68 E. D. HALL, 15 E. Z. HAMILTON, 36 G. HAMMOND, 46 M. HANEY, 123 M. M. HANKE, 10 J. HANKS, 47 C. HANNA, 63 M. D. HANNAM, 36 O. A. HANNUKSELA, 93 J. HANSON, 7 T. HARDWICK, 2 J. HARMS, 17, 18 G. M. HARRY, 124 I. W. HARRY, 38 M. J. HART, 46 C.-J. HASTER, 90 K. HAUGHIAN, 46 J. HEALY, 57 A. HEIDMANN, 70 M. C. HEINTZE, 7 H. HEITMANN, 66 P. HELLO, 28 G. HEMMING, 30 M. HENDRY, 46 I. S. HENG, 46 J. HENNIG, 46 A. W. HEPTONSTALL, 1 M. HEURS, 10, 22 S. HILD, 46 T. HINDERER, 65 D. HOAK, 30 D. HOFMAN, 26 K. HOLT, 7 D. E. HOLZ, 91 P. HOPKINS, 36 C. HORST, 21 J. HOUGH, 46 E. A. HOUSTON, 46 E. J. HOWELL, 64 A. HREIBI, 66 Y. M. HU, 10 E. A. HUERTA, 12 D. HUET, 28 B. HUGHEY, 37 S. HUSA, 102 S. H. HUTTNER, 46 T. HUYNH-DINH, 7 N. INDIK, 10 R. INTA, 83 G. INTINI, 97, 35 H. N. ISA, 46 J.-M. ISAC, 70 M. ISI, 1 B. R. IYER, 20 K. IZUMI, 47 T. JACQMIN, 70 K. JANI, 76 P. JARANOWSKI, 125 S. JAWAHAR, 62 F. JIM'ENEZ-FORTEZA, 102 W. W. JOHNSON, 2 D. I. JONES, 126 R. JONES, 46 R. J. G. JONKER, 14 L. JU, 64 J. JUNKER, 10 C. V. KALAGHATGI, 36 V. KALOGERA, 89 B. KAMAI, 1 S. KANDHASAMY, 7 G. KANG, 40 J. B. KANNER, 1 S. J. KAPADIA, 21 S. KARKI, 69 K. S. KARVINEN, 10 M. KASPRZACK, 2 M. KATOLIK, 12 E. KATSAVOUNIDIS, 15 W. KATZMAN, 7 S. KAUFER, 22 K. KAWABE, 47 F. K'EF'ELIAN, 66 D. KEITEL, 46 A. J. KEMBALL, 12 R. KENNEDY, 106 C. KENT, 36 J. S. KEY, 127 F. Y. KHALILI, 61 I. KHAN, 17, 33 S. KHAN, 10 Z. KHAN, 105 E. A. KHAZANOV, 128 N. KIJBUNCHOO, 25 CHUNGLEE KIM, 129 J. C. KIM, 130 K. KIM, 93 W. KIM, 72 W. S. KIM, 131 Y.-M. KIM, 92 S. J. KIMBRELL, 76 E. J. KING, 72 P. J. KING, 47 M. KINLEY-HANLON, 124 R. KIRCHHOFF, 10 J. S. KISSEL, 47 L. KLEYBOLTE, 34 S. KLIMENKO, 5 T. D. KNOWLES, 41 P. KOCH, 10 S. M. KOEHLENBECK, 10 S. KOLEY, 14 V. KONDRASHOV, 1 A. KONTOS, 15 M. KOROBKO, 34 W. Z. KORTH, 1 I. KOWALSKA, 73 D. B. KOZAK, 1 C. KR AMER, 10 V. KRINGEL, 10 B. KRISHNAN, 10 A. KR'OLAK, 132, 133 G. KUEHN, 10 P. KUMAR, 90 R. KUMAR, 105 S. KUMAR, 20 L. KUO, 87 A. KUTYNIA, 132 S. KWANG, 21 B. D. LACKEY, 38 K. H. LAI, 93 M. LANDRY, 47 R. N. LANG, 134 J. LANGE, 57 B. LANTZ, 51 R. K. LANZA, 15 A. LARTAUX-VOLLARD, 28 P. D. LASKY, 6 M. LAXEN, 7 A. LAZZARINI, 1 \nC. LAZZARO, 54 P. LEACI, 97, 35 S. LEAVEY, 46 C. H. LEE, 92 H. K. LEE, 135 H. M. LEE, 136 H. W. LEE, 130 K. LEE, 46 J. LEHMANN, 10 A. LENON, 41 M. LEONARDI, 110, 95 N. LEROY, 28 N. LETENDRE, 8 Y. LEVIN, 6 T. G. F. LI, 93 S. D. LINKER, 109 T. B. LITTENBERG, 137 J. LIU, 64 X. LIU, 21 R. K. L. LO, 93 N. A. LOCKERBIE, 62 L. T. LONDON, 36 J. E. LORD, 44 M. LORENZINI, 17, 18 V. LORIETTE, 138 M. LORMAND, 7 G. LOSURDO, 24 J. D. LOUGH, 10 C. O. LOUSTO, 57 G. LOVELACE, 29 H. L UCK, 22, 10 D. LUMACA, 32, 33 A. P. LUNDGREN, 10 R. LYNCH, 15 Y. MA, 48 R. MACAS, 36 S. MACFOY, 27 B. MACHENSCHALK, 10 M. MACINNIS, 15 D. M. MACLEOD, 36 I. MAGA˜NA HERNANDEZ, 21 F. MAGA˜NA-SANDOVAL, 44 L. MAGA˜NA ZERTUCHE, 44 R. M. MAGEE, 63 E. MAJORANA, 35 I. MAKSIMOVIC, 138 N. MAN, 66 V. MANDIC, 45 V. MANGANO, 46 G. L. MANSELL, 25 M. MANSKE, 21, 25 M. MANTOVANI, 30 F. MARCHESONI, 52, 43 F. MARION, 8 S. M 'ARKA, 50 Z. M'ARKA, 50 C. MARKAKIS, 12 A. S. MARKOSYAN, 51 A. MARKOWITZ, 1 E. MAROS, 1 A. MARQUINA, 100 F. MARTELLI, 121, 122 L. MARTELLINI, 66 I. W. MARTIN, 46 R. M. MARTIN, 111 D. V. MARTYNOV, 15 K. MASON, 15 E. MASSERA, 106 A. MASSEROT, 8 T. J. MASSINGER, 1 M. MASSO-REID, 46 S. MASTROGIOVANNI, 97, 35 A. MATAS, 45 F. MATICHARD, 1, 15 L. MATONE, 50 N. MAVALVALA, 15 N. MAZUMDER, 68 R. MCCARTHY, 47 D. E. MCCLELLAND, 25 S. MCCORMICK, 7 L. MCCULLER, 15 S. C. MCGUIRE, 139 G. MCINTYRE, 1 J. MCIVER, 1 D. J. MCMANUS, 25 L. MCNEILL, 6 T. MCRAE, 25 S. T. MCWILLIAMS, 41 D. MEACHER, 63 G. D. MEADORS, 38, 10 M. MEHMET, 10 J. MEIDAM, 14 E. MEJUTO-VILLA, 9 A. MELATOS, 96 G. MENDELL, 47 R. A. MERCER, 21 E. L. MERILH, 47 M. MERZOUGUI, 66 S. MESHKOV, 1 C. MESSENGER, 46 C. MESSICK, 63 R. METZDORFF, 70 P. M. MEYERS, 45 H. MIAO, 58 C. MICHEL, 26 H. MIDDLETON, 58 E. E. MIKHAILOV, 140 L. MILANO, 78, 4 A. L. MILLER, 5, 97, 35 B. B. MILLER, 89 J. MILLER, 15 M. MILLHOUSE, 101 M. C. MILOVICH-GOFF, 109 O. MINAZZOLI, 66, 141 Y. MINENKOV, 33 J. MING, 38 C. MISHRA, 142 S. MITRA, 19 V. P. MITROFANOV, 61 G. MITSELMAKHER, 5 R. MITTLEMAN, 15 D. MOFFA, 84 A. MOGGI, 24 K. MOGUSHI, 11 M. MOHAN, 30 S. R. P. MOHAPATRA, 15 M. MONTANI, 121, 122 C. J. MOORE, 13 D. MORARU, 47 G. MORENO, 47 S. R. MORRISS, 103 B. MOURS, 8 C. M. MOW-LOWRY, 58 G. MUELLER, 5 A. W. MUIR, 36 ARUNAVA MUKHERJEE, 10 D. MUKHERJEE, 21 S. MUKHERJEE, 103 N. MUKUND, 19 A. MULLAVEY, 7 J. MUNCH, 72 E. A. MU ˜NIZ, 44 M. MURATORE, 37 P. G. MURRAY, 46 K. NAPIER, 76 I. NARDECCHIA, 32, 33 L. NATICCHIONI, 97, 35 R. K. NAYAK, 143 J. NEILSON, 109 G. NELEMANS, 65, 14 T. J. N. NELSON, 7 M. NERY, 10 A. NEUNZERT, 118 L. NEVIN, 1 J. M. NEWPORT, 124 G. NEWTON, 46, † K. K. Y. NG, 93 T. T. NGUYEN, 25 D. NICHOLS, 65 A. B. NIELSEN, 10 S. NISSANKE, 65, 14 A. NITZ, 10 A. NOACK, 10 F. NOCERA, 30 D. NOLTING, 7 C. NORTH, 36 L. K. NUTTALL, 36 J. OBERLING, 47 G. D. O'DEA, 109 G. H. OGIN, 144 J. J. OH, 131 S. H. OH, 131 F. OHME, 10 M. A. OKADA, 16 M. OLIVER, 102 P. OPPERMANN, 10 RICHARD J. ORAM, 7 B. O'REILLY, 7 R. ORMISTON, 45 L. F. ORTEGA, 5 R. O'SHAUGHNESSY, 57 S. OSSOKINE, 38 D. J. OTTAWAY, 72 H. OVERMIER, 7 B. J. OWEN, 83 A. E. PACE, 63 J. PAGE, 137 M. A. PAGE, 64 A. PAI, 116, 145 S. A. PAI, 60 J. R. PALAMOS, 69 O. PALASHOV, 128 C. PALOMBA, 35 A. PAL-SINGH, 34 HOWARD PAN, 87 HUANG-WEI PAN, 87 B. PANG, 48 P. T. H. PANG, 93 C. PANKOW, 89 F. PANNARALE, 36 B. C. PANT, 60 F. PAOLETTI, 24 A. PAOLI, 30 M. A. PAPA, 38, 21, 10 A. PARIDA, 19 W. PARKER, 7 D. PASCUCCI, 46 A. PASQUALETTI, 30 R. PASSAQUIETI, 23, 24 D. PASSUELLO, 24 M. PATIL, 133 B. PATRICELLI, 146, 24 B. L. PEARLSTONE, 46 M. PEDRAZA, 1 R. PEDURAND, 26, 147 L. PEKOWSKY, 44 A. PELE, 7 S. PENN, 148 C. J. PEREZ, 47 A. PERRECA, 1, 110, 95 L. M. PERRI, 89 H. P. PFEIFFER, 90, 38 M. PHELPS, 46 O. J. PICCINNI, 97, 35 M. PICHOT, 66 F. PIERGIOVANNI, 121, 122 V. PIERRO, 9 G. PILLANT, 30 L. PINARD, 26 I. M. PINTO, 9 M. PIRELLO, 47 M. PITKIN, 46 M. POE, 21 R. POGGIANI, 23, 24 P. POPOLIZIO, 30 E. K. PORTER, 39 A. POST, 10 J. POWELL, 46, 149 J. PRASAD, 19 J. W. W. PRATT, 37 G. PRATTEN, 102 V. PREDOI, 36 T. PRESTEGARD, 21 M. PRIJATELJ, 10 M. PRINCIPE, 9 S. PRIVITERA, 38 G. A. PRODI, 110, 95 L. G. PROKHOROV, 61 O. PUNCKEN, 10 M. PUNTURO, 43 P. PUPPO, 35 M. P URRER, 38 H. QI, 21 V. QUETSCHKE, 103 E. A. QUINTERO, 1 R. QUITZOW-JAMES, 69 F. J. RAAB, 47 D. S. RABELING, 25 H. RADKINS, 47 P. RAFFAI, 55 S. RAJA, 60 C. RAJAN, 60 B. RAJBHANDARI, 83 M. RAKHMANOV, 103 K. E. RAMIREZ, 103 A. RAMOS-BUADES, 102 P. RAPAGNANI, 97, 35 V. RAYMOND, 38 M. RAZZANO, 23, 24 J. READ, 29 T. REGIMBAU, 66 L. REI, 59 S. REID, 62 D. H. REITZE, 1, 5 W. REN, 12 S. D. REYES, 44 F. RICCI, 97, 35 P. M. RICKER, 12 S. RIEGER, 10 K. RILES, 118 M. RIZZO, 57 N. A. ROBERTSON, 1, 46 R. ROBIE, 46 F. ROBINET, 28 A. ROCCHI, 33 L. ROLLAND, 8 \nJ. G. ROLLINS, 1 V. J. ROMA, 69 J. D. ROMANO, 103 R. ROMANO, 3, 4 C. L. ROMEL, 47 J. H. ROMIE, 7 D. ROSI 'NSKA, 150, 56 M. P. ROSS, 151 S. ROWAN, 46 A. R UDIGER, 10 P. RUGGI, 30 G. RUTINS, 27 K. RYAN, 47 S. SACHDEV, 1 T. SADECKI, 47 L. SADEGHIAN, 21 M. SAKELLARIADOU, 152 L. SALCONI, 30 M. SALEEM, 116 F. SALEMI, 10 A. SAMAJDAR, 143 L. SAMMUT, 6 L. M. SAMPSON, 89 E. J. SANCHEZ, 1 L. E. SANCHEZ, 1 N. SANCHIS-GUAL, 85 V. SANDBERG, 47 J. R. SANDERS, 44 B. SASSOLAS, 26 B. S. SATHYAPRAKASH, 63, 36 P. R. SAULSON, 44 O. SAUTER, 118 R. L. SAVAGE, 47 A. SAWADSKY, 34 P. SCHALE, 69 M. SCHEEL, 48 J. SCHEUER, 89 J. SCHMIDT, 10 P. SCHMIDT, 1, 65 R. SCHNABEL, 34 R. M. S. SCHOFIELD, 69 A. SCH ONBECK, 34 E. SCHREIBER, 10 D. SCHUETTE, 10, 22 B. W. SCHULTE, 10 B. F. SCHUTZ, 36, 10 S. G. SCHWALBE, 37 J. SCOTT, 46 S. M. SCOTT, 25 E. SEIDEL, 12 D. SELLERS, 7 A. S. SENGUPTA, 153 D. SENTENAC, 30 V. SEQUINO, 32, 33, 17 A. SERGEEV, 128 D. A. SHADDOCK, 25 T. J. SHAFFER, 47 A. A. SHAH, 137 M. S. SHAHRIAR, 89 M. B. SHANER, 109 L. SHAO, 38 B. SHAPIRO, 51 P. SHAWHAN, 75 A. SHEPERD, 21 D. H. SHOEMAKER, 15 D. M. SHOEMAKER, 76 K. SIELLEZ, 76 X. SIEMENS, 21 M. SIENIAWSKA, 56 D. SIGG, 47 A. D. SILVA, 16 L. P. SINGER, 79 A. SINGH, 38, 10, 22 A. SINGHAL, 17, 35 A. M. SINTES, 102 B. J. J. SLAGMOLEN, 25 B. SMITH, 7 J. R. SMITH, 29 R. J. E. SMITH, 1, 6 S. SOMALA, 154 E. J. SON, 131 J. A. SONNENBERG, 21 B. SORAZU, 46 F. SORRENTINO, 59 T. SOURADEEP, 19 A. P. SPENCER, 46 A. K. SRIVASTAVA, 105 K. STAATS, 37 A. STALEY, 50 D. STEER, 39 M. STEINKE, 10 J. STEINLECHNER, 34, 46 S. STEINLECHNER, 34 D. STEINMEYER, 10 S. P. STEVENSON, 58, 149 R. STONE, 103 D. J. STOPS, 58 K. A. STRAIN, 46 G. STRATTA, 121, 122 S. E. STRIGIN, 61 A. STRUNK, 47 R. STURANI, 155 A. L. STUVER, 7 T. Z. SUMMERSCALES, 156 L. SUN, 96 S. SUNIL, 105 J. SURESH, 19 P. J. SUTTON, 36 B. L. SWINKELS, 30 M. J. SZCZEPA 'NCZYK, 37 M. TACCA, 14 S. C. TAIT, 46 C. TALBOT, 6 D. TALUKDER, 69 D. B. TANNER, 5 M. T'APAI, 117 A. TARACCHINI, 38 J. D. TASSON, 71 J. A. TAYLOR, 137 R. TAYLOR, 1 S. V. TEWARI, 148 T. THEEG, 10 F. THIES, 10 E. G. THOMAS, 58 M. THOMAS, 7 P. THOMAS, 47 K. A. THORNE, 7 E. THRANE, 6 S. TIWARI, 17, 95 V. TIWARI, 36 K. V. TOKMAKOV, 62 K. TOLAND, 46 M. TONELLI, 23, 24 Z. TORNASI, 46 A. TORRES-FORN'E, 85 C. I. TORRIE, 1 D. T OYR A, 58 F. TRAVASSO, 30, 43 G. TRAYLOR, 7 J. TRINASTIC, 5 M. C. TRINGALI, 110, 95 L. TROZZO, 157, 24 K. W. TSANG, 14 M. TSE, 15 R. TSO, 1 L. TSUKADA, 81 D. TSUNA, 81 D. TUYENBAYEV, 103 K. UENO, 21 D. UGOLINI, 158 C. S. UNNIKRISHNAN, 119 A. L. URBAN, 1 S. A. USMAN, 36 H. VAHLBRUCH, 22 G. VAJENTE, 1 G. VALDES, 2 N. VAN BAKEL, 14 M. VAN BEUZEKOM, 14 J. F. J. VAN DEN BRAND, 74, 14 C. VAN DEN BROECK, 14 D. C. VANDER-HYDE, 44 L. VAN DER SCHAAF, 14 J. V. VAN HEIJNINGEN, 14 A. A. VAN VEGGEL, 46 M. VARDARO, 53, 54 V. VARMA, 48 S. VASS, 1 M. VAS'UTH, 49 A. VECCHIO, 58 G. VEDOVATO, 54 J. VEITCH, 46 P. J. VEITCH, 72 K. VENKATESWARA, 151 G. VENUGOPALAN, 1 D. VERKINDT, 8 F. VETRANO, 121, 122 A. VICER'E, 121, 122 A. D. VIETS, 21 S. VINCIGUERRA, 58 D. J. VINE, 27 J.-Y. VINET, 66 S. VITALE, 15 T. VO, 44 H. VOCCA, 42, 43 C. VORVICK, 47 S. P. VYATCHANIN, 61 A. R. WADE, 1 L. E. WADE, 84 M. WADE, 84 R. WALET, 14 M. WALKER, 29 L. WALLACE, 1 S. WALSH, 38, 10, 21 G. WANG, 17, 122 H. WANG, 58 J. Z. WANG, 63 W. H. WANG, 103 Y. F. WANG, 93 R. L. WARD, 25 J. WARNER, 47 M. WAS, 8 J. WATCHI, 98 B. WEAVER, 47 L.-W. WEI, 10, 22 M. WEINERT, 10 A. J. WEINSTEIN, 1 R. WEISS, 15 L. WEN, 64 E. K. WESSEL, 12 P. WESSELS, 10 J. WESTERWECK, 10 T. WESTPHAL, 10 K. WETTE, 25 J. T. WHELAN, 57 S. E. WHITCOMB, 1 B. F. WHITING, 5 C. WHITTLE, 6 D. WILKEN, 10 D. WILLIAMS, 46 R. D. WILLIAMS, 1 A. R. WILLIAMSON, 65 J. L. WILLIS, 1, 159 B. WILLKE, 22, 10 M. H. WIMMER, 10 W. WINKLER, 10 C. C. WIPF, 1 H. WITTEL, 10, 22 G. WOAN, 46 J. WOEHLER, 10 J. WOFFORD, 57 K. W. K. WONG, 93 J. WORDEN, 47 J. L. WRIGHT, 46 D. S. WU, 10 D. M. WYSOCKI, 57 S. XIAO, 1 H. YAMAMOTO, 1 C. C. YANCEY, 75 L. YANG, 160 M. J. YAP, 25 M. YAZBACK, 5 HANG YU, 15 HAOCUN YU, 15 M. YVERT, 8 A. ZADRO˙ZNY, 132 M. ZANOLIN, 37 T. ZELENOVA, 30 J.-P. ZENDRI, 54 M. ZEVIN, 89 L. ZHANG, 1 M. ZHANG, 140 T. ZHANG, 46 Y.-H. ZHANG, 57 C. ZHAO, 64 M. ZHOU, 89 Z. ZHOU, 89 S. J. ZHU, 38, 10 X. J. ZHU, 6 A. B. ZIMMERMAN, 90 M. E. ZUCKER, 1, 15 AND J. ZWEIZIG 1 \nTHE LIGO SCIENTIFIC COLLABORATION AND THE VIRGO COLLABORATION \nR. J. FOLEY, 161 D. A. COULTER, 161 M. R. DROUT, 162, 163 D. KASEN, 164, 165 C. D. KILPATRICK, 161 B. F. MADORE, 162 A. MURGUIA-BERTHIER, 161 Y.-C. PAN, 161 A. L. PIRO, 162 J. X. PROCHASKA, 161 \n- E. RAMIREZ-RUIZ, 161, 166 A. REST, 167 C. ROJAS-BRAVO, 161 B. J. SHAPPEE, 162, 168, 169 M. R. SIEBERT, 161 J. D. SIMON, 162 AND N. ULLOA 170", 'THE 1M2H COLLABORATION': "J. ANNIS, 171 M. SOARES-SANTOS, 172, 171 D. BROUT, 173 D. SCOLNIC, 174 H. T. DIEHL, 171, 171 J. FRIEMAN, 171, 174 E. BERGER, 175 K. D. ALEXANDER, 175 S. ALLAM, 171, 171 E. BALBINOT, 176 P. BLANCHARD, 177 R. E. BUTLER, 178, 171 R. CHORNOCK, 179 E. R. COOK, 180, 181 P. COWPERTHWAITE, 175 A. DRLICA-WAGNER, 171, 171 M. R. DROUT, 163, 182 F. DURRET, 183 T. EFTEKHARI, 177 D. A. FINLEY, 171 W. FONG, 184, 185 C. L. FRYER, 186 J. GARC'IA-BELLIDO, 187 M. S .S. GILL, 188 R. A. GRUENDL, 189, 190 C. HANNA, 191, 190 W. HARTLEY, 192, 193 K. HERNER, 171 D. HUTERER, 194 D. KASEN, 195 R. KESSLER, 174 T. S. LI, 171 H. LIN, 171, 171 P. A. A. LOPES, 196 A. C. C. LOURENC¸O, 196 R. MARGUTTI, 197 J. MARRINER, 171 J. L. MARSHALL, 180, 198 T. MATHESON, 199 G. E. MEDINA, 200 B. D. METZGER, 201 R. R. MU ˜NOZ, 200 J. MUIR, 202 M. NICHOLL, 175 P. NUGENT, 203 A. PALMESE, 192 F. PAZ-CHINCH 'ON, 190, 190 E. QUATAERT, 204 M. SAKO, 173 M. SAUSEDA, 180 D. J. SCHLEGEL, 205 L. F. SECCO, 173 N. SMITH, 206 F. SOBREIRA, 207, 208, 207, 208 A. STEBBINS, 171 V. A. VILLAR, 177 A. K. VIVAS, 209 W. WESTER, 171 P. K. G. WILLIAMS, 177 B. YANNY, 171 A. ZENTENO, 209 T. M. C. ABBOTT, 209 F. B. ABDALLA, 192, 210 K. BECHTOL, 181 A. BENOIT-L'EVY, 211, 192, 212 E. BERTIN, 211, 212 S. L. BRIDLE, 213 D. BROOKS, 192 E. BUCKLEY-GEER, 171 D. L. BURKE, 214, 188 A. CARNERO ROSELL, 208, 215 M. CARRASCO KIND, 189, 190 J. CARRETERO, 216 F. J. CASTANDER, 217 C. E. CUNHA, 214 C. B. D'ANDREA, 173 L. N. DA COSTA, 208, 215 C. DAVIS, 214 D. L. DEPOY, 198 S. DESAI, 218 J. P. DIETRICH, 219, 220 J. ESTRADA, 171 E. FERNANDEZ, 216 B. FLAUGHER, 171 P. FOSALBA, 217 E. GAZTANAGA, 217 D. W. GERDES, 221, 194 T. GIANNANTONIO, 222, 223, 224 D. A. GOLDSTEIN, 225, 203 D. GRUEN, 214, 188 G. GUTIERREZ, 171 W. G. HARTLEY, 192, 193 K. HONSCHEID, 226, 227 B. JAIN, 173 D. J. JAMES, 228 T. JELTEMA, 229 M. W. G. JOHNSON, 190 S. KENT, 171, 174 E. KRAUSE, 214 R. KRON, 171, 174 K. KUEHN, 230 S. KUHLMANN, 231 N. KUROPATKIN, 171 O. LAHAV, 192 M. LIMA, 232, 208 M. A. G. MAIA, 208, 215 M. MARCH, 173 C. J. MILLER, 221, 194 R. MIQUEL, 233, 216 E. NEILSEN, 171 B. NORD, 171 R. L. C. OGANDO, 208, 215 A. A. PLAZAS, 234 A. K. ROMER, 235 A. ROODMAN, 214, 188 E. S. RYKOFF, 214, 188 E. SANCHEZ, 236 V. SCARPINE, 171 M. SCHUBNELL, 194 I. SEVILLA-NOARBE, 236 M. SMITH, 237 R. C. SMITH, 209 E. SUCHYTA, 238 G. TARLE, 194 D. THOMAS, 239 R. C. THOMAS, 203 M. A. TROXEL, 226, 227 D. L. TUCKER, 171 V. VIKRAM, 231 A. R. WALKER, 209 J. WELLER, 219, 240, 224 AND Y. ZHANG 171 \nTHE DARK ENERGY CAMERA GW-EM COLLABORATION AND THE DES COLLABORATION \nJ. B. HAISLIP, \n241 \nV. V. KOUPRIANOV, \n241 \nD. E. REICHART, \nS. VALENTI, \n243 \n241 \nAND S. YANG \nL. TARTAGLIA, \n243, 244, 245", 'THE DLT40 COLLABORATION': "IAIR ARCAVI, 246, 247 GRIFFIN HOSSEINZADEH, 246, 247 D. ANDREW HOWELL, 246, 247 CURTIS MCCULLY, 246, 247 DOVI POZNANSKI, 248 AND SERGIY VASYLYEV 246, 247 THE LAS CUMBRES OBSERVATORY COLLABORATION \nN. R. TANVIR, 249 A. J. LEVAN, 250 J. HJORTH, 251 Z. CANO, 252 C. COPPERWHEAT, 253 A. DE UGARTE-POSTIGO, 252 P.A. EVANS, 249 J.P.U. FYNBO, 251 C. GONZ'ALEZ-FERN 'ANDEZ, 254 J. GREINER, 255 M. IRWIN, 254 J. LYMAN, 250 I. MANDEL, 256 R. MCMAHON, 254 B. MILVANG-JENSEN, 251 P. O'BRIEN, 249 J. P. OSBORNE, 249 D. A. PERLEY, 253 E. PIAN, 257 E. PALAZZI, 257 E. ROL, 258 S. ROSETTI, 249 S. ROSSWOG, 259 A. ROWLINSON, 260, 261 S. SCHULZE, 262 D.T.H. STEEGHS, 250 C.C. TH ONE, 252 K. ULACZYK, 250 D. WATSON, 251 AND K. WIERSEMA 249, 250", 'THE VINROUGE COLLABORATION': 'V.M. LIPUNOV, 263, 264 E. GORBOVSKOY, 264 V.G. KORNILOV, 263, 264 N .TYURINA, 264 P. BALANUTSA, 264 D.VLASENKO, 263, 264 I.GORBUNOV, 264 R. PODESTA, 265 H. LEVATO, 266 C. SAFFE, 266 D.A.H.BUCKLEY, 267 N.M. BUDNEV, 268 O. GRESS, 268, 264 V. YURKOV, 269 R. REBOLO, 270 AND M. SERRA-RICART 270 \n242, 243 \nD. J. SAND, \n242', 'THE MASTER COLLABORATION': "- 1 LIGO, California Institute of Technology, Pasadena, CA 91125, USA\n- 2 Louisiana State University, Baton Rouge, LA 70803, USA\n- 3 Universit'a di Salerno, Fisciano, I-84084 Salerno, Italy\n- 4 INFN, Sezione di Napoli, Complesso Universitario di Monte S.Angelo, I-80126 Napoli, Italy\n- 5 University of Florida, Gainesville, FL 32611, USA\n- 6 OzGrav, School of Physics & Astronomy, Monash University, Clayton 3800, Victoria, Australia\n- 7 LIGO Livingston Observatory, Livingston, LA 70754, USA\n- 8 Laboratoire d'Annecy-le-Vieux de Physique des Particules (LAPP), Universit'e Savoie Mont Blanc, CNRS/IN2P3, F-74941 Annecy, France\n- 9 University of Sannio at Benevento, I-82100 Benevento, Italy and INFN, Sezione di Napoli, I-80100 Napoli, Italy\n- 10 Max Planck Institute for Gravitational Physics (Albert Einstein Institute), D-30167 Hannover, Germany\n- 11 The University of Mississippi, University, MS 38677, USA\n- 12 NCSA, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA\n- 13 University of Cambridge, Cambridge CB2 1TN, United Kingdom\n- 14 Nikhef, Science Park, 1098 XG Amsterdam, The Netherlands\n- 15 LIGO, Massachusetts Institute of Technology, Cambridge, MA 02139, USA\n- 16 Instituto Nacional de Pesquisas Espaciais, 12227-010 S˜ao Jos'e dos Campos, S˜ao Paulo, Brazil\n- 17 Gran Sasso Science Institute (GSSI), I-67100 L'Aquila, Italy\n- 18 INFN, Laboratori Nazionali del Gran Sasso, I-67100 Assergi, Italy\n- 19 Inter-University Centre for Astronomy and Astrophysics, Pune 411007, India\n- 20 International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India\n- 21 University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA\n- 22 Leibniz Universitat Hannover, D-30167 Hannover, Germany\n- 23 Universit'a di Pisa, I-56127 Pisa, Italy\n- 24 INFN, Sezione di Pisa, I-56127 Pisa, Italy\n- 25 OzGrav, Australian National University, Canberra, Australian Capital Territory 0200, Australia\n- 26 Laboratoire des Mat'eriaux Avanc'es (LMA), CNRS/IN2P3, F-69622 Villeurbanne, France\n- 27 SUPA, University of the West of Scotland, Paisley PA1 2BE, United Kingdom\n- 28 LAL, Univ. Paris-Sud, CNRS/IN2P3, Universit'e Paris-Saclay, F-91898 Orsay, France\n- 29 California State University Fullerton, Fullerton, CA 92831, USA\n- 30 European Gravitational Observatory (EGO), I-56021 Cascina, Pisa, Italy\n- 31\n- Chennai Mathematical Institute, Chennai 603103, India\n- 32 Universit'a di Roma Tor Vergata, I-00133 Roma, Italy\n- 33 INFN, Sezione di Roma Tor Vergata, I-00133 Roma, Italy\n- 34 Universitat Hamburg, D-22761 Hamburg, Germany\n- 35 INFN, Sezione di Roma, I-00185 Roma, Italy\n- 36 Cardiff University, Cardiff CF24 3AA, United Kingdom\n- 37 Embry-Riddle Aeronautical University, Prescott, AZ 86301, USA\n- 38 Max Planck Institute for Gravitational Physics (Albert Einstein Institute), D-14476 Potsdam-Golm, Germany\n- 39 APC, AstroParticule et Cosmologie, Universit'e Paris Diderot, CNRS/IN2P3, CEA/Irfu, Observatoire de Paris, Sorbonne Paris Cit'e, F-75205 Paris Cedex 13, France\n- 40 Korea Institute of Science and Technology Information, Daejeon 34141, Korea\n- 41 West Virginia University, Morgantown, WV 26506, USA\n- 42 Universit'a di Perugia, I-06123 Perugia, Italy\n- 43 INFN, Sezione di Perugia, I-06123 Perugia, Italy\n- 44 Syracuse University, Syracuse, NY 13244, USA\n- 45 University of Minnesota, Minneapolis, MN 55455, USA\n- 46 SUPA, University of Glasgow, Glasgow G12 8QQ, United Kingdom\n- 47 LIGO Hanford Observatory, Richland, WA 99352, USA\n- 48 Caltech CaRT, Pasadena, CA 91125, USA \n- 49 Wigner RCP, RMKI, H-1121 Budapest, Konkoly Thege Mikl'os 'ut 29-33, Hungary\n- 50 Columbia University, New York, NY 10027, USA\n- 51 Stanford University, Stanford, CA 94305, USA\n- 52 Universit'a di Camerino, Dipartimento di Fisica, I-62032 Camerino, Italy\n- 53 Universit'a di Padova, Dipartimento di Fisica e Astronomia, I-35131 Padova, Italy\n- 54 INFN, Sezione di Padova, I-35131 Padova, Italy\n- 55 Institute of Physics, Eotvos University, P'azm'any P. s. 1/A, Budapest 1117, Hungary\n- 56 Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences, 00-716, Warsaw, Poland\n- 57 Rochester Institute of Technology, Rochester, NY 14623, USA\n- 58 University of Birmingham, Birmingham B15 2TT, United Kingdom\n- 59 INFN, Sezione di Genova, I-16146 Genova, Italy\n- 60 RRCAT, Indore MP 452013, India\n- 61 Faculty of Physics, Lomonosov Moscow State University, Moscow 119991, Russia\n- 62 SUPA, University of Strathclyde, Glasgow G1 1XQ, United Kingdom\n- 63 The Pennsylvania State University, University Park, PA 16802, USA\n- 64 OzGrav, University of Western Australia, Crawley, Western Australia 6009, Australia\n- 65 Department of Astrophysics/IMAPP, Radboud University Nijmegen, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands\n- 66 Artemis, Universit'e Cˆote d'Azur, Observatoire Cˆote d'Azur, CNRS, CS 34229, F-06304 Nice Cedex 4, France\n- 67 Institut FOTON, CNRS, Universit'e de Rennes 1, F-35042 Rennes, France\n- 68 Washington State University, Pullman, WA 99164, USA\n- 69 University of Oregon, Eugene, OR 97403, USA\n- 70 Laboratoire Kastler Brossel, UPMC-Sorbonne Universit'es, CNRS, ENS-PSL Research University, Coll'ege de France, F-75005 Paris, France\n- 71 Carleton College, Northfield, MN 55057, USA\n- 72 OzGrav, University of Adelaide, Adelaide, South Australia 5005, Australia\n- 73 Astronomical Observatory Warsaw University, 00-478 Warsaw, Poland\n- 74 VU University Amsterdam, 1081 HV Amsterdam, The Netherlands\n- 75 University of Maryland, College Park, MD 20742, USA\n- 76 Center for Relativistic Astrophysics, Georgia Institute of Technology, Atlanta, GA 30332, USA\n- 77 Universit'e Claude Bernard Lyon 1, F-69622 Villeurbanne, France\n- 78 Universit'a di Napoli 'Federico II,' Complesso Universitario di Monte S.Angelo, I-80126 Napoli, Italy\n- 79 NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA\n- 80 Dipartimento di Fisica, Universit'a degli Studi di Genova, I-16146 Genova, Italy\n- 81 RESCEU, University of Tokyo, Tokyo, 113-0033, Japan.\n- 82 Tsinghua University, Beijing 100084, China\n- 83 Texas Tech University, Lubbock, TX 79409, USA\n- 84 Kenyon College, Gambier, OH 43022, USA\n- 85 Departamento de Astronom'ıa y Astrof'ısica, Universitat de Val'encia, E-46100 Burjassot, Val'encia, Spain\n- 86 Museo Storico della Fisica e Centro Studi e Ricerche Enrico Fermi, I-00184 Roma, Italy\n- 87 National Tsing Hua University, Hsinchu City, 30013 Taiwan, Republic of China\n- 88 Charles Sturt University, Wagga Wagga, New South Wales 2678, Australia\n- 89 Center for Interdisciplinary Exploration & Research in Astrophysics (CIERA), Northwestern University, Evanston, IL 60208, USA\n- 90 Canadian Institute for Theoretical Astrophysics, University of Toronto, Toronto, Ontario M5S 3H8, Canada\n- 91 University of Chicago, Chicago, IL 60637, USA\n- 92 Pusan National University, Busan 46241, Korea\n- 93 The Chinese University of Hong Kong, Shatin, NT, Hong Kong\n- 94 INAF, Osservatorio Astronomico di Padova, I-35122 Padova, Italy\n- 95 INFN, Trento Institute for Fundamental Physics and Applications, I-38123 Povo, Trento, Italy\n- 96 OzGrav, University of Melbourne, Parkville, Victoria 3010, Australia\n- 97 Universit'a di Roma 'La Sapienza,' I-00185 Roma, Italy\n- 98 Universit'e Libre de Bruxelles, Brussels 1050, Belgium\n- 99 Sonoma State University, Rohnert Park, CA 94928, USA\n- 100 Departamento de Matem'aticas, Universitat de Val'encia, E-46100 Burjassot, Val'encia, Spain\n- 101 Montana State University, Bozeman, MT 59717, USA\n- 102 Universitat de les Illes Balears, IAC3-IEEC, E-07122 Palma de Mallorca, Spain\n- 103 The University of Texas Rio Grande Valley, Brownsville, TX 78520, USA\n- 104 Bellevue College, Bellevue, WA 98007, USA\n- 105 Institute for Plasma Research, Bhat, Gandhinagar 382428, India\n- 106 The University of Sheffield, Sheffield S10 2TN, United Kingdom\n- 107 Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Universit'a di Parma, I-43124 Parma, Italy\n- 108 INFN, Sezione di Milano Bicocca, Gruppo Collegato di Parma, I-43124 Parma, Italy\n- 109 California State University, Los Angeles, 5151 State University Dr, Los Angeles, CA 90032, USA\n- 110 Universit'a di Trento, Dipartimento di Fisica, I-38123 Povo, Trento, Italy\n- 111 Montclair State University, Montclair, NJ 07043, USA\n- 112 National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan\n- 113 Observatori Astron'omic, Universitat de Val'encia, E-46980 Paterna, Val'encia, Spain\n- 114 School of Mathematics, University of Edinburgh, Edinburgh EH9 3FD, United Kingdom\n- 115 University and Institute of Advanced Research, Koba Institutional Area, Gandhinagar Gujarat 382007, India\n- 116 IISER-TVM, CET Campus, Trivandrum Kerala 695016, India\n- 117 University of Szeged, D'om t'er 9, Szeged 6720, Hungary\n- 118 University of Michigan, Ann Arbor, MI 48109, USA\n- 119 Tata Institute of Fundamental Research, Mumbai 400005, India\n- 120 INAF, Osservatorio Astronomico di Capodimonte, I-80131, Napoli, Italy\n- 121 Universit'a degli Studi di Urbino 'Carlo Bo,' I-61029 Urbino, Italy\n- 122 INFN, Sezione di Firenze, I-50019 Sesto Fiorentino, Firenze, Italy\n- 123 Physik-Institut, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland\n- 124 American University, Washington, D.C. 20016, USA\n- 125 University of Białystok, 15-424 Białystok, Poland\n- 126 University of Southampton, Southampton SO17 1BJ, United Kingdom\n- 127 University of Washington Bothell, 18115 Campus Way NE, Bothell, WA 98011, USA\n- 128 Institute of Applied Physics, Nizhny Novgorod, 603950, Russia\n- 129 Korea Astronomy and Space Science Institute, Daejeon 34055, Korea\n- 130 Inje University Gimhae, South Gyeongsang 50834, Korea\n- 131 National Institute for Mathematical Sciences, Daejeon 34047, Korea\n- 132 NCBJ, 05-400 ' Swierk-Otwock, Poland\n- 133 Institute of Mathematics, Polish Academy of Sciences, 00656 Warsaw, Poland\n- 134 Hillsdale College, Hillsdale, MI 49242, USA\n- 135 Hanyang University, Seoul 04763, Korea\n- 136 Seoul National University, Seoul 08826, Korea\n- 137 NASA Marshall Space Flight Center, Huntsville, AL 35811, USA\n- 138 ESPCI, CNRS, F-75005 Paris, France\n- 139 Southern University and A&M College, Baton Rouge, LA 70813, USA\n- 140 College of William and Mary, Williamsburg, VA 23187, USA\n- 141 Centre Scientifique de Monaco, 8 quai Antoine Ier, MC-98000, Monaco\n- 142 Indian Institute of Technology Madras, Chennai 600036, India\n- 143 IISER-Kolkata, Mohanpur, West Bengal 741252, India\n- 144 Whitman College, 345 Boyer Avenue, Walla Walla, WA 99362 USA\n- 145 Indian Institute of Technology Bombay, Powai, Mumbai, Maharashtra 400076, India\n- 146 Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126 Pisa, Italy\n- 147 Universit'e de Lyon, F-69361 Lyon, France\n- 148 Hobart and William Smith Colleges, Geneva, NY 14456, USA\n- 149 OzGrav, Swinburne University of Technology, Hawthorn VIC 3122, Australia\n- 150 Janusz Gil Institute of Astronomy, University of Zielona G'ora, 65-265 Zielona G'ora, Poland\n- 151 University of Washington, Seattle, WA 98195, USA\n- 152 King's College London, University of London, London WC2R 2LS, United Kingdom\n- 153 Indian Institute of Technology, Gandhinagar Ahmedabad Gujarat 382424, India\n- 154 Indian Institute of Technology Hyderabad, Sangareddy, Khandi, Telangana 502285, India \n- 155 International Institute of Physics, Universidade Federal do Rio Grande do Norte, Natal RN 59078-970, Brazil\n- 156 Andrews University, Berrien Springs, MI 49104, USA\n- 157 Universit'a di Siena, I-53100 Siena, Italy\n- 158 Trinity University, San Antonio, TX 78212, USA\n- 159 Abilene Christian University, Abilene, TX 79699, USA\n- 160 Colorado State University, Fort Collins, CO 80523, USA\n- 161 Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064, USA\n- 162 The Observatories of the Carnegie Institution for Science, 813 Santa Barbara Street, Pasadena, CA 91101\n- 163 Hubble and Carnegie-Dunlap Fellow\n- 164 Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA\n- 165 Departments of Physics and Astronomy, University of California, Berkeley, CA 94720, USA\n- 166 Dark Cosmology Centre, Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 2100 Copenhagen, Denmark\n- 167 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218\n- 168\n- Institute for Astronomy, University of Hawai'i, 2680 Woodlawn Drive, Honolulu, HI 96822, USA\n- 169 Hubble and Carnegie-Princeton Fellow\n- 170 Departamento de F'ısica y Astronom'ıa, Universidad de La Serena, La Serena, Chile\n- 171 Fermi National Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510, USA\n- 172 Department of Physics, Brandeis University, Waltham MA, USA\n- 173 Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA\n- 174 Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA\n- 175 Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, Massachusetts 02138, USA\n- 176 Department of Physics, University of Surrey, Guildford, GU2 7XH, UK\n- 177 Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA\n- 178 Department of Astronomy, Indiana University, 727 E. Third Street, Bloomington, IN 47405, USA\n- 179\n- 180\n- Astrophysical Institute, Department of Physics and Astronomy, 251B Clippinger Lab, Ohio University, Athens, OH 45701, USA\n- George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy, and Department of Physics and Astronomy, Texas A&M University, College Station, TX 77843, USA\n- 181 LSST, 933 North Cherry Avenue, Tucson, AZ 85721, USA\n- 182 The Observatories of the Carnegie Institution for Science, 813 Santa Barbara St., Pasadena, CA 91101, USA\n- 183 Institut d'Astrophysique de Paris (UMR7095: CNRS & UPMC), 98 bis Bd Arago, F-75014, Paris, France\n- 184 Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA) and Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208, USA\n- 185 Hubble Fellow\n- 186 Center for Theoretical Astrophysics, Los Alamos National Laboratory, Los Alamos, NM 87544\n- 187\n- Instituto de Fisica Teorica UAM/CSIC, Universidad Autonoma de Madrid, 28049 Madrid, Spain\n- 188 SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA\n- 189 Department of Astronomy, University of Illinois, 1002 W. Green Street, Urbana, IL 61801, USA\n- 190 National Center for Supercomputing Applications, 1205 West Clark St., Urbana, IL 61801, USA\n- 191 Department of Physics and Astronomy & Astrophysics,The Pennsylvania State University, University Park, PA 16802, USA\n- 192 Department of Physics & Astronomy, University College London, Gower Street, London, WC1E 6BT, UK\n- 193 Department of Physics, ETH Zurich, Wolfgang-Pauli-Strasse 16, CH-8093 Zurich, Switzerland\n- 194 Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA\n- 195 Departments of Physics and Astronomy, and Theoretical Astrophysics Center, University of California, Berkeley, CA 94720-7300, USA\n- 196 Observat'orio do Valongo, Universidade Federal do Rio de Janeiro, Ladeira do Pedro Antˆonio 43, Rio de Janeiro, RJ, 20080-090, Brazil\n- 197 Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA) and Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208\n- 198 George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy, and Department of Physics and Astronomy, Texas A&M University, College Station, TX 77843, USA\n- 199 National Optical Astronomy Observatory, 950 North Cherry Avenue, Tucson, AZ 85719, USA\n- 200 Departamento de Astronomon'ıa, Universidad de Chile, Camino del Observatorio 1515, Las Condes, Santiago, Chile\n- 201 Department of Physics and Columbia Astrophysics Laboratory, Columbia University, New York, NY 10027, USA\n- 202 Department of Physics, University of Michigan, 450 Church St, Ann Arbor, MI 48109-1040\n- 203 Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA\n- 204 Department of Astronomy & Theoretical Astrophysics Center, University of California, Berkeley, CA 94720-3411, USA\n- 205 Physics Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720-8160, USA\n- 206 Steward Observatory, University of Arizona, 933 N. Cherry Ave., Tucson, AZ 85721\n- 207 Instituto de F'ısica Gleb Wataghin, Universidade Estadual de Campinas, 13083-859, Campinas, SP, Brazil\n- 208 Laborat'orio Interinstitucional de e-Astronomia - LIneA, Rua Gal. Jos'e Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil\n- 209 Cerro Tololo Inter-American Observatory, National Optical Astronomy Observatory, Casilla 603, La Serena, Chile\n- 210 Department of Physics and Electronics, Rhodes University, PO Box 94, Grahamstown, 6140, South Africa\n- 211 CNRS, UMR 7095, Institut d'Astrophysique de Paris, F-75014, Paris, France\n- 212 Sorbonne Universit'es, UPMC Univ Paris 06, UMR 7095, Institut d'Astrophysique de Paris, F-75014, Paris, France\n- 213 Jodrell Bank Center for Astrophysics, School of Physics and Astronomy, University of Manchester, Oxford Road, Manchester, M13 9PL, UK\n- 214 Kavli Institute for Particle Astrophysics & Cosmology, P. O. Box 2450, Stanford University, Stanford, CA 94305, USA\n- 215 Observat'orio Nacional, Rua Gal. Jos'e Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil\n- 216 Institut de F'ısica d'Altes Energies (IFAE), The Barcelona Institute of Science and Technology, Campus UAB, 08193 Bellaterra (Barcelona) Spain\n- 217 Institute of Space Sciences, IEEC-CSIC, Campus UAB, Carrer de Can Magrans, s/n, 08193 Barcelona, Spain\n- 218 Department of Physics, IIT Hyderabad, Kandi, Telangana 502285, India\n- 219 Excellence Cluster Universe, Boltzmannstr. 2, 85748 Garching, Germany\n- 220 Faculty of Physics, Ludwig-Maximilians-Universitat, Scheinerstr. 1, 81679 Munich, Germany\n- 221\n- Department of Astronomy, University of Michigan, Ann Arbor, MI 48109, USA\n- 222 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK\n- 223 Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK\n- 224 Universitats-Sternwarte, Fakultat fur Physik, Ludwig-Maximilians Universitat Munchen, Scheinerstr. 1, 81679 Munchen, Germany\n- 225 Department of Astronomy, University of California, Berkeley, 501 Campbell Hall, Berkeley, CA 94720, USA\n- 226 Center for Cosmology and Astro-Particle Physics, The Ohio State University, Columbus, OH 43210, USA\n- 227 Department of Physics, The Ohio State University, Columbus, OH 43210, USA\n- 228 Astronomy Department, University of Washington, Box 351580, Seattle, WA 98195, USA\n- 229 Santa Cruz Institute for Particle Physics, Santa Cruz, CA 95064, USA\n- 230 Australian Astronomical Observatory, North Ryde, NSW 2113, Australia\n- 231 Argonne National Laboratory, 9700 South Cass Avenue, Lemont, IL 60439, USA\n- 232 Departamento de F'ısica Matem'atica, Instituto de F'ısica, Universidade de S˜ao Paulo, CP 66318, S˜ao Paulo, SP, 05314-970, Brazil\n- 233 Instituci'o Catalana de Recerca i Estudis Avanc¸ats, E-08010 Barcelona, Spain\n- 234 Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Dr., Pasadena, CA 91109, USA\n- 235 Department of Physics and Astronomy, Pevensey Building, University of Sussex, Brighton, BN1 9QH, UK\n- 236 Centro de Investigaciones Energ'eticas, Medioambientales y Tecnol'ogicas (CIEMAT), Madrid, Spain\n- 237 School of Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, UK\n- 238 Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831\n- 239 Institute of Cosmology & Gravitation, University of Portsmouth, Portsmouth, PO1 3FX, UK\n- 240 Max Planck Institute for Extraterrestrial Physics, Giessenbachstrasse, 85748 Garching, Germany\n- 241 Department of Physics and Astronomy, University of North Carolina at Chapel Hill, Chapel Hill, NC, 27599, USA\n- 242 Department of Astronomy and Steward Observatory, University of Arizona, 933 N Cherry Ave, Tucson, AZ 85719, USA\n- 243 Department of Physics, University of California, 1 Shields Avenue, Davis, CA 95616-5270, USA\n- 244 Department of Physics and Astronomy, University of Padova, Via 8 Febbraio, 2-35122 Padova, Italy\n- 245 INAF Osservatorio Astronomico di Padova, Vicolo della Osservatorio 5, I-35122 Padova, Italy\n- 246 Department of Physics, University of California, Santa Barbara, CA 93106-9530, USA\n- 247 Las Cumbres Observatory, 6740 Cortona Dr Ste 102, Goleta, CA 93117-5575, USA\n- 248 School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel\n- 249 Department of Physics and Astronomy, University of Leicester, University Road, Leicester, LE1 7RH, UK\n- 250 Department of Physics, University of Warwick, Coventry, CV4 7AL, UK\n- 251 DARK, Niels Bohr Institute, University of Copenhagen, Juliane Maries Vej 30, 2100 Copenhagen Ø, Denmark\n- 252 Instituto de Astrof'ısica de Andaluc'ıa (IAA-CSIC), Glorieta de la Astronom'ıa s/n, 18008 Granada, Spain \n- 253 Astrophysics Research Institute, Liverpool John Moores University, IC2, Liverpool Science Park, 146 Brownlow Hill, Liverpool L3 5RF\n- 254 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, United Kingdom\n- 255 Max-Planck-Institut fur extraterrestrische Physik, 85740 Garching, Giessenbachstr. 1, Germany\n- 256 Birmingham Institute for Gravitational Wave Astronomy and School of Physics and Astronomy, University of Birmingham, Birmingham, B15 2TT, UK\n- 257 INAF, Institute of Space Astrophysics and Cosmic Physics, Via Gobetti 101, I-40129 Bologna, Italy\n- 258 School of Physics and Astronomy, Monash University, VIC 3800, Australia; Monash Centre for Astrophysics, Monash University, VIC 3800, Australia\n- 259 The Oskar Klein Centre, Department of Astronomy, AlbaNova, Stockholm University, SE-106 91 Stockholm, Sweden\n- 260 Anton Pannekoek Institute, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, the Netherlands\n- 261 ASTRON, the Netherlands Institute for Radio Astronomy, Postbus 2, 7990 AA Dwingeloo, the Netherlands\n- 262 Department of Particle Physics and Astrophysics, Weizmann Institute of Science, 76100, Rehovot, Israel\n- 263 M.V.Lomonosov Moscow State University, Physics Department, Leninskie gory, GSP-1, Moscow, 119991, Russia\n- 264 M.V.Lomonosov Moscow State University, Sternberg Astronomical Institute, Universitetsky pr., 13, Moscow, 119234, Russia\n- 265 Observatorio Astronomico Felix Aguilar (OAFA) , National University of San Juan, Argentina\n- 266 Instituto de Ciencias Astronomicas,de la Tierra y del Espacio (ICATE), San Juan, Argentina\n- 267 South African Astrophysical Observatory, PO Box 9, 7935 Observatory, Cape Town, South Africa\n- 268 Irkutsk State University, Applied Physics Institute, 20, Gagarin blvd,664003, Irkutsk, Russia\n- 269 Blagoveschensk State Pedagogical University, Lenin str., 104, Amur Region, Blagoveschensk 675000\n- 270 Instituto de Astrof'ısica de Canarias Via Lactea, s/n E38205 - La Laguna (Tenerife), Spain\n- ∗ Deceased, February 2017.\n- † Deceased, December 2016."} |
2024arXiv240906441D | The issue of consistency is crucial in quantum gravity. It has recently been intensively addressed for effective symmetryreduced models. In this article we exhaustively study the anomaly freedom of effective loop quantum cosmology with generalized holonomy corrections considering loop correction of the constraints at the perturbative order. We pedagogically explain why although the holonomy correction including the details of the chosen scheme applied on the background part of the constraints is crucial it becomes irrelevant when implemented on perturbative expansions in the sense that all consequences are absorbed in the counterterms used for the regularization. The possibility of closing the algebra of constraints without counterterms is also studied. It is argued that although enforcing a firstclass algebra is a strong requirement this can be achieved in several different ways often overlooked which generates ambiguities on the restriction of the form of the generalized holonomy correction. Those ambiguities are examined in details leading to the conclusion that the consistency of the effective theory for cosmological perturbations especially when considering scalar modes cannot be achieved without counterterms. We also take the opportunity of this work to clarify as much as possible all the required steps so that future works have a clear material at disposal. In particular a highly detailed calculation of all the brackets is provided emphasizing the usually implicit assumptions hypotheses and manipulations required to ensure the closure of the algebra. Prospects for future works are underlined. | 2024-09-01T00:00:00Z | ['arXiv:2409.06441', '2024arXiv240906441D', '10.48550/arXiv.2409.06441'] | ['General Relativity and Quantum Cosmology', 'High Energy Physics - Theory'] | Anomaly freedom in effective Loop Quantum Cosmology pedagogical summary and generalized holonomy corrections | 2,024 | 174 | 0.22 | ['EPRINT_HTML', 'EPRINT_PDF'] | 0 | https://arxiv.org/pdf/2409.06441.pdf | {"Maxime De Sousa, Aur'elien Barrau, Killian Martineau": "Laboratoire de Physique Subatomique et de Cosmologie, Universit'e Grenoble-Alpes, CNRS/IN2P3 53, avenue des Martyrs, 38026 Grenoble cedex, France \nE-mail: [email protected] \nAbstract: The issue of consistency is crucial in quantum gravity. It has recently been intensively addressed for effective symmetry-reduced models. In this article, we exhaustively study the anomaly freedom of effective loop quantum cosmology with generalized holonomy corrections, considering loop correction of the constraints at the perturbative order. We pedagogically explain why, although the holonomy correction - including the details of the chosen scheme - applied on the background part of the constraints is crucial, it becomes irrelevant when implemented on perturbative expansions, in the sense that all consequences are 'absorbed' in the counter-terms used for the regularization. The possibility of closing the algebra of constraints without counter-terms is also studied. It is argued that, although enforcing a first-class algebra is a strong requirement, this can be achieved in several different ways, often overlooked, which generates ambiguities on the restriction of the form of the generalized holonomy correction. Those ambiguities are examined in details, leading to the conclusion that the consistency of the effective theory for cosmological perturbations, especially when considering scalar modes, cannot be achieved without counter-terms. We also take the opportunity of this work to clarify, as much as possible, all the required steps so that future works have a clear material at disposal. In particular, a highly detailed calculation of all the brackets is provided, emphasizing the (usually implicit) assumptions, hypotheses and manipulations required to ensure the closure of the algebra. Prospects for future works are underlined.", '1 Introduction': "The quest for a quantum theory of gravity is one of the most difficult tasks for theoretical physics (see [1, 2] for reviews of currently followed paths). No consensual approach exists today but different models are being considered. One way to compare them is obviously to carefully investigate their respective observational predictions (see, e.g., [3] and references therein). This is the most obvious approach but it is extremely difficult in practice as the Planck scale is 15 orders of magnitude beyond what is currently probed with colliders (see [4] for some thoughts on the subject). Another key aspects is the one of consistency . \nAlthough all serious models are, by construction, consistent at first sight, a careful analysis can reveal severe flaws that were not a priori obvious. This is one of the central motivation for the so-called swampland program in string theory (see [5] for a review): some theories believed to lie in the landscape cannot be pushed to high energies in the presence of gravity. This article focuses on a different aspect, namely the closure of the algebra of constraints. It is a very important consistency condition as it ensures that the vectors of evolution are parallel to the sub-manifolds of constraints. Otherwise stated, this guaranties that, starting from a physical solution and using the dynamical laws of the theory, one still gets a physical solution. \nWhen the equations of gravity - especially those describing the evolution of cosmological perturbations - are quantum corrected at the effective level it is difficult to determine whether the subtle consistency conditions embedded in the first-class nature of the algebra of constraints are still satisfied [6]. Many modern approaches to canonical quantum general relativity [7] put the emphasis on background-independence as being the conceptual core of Einstein's gravity. Those frameworks cannot take advantage of usual covariance criteria as space-time itself should, in a way, emerge from solutions to their dynamical equations. Generally speaking, gauge fixing before quantization is dangerous. Observables can be significantly different from what would have been obtained if the theory had been quantized without fixing the gauge. In gravity, dynamics is part of the gauge and one should be very careful not to fix the gauge according to transformations that have to be subsequently modified [8]. The issue of possible anomalies in the Poisson brackets between constraints in a central one in this framework. \nIn the last decade, a tremendous amount of work has been devoted to the question of anomaly freedom in effective loop quantum gravity (LQG), see, e.g. [9-12] for pedagogical introductions to LQG. Notably, great progresses were recently made in the black hole sector [13-19]. The present work focuses on cosmological aspects as it seems to us that a pedagogical and rigorous derivation of all the relevant steps was missing. Currently, anyone interested in this approach has to reinvent all the machinery, which is far from being light or trivial. The first goal of this article is therefore to explain the subtleties and technicalities - some of them being sometimes erroneously neglected in the literature - so that future works can rely on a clear basic material where hypotheses are made explicit. New results regarding the use of generalized holonomy corrections (GHC), which are currently being intensively considered (in particular, but not only, because of points made in [20-23]), are derived. We point out that although the holonomy correction - including its detailed expression - plays a crucial role when implemented at the background level (see, e.g., [24-26] for the importance of the GHC), it brings no new effect at all when implemented in the perturbative expansion of the constraints 1 . We also underline that the closure of the algebra cannot be achieved with a specific expression of the generalized holonomy correction, \nwithout the use of counter-terms. \nThe first part of this work review the basics of loop quantum cosmology for the unfamiliar reader. Then, the detailed calculation of all the Poisson brackets between constraints is performed. Finally, different ways to close the algebra are considered. Promising new directions for future works are also pointed out. Throughout all the work, assumptions that could be relaxed in future studies are underlined.", '2 Effective Loop Quantum Cosmology': "Loop Quantum Cosmology (LQC) is a framework for the description of the very early universe, mimicks quantization techniques of Loop Quantum Gravity (LQG) in the cosmological sector. It has rapidly become a valuable testing ground for addressing and evading some of the technical challenges of full LQG, mostly because the LQC model is exactly solvable [27]. Notably, in the last decades, significant insights have emerged regarding the history of the early universe, such as the fact that the matter density operator ˆ ρ has an upper bound when LQC is coupled to a scalar field φ , which indicates a resolution of the 'big-bang' singularity. Additionally, the effective constructions in LQC successfully capture the dynamics of semi-classical states [28]. In order to make this article self-contained and set the notations, a brief introduction to LQC is provided in the next subsection. For nice and more detailed reviews of LQC, we refer to [27, 29, 30]. \nSeveral ways to study cosmological perturbations in the LQG paradigm do exist. Among interesting paths, are the so called 'dressed metric' approach [31-33] and the (related) 'hybrid quantization' scheme [34, 35]. Comparisons between existing models can be found, e.g., in [36-39]. This work shall focus only on the 'deformed algebra' approach which, in a way, might capture less quantum effects but puts the emphasis on consistency, which is certainly a necessary requirement. Importantly, in a sense, it also goes beyond the LQG/LQC models and constitutes a relevant framework to investigate a wide class of possible deformations of general relativity [14, 40-42].", '2.1.1 Background variables': "The line element of a space-time M parameterized by M = R × Σ, where Σ corresponds to spatial slices, can be given by the ADM formalism [43] \nds 2 = -N 2 dη 2 + q ab ( N a dη + dx a )( N b dη + dx b ) , (2.1) \nin which N is the lapse function, N a is the shift vector and q ab is the induced spatial metric on Σ. Throughout all this work, we consider a flat FLRW space-time as the background for cosmological perturbations theory, whose line element is given by [44]: \nds 2 = a 2 ( η ) ( -dη 2 + δ ab dx a dx b ) . (2.2) \nDirect comparison of Eq. (2.1) and Eq. (2.2) leads to the following identifications: \nq ab = a 2 ( η ) δ ab , N = a ( η ) , N a = 0 . (2.3) \nThe LQG canonical quantization program is based on the so-called Ashtekar's variables [45, 46]. In this setup, the densitized triads E a i enable the building of the 3-metric q ab , such that E a i E b j δ ij def = ∣ ∣ det q ∣ ∣ q ab . At the background level, the densitized triads are given by [29]: \nE a i = a 2 ( η ) δ a i def = p ( η ) δ a i . (2.4) \nTheir canonically conjugate variables K i a are built from the extrinsic curvature tensor \nK ab def = ( 2 N ) -1 ( ∂ η q ab -2 D ( a N b ) ) = ∂ η p 2 √ p δ ab , (2.5) \nwhere D denotes the spatial covariant derivative of q ab . From the previous expression, one can construct [29], \nK i a = E bi √ ∣ det E ∣ K ab = ∂ η p 2 p δ i a def = c ( η ) δ i a , (2.6) \n∣ ∣ such that K i a and E a i are canonically conjugate. They fulfill the relations \n{ K i a ( x ) , E b j ( y ) } def = κ ∫ d z ( δK i a ( x ) δK l c ( z ) δE b j ( y ) δE c l ( z ) ) = κδ i j δ b a δ ( x -y ) , (2.7) \nwith κ = 8 πG , from which the Poisson bracket between p and c can be obtained: \n{ c , p } = κ 3 V -1 . (2.8) \nIn the previous expression V is defined as the volume of a fiducial cell used to regularize the integrals appearing in the Poisson brackets of the background variables in Eq. (2.7). \nAs usual, canonical transformations can be usefully performed in the Hamiltonian framework. In particular, the use can me made of the so-called Ashtekar's connection, defined as [7]: \nA i a = Γ i a + γK i a , (2.9) \nwhere Γ i a is the so (3)-valued spin connection compatible with the triads, and γ is the Barbero-Immirzi (free) parameter. This canonically conjugate, equivalent, phase-space is now parameterized via the densitized triads E a i and the Ashtekar's connection A i a verifying the structure \n{ A i a ( x ) , E b j ( y ) } = γκδ i j δ b a δ ( x -y ) . (2.10) \nIn addition to the gravitational sector, the energy content is assumed to be a minimally coupled scalar field φ , whose action is given by: \nS φ = ∫ d 4 x √ | det g | ( 1 2 ∇ µ φ ∇ µ φ -V [ φ ] ) , (2.11) \nwhere V [ φ ] is the potential of the scalar field. \nIn a symplectic framework, the canonical variable φ and its conjugate momentum π satisfy the relation \n{ φ ( x ) , π ( y ) } = δ ( x -y ) . (2.12) \nThe background values φ and π of those fields satisfy the standard Klein-Gordon dynamics. Their Poisson bracket is \n{ φ , π } = V -1 . (2.13) \nThe evolution of the canonical variables is driven by constraints, which will be detailed in the following.", '2.1.2 Generalized holonomy corrections': "As stated above, in canonical LQG, geometry is described by the Ashtekar connection A i a and the densitized triads E a i instead of the spatial metric q ab and its associated momentum Π ab [9-12]. This is a step towards background independence, as these fields can be smeared without referring to the metric. Specifically, the Ashtekar's connection is smeared along closed curves to produce holonomies, while the densitized triads are naturally integrated over the associated surfaces [11, 12]. The quantization ''a la Dirac' of the resulting holonomies and fluxes leads to discrete spectra for geometrical operators, such as the area [47, 48] or the volume [48, 49]. Specifically, the regularization of the constraints (see Sec. 2.3) with respect to holonomies significantly impacts the dynamics [50]. The deep reasons for the 'periodization' of the connection variable is that there is no natural quantum operator associated with the Ashtekar connection (see, e.g., [51]). At the effective level, the use of holonomies instead of the connection itself modifies the expression of the curvature tensor. This modification is described by the so-called 'holonomy correction', which consists in the replacement [27] \nc -→ sin( δ c ) δ (2.14) \nin the curvature tensor. The explicit expression of δ def = δ ( p ) depends on the scheme under consideration [27]. \nNonetheless, the quantization procedure of the Hamiltonian constraint is subject to many ambiguities [52] and, as a consequence, so is the resulting correction [20]. A tremendous amount of work has been devoted to those ambiguities in the context of LQC (see, e.g., [21, 22, 53-55]). In this work, to remain as generic as possible, we follow [23, 24] and introduce a so-called Generalized Holonomy Correction (GHC), which corresponds to the replacement: \nc -→ g ( c , p ) . (2.15) \nThere is a priori no restriction on g as long as g → c in the low curvature regime. \nThroughout this work, only the background variable c is holonomy-corrected. However, we still introduce a second correction: \nc -→ ˜ g ( c , p ) , (2.16) \nthat will be applied to the perturbed constraints. Relying on two distinct corrections will help discriminating between the effects due to the use of a GHC at the background level and those arising from the use of the GHC for the perturbations. \nAs we shall see in the following sections, this second correction appears frequently in various calculations, including anomalies and counter-terms. However, it will be shown that the resulting Hamiltonian depends functionally only on g , thereby leading to the conclusion that physical observables remain unaffected by the introduction of a GHC in the perturbed constraints. This shows that including holonomy corrections in the background part of the Hamiltonian is sufficient. In addition, in the last section, we will explain why, without counter-terms, those GHCs are actually restricted to the usual GR expressions. \nIt is important to stress that, in this work, we follow the usual approach in which the holonomy correction appears in the perturbations only through modifications of the background extrinsic curvature c . An important step further would be to perform a rigorous treatment of the perturbative expansion of the holonomy, based on the perturbed connection. This goes beyond the scope of this article but it should be considered in future works.", '2.2 Perturbations': "The study of cosmological perturbations is crucial, mainly for two reasons. Firstly, deriving the cosmological power spectrum is the key ingredient for comparison with observations. Secondly, as it will be discussed later, the gauge freedoms related to the Gauss constraint G and to the (spatial) diffeomorphism constraint D are fixed for homogeneous models ensuring consistency (see sect. 2.5). Perturbations can be considered as test fields revealing the underlying space-time structure [56], e.g. its signature, that cannot be seen at the background level. The dynamics of perturbations has to be calculated within a consistent framework. The inhomogeneities of the gravitational phase-space are parametrized by \nE a i = p δ a i + δE a i , and K i a = c δ i a + δK i a , (2.17) \nand those of the matter phase-space by \nφ = φ + δφ, and π = π + δπ. (2.18) \nMoreover, to describe perturbations of all the components of the 4-metric g µν , perturbations around the mean value of the lapse and shift functions must also be considered. Those perturbations are parametrized by \nN = N + δN, and N a = N a + δN a . (2.19) \nIn this paper, our focus lies on the evolution equations of cosmological perturbations at the linear order. The equations of motion of a phase space quantity f governed by a Hamiltonian H being given by ∂ t f = { f, H } , it is necessary to consider the perturbation of the Hamiltonian H at the second order. This is essentially implied by a 'loss of perturbation order' when the Poisson bracket is applied to the perturbed phase space. The perturbation of the Hamiltonian being pushed to the second order, a question might then arise: why are second order perturbations of the canonical variables neglected? The reason lies in the fact that, within the Hamiltonian framework at the second order , the second order perturbations of canonical variables has no consequence on the linear dynamics of perturbations. 2 The first significant contributions would be at the third order of perturbations to the Hamiltonian, corresponding to corrections with respect to the dynamics addressed in this paper.", '2.3 Gravitational constraints': "As previously discussed, this work is carried out at the effective level within the classical Hamiltonian formulation, expressed in Ashtekar's variables, of holonomy-corrected GR. As underlined by Dirac [57], the constraints play a key role in deriving physical quantities for gauge theories. Specifically, constraints \n- 1. restrict the fields to the hypersurface on which the constraints vanish,\n- 2. generate gauge transformations for the fields,\n- 3. provide the equations of motion of the fields. \nIn this section, we briefly describe each constraint, focusing on the implementation of holonomy corrections.", '2.3.1 Gauss constraints': "Due to the choice of working with Ashtekar's variables, which extend the phase space of GR, a constraint known as the Gaussian constraint do arise and is expressed as [58]: \nG def = G [Λ i ] = ( κγ ) -1 ∫ d x Λ i G i = ( κγ ) -1 ∫ d x Λ i [ ∂ a E a i + /epsilon1 l ik A k a E a l ] , (2.21) \nwhere G i are the so-called Gaussian constraints densities and /epsilon1 l ik is the Levi-Civita symbol. Gauge rotations in the internal space of phase-space functions are parameterized by the Lagrange multipliers Λ i , such that the gauge transformations are given by δ Λ ( f ) def = { f, G [Λ i ] } . \n∂ t E a i = { E a i ( x ) , H ( y ) } def = κ ∫ d z δ ( x -z ) δ a b δ j i δ H ( y ) δK j b ( z ) . (2.20) \nThus, to have ∂ t E a i at the linear order, H has to be at the second order in perturbations and include δK (otherwise it would be irrelevant for the dynamics of E a i ). The same reasoning follows for each perturbed canonical variable. \nGauge transformations of canonical variables, like the densitized triads E a i , can therefore be calculated as: \nδ Λ ( E a i ) def = { E a i ( x ) , G [Λ i ] } = γ ∫ d z δ ( x -z ) δ a b δ j i δ G [Λ i ] δA j b ( z ) = κ -1 /epsilon1 k ij Λ j E a k . (2.22) \nAfter performing some identifications with the symmetry-reduced triads Eq. (2.4), it is easy to see that no gauge freedom, with respect to the Gauss constraint, is left if cosmological symmetries are applied. Consequently, there is a single way to construct the Gauss constraint at the second order in perturbations, which is, \nwith, \nG = ( κγ ) -1 ∫ d x δ Λ i G (1) i , (2.23) \nG (1) i = γ [ p /epsilon1 a ij δK j a + c /epsilon1 j ia δE a j ] , (2.24) \nwhere X ( n ) stands the perturbed expression of the quantity X at the n -th order. \nIn LQG, the kernel of the Gauss constraints related operator is easily obtained with the spin-network decomposition (for details, see [58]). Schematically, it consists in demanding invariance of cylindrical functions under the transformations generated by classical Gauss constraints [59]. This ensures that the theory which is quantized is indeed GR, and not an extended gravity theory. One can thus study the canonical transformations and their structure modulo Gauss constraints: no quantum corrections are expected to arise here at the effective level. In particular, no GHC is to be implemented in the Gauss constraints, even though G (1) i depends on the reduced curvature c .", '2.3.2 Diffeomorphism constraints': "Diffeomorphism invariance is at the heart of GR. It reflects the independence of physical laws with respect to choices of coordinate systems. In a canonical setup, the invariance under (spatial) diffeomorphisms is guaranteed if the so-called diffeomorphism constraints are satisfied. Modulo Gauss constraints, they are given by [58]: \nD g def = D g [ N a ] = ( κγ ) -1 ∫ d x N a D g a = ( κγ ) -1 ∫ d x N a [ ( ∂ a A j b -∂ b A j a ) E b j + A j a ∂ b E b j ] , (2.25) \nwhere D g a are the (geometrical) diffeomorphism constraints densities. \nIn the flat FLRW spacetime, as described by Eq. (2.2), the lapse vector N a vanishes at the background level and the smeared (geometrical) diffeomorphism constraint can be nonambiguously expressed at the second order in the perturbed canonical variables. It is given by: \nD g = ( κγ ) -1 ∫ d x δN a D g (1) a , (2.26) \nwhere the perturbed density is \nD g (1) a = γ [ p ( ∂ a δK b b -∂ i δK i a ) -c δ j a ∂ b δE b j ] . (2.27) \nAs for G , the quantization of the diffeomorphism constraint in LQG via a group averaging procedure (see [58] for details) leads, in principle, to no quantum corrections - and, in particular, no GHCs - at the effective level. However, recent results with corrected diffeomorphism constraints, such as a study of scalar cosmological perturbations in LQC using self-dual Ashtekar's variables [60] in which the usual signature change disappears, or in spherically symmetric models [14] where a no-go theorem is established on the non -closure of the algebra of constraints, suggest that corrected diffeomorphism constraints are worth being considered, at least from a heuristic perspective. The study of corrected D g is beyond the scope of this work and left for further research 3 .", '2.3.3 Hamiltonian constraint': "To ensure the full diffeomorphism invariance of the theory 4 , another constraint, the socalled Hamiltonian (or scalar) constraint, must be added to the (spatial) diffeomorphism one. In particular, this constraint encodes the invariance of the theory under time reparametrization. When expressed in terms of Ashetekar's variables, modulo Gauss constraints, the gravitational part of the Hamiltonian constraint reads [58]: \nH g def = H g [ N ] = ( 2 κ ) -1 ∫ d x N H g = ( 2 κ ) -1 ∫ d x N E c j E d k √ ∣ ∣ det E ∣ ∣ [ /epsilon1 jk i F i cd -2 ( 1 + γ 2 ) K j [ c K k d ] ] , (2.28) \nwhere H g is the Hamiltonian (or scalar) constraint density and the field-strength F i ab of the Ashtekar's connection A i a is defined by: \nF i ab def = 2 ∂ [ a A i b ] + /epsilon1 i jk A j a A k b . (2.29) \nThe Hamiltonian density constraint perturbed at second order is decomposed as the sum of a background part, \na first order term, \nH (0) g = -6 √ pc 2 , (2.30) \nH (1) g = -4 √ p δK b b -c 2 √ p δE b b + 2 √ p ∂ a ∂ i δE a i , (2.31) \nand a second order one, \nH (2) g = √ p [ δK a b δK b a -( δK b b ) 2 ] -1 2 c 2 p 3 / 2 [ δE a b δE b a -1 2 ( δE b b ) 2 ] \nand, \nδK a b δK b a def = δ a i δ b j δK j a δK i b , δE a b δE b a def = δ i a δ j b δE a j δE b i , (2.34) \nhave been introduced for simplicity and readiness. As in [61], we also define Z cidj ab as: \nZ cidj ab def = 1 4 /epsilon1 ef k /epsilon1 k mn X mjd be X nic af -/epsilon1 ie k X kjd bd δ c a -/epsilon1 ci k X kjd ba + 1 2 δ i a /epsilon1 ce k X kjd be , (2.35) \nwith, \nX ijb ca def = /epsilon1 ij c δ b a -/epsilon1 ib c δ j a + /epsilon1 ijb δ ca + /epsilon1 ib a δ j c . (2.36) \nIt is argued in [61] that the use of the scalar-vector-tensor decomposition of cosmological perturbations [62] would greatly simplify the expression of Z cidj ab . Nonetheless, most conclusions can be reached without this step and, as far as the perturbation modes entering the calculation of the Poisson brackets are concerned, the present work remains generic. \nThe Hamiltonian constraint in the geometrical sector, perturbed at the second order, is therefore: \nIt should be noticed that, as long as background equations of motion are satisfied and for consistency of the perturbations theory, the first-order of the (geometrical) Hamiltonian constraint H (1) g vanishes identically [63]. The explicit dependence of the previous constraint on the reduced curvature c at the perturbative level should be discussed as well. This becomes particularly important when constructing the effective theory using holonomy corrections. Historically, the correction has been applied to each occurrence of the reduced curvature c - this has been the standard approach for discussing cosmological perturbation theory in the (effective) LQC context [61, 64-67]. However, recent analysis have applied the corrections only to the background, H (0) g , leaving the perturbations H (1) g and H (2) g uncorrected [24]. This needs to be clarified as this might influence cosmological observables. In this work, we make explicit the reasons why applying the holonomy correction (either usual or generalized) to the perturbation expansion of the scalar constraint has no consequence at all when considering the perturbation theory at the second order. \nH g = ( 2 κ ) -1 ∫ d x ( N [ H (0) g + H (2) g ] + δN H (1) g ) . (2.37)", '2.4 Gravity coupled to a scalar field: constraints': 'The coupling of the gravitational sector to scalar matter introduces additional contributions to the diffeomorphism and scalar constraints. \n-2 c √ p δK i a δE a i + 1 p 3 / 2 Z cidj ab ( ∂ c δE a i )( ∂ d δE b j ) . (2.32) \nIn the previous expressions, the quantities \nδK b b def = δ a i δK i a , δE b b def = δ i a δE a i , (2.33)', '2.4.1 Diffeomorphism constraints': 'Similarly to geometric phase-space variables, matter variables also transform under spatial diffeomorphisms. The invariance of the theory under those transformations is, in the canonical framework, ensured by the constraints. For a scalar field, the spatial diffeomorphism constraints are unique and expressed by: \nD m def = D m [ N a ] = ∫ d x N a D m a = ∫ d x N a ( π φ ∂ a φ ) . (2.38) \nAt the background level the shift functions N a vanish due to homogeneity and the perturbative expansion at second order of the matter diffeomorphism constraint is: \nD m = ∫ d x δN a D m (1) a , (2.39) \nwhere \nD m (1) a = π ∂ a ( δφ ) . (2.40) \nThis constraint does not depend on the reduced curvature c , so, independently of the way the diffeomorphism constraint is treated in LQG, no correction is applied to D m .', '2.4.2 Hamiltonian constraint': 'The introduction of a scalar field leads to a modification of the Hamiltonian constraint. The matter contribution is given by [8]: \nH m def = H m [ N ] = ∫ d x N H m = ∫ d x N π 2 φ 2 √ ∣ ∣ det E ∣ ∣ + E a i E b j 2 √ ∣ ∣ det E ∣ ∣ δ ij ∂ a φ∂ b φ + √ ∣ ∣ det E ∣ ∣ V ( φ ) , (2.41) \nwhere H m corresponds to the matter Hamiltonian (or scalar) constraint density. Following [8], we write this density as: \nH m = H m ,π + H m , ∇ + H m ,φ , (2.42) \nwhere each term is respectively defined by: \nH m ,π def = π 2 φ 2 √ ∣ ∣ det E ∣ ∣ , H m , ∇ def = E a i E b j 2 √ ∣ ∣ det E ∣ ∣ δ ij ∂ a φ∂ b φ, and H m ,φ def = √ ∣ ∣ det E ∣ ∣ V ( φ ) . (2.43) \nIs it important to stress that this constraint does not depend on any curvature-related quantity 5 . The matter sector therefore does not require any holonomy correction. Determinants of the inverse densitized triad are however present and inverse-volume corrections are to be expected. More precisely, H m ,π and H m , ∇ should be inverse-volume corrected, \nbut not H m ,φ . We, however, do not consider such corrections in this work. The derivation of consistency conditions with both holonomy and inverse-volume corrections has been adressed in [67] and the generalization of this work to GHCs is left for a future study. \nThe perturbative expansion of the matter Hamiltonian density H m can be readily obtained from Eq. (2.43) [8]. Specifically, at the background level, the perturbative expansions are: \nH (0) m ,π = π 2 2 p 3 / 2 , H (0) m , ∇ = 0 , and H (0) m ,φ = p 3 / 2 V [ φ ] . (2.44) \nThe first order expansions read, \nH (1) m ,π = π p 3 / 2 δπ -π 2 4 p 5 / 2 δE b b , H (1) m , ∇ = 0 , and H (1) m ,φ = p 3 / 2 ( ∂ φ ( V [ φ ] ) δφ + V [ φ ] 2 p δE b b ) , (2.45) \nand the second order expansion is, \nH (2) m ,π = 1 2 p -3 / 2 [ δπ 2 -π p δπδE b b + π 2 4 p 2 ( 1 2 ( δE b b ) 2 + δE a b δE b a )] , (2.46) \nH (2) m , ∇ = √ p 2 δ ab ∂ a ( δφ ) ∂ b ( δφ ) , (2.47) \nH (2) m ,φ = p 3 / 2 [ 1 2 ∂ 2 φ ( V [ φ ] ) δφ 2 + 1 2 p ∂ φ ( V [ φ ] ) δφδE b b + V [ φ ] 4 p 2 ( 1 2 ( δE b b ) 2 -δE a b δE b a )] . (2.48) \nThe total matter contribution to the Hamiltonian constraint perturbed at second order is given by: \nH m = ( 2 κ ) -1 ∫ d x ( N [ H (0) m + H (2) m ] + δN H (1) m ) . (2.49) \nIn perturbation theory, and as long as the background equations of motion are satisfied, the first-order expansion of the (matter) Hamiltonian H (1) m vanishes [63], similarly to the gravitational sector.', '2.5 Algebra of constraints': "The algebra of constraints encodes the deep structure of the theory. In pure GR, it reads [57]: \n{ D [ M a ] , D [ N b ] } = D [ L N b M a ] , (2.50) { H [ M ] , D [ N a ] } = H [ L N a M ] , (2.51) (2.52) \nwhere h ab is the (spatial) 3-metric, L X f denotes the Lie derivative with respect to the X vector field, D def = D g + D m , and H def = H g + H m . Holonomy-type corrections modify drastically \n{ H [ M ] , H [ N ] } = D [ h ab ( M ∇ b N -N ∇ b M ) ] , \nthe algebra of constraints and, consequently, the associated space-time structure [56]. Once effective (quantum) corrections have been applied, the Poisson brackets can be written as: \n{ C I , C J } = f IJ K C K + A { C I , C J } , (2.53) \nwhere A { C I , C J } are anomalous terms 'spoiling' the space-time structure and making the theory apparently inconsistent. To restore the consistency of the model, it is mandatory to close the algebra [56], i.e. to enforce A { C I , C J } = 0. To this end, two distinct mathematical procedures have been so far considered. We study them in details in the following. \nRecently, it has been elegantly suggested to take advantage of the a priori freedom one has in the choice of the shape of the GHC to ensure the closure of the algebra [68]. In practice, this involves deriving each A { C I , C J } and identifying a subset of GHCs that lead to A { C I , C J } = 0. In the context of cosmological perturbation theory in LQC, it was shown in [68] that the algebra can be closed for vector modes thanks to specific choices for the GHCs. This nice result is extended to scalar modes in Section (4.2). We show that the procedure unfortunately fails and that another approach is required. \nHistorically introduced in [8] for LQC, and then considered extensively (see, e.g., [24, 60, 61, 64, 65, 67, 69-71]), another procedure to ensure the closure of the algebra of constraints involves the addition of counter-terms C I ct such that C I → C I + C I ct . One then has to find a peculiar C I ct , i.e , a peculiar deformation of the constraints, to ensure that A { C I , C J } = 0. In this article, we parameterize the deformations following [24, 67]: \nH g / m → H g / m + H ct g / m , (2.54) \nwhere the additional terms are \nH ct g / m = ( 2 κ ) -1 ∫ d x ( δN H (1) ct g / m + N H (2) ct g / m ) . (2.55) \nIn the previous expression, the densities of the gravitational sector are defined as \nH (1) ct g = -4 α 1 √ p δK b b -α 2 c 2 √ p δE b b + 2 √ p α 3 ∂ a ∂ i δE a i , (2.56) \nat the first order and, \nH (2) ct g = √ p [ α 4 ( δK a b δK b a ) -α 5 ( δK b b ) 2 ] -1 2 c 2 p 3 / 2 [ α 7 ( δE a b δE b a ) -1 2 α 8 ( δE b b ) 2 ] -2 c √ p α 6 ( δK i a δE a i ) + α 9 p 3 / 2 Z cidj ab ( ∂ c δE a i )( ∂ d δE b j ) , (2.57) \nat the second order. The coefficients α i def = α i ( c , p ) appearing in the previous expressions are the so-called gravitational counter-terms. They are introduced to anticipate the appearance of anomalies that will, this way, be cancelled. They are required to have the \ncorrect classical limit and are assumed to always be factorized with terms already present in the original constraint. The variables upon which they depend is relevant since this will determine their contributions, or not, to the Poisson bracket defined on each variables of the phase-space. In [65, 67], and subsequent works, it was chosen to restrict the dependence of the counter-terms to the geometrical background phase-space variables only. We follow here this reasonable assumption. It is however important to underline that this restriction is mostly motivated by simplicity, without strong justification. This is another hypothesis that could be relaxed in future works. \nIn the matter sector, the deformation of the densities are given, at first order in perturbations, by \nH (1) ct m = β 1 π p 3 / 2 δπ -β 2 π 2 4 p 5 / 2 δE b b + p 3 / 2 ( β 3 ∂ φ ( V [ φ ] ) δφ + β 4 V [ φ ] 2 p δE b b ) , (2.58) \nand, at the second order, by \nH (2) ct m = 1 2 p -3 / 2 [ β 5 δπ 2 -β 6 π p δπδE b b + π 2 4 p 2 ( β 7 2 ( δE b b ) 2 + β 8 δE a b δE b a )] + β 9 √ p 2 δ ab ∂ a ( δφ ) ∂ b ( δφ ) + p 3 / 2 [ β 10 2 ∂ 2 φ ( V [ φ ] ) δφ 2 + β 11 2 p ∂ φ ( V [ φ ] ) δφδE b b + V [ φ ] 4 p 2 ( β 12 2 ( δE b b ) 2 -β 13 ( δE a b δE b a ) )] . (2.59) \nAs for the gravitational sector, we have defined β i def = β i ( c , p ) as functions of the geometrical background variables only.", '3.1 Bracket { G , G }': 'In the preceding sections, we have stated that the Gauss constraint undergoes no quantum corrections in LQG. Consequently, at the effective level, no anomaly is to be expected from the Poisson bracket { G , G } . Still, for the completeness of the generic aspect of this article we explicitly calculate this Poisson bracket step by step: \n{ G , G } = { G , G } c , p + { G , G } δE,δK + { G , G } φ , π + { G , G } δφ,δπ . (3.1) \nSince the perturbed Gauss constraint does not depend on matter variables, see Eq. (2.23), the Poisson brackets from the matter sector vanish identically: \n{ G , G } φ , π = 0 and { G , G } δφ,δπ = 0 . (3.2) \nThus, only Poisson brackets from the geometrical sector need to be calculated. Considering the background part, it is clear from the construction of the second-order Gauss constraint \nEq. (2.23) that derivatives of this constraint with respect to background variables would lead to second-order terms in the perturbative development. Therefore, the geometrical background part of the Poisson bracket is of order four in perturbations: \n{ G , G } c , p = o ( δδ ) , (3.3) \nwhere δδ denotes a product of first -order perturbations. As the closure of the algebra is studied at the second order, those fourth-order terms are discarded. \nFinally, the Poisson bracket of the geometrical sector at the level of the perturbed phase space must also be computed: \n{ G [Λ i ] , G [Γ j ] } δE,δK = 2 κ -1 ∫ d x [ cp ( δ ij δ Λ i δ Γ j -δ ij δ Λ i δ Γ j )] = 0 . (3.4) \nTherefore, as anticipated: \n{ G , G } = 0 . (3.5) \nHence, no anomaly emerges from the bracket between two Gauss constraints.', '3.2 Bracket { D , D }': 'Arguments similar to those established for the { G , G } bracket provide insights to the final outcome: aforementioned studies mentioned in Sec. (2.3.2) put aside, diffeomorphism constraints, both in the matter or geometric sectors, remain unaltered by holonomy corrections. Consequently, at this effective level, the classical result should be recovered, and no anomalies should arise from the Poisson bracket { D , D } . Once again, for the sake of completeness, the computation is performed in details below. \nTo begin, it is interesting to notice that the geometric diffeomorphism constraint is independent of the matter sector, and the matter diffeomorphism constraint is independent of the geometric sector 6 . This means that, in full generality, \n{ D g , D m } = 0 . (3.6) \nThe brackets of the geometrical diffeomorphism constraints also vanish: \n{ D g , D g } φ , π = 0 and { D g , D g } δφ,δπ = 0 , (3.7) \nalong with the brackets of the matter constraints: \n{ D m , D m } c , p = 0 and { D m , D m } δE,δK = 0 . (3.8) \nMoreover, in the matter sector, a comparison between the ADM and FLRW line elements presented Eqs. (2.1) and (2.2) shows that the shift functions N a vanish. Consequently, the perturbed matter diffeomorphism constraint is proportional to δN a only. The density is therefore perturbed only at the first-order and is never proportional to either the full background canonical variables φ and π or to the full perturbed canonical variables δφ and δπ . It can therefore be concluded that \n{ D m , D m } φ , π = 0 and { D m , D m } δφ,δπ = 0 . (3.9) \nFrom the perturbative construction of the (geometrical) diffeomorphism constraint at the second order, Eq. (2.26), it can be seen that: \n{ D g , D g } c , p = o ( δδ ) , (3.10) \nleading to the same conclusion than previously. As for the perturbed geometrical aspect of the phase-space, the Poisson bracket is expressed as \n{ D g [ N a 1 ] , D g [ N b 2 ] } δE,δK = κ -1 ∫ d x [ cp ( ∂ a N b 2 ∂ b N a 1 + ∂ a N a 2 ∂ b N b 1 (3.11) -∂ a N b 2 ∂ b N a 1 -∂ a N a 2 ∂ b N b 1 )] = 0 . \nThus, as anticipated, \n{ D , D } = 0 . (3.12) \nNo anomaly emerges from the bracket between two diffeomorphism constraints.', '3.3 Bracket { D , G }': 'Since the Gauss constraint G depends neither on the scalar field φ nor on its momentum π , it can immediately be concluded, for the geometrical component of the diffeomorphism constraint, that \n{ D g , G } φ , π = 0 and { D g , G } δφ,δπ = 0 , (3.13) \nand, similarly, in the matter counterpart, that, \n{ D m , G } φ , π = 0 and { D m , G } δφ,δπ = 0 . (3.14) \nFor the same reasons than for the { D , D } bracket: \n{ D m , G } c , p = 0 and { D m , G } δE,δK = 0 . (3.15) \nAnd, as for { G , G } and { D , D } , a careful examination of the second-order perturbative expansion of D g and G reveals that: \n{ D g , G } c , p = o ( δδ ) . (3.16) \nThe last Poisson bracket to evaluate can be expressed as: \n{ D g [ N a ] , G [Λ i ] } δE,δK = κ -1 ∫ d x [ -cp ( /epsilon1 a ib δ Λ i ∂ a δN b -/epsilon1 a ib δ Λ i ∂ a δN b -/epsilon1 i ji δ Λ j ∂ a δN a )] = 0 , (3.17) \nand is found to be identically zero. \nIn summary, as anticipated, the Poisson bracket { D , G } yields \n{ D , G } = 0 . (3.18) \nNo anomaly arises here.', '3.4 Bracket { H , G }': 'The most interesting brackets obviously involve the Hamiltonian constraint H , which geometrical component H g receives, in principle, both holonomy and inverse-volume corrections while the matter part H m - which does not depend upon the symmetry-reduced curvature c - undergoes only inverse-volume modifications. This work still focuses only on the holonomy corrections. \nAs before, some elementary remarks allow to reduce the number of Poisson brackets to be calculated. In particular, the Gauss constraint is independent of the matter phase-space, both at the background and at the perturbations levels. Therefore, for the geometrical part of the constraint, \n{ H g , G } φ , π = 0 and { H g , G } δφ,δπ = 0 , (3.19) \nand, for the matter component, \n{ H m , G } φ , π = 0 and { H m , G } δφ,δπ = 0 . (3.20) \nFurthermore, derivatives of the Gauss constraint with respect to background quantities lead to terms perturbed at the second order. Hence, Poisson brackets related to the background phase space are reduced to Poisson brackets between the background Hamiltonian constraint and the Gauss constraint, that is \n{ H , G } c , p = { H (0) , G } c , p + o ( δδ ) . (3.21)', '3.4.1 Bracket { H g , G }': 'Thanks to the above simplifications, the Poisson bracket on the background part of the phase space can be computed quite straightforwardly: \n{ H (0) g [ N ] , G [Λ i ] } c , p = κ -1 ∫ d xN /epsilon1 j ia δ Λ i δE a j ( g 2 2 √ p +2 √ p g∂ p g ) (3.22a) \n+ κ -1 ∫ d xN /epsilon1 a ij δ Λ i δK j a ( -2 √ p g∂ c g ) . (3.22b) \nThis is clearly not related to a constraint. We shall come back later to the way this anomaly can be treated but we proceed, for now, with the calculation of the brackets. The contribution of the perturbed phase space to the full bracket is given by \n{ H g [ N ] , G [Λ i ] } δE,δK = κ -1 ∫ d xN /epsilon1 j ai δ Λ i δE a j ( c √ p [ ˜ g + α 6 ] -1 √ p [ ˜ g 2 + α 7 ] ) (3.23a) + κ -1 ∫ d xN /epsilon1 a ji δ Λ i δK j a ( -c √ p [ 1 + α 4 ] -√ p [ ˜ g + α 6 ] ) , (3.23b) \nwhich completes the computation of { H g , G } . Gathering everything together leads to: \n{ H g [ N ] , G [Λ i ] } = κ -1 ∫ d x N 2 √ p /epsilon1 j ia δE a j δ Λ i A { H , G } 1 + κ -1 ∫ d x N √ p/epsilon1 a ij δK j a δ Λ i A { H , G } 2 , (3.24) \nwhere the anomalies are given by \nand \nA { H , G } 1 = g 2 +2 c [ ˜ g + α 6 ] -[ ˜ g 2 + α 7 ] +4 p g∂ p g, (3.25) \nA { H , G } 2 = c [ 1 + α 4 ] + [ ˜ g + α 6 ] -2 g∂ c g. (3.26)', '3.4.2 Bracket { H m , G }': 'Due to the presence of counter-terms in H m , see Eqs. (2.58) and (2.59), the geometrical parts of the bracket { H m , G } might not vanish and need to be calculated, even though the constraint undergoes no holonomy corrections. \nThe bracket for the geometrical background phase space canonical variables reads: \n{ H (0) m [ N ] , G [Λ] } c , p = ∫ d xN /epsilon1 j ia δ Λ i δE a j ( π 2 4 p 5 / 2 -√ p 2 V ( φ ) ) , (3.27) \nwhile it is, at the geometrical perturbed level, \n{ H m [ N ] , G [Λ i ] } δE,δK = ∫ d xN /epsilon1 j ia δ Λ i δE a j ( -π 2 4 p 5 / 2 + √ p 2 V ( φ ) ) (3.28a) + ∫ d x N π 2 4 p 5 / 2 /epsilon1 j ia δ Λ i δE a j A { H , G } 3 (3.28b) \n+ ∫ d x N √ p 2 V ( φ ) /epsilon1 j ia δ Λ i δE a j A { H , G } 4 . (3.28c) \nThe two new anomalies are: \nA { H , G } 3 = -β 10 and A { H , G } 4 = β 12 . (3.29) \nHence, the bracket { H , G } leads to four new anomalies A { H , G } 1 , ... , A { H , G } 4 .', '3.5 Bracket { H , D }': 'Thanks to the linearity of the Poisson brackets and using the definitions of the total Hamiltonian and diffeomorphism constraints, H and D , the full { H , D } bracket can be splitted as: \n{ H , D } = { H g , D g } + { H m , D m } + { H g , D m } + { H m , D g } . (3.30) \nThis allows some straightforward simplifications. In particular, Eqs. (2.28) and (2.38) show that the geometrical part of the Hamiltonian constraint, that is H g , depends only on the geometrical phase space whereas the matter part of the diffeomorphism constraint, that is D m , depends only on the variables related to the scalar field. Hence, \n{ H g , D m } = 0 , (3.31) \nand \n{ H m , D m } c , p = 0 , together with { H m , D m } δE,δK = 0 . (3.32) \nIn addition, as D g neither depends on the canonical variables of the scalar field, one obtains, for the geometrical sector, \n{ H g , D g } φ , π = 0 , and { H g , D g } δφ,δπ = 0 (3.33) \nand, for the matter sector, \n{ H m , D g } φ , π = 0 and { H m , D g } δφ,δπ = 0 . (3.34) \nAs a final simplification at this stage, it can also be remarked that \n{ H , D } c , p = { H (0) , D } c , p + o ( δδ ) . (3.35)', '3.5.1 Bracket { H g , D g }': 'A simple computation yields: \n{ H (0) g [ N ] , D g [ N a ] } c , p = κ -1 ∫ d xN δN a ∂ b ( δ i a δE b i ) ( -g 2 2 √ p -2 √ p g∂ p g ) (3.36a) + κ -1 ∫ d xN δN a ∂ b ( δ b i δK i a ) ( 2 √ p g∂ c g ) (3.36b) \nwhere the anomalies are: \n+ κ -1 ∫ d xN δN a ∂ a ( δK b b ) ( -2 √ p g∂ c g ) , (3.36c) \nand, for the geometrical perturbations, \n{ H g [ N ] , D g [ N a ] } δE,δK = κ -1 ∫ d x δN∂ a δN a ( -√ p [ ˜ g 2 + α 2 ] -2 √ pc [ ˜ g + α 1 ] ) (3.37a) \n+ κ -1 ∫ d xN √ p ∂ a ( δN a ) δK b b ( -c [ 1 + α 5 ] + [ ˜ g + α 6 ] ) (3.37c) \n+ κ -1 ∫ d xN √ p ∂ a ( δN b ) δ a i δK i b ( c [ 1 + α 4 ] + [ ˜ g + α 6 ] ) (3.37b) \n+ κ -1 ∫ d x N 2 √ p ∂ a ( δN b ) δ i b δE a i ( [ ˜ g 2 + α 7 ] -2 c [ ˜ g + α 6 ] ) (3.37d) \nIt is important to notice that, in the above reduced brackets of the geometrical components of the constraints, the (corrected) geometrical Hamiltonian constraint is not recovered. In other words, because of the holonomy correction, the somehow expected structure is not ensured at the perturbative level. However, if loop corrections are applied only at the background level (as in [24]), the classical result is recovered. \n+ κ -1 ∫ d x N 2 √ p ∂ a ( δN a ) δE b b ( α 8 -α 7 ) . (3.37e) \nWhen summing up everything and after integrating by parts, one is led to: \n{ H g [ N ] , D g [ N a ] } = H g [ δN∂ a δN a ] (3.38a) \n+ κ -1 ∫ d x N 2 √ p ∂ a ( δN a ) δE b b A { H , D } 2 (3.38c) \n+ κ -1 ∫ d x √ p δN∂ a ( δN a ) A { H , D } 1 (3.38b) \n+ κ -1 ∫ d xN √ p ∂ a ( δN a ) δK b b A { H , D } 3 (3.38d) \n+ κ -1 ∫ d xN √ p ∂ a ( δN b ) δ a i δK i b A { H , D } 5 , (3.38f) \nA { H , D } 1 = 3 g 2 -[ ˜ g 2 + α 2 ] -2 c [ ˜ g + α 1 ] , (3.39) A { H , D } 2 = α 8 -α 7 , (3.40) \n[ ] [ ] A { H , D } 5 = c [ 1 + α 4 ] + [ ˜ g + α 6 ] -2 g∂ c g. (3.43) \nA { H , D } 3 = c [ 1 + α 5 ] + [ ˜ g + α 6 ] -2 g∂ c g, (3.41) A { H , D } 4 = g 2 -2 c ˜ g + α 6 + ˜ g 2 + α 7 +4 p g∂ p g, (3.42) \n+ κ -1 ∫ d x N 2 √ p ∂ a ( δN b ) δ i b δE a i A { H , D } 4 (3.38e) \nIt can immediately be noticed that the anomalies A { H , G } 1 (3.25) and A { H , G } 2 (3.26) coming from the { H g , G } bracket are, respectively, equal to A { H , D } 4 (3.42) and A { H , G } 5 (3.43). In other words, the bracket { H g , G } is redundant with { H g , D g } from the viewpoint of anomaly cancellation. This is due, on the one hand, to the structural similarities of the geometrical diffeomorphism constraint D g and Gauss constraint G at the perturbative level, and, on the other hand, to the fact that none of those constraints receives loop corrections. Should the diffeomorphism constraint be corrected, this conclusion might not hold anymore.', '3.5.2 Bracket { H m , D m }': 'As discussed previously, only Poisson brackets of the scalar field part of the phase space need to be computed to get the full result for { H m , D m } . This greatly simplifies the analysis. Let us start with the background pair of canonical variables ( φ , π ): \n{ H (0) m [ N ] , D m [ N a ] } φ , π = ∫ d xN p 3 / 2 ∂ φ ( V [ φ ] ) δN a ∂ a ( δφ ) . (3.44) \nAt the level of the perturbed phase space, one gets: \n{ H m [ N ] , D m [ N a ] } δφ,δπ = ∫ d x π 2 p 3 / 2 δN∂ a ( δN a ) ( 1 + β 1 ) (3.45a) \nd x N π p 3 / 2 ∂ a δN a δπ 1 + β 5 (3.45b) \n+ ∫ d x N π 2 2 p 5 / 2 ∂ a ( δN a ) δE b b ( -1 -β 6 ) . (3.45c) \n+ ∫ ( ) ( ) \nAt this point, no obvious conclusion can be reached as it not clear whether this term is anomalous or not. In addition, contrary to what could have been naively expected, the scalar matter constraint is not fully recovered from this bracket. However, the situation is different from the one of the bracket { H g , D g } : no specific procedure is here necessary to exhibit the scalar matter constraint as it naturally appears as an interplay between the geometrical diffeomorphism constraint and the scalar matter constraint in the computation of { H m , D g } .', '3.5.3 Bracket { H m , D g }': 'This is the last step in the calculation of the full { H , D } bracket. As explained earlier, it is not necessary to compute the Poisson brackets on the matter phase space. Still, the brackets on the geometrical part of the phase space have to be calculated. Starting with the background pair ( c , p ) of canonical variables: \n+ ∫ d x N √ p 2 V [ φ ] δN a ∂ b ( δ i a δE b i ) , (3.47) \n{ H (0) m [ N ] , D g [ N a ] } c , p = -∫ d x N π 2 4 p 5 / 2 δN a ∂ b ( δ i a δE b i ) (3.46) \nwhile, at the perturbed level, one gets \n{ H m [ N ] , D g [ N a ] } δE,δK = ∫ d x π 2 2 p 3 / 2 δN∂ a ( δN a ) ( -1 -β 2 ) (3.48a) \n+ ∫ d x N π 2 4 p 5 / 2 ∂ a ( δN b ) δ i b δE a i ( -1 -β 10 ) (3.48c) \n+ ∫ d x N π p 3 / 2 ∂ a ( δN a ) δπ ( -1 -β 6 ) (3.48b) \n+ ∫ d x N √ p 2 V [ φ ] ∂ a ( δN b ) δ i b δE a i ( 1 + β 12 ) (3.48d) \n+ ∫ d x N √ p 2 V [ φ ] ∂ a ( δN a ) δE b b ( β 11 -β 12 ) (3.48f) \n+ ∫ d x N π 2 4 p 5 / 2 ∂ a ( δN a ) δE b b ( 2 + β 10 + β 9 ) (3.48e) \n+ ∫ d xN p 3 / 2 ∂ φ ( V [ φ ] ) ∂ a ( δN a ) δφ ( 1 + β 8 ) (3.48g) \nIt should be noticed that, combining Eqs. (3.45a) and (3.48a), the kinetic term of the matter scalar constraint, Eq. (2.44), is obtained, modulo an anomaly, while the potential part of the constraint is given, modulo β 4 , by Eq. (3.48h). In other words, the expected classical result for the matter sector is recovered via the bracket { H m , D g + D m } . This is a key difference with the geometrical sector where a specific procedure was necessary. \n+ ∫ d x p 3 / 2 V [ φ ] δN∂ a ( δN a ) ( 1 + β 4 ) . (3.48h) \nIt can now be concluded: \n{ H m [ N ] , D g [ N a ] + D m [ N a ] } = H m [ δN∂ a δN a ] (3.49a) + ∫ d x N π p 3 / 2 ∂ a ( δN a ) δπ A { H , D } 6 (3.49b) + ∫ d x π 2 2 p 3 / 2 δN∂ a ( δN a ) A { H , D } 7 (3.49c) + ∫ d x p 3 / 2 V [ φ ] δN∂ a ( δN a ) A { H , D } 8 (3.49d) + ∫ d x N π 2 4 p 5 / 2 ∂ a ( δN b ) δ i b δE a i A { H , D } 9 (3.49e) + ∫ d x N √ p 2 V [ φ ] ∂ a ( δN b ) δ i b δE a i A { H , D } 10 (3.49f) + ∫ d x N π 4 p 5 / 2 ∂ a ( δN a ) δE b b A { H , D } 11 (3.49g) + ∫ d x N √ p 2 ∂ φ ( V [ φ ] ) ∂ a ( δN a ) δφ A { H , D } 12 (3.49h) √ (3.49i) \n+ ∫ d x N p 2 V [ φ ] ∂ a ( δN a ) δE b b A { H , D } 13 . \nThe anomalies are defined by: \nA { H , D } 6 = β 5 -β 6 , (3.50) \nA { H , D } 8 = β 4 , (3.52) \nA { H , D } 7 = 2 β 1 -β 2 , (3.51) \nA { H , D } 9 = -β 10 , (3.53) \nA { H , D } 11 = β 10 -2 β 6 + β 9 , (3.55) \nA { H , D } 10 = β 12 , (3.54) \nA { H , D } 12 = β 8 , (3.56) \nA { H , D } 13 = β 11 -β 12 . (3.57) \nAs for the geometrical part H g of the Hamiltonian constraint H , it should be noticed that A { H , G } 3 (3.29) and A { H , G } 4 (3.29) are, respectively, equal to A { H , D } 9 (3.53) and A { H , D } 10 (3.54). Hence, there are only six new anomalies, leading to a total amount of nine anomalies arising from the bracket { H , D } . It is interesting to underline that, until now, all the anomalies are either depending on the counter-terms from the geometrical sector or on the counter-terms from the matter sector. This obviously simplifies the closure of the algebra (sometimes trivially by setting counter-terms to zero, which means that the bracket would have naturally been anomaly-free). This property is however lost when considering the anomalies appearing in the bracket { H , H } , as we will see right now.', '3.6 Bracket { H , H }': 'Let us now proceed with the calculation of the most interesting (and computationally intricate) bracket. As we shall see, all the previously considered brackets ultimately reduce to the algebra of constraints of GR. However, the { H , H } bracket introduces a fundamental deformation of the algebra, which could significantly change cosmological predictions [25, 26, 36, 65, 66]. Using the linearity of the brackets and the definition of the constraints, one straightforwardly gets 7 : \n{ H [ N 1 ] , H [ N 2 ] } = { H g [ N 1 ] , H g [ N 2 ] } + { H m [ N 1 ] , H m [ N 2 ] } + [{ H g [ N 1 ] , H m [ N 2 ] } -( N 1 ↔ N 2 ) ] . (3.58) \nFor readability purposes, we firstly define: \nand \n∆ { H g , H m } def = [{ H g [ N 1 ] , H m [ N 2 ] } -( N 1 ↔ N 2 ) ] , (3.59) \n∆( δN ) def = δN 2 -δN 1 . (3.60) \nAs before, the fact that H g (Eq. 2.28) does not depend upon the matter sector leads to: \n{ H g , H } φ , π = 0 and { H g , H } δφ,δπ = 0 . (3.61) \nConsidering the expression of H m (Eq. 2.41), it is clear that the matter sector is basically not depending on the curvature. However, due to the introduction of counter-terms, H m becomes implicitly a function of c . On the other hand, H m is not related to the perturbative expansion of the curvature δK . It can therefore be concluded that: \n{ H m , H m } δE,δK = 0 . (3.62) \nThese are the main simplifications that can be made a priori for { H , H } . All the other sub-brackets have to be explicitly calculated.', '3.6.1 Bracket { H g , H g }': 'Let us start with the gravitational part of the Hamiltonian (scalar) constraint H g . For now, we shall focus on the (geometrical) background part of the Poisson bracket: \n{ H g [ N 1 ] , H g [ N 2 ] } c , p = κ -1 ∫ d xN ∆( δN ) δK b b ( -g 2 ∂ c ( ˜ g + α 1 ) +2 g [ [˜ g + α 1 ] ∂ p g +2 p ( ∂ p ( ˜ g + α 1 ) ∂ c g -∂ p g ∂ c ( ˜ g + α 1 ) )]) (3.63a) + κ -1 ∫ d x N 2 p ∆( δN ) δE b b ( -g [ ˜ g 2 + α 2 ] ∂ c g -g 2 ˜ g∂ c ( ˜ g + α 2 ) -2 g p [ ∂ p α 2 ∂ c g + ∂ p g ∂ c α 2 +2˜ g [ ∂ p g∂ c ˜ g -∂ p ˜ g∂ c g ] ]) (3.63b) + κ -1 ∫ d x N 2 p ∂ a ( ∆( δN ) ) ∂ b ( δ i a δE b i ) ( -g 2 ∂ c α 3 -2 g [ (1 + α 3 -2 p ∂ p α 3 ) ∂ c g +2 p ∂ p g ∂ c α 3 ]) . (3.63c) \nDue to the application of the holonomy correction to the perturbative expansion of H g , the resulting expressions are more complicated than those of [24], mostly because of the non-vanishing derivative of ˜ g . \nThe (geometrical) perturbative phase space bracket reads: \n{ H g [ N 1 ] , H g [ N 2 ] } δE,δK = κ -1 ∫ d x N 2 ∆( δN ) δK b b ( 2 α 2 -α 2 α 4 +3 α 2 α 5 -˜ g 2 [ 2 + α 4 -3 α 5 ] -4 α 1 α 6 -4˜ g [ α 1 + α 6 ] ) (3.64a) + κ -1 ∫ d x N 2 p ∆( δN ) δE b b ( 2˜ g 3 + α 2 α 6 + ˜ g 2 [ α 1 + α 6 ] \n-2 α 1 α 7 +3 α 1 α 8 + ˜ g [ α 2 -2 α 7 +3 α 8 ] ) (3.64b) \n+ κ -1 ∫ d xN ∂ b ( ∆( δN ) ) ∂ a ( δ a i δK i b ) ( -[ 1 + α 3 ][ 1 + α 4 ] ) (3.64d) \n+ κ -1 ∫ d xN ∂ a ( ∆( δN ) ) ∂ a ( δK b b ) ( [ 1 + α 3 ][ 1 + α 5 ] ) (3.64c) \n+ κ -1 ∫ d x N p ∂ b ( ∆( δN ) ) ∂ a ( δ i b δE a i ) ( [ 1 + α 3 ][ ˜ g + α 6 ] ) . (3.64e) \nIn the terms (3.64c), (3.64d), and (3.64e), one recognizes the diffeomorphism constraint, Eq. (2.26), corrected by the GHC ˜ g , modulo an additional deformation due to the counterterms. This comes from the correction applied to the perturbative expansion of the (geometrical) scalar constraint H g . As for { H , D } , this must be accounted for by an anomalous term. Note that, however, such manipulation would not be needed if the (geometrical) diffeomorphism constraint was holonomy-corrected. Thus, we write: \n{ H g [ N 1 ] , H g [ N 2 ] } = [ 1 + α 3 ][ 1 + α 5 ] D g [ N p ∂ a ( ∆( δN ) ) ] (3.65a) + κ -1 ∫ d xN ∆( δN ) δK b b A { H , H } 1 (3.65b) + κ -1 ∫ d x N 2 p ∆( δN ) δE b b A { H , H } 2 (3.65c) + κ -1 ∫ d xN ∂ b ( ∆( δN ) ) ∂ a ( δ a i δK i b ) A { H , H } 3 (3.65d) + (3.65e) \nwhere the first anomalies from the { H , H } bracket are defined as \nκ -1 ∫ d x N p ∂ b ( ∆( δN ) ) ∂ a ( δ i b δE a i ) A { H , H } 4 , \nA { H , H } 1 = 2 [ ˜ g + α 1 ] ( g∂ c g -˜ g -α 6 ) -( g 2 +4 p g∂ p g ) ∂ c ( ˜ g + α 1 ) +4 p g∂ c g∂ p ( ˜ g + α 1 ) + 1 2 [ ˜ g 2 + α 2 ] ( 2 + 3 α 5 -α 4 ) , (3.66) A { H , H } 2 = [ ˜ g 2 + α 2 ] ( ˜ g + α 6 -g∂ c g ) -1 2 ( g 2 +4 p g∂ p g ) ∂ c ( ˜ g 2 + α 2 ) +2 p g∂ c g∂ p ( ˜ g 2 + α 2 ) + [ ˜ g + α 1 ] ( ˜ g 2 +3 α 8 -2 α 7 ) , (3.67) A { H , H } 3 = [ 1 + α 3 ][ α 5 -α 4 ] , (3.68) A { H , H } 4 = [ 1 + α 3 ] ( g∂ c g + [ ˜ g + α 6 ] + k [ 1 + α 5 ] ) -[ ˜ g + α 1 ][ 1 + α 9 ] + +2 p g∂ c g∂ p α 3 + ( 1 2 g 2 +2 p ∂ p g ) ∂ c α 3 . (3.69) \nThose four new terms, A { H , H } 1 , ... , A { H , H } 4 bring the total number of anomalies to 13 at this stage of the derivation. However, some brackets still need to be computed.', '3.6.2 Bracket { H m , H m }': 'As discussed in the preceding sections, the (matter) scalar constraint H m classically does not depend on the extrinsic curvature K and all the Poisson brackets related to the geometrical phase space variables are therefore expected to vanish. However, the counter-terms are functions of the reduced curvature c . Consequently, H m is implicitly dependent on c as well. The Poisson bracket between matter scalar constraints reads: \n{ H m [ N 1 ] , H m [ N 2 ] } c , p = κ ∫ d x N π 2 p ( π 2 2 p 3 -V [ φ ] ) ∆( δN ) δπ A { H , H } 5 (3.70a) \n+ κ ∫ d x N π 4 16 p 5 ∆( δN ) δE b b A { H , H } 7 (3.70c) \n+ κ ∫ d x N 2 ∂ φ ( V [ φ ] ) ( π 2 2 p -p 2 V [ φ ] ) ∆( δN ) δφ A { H , H } 6 (3.70b) \n+ κ ∫ d x N p 4 V [ φ ] 2 ∆( δN ) δE b b A { H , H } 8 (3.70d) \n+ κ ∫ d x N π 2 8 p 2 V [ φ ] ∆( δN ) δE b b A { H , H } 9 , (3.70e) \nwhere the five anomalies are given by: \nA { H , H } 5 = ∂ c β 1 , (3.71) \nA { H , H } 7 = -∂ c β 2 , (3.73) \nA { H , H } 6 = ∂ c β 3 , (3.72) \nA { H , H } 8 = -∂ c β 4 , (3.74) \nAt this stage, there are 18 anomalies in the list. It can be noticed that, at the background level, only counter-terms related to H (1) do appear in the anomalies. This is due to the background lapse for N 1 and N 2 being equal, implying in turn that, \nA { H , H } 9 = ∂ c ( β 2 + β 4 ) . (3.75) \n{ H (2) m [ N 1 ] , H (0) m [ N 2 ] } x,p = { H (0) m [ N 1 ] , H (2) m [ N 2 ] } x,p , (3.76) \nwhere x and p are respectively the canonically conjugate background variables of either the geometrical or matter sectors. This obviously remains true for the background matter sector, whose bracket is given by: \n{ H m [ N 1 ] , H m [ N 2 ] } φ , π = ∫ d xN ∂ φ ( V [ φ ] ) ∆( δN ) δπ ( 1 + β 1 ) (3.77a) \n+ ∫ d xN π ∂ 2 φ ( V [ φ ] ) ∆( δN ) δφ ( -1 -β 3 ) . (3.77c) \n+ ∫ d x N π 2 p ∂ φ ( V [ φ ] ) ∆( δN ) δE b b ( -2 -β 2 -β 4 ) (3.77b) \nAt the level of perturbations, the Poisson bracket is: \n{ H m [ N 1 ] , H m [ N 2 ] } δφ,δπ = ∫ d x N π p ∆( δN ) ∂ a ∂ a ( δφ ) ( -[ 1 + β 1 ][ 1 + β 13 ] ) (3.78a) \n+ ∫ d xN ∂ φ ( V [ φ ] ) ∆( δN ) δπ ( -[ 1 + β 3 ][ 1 + β 5 ] ) (3.78b) \n+ ∫ d x N π 2 p ∂ φ ( V [ φ ] ) ∆( δN ) δE b b ( 2 + β 1 + β 3 + β 6 (3.78d) \n+ ∫ d xN π ∂ 2 φ ( V [ φ ] ) ∆( δN ) δφ ( [ 1 + β 1 ][ 1 + β 7 ] ) (3.78c) \n+ β 3 β 6 + β 8 + β 1 β 8 ) . (3.78e) \nAfter integrating by parts, Eq. (3.78a) can be shown to be the diffeomorphism constraint associated with the matter sector appearing in Eq. (2.39). The { H m , H m } bracket therefore reads: \n{ H m [ N 1 ] , H m [ N 2 ] } = [ 1 + β 1 ][ 1 + β 13 ] D m [ N p ∂ a ( ∆( δN ) ) ] (3.79a) + ∫ d xN ∂ φ ( V [ φ ] ) ∆( δN ) δπ A { H , H } 10 (3.79b) + ∫ d xN π ∂ 2 φ ( V [ φ ] ) ∆( δN ) δφ A { H , H } 11 (3.79c) + ∫ d x N π 2 p ∂ φ ( V [ φ ] ) ∆( δN ) δE b b A { H , H } 12 (3.79d) + { H m [ N 1 ] , H m [ N 2 ] } c , p , (3.79e) \ngiving rise to 3 new anomalies, \nA { H , H } 10 = β 1 -β 3 -β 5 -β 3 β 5 , (3.80) \nA { H , H } 12 = β 1 + β 3 + β 6 + β 3 β 6 + β 8 + β 1 β 8 -β 2 -β 4 , (3.82) \nA { H , H } 11 = β 1 + β 7 + β 1 β 7 -β 3 , (3.81) \nbringing the total amount to 21 terms.', '3.6.3 Bracket { H g , H m }': 'A single bracket remains to be calculated. Given the above computations of { H g , H g } and { H m , H m } , where the expected result is recovered - although deformed by a combination of counter-terms - we expect { H g , H m } to be purely anomalous. \nThe background part of the bracket is \n∆ { H g , H m } c , p = ∫ d x N π 2 2 p 2 ∆( δN ) δK b b ∂ c ( ˜ g + α 1 ) (3.83a) \n+ ∫ d x N 2 ( π 2 2 p 3 -V [ φ ] ) ∂ a ( ∆( δN ) ) ∂ b ( δ i a δE b i ) ∂ c α 3 (3.83c) \n+ ∫ d x N p 2 V [ φ ] ∆( δN ) δK b b ∂ c ( -˜ g -α 1 ) (3.83b) \n+ ∫ d x N π 2 p 2 ∆( δN ) δπ ( 2 g∂ c g ( 3 + 3 β 1 -2 p ∂ p β 1 ) + g 2 ∂ c β 1 +4 g p ∂ p g∂ c β 1 \n+ ∫ d xN ∂ φ ( V [ φ ] ) ∆( δN ) δφ ( -g∂ c g ( 3 p [ 1 + β 3 ] +2 p 2 ∂ p β 3 ) + p 2 g 2 ∂ c β 3 +2 p 2 g∂ p g∂ c β 3 ) , \n) (3.83d) + ∫ d x N π 2 8 p 3 ∆( δN ) δE b b ( 2˜ g∂ c ˜ g -∂ c α 2 + g 2 ∂ c β 2 +4 g p ∂ p g∂ c β 2 ) +2 g∂ c g ( 5 + 5 β 2 -2 p ∂ p β 2 ) (3.83e) + ∫ d x N 4 V [ φ ] ∆( δN ) δE b b ( -2 g∂ c g ( 1 + β 4 +2 p ∂ p β 4 ) + g 2 ∂ c β 4 + ∂ c α 2 +4 g p ∂ p g∂ c β 4 ) (3.83f) (3.83g) \nwhile it becomes for perturbations: \n∆ { H g , H m } δE,δK = ∫ d x N π 2 2 p 2 ∆( δN ) δK b b ( [ 1 + β 2 ]( 2 -α 4 +3 α 5 ) ) (3.84a) + ∫ d x N p 2 V [ φ ] ∆( δN ) δK b b ( [ 1 + β 4 ]( -2 + α 4 -3 α 5 ) ) (3.84b) + ∫ d x N π 2 p 2 ∆( δN ) δπ ( -6 [ ˜ g + α 1 ][ 1 + β 6 ] ) (3.84c) + ∫ d x N π 2 4 p 3 ∆( δN ) δE b b ( α 6 [ 1 + β 2 ] + α 1 ( 5 + 2 β 10 +3 β 9 ) + ˜ g ( 6 + 2 β 10 + β 2 +3 β 9 ) ) (3.84d) + ∫ d x N 4 V [ φ ] ∆( δN ) δE b b ( α 1 ( -3 β 11 +2 β 12 -1 ) + α 6 [ 1 + β 4 ] + ˜ g ( -3 β 11 +2 β 12 + β 4 ) ) (3.84e) (3.84f) \nThis leads to: \n+ ∫ d xN ∂ φ ( V [ φ ] ) ∆( δN ) δφ ( 3 p [ ˜ g + α 1 ] +3 p [ ˜ g + α 1 ] β 8 ) . \n∆ { H g , H m } = ∫ d x N π 2 4 p 2 ∆( δN ) δK b b A { H , H } 13 (3.85a) + ∫ d x N p 2 V [ φ ] ∆( δN ) δK b b A { H , H } 14 (3.85b) + ∫ d x N 2 ( π 2 2 p 3 -V [ φ ] ) ∂ a ( ∆( δN ) ) ∂ b ( δ i a δE b i ) A { H , H } 15 (3.85c) + d x N π 2 ∆( δN ) δπ { H , H } 16 (3.85d) \n∫ 2 p A \n+ ∫ d x N π 2 8 p 3 ∆( δN ) δE b b A { H , H } 17 (3.85e) \n+ ∫ d xN ∂ φ ( V [ φ ] ) ∆( δN ) δφ A { H , H } 19 , (3.85g) \n+ ∫ d x N 4 V [ φ ] ∆( δN ) δE b b A { H , H } 18 (3.85f) \nwhere seven new anomalies appear: \nA { H , H } 13 = [ 1 + β 2 ]( 2 -α 4 +3 α 5 ) -2 ∂ c ( ˜ g + α 1 ) , (3.86) A { H , H } 14 = [ 1 + β 4 ]( -2 + α 4 -3 α 5 ) +2 ∂ c ( ˜ g + α 1 ) (3.87) A { H , H } 15 = ∂ c α 3 (3.88) A { H , H } 16 = -6 [ ˜ g + α 1 ][ 1 + β 6 ] +2 g∂ c g ( 3 + 3 β 1 -2 p ∂ p β 1 ) + g 2 ∂ c β 1 +4 g p ∂ p g∂ c β 1 , (3.89) A { H , H } 17 = 12˜ g +10 α 1 +2 α 6 +4 [ ˜ g + α 1 ] β 10 +2 [ ˜ g + α 6 ] β 2 +6˜ gβ 9 -2˜ g∂ c ˜ g -2 g∂ c g ( 5 + 5 β 2 -2 p ∂ p β 2 ) -∂ c α 2 -g 2 ∂ c β 2 -4 g p ∂ p g∂ c β 2 , (3.90) A { H , H } 18 = 2 α 1 -2 α 6 +6 [ ˜ g + α 1 ] β 11 -4 [ ˜ g + α 1 ] β 12 -2 [ ˜ g + α 6 ] β 4 +2˜ g∂ c ˜ g -2 g∂ c g ( 1 + β 4 +2 p ∂ p β 4 ) + ∂ c α 2 + g 2 ∂ c β 4 +4 g p ∂ p g∂ c β 4 , (3.91) p \nA { H , H } 19 = 3 p [ ˜ g + α 1 ] +3 p [ ˜ g + α 1 ] β 8 -g∂ c g ( 3 p [ 1 + β 3 ] +2 p 2 ∂ p β 3 ) + 2 g 2 ∂ c β 3 +2 p 2 g∂ p g∂ c β 3 . (3.92) \nThe previous Poisson bracket and associated anomalies complete the calculation of the full algebra in the presence of generalized holonomy corrections, applied either to the background or to the perturbative expansion of the scalar constraint. The next step consists in cancelling those anomalies.', '4.1 Closure via a deformation of the constraints': 'The counter-terms must now be determined so that the anomalies do vanish. The procedure we follow, initiated in particular in [8, 64], is close to the one described recently in [24]. However, as the holonomy corrections are here applied not only on the background part of the scalar constraint, but also on the perturbative expansion, the calculation needs to be explicitely performed to investigate possible new closure conditions. \nSome anomalies are related to a single counter-term and can therefore be trivially cancelled with \nβ 4 = 0 , β 8 = 0 , β 10 = 0 , and β 12 = 0 , (4.1) from, respectively, A { H , D } 8 (3.52), A { H , D } 9 (3.53), A { H , D } 10 (3.54) and A { H , D } 12 (3.56). In a way, this just means that the associated counter-terms were not necessary from the beginning. Moreover, combining β 12 = 0 and A { H , D } 13 (3.57), one obtains: \nβ 11 = 0 . (4.2) \nSince β 4 = 0, it can be concluded from A { H , H } 14 = 0 (3.87) that, \n∂ c ( ˜ g + α 1 ) = 1 2 ( 2 -α 4 +3 α 5 ) , (4.3) \nwhich implies, due to A { H , H } 13 = 0 (3.86), \nβ 2 ( 2 -α 4 +3 α 5 ) = 0 . (4.4) \nAs we shall see later in this section, the cancellation of A { H , H } 3 (3.68) requires α 4 = α 5 . If β 2 = 0 is assumed, the previous equality implies α 4 = 1, which spoils the classical limit of the counter-term. As a consequence, Eq. (4.4) implies \n/negationslash \nβ 2 = 0 . (4.5) \nIt should be noticed that this cancellation does not depend on whether corrections have been implemented on the perturbative expansion of the constraints or not. Considering both β 2 = 0 and A { H , D } 7 = 0 (3.51), one gets: \nβ 1 = 0 . (4.6) \nFrom A { H , H } 12 = 0 (3.82), the following relation can be derived: \nβ 6 = -β 3 1 + β 3 , (4.7) \nsuch that A { H , H } 16 = 0 (3.89) and β 1 = 0 imply \nβ 3 = α 1 + ˜ g g ∂ c g -1 . (4.8) \nGiven this expression, the cancellation of A { H , H } 19 (3.92) and A { H , H } 6 (3.72) implies: \n∂ p β 3 = 0 . (4.9) \nTo ensure the correct classical limit (i.e. c → 0), counter-terms cannot be independent of the non-trivial part of the extrinsic curvature c - via A { H , H } 6 (3.72) and of the non-trivial part of the densitized triads p , see Eq. (4.9). This leads to: \nβ 3 = 0 and β 6 = 0 . (4.10) \nStill, Eq. (4.8) must hold with β 3 = 0, which implies: \nα 1 = g∂ c g -˜ g. (4.11) \nThe functional dependence of this particular counter-term, α 1 , depends on whether the holonomy correction has been applied to the perturbed part of the scalar constraint or not. \nThe cancellation of A { H , D } 6 (3.50), A { H , D } 11 (3.55), and A { H , H } 11 (3.81) implies \nβ 5 = 0 , β 9 = 0 , and β 7 = 0 , (4.12) \nsuch that only β 13 remains undetermined for the matter sector. In particular, β 13 does not appear in any anomaly and will be fixed later by requiring to recover the expected, although deformed, structure of the algebra of constraint for the { H , H } bracket, namely { H , H } ∝ D . \nConsidering A { H , D } 1 = 0 (3.39) together with Eq. (4.11) leads to \nα 2 = 3 g 2 -˜ g 2 -2 c g∂ c g. (4.13) \nAgain, this expression for α 2 does depend on whether the GHC has been applied to the perturbative expansion of the geometrical scalar constraint or not. Since α 3 cannot be a constant (so as to have the correct behavior in the classical limit), it can be concluded from A { H , H } 3 = 0 (3.68) that: \nα 5 = α 4 . (4.14) \nGiven Eq. (4.3), this implies \nα 4 = ( ∂ c g ) 2 + g∂ 2 c g -1 . (4.15) \nSince α 4 and α 5 are known, A { H , D } 3 (3.41) or A { H , D } 5 (3.43) can be cancelled with: \nα 6 = 2 g∂ c g -˜ g -c [( ∂ c g ) 2 + g∂ 2 c g ] . (4.16) \nSolving A { H , D } 2 = 0 (3.40), this means that \nα 7 = α 8 . (4.17) \nConsidering A { H , D } 4 = 0 (3.42), one then gets \nα 7 = 2 c ( 2 g∂ c g -c ( ∂ c g ) 2 -c g∂ 2 c g ) -4 p g∂ p g -g 2 -˜ g 2 . (4.18) \nSubstituting Eqs. (4.11), (4.15), and (4.16) into A { H , H } 4 = 0 (3.69) and making use of A { H , H } 14 = 0 (3.87) leads to a relation between α 3 and α 9 : \ng∂ c g [ α 3 +2 p ∂ p α 3 -α 9 ] = 0 . (4.19) \nThis exhibits a freedom in the choice of α 3 and α 9 . This degeneracy is often overlooked in the literature. In [24], the choice α 3 = 0 and α 9 = 0 was made so as to ensure the compatibility with results obtained in curved space. However at this stage, it seems that they were derived with inverse-volume corrections only [71]. It looks like to us that the question therefore remains quite open and, in this work, we choose to remain as generic as possible. \nConsidering A { H , H } 15 (3.88), one obtains: \nα 3 def = f ( p ) = 1 √ p [ K + ∫ p 1 d Θ α 9 (Θ) √ Θ ] , (4.20) \nwhere K is some constant. \nThe only term which remains to be fixed is now β 13 . It can be determined by requiring the { H , H } bracket to be proportional to the full diffeomorphism constraint, implying that the structure coefficients in front of D g and D m are the same. This translates, using Eqs. (4.20) and (4.15), in: \nβ 13 = ∂ c ( g∂ c g ) [ 1 + f ( p ) ] -1 . (4.21) \nThis concludes the calculation of all the counter-terms. \nIt is however important to check that the remaining anomalies are solved using the counterterms that have just been derived. First of all, A { H , H } 1 (3.66) leads to a condition on g : \ng -2 p ∂ p g -c ∂ c g = 0 . (4.22) \nInterestingly, this is the same as the one derived in [24] where holonomy corrections were applied only to the background part of the scalar constraint and not on the perturbed expansion. Although A { H , H } 1 depends explicitly, and implicitly via the counter-terms, on the GHC ˜ g implemented at the perturbative level, the calculations show that no additional restriction appears on ˜ g when compared to the case where only the background is corrected. \nIn addition, it is easy to check that, given all counter-terms derived from the matter sector 8 , all the remaining anomalies which depend only on the matter sector counter-terms are automatically cancelled. Otherwise stated, A { H , H } 5 (3.71), A { H , H } 7 (3.73), A { H , H } 8 (3.74), A { H , H } 9 (3.75), and A { H , H } 10 (3.80) are trivially set to zero with the provided solution. Finally, the same argument applied to A { H , H } 17 (3.90) and A { H , H } 18 (3.91) leads to the consistency conditions \n12˜ g +10 α 1 +2 α 6 -10 g∂ c g -2˜ g∂ c ˜ g -∂ c α 2 = 0 , (4.23) \nand \n2 α 1 -2 α 6 +2˜ g∂ c ˜ g -2 g∂ c g + ∂ c α 2 = 0 , (4.24) \nwhich were already satisfied given Eqs. (4.11), (4.13), and (4.16). \nThis completes the procedure to ensure the closing of the algebra of constraints. The resulting structure is deformed with respect to GR, as shown by Eq. (4.21). \nTo summarize, we have: \n{ G , G } = 0 , (4.25) \n{ D , D } = 0 , (4.26) D , G = 0 , (4.27) \n{ H [ N ] , D [ N a ] } = H [ δN∂ a δN a ] , (4.29) \n{ } { H , G } = 0 , (4.28) \n{ H [ N 1 ] , H [ N 2 ] } = G (2) [ 1 + f ( p ) ] D [ N p ∂ a ( ∆( δN ) ) ] , (4.30) \nwhere we have used the same notation than in [24], i.e. \nG (2) def = G (2) ( c , p ) def = 1 2 ∂ 2 c g 2 . (4.31)', '4.2 Closure via a restriction on the GHCs': 'As discussed at the beginning of this article, it is interesting to wonder whether the a priori freedom that exists in choosing the shape of the GHC can be used to close of the algebra without any counter-term. It has been nicely shown in [68] that, focusing only on the vector modes of the cosmological perturbations [62], it is indeed possible to form a first-class algebra. We argue in this section that when scalar modes are considered as well, this interesting conclusion unfortunately does not hold anymore. \nUsing the properties of vector modes [62], namely, δN = 0, δE b b = 0, δK b b = 0, and Z cidj ab = 0, it was shown [68] that 9 : \n{ H | v g [ N ] , D | v g [ N a ] } = D D | v g [ N √ p δN a ] (4.32a) + κ -1 ∫ d x N 2 √ p ∂ a ( δN b ) δ i b δE a i A { H , D } v , (4.32b) \nwhere X | v means that properties of vector modes have been applied to the quantity X , whereas D is a deformation coefficient defined as, \nD def = c + ˜ g -2 g∂ c g, (4.33) \nand A { H , D } v is given by: \nA { H , D } v = ˜ g 2 2 + g 2 2 -c 2 -2 c ˜ g +2 g ( p ∂ p g + c ∂ c g ) . (4.34) \nSince δN = 0 for vector modes, the Poisson bracket (4.32) is the only relevant one in that case. Thus, the cancellation of the previous anomaly A { H , D } v (4.34) needed to ensure that the algebra of constraints remains first-class imposes a specific restriction on the GHC. The paramount importance of those restrictions for phenomenology was already underlined \n[25, 26, 68]. Using the detailed calculations presented in the previous sections, and forcing counter-terms to vanish (as the algebra is here closed only by the GHC), it can be seen that Eq. (4.32) and Eq. (3.38) are equivalent once symmetries have been applied: \n{ H | v g [ N ] , D | v g [ N a ] } = κ -1 ∫ d x N 2 √ p ∂ a ( δN b ) δ i b δE a i A { H , D } 4 (4.35a) + κ -1 ∫ d xN √ p ∂ a ( δN b ) δ a i δK i b A { H , D } 5 . (4.35b) \nAlthough enforcing the algebra of constraints to be first-class is a strong requirement, ambiguities remain, even in this simple situation. In particular, when focusing on vector modes only, two different scenarios lead to a closed algebra, resulting in different restrictions on the GHC. First, as discussed in [68], the restriction A { H , D } v = 0 (4.34) leads to a satisfying solution. However, the restrictions A { H , D } 4 = 0 (3.42) and A { H , D } 5 = 0 (3.43) also result in a first-class algebra. Note, however, that the solution space for g ( c , p ) under the condition A { H , D } v = 0 (4.34) is less restrictive than the one for A { H , D } 4 = 0 (3.42) and A { H , D } 5 = 0 (3.43), as the latter condition implies A { H , D } v = 0 (4.34) by construction. Otherwise stated, instead of forcing the appearance of the diffeomorphism constraint, one could simply ensure that the bracket vanishes. These ambiguities have to be explicitly addressed to extend the results of [68] to scalar modes. This is the aim of the following of this section. \nStarting with the { H , D } bracket, the above-mentioned ambiguity is elegantly fixed by the introduction of matter. The latter is mandatory for studying scalar modes. As soon as matter is considered, the diffeomorphism constraint D is given by D = D g + D m . Thus, to obtain 10 { H , D } ⊃ D , one has to investigate the { H m , D } bracket. This was done in the previous section: without counter-terms, Eq. (3.49) implies that { H m , D } = H m , ensuring that no anomaly emerges from this particular sub-bracket. This means that it is not possible to obtain { H , D } ⊃ D while remaining consistent. The ambiguity discussed above is therefore fixed by the introduction of matter contributions. \nAs for the { H , G } bracket, some simple manipulations lead, schematically, to { H , G } = G + A while Eq. (3.24) still holds 11 . Therefore, similarly to { H , D } , the { H , G } bracket is inherently ambiguous: multiple solutions do exist to the anomaly freedom requirement. However, explicit calculations from the previous section reveal that all the anomalies of the { H , G } bracket are equal to some of the anomalies of the { H , D } bracket. In other words, since these anomalies must vanish to ensure the consistency of { H , D } , the ambiguity of { H , G } is fixed as well. It should be noticed that, if the diffeomorphism constraint is holonomy corrected, the ambiguity of { H , G } cannot be fixed by the above reasoning since anomalies of { H , G } and { H , D } become independent. \nImportantly, no ambiguity arises in the computation of the { H , H } bracket. The explicit derivation leads to { H , H } ⊃ D , as expected from classical consideration. Still, a question remains: could we have { H , H } ⊃ G and/or { H , H } ⊃ H , which would lead to ambiguities? This answer is negative. By construction, one cannot consistently obtain { H , H } ⊃ G due to the peculiar summation of the Gauss constraint via the Levi-Civita tensor, see Eq. (2.23), which are not recovered in the { H , H } computation. Moreover, all anomalies in the { H , H } bracket are proportional to a perturbation of the canonical variables, i.e. δE , δK , δφ or δπ . Consequently, neither H (0) g nor H (0) m can be constructed while keeping an anomaly-free algebra. It is therefore not possible to obtain { H , H } ⊃ H . Hence, it can be concluded that no ambiguity remains for the { H , H } bracket. \nAs the ambiguities are fixed and as only a single path remains to obtain a first-class algebra of constraints, let us come back to the question of a possible closure of the algebra for scalar modes by mean of a specific GHC. As discussed above, all anomalies need to be cancelled one by one, leading to restrictions on the form of the correction. We focus on the previously derived anomalies A { H , D } 1 (3.39), A { H , D } 3 (3.41), and A { H , D } 13 (3.86). Without counter-terms, A { H , H } 13 (3.57) leads to the consistency condition: \n˜ g ( c , p ) = c + f ( p ) , (4.36) \nwhere f ( p ) can, at this stage, be any function that vanishes at the classical limit p →∞ . Given this solution and A { H , D } 1 = 0 (3.39), one obtains: \ng ( c , p ) = ± √ 1 3 ( 3 c 2 +4 c f ( p ) + f ( p ) 2 ) . (4.37) \nFinally, A { H , D } 3 = 0 (3.41) implies: \nf ( p ) = 0 . (4.38) \nThis leads to the conclusion that ˜ g ( c , p ) = c and g ( c , p ) = c . In other words, the only way to obtain an algebra of constraints which is closed, not by the use of counter-terms but by mean of a specific choice of the GHC, is to consider the classical case without holonomy correction.', '5 Conclusion and prospects': "In this article, we have provided an extensive material for the calculation of Poisson brackets and anomalies in holonomy-corrected effective loop quantum cosmology. We tried to make all hypotheses, assumptions, and manipulations explicit so that future works can rely on a clear basis without the need for re-inventing all the heavy machinery. \nWe have taken this opportunity to provide some new insights. First, we have explained why implementing the holonomy correction in the cosmological perturbations - either in \nthe usual form or through a generalized function - does not change the observational predictions of the theory. The argument is made simple and explicit. \nWe have also underlined that, because there are many different ways to get a fist class algebra, trying to restrict the kind of possible generalized holonomy correction so as to ensure the anomaly freedom without counter-terms is plagued with ambiguities. Even more importantly, a careful analysis shows that the algebra cannot be close without counter-terms. \nSeveral points still deserve a closer look in the future: \n- · there are theoretical arguments preventing effective (quantum) correction to the diffeomorphism constraint due to its handling from the LQG view-point [73] but it would still make sense at the heuristic level due to interesting results in the literature [14, 60]. Moreover, we would like to mention that it could even make sense at a deeper level [74],\n- · even when implemented in the perturbative expansion of the constraints, the holonomy correction used so far (in this work and in others) is based on a modification of the background curvature through the replacement c -→ g ( c , p ). An important step forward for the study of perturbations would be to consider a rigorous treatment of the perturbative expansion of the holonomy, based on the perturbed connection,\n- · counter-terms α i are currently added to the deformed constraints as ' α i × (term entering the classical constraint)'. This is indeed enough to close the algebra. However, following an even more general path, all the possible terms at each order of the perturbative development could be added, with the only requirement that the classical limit is recovered,\n- · it would be worth considering the possibility that the counter-terms are not only functions of the geometrical phase-space variables background parts c and p , but of the full phase space variables,\n- · inverse-volume corrections should also be added, keeping the same level of generality than for the holonomy corrections. In particular, the claimed signature change should be readdressed in this framework where its occurrence is much less obvious. Some freedom about the spacetime signature at high energy is already underlined in the present work, in which only holonomy corrections were considered, through the f function in Eq. (4.30),\n- · the cosmological observables should be exhaustively investigated, generalizing the results of [25, 26]. In particular, the freedom associated with the f function and its consequences must be studied.", 'A Anomalies: Summary of computations': 'This appendix provides the expressions for all the anomalies calculated in Section (3). The steps of the derivation can be found in the main text. \nFrom the { H , G } bracket: \nA { H , G } 1 = g 2 +2 c [ ˜ g + α 6 ] -[ ˜ g 2 + α 7 ] +4 p g∂ p g, (A.1) A { H , G } 2 = c 1 + α 4 + ˜ g + α 6 -2 g∂ c g, (A.2) \nA { H , G } 4 = β 12 . (A.4) \n[ ] [ ] A { H , G } 3 = -β 10 , (A.3) \nFor the { H , D } bracket: \nA { H , D } 1 = 3 g 2 -[ ˜ g 2 + α 2 ] -2 c [ ˜ g + α 1 ] , (A.5) A { H , D } 2 = α 8 -α 7 , (A.6) \nA \nA { H , D } 3 = c [ 1 + α 5 ] + [ ˜ g + α 6 ] -2 g∂ c g, (A.7) A { H , D } 4 = g 2 -2 c [ ˜ g + α 6 ] + [ ˜ g 2 + α 7 ] +4 p g∂ p g, (A.8) { H , D } 5 = c 1 + α 4 ] + ˜ g + α \nA \n- \n[ \n[ \n] \n{ \nH \n6 \n= \nβ \n5 \nβ \n6 \n, \n(A.10) \nA { H , D } 7 = 2 β 1 -β 2 , (A.11) \n(A.12) \nA 8 = β 4 , A { H , D } 9 = -β 10 , (A.13) \n(A.14) \n{ \nH \nH \n{ \nH \n{ \n, \n, \n, \n, \nD \nD \nD \nD \n} \n} \n} \n} \nA 10 = β 12 , A { H , D } 11 = β 10 -2 β 6 + β 9 , (A.15) \n(A.16) \nA 12 = β 8 , A { H , D } 13 = β 11 -β 12 . (A.17) \nFor the { H , H } bracket: \nA { H , H } 1 = 2 [ ˜ g + α 1 ] ( g∂ c g -˜ g -α 6 ) -( g 2 +4 p g∂ p g ) ∂ c ( ˜ g + α 1 ) +4 p g∂ c g∂ p ( ˜ g + α 1 ) + 1 2 [ ˜ g 2 + α 2 ] ( 2 + 3 α 5 -α 4 ) , (A.18) H H \n{ \n, \n} \n2 \nA \n= \n[ \n˜ \ng \nc \ng \n- \n1 \n2 \ng \n+4 \np \ng∂ \np \ng \n∂ \nc \n˜ \ng \n+ \nα \n2 \nA { H , H } 3 = [ 1 + α 3 ][ α 5 -α 4 ] , (A.20) \n2 + α 2 ] ( ˜ g + α 6 -g∂ ) ( ) ( ) +2 p g∂ c g∂ p ( ˜ g 2 + α 2 ) + [ ˜ g + α 1 ] ( ˜ g 2 +3 α 8 -2 α 7 ) , (A.19) \n6 \n- \n2 \ng∂ \np \ng, \n(A.9) \n2 \n2 \nA { H , H } 4 = [ 1 + α 3 ] ( g∂ c g + [ ˜ g + α 6 ] + k [ 1 + α 5 ] ) -[ ˜ g + α 1 ][ 1 + α 9 ] + \nA { H , H } 5 = ∂ c β 1 , \nA \n{ \nH \n6 \n= \n∂ \nc \nβ \n3 \n, \n(A.23) \nA { H , H } 7 = -∂ c β 2 , (A.24) \nA \nA \nA \nA \nA \nA \n{ \nH \n8 \nH \n{ \n9 \nH \n{ \n10 \nH \n{ \n11 \nH \n{ \n12 \nH \n{ \n13 \nH \n= \n= \n∂ \n= \nβ \n= \nβ \n= \nβ \n= \n∂ \n- \nc \n1 \nc \n( \n1 \n1 \nβ \nβ \n4 \n2 \n- \n+ \nβ \n+ \nβ \n1 + \nβ \n, \n(A.25) \n+ \nβ \nβ \n3 \n7 \n3 \n2 \n4 \n[ \n4 \n- \n+ \nβ \n+ \nβ \n]( \n2 \n) β \n, \n(A.26) \n5 \n1 \n6 \n- \n- \nβ \n7 \n+ \nβ \nα \n2 + \nα \n4 \nβ \n3 \n- \n3 \nβ \n5 \nβ \n3 \nβ \n6 \n+3 \nα \n4 \n, \n(A.27) \n, \n(A.28) \n+ \nβ \n5 \n3 \nα \n8 \n- \n5 \n) \n+ \nβ \n2 \n∂ \nc \n+2 \n∂ \n1 \n( \nβ \ng \n+ \nα \n˜ \nc \nβ \n1 \n- \n, \n(A.30) \n1 \nA 14 = [ 1 + β ]( --) ( ) A { H , H } 15 = ∂ c α 3 , (A.32) \nA \nA \n{ \n17 \nH \n{ \n18 \n2 \n) \ng \n+ \nα \n+2 \n˜ \n, \n(A.31) \nA { H , H } 16 = -6 [ ˜ g + α 1 ][ 1 + β 6 ] +2 g∂ c g ( 3 + 3 β 1 -2 p ∂ p β 1 ) + g 2 ∂ c β 1 +4 g p ∂ p g∂ c β 1 , (A.33) { H , H } = 12˜ g +10 \n- \n1 \nα \ng \n6 \n1 \n( \ng \n+ \nα \np \n+4 \n6 \n2 \n- \n2 \np \n∂ \n1 \n+2 \nα \n5 + 5 \nβ \n+6 \n1 \n) \n] \nβ \n10 \n- \n4 \n+ \n∂ \n= 2 \nα \n2 \ng∂ \n- \n2 \nα \n˜ \ng \n+ \nα \n+2 \np \n∂ \nβ \n11 \np \nβ \n˜ \ng \n+ \nα \nc \n2 \n˜ \ng \n+ \nα \n- \n1 \n+ \ng \ng \nβ \n2 \n12 \n2 \n∂ \n] \n6 \nβ \n∂ \nc \n- \nc \nβ \n2 \n2 \n4 \nβ \n2 \n- \n[ \ng \n+ \nα \n˜ \n+4 \ng \np \n∂ \n+6˜ \n4 \ng \np \n∂ \n6 \np \ngβ \np \n9 \n- \ng∂ \nβ \n4 \ng∂ \nc \nc \n2˜ \nβ \ng∂ \n2 \n, \n(A.34) \ng∂ \n+2˜ \nβ \n4 \n, \n(A.35) \nc \n˜ \ng \nc \n˜ \ng \n-2 g∂ c g ( 1 + β 4 ) A { H , H } 19 = 3 p [ ˜ g + α 1 ] +3 p [ ˜ g + α 1 ] β 8 -g∂ c g ( 3 p [ 1 + β 3 ] +2 p 2 ∂ p β 3 ) + p 2 g 2 ∂ c β 3 +2 p 2 g∂ p g∂ c β 3 . (A.36) \nα \n2 \n] \n[ \n] \n[ \n∂ \n[ \n[ \n] \nβ \n2 \n- \n4 \nc \nc \nα \n˜ \n, \n, \n, \n, \n, \n, \n, \n, \nH \nH \nH \nH \nH \nH \nH \nH \n} \n} \n} \n} \n} \n} \n} \n} \n, \nH \n} \n+2 p g∂ c g∂ p α 3 + ( 1 2 g 2 +2 p ∂ p g ) ∂ c α 3 ., (A.21) \n(A.22) \n8 \n- \nβ \n4 \n, \n(A.29)', 'B Anomalies: Table of correspondence': "Some notations differ between previous works, such as [24, 65], and this study. To help the unfamiliar reader and to allow an easy comparison between approaches 12 , 13 , we provide in this appendix a table of correspondence for the anomalies (N.A. stands for 'non available'). \nTable 1 : { H , G } bracket - Table of correspondenceTable 2 : { H , D } bracket - Table of correspondence \nA \nA \nTable 3 : { H , H } bracket - Table of correspondence", 'References': "- [1] C. Kiefer, Quantum gravity: General introduction and recent developments , Annalen Phys. 15 (2005) 129 [ gr-qc/0508120 ].\n- [2] D. 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2024JHEP...01..112O | The local twodimensional Poincar algebra near the horizon of an eternal AdS black hole or in proximity to any bifurcate Killing horizon is generated by the Killing flow and outward null translations on the horizon. In holography this local Poincar algebra is reflected as a pair of unitary flows in the boundary Hilbert space whose generators under modular flow grow and decay exponentially with a maximal Lyapunov exponent. This is a universal feature of many geometric vacua of quantum gravity. To explain this universality we show that a twodimensional Poincar algebra emerges in any quantum system that has von Neumann subalgebras associated with halfinfinite modular time intervals modular future and past subalgebras in a limit analogous to the nearhorizon limit. In ergodic theory quantum dynamical systems with future or past algebras are called quantum Ksystems. The surprising statement is that modular Ksystems are always maximally chaotic.Interacting quantum systems in the thermodynamic limit and large N theories above the HawkingPage phase transition are examples of physical theories with futurepast subalgebras. We prove that the existence of modular futurepast von Neumann subalgebras also implies a second law of modular thermodynamics and the exponential decay of modular correlators. We generalize our results from the modular flow to any dynamical flow with a positive generator and interpret the positivity condition as quantum detailed balance. | 2024-01-01T00:00:00Z | ['2024JHEP...01..112O', '2023arXiv231013736O', 'arXiv:2310.13736', '10.1007/JHEP01(2024)112', '10.48550/arXiv.2310.13736'] | ['AdS-CFT Correspondence', 'Space-Time Symmetries', 'Black Holes', 'Quantum Dissipative Systems', 'High Energy Physics - Theory', 'General Relativity and Quantum Cosmology', 'Mathematical Physics', 'Nonlinear Sciences - Chaotic Dynamics'] | Local Poincar algebra from quantum chaos | 2,024 | 174 | 0.21 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML'] | 8 | https://arxiv.org/pdf/2310.13736.pdf | {"Local Poincar'e Algebra from Quantum Chaos": "Shoy Ouseph, Keiichiro Furuya, Nima Lashkari, Kwing Lam Leung, Mudassir Moosa \nDepartment of Physics and Astronomy, Purdue University, West Lafayette, IN 47907, USA E-mail: [email protected] , [email protected] , [email protected] , [email protected] , [email protected] \nAbstract: The local two-dimensional Poincar'e algebra near the horizon of an eternal AdS black hole, or in proximity to any bifurcate Killing horizon, is generated by the Killing flow and outward null translations on the horizon. In holography, this local Poincar'e algebra is reflected as a pair of unitary flows in the boundary Hilbert space whose generators under modular flow grow and decay exponentially with a maximal Lyapunov exponent. This is a universal feature of many geometric vacua of quantum gravity. To explain this universality, we show that a two-dimensional Poincar'e algebra emerges in any quantum system that has von Neumann subalgebras associated with half-infinite modular time intervals (modular future and past subalgebras) in a limit analogous to the near-horizon limit. In ergodic theory, quantum dynamical systems with future or past algebras are called quantum K-systems. The surprising statement is that modular K-systems are always maximally chaotic. \nInteracting quantum systems in the thermodynamic limit and large N theories above the Hawking-Page phase transition are examples of physical theories with future/past subalgebras. We prove that the existence of (modular) future/past von Neumann subalgebras also implies a second law of (modular) thermodynamics and the exponential decay of (modular) correlators. We generalize our results from the modular flow to any dynamical flow with a positive generator and interpret the positivity condition as quantum detailed balance. \nKeywords: Future/Past subalgebras, Emergent Poincar'e group, Second Law, Quantum Dynamical Systems, Quantum Ergodicity, Quantum K-systems, Quantum Anosov systems", '1 Introduction': 'A theory of quantum gravity in the small G N limit admits a large set of geometric states that host fluctuating quantum fields. To every smooth geometry, we associate a perturbative semiclassical Hilbert space of states of curved spacetime quantum field theory (QFT). The smoothness of the geometry implies that quantum fields in the vicinity of any point transform under a local Poincar\'e algebra corresponding to the local Minkowski space. The emergence of these local Poincar\'e groups is a curious universal feature of many vacua in any theory of quantum gravity. In this work, we identify an origin for the emergence of this universality based on the ergodic properties of modular flows in observable algebras of quantum gravity. \nIn a general QFT in curved spacetime, there is no globally time-like Killing vector; therefore, it is not clear how to impose a condition similar to the positivity of the Hamiltonian, and the analytic properties of vacuum correlators in complex coordinates. To overcome this, we focus on QFT in spacetimes with a bifurcate Killing horizon in the Hartle-Hawking state where the KMS condition provides analytic properties of correlators similar to the positivity of the Hamiltonian [1]. Similar to boost in Rindler space, a spacetime with a bifurcate Killing horizon has a global Killing vector B which splits it into four regions; see Figure 1. The future (past) Killing horizon H + ( H -) is a null surface with enhanced symmetries. The Killing vector B and null translation P + ( P -) generate isometries of these null surfaces, respectively. 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The Killing vector B of a spacetime with a bifurcate Killing horizon splits it into four regions, similar to boosts in Minkowski space. σ is the bifurcation surface and the null translations P ± on the future and past horizons H ± are isometries of these surfaces. They can be extended to G ± in the near horizon regions. \n<!-- image --> \ns 0 → e t s 0 + s ( s 0 → e -t s 0 + s ) and Lie algebraic relations 1 \nH + : e itB e isP + e -itB = e ise t P + , [ P + , B ] = iP + H -: e itB e isP -e -itB = e ise -t P -, [ P -, B ] = -iP -. (1.1) \nFor a general space-time with a bifurcate Killing horizon, the translations P ± are isometries only when restricted to the future and past Killing horizons. In the vicinity of the Killing horizon H + ( H -), we extend the null translations P + ( P -) to the vector fields G + ( G -) in normal Riemann coordinates. From the point of view of a perturbation whose proper distance to the bifurcate surface σ is much smaller than the curvature length at the horizon these operators commute, i.e. [ G + , G -] ≃ 0. This commutation relation together with (1.1) gives a local two-dimensional Poincar\'e algebra. As we move further away from the horizon, the commutator [ G + , G -] is decided by the curvature of the near-horizon geometry [2]. 2 \nIn a theory of quantum gravity, we associate von Neumann algebras with regions of spacetime that are diffeomorphism-invariant, such as the right wedge W (region III in Figure 1) in spacetime with a bifurcate Killing horizon. 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sha1\\_base64="wW/FYuM+uQPEIsfDWGUcEmwyeqw=">AAAB6HicbVDLSgNBEOz1GeMr6tHLYBA8hV3xdQx68ZiAeUCyhNlJbzJmdnaZmRXCki/w4kERr36SN//GSbIHTSxoKKq66e4KEsG1cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqhg0Wi1i1A6pRcIkNw43AdqKQRoHAVjC6m/qtJ1Sax/LBjBP0IzqQPOSMGivVm71S2a24M5Bl4uWkDDlqvdJXtx+zNEJpmKBadzw3MX5GleFM4KTYTTUmlI3oADuWShqh9rPZoRNyapU+CWNlSxoyU39PZDTSehwFtjOiZqgXvan4n9dJTXjjZ1wmqUHJ5ovCVBATk+nXpM8VMiPGllCmuL2VsCFVlBmbTdGG4C2+vEya5xXvqnJZvyhXb/M4CnAMJ3AGHlxDFe6hBg1ggPAMr/DmPDovzrvzMW9dcfKZI/gD5/MHtfGM5A==</latexit> \nFigure 2 . The wedge algebras in a spacetime with bifurcate Killing horizons. (a) W ( z + ) is the von Neumann subalgebra of all observables the observer (dashed line) has access to from a particular moment in time (black dot) until eternity forms, whereas W ( z -) is the von Neumann subalgebra of all observables the observer had access to since past infinity until now. (b) Given a point ( z + , z -), we define a right wedge W ( z + , z -) = ( U > z + , V > z -) and a left wedge W \' ( z + , z -) = ( U < z + , V < z -). \n<!-- image --> \ngravity. In particular, in the Hilbert space of quantum gravity states dual to the excitations of the Hartle-Hawking vacuum of a spacetime with a bifurcate Killing horizon, there must be an emergent Poincar\'e symmetry in a regime analogous to the limit that localizes excitations near the bifurcate Killing horizon. The first clue to the origin of these symmetry relations on the boundary comes from the observation that the Killing field B generates the modular flow of the observable algebra of the right wedge: ∆ it W = e -2 πiBt (see Theorem 5). The modular flow of an observable algebra is an example of a state-preserving quantum dynamical system that we call modular dynamical system . Consider an observer at a point in the right wedge W who moves along the Killing vector B (see Figure 2). They perceive the modular flow of the right wedge W as time evolution. The set of all observables they have access to from a particular moment in time until eternity forms a von Neumann subalgebra of all observables of the right wedge that we call the modular future subalgebra , e.g. W ( z + ) in Figure 2. Similarly, the set of all observables they had access to from past infinity until now forms the modular past subalgebras , e.g. W ( z -) in Figure 2. Since the modular flow of W is the Killing flow of B , we find the half-sided modular inclusion relations \n∀ t > 0 : ∆ ∓ it W A ( W ( z ± ))∆ ± it W ⊂ A ( W ( z ± )) . (1.2) \nIn this work, we show that, in an arbitrary quantum system with a von Neumann algebra of observables R (analog of the algebra of the right wedge A ( W )), the existence of modular future and past subalgebras A + and A -(analogs of A ( W ( z + )) and A ( W ( z -))), respectively, is sufficient for a \'local\' Poincar\'e algebra to emerge in a certain scaling regime analogous to the near-horizon limit. There are two steps to our argument. The first step relies on a key result of quantum ergodic theory (The Half-sided Modular Inclusion Theorem 16) which says that the modular past and future subalgebra relations A ± ⊂ R : \n∀ t > 0 : ∆ ∓ it R A ± ∆ ± it R ⊂ A ± (1.3) \nimply that the operators \n± G ± := K R -K A ± K = -1 2 π log ∆ (1.4) \nare positive, generate a symmetry G ± | Ω ⟩ = 0, and satisfy the algebraic relations \n∆ -it R e isG ± ∆ it R = e ie ± 2 πt sG ± . (1.5) \nIn the example of the quantum gravity algebra dual to the right wedge of the spacetime with bifurcate Killing horizon, the generators G ± do not have a nice geometric dual in terms of local spacetime transformation, except for their restrictions to the horizons H ± which are proportional to null translations P ± . \nIn general, there is no reason for [ G + , G -] to be small unless we introduce a new scaling parameter. The second part of our argument is a scaling limit that mimics the near-horizon \nlimit by defining the algebras and generators \nA ± ( s ) := ∆ -is R A ± ∆ is R ± G ± ( s ) := K R -K A ± ( s ) . (1.6) \nIn Theorem 10, we show that [ G + ( -s ) , G -( s )] = O ( e -4 πs ), therefore in the limit of large s we have an emergent Poincar\'e algebra. This limit is the analog of zooming in near the horizon to proper distances much smaller than the local curvature length scale. \nAs we review in Section 4, the relations in (1.5) constitute a special example of a class of quantum ergodic systems called quantum Anosov systems . A quantum Anosov system is a quantum dynamical system with unitary \'time-evolution\' e iKt that admits other unitary flows generated by self-adjoint operators G α such that \ne iKt e isG α e -iKt = e ise -λαt G α (1.7) \nand the constants λ α which are positive (negative) for decaying (growing) modes are often called the Lyapunov coefficients. It has been argued when the dynamics is modular flow ( e iKt = ∆ -it ) there is a universal bound on Lyapunov exponents | λ | ≤ 2 π [2, 4]. Since the algebraic relations in (1.5) saturate this bound following we will refer to the algebraic relations in (1.5) as maximal modular chaos . Following [2], we will call G ± the modular scrambling modes . \nThe AdS/CFT correspondence provides a concrete realization of quantum gravity in asymptotically Anti-de Sitter (AdS) spacetimes. A large N 2 ∼ 1 /G N theory living on the boundary of AdS is dual to curved spacetime QFT in the bulk. In the strict N → ∞ limit, the boundary is described by Generalized Free Fields (GFF) [5]. A GFF algebra is fixed by two choices of functions corresponding to the two-point correlation function and the commutator of the fundamental field [6]. 3 In a KMS state (finite temperature state), the two functions are related and we only need to fix one function, namely the spectral density (the Fourier transform of the commutator). In holographic GFF [8], assuming spherical symmetry, effectively, we reduce the bulk to 1 + 1-dimensions and the GFF to a 0 + 1 dimensional theory. 4 The relation between the bulk fields and boundary fields is explicit at the level of the one-particle Hilbert space. Given a boundary function f ( t ) we can find the bulk solution ˜ f ( t, r ) to the field equations of motion on geometry g µν that asymptotically relates to the boundary function f ( t ). This is sometimes called the HKLL bulk reconstruction [9, 10]. The KMS state of holographic GFFs exhibits a special phase transition, called the Hawking-Page phase transition. Below the critical temperature, the bulk is the thermal state of Anti-de Sitter space, whereas above the critical temperature, the geometry is an AdS Schwarzschild black hole. If we think in terms of the thermofield double (the canonical purification of the thermal state), below the transition point, the dual geometry is two disjoint copies of thermal AdS, whereas above the transition, the dual geometry is two black holes sewn together to form a two-sided wormhole (eternal AdS \n3 \nSee [7] for a review. \nblack hole) [11]. Recently, [12, 13] showed that above the Hawking-Page phase transition, there are boundary GFF von Neumann algebras associated with finite time intervals. In particular, there are future and past subalgebras associated with half-infinite time intervals ( t 1 , ∞ ) and ( -∞ , t 2 ). In the bulk, the Hawking-Page phase transition point is associated with the emergence of the following physical phenomena: \n- (1) Strong Mixing: Below the transition point, all correlators are almost periodic functions of time, whereas above the transition, all connected correlators decay to zero.\n- (2) Second Law: Below the transition point, the bulk dynamics is reversible, whereas, above the transition point, the bulk dynamics becomes irreversible because excitations can fall behind the horizon. The emergence of a monotonically increasing coarsegrained entropy (second law) is a manifestation of this irreversibility.\n- (3) Exponential decay: Above the transition point, not only do all connected correlators decay but also a large class of operators decay exponentially fast with fixed exponents called quasi-normal modes.\n- (4) Near-horizon Poincar\'e algebra: Above the transition point, a smooth horizon is formed in the bulk, and there is an emergent local two-dimensional Poincar\'e group in the vicinity of the bifurcation surface of the eternal black hole. \nProperties (1), (2) and (3) are aspects of the black hole information loss problem . However, they are not specific to black holes. The physics above the transition point occurs in most, if not all, thermalizing quantum systems in the strict thermodynamic limit of infinite finegrained entropy. These properties are not independent. In fact, as we review in section 4, they follow an ergodic hierarchy \nExponential Decay ⇒ Second Law ⇒ Strong Mixing . (1.8) \nEach of them corresponds to an important ergodic class: Exponential Decay ( Anosov systems ), Second Law (Kolmogorov systems or, in short K-systems ) and Strong Mixing systems. \nThere is an intuitive physical justification as to why we expect properties (1)-(3) to emerge in the thermodynamic limit. As we argued in [7], strong mixing follows from the physical picture that in a generic thermalizing system all non-conserved excitations should decay. In other words, at late times, the system forgets about its current state, as is manifest by the decay of all connected two-point correlators. The second law is a consequence of the physical requirement that, at late times, not only the system forgets about its current state, but it also becomes independent of the entire past of the system. Independence from the entire past (future) is what motivated Kolmogorov to introduce a class of quantum ergodic systems called K-systems which can be characterized by the existence of past (future) subalgebras. \nProperty (4), namely an emergent two-dimensional Poincar\'e group, is often considered to be purely gravitational. Our main result is that this property also emerges in modular quantum K-system. We show that all the four properties above follow from a simple feature \nthat is known to hold at finite temperature state of many thermodynamic systems, namely the existence of the future and past subalgebras :', 'Future/past subalgebras': "Local Poincar'e Group ⇒ Exponential Decay ⇒ Second Law ⇒ Strong Mixing . \n⇒ (1.9) \nThe future subalgebra is a proper subalgebra of observables generated by all perturbations (or measurements) one can make from now until future infinity in time. For instance, in a type I algebra, the set of future operators generates all operators; therefore, we say that type I algebras do not admit future subalgebras. 5 Similarly, the past subalgebra is the proper subalgebra of all perturbations (or measurements) that could have been made from past infinity in time until now. \nOur discussion generalizes from finite temperature time-evolution in two major ways: 1) Modular flow of a general state and 2) General dynamics with a positive generator. The results continue to hold in arbitrary out-of-equilibrium states, with Hamiltonian replaced with modular Hamiltonian. We show that the existence of modular future and past subalgebras implies the strong mixing of modular correlators, a modular second law, the exponential decay of modular correlators, and an emergent approximate modular Poincar'e group in a certain limit similar to the one that localizes a particle near the horizon a black hole [14]. We see in section 4, our results also generalize to any dynamical unitary flow e iKt that is state-preserving K | Ω ⟩ = 0 and has a positive generator K ≥ 0. \nIn a recent work [7], we showed that strong (modular) mixing implies that the algebra is type III 1 in a state that has a trivial centralizer. This is compatible with the fact that, in an eternal black hole, the bulk dual of each boundary algebra is an AdS Rindler wedge. We proved that GFF algebra in a KMS state with a spectral density that is Lebesgue measurable is type III 1 . Note that the bulk algebra is known to be the unique hyperfinite type III 1 algebra, and it is expected that the hyperfiniteness requires the spectral density to be a smooth and non-vanishing function of frequency, in accordance with the conjecture in [13]. The half-sided modular inclusions play a key role in our work. They were also discussed in [13] in the context of the GFF algebra with the spectral density that matches the black hole two-point function. The authors pointed out the connections to the emergence of an arrow of time (second law) and calculated the action of the modular scrambling modes G ± on the creation/annihilation operators of this GFF theory. \nOur result in (1.9) applied to holography implies that in a theory of GFF at finite temperature (KMS state), the existence of future subalgebra implies a local Poincar'e group near the horizon, the exponential decay of connected correlators of a dense set of observables, a second law of thermodynamics, and the decay of any connected correlator. However, as we remarked earlier, our results are more general, and apply to any quantum system which satisfies the following three key assumptions: 1) Symmetry: e iKt | Ω ⟩ = | Ω ⟩ , \nthat for all t > 0 ( t < 0) we have e iKt A e -iKt ⊂ A . 6 We show that the three conditions above are equivalent to the existence of modular future and past subalgebras (see Theorem 18). \nIn section 2, we review the well-known result of Bisognano-Wichmann and its generalizations to curved spacetime, showing that the modular Hamiltonian of half-space is boost, and there are future and past subalgebras with respect to the dynamical flows generated by null translations P ± . Poincar'e algebra implies that the geometric deformations along null directions grow/decay exponentially in modular time. Our work relies on converse results that prove if there are a pair of commuting dynamical flows with positive generators ± G ± and an algebra R that is a future (past) subalgebra with respect to G + ( -G -) respectively, the modular flow ∆ it R and e is ± G ± generate a two-dimensional Poincar'e group. \nIn Section 3, we give several examples of future and past algebras in physical systems and comment on the second law and the exponential decay of correlations. Section 4 reviews the key intuitions and ideas of quantum ergodic theory and the relevance of future and past subalgebras for maximal modular chaos, exponential decay of correlators, and the second law. There are many definitions of classical chaos and quantum chaos. We avoid the comparison of these definitions here and instead focus on the sharply defined ergodic hierarchy. Intuitively, one can think of classical Anosov systems as classically chaotic systems because they are ergodic and show exponential sensitivity to initial conditions, i.e. they have Lyapunov exponents. The surprise is that, in the quantum world, for modular dynamics, the ergodic hierarchy simplifies (see Corollary 17). The weaker assumption of quantum K-systems (existence of future/past algebras) becomes equivalent to maximal quantum chaos. If we then take the quantum K-systems property as the definition of quantum chaos, we conclude that modular chaos is always maximal! To explain what is special about modular dynamics, we generalize our results to arbitrary dynamical flows with a positive generator and identify the key physical insight that underlies the collapse of the ergodic hierarchy as quantum detailed balance (see Theorem 18). \nThis paper expands upon the contents of [15], which were communicated to us by its author during a conference last July. We decided to coordinate our releases.", "2.1 From Poincar'e to future/past subalgebras": "Consider QFT in Minkowski space R d, 1 with the metric \nds 2 = dx + dx -+ dx i dx i x ± = x d ± x 0 . (2.1) \nTo simplify our presentation, in most of this work, we ignore the perpendicular x i directions and focus on the two-dimensional space spanned by x ± . A pair ( z + , z -) defines a right \nwedge W ( z + , z -) = ( x + > z + , x -> z -, x i ) and a left wedge that is the causal complement W ( z + , z -) ' = ( x + < z + , x -< z -, x i ). We use the notation W ( z + ) := W ( z + , 0) and W ( z -) = W (0 , z -) and W := W (0 , 0); see Figure 2(b) with U and V replaced with x + and x -, respectively. We warn the reader that in our notation P -is a negative operator; see Figure 1. \nQFT in Minkowski space is symmetric under a CRT transformation ( C is charge conjugation, R is reflection, and T is time-reversal). 7 In the Hilbert space, the CRT transformation corresponds to an anti-unitary operator J that preserves the vacuum, i.e. J | Ω ⟩ = | Ω ⟩ , and satisfies the following commutation relation with null translation operators e iz ± P ± and fields: \nJe iz ± P ± J = e -iz ± P ± , Jϕ ( x + , x -) J = ϕ ∗ ( -x + , -x -) . (2.2) \nIn Wightman's formulation of QFT, the analytic continuations of Wightman functions are invariant under the complexified connected Poincar'e transformations. The complexified Lorentz group contains the RT transformation (reflection and time-reversal). Assuming that all fields transform as finite-dimensional representations of the Lorentz group and that every charged field is accompanied by its charge conjugate is enough to prove that CRT is a symmetry [16]. Bisognano and Wichmann gave another proof of the CRT theorem by applying modular theory 8 to local algebras built out of Wightman fields: 9 \nTheorem 1 (Bisognano-Wichmann CRT Theorem) . Consider the von Neumann algebra of observables A ( W ) associated with any wedge W in the vacuum of QFT generated by smeared functions of a Wightman field. The modular flow is boost \n∆ it W : ( x + , x -) → ( e 2 πt x + , e -2 πt x -) (2.3) \nand the modular conjugation is a CRT transformation. The wedge algebra satisfies the Haag duality condition A ( W ' ) = A ' ( W ) . \nProof. \nSee [17, 18] for a proof. \nSince the modular conjugation operator leaves the vacuum invariant, we obtain a proof of the CRT theorem in the vacuum. Borchers offered a proof of the above theorem using the axioms of algebraic QFT (AQFT) for the wedge algebras relaxing the assumption of Wightman fields. From the point of view of holography, the operator-algebraic approach has the advantage that if the boundary theory has algebras associated with time intervals they satisfy the same algebraic axioms as the algebras of the bulk QFT. \nTheorem 2 (Borchers CRT Theorem) . Consider the vacuum representation of Poincar'ecovariant QFT that satisfies Haag's duality for wedges and every double cone. The modular flow of a wedge algebra is boost, and modular conjugation is the CRT transformation. As a result,", '1. The positive generators 10': '± G ± = 1 z ± ( K W -K W ( z ± ) ) = ± P ± (2.4) \nare independent of z ± and equivalent to null momenta ± P ± .', "2. The unitary flows ∆ it W and e isG ± represent the Poincar'e group:": '∆ -it W e isG ± ∆ it W = e ie ± 2 πt sG ± (2.5) \n[ G + , G -] = 0 . (2.6) \nlim t →±∞ ∆ -it W ∆ it W ( z ± ) = e ± iz ± G ± (2.7) \nJ W J W ( z ± ) = e ± 2 iz ± G ± . (2.8) \nProof. See [19-21] for a proof. For completeness, we derive the expressions for G ± explicitly. The modular Hamiltonian K W = -1 2 π log ∆ W is proportional to the generator of the boost transformation around x ± = 0: \nK W = ∫ Σ d Σ µ B ν T µν (2.9) \nwhere B = i ( x + ∂ x + -x -∂ x -) is the boost Killing vector field and T µν is the energymomentum tensor. This integral can be performed on any Cauchy slice, Σ. A convenient choice is to integrate on the null plane x -= 0. This yields \nK W = i ∫ d d -2 x i ∫ ∞ -∞ dx + x + T ++ ( x + , x -= 0 , x i ) . (2.10) \nThe expression for K W ( z + ) is the same as the expression in (2.9) but the boost Killing vector field is now B = i (( x + -z + ) ∂ x + -x -∂ x -) . With this change, we get \nK W ( z + ) = i ∫ d d -2 x i ∫ ∞ -∞ dx + ( x + -z + ) T ++ ( x + , x -= 0 , x i ) . (2.11) \nIt follows from the definition in (2.4) that \nG + = i ∫ d d -2 x i ∫ ∞ -∞ dx + T ++ ( x + , x -= 0 , x i ) = P + . (2.12) \nis indeed independent of z + and is equal to P + . Repeating the argument for G -gives -G -= -P -. The modular Hamiltonian K W is the operator in the Hilbert space that represents B . In this work, we often say informally that the modular Hamiltonian is a boost.', '3. We obtain the relations': 'Consider three different observers in two-dimensional Minkowski space: Two observers who perceive null translations by P + ( -P -) as time evolution, and a Rindler observer localized in the right wedge W moving with finite acceleration. These setups provide the first examples of future and past (modular) subalgebras (see Figure 2 with U and V replaced with x + and x -): \nCorollary 3 (Poincar\'e to Future/Past Subalgebras) . Consider the vacuum representation of a local Poincar\'e-covariant QFT that satisfies Haag\'s duality for the algebra A ( W ) . Null translations P + and -P -are a pair of commuting positive charges that generate unitary flows satisfying \n∀ z ± > 0 , A ( W ( z ± )) = e iz ± P ± A ( W ) e -iz ± P ± ⊂ A ( W ) . (2.13) \nWhen these properties hold we say W is a future (past) subalgebra of B ( H ) with respect to the unitary flow of P + ( -P -), respectively. 11 \nCorollary 4 (Poincar\'e to Modular Future/Past Subalgebras) . Consider the vacuum representation of a local Poincar\'e-covariant QFT. Then, \n- 1. With respect to the modular flow of W (boost), the algebras of W ( z + ) and W ( z -) for z ± > 0 are, respectively, modular future and past subalgebras; i.e. \n∀ t > 0 ∆ -it W A ( W ( z + ))∆ it W ⊂ A ( W ( z + )) ∀ t < 0 ∆ -it W A ( W ( z -))∆ it W ⊂ A ( W ( z -)) . (2.14) \n- 2. The commutation relation [ G + , G -] = 0 is equivalent to the modular conjugation relation \nJ W ( z -) J W ( z + ) = J W J W ( z + ) J W ( z -) J W . (2.15) \nProof. It follows from Theorem 2 that ∆ it W , ∆ it W ( z ± ) are boosts, and J W ( z ± ) are CRT transformations. The first statement follows from the boost transformations of W ( z ± ). The second statement is equivalent to the fact that null translations commute: \n[ e iz -G -, e iz + G + ] = 0 . (2.16) \nTo see this, using J W J W ( z ± ) = e ± 2 iz ± G ± in (2.8) we repackage the commutation relation in (2.16) as \n( J W ( z -) J W )( J W J W ( z + ) ) = ( J W J W ( z + ) )( J W ( z -) J W ) . (2.17) \nFigure 3 gives a geometric picture for Corollary 4, representing Poincar\'e group relations in (2.5) and (2.7), as modular dynamics by ∆ it W , and a pair of commuting exponentially growing (decaying) modes G + ( -G -) with exponent λ = 2 π . \n<latexit sha1\\_base64="amVMrjx4gdWhZGW9EC5sLrqaRUc=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoMQL2FXfB2DXjxGNA9IljA76SRDZmeXmVkhLPkELx4U8eoXefNvnCR70MSChqKqm+6uIBZcG9f9dnIrq2vrG/nNwtb2zu5ecf+goaNEMayzSESqFVCNgkusG24EtmKFNAwENoPR7dRvPqHSPJKPZhyjH9KB5H3OqLHSQ5medoslt+LOQJaJl5ESZKh1i1+dXsSSEKVhgmrd9tzY+ClVhjOBk0In0RhTNqIDbFsqaYjaT2enTsiJVXqkHylb0pCZ+nsipaHW4zCwnSE1Q73oTcX/vHZi+td+ymWcGJRsvqifCGIiMv2b9LhCZsTYEsoUt7cSNqSKMmPTKdgQvMWXl0njrOJdVi7uz0vVmyyOPBzBMZTBgyuowh3UoA4MBvAMr/DmCOfFeXc+5q05J5s5hD9wPn8Ai+uNVA==</latexit> \n<latexit 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The two-dimensional Poincar\'e algebra can be viewed as a modular Anosov system with maximal exponents λ = 2 π (see section 4). The blue lines denote the flow generated by the Killing vector field and the red lines denote commuting exponentially growing (decaying) modes G + ( G -). \n<!-- image -->', '2.2 Modular flows in curved spacetime': 'The proofs of the CRT, Bisognano-Wichmann, and Borchers theorems in the Minkowski space crucially rely on the positivity of the Hamiltonian (the spectrum condition) because it implies the analyticity of the vacuum correlation functions in complex time domains. The lack of a globally timelike Killing vector in a general curved spacetime spoils these analytic properties. Globally hyperbolic spacetimes with a bifurcate Killing horizon are special in that null translations restricted to Killing horizons are isometries that allow us to quantize the subalgebra of QFT localized on the horizon and recover some key uniqueness and analyticity results similar to the Minkowski space [1]. \nDefinition 2.1 (Spacetime with bifurcate Killing horizon) . A spacetime with bifurcate Killing horizon is a triple ( M , g µν , B ) where \n- 1. M is a manifold with the Lorentzian metric g µν describing a globally hyperbolic spacetime.\n- 2. The vector B is an everywhere smooth Killing vector that generates a flow that is well-defined for all t ∈ R . 12\n- 3. The Killing vector B vanishes on a co-dimension two orientable submanifold called the bifurcation surface σ that belongs to some Cauchy slice of M . 13 If the restriction of the Killing vector B to this Cauchy slice is everywhere timelike, we say the Killing horizon is stationary . \nConsider the spacetime M 1 , 1 × X with a metric of the form (Kruskal-like coordinates) \nds 2 = A ( UV,x i ) dUdV + B ( UV,x i ) dx i dx i (2.18) \nwhere A and B are positive-valued smooth functions with ( U, V ) ∈ R 2 . These spacetimes have a stationary bifurcate Killing horizon with a wedge reflection symmetry ( U, V ) → ( -U, -V ); see Figure 1. The Killing vector B = i ( U∂ U -V ∂ V ) generates the isometric flow ( U, V ) → ( e 2 πt U, e -2 πt V ). Null codimension one-surfaces at UV = 0 split the spacetime into four regions similar to the Minkowski spacetime. The norm of the Killing vector vanishes on UV = 0. These null hypersurfaces are future and past Killing horizons H + and H -, respectively, and we have a bifurcate surface σ at U = V = 0. For z ± > 0 we define the right wedges W ( z + , z -) = ( U > z + , V > z -, x i ) and their causal complements the left wedges W \' ( z + , z -). Eternal black holes in both asymptotically flat and asymptotically AdS spacetimes in Kruskal coordinates have metrics of this form. For instance, the metric of an eternal AdS 3 black holes in Kruskal coordinates is \nds 2 = -4 dUdV (1 + UV ) 2 + (1 -UV ) 2 (1 + UV ) 2 d ˜ ϕ 2 . (2.19) \nIn higher dimensions, the eternal AdS d +1 black brane has the metric \nds 2 = 1 z 2 ( -h ( z ) dt 2 + h ( z ) -1 dz 2 + dx i dx i ) h ( z ) = 1 -z d /z d 0 . (2.20) \nWe make the following changes of the variables \nU = -e -d 2 z 0 ( t -z ∗ ) , V = e d 2 z 0 ( t + z ∗ ) , dz ∗ dz = h ( z ) -1 (2.21) \nwhere z ∗ is the Tortoise coordinate. Since \nz ∗ = 4 z 0 d log( -UV ) (2.22) \nthe metric in Kruskal coordinates takes the form in (2.18) for \nA = 4 d 2 UV g 1 ( UV ) , B = g 2 ( -UV ) g 1 ( -UV ) = h ( z ) ( z 0 z ) 2 , g 2 ( -UV ) = 1 z 2 . (2.23) \nIn [22] Sewel proved a generalization of Bisognano-Wichmann theorem to these spacetimes: \nTheorem 5. Consider the vacuum of QFT in a spacetime with metric (2.18). Wightman fields smeared on null hypersurfaces UV = 0 and the KMS condition define a von Neumann algebra of observables for the wedge W . Then, the action of the modular flow of W is the Killing flow generated by B = i ( U∂ U -V ∂ V ) , and the modular conjugation is CRT with RT : ( U, V ) → ( -U, -V ) . \nProof. See Theorems 4 and 8 in [22]. \n<!-- image --> \n<latexit sha1\\_base64="RVGJgKl9ueDLMIUqvV5TbW22slo=">AAAB9XicbVDLSsNAFL2pr1pfVZduBovgxpKIr2XRTZcV7APatEymk3boZBJmJkoJ/Q83LhRx67+482+cpFlo64GBwzn3cs8cL+JMadv+tgorq2vrG8XN0tb2zu5eef+gpcJYEtokIQ9lx8OKciZoUzPNaSeSFAcep21vcpf67UcqFQvFg55G1A3wSDCfEayN1O8FWI99iSdJfdY/G5QrdtXOgJaJk5MK5GgMyl+9YUjigApNOFaq69iRdhMsNSOczkq9WNEIkwke0a6hAgdUuUmWeoZOjDJEfijNExpl6u+NBAdKTQPPTKYp1aKXiv953Vj7N27CRBRrKsj8kB9zpEOUVoCGTFKi+dQQTCQzWREZY4mJNkWVTAnO4peXSeu86lxVL+8vKrXbvI4iHMExnIID11CDOjSgCQQkPMMrvFlP1ov1bn3MRwtWvnMIf2B9/gB+nZKF</latexit> \n<latexit sha1\\_base64="2wFve15prXI4PJBfb4tWTEnq3bM=">AAAB9XicbVDLSsNAFL2pr1pfVZduBosgCCURX8uimy4r2Ae0aZlMJ+3QySTMTJQS+h9uXCji1n9x5984SbPQ1gMDh3Pu5Z45XsSZ0rb9bRVWVtfWN4qbpa3tnd298v5BS4WxJLRJQh7KjocV5UzQpmaa004kKQ48Ttve5C71249UKhaKBz2NqBvgkWA+I1gbqd8LsB77Ek+S+qx/NihX7KqdAS0TJycVyNEYlL96w5DEARWacKxU17Ej7SZYakY4nZV6saIRJhM8ol1DBQ6ocpMs9QydGGWI/FCaJzTK1N8bCQ6UmgaemUxTqkUvFf/zurH2b9yEiSjWVJD5IT/mSIcorQANmaRE86khmEhmsiIyxhITbYoqmRKcxS8vk9Z51bmqXt5fVGq3eR1FOIJjOAUHrqEGdWhAEwhIeIZXeLOerBfr3fqYjxasfOcQ/sD6/AF7lZKD</latexit> \n<latexit sha1\\_base64="d99sbmCbe7NFEcnYHoW47Hxnrmw=">AAAB+XicbVDLSsNAFL2pr1pfUZduBosgCCURXxuh6MZlBdMW2hAm02k7dDIJM5NCCf0TNy4UceufuPNvnLRZaOuBC4dz7p2594QJZ0o7zrdVWlldW98ob1a2tnd29+z9g6aKU0moR2Iey3aIFeVMUE8zzWk7kRRHIaetcHSf+60xlYrF4klPEupHeCBYnxGsjRTYdiM4Q7eom2CpGeaBF9hVp+bMgJaJW5AqFGgE9le3F5M0okITjpXquE6i/Sx/j3A6rXRTRRNMRnhAO4YKHFHlZ7PNp+jEKD3Uj6UpodFM/T2R4UipSRSazgjroVr0cvE/r5Pq/o2fMZGkmgoy/6ifcqRjlMeAekxSovnEEEwkM7siMsQSE23CqpgQ3MWTl0nzvOZe1S4fL6r1uyKOMhzBMZyCC9dQhwdogAcExvAMr/BmZdaL9W59zFtLVjFzCH9gff4ADVGSqQ==</latexit> \n<latexit sha1\\_base64="vG1eTSzI3Q3O4DSicBnvfh7UKY8=">AAAB+XicbVDLSsNAFL2pr1pfUZduBovgxpKIr41QdOOygn1AG8JkOmmHTiZhZlIooX/ixoUibv0Td/6NkzYLbT1w4XDOvTP3niDhTGnH+bZKK6tr6xvlzcrW9s7unr1/0FJxKgltkpjHshNgRTkTtKmZ5rSTSIqjgNN2MLrP/faYSsVi8aQnCfUiPBAsZARrI/m23fDP0C3qJVhqhrnf8u2qU3NmQMvELUgVCjR8+6vXj0kaUaEJx0p1XSfRXpa/RzidVnqpogkmIzygXUMFjqjystnmU3RilD4KY2lKaDRTf09kOFJqEgWmM8J6qBa9XPzP66Y6vPEyJpJUU0HmH4UpRzpGeQyozyQlmk8MwUQysysiQywx0SasignBXTx5mbTOa+5V7fLxolq/K+IowxEcwym4cA11eIAGNIHAGJ7hFd6szHqx3q2PeWvJKmYO4Q+szx8R95Ks</latexit> \n<latexit sha1\\_base64="iSr1BA7jCr2WIJu3qNPi5bUmmXU=">AAAB8nicdVDLSgMxFM34rPVVdekmWARXQ6b2uSu6cVnBTgvToWTSTBuamQxJRihDP8ONC0Xc+jXu/BszbQUVPRA4nHPvzb0nSDhTGqEPa219Y3Nru7BT3N3bPzgsHR27SqSS0C4RXMh+gBXlLKZdzTSn/URSHAWc9oLpde737qlUTMR3epZQP8LjmIWMYG0kzx0kWGqGOXSHpTKya6hVbzQhshstp4YcQyrVZuWyDh0bLVAGK3SGpffBSJA0orEmHCvlOSjRfpbPI5zOi4NU0QSTKR5Tz9AYR1T52WLlOTw3ygiGQpoXa7hQv3dkOFJqFgWmMsJ6on57ufiX56U6bPoZi5NU05gsPwpTDrWA+f1wxCQlms8MwUQysyskEywx0Salognh61L4P3ErtlO3a7fVcvtqFUcBnIIzcAEc0ABtcAM6oAsIEOABPIFnS1uP1ov1uixds1Y9J+AHrLdPYxORWg==</latexit> \n<latexit sha1\\_base64="8hrcp/TZVh2lonuj3YIwkw28sJI=">AAAB8nicdVDLSgMxFM34rPVVdekmWARXQ6a1r13RjcsKTluYDiWTZtrQzGRIMkIZ+hluXCji1q9x59+YaSuo6IHA4Zx7b+49QcKZ0gh9WGvrG5tb24Wd4u7e/sFh6ei4q0QqCXWJ4EL2A6woZzF1NdOc9hNJcRRw2gum17nfu6dSMRHf6VlC/QiPYxYygrWRPHeQYKkZ5tAdlsrIRo1G1XEgsuuVaquFDGlVaqjZhI6NFiiDFTrD0vtgJEga0VgTjpXyHJRoP8vnEU7nxUGqaILJFI+pZ2iMI6r8bLHyHJ4bZQRDIc2LNVyo3zsyHCk1iwJTGWE9Ub+9XPzL81IdNv2MxUmqaUyWH4Uph1rA/H44YpISzWeGYCKZ2RWSCZaYaJNS0YTwdSn8n3QrtlO3a7eX5fbVKo4COAVn4AI4oAHa4AZ0gAsIEOABPIFnS1uP1ov1uixds1Y9J+AHrLdPWtyRVQ==</latexit> \n<latexit sha1\\_base64="wW/FYuM+uQPEIsfDWGUcEmwyeqw=">AAAB6HicbVDLSgNBEOz1GeMr6tHLYBA8hV3xdQx68ZiAeUCyhNlJbzJmdnaZmRXCki/w4kERr36SN//GSbIHTSxoKKq66e4KEsG1cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqhg0Wi1i1A6pRcIkNw43AdqKQRoHAVjC6m/qtJ1Sax/LBjBP0IzqQPOSMGivVm71S2a24M5Bl4uWkDDlqvdJXtx+zNEJpmKBadzw3MX5GleFM4KTYTTUmlI3oADuWShqh9rPZoRNyapU+CWNlSxoyU39PZDTSehwFtjOiZqgXvan4n9dJTXjjZ1wmqUHJ5ovCVBATk+nXpM8VMiPGllCmuL2VsCFVlBmbTdGG4C2+vEya5xXvqnJZvyhXb/M4CnAMJ3AGHlxDFe6hBg1ggPAMr/DmPDovzrvzMW9dcfKZI/gD5/MHtfGM5A==</latexit> \n<latexit sha1\\_base64="vOzMu3M0YAhHuheb2DFFusXaGZI=">AAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4Kon4dSx68diCaQttKJvtpF272YTdjVBKf4EXD4p49Sd589+4bXPQ1gcDj/dmmJkXpoJr47rfzsrq2vrGZmGruL2zu7dfOjhs6CRTDH2WiES1QqpRcIm+4UZgK1VI41BgMxzeTf3mEyrNE/lgRikGMe1LHnFGjZXqfrdUdivuDGSZeDkpQ45at/TV6SUsi1EaJqjWbc9NTTCmynAmcFLsZBpTyoa0j21LJY1RB+PZoRNyapUeiRJlSxoyU39PjGms9SgObWdMzUAvelPxP6+dmegmGHOZZgYlmy+KMkFMQqZfkx5XyIwYWUKZ4vZWwgZUUWZsNkUbgrf48jJpnFe8q8pl/aJcvc3jKMAxnMAZeHANVbiHGvjAAOEZXuHNeXRenHfnY9664uQzR/AHzucPtG2M4w==</latexit> \nFigure 4 . There are enhanced isometry groups on future and past horizons H ± generated by P ± and the Killing flow B that act as dilation and translations: U → e 2 πt U + U 0 and V → e -2 πt + V 0 . \n<!-- image --> \nV \nThe null hypersurfaces UV = 0 have an enhanced symmetry group because null translations U → U + z + ( V → V + z -) restricted to the hypersurface V = 0 ( U = 0) generate isometric flows; 14 see Figure 4. Sewel used this to define the observables associated with the future Killing horizon H + with the null translations ∂ U corresponding to a positive Hamiltonian. Therefore, there is a canonical choice of vacuum on this null hypersurface with correlators that can be analytically continued to complex times [1]. The state of the field is extended from horizons H ± to the right wedge W by the requirement that it enjoys the CRT symmetry in the full spacetime. In physics terminology, such a state is often called a Hartle-Hawking state [23, 24]. The physical intuition to have in mind is that modular flow is the unique state-preserving automorphism flow of a von Neumann algebra that satisfies the KMS condition [25]. The Killing flow is manifestly an automorphism of the algebra of the right wedge. The choice of the Hartle-Hawking state ensures the KMS property. Therefore, in the Hartle-Hawking state the Killing flow is the modular flow of the right wedge. \nSummers and Verch proved an analog of the theorem above in curved spacetimes with bifurcate Killing horizons in a Hartle-Hawking state without assuming Wightman fields: 15 \nTheorem 6. Consider the vacuum representation of a QFT in a curved space-time with a bifurcate Killing horizon and the metric (2.18). Modular flow ∆ it W of the Hartle-Hawking state is the Killing flow generated by B , and modular conjugation is the CRT transformation with RT : ( U, V ) → ( -U, -V ) . \nProof. See [26] for a proof. \nCorollary 7. Consider the Hartle-Hawking vacuum representation of QFT in spacetime with a bifurcate Killing horizon. With respect to the modular flow of W (generated by B ), the wedge algebras of W ( z + ) and W ( z -) are, respectively, modular future and past subalgebras (see Figure 1).', "2.3 From future/past algebras to local Poincar'e": "In previous subsections, we showed that in Minkowski space QFT the Poincar'e group implies future/past algebras with respect to null translations (Corollary 3) and modular future/past algebras with respect to boosts (Corollary 4). In curved space-times, with bifurcate Killing horizons, in the Hartle-Hawking state, we found modular future/past algebras with respect to the Killing flow by B (Corollary 7). Here, we present three results in quantum ergodicity, each of which is the converse of one of the corollaries of the previous section. The physics interpretation is that there is an emergent 'local' Poincar'e algebra in quantum systems that have modular future and past subalgebras. We postpone the systematic discussion of what quantum dynamical systems that show future/past subalgebras to section 4. \nThe converse to Corollary 3 is: \nTheorem 8 (Poincar'e Group from Future/Past Algebras) . Assume ± G ± are a pair of commuting positive operators that kill the vacuum: G ± | Ω ⟩ = 0 . If a subalgebra R is simultaneously the future subalgebra for G + and the past subalgebra for G -then the modular flow ∆ it R and e isG ± generate a two-dimensional Poincar'e group. \nProof. The proof is based on the half-sided translation theorem of Borchers (see Theorem 18) that says the existence of future (past) algebras for a flow with G ± > 0 and G ± | Ω ⟩ = 0 implies the relations \n∆ -it R e isG ± ∆ it R = e ie ± 2 πt sG ± , [ G ± , K R ] = ± iG ± . (2.24) \nSince [ G + , G -] = 0, the modular flow ∆ it R and e isG ± generate a two-dimensional Poincar'e group. \nSummers proved the converse to Corollary 4: \nTheorem 9 (Poincar'e Group from Modular Future/Past Algebras) . Consider the modular flow of a von Neumann algebra R . If A + and A -are modular future and past subalgebras, respectively, and we have \nJ A -J A + = J A J A + J A -J A . (2.25) \nThe operators ± G ± defined by \n± G ± = K R -K A ± (2.26) \nare positive, and together with ∆ it R they generate the two-dimensional Poincar'e algebra. \nProof. The proof of this is based on the half-sided modular inclusion theorem (see Theorem 16) which says that the existence of modular future (past) algebras implies that the operators G ± defined by \n± G ± = K R -K A ± ≥ 0 (2.27) \nsatisfy the relations the relations \n∀ s, t ∈ R : ∆ -it R e isG ± ∆ it R = e ie ± 2 πt sG ± [ G ± , K R ] = ± iG ± . (2.28) \nAs we argued in Corollary 4 the assumption (2.25) is equivalent to [ G + , G -] = 0, therefore we obtain a two-dimensional Poincar'e group. For more details, see [21, 27]. \nIn both the Minkowski vacuum and the Hartle-Hawking state in a space-time with a bifurcate Killing horizon, the modular flow of the right wedge W is local. However, a key difference is that P ± are Killing vectors of the full Minkowski spacetime; therefore, the modular flow of every wedge W ( z + , z -) is also local. However, in curved spacetime, it is only the restriction of the action of the modular flow of W ( z ± ) to the Killing horizons H ± that is local and geometric. In general, the modular flows of W ( z + , z -) are nonlocal. For z ± > 0, we have \n∆ -it W ( z + ) O ( U, V = 0)∆ it W ( z + ) = O ( e 2 πt U, V = 0) ∆ -it W ( z -) O ( U = 0 , V )∆ it W ( z -) = O ( U = 0 , e -2 πt V ) . (2.29) \nThe action of ∆ it W ( z ± ) away from the horizon, even though nonlocal, still satisfies the group relations t 0 → e 2 πt t 0 + s generated by the local flow ∆ it W and the non-local flow e isG ± where G ± are defined as in (2.24) as implied by the half-sided modular inclusion Theorem 16. Finally, we write a theorem that resembles a converse to Corollary 7 because it applies in situations in curved spacetimes where there is only a local Poincar'e group: \nTheorem 10 (Emergent Local Poincar'e Group) . Consider a pair of von Neumann subalgebras A ± ⊂ R such that A + and A -are modular future and past subalgebras, respectively. Define \nA ± ( s ) := ∆ -is R A ± ∆ is R ± G ± ( s ) = K R -K A ± ( s ) (2.30) \nand consider the algebra A + ( -s ) and A -( s ) , and correspondingly G + ( -s ) and G -( s ) for s ≫ 1 . Then, in the scaling limit of large enough s such that z + z -e -4 πs ≪ 1 we have an emergent Poincar'e algebra \n∆ -it R e iz ± G ± ∆ it R = e ie ± 2 πt z ± G ± e iz + G + ( -s ) e iz -G -( s ) e -iz + G + ( -s ) = e iz -G -( s )+ O ( z -z + e -4 πs ) . (2.31) \n̸ \nProof. It follows from assumptions and the half-sided modular inclusion theorem (Theorem 16) that we have the algebraic relations in (2.28) for G ± (0). In general, [ G + (0) , G -(0)] = 0, however, the half-sided modular inclusion theorem also implies that for the algebras A ± ( s ) defined in (2.30) we have \n± G ± ( s ) := K R -K A ± ( s ) = ∆ -is R ( ± G ± )∆ is R = e ± 2 πs ( ± G ± ) . (2.32) \nTherefore, we have \n[ G + ( -s ) , G -( s )] = e -4 πs [ G + (0) , G -(0)] (2.33) \nwhich means that at large s they almost commute. Using the Baker-Campbell-Hausdorff expansion we find (2.31).", '3 Future/Past Subalgebras': "In this section, we discuss two examples of physical systems with past and future subalgebras and comment on the ergodic properties of their modular flows, namely strong mixing, the second law, the exponential decay of correlators, and the emergence of an approximate Poincar'e algebra.", '3.1 Large N theories': 'Consider a large N theory in a normalization where the action has the form S = N tr( L ) with no explicit N -dependence in the Lagrangian L . In the strict N →∞ limit of a large N theory, in a KMS state (canonical ensemble) of inverse temperature β , the thermal onepoint function removed single-trace operators O - ⟨O⟩ β generate an algebra of GFF [13]. Above the Hawking-Page phase transition, the expectation value of the Hamiltonian and its fluctuations in this state are \n⟨ H ⟩ β = O ( N 2 ) , ⟨ ( H -⟨ H ⟩ β ) 2 ⟩ β = O ( N 2 ) . (3.1) \nTherefore, the time evolution operator e iHt cannot be included in the GFF algebra because of its large fluctuations [28]. 16 The operator ( H -⟨ H ⟩ β ) /N can be included in this algebra, but since it commutes with all the other GFF operators it forms a center. We denote the von Neumann algebra of noncentral single-trace operators by R . For every time interval I 12 = ( t 1 , t 2 ) we define a von Neumann GFF algebra A ( t 1 ,t 2 ) with a KMS state [7, 12, 13]. They describe all the events (perturbations) that can occur in that time interval; see Figure 5. Time evolution shifts the interval and their corresponding algebras in time: \nA ( t 1 ,t 2 ) ( s ) := e iHs A ( t 1 ,t 2 ) e -iHs = A ( t 1 + s,t 2 + s ) . (3.2) \nThe events and perturbations that can occur in the entire future (past) form the future (past) von Neumann subalgebras (see Figure 5): \nA ( t 1 , ∞ ) = ( ∨ s> 0 A ( t 1 ,t 2 ) ( s ) ) \'\' ⊂ A ( -∞ , ∞ ) A ( -∞ ,t 2 ) = ( ∨ s< 0 A ( t 1 ,t 2 ) ( s ) ) \'\' ⊂ A ( -∞ , ∞ ) . (3.3) \n<latexit sha1\\_base64="amVMrjx4gdWhZGW9EC5sLrqaRUc=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoMQL2FXfB2DXjxGNA9IljA76SRDZmeXmVkhLPkELx4U8eoXefNvnCR70MSChqKqm+6uIBZcG9f9dnIrq2vrG/nNwtb2zu5ecf+goaNEMayzSESqFVCNgkusG24EtmKFNAwENoPR7dRvPqHSPJKPZhyjH9KB5H3OqLHSQ5medoslt+LOQJaJl5ESZKh1i1+dXsSSEKVhgmrd9tzY+ClVhjOBk0In0RhTNqIDbFsqaYjaT2enTsiJVXqkHylb0pCZ+nsipaHW4zCwnSE1Q73oTcX/vHZi+td+ymWcGJRsvqifCGIiMv2b9LhCZsTYEsoUt7cSNqSKMmPTKdgQvMWXl0njrOJdVi7uz0vVmyyOPBzBMZTBgyuowh3UoA4MBvAMr/DmCOfFeXc+5q05J5s5hD9wPn8Ai+uNVA==</latexit> \n<latexit sha1\\_base64="7Nix2nM2Qvaf22OXI5C9Yy/vy+0=">AAACAHicbVDLSsNAFJ3UV62vqAsXboJFqCAlKb6WVTcuK9gHtCFMptN26GQSZm6EErLxV9y4UMStn+HOv3HSZqGtBy4czrmXe+/xI84U2Pa3UVhaXlldK66XNja3tnfM3b2WCmNJaJOEPJQdHyvKmaBNYMBpJ5IUBz6nbX98m/ntRyoVC8UDTCLqBngo2IARDFryzINegGFEME+uUy+pgOecglc7ST2zbFftKaxF4uSkjHI0PPOr1w9JHFABhGOluo4dgZtgCYxwmpZ6saIRJmM8pF1NBQ6ocpPpA6l1rJW+NQilLgHWVP09keBAqUng687sXDXvZeJ/XjeGwZWbMBHFQAWZLRrE3ILQytKw+kxSAnyiCSaS6VstMsISE9CZlXQIzvzLi6RVqzoX1fP7s3L9Jo+jiA7REaogB12iOrpDDdREBKXoGb2iN+PJeDHejY9Za8HIZ/bRHxifP87/lec=</latexit> \n<latexit sha1\\_base64="GwKiaffOiuoprGhvKyoSQZY/IXA=">AAAB6nicbVDLSgNBEOz1GeMr6tHLYBA8hd3g6xj04jGieUCyhNnJbDJkdnaZ6RXCkk/w4kERr36RN//GSbIHTSxoKKq66e4KEikMuu63s7K6tr6xWdgqbu/s7u2XDg6bJk414w0Wy1i3A2q4FIo3UKDk7URzGgWSt4LR7dRvPXFtRKwecZxwP6IDJULBKFrpAXvVXqnsVtwZyDLxclKGHPVe6avbj1kacYVMUmM6npugn1GNgkk+KXZTwxPKRnTAO5YqGnHjZ7NTJ+TUKn0SxtqWQjJTf09kNDJmHAW2M6I4NIveVPzP66QYXvuZUEmKXLH5ojCVBGMy/Zv0heYM5dgSyrSwtxI2pJoytOkUbQje4svLpFmteJeVi/vzcu0mj6MAx3ACZ+DBFdTgDurQAAYDeIZXeHOk8+K8Ox/z1hUnnzmCP3A+fwAKXI2n</latexit> \n<latexit sha1\\_base64="72oNKbybjP1FCAzIrExVLH4geVg=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgKeyKr2PQi8eI5gHJEmYns8mQ2dllplcISz7BiwdFvPpF3vwbJ8keNLGgoajqprsrSKQw6LrfTmFldW19o7hZ2tre2d0r7x80TZxqxhsslrFuB9RwKRRvoEDJ24nmNAokbwWj26nfeuLaiFg94jjhfkQHSoSCUbTSA/a8XrniVt0ZyDLxclKBHPVe+avbj1kacYVMUmM6npugn1GNgkk+KXVTwxPKRnTAO5YqGnHjZ7NTJ+TEKn0SxtqWQjJTf09kNDJmHAW2M6I4NIveVPzP66QYXvuZUEmKXLH5ojCVBGMy/Zv0heYM5dgSyrSwtxI2pJoytOmUbAje4svLpHlW9S6rF/fnldpNHkcRjuAYTsGDK6jBHdShAQwG8Ayv8OZI58V5dz7mrQUnnzmEP3A+fwAI2I2m</latexit> \n<!-- image --> \nFigure 5 . (a) The time interval algebras of GFF above the Hawking-Page phase transition. (b) The algebra A ( t 1 , ∞ ) is a future subalgebra and A ( -∞ ,t 2 ) is a past subalgebra. \n<!-- image --> \nThe future and past subalgebras A ( t 1 , ∞ ) and A ( -∞ ,t 2 ) , respectively, have the following properties: \n- 1. Ergodicity: The algebraic union of the future and the past subalgebras includes all observables on the right boundary: \nA ( -∞ ,t 2 ) ∨ A ( t 1 , ∞ ) = ∨ s ∈ R A ( t 1 ,t 2 ) ( s ) = A ( -∞ , ∞ ) . (3.4) \nThe orbit of the subalgebra gets us almost everywhere 17 . \n- 2. Strong Mixing (Information loss): The infinity limit of future algebras is trivial: \nA ∞ = ∧ s> 0 A ( t, ∞ ) ( s ) = lim t →∞ A ( t, ∞ ) = λ 1 . (3.5) \nThis property is crucial for the correlators to cluster in time (information loss). \n- 3. Half-sided Inclusions (Second law): The future and past semigroup of time evolution is the restriction map (partial trace): \n∀ s > 0 : A ( t, ∞ ) ( s ) ⊂ A ( t, ∞ ) ∀ s < 0 : A ( -∞ ,t ) ( s ) ⊂ A ( -∞ ,t ) . 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sha1\\_base64="2TyYox69prPZKG28G7NCoWz8v3M=">AAAB6HicbVDLSgNBEOz1GeMr6tHLYBA8hV3xdQx68ZiAeUCyhNnJbDJmdnaZ6RXCki/w4kERr36SN//GSbIHTSxoKKq66e4KEikMuu63s7K6tr6xWdgqbu/s7u2XDg6bJk414w0Wy1i3A2q4FIo3UKDk7URzGgWSt4LR3dRvPXFtRKwecJxwP6IDJULBKFqpjr1S2a24M5Bl4uWkDDlqvdJXtx+zNOIKmaTGdDw3QT+jGgWTfFLspoYnlI3ogHcsVTTixs9mh07IqVX6JIy1LYVkpv6eyGhkzDgKbGdEcWgWvan4n9dJMbzxM6GSFLli80VhKgnGZPo16QvNGcqxJZRpYW8lbEg1ZWizKdoQvMWXl0nzvOJdVS7rF+XqbR5HAY7hBM7Ag2uowj3UoAEMODzDK7w5j86L8+58zFtXnHzmCP7A+fwB42mNAg==</latexit> \n<latexit sha1\\_base64="GRKDZx9hXaZ+PjJ+dxzezBG4YSc=">AAAB8XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKr2PQi8cI5oHJEmYnk2TI7Owy0yuEJX/hxYMiXv0bb/6Ns8keNLFgoKjq7umuIJbCoOt+O4WV1bX1jeJmaWt7Z3evvH/QNFGiGW+wSEa6HVDDpVC8gQIlb8ea0zCQvBWMbzO/9cS1EZF6wEnM/ZAOlRgIRtFKj92YahRU9rBXrrhVdwayTLycVCBHvVf+6vYjloRcIZPUmI7nxuin2Twm+bTUTQyPKRvTIe9YqmjIjZ/ONp6SE6v0ySDS9ikkM/V3R0pDYyZhYCtDiiOz6GXif14nwcG1nwoVJ8gVm380SCTBiGTnk77QnKGcWEKZFnZXwkZUU4Y2pJINwVs8eZk0z6reZfXi/rxSu8njKMIRHMMpeHAFNbiDOjSAgYJneIU3xzgvzrvzMS8tOHnPIfyB8/kDzQ+RBA==</latexit> \nFigure 6 . Under half-sided time evolution in the boundary, we have a net of subalgebras A ( t + s, ∞ ) ⊂ A ( t, ∞ ) for s > 0. Correspondingly in the bulk, the mutual information between the future subalgebra A ( t, ∞ ) and any subsystem B in the commutant decreases monotonically in time, constituting a second law of thermodynamics. \n<!-- image --> \nStrong mixing: In a KMS state, time evolution is the modular flow. The mixing property in (3.5) implies that there are no conserved charges in the algebra (the centralizer of modular flow is trivial). In [7], we used this property to prove that the observable algebra A ( -∞ , ∞ ) is a type III 1 factor. The same argument implies that general quantum K-systems have no Poincar\'e recurrences (quasiperiodic orbits) at all. See also [29]. \nSecond law: To prove a second law of thermodynamics, we consider the thermofield double GFF dual to the eternal black holes. Consider the mutual information between the future algebra of the right boundary A ( t, ∞ ) and any subalgebra of the left boundary B ∈ A \' ( -∞ , ∞ ) as an entropy function: 18 \nS ( t ) := I ( A ( t, ∞ ) : B ) . (3.7) \nThe half-sided inclusion property implies that forward time evolution with s > 0 on the right corresponds to the restriction map sending A ( t, ∞ ) to its subalgebra A ( t + s, ∞ ) ; see Figure 6. Then, it follows from strong subadditivity of entanglement entropy (the monotonicity of mutual information under partial trace) that \n∀ t 1 ≤ t 2 : S ( t 1 ) = I ( A ( t 1 , ∞ ) : B ) ≥ I ( A ( t 2 , ∞ ) : B ) = S ( t 2 ) . (3.8) \nThis is a second law of thermodynamics. In fact, any relative entropy of the bulk QFT is monotonic under restriction. This was the idea behind the proof of the generalized second law in [30]. \n<latexit sha1\\_base64="lALSfO0MKY6L+tw32bKfa4X5nFk=">AAAB8HicbVDLTgJBEOz1ifhCPXqZSEy8QHaNryPRi0dM5GFgIbPDLEyYmd3MzJrghq/w4kFjvPo53vwbB9iDgpV0UqnqTndXEHOmjet+O0vLK6tr67mN/ObW9s5uYW+/rqNEEVojEY9UM8CaciZpzTDDaTNWFIuA00YwvJn4jUeqNIvkvRnF1Be4L1nICDZWeqCdtKTHT51St1B0y+4UaJF4GSlChmq38NXuRSQRVBrCsdYtz42Nn2JlGOF0nG8nmsaYDHGftiyVWFDtp9ODx+jYKj0URsqWNGiq/p5IsdB6JALbKbAZ6HlvIv7ntRITXvkpk3FiqCSzRWHCkYnQ5HvUY4oSw0eWYKKYvRWRAVaYGJtR3obgzb+8SOqnZe+ifH53VqxcZ3Hk4BCO4AQ8uIQK3EIVakBAwDO8wpujnBfn3fmYtS452cwB/IHz+QOG3ZA+</latexit> \n<latexit sha1\\_base64="k29M+aHCD57qvPteyFF9UdvzJJ0=">AAAB8HicbVDLSgNBEJz1GeMr6tHLYBAEMeyKr2PQi8cI5iHJJsxOepMhM7PLzKwQl3yFFw+KePVzvPk3TpI9aGJBQ1HVTXdXEHOmjet+OwuLS8srq7m1/PrG5tZ2YWe3pqNEUajSiEeqERANnEmoGmY4NGIFRAQc6sHgZuzXH0FpFsl7M4zBF6QnWcgoMVZ6gHZ6okdP7eNOoeiW3AnwPPEyUkQZKp3CV6sb0USANJQTrZueGxs/JcowymGUbyUaYkIHpAdNSyURoP10cvAIH1qli8NI2ZIGT9TfEykRWg9FYDsFMX09643F/7xmYsIrP2UyTgxIOl0UJhybCI+/x12mgBo+tIRQxeytmPaJItTYjPI2BG/25XlSOy15F6Xzu7Ni+TqLI4f20QE6Qh66RGV0iyqoiigS6Bm9ojdHOS/Ou/MxbV1wspk99AfO5w+D1ZA8</latexit> \n<latexit sha1\\_base64="amVMrjx4gdWhZGW9EC5sLrqaRUc=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoMQL2FXfB2DXjxGNA9IljA76SRDZmeXmVkhLPkELx4U8eoXefNvnCR70MSChqKqm+6uIBZcG9f9dnIrq2vrG/nNwtb2zu5ecf+goaNEMayzSESqFVCNgkusG24EtmKFNAwENoPR7dRvPqHSPJKPZhyjH9KB5H3OqLHSQ5medoslt+LOQJaJl5ESZKh1i1+dXsSSEKVhgmrd9tzY+ClVhjOBk0In0RhTNqIDbFsqaYjaT2enTsiJVXqkHylb0pCZ+nsipaHW4zCwnSE1Q73oTcX/vHZi+td+ymWcGJRsvqifCGIiMv2b9LhCZsTYEsoUt7cSNqSKMmPTKdgQvMWXl0njrOJdVi7uz0vVmyyOPBzBMZTBgyuowh3UoA4MBvAMr/DmCOfFeXc+5q05J5s5hD9wPn8Ai+uNVA==</latexit> \n<latexit sha1\\_base64="qph80ctrNuemc+a1/8j8HQy6wf0=">AAACB3icbVDLSsNAFJ3UV62vqktBBotQQUsivpZVNy4r2Ac0pUymkzp0MgkzN0IJ2bnxV9y4UMStv+DOv3HSdqGtBy4czrmXe+/xIsE12Pa3lZubX1hcyi8XVlbX1jeKm1sNHcaKsjoNRahaHtFMcMnqwEGwVqQYCTzBmt7gOvObD0xpHso7GEasE5C+5D6nBIzULe66AYF7SkRymXaT8pHLpQ/DQ+16DMhB2i2W7Io9Ap4lzoSU0AS1bvHL7YU0DpgEKojWbceOoJMQBZwKlhbcWLOI0AHps7ahkgRMd5LRHyneN0oP+6EyJQGP1N8TCQm0Hgae6cyu1tNeJv7ntWPwLzoJl1EMTNLxIj8WGEKchYJ7XDEKYmgIoYqbWzG9J4pQMNEVTAjO9MuzpHFccc4qp7cnperVJI482kF7qIwcdI6q6AbVUB1R9Iie0St6s56sF+vd+hi35qzJzDb6A+vzB8XHmUI=</latexit> \n<latexit sha1\\_base64="eavZO4AQS3wYM2txVleIgsMPOGQ=">AAACCHicbVDLSsNAFJ3UV62vqksXDhahgpZEfC2rblxWsA9oQphMJ3XoZBJmboQSunTjr7hxoYhbP8Gdf+P0sdDWAxcO59zLvfcEieAabPvbys3NLywu5ZcLK6tr6xvFza2GjlNFWZ3GIlatgGgmuGR14CBYK1GMRIFgzaB3PfSbD0xpHss76CfMi0hX8pBTAkbyi7tuROCeEpFdDvysfKTdgAE5xC6XIfQPBn6xZFfsEfAscSakhCao+cUvtxPTNGISqCBatx07AS8jCjgVbFBwU80SQnuky9qGShIx7WWjRwZ43ygdHMbKlAQ8Un9PZCTSuh8FpnN4tp72huJ/XjuF8MLLuExSYJKOF4WpwBDjYSq4wxWjIPqGEKq4uRXTe6IIBZNdwYTgTL88SxrHFeescnp7UqpeTeLIox20h8rIQeeoim5QDdURRY/oGb2iN+vJerHerY9xa86azGyjP7A+fwAkC5ls</latexit> \n<latexit sha1\\_base64="zyE58ljZGPqeqaY44IdAPnl8vAQ=">AAAB7nicbVDLSgNBEOyNrxhfUY9eBoPgxbArvo5BLx4jmAckS5idzCZDZmeXmV4hLPkILx4U8er3ePNvnCR70MSChqKqm+6uIJHCoOt+O4WV1bX1jeJmaWt7Z3evvH/QNHGqGW+wWMa6HVDDpVC8gQIlbyea0yiQvBWM7qZ+64lrI2L1iOOE+xEdKBEKRtFKrTPTDTjSXrniVt0ZyDLxclKBHPVe+avbj1kacYVMUmM6npugn1GNgkk+KXVTwxPKRnTAO5YqGnHjZ7NzJ+TEKn0SxtqWQjJTf09kNDJmHAW2M6I4NIveVPzP66QY3viZUEmKXLH5ojCVBGMy/Z30heYM5dgSyrSwtxI2pJoytAmVbAje4svLpHle9a6qlw8XldptHkcRjuAYTsGDa6jBPdShAQxG8Ayv8OYkzovz7nzMWwtOPnMIf+B8/gAG249i</latexit> \n<latexit sha1\\_base64="HHXeK6bEb5FzmGd0YEYFQiLvcoE=">AAAB7XicbVDLSgNBEOyNrxhfUY9eFoPgKeyKr2PQi8cI5gHJEmYns8mY2ZllplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrTAQ36HnfTmFldW19o7hZ2tre2d0r7x80jUo1ZQ2qhNLtkBgmuGQN5ChYO9GMxKFgrXB0O/VbT0wbruQDjhMWxGQgecQpQSs1TTdkSHrlilf1ZnCXiZ+TCuSo98pf3b6iacwkUkGM6fhegkFGNHIq2KTUTQ1LCB2RAetYKknMTJDNrp24J1bpu5HStiS6M/X3REZiY8ZxaDtjgkOz6E3F/7xOitF1kHGZpMgknS+KUuGicqevu32uGUUxtoRQze2tLh0STSjagEo2BH/x5WXSPKv6l9WL+/NK7SaPowhHcAyn4MMV1OAO6tAACo/wDK/w5ijnxXl3PuatBSefOYQ/cD5/AJycjys=</latexit> \n<latexit sha1\\_base64="GRKDZx9hXaZ+PjJ+dxzezBG4YSc=">AAAB8XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKr2PQi8cI5oHJEmYnk2TI7Owy0yuEJX/hxYMiXv0bb/6Ns8keNLFgoKjq7umuIJbCoOt+O4WV1bX1jeJmaWt7Z3evvH/QNFGiGW+wSEa6HVDDpVC8gQIlb8ea0zCQvBWMbzO/9cS1EZF6wEnM/ZAOlRgIRtFKj92YahRU9rBXrrhVdwayTLycVCBHvVf+6vYjloRcIZPUmI7nxuin2Twm+bTUTQyPKRvTIe9YqmjIjZ/ONp6SE6v0ySDS9ikkM/V3R0pDYyZhYCtDiiOz6GXif14nwcG1nwoVJ8gVm380SCTBiGTnk77QnKGcWEKZFnZXwkZUU4Y2pJINwVs8eZk0z6reZfXi/rxSu8njKMIRHMMpeHAFNbiDOjSAgYJneIU3xzgvzrvzMS8tOHnPIfyB8/kDzQ+RBA==</latexit> \n<!-- image --> \nFigure 7 . (a) Near-horizon limit in the Hartle-Hawking state of an eternal AdS black hole dual to boundary GFF time interval algebras. (b) In the vicinity of the bifurcation surfaces, we obtain an emergent Poincar\'e group which has a corresponding analog in the boundary. \n<!-- image --> \nExponential decay: The thermal one-point function of GFF in this background vanishes because the expectation value of bulk fields in an eternal black hole is zero. The connected thermal two-point functions of GFF are found from retarded Green\'s functions of the wave-equation on the black hole background [31, 32]. Perturbations can be expanded in the basis of quasinormal modes O ω whose connected correlators decay exponentially fast at late times: \n⟨O ω (0) O ω ( t ) ⟩ β ∼ e ( iω R -ω I ) t . (3.9) \nTherefore, for a dense set of observables, the connected correlator is expected to decay exponentially fast. \nLocal Poincar\'e group: We saw in Theorem 5 that in the Hartle-Hawking state of QFT in an eternal AdS black hole, the modular flow of the wedge W is the Killing flow generated by B = i ( U∂ U -V ∂ V ). Near the boundary, this Killing flow becomes the generator of the boundary time evolution β∂ t . \nWith respect to the modular flow of the right wedge W , the algebras of regions W ( z + ) and W ( z -) are the modular future and past subalgebras, respectively. In the near horizon limit, we are considering the regions W ( e -s z ± ) with large s ; see Figure 7. These algebras are dual to the boundary GFF time interval algebras ( -sβ, ∞ ) and ( -∞ , sβ ). Therefore, Theorem 10 implies that in the long time limit s ≫ 1 we obtain an approximate Poincar\'e group on the boundary that is reminiscent of the local Poincar\'e group near the bifurcate Killing horizon. \nConsider a time interval on the boundary. Without loss of generality, we will choose it to be I = ( -t, t ). The von Neumann subalgebras of GFF A ( -t,t ) are dual to the algebra of half-spaces defined by a point (sphere in higher dimensions) ( z + ( t ) , z -( t )) that is null separated from both the boundary points at ± t ; see Figure 8. The Theorem 10 does \n<latexit sha1\\_base64="4M2JV0K5xxONknKYYPIG1LTjM3s=">AAAB6HicdVDJSgNBEK1xjXGLevTSGARPYUbcjiFePCZgFkiG0NOpSdr09AzdPUIY8gVePCji1U/y5t/YWYS4PSh4vFdFVb0gEVwb1/1wlpZXVtfWcxv5za3tnd3C3n5Dx6liWGexiFUroBoFl1g33AhsJQppFAhsBsPrid+8R6V5LG/NKEE/on3JQ86osVKt0i0UvZI7BXF/kS+rCHNUu4X3Ti9maYTSMEG1bntuYvyMKsOZwHG+k2pMKBvSPrYtlTRC7WfTQ8fk2Co9EsbKljRkqi5OZDTSehQFtjOiZqB/ehPxL6+dmvDKz7hMUoOSzRaFqSAmJpOvSY8rZEaMLKFMcXsrYQOqKDM2m/xiCP+TxmnJuyid186K5co8jhwcwhGcgAeXUIYbqEIdGCA8wBM8O3fOo/PivM5al5z5zAF8g/P2CZkhjNE=</latexit> \n<latexit 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The modular future and past subalgebras, dual to boundary time intervals I ± ϵ , for wedge subregions away from the Killing horizon. \n<!-- image --> \nnot apply directly because the modular flow of the region W ( z + ( t ) , z -( t )) is no longer geometric. We choose boundary time intervals I + ϵ = ( -t (1 -ϵ ) , t ) and I -ϵ = ( -t, t (1 -ϵ )) for some ϵ ≪ 1. In the vicinity of the edge ( z + ( t ) , z -( t )), we expect the modular flow of W ( z + , z -) to be well-approximated by a local boost around this bulk point; see Figure 8. We have an approximate notion of half-sided modular inclusion relations, but with respect to the subspace of observables localized near ( z + ( t ) , z -( t )). This suggests that there might be a generalization of Theorem 10 that will explain the emergence of the local Poincar\'e algebra near any point ( z + , z -) in the geometry. We postpone further exploration of this to future work.', '3.2 Forward lightcone': 'In any QFT, there are von Neumann observable algebras associated with causal developments of ball-shaped regions D that we denote by A ( D ). Every ball-shaped region is defined in terms of its past and future tips; x µ 1 and x µ 2 such that x µ 2 -x µ 1 is time-like. We denote this algebra by A D ( x µ 1 ,x µ 2 ) ; see Figure 9. \nSecond Law: The future and past algebras of A ( D ) with respect to time evolution are C ∗ -subalgebras \nA D ( x µ 1 , ∞ ) = ∨ t> 0 A D ( x µ 1 ,x µ 2 ) ( t ) A D ( -∞ ,x µ 2 ) = ∨ t< 0 A D ( x µ 1 ,x µ 2 ) ( t ) A D ( x µ 1 ,x µ 2 ) ( t ) = e iHt A D ( x µ 1 ,x µ 2 ) e -iHt (3.10) \nwhere the algebraic union is defined by taking the closure in operator norm topology. The future/past subalgebras above are C ∗ -subalgebras. Then, by the same argument as in (3.8) the mutual information between the future algebra and any subalgebra in its commutant constitutes a second law; see Figure 9. \n<latexit sha1\\_base64="zHV+9CsjbOWbY3zLfl6InJbHX44=">AAAB+3icbVDLSsNAFJ34rPUV69LNYBEqSEnE17KoC5cV7AOaGCbTSTt0ZhJmJtIS+ituXCji1h9x5984bbPQ1gMXDufcy733hAmjSjvOt7W0vLK6tl7YKG5ube/s2nulpopTiUkDxyyW7RApwqggDU01I+1EEsRDRlrh4Gbit56IVDQWD3qUEJ+jnqARxUgbKbBLt5Vh4D56PD2BHhWRHh0HdtmpOlPAReLmpAxy1AP7y+vGOOVEaMyQUh3XSbSfIakpZmRc9FJFEoQHqEc6hgrEifKz6e1jeGSULoxiaUpoOFV/T2SIKzXioenkSPfVvDcR//M6qY6u/IyKJNVE4NmiKGVQx3ASBOxSSbBmI0MQltTcCnEfSYS1iatoQnDnX14kzdOqe1E9vz8r167zOArgAByCCnDBJaiBO1AHDYDBEDyDV/Bmja0X6936mLUuWfnMPvgD6/MHenWTdA==</latexit> \n<latexit sha1\\_base64="rGifbvIbwNj4FDyvfSGodUz5FCg=">AAAB/HicbZDLSsNAFIYnXmu9Rbt0M1iEClKS4m1Z1IXLCvYCbQyT6aQdOpmEmYk0hPoqblwo4tYHcefbOGmz0NYfBj7+cw7nzO9FjEplWd/G0vLK6tp6YaO4ubW9s2vu7bdkGAtMmjhkoeh4SBJGOWkqqhjpRIKgwGOk7Y2us3r7kQhJQ36vkog4ARpw6lOMlLZcs3RTGbv2Qy+IT+DYrWVw7Jplq2pNBRfBzqEMcjVc86vXD3EcEK4wQ1J2bStSToqEopiRSbEXSxIhPEID0tXIUUCkk06Pn8Aj7fShHwr9uIJT9/dEigIpk8DTnQFSQzlfy8z/at1Y+ZdOSnkUK8LxbJEfM6hCmCUB+1QQrFiiAWFB9a0QD5FAWOm8ijoEe/7Li9CqVe3z6tndabl+lcdRAAfgEFSADS5AHdyCBmgCDBLwDF7Bm/FkvBjvxsesdcnIZ0rgj4zPH9PJk50=</latexit> \n<latexit sha1\\_base64="amVMrjx4gdWhZGW9EC5sLrqaRUc=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoMQL2FXfB2DXjxGNA9IljA76SRDZmeXmVkhLPkELx4U8eoXefNvnCR70MSChqKqm+6uIBZcG9f9dnIrq2vrG/nNwtb2zu5ecf+goaNEMayzSESqFVCNgkusG24EtmKFNAwENoPR7dRvPqHSPJKPZhyjH9KB5H3OqLHSQ5medoslt+LOQJaJl5ESZKh1i1+dXsSSEKVhgmrd9tzY+ClVhjOBk0In0RhTNqIDbFsqaYjaT2enTsiJVXqkHylb0pCZ+nsipaHW4zCwnSE1Q73oTcX/vHZi+td+ymWcGJRsvqifCGIiMv2b9LhCZsTYEsoUt7cSNqSKMmPTKdgQvMWXl0njrOJdVi7uz0vVmyyOPBzBMZTBgyuowh3UoA4MBvAMr/DmCOfFeXc+5q05J5s5hD9wPn8Ai+uNVA==</latexit> \n<latexit sha1\\_base64="Jdg/YCgjfnrQ7g3TlX5p+0RMeQA=">AAAB7nicbVDLSgMxFL3js9ZX1aWbYBFclRnxtSy6cVnBPqAdSybNtKFJJiQZsQz9CDcuFHHr97jzb0zbWWjrgQuHc+7l3nsixZmxvv/tLS2vrK6tFzaKm1vbO7ulvf2GSVJNaJ0kPNGtCBvKmaR1yyynLaUpFhGnzWh4M/Gbj1Qblsh7O1I0FLgvWcwItk5qPnWDh45Iu6WyX/GnQIskyEkZctS6pa9OLyGpoNISjo1pB76yYYa1ZYTTcbGTGqowGeI+bTsqsaAmzKbnjtGxU3ooTrQradFU/T2RYWHMSESuU2A7MPPeRPzPa6c2vgozJlVqqSSzRXHKkU3Q5HfUY5oSy0eOYKKZuxWRAdaYWJdQ0YUQzL+8SBqnleCicn53Vq5e53EU4BCO4AQCuIQq3EIN6kBgCM/wCm+e8l68d+9j1rrk5TMH8Afe5w8ZmI9u</latexit> \n<latexit sha1\\_base64="2Tr4kdyrp92duJuJ4htuua4EiOA=">AAAB7nicbVDLSgNBEOyNrxhfUY9eBoPgKewGX8egF48RzAOSNcxOZpMhM7PLzKwYlnyEFw+KePV7vPk3ziZ70MSChqKqm+6uIOZMG9f9dgorq2vrG8XN0tb2zu5eef+gpaNEEdokEY9UJ8CaciZp0zDDaSdWFIuA03Ywvsn89iNVmkXy3kxi6gs8lCxkBBsrtZ/6tYeeSPrlilt1Z0DLxMtJBXI0+uWv3iAiiaDSEI617npubPwUK8MIp9NSL9E0xmSMh7RrqcSCaj+dnTtFJ1YZoDBStqRBM/X3RIqF1hMR2E6BzUgvepn4n9dNTHjlp0zGiaGSzBeFCUcmQtnvaMAUJYZPLMFEMXsrIiOsMDE2oZINwVt8eZm0alXvonp+d1apX+dxFOEIjuEUPLiEOtxCA5pAYAzP8ApvTuy8OO/Ox7y14OQzh/AHzucPGyCPbw==</latexit> \n<latexit sha1\\_base64="uaIHXSVs0aC2hBUjB5XE29ve2TA=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoMQL2FXfB2DXjxGNA9IljA76SRDZmeXmVkhLPkELx4U8eoXefNvnCR70MSChqKqm+6uIBZcG9f9dnIrq2vrG/nNwtb2zu5ecf+goaNEMayzSESqFVCNgkusG24EtmKFNAwENoPR7dRvPqHSPJKPZhyjH9KB5H3OqLHSQzk47RZLbsWdgSwTLyMlyFDrFr86vYglIUrDBNW67bmx8VOqDGcCJ4VOojGmbEQH2LZU0hC1n85OnZATq/RIP1K2pCEz9fdESkOtx2FgO0NqhnrRm4r/ee3E9K/9lMs4MSjZfFE/EcREZPo36XGFzIixJZQpbm8lbEgVZcamU7AheIsvL5PGWcW7rFzcn5eqN1kceTiCYyiDB1dQhTuoQR0YDOAZXuHNEc6L8+58zFtzTjZzCH/gfP4AjXCNVQ==</latexit> \n<latexit sha1\\_base64="Jdg/YCgjfnrQ7g3TlX5p+0RMeQA=">AAAB7nicbVDLSgMxFL3js9ZX1aWbYBFclRnxtSy6cVnBPqAdSybNtKFJJiQZsQz9CDcuFHHr97jzb0zbWWjrgQuHc+7l3nsixZmxvv/tLS2vrK6tFzaKm1vbO7ulvf2GSVJNaJ0kPNGtCBvKmaR1yyynLaUpFhGnzWh4M/Gbj1Qblsh7O1I0FLgvWcwItk5qPnWDh45Iu6WyX/GnQIskyEkZctS6pa9OLyGpoNISjo1pB76yYYa1ZYTTcbGTGqowGeI+bTsqsaAmzKbnjtGxU3ooTrQradFU/T2RYWHMSESuU2A7MPPeRPzPa6c2vgozJlVqqSSzRXHKkU3Q5HfUY5oSy0eOYKKZuxWRAdaYWJdQ0YUQzL+8SBqnleCicn53Vq5e53EU4BCO4AQCuIQq3EIN6kBgCM/wCm+e8l68d+9j1rrk5TMH8Afe5w8ZmI9u</latexit> \n<latexit sha1\\_base64="a6ql2g27xIRn3iVNsLZOWZX56gQ=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHaNryPRC0dI5JHAhswOvTAyO7uZmTUhhC/w4kFjvPpJ3vwbB9iDgpV0UqnqTndXkAiujet+O7m19Y3Nrfx2YWd3b/+geHjU1HGqGDZYLGLVDqhGwSU2DDcC24lCGgUCW8Hofua3nlBpHssHM07Qj+hA8pAzaqxUr/aKJbfszkFWiZeREmSo9Ypf3X7M0gilYYJq3fHcxPgTqgxnAqeFbqoxoWxEB9ixVNIItT+ZHzolZ1bpkzBWtqQhc/X3xIRGWo+jwHZG1Az1sjcT//M6qQlv/QmXSWpQssWiMBXExGT2NelzhcyIsSWUKW5vJWxIFWXGZlOwIXjLL6+S5kXZuy5f1S9LlbssjjycwCmcgwc3UIEq1KABDBCe4RXenEfnxXl3PhatOSebOYY/cD5/AKC5jNY=</latexit> \n<latexit sha1\\_base64="zHV+9CsjbOWbY3zLfl6InJbHX44=">AAAB+3icbVDLSsNAFJ34rPUV69LNYBEqSEnE17KoC5cV7AOaGCbTSTt0ZhJmJtIS+ituXCji1h9x5984bbPQ1gMXDufcy733hAmjSjvOt7W0vLK6tl7YKG5ube/s2nulpopTiUkDxyyW7RApwqggDU01I+1EEsRDRlrh4Gbit56IVDQWD3qUEJ+jnqARxUgbKbBLt5Vh4D56PD2BHhWRHh0HdtmpOlPAReLmpAxy1AP7y+vGOOVEaMyQUh3XSbSfIakpZmRc9FJFEoQHqEc6hgrEifKz6e1jeGSULoxiaUpoOFV/T2SIKzXioenkSPfVvDcR//M6qY6u/IyKJNVE4NmiKGVQx3ASBOxSSbBmI0MQltTcCnEfSYS1iatoQnDnX14kzdOqe1E9vz8r167zOArgAByCCnDBJaiBO1AHDYDBEDyDV/Bmja0X6936mLUuWfnMPvgD6/MHenWTdA==</latexit> \nFigure 9 . (a) The future algebra of the ball-shaped region D ( x µ 1 , x µ 2 ) with respect to time-evolution. (b) Time evolution is the restriction map on the future subalgebra. At finite temperature this implies that the mutual information between the subalgebra of D ( x µ 1 , ∞ ) and any subalgebra in the canonically purified copy decreases monotonically in time. \n<!-- image --> \nIn a theory of massless free fields, the future and past C ∗ -subalgebras defined above are von Neumann algebras. In this case, the commutant of the future algebra A D (0 , ∞ ) is the past algebra A D ( -∞ , 0) simply because for massive free fields time-like and spacelike commutators both vanish. In this case, the modular flow of the future algebra is dilatation D , and they satisfy the algebra [33] \ne itD e isH e -itD = e ie 2 πt sH . (3.11) \nThese relations describe a mode H that grows exponentially under modular dynamics e itD , analogous to one of the conditions in (2.24). In a massive theory, the double commutant of \nFigure 10 . Conformal diagram of the Minkowski spacetime showing that the commutant of the C ∗ -algebra D ( x µ 1 , ∞ ) in Minkowski space are operators localized on null infinity. In massive QFT, there are no such operators, therefore taking the double commutant of this C ∗ -algebra gives the von Neumann algebra of all observables on the Cauchy slice: B ( H ). \n<!-- image --> \n<latexit sha1\\_base64="as9zXpMj3VFvz4J4ZvMB57zYdGo=">AAAB+nicbVDLSsNAFJ3UV62vVJduBotQQUoivpZFXbisYB/QxDCZTtqhk0mYmagh9lPcuFDErV/izr9x2mahrQcuHM65l3vv8WNGpbKsb6OwsLi0vFJcLa2tb2xumeXtlowSgUkTRywSHR9JwignTUUVI51YEBT6jLT94eXYb98TIWnEb1UaEzdEfU4DipHSkmeWr6qPnn3nhMmhQ3mg0gPPrFg1awI4T+ycVECOhmd+Ob0IJyHhCjMkZde2YuVmSCiKGRmVnESSGOEh6pOuphyFRLrZ5PQR3NdKDwaR0MUVnKi/JzIUSpmGvu4MkRrIWW8s/ud1ExWcuxnlcaIIx9NFQcKgiuA4B9ijgmDFUk0QFlTfCvEACYSVTqukQ7BnX54nraOafVo7uTmu1C/yOIpgF+yBKrDBGaiDa9AATYDBA3gGr+DNeDJejHfjY9paMPKZHfAHxucPILaTSg==</latexit> \n<latexit 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sha1\\_base64="rWYxMc2NGPZYzc1eTYRbGknIEKc=">AAACIHicbVDJSgNBEO1xjXEb9eilMQgJapgRt2PQi8coRoVMDD2dmqRJz0J3jRCGfIoXf8WLB0X0pl9jZzkY9UHB470qqur5iRQaHefTmpqemZ2bzy3kF5eWV1bttfVrHaeKQ43HMla3PtMgRQQ1FCjhNlHAQl/Cjd89G/g396C0iKMr7CXQCFk7EoHgDI3UtI89CQEWvUAxnl0W3T24y/b2vURQ3S/1s6K7MyHsOp4S7Q6WmnbBKTtD0L/EHZMCGaPatD+8VszTECLkkmldd50EGxlTKLiEft5LNSSMd1kb6oZGLATdyIYP9um2UVo0iJWpCOlQ/TmRsVDrXuibzpBhR//2BuJ/Xj3F4KSRiShJESI+WhSkkmJMB2nRllDAUfYMYVwJcyvlHWayQpNp3oTg/n75L7neL7tH5cOLg0LldBxHjmySLVIkLjkmFXJOqqRGOHkgT+SFvFqP1rP1Zr2PWqes8cwGmYD19Q2VrqFj</latexit> \nFigure 11 . (a) The modular future/past subalgebras A ± ⊂ A D ( x µ 1 ,x µ 2 ) of the causal diamond. (b) The modular flow of the subalgebras A ± ( ∓ s ) in the large s limit for the emergent local Poincar\'e algebra. The coordinates are shown in the ( x 0 , x 1 ) basis. \n<!-- image --> \nthe future (past) C ∗ -subalgebras is the observable algebra of the whole spacetime, B ( H ). Physically, this happens because the only operators that can be in the commutant of the future algebra A (0 , ∞ ) will have to be localized on segments of null infinity; see Figure 10. However, in a massive theory, there are no such excitations. Therefore, the double commutant of the future algebra A (0 , ∞ ) in the free field theory is the whole algebra B ( H ). \nLocal Poincar\'e group: It follows from Theorem 10 that every modular flow with future and past subalgebras leads to an emergent Poincar\'e algebra. Causal developments of ballshaped regions in the vacuum of conformal field theory provide an example. Consider the algebra A D ( x µ 1 ,x µ 2 ) of a ball-shape region A of radius R centered at the origin x µ = 0 in a CFT with the past and future tips are at x µ 1 = ( -R, ⃗ 0) and x µ 2 = ( R, ⃗ 0). The modular flow of this algebra is local and generated by the conformal Killing vector [33-35]: \nK = i 2 R (( R 2 -( x 0 ) 2 -| ⃗x | 2 ) ∂ 0 -2 x 0 x i ∂ i ) . (3.12) \nx ± ( s ) = R ( 1 -e ∓ 2 πs ( R -x ± ) ( R + x ± ) ) ( 1 + e ∓ 2 πs ( R -x ± ) ( R + x ± ) ) K = i 2 R (( R 2 -( x + ) 2 ) ∂ + -( R 2 -( x -) 2 ) ∂ -) . (3.13) \nThe algebra of any ball-shaped region with the same future (past) tip and its past (future) tip inside A is a modular future (past) subalgebra. For simplicity, we consider the case of 1 + 1 d CFT, but generalization to arbitrary dimensions is straightforward. We introduce radial null coordinates x ± = x 1 ± x 0 . In these coordinates, the modular flow is given by \nNow consider another ball shaped region ˜ A centered at the origin with future tip ( R/ 2 , 0) and past tip ( -R/ 2 , 0). The modular future/past subalgebras A ± correspond to the algebra \nof ˜ A translated by ± R/ 2 in the x 0 -direction as shown in Figure 11. The generators of the modular flow for A ± are \nThe corresponding modular scrambling modes \nK ± = i R [( ( R 2 ) 2 -( x + ∓ R 2 ) 2 ) ∂ + -( ( R 2 ) 2 -( x -± R 2 ) 2 ) ∂ -] = i R (( ± Rx + -( x + ) 2 ) ∂ + -( ∓ Rx --( x -) 2 ) ∂ -) . (3.14) \n± G ± = K -K ± = i 2 R ( ( x + ∓ R ) 2 ∂ + -( x -± R ) 2 ∂ -) (3.15) \nsatisfy the commutation relations \n[ G -, G + ] = 2 iK, [ G ± , K ] = ± iG ± . (3.16) \nCorresponding to the action of the modular flow of A on A ± \nA ± ( s ) = e i 2 πsK A ± e -i 2 πsK , (3.17) \nwe have the generators of the modular flow for A ± ( s ) given by \nK ± ( s ) = e i 2 πsK K ± e -i 2 πsK . (3.18) \nFollowing Theorem 10, we consider the modular flow A ± ( ∓ s ) such that in the large s limit, both regions are almost equal to A ; see Figure 11. Under this modular flow, the ( x + , x -) coordinates of the centers of the causal developments D ( A ± ( s )) change according to ( ± R 1+ e 2 πs , ∓ R 1+ e 2 πs ) and their radii change as R 1+ e -2 πs . Therefore, \nThe corresponding modular scrambling modes evolve according to \nK ± ( ∓ s ) = i (1 + e -2 πs ) 2 R ( ( R 1 + e -2 πs ) 2 -( x + ∓ R 1 + e 2 πs ) 2 ) ∂ + -i (1 + e -2 πs ) 2 R ( ( R 1 + e -2 πs ) 2 -( x -± R 1 + e 2 πs ) 2 ) ∂ -= i 2 R (( 1 -e -2 πs ) R 2 ± 2 Re -2 πs x + -(1 + e -2 πs )( x + ) 2 ) ∂ + -i 2 R (( 1 -e -2 πs ) R 2 ∓ 2 Re -2 πs x + -(1 + e -2 πs )( x + ) 2 ) ∂ -. (3.19) \n± G ± ( ∓ s ) = K -K ± ( ∓ s ) = i 2 R ( ( R 2 -( x + ) 2 ∂ + -( R 2 -( x -) 2 ∂ -) ) -K ± ( ∓ s ) = ie -2 πs 2 R ( ( x ± ∓ R ) 2 ∂ + -( x -± R ) 2 ∂ -) = e -2 πs ( ± G ± ) . (3.20) \nThe commutator of the modular scrambling modes evolves according to \n[ G + ( -s ) , G -( s )] = e -4 πs [ G + , G -] . (3.21) \nThus in the large s limit, they approximately commute and there is an emergent Poincar\'e algebra as expected from Theorem 10.', '4 Quantum Ergodicity and Future/Past Algebras': "Up to now, we argued for the emergence of a local Poincar'e group in systems with modular future/past algebras. In the remainder of this paper, we elaborate on the ergodic properties of quantum dynamical systems with future/past subalgebras and how they lead to an exponential decay of correlators, a second law of thermodynamics, and maximal modular chaos.", '4.1 Classical ergodic hierarchy': 'Drop a handful of blueberries in a glass of water and stir with a straw. We refer to the glass of water as our system R , and denote by A the subregion where we initially dropped A . The stirring is a dynamical map T t : R → R that describes the motion of blueberries in R . We say that the system is ergodic if the blueberries visit every point in the glass. Ergodic systems mix on average, meaning that from time to time there are moments where a few blueberries are localized in a small corner of the glass, but such fluctuations average out over time. There is no particular state to which the system settles, and if we wait long enough, we will arbitrarily come close to any particular blueberry configuration in the glass. This system is almost periodic . \nIf instead of blueberries, we drop a droplet of ink, it gradually spreads until it becomes uniformly distributed over the whole glass. We refer to this final state as an equilibrium state. If the equilibrium state is unique, all states evolve to this unique equilibrium state, and we say that we have strong mixing . 19 To keep track of how the system mixes, we use the connected correlation measure 20 \nC ( A : B t ) = µ ( A ∩ B t ) -µ ( A ) µ ( B t ) = µ ( A ∩ B t ) -µ ( A ) µ ( B ) (4.1) \nwith µ the Haar measure in R and A t = T t ( A ). See appendix B for the motivation for this definition. \nPhysically, strong mixing means that events such as dropping a drop of ink in a particular corner of the glass become increasingly irrelevant to the state of the system in the distant future. In other words, the connected correlators decay at late times. Of course, since the glass of water has finite volume, even though the connected correlators decay, the eventual equilibrium cannot truly forget about the initial droplet because the total amount of ink in the initial droplet decides the final color of the water. This is because the one-point function 1 vol( R ) ∫ R ρ ( x ) is invariant with time evolution. Strictly speaking, if one wants the final state to be independent of the initial amount of ink as well, one needs an infinite water reservoir to forget about the entire past. Initially, the drop of ink spreads \nexponentially in time, and the correlations decay as e -λt . In finite volume, we can expect independence from the entire past only in the approximate sense for 1 ≪ e λt ≪ vol( R ). For larger times, the ink reaches the walls of the glass and starts folding back. \nKolmogorov introduced the class of dynamical systems, called Kolmogorov systems , or in short, K-systems , in which the system forgets its entire far past history (the K-mixing property). As we argued above, forgetting about about the entire past can only emerge in the thermodynamic limit where we first send vol( R ) → ∞ and then send t → ∞ . To understand what independence from the entire past means, we drop two droplets of ink at times 0 and t . We expect the state in the far future to forget both events. In other words, it follows from the definition in (4.1) that \nlim s →∞ C ( A (1) ∩ A (2) t : B t + s ) = 0 . (4.2) \nOne can ask what happens if we wait for an infinite amount of time in between the two droplets? The system forgets about both droplets in the future only if \nlim s →∞ lim t →∞ C ( A (1) ∩ A (2) t : B t + s ) = 0 . (4.3) \nThis is equivalent to saying that the three-region correlation measure decays away: \nlim s →∞ lim t →∞ C ( A (1) : A (2) t : B t + s ) = 0 C ( A (1) : A (2) : A (3) ) = µ ( A (1) ∩ A (2) ∩ A (3) ) -µ ( A (1) ) µ ( A (2) ) µ ( A (3) ) . (4.4) \nThe strong mixing in (4.1) is sometimes called strong 2-mixing , and the property above is called strong 3-mixing . 21 It is straightforward to generalize the definition above to strong n-mixing : \nlim t n →∞ · · · lim t 1 →∞ C ( A (1) t 1 , A (2) t 1 + t 2 , · · · , A ( n ) t 1 + ··· + t n ) = 0 , C ( A (1) , A (2) , · · · A ( n ) ) = µ ( A (1) ∩ A (2) ∩ · · · A ( n ) ) -µ ( A (1) ) · · · µ ( A ( n ) ) . (4.5) \nWe expect that dropping ink droplets is an n -mixing system for all n ∈ N . 22 This still does not mean that the system is independent of its entire past . It is possible that the effect of a perturbation at time t = 0 can be mimicked, arbitrarily well, by a countably infinite set of events in the far past. In many ergodic systems given any pair of subsystems A and B as we evolve A forward and backward in time, we will eventually end up overlapping with B . K-systems emerge when the evolution of A restricted to the past explores only a subset of all possible events. \nThe defining property of K-systems is the notion of past subalgebra (or future subalgebra ). Past subalgebra is formed by the set of all perturbations (events) that can be created in the entire far past. We define the entire past as any countable union of events in the past, i.e. ∪ t< 0 A t . More formally, we define B ( t 0 ) the σ -algebra of the past as the set generated by a set of events A t with t < t 0 that includes all countable unions and countable \nintersections of A t and is closed under complements. 23 In K-systems, the past subalgebra is a strict subalgebra of all perturbations. Famous examples of K-systems include Sinai billiards and the ideal free gas of particles with elastic collisions [37, 38]. \nWe are now ready to present some definitions: \nDefinition 4.1. A classical dynamical system is a measure space ( X, Σ , µ ) with a space X , a σ -algebra Σ 24 of measurable sets, a measure µ and a measure-preserving flow T t on X . \nDefinition 4.2. We say a classical dynamical system is \n- · Ergodic: if all correlations averaged over time decay to zero: \n∀ A,B ∈ Σ : lim T →∞ 1 T ∫ T 0 dt C ( A : B t ) = 0 . (4.6) \n- · Strong Mixing: if the correlation functions decay to zero in the far future: \n∀ A,B ∈ Σ : lim t →∞ C ( A,B t ) = 0 . (4.7) \n- · K-mixing: if the correlations with the entire past decay to zero: \n∀ A, ˜ A ∈ Σ : lim t →-∞ sup B ∈ B ( t ) | C ( A : B ) | = 0 (4.8) \nwhere B ( t ) is the σ -algebra generated by { ˜ A t \' | t \' < t } . \nFor more details on classical ergodic theory see appendix C. The definition of K-mixing above is equivalent to the definition of a K-system: \nDefinition 4.3. Consider a classical dynamical system ( X, Σ , µ ) and a measure-preserving flow T t . It is a classical K-system if there exists a σ -algebra of measurable sets Σ 0 ⊂ Σ such that \n- 1. ∨ t ∈ R T t Σ 0 = Σ.\n- 2. ∧ t ∈ R T t Σ 0 = {∅ , X } .\n- 3. For all t > 0 we have T t Σ 0 ⊂ Σ 0 . \nIn K-systems, all correlations cluster in time, but the decay can be arbitrarily slow. It is an observed fact that in many dynamical systems in physics, the correlators of relevant observables decay exponentially fast. For example, when we ring a bell, perturbations can be expanded in a basis of quasi-normal modes that decay exponentially fast. 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sha1\\_base64="1q+pbZeRTC2ru4NRXfD+8dwh+xs=">AAAB6HicbVDLSgNBEOz1GeMr6tHLYBA8hV3xdQx68ZiAeUCyhNlJbzJmdnaZmRXCki/w4kERr36SN//GSbIHTSxoKKq66e4KEsG1cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqhg0Wi1i1A6pRcIkNw43AdqKQRoHAVjC6m/qtJ1Sax/LBjBP0IzqQPOSMGivV271S2a24M5Bl4uWkDDlqvdJXtx+zNEJpmKBadzw3MX5GleFM4KTYTTUmlI3oADuWShqh9rPZoRNyapU+CWNlSxoyU39PZDTSehwFtjOiZqgXvan4n9dJTXjjZ1wmqUHJ5ovCVBATk+nXpM8VMiPGllCmuL2VsCFVlBmbTdGG4C2+vEya5xXvqnJZvyhXb/M4CnAMJ3AGHlxDFe6hBg1ggPAMr/DmPDovzrvzMW9dcfKZI/gD5/MHuPmM5g==</latexit> \nFigure 12 . In the phase space X and a dynamical flow ∂ t , there can be (a) exponentially growing modes and (b) exponentially decaying modes. \n<!-- image --> \nmodes correspond to the eigenvectors of T 1 with eigenvalues less (more) than one. In continuous time evolution, they correspond to the eigenfunctions of ∂ t operator with negative (positive) eigenvalues. Informally, one can think of classical Anosov systems as a special class of K-systems for which a dense set of correlators decay exponentially fast. Since the Anosov systems are ergodic, show exponential sensitivity to perturbations, and forget about their entire past we will refer to them as classically chaotic systems . 25 \nDynamical systems of classical physics are described in terms of flows on a phase space X . 26 For example, the phase space is the space of the position and momenta of blueberries or particles of ink. 27 Bounded measurable functions on the phase space are classical observables of the system. To every smooth observable (function) on the phase space, we can associate a flow along its integral curves that preserves the volume (measure) on the phase space. Time evolution corresponds to a distinguished smooth function h called the Hamiltonian. The observables g ± that satisfy the Poisson bracket relation \n{ g ± , h } = -λ ± g ± (4.9) \ngrow/decay exponentially in time. If we denote by G ± and H = ∂ t the vectors that generate flows along the integral curves of g ± and h , respectively, we have the commutation relation \n[ G ± , H ] = -λ ± G ± . (4.10) \nWe obtain a flow on the phase space with expanding/contracting directions; see Figure 12. \nA special class of Anosov systems that play an important role for us are those with λ = 2 π . The connection between the Poincar\'e group and the Anosov systems comes from the commutation relations \n[ B,G ± ] = ± 2 πG ± (4.11) \nwith B playing the role of H . Compare Figure 12 to Figure 3. Motivated by the modular chaos results in [2, 4] we call them maximally chaotic systems . In summary, we have a hierarchy of ergodic systems 28 \nMaximal Chaos ⊂ Anosov ⊂ Kolmogorov ⊂ Strong Mixing ⊂ Ergodic . (4.12)', '4.2 Quantum ergodic hierarchy': "As a first example of a quantum dynamical system, we start with a quantum analog of the example of glass and blueberries. We consider a d -dimensional lattice R with a qubit at each site. The lattice is to be compared to a pixelated glass of water and an excitation on a site corresponds to blueberries in a pixel. In the case of a lattice, to every subsystem A ⊂ R we associate an observable algebra, which is the quantum (non-commutative) analog of the algebra of bounded functions on A . Here, the subalgebra on A corresponds to the observable algebra B ( H A ) where H A = ⊗ x ∈ A H x and H x is the Hilbert space of the qubit at the site x ∈ R . Time evolution is given by the unitary flow e iHt where H is some Hamiltonian on the lattice that, for example, couples the nearest neighbors. The quantum analog of the example of ink droplets is the continuum limit of the lattice model above, a finite temperature quantum field that lives in some background manifold M . The von Neumann algebra of observables of the QFT is R with a state ω which is represented in the GNS Hilbert space H ω = R| Ω ⟩ . At finite temperature, the set of all observables in a Cauchy slice form a von Neumann algebra R in a KMS state represented in the double copy Hilbert space as the thermofield double vector | Ω ⟩ RR ' . If we take R to be the algebra of the right wedge, then | Ω ⟩ is the vacuum or any other CRT symmetric state of QFT. We can choose dynamics to be modular flow, or more generally, any symmetry group R ( Z ) of state-preserving transformations defines a continuous (discrete) quantum dynamical system. \nSimilar to the classical case, we define connected correlation functions for every pair of observables a, b ∈ R in the state | Ω ⟩ \nf conn ab ( t ) = ⟨ a Ω | ∆ 1 / 2+ it R b Ω ⟩ - ⟨ a Ω | Ω ⟩ ⟨ Ω | b Ω ⟩ . (4.13) \nHere, we are using a convention with a factor ∆ 1 / 2 R in the definition of the correlator to make the analytic extension of the modular correlators symmetric. They can be interpreted as Left-Right correlators [7]. In a thermalizing system, the expectation is that all connected correlators of operators (except for conserved charges) decay to zero in time. This property is called quantum strong 2 -mixing : \n∀ a, b ∈ R : lim t →∞ f conn ab ( t ) = 0 . (4.14) \nWe say a system is quantum strong n -mixing if \nlim t 1 ,t 2 ,...,t n -1 →∞ ⟨ a 1 ( t 1 ) a 2 ( t 1 + t 2 ) · · · a n -1 ( n -1 ∑ i =1 t i )Ω | ∆ 1 / 2 R b Ω ⟩ = ⟨ Ω | b Ω ⟩ n -1 ∏ i =1 ⟨ a i Ω | Ω ⟩ . (4.15) \nIf we want the late time observable to be independent of the entire past of the system, we need to generalize the notion of K-systems in (4.8) to the quantum realm. In the quantum case, the algebraic union replaces the notion of σ -algebra for classical subregions. 29 Therefore, we define the future and past subalgebras as: \nDefinition 4.4 (Future/Past subalgebras) . Consider A a proper subalgebra of a von Neumann algebra R in the standard representation H = R| Ω ⟩ . Assume that the dynamics is given by an automorphism of R realized by unitaries e iHt that leaves the state | Ω ⟩ invariant, and flows the subalgebra according to A s = e iHs A e -iHs . \n- 1. Future/Past C ∗ -subalgebras: If the closure of ∨ s<t A s in norm operator topology is a proper subalgebra of R we call it the past C ∗ -algebra of A in R . The future C ∗ -algebra of A is defined the same with s < t replaced with s > t .\n- 2. Future/Past von Neumann subalgebras: If the closure of ∨ s<t A s in weak operator topology is a proper subalgebra of R we call it the past von Neumann algebra of A in R : \nA ( -∞ ,t ) = ( ∨ s<t A s ) '' . (4.16) \nThe future von Neumann algebra of A is defined the same with s < t replaced with s > t : \nA ( t, ∞ ) = ( ∨ s>t A s ) '' . (4.17) \nFor an ergodic quantum dynamical system, we have \nR = ( A ( -∞ ,t ) ∨ A ( t, ∞ ) ) '' . (4.18) \nWe are now ready to define the following ergodicity classes: \nDefinition 4.5. We say a quantum dynamical system R with a dynamical group that is a subgroup of R is \n- · Quantum Ergodic: if all correlations averaged over time decay to zero: \n∀ a, b ∈ R : lim T →∞ 1 T ∫ T 0 dt f conn ab ( t ) = 0 . (4.19) \n- · Quantum Strong Mixing: if the correlation functions decay to zero in far future: \n∀ a, b ∈ R : lim t →∞ f conn ab ( t ) = 0 . (4.20) \n- · Quantum K-Mixing: if the correlations with the entire future decay to zero: \n∀A ⊂ R , ∀ a ∈ R : lim t →∞ sup b ∈A ( t, ∞ ) | f conn ab | = 0 . (4.21) \nSimilar to the classical case, we define quantum K-systems using the three properties we saw in the example of GFF in section 3.1: \nDefinition 4.6. Consider a von Neumann algebra R represented in a Hilbert space in the standard representation H = R| Ω ⟩ , and a unitary flow e iHt that preserves the vacuum e iHt | Ω ⟩ = | Ω ⟩ . We say this quantum dynamical system is a quantum K-system if it has a proper subalgebra A ⊂ R with the following properties: \n- 1. Ergodicity: ( ∨ s ∈ R A s ) '' = R .\n- 2. Strong Mixing: ∧ s ∈ R A s = λ 1.\n- 3. Half-sided translation: For all s > 0 (or s < 0 but not both) we have A s ⊂ A . \nA subtlety that arises is that, as opposed to the classical case, quantum K-systems, and quantum K-mixing, while intimately related, are not equivalent. 30 In this work, we focus on quantum K-systems. As we will show in Lemma 13, the assumption of half-sided translation above is equivalent to the statement that A is a future subalgebra. \nFuture subalgebras A ( s, ∞ ) correspond to all observables we can measure from time s until eternity. They are very special, as when they exist, forward time evolution acts on them as the restriction map which is a unital completely positive (CP) map (the Heisenberg picture of a quantum channel) [40] \n∀ t > 0 : e iHt A ( s, ∞ ) e -iHt ⊂ A ( s, ∞ ) . (4.22) \nThe spectrum of every unital CP map is inside the unit disk: 31 \n∀ t > 0 , ∀ a ∈ A ( s, ∞ ) : ∥ e iHt ae -itH ∥ ≤ ∥ a ∥ (4.23) \nand the map is uniquely fixed in terms of its spectrum λ ∈ C such that ℑ ( λ ) ≥ 0 \ne iHt a λ e -iHt = e iλt a λ . (4.24) \nAny mode with ℑ ( λ ) > 0 decays exponentially fast towards the future, and corresponds to a quasi-normal mode. They satisfy the algebra [ H,a λ ] = λa λ with the Hamiltonian. \nAs opposed to the classical case, in the quantum case, there is no simple way of associating 'smooth' observables with state-preserving unitary flows in the Hilbert space. In defining quantum Anosov systems, we focus on state-preserving flows (symmetries): 32 \nDefinition 4.7. We define a quantum Anosov system as a quantum K-system with expanding and contracting unitary flows that preserve the state, i.e. e isG ± | Ω ⟩ = | Ω ⟩ , such that \nU ( t ) e isG ± U † ( t ) = e ie ± λt sG ± . (4.25) \nIn the quantum world, we have a similar ergodic hierarchy \n- Q. Anosov ⊂ Q. Kolmogorov ⊂ Q. Strong Mixing ⊂ Q. Weak Mixing ⊂ Q. Ergodic \nwhere the letter Q stands for quantum. In this work, we focus on two classes of measurepreserving quantum dynamical systems: \n- 1. Modular flow.\n- 2. Flow with a positive generator: G ≥ 0. \nWe will see in Theorem 18 that modular flow K-systems are equivalent to K-systems with positive generators. In the language of ergodic theory, this means that half-sided translations imply half-sided modular inclusions and vice-versa.", '4.3 Algebra types and quantum ergodicity': 'The set of (modular) time-invariant operators generates a subalgebra of observables that we refer to as the algebra of conserved charges R ρ ⊂ R . Of course, correlators of conserved charges can never decay. To focus on ergodic properties such as strong mixing we need to project to a sector where all the charges are fixed. Fixing all charges corresponds to applying a minimal projection p ∈ R ρ to the Hilbert space H → p H . For example, if we have a set of commuting charges Q 1 , · · · Q n the projection | q 1 , · · · q n ⟩ ⟨ q 1 , · · · q n | projects to the sector with fixed charges Q i p H = q i p H . In other words, we can replace the state with ρ ( p · · · p ) so that p R p contains no non-trivial time-invariant operators. From now on, we will always project out states so that there are no conserved charges: R ρ = λ 1. 33 \nIt follows immediately that there cannot be any ergodicity in any finite-dimensional or type I quantum system. In such systems, the Hamiltonian has eigenvectors | E n ⟩ , and projections | E n ⟩ ⟨ E n | generate an algebra of time-invariant operators. If we project to a sector with fixed energy we are left with operators that are also time-invariant. To have ergodicity, we need the Hamiltonian spectrum to be continuous with no point spectrum (normalizable eigenvectors). This can be seen from the definition of quantum ergodicity in (4.19). 34 In finite-dimensional quantum systems, or more generally type I von Neumann algebras, infinite time-average decoheres the operators in the energy eigenbasis, and one cannot satisfy the condition (4.19) for all operators. In fact, a stronger statement holds: \nLemma 11. Consider a von Neumann algebra R and reversible quantum dynamics U ( t ) = e iGt that leaves the normalizable GNS (cyclic and separating) vector | Ω ⟩ invariant. The system is quantum ergodic only if both conditions below hold: \n- 1. Either the algebra is type III or type II 1 in the tracial state.\n- 2. The generator G is not a positive operator. \nProof. 1: Quantum ergodicity in (4.19) means that 35 \n∀ a, b ∈ R lim T →∞ 1 T ∫ T 0 dt ⟨ a Ω | e iGt b ⟩ = ⟨ a Ω | Ω ⟩ ⟨ Ω | b Ω ⟩ . (4.26) \nWe argued before that quantum ergodicity can happen only if the algebra is infinitedimensional. Since the integral on the left-hand-side projects to the subspace of the Hilbert space that is invariant under e iGt , and | Ω ⟩ is cyclic and separating we learn that λ | Ω ⟩ is the only ray in the Hilbert space that is invariant under e iGt . Then, it follows from Theorem 1 of [41] that R is either type III or | Ω ⟩ corresponds to a tracial state on R (see also Theorem 1 of [42]). Infinite-dimensional type II algebras are either type II 1 or II ∞ , however, the trace in type II ∞ is not normalizable. Hence, we find that either the algebra is type III or it is type II 1 in a tracial state. \n2: Consider the spectrum of G as a set of λ ∈ C . Then e iλt is in the spectrum of U ( t ) = e iGt and e -iλt is in the spectrum of U ( t ) -1 . The intersection of the spectrum of U ( t ) and U ( t ) -1 is trivial if and only if for all λ 1 , λ 2 ∈ Spec( G ) we have \ne iλ 1 t = e -iλ 2 t . (4.27) \n̸ \n̸ \nWe prove by contradiction. Assume G ≥ 0 and we have quantum ergodicity. It follows from positivity of the spectrum of G that e iλ 1 t = e -iλ 2 t for all λ 1 , λ 2 ∈ Spec( G ). We say that U ( t ) has asymmetric spectrum meaning Spec( U ) ∩ Spec( U -1 ) = { 1 } . Then, it follows from Lemma 1 of [42] that U ( t ) is an inner automorphism. However, this is in contradiction with the separating condition because an inner automorphism U ( t ) would imply \n( U ( t ) -1) | Ω ⟩ = 0 (4.28) \nand U ( t ) -1 ∈ R . As a result, G is not a positive operator. \nIt is instructive to discuss the implications of the theorem above in local many-body quantum systems. Consider an infinite lattice with a boson degree of freedom on each site (lattice scalar field theory) in a finite-temperature state of a local Hamiltonian. The vector | Ω ⟩ is the thermofield double. It is invariant under e i ( H L -H R ) t which is expected to induce ergodic and strongly mixing dynamics due to clustering in time. Then, the theorem above implies that the algebra is type III, and indeed the H L -H R operator is not positive. If we take the state to be the vacuum of QFT in Minkowski space and the algebra to be a wedge W . Then, both null translations P ± > 0 preserve the vector | Ω ⟩ , and we have strong mixing of the correlators, however, they do not generate an automorphism of the algebra. \nThe above result suggests that if we want to insist on ergodic dynamics with a positive Hamiltonian ( G ≥ 0) we need to relax the assumption of invertibility. Motivated by the case of null translations and the algebra of wedges in QFT we consider the case of semigroup of positive or negative half-line: \n∀ t > 0 , U ( t ) R U ( t ) † ⊂ R . (4.29) \nThe definition above naturally motivates future and past subalgebras. We will see in section 4.6 that such a dynamical system is equivalent to modular dynamics. \nThe failure of ergodicity means that there exists a proper subalgebra A ⊂ R that is invariant under the flow. In the case of modular flow of R this invariance implies that there exists a normalizable conditional expectation from E : R → A . Physically, one can interpret the failure of quantum ergodicity as an exact quantum error correction code [40]. \nQuantum strong mixing (the assumption in (4.20)) is much stronger than quantum ergodicity. To make this concrete, we specialize our discussion to modular flows. Strong mixing in a KMS state (or more generally, in modular time) can occur only in type III 1 von Neumann algebras: \nTheorem 12. Consider the modular flow of a von Neumann algebra R as quantum dynamical system. Strong quantum mixing implies that the algebra is type a III 1 in a state with a trivial centralizer. \nProof. See [7] for a proof. 36 \nKMS states (modular dynamics) is also special in that strong mixing implies the decay of the commutator, the Weak Asymptotic Abelianness : \n∀ a, b ∈ R : lim t →∞ ⟨ Ω | [ a ( t ) , b ] | Ω ⟩ = 0 . (4.30) \nIn a quantum system with non-modular dynamics generated by a non-positive generator, we algebras with strong mixing. Examples include translations \ncan have type III and type II 1 in lattice quantum systems. This dynamics is strong mixing if we have clustering in space. We introduced the notion of future/past algebras in connection to quantum K-systems. The motivation was to make rigorous the intuition that late-time observables are independent of the entire past of the system. Independence does not necessarily mean that the latetime observable a ( t ) commutes (or almost commutes) with the whole past algebra. Such an assumption is called the Norm Asymptotic Abelianness : \nlim t →∞ ∥ [ a ( t ) , b ] ∥ = 0 . (4.31) \nThe connection between K-systems and asymptotic Abelianness is not fully understood, however, in the case of type II 1 algebras, K-system property ensures Strong Asymptotic Abelianness \nlim t →∞ ∥ [ a ( t ) , b ] | Ω ⟩ ∥ = 0 (4.32) \nwhich guarantees the decay of the four-point functions, including the out-of-time-ordered ones [44]. For a discussion of the connection between quantum K-systems and strong asymptotic Abelianness, see appendix D.3.', '4.4 Second law in quantum K-systems': "In the thermodynamic limit, the expectation is that genuinely interacting quantum systems thermalize. In particular, besides strong mixing, we expect the emergence of a second law of thermodynamics that postulates the existence of a non-negative function of the state called entropy that grows monotonically in time. \nWe saw that every KMS state with strong modular mixing corresponds to a type III 1 algebras. Therefore, in quantum thermalizing systems, the fine-grained entropy of the system diverges. The entropy in the second law must be a coarse-grained notion. Intuitively, we can justify the emergence of the second law as follows: consider the subspace of all the observables that we can access from time t to eternity and denote it by S ( t, ∞ ) . This is a future operator system . 37 If there are no Poincar'e recurrences, this provides only partial knowledge about the state of the system, and the coarse-grained entropy in the second law is some information-theoretic measure of the amount of information the observer is missing. Forward time evolution is the restriction map on the future observables: \n∀ s > 0 : e isH S ( t, ∞ ) e -isH ⊂ S ( t, ∞ ) . (4.33) \nWe say time-evolution is a half-sided translation of the future operator system S ( t, ∞ ) . Any information-theoretic measure that satisfies the data-processing inequality decreases monotonically over time. 38 Multiplying any such measure by a minus sign, we obtain a monotonically increasing coarse-grained entropy; otherwise known as a second law of thermodynamics. \nAs a first example of an entropy function, consider \nD ( t ) = 1 -sup a ∈S ( -∞ , 0) b ∈S ( t, ∞ ) | ⟨ a | ∆ 1 / 2 | b ⟩ conn β | ∥ a ∥∥ b ∥ (4.34) \nfor any t > 0, and the past and future operator systems S ( -∞ , 0) and S ( t, ∞ ) , respectively. As we increase t the set of observables S ( t, ∞ ) shrink, therefore the supremum decreases, and D increases. In the asymptotic limit t → ∞ if there are no conserved charges we expect D → 1. More generally, for any pair of non-overlapping time intervals A and B we can define the correlation 39 \nD ( A,B ) = 1 -sup a ∈S A b ∈S B | ⟨ a | ∆ 1 / 2 b ⟩ conn β | ∥ a ∥∥ b ∥ . (4.36) \nD ( A,B ) = 1 -∥ ψ AB J -ψ A ⊗ ψ B J ∥ (4.35) \nwhere we consider the canonical purification of the state and the mirror operator B J = JBJ in the purifying copy. J is the modular conjugation operator. D ( A,B ) is monotonic under partial trace, or more generally, satisfies data processing inequality for any quantum channel. \nThe coarse-grained measures we defined above are smooth functions of the state and are always bounded by one. In thermodynamics applications, it is desirable that the coarse-grained entropy is extensive so that it can grow forever in infinite systems. When the observables S A and S B are C ∗ -algebras, a simple example of such a measure is the mutual information: \nI ( A : B ) = S ( A ) + S ( B ) -S ( AB ) = S ( ψ AB ∥ ψ A ⊗ ψ B ) (4.37) \nwhere S ( ρ AB ∥ ψ A ⊗ ψ B ) is the relative entropy of a C ∗ -algebra defined by [46] and [47], or more generally by [48] for von Neumann algebras. Once again, mutual information is monotonically decreasing under restriction. This is the celebrated strong subadditivity of von Neumann entropy: 40 \n∀ t > s : B s ⊇ B t ⇒ I ( A : B s ) ≥ I ( A : B t ) . (4.38) \nIn summary, we find that when the future and past subalgebras exist their fine-grained mutual information gives a coarse-grained entropy that satisfies a second law dictated by strong subadditivity. Note that the mutual information we defined above is a generalization of mutual information in time defined in [49]. \nMore generally, we can say any quantum K-system has a second law. The following lemma clarifies the equivalence of the assumption of future/past subalgebras, half-sided translations, and the semigroup of quantum channels that we used to prove the second law: \nLemma 13. Consider a dynamical flow e iGt acting on the Hilbert space H and a proper subalgebra A ⊂ B ( H ) . 41 The following three properties are equivalent and can be used as the definition of a future algebra: \n- 1. Future subalgebra of B : the algebra A is a proper subalgebra of B ( H ) such that there exists a subalgebra B satisfying A = ∨ t> 0 B t .\n- 2. Half-sided translations: For all s > 0 we have A s ⊂ A .\n- 3. Semigroup of Quantum Channels: For all s > t > 0 we have A s ⊂ A t . \nProof. 1 → 2,3: By definition a future/past subalgebra A is a proper subalgebra of B ( H ) such that there exists a subalgebra B satisfying A = ∨ t> 0 B t . Statements (2) and (3) follow from the definition. 2,3 → 1: This follows from the observation that if either (2) or (3) are satisfied, we have A = ∨ t> 0 A t . Therefore, A is a future algebra of itself.", '4.5 Exponential decay of correlators in quantum Anosov systems': "We motivated quantum Anosov systems as examples of quantum K-systems where a dense set of correlators decay exponentially. In this section, we review the proof of this result due to Narnhofer. We start with the following Lemma: \nLemma 14 (Theorem 3.3 of [50]) . Consider an Anosov system with two automorphism groups: first, U ( s ) = e iGs with respect to which the algebra A is a future algebra, and V ( t ) = e iKt that satisfies the Anosov relation \nV ( t ) U ( s ) V ( -t ) = U ( e λt s ) (4.39) \nfor some real λ . Then, \n- 1. The spectrum of G splits the Hilbert space into three parts H = H 0 ⊕ H + ⊕ H -corresponding to the subspace of invariant states, positive and negative parts of the spectrum. This decomposition is stable under the flow of V ( t ) and U ( s ) .\n- 2. Restricting the spectrum of V ( t ) and U ( s ) to H ± and denoting their corresponding generators with K ± and G ± . The spectrum of all four generators K ± and Λ ± = log( ± G ± ) is the entire real line R . \nProof. 1: By definition, we know that H ± and H 0 are stable under the flow by U ( s ): U ( s ) H ± = H ± . It follows from the Anosov relations that for any range I in the spectrum of G we have \nV ( t ) E G ( I ) V ( -t ) = E G ( e -λt I ) (4.40) \nwhere the projections E G ( I ) are \nE G ( I ) = ∫ τ ∈ I dP G ( τ ) G = ∫ ∞ -∞ τdP G ( τ ) . (4.41) \nTherefore, H ± and H 0 are also stable under the flow by V ( t ). \n2: On H ± the operators K ± and ± G ± are strictly positive, respectively. On these subspaces the operators Λ ± = ± log( ± G ± ) satisfy the algebra \nV ( t )Λ ± V ( -t ) = Λ ± ± λt . (4.42) \nIt follows from the Stone-von Neumann theorem that K ± and Λ ± satisfy the canonical commutation relations \n[Λ ± , K ] = ± iλ (4.43) \nand the spectrum of Λ ± is the entire real line. \nEquipped with this lemma, we are now ready to prove exponential clustering in quantum Anosov systems: \nTheorem 15 (Theorem 3.6 of [50]) . Consider an Anosov system with two automorphism groups U ( s ) = e iGs with respect to which the algebra A is a future algebra, and V ( t ) = e itK that satisfies the Anosov relations in (4.39) for some λ . Then, every operator a, b ∈ A with a in the domain of G r with r > 0 and any ϵ > 0 we have \n| ⟨ a | V ( t ) b ⟩ | ≲ ( e λt /ϵ ) r ∥ G r | a ⟩ ∥∥ | b ⟩ ∥ . (4.44) \nProof. Consider the range I = ( -∞ , -ϵ ) ∪ ( ϵ, ∞ ) in the spectrum of G . For any r > 0 we have \nV ( t ) E G ( I ) V ( -t ) = E G ( e -λt I ) ≤ ( e λt G/ϵ ) 2 r (4.45) \nwhere we have used the operator statement: \nE G ( I ) ≤ ( G/ϵ ) 2 r . (4.46) \nNow, consider the correlator \n| ⟨ a | V ( t ) E G ( I ) b ⟩ | = | ⟨ E G ( e -λt I ) a | V ( t ) b ⟩ | ≤ | ⟨ E G ( e -λt I ) a | E G ( e -λt I ) a ⟩ | 1 / 2 | ⟨ b | b ⟩ | 1 / 2 ≤ ( e λt /ϵ ) r ∥ G r | a ⟩ ∥∥ | b ⟩ ∥ . (4.47) \nwhere we have used Cauchy-Schwarz inequality in the second line. Moreover for every ϵ > 0, there exists some δ > 0 such that \n∥ E G ( -ϵ, ϵ ) | a ⟩ ∥ ≤ δ ∥ | a ⟩ ∥ . (4.48) \nTherefore we can remove the projection E G ( I ) from the correlator at the cost of a small error controlled by δ : 42 \n| ⟨ a | V ( t ) b ⟩ | ≤ | ⟨ a | V ( t ) E G ( I ) b ⟩ | + | ⟨ a | V ( t ) E G ( I ' ) b ⟩ | ≤ ( e λt /ϵ ) r ∥ G r | a ⟩ ∥∥ | b ⟩ ∥ + δ ∥ | a ⟩ ∥∥ | b ⟩ ∥ . (4.49) \nHence, as long as a is in the domain of G r the correlator decays faster than the exponent e λrt .", '4.6 Half-side translations/modular inclusions': 'A modular quantum K-system is a quantum dynamical flow with a modular future proper subalgebra A ⊂ R . Then, an important result of quantum ergodic theory is that for modular flow the ergodic hierarchy simplifies (see Corollary 17). This is based on the key theorem below: \nTheorem 16 (Half-sided modular inclusions) . Consider a von Neumann algebra in a standard GNS representation {H , | Ω ⟩ , R} . If we have an ergodic modular future subalgebra A ⊂ R , then the positive operator \nG = K R -K A ≥ 0 (4.50) \ngenerates a unitary flow U ( s ) = e isG such that: \n- 1. Maximal Chaos: The flow by U ( s ) corresponds to a growing Anosov mode with a maximal Lyapunov exponent λ = 2 π , i.e. \n∀ s, t ∈ R : ∆ -it R e isG ∆ it R = ∆ -it A e isG ∆ it A = e ie 2 πt sG . (4.51) \n- 2. Future algebra with G > 0 : The algebra R is a future algebra with respect to the dynamics U ( s ) , i.e. \n∀ s > 0 : U ( s ) R U ( s ) † ⊂ R . (4.52)', '3. Quantum detailed balance: We also have': "∀ s ∈ R : J R e isG J R = J A e isG J A = e -isG . (4.53) \nProof. \nThe proof is standard and can be found in [21, 51]. \nCorollary 17. Consider a quantum dynamical system given by the modular flow of a von Neumann algebra R . Then, \nMaximal Modular Chaos ≡ Modular K-system . (4.54) \nNext, we explain why we call the third property in Theorem 16 quantum detailed balance. We start by reviewing detailed balance in classical physics. \nThe assumption of detailed balance plays an important role in classical statistical mechanics. It was used by Boltzmann to prove a second law of thermodynamics: entropy production is positive [52]. It says that if the classical equilibrium is described by the probability vector p i = e -βE i /Z where Z = ∑ i e -βE i , and a dynamical process is the matrix T ij then \ne -βE i T ij = e -βE j T ji . (4.55) \nWritten as a matrix equation this is Te -βH = e -βH T T . In other words, it is the assumption that the matrix e βH/ 2 Te -βH/ 2 is symmetric. \nIn quantum systems, the equilibrium distribution is given by the Gibbs state ρ β ∼ e -βH that in the GNS Hilbert space is represented by the thermofield double | 1 ⟩ β (the canonical purification of the Gibbs state). A dynamical process is a general unital CP map Φ : A → A (the Heisenberg analog of a quantum channel) that acts as a linear contraction on the Hilbert space: 43 \nΦ( a ) | 1 ⟩ β = Fa | 1 ⟩ β . (4.56) \nIt is natural to define quantum detailed balance as the condition ∆ 1 / 2 F = F † ∆ 1 / 2 or the self-adjointness of ∆ 1 / 4 F ∆ -1 / 4 . 44 In other words, quantum detailed balance is the assumption \n⟨ a | ∆ 1 / 2 Φ( b ) ⟩ β = ⟨ Φ( a ) | ∆ 1 / 2 b ⟩ β . (4.57) \nWe are interested in quantum channels that correspond to unitary flows Φ t ( a ) = e iGt ae -iGt and correspond to a symmetry (state-preserving) i.e., G | 1 ⟩ β = 0. Our detailed balance condition is \nU ( -t )∆ 1 / 2 = ∆ 1 / 2 U ( t ) . (4.58) \nor equivalently \nso that \nJU ( t ) J = U ( -t ) (4.59) \nas in (4.53). \nFinally, we connect modular future/past algebras (half-sided modular inclusion) and future/past algebras with a positive generator (half-sided translations): \nTheorem 18 (Half-sided translations) . Consider a dynamical flow U ( t ) = e iGt that leaves the vacuum invariant U ( t ) | Ω ⟩ = | Ω ⟩ . If A is a future algebra ( A t ⊂ A for all t > 0 ) then the following are equivalent: \n- 1. The generator is positive: G ≥ 0 .\n- 2. The quantum detailed balance condition ∆ 1 / 2 U ( t ) = U ( -t )∆ 1 / 2\n- 3. The flow satisfies Borchers' relations ∆ -is U ( t )∆ is = U ( e 2 πs t ) .\n- 4. The algebra A t is a modular future algebra of A . \nProof. 1 → 2 & 3: This is the half-sided translation theorem of Borchers. The proofs are standard, for instance, see [21] or the original paper [19].", '3 → 4: We write': '∀ s, t ≥ 0 : ∆ -is A A t ∆ is A = ∆ -is A U ( t ) A U ( t ) † ∆ is A = U ( e 2 πs t )∆ -is A A ∆ is A U ( e 2 πs t ) † = A e 2 πs t ⊂ A t (4.60) \nwhere in going to the second line we have used 3 . \n- 4 → 1: This was the second statement in Theorem 16.\n- 2 → 1: The proof is in [53], but we reproduce it here for completeness. The idea is to first write the logarithm of the modular operator as \nK = -log ∆ 1 / 2 = ∫ ∞ 0 dβ ( 1 ∆ 1 / 2 + β -1 1 + β ) (4.61) \n[ G, -log ∆ 1 / 2 ] = ∫ ∞ 0 dβ [ G, 1 ∆ 1 / 2 + β ] . (4.62) \nTo compute this commutator we make the following formal manipulations \n[ G, 1 ∆ 1 / 2 + β ] = -2 1 (∆ 1 / 2 + β ) G ∆ 1 / 2 (∆ 1 / 2 + β ) = -2 G ∆ 1 / 2 (∆ 1 / 2 + β ) 2 +2[ G, 1 ∆ 1 / 2 + β ] ∆ 1 / 2 ∆ 1 / 2 + β (4.63) \nwhere in the first line we have used [ G, (∆ 1 / 2 + β ) -1 (∆ 1 / 2 + β )] = 0. Therefore, \n[ G, 1 ∆ 1 / 2 + β ] = ( -2 G ∆ 1 / 2 (∆ 1 / 2 + β ) 2 )( 1 -2 ∆ 1 / 2 ∆ 1 / 2 + β ) -1 = 2 G ∆ 1 / 2 (∆ 1 / 2 + β )(∆ 1 / 2 -β ) = G ( 1 ∆ 1 / 2 + β + 1 ∆ 1 / 2 -β ) . (4.64) \nThe integral expression in (4.61) requires a choice of contour for β that includes the spectrum of ∆ 1 / 2 . The spectrum of ∆ is real and positive. Therefore we choose a contour parallel to the real line slightly shifted down to the lower half plane and close the contour in the upper half plane at infinity. Since the integrand is even under β →-β , \nUsing the spectral projection ∆ 1 / 2 = ∫ dP λ λ 1 / 2 we write \n[ G, -log ∆ 1 / 2 ] = G 2 ∫ ∞-iϵ -∞-iϵ dβ ( 1 ∆ 1 / 2 + β + 1 ∆ 1 / 2 -β ) = G 2 ∫ ∞ -∞ dβ ( 1 ∆ 1 / 2 + β -iϵ + 1 ∆ 1 / 2 -β + iϵ ) . (4.65) \n[ G, -log ∆ 1 / 2 ] = G 2 ∫ dP λ ∫ ∞ -∞ dβ ( 1 λ 1 / 2 + β -iϵ + 1 λ 1 / 2 -β + iϵ ) . (4.66) \nApplying the residue theorem, only the second term has poles in the upper half plane that contribute a residue of 2 πi to the integral. Hence \n[ G,K ] = 2 iπG K = -2 log ∆ 1 / 2 . (4.67) \nIt follows from the Baker-Campbell-Hausdorff expansion that 45 \ne iGs Ke -iGs = K -2 πsG . (4.70) \nTherefore, \nAs a result \nG = 1 πs ( -log ∆ 1 / 2 +log∆ 1 / 2 s ) ≥ 0 . (4.72) \ne X Y e -X = Y +[ X,Y ] + 1 2! [ X, [ X,Y ]] + 1 3! [ X, [ X, [ X,Y ]]] + · · · . (4.68) \nIn the special case when [ X,Y ] = αX , α ∈ C we have the simplification \ne X e Y e -X = e Y + αX . (4.69) \nlog ∆ 1 / 2 s = log( e iGs ∆ 1 / 2 e -iGs ) = log ∆ 1 / 2 + πsG . (4.71) \nWe find the following corollary: \nCorollary 19. Consider a quantum dynamical system {A , H Ω , | Ω ⟩ , U ( t ) } with strongly continuous dynamical flow generated by a Hamiltonian H ≥ 0 which is ergodic (i.e. | Ω ⟩ is the unique invariant state of U ( t ) ). Then there are past (or future) subalgebras. We have a quantum K-system and the algebra A is type III 1 . \nProof. This follows from Theorem 16, Theorem 18 and Theorem 12. Also, see Theorem 4 of [42].', '5 Discussion': "In this work, we formulated an aspect of bulk locality, namely sharp horizons in spacetimes with bifurcate Killing horizons, in terms of universal ergodic properties of von Neumann observable algebras of quantum gravity. We showed that the local Poincar'e symmetry near the horizon emerges in a certain scaling limit of any quantum system with modular future and past subalgebras (modular quantum K-system). In particular, we found that any modular quantum K-system is maximally chaotic. \nIn quantum K-systems late-time observables are independent of the entire past subalgebra. This implies quantum strong mixing of all orders (see (4.15)). In a theory of GFF above the Hawking-Page phase transition, we have a maximally ergodic system. The strong two-mixing property implies that at large but finite N , in the limit 1 ≪ t/β ≪ log N , the connected correlators decay. The quantum strong n-mixing says that the connected correlator ⟨ ab 1 ( t ) b 2 (2 t ) · · · b n ( nt ) ⟩ conn β in the scaling limit 1 ≪ t/β ≪ log N vanishes. This fact was used in [54] to argue for a discrete-time second law of thermodynamics at large but finite N . \nFrom our point of view, the advantage of the operator-algebraic approach is that in holographic GFF above the Hawking-Page phase transition, the boundary von Neumann algebras associated with time intervals also satisfy the assumptions of Theorem 10. Therefore, there is an exact emergent Poincar'e algebra in the boundary GFF. In general, the modular flow of the boundary future and past algebras is highly non-local. It is only in a particular 'near-horizon' limit that this emergent Poincar'e algebra matches the approximate local Poincar'e algebra in the bulk. \nIt is worth commenting on the three key properties we assumed in our general quantum dynamical system that led to our results. We have an observable algebra A that evolves unitarily according to A ( s ) = e iKs A e -iKs . The symmetry assumption ensures that the unitary flow U ( s ) = e -iKs preserves the state. In the language of ergodic theory, we say that time evolution (dynamics) is state-preserving. The second assumption of future/past subalgebras implies that the forward time evolution of these subalgebras from any initial time t 1 to final time t 2 is given by a unital CP map Φ t 1 ,t 2 (the Heisenberg picture of a quantum channel), namely restriction to a subalgebra. We have a one-parameter family Φ t 1 ,t 2 that satisfy the following semi-group property Φ t 1 ,s · Φ s,t 2 = Φ t 1 ,t 2 for t 1 ≤ s ≤ t 2 . In other words, the forward time evolution of future subalgebras is Markovian (memoryless), \nand we have a semigroup of quantum channels. 46 The assumption of the positivity of the generator of the flow corresponds to the quantum analog of detailed balance. Even though the modular Hamiltonian is not positive, quantum detailed balance is always satisfied by modular flows. It follows from Theorem 18 that modular dynamics can be generated by a flow with the positive generator: G ∼ K ( A ) -K ( A ( s )) ≥ 0. \nFinally, we would like to make a comment in connection with cosmological spacetimes. Consider an expanding universe with a (big bang) singularity at t = 0. Let us denote the past domain of dependence of a co-moving at time t by W -( t ). Then it is clear that the algebra of observables in W -( t ) are past subalgebras of the time evolution since W -( t 2 ) ⊃ W -( t 1 ) for t 2 > t 1 . Moreover, these spacetimes satisfy a second law in terms of the monotonic increase of the area of holographic screens [55, 56]. It will be interesting to investigate the connection between the second law, the observer-dependence of holographic screens [57], and the existence of past algebras in cosmological spacetimes. \nAcknowledgments We thank Thomas Faulkner and Elliott Gesteau who pointed us to the literature on quantum Anosov systems. We also thank Yidong Chen, Hong Liu, and Sasha Zhiboedov for insightful conversations. SO is also thankful to Amit Vikram for pointing him to literature on ergodicity. The authors are grateful to the DOE that supported this work through grant DE-SC0007884 and the QuantISED Fermilab consortium.", 'A From Wightman Axioms to Local Algebras': "In this appendix, we review the construction of local observable algebras from Wightman fields. \nWe start with Wightman axioms: \n- 1. Poincar'e representation: There is a Hilbert space H and a unitary representation of the Poincar'e group U (Λ µ ν , a µ ) 47 with generator that satisfies P 0 ≥ 0 and P µ P µ ≥ 0 and a unique Poincar'e-invariant vector | Ω ⟩ ∈ H called vacuum.\n- 2. Wightman fields: Wightman fields φ a are operator-valued tempered distributions on spacetime, and the Hilbert space is spanned by the field polynomials acting on vacuum (cyclicity).\n- 3. Covariance: Wightman fields transform covariantly under Poincar'e transformations according to \nU (Λ µ ν , a µ ) φ a ( x µ ) U (Λ µ ν , a µ ) † = [ D (Λ µ ν )] b a φ b (Λ µ ν x ν + a µ ) (A.1) \nwhere D (Λ µ ν ) is a finite-dimensional representation of the Lorentz group. Note that we are assuming that fields have finite spin. \n- 4. Microscopic Causality: The (anti-)commutator of Wightman fields vanishes when they are spacelike separated. \nWightman axioms imply the axioms of local QFT in the following way: Consider any bounded open region in spacetime A and consider the subspace of all test functions supported on A . Given any such test function f A we associate the Wightman fields φ a ( f A ) = ∫ x ∈ A f A ( x ) φ a ( x ). This operator is inside the ∗ -algebra generated by φ a therefore it has a densely defined adjoint and it can be closed. We define a von Neumann algebra associated to a particular field φ a and a particular function f A as the double commutant of the algebra generated by \nA a,f A = { φ a ( f A ) † , ( φ a ( f A ) † ) † } '' . (A.2) \nIt is the smallest von Neumann algebra to which the closure of the ∗ -algebra of φ a ( f A ) is affiliated [58]. Finally, we define two von Neumann algebras for a region A , the minimal algebra A ( A ) and the maximal algebra A ( A ) as \nA ( A ) = ( ∨ a,f B ,B ⊂ A A a,f B ) '' A ( A ) = ( ∧ a,f B ,A ⊂ B A a,f B ) '' . (A.3) \nThese von Neumann algebras both satisfy the isotony property \n∀ A ⊂ B : A ( A ) ⊂ A ( B ) . (A.4) \nSince the support of a function is a closed set the algebra A ( A ) defined above are continuous from the inside that is to say for all sequence of inclusions { A 1 ⊂ A 2 ⊂ · · · } with ∪ i A i = A we have \nA ( A ) = ∨ i A ( A i ) . (A.5) \nThe minimal algebras are additive \nA ( A 1 ∨ A 2 ) = ( A ( A 1 ) ∨ A ( A 2 )) '' (A.6) \nwhereas the maximal algebras A ( A i ) satisfy Haag's duality: \nA ( A ' ) = ( A ( A ) ) ' . (A.7) \nSimilarly, the set of algebras A ( A ) are continuous from the outside that is to say for all sequences of inclusions { A 1 ⊃ A 2 ⊃ · · · } with ∩ i A i = A we have \nA ( A ) = ∧ i A ( A i ) . (A.8) \nFor A that are causal developments of balls or wedges, Haag's duality is believed to hold [59] meaning that A ( A ) = A ( A ). We will focus on this case. We also find that the algebras transform covariantly under Poincar'e transformations \nU (Λ µ ν , a µ ) A ( A ) U (Λ µ ν , a µ ) † = A ((Λ µ ν , a µ ) A ) . (A.9) \nEvery operator in a ∈ A ( A ) can be canonically decomposed into a Bose part a + and a Fermi part a -. The observable algebra of A is the von Neumann algebra generated above using bosonic generators ( a + and even powers of a -). Since the Wightman fields are unbounded, we need to assume some regularity conditions to ensure that microscopic causality implies that the observable algebras of spacelike separated regions A and B commute: 48 \n[ A ( A ) , A ( B )] = 0 . (A.10) \nWe can now collect all the properties satisfied by the algebras A ( A ) and define the axioms of algebraic QFT . To every bounded open region of spacetime A we associate a C ∗ -algebra A ( A ) that satisfies the following axioms [61]: \n- 1. Isotony: If A ⊂ B then A ( A ) ⊂ A ( B ).\n- 2. Causality: If A and B are spacelike separated, we have [ A ( A ) , A ( B )] = 0.\n- 3. Vacuum: There is a unique state | Ω ⟩ that is invariant under the covering of the Poincar'e group 49 called the vacuum and the GNS representation of the algebra A ( A ) on the vacuum generates the Hilbert space H . The Poincar'e transformations are represented using unitary operators U (Λ µ ν , a µ ) and the spectrum of the generator satisfies : P 0 ≥ 0 and P µ P µ ≥ 0.\n- 4. Relativistic Covariance: For any covering of the Poincar'e group, the algebra A ( A ) transforms covariantly: U (Λ µ ν , a µ ) A ( A ) U (Λ µ ν , a µ ) † = A ((Λ µ ν , a µ ) A ) . \nIn order to define states that are localized in a region, some authors also consider an additional axiom: \n- 5. Buchholz-Wichmann Nuclearity: For an operator a ∈ A ( A ), the representation π ( a ) = e -βP 0 a | Ω ⟩ on the GNS Hilbert space satisfies ∥ π ∥ 1 ≤ e ( c/β ) n for some n, c > 0.", 'B Classical Indepedence and Correlations': 'Consider the example of a glass R of water and a drop of ink in section 4. In statistical physics, measurable functions on R such as the density of ink molecules are observables, and the state at any time associates (expectation) values to observables. 50 An observer who has access only to a corner of the glass A ⊂ R has access to a subalgebra of observables supported in this region. The indicator function on region A : \n1 A ( x ) = { 1 if x ∈ A 0 if x / ∈ A (B.1) \n<latexit sha1\\_base64="t3P7Bo1K+RmSOF3TbywBl80y8Tw=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHaNryPqxSMk8khgQ2aHBkZmZzczsyZkwxd48aAxXv0kb/6NA+xBwUo6qVR1p7sriAXXxnW/ndzK6tr6Rn6zsLW9s7tX3D9o6ChRDOssEpFqBVSj4BLrhhuBrVghDQOBzWB0N/WbT6g0j+SDGcfoh3QgeZ8zaqxUu+kWS27ZnYEsEy8jJchQ7Ra/Or2IJSFKwwTVuu25sfFTqgxnAieFTqIxpmxEB9i2VNIQtZ/ODp2QE6v0SD9StqQhM/X3REpDrcdhYDtDaoZ60ZuK/3ntxPSv/ZTLODEo2XxRPxHERGT6NelxhcyIsSWUKW5vJWxIFWXGZlOwIXiLLy+TxlnZuyxf1M5LldssjjwcwTGcggdXUIF7qEIdGCA8wyu8OY/Oi/PufMxbc042cwh/4Hz+AJYdjM8=</latexit> \n<latexit sha1\\_base64="27aF9gcNyV8Ah3PdmkHdxk/9Dsc=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHaNryPBi0dI5JHAhswOvTAyO7uZmTUhhC/w4kFjvPpJ3vwbB9iDgpV0UqnqTndXkAiujet+O7m19Y3Nrfx2YWd3b/+geHjU1HGqGDZYLGLVDqhGwSU2DDcC24lCGgUCW8Hobua3nlBpHssHM07Qj+hA8pAzaqxUr/aKJbfszkFWiZeREmSo9Ypf3X7M0gilYYJq3fHcxPgTqgxnAqeFbqoxoWxEB9ixVNIItT+ZHzolZ1bpkzBWtqQhc/X3xIRGWo+jwHZG1Az1sjcT//M6qQlv/QmXSWpQssWiMBXExGT2NelzhcyIsSWUKW5vJWxIFWXGZlOwIXjLL6+S5kXZuy5f1S9LlWoWRx5O4BTOwYMbqMA91KABDBCe4RXenEfnxXl3PhatOSebOYY/cD5/AJehjNA=</latexit> \n<latexit sha1\\_base64="27aF9gcNyV8Ah3PdmkHdxk/9Dsc=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHaNryPBi0dI5JHAhswOvTAyO7uZmTUhhC/w4kFjvPpJ3vwbB9iDgpV0UqnqTndXkAiujet+O7m19Y3Nrfx2YWd3b/+geHjU1HGqGDZYLGLVDqhGwSU2DDcC24lCGgUCW8Hobua3nlBpHssHM07Qj+hA8pAzaqxUr/aKJbfszkFWiZeREmSo9Ypf3X7M0gilYYJq3fHcxPgTqgxnAqeFbqoxoWxEB9ixVNIItT+ZHzolZ1bpkzBWtqQhc/X3xIRGWo+jwHZG1Az1sjcT//M6qQlv/QmXSWpQssWiMBXExGT2NelzhcyIsSWUKW5vJWxIFWXGZlOwIXjLL6+S5kXZuy5f1S9LlWoWRx5O4BTOwYMbqMA91KABDBCe4RXenEfnxXl3PhatOSebOYY/cD5/AJehjNA=</latexit> \n<latexit sha1\\_base64="Ah8SWUiCptj5gmV8fKTUKTRYeOk=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHaNryORi0dI5JHAhswOvTAyO7uZmTUhhC/w4kFjvPpJ3vwbB9iDgpV0UqnqTndXkAiujet+O7m19Y3Nrfx2YWd3b/+geHjU1HGqGDZYLGLVDqhGwSU2DDcC24lCGgUCW8GoOvNbT6g0j+WDGSfoR3QgecgZNVaqV3vFklt25yCrxMtICTLUesWvbj9maYTSMEG17nhuYvwJVYYzgdNCN9WYUDaiA+xYKmmE2p/MD52SM6v0SRgrW9KQufp7YkIjrcdRYDsjaoZ62ZuJ/3md1IS3/oTLJDUo2WJRmApiYjL7mvS5QmbE2BLKFLe3EjakijJjsynYELzll1dJ86LsXZev6pelyl0WRx5O4BTOwYMbqMA91KABDBCe4RXenEfnxXl3PhatOSebOYY/cD5/AJkljNE=</latexit> \nFigure 13 . An example of A and B being independently conditioned on C . A and C are the two larger circles. B is the smaller circle with the overlapping region between A and C . The overlapping regions between any two of them are the same. \n<!-- image --> \nis called a conditional expectation because it projects the algebra of all observables in R to the subalgebra of region A . If two subregions A , B do not overlap we say they have independent subalgebras because they satisfy: \n1 A 1 B = 1 ∅ . (B.2) \nArbitrary regions depend on each other through their overlap \n1 A 1 B = 1 A ∩ B . (B.3) \nIf some other region C satisfies A ∩ C = B ∩ C = A ∩ B , intuitively, we say that A and B are independently conditioned on C , and we have (see Figure 13) 51 \n1 A 1 B ∪ C = 1 A 1 C = 1 B 1 C . (B.4) \n̸ \nIn the Heisenberg picture, instead of evolving the state, we evolve observables backward in time, and at time t the ink molecules are spread over the subregion A -t := T -t ( A ). 52 If A -t ∩ B = ∅ for some t it means that some ink molecules have made it to the subregion B , and the volume of A -t ∩ B is a measure of the percentage of the initial amount of ink that made it to B : \nvol( A -t ∩ B ) vol( A ) . (B.5) \nThe unique equilibrium state is a uniform distribution, which means the probability that an ink droplet that was added in the far past is found in any other subregion B is proportional to the volume of B : \nlim t →∞ vol( A -t ∩ B ) ∼ vol( A )vol( B ) . (B.6) \n̸ \nFor some state µ , let us normalize µ ( R ) = 1 and µ ( A ) = vol( A ) vol( R ) such that µ ( A ) is a measure of the relative size of A to R . We find that the proportionality constant above is fixed to give \nlim t →∞ µ ( A -t ∩ B ) = µ ( A ) µ ( B ) . (B.7) \nTo keep track of how the system mixes, we use the correlation measure \nC ( A : B t ) = µ ( A ∩ B -t ) -µ ( A ) µ ( B -t ) = µ ( A ∩ B -t ) -µ ( A ) µ ( B ) (B.8) \nwhere we have used the fact that µ is invariant under time evolution. 53 Note that \nµ ( A | B ) := µ ( A ∩ B ) µ ( A ) (B.9) \nhas the interpretation of conditional probability. It vanishes if and only if the subregions (events) A and B are statistically independent with respect to measure µ . In other words, A and B are uncorrelated in state µ : \nC ( A : B ) = 0 ⇒ µ ( A ∩ B ) = µ ( A ) µ ( B ) . (B.10) \nTwo events A and C might be correlated but only through a mediating event called B . This occurs if \nµ ( A | B ∪ C ) = µ ( A | B ) (B.11) \nand we say that A and C are independent conditioned on B .', 'C Classical Dynamical Systems': 'In a classical dynamical system, the state of a system is a point in some configuration space X , a state is a measure µ on X , and the observables are bounded measurable functions on X : L ∞ ( X,µ ). A state is a positive continuous map from the observables L ∞ ( X,µ ) to complex numbers (expectation values). We assume X is a locally compact manifold so that we have an invariant volume (Haar measure µ ) on X . Dynamics is given by a discrete or continuous measurable transformation T g : X → X where g ∈ G some group (or monoid if the dynamics is not invertible). 54 In this work, we will always consider G = R and T t is a continuous flow on X . We say that it the dynamics is measure-preserving if it preserves the Haar measure on X : µ · T t = µ .', 'C.1 Dynamical systems of classical physics': "In classical physics, the configuration space is the phase space (an even-dimensional symplectic manifold X ). The symplectic form endows the phase space with a volume form that defines a measure µ . The observables are bounded complex functions on the phase space, i.e. L ∞ ( X,dµ ), and the expectation values in the state dµ are \nE ( f ) = ∫ x ∈ X dµ ( x ) f ( x ) . (C.1) \nNormal states are absolutely continuous measures with respect to dµ . Radon-Nikodym derivative connects normal states and functions in L 1 ( X,dµ ). For example, consider a theory of n particles on a line with generalized coordinates { q i , p i } with i = 1 , · · · , n . The phase space is ( ⃗q, ⃗p ) ∈ R 2 n = X , and the symplectic form is ω = ∑ i dq i ∧ dp i . The volume form is vol = ω n which gives the expectation values \nIt corresponds to the Haar measure on the phase space. The observables are complex continuous functions of p i and q i , for example, the Hamiltonian h ( ⃗q, ⃗p ) = 1 2 ∑ i ( p 2 i + q 2 i + V ( p i , q i )) is an observable. More generally, if the particles live on the configuration space Y spanned by q i , then phase space is the cotangent bundle X = T ∗ Y . \nE ( f ) = ∫ f ( p i , q i ) dp 1 · · · dp n dq 1 · · · dq n . (C.2) \nDynamics is described by a continuous flow in phase space: T t : X → X that evolves observables. We often choose the flow so that it preserves the measure dµ . This means that the state corresponding to dµ is invariant under the flow (it is in equilibrium). In Hamiltonian dynamics of n particles, the dynamics is defined using a distinguished real function on the phase space, namely the Hamiltonian h : X → R . Using the symplectic two form, the one-form dh can be identified with the Hamiltonian vector field H = ⃗v h on X : \nHamilton's equations of motion set \ndh = ∑ i ( ∂h ∂q i dq i + ∂h ∂p i dp i ) H = ⃗v h = ∑ i ( ∂h ∂p i ∂ ∂q i -∂h ∂q i ∂ ∂p i ) . (C.3) \n∂h ∂p i = dq i dt ∂h ∂q i = -dp i dt . (C.4) \nTherefore, Hamilton's equations of motion will ensure that H ∼ d dt . The integral curves of the Hamiltonian vector field solve the Hamilton's equations of motion keeping the energy constant (conservation of energy). We find \ndf dt = ( df )( ⃗v h ) = { f, h } { f, g } = ( df )( ⃗v g ) = ω ( ⃗v f , ⃗v g ) ⃗v f ( · ) = ω ( · , df ) . (C.5) \nIn this formalism, the Lie derivative of ω is preserved along the Hamiltonian vector field, therefore the flow preserves the volume form ω n . This is Liouville's theorem. More generally, since we have \ndf ( h ) dt = { f ( h ) , h } = 0 (C.6) \nany state that corresponds to f ( h ) ω n is also invariant. Conserved charges correspond to the functions q : X → C that are invariant under time evolution \ndq dt = { q, h } = 0 . (C.7) \nThe algebra of functions on X with the Poisson bracket forms a Poisson algebra, which is a Lie algebra with respect to the Poisson bracket and satisfies the Leibniz rule for any three observables f, g, h \n{ f, gh } = g { f, h } + h { f, g } . (C.8) \nThe Lie bracket of the vector fields is fixed by the Poisson bracket of their corresponding functions \n[ ⃗v f , ⃗v g ] = -⃗v { f,g } . (C.9) \nConsider a Hamiltonian that has the form \nh ( ⃗ p, ⃗q ) = ∑ i γ ij ( ⃗q ) p i p j . (C.10) \nIt describes the motion of a free particle on the space Y spanned by ⃗q . Here, γ ij is the inverse of the metric tensor on Y . The Hamilton equations of motion are \ndq i dt = γ ij ( ⃗q ) p j = ∂h ∂p i dp i dt = 1 2 ∂γ jk ∂q p j p k = ∂h ∂q . \n-i -i (C.11) \nwhich are the same as the geodesic equation \nd 2 q i dt 2 +Γ i jk q j q k = 0 . (C.12) \nTherefore, the dynamics of free particles on space Y is the same as the geodesic flow on this space.", 'C.2 Some key results in classical ergodicity': "A dynamical flow in a measure space induces a flow on the functions (observables) of the measure space. Consider an observable f ∈ L ∞ ( X,µ ). The dynamics flows f t ( x ) = f ( T t x ) for x ∈ X . We define a linear map U : L ∞ ( X,µ ) → L ∞ ( X,µ ). For measure-preserving flows, the L 1 -norm is invariant under U : ∥ Uf ∥ 1 = ∥ f ∥ 1 . Thus U is an isometry in L 1 ( X,µ ) and in L 2 ( X,µ ). Assuming that the dynamics is invertible, U is an invertible isometry on the Hilbert space L 2 ( X,µ ), otherwise known as a unitary operator [62, 63]. To simplify our notation, we suppress the measure µ in our vector spaces L p ( X,µ ). \nTheorem 20 (Von Neumann ergodic theorem) . If U is an isometry on a complex Hilbert space and P is the projection on to the space of all vectors invariant under U , then 1 T ∫ T 0 U t f dt converges to Pf . \nThe proofs of the above two theorems are standard and can be found in [63]. The following two Lemmas define ergodicity in the case of finite and infinite measure spaces: \nTheorem 21 (Birkhoff's ergodic theorem) . If T is a measure-preserving (but not necessarily invertible) transformation on a space X (with possibly infinite measure) and if f ∈ L 1 ( X ) , then 1 T ∫ T 0 dt f ( T t x ) converges almost everywhere. The limit function f ∗ is integrable and invariant almost everywhere. Moreover, if µ ( X ) < ∞ , then ∫ X f ( x ) dx = ∫ X f ∗ ( x ) dx . \nLemma 22 (Ergodicity in a finite measure space) . For a measure-preserving dynamical system on a finite measure space, the following are equivalent: \n- 1. The system is ergodic (or irreducible): if A is a subregion invariant under the flow, then µ ( A ) = 0 or µ ( X -A ) = 0 .\n- 2. For an integrable function f ∈ L 1 ( X ) , the ergodic limit of the flow projects to a constant function \nf ∗ = lim T →∞ 1 T ∫ T 0 dt U t f = ⟨ f ⟩ µ µ ( X ) . (C.13) \n- 3. For two functions f, g ∈ L 1 ( X ) , the two point function clusters under the ergodic limit \nlim T →∞ 1 T ∫ T 0 dt ⟨ U t fg ⟩ µ = ⟨ f ⟩ µ ⟨ g ⟩ µ µ ( X ) . (C.14) \n- 4. In the ergodic limit, any two subregions A and B become stochastically independent \nlim T →∞ 1 T ∫ T 0 dt µ ( T -t A ∩ B ) = µ ( A ) µ ( B ) µ ( X ) . (C.15) \n- 5. It is metrically transitive: For any two non-trivial subregions A and B , there exits some t > 0 such that µ ( T -t A ∩ B ) > 0 . \nProof. (1 → 2) Using Birkhoff's ergodic theorem, the ergodic limit of the flow f ∗ exists and is an integrable function that is invariant under the flow. Since the flow is ergodic, the only functions that are invariant are constant functions. In the case of finite measure space, Birkhoff's theorem also says that ∫ X dxf ∗ ( x ) = ∫ X dxf ( x ). Thus the constant has to be equal to ⟨ f ⟩ µ µ ( X ) . \n(2 → 3) Using Fubini's theorem and dominated convergence \nlim T →∞ 1 T ∫ T 0 dt ⟨ U t fg ⟩ µ = 〈 lim T →∞ 1 T ∫ T 0 dtU t fg 〉 µ = ⟨ f ⟩ µ ⟨ g ⟩ µ µ ( X ) . (C.16) \n(3 → 4) Let us choose f and g to be the indicator functions f = 1 A and g = 1 B . Then ⟨ 1 A ⟩ µ = µ ( A ) and ⟨ 1 B ⟩ µ = µ ( B ). On the left hand side, the product of the functions 1 A ( T t x )1 B ( x ) is non-zero only when there are points in B which flow to A after some time t . Thus \n⟨ 1 A ( T t x )1 B ( x ) ⟩ µ = µ ( A ∩ T t B ) = µ ( T -t A ∩ B ) . (C.17) \n(4 → 5) The average of the integral over µ ( T -t A ∩ B ) is positive since for any two non-trivial subregions A,B , the measure µ ( A ) , µ ( B ) > 0. This implies that for some t > 0, we have µ ( T -t A ∩ B ) > 0. \n(5 → 1) Suppose there exists a non-trivial subregion A that is invariant under the ergodic flow. Then the complement of the region A ' is also invariant. Thus µ ( T -t A ∩ A ' ) = µ ( A ∩ A ' ) = 0 which is a contradiction. \nIn physics, we are often interested in systems with a non-compact phase-space X and a probability distribution ω ( x ) (a state) over the phase space. Thus, we can consider the weighted measurement space ( X,ω ) and a corresponding state | ω 1 / 2 ⟩ such that for any function f ∈ L ∞ ( X,ω ), we have ⟨ f ⟩ ω = ∫ X dxω ( x ) f ( x ). For a ω -preserving flow, U † t | ω 1 / 2 ⟩ = | ω 1 / 2 ⟩ . Therefore, ∥ U t ∥ 2 ,ω = 1 and is an isometry in the Hilbert space L 2 ( X,ω ). Since it is also invertible, it is a unitary. \nLemma 23 (Ergodicity in an infinite measure space) . For a faithful state ω and ω -preserving dynamical system on an infinite measure space, the following are equivalent: \n- 1. The system is ergodic (or irreducible): if A is a subregion invariant under the flow, then ω ( A ) = 0 or ω ( X -A ) = 0 .\n- 2. For observables f ∈ L ∞ ( X,ω ) , the ergodic limit of the flow projects to a constant function \nlim T →∞ 1 T ∫ T 0 dt ( U t f ) = ⟨ f ⟩ ω . (C.18) \n- 3. For two observables f, g ∈ L ∞ ( X,ω ) , the time-averaged point function connected correlators vanish \nlim T →∞ 1 T ∫ T 0 dt ⟨ ( U t f ) g ⟩ ω = ⟨ f ⟩ ω ⟨ g ⟩ ω . (C.19) \nProof. (1 → 2) Since U t is a unitary in the Hilbert space H ω , the von Neumann ergodic theorem says that the ergodic limit projects to the set of invariant vectors. The only invariant vector is proportional to the GNS vacuum which is the GNS representation of the identity function. The proportionality constant can be obtained by taking the expectation value in the state ω : \n〈 lim T →∞ 1 T ∫ T 0 dt U t f 〉 ω = lim T →∞ 1 T ∫ T 0 dt ⟨ U t f ⟩ ω (C.20) = lim T →∞ 1 T ∫ T 0 dt ⟨ f ⟩ ω = ⟨ f ⟩ ω \nwhere in the first line we used Fubini's theorem and dominated convergence. \n(2 → 3) The result follows by using Fubini's theorem and dominated convergence. \n〈 lim T →∞ 1 T ∫ T 0 dt U t f -⟨ f ⟩ ω 〉 ω ⟨ f ⟩ ω = 0 . (C.21) \n(3 → 2) Choosing g = ⟨ f ⟩ ω , we get \nSince ω is a faithful state and f = 0, the desired result follows. \n(2 → 1) Assume there exists a subregion A that is invariant under the flow. Then the complement A ' is also invariant. If we choose f = 1 A ( x ), for some x ∈ A ' : \n̸ \nlim T →∞ 1 T ∫ T 0 dt U t 1 A ( x ) = lim T →∞ 1 T ∫ T 0 dt 1 A ( x ) = 0 . (C.22) \nSince ω is a faithful state and 1 A is a non-zero function, this is a contradiction.", 'C.3 Examples of classical Anosov systems': "To make the ergodic hierarchy less abstract, we work out several examples of dynamical systems that are Anosov systems building up towards 'maximal' Lyapunov exponents. \nDiscrete translations on a torus (Arnold's cat map): The simplest example is Arnold's cat map. It is a dynamical system in which the configuration space (phase space) is a torus, and we have a discrete dynamics given by the transformation \n( x n +1 , y n +1 ) = ( x n , y n ) · ( 1 2 1 1 ) (C.23) \nwhere ( x n , y n ) are computed mod one. This transformation is area-preserving (its determinant is one) and it is mixing. Its unique fixed point is (0 , 0), the corner of the square. It is an Anosov system with two eigenvalues (Lyapunov exponents) λ ± = 1 2 (3 ± √ 5). The corresponding orthogonal eigenvectors are: stable mode ( x, xϕ ) and unstable mode ( x, -xϕ -1 ) where ϕ = 1 2 (1 + √ 5) is the golden ratio. More generally, any SL (2 , Z ) matrix T with tr( T ) > 2 gives an Anosov system. It is hyperbolic, meaning that its eigenvalues are 0 ≤ λ -< 1 < λ + < ∞ , and it is mixing. The eigenvectors (the principle directions) have irrational slop y n /x n , therefore, their integral curves are dense in T 2 . Any point z = ( x, y ) ∈ T 2 viewed as a vector can be decomposed in terms of v ± so that \n( z + ⃗v + + z -⃗v -) · T n = z + ( λ + ) n + z -( λ -) n . (C.24) \nWe have three automorphisms of the observable algebra L ∞ ( T 2 ). They correspond to T and σ ± which are the diffeomorphisms that move points along stable and unstable modes: σ ± ( s ) = e isv ± and v ± = ∂ x ± √ 5 ∂ y . \nFree nonrelativistic particle on a line: The phase space is a two-dimensional manifold of ( x, p ) where x is the location of the particle and p is the momentum. There is a canonical choice of symplectic form ω = dx ∧ dp . The observables are real Schwartz functions ( C ∞ 0 functions) in the phase space. To every observable f we associate a 1-form df and a vector \n⃗v f = ω ( df, · ). We have a Poisson bracket between functions that naturally decides the commutator of the integral flows of their corresponding functions \n[ ⃗v f , ⃗v g ] = -⃗v { f,g } . (C.25) \nConsider the Hamiltonian h = p 2 2 m . Functions evolve according to \n( ∂ t -p m ∂ x ) f ( x, p ) = 0 . (C.26) \nThis dynamical system generates a spatial translation ∂ t = p m ∂ x . The transformation preserves the Haar measure (volume of the phase space) but it is not state-preserving, because the volume of the phase space is infinite. \nSince translations have an entirely continuous spectrum, if the translation-invariant state is unique, strong mixing is guaranteed. There is a subalgebra of observables A A on every subregion A of the phase space. There are future and past subalgebras associated with these regions as the union of all their translations. This is a K-system. If correlation length is finite, the correlators fall off exponentially fast. If instead of translations we consider the scaling σ t ( f ( x )) = f ( e -t x ) as the dynamical system we obtain the Anosov structure because scaling and translations τ s ( f ( x )) = f ( x + s ) satisfy the algebra: \n( σ -t · τ s · σ t f )( x ) = f ( x + e t s ) . (C.27) \nWe have the symmetry group t → at + b . \nFree nonrelativistic particle on hyperbolic spaces: Consider a nonrelativistic particle on a Riemannian manifold M with metric h ab . The Hamiltonian function on the phase space ( x a , p a ) is \nH ( x, p ) = 1 2 h ab ( x ) p a p b . (C.28) \nAs we saw earlier, the equation of motion is the geodesic flow T t ( x a ) = x a ( t ) with \nd 2 x a ( t ) dt +Γ a bc dx b dt dx c dt = 0 . (C.29) \nConsider M = H , the upper half-plane ℑ ( z ) ≥ 0. At any point t on a geodesic we have two horocyclic flows T t that satisfy the Anosov relations \nT t S ± s T -t = S ± e ∓ t s . (C.30) \nThe growing and decaying modes with Lyapunov exponents λ = ± 1. It follows that the motion of a free nonrelativistic particle on any manifold X = ( H 2 / Γ) × M for some other manifold M and Γ for any Fuchsian group, is also Anosov system with, at least, one pair of growing and decaying modes. See appendix C.4 for more details on this example. \nRindler observer: Consider Minkowski spacetime R 1 , 1 and a point particle moving with constant acceleration along the path ( x + ( t ) , x -( t )) \nx ± ( t ) = x 0 e ± t . (C.31) \nFrom the point of view of this observer, time evolution is generated by boost K = x + ∂ + -x -∂ -, and the null translations P ± = ∂ ± satisfy the Anosov relations \ne -tK e -sP ± e tK = e -se ∓ t P ± [ K,P ± ] = ∓ P ± . (C.32) \nIn this Anosov system, the growing and decaying modes commute. \nAdS 2 Rindler observer: AdS 2 written in global coordinates is \nds 2 = -dT 2 + dσ 2 sin 2 σ , σ ∈ [0 , π ] . (C.33) \nwhere the isometries are time translations generated by H , radial translations generated by P and boosts generated by B \nH = ∂ T , P = -(sin T cos σ∂ T +cos T sin σ∂ σ ) B = ( -cos T cos σ∂ T +sin T sin σ∂ σ ) . (C.34) \nThey satisfy the algebra \n[ H,P ] = B, [ B,H ] = P, [ B,P ] = H . (C.35) \nThe time evolution of an AdS Rindler observer is generated by B , the AdS 2 analog of boost, and the growing and decaying modes are G ± = H ± P \n[ G ± , B ] = ∓ G ± [ G + , G -] = 2 B . (C.36) \nThis is the sl (2 , R ) Lie algebra. We have an Anosov system with the growing and decaying modes corresponding to G ± . \nLocal Poincar'e group in AdS 2 : In the example of the AdS 2 Rinlder observer, the growing and the decaying modes do not commute. However, if the observer is highly boosted (localized near the bifurcation surface), it perceives the local geometry to be Rindler space, with almost commuting generators for the growing and the decaying modes. \nNear the bifurcation surface we have T = tϵ and σ = π/ 2 -rϵ for some small ϵ . The generators of the isometries can be expanded as \nP = -∂ σ + O ( ϵ 2 ) B = ( -r∂ t + t∂ r ) + O ( ϵ 2 ) H = -ϵ -1 ∂ t . (C.37) \nIn the near horizon limit, the boost B and the generators G ± = ϵH ± P satisfy an approximate two-dimensional Poincar'e algebra.", 'C.4 Geodesic and horocycle flows': "Consider the upper half-plane H and the hyperbolic metric \nds 2 = dud ¯ u ( ℑ u ) 2 (C.38) \nand d ( u 1 , u 2 ) the hyperbolic distance between a pair of points u 1 , u 2 ∈ H . Any differentiable path between u 1 and u 2 can be parameterized as f : [0 , d ( u 1 , u 2 )] → H so that the generator of the flow along the path ( f ( t ) , f ' ( t )) is in the tangent bundle of H . The length of the path is given by \nL ( f ) ∫ 1 0 | f ' ( t ) | ℑ ( f ) dt . (C.39) \nGeodesics minimize the length between two points. All geodesics of H are vertical half-lines ℜ ( u ) = y 0 = const or upper half-circles orthogonal to the real line. \nEvery differentiable path between two points can be identified with its generating vector field. Vector fields in H are elements of its tangent bundle T H ≃ H × C . The natural inner product in the tangent plane is The isometry group of the upper half-plane is \n⟨ ⃗v | ⃗ w ⟩ u ∈ H = 1 ℑ u ⟨ ⃗v | ⃗ w ⟩ R 2 . (C.40) \nPSL (2 , R ) that acts on u as g ( u ) = au + b cu + d with g = ( a b c d ) with ad -bc = 1 and g ∼ g · ( -I ). \nIts action can be extended to the tangent bundle as \ng ( z,⃗v ) = ( g ( z ) , g ' ( z ) ⃗v ) g ' ( z ) = 1 ( cz + d ) 2 . (C.41) \nThe inner product in (C.40) has the property that under the action of PSL (2 , R ) on the tangent bundle the norm of the vectors remains unchanged \n⟨ ⃗v | ⃗v ⟩ z = ⟨ g ' ( z ) ⃗v | g ' ( z ) ⃗v ⟩ g ( z ) . (C.42) \nThe action of PSL (2 , R ) sends a unit norm vector at z to another unit norm vector at g ( z ). Furthermore, for any two points on the tangent bundle ( z 1 , ⃗v 1 ) and ( z 2 , ⃗v 2 ) with unit norm vectors there is a unique g ∈ PSL (2 , R ) such that z 2 = g ( z 1 ) and ⃗v 2 = g ' ( z ) ⃗v 1 . Therefore, there is a one-to-one map between PSL (2 , R ) and the unit norm tangent bundle T 1 H , and one can identify them. 55 For example, we can identify the identity operator in PSL (2 , R ) with the element ( i, ⃗ i ) ∈ T 1 H , and g ∈ PSL (2 , R ) with ( g ( i ) , g ' ( ⃗ i )). \nConsider the points u 1 = i and u 2 = ei . The path f ( t ) = e t i is the unique geodesic of unit speed that passes between them. Therefore, ( f ( t ) , f ' ( t )) ∈ T 1 H . It is clear that PSL (2 , R ) sends one geodesic to another. \nLemma 24. Given two points u 1 , u 2 ∈ H there is a unique geodesic path f : [0 , d ( u 1 , u 2 )] → H with unit speed between them. There is also a unique element of PSL (2 , Z ) such that f ( t ) = g ( e t i ) . \nProof. To show this first, we use g ∈ PSL (2 , R ) to map g ( u 1 ) = i and g ( u 2 ) = ei . Then, the element of PSL (2 , R ) that is g ( e t i ) generates the unique geodesic flow of unit speed between u 1 and u 2 . \nThere is a one-to-one correspondence between the points in ( u,⃗v ) ∈ T 1 H and the unit speed geodesics that go through u in the direction of ⃗v . Since moving along a geodesic ( u ( t ) , ⃗v ( t )) preserves the points in T 1 H it corresponds to a flow on PSL (2 , R ) that is multiplication on the right by \ng → g · ( e -t/ 2 0 0 e t/ 2 ) (C.43) \nAt the point ( i, ⃗ i ) ∈ T 1 H , in the direction tangent to the geodesic flow, there are two isometric flows called the horocycle flows . Since they move points on T 1 H they also correspond to flows on PSL (2 , R ). The stable horocycle corresponds to ( i, ⃗ i ) → ( i + t, ⃗ i ) that corresponds to \ng → g · ( 1 -s 0 1 ) \nwhereas the unstable one corresponds to reversing the direction of the tangent ( u,⃗v ) → ( u, -⃗v ). In PSL (2 , R ) this action corresponds to w ( u ) = -1 /u or W = ( 0 1 -1 0 ) . As a \nresult, the unstable horocycle corresponds to the flow generated by \ng → g · ( 1 0 s 1 ) . (C.44) \nConsider the action g → g · T . From the eigenvalue system \ng · ( T -λ 1) = 0 (C.45) \nit follows that \nλ 2 -tr( T ) λ +1 = 0 λ ± = 1 2 (tr T ± √ Tr( T ) 2 -4) . (C.46) \nThe elements of PSL (2 , R ) split into three groups \n- 1. Elliptic: When | tr( T ) | ≤ 2. In this case, the two eigenvalues are complex conjugates of each other λ + = λ ∗ -and have unit norm, i.e. | λ ± | = 1.\n- 2. Parabolic: When tr( T ) = 2 then the eigenvalues are λ = ± 1. For example, the elements γ ± are elliptic tr( γ ± ) = 2. \n- 3. Hyperbolic: When tr( T ) ≥ 2. As an example, the matrix γ = ( e -t/ 2 0 0 e t/ 2 ) is elliptical because Tr( γ ) = 2 cosh( t/ 2) ≥ 2. The action of these elements on the torus forms Anosov flows. \nThe geodesic flow in the upper half-plane corresponds to u → e t u , and the following flow on the unit tangent bundle g → T ( g ) = g · γ ( t ) on PSL (2 , R ): \nγ ( t ) = ( e -t/ 2 0 0 e t/ 2 ) . (C.47) \nThe translation u → u + s corresponds to the stable horocycle flow g → S + ( g ) = g · γ + ( s ) \nγ + ( s ) = ( 1 s 0 1 ) (C.48) \nwhereas the unstable horocycle flow g → S -( g ) = g · γ -( s ) is generated by \nγ -( s ) = ( 1 0 s 1 ) . (C.49) \nThey satisfy the algebraic relations \nT ( t ) S ± ( s ) T ( -t ) = S ± ( e ∓ t s ) . (C.50) \nNext, we lift these actions from the phase space T 1 H to the classical observable algebra L ∞ ( T 1 H ): \nτ ( f ( u )) = f ( T ( -u )) σ ± ( f ( u )) = f ( S ± ( -u )) (C.51) \nso that \nτ ( t ) · σ ± ( s ) · τ ( -t ) = σ ± ( e ∓ t s ) . (C.52) \nThis is a classical Anosov system.", 'D Quantum Dynamical Systems': "In this section, we briefly discuss the extension of ergodic theory to quantum systems. Every Abelian von Neumann algebra is isomorphic to the algebra L ∞ ( X,µ ) for some space X and measure µ . Therefore, the observable algebra of any classical system is always the algebra of functions on some geometry ('phase space'). This naturally connects geometry with classical dynamical systems. This connection does not generalize to non-Abelian algebras, and we cannot always associate geometry with a quantum system. Non-commutative geometry is a mathematical program that attempts at generalizing geometry and associate a non-commutative geometry to a large class of von Neumann algebras [64]. \nThe space R in our classical example of glass is a geometry with a Haar measure (volume element), and the observable algebra is the algebra of complex bounded measurable functions on R . Since the space R is measurable, it comes with a σ -algebra of measurable subsets on R . In classical ergodic theory, we can think of measurable functions on X (observables) in terms of the characteristic functions on subsets X ∈ R inside the σ -algebra. In the quantum case, there is no geometry analogous to R , and the Abelian algebra of functions is replaced by the observable algebra of a quantum system R represented in the Hilbert space H . Quantum dynamics is described by an automorphism τ g : R → R with g in some group G (or monoid if the dynamics is not reversible). Similar to the classical case, we often have a state ω : R→ C (non-commutative measure) that is preserved under the action of τ g i.e., ω · τ g = ω . \nIf the dynamical system preserves some state ω it is natural to represent the algebra in the GNS Hilbert space H ω with the identity operator corresponding to the vacuum vector | Ω ⟩ ∈ H ω . In most of the examples we consider here, the dynamical transformation group is R . Time evolution corresponds to a strongly continuous unitary flow realized in the Hilbert space by U ( t ) = e iHt with t ∈ R ; i.e. for all observables a ∈ R we have τ t ( a ) = U † ( t ) aU ( t ). 56 Note that every non-Abelian von Neumann algebra (quantum system) is a quantum dynamical system with evolution given by modular flow. In fact, modular flow ∆ it Ω is the unique automorphism of the algebra that preserves a state ω and satisfies the KMS relation [25]. Therefore, any quantum dynamical system that satisfies the KMS relation describes modular dynamics . This is the intuition that underlies the connections between modular dynamics and dynamical systems with a positive generator in Theorem 18. \nMore generally, we define a quantum dynamical system as a triple ( A , S, τ ) where A is a C ∗ -algebra, S is a locally-compact monoid that is represented as endomorphisms τ t : A → A with t ∈ S such that the three conditions \n- 1. Identity action: If e ∈ S is the identity element then τ e ( a ) = a for all a ∈ A .\n- 2. Homomorphism: Monoid multiplication is preserved α g 1 · α g 2 = α g 1 g 2 .\n- 3. Continuity: The map g → α g is continuous in operator norm topology. \nIf the algebra A is a von Neumann algebra with a state ω we can represent it on B ( H ω ). The automorphisms of A can be represented as a strongly continuous unitary flow on H ω : α t ( a ) = U ( t ) aU ( t ) † . We obtain a W ∗ -dynamical system. \nEvery automorphism of a von Neumann algebra τ s : R→R can be implemented as a unitary flow on the GNS Hilbert space: \nτ s ( a ) | Ω ⟩ = U ( s ) a | Ω ⟩ . (D.1) \nThis unitary representation is unique if we further require that \nJ R U ( s ) = U ( s ) J R . (D.2) \nThis means that it preserves the natural cone because \nU ( t ) aJa | Ω ⟩ = a t U ( t ) Ja | Ω ⟩ = a t Ja t | Ω ⟩ (D.3) \nwhich is equivalent to \n∆ 1 / 2 U ( t ) = U ( t )∆ 1 / 2 . (D.4) \nIntuitively, we can interpret the above equation as the invertibility of automorphisms with respect to an alternate inner product [40] \n( a, Φ t ( b )) := ⟨ a | ∆ 1 / 2 Φ t ( b ) ⟩ = ⟨ Φ -t a | ∆ 1 / 2 b ⟩ . (D.5) \nWith respect to this inner product, we have Φ † t = Φ -t .", 'D.1 Examples of quantum Anosov systems': 'The motivation to consider quantum Anosov systems is to have examples of systems that cluster at least exponentially fast in modular time. Note that they might cluster faster, but certainly not slower. Theorem 3.6 of [50] ensures that, in quantum Anosov systems, clustering occurs at least exponentially fast.', "Discrete translations on non-commutative torus (Quantum Arnold's Cat map):": "We start by considering a linear flow on a non-commutative torus. A non-commutative torus is a subalgebra of B ( L 2 ( R / Z )) that is generated by the following two unitaries \n( Uf )( z ) = zf ( z ) ( V f )( z ) = f ( ze -2 πiθ ) (D.6) \nfor some deformation parameter θ . It follows that \nUVU † = e -2 πiθ V . (D.7) \nThis algebra is, the Abelian algebra of continuous functions on S 1 enlarged by cross product with rotation by irrational angle θ : e 2 πiθ∂ z . This resulting cross-product is a noncommutative torus. \nConsider the parameter space of the Weyl algebra of a quantum particle. It is parameterized by a complex number z in Weyl unitary operators \nW ( z ) = e za † -z ∗ a (D.8) \nsatisfying the twisted algebra \nW ( z 1 ) W ( z 2 ) = e 2 πiθ ℑ ( z 1 ¯ z 2 ) W ( z 1 + z 2 ) . (D.9) \nWe compactify this space by identifying z ∼ z + i ∼ z + 1 to obtain a torus. Now, the fundamental domain is Z 2 . The element γ = ( e -t/ 2 0 0 e t/ 2 ) acts on the torus as an Anosov quantum system with discrete time evolution. \nFree quantum particle on a manifold M : As opposed to classical systems, quantum systems (von Neumann algebras) are dynamical systems with the flow given by the modular flow. In fact, every classical dynamical system can be viewed as a 'quantum' (noncommutative) system, as well. The idea is simply to enlarge the set of observables by including dynamical transformations in them. Taking the closure of the resulting algebra is a non-commutative algebra. In mathematics, this is called the group measure space construction. We postpone the generalities of this construction to section D.2. Here, we start with an example to build physics intuition. \nConsider a free non-relativistic quantum particle on a line. The classical observables form the Abelian algebra of bounded complex functions on the real line, i.e. L ∞ ( R , dx ), and choose translation ( T t f )( x ) = f ( x + t ) as the dynamics. The cross product of this Abelian algebra with the automorphism is L ∞ ( R , dx ) ⋊ T t R is isomorphic to the algebra of a quantum particle on a line B ( L 2 ( R , dx )) i.e., it is type I ∞ . Note that there is no invariant state. 57 The translation operator belongs to this algebra and generates an Abelian centralizer for this type I ∞ algebra. The spectrum of translation is continuous and has no normalizable eigenoperators. Therefore, this algebra is ergodic, but it is not strongly mixing because functions of momentum do not decay. \nFree quantum particle on upper half-plane In generalizing classical Anosov systems to the quantum world using the crossed product construction an important subtlety presents itself. As we saw in Lemma 14, we obtain the quantum Anosov relation only if the spectrum of the generator of the flow is complete. In the examples we considered, the dynamical flow was either generated by translations, boost, or dilatation. In Rindler space, the spectrum of boost and null translations as self-adjoint operators in the Hilbert space are entire, therefore the examples of free non-relativistic and a free particle on a line, and free relativistic particles on R 1 , 1 become quantum Anosov systems. However, in hyperbolic space (the upper half-plane) the spectrum of the Laplace-Beltrami operator is discrete. Therefore, a free non-relativistic quantum particle on hyperbolic space is not a quantum Anosov system. For more details see [50]. \nFor examples of quantum K-systems associated with quantum many body systems on a lattice see [65]. For more examples of quantum Anosov systems see [39].", 'D.2 First quantization and cross-products': 'Consider the classical dynamical system ( X,dµ,τ ) where X is a measure space with measure dµ and τ : G → Aut( X ) is the dynamical transformation. The observables of this classical system are K = L ∞ ( X,dµ ), and they act on the Hilbert space L 2 ( X,dµ ). Consider the set of G -valued vectors in L 2 ( X,dµ ): H = | Ψ; g ⟩ ∈ L 2 ( G, K ) so that the observable a acts as \nτ g ( a ) | Ψ; g ⟩ = | a Ψ; g ⟩ , (D.10) \nand we have added the dynamical transformation u g to the observables \nu h | Ψ; g ⟩ = | Ψ; hg ⟩ . (D.11) \nWe define the crossed product von Neumann algebra \nM = K ⋊ τ G (D.12) \nas the double-commutant of the resulting operators, and we have \nu g fu † g = τ g ( f ) . (D.13) \nWe can revisit the example of the classical K-system of translations on R . The action is free but not ergodic because the constant functions on R are in L ∞ ( X,dµ ). In this case, the crossed product algebra is generated by position and momentum; therefore, the observables are the type I ∞ factor B ( L 2 ( X,dµ )); i.e. a free quantum particle on a line. The dilatations f ( x ) → f ( e t x ) are inner automorphisms, and there exists no state that is invariant under them. The trace is simply integration over x . 58 The resulting Hilbert space is a one-particle Hilbert space. In physics, quite often, the one-particle observables are type I. \nMore generally, every group G acts on itself transitively by left or right translation. If G is a finite group of order n , then M is a type I n factor, and if G is infinite then M is type I ∞ . In defining ergodicity, we focus on the dynamical group R , and the time-averages were defined with respect to the Haar measure on R . However, for the group generated by translations and dilatations x → ax + b the left-Haar measure is da ∧ db a 2 , whereas the right-Haar measures is da ∧ db | a | . 59 Take a left-invariant measure µ ( h ) and translate it on the right by g , the result is another left-invariant measure \n∀ h ∈ G : µ ( g -1 h ) = ∆( g ) µ ( h ) , (D.14) \nwhere the function ∆( g ) is called the modular function of the group. 60 Consider the above action restricted to the subalgebra of functions on R + : \nσ t f ( x ) = f ( x -t )Θ( x )Θ( x -t ) (D.15) \nwhere Θ is the Heaviside function. We have \nσ t · σ -t = { id t ≥ 0 Θ( t ) t < 0 \nwhere Θ( t ) can be viewed as a conditional expectation. \nWe say the group action of G on X is free if every non-identity element of G moves every element of X . We say it is transitive if the orbit of any point gives the whole X . We say it is ergodic if it acts irreducibly. More formally, we write \nDefinition D.1. The action of G on X is free if gx = x for some x ∈ X implies that g = 1. 61 It is transitive if for any two points x, y ∈ X there exists a g ∈ G such that gx = y . The action of G on X is ergodic if the only subsets that are preserved under it are ∅ and X . \nvon Neumann and Murray proved the following result that characterizes the von Neumann algebras that can be constructed from classical dynamical systems using crossed products: \nTheorem 25. If the action of G on L ∞ ( X,dµ ) is free then the crossed product M is a factor if and only if the action of G is ergodic. \nTheorem 26. Consider M = L ∞ ( X,dµ ) ⋊ τ G where the action of G on A is free and ergodic. Then \n- · M is type I ∞ factor if and only if the action of G is transitive.\n- · M is type II 1 factor if and only if G is a discrete infinite group and there exists a finite invariant measure dν that is absolutely continuous with respect to dµ .\n- · M is type II ∞ factor if and only if the action is not transitive, and there exists a left-invariant measure dν that vanishes on the same set as dµ .\n- · M is type III factor if and only if there exists no left-invariant measure that vanishes on the same set as dµ . \nProof. For proof of both results, see chapter XIII of [66]. \nTo clarify the interpretation of the above result in terms of dynamical systems, it is helpful to discuss some examples. Consider irrational rotations on a circle: T θ e iϕ = e 2 πiϕ + θ with θ irrational. The group is Z , it acts freely but not transitively. The Lebesgue measure dϕ is invariant under these rotations. The Fourier transform of any observable is an eigenfunction of translations: \n( T θ · f ) n = e 2 πinθ f n . (D.16) \nThe only invariant observables are constant (proportional to identity), therefore the action is ergodic. This algebra is a type II 1 factor. \nLet G = R and X = T 2 (a torus: (0 , 1] × (0 , 1]). Consider the translation T s ( x + iy ) = ( x + s + y + iθs ) where θ is irrational. This action is ergodic. The invariant measure is finite, but G is not discrete, and the algebra is type II ∞ . \nFinally, consider the example X = R and G the group x → ax + b . This action is free and the Lebesgue measure dx is ergodic with respect to the subgroup of translations. There exists no nontrivial invariant measure that vanishes on the same set as dµ . Therefore, this algebra is type III.', 'D.3 K-systems, entropy and strong asymptotic Abelianness': 'For a classical K-system, there are several equivalent conditions to define a K system. One may define the K-system through the half-sided inclusions, mixing property[67]. It is interchangeable to use K-system and K-mixing in the classical case. A K-system is known to have positive Kolmogorov-Sinai entropy (KS entropy). A positive KS entropy is often taken to be an indicator of classical chaos [68]. For completeness, we review the definition of the KS entropy here: \nDefinition D.2 (Partition) . A partition of a measure space ( X, Σ , µ ) is defined as a collection α of nonempty disjoint measurable subsets such that the union of all subsets is X , i.e. \n∀ A,B ∈ α, A = B = ⇒ A ∩ B = ∅ , (D.17) \n̸ \n∪ A ∈ α A = X. (D.18) \nDefinition D.3 (Entropy of partition) . The entropy H ( α ) of a finite partition α = { A (1) , . . . , A ( r ) } is defined as \nH ( α ) = -r ∑ i =1 µ ( A ( i ) ) log( µ ( A ( i ) )) . (D.19) \nDefinition D.4 (KS entropy with automorphism) . The KS entropy h ( T ) of a given automorphism T is then defined as follow. \nh ( T ) = sup α lim n →∞ 1 n H ( α n -1 0 ) , (D.20) \nwhere the supremum is taken over all finite partitions α , and \nα n -1 0 = α ∨ T 1 α ∨ . . . ∨ T n -1 α. (D.21) \nThe KS entropy for a flow { T t } is naturally defined as h ( { T t } ) = h ( T 1 ) because h ( T t ) = | t | h ( T 1 ). It was proven in [67] that a system is a K-system if and only if lim n →∞ 1 n H ( α n -1 0 ) is positive for all non-trivial finite partitions α , thus one of the alternative ways to define a K-system. From this result, it is clear that K-systems have positive KS entropy, but the inverse may not be true. \nThe equivalent definitions of classical K-system do not generalize to the quantum (nonAbelian) systems. In [69], the authors defined the quantum K-system through algebraic relations as in Definition 4.6. This is the definition that we use in this paper. Another definition of quantum K-system was introduced in [29] that is based on the properties of entropy. It was proven that with additional assumptions of asymptotic Abelianness and KMS property, the algebraic definition implies the entropic definition [65]. However, it is unclear whether these assumptions are necessary. \nThere is an intuitive connection between K-systems and strong asymptotic Abelianness. In type II 1 algebras, we have strong asymptotic Abelianness if and only if we have \npast algebras (a quantum K-system) [70]. It was also mentioned by Narnhofer [65] that there is an example, a Fermi lattice system with shift automorphism, that is a quantum K-system but not strongly asymptotically Abelian. For a general type II or III this remains an open problem. \nAs we discussed in section 4.3, a strong mixing system with the KMS property has weak asymptotic Abelianness. Strong mixing requires the two-point function to vanish in the limit of infinite time separation, but the assumption of strong asymptotic Abelianness implies that the commutator should also vanish. This property is stable under multiplication: \nlim t →∞ [ a 1 ( -t ) a 2 ( -t ) , b ] = lim t →∞ ( a 1 ( -t )[ a 2 ( -t ) , b ] + [ a 1 ( -t ) , b ] a 2 ( -t )) = 0 . (D.22) \nTo construct the algebra of the past we need to take limits lim n a n ( -t ). Now, if the limits n and t commute strong asymptotic Abelianness implies that the whole algebra generated by operators in the past commute with b . We conclude that this is a proper subalgebra of R because otherwise, b has to be in the center of R but it is a factor. \nFinally, we point out the following algebraic implication of strong asymptotic Abelianness: \nLemma 27. 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2024arXiv240909515A | The 3Healphagamma7Be radiative capture reaction plays a key role in the creation of elements in stars as well as in the production of solar neutrinos the observation of which is one of the main tools to study the properties of our sun. Since accurate experimental measurements of this fusion cross section at solar energies are difficult due to the strong Coulomb repulsion between the reactants the onus falls on theory to provide a robust means for extrapolating from the region where experimental data is available down to the desired astrophysical regime. We present the first microscopic calculations of 3Healphagamma7Be with explicit inclusion of threenucleon forces. Our prediction of the astrophysical S factor qualitatively agrees with experimental data. We further incorporate experimental boundstate and scattering information in our calculation to arrive at a more quantitative description. This process reveals that our current model lacks sufficient repulsion in the 12 channel of our model space to simultaneously reproduce elasticscattering data. This deficit suggests that 3Healphagamma7Be probes aspects of the nuclear force that are not currently wellconstrained. | 2024-09-01T00:00:00Z | ['arXiv:2409.09515', '10.48550/arXiv.2409.09515', '2024arXiv240909515A'] | ['Nuclear Theory', 'Astrophysics - Solar and Stellar Astrophysics'] | Ab initio calculation of the 3Healphagamma7Be astrophysical S factor with chiral two and threenucleon forces | 2,024 | 174 | 0.3 | ['EPRINT_HTML', 'EPRINT_PDF'] | 0 | https://arxiv.org/pdf/2409.09515.pdf | {'Ab initio calculation of the 3 He( α, γ ) 7 Be astrophysical S factor with chiral two- and three-nucleon forces': "M. C. Atkinson a, ∗ , K. Kravvaris a , S. Quaglioni a , P. Navr'atil b \na Lawrence Livermore National Laboratory, P.O. Box 808, L-414, Livermore, CA 94551, USA b TRIUMF, 4004 Wesbrook Mall, Vancouver, British Columbia, V6T 2A3, Canada", 'Abstract': "The 3 He( α, γ ) 7 Be radiative capture reaction plays a key role in the creation of elements in stars as well as in the production of solar neutrinos, the observation of which is one of the main tools to study the properties of our sun. Since accurate experimental measurements of this fusion cross section at solar energies are difficult due to the strong Coulomb repulsion between the reactants, the onus falls on theory to provide a robust means for extrapolating from the region where experimental data is available down to the desired astrophysical regime. We present the first microscopic calculations of 3 He( α, γ ) 7 Be with explicit inclusion of three-nucleon forces. Our prediction of the astrophysical S factor qualitatively agrees with experimental data. We further incorporate experimental bound-state and scattering information in our calculation to arrive at a more quantitative description. This process reveals that our current model lacks sufficient repulsion in the 1 / 2 + channel of our model space to simultaneously reproduce elastic-scattering data. This deficit suggests that 3 He( α, γ ) 7 Be probes aspects of the nuclear force that are not currently well-constrained. \nThe 3 He( α, γ ) 7 Be radiative capture reaction, where 3 He nuclei combine with α particles ( 4 He nuclei) to form 7 Be and emit a photon, played a crucial role in the formation of the lightest elements during the early phases of the universe. The amount of 7 Li produced in the first 200 seconds after the big bang predicted by big-bang nucleosynthesis models is highly dependent on the astrophysical S factor of the 3 He( α, γ ) 7 Be reaction [1]. In addition to events far in the past, the 3 He( α, γ ) 7 Be reaction is an important part of ongoing processes occurring in young stars the size of our sun. In the second branch of the proton-proton reaction network (pp-II), the 3 He( α, γ ) 7 Be reaction is key to determining neutrino fluxes resulting from the decay of 7 Be and 8 B. In standard solar model (SSM) predictions of these neutrino fluxes, the 3 He( α, γ ) 7 Be capture rate is the largest source of uncertainty from nuclear input [2]. \nThe importance of this capture reaction has made it the focus of many experiments over multiple decades [3, 4, 5, 6, 7]. While the abundance of reactants in a stellar environment compensates for the exponential Coulomb suppression of fusion, the 3 He( α, γ ) 7 Be capture cross section at astrophysical energies is exceedingly difficult to measure in terrestrial settings. Due to this limitation, the lowest-energy S factor measurement is at a center-of-mass energy of 90 keV while the Gamow peak (the energy at which the fusion probability is maximized considering the Maxwellian velocity distribution of nuclei at solar temperatures) is around 18 keV [2]. Thus, theoretical calculations \nEmail address: [email protected] (M. C. Atkinson) \nare necessary to extrapolate the S factor down to solar energies. There have been numerous theoretical calculations of the 3 He( α, γ ) 7 Be S factor including external-capture models [8], halo-EFT approaches [9, 10], microscopic approaches [11, 12, 13, 14], and ab initio approaches [15]. By combining a subset of these theoretical predictions with a curated set of the S factor measurements, an evaluation of the S factor was performed in Solar Fusion III (SF III) resulting in S 34 (0) = 0 . 561 ± 0 . 018 (exp) ± 0 . 022 (theory) keV [2]. Not only is the theory uncertainty comparable to that from experiment, but it has grown since the previous evaluation [16]. This enduring uncertainty continues to motivate theoretical works toward more accurate predictions of S 34 ( E ) [12, 15, 17, 9, 10]. \nIn this Letter, we apply the ab initio framework of the no-core shell model with continuum (NCSMC) [18, 19, 20] to describe the 3 He+ α system and calculate the 3 He( α , γ ) 7 Be capture reaction, starting form a many-body chiral Hamiltonian with explicit inclusion of three-nucleon (3N) forces. This work is the first calculation of this reaction to include 3 N forces, which have been shown to be essential in describing big bang and solar fusion cross sections [21, 22]. The present results are an important step toward improving the evaluation of the zero-energy S factor, S 34 (0), and hence reduce the uncertainty of SSM calculations. \nThe 3 He( α, γ ) 7 Be radiative-capture cross section at as- \nophysically relevant energies can be written as [23] \nσ ( E ) = 64 π 4 4 πϵ 0 ℏ ν ∑ κλ k 2 λ +1 γ [(2 λ +1)!!] 2 λ +1 λ (1) × ∑ J i ℓ i s i ˆ J 2 f ˆ s 2 P ˆ s 2 T ˆ ℓ 2 i ∣ ∣ ∣ 〈 Ψ J π f f T f ∥ ∥ ∥ M κλ ∥ ∥ ∥ Ψ J π i i T i ℓ i s i 〉∣ ∣ ∣ 2 , \nwhere Ψ J π f f T f and Ψ J π i i T i ℓ i s i correspond to the wave functions for the final bound state ( 7 Be in this case) and the initial scattering state ( 3 He+ 4 He in this case), respectively, with corresponding quantum numbers J , ℓ , s , π , and T representing total angular momentum, orbital angular momentum, spin, parity, and isospin. s P and s T are the spin quantum numbers of the projectile ( 3 He) and target ( 4 He) nuclei, respectively, and λ is the multipolarity of the electric ( κ = E ) and magnetic ( κ = M ) transition operators, M κλ , the expressions for which can be found in, e.g., Ref. [18]. We employ the notation where ˆ s = √ 2 s +1. In our calculations, we include electric transition operators up to λ = 2 ( E 2) and magnetic transition operators up to λ = 1 ( M 1). For the energy ranges considered in this radiative-capture calculation, the contribution from the E 1 transition is dominant while those of M 1 and E 2 are almost negligible [15] (and therefore higher order terms are ignored). The capture cross section can be factored as \nσ ( E ) = S 34 ( E ) E exp { -2 πZ 1 Z 2 e 2 ℏ √ 2 E/m } , (2) \nwhere S 34 ( E ) isolates the nuclear component of σ ( E ) and 1 /E and the exponential term, respectively, account for reaction kinematics and tunneling through the Coulomb barrier. \nBoth the initial scattering and final bound states in Eq. (1) are calculated within the NCSMC framework through the explicit inclusion of 3 He+ α clustering in the manybody wave function. The ansatz for the NCSMC initial and final state is a generalized cluster expansion [18] \n∣ ∣ ∣ Ψ J π T 〉 = ∑ λ c J π T λ | AλJ π T ⟩ + ∑ ν ∫ drr 2 γ J π T ν ( r ) r ˆ A ν ∣ ∣ ∣ Φ J π T νr 〉 . (3) \nThe first term on the right-hand side of the equation is an expansion over translationally invariant eigenstates, | AJ π T ⟩ , of the aggregate system ( 7 Be in this case) calculated within the no-core shell model (NCSM) [24]. The NCSM is an ab initio many-body method for the description of static wave functions that allows for the use of both Jacobi relative coordinate [25] and single-particle Slater determinant basis states [26, 27, 28]. The second term is an expansion over fully antisymmetrized microscopic cluster basis channels which describe the 3 He and 4 He clusters in relative \nmotion [29, 30], \n∣ ∣ ∣ Φ J π T νr 〉 = [ (∣ ∣ 4 He λ 4 J π 4 4 T 4 〉 ∣ ∣ 3 He λ 3 J π 3 3 T 3 〉) ( sT ) Y ℓ (ˆ r 34 ) ] ( J π T ) × δ ( r -r 34 ) rr 34 . \nHere, ∣ ∣ 4 He λ 4 J π 4 4 T 4 〉 is an NCSM eigenstate of 4 He with energy label λ 4 , total angular momentum J 4 ( s T in Eq. (1)), parity π 4 , and isospin T 4 , ∣ ∣ 3 He λ 3 J π 3 3 T 3 〉 is analogously defined for 3 He, s is the channel spin, r 34 = r 34 ˆ r 34 is the relative radial coordinate between the centers of mass of 3 He and α , and ν is a collective index of the relevant quantum numbers. The projectile and target wave functions are once again described within the NCSM approach. \nThe discrete coefficients, c J π T λ , and continuous relativemotion amplitudes, γ J π T ν ( r ), are obtained as solutions to the coupled equations [19, 20] \n( H NCSM ¯ h ¯ h ¯ H )( c χ ) = E ( 1 ¯ g ¯ g 1 )( c χ ) . (4) \nHere, ( H NCSM ) λλ ' = E λ δ λλ ' is the expectation value of the Hamiltonian in the NCSM model space which evaluates to a diagonal matrix consisting of NCSM eigenvalues; ¯ H νν ' = ( N -1 / 2 HN -1 / 2 ) νν ' and χ ν = ( N 1 / 2 γ ) ν are the Hamiltonian kernel and relative wave functions, respectively, where N νν ' ( r, r ' ) = ⟨ Φ J π T ν ' r ' | ˆ A ν ' ˆ A ν | Φ J π T νr ⟩ and H νν ' ( r, r ' ) = ⟨ Φ J π T ν ' r ' | ˆ A ν ' ˆ H ˆ A ν | Φ J π T νr ⟩ ; ¯ g λν ( r ) and ¯ h λν ( r ) are the overlap and Hamiltonian form factors describing the coupling between the NCSM sector and the cluster sector of the full basis, respectively, proportional to ⟨ AλJ π T | ˆ A λ | Φ J π T νr ⟩ and ⟨ AλJ π T | ˆ H ˆ A λ | Φ J π T νr ⟩ . The bound states and scattering matrix (and from it any scattering observable) are then obtained by matching the solutions of Eq. (4) with the known asymptotic behavior of the wave function at large distances by means of the microscopic R-matrix method [31, 19]. \nThe microscopic A -nucleon Hamiltonian, ˆ H , adopted in the present work is built on the nucleon-nucleon ( NN ) chiral interaction at next-to-next-to-next-to leading order of Ref. [32], denoted as NN -N 3 LO, along with a threebody interaction at next-to-next-to leading order (N 2 LO) with simultaneous local and nonlocal regularization [33, 34, 35, 36]. The whole chiral interaction (two- plus threebody) will be referred to as NN -N 3 LO+3 N lnl . To accelerate the convergence of the NCSMC calculation, we first soften the chiral interaction through the similarity renormalization group (SRG) technique [37, 38, 39, 40]. We include SRG induced forces up to the three-body level. To minimize the influence of four- and higher-body induced terms, we adopt the SRG momentum scale of λ SRG =2 . 0 fm -1 [41]. Further, we choose a harmonic oscillator (HO) frequency of ℏ Ω = 20 MeV that minimizes the groundstate energies of the investigated nuclei [42]. \nThe NCSMC model space for the present calculation consists of microscopic seven-body cluster states built from the 0 + and 1 / 2 + ground states of 4 He and 3 He, respectively, and the ten lowest positive and negative parity \neigenstates of 7 Be with total angular momentum J ranging from 1 / 2 to 7 / 2. The 3 He (and even 4 He) reactants can be deformed in the reaction process, in principle requiring the inclusion of excited states of the reactants to take this deformation into account in a microscopic cluster expansion. In the NCSMC, we compensate for the omission of these excited states by including eigenstates of the 7 Be aggregate system, which help to describe short range correlations [29]. Among the twenty 7 Be states, three 1 / 2 + and 3 / 2 -states and two 1 / 2 -states are the most relevant channels in this radiative-capture process. All projectile, target, and aggregate states are calculated within the NCSM using the same ˆ H that generates the Hamiltonian kernel in Eq. (4). \nThe HO model space used to compute the NCSMC kernels is N max = 10 for 3 He, 4 He, and negative-parity 7 Be states and N max = 11 for positive-parity 7 Be states. To compute the RGM part of the NCSMC kernels, we include matrix elements of the 3N force up to a total number of single-particle quanta for the three-body basis of E 3max = 18. Beyond this mode-space size, the NCSMC calculations become computationally intractable when including 3 N forces. In such a model space, the groundstate energies of 3 He and 4 He are close to experiment ( E expt. ( 3 He) -E NCSM ( 3 He) = -0 . 10 MeV, E expt. ( 4 He) -E NCSM ( 4 He) = -0 . 016 MeV) while the bound states of 7 Be are within 1.5 MeV of experiment. Taking advantage of the convergence behavior in N max , we extrapolate our NCSM levels to N max →∞ by using an exponential decay function E ( N ) = Ae -bN + E ∞ [43]. The resulting levels (labeled ' ∞ ' in Fig. 1) are closer to experiment and demonstrate that N max = 10 NCSM eigenstates employed in the following NCSMC calculations are converged within 4% of their extrapolated values. \nFigure 1: Convergence of calculated energy levels in 7 Be for increasing basis size N max compared to experiment [44]. The horizontal dotted lines correspond to the 3 He+ α threshold at each N max . The left-half of the figure (partitioned by the vertical dashed line) contains the NCSM levels at each N max . The column labeled as ' ∞ ' corresponds to extrapolated energy levels (see text). The right-half of the figure contains NCSMC levels and resonances from solving the scattering equations. Widths of resonant states are represented by shaded regions. \n<!-- image --> \nmax \nmax \nmax \nmax \nmax \nmax \nmax \nmax \nmax \nTable 1: Bound-state properties of 7 Be generated by the NCSM and NCSMC at N max = 10 compared to experimental data [45, 46, 47] where E 3 / 2 -and E 1 / 2 -are in MeV with respect to the 3 He+ α threshold, C 3 / 2 -and C 3 / 2 -are in fm -1 / 2 , r ch is in fm, Q is in e · fm 2 , and µ is in µ N (the nuclear magneton). The column labeled 'NCSMC pheno ' is the result obtained by phenomenologically adjusting the NCSM eigenvalues employed in the NCSMC to reproduce experimental binding energies (see text). \nConcerning the NCSMC kernels in Eq. (4), the couplings between the microscopic cluster states, N νν ' ( r, r ' ) and H νν ' ( r, r ' ), are calculated using the configuration interaction framework for scattering and reactions induced by light projectiles developed in Ref. [48, 49]. The couplings between the NCSM sector and the cluster sector, ¯ g λν ( r ) and ¯ h λν ( r ), are calculated in a similar way and details will be described in a future publication. We calculate the full NCSMC wave function by solving Eq. (4) using these computed kernels. The energies computed in the NCSMCare an improvement over the NCSM thanks to the inclusion of the microscopic cluster states of 3 He+ α (see Fig. 1 and Table 1). Especially noteworthy are the 3 / 2 -and 1 / 2 -states which are now bound in the NCSMC. The bound levels calculated in the NCSMC change by less than 4% between N max = 8 and N max = 10, demonstrating that the calculation is well-converged. This convergence is further confirmed by the good agreement between the N max = 10 NCSMC energy levels and the corresponding NCSM values extrapolated to N max → ∞ . Furthermore, the correct description of the wave function at long range allows us to calculate the position and width of the 5 / 2 -and 7 / 2 -resonances which are also changing by less than 4% from N max = 8 to N max = 10. \nTo probe the radial shape of our 7 Be ground-state wave function, we calculate the charge radius ( r ch ), electric quadrupole moment ( Q ), and magnetic moment ( µ ) of the 7 Be ground state and compare to experiment (where available) in Table 1. Also included in Table 1 are the 7 Be 3 / 2 -and 1 / 2 -bound-state energies along with their corresponding asymptotic normalization coefficients (ANCs). Just as with the energy levels in Fig. 1, the inclusion of the α + 3 He cluster state improves the predictions of the NCSMC over the NCSM. The charge radius, in particular, increases by more than 10% from the NCSM prediction, leaving the NCSMCprediction less than 4% from the measured values. This improved charge radius calculation is a natural con- \nquence of representing the correct asymptotics through the inclusion of the microscopic cluster states, ∣ ∣ Φ J π T νr 〉 , in Eq. (3) [50, 51]. \nFigure 2: NCSMC calculations of S 34 ( E ) at N max = 10. The dashed line is the unshifted NCSMC result. The solid line is the NCSMC pheno result after adjusting to experimental bound-state data. The dot-dashed line is the NCSMC ∗ pheno result after adjusting to both experimental bound-state and scattering data. The measurements are from Refs. [3](triangles), [4] (squares), [52] (filled diamonds), [5] (open circles), [6] (upside-down triangles), [53] (open diamonds), [7] (stars), and the filled circle at the Gamow peak is deduced from solar-neutrino data [54]. The shaded region represents the current evaluation from SF III, the form of which is reported as a polynomial fit to halo-EFT results. The widths of the regions are based on the reported uncertainties of S (0) (with different shades representing uncertainty from theory or experiment) [2]. \n<!-- image --> \nUsing the bound- and scattering-state wave functions calculated in the NCSMC, we employ Eqs. (1) and (2) to calculate S 34 ( E ), yielding the dashed line in Fig. 2, which is in qualitative agreement with the experimental data. While our prediction is below the experimental data, the shape matches that of the data except at higher energies where we miss the peak induced by the 7 / 2 -resonance (which lies at higher energy in our calculation). To understand the origin of the difference between our prediction and the latest evaluation, we consider a series of adjustments to the NCSMC Hamiltonian kernels. \nWe apply phenomenological shifts to the 7 Be 1 / 2 -, 3 / 2 -, and 7 / 2 -NCSM energy eigenvalues in the upper left quadrant of the NCSMC Hamiltonian kernel (left-hand side of Eq. (3)) to exactly reproduce the corresponding experimental energies. We also adjust the 4 He and 3 He energies entering the definition of the total energy in the righthand side of Eq. (4) to match experiment even though these minor shifts (see Fig. 1) result in almost negligible changes to the NCSMC results. These phenomenological corrections improve not only the energies but also the radial shape of the 3 / 2 -and 1 / 2 -bound states, yielding more quantitative predictions for the observables in Table 1 and resulting in an increase of the overall normalization of S 34 ( E ). The S factor obtained after these phenomenological shifts, dubbed NCSMC pheno , is shown by the solid line in Fig. 2. The NCSMC pheno prediction quantitatively reproduces the experimental data and \nagrees with the SF III evaluation over the entire range of energies between threshold and 4.5 MeV. \nWe also probe our predictions of the scattering wave functions by calculating 3 He+ 4 He elastic-scattering differential cross sections and comparing them with measurements from Refs. [55, 56, 57]. Of particular interest is the experiment performed by the SONIK collaboration in 2022 that reaches energies as low as E lab = 239 keV [55] and provides better angular range than previous measurements. The NCSMC is in good agreement with the lowerenergy SONIK data (dashed line in panel (c) of Fig. 3), but the agreement declines at higher-energy backward angles (dashed line in panels (d) and (e) of Fig. 3). The discrepancy at backward angle is also present when comparing to older experimental data sets at θ c.m. = 104 · and θ c.m. = 106 . 4 · [56, 57] (dashed line in panel (b) of Fig. 3). \nWe note that the logarithmic scale amplifies the differences at large angles owing to the divergence of the Rutherford cross section at θ c.m. = 0 · . In fact, we find that the difference between the NCSMC results and the SONIK data is a constant (angle-independent) shift of about 10 mb. A constant shift such as this must be rooted in the s-wave channel of the scattering wave function corresponding to the 1 / 2 + 3 He+ α phase shift. Indeed, the NCSMC phase shift is less-repulsive than the data of Refs. [57, 8] (see the dashed line in panel (a) of Fig. 3), indicating that the 1 / 2 + channel of the NCSMC Hamiltonian kernel lacks sufficient repulsion. Similarly, we find that the 1 / 2 + scattering length, a 0 = 11 . 2 fm, calculated by fitting the effective range expansion to our corresponding phase shift, is much smaller than the one derived from an R -matrix fit to the SONIK elastic data [55]. This further confirms the lack of repulsion in our 1 / 2 + channel. \nTable 2: S factor of 3 He( α, γ ) 7 Be at zero energy, S 34 (0). The NCSMC pheno result is after adjusting to experimental bound-state data. The NCSMC ∗ pheno result is after adjusting to both experimental bound-state and scattering data. The quoted R -matrix value is specifically excluding the Barnard data set (see Ref. [59] for detail). \n± \n± \nIt is unlikely that the missing repulsion in the computed 1 / 2 + channel could be fully explained by a NCSMC calculation at the next largest HO model space dimension of N max = 12 / 13 (currently computationally out of reach). As discussed, we obtain well-converged bound state energies as well as resonance centroids and widths (see Fig. 1). Furthermore, the difference in the 1 / 2 + a 0 between N max = 10 and N max = 8 is 1.7 fm as opposed to the difference of over 15 fm when compared to the R \nFigure 3: This figure illustrates the effect of adding repulsion to the NCSMC Hamiltonian kernel (see Eq. (5)). Each panel contains 4 lines representing NCSMC pheno , and starting from the dashed line, each line has increasing magnitudes of added repulsion. The dashed line has no added repulsion, the dotted line has 13 MeV repulsion added, the dot-dashed line has 23 MeV repulsion added, and the solid line has 33 MeV repulsion added. (a) Reduction of the 1 / 2 + phase shift due to added repulsion. The circle points are phase shifts extracted from the elastic-scattering experiment in Ref. [58], and the square points are extracted from the elastic-scattering experiment in Ref. [57]. (b) Increased strength in the NCSMC-calculated differential cross section as a function of energy for a fixed θ c.m. = 104 · due to added repulsion. The triangle points represent data measured at θ c.m. = 104 · [56]. The square points represent data measured at θ c.m. = 106 . 4 · [57]. (c)-(f) Increased strength in the elastic-scattering angular distributions at E c.m. = 0 . 394 , 0 . 997 , 2 . 05 , 3 . 125 MeV due to added repulsion. Data from Ref. [55]. \n<!-- image --> \nmatrix fit of the SONIK data [55]. Additionally, the NCSMC investigation of S 34 ( E ) in Ref. [15], which reached the N max = 12 / 13 HO model space size using the twonucleon component of the SRG-evolved NN interaction ( NN -only), manifests a similar lack of repulsion in the 1 / 2 + channel. This was originally attributed to the lack of explicit 3 N forces in the calculation (see Fig. (3) of Ref. [15]). While the present study shows that the inclusion of 3 N forces does introduce additional repulsion, it is not sufficient to reproduce experimental data. \nFor the time being, we emulate the effect of a more repulsive nuclear force in the 1 / 2 + channel by including a nonlocal Woods-Saxon potential, V ( r, r ' ), in the microscopiccluster quadrant of the Hamiltonian kernel (in the bottomright element of the matrix on the left-hand-side of Eq. (4)) such that \nH 1 / 2 + νν ' ( r, r ' ) →H 1 / 2 + νν ' ( r, r ' ) + V ( r, r ' ) , \nwhere \nV ( r, r ' ) = V ws 1 + e ( R -r ws ) /a ws × e ( r -r ' ) 2 /a 2 ws . (5) \nHere R = r + r ' 2 , the width and radius r ws and a ws of the Wood-Saxon potential are fixed at 0 . 6 fm to resemble \nthe nonlocality and range of H 1 / 2 + νν ' ( r, r ' ), and V ws is the strength of the repulsion which we varied. As expected, the computed phase shift becomes closer to the empirical data as we increase V ws from 13 to 33 MeV (see the progression from dashed to solid lines in Fig. 3(a). Correspondingly, we observe progressively improved agreement with experiment in the elastic-scattering differential cross sections (see panels (b)-(f) of Fig. 3). This indicates that, with suitable repulsion in the 1 / 2 + channel, our prediction of the scattering wave function is realistic and can reproduce experimental elastic data. \nThe S factor obtained after this series of adjustments to the NCSMC Hamiltonian kernel (dubbed NCSMC ∗ pheno ) still quantitatively reproduces the majority of the experimental data at the bottom of their uncertainties over the entire range of energies, yet it lies below the SF III evaluation for most of the energy range (dot-dashed line in Fig. 2). It is curious that reproducing the elastic-scattering data from the SONIK collaboration causes our S factor to decrease away from the SF III evaluation. This discrepancy seems to reveal a mild tension between some of the S factor measurements (along with the SF III evaluation) and elastic-scattering measurements, similar to what is observed in Ref. [59]. Indeed, adding more repulsion to match the Barnard elastic scattering data would not only \nresult in overestimating the SONIK data, but it would further reduce the NCSMC ∗ pheno prediction in Fig. 2 (again, similar to what is observed in Ref. [59]). Even among the S factor data, the NCSMC ∗ pheno result points to a mild tension between the Weizmann [3] and Seattle [52] data sets. \nWhen compared to the halo-EFT results in Fig. 4 (which the SF III evaluation is based on), our NCSMC pheno results match at lower energies and favor a slightly larger S ( E ) at higher energies. As shown by the difference between the NN -only calculation (dash-dotted line) and the present result, the inclusion of 3 N forces lowers the absolute normalization of the S factor and molds its shape to be more consistent with the trend of the experimental data. The present results differ in both normalization and shape from those obtained in Ref. [12] (dotted line) using the fermionic molecular dynamics (FMD) approach over the entire energy range. \nAt zero energy, the NCSMC pheno yields S 34 (0) = 0 . 545(1) keV b (see Table 2), where the uncertainty is estimated as the difference between the NCSMC pheno N max = 8 and N max = 10 S 34 (0) predictions. The NCSMC pheno and NCSMC ∗ pheno results in Table 2 provide a lower and upper bound for our S 34 (0) prediction of 0.505 and 0.545 keV b, respectively, that acknowledges the lack of repulsion in our 1 / 2 + channel and the mild tension between elasticscattering and S factor measurements. The halo-EFT values quoted in Table 2 are obtained using the experimental data sets detailed in Ref. [9, 10], while the SF III value is obtained using the same halo-EFT model of Ref. [10] but employing an adjusted data set as detailed in Ref. [2]. \nIn conclusion, we have presented the first ab initio calculation of the 3 He( α, γ ) 7 Be S factor including the 3N force. We demonstrated that the ab initio calculations of capture reactions obtained with the NCSMC can be improved through the incorporation of the information provided by bound-state and scattering measurements and still maintain predictive capability. While the phenomenological shift applied to reproduce the observed energy levels in 7 Be can be interpreted as emulating the effect of an infinitely large NCSM model space (see Fig. 1), the repulsive interaction added in the 1 / 2 + channel has no analogous explanation. \nUnderstanding the origin of this lack of repulsion in the 1 / 2 + partial wave will play an important role in accurately describing the 3 He+ 4 He elastic scattering and 3 He( α, γ ) 7 Be capture cross sections simultaneously. Barring the existence of a broad 1 / 2 + resonance which could manifest with the inclusion of the p + 6 Li channel in our model, we hypothesize that the present A = 7 reaction observables probe aspects of chiral interactions that are currently not well-constrained. Looking ahead, we plan to include the p + 6 Li channel as well as conduct a systematic analysis of additional chiral interaction models to explore this puzzle in the 1 / 2 + channel. The present calculation also sets the stage for a more extensive study with different chiral interactions (at several orders) and exploiting \nFigure 4: The S factor for the radiative capture process 3 He( α, γ ) 7 Be. The solid line is the present NCSMC pheno result after adjusting to the bound-state data (see text). The dot-dashed line is the previous NCSMC result employing NN -only forces [15]. The dashed line is the result of the microscopic FMD calculation from Ref. [12]. The dotted line is a halo-EFT calculation with the updated data-constraints detailed in SF III [2]. The measurements are from Refs. [3](triangles), [4] (squares), [52] (filled diamonds), [5] (open circles), [6] (upside-down triangles), [53] (open diamonds), [7] (stars), and the filled circle at the Gamow peak is deduced from solar-neutrino data [54]. The shaded region represents the current evaluation from SF III, the form of which is reported as a polynomial fit to halo-EFT results. The widths of the regions are based on the reported uncertainties of S (0) (with different shades representing uncertainty from theory or experiment) [2]. \n<!-- image --> \ncorrelations among the measured S factor data to arrive at an accurate evaluation with reduced uncertainty in a manner similar to that of Ref. [22]. \nThe authors would like to thank Gautam Rupak for sharing his halo-EFT S factor results. We also thank Chloe Hebborn for helpful discussions. Computing support for this work came from the Lawrence Livermore National Laboratory (LLNL) institutional Computing Grand Challenge program and from an INCITE Award on the Summit and Frontier supercomputers of the Oak Ridge Leadership Computing Facility (OLCF) at ORNL. This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under Work Proposal No. SCW0498. 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2024arXiv240903828B | We develop a general framework for calculating the leadingorder fullyrelativistic contributions to the gravitational phase shift in singlephoton atom interferometers within the context of linearized gravity. We show that the atom gradiometer observable which only depends on the atom interferometer propagation phase can be written in terms of three distinct contributions the Doppler phase shift which accounts for the tidal displacement of atoms along the baseline the Shapiro phase shift which accounts for the delay in the arrival time of photons at atomlight interaction points and the Einstein phase shift which accounts for the gravitational redshift measured by the atoms. For specific atom gradiometer configurations we derive the signal and response functions for two physicallymotivated scenarios i transient gravitational waves in the transversetraceless gauge and for the first time in the proper detector frame and ii transient massive objects sourcing weak and slowvarying Newtonian potentials. We find that the Doppler contribution of realistic Newtonian noise sources e.g. a freight truck or a piece of space debris at proposed atom gradiometer experiments such as AION MAGIS and AEDGE can exceed the shot noise level and thus affect physics searches if not properly subtracted. | 2024-09-01T00:00:00Z | ['2024arXiv240903828B', 'arXiv:2409.03828', '10.48550/arXiv.2409.03828'] | ['General Relativity and Quantum Cosmology', 'Astrophysics - Cosmology and Nongalactic Astrophysics', 'High Energy Physics - Phenomenology', 'High Energy Physics - Theory', 'Physics - Atomic Physics'] | Signatures of Linearized Gravity in Atom Interferometers a Simplified Computational Framework | 2,024 | 174 | 0.18 | ['EPRINT_HTML', 'EPRINT_PDF'] | 1 | https://arxiv.org/pdf/2409.03828.pdf | {'Signatures of Linearized Gravity in Atom Interferometers: a Simplified Computational Framework': 'Leonardo Badurina, 1, ∗ Yufeng Du, 1, † Vincent S. H. Lee, 1, ‡ Yikun Wang, 1, § and Kathryn M. Zurek 1, ¶ \n1 \nWalter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125, USA (Dated: September 9, 2024) \nWe develop a general framework for calculating the leading-order, fully-relativistic contributions to the gravitational phase shift in single-photon atom interferometers within the context of linearized gravity. We show that the atom gradiometer observable, which only depends on the atom interferometer propagation phase, can be written in terms of three distinct contributions: the Doppler phase shift, which accounts for the tidal displacement of atoms along the baseline, the Shapiro phase shift, which accounts for the delay in the arrival time of photons at atom-light interaction points, and the Einstein phase shift, which accounts for the gravitational redshift measured by the atoms. For specific atom gradiometer configurations, we derive the signal and response functions for two physically-motivated scenarios: (i) transient gravitational waves in the transverse-traceless gauge and, for the first time, in the proper detector frame, and (ii) transient massive objects sourcing weak and slow-varying Newtonian potentials. We find that the Doppler contribution of realistic Newtonian noise sources ( e.g. , a freight truck or a piece of space debris) at proposed atom gradiometer experiments, such as AION, MAGIS and AEDGE, can exceed the shot noise level and thus affect physics searches if not properly subtracted.', 'I. INTRODUCTION': "Atom interferometry is a versatile and rapidly-developing experimental technique that can be used for a wide variety of precision measurements [1]. For instance, atom interferometers (AIs) have been used to measure fundamental constants [2-4], probe the foundational principles of general relativity [5-9] and quantum mechanics [10-12], and test models of dark energy and modified gravity [13-16]. Atom gradiometers (AGs), which consist of two spatially-separated AIs that are referenced by common lasers, have also been proposed to detect gravitational waves (GWs) in the unexplored 'mid-frequency band' [1721], search for violations of the universality of free-fall [22] and measure time-varying corrections to atomic transition energies induced by scalar ultralight dark matter [23-25]. \nIn recent years, a number of ambitious AG experiments have been proposed as quantum sensors for fundamental physics (see Ref. [26] for a recent review). These include large-scale terrestrial experiments, such as AION [27], MAGIS [28], MIGA [29], ELGAR [30], and ZAIGA [31], and futuristic space-based experiments, such as STE-QUEST [32] and AEDGE [33]. By overcoming a number of experimental systematics, these experiments are expected to operate at the shot-noise level. However, fluctuations in an atom's local gravitational field, e.g. due to seismic waves [34] or Newtonian noise (NN) [35], may significantly reduce the projected reach in the key (10 -3 -1) Hz frequency window. If left unmitigated, these effects will dramatically limit the physics potential of these ambitious experiments. \nInstead of cutting large sections of an experiment's time series [35], it may be possible to subtract the phase shift from transient sources of NN and recover an experiment's shot-noise limited sensitivity provided that this phase shift is known with sufficient precision. Depending on an experiment's projected reach, such a strategy may require a fully relativistic calculation. For example, a massive object traveling with speed v s will induce a phase shift by accelerating the atoms, given by ∆ ϕ non -rel ∼ k eff aT 2 [36], with k eff being the maximum momentum difference between the two arms of an interferometer, T being the interrogation time, and a being the acceleration of the atoms. This is typically calculated utilizing only classical Newtonian mechanics. However, relativistic corrections are expected. Using dimensional analysis and the invariance of general relativity under parity and time-reversal, the dominant relativistic phase shift is expected to scale as ∆ ϕ rel ∼ v s ∆ ϕ non -rel . Although useful to estimate the size of the effect, ∆ ϕ rel does not inform us of: (i) the coefficient of this phase shift term, which may be sequencedependent, and (ii) the shape of the power-spectrum associated with this phase shift, both of which can only be inferred from a general-relativistic ( i.e. general coordinate-invariant) phase shift calculation. \nSeveral coordinate-invariant formalisms have been proposed [7, 37-40] and have been used to calculate, e.g. , the phase shift induced by gravitational waves [17, 18, 41] and static potentials ( e.g. the Earth's gravitational field) [7, 37, 39, 40]. Notably, the formalism proposed in Ref. [37] defines all tunable experimental parameters in a frame-independent manner, solves for the geodesics of the freely falling \natomic wavepackets and laser pulses, and determines the momentum transferred to the atoms as a result of atom-light interactions in the atom's local interial frame. Although crucial for correctly predicting the size of relativistic contributions to the phase shift, these formalisms are computationally cumbersome when the number of interaction points exceeds O (1). This is especially relevant in proposed atom gradiometer experiments employing large momentum transfer (LMT) such as AION and MAGIS, which plan up to O (10 4 ) atom-light interactions per cycle. \nAside from these computational considerations, outstanding questions remain about the physical interpretation of existing gauge-invariant frameworks for computing gravitational phase shifts, such as the interpretation of AGs as gravitational antennas, or equivalently the mapping between AG and laser interferometer observables. For example, Ref. [37] decomposes the gauge-invariant phase shift for a single AI (and consequently for an AG) into three contributions: the phase shift associated with the free-evolution of atomic wavepackets in spacetime ( i.e. the propagation phase), the phase shift imprinted by the laser pulses during atom-light interactions ( i.e. the laser phase) and the phase shift associated with the degree to which the two spatially-separated wavepackets do not overlap at the application of the final beamsplitter pulse ( i.e. the separation phase). As shown for GWs in Ref. [42] and recently for more general metric perturbations via a proper time treatment in Ref. [43], the observable in laser interferometers can be written as a sum of three distinct (and separately not diffeomorphism-invariant) contributions: the time-delay caused by the tidal displacement of the mirrors along the baseline ( i.e. the Doppler time delay), the delay in the arrival time of photons at the mirrors ( i.e. the Shapiro time delay), and the time-delay due to the gravitational redshift measured by the beamsplitter ( i.e. the Einstein time delay). Since AGs have been proposed as gravitational wave interferometers, it should be possible to extract these three contributions from the gauge-invariant AG phase shift. This endeavor would clarify the interpretation of AGs as exquisite accelerometers [44] and time-keeping devices [45], and elucidate the origin of the relativistic phase shift contributions in different frames. \nIn this work, we address these points by developing from first principles a simplified general coordinate-invariant framework for calculating phase shifts in single-photon AGs. Since the metric perturbation in the problems of interest is very small, we will work within the context of linearized gravity. Additionally, as the motion of the atoms relative to the laser sources is highly non-relativistic, we work to leading order in the atom velocity. Notably, for the case of AGs, we express the differential phase shift in terms of contributions which are in one-to-one correspondence with the time-delays that enter the laser interferometer observable: the phase shifts associated with the Doppler, Shapiro and Einstein time delays. Equipped with this formalism, we compute the signal and response function induced by two well-motivated physical scenarios: transient gravitational waves (GWs) and weak and slow-varying Newtonian potentials sourced by transient massive objects. Since the form of the response function depends on the pulse sequence, we will perform explicit calculations for gradiometers employing \nMach-Zehnder and LMT configurations. Importantly, our examples highlight the gauge-invariance of our framework (as explicitly shown in the GW calculation, which is performed in both the transversetraceless and the proper detector frame) and the accuracy of our formalism in reproducing existing results in the literature in a more physically and computationally transparent fashion. \nThis paper is structured as follows. After reviewing the basics of atom interferometry, in section II we introduce our general coordinate-invariant framework for computing gradiometer phase shifts in linearized gravity. After deriving the basis for our formalism in section II A, in section II B we introduce the gradiometer observable and provide expressions for Mach-Zehnder and LMT gradiometer configurations. As example applications, in section III we compute the phase shifts induced by gravitational waves and slow-varying weak Newtonian potentials. In section IV we summarize the key results of this paper. Appendices A-C support the calculations in sections II-III.", 'II. DERIVATION OF THE LEADING-ORDER GRADIOMETER PHASE SHIFT IN LINEARIZED GRAVITY': "Schematically, atom interferometers (AIs) utilize matter wave interference to detect the phase difference between two coherent atomic states in a spatial superposition. In order for an atom to be prepared in a spatial superposition and then measured via matter wave interference, the trajectories of the atomic wavepackets are manipulated using laser pulses. Most experiments rely on a two-level system where the external momentum and the internal energy state of the atoms can be manipulated via Rabi oscillations [46]. A laser pulse that interacts with the atom over a quarter of the Rabi cycle (a π/ 2-pulse) takes an atom in one state to an equal superposition of two states, thus acting as a 'beamsplitter'; a pulse over half of the Rabi cycle (a π -pulse) reverses the state of the atom, thus acting as a 'mirror' [47]. In this paper, we focus on experiments that rely on two spatially-separated AIs operating common lasers and single-photon transitions with energy separation ω a . We refer to these configurations as single-photon atom gradiometers (AGs). \nIn order to correctly capture relativistic effects, which manifest through the dependence of dynamics on the curvature of spacetime, it is of paramount importance to describe the interferometer sequence and all experimental quantities in a frame-independent manner. Importantly, this guarantees that the observable is free of gauge artifacts. Let us consider the description of atomic fountains (which we assume in this work) in the language of general relativity ( e.g. Ref. [37]). In these AI experiments, the atoms are in free-fall. Provided that the radius of curvature is much larger than the wavepacket size, a semi-classical treatment is sufficient. In this regime, atom trajectories are described in terms of timelike geodesics, while photon trajectories are described in terms of null geodesics. The spacetime points at which atom-light interactions occur can be solved in coordinate time and position. Furthermore, for the \nphase shift to be frame-independent, the coordinate times at which the laser pulses are emitted must be related to the established time difference measured by an observer traveling along the laser's worldline. Finally, the four momentum of a pulse must be related to the frequency of the pulse at emission; this four momentum is then evolved from emission to the designated atom-light interaction point, where the atom's recoil is computed in a local inertial frame. \nInspired by this description, in the following sections we provide a detailed derivation from first principles of the gauge-invariant gradiometer phase shift. Importantly, our framework is valid to leading order in a generic metric perturbation and correctly predicts the coefficients of phase shift terms that are linear in the atom's recoil velocity. Starting from the phase shift for a single AI, we show that the single-photon AG observable can be reinterpreted in terms of coordinate time-delays that are well understood in the context of laser interferometers: the Doppler, Shapiro and Einstein time-delays.", 'A. Propagation Phase Shift': "Let us consider the semi-classical evolution of an atomic wavepacket | ψ ⟩ in spacetime. In this regime, the atom's dynamics can be described in terms of the evolution of the atom's center-of-mass (c.o.m.). Consequently, solving the Schrodinger equation, | ψ ⟩ ∝ exp(i S ), where S is the action of the atom's c.o.m. [48]. In general relativity, the action of a massive point-like particle can be expressed in terms of the proper time elapsed along particle's timelike worldline [49]. Therefore, the phase difference associated with the spacetime propagation of two spatially-separated wavepackets corresponds to the difference in the actions evaluated along the worldlines of the two wavepackets' c.o.m. from state initialization to measurement. For a spacetime with metric g µν , the propagation phase is \n∆ ϕ = ∮ C mdτ = ∮ C m √ -g µν dx µ dt dx ν dt dt , (1) \nwhere m is the path-dependent mass of the atom, with m = m o and m = m o + ω a for ground and excited states, respectively. 1 The proper time along the atom's worldline is defined as τ , which is not to be confused with the coordinate time x 0 = t , and the atom's path-dependent four coordinate-velocity is defined as dx µ /dt . We parameterize the atom's evolution with respect to t and perform the loop integral over the closed semi-classical path C . This path depends on: (i) the free evolution of the atomic wavepackets between atom-light interaction points, (ii) the arrival time of laser pulses at atom-light interaction points and (iii) the metric-dependent correction to the laser beam's wave-vector, which leads to a correction in the recoil of the atoms after atom-light interactions. Here, we assume that C is initiated when the superposition of states is created ( i.e. at the initial beamsplitter pulse) and closes when the \nFIG. 1: Schematic spacetime diagram showing the atom trajectories used in deriving the framework. The dotted yellow lines denote hypersurfaces defined by the initial beam-splitter pulse, the final beamsplitted pulse, and the measurement. The unperturbed atom trajectory C o is schematically shown as the dashed black lines. In the presence of a metric perturbation, the atom follows a geodesic C (purple), which can be deformed into another geodesic ˜ C (blue) that closes at the final beam-splitter. ∆ x µ is the separation of path C at the final beam-splitter pulse. λ parameterizes the atom geodesics, and the deformation ξ ( λ ) is of order O ( h ) and only enters into the observable at O ( h 2 ). At O ( hv ), ˜ C is a good approximation for C in calculating the propagation phase shift. \n<!-- image --> \ninterference pattern is measured. The latter should not be confused with the final beamsplitter used to redirect the atomic states to the measurement ports. \nNote that the full AI phase observable, ∆ ϕ ∣ ∣ AI , also depends on the laser phase ∆ ϕ laser , which is imprinted onto the atoms during atom-light interaction and arises from the linear coupling between the photon field and the atom's electric dipole [50]: \n∆ ϕ ∣ ∣ AI = ∆ ϕ +∆ ϕ laser . (2) \nIn gradiometer setups featuring more than one AI referenced by the same laser, the AIs experience a common laser phase which cancels out in a differential measurement. 2 Since we focus on gradiometer observables, we do not consider this effect further, though it is important to note that the laser phase contribution is generally non-zero in single-AI setups. \nFor the relativistic effects considered in this work, it is sufficient to linearize the metric tensor, i.e. g µν = η µν + h µν , with h µν ≪ 1. By working at leading order in h µν and relating C to ancillary paths, which we schematically depict in Fig. 1, a number of simplifications will arise. First, in the absence of \n̸ \nmetric perturbations, g µν = η µν , we assume that the atomic wavepackets travel along trajectories that recombine at the application of the final beamsplitter pulse; we will refer to this path as C o . Second, as alluded to above, we define C as the closed path traced by the atomic wavepackets in the presence of a non-zero metric perturbation h µν ; however, while C is closed at measurement, it is no longer closed at the final beamsplitter pulse. Finally, we define a third and fictitious path ˜ C , constructed from C assuming the same metric perturbation h µν ; the initial conditions ( e.g. atom velocity) are adjusted for the path to close at the final beamsplitter pulse. Hence, generally C = C . \n˜ The advantage of introducing these ancillary paths is that, at leading order in the metric perturbation, the phase shift along the true path C is equivalent to the phase shift computed along the unphysical path ˜ C . This can be seen simply as follows. Let ξ µ ( λ ) be the coordinate separation between the atom geodesics in C and ˜ C , where λ ∈ [0 , 1] parameterizes the atom worldlines and ξ µ (0) = ξ µ (1) = 0, since both C and ˜ C are closed with respect to state initialization and state measurement. Expanding in the coordinate deviation ξ and defining L = √ g µν ˙ x µ ˙ x ν , where ˙ x µ = dx µ /dλ , the phase shift over the true path C takes the form 3 \n∆ ϕ = ∮ C mdτ = ∮ ˜ C mdτ + ∫ 1 0 ξ µ ( d dt ∂L ∂ ˙ x µ -∂L ∂x µ ) ∣ ∣ ∣ ∣ ∣ R dλ -∫ 1 0 ξ µ ( d dt ∂L ∂ ˙ x µ -∂L ∂x µ ) ∣ ∣ ∣ ∣ ∣ L dλ + O ( ξ 2 ) = ∮ ˜ C mdτ + O ( ξ 2 ) , (3) \nwhere the left and right trajectories are marked by the L and R subscripts, respectively. The second and third terms in the first line vanish because the atom worldlines in ˜ C are geodesics, and hence extrema of the action. \nWe next expand Eq. (3) to leading order in the metric perturbation, utilizing ˜ C = C o + δC with δC = O ( h ): \n∆ ϕ = ∮ ˜ C m √ -η µν dx µ dt dx ν dt dt -1 2 ∮ C o m √ -η µν dx µ dt dx ν dt h µν dx µ dt dx ν dt dt + O ( h 2 ) . (4) \nWriting the phase in Eq. (4) to leading order in O ( hv ), we find 4 \n∆ ϕ = ∮ ˜ C mdt -∮ C o mv i δv i dt -1 2 ∮ C o mh 00 dt -∮ C o mh 0 i v i dt + O ( h 2 , v 2 ) , (5) \nwhere δv i is the O ( h ) correction to the atom three velocity arising from free-falling atoms, obtained by \nsolving geodesic equations 5 \nδv i ( t, x ) = -∫ t t 0 dt ' Γ i 00 ( x, t ' ) = -η ij ∫ t t 0 dt ' ( ∂ 0 h j 0 -1 2 ∂ j h 00 ) + O ( h 2 , v ) , (6) \nwhere Γ µ αβ is the Christoffel symbol in linearized gravity and t 0 is the time at which the experiment is initialized. We neglect the contribution to the atom's recoil velocity after atom-light interactions that arises from the O ( h ) correction to the photon's wave-vector, since the leading-order correction is proportional to v and therefore enters the observable at next-to-leading order. \nThe first term on the RHS of Eq. (5) corresponds to phase shift due to the deformation of the atom trajectories from C 0 to ˜ C . We allow for multiple pulses to divide each interferometer arm into path segments labeled by k , so that this term can be written as \n∮ ˜ C mdt ⊃ ∮ δC mdt = ( ∑ k ∈ E ω a ∆ t ( k ) )∣ ∣ ∣ ∣ ∣ R -( ∑ k ∈ E ω a ∆ t ( k ) )∣ ∣ ∣ ∣ ∣ L , (7) \nwhere the leading order term is proportional to ω a since the rest mass contribution vanishes under a loop-integral, i.e. ∮ ˜ C m o dt = 0; 6 hence we only sum over the excited segments, denoted as the set E . As a result, only excited state path segments contribute to the phase shift in Eq. (7); this will be very important in the calculations below. In Eq. (7), ∆ t ( k ) is the O ( h ) coordinate time duration of the k -th path segment, and path segments on both the right ( R ) and left arm ( L ) are summed over. The overall minus sign in the left arm contributions originates from the loop integral. The coordinate time corrections can be computed by solving for the intersection of atom and photon worldlines. Importantly, these atom-light interaction points define the path segments. We label each path segment k by the laser pulse that starts the sequence. The k -th segment is ended by the laser pulse labeled as k + 1, which starts the subsequent segment k + 1. Denoting the perturbed initial and final times of the k -th path segment as ¯ t ( k ) + δt ( k ) and ¯ t ( k +1) + δt ( k +1) , the perturbed atom worldline as ¯ x i ( t ) + δx i ( t ), the worldlines of the photons as ¯ x i γ ( k ) ( t ) + δx i γ ( k ) ( t ) and ¯ x i γ ( k +1) ( t ) + δx i γ ( k +1) ( t ), with unperturbed quantities denoted by overbars, and n i as the unit vector pointing along the baseline (considering the motion of both the photons and atoms in a single spatial direction), the corresponding equation to solve is \nn i [ ¯ x i ( ¯ t ( k ) + δt ( k ) ) + δx i ( ¯ t ( k ) )] = n i [ ¯ x i γ ( k ) ( ¯ t ( k ) + δt ( k ) ) + δx i γ ( k ) ( ¯ t ( k ) )] . (8) \nExpanding Eq. (8) to O ( h ) and neglecting the unperturbed atom velocity (as the prefactor ω a in Eq. (7) \nis parametrically O ( v )), we find \nδt ( k ) = ( ± ) ( k ) n i [ δx i ( t ( k ) ) -δx i γ ( k ) ( t ( k ) )] , (9) \nwhere ( ± ) ( k ) is taken to be +1 or -1 for outgoing or incoming photons ( i.e. parallel or anti-parallel to the baseline) interacting with the atom, respectively. Here we drop the overbars, which is valid to O ( h ). The perturbed atom positions, δx i ( t ), are given by integrating Eq. (6), i.e. \nδx i ( t, x ) = ∫ t t 0 δv i ( t ' , x ) dt ' , (10) \nand the perturbed photon trajectories, δx i γ ( k ) ( t ), are given by solving the null geodesic condition, ds 2 = g µν dx µ dx ν = 0, leading to \nn i d dt δx i γ ( k ) ( t ) = ( ∓ ) ( k ) H ( ± ) ( k ) , (11) \nwhere we defined \nH ± ≡ 1 2 ( h 00 ± 2 h 0 i n i + h ij n i n j ) . (12) \nPutting Eq. (10) and Eq. (11) into Eq. (9), the time shifts, ∆ t ( k ) , can thus be expressed as the sum of Doppler and Shapiro time delays, \n∆ t ( k ) = δt ( k +1) -δt ( k ) ≡ ∆ t ( k ) D +∆ t ( k ) S . (13) \nThe Doppler term is due to the atom's motion under the metric perturbation as derived in Eq. (10): \n∆ t ( k ) D = n i [ ( ± ) ( k +1) δx i ( t ( k +1) , x ( k +1) ) -( ± ) ( k ) δx i ( t ( k ) , x ( k ) )] , (14) \nwith x ( k ) and x ( k +1) being the unperturbed initial and final atom positions of the k -th and ( k +1)-th path segment. The Shapiro term, which corresponds to the time delay accrued by the photon along its geodesic as derived in Eq. (11), is given by \n∆ t ( k ) S = ∆ T ( ± ) ( k +1) S ( t ( k +1) L , x ( k +1) L , x ( k +1) ) -∆ T ( ± ) ( k ) S ( t ( k ) L , x ( k ) L , x ( k ) ) +∆ t ( k ) laser , (15) \nwhere we defined \n∆ T ± S ( t, x 1 , x 2 ) ≡ ± ∫ x 2 x 1 H ± ( t ± ( x ' -x 1 ) , x ' ) dx ' . (16) \nHere, the photons which interact with the atom at the k -th and ( k +1)-th intersection are emitted from the lasers at x ( k ) L and x ( k +1) L , and t ( k ) -t ( k ) L ≡ ( ± ) ( k ) ( x ( k ) -x ( k ) L ) with t ( k ) L the corresponding photon emission time. The quantity ∆ t ( k ) laser depends on the O ( h ) correction to the photon's emission spacetime points and cancels out in a differential measurement, as discussed in Appendix A. \nThe remaining three terms on the RHS of Eq. (5) only depend on the unperturbed path C o . By using Eq. (6), these terms can be rewritten as \n∆ ϕ E ≡ -∮ C o mv i δv i dt -1 2 ∮ C o mh 00 dt -∮ C o mh 0 i v i dt = -1 2 ∮ C o dt m ( 1 + v i ∫ t t 0 dt ' ∂ i ) h 00 . (17) \nThese contributions only depend on h 00 , and are thus identified as the Einstein term ( i.e. time dilation as measured by the atoms). We note that this phase shift can be significantly simplified if the metric perturbation is spatially slow varying over C o , which allows it to be expanded as \nh µν ( t, x AI + x a ) = h µν ( t, x AI ) + x i a ∂ i h µν ( t, x AI ) + O ( v 2 ) . (18) \nHere, the distance traveled by the atoms from their initial position is x i a ( t ) ∝ v i and x AI is the unperturbed position of the AI at the start of the sequence. The expansion is valid when ( vT ) ∂ i h µν ≪ h µν , where T is the interrogation time, and holds for all examples considered in this work. 7 With this approximation and integrating by parts, Eq. (17) simplifies to \n∆ ϕ E = ( ∑ k ∈ E ω a ∆ t ( k ) E )∣ ∣ ∣ ∣ ∣ R -( ∑ k ∈ E ω a ∆ t ( k ) E )∣ ∣ ∣ ∣ ∣ L , (19) \nwhere the non-vanishing contribution comes from the excited state segments, as the ground state contribution sums to zero over the closed loop, and \n∆ t ( k ) E = -1 2 ∫ C ( k ) o h 00 ( t, x AI ) dt + O ( v 2 ) (20) \ndefines the Einstein time delay. \nIn summary, the propagation phase shift in Eq. (5) can be written schematically as \n∆ ϕ = ∆ ϕ D +∆ ϕ S +∆ ϕ E + O ( h 2 , v 2 ) , (21) \nwhere the Doppler and Shapiro phase shifts originate from Eq. (7) with ∆ t ( k ) given in Eq. (13), while the Einstein phase shift originates from Eq. (17). The expressions of the Doppler, Shapiro and Einstein time \ndelays are given in Eqs. (14), (15) and (20), respectively. Together with Eq. (19), Eq. (21) facilitates the direct comparison between atom interferometer experiments, such as AION and MAGIS, and laser interferometers, such as LIGO [52] and GQuEST [53]. Indeed, the time-delays that enter the atom interferometer propagation phase also appear in laser interferometer calculations. The only difference between the two propagation phase shifts is the physical origin of the frequency: in the case of atom interferometers, the phase shift is proportional to the energy difference between the ground and excited state; in the case of laser interferometers, the phase shift is proportional to the frequency of the laser pulse traveling along the baseline. More broadly, the appearance of the Doppler and Shapiro time-delays facilitates the analogy between the atom interferometers and the mirrors of a one-dimensional laser interferometer. Furthermore, the appearance of the Einstein time-delay clarifies the analogy between an atom interferometer and the beamsplitter in a laser interferometer. Finally, since we started with the action for massive point-like particles ( cf . Eq. (1)), the phase shift is manifestly invariant under diffeomorphisms.", 'B. Doppler, Shapiro, and Einstein Phase Shifts for Atom Gradiometers': "In an AG, the observable is the difference between the phase shifts measured by each AI, i.e. ∆ ϕ grad ≡ ∆ ϕ ∣ ∣ AI 1 -∆ ϕ ∣ ∣ AI 2 . Since the transitions in each AI are driven by common laser pulses, the gradiometer observable exclusively depends on the difference between the propagation phase shifts of each AI and, subsequently, on the difference between the coordinate time corrections between path segments, which we refer to as gradiometer time delays. Without loss of generality, we can write the gradiometer time delays for each segment k because the pair of laser pulses that start and end the path segment in a given AI also start and end a path segment in the other AI. Restricting our attention to such configurations, the O ( h ) correction to the laser's motion cancels, as we carefully show in Appendix A. Pulling all parts of the derivation together, the gradiometer phase shift for experiments using single-photon transitions can be schematically expressed as \n∆ ϕ grad = ∆ ϕ grad , D +∆ ϕ grad , S +∆ ϕ grad , E + O ( h 2 , v 2 ) , ∆ ϕ grad , D , S , E = ∑ k ∆ ϕ ( k ) grad , D , S , E = ( ∑ k ∈ E ω a ∆ t ( k ) grad , D , S , E )∣ ∣ ∣ ∣ ∣ R -( ∑ k ∈ E ω a ∆ t ( k ) grad , D , S , E )∣ ∣ ∣ ∣ ∣ L , (22) \nwhere the gradiometer time delay is expressed in terms of the single AI time delays ∆ t ( k ) D , S , E ( cf. Eqs. (14), (15) and (20)) as \n∆ t ( k ) grad , D , S , E ≡ ∆ t ( k ) D , S , E ∣ ∣ ∣ AI 1 -∆ t ( k ) D , S , E ∣ ∣ ∣ AI 2 . (23) \nSince the ground state contribution vanishes over the loop, we only sum over the path segments where the atoms are in the excited state. \nExplicit expressions for the Doppler, Shapiro, and Einstein phase shifts can be extracted for particular configurations. In what follows, we provide analytical phase shift expressions for two popular single-photon gradiometer designs: the Mach-Zehnder (MZ) and the large-momentum-transfer (LMT) gradiometers. For convenience, we will describe both experiments in a frame where the lasers are at rest in the absence of a metric perturbation. 8 By convention, the right arm receives the initial momentum deposition.", '1. Mach-Zehnder Gradiometer': "As we showed in section II B 1, the Einstein, Doppler and Shapiro gradiometer phase shift contributions for a MZ gradiometer initiated at time t 0 may be expressed as \n∆ ϕ MZ grad , E , D , S ( t 0 ) = ω a ( ∆ t MZ grad , E , D , S ( t 0 ) -∆ t MZ grad , E , D , S ( t 0 + T ) ) , (B1) \nwhere the time-delays are defined as \n∆ t MZ grad , E ( t ) = -1 2 [∫ t + T t h 00 ( t ' , x AI 1 ) dt ' -∫ t + T + L t + L h 00 ( t ' , x AI 2 ) dt ' ] , ∆ t MZ grad , D ( t ) = n i [ δx i ( t + T, x AI 1 ) -δx i ( t, x AI 1 ) + δx i ( t + L, x AI 2 ) -δx i ( t + T + L, x AI 2 ) ] , ∆ t MZ grad , S ( t ) = ∫ x AI 2 x AI 1 H + ( t ( x ' ) , x ' ) dx ' -∫ x AI 2 x AI 1 H + ( t ( x ' ) + T, x ' ) dx ' , (B2) \nwith t ( x ' ) ≡ t + x ' -x AI 1 in the limit where the distance between the laser source and the first AI can be neglected. Using the Fourier transform convention \n˜ x ( ω ) = ∫ ∞ -∞ dt x ( t ) e -i ωt , (B3) \nand performing the time shift t 0 → t 0 -T , each gradiometer phase shift contribution may be expressed in the frequency domain as \n∆ ˜ ϕ MZ grad , E , D , S ( ω ) = ω a ( 1 -e i ωT ) ∆ ˜ t MZ grad , E , D , S ( ω ) . (B4) \nSimilarly, the Fourier transform of the Doppler gradiometer time delay can be expressed as \n∆ ˜ t MZ grad , D ( ω ) = ( e i ωT -1 ) [ n i δ ˜ x i ( ω, x AI ) e i ω ( x AI -x AI 1 ) ] ∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 . (B5) \nFor convenience, let us rewrite the Shapiro gradiometer time-delay in terms of the Shapiro time-delay computed between the laser located at x L and a particular AI ( cf . Eq. (16)). Explicitly, \n∆ t MZ grad , S ( t 0 ) = [ ∆ T + S ( t 0 + T, x AI 1 , x AI ) ] ∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 -[ ∆ T + S ( t 0 , x AI 1 , x AI ) ] ∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 , (B6) \nwith \n∆ T + S ( t 0 , x AI 1 , x AI ) ≡ ∫ x AI x AI 1 H + ( t 0 +( x ' -x AI 1 ) , x ' ) dx ' . (B7) \nEmploying this decomposition and adopting the procedure for deriving the MZ gradiometer phase shift, the Fourier transform of the Shapiro gradiometer time delay takes the following compact form: \n∆ ˜ t MZ grad , S ( ω ) = ( e i ωT -1 ) [ ∆ ˜ T + S ( ω, x AI 1 , x AI ) ] x AI 1 x AI 2 . (B8) \nTo express the Einstein gradiometer time delay in the frequency domain, we use the following identity: \nFT t → ω {∫ t -∞ f ( t ' ) dt ' } = ˜ f ( ω ) i ω +2 π ˜ f (0) δ ( ω ) , (B9) \nwhere δ ( ω ) is the Dirac delta function [62]. Consequently, for constant t A and t B , we find \nFT t → ω {∫ t + t B t + t A f ( t ' ) dt ' } = ( ˜ f ( ω ) i ω +2 π ˜ f (0) δ ( ω ) ) ( e i ωt B -e i ωt A ) = ˜ f ( ω ) i ω ( e i ωt B -e i ωt A ) , (B10) \nwhere the last equality follows from the definition of δ ( ω ). Using Eq. (B10), the Einstein gradiometer time delay takes the form \n∆ ˜ t MZ grad , E ( ω ) = ( e i ωT -1 ) [ -1 2i ω ˜ h 00 ( ω, x AI ) e i ω ( x AI -x AI 1 ) ] ∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 . (B11) \nFinally, noting that ( e i ωT -1 ) 2 = -4 e i ωT sin 2 ( ωT/ 2) = -ω 2 T 2 e i ωT sinc 2 ( ωT/ 2) and using Eqs. (B1), (B5), (B8) and (B11), the three gradiometer phase shift contributions take the form \n∆ ˜ ϕ MZ grad , E ( ω ) = ω a T 2 ω 2 K MZ ( ω ) [ -1 2i ω ˜ h 00 ( ω, x AI ) e i ω ( x AI -x AI 1 ) ] ∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 , ∆ ˜ ϕ MZ grad , D ( ω ) = ω a T 2 ω 2 K MZ ( ω ) [ n i δ ˜ x i ( ω, x AI ) e i ω ( x AI -x AI 1 ) ] ∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 , ∆ ˜ ϕ MZ grad , S ( ω ) = ω a T 2 ω 2 K MZ ( ω ) [ ∆ ˜ T + S ( ω, x AI 1 , x AI ) ] ∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 , (B12) \nwhere we defined the MZ response function as K MZ ( ω ) = e i ωT sinc 2 ( ωT/ 2).", '2. Large-momentum-transfer Gradiometer': "The derivation of the Einstein, Doppler and Shapiro gradiometer phase shift contributions for a LMT gradiometer follows in a similar fashion. As we showed in section II B 2, the gradiometer phase shift contributions take the form \n∆ ϕ LMT grad , E , D , S ( t 0 ) = ω a n -2 ∑ k =0 even ( ∆ t ( k ) grad , E , D , S ( t 0 ) ∣ ∣ ∣ ∣ L 1 +∆ t ( k ) grad , E , D , S ( t 0 + T -( n -1) L ) ∣ ∣ ∣ ∣ L 2 -∆ t ( k ) grad , E , D , S ( t 0 + T ) ∣ ∣ ∣ ∣ L 1 -∆ t ( k ) grad , E , D , S ( t 0 +2 T -( n -1) L ) ∣ ∣ ∣ ∣ L 2 ) , (B13) \nHere we sum over only even k because only excited states segments contribute to the observable phase shift. We quote Eq. (28)-(29) from Sec. B 2 for the expressions of ∆ t ( k ) grad , E , D , S ( t ) ∣ ∣ L 1 and ∆ t ( k ) grad , E , D , S ( t ) ∣ ∣ L 2 to start our derivation: \n∆ t ( k ) grad , E ( t ) ∣ ∣ ∣ L 1 , L 2 = -1 2 [ ∫ t k +1 ( t,x AI ) t k ( t,x AI ) h 00 ( t ' , x AI ) dt ' ]∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 , ∆ t ( k ) grad , D ( t ) ∣ ∣ ∣ L 1 , L 2 = n i [ ( ± ) ( k +1) δx i ( t k +1 ( t, x AI ) , x AI ) -( ± ) ( k ) δx i ( t k ( t, x AI ) , x AI ) ] ∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 , ∆ t ( k ) grad , S ( t ) ∣ ∣ ∣ L 1 , L 2 = [ ∆ T ( ± ) ( k +1) S ( t +( k +1) L, x ( k +1) L , x AI ) -∆ T ( ± ) ( k ) S ( t + kL, x ( k ) L , x AI ) ] ∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 , (B14) \nwhere t k ( t, x ) is defined as \nt k ( t, x ) = t + kL +( ± ) ( k ) ( x -x ( k ) L ) . (B15) \nWe note that x ( k ) L is the location of the laser source that emits the k -th pulse, and ( ± ) ( k ) corresponds to the direction of the k -th laser pulse. An 'outgoing' pulse is defined to be parallel to the AG baseline ˆ n and takes the sign '+', while an 'incoming' pulse is anti-parallel to the baseline and takes the sign ' -'. In an LMT sequence, a pair of laser pulses that define a path segment always consists of two consecutive pulses from opposite ends of the baseline. There are two possible combinations of laser pulse pairs, ' > ' consists of an outgoing-incoming pair, and ' < ' consists of an incoming-outgoing pair. \nFor the sake of clarity, in this section, we spell out all the x ( k ) L and ( ± ) ( k ) for different path segments in an LMT sequence, and write down explicit equations for the gradiometer time delays for the two kinds of segments. We define the gradiometer time delays for a path segment initiated by a pulse from x L 1 ( i.e. a ' > ' segment) to be ∆ t ( k ) grad , D , S , E ∣ ∣ L 1 . In this case, pulse k is emitted from x (0) L = x L 1 and is outgoing ( i.e. ( ± ) ( k ) = +1), while pulse k +1 is emitted from x ( k +1) L = x L 2 and is incoming ( i.e. ( ± ) ( k +1) = -1). We can write down the gradiometer time delays for excited state segments started by an outgoing pulse \nfrom x L 1 : \n∆ t ( k ) grad , E ( t ) ∣ ∣ ∣ L 1 = -1 2 [ ∫ t k +1 ( t,x AI ) t k ( t,x AI ) h 00 ( t ' , x AI ) dt ' ]∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 , ∆ t ( k ) grad , D ( t ) ∣ ∣ ∣ L 1 = -n i [ δx i ( t k ( t, x AI ) , x AI ) + δx i ( t k +1 ( t, x AI ) , x AI ) ] ∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 , ∆ t ( k ) grad , S ( t ) ∣ ∣ ∣ L 1 = -[ ∫ x AI x L 1 H + ( t k ( t, x ' ) , x ' ) dx ' + ∫ x AI x L 2 H -( t k +1 ( t, x ' ) , x ' ) dx ' ]∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 , (B16) \nwhere we have for even k : \nt k ( t, x ) = t + kL +( x -x L 1 ) , for | g ⟩ → | e ⟩ , t k +1 ( t, x ) = t +( k +1) L -( x -x L 2 ) , for | e ⟩ → | g ⟩ . (B17) \nSimilarly, the gradiometer time delays for a path segment initiated by a pulse from x L 2 ( i.e. a ' < ' segment) is written as ∆ t ( k ) grad , D , S , E ∣ ∣ L 2 . In this case, pulse k is emitted from x ( k ) L = x L 2 and is incoming ( i.e. ( ± ) ( k ) = -1), while pulse k +1 is emitted from x ( k +1) L = x L 1 and is outgoing ( i.e. ( ± ) ( k +1) = +1). The gradiometer time delays for excited state segments started by an incoming pulse from x L 2 can be written as: \n∆ t ( k ) grad , E ( t ) ∣ ∣ ∣ L 2 = -1 2 [ ∫ t k +1 ( t,x AI ) t k ( t,x AI ) h 00 ( t ' , x AI ) dt ' ]∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 , ∆ t ( k ) grad , D ( t ) ∣ ∣ ∣ L 2 = n i [ δx i ( t k ( t, x AI ) , x AI ) + δx i ( t k +1 ( t, x AI ) , x AI ) ] ∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 , ∆ t ( k ) grad , S ( t ) ∣ ∣ ∣ L 2 = [ ∫ x AI x L 1 H + ( t k +1 ( t, x ' ) , x ' ) dx ' + ∫ x AI x L 2 H -( t k ( t, x ' ) , x ' ) dx ' ]∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 , (B18) \nwhere we have for even k : \nt k ( t, x ) = t + kL -( x -x L 2 ) , for | g ⟩ → | e ⟩ , t k +1 ( t, x ) = t +( k +1) L +( x -x L 1 ) , for | e ⟩ → | g ⟩ . (B19) \nIn the frequency domain, using the same procedure as before, Eq. (B13) can be rewritten as \n∆ ˜ ϕ LMT grad , E , D , S ( ω ) = ω a ( 1 -e i ωT ) n -2 ∑ k =0 even ∆ ˜ t ( k ) grad , E , D , S ( ω ) ∣ ∣ ∣ ∣ L 1 + ω a e i ω ( T -( n -1) L ) ( 1 -e i ωT ) n -2 ∑ k =0 even ∆ ˜ t ( k ) grad , E , D , S ( ω ) ∣ ∣ ∣ ∣ L 2 . (B20) \nUsing Eq. (B10) and the steps shown in Appendix B 1, and setting x L 2 = x L 1 + L , the Fourier transforms of the Einstein gradiometer time delays for a particular value of k take the form \n∆ ˜ t ( k ) grad , E ( ω ) ∣ ∣ ∣ L 1 = e i ωkL [ -1 2i ω ˜ h 00 ( ω, x AI ) ( e i ωL e i ω ( x L 2 -x AI ) -e i ω ( x AI -x L 1 ) ) ] ∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 , ∆ ˜ t ( k ) grad , E ( ω ) ∣ ∣ ∣ L 2 = e i ωkL [ -1 2i ω ˜ h 00 ( ω, x AI ) ( e i ωL e i ω ( x AI -x L 1 ) -e i ω ( x L 2 -x AI ) ) ] ∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 , (B21) \nwhile the Doppler contributions can be expressed as \n∆ ˜ t ( k ) grad , D ( ω ) ∣ ∣ ∣ L 1 = e i ωkL [ -n i δ ˜ x i ( ω, x AI ) ( e i ω ( x AI -x L 1 ) + e i ωL e i ω ( x L 2 -x AI ) )] ∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 , ∆ ˜ t ( k ) grad , D ( ω ) ∣ ∣ ∣ L 2 = e i ωkL [ n i δ ˜ x i ( ω, x AI ) ( e i ω ( x L 2 -x AI ) + e i ωL e i ω ( x AI -x L 1 ) )] ∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 . (B22) \nTo achieve compact closed form expressions in the frequency domain, we express the Shapiro gradiometer time delay in terms of Eqs. (16). Explicitly, in terms of x L 1 and x L 2 , we find \nwhere \n∆ t ( k ) grad , S ( t ) ∣ ∣ ∣ L 1 = [ ∆ T -S ( t +( k +1) L, x L 2 , x AI ) -∆ T + S ( t + kL, x L 1 , x AI ) ] ∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 , ∆ t ( k ) grad , S ( t ) ∣ ∣ ∣ L 2 = [ ∆ T + S ( t +( k +1) L, x L 1 , x AI ) -∆ T -S ( t + kL, x L 2 , x AI ) ] ∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 , (B23) \n∆ T + S ( t, x L 1 , x AI ) = ∫ x AI x L 1 H + ( t -x L 1 + x ' , x ' ) dx ' , ∆ T -S ( t, x L 2 , x AI ) = -∫ x AI x L 2 H -( t + x L 2 -x ' , x ' ) dx ' . (B24) \nIn the frequency domain, Eqs. (B23) can be compactly expressed as \n∆ ˜ t ( k ) grad , S ( ω ) ∣ ∣ ∣ L 1 = -e i ωkL [ ∆ ˜ T + S ( ω, x L 1 , x AI ) -e i ωL ∆ ˜ T -S ( ω, x L 2 , x AI ) ] ∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 , ∆ ˜ t ( k ) grad , S ( ω ) ∣ ∣ ∣ L 2 = e i ωkL [ e i ωL ∆ ˜ T + S ( ω, x L 1 , x AI ) -∆ ˜ T -S ( ω, x L 2 , x AI ) ] ∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 . (B25) \nNote that all of the gradiometer time-delay expressions in frequency space appear with a factor of exp(i ωkL ), where k is summed from zero to n -2 with only the even k 's. Since the k -dependence of all \ncontributions can be isolated in this term, the gradiometer phase shifts appear with an overall factor \nn -2 ∑ k =0 even e i ωkL = n 2 e i ω ( n -2) L/ 2 sinc( nωL/ 2) sinc( ωL ) . (B26) \nMaking use of Eqs. (B20), (B21), (B22), (B25) and (B26), the gradiometer phase shift contributions in frequency space may be rewritten compactly as \n∆ ˜ ϕ LMT grad , E ( ω ) = 1 2 k eff T 2 ω 2 K MZ ( ω ) × [ -1 2i ω ˜ h 00 ( ω, x AI ) ( K + LMT ( ω ) e i ω ( x AI -x L 1 ) -K -LMT ( ω ) e i ω ( x L 2 -x AI ) ) ] ∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 , ∆ ˜ ϕ LMT grad , D ( ω ) = 1 2 k eff T 2 ω 2 K MZ ( ω ) × [ n i δ ˜ x i ( ω, x AI ) ( K + LMT ( ω ) e i ω ( x AI -x L 1 ) + K -LMT ( ω ) e i ω ( x L 2 -x AI ) )] ∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 , ∆ ˜ ϕ LMT grad , S ( ω ) = 1 2 k eff T 2 ω 2 K MZ ( ω ) × [ K + LMT ( ω )∆ ˜ T + S ( ω, x L 1 , x AI ) -K -LMT ( ω )∆ ˜ T -S ( ω, x L 2 , x AI ) ] ∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 , (B27) \nwith k eff = nω a and the LMT-specific response functions defined as \nK ± LMT ( ω ) = sinc( nωL/ 2) sinc( ωT/ 2) sinc( ω ( T -( n -2) L ) / 2) sinc( ωL ) ( 1 -( n -2) L T ) for outgoing photons (+) , sinc( ω ( T -nL ) / 2) sinc( ωL ) ( 1 -nL T ) for incoming photons ( -) . (B28)", 'III. APPLICATIONS': 'Equipped with the formalism introduced in the previous section, we now compute the leading order phase shift in both MZ and LMT configurations for two well-motivated examples: a transient gravitational wave (GW) and a slow-varying weak Newtonian potential.', 'A. Gravitational Waves': 'The first application that we consider in this work is the signal induced by a transient GW. By working in different frames and with different single-photon gradiometer configurations, we demonstrate that our result: (i) is invariant under general coordinate transformations and (ii) accurately reproduces results from the literature. Here, we note that an analogous calculation in these two gauges, within the context of a laser interferometer, is well documented in the literature (see, e.g. , Ref. [56]).', '1. Transverse-traceless Gauge': "Firstly we compute the phase shift in the transverse-traceless (TT) gauge. Assuming that the GW propagates in the z direction, in TT-gauge the metric tensor perturbation takes the form \nh TT ij ( t, z ) = h + h × 0 h × -h + 0 0 0 0 ij cos ( ω g ( t -z ) + θ ) . (33) \nHere, i, j ∈ { 1 , 2 , 3 } , h × and h + are the amplitudes of the two polarizations, ω g is the GW angular frequency, and θ is the phase of the GW at ( t, z ) = (0 , 0) [56]. Without loss of generality, we set the gradiometer at z = 0 and the baseline in the x -direction. To make contact with the literature, we work in the very long-baseline limit, i.e. L = x L 2 -x L 1 ≈ x AI 2 -x AI 1 . In this gauge, the Einstein phase shift, which only depends on h 00 , vanishes ( cf. Eq. (17)). The Doppler phase shift, which depends on both h i 0 and h 00 through the atom's geodesic equation, also vanishes ( cf. Eqs. (6), (10) and (14)), regardless of the gradiometer configuration. Hence, in this gauge, the phase shift can only depend on the Shapiro time delay. \nLet us first focus on the simpler MZ gradiometer configuration. Making use of Eqs. (24)-(25), the leading order phase shift in this gauge takes the simple form \n∆ ϕ MZ grad ( t 0 ) = -2 h + Lω a sinc ( ω g L 2 ) sin 2 ( ω g T 2 ) cos ( ω g t 0 + ω g ( T + L 2 ) + θ ) , (34) \nwhose amplitude can be rewritten as \n∣ ∣ ∆ ϕ MZ grad ∣ ∣ = 2 h + Lω a ∣ ∣ ∣ ∣ sinc ( ω g L 2 )∣ ∣ ∣ ∣ sin 2 ( ω g T 2 ) . (35) \nSimilarly, using Eqs. (28)-(30), the phase shift induced by a transient GW in an LMT gradiometer takes the form \n∆ ϕ LMT grad ( t 0 ) = -2 h + Lk eff sinc ( nω g L 2 ) sin ( ω g T 2 ) sin ( ω g ( T -( n -1) L ) 2 ) × cos ( ω g t 0 + ω g ( T + L 2 ) + θ ) , (36) \nwhere k eff = nω a . The amplitude of the signal can be rewritten as \n∣ ∣ ∆ ϕ LMT grad ∣ ∣ = 2 h + Lk eff ∣ ∣ ∣ ∣ sinc ( nω g L 2 ) sin ( ω g T 2 ) sin [ ω g ( T -( n -1) L ) 2 ]∣ ∣ ∣ ∣ . (37) \nNotably, Eqs. (35) and (37) agree with previous results from the literature [18, 41]. Furthermore, our result highlights the origin of the effect in TT-gauge: since test masses in free-fall are unaffected by a transient GW in TT-gauge, the effect of interest appears solely through the delay in the arrival time of photons between the two AIs. \nThis analysis can also be repeated in the frequency domain. Using the master equation for the MZ configuration ( cf . Eq. (26)), the Fourier transform of the Shapiro time delay ( cf . Eq. (16)) and the time-dependence of the GW's phase, we find that the MZ phase shift takes the form \n∆ ˜ ϕ MZ grad ( ω ) = -ω a T 2 ω 2 K MZ ( ω )∆ ˜ T + S ( ω, 0 , L ) = -π 2 ω a h + Lω 2 T 2 K MZ ( ω ) sinc ( ωL 2 ) e i θ e i ωL/ 2 δ ( ω -ω g ) , (38) \nwhere we restricted our attention to positive frequencies. Similarly, using the master equation for the LMT configuration ( cf . Eq. (31)) the gradiometer phase shifts in the frequency domain can be expressed as \n∆ ˜ ϕ LMT grad ( ω ) = -1 2 k eff T 2 ω 2 K MZ ( ω ) ( K + LMT ( ω )∆ ˜ T + S ( ω, 0 , L ) + K -LMT ( ω )∆ ˜ T -S ( ω, L, 0) ) = -π 4 k eff h + Lω 2 T 2 sinc ( ωL 2 ) e i θ e i ωL/ 2 K MZ ( ω ) [ K + LMT ( ω ) + K -LMT ( ω ) ] δ ( ω -ω g ) . (39) \nImportantly, after some algebra, one can show that Eqs. (38) and (39) agree exactly with the Fourier transform of Eqs. (34) and (36), respectively, thus validating the form of the master equations in section II B. \nTo explicitly verify that our result is invariant under general coordinate transformations, we now repeat the GW calculation in the gradiometer's proper detector frame. The coordinate system in this frame is constructed along the worldline of a freely-falling ( i.e. non-accelerating) observer by extending spacelike vectors orthogonal to the observer's four-velocity [57]. In this frame, the metric takes the form \nds 2 = ( -1 + ̂ h 00 ) dt 2 + ̂ h 0 i dtdx i +(1 + ̂ h ij ) dx i dx j , (40) \nwhere the components of the metric perturbation tensor are defined as \n̂ h 00 = -∞ ∑ r =0 2 ( r +2)! ̂ R 0 m 0 n,k 1 ··· k r x m x n x k 1 · · · x k r , ̂ h 0 i = -∞ ∑ r =0 2( r +2) ( r +3)! ̂ R 0 min,k 1 ··· k r x m x n x k 1 · · · x k r , ̂ h ij = -∞ ∑ r =0 2( r +1) ( r +3)! ̂ R imjn,k 1 ··· k r x m x n x k 1 · · · x k r . (41) \nHere, the comma preceding the indices k 1 · · · k r denotes differentiation with respect to these coordinates and the hat above any function implies that this function is evaluated along the observer's worldline x obs e.g. , \nˆ R µνρσ,k 1 ...k r ≡ ∂ r R µνρσ ∂x k 1 . . . ∂x k r ∣ ∣ ∣ ∣ x = x obs , (42) \nwith R µνρσ = ( ∂ ν ∂ ρ h µσ + ∂ µ ∂ σ h νρ -∂ µ ∂ ρ h νσ -∂ ν ∂ σ h µρ ) / 2 the Riemann tensor in linearized gravity [58]. In this frame, x i is the spacelike separation between the observer's worldline and an arbitrary spacetime point at fixed time; setting x i = 0 we recover the metric of flat spacetime. Importantly, the Riemann tensor is invariant (element-wise) under gauge transformations to leading order in the metric perturbation. Therefore, without loss of generality, we may evaluate it in TT-gauge. Since the gradiometer probes test mass dynamics along a single spatial component, and we assumed that the GW travels in the z -direction with the atoms at z = 0 and moving only along the x -direction ( i.e. i = 1), one can easily show that the metric tensor perturbation in this frame takes the especially simple form [59] \n̂ h µν = ̂ h TT 00 , ̂ h TT 00 = -( x i ) 2 ̂ R TT 0 i 0 i = x 2 2 ∂ 2 0 h TT 11 = -ω 2 g x 2 2 h + cos( ω g ( t -z ) + θ ) . (43) \nFrom the results of section B, we observe that in this frame all contributions to the gradiometer phase shift are non-zero. \nUsing Eq. (43), we compute the phase shift for the MZ and LMT pulse sequences in the proper detector frame. Using our master equations and working in the limit x AI 2 -x AI 1 ≈ L , the Doppler, Shapiro and Einstein phase shift contributions to the MZ observable take the form \n∆ ϕ MZ grad , D ( t 0 ) = -2 h + Lω a sin 2 ( ω g T 2 ) cos ( ω g t 0 + ω g ( T + L ) + θ ) , ∆ ϕ MZ grad , S ( t 0 ) = + 2 h + Lω a sin 2 ( ω g T 2 ) cos ( ω g t 0 + ω g ( T + L ) + θ ) -2 h + Lω a sinc ( ω g L 2 ) sin 2 ( ω g T 2 ) cos ( ω g t 0 + ω g ( T + L 2 ) + θ ) + h + L 2 ω a ω g sin 2 ( ω g T 2 ) sin ( ω g t 0 + ω g ( T + L ) + θ ) , ∆ ϕ MZ grad , E ( t 0 ) = -h + L 2 ω a ω g sin 2 ( ω g T 2 ) sin ( ω g t 0 + ω g ( T + L ) + θ ) . (44) \nAdding the three contributions, we exactly recover Eq. (34), as desired. To gain a better understanding of the effects induced by a transient gravitational wave on the AG, it is advantageous to take the limit ω g L ≪ 1 and isolate the leading order phase shifts in Eq. (44). Explicitly, \n∆ ϕ MZ grad , D ( t 0 ) = -2 h + Lω a sin 2 ( ω g T 2 ) cos ( ω g t 0 + ω g T + θ ) + O ( ω g L ) , ∆ ϕ MZ grad , S ( t 0 ) = 1 3 h + L 3 ω a ω 2 g sin 2 ( ω g T 2 ) cos( ω g t 0 + ω g T + θ ) + O ( ω 3 g L 3 ) , ∆ ϕ MZ grad , E ( t 0 ) = -h + L 2 ω a ω g sin 2 ( ω g T 2 ) sin ( ω g t 0 + ω g T + θ ) + O ( ω 2 g L 2 ) . (45) \nFrom Eq. (45), the hierarchy of contributions is evident: in this frame, the Doppler phase shift dominates, while the Einstein and Shapiro phase shift are subdominant, with the latter being further suppressed with respect to the leading order Einstein phase shift. This can be understood as follows. In the PD frame, the effect of a GW can be described in terms of Newtonian forces. Indeed, ∂ 2 0 x i = h TT ij x j / 2, which follows from the geodesic deviation equation [56]; in turn this implies that ˆ h 00 = x i a i , where a i ≡ ∂ 2 0 x i is the acceleration induced by the gravitational wave on photons and atoms. In NR computational frameworks ( e.g. [60]), the leading order phase shift due to forces acting on atoms scales as k eff aT 2 . Hence, the Doppler phase shift contribution, which precisely accounts for the O ( h ) acceleration of the atoms in the AG, recovers this term at leading order. 11 The Einstein and Shapiro contributions are pure GR effects, since the latter depends on the rest mass correction of excited state path segments while the former depends on the finite speed of light. It is therefore unsurprising that these effects are parametrically suppressed with respect to the leading order Doppler phase shift. \nThe calculation for the LMT pulse configuration follows analogously. Using the definitions in sec- \ntion II A, the gradiometer phase shift contributions take the form \n∆ ϕ LMT grad , D ( t 0 ) = 2 h + Lω a sin ( ω g L ) sin ( nω g L 2 ) sin ( ω g T 2 ) × { sin ( ω g t 0 + ω g ( nL + T 2 ) + θ ) -cos( ω g L ) sin ( ω g t 0 + ω g ( 3 T -( n -2) L 2 ) + θ )} , ∆ ϕ LMT grad , S ( t 0 ) = -2 h + Lk eff sinc ( nω g L 2 ) sin ( ω g T 2 ) sin ( ω g ( T -( n -1) L ) 2 ) cos ( ω g t 0 + ω g ( T + L 2 ) + θ ) + h + L 2 ω g ω a sin ( nω g L 2 ) sin ( ω g T 2 ) sin ( ω g t 0 + ω g ( 3 T -( n -2) L 2 ) + θ ) -2 h + Lω a sin ( ω g L ) sin ( nω g L 2 ) sin ( ω g T 2 ) × { sin ( ω g t 0 + ω g ( nL + T 2 ) + θ ) -cos( ω g L ) sin ( ω g t 0 + ω g ( 3 T -( n -2) L 2 ) + θ )} , ∆ ϕ LMT grad , E ( t 0 ) = -h + L 2 ω g ω a sin ( nω g L 2 ) sin ( ω g T 2 ) sin ( ω g t 0 + ω g ( 3 T -( n -2) L 2 ) + θ ) . (46) \nAs for the MZ case, the sum of the terms in Eq. (46) exactly recovers the TT-gauge result in Eq. (36). In the long wavelength limit, the result exhibits interesting features. Explicitly, \n∆ ϕ LMT grad , D ( t 0 ) = -2 h + Lk eff sin 2 ( ω g T 2 ) cos ( ω g t 0 + ω g T + θ ) + O ( ω g L ) , ∆ ϕ LMT grad , S ( t 0 ) = 1 3 h + L 3 ω 2 g k eff sin 2 ( ω g T 2 ) cos( ω g t 0 + ω g T + θ ) + O ( ω 3 g L 3 ) , ∆ ϕ LMT grad , E ( t 0 ) = -1 2 h + L 3 k eff ω 2 g sin ( ω g T 2 ) sin ( ω g t 0 + 3 ω g T 2 + θ ) + O ( ω 3 g L 3 ) . (47) \nWhile the Shapiro and Doppler contributions are structurally equivalent to their respective contributions in Eq. (45), the leading-order LMT Einstein phase shift term is further suppressed by a factor of L/T in the limit ωT ≪ 1, as anticipated at the end of section II B for a generic metric perturbation. \nIn summary, we explicitly showed that, for a transient GW, the phase shift obtained in TT-frame agrees with the observable computed in the PD frame. Furthermore, the results obtained using our simplified framework agree with previous work. This confirms the gauge-invariant nature of the gradiometer observable derived in section II.", 'B. Slow-varying Weak Newtonian Potential': "It is challenging to correctly account for the GR contributions to the gradiometer phase shift induced by a moving point mass, as a fully covariant treatment is often algebraically involved, and unfeasible for complicated configurations such as the LMT sequence. However, with the simplified formalism derived in Sec. II, we can obtain the leading order O ( hv ) contributions to the gradiometer phase shift arising from a slow-varying and weak Newtonian potential in a tractable manner, which can be used to model \nNewtonian noises from passing planes or trains, as well as to study transient signals from new physics. \nIn the presence of a weak Newtonian potential Φ and in the rest frame of the source, the metric in harmonic gauge takes the form \nds 2 = -(1 + 2Φ) dt 2 +(1 -2Φ) dx i dx i , (48) \nwhere i = 1 , 2 , 3 and we make use of the Einstein summation convention. In a frame where the source is moving at a constant velocity v s with its amplitude v s ≪ 1, to leading order the metric takes the form \nds 2 = -(1 + 2Φ) dt 2 -8Φ v s,i dx i dt +(1 -2Φ) dx i dx i , (49) \n̸ \nwhich differs from Eq. (48) in so far as h 0 i = h i 0 = -4Φ v s,i = 0. For a derivation, see Appendix C. In the weak field limit, potentials that are sourced by different bodies can be added linearly. This allows us to conveniently choose an inertial frame at rest in the Earth's gravitational field. \nUnlike signals from periodic sources ( e.g. , GW, ultralight bosonic fields), transient signals from a moving Newtonian potential do not factorize into signal strength and response function in the time domain, making it challenging to understand the signal frequency dependence and analyze the detector sensitivity. Therefore, we directly work in the Fourier domain, where the factorization of the detector response is more straightforward. As the source of the Newtonian potential can move in a general direction, it is necessary to express the result in three spatial dimensions. For simplicity, in this section, we also assume that the two AIs are located at two ends of the baseline, i.e. x AI 1 = x L for the MZ sequence, and x AI 1 = x L 1 , x AI 2 = x L 2 for the LMT sequence. The direction of the baseline is defined by n . \nLet us first focus on the MZ configuration. Following Eq. (26), the total phase shift in Fourier space reads \n∆ ˜ ϕ MZ grad ( ω ) = ω a T 2 K MZ ( ω ) { -ω 2 ∆ ˜ T + S ( ω, x AI 1 , x AI 2 ) ︸ ︷︷ ︸ Shapiro + [ n · ( ∇-4 iω v s ) ︸ ︷︷ ︸ Doppler -iω ] ︸ ︷︷ ︸ Einstein [ ˜ Φ( ω, x AI 1 ) -e iωL ˜ Φ( ω, x AI 2 ) ] } . (50) \nHere, the Shapiro time delay is expressed in terms of Eq. (16) with the metric perturbation introduced \nin Eq. (49), i.e. \n∆ T + S ( t, x AI 1 , x AI 2 ) = -2(1 + 2 v s · n ) ∫ L/ 2 -L/ 2 Φ ( t + L 2 + x ' , x mid + x ' n ) dx ' ∆ T -S ( t, x AI 2 , x AI 1 ) = -2(1 -2 v s · n ) ∫ L/ 2 -L/ 2 Φ ( t + L 2 -x ' , x mid + x ' n ) dx ' , (51) \nwhere we have changed the integration variable and defined x mid ≡ ( x AI 1 + x AI 2 ) / 2 as the midpoint of the baseline. 12 Inspecting the frequency dependence in Eq. (50), at low frequencies ( i.e. ωT, ωL ≪ 1), the signal is dominated by the difference in the gradient of the potential evaluated at the location of the two AIs, i.e. the acceleration gradient. Recalling the asymptotic behavior of the MZ response function, the total phase shift is approximately \n∆ ˜ ϕ MZ grad ( ω ) ≈ ω a T 2 n · [ ∇ ˜ Φ( ω, x AI1 ) -∇ ˜ Φ( ω, x AI2 ) ] , (52) \nwhich agrees with the familiar leading order non-relativistic phase shift [36]. \nSimilarly, for an LMT sequence, following Eq. (31), the total phase shift in the Fourier space induced by a slow-varying Newtonian potential reads \n∆ ˜ ϕ LMT grad ( ω ) = k eff T 2 K MZ ( ω ) 1 2 { -ω 2 [ K + LMT ∆ ˜ T + S ( ω, x AI 1 , x AI 2 ) + K -LMT ∆ ˜ T -S ( ω, x AI 2 , x AI 1 ) ] ︸ ︷︷ ︸ Shapiro -iω [ ( K + LMT -e iωL K -LMT ) ˜ Φ( ω, x AI 1 ) -( K + LMT -e -iωL K -LMT ) e iωL ˜ Φ( ω, x AI 2 ) ] ︸ ︷︷ ︸ Einstein + n · ( ∇-4 iω v s ) [ ( K + LMT + e iωL K -LMT ) ˜ Φ( ω, x AI 1 ) -( K + LMT + e -iωL K -LMT ) e iωL ˜ Φ( ω, x AI 2 ) ] ︸ ︷︷ ︸ Doppler } , (53) \nwhere it is understood that K ± LMT are functions of frequency as given in Eq. (32). Notice that \nlim ω → 0 K + LMT ( ω ) = 1 -( n -2) L T , lim ω → 0 K -LMT ( ω ) = 1 -nL T . (54) \nAt low frequencies ( i.e. ωT, nωL ≪ 1), the phase shift is therefore dominated by \n∆ ˜ ϕ LMT grad ( ω ) ≈ k eff T 2 ( 1 -( n -1) L T ) n · [ ∇ ˜ Φ( ω, x AI1 ) -∇ ˜ Φ( ω, x AI2 ) ] . (55) \nNotice that this result differs from the MZ phase shift by a factor of (1 -( n -1) L/T ). This can be attributed to the effective spacetime area enclosed by a single AI. In an LMT sequence, because of \nFIG. 3: The square root of the power spectral density, √ S n ( ω ) ≡ √ ω/ 2 π | ∆ ϕ grad ( ω ) | , induced by a moving Newtonian noise source. The left panel shows a freight truck with mass M = 10 3 kg, velocity v s = 25 km/h, and impact parameter b = 100 m, moving near a terrestrial, single-photon atom gradiometer (AG) utilizing the large-momentum-transfer (LMT) pulse sequence with baseline L = 1 km, interrogation time T = 1 . 7 s, and n = 2500. Experimental parameters align with advanced designs potentially used in future vertical gradiometers such as AION-100, MAGIS-100, and AION-km (see Table 1 in Ref. [34]). The right panel depicts a piece of space debris with mass M = 10 4 kg, velocity v s = 10 m/s, and impact parameter b = 100 m, passing by a space-based AG, also utilizing the LMT sequence, with baseline L = 4 . 4 × 10 7 m, interrogation time T = 150 s, and n = 250. These parameters match the proposed AEDGE experiment [33]. Both setups employ the clock transition (5 s 21 S 0 ↔ 5s5p 3 P 0 ) in Sr87 with an angular transition frequency ω a = 2 . 70 × 10 15 Hz. The source's impact parameter spans an angle of 45 · with the baseline. Individual contributions from the Einstein (green), Doppler (blue), and Shapiro (orange) effects, as derived in Eq. (53), are illustrated. The estimated level of shot noise, √ S n ( f ) ∼ 10 -5 Hz -1 / 2 , is plotted in grey for comparison. The low-frequency behaviors of the dominant contributions are shown in sky blue. Finally, the characteristic frequencies of the noise spectrum and the response functions, ω = v s /b and ω = 2 π/T , are plotted in black for reference. \n<!-- image --> \nthe incremental velocity boost, the maximum separation between atom paths is ∆ x ∝ k eff ( T -( n -1) L ) instead of ∆ x ∝ k eff T . Hence, the enclosed spacetime area receives a correction proportional to ( n -1) L/T with respect to the MZ case. \nTo better understand the impact of Newtonian noise on the spectrum of an AG experiment, we use the results from this section to compute the power spectral density of the gradiometer phase, which is defined as S n ( ω ) ≡ √ ω/ (2 π ) | ∆ ˜ ϕ grad ( ω ) | 2 [55]. As an example, in the left panel of Fig. 3, we plot the square root of the PSD induced by a freight truck with a mass M = 10 3 kg, an impact parameter b = 100m from the nearest AI, and a constant velocity v s = 25km/h. Experimental parameters are derived from Table 1 in [34] to align with advanced designs that are potentially achievable by future experiments employing the LMT pulse sequence, such as AION-km. The estimated shot noise level is shown for comparison. \nThe contributions from Doppler, Shapiro and Einstein term are individually displayed. As expected, the spectrum is dominated by the Doppler term, which far exceeds the noise floor for ω ≲ 2 π × 10 -2 Hz and would therefore impact a terrestrial experiment's projected reach, e.g. , to mid-frequency GWs. At low frequencies, the dominant phase shift contribution of the Doppler phase shift originates from the first AI as | ∆ ϕ grad | ∼ k eff T 2 (2 GM/bv s ) | ˆ b · ˆ n | , where (2 GM/bv s ) is the characteristic amplitude of the acceleration caused by a transiting source with impact parameter b , as indicated in Fig. 3, where we show the analytic scaling with a cyan dotted curve. The Shapiro contribution is the most subdominant and is parametrically suppressed by v 2 s compared to the Doppler term due to its nature as a relativistic correction. Finally, the Einstein contribution is parametrically suppressed by ∼ v s ( L/T ) compared to the Doppler term. All contributions scale as power laws at low frequency, and are exponentially supressed at high frequency above ω ∼ v s /b , which is the (inverse) characteristic time-scale of the moving source. \nIt is worth exploring whether the Einstein and Shapiro terms can surpass the shot noise floor in a more futuristic setup. In the right panel of Fig. 3, we show the PSD for a piece of space debris with mass M = 10 4 kg, an impact parameter b = 100 m from the nearest AI, and a constant velocity v s = 10 m/s relative to a space-based AG. The small relative velocity could arise from both the debris and the AG orbiting at similar altitudes above Earth's surface. Experimental parameters are chosen to match the AEDGE proposal [33]. As indicated by Eq. (53), the Einstein spectrum peaks at ω ∼ v s /b , with ∆ ϕ E ∼ 2 k eff LTω ( GM/v s ). An estimate with this formula naively suggests that the Einstein contribution might exceed the noise floor for certain realistic Newtonian noise parameters. However, the response functions, K MZ ( ω ) and K ± LMT ( ω ) as defined in Eq. (27) and Eq. (32), introduce additional suppression factors of ( ωT ) -1 for frequencies beyond ω ∼ 2 π/T , which suppresses all contributions at the naively expected peak spectrum frequency. The sin( ωT ) term in the response functions is also responsible for the choppiness in the shape of the spectrum for frequencies ω ≳ 2 π/T . Consequently, it is unlikely that the Einstein contribution to a realistic Newtonian noise will be significant in any experimental search. A similar conclusion can be drawn for the Shapiro contribution, which, although not suppressed by factors of L/T , is further suppressed by a factor of v s . It is important to note that while noise sources with a larger v s could be considered, the peak frequency of these sources is greater than 2 π/T , thereby further suppressing the noise amplitude, and thus can be neglected.", 'IV. DISCUSSION & CONCLUSIONS': "In this work, we have developed a simplified gauge-invariant formalism for calculating the leadingorder gravitational phase shift at O ( hv ) in a single-photon atom gradiometer under the influence of a generic metric perturbation. We have found that the leading-order contributions can be fully accounted for by the rest mass correction of the atom excited states and can be conveniently written in terms of \nDoppler, Shapiro, and Einstein time delays in analogy with laser interferometers. \nWe applied this formalism to MZ and LMT gradiometer configurations, and we calculated the gravitational signals from GWs and slow-varying weak Newtonian potentials in these experiments. Our calculations of the GW phase shift are performed in both the TT gauge and the proper detector frame. Our result confirms the gauge-invariance of the leading-order phase shift and agrees with the literature [18, 41]. In the proper detector frame, the Doppler phase shift gives the familiar ∆ ϕ ∝ k eff aT 2 with an acceleration induced by GW in the long wavelength limit ω g L ≪ 1. We also derived the signal of a slow-varying weak Newtonian potential in the frequency domain and calculated the response functions for both the MZ and LMT configurations. We found that in the laser's rest frame, the dominant effect comes from the Doppler phase shift, which gives the k eff aT 2 scaling in the low frequency regime; the Einstein contribution receives a v s L/T suppression. \nThe formalism presented in this paper is valid to O ( hv ). Extending the range of validity to higher order in the atom velocity and the metric perturbation requires a punctilious analysis, which we leave to future work. For instance, to calculate effects at O ( v 2 ) or O ( h 2 ), one would need a more careful and precise treatment of the separation phase and atom paths, as well as the momentum transferred in atom-light interactions. In our derivation, we assumed that the physical path C is closed; to achieve this in the laboratory, prior knowledge of the metric is required. In practice, however, this is not possible for the type of searches envisaged in this work. Consequently, the atom's worldlines are not expected to overlap at the moment the measurement is performed. In this case, the phase shift for each AI in the gradiometer set-up receives a correction ∼ ¯ p µ ∆ X µ commonly known as the separation phase. Here, ¯ p µ is the average four momentum of the two wavepackets being recombined at a particular measurement port and ∆ X µ is the coordinate separation between the two wavepackets on the null hypersurface defining the measurement. Following the same arguments from section II A, ∆ X µ is zero in the absence of a metric perturbation, and both ∆ X µ and ¯ p µ vanish in the limit of zero recoils. Hence, the correction from this separation phase enters the observable at O ( hv 2 ). Furthermore, the corrections to the atom's initial conditions in ˜ C also enter the observable at O ( hv 2 ), as do the corrections to the atom recoil velocities due to the photons' propagation in curved spacetime and the initial launch velocity of the atoms. Finally, the equivalence between the phase shift evaluated over C and ˜ C only holds at leading order in the metric, so an extension of our formalism to O ( h 2 ) would require a careful treatment order by order in perturbation theory. \nAlthough we have restricted our attention to vertical gradiometers employing single-photon transitions, our results can also be modified to predict the observable in AG experiments which utilize horizontal baselines and two-photon transitions ( e.g. MIGA, ELGAR and ZAIGA). In these types of experiments, the baseline is perpendicular to the initial launch velocity of the atoms and the transition from ground to excited state (and vice versa) is driven by two counter-propagating laser beams which in- \nteract with a particular atomic wavepacket almost synchronously. As a result, the emission times of two pulses are interrelated. However, following the same arguments presented in Appendix A, the observable is not sensitive to O ( h ) corrections to the laser pulse emission times; therefore, only the unperturbed emission times would have to be carefully chosen. \nIn conclusion, the formalism developed in this paper opens exciting new avenues for future studies. For example, our simplified framework, and especially the gradiometer phase shift expressions presented in this work, can assist with the development of Wiener filters in atom gradiometer experiments to mitigate Newtonian noise from nearby traffic (air crafts, trains, etc.) and well-monitored seismic activity. Additionally, because of the generality of our framework insofar as the form of the metric perturbation is concerned, the key results of our work can be used to study signatures of new physics, such as macroscopic dark matter, which we explore in a companion paper [61], and more exotic signatures, such as violations of general coordinate invariance.", 'ACKNOWLEDGMENTS': "We thank Albert Roura and Allic Sivaramakrishnan for useful discussions. L.B., Y.D., V.L., Y.W. and K.Z. are supported by the U.S. Department of Energy, Office of Science, Office of High Energy Physics under Award Number DE-SC0011632, and by the Walter Burke Institute for Theoretical Physics. KZ is also supported by a Simons Investigator award, and by Heising-Simons Foundation 'Observational Signatures of Quantum Gravity' collaboration grant 2021-2817.", 'Appendix A: Shapiro and Doppler Phase Shifts in Single-Photon Atom Gradiometry': 'In atom gradiometers utilizing single-photon transitions, the Shapiro and Doppler phase shifts are manifestly independent of the dynamics of the laser. This is due to the fact that a laser pulse that starts an excited state segment of one atom interferometer will also start the excited state of the other atom interferometer, and those excited state segments are ended by another common laser pulse. To see this explicitly, we consider the Doppler and Shapiro phase shifts for an arbitrary excited state path segment. \nThroughout this appendix, we use overbars to identify unperturbed quantities, and denote O ( h ) corrections to coordinates by δt and δx . Let us first consider a gradiometer configuration with a single laser ( e.g. a Mach-Zehnder gradiometer). For simplicity, let us set the baseline along the x -direction ( i.e. i = 1). Let ( t I L , x I L ) = ( ¯ t I L + δt I L , ¯ x I L + δx I L ) be the spacetime point at which the laser pulse that initialises the excited state paths in both AIs is emitted. This pulse will drive the transition from ground to excited state in the first and second AI at ( t I 1 , x I 1 ) = ( ¯ t I 1 + δt I 1 , ¯ x I 1 + δx I 1 ) and ( t I 2 , x I 2 ) = ( ¯ t I 2 + δt I 2 , ¯ x I 2 + δx I 2 ), respectively. Let ( t F L , x F L ) = ( ¯ t F L + δt F L , ¯ x F L + δx F L ) be the spacetime point at which the laser pulse that drives the transition from excited to ground state is emitted. This second pulse will drive the transition from excited to ground state in the first and second AI at ( t F 1 , x F 1 ) = ( ¯ t F 1 + δt F 1 , ¯ x F 1 + δx F 1 ) and ( t F 2 , x F 2 ) = ( ¯ t F 2 + δt F 2 , ¯ x F 2 + δx F 2 ), respectively. We show this diagrammatically in the left panel of Fig. 4. Since photons travel on null geodesics, the O ( h ) corrections to the coordinate time at which the atom and laser worldlines intersect are given by \nδt I,F 1 , 2 = δt I,F L + ∫ δx I,F 1 , 2 δx I,F L dx \' + ∫ ¯ x I,F 1 , 2 ¯ x I,F L H + ( ¯ t I,F L -¯ x I,F L + x \' , x \' ) dx \' + O ( h 2 ) , (A1) \nwhere we introduced H + in Eq. (12) with n = ˆ x . Note the appearance of the O ( h ) correction to the spacetime points at which the photons are emitted, which in turn gives rise to the ∆ t ( k ) laser term in Eq. (15). From Eq. (22), the sum of the Shapiro and Doppler phase shifts for this particular excited state path segment is given by \n∆ ϕ ( ⋆ ) grad , D +∆ ϕ ( ⋆ ) grad , S = ω a ( δt F 1 -δt I 1 ) -ω a ( δt F 2 -δt I 2 ) = ω a ( δt I 2 -δt I 1 ) -ω a ( δt F 2 -δt F 1 ) . (A2) \nCombining Eq. (A1) with Eq. (A2), it immediately follows that the δt L and δx L terms cancel out (i.e. the gradiometer observable is insensitive to ∆ t ( ∗ ) laser ). Furthermore, from the linearity of integration, it also follows that the integration limits do not depend on δx L . Explicitly, \nδt I,F 2 -δt I,F 1 = ∫ δx I,F 2 δx I,F 1 dx \' + ∫ ¯ x I,F 2 ¯ x I,F 1 H + ( ¯ t I,F L -¯ x I,F L + x \' , x \' ) dx \' . (A3) \n<latexit sha1\\_base64="wwlTEqLfXHYnVEYj5S49pMifVgc=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgKeyKr2PQi8eI5gHJEmYns8mQ2dllplcMSz7BiwdFvPpF3vwbJ8keNFrQUFR1090VJFIYdN0vp7C0vLK6VlwvbWxube+Ud/eaJk414w0Wy1i3A2q4FIo3UKDk7URzGgWSt4LR9dRvPXBtRKzucZxwP6IDJULBKFrp7pFgr1xxq+4M5C/xclKBHPVe+bPbj1kacYVMUmM6npugn1GNgkk+KXVTwxPKRnTAO5YqGnHjZ7NTJ+TIKn0SxtqWQjJTf05kNDJmHAW2M6I4NIveVPzP66QYXvqZUEmKXLH5ojCVBGMy/Zv0heYM5dgSyrSwtxI2pJoytOmUbAje4st/SfOk6p1Xz25PK7WrPI4iHMAhHIMHF1CDG6hDAxgM4Ale4NWRzrPz5rzPWwtOPrMPv+B8fAMUwY2u</latexit> \nxt \n<latexit sha1\\_base64="g2EL6HCcpJz8bLtJCSH2Ns283ak=">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</latexit> \n<latexit sha1\\_base64="g2EL6HCcpJz8bLtJCSH2Ns283ak=">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</latexit> \n<latexit sha1\\_base64="7NaxWW4epP/psj9P5Wnex1Glsbk=">AAAB83icbVDLSgMxFL1TX7W+qi7dBIvgqsyIVJdFNy4r2Ad0hpJJ0zY0kxmSO2IZ+htuXCji1p9x59+YtrPQ1gOBwzn3lRMmUhh03W+nsLa+sblV3C7t7O7tH5QPj1omTjXjTRbLWHdCargUijdRoOSdRHMahZK3w/HtzG8/cm1ErB5wkvAgokMlBoJRtJLvI3/CTNoBetorV9yqOwdZJV5OKpCj0St/+f2YpRFXyOwI0/XcBIOMahRM8mnJTw1PKBvTIe9aqmjETZDNb56SM6v0ySDW9ikkc/V3R0YjYyZRaCsjiiOz7M3E/7xuioPrIBMqSZErtlg0SCXBmMwCIH2hOUM5sYQyLeythI2opgxtTCUbgrf85VXSuqh6tWrt/rJSv8njKMIJnMI5eHAFdbiDBjSBQQLP8ApvTuq8OO/Ox6K04OQ9x/AHzucPw3iSLQ==</latexit> \n( \n) \n( x L 2 ,t L 2 ) ( x F 1 ,t F 1 ) ( x I 1 ,t I 1 ) ( x F 2 ,t F 2 ) ( x I 2 ,t I 2 ) 2 2 ( x I 1 ,t I 1 ) ( x F 2 ,t F 2 ) ( x I 2 ,t I 2 ) FIG. 4: Schematic representation of an arbitrary excited state path segment (dashed red line) in a singlephoton atom gradiometer. On the left, we show the relevant spacetime diagram for a MZ gradiometer sequence, where the transitions are driven by a single laser source. On the right, we show the relevant spacetime diagram for a LMT gradiometer sequence, where the transitions are driven by two laser sources \n( \nx \n,t \n) \n) \n) \n) \n( \n) \n( \nx \nF \nL \nx \nx \nI \nF \nL \nI \nF \n( \nF \n<!-- image --> \nI \n2 \n,t \nxt \nI \n2 \n) \n( \n) \n( \n) \nAfter some algebra, the terms of the LHS of Eq. (A2) may be rewritten using Eq. (A3) as \n∆ ϕ ( ⋆ ) grad , D = ω a ( ∫ δx I 2 δx I 1 dx \' -∫ δx F 2 δx F 1 dx \' ) = ω a ( δx I 2 + δx F 1 -δx I 1 -δx F 2 ) , ∆ ϕ ( ⋆ ) grad , S = ω a ( ∫ ¯ x I 2 ¯ x I 1 H + ( ¯ t I L -¯ x I L + x \' , x \' ) dx \' -∫ ¯ x F 2 ¯ x F 1 H + ( ¯ t F L -¯ x F L + x \' , x \' ) dx \' ) . (A4) \nIn this example, we assumed that the excited state path segment is on the right arm of the interferometer. If this segment were on the left arm of the interferometer, one can easily show that the Doppler and Shapiro would gain an overall minus sign. \nLet us now show the same result for a configuration employing two laser sources ( e.g. a LMT gradiometer configuration). Let ( t L 1 , x L 1 ) = ( ¯ t L 1 + δt L 1 , ¯ x L 1 + δx L 1 ) be the spacetime point at which the first laser pulse that initialises the excited state paths in both AIs is emitted. This pulse will drive the transition from ground to excited state in the first and second AIs at spacetime points ( t I 1 , x I 1 ) = ( ¯ t I 1 + δt I 1 , ¯ x I 1 + δx I 1 ) and ( t I 2 , x I 2 ) = ( ¯ t I 2 + δt I 2 , ¯ x I 2 + δx I 2 ), respectively. Similarly, let ( t L 2 , x L 2 ) = ( ¯ t L 2 + δt L 2 , ¯ x L 2 + δx L 2 ) be the spacetime point at which the second laser pulse that drives the transition from excited to ground state is emitted, so that the spacetime endpoints of the excited state paths of the AIs are ( t F 1 , x F 1 ) = ( ¯ t F 1 + δt F 1 , ¯ x F 1 + δx F 1 ) and ( t F 2 , x F 2 ) = ( ¯ t F 2 + δt F 2 , ¯ x F 2 + δx F 2 ). We show this diagrammatically in the right panel of Fig. 4. The O ( h ) corrections to the coordinate time at which the atom and laser worldlines intersect now depend on whether the pulse was emitted from the first or second laser pulse. \nx \n( \n<latexit sha1\\_base64="g2EL6HCcpJz8bLtJCSH2Ns283ak=">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</latexit> \n<latexit 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sha1\\_base64="g2EL6HCcpJz8bLtJCSH2Ns283ak=">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</latexit> \n<latexit 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sha1\\_base64="HlgHdxzAj5eIUNv7En+5LAOI49A=">AAAB+HicbVC7SgNBFJ31GeMjq5Y2g0GwCrtBomXQxjKCeUCyhNnJ3WTI7IOZu2Jc8iU2ForY+il2/o2TZAtNPDBwOOe+5viJFBod59taW9/Y3Nou7BR39/YPSvbhUUvHqeLQ5LGMVcdnGqSIoIkCJXQSBSz0JbT98c3Mbz+A0iKO7nGSgBeyYSQCwRkaqW+XegiPmEkzQk17tNq3y07FmYOuEjcnZZKj0be/eoOYpyFEyM0U3XWdBL2MKRRcwrTYSzUkjI/ZELqGRiwE7WXzw6f0zCgDGsTKvAjpXP3dkbFQ60nom8qQ4UgvezPxP6+bYnDlZSJKUoSILxYFqaQY01kKdCAUcJQTQxhXwtxK+YgpxtFkVTQhuMtfXiWtasWtVWp3F+X6dR5HgZyQU3JOXHJJ6uSWNEiTcJKSZ/JK3qwn68V6tz4WpWtW3nNM/sD6/AHCX5Mq</latexit> \n<latexit sha1\\_base64="wwlTEqLfXHYnVEYj5S49pMifVgc=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgKeyKr2PQi8eI5gHJEmYns8mQ2dllplcMSz7BiwdFvPpF3vwbJ8keNFrQUFR1090VJFIYdN0vp7C0vLK6VlwvbWxube+Ud/eaJk414w0Wy1i3A2q4FIo3UKDk7URzGgWSt4LR9dRvPXBtRKzucZxwP6IDJULBKFrp7pFgr1xxq+4M5C/xclKBHPVe+bPbj1kacYVMUmM6npugn1GNgkk+KXVTwxPKRnTAO5YqGnHjZ7NTJ+TIKn0SxtqWQjJTf05kNDJmHAW2M6I4NIveVPzP66QYXvqZUEmKXLH5ojCVBGMy/Zv0heYM5dgSyrSwtxI2pJoytOmUbAje4st/SfOk6p1Xz25PK7WrPI4iHMAhHIMHF1CDG6hDAxgM4Ale4NWRzrPz5rzPWwtOPrMPv+B8fAMUwY2u</latexit> \nExplicitly, \nδt I 1 , 2 = δt L 1 + ∫ δx I 1 , 2 δx L 1 dx \' + ∫ ¯ x I 1 , 2 ¯ x L 1 H + ( ¯ t L 1 -¯ x L 1 + x \' , x \' ) dx \' + O ( h 2 ) , δt F 1 , 2 = δt L 2 -∫ δx F 1 , 2 δx L 2 dx \' -∫ ¯ x F 1 , 2 ¯ x L 2 H -( ¯ t L 2 + ¯ x L 2 -x \' , x \' ) dx \' + O ( h 2 ) . (A5) \nMaking use of Eq. (A5) to write \nδt I 2 -δt I 1 = ∫ δx I 2 δx I 1 dx \' + ∫ ¯ x I 2 ¯ x I 1 H + ( ¯ t L 1 -¯ x L 1 + x \' , x \' ) dx \' , δt F 2 -δt F 1 = -∫ δx F 2 δx F 1 dx \' -∫ ¯ x F 2 ¯ x F 1 H -( ¯ t L 2 + ¯ x L 2 -x \' , x \' ) dx \' , (A6) \nthe Doppler and Shapiro gradiometer phase shift terms ( cf . Eq. (A2)) for this path segment can be rewritten as \n∆ ϕ ( ⋆ ) grad , D = ω a ( δx I 2 + δx F 2 -δx I 1 -δx F 1 ) , ∆ ϕ ( ⋆ ) grad , S = ω a ( ∫ ¯ x I 2 ¯ x I 1 H + ( ¯ t L 1 -¯ x L 1 + x \' , x \' ) dx \' + ∫ ¯ x F 2 ¯ x F 1 H -( ¯ t L 2 + ¯ x L 2 -x \' , x \' ) dx \' ) . (A7) \nHere, we assumed that the excited state path segment is on the right arm of the AI. The Doppler and Shapiro contributions from left-arm paths gain an overall minus sign. Furthermore, we assumed that the first pulse originated from the end of the baseline closest to the first AI. When the first pulse originates from the opposite end of the baseline, the Doppler and Shapiro phase shifts are given by \n∆ ϕ ( ⋆ ) grad , D = -ω a ( δx I 2 + δx F 2 -δx I 1 -δx F 1 ) , ∆ ϕ ( ⋆ ) grad , S = -ω a ( ∫ ¯ x I 2 ¯ x I 1 H -( ¯ t L 2 + ¯ x L 2 -x \' , x \' ) dx \' + ∫ ¯ x F 2 ¯ x F 1 H + ( ¯ t L 1 -¯ x L 1 + x \' , x \' ) dx \' ) , (A8) \nfor right-arm paths. The contribution from left-arm segments differs by a relative sign. Note, also, that Eqs. (A4), (A7) and (A8) can be expressed in terms of the gradiometer time-delays introduced in sections II. Indeed, the ω a -independent piece in Eq. (A4) is the Doppler/Shapiro MZ gradiometer time delay, while the ω a -independent piece in Eqs. (A7) and (A8) is the Doppler and Shapiro LMT gradiometer time-delay of \' > \' or \' < \' excited state path segments, respectively ( cf . section II B).', 'Appendix B: Derivation of the Phase Shift Formulae in the Frequency Domain': 'In this appendix, we derive the form of the gradiometer phase shift inthe frequency domain. We specialise to the MZ and LMT configurations defined in section II B.', 'Appendix C: Tools for Computing the Gradiometer Phase Shift Induced by a Slow-varying Weak Newtonian Potential': "In this appendix we provide tools for computing the single-photon gradiometer phase shift induced by a slow-varying weak Newtonian potential. We first derive the metric sourced by a massive point-like object moving at constant velocity v s ≡ v i s . Finally, we collect key results which are necessary for computing the gradiometer phase shift induced by such an object ( e.g. expressions for the potential in the frequency-domain, geometric factors, etc.). \nIn the rest frame of the source, spacetime can be described by the Schwarzschild metric. In the weak-field limit, the Schwarzschild metric in isotropic coordinates can be written as [63]: \nds 2 = ( -1 -2Φ) dt 2 +(1 -2Φ) dx i dx i , (C1) \nwhere Φ( r ) ≡ -GM/r = -GM/ √ x i x i is the gravitational potential sourced by the massive point-like object. Boosting to the laboratory frame, the metric becomes \nds 2 = [ -1 -2 ( 1 + 2 v 2 s 1 -v 2 s Φ v s )] dt 2 + [ 1 -2 ( 1 + 2 v 2 s,i 1 -v 2 s ) Φ v s ] dx i dx i -8 v s,i 1 -v 2 s Φ v s dtdx i -8 v s,i v s,j 1 -v 2 s Φ v s dx i dx j , (C2) \n̸ \nwhere Φ v s = -GM/ | γ ( r -v t ) -( γ -1)( r -( r · n ) n ) | is the boosted potential, with γ = 1 / √ 1 -v 2 s , and the last term is evaluated for i = j . For a slow moving source, we can express the metric to O ( v s ), i.e. \nds 2 = -(1 + 2Φ) dt 2 +(1 -2Φ) dx i dx i -8Φ v s,i dtdx i . (C3) \nWe note that the diagonal terms reproduce the static weak field metric in the isotropic coordinates, and the leading order relativistic corrections appears in g 0 i . \nWe now derive the Fourier transform of the Newtonian potential due to a moving point source, which enters the Fourier-transformed signal in Eqs. (50)-(55). The trajectory of a point mass moving with a constant velocity v s can be written as r s = b + v s ( t -t s ), where b is its impact parameter (defined with respect to the origin), and t s is the time when it is closest to the origin. The potential evalauted at ( t, x ) is then given by \nΦ( t, x ) = -GM | b + v s ( t -t s ) -x | , (C4) \nwhere M is the mass of the object. The Fourier transform of the potential is \n˜ Φ( ω, x ) = -2 GM v s e -iωt s e iω ˆ v s · x vs K 0 ( r ⊥ v s ω ) , (C5) \nwhere we have defined the transverse component r ⊥ and its magnitude to be \nr ⊥ ( x ) ≡ b -x +( ˆ v s · x ) ˆ v s , (C6) \nr ⊥ ( x ) ≡ √ | b -x | 2 -( ˆ v s · x ) 2 , (C7) \nand v s ≡ | v s | . The gradient of the potential, which induces accelerations to test masses, takes the following form in Fourier space \n∇ ˜ Φ( ω, x ) = -2 GM v 2 s e -iωt s e iω ˆ v s · x vs ω [ ˆr ⊥ K 1 ( r ⊥ v s ω ) + i ˆ v s K 0 ( r ⊥ v s ω )] . (C8) \nThe argument x is the location of the AI that measures the phase shift. For gradiometer configurations, these functions should be evaluated at the two baseline-separated AIs, with the difference evaluated according to, e.g. , Eq. (50) and Eq. (53). 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2023A&A...673A..32M | Context. Exoplanets with orbital periods of less than one day are known as ultrashort period USP planets. They are relatively rare products of planetary formation and evolution processes but especially favourable for characterisation with current planet detection methods. At the time of writing 125 USP planets have already been confirmed. BR Aims Our aim is to validate the planetary nature of two new transiting planet candidates around M dwarfs announced by the NASA Transiting Exoplanet Survey Satellite TESS registered as TESS Objects of Interest TOIs TOI1442.01 and TOI2445.01. BR Methods We used TESS data groundbased photometric light curves and SubaruIRD spectrograph radial velocity RV measurements to validate both planetary candidates and to establish their physical properties. BR Results TOI1442 bis a validated exoplanet with an orbital period of P 0.4090682 0.0000004 day a radius of RSUBpSUB 1.15 0.06 RSUBSUB and equilibrium temperature of TSUBpeqSUB 1357SUB42SUBSUP49SUP K. TOI2445 b is also validated with an orbital period of P 0.3711286 0.0000004 day a radius of RSUBpSUB 1.33 0.09 RSUBSUB and equilibrium temperature of TSUBpeqSUB 1330SUB56SUBSUP61SUP K. Their physical properties align with current empirical trends and formation theories of USP planets. Based on the RV measurements we set 3 upper mass limits of 8 MSUBSUB and 20 MSUBSUB thus confirming the nonstellar subJovian nature of both transiting objects. More RV measurements will be needed to constrain the planetary masses and mean densities and the predicted presence of outer planetary companions. These targets extend the small sample of USP planets orbiting around M dwarfs up to 21 members. They are also among the 20 most suitable terrestrial planets for atmospheric characterisation via secondary eclipse with the James Webb Space Telescope according to a widespread emission spectroscopy metric. | 2023-05-01T00:00:00Z | ['2022arXiv220113274M', '10.48550/arXiv.2201.13274', 'arXiv:2201.13274', '2023A&A...673A..32M', '10.1051/0004-6361/202243592'] | ['planetary systems', 'planets and satellites: individual: TOI-1442 b', 'planets and satellites: individual: TOI-2445 b', 'techniques: photometric', 'techniques: spectroscopic', 'methods: observational', 'Astrophysics - Earth and Planetary Astrophysics'] | TOI1442 b and TOI2445 b Two potentially rocky ultrashort period planets around M dwarfs | 2,023 | 174 | 0.52 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF'] | 9 | https://arxiv.org/pdf/2201.13274.pdf | {'TOI-1442 b and TOI-2445 b: two potentially rocky ultra-short period planets around M dwarfs': 'G. Morello 1 ; 2 ; 3 ; 4 , H. Parviainen 1 ; 2 , F. Murgas 1 ; 2 , E. Pallé 1 ; 2 , M. Oshagh 1 ; 2 , A. Fukui 6 ; 1 , T. Hirano 7 ; 13 , H. T. Ishikawa 7 ; 13 , M. Mori 14 , N. Narita 6 ; 7 ; 1 , K. A. Collins 8 , K. Barkaoui 26 ; 27 ; 1 , P. Lewin 9 , C. Cadieux 10 , J. P. de Leon 14 , A. Soubkiou 29 ; 30 ; 31 , N. Abreu Garcia 1 ; 2 , N. Crouzet 12 , E. Esparza-Borges 1 ; 2 , G. E. Fernández Rodríguez 1 , D. Galán 1 ; 2 , Y. Hori 7 ; 13 , M. Ikoma 25 , K. Isogai 15 ; 16 , T. Kagetani 16 , K. Kawauchi 1 ; 2 ; 6 , T. Kimura 24 , T. Kodama 6 , J. Korth 18 ; 19 ; 20 , T. Kotani 7 ; 13 ; 17 , V. Krishnamurthy 7 ; 13 , S. Kurita 24 , A. Laza-Ramos 1 , J. H. Livingston 14 , R. Luque 21 , A. Madrigal-Aguado 1 ; 2 , T. Nishiumi 17 ; 7 ; 16 , J. Orell-Miquel 1 ; 2 , M. Puig-Subirà 1 ; 2 , M. Sánchez-Benavente 1 ; 2 , M. Stangret 5 ; 1 ; 2 , M. Tamura 14 ; 7 ; 13 , Y. Terada 22 ; 23 , N. Watanabe 16 , Y. Zou 16 , Z. Benkhaldoun 29 , K. I. Collins 32 , R. Doyon 11 , L. Garcia 26 , M. Ghachoui 26 ; 29 , M. Gillon 26 , E. Jehin 28 , F. J. Pozuelos 21 ; 26 , R. P. Schwarz 8 , and M. Timmermans 26 \n(A GLYPH<14> liations can be found after the references) \nNovember 21, 2024', 'ABSTRACT': 'Context. Exoplanets with orbital periods of less than one day are known as ultra-short period (USP) planets. They are relatively rare products of planetary formation and evolution processes, but especially favourable for characterisation with current planet detection methods. At the time of writing, 125 USP planets have already been confirmed. \nAims. Our aim is to validate the planetary nature of two new transiting planet candidates around M dwarfs announced by the NASA Transiting Exoplanet Survey Satellite (TESS), registered as TESS Objects of Interest (TOIs) TOI-1442.01 and TOI-2445.01. \nMethods. Weused TESS data, ground-based photometric light curves, and Subaru / IRD spectrograph radial velocity (RV) measurements to validate both planetary candidates and to establish their physical properties. \nResults. TOI-1442 b is a validated exoplanet with an orbital period of P = 0 : 4090682 GLYPH<6> 0 : 0000004 d , a radius of R p = 1 : 15 GLYPH<6> 0 : 06 R GLYPH<8> , and equilibrium temperature of T p ; eq = 1357 + 49 GLYPH<0> 42 K . TOI-2445 b is also validated with an orbital period of P = 0 : 3711286 GLYPH<6> 0 : 0000004 d , a radius of R p = 1 : 33 GLYPH<6> 0 : 09 R GLYPH<8> , and equilibrium temperature of T p ; eq = 1330 + 61 GLYPH<0> 56 K . Their physical properties align with current empirical trends and formation theories of USP planets. Based on the RV measurements, we set 3 GLYPH<27> upper mass limits of 8 M GLYPH<8> and 20 M GLYPH<8> , thus confirming the non-stellar, sub-Jovian nature of both transiting objects. More RV measurements will be needed to constrain the planetary masses and mean densities, and the predicted presence of outer planetary companions. These targets extend the small sample of USP planets orbiting around M dwarfs up to 21 members. They are also among the 20 most suitable terrestrial planets for atmospheric characterisation via secondary eclipse with the James Webb Space Telescope, according to a widespread emission spectroscopy metric. \nKey words. planetary systems - planets and satellites: individual: TOI-1442 b, TOI-2445 b - techniques: photometric - techniques: spectroscopic - methods: observational', '1. Introduction': 'The main theories of planetary formation based on our Solar System did not predict the existence of planets on orbits much narrower than that of Mercury (e.g. Lissauer 1993; Lin et al. 1996; Bodenheimer et al. 2000). However, since the discovery of the first hot Jupiter (Mayor & Queloz 1995), many exoplanets have been detected with orbital periods of just a few days. Among the close-in planet population, an ultra-short period (USP) planet is defined as one that completes its entire orbit in less than one day (Sahu et al. 2006). These extreme cases may provide some of the most revealing insights into the formation and evolution processes of planetary systems (Winn et al. 2018). For example, USP planets are important to constrain the slope of the radius valley of close-in small transiting planets (e.g. Fulton et al. 2017; Gupta et al. 2022; Luque & Pallé 2022). They are also excellent test beds for studying star-planet interactions, such as the e GLYPH<11> ects of tidal forces and atmospheric erosion processes (e.g. Lopez 2017; Hamer & Schlaufman 2020; AlvaradoMontes et al. 2021). \nAll other conditions being equal, USP planets are the easiest to detect and characterise by means of occultations and radial velocity (RV) measurements (Gaudi et al. 2005). Despite this strong selection bias, the observed period distribution of exoplanets drops at P . 4 d, indicating that planets with shorter orbital periods are increasingly rare. Recent estimates of the occurrence rate of USP planets are GLYPH<24> 0.5% (e.g. Sanchis-Ojeda et al. 2014; Winn et al. 2018; Zhu & Dong 2021; Uzsoy et al. 2021), which is comparable with the frequency of hot Jupiters (e.g. Howard et al. 2012; Fressin et al. 2013; Wang et al. 2015; Zhu & Dong 2021). Additionally, more than 80% of the known USP population have radii R p < 2 R GLYPH<8> . There are only eight USP Jupiter-size and / or Jupiter-mass planets with 0 : 75 < P < 1 d. Both the frequency and the small size of USP planets have suggested that they could be the remnant rocky cores of evaporated hot Jupiters (e.g. Valsecchi et al. 2014; Königl et al. 2017). The lack of a metallicity trend for the USP planet host stars, unlike hot Jupiters, suggests that the USP planet population does not coincide with that of hot Jupiters in a later evolutionary stage (e.g. Fischer & Valenti 2005; Winn et al. 2017). Other proposed pathways include photoevaporation of gaseous sub-Neptunes (e.g. \nLundkvist et al. 2016; Lee & Chiang 2017), in situ formation (Chiang & Laughlin 2013), and inward migration of rocky planets (Petrovich et al. 2019). The last hypothesis finds empirical support as several USP planets are part of compact systems showing a broad range of mutual inclinations (Petrovich et al. 2020). \nIn this paper, we report the discovery of two potentially rocky USP planets around M dwarfs from the Transiting Exoplanet Survey Satellite (TESS) and ground-based follow-up programs. During the preparation of this manuscript, Giacalone et al. (2022) published another paper validating TOI-1442 b and TOI-2445 b, among other targets. Here we provide additional evidence towards the confirmation of both planets through colour contamination analysis of the photometric time series, and measurements of the stellar RVs and cross-correlation functions (CCFs) obtained from new spectroscopic datasets. \nThis paper is structured as follows. Section 2 describes the observations. In particular, Section 2.1 describes the TESS observations, Section 2.2 the ground-based transit photometry, and Section 2.3 the spectroscopic observations. Section 3 discusses the characterisation of the host stars. Sections 4 and 5 explain the methods used to analyse the transit photometric observations and spectroscopic RVs. Section 6 reports the results of our analyses. Section 7 puts them into scientific context. Section 8 summarises our findings.', '2.1. TESS photometry': "TOI-1442 (TIC 235683377) was observed by TESS in 2 min integrations during sectors 14 to 26 (2019 July 18-2020 July 04) and 40 to 41 (2021 June 25-2021 August 20). It was announced on 2019 November 14 as a TESS object of interest (TOI) by the Science Processing Operations Center (SPOC) pipeline at NASA Ames Research Center. \nTOI-2445 (TIC 439867639) was observed by TESS in 2 min integrations during sectors 4 (2018 October 19-2018 November 14) and 31 (2020 October 22-2020 November 16). It was announced on 2021 January 6 as a TOI also detected by SPOC. \nWe downloaded the data from the Mikulski Archive for Space Telescopes 1 (MAST) and extracted the Pre-search Data Conditioned Simple Aperture Photometry (PDCSAP) from the light curve files (extension: '\\_lc.fits'). The PDCSAP time series were computed by the SPOC pipeline (Jenkins et al. 2016), which calibrates the image data, performs quality control (e.g. flags bad data), calculates the flux for each target in the field of view through simple aperture photometry (SAP module, Morris et al. 2020), and corrects for instrumental systematic e GLYPH<11> ects (PDC module, Smith et al. 2012; Stumpe et al. 2014). Figure 1 shows the target pixel files (TPFs) and aperture masks used to extract the SAP flux, created using tpfplotter 2 (Aller et al. 2020). There are no known contaminants falling within the aperture masks of either target in most sectors with contrast magnitude GLYPH<1> m < 6 mag, except for a fainter source with GLYPH<1> m = 3 : 04 mag with respect to TOI-1442 in sectors 15, 18, 21, 26, and 41. This contamination was already removed by the PDC pipeline (Smith et al. 2012; Stumpe et al. 2014).", '2.2. Ground-based photometry': "The TESS Follow-up Observing Program (TFOP) is a network of observatories and researchers whose aims are to validate the planetary nature of transit-like signals tagged as TOIs, and to measure the masses and radii of the planets, the orbital properties, and the stellar host parameters. We make use of transit photometry data acquired under the TFOP to confirm that the planetary transits occur on the targeted stars, to rule out some false positive scenarios (e.g. blended eclipsing binaries), and to refine the planet's radius measurement. The TFOP data are available to the working group members on the Exoplanet Follow-up Observing Program for TESS (ExoFOP-TESS) 3 website. In Table 1 we summarise the ground-based photometric observations analysed in this paper.", '2.2.1. MuSCAT': 'The Multi-colour Simultaneous Camera for studying Atmospheres of Transiting planets (MuSCAT, Narita et al. 2015) is a three-band imager mounted on the 188 cm telescope of National Astronomical Observatory of Japan in Okayama, Japan. MuSCAT is equipped with three 1024 GLYPH<2> 1024 pixel CCDs with a pixel scale of 0 00 .358 pixel GLYPH<0> 1 . The field of view of MuSCAT is 6 0 .1 GLYPH<2> 6 0 .1. Each CCD is coupled with an Astrodon Photometrics Generation 2 Sloan filter. The three filter bands are g (400-550 nm), r (550-700 nm) and zs (820-920 nm). \nWe observed a full transit of TOI-2445 b on 2021 February 7 UT with MuSCAT. The exposure times were 30, 20, and 35 s for the g , r , and zs bands, respectively. After performing dark and flat-field calibrations, we extracted light curves by aperture photometry with radii of 3 00 .6, 4 00 .3, and 3 00 .6 for the respective filters using a custom data reduction pipeline (Fukui et al. 2011). The resulting photometric dispersion per exposure (i.e. the root mean square of the light curve fitting residuals) are 1.44%, 0.71%, and 0.15%.', '2.2.2. MuSCAT2': 'MuSCAT2 (Narita et al. 2019) is a four-band imager mounted on the 152 cm Telescopio Carlo Sánchez (TCS) at Teide Observatory in Spain. MuSCAT2 is equipped with four 1024 GLYPH<2> 1024 pixel CCDs with a pixel scale of 0 00 .435 pixel GLYPH<0> 1 . The field of view of MuSCAT2 is 7 0 .4 GLYPH<2> 7 0 .4. The four filter bands are g (400-550 nm), r (550-700 nm), i (700-820 nm), and zs (820-920 nm). The g , r , and zs filters are identical to those adopted by MuSCAT. The i filter was custom-ordered and manufactured by Asahi Spectra Co., Ltd. \nWe observed a partial transit of TOI-2445 b on 2021 August 6 UT and a full transit on 2021 September 14 UT with MuSCAT2. The r filter was not available during the last observation. The exposure times were set between 50 and 100 s, depending on the band, instrument, and observing conditions. We extracted photometry with aperture radii of 10 00 .875. The resulting photometric dispersion per exposure are 0.52-0.39% in g , 0.28% in r , 0.30-0.25% in i , and 0.23-0.25% in zs for the two nights, respectively.', '2.2.3. MuSCAT3': "MuSCAT3 (Narita et al. 2020) is another four-band imager with g , r , i , and zs filters, mounted on the 2 m Faulkes Telescope North \nFig. 1. TESS target pixel file images of TOI-1442 (top 15 panels) and TOI-2445 (last 2 panels). The pixels highlighted in red denote the aperture mask used to calculate the SAP. The red circles represent neighbouring sources listed in Gaia DR2; the target star is shown with a white 'x' and identifier 1. The size of the red circles is inversely proportional to the apparent magnitude di GLYPH<11> erence with respect to the target star. The maximum contrast magnitude of the plotted sources is GLYPH<1> m = 6 mag. \n<!-- image --> \nTable 1. List of ground-based transit observations, their main characteristics, and auxiliary parameters used for detrending. \nNotes. ( a ) The rms of fitting residuals. ( b ) Names of the linear detrending parameters, as given in the data files from ExoFOP-TESS. \n(FTN) of Las Cumbres Observatory at the Haleakala observatory in Hawaii. The 2048 GLYPH<2> 2048 pixel CCDs, with a pixel scale of 0 00 .266 pixel GLYPH<0> 1 , were manufactured by Teledyne Princeton Instruments. The field of view of MuSCAT3 is 9 0 .1 GLYPH<2> 9 0 .1. \nWe observed three transits of TOI-1442 b on 2021 May 05, 2021 June 06, and 2021 June 17 UT with MuSCAT3. We discarded the g -band data from the first two observations because of too high scatter. The exposure times were set to 240, 67, 30, and 29 s for the g , r , i , and zs bands, respectively. We extracted photometry with aperture radii of 2 00 .65, 7 00 .95, and 4 00 .68 for the respective filters. The resulting photometric dispersion per exposure are 0.14% in g , 0.13-0.16% in r , 0.14-0.22% in i , and 0.14-0.21% in zs .", '2.2.4. TRAPPIST-South': 'The TRAnsiting Planets and PlanetesImals Small TelescopeSouth (TRAPPIST-South, Jehin et al. 2011; Gillon et al. 2011) is a 60 cm telescope at La Silla Observatory in Chile. It is equipped with a thermoelectrically cooled 2 K GLYPH<2> 2 K pixel FLI Proline PL3041-BB CCD with a pixel scale of 0 00 .64 pixel GLYPH<0> 1 , resulting in a field of view of 22 0 GLYPH<2> 22 0 . \nWe observed two transits of TOI-2445 b on 2021 January 8 and 14 in the I + z filter with an exposure time of 50 s. We used the TESS Transit Finder tool, 4 which is a customised version of \nthe Tapir software package (Jensen 2013), to schedule the photometric time series. Data calibration and photometric measurements were performed using the PROSE 5 pipeline (Garcia et al. 2021). The resulting photometric dispersions per exposure are 0.42-0.37% for the two nights.', '2.2.5. LCOGT 1m': 'We observed four full transits of TOI-1442 b from the Las Cumbres Observatory Global Telescope (LCOGT; Brown et al. 2013) 1.0 m network node at McDonald Observatory. Observations on 2020 August 14 and 2020 August 30 were conducted in I band with exposure times of 100 s , and observations on 2020 September 26 and 2020 October 21 were conducted in Sloan i 0 band with exposure times of 90 s . We used the TESS Transit Finder to schedule our transit observations. The 1 m telescopes are equipped with 4096 GLYPH<2> 4096 pixel SINISTRO cameras with an image scale of 0 00 : 389 per pixel, resulting in a 26 0 GLYPH<2> 26 0 field of view. The images were calibrated by the standard LCOGT BANZAI pipeline (McCully et al. 2018), and photometric data were extracted using AstroImageJ (Collins et al. 2017). The images were focused and have mean stellar point spread functions with a FWHM of GLYPH<24> 2 00 , and circular photometric apertures with radius GLYPH<24> 4 00 were used to extract the di GLYPH<11> erential photometry. The apertures exclude flux from the nearest Gaia EDR3 and \nTESS Input Catalog neighbour (TIC 1718221659) 13 00 west of the target. The resulting photometric dispersions per exposure are in the range 0.13-0.18%.', '2.2.6. OMM/PESTO': 'We observed a full transit of TOI-1442 b at Observatoire du Mont-Mégantic, Canada, on 2020 February 9. The observations were made in the i 0 filter using the 1.6 m telescope of the observatory equipped with the PESTO camera. The adopted exposure times were of 30 s . The light curve extraction via di GLYPH<11> erential photometry was accomplished with AstroImageJ , which was also used for image calibration (bias subtraction and flatfield division). The images have typical stellar point spread functions with a FWHM of GLYPH<24> 2 00 , and circular photometric apertures with radius GLYPH<24> 5 00 were used to extract the di GLYPH<11> erential photometry, excluding flux from the nearest Gaia EDR3 neighbour. The resulting photometric dispersion per exposure is 0.24%.', '2.2.7. MLO/Lewin': 'We observed a full transit of TOI-2445 b in I band on 2021 January 10 from the Maury Lewin Astronomical Observatory 0.36 m telescope near Glendora, CA. The telescope is equipped with 3326 GLYPH<2> 2504 pixel SBIG STF8300M camera having an image scale of 0 00 : 84 per pixel, resulting in a 23 0 GLYPH<2> 17 0 field of view. The adopted exposure times were of 75 s . The images were calibrated and the photometric data were extracted using AstroImageJ . The images have typical stellar point spread functions with a FWHM of GLYPH<24> 5 00 , and circular photometric apertures with radius GLYPH<24> 7 00 were used to extract the di GLYPH<11> erential photometry, which excluded most of the flux from the nearest Gaia EDR3 neighbour. The resulting photometric dispersion per exposure is 1.30%.', '2.3. Spectroscopic observations with Subaru/IRD': "The InfraRed Doppler (IRD) instrument on the 8.2m Subaru telescope is a fibre-fed spectrograph covering the wavelength range 930-1740 nm with a spectral resolution of R GLYPH<24> 70 000 (Tamura et al. 2012; Kotani et al. 2018). We obtained 20 IRD spectra of TOI-1442 on 11 nights between 2020 September 27 and 2021 October 22 UT. We also obtained 12 IRD spectra of TOI-2445 on five nights between 2021 September 9 and 2021 November 12 UT. The exposure times were set to 1200-1800 s for TOI-1442 and 900-1800 s for TOI-2445, depending on the observing conditions. For all scientific exposures, we also injected the laser-frequency comb into the spectrograph, whose spectra were used for the simultaneous wavelength calibrations. \nWe reduced raw IRD data and extracted one-dimensional stellar and reference (laser-frequency comb) spectra as described in Hirano et al. (2020). Wavelengths were calibrated by using the data of the Thorium-Argon hollow-cathode lamp as well as the laser-frequency comb taken during the daytime. Using IRD's standard analysis pipeline (Hirano et al. 2020), we measured RVs for both TOI-1442 and TOI-2445. In short, we extracted a template spectrum for each target that is free from telluric features and instrumental broadening based on multiple observed frames, and relative RVs were measured with respect to this template by the forward modelling technique taking into account the instantaneous instrumental profile of the spectrograph and telluric absorptions. The resulting RV precisions (internal errors) \nwere typically 4 GLYPH<0> 5 m s GLYPH<0> 1 for TOI-1442 and 5 GLYPH<0> 8 m s GLYPH<0> 1 for TOI-2445.", '3. Stellar characterisation': 'We applied the following procedure to determine the main stellar parameters relevant to the light curve analysis. Firstly, we derived the e GLYPH<11> ective temperature ( T GLYPH<3> ; e GLYPH<11> ), iron abundance ([ Fe = H ] GLYPH<3> ), and overall metallicity ([ M = H ] GLYPH<3> ) from the telluricfree template IRD spectra by following the same analysis as in Ishikawa et al. (2022). We note that for TOI-2445, we risked using the spectrum without instrumental-profile deconvolution. This is because the signal-to-noise ratio (S / N) per frame of TOI2445 is so low that the noise is amplified during the deconvolution, especially in the Y band. \nWe used the 47 FeH molecular lines in the Wing-Ford band at 990 GLYPH<0> 1020 nm to estimate T GLYPH<3> ; e GLYPH<11> . We note that the errors given in Tables 2 and 3 are dominated by systematic errors, which are much larger than the standard deviation ( GLYPH<27> ) of estimates based on individual FeH lines divided by the square root of the number of lines ( GLYPH<27>= p N ). More details can be found in Ishikawa et al. (2022). For elemental abundances, we used 34 and 30 neutral atomic lines for TOI-1442 and TOI-2445, respectively. The atomic species responsible for these lines are Na, Mg, Ca, Ti, Cr, Mn, Fe, and Sr. The analyses are based on the equivalent width comparison of individual absorption lines between the calculated model spectra and the observed spectra. The detailed procedures to determine individual elemental abundances and their errors are described in Ishikawa et al. (2020). \nWe iterated the T GLYPH<3> ; e GLYPH<11> estimation and the abundance analysis as described below. Firstly, we derived a provisional T GLYPH<3> ; e GLYPH<11> adopting the solar metallicity ([ Fe = H ] GLYPH<3> = 0), and then we determined the individual abundances of the eight elements [ X = H ] GLYPH<3> using the T GLYPH<3> ; e GLYPH<11> . Secondly, we redetermined the T GLYPH<3> ; e GLYPH<11> adopting the iron abundance [ Fe = H ] GLYPH<3> as the input metallicity, and then we redetermined the abundances using the new T GLYPH<3> ; e GLYPH<11> . We iterated the estimation of T GLYPH<3> ; e GLYPH<11> and [ Fe = H ] GLYPH<3> until the final results and the results of the previous step agreed within the error margin. From the final results of the abundances of the eight elements, [ M = H ] GLYPH<3> was determined by calculating the average weighted by the inverse of the square of their estimated errors. \nWe performed spectral energy distribution (SED) fitting on the photometric magnitudes of Gaia EDR3 G , BP , and RP bands (Gaia Collaboration et al. 2021); 2MASS J , H , and Ks bands (Skrutskie et al. 2006); and WISE W 1, W 2, and W 3 bands (Cutri et al. 2021), applying the [ M = H ] GLYPH<3> prior from the spectral analysis. Hence, we obtained an independent estimate of T GLYPH<3> ; e GLYPH<11> , along with the stellar radius ( R GLYPH<3> ) and luminosity ( L GLYPH<3> ). The BT-Settl synthetic spectra (Allard 2014) were then fitted to the SED with the parameters of T GLYPH<3> ; e GLYPH<11> , [ M = H ] GLYPH<3> , log surface gravity (log g GLYPH<3> ), and log( R GLYPH<3> = D ), where D is the distance to the system. We calculated the posterior probability distributions of these parameters using the Markov chain Monte Carlo (MCMC) method implemented in the Python package emcee (Foreman-Mackey et al. 2013). At each MCMC step, a synthetic spectrum was calculated by linearly interpolating the model grid for a given set of parameters. A Gaussian prior from IRD was applied for [ M = H ] GLYPH<3> . A white noise jitter term, GLYPH<27> jitter, was also fitted for each of the Gaia EDR3, 2MASS, and WISE datasets such that the magnitude uncertainty was given by q GLYPH<27> 2 cat + GLYPH<27> 2 jitter , where GLYPH<27> cat is the catalogued uncertainty in magnitude. Using the obtained posteriors of log( R GLYPH<3> = D ) and T GLYPH<3> ; e GLYPH<11> , we also derived the posteriors of R GLYPH<3> and L GLYPH<3> applying the distance from Bailer-Jones et al. (2021) \nTable 2. Stellar properties of TOI-1442. \nNotes. ( a ) References: (1) Gaia EDR3 (Gaia Collaboration et al. 2021), (2) Bailer-Jones et al. (2021), (3) TIC v8.2 (Stassun et al. 2019), (4) 2MASS (Skrutskie et al. 2006). \nfor D , which are based on the Gaia EDR3 parallaxes (Lindegren et al. 2021). Finally, we estimated the stellar mass ( M GLYPH<3> ) and radius from [ Fe = H ] GLYPH<3> and the absolute Ks magnitude via the empirical relations of Mann et al. (2019) and Mann et al. (2015), respectively. \nTables 2 and 3 report the stellar parameters obtained with the three methods and values adopted from the literature. The parameter values obtained with multiple methods are consistent within 1 GLYPH<27> . The log g GLYPH<3> is weakly constrained by SED fitting. The empirical relations provide the most precise determinations of M GLYPH<3> , GLYPH<26> GLYPH<3> , and log g GLYPH<3> ; therefore, we adopted these values to compute the planetary parameters.', '4.1. Contamination transit models': 'In order to validate the planetary nature of our TOIs, we fitted simultaneously the TESS and ground-based photometric light curves by modelling planetary transits with third-light contamination. Our approach is similar to that described by Parviainen et al. (2019, 2020, 2021) and Esparza-Borges et al. (2022). \nThe transit models were generated with a customised version of PYLIGHTCURVE 6 (Tsiaras et al. 2016), which implements the analytic formulae derived by Pál (2008). Conventionally, the model light curves are normalised so that the out-of-transit flux is unity, and the flux drop during transit corresponds to the fraction of stellar flux occulted by the transiting planet. We fitted the following transit parameters: planet-to-star radius ratio ( p = R p = R GLYPH<3> ), orbital period ( P ), epoch of transit ( T 0), total transit duration ( T 14), stellar mean density ( GLYPH<26> GLYPH<3> ), and two limbdarkening coe GLYPH<14> cients (LDCs, q 1 and q 2). We adopted the power2 law to approximate the stellar limb-darkening profile (Hestroffer 1997), as recommended by Morello et al. (2017), especially for M dwarfs. Additionally, we implemented optimal sampling by means of the transformed LDCs, q 1 and q 2, derived by Short et al. (2019) following the procedure of Kipping (2013). \nThird light generally means any contribution to the flux from sources outside the star-planet system, such as blended or nearby stars a part of whose photons hit the selected photometric aperture of the target. In this work, we define the relative third-light \nTable 3. Stellar properties of TOI-2445. \nNotes. ( a ) References: (1) Gaia EDR3 (Gaia Collaboration et al. 2021), (2) Bailer-Jones et al. (2021), (3) TIC v8.2 (Stassun et al. 2019), (4) 2MASS (Skrutskie et al. 2006). \nflux as \nGLYPH<12> = F c F GLYPH<3> + F c ; (1) \nwhere F GLYPH<3> and F c denote the flux from the target star and contaminating sources, respectively. Thus, the contamination transit model can be expressed as \nˆ F ( t ) = (1 GLYPH<0> GLYPH<12> )(1 GLYPH<0> GLYPH<3> ( t )) + GLYPH<12>; (2) \nwhere ˆ F ( t ) is the normalised astrophysical flux, and 1 GLYPH<0> GLYPH<3> ( t ) is the pure planetary transit model. We further assumed that the contaminating flux comes from only one blended star, except for the TESS observations, due to the significantly larger pixel scale of TESS compared to that of ground-based detectors. The planet self-blend e GLYPH<11> ect is negligible in the analysed datasets (Kipping & Tinetti 2010; Martin-Lagarde et al. 2020). We fitted the photospheric parameters of the contaminating star ( T c ; e GLYPH<11> and log g c), a flux scaling factor ( f c) to account for, for example, di GLYPH<11> erent distances of the target and contaminating stars, and an independent blend TESS factor ( GLYPH<12> TESS ). The passband-integrated fluxes of the target and hypothetical contaminant stars were computed \nusing ExoTETHyS.BOATS 7 (Morello et al. 2021), based on a precomputed grid of PHOENIX stellar spectra (Claret 2018).', '4.2. Baseline models': 'In addition to planetary transit and possible third-light contamination, other signals are present in the observed light curves, of astrophysical and instrumental origin. We modelled the modulations present in the TESS time series by using Gaussian processes (GPs), which provide a flexible non-parametric method to approximate stochastic trends in various types of datasets (Rasmussen & Williams 2006; Roberts et al. 2012). In the field of astrophysics, GPs are often used to filter out stellar variability and instrumental systematic e GLYPH<11> ects in photometric time series (e.g. Gibson et al. 2012; Evans et al. 2015; Barros et al. 2020). In this paper we computed the TESS GPs using celerite 8 (ForemanMackey et al. 2017) with the Matern-3 / 2 kernel: \nk ( GLYPH<28> ) = GLYPH<27> 2 GP " 1 + 1 GLYPH<15> ! e GLYPH<0> (1 GLYPH<0> GLYPH<15> ) p 3 GLYPH<28>=GLYPH<26> GP + 1 GLYPH<0> 1 GLYPH<15> ! e GLYPH<0> (1 + GLYPH<15> ) p 3 GLYPH<28>=GLYPH<26> GP # : (3) \nTable 4. Prior probability distributions of the fitted parameters. \nNotes. U ( a ; b ) denotes a uniform prior delimited by a and b ; N ( GLYPH<22>; GLYPH<27> ) denotes a normal prior with GLYPH<22> mean and GLYPH<27> width. \nHere GLYPH<28> = j ti GLYPH<0> t j j is the time interval between two points, GLYPH<15> = 0 : 01, and GLYPH<27> GP and GLYPH<26> GP are the characteristic amplitude and timescale of the modulations. The choice of GP Matern-3 / 2 kernel has proven e GLYPH<11> ective to detrend TESS photometry (e.g. GonzálezÁlvarez et al. 2021; Kossakowski et al. 2021; Murgas et al. 2021). \nWe detrended the ground-based light curves by fitting linear models with maximum two auxiliary parameters, as recommended in the data reports uploaded by the relevant observing teams on the ExoFOP website. The last column of Table 1 lists the names of the auxiliary parameters used for detrending, as they are given in the original reports.', '4.3. Bayesian priors': 'Wemade use of the emcee package to sample the posterior probability distributions of the astrophysical parameters associated \nwith the contamination transit models (Section 4.1) and with the baseline model parameters (Section 4.2), simultaneously. The adopted prior distributions were generally broader than the potential constraints available from ancillary observations (e.g. the stellar spectra). They were chosen to discard unphysical solutions without biasing or boosting the inferences from the colour contamination analysis. Table 4 lists the Bayesian priors for all parameters. \nWe selected uniform priors on the radius ratio and total transit duration to let them be fully constrained by the transit light curves themselves. These parameters were shared by all observations and passbands; second-order e GLYPH<11> ects such as the wavelength-dependent absorption of the planetary atmospheres and possible changes in orbital inclinations were neglected. We also assumed linear ephemerides with Gaussian priors on the orbital period and epoch of transits, centred on the ExoFOP values, but their GLYPH<27> widths were conservatively enhanced by a factor of \nTOI-1442 bFig. 2. Photometric observations of TOI-1442 b. Top-left panel: TESS phase-folded light curve of TOI-1442 b with bin factor of 20 (gray) and best-fitting transit model (black). Other panels: Ground-based light curves after data detrending (lighter dots), with 5-min bins (darker dots) and corresponding error bars, and best-fitting transit models (solid lines). \n<!-- image --> \n3. For the stellar mean density, we used our estimates from empirical relations (Tables 2 and 3) with a factor of 1.5 on the error bars. The orbital eccentricity was fixed to zero, as expected for USP planets. \nThe stellar LDCs were computed with ExoTETHyS.SAIL (Morello et al. 2020b,a), based on the photospheric parameters in Tables 2 and 3, then transformed into q 1 and q 2 (see Section 4.1). We adopted broad Gaussian priors centred on the theoretical values of q 1 and q 2 with widths GLYPH<27> = 0 : 1, that largely encompass all plausible values derived from stellar models with consistent photospheric parameters. \nTo compute the physical contamination model, we fixed the e GLYPH<11> ective temperature and surface gravity of the target star, and assumed uniform priors for all the contaminant parameters, including the blend TESS factor. The choice of uniform priors is a conservative one as it constrains the possible flux contamination based only on the photometric time series. The previous knowledge of the field around the target stars (see Section 2.1 and Figure 1), as well as the high resolution images analysed by Giacalone et al. (2022), point towards a low probability and / or amount of blending between sources. \nFinally, we selected uninformative prior distributions for all data detrending parameters and normalisation factors. In particu- \nTOI-2445 bFig. 3. TESS phase-folded and ground-based light curves of TOI-2445 b. Analogous to Figure 2. \n<!-- image --> \nlar, we adopted log-uniform priors for the TESS GP parameters, that were shared over all sectors for the same target. We adopted uniform priors for the coe GLYPH<14> cients of the linear detrending models of the ground-based observations and for the normalisation factors. All data detrending and astrophysical parameters were fitted simultaneously. The final emcee fit for TOI-1442 had 68 parameters, 300 walkers, and 300 000 iterations. The final emcee fit for TOI-2445 had 48 parameters, 200 walkers, and 300 000 iterations. We applied a conservative burn-in of 100 000 iterations, which is much longer than the autocorrelation lengths of each parameter chain. Figures 2 and 3 show the TESS phase-folded and ground-based light curves, after data detrending, and max- \nimum likelihood contamination transit models for TOI-1442 b and TOI-2445 b, respectively.', '5.1. Cross-correlation functions': 'We computed the CCFs of IRD spectra of TOI-1442 and TOI2445 with that of the star TOI-1634, to check whether our targets are double-lined spectroscopic binaries. We selected TOI-1634 as a spectral template as it is a well-studied star (Hirano et al. 2021) that shows single-peaked spectra, and it is of similar spectral type ( GLYPH<24> M3 dwarf). We followed the same method to compute the CCFs, as described by Mori et al. (2022) in their Section \nTable 5. Final system parameters \nNotes. Values preceded by < report 3 GLYPH<27> upper limits. \n- ( a ) Orbital semi-major axis relative in units of the stellar radius. \n3.2. In short, we first computed the CCFs for five spectral segments that are less a GLYPH<11> ected by telluric absorption (i.e. 988-993, 995-1000, 1009-1014, 1016-1021, and 1023-1028 nm), and then took their median. \nIf there are any nearby sources, they could cause two peaks or broadening in the CCFs, if the flux of both stars are detectable. Another possible scenario is a bright source A with a fainter physically bound eclipsing binary BC; the CCFs from both A and BC would show up as a single peak. In this case the fainter CCF of BC would wobble by several km / s, but will not be noticeable in the composed CCF, which would only wobble by a few m / s. However, this scenario could be unraveled by the correlation between the FWHM of the CCF and the RV.', '5.2. Radial velocities': 'We fitted the IRD RV data using radvel 9 (Fulton et al. 2018). We adopted Gaussian priors on the orbital period and epoch of transits, based on the results of the photometric analysis reported in Table 5, and uniform prior with upper bound of 300 m s GLYPH<0> 1 on the RV semi-amplitude ( K p). We also included the jitter term with uniform prior between 0 and 100.', '6.1. From the light curve analysis': 'Figures 4 and 5 show the posterior distributions and mutual correlations of the contamination transit model parameters for TOI1442 b and TOI-2445 b, respectively. Table 6 lists the corresponding median and 1 GLYPH<27> intervals of the posteriors and those of the derived parameters. We cannot completely rule out some degree of contamination, but we do limit contamination from sources with a colour di GLYPH<11> erence. These partial inferences are expected due to the low S / N of the observed light curves (Parviainen et al. 2019). In particular, we note that the di GLYPH<11> erential third-light fraction \nGLYPH<1> GLYPH<12> pass = GLYPH<12> pass GLYPH<0> GLYPH<12> zs ; (4) \nwith GLYPH<12> defined as in Equation 1, is consistent with zero within 1.3 GLYPH<27> for all passbands. Equivalently, the T c ; e GLYPH<11> of a hypothetical blended source must have similar or cooler values than T GLYPH<3> ; e GLYPH<11> . The posterior distributions of T c ; e GLYPH<11> are bimodal with a peak towards the lower temperature limit of the PHOENIX stellar models grid (2300 K ), and a second peak at T c ; e GLYPH<11> GLYPH<24> 3000 K . Overall, we pose 3 GLYPH<27> upper limits at T c ; e GLYPH<11> < 4100 K and 5900 K to TOI-1442 b and TOI-2445 b, respectively. \nThe radius ratios turned out to be p = 0 : 0375 + 0 : 0056 GLYPH<0> 0 : 0026 for TOI-1442 b and p = 0 : 0510 + 0 : 0353 GLYPH<0> 0 : 0067 for TOI-2445 b. Multiplying these values by R GLYPH<3> from Tables 2 and 3, the radii of the planetary candidates are R p = 1 : 27 + 0 : 23 GLYPH<0> 0 : 12 R GLYPH<8> and R p = 1 : 50 + 1 : 08 GLYPH<0> 0 : 24 R GLYPH<8> , \nFig. 4. Cornerplot showing the posterior distributions and mutual correlations of the transit model parameters for TOI-1442. The histograms along the diagonal give the median value 1 GLYPH<27> error bars (absolute di GLYPH<11> erences between the medians and the 16th and 84th quantiles) of the distributions. This plot was generated by using the corner Python package (Foreman-Mackey 2016). \n<!-- image --> \nrespectively. We also report 3 GLYPH<27> upper limits of R p < 3 : 22 R GLYPH<8> and R p < 5 : 97 R GLYPH<8> . The 3 GLYPH<27> upper limits on the radii are smaller than the minimum radius of a brown dwarf (Burrows et al. 2001), even in the unlikely cases of strong third-light contamination. The corresponding false alarm probabilities are < 10 GLYPH<0> 6 (TOI1442 b) and 1 : 6 GLYPH<2> 10 GLYPH<0> 4 (TOI-2445 b), thus validating the planetary nature of both transiting objects. \nThe radius ratios are largely degenerate with the third-light contamination parameters, leading to heavy tails on the right side of the posterior distributions. We repeated the light curve fits assuming that the third-light contamination is negligible to measure the planetary radii with greater precision. This assump- \ntion is supported by various pieces of evidence coming from the above colour contamination analysis, the spectral CCFs analysis (see Sections 5.1 and 6.2), the Gaia DR2 (see Figure 1) and high-resolution imaging (Giacalone et al. 2022). We obtained p = 0 : 03405 GLYPH<6> 0 : 00077 and R p = 1 : 15 GLYPH<6> 0 : 06 R GLYPH<8> for TOI-1442 b, and p = 0 : 0453 GLYPH<6> 0 : 0018 and R p = 1 : 33 GLYPH<6> 0 : 09 R GLYPH<8> for TOI-2445 b. According to the classification of small planets proposed by Luque & Pallé (2022), TOI-1442 b and TOI-2445 b belong to the rocky planet population. \nFig. 5. Cornerplot showing the posterior distributions and mutual correlations of the transit model parameters for TOI-2445 (analogous to Figure 4). \n<!-- image --> \nWe used Forecaster 10 to predict the planetary masses, obtaining M p = 1 : 56 + 1 : 07 GLYPH<0> 0 : 52 M GLYPH<8> (for TOI-1442 b) and M p = 2 : 33 + 1 : 76 GLYPH<0> 0 : 80 M GLYPH<8> (for TOI-2445 b), based on the probabilistic massradius relations from Chen & Kipping (2017). The corresponding RV amplitudes are K p = 3 : 1 + 2 : 1 GLYPH<0> 1 : 0 ms GLYPH<0> 1 and K p = 5 : 3 + 4 : 0 GLYPH<0> 1 : 8 ms GLYPH<0> 1 . We also determined physical 3 GLYPH<27> upper limits of M p < 8 M GLYPH<8> and 18 M GLYPH<8> respectively, based on the pure iron composition model of Zeng & Sasselov (2013). \nWe estimated the equilibrium temperatures to be T p ; eq = 1357 + 49 GLYPH<0> 42 K (TOI-1442 b) and 1330 + 61 GLYPH<0> 56 K (TOI-2445 b), assuming zero albedo and no heat redistribution. The equilibrium temper- \natures are above the 880 K limit to melt rocks and metals on the surface of the dayside hemisphere (McArthur et al. 2004).', '6.2. From the spectral CCFs': 'We did not find any suspicious secondary peaks in any of the spectral CCFs analysed in Section 5.1 (see also Figure 6). Thus, we conclude that there is a small possibility that the targets have blended sources, or that they must be significantly fainter than the target stars. We also did not find statistically significant correlations between the FWHM of the CCFs and the RVs, although the sample sizes may be too small for this test. With the avail- \nTable 6. Best-fit transit and contamination parameters. \nNotes. Values preceded by < report 3 GLYPH<27> upper limits. \nFig. 6. Examples of CCFs. Left panel: Calculated CCF of the IRD spectrum of TOI-1442 (blue) taken on UT 2021 June 25 at the orbital phase 0.33, to the template spectrum of TOI-1634, exhibiting a single peak with width of GLYPH<24> 10 km s GLYPH<0> 1 . The autocorrelation function of the TOI-1634 spectrum is overplotted as a reference. Right panel: Analogous plot with the CCF of the IRD spectrum of TOI-2445 (blue) taken on UT 2021 October 27 at the orbital phase 0.75. \n<!-- image --> \nable data there is no evidence for a bright source A with a fainter physically bound eclipsing binary BC.', '6.3. From the RVs': 'The radvel fits failed to provide a good match to the observed RV data, as can be seen in Figure 7. The maximum likelihood Keplerian models are not statistically favoured over flat lines, but the standard deviation of the residuals are GLYPH<24> 2.6 times (TOI- \n1442 b) and 3.8 times (TOI-2445 b) larger than the nominal error bars. We also checked that alternative configurations with a linear trend do not provide significant improvements. \nThere are at least three possible explanations for such a poor agreement between the RV data and our simple RV models. Firstly, the RV measurements can be dominated by random noise and systematic e GLYPH<11> ects as their overall ranges of variations are comparable with the potential instrumental o GLYPH<11> sets of GLYPH<24> 10 m s GLYPH<0> 1 (Delrez et al. 2022; Mori et al. 2022). Secondly, strong stellar \nFig. 7. Radial velocity data and maximum likelihood radvel models for TOI-1442 (left) and TOI-2445 (right). The top panels display the full timeline data, while the bottom panels show the phase-folded data using the best-fit orbital parameters. The error bars do not account for systematic o GLYPH<11> sets. \n<!-- image --> \nactivity could also cause similar o GLYPH<11> sets. Another possibility is that TOI-1442 and / or TOI-2445 host additional planets with detectable RV signals, but the current data are insu GLYPH<14> cient to support this scenario. \nNonetheless, the current RV measurements have a good phase coverage, so that we can constrain K p. From the radvel MCMC fits, we derive 3 GLYPH<27> upper limits of K p < 15 m s GLYPH<0> 1 for TOI-1442 b and K p < 43 m s GLYPH<0> 1 for TOI-2445 b. Using M GLYPH<3> from Tables 2 and 3, we infer 3 GLYPH<27> upper limits on the (projected) planet masses of M p sin i GLYPH<25> M p < 8 M GLYPH<8> and 20 M GLYPH<8> , respectively. These mass upper limits are more than 100 times smaller than the accepted minimum mass of a brown dwarf (Burrows et al. 1997), thus suggesting the planetary nature of the transiting companions.', '7.1. Comparison with previously published parameters': 'Table 5 reports our final system parameters. Our stellar, orbital, and planetary parameters are consistent within 1 GLYPH<27> with those published by Giacalone et al. (2022). The only apparent discrepancy is an o GLYPH<11> set of nearly + 300 K on the planetary equilibrium temperatures from Table 5 compared to those reported by Giacalone et al. (2022). However, this is actually a matter of different definitions adopted. The previous study assumed full atmospheric circulation e GLYPH<14> ciency. Instead, we calculated the dayside temperature assuming no heat redistribution. This second scenario is closer to what is inferred from the phase-curves of other close-in planets, including USP planets (e.g. Demory et al. 2016; Morello et al. 2019; Zieba et al. 2022). The two mathematical definitions of equilibrium temperature di GLYPH<11> er by a factor of GLYPH<24> 1.28. Dividing the temperatures from Table 5 by this factor, we recover the previously published values within 1 GLYPH<27> . \nMost datasets are shared between this work and that of Giacalone et al. (2022). We performed almost simultaneous analyses using di GLYPH<11> erent validation methods, as well as di GLYPH<11> erent data detrending techniques, transit light curve parametrisations, and software modelling tools. The 1 GLYPH<27> consistency confirms the reliability of both sets of results for each of the TOI-1442 and TOI2445 systems. \nIn addition, we analysed new high-resolution spectra obtained with Subaru / IRD. The IRD datasets enabled us to set more precise stellar parameters than those reported in the previous literature. In particular, we could shrink the error bars on the stellar masses by a factor of 2-3, on the radii by a factor of 1.1-1.3, and on the e GLYPH<11> ective temperatures by a factor of 1.6. The planetary parameter error bars are comparable between the two studies, probably being compensated by the use of GPs and / or broader Bayesian priors in our analysis.', '7.2. Comparison with other USP planets': 'Figure 8 shows the radius versus orbital period distribution of the known USP planets. TOI-1442 b and TOI-2445 b are among the 12 validated USP planets with the shortest orbital periods, and likewise their radii are smaller than 2 R GLYPH<8> . If we consider the known sample of 21 USP planets around M dwarfs, TOI-1442 b and TOI-2445 b have the third and the fifth shortest periods, respectively. They also have the seventh and eighth highest equilibrium temperatures. All the USP planets around M dwarfs have radii smaller than 2 R GLYPH<8> , except K2-22 b (Sanchis-Ojeda et al. 2015). The mass upper limits of M p < 8 M GLYPH<8> for TOI-1442 b and M p < 20 M GLYPH<8> for TOI-2445 b confirm the sub-giant nature. More RV measurements are desirable to place significant constraints on their masses and mean densities, hence their chemical compositions. \nWe note that the standard deviation in the RV datasets are 2.6-3.8 times larger than the respective mean error bars. The \nFig. 8. Planetary radius vs orbital period for the known USP planets, based on NASA Exoplanet Archive data as of 2022 November 17. Planets around M dwarfs are colored in red. The green and orange stars correspond to TOI-1442 b and TOI-2445 b, based on the final results of our analysis. The horizontal dashed lines delimit the regions R p < 2 R GLYPH<8> , encompassing 80% of the USP population, and R p < 4 R GLYPH<8> , which approximates the sub-Neptunes. \n<!-- image --> \ndispersion in our RV measurements could be due to instrumental systematic o GLYPH<11> sets, stellar activity, or the possible presence of additional non-transiting planets. Although only a small fraction of USP planets have been detected in multiplanet systems, at least 6 out of the 19 USP planets previously reported around M dwarfs are members of multiplanet systems. These systems are Kepler32 (Fabrycky et al. 2012), Kepler-42 (Muirhead et al. 2012), Kepler-732 (Morton et al. 2016), LTT-3780 (Cloutier et al. 2020; Nowak et al. 2020), LP 791-18 (Crossfield et al. 2019), and LHS1678 (Silverstein et al. 2022). Another peculiarity of these six systems is that their orbits are aligned so that the outer planets are also transiting. We could also add TOI-1238 to this group; it is a K7-M0 dwarf with two confirmed transiting planets, one of which has an USP (González-Álvarez et al. 2021). For TOI1238 and LHS-1678, the discovery papers also reported evidence of a non-transiting companion, likely a giant planet or a brown dwarf, in a wide orbit with a period of years. Among the other M dwarf hosts of a transiting USP planet, non-transiting planet candidates have been identified from RV measurements of TOI1685 (Bluhm et al. 2021; Hirano et al. 2021), TOI-1634 (Hirano et al. 2021; Luque & Pallé 2022), and GJ-1252 (Luque & Pallé 2022). The new RV measurements will also be useful to assess the architecture of the planetary systems around TOI-1442 and TOI-2445, which is important to validate formation theories for USP planets around M dwarfs and di GLYPH<11> erences with those around later-type stars (e.g. Petrovich et al. 2020). \nBoth TOI-1442 b and TOI-2445 b are suitable targets to observe their thermal emission spectra. We estimated their emission spectroscopy metric (ESM) to be 9 : 0 + 1 : 1 GLYPH<0> 1 : 0 and 11 : 1 + 1 : 7 GLYPH<0> 1 : 5 , ac- \ncording to the definition given by Kempton et al. (2018). Given their ESM > 7.5, both TOI-1442 b and TOI-2445 b should be among the top 20 terrestrial targets to be observed in eclipse with the James Webb Space Telescope (JWST) / Mid-InfraRed Instrument (MIRI). We note that the ESM is an estimate of the S / N on the white light eclipse as it would be observed with JWST / MIRI. Such observations can clarify whether these USP planets are bare rocks stripped of their primordial atmospheres, or whether they have retained substantial gaseous envelopes, and can help us characterise their surface and gas composition.', '8. Conclusions': "We validate the planetary nature of TOI-1442 b and TOI-2445 b, two USP planets with M dwarf stellar hosts. TOI-1442 b has an orbital period of P = 0 : 4090682 GLYPH<6> 0 : 0000004 d , a radius of R p = 1 : 15 GLYPH<6> 0 : 06 R GLYPH<8> , and an equilibrium temperature of T p ; eq = 1357 + 49 GLYPH<0> 42 K . TOI-2445 b has an orbital period of P = 0 : 3711286 GLYPH<6> 0 : 0000004 d , a radius of R p = 1 : 33 GLYPH<6> 0 : 09 R GLYPH<8> , and equilibrium temperature of T p ; eq = 1330 + 61 GLYPH<0> 56 K . We report 3 GLYPH<27> upper limits on their masses of M p < 8 M GLYPH<8> and M p < 18 M GLYPH<8> , respectively. The upper mass limits are obtained by assuming a pure iron composition. We also provide precise stellar parameters from previously unpublished high-resolution spectra. \nIt would be interesting to follow-up on these targets with high-precision RV facilities to improve their planetary mass measurements (to constrain their bulk compositions) and possibly detect other planetary companions. They are also suitable targets for emission spectroscopy with JWST. \nAcknowledgements. G. M. has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie SkłodowskaCurie grant agreement No. 895525, and the Ariel Postdoctoral Fellowship Program. This work is partly financed by the Spanish Ministry of Economics and Competitiveness through grants PGC2018-098153-B-C31. This work was partly supported by MEXT / JSPS KAKENHI Grant Numbers JP17H04574, JP18H05439, JP20J21872, JP18H05439, JP20K14521, JP20K14518, JP21K20376, JP21K13975, JP18H05439, JP18H05442, JP15H02063, JP22000005, JP18H05439, JP19K14783, JP21H00035, JP21K20388, JP21K13955, JST CREST Grant Number JPMJCR1761, Astrobiology Center SATELLITE Research project AB022006, and Astrobiology Center PROJECT Research AB031014. R.L. acknowledges funding from University of La Laguna through the Margarita Salas Fellowship from the Spanish Ministry of Universities ref. UNI / 551 / 2021-May 26, and under the EU Next Generation funds. J. K. gratefully acknowledges the support of Swedish National Space Agency (SNSA; DNR 2020-00104) and of the Swedish Research Council (VR: Etableringsbidrag 2017-04945) M.S. acknowledges the support of the Italian National Institute of Astrophysics (INAF) through the project 'The HOT-ATMOS Project: characterizing the atmospheres of hot giant planets as a key to understand the exoplanet diversity' (1.05.01.85.04). This paper is based on data collected at the Subaru Telescope, which is located atop Maunakea and operated by the National Astronomical Observatory of Japan (NAOJ). We wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Maunakea has always had within the indigenous Hawaiian community. A part of our data analysis was carried out on common use data analysis computer system at the Astronomy Data Center, ADC, of the National Astronomical Observatory of Japan. The research leading to these results has received funding from the ARC grant for Concerted Research Actions, financed by the Wallonia-Brussels Federation, and from the French Community of Belgium in the context of the FRIA Doctoral Grant awarded to MT. TRAPPIST is funded by the Belgian Fund for Scientific Research (Fond National de la Recherche Scientifique, FNRS) under the grant PDR T.0120.21. MG and EJ are F.R.S.-FNRS Senior Research Associate. This work makes use of observations from the LCOGT network. Part of the LCOGT telescope time was granted by NOIRLab through the Mid-Scale Innovations Program (MSIP). MSIP is funded by NSF. 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N., Sanchis-Ojeda, R., Rogers, L., et al. 2017, AJ, 154, 60 Zeng, L. & Sasselov, D. 2013, PASP, 125, 227 Zhu, W. & Dong, S. 2021, arXiv e-prints, arXiv:2103.02127 \nZieba, S., Zilinskas, M., Kreidberg, L., et al. 2022, A&A, 664, A79 \n- 1 Instituto de Astrofísica de Canarias (IAC), 38205 La Laguna, Tenerife, Spain\n- 2 Departamento de Astrofísica, Universidad de La Laguna (ULL), 38206, La Laguna, Tenerife, Spain\n- 3 INAF- Palermo Astronomical Observatory, Piazza del Parlamento, 1, 90134 Palermo, Italy\n- 4 Department of Space, Earth and Environment, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden\n- 5 INAF -Osservatorio Astronomico di Padova, Vicolo dell'Osservatorio 5, 35122, Padova, Italy\n- 6 Komaba Institute for Science, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan\n- 7 Astrobiology Center, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan\n- 8 Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA\n- 9 The Maury Lewin Astronomical Observatory, Glendora, California, 91741, USA\n- 10 Université de Montréal, Département de Physique, IREX, Montréal, QC H3C 3J7, Canada\n- 11 Observatoire du Mont-Mégantic, Université de Montréal, Montréal, QC H3C 3J7, Canada\n- 12 European Space Agency (ESA), European Space Research and Technology Centre (ESTEC), Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands\n- 13 National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan\n- 14 Department of Astronomy, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan\n- 15 Okayama Observatory, Kyoto University, 3037-5 Honjo, Kamogatacho, Asakuchi, Okayama 719-0232, Japan\n- 16 Department of Multi-Disciplinary Sciences, Graduate School of Arts and Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan\n- 17 Department of Astronomical Science, The Graduated University for Advanced Studies, SOKENDAI, 2-21-1, Osawa, Mitaka, Tokyo, 181-8588, Japan\n- 18 Department of Space, Earth and Environment, Astronomy and Plasma Physics, Chalmers University of Technology, 412 96 Gothenburg, Sweden\n- 19 Division of Astrophysics, Department of Physics, Lund University, Box 43, 22100 Lund, Sweden\n- 20 Lund Observatory, Department of Astronomy and Theoretical Physics, Lund University, Box 43, SE-221 00 Lund, Sweden\n- 21 Instituto de Astrofísica de Andalucía (IAA-CSIC), Glorieta de la Astronomía s / n, 18008 Granada, Spain\n- 22 Institute of Astronomy and Astrophysics, Academia Sinica, P.O. Box 23-141, Taipei 10617, Taiwan, R.O.C.\n- 23 Department of Astrophysics, National Taiwan University, Taipei 10617, Taiwan, R.O.C.\n- 24 Department of Earth and Planetary Science, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan\n- 25 National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan\n- 26 Astrobiology Research Unit, Université de Liège, 19C Allée du 6 Août, 4000 Liège, Belgium\n- 27 Department of Earth, Atmospheric and Planetary Science, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA\n- 28 Space sciences, Technologies and Astrophysics Research (STAR) Institute, Université de Liège, Belgium\n- 29 Oukaimeden Observatory, High Energy Physics and Astrophysics Laboratory, Cadi Ayyad University, Marrakech, Morocco\n- 30 Departamento de Fisica e Astronomia, Faculdade de Ciencias, Universidade do Porto, Rua do Campo Alegre, 4169-007 Porto, Portugal\n- 31 Instituto de Astrofisica e Ciencias do Espaco, Universidade do porto, CAUP, Rua das Estrelas, 150-762 Porto, Portugal\n- 32 George Mason University, 4400 University Drive, Fairfax, VA, 22030 USA"} |
2024arXiv240907531K | In the present work we introduce a holographic prescription to independently describe sea and valence quarks in the context of the gaugegravity correspondence. We use such prescription to perform an initial calculation that permits us to compare our results with those obtained through lattice techniques when studding magnetic catalysis and its inverse. We find in agreement with previous studies that the elaborated behavior of the condensate is mostly attributable to the sea quarks rather than the valence which show a quite featureless participation. | 2024-09-01T00:00:00Z | ['2024arXiv240907531K', '10.48550/arXiv.2409.07531', 'arXiv:2409.07531'] | ['High Energy Physics - Theory', 'General Relativity and Quantum Cosmology'] | Identifying sea and valence quarks in a magnetically driven catalysis | 2,024 | 175 | 0.17 | ['EPRINT_HTML', 'EPRINT_PDF'] | 0 | https://arxiv.org/pdf/2409.07531.pdf | {'Identifying sea and valence quarks in a magnetically driven catalysis.': 'Daniel Kosoi 1, ∗ and Leonardo Patiño 1, ‡ \n1 Departamento de Física, Facultad de Ciencias, Universidad Nacional Autónoma de México, Apartado Postal 70-542, CP 04510, Ciudad de México, México. \nIn the present work we introduce a holographic prescription to independently describe sea and valence quarks in the context of the gauge/gravity correspondence. We use such prescription to perform an initial calculation that permits us to compare our results with those obtained through lattice techniques when studding magnetic catalysis and its inverse. We find, in agreement with previous studies, that the elaborated behavior of the condensate is mostly attributable to the sea quarks, rather than the valence which show a quite featureless participation.', 'I. INTRODUCTION AND MAIN RESULTS': 'Since its first form, speculated in the 1970s, the phase diagram of Quantum Chromodynamics (QCD) has become increasing more complex, considering dependencies on parameters other than the temperature and the baryon chemical potential, like the masses of the quarks or separated chemical potential for each flavor. These and other elements have been incorporated [1-3] into the description that is conjectured to be provided by the gauge/gravity correspondence [4]. The motivation to use such tool is that there are regions of the aforementioned phase diagram that escape the usual perturbative treatment of quantum field theory, making the so called holographic calculations an appealing alternative. Another agent that has proven to have a relevant influence on the shape of the phase diagram is an intense external magnetic field, expected to be generated in non-central high energy collision experiments where the state of matter known as quark gluon plasma (QGP) is produced. Even the most conservative estimations predict an intensity of 10 -1 m π 2 for this magnetic field, while the more extreme go as high as 15 m π 2 [5]. Regardless of the specific value, any intensity in this range is guarantied to have consequences not only on the phase diagram, but also importantly, on the interpretation of experimental results, so a lot of effort has been placed to understand the related phenomenology. \nA five dimensional background was constructed in [6] to model a strongly coupled plasma subjected to an external magnetic field, and it was latter uplifted to ten dimensions [7] so that fields in the fundamental representation could be incorporated. In this latter work we showed that the quasinormal modes of a fundamental scalar operator accommodated themselves in Landau like levels in the reference frame of the plasma. In the present letter we step away from the rest frame of the plasma so that a dispersion relation can be extracted, exhibiting that these modes indeed behave like quasi-particles, and letting us evaluate the impact of the magnetic field over their condensate at either constant kinetic momentum \nor magnetic field. The benefit of independently controlling these two parameters is that sea quarks and valence quarks react differently to the presence of an intense magnetic field [8]. The field theory expressions in this latter article allow us to identify the gravitational dual to the condensate of both kinds of quarks, and use this to show that the valence condensate in particular depends solely on the combination of canonical momentum and magnetic field specifically given by the kinetic momentum. \nThe final result turns out to be holographically quite intuitive, indicating that the effect of the magnetic field over the sea quarks is codified in the back reaction of the geometry to its presence, while that over the valence quarks is reflected on the impact over the perturbations of the flavor branes. \nSince our analysis is done using the behavior of the quark condensate as characterizing element, we seize the opportunity to exhibit the existence of inverse magnetic catalysis and magnetic catalysis for different ranges of the magnetic field, consistently with the results reported in [9]. \nAlong the way we supplement the study of the quasiparticles displaying their width as a function solely of either the magnetic field or the kinetic momentum, showing that the former is always destabilizing, while the latter has the opposite effect.', 'II. GRAVITATIONAL MODEL AND PREVIOUS RESULTS': "To provide a gravitational dual of a strongly coupled plasma with degrees of freedom in the fundamental representation subject to a constant magnetic field, in [7] we constructed the ten dimensional uplift of the five dimensional background introduced in [6], and embedded a D7-brane on it. This was done in such a manner that the fundamental degrees of freedom were massless, and left us proceeded to study the perturbations of the brane that are dual to scalar excitations of said fundamental fields in the gauge theory. \nOur results in [7] were obtained using either gauge, Landau A = Bxdy or symmetric A = B/ 2( xdy -y dx ) , to introduce the magnetic field B = d A , but for the sake of concreteness, we will employ the former in what follows. Once Ladau gauge has been adopted, the ten \ndimensional metric, consistent with the symmetries introduced in [6], reads \nds 2 10 = -U ( r ) dt 2 + 1 U ( r ) dr 2 + V ( r ) ( dx 2 + dy 2 ) + W ( r ) dz 2 + [ dθ 2 +sin 2 θd ˜ ϕ 2 1 +cos 2 θ ( dϑ 2 +sin 2 ϑd ˜ ϕ 2 2 +cos 2 ϑd ˜ ϕ 2 3 )] , (1) \nwith d ˜ ϕ i = dϕ i + 2 √ 3 Bxdy . The directions t, x, y and z are dual to those of the spacetime where the field theory lies, while r is the radial holographic coordinate, on which the functions U ( r ) , V ( r ) , and W ( r ) , depend solely, and the compact directions are described by the second line in (1). \nFor stability reasons [7], the brane is embedded to extend in the r, t, x, y, z directions and wrap the 3-cycle at the end of (1) given by ϑ, ϕ 2 , and ϕ 3 . In this manner, the embedding can be described placing the brane at constant ϕ 1 and writing θ as a function of r . From (1) we notice that as usual in this descriptions, it is convenient to use χ ( r ) = sin[ θ ( r )] to study the profile of the brane. The object of interest to our work are the perturbations δχ to the aforementioned embedding, of which in [7] we took the particular case χ ( r ) = 0 so that the resulting equation for the brane perturbations, when expressed as the product δχ ( r, t, x, y, z ) = χ t ( t ) χ x ( x ) χ y ( y ) χ z ( z ) χ r ( r ) /χ r ( r ∞ ) , reduced to \n[ 3 UVW ' χ ' r +6 W ( V U ' χ ' r + UV ' χ ' r + UVχ '' r ) (2) \n+6 V W ( 3 -∂ 2 t χ t Uχ t + ∂ 2 y χ y V χ y + ∂ 2 z χ z Wχ z ) χ r ] χ x + Wχ r ( 6 ∂ 2 x χ x -8 B 2 x 2 χ x ) = 0 , \nwith a general solution that includes the factor χ t ( t ) χ y ( y ) χ z ( z ) = e -i ( ωt -k y y -k z z ) , and which can be further separated as \n( 3 + ω 2 U -k 2 y V -k 2 z W ) V Wχ r + 1 2 UVW ' χ ' r + W ( V U ' χ ' r + UV ' χ ' r + UVχ '' r ) = 2 E Wχ r , (3) \nand \n1 2 [ -∂ 2 x χ x + e 2 B 2 x 2 χ x ] = E χ x , (4) \nIn the expressions above e = 2 √ 3 , and we have taken into account that U ( r ) , V ( r ) , and W ( r ) , are only functions of the radial coordinate r , denoting the differentiation with respect to the latter by a prime. \nThe details of the ten dimensional background, and the embedding of the brane, can be found in [7], but for completeness we should mention that there is a 5form that plays a relevant role in the construction of the uplift, but has no effect on the embedding nor on our current calculations, and that the 1-forms d ˜ ϕ i = dϕ i + \n2 √ 3 Bxdy in (1) show that, as the U (1) field is encoded in these compact directions, the quantity e = 2 √ 3 correctly represents the charge with respect to such fields. \nWe began our previous study of the equations above noticing that for the solutions to describe acceptable embeddings, the separation constant E had to take values on the discrete spectrum \nE n = ( n + 1 2 ) ω c , (5) \nwhere ω c = e B was identified as the cyclotron frequency and the integer n with the Landau level number. \nWe continued by focusing on the k y = k z = 0 case, and looked for the complex values of ω for which the solutions to (3) were ingoing and had a normalizable profile in the radial direction, since these are dual to quasinormal modes of the scalar excitations. \nIt was the energy obtained from the latter frequencies that we proved to closely follow the dispersion relation characteristic of Landau levels, presenting a small deviation as expected from modes that are not fully stable. Furthermore, when we studied the ratio of the with of the states over their energy as a function of the magnetic field, we observed a behavior reminiscent of magnetic catalysis for large intensities of such field, and inverse magnetic catalysis for small ones, all with respect to an identifiable critical intensity.", 'III. ROLL OF THE SEA AND VALENCE QUARKS IN (INVERSE) MAGNETIC CATALYSIS': 'The difference between the rolls that sea and valence quarks play for (inverse) magnetic catalysis has been pointed out in investigations concerning the origin of such phenomenon. The aim of the present letter is to exploit this difference to identify the elements of our gravitational construction that are dual to each of these types of quarks, or at least, that codify the two different effects. \nIsolating the effect of the magnetic field was not possible in our previous work, where the calculations were made in the rest frame of the plasma and as the intensity of said field became larger, its direct impact was certainly augmented, but also unavoidably increased the kinetic momentum associated to each Landau level. \nIn what follows we will consider more general frames by allowing nonzero values of k z in (3). Since the square of the kinetic momentum of the n th Landau level in this scenario is given by k K 2 = 2 e b ( n +1 / 2)+ k 2 z , introducing a non-vanishing k z permits the exploration of the properties of the quasinormal modes as functions of the intensity of the magnetic field in a range 0 ≤ b ≤ k K / 2 e ( n +1 / 2) , while the kinetic momentum is kept constant by varying k z from k K to 0. \nSince b explicitly appears in (3), it is clear that changing its value has a direct impact on this equation, but \njust as important is the implicit modification due to the fact that the form that U ( r ) , V ( r ) , and W ( r ) have as functions of r depends on the intensity of the magnetic field. Since this metric functions are only known to us numerically, we need to resource to such methods to solve Eq. (3).', 'IV. FINDING THE QUASINORMAL FREQUENCIES': "As can be consulted in [7], close to the horizon the metric functions are given by the expansions \nU ( r ) = 6 r h ( r -r h ) + ∞ ∑ i =2 U i ( r -r h ) i , V ( r ) = ∞ ∑ i =0 V i ( r -r h ) i , W ( r ) = 3 r h 2 ( r -r h ) 0 + ∞ ∑ i =1 W i ( r -r h ) i , (6) \nwhich through Eq. (3) show that the radial profile of the embedding behaves like ( r -r h ) iα , with α = ± ω 4 πT = ± ω 6 r h , in the r → r h limit. The ingoing wave requirement is imposed by choosing the negative sign for α and approximating χ ( ω,n ) r ( r ) near the horizon through the resulting series \nχ ( ω,n ) r ( r ) ≃ ( r -r h ) -i ω 6 r h χ (0) r [ 1 + C (1) ( r,ω,n ) ( r -r h ) + C (2) ( r,ω,n ) ( r -r h ) 2 + O ( r -r h ) 3 ] , (7) \nwhere χ (0) r is a free global factor due to the linear character of (3), while \nC (1) ( r,ω,n ) = 1 108 r h 2 V ( r h ) 2 (3 r h -iω ) { b 2 ω (5 ω -3 ir h ) + 18 k 2 z V ( r h ) 2 -6 V ( r h ) [ -18( n +1 / 2) e br h 2 + V ( r h ) ( 27 r h 2 -15 ir h ω + ω 2 )] } C (2) ( r,ω,n ) = 1 23328 r h 4 V ( r h ) 4 (18 r h 2 -9 ir h ω -ω 2 ) { b 4 ω ( -252 r h 2 ω -468 ir h 3 +35 ir h ω 2 +25 ω 3 ) +324 k 4 z V ( r h ) 4 +12 b 2 V ( r h ) [ 18( n +1 / 2) e br h 2 ( 36 r h 2 -12 ir h ω +5 ω 2 ) + V ( r h ) ( -459 r h 2 ω 2 -288 ir h 3 ω +972 r h 4 +97 ir h ω 3 -5 ω 4 ) ] + 36 V ( r h ) 2 [ 324( n +1 / 2) 2 e 2 b 2 r h 4 -36( n +1 / 2) e br h 2 V ( r h ) ( 99 r h 2 -24 ir h ω + ω 2 ) + V ( r h ) 2 ( -126 r h 2 ω 2 -2268 ir h 3 ω +2673 r h 4 -45 ir h ω 3 + ω 4 ) ] -36 k 2 z V ( r h ) 2 [ B 2 (72 r 2 h -24 ir h ω -5 ω 2 ) +6 V ( r h )( -18( n + 1 2 ) ebr 2 h + V ( r h )(99 r 2 h -24 ir h ω + ω 2 ))] } . (8) \nIn the asymptotic region r →∞ the metric functions \nare described by \nU ( r ) = r 2 + U 1 r + U 2 1 4 + 1 r 2 ( U -2 -2 3 b 2 log r ) + U 1 1 r 3 ( -U -2 -1 3 b 2 + 2 3 b 2 log r ) + O ( 1 r 4 ) , V ( r ) = r 2 + U 1 r + U 2 1 4 + 1 r 2 ( V -2 + 1 3 b 2 log r ) + U 1 1 r 3 ( -V -2 + 1 6 b 2 -1 3 b 2 log r ) + O ( 1 r 4 ) , W ( r ) = r 2 + U 1 r + U 2 1 4 + 1 r 2 ( -2 V -2 -2 3 b 2 log r ) + U 1 1 r 3 ( 2 V -2 -1 3 b 2 + 2 3 b 2 log r ) + O ( 1 r 4 ) , (9) \nand the radial profile therefore by \nχ ( ω,n ) r ( r ) ≃ χ r ( -1) [ 1 r -U 1 2 r 2 + ( ω 2 -k 2 z -2( n +1 / 2) e b ) log r 2 r 3 -3 U 1 ( ω 2 -k 2 z -2( n +1 / 2) e b ) log r 4 r 4 + U 1 ( ω 2 -k 2 z -2( n +1 / 2) e b + U 1 2 ) 1 4 r 4 ] + χ r ( -3) [ 1 r 3 -6 U 1 1 4 r 4 ] + O ( 1 r 5 ) , (10) \nas can be verified by substituting (9) in (3). \nThe expansion coefficients χ ( -1) r and χ ( -3) r are respectively related to the source and the vacuum expectation value of the excitation dual to χ . The quasinormal modes of such excitation are given by the normalizable solutions, identified as those for which χ ( -1) r vanishes, since as can be seen in (10), this is the coefficient that multiplies the no-normalizable part of the radial profile. \nThe quest now is to obtain the quasinormal frequencies as functions of the dimensionless parameters b/T 2 and k K 2 /T 2 . To this end, we notice that the specific behavior of χ ( r ) over any member of our family of backgrounds, at a given temperature T and intensity b of the magnetic field, is parametrized by ω, k y , k z , and n . We fix our attention on the lowest Landau level set by n = 0 in (5), and chose k y = 0 in (3) while employing k z as described above to select a value for k K 2 = e b + k z 2 . Numerically solving the latter equation under these circumstances and near horizon conditions given by (7) for a particular value of ω , permits us to use the asymptotic behavior of the solution χ ( r ) to extract the coefficients χ ( -1) r ( ω ) and χ ( -3) r ( ω ) corresponding to this frequency. Performing such integration for values of ω that explore the complex plane, starting at the origin and searching for the nearest locus Re[ χ ( -1) r ( ω ) ]=0 and Im[ χ ( -1) r ( ω ) ]=0, leads us to extract the frequency of the quasinormal mode as the value at which these lines intersect. Repeating the procedure above for several k K 's \nover a series of backgrounds with a range of temperatures and intensities for the magnetic field renders the desire function ω ( b/T 2 , k K 2 /T 2 ) .", 'V. THE COMPLEX CONDENSATE': "Once the quasinormal frequencies have been found, they can be used to determine the associated condensate dual to the coefficient χ ( -3) r . The calculation results in a complex function of b/T 2 and k K 2 /T 2 . \nTo understand the meaning of both parts, real and imaginary, in the dual gauge theory, we first note that even if we have kept our focus on δχ ( r, x, y, z ) = sin [ δθ ( r, x, y, z )] , the embedding of the D7-brane also accepts perturbations δϕ ( r, x, y, z ) over its position in ϕ 1 that we fixed as part of our construction. The holographic dictionary developed in Appendix A of [10] describes how δχ (equal there to -δχ ) and δϕ are respectively related to scalar and pseudoscalar excitations in the gauge theory. To exhibit the correspondence stated above, the authors in [10] use the near boundary expansion (10) and the fact that the lowest component of the massless modes Φ 7 , 7 of the open string sector stretching from the D7-brane to itself, is a complex scalar Φ 0 7 , 7 = 1 √ 2 ( X 8 + iX 9 2 πℓ 2 s ) that describes the fluctuations X 8 and X 9 of such brane in the 2-dimensional space perpendicular to it. If the fiducial embedding is taken at ϕ 1 = 0 , δϕ corresponds precisely to the phase of the scalar field Φ 0 7 , 7 , while if δχ remains real it can be roughly thought as the modulus. \nWe have indeed set ϕ 1 = 0 because, as described in [7], a consistent solution to the equations of motion can be found by studying the δχ mode with the δϕ mode turned off. Even though this certainly restricts the leading order of the perturbation near the boundary to only capture information of the dual scalar excitation, the complex nature of the radial equation leads to solutions χ ( ω,n ) r ( r ) that become complex as soon as they enter the bulk. This is why the coefficient χ ( -3) r is a complex number, which imaginary part is dual to the condensate of the pseudoscalar excitation in the gauge theory, because it constitutes the right subleading perturbation to the phase of the scalar field Φ 0 7 , 7 = 1 √ 2 ( X 8 + iX 9 2 πℓ 2 s ) that δϕ would directly produce. The reading of the condensate of the dual scalar excitation as the real part of -√ λN f N c T 3 χ ( -3) r / 8 , where λ is 't Hooft coupling, N f the number of D7branes, N C the number of black D3-branes, and T the temperature, is unchanged from the one in [10], while the one we have just provided about the imaginary component of the same expression is obtained from the variation, consisting of a phase rotation, of its respective Lagrangian.", 'VI. VALENCE AND SEA CONTRIBUTIONS TO THE CONDENSATE': "As we mentioned previously, there have been several works studying the valence and sea quarks effects in (inverse) magnetic catalysis, and particularly in [8] a way to separate and explore each effect by itself is presented. Unlike our case, in said work the field under consideration is fermionic, and its condensate is determined to be given by \nψψ ( b ) = 1 Z ( b ) ∫ D Ue -S g det ( / D ( b ) + m ) Tr ( / D ( b ) + m ) -1 , \nwhere Z ( b ) = ∫ D Ue -S g det ( / D ( b ) + m ) . (11) \nFrom this expression the valence and sea condensates are defined as \nψψ val ( b ) = 1 Z (0) ∫ D Ue -S g det ( / D (0) + m ) Tr ( / D ( b ) + m ) -1 , ψψ sea ( b ) = 1 Z ( b ) ∫ D Ue -S g det ( / D ( b ) + m ) Tr ( / D (0) + m ) -1 , (12) \nreflecting that in the presence of a magnetic field, the valence effect is codified in a shift of the lower mode of the Dirac operator, while the sea quarks affect the condensate through the fermion determinant. It is also shown, using lattice QCD techniques, that the valence quarks only contribute to magnetic catalysis by enhancing the condensate, but the sea contribution suppresses it, promoting quiral symmetry restoration and presenting inverse magnetic catalysis. This last phenomenon being more noticeable when near the critical temperature and independent of the central value around which the gauge field is explored. \nMotivated by the distinction above to create an entry of the holographic dictionary, we will employ our construction to identify the shift in spectrum as the effect on the perturbation δχ caused by the explicit appearance of b in (3), and the change in the integration measure as dual to the backreaction of the bulk geometry (1) to such magnetic field. \n̸ \nIn the scalar case that we are working on this can be concretely implemented by holographically computing the valence condensate < O m > val ( b ) using the coefficient χ ( -3) ( b ) extracted from the solution to (3) at b = 0 , but with U 0 , V 0 , and W 0 the metric functions of a black D3-brane at b = 0 . \nConversely, for the sea condensate < O m > sea ( b ) the coefficient χ ( -3) ( b ) that should be used in our holographic calculation is the one extracted from the solution \nto (3) at b = 0 , but with U, V, and W the metric functions that have fully backreacted to the presence of the magnetic field. \nGiven that the valence condensate is inherently easier to calculate, we follow the same strategy as in [8] and study it along with the complete condensate. The procedure described above to compute the valence condensate simplifies (3) to \n[3 r 3 ( r 4 -1) + r 5 ω 2 ] χ r +(1 -6 r 4 +5 r 8 ) χ ' r + r ( r 4 -1) 2 χ '' r = r ( r 4 -1)(2 E + k z 2 ) χ r , (13) \nwhere the metric functions of the black D3-brane have been used and k y has been set to zero as in previous sections. \nFrom (5) we see that the quantity 2 E + k z 2 in parenthesis on the right hands side of (13) is the kinetic momentum k K 2 = eb + k z 2 for n = 0 , showing that the valence condensate depends on k z and b only through k K , and not on their independent values.", 'VII. RELATION TO THE QUASI-PARTICLE MODELS': 'A quasi-particle model of the quark-gluon plasma [11] has been successfully used to describe certain aspects of it [12], including scenarios where it is magnetized [13]. \nBefore we study the behavior of the condensate, it is interesting to see that the quasinormal modes in our work follow a dispersion relation consistent with said model. To this end, we would like to display the energy of the mode as a function of the kinetic momentum at constant magnetic field. Since the kinetic momentum k K is defined by the relation k K 2 = eb + k z 2 , isolating the effect that it has on the quasinormal modes is a simple task, as it suffice to explore the result of changing k z while keeping b constant. To graphically compare the results for different values of b/T 2 , it turns out to be convenient to compensate for the shift that the field introduces, so the results will be plotted as functions of ( k K 2 -eb ) /T 2 . \nIn figure 1 we show the energy of the mode, given by the real part of the quasinormal frequency squared, as a function of the kinetic momentum. We notice that it is monotonically increasing and becomes linear for large values of k K 2 /T 2 . One can observe that the dispersion relation for a massless transversal field in a thermal theory [14], characterized by the asymptotic behaviors \nRe[ ω 2 ] ≃ ω p 2 + αk K 2 for k K ≪ eT, and Re[ ω 2 ] ≃ m T 2 + k K 2 for k K ≫ eT, (14) \nis recovered. This behavior is a strong indication that the quasinormal modes are indeed acting like quasi-particles in the plasma. \nIn figure 2 we present the width m 0 Γ = 2 Re [ ω ] Im [ ω ] of the mode over its energy E = √ Re [ ω 2 ] . The curves \nFIG. 1: Real part of the squared quasinormal frequency as a function of the kinetic momentum k K 2 at fixed magnetic field intensity. The dotted, dot dashed, dashed (medium), dashed (large) and continuous curves correspond to b/T 2 = { 0 , 4 π 2 / 9 , 8 π 2 / 3 , 40 π 2 / 9 , 64 π 2 / 9 } respectively. \n<!-- image --> \nFIG. 2: Width of the unstable states dual to the quasinormal modes as a function of the kinetic momentum ( k K 2 -eb ) /T 2 at fixed magnetic field intensity. The dotted, dot dashed, dashed (medium), dashed (large) and continuous curves correspond to b/T 2 = { 0 , 4 π 2 / 9 , 16 π 2 / 3 , 40 π 2 / 9 , 64 π 2 / 9 } respectively. \n<!-- image --> \ndecrease with increasing kinetic momentum, consistent with the expected stabilizing effect. We also observe that the plots accommodate from bottom to top for increasing values of the magnetic field, suggesting a destabilizing effect that shall become clear when inspecting the behavior as a function of b . \nAs mentioned before, for any given value k K of the ki- \nnetic momentum and not to exceed it, the magnetic field intensity can only be in the interval 0 ≤ b ≤ k K /e . To be able to make comparisons for plots at different values of fixed kinetic momentum, we normalize the horizontal axis by the maximum value that b can take. \nIn figure 3 we show the dependence of Γ /E on b/b max at fixed k K 2 /T 2 . The curves are monotonically increasing for every value of the momentum, reinforcing that the magnetic field has a destabilizing effect on the quasiparticles. It is of note that as k K 2 grows, the graphs accommodate form top to bottom, consistently with the notion that the kinetic momentum has a stabilizing effect. \nFIG. 3: Width of the unstable states dual to the quasinormal modes as a function of the magnetic field b/T 2 at fixed kinetic momentum. The dotted, dot dashed, dashed (medium), dashed (large) and continuous curves correspond to k K 2 /T 2 = { 2 π 2 / 45 , 4 π 2 / 9 , 8 π 2 / 3 , 40 π 2 / 9 , 64 π 2 / 9 } respectively. \n<!-- image --> \nPart of our motivation to work at fixed kinetic momentum becomes apparent on the light of the conclusions above, since had we increased b without using k z to compensate the change in k K , we would have seen the result of competing destabilizing and stabilizing effects, given that b also augments the kinetic momentum.', 'VIII. SCALAR CONDENSATE': 'We now present the results for the scalar condensate. In figure 4 we display the dimensionless [10] quantity -8 ⟨ O m ⟩ √ λN f N c T 3 at fixed b/T 2 as a function of ( k K 2 -eb ) /T 2 , where the choice of variables is for the same reasons as above. The curves are all monotonically increasing, showing that the kinetic momentum promotes spontaneous symmetry breaking for the scalar excitation when both, valence and sea contributions are considered. \nIn figure 5 the full scalar condensate is shown as a function of the magnetic field at constant kinetic momentum. \nWe can see that for certain values of k K 2 /T 2 , there is an inflection point as b/T 2 gets closer to its maximal value, and the condensate begins to decrease. The fact that this phenomenon appears only for the plots at larger k K 2 /T 2 indicates that such inverse magnetic catalysis is associated to large magnetic fields, that as we explained, are only achievable at large kinetic momentum. \nTo separate now the valence condensate we follow the ideas developed in section VI, where we proposed for the valence contribution to be calculated by turning off the magnetic field in the background, while leaving it intact in the equations for the D7-brane perturbation. As we saw, this made the quasinormal modes to depend only on the kinetic momentum. To verify that we had indeed set to zero the field in every place of our code where it was necessary, we numerically confirmed that the same results were obtained for any given k K 2 /T 2 , regardless of the value of b leading to it and imputed in (13) through the substitution of E = 1 2 eb . \nThe observation in the previous paragraph means that the behavior of the valence condensate of the scalar excitation as a function of the kinetic momentum at any constant magnetic field, is fully captured by the b/T 2 = 0 case in figure 4. This plot recovers the result in [8] that determines the effect of increasing the kinetic momentum to be an enhancement of the valence condensate, promoting spontaneous symmetry breaking. \nIt is important to acknowledge that in contrast to the results reported in [8], where the valence condensate is a convex function of the kinetic momentum, our curve is concave instead. We believe this difference to arise from the sensibility of the calculations to the value of the quark mass, since while in the aforementioned paper a small mass approximation is used, we work on the quiral limit where m q = 0 . \nFIG. 4: Scalar condensate as a function of the momentum ( k K 2 -eb ) /T 2 at fixed magnetic field intensity b/T 2 . The dotted, dot dashed, dashed (medium), dashed (large) and continuous curves correspond to \n<!-- image --> \nb/T 2 = { 0 , 4 π 2 / 9 , 16 π 2 / 3 , 40 π 2 / 9 , 64 π 2 / 9 } respectively. \nFigure 7 shows the pseudoscalar condensate as a function of the magnetic field at fixed kinetic momentum. For the plots at lower fixed values of k K 2 /T 2 the only effect that we observe is enhancement, in the sense that the magnitude of the condensate grows when increasing the intensity of the field, preventing restoration of the quiral symmetry. Nonetheless, for larger values of k K 2 /T 2 there is a point after which increasing the magnetic field reduces the magnitude of the condensate, getting closer to an spontaneous restoration of quiral symmetry, and thus presenting inverse magnetic catalysis. \n<!-- image --> \nFIG. 5: Scalar condensate as a function of the magnetic field intensity b/T 2 at fixed kinetic momentum k K 2 /T 2 . The dotted, dot dashed, dashed (medium), dashed (large) and continuous curves correspond to k K 2 /T 2 = { 2 π 2 / 45 , 4 π 2 / 9 , 8 π 2 / 3 , 40 π 2 / 9 , 64 π 2 / 9 } respectively. \n<!-- image -->', 'IX. PSEUDOSCALAR CONDENSATE': 'FIG. 6: Pseudoscalar condensate as a function of the momentum ( k K 2 -eb ) /T 2 at fixed magnetic field intensity b/T 2 . The dotted, dot dashed, dashed (medium), dashed (large) and continuous curves correspond to b/T 2 = { 0 , 8 π 2 / 9 , 16 π 2 / 3 , 40 π 2 / 9 , 64 π 2 / 9 } respectively. \n<!-- image --> \nThe last result we present before the discussion of our findings is what we concluded in section V to be the pseudoscalar condensate, graphed below in figure 6 as a function of ( k K 2 -eb ) /T 2 for fixed b/T 2 . We see that the value of this condensate starts negative, signaling an instability in the brane embedding [15], at some point it starts growing with ( k K 2 -eb ) /T 2 until it becomes positive, and then keeps rising. This again shows that the kinetic momentum has a stabilizing effect over the quasiparticles. As shown in figure 6, for the lower and higher values of b/T 2 that we explored, b/T 2 = { 0 , 4 π 2 / 9 , 40 π 2 / 9 } , this \nFIG. 7: Pseudoscalar condensate as a function of the magnetic field intensity b/T 2 at fixed kinetic momentum k K 2 /T 2 . The dotted, dot dashed, dashed (medium), dashed (large) and continuous curves correspond to k K 2 /T 2 = { 2 π 2 / 45 , 4 π 2 / 9 , 8 π 2 / 3 , 40 π 2 / 9 , 64 π 2 / 9 } respectively \npseudoscalar condensate has an inflection point now close to ( k K 2 -eb ) /T 2 = 0 , that is, it decreases at first with ( k K 2 -eb ) /T 2 , and after a certain value of this variable it begins to increase monotonically. For the intermediate magnitudes, b/T 2 = { 16 π 2 / 9 , 8 π 2 / 3 } , the value of such condensate always increases with the kinetic momentum. \nAs in the scalar case, the valence condensate depends solely on the kinetic momentum, therefore changing b while keeping k K fixed will have no effect, and the behavior with the latter as a variable is fully capture by the plot at b = 0 in figure 6.', 'X. DISCUSSION AND FUTURE WORK': 'The main objective of the preset work was to take advantage of a background with a fully backreacted magnetic field, and use the relevance that the presence of such field has in high energy collisions, to introduce a holographic prescription that distinguishes valence from sea quarks in the QGP, and perform a first test of such prescription. \nAs part of the construction we verified in section VII that the modes used in our study followed the dispersion relation expected for a massless transversal field in a \nthermal theory, allowing them to be interpreted as quasiparticle states. In passing we showed that for these quasiparticles, momentum has an stabilizing effect while the magnetic field destabilizes them. \nIntended to use it as an analyzing tool, we stopped and noticed that the condensate of both the scalar and pseudoscalar excitations have a nonzero value. In section V we pointed out that from the perspective of the gravity side, this happens because of the complex nature of the perturbation equations of an embedded brane that crosses the horizon of the background. It is interesting now to interpret the above in the field theory model of pseudo-particle as a consequence of their finite width, causing that even if the excitation is exclusively done on the scalar, it induces a nonzero value for both scalar and pseudoscalar condensates. \nIt is at this point that the importance of using our gravitational configuration becomes evident. It comes from our holographic prescription which assumes that the physics of the sea quarks is captured by the effect of the background on the embedding, while that of the valence quarks is encoded explicitly in the perturbation equations, provoking the necessity to create a scenario where such entities can be modified independently. That is exactly what a brane embedded in a background that includes a fully backreacted magnetic field provides, since it enabled us to change the background to independently probe its effect on the perturbations of the embedding, or modify the value of the intensity of the field directly in the embedding equation to see what results of this action. \nThe procedure above lead us to the particular result of an equation to describe the valence excitations where the canonical momentum and the magnetic field exclusively enter through their combination given by the kinematic momentum k K 2 = eb + k z 2 of the lowest level n = 0 . This is one of the main reasons why among our goals was to have control over k K and generate data keeping it constant while varying b . \nWith all of the above in hand, the numerical results for the scalar condensate showed that when in a constant magnetic field, the effect of the momentum was to drag the system away from quiral symmetry restoration. Differently, when working at fixed k K 2 /T 2 , and for high enough values of such constant, the curves exhibited both, enhancement of the condensate at low values of b in comparison to its maximal value b max = k K 2 /e , and suppression, with respect to the maximal point, as the intensity of the field approaches such upper limit. This increasing and decreasing behavior when augmenting the intensity of the field is precisely what is respectively termed magnetic catalysis and inverse magnetic catalysis, but before we say more about it, let us revisit another result. \nAs mentioned earlier, the fact that the valence condensate depends only on k K means that the numerical results for its scalar part are described by the curve at b = 0 in figure (4). What is also true is that therefore this curve can be seen equally well as depicting the valence con- \nFIG. 8: Total (dotted) and valence (points) scalar condensates as a function of the magnetic field intensity b/T 2 in the rest frame of the plasma. The thick line corresponds to the best fitting second order polynomial for the valence condensate data \n<!-- image --> \n6 . 05 × 10 -2 +1 . 86 × 10 -4 ( b/T 2 ) -4 . 86 × 10 -7 ( b/T 2 ) 2 \ndensate calculated in the rest frame as a function of the magnetic field, making it actually usable for comparisons with measurements performed in such frame. Furthermore, since the curve so obtained only presents magnetic catalysis, we conclude that any inverse catalysis of this type for a scalar condensate should be attributed to the sea quarks, consistently with the conclusion in [8] and providing a first corroborating test for our holographic prescription. The comparison of the full scalar condensate and the valence contribution to it, both in the rest frame is presented in figure 8. \nConcerning the pseudoscalar condensate, our numerical results show that it is negative over a range of values for the momentum and magnetic field. As a function of k K alone, it starts negative close to the minimum value that the constant magnetic field permits, and evolves in such a manner that becomes positive as k K 2 grows. Since a negative condensate indicates an instability of the brane embedding in the gravity side, this result is another confirmation that the momentum has a stabilizing effect on the system. \nWe already notice that when plotted as a function of the intensity of the magnetic field at any constant kinetic momentum, the magnitude of the pseudoscalar condensate increases with b when such quantity is far bellow the maximum value it can take. We also pointed out in passing that for curves traced at larger values of k K , this tendency is reversed and the magnitude of this condensate begins to decrease as the field approaches its maximum intensity for such kinetic momentum. We would like to add two further observations, one being that even if the way we present our results is suggestive about the transition to inverse magnetic catalysis mentioned above happening because of the large kinetic momentum at which certain curves are traced, it would probably be more ac- \nFIG. 9: Total (dotted) and valence (dots) pseudoscalar condensate as a function of the magnetic field intensity b/T 2 in the rest frame of the plasma. The thick line corresponds to the best fitting second order polynomial for the valence condensate data \n<!-- image --> \n-8 . 23 × 10 -3 +2 . 08 × 10 -5 ( b/T 2 ) + 2 . 30 × 10 -7 ( b/T 2 ) 2 \ncurate to think of it as occurring for very large intensities of the magnetic field. The need to clarify this comes from using b/b max as a variable so that the full range of b would be covered from 0 to 1, and the plots could be compared regardless of how largely different their maximum intensities were. A side effect of this is not making it visually clear how large the direct intensity is at which the magnitude of the condensate begins to decrease. The correctness of the latter point of view is supported by the plot of the pseudoscalar condensate as a function of b in the rest frame of the plasma shown in figure 9, where the field can take arbitrarily high intensities and we indeed see how inverse magnetic catalysis appears as that happens. For comparison, we have included in the same figure 9 the plot of the valence contribution to the pseudoscalar condensate, as obtained from the trace at b = 0 in figure 6 by the considerations mentioned above. The other observation is that from the plots included in figure 7, it would be imaginable that for other graphs, done at larger values of k K , the change in tendencies for b close to b max would be intense enough to make the condensate positive. Even if a full plot done at such high values of k K is too computationally demanding just to confirm this perception, we have actually verified that at b/b max = 0 . 91 for k K 2 /T 2 = 152 π 2 / 9 , the pseudoscalar condensate is indeed positive. \nNow that figures 8 and 9 have been presented, it is time to notice that all the distinctive features of the behavior of both condensates, including magnetic catalysis and its inverse, seem to be attributable to the sea quarks, since either of the valence contributions is well approximated by a simple function of second order in b . We do not think much should be made out of the particular fittings, but it is interesting to see how featureless the valence plots turn out to be in comparison with those of the full condensate. \nEven if the analysis above exhausts the scope of the current work, it also indicates some elements that require further investigation. \nOne example is that the curves traced in figure 6 at the lower and higher values of b present an inflection point close to k K 2 -eb = 0 , while those at intermediate fixed intensities do not. We currently do not posses a clear understanding of the reason behind this, and believe it deserves further clarification. \nAlso, equations (11) are presented in [8] as an approximation, in the sense that the addition of both contributions is not expected to match the value of the condensate computed using all the elements. Our holographic construction presents the opportunity to verify how accurate this approximation is, but obtaining the sea condensate demands so much computational time that it has to be addressed on its own. As a first glance of the interest that such an investigation would bear, we present the plots in figure 10, where the subtraction of total minus valence, that should approximate the sea contribution, is plotted for both condensates. It stands out that the scalar and pseudoscalar are almost reflections of each other with respect to the intermediate line we have drawn in between them. \nFIG. 10: Difference between total and valence condensates for the scalar (dashed) and pseudoscalar (dot dashed) case as a function of the magnetic field intensity b/T 2 in the rest frame of the plasma. \n<!-- image --> \nOne last topic that can be addressed using our construction is the effect of a magnetic field over the thermal mass [16], in our case, of the quasi-particles dual to the modes we studied. This could be done performing a very precise calculation to extract the value of m T that better fits the asymptotic behavior in (14) for different intensities of the magnetic field.', 'XI. ACKNOWLEDGMENTS': "We are very thankful to Alejandro Ayala for meaningful discussions about the thermal dispersion relations. \nThe work of D.K. is supported by CONAHCyT MSc. \n- [1] Andreas Karch and Emanuel Katz. Adding flavor to AdS / CFT. JHEP , 06:043, 2002.\n- [2] Shinpei Kobayashi, David Mateos, Shunji Matsuura, Robert C. Myers, and Rowan M. Thomson. Holographic phase transitions at finite baryon density. JHEP , 02:016, 2007.\n- [3] Johanna Erdmenger, Matthias Kaminski, Patrick Kerner, and Felix Rust. Finite baryon and isospin chemical potential in AdS/CFT with flavor. JHEP , 11:031, 2008.\n- [4] Juan Martin Maldacena. The Large N limit of superconformal field theories and supergravity. Int. J. Theor. Phys. , 38:1113-1133, 1999. [Adv. Theor. Math. Phys.2,231(1998)].\n- [5] V. Skokov, A. Yu. Illarionov, and V. Toneev. Estimate of the magnetic field strength in heavy-ion collisions. Int. J. Mod. Phys. , A24:5925-5932, 2009.\n- [6] Eric D'Hoker and Per Kraus. Magnetic Brane Solutions in AdS. JHEP , 10:088, 2009.\n- [7] Uriel Elinos and Leonardo Patino. Fundamental Landau levels in a strongly coupled plasma. Phys. Rev. D , 105(12):126016, 2022.\n- [8] Falk Bruckmann, Gergely Endrődi, and G. Tamás Kovács. Inverse magnetic catalysis and the Polyakov Loop. JHEP , 2013(4), 2013. \ngrant. All the plots in this paper were generated using Wolfram Mathematica. \n- [9] Alejandro Ayala, M. Loewe, C. Villavicencio, and R. Zamora. On the magnetic catalysis and inverse catalysis of phase transitions in the linear sigma model. Nucl. Part. Phys. Proc. , 258-259:209-212, 2015.\n- [10] Robert C. Myers, Andrei O. Starinets, and Rowan M. Thomson. Holographic spectral functions and diffusion constants for fundamental matter. JHEP , 11:091, 2007.\n- [11] A. Peshier, B. Kämpfer, O. P. Pavlenko, and G. Soff. A Massive quasiparticle model of the SU(3) gluon plasma. Phys. Rev. D , 54(3):2399-2402, 1996.\n- [12] W. Cassing and E. L. Bratkovskaya. Parton transport and hadronization from the dynamical quasiparticle point of view. Phys. Rev. C , 78(3), 2008.\n- [13] M. Kurian, S. Mitra, S. Ghosh, and V. Chandra. Transport coefficients of hot magnetized QCD matter beyond the lowest Landau level approximation. Eur. Phys. J.C , 79(134), 2019.\n- [14] Michel Le Bellac. Thermal Field Theory . Cambridge Monographs on Mathematical Physics. Cambridge University Press, 3 2011.\n- [15] Martin Kruczenski, David Mateos, Robert C. Myers, and David J. Winters. Towards a holographic dual of large N(c) QCD. JHEP , 05:041, 2004.\n- [16] Sebastian Koothottil and M. Vishnu Bannur. Thermodynamic Behaviour of Magnetized QGP within the SelfConsistent Quasiparticle Model. Phys. Rev. C , 99(3), 2019."} |
2024A&A...690A.396L | Context. Most massive galaxies host a supermassive black hole at their centre. Matter accretion creates an active galactic nucleus AGN forming a relativistic particle wind. The wind heats and pushes the interstellar medium producing galacticwide outflows. Fast outflows remove the gas from galaxies and quench star formation and while slower lt 500 km sSUP1SUP outflows are ubiquitous their effect is less clear but can be both positive and negative. Aims. We wish to understand the conditions required for positive feedback. We investigated the effect that slow and warmhot outflows have on the dense gas clouds in the host galaxy. We aim to constrain the region of outflow and cloud parameter space if any where the passage of the outflow enhances star formation. Methods. We used numerical simulations of virtual wind tunnels to investigate the interaction of isolated turbulent spherical clouds 10SUP345SUP MSUBSUB with slow outflows 10 km sSUP1SUP SUBoutSUB 400 km sSUP1SUP spanning a wide range of temperatures 10SUP456SUP K. We modelled 57 systems in total. Results. We find that warm outflows compress the clouds and enhance gas fragmentation at velocities 200 km sSUP1SUP while hot TSUBoutSUB 10SUP6SUP K outflows increase fragmentation rates even at moderate velocities of 400 km sSUP1SUP. Cloud acceleration on the other hand is typically inefficient with dense gas only attaining velocities of lt0.1 SUBoutSUB. Conclusions. We suggest three primary scenarios where positive feedback on star formation is viable stationary cloud compression by slow outflows in lowpowered AGN sporadic enhancement in shear flow layers formed by luminous AGN and selfcompression in fragmenting AGNdriven outflows. We also consider other potential scenarios where suitable conditions arise such as compression of galaxy discs and supernova explosions. Our results are consistent with current observational constraints and with previous works investigating triggered star formation in these disparate domains. | 2024-10-01T00:00:00Z | ['2024A&A...690A.396L', '10.48550/arXiv.2409.13234', '2024arXiv240913234L', 'arXiv:2409.13234', '10.1051/0004-6361/202450286'] | ['ISM: clouds', 'ISM: jets and outflows', 'galaxies: active', 'galaxies: ISM', 'Astrophysics - Astrophysics of Galaxies'] | Slow and steady does the trick Slow outflows enhance the fragmentation of molecular clouds | 2,024 | 175 | 0.5 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF'] | 1 | https://arxiv.org/pdf/2409.13234.pdf | {'Slow and steady does the trick: Slow outflows enhance the fragmentation of molecular clouds': 'M. Laužikas 1 and K. Zubovas 1 , 2 \n- 1 Center for Physical Sciences and Technology, Saul˙etekio al. 3, Vilnius LT-10257, Lithuania e-mail: [email protected]\n- 2 Astronomical Observatory, Vilnius University, Saul˙etekio al. 3, Vilnius LT-10257, Lithuania \nReceived; accepted', 'ABSTRACT': "Context. Most massive galaxies host a supermassive black hole at their centre. Matter accretion creates an active galactic nucleus (AGN), forming a relativistic particle wind. The wind heats and pushes the interstellar medium, producing galactic-wide outflows. Fast outflows remove the gas from galaxies and quench star formation, and while slower ( v < 500 km s -1 ) outflows are ubiquitous, their e ff ect is less clear but can be both positive and negative. \nAims. Wewish to understand the conditions required for positive feedback. We investigated the e ff ect that slow and warm-hot outflows have on the dense gas clouds in the host galaxy. We aim to constrain the region of outflow and cloud parameter space, if any, where the passage of the outflow enhances star formation. \nMethods. We used numerical simulations of virtual 'wind tunnels' to investigate the interaction of isolated turbulent spherical clouds (10 3;4;5 M ⊙ ) with slow outflows (10 km s -1 ≤ v out ≤ 400 km s -1 ) spanning a wide range of temperatures (10 4;5;6 K). We modelled 57 systems in total. \nResults. We find that warm outflows compress the clouds and enhance gas fragmentation at velocities ≤ 200 km s -1 , while hot ( T out = 10 6 K) outflows increase fragmentation rates even at moderate velocities of 400 km s -1 . Cloud acceleration, on the other hand, is typically ine ffi cient, with dense gas only attaining velocities of < 0 . 1 v out. \nConclusions. We suggest three primary scenarios where positive feedback on star formation is viable: stationary cloud compression by slow outflows in low-powered AGN, sporadic enhancement in shear flow layers formed by luminous AGN, and self-compression in fragmenting AGN-driven outflows. We also consider other potential scenarios where suitable conditions arise, such as compression of galaxy discs and supernova explosions. Our results are consistent with current observational constraints and with previous works investigating triggered star formation in these disparate domains. \nKey words. Galaxies: active - ISM: clouds - ISM: jets and outflows - Hydrodynamics", '1. Introduction': "It is by now well established that centres of massive galaxies are hosts to super-massive black holes (SMBHs). Accretion onto an SMBH creates an active galactic nucleus (AGN) (Ho 2008) and often launches a quasi-relativistic particle wind. The wind interacts with the di ff use interstellar medium (ISM), forming shocks and discontinuities and driving massive large-scale outflows (King & Pounds 2015). Over the past few decades, outflows have been detected in galaxies at various stages of evolution (for a comprehensive review, see Veilleux et al. 2020; Laha et al. 2021), suggesting numerous scenarios for feedback on the surrounding ISM. It has been argued that outflows play a major role in gas transport, regulating star formation and therefore establishing the observed SMBH-galaxy scaling relations, such as M -σ (Ferrarese & Merritt 2000; Zubovas & King 2019). \nMassive outflows, common in active galaxies, are typically found within 10 kpc from the AGN, with velocities of 100 to 1000 km s -1 (Veilleux et al. 2020; Laha et al. 2021). These values match the predictions of the semi-analytic energydriven outflow model derived by Zubovas & King (2016). It has been suggested that such outflows have a negative feedback e ff ect over the long term ( ∼ 100 Myr), as they remove the gas of a galaxy, thus stifling star formation, further inhibiting SMBH growth, and leaving galaxies 'red and dead' (Schawinski \net al. 2014). However, observations have also indicated simultaneous negative and positive feedback at the boundary of outflowblown cavities (Cresci et al. 2015). Moreover, local galaxies have shown a positive correlation between AGNs luminosity and star formation rates in circumnuclear regions (Dahmer-Hahn et al. 2022), raising the possibility that an AGN can facilitate star formation. The predictions of semi-analytic models are in partial agreement with observations - outflows can produce external pressure, compressing the cold gas and thus enhancing star formation (Silk 2013; Zubovas et al. 2013b). Positive feedback is also possible within the outflow itself as the gas cools down (Zubovas & King 2014; Thompson et al. 2016). This prediction has been confirmed by observations (Maiolino et al. 2017; Gallagher et al. 2019). These pieces of evidence indicate a diverse e ff ect of AGN activity on the surrounding medium. Nevertheless, the spatial and temporal scales for AGN-enhanced star formation remain uncertain. The uncertainty is not surprising, however, as semi-analytical models ignore various complications found in real galaxies, such as the presence of dense gas structures. The structures alter the flow dynamics and induce mixing, leading to a reduction of typical outflow velocities (Fluetsch et al. 2021). Such slow outflows can enhance star formation locally, as seen in numeric 3D simulations, while faster outflows provide a negative global e ff ect (e.g. Bieri et al. 2016; Mercedes- \n<!-- image --> \nFeliz et al. 2023). However, the aforementioned simulations rely on sub-resolution methods to account for the small-scale e ff ects of the mixing between the molecular cloud and the outflow. The detailed results of such simulations depend on the selected subresolution prescription (Wurster & Thacker 2013; Valentini et al. 2017), raising the question of whether such methods are su ffi -ciently robust for realistic environments. \nIndividual cloud-outflow interactions have been studied extensively both analytically and with numerical simulations. Early studies found that supersonic outflows disperse molecular clouds (Klein et al. 1994). However, they exclude several crucial physical processes, such as radiative cooling, turbulence, and selfgravity. A similar approach was taken by Cooper et al. (2009), who included density gradients in the clouds and cooling but did not account for turbulent velocities. Currently, there is little doubt that fast galactic outflows, with velocities v out > ∼ 1000 km s -1 , disperse molecular clouds and quench star formation (Pittard et al. 2010; Hopkins & Elvis 2010). On the other hand, slower outflows can compress the clouds without dispersing them, resulting in positive feedback. For example, Zubovas et al. (2014) and Dugan et al. (2017) have independently showed, using simulations with di ff erent cloud density profiles and turbulent field structures, that outflows with a velocity of 300 km s -1 or lower can enhance star formation. However, despite the plethora of studies (conveniently summarised in BandaBarragán et al. (2016, Table 1), Dugan et al. (2017, Table 1)), there is no clear answer regarding the parameters that determine whether the clouds are destroyed or their fragmentation is enhanced and how rapidly the two outcomes are achieved. \nIn this paper, we aim to identify and constrain the outflow properties required for positive AGN feedback. We focus primarily on slower (i.e. with radial velocity or velocity di ff erence between cold and warm phases < 500 km s -1 ) outflows with a radial distance from the AGN on the order of kiloparsecs. We used an enhanced version of the public SPH / N-body code G adget 4 supplemented with a radiative cooling prescription to model the interaction between individual turbulent molecular clouds and warm-hot galactic outflows. We simulated 57 systems - virtual 'wind tunnels' - and we aim to identify a region, or regions, in the parameter space of cloud mass, outflow velocity, and temperature where star formation is enhanced or quenched. We considered three values of molecular cloud mass, M cl = 10 3 , 10 4 , 10 5 M ⊙ ; six values of outflow velocity v out ≤ 400 km s -1 ; and three values of outflow temperature, T out = 10 4 , 10 5 , 10 6 K. We find that 10 6 Koutflows compress the clouds and provide positive feedback throughout the investigated velocity range, while at lower outflow temperatures, the velocity threshold value for positive feedback is reduced to < ∼ 200 km s -1 . We propose that star formation-enhancing regions are likely to develop in gas-rich galaxies within the outflows and their surroundings. We do not exclude outflows faster than those considered in this work, as they can also sporadically form bursts of star formation where shear flow develops. Notably as well, the outflow itself can fragment and thus form the stars within. \nThe paper is structured as follows. In Sect. 2, we introduce the theoretical background and relevant evolutionary timescales. In Sect. 3, we describe the numerical methods used and present the results in Sect. 4. We discuss the applicability of our results, peculiarities of outflow-enhanced star formation, and caveats of our models in Sect. 5, and we conclude in Sect. 6.", '2.1. AGN-driven outflows': 'Several mechanisms can transfer the energy liberated during matter accretion on to the SMBH to the surrounding matter. Outflows can form via the action of relativistic winds 1 (King & Pounds 2015), radiation pressure on dusty medium (Arakawa et al. 2022) or jets (Fabian 2012). The relativistic wind, when shocking against the surrounding gas, can provide significant momentum boosts by a factor of ∼ 20 and so is probably the primary contributor to galactic-wide outflows. Wind interaction with the ISM is further divided into two major sub-mechanisms: momentum-driven and energy-driven outflows (for a summary see Zubovas & King 2012b, Fig. 1). Energy-driven outflows form in di ff use gas, with the main driver being the adiabatic expansion of the shocked wind. In contrast, when the relativistic wind interacts with dense gas, cooling is e ffi cient, leading to momentum-driven outflows. In the case of a multiphase gas, as typically found in galaxies, an energy-driven outflow can form and envelop the embedded cold molecular clouds that are only accelerated by the wind momentum. In the latter case, due to the high density contrast, cloud acceleration is ine ffi cient; the clouds are either compressed or dispersed and create mixing instabilities agitating the flow. As instabilities grow and shocked wind mixes with the shocked ISM, complex multiphase outflows with a wide range of densities, temperatures, and relative velocities between the phases develop. We show a schematic overview of such a multiphase system in Fig. 1, where we also mark potential regions of star formation enhancement (see Sect. 5.1).', '2.2. Outflow radial properties': 'Galaxy-wide outflows are primarily categorised by the prevalent gas phase, although they are always multiphase. The ratio of the mass contained in di ff erent phases and the relative velocity between them depend on AGN luminosity, distance from the nucleus, properties of the surrounding medium and the evolutionary stage of the outflow. We show a compilation (admittedly biased and incomplete) of outflow properties from Fiore et al. (2017); Fluetsch et al. (2019); Lutz et al. (2020); Zubovas et al. (2022) in Figs. 2 and 3. 2 The first figure shows the relation between outflow velocity and radius, with symbol size proportional to the logarithm of the mass outflow rate, symbol type showing the observed outflow phase and colour representing AGN luminosity. In the second figure, we show the dependence of outflow velocity on AGN luminosity, with symbol type and size the same as in the first, while the symbol colour now represents the radius of the outflow. In addition, we plot three lines representing theoretical predictions of outflow velocity under continuous driving, assuming di ff erent Eddington ratios ( l ≡ L AGN / L Edd) and gas fractions ( f g ≡ ρ g /ρ total) in an isothermal potential. The equation plotted is adapted from Zubovas & King (2012a, Eq. 8). We use the observed M -σ relation (Ferrarese & Merritt 2000; Kormendy & Ho 2013) M ≃ 3 × 10 8 GLYPH<16> σ/ 200 km s -1 GLYPH<17> α M ⊙ with α ≃ 4 . 4 to eliminate σ and the Eddington ratio l to eliminate the \nFig. 1. Diagram of star-formation-enhancing regions. Here we show a schematic of an AGN-driven outflow in a multiphase medium, with the wind filling a cavity surrounded by the shocked wind and shocked ISM. The three possible regions of star formation enhancement are named and indicated by arrows. Outflows can compress stationary clouds directly (for discussion see Sect. 5.1.1). Star formation can be sporadically enhanced in turbulent shear flow between the outflow and the galactic disc (see Sect. 5.1.2). Finally, outflows can fragment by themselves (see Sect. 5.1.3). \n<!-- image --> \nSMBH mass, leading to a final form \nv out ≃ 1438 L AGN 2 . 3 × 10 47 erg s -1 ! 2 / (3 α ) l ( α -2) / (3 α ) f g 0 . 1 ! -1 / 3 kms -1 . (1) \nWe note that the dependence of this expression on α is very weak. \nThe plots show several properties of the observed outflow population relevant to our study. First, there is an overlap between the molecular and ionised outflow velocities and radial distances. The average velocity di ff erence between the phases is of the order of several hundred km s -1 , similar to the average velocity of cold outflows; both are also co-spatial within 0 . 3-10 kpc. In this work, we consider outflows with the maximum velocity v out = 400 km s -1 (or, equivalently, velocity difference between the hot and cold phases of ∆ v out = 400 km s -1 ). We mark this limit with a horizontal dashed line at v out in Fig. 2 and as a slanted line showing v out , mol + ∆ v out. Our choice of the maximum upper velocity is arbitrary; however, we infer cloud dispersal and star formation quenching at higher v out or ∆ v out. Secondly, slow outflows can be driven even at moderately large AGN luminosities < ∼ 3 × 10 46 erg s -1 . The presence of slow outflows in these AGN hosts can be explained by them having low Eddington ratios (which means they are powered by very massive black holes and reside in very massive galaxies with a strong gravitational potential), the host galaxy being very gasrich, and / or the AGN driving being intermittent. This last possibility leads to the outflow being driven by an e ff ective luminosity equal to the long-term average AGN luminosity, which may be much lower than the observed instantaneous value (Zubovas \nFig. 2. Compilation of observed outflow velocities and radial distances. The symbol indicates the detected gas phase, symbol size - mass outflow rate ( ˙ M ). The colour indicates the AGN luminosity. The oblique continuous line shows the average ˙ M -weighted velocity of molecular outflows shifted by ∆ v out = 400 km s -1 . The horizontal dashed line shows the upper velocity limit for outflows simulated in this work. Data aggregated from compilations in Fiore et al. (2017); Fluetsch et al. (2019); Lutz et al. (2020); Zubovas et al. (2022). \n<!-- image --> \nFig. 3. Same as Fig. 2 but showing outflow velocity against AGN luminosity. The radius is shown by the colour of the symbols. Oblique lines show the predicted upper outflow velocity limit for SMBH emitting at Eddington ratio l (Eq. 1) . \n<!-- image --> \n& Nardini 2020). The theoretical model we used to draw the lines is based on the assumption of an isothermal density distribution of the gas, which is approximately correct for galactic bulges. In this case, outflow velocity remains constant with radius. In the case of an NFW potential, the velocity first decreases and later increases as the density drops significantly (Zubovas & King 2016). This can allow slow outflows to exist in a narrow radial range where the e ff ective velocity dispersion is the greatest. \nTheoretical models provide robust predictions for global structure and radial properties of outflows in smooth medium. However, outflowing material density and temperature on intermediate scales (i.e. tens of parsecs) are less certain as they are determined by the cooling and mixing rates between the gas phases. The sharp discontinuity between shocked wind and shocked ISM might not exist, as it is unstable to KH instabilities (Zubovas & King 2014), especially if the medium is inhomogeneous (Ward et al. 2024). Furthermore, the region close to the discontinuity is heated by thermal conduction, which when combined with mixing causes the layer to swell (Weaver et al. 1977). We estimated the density range of the mixed gas in the layer. For an isothermal sphere, the particle number density is \nn is = σ 2 f g 2 π Gr 2 ≈ 36 f g cm -3 M SMBH 10 8 M ⊙ ! 2 /α r kpc ! -2 , (2) \nwhere r is the distance from the AGN. Here we again used the M -σ relation. First we consider galaxies with low bulge gas mass fraction. At a several hundred parsecs from the AGN, outside the dense disc, number density of ISM is several particles per cm 3 . The shocked ISM density is several times above the undisturbed one, of the order of 10 cm -3 . The shocked ISM cooling time is longer than the dynamical time, it is Jeans stable and does not fragment. As such, the layer can be considered an extended warm outflow enveloping the molecular clouds. On the contrary, in gas rich galaxies, ambient density is > 10 cm -3 . The shocked ISM cools and forms a thin, dense layer in which the molecular phase precipitates from the hot outflow (Richings & Faucher-Giguère 2018; Costa et al. 2020). The shocked and fragmented ISM reduces the covering factor from the position of the AGN, allowing confined high pressure shocked wind to leak and mix with cool gas. In both gas-rich and gas-poor cases, low-density shocked hot wind ( T > 10 9 K, n ≪ 1 mp cm -3 ) (Faucher-Giguère & Quataert 2012) mixes with cooler ISM regardless of the density of the surrounding medium. The density of the mixed material should be somewhere in between that of the hot wind and the warm envelopes of molecular clouds. In the absence of detailed models, we assumed this density to be ∼ 1 mp cm -3 and have a wide temperature range of 10 4 -10 6 K. It ablates embedded molecular clumps either compressing or dispersing them. For a more thorough discussion of where and when slow outflows occur (see Sect. 5.1).', '2.3. Molecular clouds': 'The vast majority of stars in galaxies form in molecular clouds. In the Milky Way, typical cloud masses range between 10 2 -10 6 M ⊙ . Smaller clouds are found throughout the periphery of the disc and the bulge, with larger masses towards the disc midplane (Miville-Deschênes et al. 2017). The cloud mass function at the high end is a power law: \ndN d ln M ∝ M -β , (3) \nwhere N , M are the number and mass of molecular clouds, respectively, and the exponent β = 2 . 0 ± 0 . 1. Despite lower mass \nclouds being more numerous, the majority of the cold gas in the galaxies is contained within massive M ≥ 10 4 M ⊙ molecular clouds. \nThe properties of many individual clouds are known robustly. Their masses, linear sizes R , number densities n , and velocity dispersions σ tend to follow approximate power-law relations known as Larson laws (Larson 1981; Miville-Deschênes et al. 2017; Sun et al. 2018): \nn ∝ R -p , M ∝ R q , σ ∝ R r , (4) \nwith p ∼ 1 . 0 , q ∼ 2 . 1 , r ∼ 0 . 4. In our simulations, we assumed that molecular clouds follow these scaling relations exactly (see Sect. 3.1).', '2.3.1. Cloud compression': 'In quiescent galaxies, molecular clouds are in partial equilibrium with the surrounding medium. An outflow surrounding the cloud acts as external pressure and compresses the cold gas. To estimate this e ff ect analytically, we started with a spherical cloud in equilibrium as outlined by Bertoldi & McKee (1992), but excluded the magnetic field contribution. We further assumed a constant mean molecular weight µ and constant specific heat ratio γ . \nThe equilibrium condition is defined via the virial equation \n2( T - T 0) + W = 0 , (5) \nwhere T , W are the total kinetic and gravitational potential energies of the system, respectively. External pressure is included via the extra term T 0 = 3 2 P 0 V cl. The virial parameter for a spherical cloud is defined as \nα = 2 T | W | = 5 R σ 2 GM ≈ 1 . 2 R pc ! M 10 3 M ⊙ ! -1 GLYPH<18> σ km s -1 GLYPH<19> 2 . (6) \nHere, R , M , and σ are the radius, mass, and velocity dispersion of the cloud, respectively. With α ∼ 1 turbulence supports the cloud against gravitational collapse, while clouds with α < 1 do not have su ffi cient kinetic energy and collapse. On the contrary, clouds with α > 1 disperse, unless they are bound by external pressure. \nThe sources of external pressure can be the surrounding ISM or the outflow. Outflows contribute to external pressure via ram and thermal pressure. Their ratio is \nP ram P th = Kv 2 out µ cos 2 θ k B T out = K cos 2 θ M 2 . (7) \nHere, M is the outflow Mach number, K ( M ) < ∼ 1 accounts for the peculiarities of sub, trans, and supersonic regimes (Spreiter et al. 1966), and θ is the angle between the flow direction and the cloud surface normal. \nThe outflow ram pressure contribution to cloud compression decreases towards the edges of the cloud. In the simplest subsonic case ( M < 1) the outflow impinges on the cloud directly. However, at higher outflow velocities the shock structure becomes more complex. Supersonic interaction ( M > 1) leads to the formation of discontinuities in the outflow, resulting in a bow shock upstream of the cloud (Landau & Lifshitz 1959). In a strong shock ( M≫ 10), the evolution becomes practically independent of Mach number (Klein et al. 1994) - cloud compression is determined by the gas properties near the cloud surface. \nOutflows also drive shocks into the cloud. The shocked gas is compressed, cools rapidly and becomes unstable to fragmentation, resulting in multiple secondary (reflected and refracted) shocks (for shock propagation in clumpy environments see Poludnenko et al. 2002). The resulting cloud and cloudlet compression is non-isotropic and non-homogeneous and depends on the shock strength in the outflow.', '2.3.2. Cloud evolution timescales': "In the simplest case, without external or internal pressure, a uniform gas sphere collapses under its gravity in a free-fall time that depends only on cloud density ρ : \nt ff = s 3 π 32 G ρ ≈ 1 . 6 Myr GLYPH<18> n 10 3 cm -3 GLYPH<19> -1 / 2 . (8) \nWhen external pressure is negligible, real clouds evolve on timescales longer than the free-fall time due to internal pressure, which arises due to turbulence, magnetic fields, collapse-induced heating, and (proto-)stellar feedback. As density increases, the free-fall time decreases and locally collapsing regions develop. The typical mass of a collapsing fragment is the Jeans' mass \nM J = π c 3 s 6 G 3 / 2 ρ 1 / 2 ≈ 2 M ⊙ GLYPH<18> c s 0 . 2 km s -1 GLYPH<19> 3 GLYPH<18> n 10 3 cm -3 GLYPH<19> -1 / 2 . (9) \nHere, c s is the speed of sound. Turbulence-induced overdensities cause further instabilities, so the collapse is not spatially uniform. Cloud collapse and formation of Jeans-unstable gas is further accelerated by external pressure: the external surface of the cloud is compressed first and is susceptible to fragmentation. \nThe time required for the outflow-induced shock to traverse the cloud is known as the cloud-crushing time: \nt cc = χ 1 / 2 R v out ≈ 9 . 6 Myr GLYPH<18> χ 100 GLYPH<19> 1 / 2 R pc ! GLYPH<18> v out km s -1 GLYPH<19> -1 , (10) \nwhere χ is the ratio of cloud density to that of the surrounding medium. As the shock propagates, Kelvin-Helmholtz (KH), Rayleigh-Taylor (RT) and Richtmyer-Meshkov (RM) instabilities form in the interface between the cloud and the ISM (Klein et al. 1994; Zhou et al. 2021). The resulting vortices mix the two gas phases, leading to temperature increase and erosion of the cloud's outer layers. The mixing is most prominent where the outflow is tangential to the surface of the cloud. The KH instability growth timescale is approximately \nt KH ∼ t cc kR , (11) \nand the RT instability growth timescale is \nt RT ∼ t cc ( kR ) 1 / 2 , (12) \nwhere k and R are the wave number and cloud radius, respectively. In both cases, cloud destruction is dominated by the largest ( kR > ∼ 1) wavelengths, which have a characteristic mixing time comparable to the cloud crushing time. \nHowever, this mixing-induced destruction time estimate is not accurate for turbulent clouds undergoing rapid compression. The growth of dense cloudlets is primarily determined by the cooling time \nt cool = 2 3 k B T n Λ , (13) \nwhere Λ is the volumetric cooling rate. Due to the nonlinearity of the cooling function and outflow pressure anisotropy, we cannot analyse cloud evolution analytically and turn to numerical simulations.", '2.3.3. Star formation': 'As the molecular cloud fragments and density of the resulting cloudlets increases, self-gravity becomes the dominant force. The collapse leads to the formation of gravitationally bound dense cores. The characteristic cooling time - and hence evolution - of the cores is < 0 . 1 Myr. After that, star formation begins (Chevance et al. 2020). The star formation e ffi ciency per free-fall time is defined as \nϵ ff = ˙ M ∗ M cl + M ∗ t ff , (14) \nwhere ˙ M ∗ is the star formation rate and M cl is the cloud gas mass. Typical values of ϵ ff range between 0 . 001-0 . 1 depending on spatial scale, with higher values in smaller clouds and / or fragments. When collapsing cores reach density values 10 6 . 5 cm -3 , ϵ ff values increase rapidly, approaching unity (Khullar et al. 2019). Gas above the density threshold produces stars until eventually stellar feedback and turbulence disperse the cloud (for review, see for example Chevance et al. (2020)). The fraction of initial cloud mass converted into stars, known as the integrated star formation e ffi ciency, is ϵ int = M ∗ / ( M cl + M ∗ ) ≃ 0 . 1 in GMCs in quiescent galaxies (Murray 2011; Kennicutt & Evans 2012). \nIn non-quiescent galaxies, the values of ϵ ff , ϵ int have even wider uncertainty. In addition to self-regulation by stellar feedback, outflows supplement stellar feedback with Mach-numberdependent external pressure (Eq. 7). Outflows confine, mix and compress the cloud, increasing the variation in ϵ . Due to the complexity of stellar feedback, we limit our investigation to the initial stages of star formation and aim to measure the e ff ects of AGN feedback on the onset of star formation.', '3. Numeric methods': 'We used G adget 4, a public hybrid SPH / N-body code (Springel et al. 2021) with the pressure-entropy formulation of the equation of motion (Hopkins 2013) and time-dependent artificial viscosity. The code was chosen due to its scalability and ability to simulate an elongated volume with periodic boundary conditions and gravity. We modelled hydrodynamicsand gravity; radiative processes were simulated via the cooling function. We assumed fully ionised, monoatomic gas and used a specific heat ratio γ = 5 / 3 with constant mean molecular weight µ = 0 . 63 mp. We chose a Wendland C4 kernel (Dehnen & Aly 2012) with 256 neighbours due to its good performance in analytic tests and modest requirements for computational resources.', '3.1. Initial conditions': "We simulated a virtual 'wind tunnel' - an elongated box with periodic boundary conditions in all directions. Each simulated system was composed of a stationary cloud and an enveloping outflow. Contrary to the common direct wind or momentum injection into the initially stationary ambient medium, 3 we set a uniform initial outflow velocity and artificially heat the ambient gas to match the post-shock conditions. \n<!-- image --> \nTo investigate the properties of mixed gas, and to enable the analysis of the long-term evolution of the system, we set the length of the box to l box ≃ v out t evo, where t evo is the total evolution time. 4 The length of the sides perpendicular to the outflow direction was chosen to prevent artificial pressure buildup upstream of the cloud. We set a minimum distance of 30 pc from the edge of the cloud to the boundary of the box; a complete list of simulated volumes is provided in Appendix A. \nThe cloud is a turbulent sphere, with an initially uniform density and temperature of 25 K. We chose three masses of molecular clouds - 10 3 , 10 4 , 10 5 M ⊙ , and used the Larson relations to determine cloud radii: R cl = 0 . 11 ( M / M ⊙ ) 0 . 48 pc. Combined with n cl = 1300 GLYPH<0> R / pc GLYPH<1> -1 cm -3 , the initial cloud density values are n cl = 433 , 143 , 47 cm -3 , respectively. Due to the nature of the mass-radius relationship, each cloud has almost the same column density Σ cl ≃ 40 M ⊙ pc -2 . We set the initial turbulent velocities using σ = 0 . 27 ( M / M ⊙ ) 0 . 19 km s -1 , leading to mean values of 1 , 1 . 6 , 2 . 4 km s -1 , respectively. The mass of a single SPH particle was set to 0 . 1 M ⊙ , resulting in clouds composed of approximately 10 4 , 10 5 , 10 6 particles, respectively. This guaranteed su ffi cient resolution for mixing with moderate requirements for computational resources. \nThe rest of the box was filled with a hot, homogeneous outflow with a number density of 1 cm -3 ; it was simulated with SPH particles of the same mass as the clouds. The outflow has a uniform initial velocity in the x direction (from left to right in figures). We modelled outflows with velocities of 10 , 30 , 60 , 100 , 200 , 400 km s -1 and temperatures of 10 4 , 10 5 , 10 6 K. These temperatures correspond to sound speeds c s ∼ 15 , 47 , 148 km s -1 , respectively, so for each outflow temperature, we have both subsonic and supersonic outflows. \nAdditionally, for each cloud mass, we created a 'control' simulation in which the medium surrounding the cloud was stationary and had a temperature of 10 4 K. Both cloud and outflow parameters were selected to densely fill the parameter space and identify star formation-inducing and quenching regions. In total, we modelled 57 systems (Table 1). Each system was evolved until fragmentation time (Sect. 3.4) or until the cloud was dispersed.", '3.2. Turbulence': 'To generate a realisation of turbulent velocities, we started with a uniformly spaced lattice, totalling 256 3 points in Fourier ( k ) space. At each point, the amplitudes of the generating field are determined by the power law \nP ≡ ⟨| vk | 2 ⟩ ∝ k -11 / 3 . (15) \nFor each grid point, we sampled a random complex number from a bi-variate Gaussian distribution as outlined in Dubinski et al. (1995). The resulting field amplitudes at a point are Rayleigh distributed with uniform phase distribution from 0 to 2 π . \nThe resulting velocity field, formed by such a process, is purely compressive. However, various compositions of the turbulent field can be recovered using a projection operator (Federrath et al. 2008): \nV τ i j ( k ) = τ V ⊥ i j + (1 -τ ) V ∥ i j = τδ i j + (1 -2 τ ) kikj | k | 2 . (16) \nHere, V τ i j is a projection tensor in the Fourier domain, and τ ∈ [0-1] is the ratio of solenoidal turbulent energy to total. In the \nFig. 4. Heating and cooling (negative) times. Artificially heated outflow regions are shown as oblique grey stripes, with outflow temperatures indicated on the plot. \n<!-- image --> \nsecond part of Eq. 16, we decomposed the projection tensor into a sum of transversal V ⊥ i j and longitudinal V ∥ i j operators and finally rewriten them as the sum for vector components in Fourier space, where δ i j is the Kronecker delta as usual. We tested the e ff ect of τ on fragmentation and found no significant di ff erences for an expected range 0 . 6 ≤ τ ≤ 1 . 0 in realistic clouds (Ginsburg et al. 2013). As a result, we selected purely solenoidal turbulence ( τ = 1) for this work. \nFinally, the velocity field components ( vx , vy , vz ) were calculated by an inverse Fourier transform. The turbulent velocity is the real part of the transformation. Velocity components of individual SPH particles were calculated by linear interpolation.', '3.3. Cooling function': 'We estimated radiative cooling rates with a cooling function as presented in Kakiuchi et al. (2024). For low temperatures (i.e. T < 10 4 K) it uses the results by Koyama & Inutsuka (2000, 2002), whereas at higher temperatures, it uses a fit to Sutherland & Dopita (1993). (For the explicit form of the cooling function, see Appendix E.) \nWe neglected thermal conduction, assuming it is either small compared to radiative cooling, or the flux is saturated, or evaporation times in the classical regime are long compared to the evolution times of our systems (Spitzer 1978). \nReal outflows are expected to maintain approximately constant temperature, at least during the AGN phase. To approximate this, we artificially heated the ambient gas: for gas particles with number densities n < n thresh = 10 cm -3 and temperatures in the range of T out / 2 < T < T out, the value of Λ (Eq. E.3) was set to zero. We tested several sets of density and temperature thresholds with little impact on the results (see also Appendix C). The modified cooling function ensures a warm-to-hot outflow with constant temperature and pressure and has little e ff ect on the total mass of warm gas produced by mixing. \nWe show the heating and cooling time dependence on gas density and pressure in Fig. 4. The modified cooling regions \n<!-- image --> \nTable 1. Summary of the physical parameters. \nare highlighted in grey, with corresponding outflow temperature labels. The thin grey curve starting around n ∼ 1 cm -3 and P / k B ∼ 5 × 10 3 K cm -3 is the equilibrium state of the gas.', '3.4. Star formation': "We adopted a simple prescription to track star formation: we assumed Jeans unstable gas with number densities n > 10 6 cm -3 rapidly collapses into protostellar fragments. As soon as an SPH particle satisfied both conditions, it was instantaneously converted to a star particle and subsequently interacted with the rest of the system only through gravity. We evolved each system until the total mass of star particles reached f ∗ = 0 . 04 of the initial cloud mass. We labelled this moment as the 'fragmentation time', t frag. \nTo account for di ff erent evolution timescales of clouds with di ff erent masses, we defined a normalised dimensionless fragmentation time as \nt norm = t frag t frag(ref) = t [ M cl , v out , T out] t [ M cl , 0 km s -1 , 10 4 K] , (17) \nwhere t frag is the fragmentation time of a given system, while t frag(ref) is the fragmentation time of the control simulation - a cloud of the same mass in stationary 10 4 K ambient medium.", '4. Results': 'Wefirst present the evolution of gas morphology, followed by the evolution of the mass of gas of di ff erent phases and finally move on to the onset of fragmentation and initial fragmentation rates. We use a Cartesian coordinate system with outflow velocity in the positive x direction in all density maps.', '4.1. Morphology': 'The outflow and cloud interaction can be split into four phases. First, as the outflow impacts the cloud, an inner cloud shock forms. During the second stage, the cloud is compressed, mostly in the direction of the outflow. As the shock traverses the cloud, fragmentation begins. The final stage is cloud destruction due to the mixing of cloud and outflow material, which allows instabilities to develop and produces eddies. The interaction of subsonic and supersonic outflows with the cloud di ff er in their details - we present them below.', '4.1.1. Subsonic outflow': "In subsonic interaction (i.e. simulations with v out < c out; top two rows of Fig. 5), the flow over the surface of the cloud is smooth and follows the contours of the cloud. The outflow ram pressure is low, so pressure anisotropy is weak. The cloud is more strongly deformed by the Venturi e ff ect, that is, lower pressure caused by accelerated flow along the edges of the cloud. However, this does not result in major morphological changes and cloud compression can be considered isotropic. A dense collapsing shell forms, which sweeps the turbulent cold cloud gas, leading to accelerated growth of overdensities in the shell. \nThe dense collapsing shell also inhibits large-scale instability-induced mixing. On smaller scales, the large density contrast and e ffi cient cooling reduces shear flow mixing even further. Therefore, the cloud is confined and retains the majority of its initial mass. The cloud's interior is isolated from the outflow and evolves independently. 5 Turbulence causes prominences to extend from the cloud, which can be blown away by the outflow. However, their removal is also ine ffi cient due to the high-density contrast and so the resulting mass loss is negligible. At later stages, the shell fragments due to self-gravity and is no longer able to shield the cloud interior from the flow. As the outflow pierces the cloud shell, the mass of the turbulence-induced clumps ( > 50 M ⊙ ) determines whether they are dispersed. Eventually, the clumps cool and contract due to self-gravity, their density reaches the star formation threshold and star particles begin to form (Sect. 3.4).", '4.1.2. Supersonic outflow': "In the supersonic models (third and lower rows of Fig. 5), a bow shock upstream of the cloud is essentially indistinguishable from the surface of the cloud due to the fast cooling of shocked material. In hot outflows T = 10 6 K, cooling is less e ffi cient, thus a prominent shock forms upstream of the cloud (see Appendix B). The reverse shock, compressing the opposite side of the cloud is observed only in the low supersonic regime. The cloud shock produces fragmented multi-phase gas with t ff ≃ t cool < t KH < t cc in the densest clumps. The clumps obstruct the flow, however, \nFig. 5. Particle density slices of a 10 5 M ⊙ molecular cloud embedded in a 10 5 K outflow. The left column shows the system's state after 0.5 Myr, the right one - at t frag. Outflow velocity (Mach number), indicated at the top left of each panel is 30 km s -1 ( M = 0 . 64) in the top row and 200 km s -1 ( M = 4 . 28) in the bottom one. Coordinates are centred on the cloud centre of mass, with outflow velocity in the positive x direction. For a complete set of clouds in 10 5 K outflows and equivalent plots of clouds in 10 4 K and 10 6 K outflows, see Appendix B. \n<!-- image --> \nthe ram pressure is high enough to pierce low-density regions resulting in the formation of dense pillars in the direction of the outflow. \nFragmentation of the shell and shocked cloud material produces multiple weaker shocks and mixing instabilities inside the cloud, with a typical size of a few parsecs. Simultaneously, as the shock traverses the cloud, the dense pillars and di ff use channels penetrate within until they traverse the cloud completely. When this happens, fast outflow material starts streaming through the cloud. Ablation by the outflow results in an increased area of contact between the outflow and cloud material, further increasing their mixing. The relative timing of fragmentation and channel formation determines the ultimate fate of the system. If the channels penetrate the cloud before the formation of dense selfgravitating cloudlets, the cloud is dispersed and star formation quenched. Conversely, if self-gravitating cloudlets form earlier, the outflow material filling the channels further compresses the cloudlets and star formation is enhanced. \nAt the edges of the cloud, KH and RT instabilities form large eddies with a circulation diameter comparable to the radius of the cloud. These eddies produce mixing regions behind the cloud, separated by a low-density cavity in between (bottom three rows of Fig. 5, left column; see also Sect. 4.2). These largescale mixing processes are suppressed in later stages by outflow material piercing the cloud, filling the cavity behind the cloud, and inhibiting further growth of mixing instabilities (right panels of Fig. 5).", '4.2. Mass and dynamics of cool gas': "The competing processes of shock-induced heating and radiative cooling of the dense material lead to constant changes in the material comprising the cloud. In addition, ram pressure and self- \ngravity compete to move the cloud material in di ff erent directions, facilitating or impeding cloud destruction. The net e ff ect of all these processes can be seen by considering the evolution of the cold gas mass. \nWe define cold gas, as that with a number density > 10 cm -3 and temperature < 10 3 K, while the rest is considered to be part of the outflow. Both thresholds are chosen to be consistent with the threshold values in the cooling function (Sect. 3.3). The precise threshold values have little impact on our results, since in most cases the cloud and outflow phases are well separated both in density and in temperature (see Appendix C). \nFigure 6 shows the evolution of cloud mass (solid lines), and the di ff erence between converging and diverging cold gas mass (dashed lines) of M cl = 10 5 M ⊙ simulations. Both quantities are given as ratios with the initial cloud mass. Cloud mass decreases due to compression heating and evaporation. Conversely, e ffi -cient radiative cooling results in the growth of cold gas mass. Cold mass also grows due to thermal instability in the swept-up hot outflow medium, provided the gas can cool e ffi ciently. Measuring the di ff erence between converging and diverging cold gas mass helps elucidate which process dominates. Positive values of the di ff erence mean that the cloud itself is growing (the net flux of cool gas is towards the cloud's centre of mass), while negative values indicate cloud dispersal. \nThe mass of the cold gas remains approximately constant for velocities ≤ 100 km s -1 independently of outflow temperature. Faster outflows, produce two stages in mass evolution. Initially, the mass decreases as cloud material is heated above the cold gas threshold due to compression by the propagating shock wave. The shocked gas fragments and is accelerated by the outflow, increasing the mass of the in-falling gas. After the cloud shock traverses the cloud's centre of mass (a peak in the converging / diverging mass graphs in Fig. 6), the second stage begins. \nFig. 6. Evolution of cold gas mass. We show the ratio of cold gas mass to initial cloud mass (solid line) and the ratio of the di ff erence between converging and diverging cold gas mass to the initial cloud mass (dashed line) for the M init = 10 5 M ⊙ simulations. The outflow velocity increases from left to right; colours show the outflow temperature as indicated in the legend. (For equivalent plots of clouds with lower masses, see Appendix D.) \n<!-- image --> \nDuring this stage, the fragmented gas mixes with the outflow, and the in-falling gas fraction decreases. Some of the mixed gas cools down, adding to the cold gas mass. As the shock fully traverses the cloud (an increase in the mass ratio starting ∼ 1 Myr at 400 km s -1 ), the cold gas mass begins to decrease again as the cloud starts to disperse. This e ff ect is absent in the 10 6 K outflows, where the cold gas mass remains fairly constant due to the high thermal pressure confining the cloud, thus preventing dispersal. \nAs the shock propagates through the cloud and the gas fragments, outflow material streams through the clouds, enhancing mixing and dispersing the cloud. This is indicated by negative values of the converging-diverging gas mass di ff erence. The destruction of the cloud produces elongated di ff use warm clouds that retain a significant fraction of the initial molecular cloud mass. \nAt supersonic velocities, a cavity with a very low number density ( n ≪ 1 cm -3 ) forms behind the cloud. Outflows with temperatures > ∼ 10 5 K eventually crush the cavity behind the cloud leading to thermal instability that causes cool gas to 'precipitate' (Inoue & Inutsuka 2008). This e ff ect is most prominent at 100-400 km s -1 outflow velocities and 10 5 , 10 6 K outflow temperatures. The velocity of the cloudlets is similar to the velocity of the outflow, and therefore it increases the mass of the diverging gas and simultaneously contributes to the mass growth of the cold gas. The gas in the cloudlets does not fragment (see Sect. 4.3) but initiates further thermal instabilities in the outflow.", '4.3. Fragmentation': "As explained in Sect. 3.4, we use a normalised fragmentation time (Eq. 17) as the primary metric for star formation quenching or enhancement. The fragmentation times in the three 'control' simulations with stationary, warm (10 4 K) ambient medium are 2 . 43 , 5 . 50 and 7 . 18 Myr for 10 3 , 10 4 and 10 5 M ⊙ clouds, respectively. \nFragmentation occurs either in the shocked cloud gas or in the undisturbed cloud within dense turbulence-induced filament knots. Shocked gas is heated to several thousand K and fragments as it cools, and unshocked cloud gas is compressed by turbulent velocities before the arrival of shock. Lower shock velocity and larger cloud radius prolong the initial turbulence- \nFig. 7. Normalised fragmentation times. Symbol size shows cloud mass, and colours show the outflow temperature, as given in the legend. Circles show systems that reached the fragmentation time, and crosses indicate systems where the cloud is dispersed before forming any star particles. Star symbols represent systems where a few star particles form but the cloud is dispersed before reaching the adopted fragmentation criterion. The vertical position of the symbols for non-fragmented systems is arbitrary. The dashed horizontal line separates star formation enhancing ( t norm < 1) and quenching regions. \n<!-- image --> \ninduced filament formation phase resulting in denser clumps. 6 In 10 4 , 10 5 K outflows, the mean mass of the resulting cloudlets is ∼ 300 M ⊙ with a mean radius of ∼ 1 . 0 pc. Clouds of initial mass 10 5 M ⊙ produce ∼ 100 cloudlets with a few reaching 10 4 M ⊙ , fragmentation occurs primarily in the most massive ones. The rest of the gas in the cloud is too di ff use and uniform to be considered a cloudlet. Fragmentation of the 10 4 M ⊙ \nFigure 7 shows the normalised fragmentation times for all simulations. In the vast majority of our simulations, fragmentation is enhanced, that is, the compressive e ff ect of outflow temperature is higher than the disruptive e ff ect of its kinematics. As expected, higher outflow temperature results in stronger enhancement. In extreme cases, for the smallest clouds embedded in hot (10 6 K) outflows, the fragmentation time is up to a factor ∼ 10 shorter than in the control runs, with more massive cloud fragmentation enhanced by a factor ∼ 5. Even at the highest velocities we have simulated, the thermal pressure of hot outflow crushes the cloud, preventing dispersal. (For a full list of numerical values, see Appendix A.) \n<!-- image --> \nclouds is similar, except for the lower overall count ( ∼ 30) and lower mass (10 3 M ⊙ ) of the largest cloudlet. The lowest-mass clouds hardly fragment at all, usually collapsing into a single or several cloudlets with a cumulative mass of 500-800 M ⊙ . Despite the di ff erences in the initial density, the fragmentation of the clouds is self-similar, and the minimum size of the cloudlets formed is determined by the cooling rates. Contrary to the cooler ones, 10 6 K outflows rapidly compress the cloud, and the turbulence-induced overdensities are aggregated into several massive ( > 10 4 M ⊙ ) clumps. At supersonic velocities, the clouds shatter - the clumps completely separate from each other. \nIn 10 5 K outflow simulations, cloud compression is less effective, roughly by a factor of 2 to 3. Again, clouds of lower mass are more strongly a ff ected and fragment independently of the kinematic pressure. At ≥ 100 km s -1 , more massive clouds are broken up into distinct components with masses similar to the clouds of the lowest mass, which then evolve roughly independently of one another. At the highest outflow velocity, massive clouds are dispersed before reaching the fragmentation threshold. The lowest mass clouds survive and fragment at all velocities. All clouds form some fragments before being dispersed (indicated by star symbols in Fig. 7). \nIn warm outflows (10 4 K), the outflow thermal pressure is similar to that of the undisturbed ISM - in other words, the outflow is close to pressure equilibrium with the cloud. This allowed us to e ff ectively isolate the influence of ram pressure even at velocities as low as 30 km s -1 . In such low-velocity simulations, fragmentation is enhanced by up to a factor of 1 . 5 in massive clouds. The enhancement is even stronger at v out = 100 km s -1 . At 200 km s -1 , the normalised fragmentation time of the massive clouds approaches unity, indicating a transition to a cloud dispersal regime. At 400 km s -1 , all but the lowest-mass clouds are completely dispersed and do not fragment. \nIn the majority of simulations with v ≤ 100 km s -1 , fragmentation is rapid, lasting ∼ 0 . 1 Myr from the moment the first fragments appear until fragmentation time is reached. However, at higher outflow velocities ( > 100 km s -1 ), there is competition between cloudlet destruction and fragmentation, leading to prolonged fragmentation spanning > 1 Myr at the highest velocities. \nOverall, our results show that there is a region of outflow and cloud parameter space where outflow passage enhances fragmentation (and, most likely, subsequent star formation). This parameter space is defined by threshold velocities of > 400 km s -1 at an outflow temperature of 10 6 K, and < 400 km s -1 at 10 4 , 10 5 K.", '5. Discussion': 'We simulated the interaction of an isolated turbulent molecular cloud and a warm outflow, and identified a region in the parameter space of cloud mass, outflow velocity, and temperature in \nwhich gas fragmentation, and presumably subsequent star formation, is enhanced. Slow outflows with velocities ≤ 200 km s -1 induce or enhance star formation in a wide temperature range, increasing gas consumption rates. Faster outflows disperse the massive clouds and quench star formation unless the outflow temperature reaches 10 6 K, in which case the velocity threshold for destruction rises to > 400 km s -1 . \nWe observed, that star formation enhancement is likely in regions where outflows are moving at low-to-moderate velocities relative to the dense gas clouds. In the next section, we provide an overview of such scenarios. We then discuss contributions from other star-formation-enhancing or quenching processes that can mask the e ff ects of positive AGN feedback. Later, in Sect. 5.2, we briefly overview the expected kinematics of stellar populations formed in this regime, then compare our results with several other works in Sect. 5.3 and lastly address the shortcomings of our models in Sect. 5.4.', '5.1.1. Compression of stationary clouds in continuous outflows': 'We first consider the compression of clouds with initially negligible radial velocity by the passage of an AGN-driven outflow. The simplest viable model of AGN outflow propagation is a spherically symmetric energy-driven wind model (e.g. King & Pounds 2015). Assuming constant AGN luminosity and an isothermal density profile, the outflow velocity quickly reaches a constant value and maintains it at least until it clears the bulge. The velocity also determines whether this duration is long enough to compress the cloud and significantly enhance its fragmentation rate. In the simplest, fully adiabatic, case, the thickness l out of the shocked ISM layer is ∼ 1 / 4 of the total outflow radius (Zubovas & King 2014). So the cloud is compressed for a duration t compr ∼ l out / v out ∼ R out / (4 v out). Compression is significant only when it lasts for at least ∼ t frag and preferably longer. In our simulations, t frag ranges from ∼ 0 . 3 Myr for the smallest highly compressed clouds to > 3 Myr for the slightly compressed most massive clouds. Considering a range of v out between 10-400 km s -1 , we get a minimum thickness required for sustained compression to be 0 . 005 kpc < l out < 1 . 2 kpc. Equivalently, the required outflow radii have a range of 0 . 02 kpc < R out < 4 . 8 kpc. \nObservational evidence (see Fig. 2) suggests, that the upper half of this range may occur only rarely, if ever. The maximum radii of outflows with v out < 400 km s -1 are ∼ 2 . 5 kpc, and the majority of such outflows are found within the central kiloparsec. In all cases, such outflows only exist in AGNs with luminosity L AGN < 10 46 erg s -1 (Fig. 3). This suggests that star formation enhancement should also be expected mainly in the central parts of galaxies with low-power AGNs, rather than further out - in the halo. \nIf we consider a more realistic setup, the picture described above becomes more complicated (see Fig. 1). Di ff erent gas density profiles and AGN luminosity variability on timescales comparable to outflow expansion induce changes in outflow velocity and / or temperature. Additionally, even if the outflow expands at a constant velocity, its density, and hence pressure, decreases with time (Zubovas & King 2016). We can expect that a decreasing pressure would have a lower e ff ect on the cloud than a constant one, but we refrain from further speculation since we did not simulate such a scenario. We intend to investigate how a varying outflow a ff ects the star formation rate in future work. \n<!-- image --> \n<!-- image --> \nOne more complication is the cooling of the shocked ISM. If it is rapid, the predicted outflow thickness decreases, potentially by a large factor (Richings & Faucher-Giguère 2018). In that case, a typical stationary cloud first traverses the ISM shock and, soon after, the contact discontinuity. The passage is less disruptive since the pressure is the same on both sides. However, after traversal, the cloud is embedded in the shocked AGN wind, with extremely high temperatures reaching > 10 9 K. As the shocked fragmented cloud is engulfed in the hot wind, thermal conduction can no longer be ignored, leading to evaporation of cloudlets in ≪ 1 Myr (Cowie & McKee 1977). From this, we conclude that rapidly cooling outflows are unlikely to enhance star formation in dense pre-existing ISM clouds. However, they can lead to star formation by fragmenting themselves (see Sect. 5.1.3 below).', '5.1.2. Sporadic fragmentation in clumpy galactic discs': "We now discuss viable regions in or near the galactic disc where the velocity di ff erence between hot and cold phases falls within the star-formation-enhancing region of the parameter space considered in this work. This mode depends on many circumstances of ISM distribution and instabilities along the sides of the outflow, so we call it 'sporadic' (see Fig. 1). \nThe vast majority of outflows in active galaxies are bi-cones (Nevin et al. 2018). The dense disc gas collimates the outflow and a layer with density and velocity gradients between the conical outflow and its surroundings forms. The shear flow in these layers creates hydrodynamical instabilities, increases turbulence and thus can compress the clouds. This happens essentially independently of the geometry of the outflow as it is being launched or independently of the driving mechanism. Due to the collimation, as well as the increasing mass of swept-up and mixed gas, the outflows have diminishing e ff ects on the galactic disc with increasing radial distance. This has been confirmed by the results from the MaNGA survey (Ilha et al. 2019): they show that low luminosity AGNs alter gas kinematics only within 1 to 2 kpc and does not a ff ect the gas outside this region. The intrinsic velocity of the gas in the cones is ∼ 300 km s -1 and the cones extend to 3 . 4 ± 1 . 8 kpc. The observed velocities fall within the star-formation-enhancing region of our simulations. Therefore, star formation is likely enhanced along the cone boundary. \nSuch a sporadic star formation may be the initial link in the positive feedback chain as outflows can also enhance star formation by compressing the galactic disc as proposed by Zubovas et al. (2013b). Such action extends the range of positive feedback from shear flow layers of the outflow to the galactic plane, possibly increasing star formation rates by up to an order of magnitude (Bieri et al. 2016). While such proposed star-formationenhancing modes are expected from theoretical and numeric models, the observational evidence remains unclear. This discrepancy can be explained by the high-Eddington ratio AGN phase being short compared to the outflow dynamic time. We address the implication of di ff erent timescales in Sect. 5.1.4.", '5.1.3. Fragmentation of clumpy outflows': 'Outside the galactic disc, outflows sweep and compress tenuous ISM and form shells of outflowing material. In gas-rich ( f g > ∼ 0 . 1) systems, some fragmentation of the shell appears inevitable (Nayakshin & Zubovas 2012; Scannapieco 2017; Richings & Faucher-Giguère 2018); this is accompanied by rapid cooling of the fragments (Zubovas & King 2014). Observations support the fragmenting outflow scenario as most AGN-driven \noutflows are dominated by molecular gas at a lower bolometric luminosity L AGN < 10 46 erg s -1 . Higher luminosity AGNs produce outflows with lower cold gas fractions, with the molecular phase found closer to the nucleus than the ionised one (Fiore et al. 2017; Fluetsch et al. 2019). \nThe formation of molecular clumps within outflows reduces the outflow covering fraction as seen from the SMBH. This allows the di ff use hot gas to escape from the bubble and expand further. As it does so, the dense clumps are e ffi ciently compressed by high thermal pressure. The clumps move essentially ballistically and slow down as they climb out of the gravitational potential of the galaxy, while the di ff use gas maintains its velocity. This leads to a small velocity gradient between the cold and hot outflow phases. Such multiphase regions are within the parameter space considered in this work and can elevate star formation in a galaxy outside the galactic plane. This scenario is consistent with observational evidence of star formation in a significant fraction of outflows (Maiolino et al. 2017; Gallagher et al. 2019). Moreover, in the Teacup galaxy (QSO J1430 + 1339), Venturi et al. (2023) found a young stellar population and multiphase outflows with velocity dispersion of ≥ 300 km s -1 . The detected stars coincide with the edges of outflow-blown cavities suggesting positive feedback. \nThe corollary of this enhancement mode is that the presence of massive molecular clouds is not required for a burst of star formation. Even di ff use, initially almost homogeneous gas can form clumpy, star-forming outflows. If the velocity di ff erence between the clumps and the hot phase is below the threshold limit of cloud dispersal, a starburst is almost guaranteed to occur. \nFurthermore, gas-poor ( f g < ∼ 0 . 1) systems (e.g. regions where gas has been partially expelled or consumed by stars) can also enhance star formation. The outflows themselves would not necessarily fragment and have velocities above the cloud destruction threshold, the dispersed clouds can mass-load the outflow, increasing the average density of the gas (Fig. 6), consistent with findings of Banda-Barragán et al. (2019); Girichidis et al. (2021). As galactic activity is hierarchically clustered in time (Hopkins et al. 2005; Zubovas et al. 2022), we expect subsequent AGN-driven outflows to encounter the leftover clumps from previous episodes, still travelling outward with lower relative velocity (i.e. < 400 km s -1 ) with respect to the new outflow. The compression by the subsequent activity episodes seems inevitable although it can be delayed by several flow crossing times.', '5.1.4. Cloud compression by fossil outflows': 'So far we focused on local e ff ects spanning several dynamic timescales of the clouds. We now address the e ff ects of AGN episode length and variability on cloud compression as it may explain the elusiveness of positive feedback. Typical AGNs flicker between high- and low-Eddington phases, with each cycle lasting approximately 0 . 1 Myr (Schawinski et al. 2015; King &Nixon 2015; Zubovas et al. 2022), orders of magnitude shorter than the dynamical time of an outflow. These cycles last for the total duration of an episode, which can be 1 to 10 Myr (Hopkins et al. 2005) comparable to the dynamical time of the molecular clouds. During low-Eddington phases and after the end of the whole episode, outflows persist without obvious nuclear activity and can be seen as fossils. Such coasting outflows are expected to expand for two to three times longer than the driving AGN episode (Zubovas & Maskeli¯unas 2023). As the fossil outflow expands in a gas-rich environment, its velocity and pressure gradually decrease, inevitably reaching velocities considered in this work and as such enhancing star formation. The lower pres- \n<!-- image --> \nure, both thermal and kinematic, of the fossil outflows means that star formation enhancement is less e ff ective and is limited to a narrower velocity range (see Sect. 4.3). This mode of star formation would be di ffi cult to investigate since outflows may be misclassified as driven by star formation and generally di ffi -cult to detect due to low velocity (Zubovas et al. 2022). Future detection and identification of fossil outflows might reveal the level of star formation enhancement in them.', '5.1.5. Supernova-driven outflows': 'Galaxy-scale outflows are not exclusively driven by AGN, they can also be powered by stellar winds and supernovae. Although typical values of stellar-feedback-driven outflows tend to be lower than those of AGN-driven ones, both samples have overlapping properties. The existence of very high pressure (5 . 6 × 10 7 K cm -3 ) fast (600-2000 km s -1 ) ionised starburst-driven outflows has been predicted for M82 (Chevalier & Clegg 1985). At the other end of the parameter range, Perrotta et al. (2023) shows the presence of massive cool T ∼ 10 4 K outflows in starburst galaxies with velocities and mass transfer rates comparable to the sample presented in Sect. 2.2 with the majority of them detected within the central kiloparsec. Similarly, massive cold molecular outflows have been observed in starburst galaxies as well (Bolatto et al. 2013) exhibiting conical geometry (Rubin et al. 2014; Bizyaev et al. 2019). Finally, as we show in this work, AGN outflows can enhance star formation and so induce supernova-driven outflows. \nAdditionally, starbursts can form superbubbles comparable in size, velocity and thermal pressure to the AGN-driven fragmenting outflows. For example, the pressure in the NGC 3628 bubble is high ( P / k B ∼ 10 6 -8 K cm -3 ) and it is surrounded by slow (90 ± 10 km s -1 ) molecular gas at 1 kpc from the centre (Tsai et al. 2012). It is also well-established that expanding swept-up ISM bubbles can lead to self-propagating star formation via fragmentation of their shells (Whitworth et al. 1994), although this probably requires a threshold luminosity (see e.g. Whitworth & Francis 2002, and references therein) and the scale of the e ff ect may be limited by interstellar turbulence (Nomura & Kamaya 2001). The observed values of superbubble pressure overlap with the upper portion of the range of pressures of our simulated outflows that lead to enhanced star formation. So our results can be seen as complementary to the self-propagating star formation scenario, but also reveal the possibility of star formation enhancement via compression of pre-existing clouds rather than just those arising from fragmentation of the shell.', '5.2. Kinematics of newly formed stars': "Wherever the star formation enhancement occurs (see the previous section), the stars form from significantly perturbed gas. At least part of the perturbation is directional, that is, the cloud is pushed in the direction of the outflow, and the newly formed stars retain some of that momentum. Multiplied by the duration of cloud fragmentation time and the duration of the pre-stellar evolution phase, this can lead to significant displacement of the newly formed stars. Thus, stellar kinematics can be used to identify relatively recent episodes of enhanced star formation. \nStars formed in the 'stationary' scenario (Sect. 5.1.1) have a low radial velocity < 0 . 1 v out due to long cloud acceleration times. Given that star formation is quenched at outflow velocities above several hundred km s -1 , we expect stars formed in this scenario to have radial velocities of order 40 km s -1 or less. This scenario \nis compatible with stars being dynamically colder compared to gas (Gallagher et al. 2019; Oh et al. 2022). However, the cumulative mass of formed stars is limited by the relatively narrow outflow velocity window capable of enhancing star formation. The window is even narrower at larger radial distances where the molecular clouds are generally smaller and easier to disperse. \nContrary to stationary cloud compression, fragmenting outflows (Sect. 5.1.3) result in lower relative velocities between the hot and molecular phases. Gallagher et al. (2019) show that the enhanced star formation 'in situ' (i.e. inside the outflow) is common in AGN host galaxies with rates up to 0 . 3 of total SFR, and might even dominate in the central kiloparsec. Outflow-formed stars initially have velocities of the outflowing gas 7 and lose their kinetic energy by doing work against gravity. Therefore, the kinematics of the newly formed stars is coupled with the driving pressure, the surrounding gas density and the host galaxy's gravitational potential. If the driving pressure is decreasing (e.g. after the end of an activity episode), the outflow coasts. Contrary to the stationary case, the outflow-formed stars are dynamically hotter as they overtake the outflow. This suggests the stars form in the early stages of the outflow propagation and lose radial momentum rapidly due to gravity. \nIn high-luminosity AGNs, in-situ formation from fragmenting outflows is the only viable mechanism of star formation enhancement, since initially stationary clouds are dispersed by the outflow kinematic pressure. The requirement of fragmentation necessitates e ffi cient cooling, which can be negated by the powerful radiation of the AGN itself. As a result, enhanced star formation appears more likely in fossil outflows, where the AGN has already shut down. In this scenario, the newly formed stars have lower transverse velocities than the ones formed from stationary clouds. As pointed out by Zubovas et al. (2013a), the velocities can be high enough to escape the bulge and become a part of the galactic halo. Stars with such elongated orbits have been detected in the solar neighbourhood (Belokurov et al. 2020) and may indicate an early Galactic activity episode. As highluminosity AGNs can produce massive outflow rates reaching 10 3 -10 4 M ⊙ yr -1 they can easily be the primary source of positive feedback. \nAs an intermediate case, the kinematics of stars formed via sporadic enhancement in the surroundings of the conical outflows (Sect. 5.1.2) is least certain. The stars retain dynamics of outflow-induced eddies and can have a wide range of radial velocities. The net e ff ect should be an enhancement of the radial anisotropy of stellar velocities in the bulge. However, the properties of these stars potentially resemble both the disc and halo populations and hence makes them very di ffi cult to distinguish and investigate.", '5.3. Comparison with other works': 'There has been a large number of works, spanning decades, investigating the conditions under which star formation may be enhanced by collisions, outflows or other scenarios of increased external pressure (for a compilation of works see Dugan et al. 2017, Table 1 and Banda-Barragán et al. 2019, Table A1). More recent works explore the destruction threshold values of for cold clouds (Li et al. 2020; Farber & Gronke 2022), or even the crushing of multi-cloud systems (Villares et al. 2024). Here we com- \n<!-- image --> \n<!-- image --> \nare our results with several works that use outflow velocity and density ranges overlapping with ours. \nOur results match those of AGN outflow shocks on BonnorEbbert spheres explored by Dugan et al. (2017). Despite di ff erent cloud density profiles compared to our models, and cloud masses comparable to those of cloudlets of our models, they find that outflows of 300 km s -1 , with a temperature of several million K and particle densities of 0 . 5 , 1 . 0 , 3 . 0 cm -3 , compress the clouds, enhancing star formation. Higher velocity outflows disperse the clouds or produce a negligible mass of stars. They find threshold values for cloud destruction P / k B = 1 . 2 × 10 8 Kcm -3 . These values are several times higher compared to our work, due to differences in the adopted survival criteria. If we assume the cloud survival without meeting fragmentation criteria, the cloud destruction threshold pressure is comparable. In our models, there are several surviving cloudlets in the most massive clouds of 10 5 M ⊙ , and even the fastest (400 km s -1 ) outflows do not disperse them. \nZubovas et al. (2014) analyse similar systems to the ones in this work; they simulate the impact of slow (30 , 100 , 300 km s -1 ) outflows on turbulent molecular clouds. They model a spherical turbulent cloud of M = 10 5 M ⊙ , with density contrast χ = 380. The cloud density is higher, comparable to lower mass clouds in this work, with turbulent velocity dispersion several times higher. They find clouds are compressed by external thermal pressure, but outflow velocity has little e ff ect on cloud fragmentation. The fragmentation times are ∼ 1 . 5 Myr in 10 5 Koutflows and vary little with outflow velocity. We note the di ff erence in the definition of fragmentation time: we end our simulations when the mass of the formed stars reaches 0 . 1 Mcloud , init, while Zubovas et al. (2014) run theirs until the fraction is ten times higher. Therefore, it is more appropriate to compare our t frag with t sink - the appearance of first sink (stellar) particles. Despite the differences in density, the qualitative behaviour is similar to our models of the same temperature. If we normalise the timescales by the free-fall time of the clouds, they become similar in both works. They also find no significant di ff erences in fragmentation time in rotating clouds or clouds with high turbulent velocities ∼ 10 km s -1 . \nArecent study by Mandal et al. (2024) explore a similar scenario. They embed the cloud into a hot ambient medium of 10 6 K and n = 0 . 1 cm -3 and model propagating wind of ≥ 400 km s -1 and n = 0 . 01 cm -3 , that is, a non-relativistic low-density outflow with moderate velocities. The propagating wind interacts with massive GMCs with particle densities of 20 and 200 cm -3 . As in our work, the authors include self-gravity and radiative cooling. Despite the di ff erences in the models, authors also lean towards positive, although delayed, feedback even at moderate velocity. Moreover, they show that AGN wind produces multiphase outflows with velocities of 100-1000 km s -1 and a wide temperature range (10 2 -10 7 K). As a result, positive AGN feedback occurs in two situations. Initially, the low-density wind compresses clouds closer to the AGN, while later, stationary clouds further away are compressed by the dispersed cloud material mixed with the wind, that is, a slow and warm-to-hot outflow. These findings suggest that star formation enhancement is viable as long as the external pressure is maintained. \nCooper et al. (2009) investigate cloud compression by starformation-driven outflows and found similar results to ours. Although they find that star formation is enhanced at a higher velocity of 1200 km s -1 , their simulations have ten times lower outflow density. The ram and thermal pressures are, in fact, comparable to those in our simulations with an outflow temperature \nof 10 6 K. They also find a similar morphology - a fragmented cloud with surviving dense cloudlets embedded in the outflow. \nAt the opposite extreme, there is little doubt that highly supersonic ( M≥ 20) flows with higher density contrasts compared to our models disperse molecular clouds (e.g. Orlando et al. 2005; Hopkins & Elvis 2010; Scannapieco & Brüggen 2015). This suggests that there probably are no other regions of the parameter space where star formation is enhanced.', '5.4.1. Wind or shock': 'There are two commonly used approaches for simulating the compression of a molecular cloud by an outflow: shock propagation (e.g. Pittard & Parkin 2016; Goldsmith & Pittard 2017) and continuous wind (e.g. Banda-Barragán et al. 2016; Sparre et al. 2019; Li et al. 2020). In the first case, a stationary cloud embedded in the pre-shock medium is struck with the propagating shock that engulfs the cloud. In the continuous wind scenario, a stationary cloud is immersed in uniform gas with velocity and temperature matching those of the post-shock gas. We selected the latter approach due to the simplicity of implementation. However, we note that clouds embedded in a continuous wind experience less compression due to cavities formed in the wake of the cloud and hence have longer evolutionary times comparedk-cloud interaction (Goldsmith & Pittard 2017). We mitigated the discrepancies by ending the simulations at fragmentation time and scaling the results to those of a control simulation. Nevertheless, our approach may underestimate the compressive e ff ect that relatively slow outflows have on interstellar clouds.', '5.4.2. Viability of the cooling function': 'We estimated the radiative cooling rates via a simplistic cooling function. We assumed an optically thin medium, a constant mean molecular weight, and a constant specific heat ratio for all the gas. Application of a more realistic thermodynamic prescription, especially to the non-equilibrium conditions occurring in the shocks, may alter the results. To justify the validity of our approach we consider the shocks with a velocity of ∼ 50 km s -1 , which is the approximate value of cloud shock in 400 km s -1 outflows and the highest shock velocity in our simulations. Such shocks in dense molecular clouds are of continuous type (C-type) - they have no discontinuities (for review see e.g. Draine & McKee 1993). A corollary is that such shocks are nondissociative, and both molecules and dust survive the passage (Draine et al. 1983). Moreover, in slow to moderate outflows, the dust grains enable the formation of additional coolant molecules, thus preventing a complete shutdown of the cooling channels (Hollenbach & McKee 1979; Neufeld 1990). We observed that in our models, the shocked cloud gas is capable of rapid cooling, so ignoring complex shock astrochemistry and optical depth e ff ects is justified.', '5.4.3. Initial conditions and turbulence': 'We start our simulations with spherical clouds of uniform density. The initially smooth cloud gas develops stochastic overdensities due to the turbulent velocity field. While such a choice is simple to implement and reduces the parameter space it only partially accounts for the shape of the clouds, as they remain quasispherical throughout the control simulations. Realistic clouds are \n<!-- image --> \nsurrounded by di ff use gas envelopes without sharp density or temperature gradients. The presence of such a layer, primarily composed of low-density n > 10 cm -3 HI (Heiles & Troland 2003), can reduce the overpressure of propagating shock waves. \nOur choice of initial conditions somewhat accounts for the presence of a surrounding gas envelope. The initial density contrast χ ≃ 47-433 is low enough that it is similar to the contrast between the warm and cold neutral gas phases expected in a real system (Heiles & Troland 2003). Having such a contrast allows for a realistic development of the internal overdensity structure, so we did not expect major changes in evolution with the inclusion of an additional external di ff use gas layer. \nThe geometry of the clouds, however, poses more issues. We expect that the column density of the cloud in the direction parallel to outflow velocity is a key parameter determining how susceptible the cloud is to destruction. If the cloud is strongly nonspherical, its orientation to the outflow becomes important in determining its evolution. We expect to test these issues in future work with a more realistic initial setup.', '6. Summary and conclusions': 'In this work, we have investigated the interaction of hot galactic outflows with isolated molecular clouds and determined the temperature and velocity threshold values for enhancement of cloud fragmentation by the outflow. We carried out SPH simulations of clouds with masses M cloud = 10 3;4;5 M ⊙ a ff ected by outflows with a constant number density of 1 cm -3 , temperatures of T out = 10 4;5;6 K, and velocities v out = 10 , 30 , 60 , 100 , 200 , 400 km s -1 in order to analyse the e ff ect of di ff erent thermal and kinematic pressures. We find that slow GLYPH<16> v out < 400 km s -1 GLYPH<17> outflows can compress the clouds and induce or enhance ongoing star formation. \nThe main results of this work are as follows: \n- -Wefindasingle, well-defined region in the cloud and outflow parameter space where star formation is enhanced.\n- -Warm outflows of 10 4 K with velocities of 60 , 100 , 200 km s -1 enhance star formation. At lower velocities, the outflow has a negligible e ff ect, while faster outflows disperse the massive clouds completely. Star formation in low-mass (10 3 M ⊙ ) clouds is enhanced at all explored velocities.\n- -At higher outflow temperatures of 10 5 K, thermal pressure compresses the clouds, shortening the fragmentation several times compared to the cooler simulations. The kinematic pressure has little e ff ect on cloud compression in outflows below ∼ 200 km s -1 , and it disperses massive molecular clouds at higher velocities, similar to the lower temperature outflows.\n- -In hot outflows of 10 6 K, the clouds are rapidly compressed, and the fragmentation time is reduced by an order of magnitude in the smallest clouds. Kinematic pressure has little e ff ect on cloud evolution, all clouds survive and fragment. \nWe suggest three primary scenarios where star formation enhancement is viable: \n- -Stationary molecular cloud compression in low-powered AGNs where slow outflows develop. This scenario requires both a low velocity and a sustained high outflow temperature to produce positive feedback. It is limited by the availability of cold gas with a low radial velocity.\n- -Sporadic enhancement in shear flow layers surrounding the conical outflows provides a positive feedback mechanism in gas-rich, luminous AGNs.\n- -Fragmenting multiphase outflows create cold gas with a low velocity ( < 400 km s -1 ) relative to the hot phase. As massive molecular AGN-driven outflows are prevalent, they are the main source of positive feedback. \nThe scenarios are consistent with current observational constraints and with previous works investigating triggered star formation in these disparate domains. Moreover, all three scenarios can occur during and after AGN episode. In the fossil case, positive feedback on star formation occurs without corresponding activity in the nucleus. As a result, enhanced star formation should only weakly correlate with present-day AGN luminosity, except for the central regions of galaxies where the outflow expansion time is short. This temporal discrepancy between cause and e ff ect makes interpretation of observations more di ffi cult. On the other hand, it provides an opportunity to use enhanced star formation as a tool to probe recent (several ayears) nuclear activity histories. \nPeculiar stellar kinematics (see Sect. 5.2) can help with this interpretation as well. In extreme cases, massive AGN-driven molecular outflows can create fountain-like streams of stars. Detection of such stars can be used to investigate former AGN episodes, outflow properties, and dynamics. Additionally, such stars can form peculiar sub-populations in galactic discs. \nFinally, we note that star formation enhancement and outflows from supernovae occur concurrently with AGN-driven outflows. The understanding of gas dynamics in such cycles and the interplay between thermal and kinematic pressure can help in investigations of the di ff erence in the e ff ects of wind, radiation, jet feedback, and / or star formation and can provide a comprehensive view of AGN feedback in general. \nAcknowledgements. We thank the anonymous referee for their insights and improvements to this work. This research was funded by Research Council Lithuania grant no. S-MIP-24-100. The simulations were performed on the supercomputer GALAX of the Center for Physical Sciences and Technology, Lithuania.', 'References': "Agertz, O., Moore, B., Stadel, J., et al. 2007, MNRAS, 380, 963 \nArakawa, N., Fabian, A. C., Ferland, G. J., & Ishibashi, W. 2022, MNRAS, 517, \n5069 \n- Banda-Barragán, W. E., Parkin, E. R., Federrath, C., Crocker, R. M., & Bicknell, G. V. 2016, MNRAS, 455, 1309\n- Banda-Barragán, W. E., Zertuche, F. J., Federrath, C., et al. 2019, MNRAS, 486, 4526\n- Belokurov, V., Sanders, J. 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While we freely choose the size of the box perpendicular to the flow ( y = z ), the code we used limits the choice of length ratios to powers of two ( x = 2 n × y ). We note the lengths of some boxes are comparable to the sizes of bulges in small galaxies. The last two columns show the main results of this work - fragmentation times, in both non-normalised and normalised (Eq. 17) forms. Missing entries here indicate the models where clouds were dispersed without reaching fragmentation criteria (see Sect. 3.4). \nTable A.1. Summary of the simulations. \nNotes. Columns 1-3 show cloud mass, outflow temperature and velocity, followed by χ - density contrast; M - Mach number; cloud crushing ( t cc) and acceleration ( t acc) times; size of the simulated 'box' in y , z , x ; fragmentation and normalised fragmentation times t frag, t norm. Missing entries in the last two columns show non-fragmented systems.", 'Appendix B: Density maps': "Here we provide complementary (see Fig. 5) density maps for the 10 5 M ⊙ cloud simulations with outflow temperatures of 10 4 , 10 5 , and 10 6 K. We briefly address the evolutionary di ff erences from outflows of intermediate temperature (see Sect. 4.1). In 10 4 K outflows (Fig. B.2), all but the lowest velocities result in supersonic interaction. The thermal pressure is the lowest of all simulations, and therefore compression by ram pressure is more pronounced. The dense outer shell of shocked gas does not form, increasing the cloud surface erosion and dispersal. In contrast, in 10 6 K outflows, thermal pressure is dominant, and it rapidly compresses the cloud (Fig. B.3). For the hot outflow case, only the last two rows are supersonic. \nFig. B.1. Particle density slices of a 10 5 M ⊙ molecular cloud embedded in a 10 5 Koutflow. The left column shows the system's state after 0.5 Myr, the right one - at t frag. Outflow velocity (Mach number), indicated at the top left of each panel, increase from 10 km s -1 ( M = 0 . 21) in the top row to 400 km s -1 ( M = 8 . 56) in the bottom one. Coordinates are centred on the cloud centre of mass, with outflow velocity in the positive x direction. \n<!-- image --> \nFig. B.2. Same as Fig. B.1 but for T out = 10 4 K. \n<!-- image --> \n<!-- image --> \nn \n[cm \n] \nFig. B.3. Same as Fig. B.1 but for T out = 10 6 K. \n<!-- image --> \nn \n[cm \n]", 'Appendix C: Gas state histograms': 'In Sects. 3.3 and 4.2 we introduce the selection criteria for gas to be considered part of the molecular cloud ( n > 10 cm -3 and T < 1000 K). In Fig. C.1 we show 2D histograms binned by temperature and density. The molecular cloud is composed of material contained within the grey box. We note that the precise choice of selection threshold values has little impact on the total mass of the cloud, since the majority of its mass has far more extreme values of both density and temperature. \nFig. C.1. Temperature, number density histograms of 10 5 M ⊙ clouds at fragmentation time (given in the upper right corner) with outflow temperatures of 10 4 , 10 5 , 10 6 K (from left to right). Outflow velocity increases in rows - 30 , 100 , 400 km s -1 . The rightmost column shows the total gas mass in each temperature bin. The gas in the grey area is considered to comprise the molecular cloud. \n<!-- image -->', 'Appendix D: Evolution of cold gas': 'In Sect. 4.2 we presented cold gas evolution for the 10 5 M ⊙ cloud. Here we provide a brief overview for lower-mass clouds. The intermediate-mass cloud (Fig. D.1) evolution is similar to the 10 5 M ⊙ cloud (see Fig. 6) but with the main di ff erence being shorter evolutionary times. Several models did not fragment (the ones that reach 4 Myr in the rightmost panel). Despite not meeting fragmentation criteria, the clouds are not dispersed and retain a significant mass of cold gas. For the lowest mass clouds (Fig. D.2) the evolution times are even shorter, and in none of the models we observe the growth of cold gas mass. \nFig. D.1. Evolution of cold gas mass. We present the ratio of cold gas mass to the initial cloud mass (solid line) and the ratio of the di ff erence between converging and diverging cold gas mass (dashed line) to the initial cloud mass ( M init) for the M cl = 10 4 M ⊙ simulations. Outflow velocity increases from left to right, the colours indicate the outflow temperature, as given in the legend. \n<!-- image --> \nFig. D.2. Same as in Fig. D.1 but for M cl = 10 3 M ⊙ . \n<!-- image -->', 'Appendix E: Cooling function': 'We used a cooling function as outlined by Kakiuchi et al. (2024). We assumed low optical depth, a constant mean molecular weight and solar metallicity. The generic form of the cooling function is \nρ n L = n ( -Γ + n Λ ) . (E.1) \nHere, ρ n and L are the density and total loss rate of internal energy, while n is the particle number density, and Λ , Γ are volumetric cooling and heating rates, respectively. The heating rate depends only on the temperature T and cut-o ff temperature T cut: \nΓ = 2 × 10 -26 exp -T T cut ! erg s -1 . (E.2) \nWechoose T cut = 5 × 10 4 K. The cooling function is a piecewise combination of low- and high-temperature regions. At temperatures below T < 10 4 K, the cooling rate is \nΛ l = 2 × 10 -19 exp -118400 T + 1000 ! + 2 . 8 × 10 -28 √ T exp -92 T ! erg cm -3 s -1 , (E.3) \nand at higher temperatures, \nlog 10 Λ h = -156 . 919 + 84 . 2271(log 10 T ) -19 . 0317(log 10 T ) 2 + 1 . 85211(log 10 T ) 3 -0 . 0658615(log 10 T ) 4 . (E.4) \nTo get the total cooling rate over the whole temperature range, we smoothly connected the cooling functions at a crossover point T b: \nΛ = 0 . 5 ( Λ l(1 -f ) + Λ h(1 + f )) , where f = tanh log 10 T -log 10 T b log 10 ∆ T b ! . (E.5) \nFunction f smooths the connection point between the two temperature regions. We set T b = 10 4 K and log 10 ∆ T b = 0 . 1.'} |
2023ApJS..269...33N | We present the evolution of the massmetallicity MZ relation at z 410 derived with 135 galaxies identified in JWSTNIRSpec data taken from the three major public spectroscopy programs of ERO GLASS and CEERS. Because there are many discrepancies between the flux measurements reported by the early ERO studies we first establish our NIRSpec data reduction procedure for reliable emissionline flux measurements and errors successfully explaining Balmer decrements with no statistical tensions thorough comparisons with the early ERO studies. Applying the reduction procedure to the 135 galaxies we obtain emissionline fluxes for physical property measurements. We confirm that 10 out of the 135 galaxies with O III 4363 lines have electron temperatures of 1.12.3 10SUP4SUP K similar to lowerz starforming galaxies which can be explained by heating by young massive stars. We derive the metallicities of the 10 galaxies by a direct method and the rest of the galaxies with strong lines using the metallicity calibrations of Nakajima et al. applicable for these lowmass metalpoor galaxies anchoring the metallicities with the directmethod measurements. We thus obtain the MZ relation and star formation rate SFRMZ relation over z 410. We find that there is a small evolution of the MZ relation from z 23 to z 410 while interestingly the SFRMZ relation shows no evolution up to z 8 but a significant decrease at z gt 8 beyond the errors This SFRMZ relation decrease at z gt 8 may suggest a break of the metallicity equilibrium state via star formation inflow and outflow while further statistical and localbaseline studies are needed for a conclusion. | 2023-12-01T00:00:00Z | ['arXiv:2301.12825', '10.3847/1538-4365/acd556', '10.48550/arXiv.2301.12825', '2023arXiv230112825N', '2023ApJS..269...33N'] | ['Chemical abundances', 'Galaxy chemical evolution', 'Galaxy evolution', 'James Webb Space Telescope', 'High-redshift galaxies', '224', '580', '594', '2291', '734', 'Astrophysics - Astrophysics of Galaxies'] | JWST Census for the MassMetallicity Star Formation Relations at z 410 with Selfconsistent Flux Calibration and Proper Metallicity Calibrators | 2,023 | 175 | 0.68 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF'] | 192 | https://arxiv.org/pdf/2301.12825.pdf | {'JWST Census for the Mass-Metallicity Star-Formation Relations at z = 4 -10 with the Self-Consistent Flux Calibration and the Proper Metallicity Calibrators': 'Kimihiko Nakajima , 1 Masami Ouchi , 1, 2, 3 Yuki Isobe , 2, 4 Yuichi Harikane , 2 Yechi Zhang , 2, 4 Yoshiaki Ono , 2 Hiroya Umeda, 2, 4 and Masamune Oguri 5, 6 \n1 National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan 2 Institute for Cosmic Ray Research, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8582, Japan 3 Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, Kashiwa, Chiba 277-8583, Japan 4 Department of Physics, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan 5 Center for Frontier Science, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522, Japan 6 Department of Physics, Graduate School of Science, Chiba University, 1-33 Yayoi-Cho, Inage-Ku, Chiba 263-8522, Japan \nSubmitted to ApJ Supplement on Jan 30, 2023 / Accepted on May 8, 2023', 'ABSTRACT': 'We present the evolution of the mass-metallicity (MZ) relations at z = 4 -10 derived with 135 galaxies identified in the JWST/NIRSpec data taken from the three major public spectroscopy programs of ERO, GLASS, and CEERS. Because there are many discrepancies between flux measurements reported by early ERO studies, we first establish our NIRSpec data reduction procedure for reliable emission-line flux measurements and errors successfully explaining Balmer decrements with no statistical tensions via thorough comparisons of the early ERO studies. Applying the reduction procedure to the 135 galaxies, we obtain emission-line fluxes for physical property measurements. We confirm that 10 out of the 135 galaxies with [O iii ] λ 4363-lines have electron temperatures of /similarequal (1 . 1 -2 . 3) × 10 4 K, similar to lowerz star-forming galaxies, that can be explained by heating of young massive stars. We derive metallicities of the 10 galaxies by the direct method and the rest of the galaxies with strong lines by the metallicity calibrations of Nakajima et al. (2022) applicable for these low-mass metal-poor galaxies, anchoring the metallicities with the direct-method measurements. We thus obtain MZ relations and star-formation rate (SFR)-MZ relations over z = 4 -10. We find that there is a small evolution of the MZ relation from z ∼ 2 -3 to z = 4 -10, while interestingly that the SFR-MZ relation shows no evolution up to z ∼ 8 but a significant decrease at z > 8 beyond the error. This SFR-MZ relation decrease at z > 8 may suggest a break of the metallicity equilibrium state via star-formation, inflow, and outflow, while further statistical and local-baseline studies are needed for a conclusion. \nKeywords: Chemical abundances(224) - Galaxy chemical evolution(580) - Galaxy evolution(594) High-redshift galaxies(734) - James Webb Space Telescope(2291)', '1. INTRODUCTION': "The James Webb Space Telescope (JWST) has dramatically advanced the exploration of the high redshift universe. One important advancement for the high redshift community is provided by the high sensitivity of the Near Infrared Spectrograph (NIRSpec; Jakobsen et al. \nCorresponding author: Kimihiko Nakajima \[email protected] \n2022) at λ /similarequal 2 -5 µ m, allowing us to directly address the key questions relating to the physical conditions of inter-stellar medium (ISM) and the nature of the ionizing spectrum for galaxies in the early universe. The properties of the hot ISM, especially the gasphase metallicity determined by the oxygen abundance (12 + log(O / H)), can be accurately constrained using rest-frame optical emission lines. These emission lines have been well calibrated and understood over decades, as discussed in a recent review by Maiolino & Mannucci (2019). Gas-phase metallicities provide a crucial exper- \nental tool for studying the early chemical enrichment and feedback processes in galaxy evolution. These measurements can be used in conjunction with cosmological simulations (e.g., Finlator & Dav'e 2008; Ma et al. 2016; Torrey et al. 2019; Langan et al. 2020; Ucci et al. 2021) and chemical evolution models (e.g., Dav'e et al. 2012; Lilly et al. 2013; Dayal & Ferrara 2018) to gain insights into galaxy formation and evolution. Moreover, the efficiency of ionizing photon production, as quantified by ξ ion (the production rate of hydrogen ionizing photons per unit luminosity in the UV-continuum), can be used to study the nature of the ionizing spectrum. The Hydrogen Balmer emission via recombination physics provides the best constraints on ξ ion , while high ionization lines such as He ii offer key spectroscopic tools to investigate the hardness of the ionizing spectrum. These approaches are crucial for understanding the stellar population of early galaxies, diagnosing the presence of accreting black holes in the system, and detecting galaxies hosting the very first generation of stars in the early universe (e.g., Schaerer 2003; Inoue 2011; Kewley et al. 2013; Bouwens et al. 2016; Nakajima & Maiolino 2022; Katz et al. 2022; Trussler et al. 2022). \nDeterminations of these properties ideally require key optical emission lines such as [O iii ] λλ 5007 , 4959, [O iii ] λ 4363, [O ii ] λλ 3726 , 3729 1 , and the Balmer lines. In particular, [O iii ] λ 4363 is crucial for gas-phase metallicity studies based on electron temperature ( T e ). However, these lines are limited in their applicability to galaxies up to z /lessorsimilar 3 in the pre-JWST era, with the detection of [O iii ] λ 4363 from sources at z /greaterorsimilar 1 still challenging due to faintness (e.g., Christensen et al. 2012; Jones et al. 2015; Sanders et al. 2020, 2021). These lines are beyond the reach of ground-based telescopes at higher redshift. Now JWST/NIRSpec has become online to directly examine these key emission lines in detail for high redshift sources. \nUsing the Early Release Observations (ERO) of NIRSpec within one month after the data release, several studies report a detection of [O iii ] λ 4363 from three objects at z = 7 . 6 -8 . 5 which are magnified by the strong lensing galaxy cluster SMACS J0723.3-7327 (Schaerer et al. 2022; Curti et al. 2023a; Trump et al. 2022; Rhoads et al. 2023; Arellano-C'ordova et al. 2022; Brinchmann 2022). These objects have provided the first reliable metallicity determinations at high redshift based on the direct T e measurement. The metallicity values from the different teams are overall consistent to each other for each source, resulting in a mass- \ntallicity relation which is scattered around the extrapolation of the z /similarequal 2 -3 relation towards the lower mass end ( M /star = 10 7 . 5 -10 9 M /circledot ). Curti et al. (2023a) also examine the mass-metallicity-star formation rate (SFR) relation as indicated to be a fundamental relation from z = 0 to z /lessorsimilar 2 -3 (Mannucci et al. 2010; Maiolino & Mannucci 2019; Sanders et al. 2021, and the references therein), suggesting that no clear evolution is seen from z = 0 -3 to z = 7 . 6 -8 . 5, particularly excepting for the z = 8 . 5 object (see below). However, the sample size is obviously too small to conclude the evolution on the mass-metallicity relation and its SFR dependence. \nMoreover, several pieces of evidence in the previous ERO studies imply that the early release of the NIRSpec products contain some issues. At the early stage of the ERO data release, the reductions including the flux calibration are partly based on the predicted preflight data such as the throughputs of the spectrograph, the optical telescope element, and the NIRSpec fore optics. A sensitivity function, whose shape is strongly dependent on the wavelength, is necessary for appropriate flux measurements as adopted in Curti et al. (2023a) using standard stars taken during the commissioning. Even after the flux-calibration, there remains some tensions for faint emission lines. A notable issue is seen in the Balmer decrements such as H γ /H β and H δ /H β which are (not always but sometimes) unexplainable in a consistent way within the uncertainties according to the Case B recombination. Such tensions, especially seen in the faint emission lines, may be caused by background residuals and/or hot pixels that persist in the released final spectra. Interestingly, the z = 8 . 5 object (ID:04590) is suggested to present a high [O iii ] λλ 4363 / 5007 line ratio, and accordingly a very high T e /greaterorsimilar 25000K, which is hard to be explained by heating of young massive stars alone (Katz et al. 2023). The low metallicity indicates it falls significantly below the mass-metallicity-SFR relation (Curti et al. 2023a) and can be in an early stage of galaxy evolution. Such a high electron temperature and low metallicity need to be confirmed by a more carefully reduced spectrum free from any tensions. The NIRSpec data reduction procedure will be revisited as one of the key parts of this paper for reliable emission-line flux measurements and errors. \nDespite these possible uncertainties, recent NIRSpec observations are now providing increasing numbers of spectra from high-redshift objects, and the reports of the identifications of strong emission lines such as [O iii ] λλ 5007 , 4959+H β from objects at z > 7 -8 follows. Notable results has come from the Early Release Science (ERS) observations of GLASS (Treu et al. \n2022; Morishita et al. 2022; Mascia et al. 2023) and CEERS (Finkelstein et al. 2022; Bagley et al. 2022; Sanders et al. 2023; Tang et al. 2023; Fujimoto et al. 2023; Shapley et al. 2023), two Director's Discretionary Time (DDT) programs (Williams et al. 2022; Heintz et al. 2022; Wang et al. 2022; Langeroodi et al. 2022), and the Guaranteed Time Observations of JADES (Robertson et al. 2022; Curtis-Lake et al. 2022; Bunker et al. 2023; Cameron et al. 2023b; Saxena et al. 2023; Curti et al. 2023b). Regarding the ISM properties, Sanders et al. (2023) examine the ionization properties of ∼ 160 galaxies at z = 2 -9 from CEERS by using high-to-low ionization emission line ratios such as [O iii ] λ 5007 / [O ii ] λ 3727, and suggest that galaxies tend to present hard ionizing spectra at high-redshift. The metallicities at z > 6 . 5 are suggested to be sub-solar on average (see also Tang et al. 2023; Shapley et al. 2023). A similarly high ionization state and modest metallicity is also suggested in 29 galaxies at z = 4 . 8 -8 from GLASS (Mascia et al. 2023) and in 26 galaxies at z = 5 . 5 -9 . 5 from JADES (Cameron et al. 2023b, see also Saxena et al. 2023). Fujimoto et al. (2023) explicitly derive metallicities of ∼ 10 z = 8 -9 CEERS galaxies using a metallicity indicator of [O iii ] λ 5007/H β , suggesting that these high-redshift galaxies have a lower metallicity than z = 0 -3 galaxies for a given stellar mass. More extremely, Williams et al. (2022) report a z = 9 . 5 galaxy with [O iii ]+H β , revealing a very high [O iii ] / [O ii ] ratio and a modestly low-metallicity, 12 + log(O / H) = 7 . 48 ± 0 . 08, falling below the z = 0 mass-metallicity relation but still consistent within 2 σ . Including the z = 9 . 5 object, Heintz et al. (2022) analyzed five objects at z > 7 . 8 from the DDT programs and suggest a systematic decrease of metallicity at z > 7 . 8 on the mass-metallicity-SFR relation found in lowerredshift, as seen in the ERO z = 8 . 5 object. A similar conclusion is also derived by Langeroodi et al. (2022). Boyett et al. (2023) report another z = 9 galaxy from GLASS whose [Ne iii ] λ 3869/[O ii ] ratio implies a subsolar metallicity. Finally, Bunker et al. (2023) report the astonishing spectrum of GN-z11 at z = 10 . 6 using the deep JADES observations. The author use several metal lines such as [Ne iii ], [O ii ], as well as the UV lines of N iv ] λ 1486, N iii ] λ 1748, and C iii ] λ 1909 and imply an unusually high nitrogen-to-oxygen abundance ratio for its modest ( ∼ sub-solar) oxygen abundance (see also Cameron et al. 2023a). We note that these metallicities are based on the strong emission line ratios which are empirical indicators and calibrated mostly in the local universe (e.g., Maiolino & Mannucci 2019). The applicability of these methods at high-redshift need to be carefully confirmed (e.g., Curti et al. 2023a), partic- \nindication of different degrees of ionization in the ISM of galaxies between z = 0 and highredshift for a fixed metallicity (Sanders et al. 2023). \nIn this paper, we provide a summary of three major public NIRSpec observation programs of ERO, GLASS, and CEERS, aimed at characterizing the massmetallicity relation at z = 4 -10 and examining its evolution over cosmic time. We establish a reliable data reduction procedure for emission-line flux measurements and errors using NIRSpec data, which is applied to construct a large sample of galaxies at z = 4 -10 as detailed in Section 2. In Section 3, we utilize key emission line ratios to derive metallicities for the JWST objects with improved NIRSpec spectra. We then investigate the mass-metallicity relation and its dependence on SFR at high redshift. We begin with the 10 objects in which [O iii ] λ 4363 is identified and metallicity is determined using the direct T e method. The empirical metallicity indicators validated with these 10 objects are then applied to estimate metallicities for the remaining JWST objects. We analyze the mass-metallicity and mass-metallicity-SFR relations at z = 4 -10 using the metallicity measurements obtained for the JWST objects. The results and implications of these metallicity relationships are discussed in Section 4, and we provide a summary of our conclusions. Throughout the paper we adopt a standard ΛCDM cosmology with Ω Λ = 0 . 7, Ω m = 0 . 3, and H 0 = 70kms -1 Mpc -1 .", '2. DATA': 'We assemble the publicly available NIRSpec data of ERO and ERS, re-reduce and analyze the spectra, and identify high-redshift galaxies with the rest-frame optical emission lines to discuss their nebular properties. In the following subsections, we detail our improved reduction procedures, particularly for the ERO data to demonstrate the improvements by comparing with the early results.', '2.1. ERO Data': "2.1.1. NIRSpec observations and Data reduction \nThe JWST/NIRSpec ERO observations were undertaken on UT 2022 June 30, targeting the SMACS 0723 lensing cluster field (Pontoppidan et al. 2022; Proposal ID: 2736). Multi-slit spectroscopy was taken using the micro-shutter assembly (MSA), with the medium resolution ( R ∼ 1000) gratings/filters of G235M/F170LP and G395M/F290LP sampling the wavelength range of 1 . 7 -3 . 1 and 2 . 9 -5 . 1 µ m, respectively. Two observing blocks were carried out, o007 and o008, each of which consisted of three exposures of 2918 sec in G395M and F235M each, with a 3-shutter slitlet nod pattern, i.e., \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 1. Observed NIRSpec spectra for the five z = 5 -8 . 5 EROobjects. For each object: (top) 2D spectra for the six individual exposures (three nods each in o007 and o008). (bottom) 1D composite spectrum. We generate the 1D composite spectrum by first median-stacking the individual 2D spectra and then extracting the 1D spectrum with a small aperture (see text). This procedure allows us to minimize the possible impacts of hot pixels and to rescue some individual 2D frames where the spectrum is partly out of the slit (e.g., two nod2 frames of 10612). \n<!-- image --> \nslightly nodded positions, nod1, nod2, and nod3, in different shutters within the slitlets. The total integration time was 4 . 86hr in G235M and G395M each. \nBy visually checking all of the 1D and 2D composite spectra that were publicly available, we identified 5 objects whose [O iii ] λλ 5007 , 4959+H β were clearly detected and that were useful for the investigation of the nebular properties at high-redshift z > 5, with IDs 04590, 05144, 06322, 08140, and 10612. Although there were additional two sources in the ERO sample that were tentatively reported at z = 5 -6 (Brinchmann 2022; Mahler et al. 2022), they were not included in the following analysis because of their lines' poorer confidence levels. \nFor the five sources, data reduction was re-performed for a more careful analysis and flux calibration. Starting with the Level1 products provided by the observatory, we carried out the Spec2 and Spec3 pipelines per nod and per observing block using the Python library for JWST observations by STScI (ver.1.8.5). We used the latest reference files stored in a pmap file of either 1028 or 1027, where notably the flat files were created from in-flight data for all MOS data taken after the launch, resolving the MOS flux calibration issues reported in the early report of the NIRSpec ERO data (see below for more details about the flux calibration). According to the original reduction procedure for NIRSpec MSA data provided by the CEERS team 2 , during the Spec2 pipeline, a single science image for a particular nodding position was background-subtracted by using the other nods' images. However, we realized some hot pixels including uncleaned cosmic rays in one of the background images fell on the spatial position of the science objects and badly influenced the final 1D spectrum. To avoid such additional noises, we carefully checked the background-subtracted science images and removed one of the other nods' images, if necessary, for the background images and re-ran the Spec2 pipeline in an iterative manner. Furthermore, we subtracted the local residual background at each spectral pixel for each science image by masking the spatial position of the object and taking the median value of the pixels in the range [ -10pix:+10pix] along the spectral direction 3 . We performed the Spec2 pipeline assuming that all of the five sources were point sources, as they all looked compact in the NIRCam images with the half- \nlight radii smaller than the PSF (e.g., Harikane et al. 2022a; Tacchella et al. 2022; see also Ono et al. 2022 for the size evolution at high-redshift). This assumption affected how the pathloss corrections, which accounted for various types of signal loss in NIRSpec data, were calculated and applied. The correction depended on the position of the source within the configured MSA shutter, the aperture used for the observations, the wavelength, as well as the type of the object. We used the original MSA metadata files to refer to the relative placement of the sources within the MSA shutters assuming the pointing uncertainties during the observations were negligible, but manually changed the values in the column STELLARITY to force the objects to be treated as point sources. Figure 1 summarizes the six 2D outcomes (3 nods for 2 observing blocks) for each of the ERO sources. \nChecking the six single exposures in both 1D and 2D for each of the sources, we confirmed no signal was detected in the nod2/o007 observing block for 04590, as already pointed out by the earlier studies (e.g., Curti et al. 2023a; Arellano-C'ordova et al. 2022). This exposure obviously needed to be removed from the final composition. Moreover, we noticed the spectra of nod2/o008 of 05144 were badly affected especially around the H γ +[O iii ] λ 4363 wavelength regime. To be conservative, we decided to remove the visit for 05144 to make a composite. In short, the integration time we used for the composite spectra of 04590 and 05144 was 4 . 05hr, and 4 . 86hr for 06355, 08140, and 10612. \nThe composition was done by median-stacking the available 5-6 2D spectra. The individual 2D spectra were residual background-subtracted, and shifted to have common spatial and spectral ranges for the composition. All the shifts were integer movements. We adopted the 2D median-stacking procedure to further alleviate the possible impacts of hot pixels that persisted in the final spectra, which were produced by the standard procedure by the Spec3 pipeline. Moreover, the 2D median-stacking method allowed us to rescue the frames where the spectrum was partly out of the slit. Examples were found in the two nod2 positions for 10612 whose losses of the fluxes were > 3 σ level if extracted in individual frames and compared with the other nods' 1D. Our method treated the outer regions of the slit as NaN and calculated the median value without NaN at each pixel in 2D. Using the 2D composite, one-dimensional spectrum was produced for each source and grating via the summation of 3 pixels (= 0 . '' 3 /similarequal 2 × FWHM) along the spatial direction centered on the spatial peak position. We adopted the narrow extraction aperture as compared to the previous studies to minimize the effects coming from the noisy regions close to the edge, partic- \nfor 04590, 08140, and 10612 that were observed with a short MSA slitlet. \nThe flux calibration was one key procedure to be improved for the early NIRSpec studies, as some of the reference files needed for the successful flux calibration, such as the flat frames for the spectrograph and the fore optics, were based on the predicted pre-flight throughputs. As of the pmap file of 1022, these flat files have been updated and created from in-flight data for all MOS data. The MOS flux calibration accuracy is now estimated at approximately 5 percent or less over the full wavelength range 4 . We therefore used the flux solutions provided by the spec2pipeline together with the latest reference files. Finally, we scaled the spectrum to match the broadband photometry of JWST/NIRCam to combine the spectroscopic and photometric measures based on a common aperture to derive the quantities such as equivalent widths of the optical emission lines and ξ ion . We used the NIRCam broadband that covered the [O iii ]+H β emission for the scaling. The final composite 1D spectrum is presented in the bottom panel for each of the sources in Figure 1. \nTo evaluate the noise level, we used the 2D noise frames of read-out noise ( VAR\\_RNOISE ) and Poisson noise ( VAR\\_POISSON ) outputted in the Spec2 procedure for each object and nod. In addition, we added the standard deviation of the residual background at each spectral bin in quadrature. We then averaged the noise frames of the available nods and extracted the 1D noise spectra in quadrature by using the same spectral and spatial ranges as adopted for the science spectra. Finally the scaling was performed in the same manner as done for the science spectra.", '2.1.2. Emission line flux measurements': "We measured the key optical emission lines of hydrogen Balmer lines, [O ii ] λ 3727, [Ne iii ] λ 3869, [O iii ] λ 4363, and [O iii ] λλ 5007 , 4959 by performing a Gaussian profile fitting to each line, and used the noise spectrum to weight the fit. In the fitting procedure for the relatively faint objects of 04590, 05144, and 08140, the redshift of the [O iii ] λ 5007 (i.e., the strongest emission line) was adopted as Gaussians for the other lines. We perform all wavelength measurements on the vacuum wavelength scale, although we follow a longstanding convention of specifying the emission lines by their wavelength in air. The error of each flux measurement was calculated by adding the noise levels of the spectral bins in quadrature within the wavelength range of ± FWHM centered on the Gaussian peak. Using the flux measure- \nFigure 2. Balmer emission line ratios of H γ /H β and H δ /H β for the ERO objects (04590 in red, 05144 in emerald green, 06355 in orange, and 10612 in green) in the main panel. The colored filled circles represent the new emission line flux measurements and errors with our improved reduction, while the other symbols show the measurements based on the early data release, as shown in the legend (Curti et al. 2023a; Rhoads et al. 2023; Arellano-C'ordova et al. 2022). For the measurements of Arellano-C'ordova et al. (2022), we adopt the average values based on the two spectra (o007 and o008) for 06355 and 10612, but refer only to the o008 value for 04590. For those with line ratios outside the ranges of the plot, we place them close to the edge of the plot with an arrow to indicate the direction toward the actually reported line ratio. The lines show expected Balmer decrements as a function of E(B -V) for the three different attenuation curves (Cardelli et al. 1989 with dashed, Calzetti et al. 2000 with dotted, and SMC (Gordon et al. 2003) with dot-dashed) and the two different electron temperatures ( T e = 10000 K in gray and 25000 K in magenta), although the different attenuation curve and T e have a small impact on this diagram. The deviation of the H δ /H β ratio from the theoretically expected value based on the H γ /H β ratio (∆ H δ /H β = ((H δ /H β )obs -(H δ /H β )expected) normalized by (H δ /H β ) expected ) is highlighted in the upper panel, and that of the H γ /H β ratio based on the H δ /H β ratio in the left panel, for each symbol as in the main panel. The corresponding E(B -V) values are printed in these sub-panels, where the Calzetti et al. 2000 attenuation curve and T e = 17500 K are adopted for reference. Our measurements all follow the sequence of the expected Balmer decrements within the uncertainties, while quite a few of the measurements from the earlier studies show the ratios that are not physically and/or consistently explainable. \n<!-- image --> \nnd the error evaluations, we confirm that four of the ERO objects, 04590, 05144, 06355, and 10612 present a > 3 σ detection of [O iii ] λ 4363. We also derived equivalent widths of the optical emission lines by combining the fluxes with the continuum level provided from a best-fit SED to the appropriate rest-frame optical broad-band photometry (Section 2.1.3). We summarize the key emission line measurements in Table 1 for the [O iii ] λ 4363-detected sources to discuss the metallicity indicators (Section 3.2). For the full sample, we list the metallicity diagnostic emission line ratios in Table D1 in Appendix D together with the other physical quantities. \nTo test our new flux measurements with the improved reduction and calibration of the NIRSpec spectra, we show in Figure 2 the Balmer decrements of H γ /H β and H δ /H β for the four ERO objects with a significant detection of the three lines, and compare them with the prediction of Case B recombination. Observed Balmer line ratios should follow the sequence of the expected Balmer decrements as a function of dust attenuation, as shown with the curves adopting different reddening laws (Cardelli et al. 1989; Calzetti et al. 2000; Gordon et al. 2003) and electron temperatures ( T e = 10000 and 25000K). The four ERO objects all follow the sequence and present physically-reasonable Balmer decrements within the uncertainties. The other object (08140) is not detected with H γ and H δ but the non-detections as well as the H α /H β line ratio are consistent with the theoretically expected line ratios at the /lessorsimilar 3 σ level. In contrast, previous studies claim some tensions of the observed Balmer decrements based on the early release products which show inconsistent with the expected line ratios, as referenced with the open and the cross symbols in Figure 2. We speculate such inconsistencies have been improved by implementing the residual background subtractions and the 2-dimensional stacking procedure, where we can ease the potential impacts of hot pixels that could have persisted in the final spectra in the early release. We note that the error-bars of our measurements are slightly larger than those in the previous studies (Curti et al. 2023a; Arellano-C'ordova et al. 2022), mostly due to the addition of the uncertainty of the subtracted residual background into the noise levels. \nPrior to quantitative analysis, it is necessary to consider corrections for dust reddening. We use the observed Balmer decrement of H γ /H β for the four objects to estimate the dust attenuation assuming the Calzetti et al. (2000) attenuation curve and a Case B recombination with T e = 17500K. Adding a lower signalto-noise ratio emission of H δ does not change the estimate as indicated in Figure 2. For 08140, we use the H α /H β ratio instead. We adopt E(B -V) = 0 . 22 for \n04590, = 0 . 03 for 05144, = 0 . 08 for 10612, and = 0 . 00 for 08140 and 06355, and do not propagate the uncertainty in E(B -V) in the following analysis.", '2.1.3. NIRCam photometry and SED fit': "All five ERO objects have been observed with JWST/NIRCam in the F090W, F150W, F200W, F277W, F356W, and F444W filters. These photometric data are crucial for deriving stellar properties such as stellar mass, and for evaluating the continuum level to correct for slit-loss in the NIRSpec observations. The derived corrections are important for obtaining key properties, such as the total SFR via recombination physics, ξ ion , and EWs of optical emission lines such as EW(H β ). \nWe refer to the catalog provided by Harikane et al. (2022a) to extract the necessary photometric data. We adopt the total magnitudes that are estimated from the 0 . '' 3-diameter aperture magnitudes with the aperture corrections. The uncertainties of the magnitudes include the photometric errors from the images as well as the 10 % error floor to account for systematic uncertainties arising from e.g., the zero-point corrections. \nUsing the total magnitudes and the errors of NIRCam, we perform the Bayesian spectral energy distribution (SED) fitting with Prospector (Johnson et al. 2021) to derive the stellar population properties. The procedure of the SED fitting is the same as that of Harikane et al. (2022a), except for the fixed redshifts based on the NIRSpec emission line measurements. We use the stellar population synthesis package, Flexible Stellar Population Synthesis (FSPS; Conroy et al. 2009; Conroy & Gunn 2010) for stellar SEDs, and include nebular emission from the photoionization models of Cloudy (Byler et al. 2017). We assume the stellar initial mass function (IMF) of Chabrier (2003), the intergalactic medium (IGM) attenuation model of Madau (1995), the Calzetti et al. (2000) dust attenuation law, and a fixed metallicity of 0 . 2Z /circledot . We choose a flexible star formation history as adopted in Harikane et al. (2022a). \nThe stellar masses we obtain from the SED fitting are the apparent values, i.e., after experiencing the strong lensing magnifications for the ERO objects. To correct for the lensing effects, we refer to the mass model of SMACS J0723 constructed with glafic (Oguri 2010, 2021) updated with the new JWST ERO data, as adopted in Harikane et al. (2022a). The adopted magnification factors as well as the stellar masses after the lensing magnification correction are summarized in Table D1. The errors of the best-fit parameters from the SED fitting correspond to the 1 σ confidence interval (∆ χ 2 < 1) for each parameters. \n2.2. ERS GLASS Data \nFigure 3. Redshift distribution of the high-redshift galaxy samples used in this work. The red histogram presents the JWST sample constructed in this study using our analyzed ERO, GLASS, and CEERS spectra. The blue histogram shows the samples compiled from the literature (see Section 3.2.3). The sum of these samples, represented by the gray histogram, is used to investigate the mass-metallicity relationships at z = 4 -10 in this work. \n<!-- image --> \nThe ERS GLASS NIRSpec MSA observations were carried out on UT 2022 Nov 10-11 for objects behind the galaxy cluster Abell 2744 (Proposal ID: 1324; Treu et al. 2022). They adopted the R ∼ 2700 high resolution gratings/filters of G140H/F100LP, G235H/F170LP, and G395H/F290LP sampling the wavelength range of 1 . 0 -1 . 9, 1 . 7 -3 . 2, and 2 . 8 -5 . 3 µ m, respectively. Twelve exposures of 1473 sec were conducted in one pointing of NIRSpec MSA for each spectral configuration with the 3-shutter slitlet nod pattern, totaling the on-source integration of 4 . 9hr in each of G140H, G235H, and G395H. \nThrough the initial screening of all of the public 1D and 2D composite spectra, we identified 15 objects at z > 4 with [O iii ]+H β 5 . The sample contains the members of the protocluster at z = 7 . 89 as presented by Morishita et al. (2022). We re-reduced the spectra of the 15 objects with our custom procedure as described in Section 2.1.1. We carefully checked all of the 12 nods' spectra for each objects to determine the extraction aperture and any nods to be removed from the com- \nosition. For 10005, the light from another member of the z = 7 . 89 protocluster partly came into the same slit at a spatially-offsetted position. We changed the background frames during the spec2 procedure so that the signal from 10005 was not over-subtracted by the neighboring object in different nods. \nThe emission line flux measurements were performed in the same manner as explained in Section 2.1.2. Among the 15 objects, we identified 5 with [O iii ] λ 4363. One of the [O iii ] λ 4363-detected source, 40066, lacked the [O ii ] λ 3727 emission which fell in the detector gap. The other 4 objects, whose flux measurements are given in Table 1, are thus used for the metallicity measurements with the direct T e method (Section 3.2). The nebular dust attenuation was evaluated using H α , H β , and/or H γ . \nThe NIRCam multi-band photometric catalog was generated in the same manner as in Section 2.1.3 based on the images publicly available from the GLASS and UNCOVER (Proposal ID: 2561) programs. We use the data of F115W, F150W, F200W, F277W, F356W, and F444W for the subsequent SED fitting analysis. Note that one of 15 objects (160122) is not covered by NIRCam, and its stellar mass are not determined. \nAll of the GLASS objects are strongly lensed. To correct for the magnification factors, we use the glafic (Oguri 2010, 2021) mass model of Kawamata et al. (2018) with a minor update based on the Multi Unit Spectroscopic Explorer (MUSE) follow-up spectroscopy (Bergamini et al. 2022). The updated mass model uses 45 multiple image systems, 27 of which have spectroscopic redshifts. This model reproduces the observed multiple image positions with an accuracy of 0 . '' 42. The derived magnification factors, the stellar masses after the lensing magnification correction, as well as the other key quantities for the entire GLASS sample are summarized in Table D1 in Appendix D.", '2.3. ERS CEERS Data': "The ERS CEERS NIRSpec MSA observations were carried out on UT 2022 Dec 20-22, and 24 in the blank field of EGS (Finkelstein et al. 2022; Fujimoto et al. 2023; Proposal ID: 1345). They used both of the Prism ( R ∼ 100) and the medium resolution ( R ∼ 1000) gratings to cover from 1 . 0 to 5 µ min 6 pointings (P4, P5, P7, P8, P9, and P10) with some overlaps of objects. Note that two of the Prism pointings, P9 and P10, were not fully completed. To compensate the two Prism pointings, additional observations were performed on UT 2023 Feb 9-10 (P11 and P12; see also Arrabal Haro et al. 2023). These additional data were also analyzed and presented in this paper to provide a complete spectro- \n<!-- image --> \nFigure 4. O32 vs. R23 diagram comparing the JWST sample at z = 4 -9, analyzed in this work, with lower redshift samples at z = 2 -3. The JWST objects are shown as red filled symbols, with circles representing ERO, pentagons representing GLASS, and diamonds representing CEERS. Objects with metallicity measurements obtained using the direct T e method are indicated with red open circles. The gray open symbols represent the averages of lower-redshift samples of continuum-selected galaxies (Nakajima & Ouchi 2014; Troncoso et al. 2014; Sanders et al. 2016; Onodera et al. 2016; Strom et al. 2017) and Ly α emitting galaxies (LAEs; Nakajima & Ouchi 2014; Erb et al. 2016; Nakajima et al. 2020), as indicated in the legend. Arrows indicate 3 σ lower limits. Gray shading illustrates the equivalent distribution for nearby SDSS galaxies. \n<!-- image --> \nc sample of CEERS to the community. Three exposures of 1036 sec were conducted for each spectral configuration with the 3-shutter slitlet nod pattern, totaling the on-source integration of 0 . 9hr in each Prism, G140M, G235M, and G395M, excepting for P9 and P11 where another set of three exposures of 1036sec were repeated in the Prism observations. \nAt redshifts z > 4, we detected a total of 147 Prism spectra and 77 medium grating spectra with [O iii ]+H β emission lines. Among these, three objects (IDs 00618, 00717, and 01027) were observed with two different Prism and medium grating pointings each, resulting in four duplicate observations. Additionally, one object (ID 00603) was observed with two different Prism pointings and one medium grating pointing, resulting in three duplicate observations. Furthermore, 50 objects were observed with both Prism and medium grating, resulting in two duplicate observations with one pointing each. In summary, a total of 163 objects were identified through the ERS CEERS NIRSpec observations, as summarized in Table D1 in Appendix D. For the 163 objects, we performed our re-reduction procedure as described in Section 2.1.1. \nThe same emission line flux measurements were performed as detailed in Section 2.1.2. We identified two out of the 163 objects with [O iii ] λ 4363 among the \nCEERS sample. Their flux measurements are added in Table 1. The nebular dust attenuation was evaluated using H α , H β , and/or H γ depending on the detections and the redshift. \nThe CEERS NIRCam observations consists of 10 pointings, four of which were taken in Jun 2022 and the rest in Dec 2022 (Bagley et al. 2022). The early observations of 4 pointings are presented in Harikane et al. (2022a). We reduce the latter observations' data in the same manner, and generate the NIRCam full photometry catalog, including the seven bands of F115W, F150W, F200W, F277W, F356W, F410M, and F444W (Y. Harikane et al. in prep.). One caveat is that not all of the NIRSpec CEERS objects are covered by NIRCam. We find 91 of the 163 CEERS objects are in the NIRCam coverage. For the 91 objects, we perform the SED fitting as in Section 2.1.3 to derive the stellar masses, and scale the NIRSpec spectra to be consistent with the NIRCam total magnitude photometry (i.e., slit-loss correction). For the others except for a single object (ID 01115), we rely on the Hubble Space Telescope (HST) WFC3 photometry (Stefanon et al. 2017), as detailed in Isobe et al. (2023). Briefly, we derive the UV absolute magnitudes ( M UV ) from the HST/WFC3 photometry, use the empirical relationships between stellar mass and M UV at z = 4 -8 (Song et al. 2016) to estimate the \nstellar masses, and rescale them to the Chabrier (2003) IMF. We exclude the single object with ID 01115 from the following analysis as its HST photometry is not given. For those not detected in the HST bands, we translate the corresponding 5 σ limiting magnitudes to the 5 σ upper-limits of stellar mass. While acknowledging that the empirical method utilizing M UV for estimating stellar mass may be less reliable compared to the method involving an SED fit to NIRCam photometry, we demonstrate in the subsequent sections (Sections 3.3 and 3.4) that our main findings remain unchanged even when excluding objects without NIRCam photometry. This is partly because the fraction of such objects is small ( ∼ 28%) in deriving the average relationships. Furthermore, we have found reasonable consistency in stellar mass estimates between the two methods by using objects with both NIRCam and HST photometry. Details are described in Appendix A. Note that we cannot correct for the slit-loss of the NIRSpec spectra for the objects without NIRCam photometry. This will not affect the metallicity measurements as we only use the line ratios. However, that does prevent us from deriving the equivalent widths of the optical emission lines such as EW(H β ), and the SFR based on the total H β luminosity. \nIn summary, we construct a JWST sample of 182 objects at z = 3 . 8 -8 . 9. The redshift distribution of the sample is shown in red in Figure 3.", '3.1. Emission line ratios': "To examine the nebular properties of JWST objects at z = 4 -9 and compare with those of lower-redshift galaxies, we plot them in the line ratio diagnostic diagram of ([O iii ] λλ 5007 , 4959+[O ii ] λ 3727) / H β (R23) and [O iii ] λ 5007 / [O ii ] λ 3727 (O32) in Figure 4 (see also Mascia et al. 2023; Tang et al. 2023; Sanders et al. 2023). This diagram has been widely used to investigate the metallicity and ionization state in the local universe and up to z = 2 -3 (e.g., Kewley & Dopita 2002; Maiolino et al. 2008; Nakajima & Ouchi 2014; Troncoso et al. 2014; Sanders et al. 2016; Onodera et al. 2016; Strom et al. 2017; Nakajima et al. 2020). When compared to the local Sloan Digital Sky Survey (SDSS) galaxies, which are free from AGN contamination as determined by the BPT diagram (e.g., Baldwin et al. 1981; Kauffmann et al. 2003; see below for more details), we observe that the z = 2 -3 galaxies exhibit a higher O32 line ratio, indicating an overall increase in ionization parameter at higher redshifts. Figure 4 confirms the trend, and the z = 4 -9 JWST objects show a fur- \nFigure 5. The [N ii ] BPT diagram. Red symbols represent the JWST objects, as shown in Figure 4, whose H α +[N ii ] lines are covered in the NIRSpec spectra. Arrows indicate 3 σ limits. Two popular demarcation curves between AGNs and star-forming galaxies, from Kewley et al. (2001) (dotlong dashed; 'Kew01') and Kauffmann et al. (2003) (short dash-long dashed; 'Kau03'), are also shown. None of the objects fall above the demarcation curves significantly beyond the measurement uncertainties. \n<!-- image --> \nigh O32 line ratio on average than typically seen in z = 2 -3 continuum-selected galaxies. Remarkably, one of the ERO objects, ERO 04590 at z = 8 . 5 presents a stringent lower-limit of O32 as > 14 . 8 (3 σ ), which corresponds to an ionization parameter of log U > -2 . 21 according to the prescription of Kewley & Dopita (2002) (see also Fujimoto et al. 2022 6 ). The lower-limit is a factor of ∼ 10 higher value than typically seen in the SDSS galaxies. \nAt z = 2 -3, low-mass Ly α emitting galaxies (LAEs) are suggested to have a remarkably high O32 line ratio, which would be achieved by a hard ionizing radiation field and/or a low metallicity (Nakajima et al. 2016; Trainor et al. 2016; Erb et al. 2016). Its implication regarding the escape of ionizing photons (and that of Ly α photons) are also discussed (e.g., Nakajima & Ouchi 2014; Izotov et al. 2016, 2018; Verhamme et al. 2017; Steidel et al. 2018; Fletcher et al. 2019; Erb et al. 2019; Nakajima et al. 2020; Katz et al. 2020; Flury et al. 2022). The extreme O32 line ratios seen in the z = 4 -9 JWST objects are comparable to those in z = 2 -3 lowmass LAEs and z ∼ 0 Lyman-continuum (LyC) leaking objects, suggesting that high-redshift sources have a \nnebular and ionization condition of gas that is similar to those typically found in lower-redshift low-mass galaxies having a strong Ly α and/or LyC escape with a hard ionizing spectrum. A similar finding is also discussed by Mascia et al. (2023) and Sanders et al. (2023), and notably by Tang et al. (2023), where the strong Ly α emitting galaxies at z > 7 are directly examined with JWST and suggested to present the largest O32 line ratios. \nOne uncertainty in interpreting O32 arises as it also depends on the metallicity. In the next section, we present our method to derive the metallicities for the JWST objects using several indicators including R23, particularly by referring to the objects whose [O iii ] λ 4363 is detected and metallicity is precisely determined with electron temperature. \nBefore moving to the metallicity results, we also examine our objects using another popular emission line diagram, the [N ii ] BPT diagram, which plots the ratios of [O iii ] λ 5007 / H β and [N ii ] λ 6584 / H α (Baldwin et al. 1981). This diagram is commonly used to diagnose the presence of an active galactic nucleus (AGN) and is applicable to our JWST objects up to a redshift of z < 6 . 9, where the H α +[N ii ] emission is covered by NIRSpec spectra. Figure 5 shows the [N ii ] BPT diagram for our JWST objects at z < 6 . 9, along with two curves that discriminate between sources dominated by AGNs and stars (Kewley et al. 2001; Kauffmann et al. 2003). Sources located above the curves are classified as AGNdominated. \nFigure 5 shows that most JWST sources have an upper limit of [N ii ]. The [N ii ]-detected objects fall either below or on the demarcation curves within the measurement uncertainties. None of the JWST objects are thus classified as obvious AGNs. However, it is worth noting that low-metallicity AGNs with Z /lessorsimilar 0 . 5Z /circledot may contaminate the star-forming galaxy region on the diagram (e.g., Kewley et al. 2013; Nakajima & Maiolino 2022). We cannot fully rule out the presence of such low-metallicity AGNs in our sample based on the [N ii ] BPT result, but we suggest an absence of evolved AGNs as far as we explore ( z < 6 . 9). Accordingly, we do not remove any objects from the sample for the following results based on the [N ii ] BPT diagram. Although there are some objects whose [N ii ] upper-limits are too weak to conclude their BPT diagnostics, we mention in the following sections (Sections 3.3 and 3.4) that our main results are not changed by excluding these unclear objects. \nAs a complementary approach to our method using the [N ii ] BPT diagram, a companion study by Harikane et al. (2023) conducts a search for faint AGNs \nFigure 6. Comparison of literature determinations of electron temperature ( T e ) for four ERO galaxies (04590 in red, 06355 in orange, 10612 in green, 05144 in emerald green) where [O iii ] λ 4363 is identified. Different symbols indicate different publications, as shown in the legend. For the measurements of Arellano-C'ordova et al. (2022), we show the average values based on the two spectra (i.e., two observing blocks: o007 and o008) for 06355 and 10612, but refer only to the o008 value for 04590. For each object, the symbols are slightly offset in the horizontal direction for display purposes. Some of the measurements in the literature do not present uncertainties and are plotted without error bars in the vertical direction. \n<!-- image --> \nin our full JWST spectroscopic sample by examining the broad component (FWHM > 1000kms -1 ) around the H α emission line. The authors identify 10 objects with signatures of AGNs at redshifts z = 4 . 0 -6 . 9 (see also Kocevski et al. 2023; Ubler et al. 2023). To avoid potential biases in our mass-metallicity relations, we remove these 10 objects from our sample when we examine the mass-metallicity relations. These objects are flagged in Table D1.", '3.2. Gas-phase Metallicity': "3.2.1. Direct T e method \nTen objects (4 from ERO, 4 from GLASS, and 2 from CEERS) present a significant detection of [O iii ] λ 4363 allowing us to reliably determine the oxygen abundance with the direct T e method at z = 4 . 0 -8 . 5 and use it as a proxy for gas-phase metallicity. We follow the procedure as summarized in Nakajima et al. (2022) to derive T e -based metallicities. Briefly, we first estimate the electron temperature of O 2+ zone ( T e ([O iii ])) using \nTable 1. Summary of JWST Objects with Direct T e Method \nNote -( /star ) Observed flux ratios relative to H β ( × 100). Upper-limit values at the 1 σ level. ( † ) The presence of high ionization line of [Ne iv ] is seen (Brinchmann 2022). ( ‡ ) The presence of broad H α is indicated (Harikane et al. 2023) \nthe reddening-corrected [O iii ] λλ 4363 / 5007 ratio and an assumed electron number density of 100cm -3 with the task getTemDen from PyNeb (Luridiana et al. 2015). Different densities such as 10 -1000cm -3 do not significantly change the results (see Isobe et al. 2023 for the typical electron density in high-redshift galaxies; see also Fujimoto et al. 2022). The temperature of O + zone, T e ([O ii ]), is extrapolated from T e ([O iii ]) employing the prescription of Izotov et al. (2006) (cf., Brinchmann 2022 for a different assumption). We derive the abundance of O 2+ / H + with the fluxes of [O iii ] λλ 4959 , 5007 to H β and T e ([O iii ]), and O + / H + with the [O ii ] to H β reddening-corrected ratio and T e ([O ii ]) using the PyNeb package getIonAbundance . We do not take into account a higher ionization abundance of O 3+ / H + , as we follow the approximation given by Izotov et al. (2006) and confirm no clear presence of the He ii λ 4686 emission line in the spectra. In the spectrum of ERO 04590, the [O ii ] doublet is not detected at the 3 σ level. We therefore count the O 2+ component alone for its oxygen abundance. We have checked that the 3 σ upper-limit of [O ii ] would increase the O/H value by 0 . 05dex, which \nis small enough with respect to the measurement error ( /similarequal 0 . 15dex). The measured oxygen abundances as well as T e ([O iii ]) are summarized in Table 1. \nFor the ERO objects, there exist several earlier studies that report the metallicities with the electron temperatures (Schaerer et al. 2022; Curti et al. 2023a; Trump et al. 2022; Rhoads et al. 2023; Arellano-C'ordova et al. 2022; Brinchmann 2022). It is practically useful to compare our measurements with them. Figure 6 compares the T e ([O iii ]) values based on our measurements with the earlier studies for the four [O iii ] λ 4363-detected ERO objects. Different colors correspond to different objects. In some previous studies, the z = 8 . 5 object (ERO 04590 ; red-marks in the plot) is claimed to present an exceptionally high electron temperature at T e > 2 . 5 × 10 4 K, which is not usually observed in the local universe and may require some additional explanations (e.g., Katz et al. 2023; Rhoads et al. 2023). Our re-measurement confirms a high, but not extremely high, electron temperature for ERO 04590, T e = (2 . 08 ± 0 . 26) × 10 4 K, which is explainable by heating of young massive stars without \nFigure 7. Comparison of metallicity relationships with strong line ratios (R23, R3, R2, O32, and Ne3O2 from top left to bottom right) for ERO (circles), GLASS (pentagons), and CEERS (diamonds) galaxies with metallicity measurements based on the direct T e method. Each data point is color-coded by its EW(H β ) value, if available, as shown in the legend. The black dashed, long-dashed, and dot-dashed curves represent the metallicity relationships from Maiolino et al. (2008), Curti et al. (2017, 2020), and Bian et al. (2018), respectively. The blue and orange curves show the functions derived in Nakajima et al. (2022) for lowmetallicity galaxies, with blue representing high-ionization galaxies (EW(H β ) > 200 ˚ A) and orange representing low-ionization galaxies (EW(H β ) < 100 ˚ A). Each curve is shown only within the metallicity range explored in the original paper. \n<!-- image --> \nFigure 8. Empirical metallicity indicator using both R23 and O32 as proposed by Izotov et al. (2019, 2021) in the low-metallicity regime (solid) and its extrapolation (dotted). The red symbols show the JWST objects as in Figure 7. \n<!-- image --> \nany special mechanisms. It is thus important to carefully reduce and combine the spectra, as well as to extract the 1D to reliably discuss the nebular properties with the measurements of faint emission lines. For the other objects, their electron temperatures are modest, T e = (1 . 1 -2 . 3) × 10 4 K, fairly consistent with the values in the earlier literature and similar to the electron temperatures seen in lower-redshift star-forming galaxies. We will also discuss comparisons of metallicity determinations later on the mass-metallicity relationship. \nThe four ERO objects as well as the 6 newly identified GLASS and CEERS objects with [O iii ] λ 4363 provide the opportunity to test the empirical metallicity indicators at such high-redshift of z > 4 (Curti et al. 2023a). In Figure 7, we examine the following five popular metallicity indicators: [O iii ] λ 5007 / H β (R3), [O ii ] λ 3727 / H β (R2), [Ne iii ] λ 3869 / [O ii ] λ 3727 (Ne3O2), as well as R23 and O32 by comparing with the empirical indicators pro- \ny Maiolino et al. (2008) (see also Nagao et al. 2006), Curti et al. (2017, 2020), Bian et al. (2018), and Nakajima et al. (2022). We draw each empirical relationship only in the calibrated metallicity range without showing any extrapolation. For the relationships of Nakajima et al. (2022), two curves are depicted in each panel, showing the dependence of EW(H β ) on the indicators. The authors use EW(H β ) as a proxy to correct for the degree of ionization state of the gas in each galaxy, because EW(H β ) is sensitive to the current massive-star formation efficiency and known to be well-correlated with the ionization state as probed by e.g., O32 (e.g., Nakajima & Ouchi 2014; Mingozzi et al. 2020; Nakajima et al. 2022). To examine the dependencies of EW(H β ) on the indicators, the JWST objects are color-coded by EW(H β ) except for CEERS 01536, whose EW(H β ) is not derived due to the lack of NIRCam images (Section 2.3). \nThe JWST objects tend to present high ionization emission lines such as [O iii ] and [Ne iii ] stronger (and low ionization lines such as [O ii ] weaker) than expected from the local empirical relationships as defined by Maiolino et al. (2008) and Curti et al. (2017, 2020). Rather, they present a fairly good agreement with the relationships proposed by Bian et al. (2018) and Nakajima et al. (2022) for the large EW(H β ) objects. Because the relationships of Bian et al. (2018) are based on the highly-ionized objects as typically found at z ∼ 2 -3 on the [N ii ] BPT diagram (e.g., Baldwin et al. 1981; Steidel et al. 2014; Shapley et al. 2015), those consistencies suggest the JWST objects at z = 4 -8 . 5 are highly ionized systems. This trend is consistent with the analysis by Sanders et al. (2023). One caveat is that there is one JWST objects whose EW(H β ) is smaller than 100 ˚ A, GLASS 10021, falling close to the large EW(H β ) relationships of Nakajima et al. (2022) rather than the small EW ones like the other JWST objects despite the relatively small EW(H β ). This suggests we need to test whether the prescriptions work for low ionization sources as well with a larger sample at high-redshift. \nIn summary, although we cannot fully test the indicators for low ionization sources, the current results suggest that the degree of ionization significantly influences the resulting metallicity value if one relies on an empirical metallicity indicator, and that no strong redshift evolution is seen in the strong line ratios as a function of metallicity at least for highly ionized galaxies. Ionization-corrections, such as those proposed in Nakajima et al. (2022), are thus crucial for metallicity estimations with the strong line methods. In particular, we find R23, R3, and R2 show a good agree- \nment with the observations and the empirical relationships, and confirm that they are sensitive to a small change in metallicity. The indicators of O32 and Ne3O2 look highly dependent on the ionization state and their plateau-like behaviors against metallicity prevent us from deriving a stable metallicity solution at the metalpoor regime. \nFigure 8 shows another metallicity indicator proposed by Izotov et al. (2019, 2021) which combines R23 and O32 to correct for the ionization state to improve the accuracy of metallicity in the low-metallicity regime. Unfortunately there is only one object, ERO 04590, whose metallicity is low enough to be fairly compared with the method, and whose non-detection of [O ii ] prevents us from confirming the reliability and accuracy of the method. This is exactly the situation Nakajima et al. (2022) have anticipated, i.e., not all of the lines are spectroscopically available at high redshift, and it is practically useful to use EW(H β ) as a probe of ionization state. Still, many of the JWST objects are located along the simple extrapolation of the relationship, indicating the method can be useful even up to 12+log(O / H) ∼ 7 . 8. \nOne caveat should be noted for ERO 06355, which has an oxygen abundance of 12+log(O / H) = 8 . 3 and shows an unusually strong [O iii ] line. Curti et al. (2023a) have used the [O iii ] / H β vs. [O ii ] / H β diagram and found that this object is slightly above the limit that can be explained by star-forming galaxies alone. This indicates the possibility of a hidden high-energy ionizing source. In addition, Brinchmann (2022) tentatively suggest the presence of the high ionization line [Ne iv ] λλ 2422 , 2424, indicating a hard ionizing spectrum up to ∼ 60 eV, which cannot be fully explained by a conventional stellar population. These pieces of evidence suggest that the anomalous nature of this object may be explained by the presence of a high-energy ionizing source in the system, which may account for its deviation from the empirical metallicity relationships. Another note is for GLASS 160133 and GLASS 150029 that have a broad component associated with their H α emission line, indicating the presence of faint AGNs in the systems. However, they still present metallicity diagnostic line ratios and EW(H β ) that agree reasonably well with the relationships for star-forming galaxies, implying a minor contribution of AGN to the total emission strengths as well as to the optical continuum for these two objects. In any case, one would need higher ionization lines (e.g., [Ne v ]; Cleri et al. 2023) to further discuss the presence of hard radiation components in these systems. \nFor the other JWST objects without a T e -based metallicity ( N o = 182 -10 = 172), we adopt the empirical metallicity indicators which have been tested and confirmed to work at high-redshift (Section 3.2.1) to estimate metallicities. Prior to estimating metallicities empirically, we require that the objects have coverage of the [O iii ]+H β emission lines in their spectra, and that the [O iii ] λλ 5007 / 4959 doublet line ratio is consistent with the theoretical value (2 . 98; Storey & Zeippen 2000) at a 2 . 5 σ significance level. We only work with objects that meet these criteria and have appropriate flux measurements. As a result, 11 objects were removed from the subsequent analysis. \nFor the remaining 161 objects, we use the R23-index as a primary indicator, following the overall consistency of the metallicity indicator at high-redshift as shown in Section 3.2.1. One caveat in using R23 is that two solutions are derived for a given R23 value. We use the prescription of Nakajima et al. (2022) with EW(H β ) for the low-metallicity solutions (12 + log(O / H) /lessorsimilar 8 . 0), and the average relationship (without any ionizationcorrection) for the high-metallicity ones ( /greaterorsimilar 8 . 0). The latter is almost consistent with the indicators presented in Curti et al. (2017, 2020). If the two metallicity solutions are not significantly separated, i.e., the observed R23 spans the peak value that appears around 12 + log(O / H) = 8 . 0 within the uncertainty, we adopt the 1 σ lower-limit of the low-metallicity solution as the lowerlimit, and the upper-limit of the high-metallicity solution as the upper-limit. If there are two distinct metallicity solutions, we rely on the O32 line ratios to distinguish between them. It should be noted that O32 is used only for the purpose of distinguishing between the two branches, and not for calculating the actual metallicity value. Specifically, we choose the R23-based metallicity solution whose expected O32 line ratio according to the O32-metallicity indicator of Nakajima et al. (2022) is more close to the observed O32 than the other. If the [O ii ] is not detected and the resulting lower-limit on O32 is higher than the expected O32 at the highmetallicity branch at the > 3 σ level, we choose the lowmetallicity solution. If the lower-limit on O32 is not high enough to distinguish, we translate EW(H β ) to O32 using the average relationship found in Nakajima et al. (2022): log EW(H β ) = 0 . 64 × log O32+1 . 68, and choose the solution. Finally, the systematic uncertainty of the empirical method is added in quadrature to the metallicity errors. For the other cases, we cannot reliably distinguish between the two solutions and thus leave its metallicity unconstrained based on R23. \nWe are unable to derive a R23-based metallicity for objects that lack [O ii ] coverage and/or appropriate \ndust correction due to the absence of multiple Balmer emission lines. For such objects, we estimate metallicity using the R3 indicator, as it does not require a reddening correction. We follow a similar procedure as for R23 to account for the two-branch nature of the R3-index and estimate metallicities. As O32 is not applicable for these objects, we rely on EW(H β ) to correct for the ionization state and attempt to differentiate between the two solutions. \nIn summary, our sample consists of 86 objects with an R23-based metallicity, 49 objects with an R3-based metallicity, and 26 objects without a metallicity constraint. We note that for the objects in the CEERS sample that have multiple spectra taken with different gratings and/or pointings, we estimate the metallicity for each spectrum and calculate the average value and its uncertainty by combining the individual metallicity measurements. While we understand that some of the spectra (P11 and P12) were obtained with different position angles compared to the others, we assume that the objects are compact and the different observations targeting the same objects at the center of the slit probe the same gas properties. After excluding 10 objects with spectroscopic signatures of AGNs (Section 3.1), comprising of 8 objects with R23-based metallicity and 2 objects with T e -based metallicity, we use 127 objects with an R23- or R3-based metallicity, along with 8 objects with direct T e method, for the subsequent analysis and discussion of mass-metallicity relationships.", '3.2.3. Other high-redshift galaxies from the literature': "In addition to our JWST sample of 135 objects constructed with the ERO, GLASS, and CEERS programs, we have compiled reports of metallicity at highredshift from the literature to discuss with a larger sample. Our compilation includes three objects at z = 8 . 1 -9 . 5 identified with the two DDT programs using the prism grating of NIRSpec (Proposal IDs: DD-2756 and DD-2767; (Williams et al. 2022; Heintz et al. 2022; Langeroodi et al. 2022; Wang et al. 2022)), one stacked point of 117 EIGER objects at z = 5 . 3 -6 . 9 whose spectra are taken with the NIRCam slitless spectroscopy (Proposal ID: 1243; Matthee et al. 2022; Kashino et al. 2022), four objects at z = 6 . 1 -6 . 4 with commissioning data of NIRCam slitless spectroscopy (Sun et al. 2022a,b), and four ALMA objects at z = 7 . 2 -9 . 1 whose metallicity is measured with [O iii ]88 µ m (Jones et al. 2020). Their redshift distributions are illustrated in blue in Figure 3. Note that the EIGER stacked object is treated as a single object in the histogram as well as in the following figures. \nFor a fair comparison of metallicity in the following analysis, it is important to have metallicities that are estimated in a consistent manner, as done for our sample. All of the measurements compiled above are based on empirical indicators using strong emission lines that are calibrated with the direct T e method. For the NIRSpec DDT objects, five objects at z = 7 . 9 -9 . 5 have reported metallicity measurements in several studies (Williams et al. 2022; Heintz et al. 2022; Langeroodi et al. 2022; Wang et al. 2022). Despite initially adopting the O32-index as the metallicity indicator in the original manuscript, Heintz et al. (2022) have updated their results, now primarily using the R3-index of Nakajima et al. (2022) with correction for the ionization state for large EW(H β ) ( ≥ 200 ˚ A) objects. This approach is supported by Figure 7 in this paper and has been adopted for some of our JWST objects. It should be noted, however, that it would be more appropriate to adopt the EW(H β )-dependence of the indicator following the variation of EW(H β ) as observed in the DDT sample, instead of fixing the relation for large EW(H β ) (Heintz et al. 2022). Large EW(H β ) of ∼ 180 -250 ˚ A are confirmed for two of the five objects (Williams et al. 2022; Wang et al. 2022). Among the five objects reported in Heintz et al. (2022), we use three objects, RXJ-z9500, RXJ-z8152, and RXJ-z8149, in this paper. Their metallicities are estimated to be 12+log(O / H) = 7.56 (+0.16/-0.17), 7.68 (+0.18/-0.19), and 7.29 (+0.22/-0.28), respectively. One of the objects not included in our compilation, Abell-z7878, was originally suggested to have an [O iii ] λλ 5007 / 4959 doublet ratio that is significantly smaller than the theoretical value. We remove this object from our compilation following the same approach used for our JWST sample. The remaining object, Abell-z7885, still relies on the O32-index for estimating its metallicity. We have decided not to include it in our compilation due to several uncertainties associated with using the O32-index for metallicity estimations, such as the strong dependence of O32 on ionization state (Figure 7) and the accuracy of dust reddening correction, as we were careful regarding the results presented in the original manuscript of Heintz et al. (2022). \nThe metallicity values for RXJ-z9500, RXJ-z8152, and RXJ-z8149 in Heintz et al. (2022) are consistent with those reported in other studies. The metallicity of RXJz9500 is first reported by Williams et al. (2022), where it was estimated to be 12+log(O / H) = 7 . 48 ± 0 . 08 using the R23+O32 method of Izotov et al. (2019, 2021) (Figure 8). An ionization correction is thus taken into account for the metallicity measurement there. Similarly, \nFigure 9. Comparison between the mass-metallicity relation obtained in this study for ERO objects at z = 6 . 3 -8 . 5 using T e -based metallicities (colored circles) and those reported in the literature. Different symbol colors indicate different objects, as shown in Figure 6, and different symbols represent different publications, as indicated in the legend. For the metallicities from Arellano-C'ordova et al. (2022), we show the average values based on the two spectra (i.e., two observing blocks: o007 and o008) for 06355 and 10612, but refer only to the o008 value for 04590. For the relationships by Schaerer et al. (2022) and Curti et al. (2023a), the stellar masses are adopted as originally estimated in each paper. For the others where stellar masses are referenced from other studies, we show the two stellar masses from Carnall et al. (2023) and Tacchella et al. (2022) for each object (only Carnall et al. (2023) for 05144). The comparison confirms a good agreement between our measurements and earlier results on the MZ relation. \n<!-- image --> \nLangeroodi et al. (2022) estimate metallicities for RXJz8152 and RXJ-z8149 (RX2129-ID11002 and RX2129ID11022 in Langeroodi et al.) using the R23+O32 method (Izotov et al. 2019, 2021), and report 12 + log(O / H) = 7 . 65 ± 0 . 07 and < 7 . 51 (1 σ ), respectively. Wang et al. (2022) independently suggest a consistent metallicity for RXJ-z8152, although the ionization correction is not fully taken into account. We use the properties, including metallicities, that are calculated by averaging the values from Williams et al. (2022), Heintz et al. (2022), and Langeroodi et al. (2022) for the three DDT objects in our compilation. \nFor the other compilation, we adopt the metallicity values as derived in the original papers. The stacked EIGER object has already derived a metallicity fairly consistent with the value based on our method using R23. The four NIRCam objects are all located in the high-metallicity branch based on the [N ii ] λ 6564 detection, and the indicator of Bian et al. (2018) is used. We also note that the stellar masses and SFRs are all corrected for different IMFs to have the same \nFigure 10 illustrates the relation between stellar mass and metallicity (MZ) with the full sample at z = 4 -10. The MZ relations determined at z = 0, /similarequal 2 . 3, and /similarequal 3 . 3 with the direct T e method are also plotted \n<!-- image --> \nFigure 10. The relationship between stellar mass and metallicity. The red points represent galaxies at z = 4-10, including the filled circles, pentagons, and diamonds, which represent the ERO, GLASS, and CEERS objects analyzed in this paper. Red circles denote galaxies whose metallicity is determined using the direct T e method. The large red stars represent the average relationships for the ERO, GLASS, and CEERS objects in three equally separated mass ranges ( M /star =10 7 -10 8 , 10 8 -10 9 , and 10 9 -10 10 M /circledot ), along with the best-fit function shown as a thick long-dashed line (Equation 1). Emerald green open pentagons show the average relations at z = 3 -10 based on the JADES+CEERS sample (Curti et al. 2023b). Other red symbols represent high-redshift objects compiled from the literature, including open right-pointing triangles for NIRSpec DDT objects (Williams et al. 2022; Heintz et al. 2022; Langeroodi et al. 2022), left-pointing triangles for four NIRCam objects (Sun et al. 2022a,b), open hexagram for the stacked EIGER object (Matthee et al. 2022), and crosses for ALMA objects (Jones et al. 2020). Additionally, relationships at lower redshifts are displayed, including SDSS stacked galaxies in the local universe shown as a gray dashed curve (Andrews & Martini 2013), and those at z ∼ 2 . 3 with gray open triangles and ∼ 3 . 3 with gray open squares (best-fits shown as gray dot-dashed curve; Sanders et al. 2021). The curves are displayed in the mass ranges explored in the original papers. These low-redshift metallicities are based on the direct T e method. \n<!-- image --> \nChabrier (2003) IMF using the conversion factors shown in Madau & Dickinson (2014). \nIn summary, our total sample consists of 147 galaxies (135 from our reduction of ERO, GLASS, and CEERS, and 12 from the compilation) with metallicity measurements at z = 4 -10. This sample is used in the following analysis of the metallicity relationships.", '3.3. Mass-Metallicity Relation': 'The main focus of this paper is to discuss the evolution of metallicity in star-forming galaxies. In this section, we specifically present the stellar mass-metallicity relation at high redshift using the JWST observations, as well as data compiled from relevant literature. \nBefore presenting the full results we firstly focus on the four ERO objects and compare in Figure 9 our mass-metallicity relation with those presented in the earlier studies (Schaerer et al. 2022;', 'Curti et al. 2023a; Trump et al. 2022; Rhoads et al.': "2023; Arellano-C'ordova et al. 2022; Brinchmann 2022). The metallicities are all derived based on the direct T e method as summarized in Figure 6. The stellar masses are corrected for different IMFs to have the same Chabrier (2003) IMF. Schaerer et al. (2022) and Curti et al. (2023a) derive the masses in their own papers, while the others refer to either Carnall et al. (2023) or Tacchella et al. (2022) and hence both values are shown in the plot. We confirm our mass-metallicity relations for the four ERO objects overall show a good agreement with the earlier results. One note is that we confirm the metallicity of ERO 04590 is not extremely low, 12 + log(O / H) = 7 . 26 (+0.15/-0.13) for its stellar mass. \nTable 2. Average values of stellar mass, metallicity and SFR for our JWST sample \nNote -The first half of this table displays the average masses and metallicities of the samples used for the MZ relations, while the second half shows the samples used for the SFR-MZ relations. The second half samples exclude objects whose SFRs are not measured or constrained (Section 3.4), resulting in a smaller sample size compared to the first half. \n(Andrews & Martini 2013; Sanders et al. 2021; see also Nishigaki et al. 2023 that the T e -based MZ relation at z = 0 continues decreasing down to M /star ∼ 10 5 M /circledot ). The red symbols denote the galaxies at z = 4 -10, including our 135 ERO, GLASS, and CEERS galaxies, 3 NIRSpec DDT objects, as well as the massive galaxies provided by NIRCam slitless and ALMA spectroscopy, and the low-mass stacked object as presented in Section 3.2.3. We note again that the 10 objects with spectroscopic signatures of AGNs (Harikane et al. 2023) have been excluded here to conservatively discuss the results free from any AGN biases (Section 3.1). The objects analyzed using the direct T e method are marked with a red open circle, suggesting a positive correlation between mass and metallicity in place in the redshift range z = 5 -8 . 5. Furthermore, we divide our full sample of ERO, GLASS, and CEERS galaxies into three according to the stellar mass: M /star = 10 7 -10 8 , 10 8 -10 9 , and 10 9 -10 10 M /circledot , and obtain the average MZ relations as \nshown with the large open stars in Figure 10 and as given in Table 2. Note that the compiled objects, as well as the CEERS objects with only an upper-limit on M /star , are not used in deriving the average relations. Following the single power law form of Sanders et al. (2021), the average MZ relation of the z = 4 -10 galaxies can be approximated as: \n12 + log(O / H) = Z 10 + γ log( M /star / 10 10 M /circledot ) (1) \nwith the best-fit parameters of Z 10 = 8 . 24 ± 0 . 05 and γ = 0 . 25 ± 0 . 03 in the mass range of M /star ∼ 10 7 . 5 -10 9 . 5 M /circledot , as shown with a gray long-dashed line. The parameter γ corresponds to the slope of the MZ relation, and its best-fit value and uncertainty confirm an increasing trend of metallicity with stellar mass, as tentatively seen with the three ERO objects (e.g., Schaerer et al. 2022; Curti et al. 2023a; Trump et al. 2022), and as widely known at low-redshift. \nCompared to the z = 0 MZ relation, these highredshift galaxies clearly present a metallicity lower than typical galaxies at z = 0 for a given stellar mass. The decrease is typically ∼ 0 . 5dex around M /star ∼ 10 9 M /circledot , but it becomes smaller at the low-mass end ( ∼ 0 . 3dex). Interestingly, a similar offset of ∼ 0 . 3dex is observed between z = 0 and z ∼ 2 -3, suggesting that the evolution of MZ relation is small from z ∼ 2 -3 to z = 4 -10. Although there may be a decrease of ∼ 0 . 2dex in the typical metallicity at the high-mass end of M /star ∼ 10 9 . 5 M /circledot from z ∼ 2 -3 to z = 4 -10, no strong evolution is found beyond the error. The same conclusion can be drawn from comparisons with the MZ relations at z /lessorsimilar 4 whose metallicities are empirically estimated with the strong line indicators, as presented in Appendix B. \nFigure 10 also includes a comparison with the latest results from the JADES observations. Curti et al. (2023b) very recently report on the metallicity measurements of galaxies at z = 3 -10 based on deep JADES spectroscopic observations (see also Cameron et al. 2023b). In the figure, their MZ relation is shown using emerald green open pentagons, which represent the combination of the low-mass JADES sample with the high-mass CEERS sample, latter of which is provided in this paper. We find that their MZ relation, despite covering slightly different redshift ranges, agrees well with ours (Eq. 1). It is not surprising to see a good agreement between our results and those reported by Curti et al. (2023b) in the high-mass end ( M /star /greaterorsimilar 10 8 M /circledot ), since the regime is dominated by the CEERS objects presented in this paper. The JADES objects confirm that the MZ relation we obtained in Eq. 1 continues down to M /star /similarequal 10 7 M /circledot on average. \nFigure 11. The MZ relation in three different redshift bins: (a) z = 4-6, (b) z = 6-8, and (c) z = 8-10. The red symbols and the best-fit function (Eq. 1) are as shown in Figure 10. The large stars represent the average MZ relations re-derived in each redshift bin, by splitting the sample into two groups based on stellar mass to have the equal numbers of galaxies. For the JADES+CEERS relations (Curti et al. 2023b in emerald green), we plot their z = 3 -6 sub-sample's relations in Panel (a), while adopt the z = 6 -10 relations in Panels (b) and (c). In addition, the cosmological simulation results at z = 5, 7, and 9 are displayed in Panel (a), (b), and (c), respectively; FIRE in black (Ma et al. 2016), IllustrisTNG in green (Torrey et al. 2019), FirstLight in blue (Langan et al. 2020 in the low-mass regime with solid curves, and Nakazato et al. 2023 in the high-mass regime with shades), Astraeus in yellow (Ucci et al. 2021), and FLARES in magenta (Wilkins et al. 2022). Some extrapolations are applied, as detailed in the text. \n<!-- image --> \nIn Appendix A, we investigate our MZ relation using only galaxies for which the stellar mass is welldetermined through SED fitting to the JWST/NIRCam photometry, after excluding CEERS objects whose masses are estimated empirically from M UV (as discussed in Section 2.3). Because we have already excluded the CEERS objects with only an upper-limit on M /star , all of which are M UV -based, in deriving the average relations, the fraction of objects without NIRCam photometry is small ( ∼ 28%). Indeed, our conclusions remain unchanged when considering only the objects with reliable measures of M /star , as demonstrated in the figures in Appendix A. Moreover, we mention in the [N ii ] BPT diagram in Section 3.1 that some of the JWST objects have [N ii ] upper-limits that are too weak to conclude the ionization nature (stars vs. AGNs). By removing these unclear objects and using only galaxies that are surely diagnosed as star-forming galaxies in the [N ii ] BPT diagram (i.e., those with an (upper-limit on) [N ii ]/H α falling below the demarcation curve), we obtain a fully consistent MZ relation as found in the full sample (Figure 10). This implies a negligible contribution of AGNs in our sample. \nTo further examine any redshift evolution among our sample from z = 4 to 10, we plot in Figure 11 the MZ relations for the three different redshift bins, z = 4 -6, 6 -8, and 8 -10. In each panel, the sub-sample is further divided into two groups based on their masses, and their average values are shown with large stars as summarized in Table 2. Although there are large error bars, these average points suggest that the slope and normalization of the MZ relation in different redshift bins are consistent with those based on the full sample at z = 4 -10 (Eq. 1). Figure 11 indicates that there is no significant evolution in the MZ relation among our z = 4 -10 sample. We note that there may be a weak trend towards lower metallicity in the highest-redshift bin, albeit with a small sample size. Curti et al. (2023b) also tentatively suggest a similar evolution. That will be further explored in the discussion. \nThe no/weak evolution is consistent with the predictions of some cosmological simulations, as presented and compared in Figure 11. We compile the hydrodynamic, N -body, and/or semi-numerical simulations showing the MZ relation at high-redshift; FIRE by Ma et al. (2016) in black, IllustrisTNG by Torrey et al. (2019) in green, FirstLight by Langan et al. (2020) and Nakazato et al. (2023) in blue (see also Ceverino et al. 2017), Astraeus by Ucci et al. (2021) in yellow, and FLARES by Wilkins et al. (2022) in magenta (see also Lovell et al. 2021). We plot the theoretical predictions \nat z = 5, 7, and 9 for the redshift bin of z = 4 -6, 6 -8, and 8 -10, respectively. For the FIRE curves, we extrapolate the result at z = 6 to z = 7 and 9 using their redshift evolution function, although the evolution is tiny (∆ log (O / H) = -0 . 025dex from z = 6 to 7, and -0 . 05dex from z = 6 to 9). Similarly, the IllustrisTNG predictions at z = 5, z = 7, and z = 9 are extrapolated from the results at z = 4, z = 6, and z = 6, respectively, using their redshift evolution function (∆log(O / H) = -0 . 08dex from z = 4 to 5, -0 . 075dex from z = 6 to 7, and -0 . 1dex from z = 6 to 9). For the FirstLight results, we refer to Langan et al. (2020) and Nakazato et al. (2023) to prove the low-mass and the massive regime, respectively. We assume the z = 8 relation of Langan et al. (2020) in panel (c), and the z = 6 relation of Nakazato et al. (2023) in panel(a), assuming no strong evolution at z = 8 -9 and z = 5 -6, respectively. For the FLARES results, we adopt the stellar metallicities of only young ( < 10Myr) star particles and assume the stellar and gas-phase metallicities in the region of massive-star formation are comparable. Likewise, the metallicities for the IllustrisTNG and FirstLight model are SFR-weighted and massweighted of young ( < 100Myr) star particles, respectively, allowing a fair comparison with our metallicity measurements based on the nebular emission lines. On the other hand, we note that the FIRE model adopts the mass-weighted metallicity of all gas particle that belong to the ISM, and the Astraeus model counts the oxygen mass in the halo without any weighting which could result in an inequitable comparison with the observations. \nComparing the observations with the simulations in Figure 11, we find the observed MZ relation and its weak evolution over z = 4 -10 are in good agreement particularly with IllustrisTNG , FirstLight , and FLARES in the mass range of M /star = 10 8 -10 9 . 5 M /circledot . Notably, the slope of the MZ-relation found in the IllustrisTNG results is likely coincide with the observations. On the other hand, the simulations of FIRE and Astraeus are generally suggested to under-predict metallicities except for some metal-poor galaxies such as ERO 04590 found in the highest-redshift bin at z > 8. That is probably due to their implementation of feedbacks, i.e., galaxies eject too many metals from galaxies and/or accreting gas is too efficient in lowering the metal contents in galaxies (Ma et al. 2016; Ucci et al. 2021). Finally, we note that the average metallicity measured for low-mass galaxies below M /star < 10 8 M /circledot can be slightly higher than the existing predictions by ∼ 0 . 2dex. It is thus necessary to increase the sample size to determine the MZ relation in the low-mass end as well as to extend the \nFigure 12. The relationship between stellar mass and SFR, with SFR derived based on the total H β luminosity (as described in the text) for use in the SFR-MZ relation. The symbols used are consistent with those in Figure 10. The background lines represent sSFR values ranging from 10 -9 to 10 -6 yr -1 , from bottom to top. The JWST objects are distributed along the sequence of sSFR values around 10 -8 -10 -7 yr -1 . \n<!-- image --> \ntheoretical predictions such as IllustrisTNG towards lower-mass to conclude the consistency and to betterunderstand the early chemical enrichment in the lowmass systems at high-redshift.", '3.4. Mass-Metallicity-SFR Relation': "The next important aspect is the SFR-dependence of the MZ relation to discuss the chemical evolution, given the claim of a redshift-invariant fundamental relation between mass, metallicity, and SFR (SFR-MZ relation) out to z ∼ 2 -3 (Mannucci et al. 2010; Sanders et al. 2021). There are several expressions to describe the mutual dependencies between the three quantities (Mannucci et al. 2010; Lara-L'opez et al. 2010; Andrews & Martini 2013; Sanders et al. 2017; Curti et al. 2020; Sanders et al. 2021). In this paper, we primarily use the variable µ α originally proposed by Mannucci et al. (2010): µ α = log(M /star ) -α log(SFR), with α = 0 . 66 being adopted here that minimizes the scatter of the local low-metallicity galaxies with a direct T e -based metallicity in the µ α -metallicity plane (Andrews & Martini 2013) 7 : \n12 + log(O / H) = 0 . 43 × µ 0 . 66 +4 . 58 . (2) \nFigure 13. (a: Left:) Star-formation rate dependence of the Mass-Metallicity (SFR-MZ) relation. The symbols and curves are consistent with those in Figure 10, including the average points which are based on the three mass ranges that are equally separated. The coefficient of 0 . 66 for the combination of stellar mass and SFR is as indicated by Andrews & Martini (2013) in the local universe, down to M /star ∼ 10 7 . 4 M /circledot . Stacked galaxies at z = 2 -3 with T e -based metallicities are also suggested to fall on the z ∼ 0 relation in this parameter space (Sanders et al. 2021). Notably, the JWST objects analyzed in this study at z = 4 -10 (red) also fall on the same SFR-MZ relation on average. A consistent view is obtained with the JADES+CEERS average points at z = 3 -10 (emerald green; Curti et al. 2023b). (b: Right:) Metallicity difference from the SFR-MZ relation of Andrews & Martini (2013) (∆ log(O/H) = observed -predicted metallicity) for different redshift sources. The symbols are consistent with Panel (a), except for the average points (large red stars), which are recalculated on this panel to represent the average ∆ log(O/H) values for the objects found in three different redshift bins, z = 4 -6, 6 -8, and 8 -10. For the JADES+CEERS sample, average points of three mass bins are plotted for the two sub-samples at z = 3 -6 and 6 -10, as found in Figure 11. This plot illustrates that no evolution is seen up to z ∼ 8, but a significant decrease in metallicity is observed beyond z > 8, albeit with a small sample size in this highest-redshift bin. \n<!-- image --> \nThis is advantageous to be directly compared with a majority of the JWST objects down to M /star ∼ 10 7 . 4 M /circledot . Interestingly, the z /similarequal 2 . 3 and /similarequal 3 . 3 stacked objects of Sanders et al. (2021), whose metallicities are reliably determined with the direct T e method, fall directly on the same relationship. Another popular form of the SFR-MZ relation is provided by Curti et al. (2020). Our choice of the relation of Andrews & Martini (2013) against that of Curti et al. (2020) will be revisited later. \nTo estimate SFRs for the JWST objects, we adopt the total, reddening-corrected H β luminosity for each object, as it is the best indicator for the on-going ( /lessorsimilar 10Myr) star-formation activity. The SFR relation of Kennicutt (1998) is adopted for the Balmer line with a correction of IMF to Chabrier (2003) using the conversion factor of (Madau & Dickinson 2014). Accordingly, the SFR-MZ relation is examined only for the JWST objects whose reddening correction is successfully applied and whose spectrum is slit-loss corrected. The latter constraint depends on whether the object has the NIRCam coverage (only in the CEERS field; Section 2.3). Among the CEERS objects lacking a slitloss correction, we rescue 42 objects whose UV stellar emission is constrained with HST. For these objects, we \ntranslate M UV into SFR(H β ) assuming a typical ionizing photon production efficiency as indicated at highredshift (log ξ ion = 25 . 6 ± 0 . 2; e.g., Endsley et al. 2022; Matthee et al. 2022) and the factor of 0 . 44 difference between the reddening E(B -V) for the nebular and stellar emission (Calzetti et al. 2000). The assumption of ξ ion will be revisited elsewhere (K. Nakajima et al. in prep.) using the JWST sample presented in this paper. The objects lacking a proper dust reddening correction or having just upper-limits on both M /star and SFR are excluded. In short, 96 out of the 147 objects are used for the SFR-MZ relation at high-redshift. The second half of Table 2 presents the average SFR values for different masses and redshifts. \nIn Figure 12, we show the individual and average distributions of the JWST objects on the stellar massSFR plane using the SFR values as derived above (i.e., mainly from H β ). The high-redshift objects are distributed along the sequence of sSFR = 10 -8 -10 -7 yr -1 . This is overall consistent with the starformation main sequence of galaxies at z = 4 -7 where sSFR of 10 -8 . 5 -10 -7 . 5 yr -1 is typically suggested (e.g., Stark et al. 2013; de Barros et al. 2014; Santini et al. 2017; Popesso et al. 2023). We note that some of our \nFigure 13(a) show the SFR-MZ relation of the z = 4 -10 galaxies and their average points as shown in Figure 10 on the µ 0 . 66 -metallicity plane, together with the z = 0 average relation (Eq. 2) and the data-points of z /similarequal 2 . 3 and 3 . 3 stacked galaxies (Sanders et al. 2021). Adopting the best-fit parameter α = 0 . 66 found in the low-redshift metal-poor star-forming galaxies (Andrews & Martini 2013), the z = 4 -10 objects interestingly fall on the same SFR-MZ relationship. The JADES+CEERS sample, notably including lower-mass galaxies than ours, is also illustrated to show a consistent SFR-MZ relation on average. \n<!-- image --> \nFigure 14. (a: Left:) Similar to the left panel of Figure 13, but with the regions of µ 0 . 66 explored and unexplored by the original paper of Andrews & Martini (2013) highlighted. The blue solid curve represents the best-fit 2D projection of the SFR-MZ relation by Andrews & Martini (2013), determined down to µ 0 . 66 ∼ 7 . 5, with extrapolation towards lower mass shown by the dotted curve. Among the JWST objects, 83 % have µ 0 . 66 > 7 . 5 and can be directly compared with Andrews & Martini's relation. (b: Right:) Similar to the left panel, but with a different coefficient of 0 . 55 adopted for the combination of mass and SFR on the abscissa axis, as presented by Curti et al. (2020). The blue solid curve represents the best-fit 2D projection of Curti et al. (2020)'s SFR-MZ relation, determined down to µ 0 . 55 ∼ 8 . 5, and extrapolated towards lower mass with the dotted curve. Only approximately 15 % of the JWST objects have µ 0 . 55 > 8 . 5, and as such, the vast majority of the sample occupies a parameter space that is not explored by Curti et al. (2020). \n<!-- image --> \ngalaxies are above sSFR /greaterorsimilar 10 -7 . 5 yr -1 , in particular in the low-mass regime. This can be because the current spectroscopically-confirmed sample is partly biased toward actively star-forming systems, and/or because the sample contains higher-redshift objects at z > 7. \nWe adopt the SFR-MZ relation of Andrews & Martini (2013) at z = 0 that is compared with the relation at z = 4 -10 because the sample of Andrews & Martini (2013) contains comparably low-mass, actively starforming galaxies. Figure 14(a) clarifies the parameter space that is explored by Andrews & Martini (2013), demonstrating that the majority (83 %) of the JWST objects have µ 0 . 66 > 7 . 5 and can be directly compared with the relation of Andrews & Martini (2013). Note that local extremely metal-poor galaxies with M /star \n∼ 10 5 -10 7 M /circledot and 12 + log(O / H) = 7 . 0 -7 . 7 are suggested to be reproduced by the chemical evolution models (Lilly et al. 2013) with the same parameters as found for the Andrews & Martini's galaxies (Nishigaki et al. 2023). This evidence supports the simple extrapolation of the relation towards lower masses. On the other hand in Figure 14(b), we show another form of the SFR-MZ relation derived by Curti et al. (2020). The authors find another best-fit α = 0 . 55 as the best 2D projection of the SFR-MZ relation on the µ α -metallicity plane by using the global sample of stacked SDSS galaxies. The relation is determined over µ 0 . 55 ∼ 8 . 5 -11 . 0. Because ∼ 85% of the JWST sample have µ 0 . 55 below 8 . 5, most of the z = 4 -10 galaxies occupies the parameter space that is not explored by Curti et al. (2020). Moreover, Curti et al. (2020) clarify a different degree of SFR-dependence of the MZ relation for the low- and high-sSFR subsamples in the local universe. For the low-sSFR subsample (sSFR < 10 -9 . 5 yr -1 ), a weaker SFR dependence is indicated ( α = 0 . 22). On the other hand, the high-sSFR subsample with sSFR > 10 -9 . 5 yr -1 presents a stronger SFR dependence with α = 0 . 65, in close agreement with the best-fit value found by Andrews & Martini (2013). Given the high sSFRs for the JWST objects (sSFR ∼ 10 -7 -10 -8 yr -1 ; see Figure 12) and also for the z = 2 -3 galaxies (sSFR ∼ 10 -8 . 5 yr -1 , we conclude the strong SFR dependence \nFigure 13(b) clarifies the redshift evolution on the SFR-MZ relation of Andrews & Martini (2013) by showing the residual metallicity for a given µ 0 . 66 with respect to the z = 0 relation: ∆ log(O / H) = 12 + log(O / H) obs -12+log(O / H)( µ 0 . 66 ; Eq. 2) for each galaxy's redshift. In this panel, the average points are derived for the objects found in the different redshift bins, at z = 4 -6, 6 -8, and 8 -10. \n<!-- image --> \n<!-- image --> \nFigure 15. Metallicity difference from the SFR-MZ relation of Andrews & Martini (2013) as a function of stellar mass in three different redshift bins: (a) z = 4 -6, (b) z = 6 -8, and (c) z = 8 -10. The symbols are consistent with those in Figure 11. No clear dependence of ∆ log(O/H) on stellar mass is evident beyond the statistical errors. \n<!-- image --> \nof α = 0 . 66 as found by Andrews & Martini (2013) is \npreferred and adopted in this paper to be compared with the high-redshift galaxies. We discuss the evolution of the Curti et al.'s SFR-MZ relation later in Section 4. \nTwo interesting results arise in Figure 13(b). One is that the SFR-MZ relation shows no evolution up to z ∼ 8 within ∆ log O / H /similarequal 0 . 3dex. Another is that a significant decrease of metallicity is found beyond the error at z > 8. The decrease at the highest-redshift bin is not visible in Panel (a) due to the small sample size, but hinted by the MZ relation at z = 8 -10 (Figure 11c). To further examine the evolution of the SFR-MZ relation, we plot in Figure 15 the residual metallicity as a function of stellar mass for each of the three redshift bins. According to the two average points probing different mass ranges, no significant dependency of the residual metallicity on stellar mass is indicated in each redshift bin with the current sample. For the highest redshift bin, the decrease of metallicity is suggested for individual galaxies regardless of their stellar masses. Although the JADES+CEERS sub-sample at z = 6 -10 appears to exhibit a mass dependence in the residual metallicity, we suggest this would be caused by a bias introduced by the dominance of lower-redshift galaxies (i.e., z = 6 -8) in the lower-mass regime. This is consistent with the MZ plane in Figure 11, where the lowest-mass point of the JADES+CEERS sub-sample at z = 6 -10 is more consistent with our z = 6 -8 relation rather than the z = 8 -10 one. These results suggest that chemical properties of star-forming galaxies up to z ∼ 8 are very similar to those of local and z = 2 -3 counterparts, while a break of the metallicity equilibrium state may be in place beyond z ∼ 8. More discussions follow in Section 4. \nFinally, we have conducted several tests to demonstrate the robustness of the results presented in this section. In Appendix A, we examine the SFR-MZ relation using only the JWST galaxies whose stellar mass is welldetermined with SED fit to the NIRCam photometry and whose SFR is obtained with the slit-loss corrected H β . Our conclusions remain unchanged even with the smaller sample that has good measurements of M /star and SFR, although the scatter of individual data points, as seen in Figure 15 for example, becomes less prominent. In Appendix C, we further investigate the SFR-MZ rela- \nion by using only the JWST galaxies with µ 0 . 66 > 7 . 5, where the relation is explored by Andrews & Martini (2013), allowing for a more robust comparison with highredshift galaxies to discuss their evolution. We confirm that almost the same SFR-MZ relation and its redshift evolution are obtained based on this subsample, suggesting that the extrapolation of the relation at µ 0 . 66 ∼ 7 -7 . 5 for the low-mass end galaxies in our sample does not introduce biases. Furthermore, we note that a consistent view of the SFR-MZ relations is also supported by using only galaxies that are surely diagnosed as star-formation dominated in the [N ii ] BPT diagram (see also Section 3.3).", '4. DISCUSSION AND SUMMARY': "We have conducted a (re-)analysis of the JWST/NIRSpec ERO data, as well as the ERS data of GLASS and CEERS, to investigate the chemical properties of galaxies at redshifts ranging from z = 4 to 10. Our analysis includes the use of 135 JWST objects based on our improved reduction and calibration of the NIRSpec data, as well as the compilation of 12 objects from the other recent JWST observations and previous literature. We confirm that our new emissionline flux measurements and errors successfully address the issues reported in the literature related to Balmer decrements and electron temperatures. The estimated electron temperatures for the four ERO objects, along with the 6 GLASS + CEERS objects with [O iii ] λ 4363 at z = 4 -8 . 5 range from T e = 1 . 1 × 10 4 to 2 . 3 × 10 4 K. These temperatures are similar to those found in lowerredshift star-forming galaxies and can be fully explained by heating of young massive stars, without the need for additional ionizing mechanisms. This is particularly evident for the z = 8 . 5 object, whose T e has been updated from /greaterorsimilar 25000K to (2 . 08 ± 0 . 26) × 10 4 K. \nWe have determined the mass-metallicity (MZ) relation for galaxies at z = 4 -10 and find no significant evolution compared to z ∼ 2 -3 when extrapolating to the low-mass regime. This result is consistent with the early JWST study results reported shortly after the ERO data release, except for the z = 8 . 5 object whose deviation from the others on the MZ relation becomes small after re-measurement of T e . Furthermore, our compilation of results does not show any significant evolution among the z = 4 -10 sample. Theoretical simulations also predict a similar trend of small redshift evolution in the MZ relation at z ∼ 5 -9 (∆log(O / H) /lessorsimilar 0 . 1dex; Ma et al. 2016; Langan et al. 2020; Ucci et al. 2021; Nakazato et al. 2023), with some simulations suggesting a potentially observable decrease in metallicity with redshift for a given mass ( ∼ 0 . 26dex \nfrom z = 5 to 9; Torrey et al. 2019). Although there may be a weak evolution towards lower metallicity at z > 8, the current sample has relatively large uncertainties in metallicity (∆ log(O/H) ∼ 0 . 3dex), which makes it difficult to fully distinguish between different predictions of redshift evolution at high redshifts z > 4, especially for low-mass galaxies with M /star < 10 8 M /circledot . Further detailed observational and theoretical studies are needed to address the apparent discrepancy and understand the early chemical enrichment in low-mass systems at high redshifts. \nWe have also investigated the SFR dependence of the MZ relation (SFR-MZ relation) at high redshifts, finding (i) no evolution from z = 0 to z = 4 -8 within ∆log(O / H) = 0 . 3dex, and (ii) a significant decrease of metallicity at z > 8. The former finding supports the idea that the SFR-MZ relation, also known as the Fundamental Metallicity Relation (FMR), can indeed describe the properties of galaxies with no redshift evolution. This suggests the existence of a metallicity equilibrium state via mechanisms such as star-formation, metal-poor gas inflow, and outflow (e.g., Lilly et al. 2013) that persists up to z ∼ 8. This sounds inconsistent with the potential deviation from the FMR at z /greaterorsimilar 2 . 5 as originally indicated by Mannucci et al. (2010). The authors have suggested, based on metallicities estimated using local empirical relations, that the same SFR-MZ relation shows no redshift evolution up to z ∼ 2 . 5. They claimed that galaxies at z ∼ 3 fall below the relation by approximately ∼ 0 . 6dex, with a combination of M /star and SFR of α = 0 . 32 in µ α . However, when using T e -based metallicities from Sanders et al. (2021) and adopting a different parameterization of α = 0 . 66 in µ α from Andrews & Martini (2013), no clear evolution from z = 0 up to z /similarequal 3 . 3 is found. Our findings, which utilize the parameterization of α = 0 . 66, support the claim of no average evolution of the SFRMZ relation up to z ∼ 8. This apparent inconsistency is likely attributed to the differences in metallicity estimations. High-redshift galaxies are known to have a higher ionization state of gas, parameterized by the ionization parameter, compared to local galaxies (Figure 4; see also e.g., Nakajima & Ouchi 2014; Sanders et al. 2023). Consequently, local empirical relations that assume a relatively low ionization parameter may result in biased metallicity estimates for high-redshift galaxies, particularly at z /greaterorsimilar 2 . 5 where the H α +[N ii ] λ 6584 lines are not available in the pre-JWST era. In fact, Figure 7 demonstrates that JWST objects with a large EW(H β ) exhibit a systematic offset from the empirical relations of Maiolino et al. (2008), which is used for metallicity estimations in Mannucci et al. (2010). Therefore, accu- \nFigure 16. Similar to the right panel of Figure 13, but employing the SFR-MZ relation of Curti et al. (2020). As similarly found in Figure 13, our analysis (red) and the JADES+CEERS study by Curti et al. (2023b) (emerald green) suggest a consistent evolution of the SFR-MZ relation at z = 4 -10 on average. If we adopt the formalism of Curti et al. (2020) which is defined at z = 0, our JWST objects begin to deviate from the SFR-MZ relation at z ∼ 6. This finding contrasts with the use of another z = 0 SFR-MZ relation of Andrews & Martini (2013), which suggests deviations starting at z ∼ 8 (Figure 13b). It is worth noting, however, that a simple extrapolation of the relation towards lower mass and higher SFR is assumed for the majority of the JWST objects (see Figure 14(b)). \n<!-- image --> \nations using the reliable direct T e method, or accounting for the ionization state evolution as prescribed in Nakajima et al. (2022) and Izotov et al. (2019, 2021), are essential for future metallicity studies of the high-redshift universe with JWST. \nThe latter finding of the potential evolution of the SFR-MZ relation beyond z ∼ 8 is intriguing. A similar result is supported by Heintz et al. (2022) as independently illustrated with the three DDT objects in our figures (red open right-pointing triangles), and also by the recent study of Curti et al. (2023b) based on the deep JADES observations. However, in contrast to our results using the SFR-MZ relation of Andrews & Martini (2013), Curti et al. (2023b) argue that galaxies begin to deviate at z ∼ 6, if they adopt the SFR-MZ relation of Curti et al. (2020). Because we confirm that our results and the JADES study yield the similar MZ and SFRMZ relations, as evidenced by the consistency between the red and emerald green symbols in Figures 10, 11, and 13, we attribute the redshift differences where deviations occur to the different functional forms of the \nSFR-MZ relation defined at z = 0. This is also supported by the evolution of the SFR-MZ relation based on the Curti et al. (2020)'s formalism for our JWST objects, as shown in Figure 16, which indicates a consistent evolution with deviations starting at z ∼ 6 as reported by Curti et al. (2023b). While the use of a 3D SFR-MZ relation can account for the SFR-dependence of the MZ relation's slope and capture the metallicity variations more accurately than a simple 2D projection using µ α as assumed in the Andrews & Martini's relation (e.g., Curti et al. 2020), this is probably not an essential issue in this study. This is because the JWST objects analyzed so far present a narrow distribution of sSFR (Figure 12; see also Figure 4 of Curti et al. 2023b). We acknowledge that extended local-baseline studies that cover low-mass and high sSFR galaxies at z = 0 are necessary to discuss the chemical evolution, including the fundamental SFR-MZ relation, and determine when and if high-redshift galaxies begin to deviate from the relation. In this work, however, we regard that the best estimates are the results based on the SFR-MZ relation by Andrews & Martini (2013) as it provides a more appropriate comparison by covering the low-mass regime where most of the JWST objects are found (Section 3.4 and Figure 14). \nIf we assume that the first star formation began to occur at z = 16 -27 (Harikane et al. 2022b; see also e.g., Abel et al. 2002; Bromm et al. 2002; Dayal & Ferrara 2018), galaxies at z = 8 -10 would be at most ∼ 200 -500Myr old. During such a short timescale after the Big Bang, there could be a higher probability for galaxies at higher redshifts to be in the early stages of formation, where they have not yet reached the metallicity equilibrium state via processes such as star formation, inflow, and outflow. An alternative explanation could be varying degrees of feedback processes, including inflow and outflow, in the early universe. As evident from the comparison of cosmological simulation results in Figure 11, galaxies could exhibit lower metallicities if metals are efficiently ejected from galaxies or diluted by gas accretion from inflow of pristine gas, as observed in simulations such as FIRE and Astraeus . Furthermore, other events such as galaxy mergers and AGN activities may also play a role in modulating star-formation, causing metal redistribution, and potentially influencing the SFR-MZ relationship at highredshift (e.g., Springel et al. 2005; Torrey et al. 2012; Weinberger et al. 2018). Due to the limited sample size and associated metallicity errors, particularly at z > 8, it is challenging to make definitive conclusions from the current study. Further statistical investigations of metallicity are crucial to confirm the suggested lack of \nevolution in the SFR-MZ relation up to z ∼ 8, followed by a decrease at higher redshifts. These findings need to be rigorously compared with theoretical studies to better understand the underlying physics that govern early galaxy evolution.", 'ACKNOWLEDGEMENTS': 'We are grateful to Hidenobu Yajima, Hajime Fukushima, Chris Lovell, Yurina Nakazato, and the anonymous referee for useful comments and discussions that improved our manuscript. This work is based on observations made with the NASA/ESA/CSA James Webb Space Telescope. The data were obtained from the Mikulski Archive for Space Telescopes at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-03127 for JWST. These observations are associated with program #1324 (ERSGLASS), #1345 (ERS-CEERS), #2561 (UNCOVER), and #2736 (ERO). The authors acknowledge the teams \nof JWST commissioning, ERO, GLASS, UNCOVER, and CEERS for developing their observing programs with a zero-exclusive-access period. Moreover, this work is based in part on observations taken by the CANDELS Multi-Cycle Treasury Program with the NASA/ESA HST, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. This paper is supported by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan, as well as the joint research program of the Institute of Cosmic Ray Research (ICRR), the University of Tokyo. This work is supported by KAKENHI (JP19H00697, JP20H00180, and JP21H04467) Grant-in-Aid for Scientific Research (A) through the Japan Society for the Promotion of Science. In addition, KN acknowledges support from JSPS KAKENHI Grant JP20K22373. YI is supported by JSPS KAKENHI Grant JP21J20785, and also acknowledges funding from the Hayakawa Satio Fund awarded by the Astronomical Society of Japan. YH acknowledges support from JSPS KAKENHI Grant JP21K13953.', 'REFERENCES': "Abel, T., Bryan, G. L., & Norman, M. 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S. 1989, ApJ, 345, 245, doi: 10.1086/167900 Carnall, A. C., Begley, R., McLeod, D. J., et al. 2023, MNRAS, 518, L45, doi: 10.1093/mnrasl/slac136 Ceverino, D., Glover, S. C. O., & Klessen, R. S. 2017, MNRAS, 470, 2791, doi: 10.1093/mnras/stx1386 Chabrier, G. 2003, PASP, 115, 763, doi: 10.1086/376392 Christensen, L., Laursen, P., Richard, J., et al. 2012, MNRAS, 427, 1973,\n``` \ndoi: 10.1111/j.1365-2966.2012.22007.x", 'A. MZ AND SFR-MZ RELATIONS BASED ONLY ON GALAXY PROPERTIES DERIVED FROM SED FITTING TO NIRCAM PHOTOMETRY': 'In the main text, we present the mass-metallicity (MZ) and the SFR-MZ relations for the full JWST sample with the NIRSpec spectra. Out of the 135 JWST objects with metallicity measurements, 81 objects have NIRCam imaging data, and their stellar masses M /star are derived based on SED fitting to the NIRCam photometry. The remaining 54 objects, all of which are in the CEERS field, are not covered by NIRCam, and their M /star is empirically estimated using UV luminosities from HST photometry (Section 2.3). Additionally, their SFRs are estimated from M UV with some assumptions for comparison with SFR(H β ) (Section 3.4). We note that 23 out of the 54 objects are not detected in HST and have only upper limits on M /star and SFR. Since objects with only an upper limit on M /star are not considered when deriving the average MZ relations and best-fit, there are practically 31 objects (i.e., = 54 -23), accounting for ∼ 28% of the sample, which may introduce bias due to different methods used for M /star and SFR estimation. In this appendix, we present the main figures with only the JWST objects having NIRCam photometry and properties derived from SED fitting, in order to examine how the main results can be affected by excluding the 31 objects with less certain estimations of M /star and SFR. \nFigure A1 presents the MZ relation using only the JWST objects with M /star based on NIRCam photometry. Their average points are shown with the inverted purple stars. The red stars and the gray long-dashed line represent the average relations based on the full sample of the JWST objects as shown in Figure 10. We confirm that the difference between the two is negligibly small, and the MZ relation based only on the JWST objects with reliable M /star estimates is fully consistent with that based on the full sample, within the uncertainty. \nLikewise, Figures A2, A3, and A4 are regenerated plots of Figures 11, 13, and 15 from the main text, respectively, using only the JWST objects with NIRCam photometry. These figures confirm that our results remain unchanged when adopting the subsample, indicating that no clear biases are introduced by using the 31 objects with M /star and SFR estimated in a different manner. We have also checked the consistency of M /star between the two methods by comparing the estimates from the 81 CEERS objects with both measurements, and found that ∼ 90% of them have consistent M /star values at the 2 σ level. The remaining 8 objects show overestimated M UV -based M /star values compared to NIRCam-based M /star at > 2 σ level. Although such a small fraction of outliers may exist among the objects lacking NIRCam photometry and introduce additional scatter in the MZ and MZ-SFR relations (e.g., Fig 11(a) versus Fig A2(a) at the high-mass region), the impact is minor as demonstrated in this appendix. \nFigure A1. Similar to Figure 10, but excluding the individual JWST objects whose stellar masses are empirically derived, so all the plotted data-points have stellar masses that are derived based on SED fitting to NIRCam photometry. The average of the JWST objects plotted on this figure are represented by purple inverted stars. As a comparison, the average relations based on the full sample are shown with the red stars and the gray long-dashed line, as presented in 10, which are positioned almost behind the purple inverted stars. This confirms that our results remain unchanged when adopting the subsample with certain estimations of M /star . \n<!-- image --> \nFigure A2. Similar to Figure 11, but excluding the individual JWST objects without JWST/NIRCam photometry. The average relations of the JWST objects in each panel are represented by purple inverted stars. \n<!-- image --> \nFigure A3. Similar to Figure 13, but excluding the individual JWST objects without JWST/NIRCam photometry. The average relations of the JWST objects in each panel are represented by purple inverted stars. \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure A4. Similar to Figure 15, but excluding the individual JWST objects without JWST/NIRCam photometry. The average relations of the JWST objects in each panel are represented by purple inverted stars. In the z = 8 -10 panel, only a single average point is displayed due to the limited sample size. \n<!-- image -->', 'B. COMPARING WITH EMPIRICALLY ESTIMATED METALLICITY RELATIONS AT LOW-REDSHIFTS USING STRONG LINE INDICATORS': 'In the main text we refer to Andrews & Martini (2013) and Sanders et al. (2021) to compare our results with the MZ relations at lower redshifts. This is because their metallicities are reliably determined with the direct T e method. Moreover, the z = 0 relation of Andrews & Martini (2013) is explored down to M /star ∼ 10 7 . 4 M /circledot and can be directly compared with most of the JWST objects. In this appendix, we also present comparisons with the other MZ relations at low-redshifts whose metallicities are empirically estimated with the strong line indicators. \nFigure B5 shows the strong line-based MZ relations at z /lessorsimilar 3 in addition to our results (the average points and the best-fit) and the T e -based MZ relations at z = 0 -3 (Andrews & Martini 2013; Sanders et al. 2021) as in Figure 10. The strong line-based MZ relations include those for z ∼ 0 . 7 galaxies (Savaglio et al. 2005), z ∼ 1 -2 (Papovich et al. 2022), z ∼ 2 -3 (Erb et al. 2006; Steidel et al. 2014), and z ∼ 3 -4 (Troncoso et al. 2014 (see also Maiolino et al. 2008; Mannucci et al. 2009), Onodera et al. 2016). The results by Savaglio et al. (2005) and Erb et al. (2006) are revisited by Maiolino et al. (2008) and their metallicities are re-measured. We also apply corrections to the mass scale to have the same Chabrier (2003) IMF. For the z = 2 -3 galaxies the [N ii ] / H α index is mainly used for the metallicity estimations, while for the other redshift studies combinations of [O ii ], [O iii ] and H β (i.e., R23-index and O32-index in practice) are utilized, in addition to [Ne iii ] when available. Although there are some differences in the MZ relations between different studies and methods for a similar redshift, several factors account for the scatter, such as different SFR activities for the different samples and variations of the metallicity indicators. Despite the slight differences, these overall confirm a redshift evolution of MZ relation, i.e., a decreasing trend of metallicity in the mass range of 10 9 -10 11 M /circledot . Comparing with these strong line-based MZ relations, our main conclusion does not change that the JWST objects at z = 4 -10 are distributed along the simple extrapolation of the MZ relations at z ∼ 2 -3 towards lower mass. \nFigure C6 shows the redshift evolution of the SFR-MZ relation by only using the JWST objects having µ 0 . 66 > 7 . 5. Their average are represented by the inverted blue stars. As a comparison, the average relations based on the full sample are shown with the red stars, as presented in Figure 13, which are positioned almost behind the blue stars. This confirms that the same results arise from the subsample as discussed in the main text, i.e., (i) the same Andrews & Martini \n<!-- image --> \nFigure B5. Similar to Figure 10, but excluding the individual JWST objects, and instead focusing on comparisons with low-redshift MZ relations based on empirically estimated metallicities using strong line indicators, as indicated in the legend. Each MZ relation is depicted in the mass range explored in the original paper. \n<!-- image -->', 'C. EVOLUTION OF THE SFR-MZ RELATION WITHOUT THE LOW-MASS END GALAXIES': 'Figure 14(a) highlights that the majority of the JWST sample (83 %) has µ 0 . 66 > 7 . 5 and is directly compared with the lower-redshift galaxies in the SFR-MZ relation of Andrews & Martini (2013). Still, we rely on the simple extrapolation of the relation towards lower µ 0 . 66 for the remaining galaxies and discuss the evolution of the SFR-MZ relation together with the direct comparisons of galaxies at µ 0 . 66 > 7 . 5. In this appendix, we present how the low-mass end galaxies at µ 0 . 66 < 7 . 5 can impact the view of the evolution of the SFR-MZ relation. \n(2013) relation between mass, metallicity, and SFR can explain the properties of galaxies up to z ∼ 8, and (ii) a deficit in metallicity is indicated at z > 8. We therefore conclude that there is no critical impact of the low-mass end galaxies and the extrapolation of the relation at least in the range µ 0 . 66 ∼ 7 -7 . 5 for the discussion of the evolution of the SFR-MZ relation. \nFigure C6. Similar to the right panel of Figure 13, but excluding the individual JWST objects at µ 0 . 66 < 7 . 5. The average relations of the subsample are denoted with the blue inverted stars. As a comparison, the average relations based on the full sample are shown with the red stars, as presented in Figure 13, which are positioned almost behind the blue inverted stars. This confirms that our results on the redshift evolution of the SFR-MZ relation remain unchanged when considering only the objects at µ 0 . 66 > 7 . 5. \n<!-- image -->', 'D. SUMMARY OF PROPERTIES FOR JWST OBJECTS IN A TABULAR FORMAT': 'In Table D1, we have compiled the essential properties of the complete sample of JWST objects presented in this paper, based on our improved reduction and calibration of the NIRSpec data from ERO, ERS-GLASS, and ERSCEERS. \n34 \nNakajima et al. \nd \na \no \nr \nB \n, \nd \na \no \nr \nB \n, \n1 . 8 6 1 . 3 8 1 . 4 1 1 . 3 4 1 . 3 3 1 . 7 2 2 . 0 6 1 . 7 6 1 . 4 0 1 . 1 9 1 . 7 0 7 . 9 1 1 . 4 8 1 . 4 0 1 . 5 4 2 . 3 9 \n= \n0 \n7 \n. \n8 \n= \n8 \n1 \n. \n3 \n= \n8 \n7 \n. \n1 \n= \n3 \n2 \n. \n3 \n= \n= \n= \n= \n= \n= \n= \n= \n= \n= \n= \n= \n= \n= \n= \n= \n2 \n= \no \n2 \n= \no \n2 \n= \no \n2 \n= \no \nd \na \no \nr \nB \n, \n2 \n= \no \n2 \n= \no \n1 \n0 \n9 \n5 \n4 \n0 \nO \nR \nE \n1 \n4 \n4 \n1 \n5 \n0 \nO \nR \nE \n1 \n5 \n5 \n3 \n6 \n0 \nO \nR \nE \n1 \n0 \n4 \n1 \n8 \n0 \nO \nR \nE \n1 \n2 \n1 \n6 \n0 \n1 \nO \nR \nE \n2 \n3 \n0 \n0 \n0 \n0 \nS \nR \nE \nE \nC \n2 \n7 \n0 \n0 \n0 \n0 \nS \nR \nE \nE \nC \n2 \n4 \n4 \n0 \n0 \n0 \nS \nR \nE \nE \nC \n2 \n7 \n6 \n0 \n0 \n0 \nS \nR \nE \nE \nC \n2 \n4 \n1 \n3 \n0 \n0 \nS \nR \nE \nE \nC \n2 \n3 \n2 \n3 \n0 \n0 \nS \nR \nE \nE \nC \n2 \n5 \n5 \n3 \n0 \n0 \nS \nR \nE \nE \nC \n2 \n6 \n5 \n3 \n0 \n0 \nS \nR \nE \nE \nC \n2 \n2 \n6 \n3 \n0 \n0 \nS \nR \nE \nE \nC \n2 \n1 \n8 \n3 \n0 \n0 \nS \nR \nE \nE \nC \n2 \n6 \n8 \n3 \n0 \n0 \nS \nR \nE \nE \nC \n2 \n7 \n9 \n3 \n0 \n0 \nS \nR \nE \nE \nC \n2 \n3 \n0 \n4 \n0 \n0 \nS \nR \nE \nE \nC \n2 \n7 \n0 \n4 \n0 \n0 \nS \nR \nE \nE \nC \n2 \n9 \n3 \n4 \n0 \n0 \nS \nR \nE \nE \nC \ne \nt \no \nN \nD \nI \n) \n¶ \n( \n0 \n0 \n0 \n0 \n1 \nS \nS \nA \nL \nG \n1 \n0 \n0 \n0 \n0 \n1 \nS \nS \nA \nL \nG \n3 \n0 \n0 \n0 \n0 \n1 \nS \nS \nA \nL \nG \n5 \n0 \n0 \n0 \n0 \n1 \nS \nS \nA \nL \nG \n1 \n2 \n0 \n0 \n1 \nS \nS \nA \nL \nG \n8 \n0 \n0 \n0 \n5 \n1 \nS \nS \nA \nL \nG \n8 \n0 \n0 \n2 \n2 \n2 \n0 \n9 \n2 \n0 0 3 1 0 0 6 0 1 0 0 0 0 0 7 5 0 0 \n5 \nS \nS \nA \nL \nG \n1 \nS \nS \nA \nL \nG \n1 \nS \nS \nA \nL \nG \n5 \nS \nS \nA \nL \nG \n8 \nS \nS \nA \nL \nG \n1 \nS \nS \nA \nL \nG \n6 \n6 \n0 \n0 \n4 \nS \nS \nA \nL \nG \n3 \n3 \n1 \n0 \n6 \n1 \nS \nS \nA \nL \nG \n9 \n2 \n0 \n0 \n8 \nS \nS \nA \nL \nG \nD \nI \nMass-Metallicity Star-Formation Relations at \nz \n= 4 \n- \n10 \n35 \na d a d \no \nr \no \nr \n8 \n9 \n4 \n0 \n0 \nS \nR \nE \nE \nC \n5 \n1 \n5 \n0 \n0 \nS \nR \nE \nE \nC \n4 \n3 \n5 \n0 \n0 \nS \nR \nE \nE \nC \n2 \n4 \n5 \n0 \n0 \nS \nR \nE \nE \nC \n5 \n4 \n5 \n0 \n0 \nS \nR \nE \nE \nC \n7 \n7 \n5 \n0 \n0 \nS \nR \nE \nE \nC \n3 \n0 \n6 \n0 \n0 \nS \nR \nE \nE \nC \n8 \n1 \n6 \n0 \n0 \nS \nR \nE \nE \nC \n9 \n6 \n6 \n0 \n0 \nS \nR \nE \nE \nC \n0 \n7 \n6 \n0 \n0 \nS \nR \nE \nE \nC \n2 \n7 \n6 \n0 \n0 \nS \nR \nE \nE \nC \n6 \n8 \n6 \n0 \n0 \nS \nR \nE \nE \nC \n9 \n8 \n6 \n0 \n0 \nS \nR \nE \nE \nC \n8 \n9 \n6 \n0 \n0 \nS \nR \nE \nE \nC \n7 \n0 \n6 \n1 \n7 \n1 \n6 \n4 \n2 \n9 \n9 \n2 \nS \nR \nE \nE \nC \nS \nR \nE \nE \nC \nS \nR \nE \nE \nC \nS \nR \nE \nE \nC \nS \nR \nE \nE \nC \nS \nR \nE \nE \nC \n3 \n3 \n9 \n0 \n0 \nS \nR \nE \nE \nC \n9 \n1 \n0 \n1 \n0 \nS \nR \nE \nE \nC \n3 \n2 \n0 \n1 \n0 \nS \nR \nE \nE \nC \n5 \n2 \n0 \n1 \n0 \nS \nR \nE \nE \nC \n7 \n2 \n0 \n1 \n0 \nS \nR \nE \nE \nC \n9 \n2 \n0 \n1 \n0 \nS \nR \nE \nE \nC \n8 \n3 \n0 \n1 \n0 \nS \nR \nE \nE \nC \n4 \n6 \n0 \n1 \n0 \nS \nR \nE \nE \nC \n5 \n6 \n0 \n1 \n0 \nS \nR \nE \nE \nC \n2 \n0 \n1 \n1 \n0 \nS \nR \nE \nE \nC \n5 \n1 \n1 \n1 \n0 \nS \nR \nE \nE \nC \n2 \n4 \n1 \n1 \n0 \nS \nR \nE \nE \nC \n3 \n4 \n1 \n1 \n0 \nS \nR \nE \nE \nC \n9 \n4 \n1 \n1 \n0 \nS \nR \nE \nE \nC \n0 \n6 \n1 \n1 \n0 \nS \nR \nE \nE \nC \n3 \n6 \n1 \n1 \n0 \nS \nR \nE \nE \nC \n3 \n7 \n1 \n1 \n0 \nS \nR \nE \nE \nC \n7 \n0 \n2 \n1 \n0 \nS \nR \nE \nE \nC \n36 \nNakajima et al. \nD \nI \nd \na \no \nr \nd \na \no \nr \nd \na \no \nr \n7 \n1 \n2 \n1 \n0 \nS \nR \nE \nE \nC \n6 \n3 \n2 \n1 \n0 \nS \nR \nE \nE \nC \n4 \n4 \n2 \n1 \n0 \nS \nR \nE \nE \nC \n7 \n6 \n2 \n1 \n0 \nS \nR \nE \nE \nC \n9 \n8 \n2 \n1 \n0 \nS \nR \nE \nE \nC \n4 \n9 \n2 \n1 \n0 \nS \nR \nE \nE \nC \n4 \n2 \n3 \n1 \n0 \nS \nR \nE \nE \nC \n4 \n3 \n3 \n1 \n0 \nS \nR \nE \nE \nC \n8 \n5 \n3 \n1 \n0 \nS \nR \nE \nE \nC \n5 \n6 \n3 \n1 \n0 \nS \nR \nE \nE \nC \n4 \n7 \n3 \n1 \n0 \nS \nR \nE \nE \nC \n8 \n8 \n3 \n1 \n0 \nS \nR \nE \nE \nC \n5 \n9 \n3 \n1 \n0 \nS \nR \nE \nE \nC \n0 \n0 \n4 \n1 \n0 \nS \nR \nE \nE \nC \n1 \n0 \n0 \n1 \n0 \n2 \n3 \n3 \n9 \n4 \n2 \n5 \n0 1 4 0 1 4 0 1 4 0 1 4 0 1 4 0 1 4 \nS \nR \nE \nE \nC \nS \nR \nE \nE \nC \nS \nR \nE \nE \nC \nS \nR \nE \nE \nC \nS \nR \nE \nE \nC \nS \nR \nE \nE \nC \n5 \n6 \n4 \n1 \n0 \nS \nR \nE \nE \nC \n7 \n6 \n4 \n1 \n0 \nS \nR \nE \nE \nC \n7 \n7 \n4 \n1 \n0 \nS \nR \nE \nE \nC \n8 \n1 \n5 \n1 \n0 \nS \nR \nE \nE \nC \n4 \n3 \n5 \n1 \n0 \nS \nR \nE \nE \nC \n6 \n3 \n5 \n1 \n0 \nS \nR \nE \nE \nC \n9 \n3 \n5 \n1 \n0 \nS \nR \nE \nE \nC \n4 \n4 \n5 \n1 \n0 \nS \nR \nE \nE \nC \n1 \n6 \n5 \n1 \n0 \nS \nR \nE \nE \nC \n5 \n6 \n5 \n1 \n0 \nS \nR \nE \nE \nC \n5 \n0 \n6 \n1 \n0 \nS \nR \nE \nE \nC \n7 \n1 \n6 \n1 \n0 \nS \nR \nE \nE \nC \n0 \n2 \n6 \n1 \n0 \nS \nR \nE \nE \nC \n6 \n2 \n6 \n1 \n0 \nS \nR \nE \nE \nC \n4 \n3 \n6 \n1 \n0 \nS \nR \nE \nE \nC \n1 \n5 \n6 \n1 \n0 \nS \nR \nE \nE \nC \n8 \n5 \n6 \n1 \n0 \nS \nR \nE \nE \nC \n5 \n6 \n6 \n1 \n0 \nS \nR \nE \nE \nC \nD \nI \nMass-Metallicity Star-Formation Relations at \nz \n= 4 \n- \n10 \n37 \nd \na \no \nr \n2 \n7 \n6 \n1 \n0 \nS \nR \nE \nE \nC \n7 \n7 \n6 \n1 \n0 \nS \nR \nE \nE \nC \n1 \n9 \n6 \n1 \n0 \nS \nR \nE \nE \nC \n9 \n9 \n6 \n1 \n0 \nS \nR \nE \nE \nC \n6 \n0 \n7 \n1 \n0 \nS \nR \nE \nE \nC \n2 \n3 \n7 \n1 \n0 \nS \nR \nE \nE \nC \n6 \n4 \n7 \n1 \n0 \nS \nR \nE \nE \nC \n6 \n5 \n7 \n1 \n0 \nS \nR \nE \nE \nC \n9 \n5 \n7 \n1 \n0 \nS \nR \nE \nE \nC \n7 \n6 \n7 \n1 \n0 \nS \nR \nE \nE \nC \n6 \n3 \n8 \n1 \n0 \nS \nR \nE \nE \nC \n2 \n1 \n9 \n1 \n0 \nS \nR \nE \nE \nC \n3 \n5 \n9 \n1 \n0 \nS \nR \nE \nE \nC \n0 \n0 \n0 \n2 \n0 \nS \nR \nE \nE \nC \n6 \n3 \n9 \n8 \n6 \n1 \n3 \n2 \n0 \n4 \n8 \n6 \nS \nR \nE \nE \nC \nS \nR \nE \nE \nC \nS \nR \nE \nE \nC \nS \nR \nE \nE \nC \nS \nR \nE \nE \nC \nS \nR \nE \nE \nC \n4 \n7 \n1 \n2 \n0 \nS \nR \nE \nE \nC \n2 \n6 \n3 \n2 \n0 \nS \nR \nE \nE \nC \n2 \n8 \n7 \n2 \n0 \nS \nR \nE \nE \nC \n4 \n8 \n5 \n3 \n0 \nS \nR \nE \nE \nC \n4 \n4 \n1 \n4 \n0 \nS \nR \nE \nE \nC \n6 \n9 \n1 \n4 \n0 \nS \nR \nE \nE \nC \n0 \n1 \n2 \n4 \n0 \nS \nR \nE \nE \nC \n8 \n8 \n2 \n8 \n0 \nS \nR \nE \nE \nC \n5 \n9 \n9 \n0 \n1 \nS \nR \nE \nE \nC \n7 \n1 \n1 \n1 \n1 \nS \nR \nE \nE \nC \n3 \n8 \n3 \n1 \n1 \nS \nR \nE \nE \nC \n4 \n2 \n6 \n1 \n1 \nS \nR \nE \nE \nC \n6 \n7 \n6 \n1 \n1 \nS \nR \nE \nE \nC \n8 \n2 \n7 \n1 \n1 \nS \nR \nE \nE \nC \n1 \n2 \n2 \n2 \n1 \nS \nR \nE \nE \nC \n6 \n9 \n4 \n2 \n1 \nS \nR \nE \nE \nC \n7 \n7 \n7 \n4 \n1 \nS \nR \nE \nE \nC \n9 \n2 \n3 \n1 \n3 \nS \nR \nE \nE \nC \n38 \nNakajima et al. \n0 \n9 \n5 \n4 \n1 \n9 \n8 \n. \n4 \n1 \n2 \n7 \n9 \n6 \n7 \n3 \nS \nR \nE \nE \nC \n0 \n0 \n5 \n8 \n0 \n9 \n8 \n. \n4 \n1 \n2 \n2 \n7 \n0 \n0 \n8 \nS \nR \nE \nE \nC \n0 \n6 \n7 \n2 \n1 \n6 \n9 \n. \n4 \n1 \n2 \n3 \n8 \n0 \n0 \n8 \nS \nR \nE \nE \nC \n0 \n4 \n5 \n0 \n6 \n9 \n8 \n. \n4 \n1 \n2 \n9 \n3 \n2 \n0 \n8 \nS \nR \nE \nE \nC \n0 \n8 \n9 \n7 \n7 \n2 \n9 \n. \n4 \n1 \n2 \n2 \n7 \n3 \n0 \n8 \nS \nR \nE \nE \nC \n0 \n4 \n7 \n0 \n8 \n9 \n8 \n. \n4 \n1 \n2 \n4 \n7 \n3 \n0 \n8 \nS \nR \nE \nE \nC \n0 \n6 \n5 \n0 \n2 \n1 \n8 \n. \n4 \n1 \n2 \n2 \n3 \n4 \n0 \n8 \nS \nR \nE \nE \nC \n0 \n5 \n1 \n1 \n3 \n4 \n8 \n. \n4 \n1 \n2 \n5 \n4 \n4 \n0 \n8 \nS \nR \nE \nE \nC \n0 \n4 \n2 \n9 \n3 \n7 \n7 \n. \n4 \n1 \n2 \n3 \n7 \n5 \n0 \n8 \nS \nR \nE \nE \nC \n0 \n7 \n2 \n6 \n9 \n6 \n7 \n. \n4 \n1 \n2 \n6 \n7 \n5 \n0 \n8 \nS \nR \nE \nE \nC \n0 \n5 \n6 \n8 \n1 \n7 \n7 \n. \n4 \n1 \n2 \n6 \n9 \n5 \n0 \n8 \nS \nR \nE \nE \nC \n0 \n5 \n8 \n9 \n4 \n8 \n8 \n. \n4 \n1 \n2 \n0 \n1 \n7 \n0 \n8 \nS \nR \nE \nE \nC \n0 \n0 \n3 \n6 \n1 \n9 \n8 \n. \n4 \n1 \n2 \n6 \n1 \n9 \n0 \n8 \nS \nR \nE \nE \nC \n0 \n3 \n8 \n1 \n8 \n0 \n6 \n0 \n9 \n0 \n1 \n0 \n5 \n0 \n2 \n0 \n0 \n1 \n9 \n. \n4 \n1 \n2 \n4 \n5 \n9 \n0 \n8 \nS \nR \nE \nE \nC \n5 4 7 0 4 4 9 8 4 2 6 2 9 8 2 9 1 1 \n0 \n8 \n. \n4 \n1 \n2 \n8 \n1 \n0 \n1 \n8 \nS \nR \nE \nE \nC \n0 \n8 \n. \n4 \n1 \n2 \n2 \n2 \n0 \n1 \n8 \nS \nR \nE \nE \nC \n0 \n8 \n. \n4 \n1 \n2 \n6 \n2 \n0 \n1 \n8 \nS \nR \nE \nE \nC \n1 \n8 \n. \n4 \n1 \n2 \n2 \n3 \n0 \n1 \n8 \nS \nR \nE \nE \nC \n8 \n7 \n. \n4 \n1 \n2 \n9 \n4 \n0 \n1 \n8 \nS \nR \nE \nE \nC \n9 \n7 \n. \n4 \n1 \n2 \n3 \n6 \n0 \n1 \n8 \nS \nR \nE \nE \nC \n0 \n7 \n0 \n5 \n0 \n2 \n8 \n. \n4 \n1 \n2 \n8 \n6 \n0 \n1 \n8 \nS \nR \nE \nE \nC \n0 \n6 \n8 \n9 \n9 \n1 \n7 \n. \n4 \n1 \n2 \n3 \n4 \n0 \n2 \n8 \nS \nR \nE \nE \nC \n0 \n1 \n6 \n5 \n6 \n6 \n7 \n. \n4 \n1 \n2 \n2 \n5 \n0 \n2 \n8 \nS \nR \nE \nE \nC \n0 \n7 \n5 \n5 \n0 \n0 \n9 \n. \n4 \n1 \n2 \n0 \n0 \n3 \n2 \n8 \nS \nR \nE \nE \nC \n0 \n3 \n1 \n2 \n3 \n5 \n8 \n. \n4 \n1 \n2 \n7 \n0 \n5 \n2 \n8 \nS \nR \nE \nE \nC \n0 \n9 \n0 \n4 \n0 \n0 \n9 \n. \n4 \n1 \n2 \n8 \n9 \n3 \n3 \n8 \nS \nR \nE \nE \nC \n0 \n0 \n7 \n0 \n8 \n3 \n9 \n. \n4 \n1 \n2 \n9 \n3 \n4 \n3 \n8 \nS \nR \nE \nE \nC \n0 \n7 \n4 \n8 \n5 \n0 \n9 \n. \n4 \n1 \n2 \n2 \n0 \n5 \n3 \n8 \nS \nR \nE \nE \nC \n0 \n3 \n9 \n6 \n6 \n5 \n9 \n. \n4 \n1 \n2 \n2 \n9 \n5 \n3 \n8 \nS \nR \nE \nE \nC \n0 \n7 \n2 \n7 \n5 \n8 \n7 \n. \n4 \n1 \n2 \n2 \n7 \n7 \n3 \n8 \nS \nR \nE \nE \nC \n0 \n7 \n1 \n4 \n1 \n2 \n8 \n. \n4 \n1 \n2 \n9 \n7 \n7 \n3 \n8 \nS \nR \nE \nE \nC \n0 \n2 \n2 \n0 \n3 \n0 \n8 \n. \n4 \n1 \n2 \n6 \n5 \n8 \n3 \n8 \nS \nR \nE \nE \nC \n0 \n8 \n7 \n3 \n8 \n2 \n8 \n. \n4 \n1 \n2 \n0 \n6 \n8 \n3 \n8 \nS \nR \nE \nE \nC \n0 \n4 \n0 \n2 \n1 \n3 \n8 \n. \n4 \n1 \n2 \n6 \n3 \n8 \n5 \n8 \nS \nR \nE \nE \nC \n. \nA \n. \nR \nD \nI \n) \ng \ne \nd \n( \n) \nd \ne \nu \nn \ni \nt \nn \no \nc \n( \n1 \nD \ne \nl \nb \na \nT \n3 \n2 \nR \n3 \nR \n) \nβ \nH \n( \nW \nE \nR \nF \nS \ng \no \nl \n/star \nM \ng \no \nl \nV \nU \nM \nt \nf \ni \nh \ns \nd \ne \nR \n. \nl \nc \ne \nD \n. \nA \n. \nR \nD \nI \n) \n/star \n( \n) \n/star \n( \n) \n˚A \n( \n) \n1 \n- \nr \ny \n/circledot \nM \n( \n) \n/circledot \nM \n( \n) \ng \na \nm \n( \n) \ng \ne \nd \n( \n) \ng \ne \nd \n( \nr \no \nf \n) \nd \ne \nt \na \nm \ni \nt \ns \ne \nt \no \nn \ne \nr \na \ns \ne \ni \nt \ni \nc \ni \nl \nl \na \nt \ne \nm \ne \nh \nt \ne \nc \nn \ne \nh \nd \nn \na \n( \nn \nw \no \nh \ns \nt \no \nn \ne \nr \na \ns \ne \nu \nl \na \nv \ne \nh \nT \n. \nl \ne \nv \ne \nl \nσ \n3 \ne \nh \nt \nt \na \ns \ne \nu \nl \na \nv \nt \ni \nm \ni \nl \n- \nr \ne \nw \no \nL \n) \n/star \n( \n- \ne \nt \no \nN \na \ns \ne \ns \ns \na \nm \nr \na \nl \nl \ne \nt \ns \ne \nh \nt \ns \nu \nh \nt \nd \nn \na \n, \ny \nr \nt \ne \nm \no \nt \no \nh \np \nm \na \nC \nR \nI \nN \ne \nh \nt \ne \nv \na \nh \n1 \n= \ng \na \nfl \nh \nt \ni \nw \ns \nt \nc \ne \nj \nb \nO \n. \ng \na \nfl \ny \nr \nt \ne \nm \no \nt \no \nh \nP \n) \n† \n( \n. \nd \ne \nt \na \nu \nl \na \nv \ne \ny \nl \nr \ne \np \no \nr \np \ne \nm \ne \nT \nt \nc \ne \nr \ni \nd \ne \nh \nt \n; \nd \ne \nt \np \no \nd \na \ns \ni \nd \no \nh \nt \ne \nm \nh \nc \ni \nh \nw \ng \nn \ni \ny \nf \ni \nc \ne \np \ns \n, \ng \na \nfl \ny \ni \nt \ni \nc \nl \nl \na \nt \ne \nM \n) \n‡ \n( \n. \nT \nS \nH \nf \no \ns \ne \ni \nt \ni \ns \no \nn \ni \nm \nu \nl \nV \nU \ne \nh \nt \nh \nt \ni \nw \nd \ne \nt \na \nm \ni \nt \ns \ne \nMass-Metallicity Star-Formation Relations at \nz \n= 4 \n- \n10 \n39 \n) \nO 3 2 l o g ( O / H ) F l a g s ( /star ) + 1 2 P ( † ) M ( ‡ t h o s e w h o s e m e a s u r e d f t h e d u s t -r e d d e n i n g i s r e r e l i a b l y d e t e r m i n e d m a s s e s a n d S F R s a r e t h o d o r t h e e m p i r i c a l t s , t h e v a l u e s o f M U V , p o i n t i n g s / g r a t i n g s , t h e e c t s w i t h s p e c t r o s c o p i c r d e r i v i n g t h e M Z a n d t i m a t i o n b a s e d o n t h e \ni \nn \ne \nv \ni \ng \ne \nr \na \n2 \n3 \nO \nd \nn \na \n3 \n2 \nR \n. \nl \ne \nv \ne \nl \nσ \n5 \n. \n2 \ne \nh \nt \nt \na \n8 \n9 \n. \n2 \nf \no \ne \nu \nl \na \nv \nl \na \nc \ni \nt \ne \nr \no \ne \nh \nt \ne \nh \nt \nh \nt \ni \nw \nt \nn \ne \nt \ns \ni \ns \nn \no \nc \nt \no \nn \ns \ni \no \ni \nt \na \nr \ne \nn \ni \nl \n9 \n5 \n9 \n4 \n/ \n7 \n0 \n0 \n5 \nλ \nλ \n] \ni \ni \ni \nO \n[ \nr \ni \ne \nh \nt \n, \n0 \n= \ng \na \nfl \nh \nt \ni \nw \ns \nt \nc \ne \nj \nb \no \ne \nh \nt \nr \no \nF \n. \ns \ne \ni \nt \ni \ns \no \nn \ni \nm \nu \nl \nβ \nH \nd \ne \nt \nc \ne \nr \nr \no \nc \ns \ns \no \nl \n- \nt \ni \nl \ns \ne \nh \nt \nh \nt \ni \nw \nd \ne \nv \ni \nr \ne \nd \ne \nr \na \ns \nR \nF \nS \nd \nn \na \nt \nfi \nD \nE \nS \ne \nh \nt \nh \nt \ni \nw \nj \nb \no \ne \nh \nT \n. \nn \ne \nv \ni \ng \ne \nr \na \ny \nt \ni \nc \ni \nl \nl \na \nt \ne \nm \nd \nn \na \n, \ns \no \ni \nt \na \nr \ne \nn \ni \nl \n, \n) \nβ \nH \n( \nW \nE \nf \no \ns \ne \nu \nl \na \nv \nd \ne \ng \na \nr \ne \nv \na \nr \ni \ne \nh \nt \nd \nn \na \n, \nd \ne \nt \no \nn \ns \ni \ns \ng \nn \ni \nt \na \nr \ng \n/ \ns \ng \nn \ni \nt \nn \ni \no \np \nf \no \nr \ne \nb \nm \nu \nn \no \nf \nd \ne \ns \nu \nt \no \nn \ne \nr \na \ns \nt \nc \ne \nj \nb \no \nd \na \no \nr \nB \ne \ns \ne \nh \nT \n. \nd \na \no \nr \nB \nh \nt \ni \nw \nd \ne \nk \nr \na \nm \ne \nr \na \n) \n3 \n2 \n0 \n2 \n. \nl \na \nt \ne \ne \nn \na \nk \ni \nr \na \nH \n( \ny \nb \nd \ne \nt \na \nc \ni \nd \nn \ni \ns \na \ns \nN \nG \nA \nf \no \ns \ne \nr \nu \nt \na \nn \ng \ni \ns \nc \ne \nj \nb \no \nd \ne \ns \nn \ne \nl \ne \nh \nt \nr \no \nF \n. \nd \ne \nt \no \nn \ns \ni \nµ \nr \no \nt \nc \na \nf \nn \no \ni \nt \na \nc \nfi \ni \nn \ng \na \nm \ne \nh \nt \n, \ns \nt \nc \ne \nj \nb \no \nd \ne \ns \nn \ne \nl \ny \nl \nl \na \nn \no \ni \nt \na \nt \ni \nv \na \nr \ng \ne \nh \nt \nr \no \nF \n) \n¶ \n( \n. \n3 \nR \nr \no \n3 \n2 \nR \nf \no \nr \no \nt \na \nc \ni \nd \nn \ni \nl \na \nr \ne \nv \ne \ns \nn \ni \nd \ne \nv \nr \ne \ns \nb \no \ny \nl \np \ni \nt \nl \nu \nm \ne \nr \na \nt \na \nh \nt \ns \nt \nc \ne \nj \nb \no \nS \nR \nE \nE \nC \ne \nh \nt \nr \no \nF \n. \nn \no \ni \nt \na \nc \nfi \ni \nn \ng \na \nm \ne \nh \nt \nr \no \nf \nd \ne \nt \nc \ne \nr \nr \no \nc \ny \nd \na \ne \nr \nl \na \ne \nr \na \nR \nF \nS \nd \nn \na \ns \ns \na \nm \ns \ne \ns \ns \na \nm \nr \na \nl \nl \ne \nt \ns \ne \nh \nt \nn \ni \ny \nt \nn \ni \na \nt \ne \nc \nn \nu \n, \n. \ne \n. \ni \n( \nl \ne \nv \ne \nl \nm \nu \nu \nn \ni \nt \nn \no \nc \ne \nh \nt \no \nt \ns \nN \nG \nA \nf \no \nn \no \ni \nt \nu \nb \ni \nr \nt \nn \no \nc \nn \nw \no \nn \nk \nn \nu \ne \nh \nt \no \nt \ne \nu \nd \ns \nn \no \ni \nt \na \nl \ne \nr \nZ \nM \n- \nR \nF \nS \n. \n) \ng \nn \ni \nt \nt \nfi \nD \nE \nS'} |
2024A&A...691A.281K | Context. Very Low Luminosity Objects VeLLOs are deeply embedded and extremely faint objects LSUBintSUB lt 0.1 LSUBSUB and are thought to be in the quiescent phase of the episodic accretion process. They fill an important gap in our understanding of star formation. Aims. The VeLLO in the isolated DC327218 cloud has undergone an outburst in the past 10SUP4SUP yr and is thus an ideal target for investigating the chemical inventory in the gas phase of an object of its type. The aim of this study is to investigate the direct impact of the outburst on the chemical processes in the object and identify molecules that can act as tracers of past heating events. Methods. Observations with the Atacama Pathfinder EXperiment APEX in four spectral windows in the frequency range of 213.6272.4 GHz have been carried out to identify molecules that can be directly linked to the past outburst to utilize the line fluxes column densities and the abundance ratios of the detected species to characterize the different physical components of the VeLLO and to probe for the presence of complex organic molecules. Results. Nitric oxide NO is detected for the first time in a source of this type and its formation could be induced by the sublimation of grainsurface species during the outburst. In addition the observations securely detect CHSUB3SUBOH HSUB2SUBCO DSUB2SUBCO SO SOSUB2SUB CO SUP13SUPCO CSUP18SUPO NSUB2SUBDSUPSUP HCOSUPSUP DCOSUPSUP HCN DCN HNC cCSUB3SUBHSUB2SUB and CSUB2SUBD. The upper state energies of the securely detected lines and their derived line intensity ratios indicate that most of the probed material stems from regions of cold gas in the envelope enshrouding the VeLLO in the DC327218 cloud with a temperature of 10 K. In addition cCSUB3SUBHSUB2SUB traces a second warmer gas reservoir with a temperature of 35 K. The high DH ratio derived from DSUB2SUBCO points toward its origin from the prestellar stage while deuteration of the gasphase species DCOSUPSUP DCN and CSUB2SUBD could still be ongoing in the gas in the envelope. Conclusions. The gas probed by the observations already cooled down after the past heating event caused by the outburst but it still has lasting effects on the chemistry in the envelope of the VeLLO. CHSUB3SUBOH HSUB2SUBCO SO SOSUB2SUB and CO sublimated from grains during the outburst and have not fully frozen out yet which indicates that the outburst took place lt 10SUP4SUP yr ago. A pathway to form NO directly in the gas phase is from the photodissociation products created after the sublimation of HSUB2SUBO and NHSUB3SUB from the ices. While the present time water snowline has likely retreated to a preoutburst small radius the volatile NO species is still extensively present in the gas phase as is evident by its high column density relative to methanol in the observations. This suggests that NO could be potentially used to trace the water snowline in outbursting sources. In order to rule out nonthermal desorption processes that could also have led to the formation of NO this proposition has to be verified with future observations at a higher spatial resolution and by searching for NO in additional targets. | 2024-11-01T00:00:00Z | ['2024A&A...691A.281K', '2024arXiv240904575K', 'arXiv:2409.04575', '10.1051/0004-6361/202450792', '10.48550/arXiv.2409.04575'] | ['astrochemistry', 'stars: formation', 'stars: low-mass', 'ISM: abundances', 'ISM: individual objects: DC3272+18', 'Astrophysics - Solar and Stellar Astrophysics', 'Astrophysics - Astrophysics of Galaxies'] | Postoutburst chemistry in a Very LowLuminosity Object Peculiar high abundance of nitric oxide | 2,024 | 175 | 0.52 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF'] | 1 | https://arxiv.org/pdf/2409.04575.pdf | {'Peculiar High Abundance of Nitric Oxide': "B. M. Kulterer 1 , 2 , S. F. Wampfler 2 , N. F. W. Ligterink 3 , 4 , N. Murillo 5 , 6 , T.-H. Hsieh 7 , M. K. McClure 8 , A. Boogert 9 , K. Kipfer 3 , P. Bjerkeli 10 , and M. N. Drozdovskaya 2 , 11 \n- 1 Center for Astrophysics | Harvard & Smithsonian, Cambridge, MA 02138, USA e-mail: [email protected]\n- 2 Center for Space and Habitability, Universität Bern, Gesellschaftsstrasse 6, 3012 Bern, Switzerland\n- 3 Space Research & Planetary Sciences, Physics Institute, University of Bern, 3012 Bern, Switzerland\n- 4 Faculty of Aerospace Engineering, Delft University of Technology, Delft, The Netherlands\n- 5 Instituto de Astronomía, Universidad Nacional Autónoma de México, AP106, Ensenada CP 22830, B. C., México\n- 6 Star and Planet Formation Laboratory, RIKEN Cluster for Pioneering Research, Wako, Saitama 351-0198, Japan\n- 7 Max Planck Institut für Extraterrestrische Physik (MPE), Giessenbachstrasse 1, 85748 Garching, Germany\n- 8 Leiden Observatory, Leiden University, PO Box 9513, NL-2300 RA Leiden, The Netherlands\n- 9 Institute for Astronomy, University of Hawai'i at Manoa, 2680 Woodlawn Drive, Honolulu, HI 96822, USA\n- 10 Chalmers University of Technology, Department of Space, Earth and Environment, 412 96 Gothenburg, Sweden\n- 11 Physikalisch-Meteorologisches Observatorium Davos und Weltstrahlungszentrum (PMOD / WRC), Dorfstrasse 33, CH-7260, Davos Dorf, Switzerland \nReceived -; accepted -", 'ABSTRACT': 'Context. Very Low Luminosity Objects (VeLLOs) are deeply embedded, and extremely faint objects (Lint < 0.1 L ⊙ ), and are thought to be in the quiescent phase of the episodic accretion process. They fill an important gap in our understanding of star formation. Aims. The VeLLO in the isolated DC3272 + 18 cloud has undergone an outburst in the past ∼ 10 4 yr, and is thus an ideal target for investigating the chemical inventory in the gas phase of an object of its type. The aim of this study is to investigate the direct impact of the outburst on the chemical processes in the object and identify molecules that can act as tracers of past heating events. Methods. Observations with the Atacama Pathfinder EXperiment (APEX) in four spectral windows in the frequency range of 213.6272.4 GHz have been carried out to identify molecules that can be directly linked to the past outburst, utilize the line fluxes, column densities, and the abundance ratios of the detected species to characterize the di ff erent physical components of the VeLLO, and probe for the presence of complex organic molecules. \nResults. Nitric oxide (NO) is detected for the first time in a source of this type, and its formation could be induced by the sublimation of grain-surface species during the outburst. In addition, the observations securely detect CH3OH, H2CO, D2CO, SO, SO2, CO, 13 CO, C 18 O, N2D + , HCO + , DCO + , HCN, DCN, HNC, c-C3H2, and C2D. The upper state energies of the securely detected lines and their derived line intensity ratios indicate that most of the probed material stems from regions of cold gas in the envelope enshrouding the VeLLO in the DC3272 + 18 cloud with a temperature of ∼ 10 K. In addition, c-C3H2 traces a second, warmer gas reservoir with a temperature of ∼ 35 K. The high D / H ratio derived from D2CO points towards its origin from the prestellar stage, while deuteration of the gas-phase species DCO + , DCN, and C2D could still be on-going in the gas in the envelope. \nConclusions. The gas probed by the observations has already cooled down after the past heating event caused by the outburst, but it still has lasting e ff ects on the chemistry in the envelope of the VeLLO. CH3OH, H2CO, SO, SO2, and CO have sublimated from grains during the outburst and did not fully freeze out yet, which indicates that the outburst took place < 10 4 yr ago. A pathway to form NO directly in the gas phase is from the photodissociation products created after the sublimation of H2O and NH3 from the ices. While the present time water snowline has likely retreated to pre-outburst small radius, the volatile NO species is still extensively present in the gas phase, as evident by its high column density relative to methanol in the observations. This suggests that NO could be potentially used to trace the water snowline in outbursting sources. In order to rule out non-thermal desorption processes that could also have led to the formation of NO, this proposition has to be verified with future observations at higher spatial resolution, and by searching for NO in additional targets. \nKey words. Astrochemistry - ISM: abundances - ISM: individual objects: DC3272 + 18 - Stars: formation - Stars: low-mass', '1. Introduction': 'Very Low Luminosity Objects (VeLLOs) have first been identified by the "Cores to Disk" (c2D) legacy survey of the Spitzer Space Telescope (Evans et al. 2003; Young et al. 2004), and have been classified as protostellar objects that are deeply embedded in molecular clouds with an internal luminosity of ≤ 0.1 L ⊙ \n(Young et al. 2004; di Francesco et al. 2007). The nature and the future evolution of VeLLOs are currently under debate, as only a handful of sources have been studied in detail, because their low luminosity makes observations challenging. Their low internal luminosity suggests that they could be either young Class 0 sources (e.g., IRAM 04191: André et al. (1999); Dunham et al. (2006); L1521F: Bourke et al. (2006); Cha-MMS1: \nBelloche et al. (2006); Väisälä et al. (2014); IRAS 16253: Hsieh et al. (2015, 2018)), or even extremely low-mass protostars or proto-brown dwarf candidates (e.g., L328: Lee et al. (2009, 2013) ; L1148: Kau ff mann et al. (2011); IC 348-SMM2E: Palau et al. (2014)). One hypothesis is that VeLLOs are objects in the quiescent phase of the episodic accretion process. Studies as carried out for L673-7 (Dunham et al. 2010b), and for L1521F (Takahashi et al. 2013) have derived average mass accretion rates from the outflows of these objects that are a few times higher than their current internal luminosity allows for. Moreover, the aforementioned studied objects have bolometric temperatures and spectral energy distributions that are similar to protostars. However, their luminosities are significantly fainter than the luminosity that would be expected for the least massive protostars in standard star formation models with a constant mass accretion rate (Shu et al. 1987; Dunham et al. 2006). Episodic accretion could bypass this issue, if outbursts boost the accretion rate for short periods of time, before the objects return back to a more quiescent phase which is accompanied by less e ffi cient mass accretion. The internal average luminosity of such objects would thus be smaller than an object with a constant mass accretion rate (Kenyon & Hartmann 1995; Dunham et al. 2010a). \nWhile detailed studies of the physical properties of VeLLOs are rare, even less is known about the chemistry in sources of this type. VeLLOs are faint objects, they are usually studied with typical molecular tracers such as CO and its isotopologues, N2H + or HCN to quantify their large-scale structures and system properties (e.g., Hsieh et al. 2018; Kim et al. 2019, 2021). Species with a higher degree of complexity are rarely observed. Complex organic molecules (COMs; carbon-bearing species consisting of at least 6 atoms; Herbst & van Dishoeck 2009) are readily detected in cores at the prestellar stage (e.g., Bacmann et al. 2012; Jiménez-Serra et al. 2016, 2021; Lattanzi et al. 2020; Scibelli & Shirley 2020; Megías et al. 2023), and are also commonly observed around low-mass protostars (e.g., Jørgensen et al. 2016; Yang et al. 2021). In order to follow the trail of chemical complexity in space, the study of the chemistry of VeLLOs is a crucial step to close the gap between pre- and protostellar objects. To date, only Favre et al. (2020) have discovered emission of methanol, the simplest COM, at a distance of ∼ 1 000 au of the VeLLO in L1521F. A search for additional COMs in that source has not been successful to date, due to a lack of sensitivity. To complete our understanding of the complex chemistry that is occurring in these types of objects, more VeLLOs like L1521F need to be studied in detail. The study of these young objects is also important in terms of understanding the composition of forming planets. Substructures in the dust of more evolved Class II objects have been linked to the formation of gas giant planets (e.g., Andrews et al. 2018; van der Marel et al. 2019), but signs of on-going planet formation such as rings and gaps have already been detected in a system with the age of ∼ 500 kyr (Segura-Cox et al. 2020), and most recently Maureira et al. (2024) have reported evidence for annular substructures in a Class 0 source, which suggests that planets can already form earlier in the protostellar evolution (Tychoniec et al. 2020). The composition of planets is set by the composition of the gas and solids they are accreting, and thus the presence of molecules at their location in the disk. The temperature determines which molecules are confined to the solid phase, and which molecules are present in the gas interior to their snowlines (Minissale et al. 2022; Ligterink & Minissale 2023). A change in the temperature, for instance due to an outburst, can temporarily change the composition of the gas due \nto the shifting of the snowlines and the associated sublimation of icy species, and by the formation of new gas-phase molecules induced by the species sublimating from the ice. The chemical response time (for e.g., freeze-out) is slower than the shift of physical conditions from the outburst to the quiescent stage. The duration between outbursting events can be estimated from observations and is ∼ 5 000-50 000 yr (Scholz et al. 2013; Hsieh et al. 2019), however most observational tracers that are enhanced due to the outburst remain altered for only ∼ 10-100 yr into the quiescent phase (Zwicky et al. 2024), exceptions being e.g., CO, which can stay enhanced for up to 10 4 yr before it has frozen out again at typical envelope densities (Visser et al. 2015; Frimann et al. 2017). The emission of the freshly sublimated molecules is expected to be brighter and farther extended than if the object would be quiescent, which makes them easier to detect. Another possibility is to look for species whose formation can be attributed to chemistry that has occurred due to the outburst, and that remain present in the gas phase throughout the quiescent phase. \nThe aim of this study is to search for signposts of active chemistry that have been induced by the change of physical conditions in the VeLLO in the DC3272 + 18 cloud during its most recent outburst. The detection of NO for the first time in a source of this type is potentially directly tied to the outburst, and it is proposed that it could act as a tracer of the water snowline in outbursting sources. In addition, this work utilizes molecular tracers for di ff erent components to gain a better understanding of the structure of the system. \nThis paper is structured as follows: the targeted object and the observations are described in Section 2. The detected lines and their analysis are detailed in Section 3 and discussed in Section 4. The conclusions are stated in Section 5.', '2.1. Targeted VeLLO in DC3272+18': "This study investigates the chemistry toward the isolated VeLLO in the DC3272 + 18 region, which is located at a distance of 250 ± 50 pc (Kim et al. 2019). There are no massive stars nearby, so external irradiation influences are minimal for this object. The VeLLO has an internal luminosity (Lint) of 0.04 ± 0.02 L ⊙ , and an envelope mass (Menv) of 0.04 ± 0.08 M ⊙ (Hsieh et al. 2018; Kim et al. 2019). Its bolometric luminosity (Lbol) of 0.06 ± 0.01 L ⊙ and its bolometric temperature (Tbol) of 105 ± 3 K classify it as a late Class 0 object (Dunham et al. 2008; Enoch et al. 2009; Hsieh et al. 2018; Kim et al. 2019). Observations with the Atacama Submillimeter Experiment (ASTE) have found proof of an outflow in the J = 3 -2 transitions of 12 CO, 13 CO, and C 18 O, showing blue-shifted asymmetries and wings in the line profiles, mapping of an area of 120 '' × 140 '' with the same facility has revealed outflow lobes as well (Kim et al. 2019). The physical properties of the outflow, such as its dynamical time, were quantified from the bipolarity of the outflow and used to infer a current accretion rate from the outflow, ˙ M acc, of 0.09 ± 0.01 × 10 -6 M ⊙ yr -1 (Kim et al. 2019). Over the estimated dynamical time of the outflow, the VeLLO will be able to accrete 0.02 ± 0.01 M ⊙ , which classifies it as a proto-brown dwarf according to Kim et al. (2019). The emission of the J = 3 -2 transition of 13 COpeaks at a distance of 2.4 '' from the continuum emission peak, which is likely due to the outflow reported in Kim et al. (2019). Observations with the Atacama Large Millimeter / submillimeter Array (ALMA) presented in \nHsieh et al. (2018) have inferred from maps of 13 CO, C 18 O, and N2H + that the VeLLO in DC3272 + 18 has recently undergone an outburst. The observations reveal that the center is devoid of N2H + emission where CO evaporates, because gaseous CO destroys N2H + via the reaction N2H + + CO -→ HCO + + N2. Thus, the location of the N2H + emission can be used to infer the snowline of CO, the location where it freezes out onto the grains. Chemical models by Hsieh et al. (2018) have shown that the CO sublimation radius at T = 20 K should be located at 102-164 au, based on its current internal and bolometric luminosity. However, the current sublimation radius of CO is found to be at a distance of 275-311 au. Moreover, the peak of the N2H + emission is located at a distance of 775-1149 au. This is further evidence that the VeLLO has undergone a recent outburst, which has moved the CO snowline outwards, closer to the current peak abundance position of N2H + (Hsieh et al. 2018). The study of Hsieh et al. (2018) also conducted modeling in order to constrain the outburst luminosity that is required to lead to the peak position of N2H + and concluded that it can be explained if Lburst was 1-4 L ⊙ , leading to a mass accretion rate of 6 ± 4 × 10 -6 M ⊙ yr -1 . This is one order of magnitude higher than what is derived from the outflow by Kim et al. (2019). The models of Hsieh et al. (2018) put the CO sublimation radius during the outburst at a distance of 633-836 au.", '2.2. APEX observations': "Single-pointing observations with the Atacama Pathfinder EXperiment (APEX) were centered on the VeLLO in the DC3272 + 18 cloud ( α 2000 = 15 h 42 m 16 s .99, δ 2000 = -52 · 48 ' 02 '' .2) with the nFLASH230 instrument (project-ID O-0109.F-9305A2022, PI: N. F. W. Ligterink). Two spectral settings were centered on frequencies of 219.6 and 254.4 GHz. The FFTS spectrometer was used as the backend, which provided a bandwidth of 32 GHz observed across 8 × 65 536 channels with a spectral resolution of 61 kHz (0.07-0.08 km s -1 ). This resulted in spectral windows of 213.6-221.6 GHz and 229.6-237.6 GHz for setup 1, and 248.4-256.4 GHz and 264.4-272.4 GHz for setup 2. The observations were conducted on April 21-22 and April 28 - May 1 2022 with a precipitable water vapor (pwv) between 0.6 and 1.8 mm. Inspection of the data revealed the presence of prominent atmospheric features, but at frequencies that do not concern the transitions discussed in this work, the meteorological data point towards somewhat unstable conditions during the observations. The typical noise range of the observations is 310 mK (rms) for velocity resolutions of 0.07-0.08 km s -1 . An outlier is a noise of 140 mK for the CO line at 230.538 GHz (Section 3.2.3). Typical system temperatures during the observations ranged from 76 to 152 K. The main beam e ffi ciency ( η mb) for observations at 230 GHz is listed by the APEX website 1 as 0.81, and the half power beam width (HPBW) as 26.2 '' for April and May 2022. The observations were conducted in wobblerswitching mode during the first observing session. However, the wobbler position showed contamination of the CO J = 2 -1 transition at 230.538 GHz from the o ff position, as the o ff position was not emission-free from CO. The remaining observations were carried out in position-switching mode with the o ff position at α 2000 = 15 h 37 m 00 s , δ 2000 = -51 · 10 ' 00 '' . Thus, observations of the CO line from April 21 were not taken into account for the data analysis. For all other lines, the wobbler-switched \nand position-switched data were examined, found to be free from contamination from the wobbler-switched o ff positions, and thus considered for the data analysis. Dedicated calibration uncertainties for nFLASH230 are not published, thus the calibration uncertainties of 10% as published for the previous APEX-1 and APEX-2 receivers (Dumke & Mac-Auli ff e 2010) are assumed. Adopting a distance of 250 pc (Kim et al. 2019) to the source and at the beam size of 26.2 '' , the observations probe spatial scales of 6 550 au in DC3272 + 18.", '3.1. Detected molecules': 'Line identification and baseline fitting were carried out with the class package of the gildas 2 software. Line lists were taken from the Jet Propulsion Laboratory (JPL) catalog (Pickett et al. 1998), and the Cologne Database for Molecular Spectroscopy (CDMS; Müller et al. 2005; Endres et al. 2016), the references to the spectroscopic parameters of the molecules that were fit in this work can be found in the Appendix in Table C.1. Identified spectral lines were then fitted with a Gaussian profile with the same program that calculates the peak temperature (Tpeak), the noise (Trms), the line width ( δ v), and the line velocity (vlsr). The securely and tentatively detected transitions and the results of the Gaussian fits are listed in Table A.1, their spectra are shown in the appendix (Figs. A.1, A.2). Calibration errors are assumed to be 10% (Section 2.2), thus the uncertainty of the peak in- \ntensities is derived as q (0 . 1 T peak) 2 + ( T peak) 2 . If the Tpeak of a line is ≥ 3 × rms, the line is identified as a detection, if it is ∼ 3 × rms, it is identified as a tentative detection. The rms is calculated with the baseline routine in class , Gaussian profiles were fit with class and the curve \\_ fit package of scipy (Virtanen et al. 2020). \nMost of the detected molecules (Table A.1) are species that are commonly found around protostars. CO, and its 13 C- and 18 Oisotopologues are securely detected, its minor isotopologue of 13 C 17 O is not detected (Fig. B.1, Table B.1). Kim et al. (2019) found that CO traces the outflow in this VeLLO; in this case one could expect to detect multiple velocity components. While the CO emission is indeed double-peaked, which can hint that the material traces di ff erent velocity components, its line shape is consistent with self-absorption. It is to be noted that CO and also 13 CO are optically thick, thus self-absorption can indeed contribute to the observed line shapes. This is supported by the finding that the CO absorption peaks at the systemic velocity of the cloud at ∼ -0.1 km / s (Hsieh et al. 2018), and therefore one could conclude that the bulk of the emission traced by our observations stems from the on-source position covering the envelope enshrouding the VeLLO instead of the outflow. Fitting CO with two line components did not lead to a good match. The fit of CO in Fig. A.1 was obtained by masking the region that shows self-absorption between -1.7 and 0.75 km / s, and fitting the line with one component. On the other hand, 13 CO can be fit well with two components (Fig. A.1), their respective line widths of 0.8 and 0.5 km / s are slightly broadened compared to the rest of the molecules, which predominantly exhibit line widths between 0.35 and 0.5 km / s. This can be seen as a hint that some of 13 CO that is observed stems from the outflow reported in Kim et al. (2019), which would explain the broader line widths. Alternatively, this could also be due to the lines being optically thick. \nThe J = 3 -2 transitions of HCO + , HCN, and HNC, are also securely detected. Moreover, species that are commonly found in UV-irradiated cavity walls carved out by the outflow (Tychoniec et al. 2021), such as c-C3H2 are detected. SO2, a species that is associated with shocks, outflows and sublimation products (Tabone et al. 2017), is securely detected in one transition, and tentatively detected in a second transition. In addition, H2CO (formaldehyde), CH3OH (methanol) and SO are detected towards the VeLLO. Deuterated molecules, namely DCO + , DCN, N2D + , one line of D2CO, and C2D are found as well. The latter is also a tracer of the cavity walls. Two CH2DOH lines with favorable upper state energies ( < 25 K) and Einstein A coe ffi cients ( ∼ 10 -5 s -1 ) are below the detection threshold for a tentative detection (Fig. B.1, Table B.1). This is also true for the H 13 2 COtransition at 219.909 GHz. H2CO, CH3OH and SO are species that are typically associated with the emission of warm gas close to a protostar and / or the envelope (e.g., Ceccarelli et al. 2007; Tychoniec et al. 2021). However, the upper energies of the securely detected lines are ∼ 20-44 K. Other H2CO, CH3OH and SO transitions covered by the observations with Eup ∼ 50-90 K are not detected. This points toward their origin from the cold, extended envelope. With the exception of CO and 13 CO that show multiple line components, the detected molecules have line widths of ∼ 0.35-0.50 km / s, and there is no clear trend that can attribute di ff erent molecules to di ff erent structures of the system based on their line widths and the vlsr of the di ff erent lines. All of the so far mentioned species are routinely observed towards protostars, but the molecular inventory probed by the APEX observations also reveals multiple transitions of nitric oxide (NO) at an intensity of 3-17 × rms, and line widths of ∼ 0.4 km / s in the VeLLO, which adds this source to the small list of objects with reported NO detections in the literature (Section 3.3). \nThe SiO J = 50 -40 transition is targeted in the observations, but not detected (Fig. B.1). SiO is a species that is commonly associated with shocks, but the transition at 217.1050 GHz (Eup = 31.3 K, Aij = 5.2 × 10 -4 s -1 ; Fig. B.1) is not detected in the spectrum. This could well be due to beam dilution e ff ects, as SiO is a tracer of shocked regions that are more compact (Tychoniec et al. 2021). It is also to note that no species with a higher degree of chemical complexity than methanol are detected in the data. This is also true for species with upper level energies that point toward an origin in the hot gas in close proximity to the VeLLO. This does not rule out their presence, it could also be that the low spatial resolution of the data is not su ffi cient to detect them due to beam dilution. The highest upper energy for a transition with a secure detection is 48 K, tentative detections have upper level energies of 41-131 K (Table A.1).', '3.2. Constraints on the excitation temperature and column density': 'Subsequent analysis of the detected lines to determine their excitation temperatures and column densities was carried out under the assumption of local thermal equilibrium (LTE). It is assumed that the beam size equals the source size and that both distributions are Gaussian, which leads to a beam filling factor of 0.5. This choice is made based on the assumption that the envelope material fills the beam. The beam-filling factor a ff ects the column density, but as the detected lines are mainly optically thin, the derived column densities would be scaled by the beam-filling factor. If the molecules emit from the same region, the abundance ratios are independent of the choice of filling factor. \nTable 1: Column densities for the detected species. \nNotes. The column densities were derived via the grid-fitting method for Tex of 10, 30, and 50 K with the exception of CH3OH and H2CO, where only the values calculated at 10 K are shown (see text for discussion).', '3.2.1. Rotational diagram analysis of c-C 3 H 2': 'Under the assumption that LTE is applicable, the distribution of the molecular energy levels follows a Boltzmann distribution, and the upper state level population (Nu) divided by the statistical weight (gu) of the molecular energy levels can be related to their upper level energies (Eu) by the Boltzmann equation: \nNu gu = Ntot Q ( Trot ) e -Eu / kTrot (1) \nwhere k is the Boltzmann constant, Ntot is the total column density, and Q corresponds to the partition function value at the corresponding rotational temperature, Trot. In a conventional rotational diagram analysis (e.g., Blake et al. 1987; Goldsmith & Langer 1999), taking the logarithm of Eq. 1 allows for a linear least squares regression: \nln Nu gu = lnNtot -lnQ ( Trot ) -Eu kTrot . (2) \n̸ \nIn LTE, the rotational temperature Trot corresponds to the excitation temperature Tex. By constructing a semi-log plot of Nu / gu against Eu, the rotational temperature, Trot, and the total column density, Ntot, can be derived from the best-fit slope and intercept, respectively. As the optical depth of the transitions is unknown, an optical depth correction factor, C τ , needs to be applied, to account for cases in which the optical depth is τ ≪ 1 \nC τ = τ 1 -e -τ . (3) \nTherefore, the right-hand side of Eq. 2 is rewritten as: \nln ( Ntot ) -lnQ ( Trot ) -lnC τ -Eu kTrot . (4) \nThe optical depth of each transition is calculated via \nτ ul = Aulc 3 8 πν 3 δ v Nu ( e h ν kTrot -1) . (5) \nAul corresponds to the Einstein A coe ffi cient, c to the speed of light, ν to the frequency, and δ v to the line width. This allows to re-write C τ as a function of Nu, and substitute it into Eq. 4 \nto construct a likelihood function L (Nu, Trot) for χ 2 minimization. This likelihood function is used with the a ffi ne-invariant Markov Chain Monte Carlo (MCMC) code emcee (ForemanMackey et al. 2013) to determine Nu and Trot. \nThis method requires multiple transitions per molecule to ensure a meaningful fit, thus this method is only applied to c-C3H2. The result is obtained with two calculations. First, a broad parameter space for the total column density Ntot = 10 7 -10 14 cm -2 and Trot = 10-500 K is explored with 50 walkers and 500 steps. Those results are used to narrow down the parameter space to Ntot = 10 11 -10 13 cm -2 and Trot = 10-100 K. After 2000 steps with 50 walkers, a column density of Ntot = 3.78 + 0 . 37 -0 . 40 × 10 11 cm -2 and a Trot of 35.4 + 3 . 2 -4 . 9 K are obtained. Parameters and uncertainties correspond to the 50th, 16th, and 84th percentiles from the marginalized posterior distributions. The resulting population diagram is plotted in Fig. 1 and reveals that c-C3H2 traces a warm gas component in the VeLLO in DC3272 + 18. This species is commonly associated with the cavities carved by protostellar outflows (e.g., Murillo et al. 2018; Tychoniec et al. 2021), and it is likely that it indeed traces the cavities of the outflow that Kim et al. (2019) have detected in CO. The opacities of the c-C3H2 transitions are in the range of 0.002-0.006.', '3.2.2. Constraining the excitation temperature of methanol and formaldehyde': 'Methanol and formaldehyde emit from a cold gas reservoir in the envelope surrounding the VeLLO. In DC3272 + 18 one H2CO transition at 218.2222 GHz with an upper level energy of 20.96 K is detected (Fig. A.1). However, two additional transitions at 218.4756 GHz ( J = 32 , 2 -22 , 1) and 218.7601 GHz ( J = 32 , 1 -22 , 0), both with Eup of 68.1 K, are not detected. Assuming that intensities > 3 × rms correspond to a detection, an LTE model that is based on the formalism of the cassis 3 (Vastel et al. 2015) software was utilized to explore column densities in combination with excitation temperatures of 10-50 K in 5 K steps that simultaneously match the Tpeak value of the detection at 218.222 GHz and the non-detection of the two other transitions. First, the fixed excitation temperature was used to find a column density that can reproduce the line intensity of the observed H2CO transition. In a second step, this column density and excitation temperature was used to obtain the peak intensity of the two other H2CO transition. Only at Tex = 10 K the two other transitions remain undetected. \nThis was also tested for 12 additional transitions of CH3OH with Eup in the range of ∼ 50-90 K in addition to the two lines that are detected in DC3272 + 18. As for H2CO, it was not possible to reproduce the detected lines without overproducing the intensities of the non-detections at Tex > 10 K. Thus, the H2CO and CH3OH detected in the APEX data stem from a cold gas component and not from a potential hot corino, which are regions close to the protostar that have T > 100 K and are thus associated with the sublimation of COMs (Ceccarelli et al. 2007). If a second, warmer gas reservoir is present, the spatial resolution of the observations is not su ffi cient to detect this emission.', '3.2.3. Grid-fitting of species tracing the cold gas': 'Species such as N2D + , HNC, HCN, DCN, HCO + , and DCO + are commonly associated with the cold envelope (e.g., Tychoniec et al. 2021), and thus likely emit from a region with conditions similar to H2CO and CH3OH. In order to determine the column \nFig. 1: Population diagram for c-C3H2. The shaded region in purple marks the region between the 16th and 84th percentile. Here, NT corresponds to the total column density. \n<!-- image --> \ndensities of these species, the excitation temperature is set to a fixed value of 10 K, which corresponds to Tex derived from H2CO and CH3OH, but it is also the value used by Hsieh et al. (2018) to fit the column densities of N2H + and the CO isotopologues. cassis is used to fit over a grid of Ntot assuming LTE conditions. The position of the lines is set to the source vLSR of -0.1 km / s, the full width half maximum (FWHM) are set to the values given in Table A.1, and the source size is set to match the beam size. Model spectra are computed with a step size of 0.01 in logarithmic space to explore column densities in the range of 10 10 -10 15 cm -2 . The best-fit column densities and their 2 σ uncertainties for each of the above-mentioned molecules, as well as NO, SO, and D2CO are listed in Table 1. CO and its isotopologues were excluded from this process. CO and 13 CO do su ff er from optical depth issues. C 18 O is potentially a ff ected as well, while the rms at the position of the covered 13 C 17 O transition is only ∼ 2 × rms, and therefore counted as a non-detection. Optical depth is not an issue for the remaining molecules, as τ is found to be < 1.', '3.3. Detection of NO': 'Seven transitions of NO (six secure, one tentative) are detected around the DC3272 + 18 VeLLO, which adds it to a small list of di ff erent types of sources that have reported detections of this molecule. It has been detected in the dark clouds TMC-1 and L134N (McGonagle et al. 1990; Gerin et al. 1993), around the low-mass protostars NGC 1333 IRAS 4A, SVS13-A, and IRAS 16293-2422 B, in the shock of L1157-B1 (Yıldız et al. 2013; Codella et al. 2018; Ligterink et al. 2018), in the high-mass starforming region Sgr B2 toward Sgr B2(M) and Sgr B2(N) (Liszt &Turner 1978; Ziurys et al. 1994; Halfen et al. 2001), and most recently in the dust trap of the Oph-IRS 48 disk (Brunken et al. 2022; Leemker et al. 2023). For Oph-IRS 48, models have shown that its abundance can be increased when ice sublimation boosts the abundance of H2O and NH3 (Leemker et al. 2023). The main formation pathway of NO is the gas-phase neutral-neutral reaction \nN + OH -→ NO + H \nin oxygen-enriched gas, where OH is a product of the photodissociation of water. \nIt can also form via a second neutral-neutral reaction, namely \nNH + O -→ NO + H (7) \n(Millar et al. 1991; Baulch et al. 2005; Wakelam et al. 2012). In addition, photodissociation of species such as HNO, N2O, and NO2 can also produce NO (van Dishoeck et al. 2006; Heays et al. 2017), but these species are usually neither abundant nor commonly detected around protostars, so their contribution to the abundance of NO is presumably small. After its formation NO can deplete from the gas by accreting onto dust grains, where it gets hydrogenated to form species such as NH2OH (Congiu et al. 2012; Fedoseev et al. 2012). The formation of N2O is a byproduct of the hydrogenation process of NO to NH2OH. A ratio < 1 for NO / N2O would indicate that some of the NO is lost to this formation channel. However, the observed frequency range only covers one N2O transition at 251.2216 GHz that is not detected, likely due to its high Eup (63.3 K) and low Aij coe ffi cient (2.3 × 10 -6 s -1 ), so it is not possible to assess whether some NO has already been frozen out and has been incorporated in grain surface processes via this pathway. If returned back into the gas phase, the photodissociation of NH2OH can also form NO, but this destruction channel is not e ffi cient (Gericke et al. 1994; Fedoseev et al. 2016). The predominant product channel from photodissociated NH2OH leads to NH2 + OH, which is not a direct parent species for NO (Betts & Back 1965). In addition to freezing out, NO can be directly destroyed in the gas phase via photodissociation reactions, or the reaction N + NO -→ N2 + O(Millar et al. 1991). However, photodissociation induced by internal UV photons from the VeLLO during its quiescent phase is less e ff ective than during the outburst, so while increased UV flux during the outburst can indeed boost the photodissociation of species such as H2O and NH3, which would deliver the reactants to form NO from the two neutral-neutral reactions in Eq. 6 and 7, the photodissociation of NO after the VeLLO has returned to its quiescent phase does play a smaller role. NO has a low binding energy ( ∼ 1 600 K; Wakelam et al. 2017), and thus does not freeze out rapidly after the gas in the envelope cools down to its pre-outburst temperature. The sublimation temperature of NO is ∼ 30-50 K, so in current envelope conditions its presence in the gas phase cannot be explained by thermal desorption. In addition, internal UV photons from the quiescent VeLLO do not lead to significant photodissociation, while its isolated position in the DC3272 + 18 region distances it from external UV flux. On the other hand, H2O has a binding energy of ∼ 5 640 K, which requires temperatures between 90 and 140 K in the interstellar medium (ISM) to sublimate, and already freezes out after 10 2 -10 3 yr at typical envelope densities (Minissale et al. 2022). \nAs for the other species, the beam size of the observations does not allow the location of the NO emission to be spatially resolved. Under the assumption that the molecular emission is cospatial, the column density of NO is e.g., 10 times higher than that of CH3OH, which is a species that freezes out on similar time scales as water (e.g., Collings et al. 2004), so this indicates that the freeze-out process of NO has not progressed as far as for e.g., CH3OH, yet.', '3.4. D/H ratios': "The average ratio of deuterium over hydrogen (D / H ratio) in the local ISM is ∼ 2 × 10 -5 (Linsky et al. 2006; Prodanovi'c et al. 2010). Molecules with an enrichment in deuterium by up to 4 \norders of magnitude compared to the interstellar average are often linked to a formation before the protostellar stage, as temperatures < 20 K boost deuterium fractionation (Caselli & Ceccarelli 2012). The D / Hratios for the deuterated molecules in Table 2 have been calculated based on the column densities in Table 1, but it is to note that all ratios have been derived based on one transition per molecule, and therefore, the presented values should be interpreted with caution. All D / H ratios are in the percent range (17-23%), thus pointing to an origin from a cold environment, potentially before the formation of the VeLLO. DCO + and DCN formation occurs in the gas phase (Willacy 2007), while D2CO forms via grain-surface reactions at temperatures < 20 K (Nagaoka et al. 2007; Hidaka et al. 2009). The derived D / H ratios in the VeLLO are comparable to findings in prestellar cores and around low-mass protostars (Bizzocchi et al. 2014; Chacón-Tanarro et al. 2019; Ambrose et al. 2021; Drozdovskaya et al. 2022; Lin et al. 2023) and thus favor their prestellar origin and subsequent inheritance to the protostellar stage. \nTable 2: D / H ratios calculated with a fixed Tex of 10 K.", '3.5. Physical conditions implied by line intensity ratios': "The line intensity ratios of emission lines can be utilized to probe the physical conditions in the ISM. Hacar et al. (2020) demonstrated that the intensity ratio of the J = 1 -0 transitions of HCN and HNC can be used to infer the kinetic gas temperature and derived a two-component scaling relation from large-scale observations in Orion. This scaling relation is attributed to di ff erent HNC destruction pathways whose e ffi ciency depends on the temperature. Hacar et al. (2020) deduce the scaling relation from the J = 1 -0 transitions of HCN and HNC. As a test in this work, it is assumed that the relation also holds true for the J = 3 -2 transitions that are detected in DC3272 + 18. The peak intensities listed in Table A.1 assume a flux calibration uncertainty of 10%. However, both HCN and HNC are observed in the same sideband, thus a relative calibration uncertainty within the band of \n1% is assumed, resulting in an error of q (0 . 01 T peak) 2 + ( T rms) 2 . The line intensity ratio I HCN / I HNC derived from the Tpeak values in DC3272 + 18 is 0.27 ± 0.07. Thus, the scaling relation for I HCN / I HNC < 4 given in Eq. 3 in Hacar et al. (2020) would apply, which yields Tkin (K) = 10 × I HCN / I HNC = 2.68 ± 0.7 K, which would put the kinetic temperature below the cosmic background temperature if this scaling law holds true. However, Hacar et al. (2020) point out that this relation is only valid down to kinetic temperatures of 15 K, below the errors become large, e.g., errors in Tkin are > 5 K for intensity ratios < 1. Therefore, this method cannot be used to derive the kinetic temperature, but it reveals that it must be cold, likely below the 15 K threshold, where the scaling law is not valid anymore. This is also supported by additional line ratios that o ff er tighter constraints. \nMurillo et al. (2018) conducted a survey of twelve low-mass protostellar systems in Perseus with APEX. The frequency range of their observations partially overlaps with the range observed in this work, thus the line intensity ratios of the two works can be compared. The first intensity ratio that is considered is I DCN / I DCO + . This is due to the D / Hratio of both molecules, which \nis tied to chemical pathways leading to their deuteration at di ff erent temperatures. Both transitions discussed here were observed in the same sideband, thus a flux calibration uncertainty of 1% is assumed. This ratio can be used as a proxy for the gas temperature. While ∼ 2 × 10 -5 corresponds to the average D / Hratio in the local ISM (Linsky et al. 2006; Prodanovi'c et al. 2010), species get enriched in deuterium when the deuterium fractionation becomes e ffi cient once temperatures drop below 20 K and gas densities reach 10 4 cm -3 via the reaction \nH + 3 + HD ⇌ H2D + + H2 + ∆ E , (8) \nwhere ∆ E = 232 K (Watson 1974). DCO + subsequently forms from the reaction of H2D + + CO, thus it has been deemed a suitable tracer for the cold gas (Jørgensen et al. 2005; Murillo et al. 2015). Then, DCO + can propagate its deuteration to DCN (Willacy 2007) via \nDCO + + HNC -→ HNCD + + CO (9) \nand \nHNCD + + e --→ DCN + H . (10) \nOnce the temperature increases to > 30 K, a second pathway leading to deuteration fractionation starts to dominate, namely via the reactions \nCH + 3 + HD ⇌ CH2D + + H2 + ∆ E (11) \nand \nC2H + 2 + HD ⇌ C2HD + + H2 + ∆ E (12) \n(Millar et al. 1989; Roue ff &Lique 2013), where ∆ E is ∼ 390 K for the first reaction (Asvany et al. 2004), and ∼ 550 K for the second reaction (Herbst et al. 1987). \nCH2D + is the starting point for DCO + (via CH2D + + CO -→ DCO + + CH2; Favre et al. 2015) and DCN formation (via CH2D + + e --→ CHD + H followed by CHD + N -→ DCN + H; Millar et al. 1989). Both routes are most e ffi cient for temperatures of ∼ 70-100 K (Favre et al. 2015). The intensity ratio of I DCN / I DCO + in DC3272 + 18 is 0.050 ± 0.005, which is lower by a factor of ≥ 2 than the values that Murillo et al. (2018) derive for their sample of low-mass protostars. The low intensity of DCN does not suggest significant contribution from the warm formation pathway. It is also most likely that DCO + forms solely from the cold formation pathway, and has not proceeded to form a substantial amount of DCN, yet. Thus, the combination of the low value derived for I HCN / I HNC and the high intensity of DCO + strongly hints that the probed material around the VeLLO is predominately cold. \nThis is also concluded from two additional sets of line ratios that are calculated to probe the gas temperature from c-C3H2 transitions. The 60 , 6 - 51 , 5 and 33 , 0 - 22 , 1 transitions of c-C3H2 have Eup of 38.6 and 19.5 K, respectively, thus their intensity ratio can give hints about the gas temperature, too. This also applies to the ratio of the 51 , 4 - 42 , 3 transition (Eup = 35.4 K) over the 33 , 0 - 22 , 1 (Eup = 19.5 K) transition. As for the other two ratios discussed above, all transitions were observed in the same sideband, a flux uncertainty of 1% is therefore assumed. The derived intensity ratios are 0.297 ± 0.040, and 0.173 ± 0.014, respectively, which yet again indicates that the warmer transitions are less excited as the gas probed by the observations is cold. The line ratios obtained for the VeLLO are also on the lower end of the range of ratios that Murillo et al. (2018) derive for their sample of low-mass protostars, which are ∼ 0.38-1.15 and 0.3-0.92 \n<!-- image --> \nFig. 2: Line intensity ratios of c-C3H2 transitions for the VeLLO, and the sample of Murillo et al. (2018) that consists of single protostars, close binary systems, and wide binaries. Close binaries in Murillo et al. (2018) are separated by < 2 '' , the wide binaries in this work have separations in the range of 7-46 '' . The ratios are plotted against the bolometric luminosity, Lbol in the upper panel, and against the envelope mass, Menv in the lower panel. \n<!-- image --> \nfor the two c-C3H2 ratios, respectively. The ratios do not seem to depend on the mass of the envelope, but they do scale with the bolometric luminosity (Fig. 2), as lower ratios are found for the faintest sources. \nThus, even though it is not possible with the current data to derive an exact value for the excitation temperature of the detected molecules, the line intensity ratios also strongly suggest that the probed material is predominantly cold. The low ratio of I DCN / I DCO + suggests a kinetic gas temperature of < 30 K, as otherwise DCN should be more abundant, and the ratio of I HCN / I HNC suggests an even lower value of ≤ 15 K. These findings are in line with the excitation temperature that was derived for formaldehyde and methanol (Section 3.2.2).", '3.6. Line emission and system parameters': "The large beam of the observations (26.2 '' ) does not allow to spatially resolve di ff erent physical components of the system. However, the variety of the detected species and specifically their detected transitions can constrain the physical conditions and structures around the VeLLO. The discussed line ratios in Section 3.5, and the constraint on Tex of 10 K for species such as H2COand CH3OH imply that most of the gas around the VeLLO is cold. This is supported by the detection of molecular tracers that are commonly associated with the cold envelope, such as DCO + and N2D + (Tychoniec et al. 2021). In addition to their excitation temperature, their narrow line width of ∼ 0.5 km / s points to its origin from the cold envelope as well. This is also likely for the species NO, SO, H2CO, and CH3OH. In contrast, c-C3H2 is a molecule commonly associated with the cavity walls carved out by the outflow. This is strongly supported by the population diagram fit that derives a rotational temperature of ∼ 35 K for c-C3H2, indicating that it traces a second, warm gas reservoir around the VeLLO. The presence of an outflow cavity is also supported by the detection of multiple C2D lines (Tychoniec et al. 2021). SiO is commonly used to trace shocks around protostars, as it is linked to the destruction of Si-rich grains due to the high temperatures induced by shocks. The non-detection of SiO in the DC3272 + 18 cloud suggests that either no shock is present in this source, or that it is too weak to destroy dust grains. The presence of SO2 does not clearly support the detection of a shock either. While it can be a tracer of shocked gas, it is also often associated with sublimated grain-surface species (Tychoniec et al. 2021). The 42 , 2 - 31 , 3 transition with an upper energy of 19.0 K is securely detected, and most likely traces the same gas reservoir as species like NO and CH3OH, namely the cold envelope. A second SO2 transition with Eup of 130.7 K is only tentatively detected, a warmer gas reservoir, e.g., one that is heated by a shock, would be necessary to excite this line. However, no other shock tracers such as H2CO, CO and SO transitions with high Eup are detected (Tabone et al. 2020; Tychoniec et al. 2021), so it is unlikely that the APEX observations have picked up signatures of shocked material. In contrast, tracers of the outflow from the VeLLO are detected. Previous observations of Kim et al. (2019) had already confirmed the presence of an outflow in 12 CO, 13 CO, and C 18 O, and the line widths of those molecules are broadened by ∼ 20% compared to species such as C2D or CH3OH in the APEX data. \nThe here presented observations indicate that the majority of the probed material stems from the cold envelope. Molecular tracers attributed to the outflow, and the cavity walls have also been detected.", '4.1. Does NO trace the fossilized water snowline?': 'The high column density of NO relative to CH3OH can be interpreted as a chemical signpost of the past outburst. If NO and methanol indeed probe the same region, with all evidence pointing to it being the cold envelope, their column density ratio at Tex of 10 K comes to ∼ 10 (Table 1). Thus, NO is estimated to be more abundant than CH3OH by one order of magnitude. This is in contrast to all NO detections that have been reported so far. NO is 1-3 orders of magnitudes less abundant in the lowmass protostars IRAS 16293-2422 B (Jørgensen et al. 2018; Ligterink et al. 2018) and SVS-13A (Bianchi et al. 2017), and the shock in L1157-B1 (Codella et al. 2018). Grain-surface formation of NO and CH3OH and subsequent thermal sublimation dur- \nthe quiescent phase can be ruled out. NO has a binding energy of 1 600 K and thermally desorbs at ∼ 30-50 K (Collings et al. 2004; Wakelam et al. 2017), CH3OH has a binding energy of 5 500-6 600 K, which leads to a desorption temperature of ∼ 100 K (Collings et al. 2004; Sakai & Yamamoto 2013; Minissale et al. 2022). Thus, the current temperature in the envelope would not allow for e ffi cient thermal desorption of either of the two species. In addition, NO is hydrogenated to form NH2OH once it freezes out onto the grain surfaces (Fedoseev et al. 2016), therefore it is not likely that abundant NO is sublimated directly from the grain surfaces. It is more likely that the high NO column density is linked to its e ffi cient formation after the outburst in the gas phase. Its main formation pathway in the gas phase (Eq. 6) requires N and OH. The increased temperature during the heating event leads to the sublimation of icy material containing species such as CH3OH, but more importantly H2O and NH3, which all desorb at temperatures of ∼ 100 K (Collings et al. 2004; Wakelam et al. 2017; Minissale et al. 2022). Photodissociated products of the major ice constituents H2O and NH3 (Boogert et al. 2015) are then consumed to form NO at the location where the water snowline got shifted to during the outburst. Due to the increased UV flux from the VeLLO during the outburst the e ffi ciency of photodissociation is elevated, which provides the necessary ingredients to form abundant NO in the gas phase. Thus, the presence of NO at a high column density relative to methanol could be a direct consequence of the outburst of the VeLLO. After the end of the accretion outburst the temperature will decrease again as the VeLLO reaches its quiescent phase. Subsequently, species that have been evaporated during the outburst will start to freeze out again, that order depends on the binding energies of the molecules. Chemical models by Visser et al. (2015) have shown that the water snowline moves outwards by a factor of ∼ 10 during the outburst and that H2O abundances remain enhanced for 10 2 -10 3 yr after the outburst. In the case of CO the enhancement is seen for 10 3 -10 4 yr, so while it is not possible to pinpoint the exact point in time of the outburst, the extended CO snowline compared to its quiescent luminosity (Hsieh et al. 2018) is evidence that the outburst happened less than 10 4 yr ago. The results presented in this paper show that NO could be another molecule that serves as a tracer that is available for longer timescales than the typical 10 2 -10 3 yr of most direct tracers of the water snowline. Its longevity as a tracer is reflected in the elevated NO / CH3OH ratio. Due to its higher binding energy, CH3OH starts its freeze-out in a point in time closer to the outburst, while NO still remains in the gas phase. If NO indeed forms from the reaction in Eq. 6, it can be used to expose the location of the position of the shifted water snowline during the outburst, as this proposed formation pathway is tightly linked to species that sublimate / freeze out at the water snowline. This would add NO to the list of tracers of the water snowline that is more commonly probed via HCO + , CH3OH, HDO and H 18 2 O (Hsieh et al. 2019; Tobin et al. 2023). Observations with a better spatial resolution are required to confirm the suspected co-spatial origin of the NO and CH3OH emission.', '4.2. Post-outburst chemistry, non-thermal desorption, or outflows?': 'While an outburst that leads to sublimation of grain-surface species due to its accompanied temperature increase in the envelope followed by boosted photodissociation due to its higher UV flux during its active phase is a possibility to explain the peculiarly high NO abundance in the VeLLO, other options also need to be considered. \nAnother mechanism that can release grain surface species into the gas phase and that consequently has to be considered as an alternative explanation is reactive desorption. Reactive desorption is a non-thermal desorption process where the energy of chemical reactions on the grains is not fully absorbed by the grain, and the remaining energy causes the ejection of a percentage of the newly formed product in the gas-phase (e.g., Vasyunin et al. 2017; Pantaleone et al. 2020). This process is responsible for the presence of COMs such as CH3OH in the gas in prestellar cores (e.g., Jiménez-Serra et al. 2016), and could also contribute to the molecular abundances in the envelope around the VeLLO. Chemical models find that 1-10% of the formed grain surface product gets ejected into the gas phase via this desorption process (e.g., Garrod et al. 2008). Thus, reactive desorption processes could have released CH3OH into the gas phase, which would be an alternative explanation to thermal desorption induced by the outburst. This is also true for H2O and NH3. However, without an outburst that increases the internal UV flux and photodissociates H2O and NH3 it is unlikely that the reactants required to form NO via Eq. 6 and 7 are abundant enough to result in a column density ratio of 10 for NO / CH3OH. \nAnother possibility is that the outflow emanating from the VeLLO sublimates grain-surface species locally. Yet again, this option could therefore explain the presence of CH3OH in the gas. However, there are two caveats coming with this option. First, lines of molecules in the outflow are usually broadened, as it is seen for CO and 13 CO, but not for CH3OH. Second, as for the explanation with reactive desorption, this option does not present an e ff ective path towards photodissociation of H2O and NH3. Thus, it is unlikely that the outflow played a major role in the formation of NO, and the presence of gaseous CH3OH in this source. If it contributes marginally can only be assessed with observations with a higher spatial resolution. If those species are originating from the outflow cavity wall, a shift in their vvlsr would be expected. \nThus, an outburst is the most likely explanation for the high abundance of NO in the VeLLO, but based on the available data it is not possible to fully rule out other options or a combination of multiple scenarios.', '4.3. Limits on Complex Organic Molecules': 'COMsare precursors of prebiotic molecules, thus revealing their formation pathways in space is a central interest of astrochemical studies. Around protostars they are often associated with hot cores or hot corinos, which is a region in close proximity to the protostar that has warmed up su ffi ciently to sublimate COMs from icy grains and form them actively in the gas phase (Ceccarelli et al. 2007; Herbst & van Dishoeck 2009). Nevertheless, they are also readily detected in cold, prestellar cores, where their presence in the gas phase is attributed to non-thermal desorption processes such as reactive desorption (Garrod & Herbst 2006; Vasyunin et al. 2017), or cosmic ray-induced desorption (Sipilä et al. 2021). The most complex species detected in the VeLLO is CH3OH, the question now becomes whether the VeLLO is COM-poor, or if the current sensitivity is not su ffi -cient to detect more complex species, or if the beam dilution effects hamper the chance of a detection. This has been tested for three COMs that are routinely detected around protostars and in cores, namely methyl formate (CH3OCHO), dimethyl ether (CH3OCH3), and acetaldehyde (CH3CHO). Their abundance relative to methanol is found to be in the range of 1-40% for low-mass protostars (Jørgensen et al. 2018; Belloche et al. 2020; Yang et al. 2021). In the well-studied prestellar core L1544, \nJiménez-Serra et al. (2016) find an abundance of 2.0-2.5% for dimethyl ether relative to methanol, 5.9-7.4% for methyl formate, and 2.0-8.2% for acetaldehyde for two positions in this core. In addition, Scibelli & Shirley (2020) detect acetaldehyde in 22 prestellar cores, with a ratio of CH3CHO / CH3OH of 226%. As the majority of the probed material in the VeLLO is cold, and it is likely that potential COM emission from a warm gas component close to the VeLLO will not be detected due to the large beam of the observations, an excitation temperature of 10 K was chosen to test which ratio of COM / CH3OH would have led to a detection in the APEX data. The rms of the data is typically 3-5 mK (Table A.1), and the same LTE model that was used to constrain the excitation temperature of H2CO and CH3OH was used to derive column densities that would lead to Tpeak intensities of 3 × rms for transitions covered in the data that have Eup of ≤ 50 K. In the case of acetaldehyde, a ratio of 30-40% compared to methanol would be required for a detection in the APEX observations. For acetaldeyde and methyl formate this value is > 60%. Thus, it is very unlikely that they could have been detected in the current observational data.', '5. Conclusions': 'This work presents APEX observations of the VeLLO in the isolated DC3272 + 18 cloud that has undergone an outburst that likely occurred less than 10 4 yr ago. The presence of molecules such as CH3OH, H2CO, SO, SO2, and for the first time in a source of this type, NO, is likely tied to the past heating event. Moreover, typical chemical tracers of multiple physical components of the protostellar system are detected. Line intensity ratios and column densities are utilized to constrain the kinetic gas temperature of the system and the excitation temperature of the detected species. The main findings are listed below: \n- -For the first time, NO is detected in a VeLLO. The most likely explanation for its high column density is its formation after species such as H2O and NH3 have been sublimated and subsequently photodissociated after the outburst. If this proposition holds true, it could be used to trace the position of the extended water snowline during the outburst. Due to its high volatility it remains enhanced in the gas phase long after the central object has returned to its quiescent stage. Its potential as a tracer for past outbursts has to be tested with observations with a higher spatial resolution for this source, and for a larger sample of sources as well. This is especially interesting to test for Class I and II objects when planet formation is already an on-going process.\n- -The detection of CH3OH and H2CO with an excitation temperature of ∼ 10 K suggests that they stem from the cold envelope and have sublimated from the grains during the outburst. The securely detected transitions of SO with Eup < 44 K most likely originate from the same gas reservoir. The high column density ratio of NO / CH3OH is attributed to di ff erent freeze-out timescales of molecules related to their binding energies. Observations with a higher spatial resolution are required to confirm that those molecules are indeed co-spatial.\n- -The low line intensity ratio of I HCN / I HNC indicates that the gas kinetic temperature is ≤ 15 K. A low kinetic temperature is also supported by the low line intensity of the I DCN / I DCO + ratio. When compared with APEX observations of a sample of low-mass protostars in Perseus (Murillo et al. 2018), the ratios of the VeLLO are low compared to the other objects, in line with its low bolometric luminosity and envelope mass compared to the rest of the sample. \n- -The detections of c-C3H2 and C2D are consistent with the presence of cavity walls carved out by the outflow driven by the VeLLO. The population diagram analysis of the c-C3H2 transitions calculates a rotational temperature of 35 + 3 -5 K, and thus reveals a second, warmer layer of gas around the VeLLO. However, ratios of the 6-5 / 3-2 and 5-4 / 3-2 transitions of c-C3H2 are also lower than what is found in the sample by Murillo et al. (2018), so while c-C3H2 traces a reservoir that is warm compared to the rest of the probed material in the VeLLO, it is still colder compared to the material that is found around low-mass protostars.\n- -The detections of three deuterated species, namely DCO + , DCN, D2CO, and their respective D / H ratios reveal that deuteration has been e ff ective in the past and potentially also in the present. C2D is also detected in the data, but C2H is not covered, so its D / H ratio cannot be calculated. D2CO is a product of grain-surface formation and has sublimated into the gas phase during the outburst. DCO + , DCN, and C2D are products of gas-phase chemistry, their deuterium enhancement could still be on-going in the cold layers of the envelope. \nThe study of outbursting objects is crucial to understand to which extent bursts sublimate and reset icy grain mantles and influence the ice and gas-phase chemical inventory of future planetforming material by sublimating species that would otherwise remain on the icy grains. These kind of bursts may completely alter the balance between inheritance and reconstitution of the volatile reservoirs during the formation of protoplanetary systems. Investigating the chemical composition of VeLLOs allows the study of objects at the low luminosity / low mass end of star formation, which has been understudied so far. The APEX data have revealed that VeLLOs, at least the one in DC3272 + 18, is rich in molecules. To fully understand their chemical composition it is necessary to obtain observations with a higher spatial resolution with facilities like ALMA. This will allow to spatially resolve emission of molecules, and also to probe for the existence of a hot corino, where COMs are expected to be in the gas phase, if the chemical composition in VeLLOs is in line with what is found around low-mass protostars, just at closer distances to the central star due to its lower luminosity. Complementary observations with the James Webb Space Telescope would o ff er the opportunity to study the level of reprocessing of the ices induced by the past outburst. \nAcknowledgements. This publication is based on data acquired with the Atacama Pathfinder Experiment (APEX) under programme ID O-0109.F-9305A-2022. APEX is a collaboration between the Max-Planck-Institut fur Radioastronomie, the European Southern Observatory, and the Onsala Space Observatory. We want to thank the APEX sta ff for support with these observations. B.M.K acknowledges the SNSF Postdoc.Mobility stipend P500PT\\_214459. B.M.K and M.N.D acknowledge the Swiss National Science Foundation (SNSF) Ambizione grant no. 180079. M.N.D. acknowledges the Holcim Foundation Stipend. S.F.W. acknowledges the financial support of the SNSF Eccellenza Professorial Fellowship (PCEFP2\\_181150). N.F.W.L. and K.A.K. acknowledge support from the Swiss National Science Foundation (SNSF) Ambizione grant 193453. T.H.H. acknowledges the support by the Max Planck Society. P.B. acknowledges the support of the Swedish Research Council (VR) through contract 20170492. 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In Fig. A.1, velocity binning over two channels was applied to the D2CO line, and the SO2 line at 235.151 GHz, and for the NO transition at 250.475 GHz and the SO2 transition at 236.217 GHz in Fig. A.2.', 'Appendix B: Spectra of notable non-detections': 'Fig. B.1 shows the spectral window with the non-detections of the SiO line at 217.105 GHz, the 13 C 17 O line at 214.574 GHz, the H 13 2 CO line at 219.909 GHz, and two CH2DOH transitions at 214.702 and 253.629 GHz that were searched for in the data based on their similar upper state energies, and line strengths compared to detected species. Velocity-binning over two channels was applied to both CH2DOH transitions and the H 13 2 CO line.', 'Appendix C: Spectroscopic parameters': "Table C.1 lists whether the spectroscopic data was taken from JPL or CDMS, and lists the papers that the catalog entries are based on. \nB. M. Kulterer et al.: Post-Outburst Chemistry in a Very Low-Luminosity Object \nTable A.1: Observed molecular transitions in this work. \nNotes. Tentative detections are marked with an asterisk in front of the molecule name. The spectra and their fits are plotted in Figs. A.1 and A.2. (a) 13 CO was fit with two Gaussians (b) The peak temperature, Tmb , peak, the line width of the spectral lines, ∆ v, and the vLSR, and the Trms of the data of the spectral lines were determined with a Gaussian fit with the class package of the gildas software and the curve \\_ fit module of scipy (Virtanen et al. 2020); (c) hyperfine components were detected; (d) the position of the hyperfine transitions is given relative to the strongest transition (Fig. A.1). \nFig. A.1: Spectra of the securely detected transitions (Table A.1) overlaid with the Gaussian best-fit model in red (Table A.1). Not all lines are perfectly fit by a Gaussian, kinematics or line broadening are factors that can influence the line shape, in the case of CO and 13 CO two components are used to fit the line. It is also to note that due to the large beam size of 26.2 '' di ff erent physical components are not resolved, and emission is picked up from multiple components. The dashed, dark gray line indicates the 3 × rms value, the dashed, light gray line indicates the rms value. \n<!-- image --> \nFig. A.1: (continued) \n<!-- image --> \nFig. A.2: Spectra of the tentatively detected transitions (Table A.1) overlaid with the Gaussian fit in red (Table A.1). The dashed, dark gray line indicates the 3 × rms value, the dashed, light gray line indicates the rms value. \n<!-- image --> \n10.0 \nFig. B.1: Notable non-detections in the data. The 3 × rms value is indicated by the dashed, light gray line, the rms value is indicated by the dashed, dark gray line. The position of the targeted SiO transition is indicated by the red line, for the remaining lines the attempt to fit a Gaussian to the data is displayed in red. \n<!-- image --> \nTable B.1: Notable non-detections. \nTable C.1: References to the spectroscopic data of all the molecules that were detected or searched for in the VeLLO in the DC3272 + 18 cloud and discussed in this work."} |
2024Univ...10..428O | In this paper we investigate the vacuum bosonic current density induced by a carryingmagneticflux cosmic string in a inlineformulammlmath idmm1mmlsemanticsmmlmrowmmlmommlmommlmiDmmlmimmlmommlmommlmn1mmlmnmmlmommlmommlmrowmmlsemanticsmmlmathinlineformulade Sitter spacetime considering the presence of two flat boundaries perpendicular to it. In this setup the Robin boundary conditions are imposed on the scalar charged quantum field on the boundaries. The particular cases of Dirichlet and Neumann boundary conditions are studied separately. Due to the coupling of the quantum scalar field with the classical gauge field corresponding to a magnetic flux running along the strings core a nonzero vacuum expectation value for the current density operator along the azimuthal direction is induced. The two boundaries divide the space in three regions with different properties of the vacuum states. In this way our main objective is to calculate the induced currents in these three regions. In order to develop this analysis we calculate for both regions the positive frequency Wightman functions. Because the vacuum bosonic current in dS space has been investigated before in this paper we consider only the contributions induced by the boundaries. We show that for each region the azimuthal current densities are odd functions of the magnetic flux along the string. To probe the correctness of our results we take the particular cases and analyze some asymptotic limits of the parameters of the model. Also some graphs are presented exhibiting the behavior of the current with relevant physical parameter of the system. | 2024-11-01T00:00:00Z | ['2024arXiv240905691O', 'arXiv:2409.05691', '2024Univ...10..428O', '10.48550/arXiv.2409.05691', '10.3390/universe10110428'] | ['cosmic string', 'magnetic flux', 'de Sitter spacetime', 'flat boundaries', 'High Energy Physics - Theory', 'General Relativity and Quantum Cosmology'] | Current Density Induced by a Cosmic String in de Sitter Spacetime in the Presence of Two Flat Boundaries | 2,024 | 175 | 0.28 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML'] | 1 | https://arxiv.org/pdf/2409.05691.pdf | {'Current density induced by a cosmic string in de Sitter spacetime in the presence of two flat boundaries': "W. Oliveira dos Santos 1 ∗ , H. F. Santana Mota 1 † and E. R. Bezerra de Mello 1 ‡ 2 Departamento de F'ısica, Universidade Federal da Para'ıba 58.059-970, Caixa Postal 5.008, Jo˜ao Pessoa, PB, Brazil \nDecember 5, 2024", 'Abstract': "In this paper, we investigate the vacuum bosonic current density induced by a carrying-magneticflux cosmic string in a ( D +1)-de Sitter spacetime considering the presence of two flat boundaries perpendicular to it. In this setup, the Robin boundary conditions are imposed on the scalar charged quantum field on the boundaries. The particular cases of Dirichlet and Neumann boundary conditions are studied separately. Due to the coupling of the quantum scalar field with the classical gauge field, corresponding to a magnetic flux running along the string's core, a nonzero vacuum expectation value for the current density operator along the azimuthal direction is induced. The two boundaries divide the space in three regions with different properties of the vacuum states. In this way, our main objective is to calculate the induced currents in these three regions. In order to develop this analysis we calculate, for both regions, the positive frequency Wightman functions. Because the vacuum bosonic current in dS space has been investigated before, in this paper we consider only the contributions induced by the boundaries. We show that for each region the azimuthal current densities are odd functions of the magnetic flux along the string. To probe the correctness of our results, we take the particular cases and analyze some asymptotic limits of the parameters of the model. Also some graphs are presented exhibiting the behavior of the current with relevant physical parameter of the system. \nKeywords: Cosmic string, magnetic flux, de Sitter spacetime, flat boundaries", '1 Introduction': "De Sitter (dS) space is solution of the Einstein equation in the presence of positive cosmological constant. Although being a curved spacetime it enjoys the same degree of symmetry as the Minkowski one [1], so several physical problem can have exact solutions on this background; in addition, the relevance of these theoretical analysis has received great attention due to the appearance of the inflationary cosmology scenario [2]. In many inflationary models, an approximate de Sitter (dS) spacetime is used to address relevant problems in standard cosmology. During an inflationary epoch, quantum fluctuations in the inflaton field generate inhomogeneities that can influence the transition to the true vacuum. These fluctuations play a crucial role in the formation of cosmic structures originating from inflation. Specifically the problem of particle creation in the inflationary phase of the Universe, was analyzed in [3] considering de Sitter space. There it is was calculated the energy momentum of the \ncreated particles during the inflation, by computing the difference between the in - and out -vacuum states \nCosmic strings are linear gravitational topological defects which may have been formed in the early Universe as consequence of phase transitions in the context of the standard gauge field theory of elementary particle physics [4, 5, 6]. Although the observations of anisotropies in the Cosmic Microwave Background Radiation by COBE, WMAP and more recently by the Planck Satellite have ruled out cosmic strings as the primary source for primordial density perturbations, they give rise to a number of interesting physical effects such as the emission of gravitational waves and the generation of high-energy cosmic rays (see, for instance, [7]-[9]). \nThe geometry of the spacetime produced by an idealized cosmic string, i.e., infinitely long and straight, is locally flat, but topologically conical. It presents a planar angle deficit on the two-surface orthogonal to the string. This object was first introduced in the literature as solution of the Einstein equation in the presence of a Dirac-delta type distribution of energy and axial stress along a straight infinity line. However this spacetime can also be obtained in the context of Classical Field Theory, by coupling the energy-momentum tensor associated with a vortex field configuration proposed by Nielsen and Olesen in [10], with the Einstein's equation, as investigated in [11] and [12]. In both publications, the authors have shown that a planar angle deficit arises on the two-surface perpendicular to a string, as well as a magnetic flux running through its core. The conical geometry in the spacetime produced by a cosmic string has been considered in different lines of research, since the 80 ' s of the last century. Gott III in [13], proposed that cosmic strings can produce double images serving as gravitational lens. Also Linet [14] has showed that a test charged particle place at rest in the region outside the cosmic string becomes subjected to repulsive electrostatic self-interaction. In addition, Smith in [15], also has found a similar phenomenon considering gravitational effect in the Newtonian limit. He sowed that a test massive particle place at rest in the neighborhood of a cosmic string becomes subjected to an attractive self-interaction. The reason for the last two phenomena resides in the fact that the conical topology produced by a cosmic string distorts the particles fields. \nThe analysis of the combined effects of the curvature of the dS background and the conical topology produced by the cosmic string in the vacuum expectation value (VEV) of the induced azimuthal current, ⟨ j ϕ ⟩ , associated with a charged scalar field was presented in [16]. Another type of vacuum polarization arises when boundaries are considered in the system. The imposition of boundary conditions on quantum fields changes the vacuum fluctuations, and result in additional shifts in the VEV of physical quantities, such as the energy-momentum tensor. In this sense, the investigation of the VEV of the energy-momentum tensor and the field squared, associated with a charged massive scalar quantum field in the dS background considering the presence of a cosmic string and just one flat plate perpendicular to it, has been developed in [17]. \nIn [18], the authors have calculated the VEV of the energy-momentum tensor and the field squared, associated with a massive scalar quantum field propagating in dS spacetime considering the presence of two parallel flat plates. The authors imposed that on the plates, the scalar field obeys Robin boundary condition. Considering this approach they calculated the contributions to these observables, energymomentum tensor and field squared, induced by the presence of the plates in the region between them. With the objective to extend these analyses, we decided to consider in this present work the presence of a carrying-magnetic-flux cosmic string in dS spacetime perpendicular to the two flat plates, and calculate the induced vacuum current associated with a quantum massive charged scalar field propagating in this manifold. Because the analysis of vacuum bosonic current induced by a cosmic string in the dS spacetime in absence of flat plates has been developed previously, our focus here is to investigate the contributions induced by the plates. In order to develop these analyses we calculate the Wightman function in this manifold, considering that the bosonic modes are prepared in Bunch-Davies vacuum. Following a procedure similar to [18], we decompose this Wightman function in three distinct contributions. One corresponding to the function induced by the cosmic string in dS in absence of plates, plus other two terms induced by the presence of one plate and two plates, separately. \nAs explained previously, our focus here is to consider the contributions induced by the plates. In this sense we analyze in detail, considering some limiting situations, the only non-vanishing azimuthal components of the boundary induced currents in the three different regions of the space: considering first the Wightman function induced by each plate separately, we obtain the induced current for the corresponding regions outside the plates, and considering the Wightman function induced by the two plate, we calculate the current in the region between the plates. These currents correspond to a Casimir-like effect, i.e., they are induced due to the boundary condition imposition on the quantum fields on the two flat planes, and as we will see, their intensities decay with the distance from the plane, representing a typical Casimir-like effect. \nThe plan of this work is as follows: In the Section 2 we present the geometry of the spacetime that we want to consider, the Klein-Gordon equation obeyed by the charged massive quantum field operator, and the boundary condition that the field has to satisfies on the flat plates. The complete set of normalized positive energy solutions of the Klein-Gordon equation in the region between two parallel flat plates considering that the field obeys the Robin boundary condition on them is presented in Section Bpsonic Modes. Having obtained this set of bosonic modes, in the Section 3 we calculate the corresponding Wightmann function, by adopting the mode sum formula approach. Because the momentum along the direction of cosmic string is discretized, we use the Abel-Plana summation formula to obtain the sum over this quantum number. Doing this procedure, the Wightmann function is expressed as the sum of three contributions, the first one associated with the presence of a cosmic string in the dS space without plates, the second induced by the presence of a single plate, and the third induced by the two plates. The expressions for second function is applied for each plate separately, and the third function only for the region between the plates. In Section 4 we present formally the complete decomposition of the induced bosonic current, ⟨ j ϕ ⟩ . The contribution induced by a single plate is developed in subsection 4.1, and in the region between plates in 4.2. Also in these subsections, various asymptotic limits of the currents are considered and numerical results are presented. In Section 5 we summarize the most relevant results obtained. Throughout the paper, we use natural units G = ℏ = c = 1.", '2 Background Geometry and Matter Field Content': 'The line element describing the geometry produced by a cosmic string in (1 + D ) -de Sitter spacetime is given by the following expression: \nds 2 = dt 2 -e 2 t/a ( dr 2 + r 2 dϕ 2 + dz 2 + D -3 ∑ i =1 dx 2 i ) , (2.1) \nwhere r ≥ 0 and ϕ ∈ [0 , 2 π/q ] define the coordinates on the conical geometry ( q ≥ 1 encodes the angle deficit), ( t, z, x i ) ∈ ( -∞ , ∞ ) and α stands for the length scale of dS spacetime and it is related with the cosmological constant and the curvature scalar, R , by the following relations: \nΛ = D ( D -1) 2 α 2 , R = D ( D +1) α 2 . (2.2) \nFor convenience of the discussion that follows below the line element (2.1), written in synchronous time coordinate, can be expressed in a conformal form by introduction of the conformal time coordinate, τ , defined as τ = -αe -t/α with τ ∈ ( -∞ , 0]. By doing so, we get \nds 2 = ( α τ ) 2 ( dτ 2 -dr 2 -r 2 dϕ 2 -dz 2 -D -3 ∑ i =1 dx 2 i ) . (2.3) \nNote that the line element inside brackets describes an idealized cosmic string in Minkowski spacetime. \nIn this paper, we want to analyze the vacuum effects due to a propagating charged scalar field in the dS spacetime with a magnetic-carrying-flux cosmic string and in the presence of two flat boundaries. For this purpose we consider the following Klein-Gordon field equation: \n( g µν D µ D ν + m 2 + ξR ) φ ( x ) = 0 , (2.4) \nwhere D µ = ∂ µ + ieA µ and m is the mass of the scalar field. In addition, in the expression above, we have considered the nonminimal coupling between the background curvature and the scalar field through the term ξR , where ξ is the curvature coupling constant and R denotes the Ricci scalar. The magnetic flux along the string axis is introduced through the vector potential A µ = A ϕ δ ϕ µ , where A ϕ = -q Φ / 2 π is constant and Φ represents the magnetic flux along the string. \nIn order to consider the two flat boundaries, we impose that solutions of the Klein-Gordon equation (2.4) satisfy the Robin boundary conditions given by \n(1 + β j n l ∇ l ) φ ( x ) = 0 , z = a j , j = 1 , 2 , (2.5) \nwhere β j are constant coefficients (in particular, for β j = 0 and β j = ∞ , the Robin boundary conditions are reduced to the Dirichlet and Neumann boundary conditions, respectively), and n l represents the normal vectors to the boundaries. In the region between the two plates one has n l = ( -1) j -1 δ l z . According to the notation above, the two flat boundaries are located at z = a 1 and z = a 2 , with a 1 < a 2 . Moreover, note that in our setup problem the cosmic string is perpendicular to the two boundaries, since it is located along the z -axis.', 'Bosonic Modes': "In this subsection, our aim is to determine the complete set of normalized solutions for the KleinGordon equation (2.4). \nIn the spacetime geometry given by (2.3) and with the gauge field A µ = A ϕ δ ϕ µ , the Klein-Gordon equation simplifies to \n[ ∂ 2 ∂τ 2 + (1 -D ) τ ∂ ∂τ + D ( D +1) ξ +( mα ) 2 τ 2 -∂ 2 ∂r 2 -1 r ∂ ∂r -1 r 2 ( ∂ ∂ϕ + ieA ϕ ) 2 -∂ 2 ∂z 2 -D -4 ∑ i =1 ∂ 2 ∂ ( x i ) 2 ] φ ( x ) = 0 . (2.6) \nThe equation is completely separable and in accordance to the symmetries present in the geometry under consideration; so, we propose the following Ansatz: \nφ ( x ) = f ( τ ) R ( r ) h ( z ) e iqnϕ + i ⃗ k · ⃗x || , (2.7) \nwhere ⃗x || denotes the coordinates along the ( D -3) extra dimensions, with ⃗ k representing the corresponding momenta. The function h ( z ) will be determined by the Robin boundary conditions that the scalar field satisfies on both flat boundaries placed at z = a 1 and z = a 2 . \nTaking the Ansatz proposed above into (2.6) and admitting that \n∂ 2 h ( z ) ∂z 2 = -k 2 z h ( z ) , (2.8) \nwe obtain the following differential equations for the functions f ( τ ) and R ( r ): \n[ ∂ 2 ∂τ 2 + (1 -D ) τ ∂ ∂τ + D ( D +1) ξ +( mα ) 2 τ 2 + λ 2 ] f ( τ ) = 0 , (2.9) \nand \nwith \n[ ∂ 2 ∂r 2 + 1 r ∂ ∂r -q 2 ( n + α ) 2 r 2 + p 2 ] R ( r ) = 0 , (2.10) \nλ = √ p 2 + k 2 z + ⃗ k 2 (2.11) \nand the notation \nα = eA ϕ q = -Φ Φ 0 , (2.12) \nbeing Φ 0 = 2 π/e the quantum flux. \nThe solution for (2.10) that is regular at r = 0 is given by \nR ( r ) = J q | n + α | ( pr ) , (2.13) \nwhere J µ ( x ) denotes the Bessel function of first kind [19]. The solution for the time-dependent equation is expressed by the linear combination of Hankel functions: \nf ( τ ) = η D/ 2 ( c 1 H (1) ν ( λη ) + c 2 H (2) ν ( λη )) , (2.14) \nwith the order given by \nν = √ D 2 / 4 -m 2 a 2 -ξD ( D +1) . (2.15) \nAdditionally, in (2.14) we have defined the variable η = | τ | , and H ( l ) ν ( x ) denotes the Hankel function [19]. Different choices of the coefficients c 1 , 2 in (2.14) lead to different choices of the vacuum state. In this paper we consider the Bunch-Davies vacuum, corresponding to the choice c 2 = 0. \nAs to the solution of the z -dependent equation, it is constrained in the region a 1 < z < a 2 by the Robin boundary conditions (2.5) on the two flat boundaries. For the plate at z = a 1 , we have \nh ( z ) = cos[ k z ( z -a 1 ) + α 1 ( k z )] , (2.16) \nwith the notation \ne 2 iα 1 ( x ) = iβ 1 x -1 iβ 1 x +1 . (2.17) \nFrom the boundary condition on the second plate z = a 2 , we get the following equation: \n(1 -b 1 b 2 v 2 ) sin( v ) -( b 1 + b 2 ) v cos( v ) = 0 , v = k z ˜ a , (2.18) \nwith ˜ a = a 2 -a 1 and b j = β j / ˜ a . We will denote the solutions of (2.18) by v = v l , with l = 1 , 2 , 3 ... These solutions constrain the eigenvalues k z through the relation k z = v l / ˜ a . \nFinally, combining (2.13), (2.14) and (2.16), we obtain the mode functions that satisfy both the Klein-Gordon equation (2.4) and the Robin boundary conditions (2.5) on the plates: \nφ σ ( x ) = C σ η D/ 2 H (1) ν ( λη ) J q | n + α | ( pr ) cos[ k z ( z -a 1 ) + α 1 ( k z )] e iqnϕ + i ⃗ k · ⃗x || , (2.19) \nwhere σ = { λ, p, n, k z , ⃗ k } represents the set of quantum numbers that specify each mode of the field. The coefficient C σ is fixed by the orthonormalization condition \n-i ∫ d D -1 x ∫ a 2 a 1 dz √ | g | g 00 [ φ σ ( x ) ∂ t φ ∗ σ ' ( x ) -φ ∗ σ ' ( x ) ∂ t φ σ ( x )] = δ σ,σ ' , (2.20) \nwhere the integral is evaluated over the spatial hypersurface τ = const, and δ σ,σ ' represents the Kronecker-delta for discrete indices and Dirac-delta function for continuous ones. Applying the normalization condition to the mode functions in (2.19) gives \n| C σ | 2 = (2 π ) 3 -D α 1 -D qpe i ( ν -ν ∗ ) π/ 2 4˜ a { 1 + cos[ v l +2 α 1 ( v l / ˜ a )] sin( v l ) /v l } . (2.21)", '3 Wightman Function': "In this paper, our objective is to examine the vacuum polarization effects arising from the background setup described in the previous section. To achieve this, we will utilize the Wightman function, which is particularly useful for calculating vacuum expectation values of physical observables dependent on bilinear field operators. Specifically, the vacuum properties can be characterized by the positivefrequency Wightman function, W ( x, x ' ) = ⟨ 0 | ˆ φ ( x ) ˆ φ ∗ ( x ' ) | 0 ⟩ , where | 0 ⟩ denotes the vacuum state. To evaluate this function, we will use the mode-sum technique, expressing the Wightman function in the form: \nW ( x, x ' ) = ∑ σ φ σ ( x ) φ ∗ σ ( x ' ) . (3.1) \nwhere ∑ σ denotes the summation over both discrete and continuous quantum numbers, with σ = { λ, p, n, k z , ⃗ k } . \nTaking (2.19), along with the coefficient (2.21), into (3.1), we obtain \nW ( x, x ' ) = 4 q ( ηη ' ) D/ 2 (2 π ) D -1 α D -1 ˜ a ∞ ∑ n = -∞ e inq ∆ ϕ ∫ ∞ 0 dppJ q | n + α | ( pr ) J q | n + α | ( pr ' ) × ∫ ∞ -∞ d ⃗ ke i ⃗ k · ∆ ⃗x || ∞ ∑ l = -∞ K ν ( e -πi/ 2 ηλ l ) K ν ( e πi/ 2 η ' λ l ) × cos[ v l ( z -a 1 ) / ˜ a + α 1 ( v l / ˜ a )] cos[ v l ( z ' -a 1 ) / ˜ a + α 1 ( v l / ˜ a )] 1 + cos[ v l +2 α 1 ( v l / ˜ a )] sin( v l ) /v l , (3.2) \nwhere ∆ ϕ = ϕ ' -ϕ and ∆ ⃗x || = ⃗x ' || -⃗x || . Moreover, to obtain the expression above we have introduced the notation λ l = √ p 2 + v 2 l / ˜ a 2 + ⃗ k 2 and used the identity [20], \ne i ( ν -ν ∗ ) π/ 2 H (1) ν ( λη ) [ H (1) ν ( λη ' ) ] ∗ = 4 π 2 K ν ( -iλη ) K ν ( iλη ' ) . (3.3) \nTo develop the sum over the quantum number l , we adopt a variant of the Abel-Plana summation formula [18], \n∞ ∑ l =1 πv l f ( v l ) v l +sin( v l ) cos[ v l +2 α 1 ( v l / ˜ a )] = -π 2 f (0) 1 -b 1 -b 2 + ∫ ∞ 0 dyf ( y ) + i ∫ ∞ 0 dy f ( iy ) -f ( -iy ) ( b 1 y -1) b 1 +1 ( b 2 y -1) b 2 +1 e 2 y -1 . (3.4) \nFor our case, \nf ( y ) = K ν ( e -πi/ 2 ηλ l ) K ν ( e πi/ 2 η ' λ l ) cos[ y ( z -a 1 ) / ˜ a + α 1 ( y/ ˜ a )] cos[ y ( z ' -a 1 ) / ˜ a + α 1 ( y/ ˜ a )] . (3.5) \nIn accordance with the formula above, the Wightman function can be decomposed as \nW ( x, x ' ) = W 1 ( x, x ' ) + ∆ W ( x, x ' ) , (3.6) \nwhere the first term corresponds to the contribution to a single plate in z = a 1 with a cosmic string perpendicular to it and has been considered in [17] in the analysis of VEV of the bosonic energymomentum tensor. It reads, \nW 1 ( x, x ' ) = 8 q ( ηη ' ) D/ 2 (2 π ) D a D -1 ∞ ∑ n = -∞ e inq ∆ ϕ ∫ ∞ 0 dppJ q | n + α | ( pr ) J q | n + α | ( pr ' ) ∫ d ⃗ ke i ⃗ k · ∆ ⃗x || × ∫ ∞ 0 duK ν ( e -πi/ 2 η √ u 2 + p 2 + k 2 ) K ν ( e πi/ 2 η ' √ u 2 + p 2 + k 2 ) × cos[ u ( z -a 1 ) + α 1 ( u )] cos[ u ( z ' -a 1 ) + α 1 ( u )] . (3.7) \nThe second contribution in (3.6), ∆ W ( x, x ' ) is the interference term and it is given by \n∆ W ( x, x ' ) = 2 q (2 π ) D -1 a D -1 ∞ ∑ n = -∞ e inq ∆ ϕ ∫ ∞ 0 dppJ q | n + α | ( pr ) J q | n + α | ( pr ' ) ∫ d ⃗ ke i ⃗ k · ∆ ⃗x || × ∫ ∞ √ p 2 + k 2 du c 1 ( u ) c 2 ( u ) e 2˜ au -1 cos[ u ( z -a 1 ) + ˜ α 1 ( u )] cos[ u ( z ' -a 1 ) + ˜ α 1 ( u )] × y -D [ ˜ K ν ( ηy ) ˜ I ν ( η ' y ) + ˜ I ν ( ηy ) ˜ K ν ( η ' y )] ∣ ∣ ∣ y = √ u 2 -p 2 -k 2 , (3.8) \nwhere ˜ α 1 ( u ) is given by the relation e 2˜ α 1 ( u ) = c 1 ( u ) and the following notations were introduced: \n˜ K ν = y D/ 2 K ν ( y ) , ˜ I ν = y D/ 2 [ I ν ( y ) + I -ν ( y )] , (3.9) \nand \nc j ( u ) = β j u -1 β j u +1 . (3.10) \nFor further convenience, the contribution induced by a single plate given in (3.7) can be generalized as \nW j ( x, x ' ) = W dS , cs ( x, x ' ) + W (1) j ( x, x ' ) , (3.11) \nwhere for j = 1 or j = 2 it is induced by a single plate at z = a 1 or z = a 2 , respectively. Moreover, the first term in the above expression contains two contributions: one induced in pure dS spacetime, i.e., in the absence of cosmic string, and the other one induced by it. This contribution reads, \nW dS , cs ( x, x ' ) = 4 q ( ηη ' ) D/ 2 (2 π ) D a D -1 ∞ ∑ n = -∞ e inq ∆ ϕ ∫ ∞ 0 dppJ q | n + α | ( pr ) J q | n + α | ( pr ' ) ∫ d ⃗ ke i ⃗ k · ∆ ⃗x || × ∫ ∞ 0 duK ν ( e -πi/ 2 η √ u 2 + p 2 + k 2 ) K ν ( e πi/ 2 η ' √ u 2 + p 2 + k 2 ) cos( u ∆ z ) . (3.12) \nIn [17] this function has been explicitly developed. \nOur aim in this work is to investigate the current induced by the plates. So, let us consider first the second contribution in (3.11), which is induced by a single plate at z = a j and it is given by \nW (1) j ( x, x ' ) = qa 1 -D 2(2 π ) D -1 ∞ ∑ n = -∞ e inq ∆ ϕ ∫ ∞ 0 dppJ q | n + α | ( pr ) J q | n + α | ( pr ' ) ∫ d ⃗ ke i ⃗ k · ∆ ⃗x || × ∫ ∞ √ p 2 + k 2 du e -u | z + z ' -2 a j | c j ( u ) y -D [ ˜ K ν ( ηy ) ˜ I ν ( η ' y ) + ˜ I ν ( ηy ) ˜ K ν ( η ' y )] ∣ ∣ ∣ y = √ u 2 -p 2 -k 2 . (3.13) \nThe decomposition in (3.11), allow us rewrite the Wightman function in the more symmetric form: \nW ( x, x ' ) = W dS , cs ( x, x ' ) + ∑ j =1 , 2 W j ( x, x ' ) + q 2(2 π ) D -1 a D -1 ∞ ∑ n = -∞ e inq ∆ ϕ × ∫ ∞ 0 dppJ q | n + α | ( pr ) J q | n + α | ( pr ' ) ∫ d ⃗ ke i ⃗ k · ∆ ⃗x || ∫ ∞ √ p 2 + k 2 du c 1 ( u ) c 2 ( u ) e 2˜ au -1 × 2 cosh( u ∆ z ) + ∑ j =1 , 2 e -u | z + z ' -2 a j | /c j ( u ) × y -D [ ˜ K ν ( ηy ) ˜ I ν ( η ' y ) + ˜ I ν ( ηy ) ˜ K ν ( η ' y )] ∣ ∣ ∣ y = √ u 2 -p 2 -k 2 . (3.14) \nwhere the last term is interference part, ∆ W ( x, x ' ), that is induced by the two plates. For further convenience, we will examine the problem in the particular cases of the well known Dirichlet and Neumann boundary conditions, separately, corresponding to β j → 0 and β j →∞ , respectively. This allow us to rewrite the interference part as \n∆ W ( J ) ( x, x ' ) = q 2(2 π ) D -1 a D -1 ∞ ∑ n = -∞ e inq ∆ ϕ ∫ ∞ 0 dppJ q | n + α | ( pr ) J q | n + α | ( pr ' ) × ∫ d ⃗ ke i ⃗ k · ∆ ⃗x || ∫ ∞ √ p 2 + k 2 du e 2˜ au -1 2 cosh( u ∆ z ) + δ ( J ) ∑ j =1 , 2 e -u | z + z ' -2 a j | × y -D [ ˜ K ν ( ηy ) ˜ I ν ( η ' y ) + ˜ I ν ( ηy ) ˜ K ν ( η ' y )] ∣ ∣ ∣ y = √ u 2 -p 2 -k 2 , (3.15) \nwhere J = D for Dirichlet BC, δ ( D ) = -1, and J = N for Neumann BC, δ ( N ) \nNow, introducing a new variable v = √ u 2 -p 2 -k 2 and using the identity ( e 2˜ au -1) -1 ∑ ∞ l =1 e -2 u ˜ al , we have \n= 1. = \n∆ W ( J ) ( x, x ' ) = q 2(2 π ) D -1 a D -1 ∞ ∑ n = -∞ e inq ∆ ϕ ∫ ∞ 0 dppJ q | n + α | ( pr ) J q | n + α | ( pr ' ) ∫ d ⃗ ke i ⃗ k · ∆ ⃗x || × ∫ ∞ 0 dvv √ v 2 + p 2 + k 2 ∞ ∑ l =1 e -2˜ al √ v 2 + p 2 + k 2 [ 2 cosh(∆ z √ v 2 + p 2 + k 2 ) + δ ( J ) ∑ j =1 , 2 e -| z + z ' -2 a j | √ v 2 + p 2 + k 2 ] v -D [ ˜ K ν ( ηv ) ˜ I ν ( η ' v ) + ˜ I ν ( ηv ) ˜ K ν ( η ' v )] . (3.16) \nWe now proceed by using the identity [19], \ne -ab a = 2 √ π ∫ ∞ 0 e -a 2 s 2 -b 2 / (4 s 2 ) , (3.17) \nwhich allow us to perform the integration over p and ⃗ k variables in (3.16), yielding the following result: \n∆ W ( J ) ( x, x ' ) = 4 q (4 π ) D/ 2+1 a D -1 ∞ ∑ l =1 ∫ ∞ 0 ds s D -1 e -( r 2 + r ' 2 +∆ ⃗x 2 ∥ ) / (4 s 2 ) ∫ ∞ 0 dvve -s 2 v 2 × [ ∑ ϵ = ± 1 e -(2˜ al + ϵ ∆ z ) 2 / (4 s 2 ) + δ ( J ) ∑ j =1 , 2 e -(2˜ al + | z + z ' -2 a j | ) 2 / (4 s 2 ) ] × v -D [ ˜ K ν ( ηv ) ˜ I ν ( η ' v ) + ˜ I ν ( ηv ) ˜ K ν ( η ' v )] ∞ ∑ n = -∞ e inq ∆ ϕ I q | n + α | ( rr ' 2 s 2 ) . (3.18) \nNow in order to continue our calculation, we develop the sum over n . The parameter α in Eq. (2.12) can be written in the form \nα = n 0 + α 0 , with | α 0 | < 1 2 , (3.19) \nbeing n 0 an integer number. This allow us to sum over the quantum number n in (3.18) by using the formula obtained in [21]: \n∞ ∑ n = -∞ e iqn ∆ ϕ I q | n + α | ( x ) = 1 q ∑ k e x cos(2 πk/q -∆ ϕ ) e iα 0 (2 πk -q ∆ ϕ ) -e -iqn 0 ∆ ϕ 2 πi ∑ j = ± 1 je jiπq | α 0 | ∫ ∞ 0 dy cosh [ qy (1 -| α 0 | )] -cosh ( | α 0 | qy ) e -iq (∆ ϕ + jπ ) e x cosh ( y ) [ cosh ( qy ) -cos ( q (∆ ϕ + jπ )) ] , (3.20) \nwhere k is an integer number varying in the interval \n-q 2 + ∆ ϕ Φ 0 ≤ k ≤ q 2 + ∆ ϕ Φ 0 . (3.21) \nThe substitution of (3.20) in (3.18), allow us to write \n∆ W ( J ) ( x, x ' ) = 4 (4 π ) D/ 2+1 a D -1 ∞ ∑ l =1 ∫ ∞ 0 ds s D -1 { ∑ k e iα 0 (2 πk -q ∆ ϕ ) × [ ∑ ϵ = ± 1 e -[ ρ kl +(2˜ al + ϵ ∆ z ) 2 ] / (4 s 2 ) + δ ( J ) ∑ j =1 , 2 e -[ ρ klj +(2˜ al + | z + z ' -2 a j | ) 2 ] / (4 s 2 ) ] -e -iqn 0 ∆ ϕ 2 πi ∑ j = ± 1 je jiπq | α 0 | ∫ ∞ 0 dy cosh [ qy (1 -| α 0 | )] -cosh ( | α 0 | qy ) e -iq (∆ ϕ + jπ ) cosh ( qy ) -cos ( q (∆ ϕ + jπ )) × [ ∑ ϵ = ± 1 e -[ ρ yl +(2˜ al + ϵ ∆ z ) 2 ] / (4 s 2 ) + δ ( J ) ∑ j =1 , 2 e -[ ρ ylj +(2˜ al + | z + z ' -2 a j | ) 2 ] / (4 s 2 ) ]} × ∫ ∞ 0 dve -s 2 v 2 v 1 -D [ ˜ K ν ( ηv ) ˜ I ν ( η ' v ) + ˜ I ν ( ηv ) ˜ K ν ( η ' v )] , (3.22) \nwhere the following notation was introduced \nρ kl = r 2 + r ' 2 -2 rr ' cos(2 πk/q -∆ ϕ ) + ∆ ⃗x 2 ∥ , ρ yl = r 2 + r ' 2 +2 rr ' cosh( y ) + ∆ ⃗x 2 ∥ . (3.23)", '4 VEV of the Current Density': "The VEV of the bosonic current density is formally calculated using the Wightman function through the formula \n⟨ j µ ( x ) ⟩ = ie lim x ' → x ( ∂ µ -∂ µ ' ) W ( x, x ' ) + 2 ieA µ W ( x, x ' ) . (4.1) \nThe only non vanishing component in the setup problem under consideration is the one along the azimuthal direction. According to the decomposition made in (3.14), we have \n⟨ j ϕ ⟩ = ⟨ j ϕ ⟩ dS , cs + ∑ j =1 , 2 ⟨ j ϕ ⟩ ( j ) ( J ) +∆ ⟨ j ϕ ⟩ ( J ) . (4.2) \nThis component is induced by the presence of the constant potential vector component along the angular direction, A ϕ , interacting with the scalar field. Although the corresponding field strength vanishes, the nontrivial topology of the string gives rise to Aharonov-Bohm-like effect on the current density along azimuthal direction. As to the other components of the current density, it can be easily checked that they trivially vanish. \nLet us develop and study each term of (4.2) individually.", '4.1 Azimuthal Current in the Presence of a Single Plate': "The first term on the right-hand side is induced by the string, which is obtained by taking (3.12) into (4.1): \n⟨ j ϕ ⟩ dS , cs = -8 q 2 eη D (2 π ) D a D -1 ∞ ∑ n = -∞ ( n + α ) ∫ ∞ 0 dpp ( J q | n + α | ( pr )) 2 ∫ d ⃗ k × ∫ ∞ 0 duK ν ( e -πi/ 2 η √ u 2 + p 2 + k 2 ) K ν ( e πi/ 2 η √ u 2 + p 2 + k 2 ) . (4.3) \nThis term has been already analyzed in [16] for (1 + 3)-dimensions, considering that the string is compactified to a circle. Our aim in this paper, however, is the study of the contributions engendered in the presence of the plates. \nFor the contribution induced in the presence of a single plate, we use the representation of the Wightman function (3.13) for Dirichlet ( c j ( u ) = -1) and Neumann ( c j ( u ) = 1): \n⟨ j ϕ ⟩ ( j ) ( J ) = -2 δ ( J ) q 2 eη D (2 π ) D -1 a D -1 ∞ ∑ n = -∞ ( n + α ) ∫ ∞ 0 dpp ( J q | n + α | ( pr )) 2 ∫ d ⃗ k × ∫ ∞ √ p 2 + k 2 due -2 u | z -a j | K ν ( ηy )[ I ν ( ηy ) + I -ν ( ηy )] ∣ ∣ ∣ y = √ u 2 -p 2 -k 2 . (4.4) \nIntroducing a new variable v = √ u 2 -p 2 -k 2 and using the identity given in (3.17), we get \n⟨ j ϕ ⟩ ( j ) ( J ) = -4 δ ( J ) q 2 eη D √ π (2 π ) D -1 a D -1 ∞ ∑ n = -∞ ( n + α ) ∫ ∞ 0 dse -( z -a j ) 2 / (2 s 2 ) ∫ ∞ 0 dppe -s 2 p 2 ( J q | n + α | ( pr )) 2 × ∫ d ⃗ ke -s 2 k 2 ∫ ∞ 0 dvve -s 2 p 2 K ν ( ηv )[ I ν ( ηv ) + I -ν ( ηv )] . (4.5) \nWe can now perform the integrals over ⃗ k , p and v . In ⃗ k we have ( D -3) Gaussian integrals and the integrals over p and v are performed by using the formulas [19]: \n∫ ∞ 0 dppe -s 2 p 2 ( J γ ( pr )) 2 = e -r 2 / (4 s 2 ) 2 s 2 I γ ( r 2 / (2 s 2 )) (4.6) \nand \n∫ ∞ 0 dvve -s 2 v 2 K ν ( ηv )[ I ν ( ηv ) + I -ν ( ηv )] = e η 2 / (2 s 2 ) 2 s 2 K ν ( η 2 / (2 s 2 )) . (4.7) \nSubstituting the results of these integrations into (4.5) we get, \n⟨ j ϕ ⟩ ( j ) ( J ) = -δ ( J ) q 2 eη D (2 π ) D/ 2+1 a D -1 ∫ ∞ 0 dχχ D/ 2 -1 e -χ [ r 2 +( z -a j ) 2 -η 2 ] /η 2 K ν ( χ ) × ∞ ∑ n = -∞ ( n + α ) I q | n + α | ( χr 2 /η 2 ) . (4.8) \nThe summation over n has been obtained in [22] and it is given by the representation \n∞ ∑ n = -∞ ( n + α ) I q | n + α | ( x ) = 2 x q 2 [ q/ 2] ∑ ' k =0 sin(2 πk/q ) sin(2 πkα 0 ) e x cos(2 πk/q ) + x qπ ∫ ∞ 0 dy sinh( y ) e -x cosh( y ) g ( q, α 0 , y ) cosh ( qy ) -cos ( qπ ) , (4.9) \nwith the function \ng ( q, α 0 , y ) = sin( α 0 qπ ) sinh[(1 -| α 0 | ) qy ] -sinh( qα 0 y ) sin[(1 -| α 0 | ) qπ ] . (4.10) \nIn addition, the notation [ q/ 2] denotes the integer part of q/ 2, and the prime on the summation symbol over k indicates, that for even values of q , the term k = q/ 2 should be taken with the coefficient 1 / 2. \nThe integration over χ can be performed by using the formula \n∫ ∞ 0 dxx µ -1 e -vx K ν ( x ) = √ π 2 ν Γ( µ -ν )Γ( µ + ν ) Γ( µ +1 / 2)( v +1) µ + ν F ( µ + ν, ν + 1 2 ; µ +1 / 2; v -1 v +1 ) , (4.11) \nwhere F ( a, b ; c, z ) represents the hypergeometric function [19]. \nThus, taking (4.9) into (4.8), we integrate over χ obtaining the following result \n⟨ j ϕ ⟩ ( j ) ( J ) = -2 δ ( J ) e (2 π ) ( D +1) / 2 a D +1 [ [ q/ 2] ∑ ' k =1 sin(2 πk/q ) sin(2 πkα 0 ) F D/ 2+1 ν ( u kj ) + q π ∫ ∞ 0 dy sinh( y ) g ( q, α 0 , y ) cosh ( qy ) -cos ( qπ ) F D/ 2+1 ν ( u yj ) ] , (4.12) \nwhere we have introduced the function \nF D/ 2+1 ν ( x ) = 2 ν +1 / 2 Γ( D/ 2 -ν +1)Γ( D/ 2 + ν +1) Γ( D/ 2 + 3 / 2)( x +1) D/ 2+ ν +1 F ( D 2 + ν +1 , ν + 1 2 ; D +3 2 ; x -1 x +1 ) , (4.13) \nvariables \nu kj = 2 r 2 p s 2 k +2( z p -a j /η ) 2 -1 , u yj = 2 r 2 p c 2 y +2( z p -a j /η ) 2 -1 . (4.14) \nand the notations \ns k = sin( πk/q ) , c y = cosh( y ) . (4.15) \nMoreover, in (4.12), r p = r/η and z p = z/η are the proper distances from the string and the plate, respectively, in unities of the dS spacetime curvature, a . From (4.12), we can see that ⟨ j ϕ ⟩ ( j ) ( J ) is an odd function of α 0 with period equal to quantum flux, Φ 0 = 2 π/e ; moreover, for 1 ≤ q < 2, the first term on the right-hand side of (4.12) is absent. \nLet us now analyze the behavior of this VEV in some limiting cases. In the conformal coupled massless scalar field case, the function F D/ 2+1 ν ( x ) takes the form [17]: \nF D/ 2+1 ν ( x ) = Γ( D/ 2 + 1 / 2) ( x +1) ( D +1) / 2 . (4.16) \nTherefore, the azimuthal current density (4.12) in this case reads \n⟨ j ϕ ⟩ ( j ) ( J ) = -2 δ ( J ) e Γ( D/ 2 + 1 / 2) (4 π ) ( D +1) / 2 a D +1 [ [ q/ 2] ∑ ' k =1 sin(2 πk/q ) sin(2 πkα 0 ) [ r 2 p s 2 k +( z p -a j /η ) 2 ] ( D +1) / 2 + q π ∫ ∞ 0 dy sinh( y ) g ( q, α 0 , y ) [cosh ( qy ) -cos ( qπ )][ r 2 p c 2 y +( z p -a j /η ) 2 ] ( D +1) / 2 ] . (4.17) \n̸ \nWe now want to study the asymptotic behavior of the azimuthal current density in the limits near and distant points from the core of the string, located at r = 0. For points outside of the plate, z = a j , the VEV of the current density on the string is finite for q | α 0 | > 1 and can be obtained directly by putting r = 0 in (4.12). On the other hand, for q | α 0 | < 1, the VEV diverges near the string as \n⟨ j ϕ ⟩ ( j ) ( J ) ≈ -2 q | α 0 |-1 / 2 δ ( J ) qe Γ( 3 2 -q | α 0 | ) (2 π ) ( D +3) / 2 a D +1 r 2(1 -q | α 0 | ) p F D/ 2+ q | α 0 |-1 / 2 ν (2( z p -a j /η ) 2 -1) . (4.18) \nFor distant points from the string, r ≫ η, | z -a j | , we use the corresponding asymptotic expression for function F D/ 2+1 ν ( x ) [17]: \nF D/ 2+1 ν ( x ) ≈ 2 ν -1 / 2 √ π Γ( ν )Γ( D/ 2 -ν +1) x -D/ 2 -1+ ν , (4.19) \nwhich taken into (4.12) gives \n⟨ j ϕ ⟩ ( j ) ( J ) ≈ -2 2 ν -D -1 δ ( J ) e Γ( ν )Γ( D/ 2 -ν +1) π D/ 2+1 a D +1 r D +2 -2 ν p [ [ q/ 2] ∑ ' k =1 sin(2 πk/q ) sin(2 πkα 0 ) s D +2 -2 ν k + q π ∫ ∞ 0 dy sinh( y ) g ( q, α 0 , y ) c D +2 -2 ν y [cosh ( qy ) -cos ( qπ )] ] . (4.20) \n̸ \nWe now turn to the investigation of the behavior of azimuthal current density for points close and far from the plate at z = a j . On the surface of the plate, z = a j , the VEV induced by the plate is finite for points outside the string, r = 0. However, for points on the string's, r = 0, and q | α 0 | > 1, the VEV diverges as \n⟨ j ϕ ⟩ ( j ) ( J ) ≈ -4 δ ( J ) e Γ( D +1 2 ) (4 π ) ( D +1) / 2 a D +1 | z p -a j /η | D -1 [ [ q/ 2] ∑ ' k =1 sin(2 πk/q ) sin(2 πkα 0 ) + q π ∫ ∞ 0 dy sinh( y ) g ( q, α 0 , y ) cosh ( qy ) -cos ( qπ ) ] , (4.21) \nand for q | α 0 | < 1, it diverges as \n⟨ j ϕ ⟩ ( j ) ( J ) ≈ -4 δ ( J ) qe Γ( D +1 2 )Γ( 3 2 -q | α 0 | ) (2 π ) ( D +3) / 2 a D +1 r 2(1 -q | α 0 | ) p | z p -a j /η | D +2 q | α 0 |-4 . (4.22) \nFor distant regions from the plate, | z -a j | ≫ η, r , we use again the formula given in (4.19), obtaining \n⟨ j ϕ ⟩ ( j ) ( J ) ≈ -2 2 ν -D -1 δ ( J ) e Γ( ν )Γ( D/ 2 -ν +1) π D/ 2+1 a D +1 | z p -a j /η | D +2 -2 ν [ [ q/ 2] ∑ ' k =1 sin(2 πk/q ) sin(2 πkα 0 ) + q π ∫ ∞ 0 dy sinh( y ) g ( q, α 0 , y ) [cosh ( qy ) -cos ( qπ )][1 + r 2 p c 2 y / ( z p -a j /η ) 2 ] D/ 2+1 -ν ] . (4.23) \nWe also consider the Minkowskian limit, a →∞ , while keeping t fixed. In this limit, the geometry under consideration simplifies to that of a cosmic string in the background of ( D + 1)-dimensional Minkowski spacetime. For the analysis of this limit, the representation for the VEV in the presence of one plate, given in (4.12), is not convenient. Therefore, for this end, we return to the representation \npresent in (4.8) with the series over n given by (4.9). For the coordinate η in the arguments of the modified Bessel function, we have η ≈ | t -a | . In this limit, ν ≫ 1 and, according to (2.15), we have ν ≈ ima . Using the uniform asymptotic expansion for the Macdonald function of imaginary order as provided in [23], substituting it into (4.8), and following some intermediate steps, we obtain \n⟨ j ϕ ⟩ ( j ) , ( M ) ( J ) = -4 δ ( J ) em D +1 (2 π ) ( D +1) / 2 [ [ q/ 2] ∑ ' k =1 sin(2 πk/q ) sin(2 πkα 0 ) f D +1 2 (2 m √ r 2 s 2 k +( z -a j ) 2 ) + q π ∫ ∞ 0 dy sinh( y ) g ( q, α 0 , y ) cosh ( qy ) -cos ( qπ ) f D +1 2 (2 m √ r 2 c 2 y +( z -a j ) 2 ) ] , (4.24) \nwhere we have introduced the notation \nf µ ( x ) = K µ ( x ) x µ , (4.25) \nbeing K µ ( x ) the Macdonald function. \nIn Fig. 1 is exhibited the behavior of the azimuthal current density induced by a single plate at a j = 0, as function of the the proper distances from the string, r p , (top panel) and the plate, z p , (bottom panel), in unities of the dS spacetime curvature, a . We consider Dirichlet and Neumann boundary conditions for various values of the parameter q , which is associated with the deficit angle. From the top panel we can see that the VEV of the azimuthal current density is is finite on the string and rapidly tends to zero as r p increases. From the bottom panel, we observe that the VEV is finite on the plate location and rapidly goes to zero as z p goes large, in accordance to our asymptotic analysis. Moreover, note that in both plots the intensities increase with q and are higher for Dirichlet BC, compared with Neumann BC, near the string or the plate.", '4.2 Azimuthal Current in the Region between the Plates': "Let us analyze now the contribution induced in the region between the plates, a 1 < z < a 2 . To this end, we take (3.18) into (4.1), obtaining the expression: \n∆ ⟨ j ϕ ⟩ ( J ) = -16 eq 2 η 2 (4 π ) D/ 2+1 a D -1 ∞ ∑ l =1 ∫ ∞ 0 ds s D -1 e -r 2 /s 2 × [ 2 e -(˜ al ) 2 /s 2 + δ ( J ) ∑ j =1 , 2 e -(˜ al + | z -a j | ) 2 /s 2 ] ∫ ∞ 0 dvve -s 2 v 2 × K ν ( ηv )[ I ν ( ηv ) + I -ν ( ηv )] ∞ ∑ n = -∞ ( n + α ) I q | n + α | ( r 2 2 s 2 ) . (4.26) \nThe next step is to integrate over v by using (4.7): \n∆ ⟨ j ϕ ⟩ ( J ) = -2 q 2 e (2 π ) D/ 2+1 a D -1 ∞ ∑ l =1 ∫ ∞ 0 dχχ D/ 2 -1 e -( r 2 /η 2 -1) χ K ν ( χ ) × [ 2 e -2 χ (˜ al/η ) 2 + δ ( J ) ∑ j =1 , 2 e -2 χ (˜ al + | z -a j | ) 2 /η 2 ] ∞ ∑ n = -∞ ( n + α ) I q | n + α | ( χr 2 /η 2 ) , (4.27) \nwhere we have introduced the variable χ = η 2 / (2 s 2 ). By using the formula (4.9) for the sum over n , we can integrate over χ with the help of (4.11). The result is the following expression: \n<!-- image --> \nFigure 1: The VEV of the azimuthal current density induced by a single plate, located at z = a j , is plotted as function of the proper distance from the string, r p , (top panel) and the proper distance from the plate, z p , (bottom panel), in units of a . In both plots, we consider Dirichlet and Neumann boundary conditions and various values of q . Both graphs are plotted for D = 3, α 0 = 0 . 25, ξ = 0, ma = 1 . 5 and a j = 0. Moreover, in the top panel we have fixed z p = 0 and in the bottom one, r p = 1. \n<!-- image --> \n∆ ⟨ j ϕ ⟩ ( J ) = -2 e (2 π ) ( D +1) / 2 a D +1 ∞ ∑ l =1 { [ q/ 2] ∑ ' k =1 sin(2 πk/q ) sin(2 πkα 0 ) × [ 2 F D/ 2+1 ν ( v kl ) + δ ( J ) ∑ j =1 , 2 F D/ 2+1 ν ( v kjl ) ] + q π ∫ ∞ 0 dy sinh( y ) g ( q, α 0 , y ) cosh ( qy ) -cos ( qπ ) × [ 2 F D/ 2+1 ν ( v yl ) + δ ( J ) ∑ j =1 , 2 F D/ 2+1 ν ( v yjl ) ]} , (4.28) \nwhere we have introduced the variables \nv kl = 2 r 2 p s 2 k +2(˜ al/η ) 2 -1 , v yl = 2 r 2 p c 2 y +2(˜ al/η ) 2 -1 , (4.29) \nand \nv kjl = 2 r 2 p s 2 k +2(˜ al/η + | z p -a j /η | ) 2 -1 , v yjl = 2 r 2 p c 2 y +2(˜ al/η + | z p -a j /η | ) 2 -1 . (4.30) \nLet us now study the behavior of this VEV in some limiting situations. In the conformal coupled massless scalar field case, the function F D/ 2+1 ν ( u ) has the simple form given in (4.16). Thus, in this case, the VEV induced in the region between the plates reads: \n∆ ⟨ j ϕ ⟩ ( J ) = -2 e Γ( D/ 2 + 1 / 2) (4 π ) ( D +1) / 2 a D +1 ∞ ∑ l =1 { [ q/ 2] ∑ ' k =1 sin(2 πk/q ) sin(2 πkα 0 ) [ 2 [ r 2 p s 2 k +(˜ al/η ) 2 ] ( D +1) / 2 + ∑ j =1 , 2 δ ( J ) [ r 2 p s 2 k +(˜ al/η + | z p -a j /η | ) 2 ] ( D +1) / 2 ] + q π ∫ ∞ 0 dy sinh( y ) g ( q, α 0 , y ) cosh ( qy ) -cos ( qπ ) × [ 2 [ r 2 p c 2 y +(˜ al/η ) 2 ] ( D +1) / 2 + ∑ j =1 , 2 δ ( J ) [ r 2 p c 2 y +(˜ al/η + | z p -a j /η | ) 2 ] ( D +1) / 2 ]} . (4.31) \nWe now consider the limit of large values of the distance between the plates, ˜ a ≫ r, | z -a j | . In this case, we can not neglect the term 2 r 2 p c 2 y in the arguments of the functions F D/ 2+1 ν ( u ), since it is essential for the convergence of the integral over y . Therefore, in this limit, we get \n∆ ⟨ j ϕ ⟩ ( J ) ≈ 2 2 ν -D e Γ( ν )Γ( D/ 2 -ν +1) π D/ 2+1 a D +1 (˜ a/η ) D +2 -2 ν ∞ ∑ l =1 { [ q/ 2] ∑ ' k =1 sin(2 πk/q ) sin(2 πkα 0 ) × 2 l D +2 -2 ν + ∑ j =1 , 2 δ ( J ) ( l + | z -a j | / ˜ a ) D +2 -2 ν + q π ∫ ∞ 0 dy sinh( y ) g ( q, α 0 , y ) [cosh ( qy ) -cos ( qπ )] × 2 [( rc y / ˜ a ) 2 + l 2 ] D/ 2+1 -ν + ∑ j =1 , 2 δ ( J ) [( rc y / ˜ a ) 2 +( l + | z -a j | / ˜ a ) 2 ] D/ 2+1 -ν } . (4.32) \nThis result show us that the VEV of the azimuthal current density decays as the separation between the plates increases. \nFor distant points from the string and fixed distances from the plates, r ≫ η, | z -a j | , we have v kjl ≈ v kl and v yjl ≈ v yl , according to (4.29) and (4.30). Therefore, using the corresponding asymptotic expression for the function F D/ 2+1 ν ( u ) given in (4.19), we obtain the following result: \n∆ ⟨ j ϕ ⟩ ( J ) ≈ -2 2 ν +1 -D (1 + δ ( J ) ) e Γ( ν )Γ( D/ 2 -ν +1) π D/ 2+1 a D +1 r D +2 -2 ν p ∞ ∑ l =1 { [ q/ 2] ∑ ' k =1 sin(2 πk/q ) sin(2 πkα 0 ) [ s 2 k +( l ˜ a/r ) 2 ] D/ 2+1 -ν + q π ∫ ∞ 0 dy sinh( y ) g ( q, α 0 , y ) [cosh ( qy ) -cos ( qπ )][ c 2 y +(˜ al/r ) 2 ] D/ 2+1 -ν } . (4.33) \nFinally, we consider the Minkowskian limit, a → ∞ , with a fixed value of t . Following the same procedure adopted for the contribution induced by a single plate, the VEV of the azimuthal current density induced in the region between the plates reads: \n∆ ⟨ j ϕ ⟩ ( M ) ( J ) = -4 em D +1 (2 π ) ( D +1) / 2 a D +1 ∞ ∑ l =1 { [ q/ 2] ∑ ' k =1 sin(2 πk/q ) sin(2 πkα 0 ) × [ 2 f D +1 2 (2 m √ r 2 s 2 k +( z -a j ) 2 ) + δ ( J ) ∑ j =1 , 2 f D +1 2 (2 m √ r 2 p s 2 k +(˜ al/η + | z p -a j /η | ) 2 ) ] + q π ∫ ∞ 0 dy sinh( y ) g ( q, α 0 , y ) cosh ( qy ) -cos ( qπ ) [ 2 f D +1 2 (2 m √ r 2 c 2 y +( z -a j ) 2 ) + δ ( J ) ∑ j =1 , 2 f D +1 2 (2 m √ r 2 p c 2 y +(˜ al/η + | z p -a j /η | ) 2 ) ]} , (4.34) \nwith the function f µ ( x ) defined in (4.25). \nIn Fig. 2 is displayed the dependence of the VEV of the azimuthal current density in the region between the plates as function of the proper distance from the string, r p , considering z p = 0 . 2 (top panel) and z p = 0 . 5 (bottom panel). This is presented in unities of the dS spacetime curvature, a . In both plots, we consider Dirichlet and Neumann boundary conditions for various values of the parameter associated with the deficit angle, q . We observe that the current density in the region between the plates is finite on the string and rapidly goes to zero as the proper distance from the string, r p , increases. \n<!-- image --> \nFigure 2: The VEV of the azimuthal current density induced between the plates is plotted as function of the proper distance from the string, r p . In the top panel we consider z p = 0 . 2 and in the bottom panel, z p = 0 . 5. In both plots we consider Dirichlet and Neumann boundary conditions and different values of q . Both graphs are plotted for D = 3, α 0 = 0 . 25, ξ = 0, ma = 1 . 5. The positions of the plates are in both plots at a 1 = 0 and a 2 = 1. \n<!-- image --> \nIn Fig. 3 we display the behavior of the VEV of the azimuthal current density in the region between the plates as function of z p . In the top panel we consider r p = 0 . 1 and in the bottom panel, r p = 0 . 5. Here also, we assume Dirichlet and Neumann boundary conditions and different values for q . We can observe from both plots that the VEV is finite on the plates at z = 0 and z = 1, being symmetric with respect to the midpoint between the plates at z = 0 . 5. Moreover, in both plots the intensities increase with the parameter q and are higher for Dirichlet BC compared with the Neumann BC for the same values of q . \n<!-- image --> \nFigure 3: The VEV of the azimuthal current density induced between the plates is plotted as function of z p . In the top panel we assume r p = 0 . 1, and in the bottom, r p = 0 . 5. For both plots we also assume D = 3, α 0 = 0 . 25, ξ = 0, ma = 1 . 5. \n<!-- image -->", '5 Conclusions': '̸ \nThe main objective of this work was to investigate the vacuum bosonic current induced by the presence of a carrying-magnetic-flux cosmic string in a ( D +1)-de Sitter spacetime considering the presence of two flat boundaries perpendicular to it. In this setup, we impose that the scalar charged quantum field obeys the Robin boundary conditions on the two flat boundaries. The particular cases of Dirichlet and Neumann boundary conditions are study separately. In order to develop this analysis, we presented the the Wightman function in (3.14) in a more symmetric for, i.e., decomposed in a part associated with the presence of the string in dS space only, plus the contributions induced by just one flat plane followed by other induced by two flat planes. Because the current induced by a cosmic string have been calculated before, our focuses were in the obtainment of the azimuthal component induced by a single plate, develop in subsection 4.1. The contravarient component, ⟨ j ϕ ⟩ , was presented in (4.12) combined with (4.13), (4.14) and (4.15). Some limiting cases for this current have been presented. In the massless conformal coupled was given in (4.17). For points near the string, and considering z = a j , we have shown that for q | α 0 | > 1 this component is finite, and we can take r = 0 in (4.12); however for q | α 0 | < 1 this VEV diverges with r -2(1 -q | α 0 | ) p as shown in (4.18). For points far from the string, (4.12) decays with r -( D +2 -2 ν ) p . For points close to the plate, z = a j , but outside the string, r = 0, the VEV is finite; however on the string, r = 0, and q | α 0 | > 1 the VEV diverges as exhibited in (4.21). The Minkowskian limit, i.e., a →∞ and fixed value of t , has been also considered and is given in (4.24). Finally for points distant from the plate, | z -a j | ≫ η, r , the current decays as | z p -a j /η | -( D +2 -2 ν ) . Also in the subsection 4.1, we have presented two plots, in Fig. 1, exhibiting the behavior of ⟨ j ϕ ⟩ as function of r p (top panel) and z p (bottom panel), considering separately the Dirichlet and Neumann BC and different values attributed to q . We can observe that these plots are in accordance with our asymptotic analysis. \n̸ \nThe analysis of the VEV of azimuthal current in the region between the plates, has been developed in subsection 4.2. The complete expression for this VEV is given in (4.28), combined with (4.29) and (4.30). Some limiting cases for this contribution has have been analyzed. For a conformal coupled massless scalar field case, the VEV takes a simpler form given by (4.31). In the asymptotic limit of large values of the distance between the plates, ˜ a = | a 1 -a 2 | ≫ r, | z -a j | , it decays with the inverse of (˜ a/η ) D +2 -2 ν as shown in (4.32). For large distances from the string and considering fixed distances from the plates, r ≫ η, | z -a j | , the corresponding asymptotic formula is present in (4.33) and shows that the VEV induced in the region between the plates decays as 1 /r D -2+2 ν p . The Minkowskian limit has been also analyzed for this contribution and it is presented in (4.34). The behavior of the VEV of the azimuthal current density in this region, as function of the proper distance from the string, r p , considering Dirichlet and Neumann boundary conditions with different values of q , are exhibited in \nFig. 2. In the same region, in Fig. 3 we have plotted the behavior of the azimuthal current density the proper distance from the plates, z p , considering also the same boundary conditions and different values of q . Like in the previous graphs, the plots confirm the analytical asymptotic behaviors. \nTo finish this paper we want to say that in our analysis we have considered the spacetime fixed. In this sense we have quantized only the matter field. Here, the charged bosonic field. The induced azimuthal current can be considered as the source in the semiclassical formulation of the Maxwell equations. By its turn, the energy density present in the corresponding electromagnetic field can also be considered as source in the Einstein equation in a back-reaction approach, providing corrections on the metric tensor of spacetime background. The calculations of these corrections correspond in fact a hard work that can be developed in new project.', 'Acknowledgments': "We want to thank A. A. Saharian for helpful discussions. W.O.S. is supported under grant 2022/2008, Para'ıba State Research Foundation (FAPESQ). H.F.S.M. is partially supported by CNPq under Grant No. 308049/2023-3.", 'References': '- [1] Birrell, N.D.; Davies, P.C.W. Quantum Fields in Curved Space ; Cambridge University Press: Cambridge, UK, 1982.\n- [2] Linde, A.D. Particle Physics and Inflationary Cosmology ; Harwood Academic Publishers: Chur, Switzerland, 1990.\n- [3] Mottola, E. Particle creation in de Sitter space. Phys. Rev. D 1985 , 31 , 754.\n- [4] Vilenkin, A.; Shellard, E.P.S. Cosmic Strings and Other Topological Defects ; Cambridge monographs on mathematical physics; Cambridge Univ. Press: Cambridge, UK, 1994.\n- [5] Hindmarsh, M.; Kibble, T. Cosmic strings. Rept. Prog. Phys. 1995 , 58 , 477-562.\n- [6] Hyde, J.M.; Long, A.J.; Vachaspati, T. Dark Strings and their Couplings to the Standard Model. Phys. Rev. D 2014 , 89 , 065031.\n- [7] Damour, T.; Vilenkin, A. Gravitational Wave Bursts from Cosmic Strings. Phys. Rev. Lett. 2000 , 85 , 3761.\n- [8] Battacharjee, P.; Sigl, G. Origin and propagation of extremely high-energy cosmic rays Phys. Rep. 2000 , 327 , 109.\n- [9] Berezinski, V.; Hnatyk, B.; Vilenkin, A. Gamma ray bursts from superconducting cosmic strings Phys. Rev. D 2001 , 64 , 043004.\n- [10] Nielsen, H.B.; Olesen, P. Vortex Line Models for Dual Strings. Nucl. Phys. B 1973 , 61 , 45-61.\n- [11] Garfinkle, D. General Relativistic Strings. Phys. Rev. D 1985 , 32 , 1323-1329.\n- [12] Linet, B. A Vortex Line Model for Infinite Straight Cosmic Strings. Phys. Lett. A 1987 , 124 , 240.\n- [13] Gott, J.R., III. Gravitational lensing effects of vacuum strings-Exact solutions. Astrophys. J. 1985 , 288 , 422-427'} |
2024MNRAS.534.2168N | The mechanism of Xray outbursts in Be Xray binaries remains a mystery and understanding their circumstellar discs is crucial for a solution of the masstransfer problem. In particular it is important to identify the Be star activities e.g. pulsations that cause mass ejection and hence disc formation. Therefore we investigated the relationship between optical flux oscillations and the infrared IR excess in a sample of five Be Xray binaries. Applying the LombScargle technique to highcadence optical light curves from the Transiting Exoplanet Survey Satellite inlineformulatexmath idTM0001 notationLaTeXit TESStexmathinlineformula we detected several significant oscillation modes in the 324 h period range for each source. We also measured the IR excess a proxy for disc growth of those five sources using Jband light curves from Palomar GattiniIR. In four of the five sources we found anticorrelations between the IR excess and the amplitude of the main flux oscillation modes. This result is inconsistent with the conventional idea that nonradial pulsations drive mass ejections. We propose an alternative scenario where internal temperature variations in the Be star cause transitions between pulsationactive and massejectionactive states. | 2024-11-01T00:00:00Z | ['2024arXiv240909581N', '10.1093/mnras/stae2160', 'arXiv:2409.09581', '2024MNRAS.tmp.2138N', '10.48550/arXiv.2409.09581', '2024MNRAS.534.2168N'] | ['Astrophysics - Solar and Stellar Astrophysics', 'Astrophysics - High Energy Astrophysical Phenomena'] | Possible anticorrelations between pulsation amplitudes and the disc growth of Be stars in giantoutbursting Be Xray binaries | 2,024 | 175 | 0.56 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF'] | 0 | https://arxiv.org/pdf/2409.09581.pdf | {'Possible anti-correlations between pulsation amplitudes and the disk growth of Be stars in giant-outbursting Be X-ray binaries': "Masafumi Niwano, 1 ★ Michael M. Fausnaugh, 2 Ryan M. Lau, 3 Kishalay De, 4 Roberto Soria, 5 , 6 , 7 George R. Ricker, 4 Roland Vanderspek, 4 Michael C. B. Ashley, 8 Nicholas Earley, 9 Matthew J. Hankins, 10 Mansi M. Kasliwal, 9 Anna M. Moore, 11 Jamie Soon, 11 Tony Travouillon, 11 Mahito Sasada, 12 Ichiro Takahashi, 1 Yoichi Yatsu, 1 and Nobuyuki Kawai 1 \n1 Department of Physics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan \n2 Department of Physics & Astronomy, Texas Tech University, Lubbock, TX 79410-1051, USA \n3 NSF's NOIRLab, 950 N. Cherry Avenue, Tucson, AZ 85719, USA \n- 4 MIT-Kavli Institute for Astrophysics and Space Research, 77 Massachusetts Avenue, Cambridge, MA 02139, USA \n5 \n6 \nCollege of Astronomy and Space Sciences, University of the Chinese Academy of Sciences, Beijing 100049, China \nINAF-Osservatorio Astrofisico di Torino, Strada Osservatorio 20, I-10025 Pino Torinese, Italy \n- 7 Sydney Institute for Astronomy, School of Physics A28, The University of Sydney, Sydney, NSW 2006, Australia \n8 \nSchool of Physics, University of New South Wales, Sydney NSW 2052, Australia \n- 9 Cahill Center for Astrophysics, California Institute of Technology, 1200 E. California Blvd. Pasadena, CA 91125, USA 10\n- Arkansas Tech University, Russellville, AR 72801, USA\n- 11 Research School of Astronomy and Astrophysics, Australian National University, Canberra, ACT 2611, Australia\n- 12 Institute of Innovative Research, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan \nAccepted XXX. Received YYY; in original form ZZZ", 'ABSTRACT': 'The mechanism of X-ray outbursts in Be X-ray binaries remains a mystery, and understanding their circumstellar disks is crucial for a solution of the mass-transfer problem. In particular, it is important to identify the Be star activities (e.g., pulsations) that cause mass ejection and, hence, disk formation. Therefore, we investigated the relationship between optical flux oscillations and the infrared (IR) excess in a sample of five Be X-ray binaries. Applying the Lomb-Scargle technique to high-cadence optical light curves from the Transiting Exoplanet Survey Satellite ( TESS ), we detected several significant oscillation modes in the 3 to 24 hour period range for each source. We also measured the IR excess (a proxy for disk growth) of those five sources, using J-band light curves from Palomar Gattini-IR. In four of the five sources, we found anti-correlations between the IR excess and the amplitude of the main flux oscillation modes. This result is inconsistent with the conventional idea that non-radial pulsations drive mass ejections. We propose an alternative scenario where internal temperature variations in the Be star cause transitions between pulsation-active and mass-ejection-active states. \nKey words: stars: early-type - stars: emission-line, Be - stars: oscillations - stars:rotation - binaries: close - X-rays:binaries', '1 INTRODUCTION': 'A Be X-ray binary (BeXB) is a binary system composed of a compact object (a neutron star in most cases) and a Be star, an early type star with a circumstellar disk (Reig 2011). In addition to standard observational features of Be stars such as double-peaked emission lines and infrared (IR) excess (the latter usually explained as thermal bremsstrahlung from the disk), BeXBs exhibit various peculiar phenomena caused by the interaction of the compact object with the circumstellar disk. The most notable of them are normal (Type-I) and giant (Type-II) outbursts. Normal outbursts occur regularly and (quasi-) periodically, and are caused by the compact object capturing mass from the circumstellar disk when passing around the periastron. Their typical duration and luminosity are about 0.2-0.3 times the or- \nbital period and ≲ 10 37 erg s -1 , respectively. On the other hand, giant outbursts are rarer, more luminous and unpredictable; their mechanism is not well understood. They last several times the orbital period, and their luminosity often exceeds ∼ 10 38 erg s -1 (Reig & Nespoli 2013), the Eddington limit of a neutron star. In at least three well-studied cases (SMC X-3: Tsygankov et al. 2017, Townsend et al. 2017; Swift J0243.6+6124: Wilson-Hodge et al. 2018, AlfonsoGarzón et al. 2024; RX J0209.6-7427: Vasilopoulos et al. 2020, Hou et al. 2022, West et al. 2024), the peak X-ray luminosity reached ≈ 2 × 10 39 erg s -1 , placing those BeXBs into the ultra-luminous Xray (ULX) pulsar class. Therefore, giant-outbursting BeXBs might be a significant component of the observed ULX population in the local universe (Earnshaw et al. 2018; Kuranov et al. 2020; Gúrpide et al. 2021; Misra et al. 2024). In particular, luminous BeXBs can be the dominant component of the X-ray binary population in galaxies where star formation has peaked ≈ 25-60 Myr ago (An- \nl. 2010; Antoniou & Zezas 2016), as opposed to more recent (dominated by high-mass X-ray binaries) or older (dominated by low-mass X-ray binaries) star-forming environments. Thus, understanding the connection between donor-star variability and giant outbursts in BeXBs is an important step for the modelling of X-ray binary populations in external galaxies. Such studies also provide observational constraints to theoretical model of super-critical accretion onto magnetised neutron stars (Tsygankov et al. 2017; Mushtukov et al. 2017, 2019). \nSeveral studies (Martin et al. 2011, 2014; Okazaki et al. 2013; Reig & Blinov 2018) suggest that giant outbursts happen when the outer radius of the circumstellar disk (possibly warped, inclined or eccentric) reaches the orbit of the compact object and the latter captures a large amount of mass in a short time. A circumstellar disk is formed via outwards viscous diffusion of mass ejected from the stellar surface (Rivinius et al. 2013; Rímulo et al. 2018). However, the mass ejection mechanism is still unclear. Although Be stars are generally rapid rotators, their rotation velocities are typically only about 70 per cent of their critical velocity (Rivinius et al. 2013). This means that their rotation alone cannot explain mass ejections. Nonradial pulsations are an alternative mechanism proposed to explain mass ejections (Bagnulo et al. 2012). In support of this scenario, observations of the B2Ve star 𝜇 Cen showed a coincidence of mass ejections with constructive interference of the pulsational velocity fields (Rivinius et al. 1998). However, theoretically, non-radial pulsations limited by the speed of sound seem to be insufficient for disk formation (Smith et al. 2012; Torrejón et al. 2012). Addressing this mass ejection problem is the main objective of this work. \nThe operational launch of the Transiting Exoplanet Survey Satellite ( TESS ; Ricker et al. 2015) in 2018 was a notable event for the asteroseismology of massive stars, including Be stars. Although the main purpose of TESS is to explore exoplanets, its high-precision and high-cadence continuous light curves are very useful for the analysis of periodic flux oscillations of massive stars due to their pulsations and rotation. Several TESS studies of Be stars and BeXBs have already shed new light in this field. Balona & Ozuyar (2020) examined TESS light curves of 57 Be stars, and found that 74 per cent of their targets show a single or harmonic double peak in the 0.5-3 d -1 frequency range of their power spectra. They argued that such peaks are caused by stellar rotation. Ramsay et al. (2022) investigated TESS light curves of 23 high-mass X-ray binaries, and confirmed the detection of 0.1-1 d -1 oscillations in the power spectra of all their targets. Reig & Fabregat (2022) used TESS to study the power spectrum morphology of 22 BeXBs, and found that BeXBs and classical Be stars are indistinguishable in terms of pulsational characteristics. \nIn this work, we analyse the TESS light curves of BeXBs to investigate long-term variations in the activity level of Be donor stars. We compare the results with the long-term multi-wavelength light curves in order to constrain the mass ejection mechanism of the Be donor star. This article is composed as follows. In Section 2, we provide a brief description of the target selection. Section 3 describes the observational data used in this work: TESS , Swift , MAXI, ZTF, ATLAS, and Gattini-IR. We explain our Lomb-Scargle analysis of the TESS light curves and the IR excess evaluation in Section 4. Section 5 shows long-term multi-wavelength light curves and amplitude spectra of the target sources. We highlight the discovery of an anticorrelation between the IR excess and the amplitude of the main flux oscillation modes. In Section 6, we discuss possible interpretations of this observed anti-correlation. We propose that internal temperature variations in the Be star cause transitions between pulsation-active and mass-ejection-active states. Finally, Section 7 concludes this article.', '2 TARGET SELECTION': 'For BeXBs which have an optical magnitude of ≲ 14 mag and available TESS light curves, and are in the northern sky, we checked multi-wavelength light curves (cf. Section 3.2) in the modified Julian date (MJD) range of 58400-60000 and found that five of them showed significant variations in X-ray, optical and near-infrared (NIR) band (cf. Section 5.1). In most cases, optical and infrared (OIR) variations of Be stars are derived by the growth or the decay of their circumstellar disks. Thus we selected these five BeXBs as the targets for this study. Table 1 describes the selected five sources: 4U0115+634, GRO J2058+42, RX J0440.9+4421, SAX J2103.5+4545, and V0332+53. \nThese are already known to exhibit long-term multi-wavelength activities such as giant outbursts, disk-derived OIR variations. For 4U0115+634, long-term variations have been studied in Negueruela & Okazaki (2001) and Reig et al. (2007), and they reported quasiperiodic disk variations with an approximately 5-year cycle and giant outbursts linked to them. Furthermore, variations in polarization were observed that appear to be related to disturbances in the disk during the giant outburst (Reig & Blinov 2018). For GRO J2058+42, Reig et al. (2023) confirmed sinusoidal variations in optical light curves and the H 𝛼 equivalent width with a period of about 9.5 years and occurrences of giant outbursts related to them, and stated that the symmetry of the double-peaked emission lines suggested that the disk was not warped. Reig et al. (2005) reported long-term variability in OIR light curves and H 𝛼 equivalent width of RX J0440.9+4431, and suggested that disturbances in the disk due to interactions with neutron stars can only occur if the disk is sufficiently developed, based on V/R 1 variations. Long-term activities of SAX J2103.5+4545 are studied in Reig et al. (2010) and Camero et al. (2014). This source shows highly correlated X-ray and OIR variability, and X-ray flares linked to peaks of OIR light curves. One notable feature is that the orbital period of this source ( 𝑃 orb = 12 . 67 d; Camero Arranz et al. 2007) is particularly short for known BeXBs. Caballero-García et al. (2016) carried out a long-term X-ray and OIR observation of V0332+53, and suggested that the inner regions surrounding the magnetosphere was visualized during the lowest flux state based on the presence of X-ray quasi-periodic oscillation at that time.', '3 OBSERVATIONS': 'Weutilised high-cadence optical light curves of TESS , and long-term multi-wavelength light curves.', '3.1 Transiting Exoplanet Survey Satellite': 'Transiting Exoplanet Survey Satellite ( TESS ; Ricker et al. 2015) is a optical space telescope operated by National Aeronautics and Space Administration (NASA) and Massachusetts Institute of Technology (MIT), and its main purpose is to explore exoplanets by observing their transits. Full frame images (FFIs) at 10 or 30-minutes intervals are available from the Mikulski Archive for Space Telescopes (MAST). \nWe carried out the photometry of FFIs with the pipeline of TESS Transients project (Fausnaugh et al. 2021, 2023). This pipeline evaluates a brightness of the star with difference imaging and fitting the point-spread function, and is expected to be more accurate than a simple aperture photometry for fainter sources ( TESS magnitude ≳ 12). FFIs record photo-electron count rate, and we converted it \nTable 1. Target BeXBs. \n- [a] Wenger et al. (2000); [b] Paegert et al. (2022); [c] Reig & Fabregat (2015); [1a] Tamura et al. (1992); [1b] Negueruela & Okazaki (2001);\n- [2a] Wilson et al. (2005); [2b] Kızıloˇglu et al. (2007); [3a] Ferrigno et al. (2013); [3b] Reig et al. (2005); [4a] Camero Arranz et al. (2007);\n- [4b] Reig et al. (2004); [5a] Negueruela et al. (1999); ∗ Distances are calculated from Gaia DR3 parallaxes (Gaia Collaboration et al. 2023). \nFigure 1. Energy bands of multi-wavelength light curves. Data of ZTF and ATLAS filters were obtained from the SVO Filter Profile Service ( http://svo2.cab.inta-csic.es/svo/theory/fps/index.php ). As for Gattini-IR, it is drawn with the value of the Cousins J-band filter. \n<!-- image --> \nto energy flux density using the following relationship based on a documentation in an official web-page of TESS Transients project 2 : \n𝐹 𝐼 = 1 . 61 × 10 -5 GLYPH<20> Jy counts s -1 GLYPH<21> , (1) \nwhere 𝐼 is the photo-electron count rate, and 𝐹 is an energy flux density in the TESS pass-band. The interval of FFIs was changed from 30 to 10 minutes at the end of the prime mission in July 2020 (Ricker 2021). We unified the sampling rate of the light curves by binning at 30-minutes widths during the analysis process (cf. Section 4.3.1). In addition, we obtained TESS magnitude (Tmag) of target sources from TESS Input Catalog (Stassun et al. 2018, 2019).', '3.2 Long-term multi-wavelength light curves': 'Weutilised X-ray light curves of Swift /BAT and MAXI/GSC, optical light curves of ZTF and ATLAS, and NIR light curves of GattiniIR, covering MJD range of ∼ 58400-60000. Figure 1 shows energy bands of multi-wavelength light curves.', '3.2.1 Neil Gehrels Swift Observatory': 'Neil Gehrels Swift Observatory ( Swift ; Gehrels et al. 2004) is a gamma-ray burst survey satellite operated by NASA. We downloaded \nthe Burst Alert Telescope (BAT) 15-50 keV daily light curves of target BeXBs, from Swift /BAT Hard X-ray Transient Monitor webpage (Krimm et al. 2013). Seven days binning was performed to improve signal-to-noise ratio for each light curves.', '3.2.2 Monitor of All-sky X-ray Image': 'Monitor of All-sky X-ray Image (MAXI; Matsuoka et al. 2009) is an X-ray camera installed onboard the International Space Station (ISS). We obtained weekly-averaged Gas Slit Camera (GSC) 2-20 keV light curves covering 58000-60000 MJD for target BeXBs, via MAXI on-demand web interface (Nakahira et al. 2012).', '3.2.3 Zwicky Transient Facility': "Zwicky Transient Facility (ZTF; Bellm et al. 2019) is an optical widefield survey project using a CCD camera array attached to Samuel Oschin telescope at Palomar Observatory in California, operated by California Institute of Technology (Caltech). We utilised 𝑔 ' and 𝑟 ' -band light curves of the targets acquired from ZTF Public Data Release 15 (DR15; Masci et al. 2019), which covers the epoch of ∼ 58250-59900 MJD. The light curve is not available only for RX J0440.9+4431 due to its brighter optical magnitude than the ZTF saturation magnitude ( ∼ 12.5 AB).", '3.2.4 Asteroid Terrestrial-impact Last Alert System': 'Asteroid Terrestrial-impact Last Alert System (ATLAS; Tonry et al. 2018) is a high-cadence optical all-sky survey project using four 50 cm telescope systems located in Haleakala, Mauna Loa (Hawaii), El Sauce (Chile), and Sutherland (South Africa). We obtained 𝑐 and 𝑜 -band light curves covering 58000-60000 MJD via ATLAS Forced Photometry web interface (Smith et al. 2020). The saturation magnitude of ALTAS is ∼ 12.5 AB, about same as ZTF. Thus, the light curve for RX J0440.9+4431, although it can be created, is not reliable.', '3.2.5 Gattini-IR': 'Gattini-IR (De et al. 2020) is a NIR wide-field survey project using 30 cm robotic 𝐽 -band telescope system at Palomar Observatory, operated by Caltech. Gattini-IR sweeps ∼ 7500 deg 2 of the sky at one night with an upper-limit magnitude of ∼ 15.7 (AB), and scans the northern-sky observable from Palomar (Dec. ≳ -30 · ) with a cadence of ∼ 2 days. The system is designed for the NIR-transient detection, and a real-time data processing pipeline (Gattini Data Processing System; GDPS) analyses observed images and detects the \nTable 2. Coefficients of Equations (3) and (4) (Torres et al. 2010). \ntransients. We obtained the 𝐽 -band light curves of the targets generated by GDPS. The data-set covers the epoch of ∼ 58400-59900 MJD.', '4 DATA ANALYSIS': 'We first estimated Be donor properties to evaluate the IR excess and the critical velocity. Next, the IR excess was evaluated to investigate the evolution of the disks. Finally, a flux periodicity was analysed applying the Lomb-Scargle technique (Lomb 1976; Scargle 1982) to TESS light curves. We utilised curve\\_fit function of scipy.optimize library (Virtanen et al. 2020) for fitting, and LombScargle class of astropy.timeseries library (Astropy Collaboration et al. 2013, 2018, 2022) as an implementation of the Lomb-Scargle technique.', '4.1 Estimation of Be donor properties': 'We estimated an effective temperature 𝑇 eff , radius 𝑅 ∗ , and mass 𝑀 ∗ for each Be donor of target BeXBs by fitting of an optical spectral energy distribution (SED) with a simple blackbody model. First, the optical SED of the target was created using catalogued optical magnitudes, a color excess 𝐸 ( 𝐵 -𝑉 ) , and a distance. The reason for not using the NIR data (e.g. 2MASS) is that it is more susceptible to the IR excess than the optical band and the blackbody radiation may not be dominant. We used the Cardelli et al. (1989) extinction function to derive extinctions for each band. This SED was then fitted with a model spectrum of a spherical blackbody to obtain 𝑇 eff and 𝑅 ∗ . Finally, 𝑀 ∗ was calculated using the empirical relations of Torres et al. (2010) with some modifications: \n𝑋 = log GLYPH<18> 𝑇 eff 12 . 6 [ kK ] GLYPH<19> , (2) \nlog GLYPH<18> 𝑀 ∗ 𝑀 ⊙ GLYPH<19> = 𝑎 1 + 𝑎 2 𝑋 + 𝑎 3 𝑋 2 + 𝑎 4 𝑋 3 + 𝑎 5 ( log 𝑔 ) 2 + 𝑎 6 ( log 𝑔 ) 3 , \n(3) \nlog GLYPH<18> 𝑅 ∗ 𝑅 ⊙ GLYPH<19> = 𝑏 1 + 𝑏 2 𝑋 + 𝑏 3 𝑋 2 + 𝑏 4 𝑋 3 + 𝑏 5 ( log 𝑔 ) 2 + 𝑏 6 ( log 𝑔 ) 3 , (4) \nwhere 𝑔 is a surface gravity normalised in unit of cm s -2 , 𝑀 ⊙ and 𝑅 ⊙ are the solar mass and radius, respectively, and 𝑎 1-6 and 𝑏 1-6 are coefficients. The base of the logarithm here is 10. Values of 𝑎 1-6 and 𝑏 1-6 are listed in Table 2. Specifically, we solved Equation (4) with a numerical technique to obtain log 𝑔 from 𝑇 eff and 𝑅 ∗ , and calculated 𝑀 ∗ by substituting 𝑇 eff and log 𝑔 into Equation (3). Note that we ignored metallicity [Fe/H] terms, which are in original equations, since all targets in this study are galactic sources. We obtained the optical multi-band magnitudes from APASS DR10 (Henden et al. 2018), PS1 DR1 (Flewelling et al. 2020), or Table 2 of Reig & \nFabregat (2015). The indeterminacy of the result was estimated by the bootstrap re-sampling. If multiple results were obtained by multiple catalogues, we selected the one for which 𝑇 eff is most plausible to the typical value of its spectral type (cf. Table A1). \nWeestimated a blackbody 𝐽 -band magnitude 𝑚 bb 𝐽 , critical velocity v b , and critical frequency 𝜈 b . 𝑚 bb 𝐽 was obtained by extrapolating the obtained blackbody SED model to the 𝐽 -band. This is the expected 𝐽 -band magnitude of the Be donor itself excluding the IR excess. 𝜈 b is a rotation frequency when spinning at v b . We calculated v b and 𝜈 b assuming a sphere: \nv b = √︂ 𝐺𝑀 ∗ 𝑅 ∗ , (5) 𝜈 b = v b 𝜋𝑅 , (6) \n2 ∗ \nwhere 𝐺 is the gravitational constant. Since the rotation of the Be star does not reach v b in most cases (Rivinius et al. 2013), 𝜈 b can be regarded as the upper limit of the rotational frequency. If the contribution of the IR excess to the catalog optical magnitudes is not negligible, it causes the color to be redder and the magnitude to be larger, and results in underestimation of 𝑇 eff and overestimation of 𝑅 ∗ and 𝑚 bb 𝐽 . Estimated properties are summarised in Table 3. At temperature 𝑇 ∼ 30000 K, the Planck distribution peak is at ultraviolet-band ( 𝜈 ∼ 10 15 Hz), and 𝑇 change produces only a small difference in slope in optical band. Therefore, the temperature estimation in this method is highly sensitive to small differences in the slope of the SED, resulting in a large error of 𝑇 eff due to a slight uncertainty of 𝐸 ( 𝐵 -𝑉 ) . Note that v b is consistent with the literature value of v sin 𝑖 (i.e. v b ≳ v sin 𝑖 ) for all sources (cf. Table 1).', '4.2 IR excess evaluation': 'We evaluated the IR excess as a measure of disk development. The 𝐽 -band flux observed by Gattini-IR was averaged per sector and converted into magnitudes, defined as 𝑚 𝐽 . We used 𝑚 𝐽 -𝑚 bb 𝐽 to evaluate the amount of the IR excess (Figure 2). The smaller the value of 𝑚 𝐽 -𝑚 bb 𝐽 , the greater the IR excess. The Gattini-IR magnitudes were systematically dimmer than 𝑚 bb 𝐽 in 4U0115+634. This can be qualitatively explained by overestimation of 𝑚 bb 𝐽 due to some factors such as underestimation of 𝑇 eff (cf. Section 4.1). In any case, the IR excess evaluated in this way can be used to discuss long-term variations of one source, but not for cross-source comparisons.', '4.3 Analysis of flux periodicity': 'We made and analyzed amplitude spectra of TESS light curves for each sector of each targets to investigate the flux periodicity. This analysis was carried out with the following procedure: \n- (i) Data reduction to raw TESS light curves,\n- (ii) Amplitude spectrum estimation using the Lomb-Scargle technique,\n- (iii) Decomposition of the spectrum into peaks and the red noise.', '4.3.1 Data reduction': "The data reduction process consists of three stages: detrending, 5sigma clipping, and 30-min binning. First, we subtracted the moving averaged flux from the light curve to remove the long-term flux variation in time scale of ≳ several days. For convenience, we call this process as 'detrending'. The width of the detrending window was set \nTable 3. Estimated stellar properties of target BeXBs. \n- ∗ Sources of the optical magnitudes used to derive the properties. [R&F] means Reig & Fabregat (2015). \nFigure 2. IR excess evaluation of 4U0115+634. Gray square and blue circular markers correspond to the raw Gattini-IR 𝐽 -band light curve and the sectorby-sector averaged light curve, respectively. The horizontal dashed line is 𝑚 bb 𝐽 which is the estimated IR excess excluded magnitude, and the blue shaded regions represent sectors where TESS observed this source. We used the difference between the blue points and 𝑚 bb 𝐽 line as a measure of the IR excess. The fact that the light curve is sometimes systematically below 𝑚 bb 𝐽 (e.g. 58800-59500 MJD) may be due to an overestimation of 𝑚 bb 𝐽 . \n<!-- image --> \nto 3 days except for SAX J2103.5+4545. For SAX J2103.5+4545, the width was set to 1 day because the 3-day detrending was insufficient to remove long-term variability. Next, 5-sigma clipping was performed to remove outliers. Finally, we binned light curves in 1/48 days ( ∼ 30 min) width to minimise the effect of differences in the sampling rate of FFIs between the prime and extended missions (Ricker 2021). Figure 3 shows the data reduction process to the TESS light curve of V0332+53, sector 59.", '4.3.2 Amplitude spectrum': 'We applied the Lomb-Scargle technique to the reduced light curves. Note that we did not use a power spectrum (periodogram), but an amplitude spectrum calculated by the following transformation (cf. Aerts 2021): \n𝐴 ( 𝜈 ) = √︂ 4 𝑁 𝑃 ( 𝜈 ) , (7) \nwhere 𝜈 is a frequency, 𝐴 ( 𝜈 ) is the amplitude spectrum, 𝑁 is a number of points in the light curve, and 𝑃 ( 𝜈 ) is the power spectrum. We first made dynamic amplitude spectra. A dynamic amplitude spectrum is obtained by taking a portion of the light curve with an appropriate window and obtaining its spectrum by applying the Lomb-Scargle while moving the window. That shows the variation of \nFigure 3. Data reduction to the TESS light curve of V0332+53, sector 59. The three panels correspond to the 3-day detrending, 5-sigma clipping, and 30-min binning processes, respectively. \n<!-- image --> \nthe spectrum. The window width was set to 5 days and moved by 1/48 days. Note that oscillations with a period equal to or greater than the width of the detrending window had removed in the data reduction process. Figure 4 is a dynamic amplitude spectrum of V0332+53, sector 59. If the number of points in the extracted light curve was less than 160, the obtained spectrum was considered to be unreliable, where the number 160 is 2/3 of the number of points expected from the sampling rate (48 d -1 ) and the window width (5 days). At the same time, we evaluated the contribution of photometric errors to the spectrum by creating mock light curves which fluctuate randomly with a Gaussian-distribution over the width of the photometric error and calculating their spectrum. Note that the photometry pipeline we used has a tendency to underestimate errors by about 10 per cent and up to 50 per cent, compared with estimated from the light curve fluctuations. Finally, we averaged each dynamic spectrum along the time-axis direcion (Figure 5). In this calculation, spectrum with points fewer than 160 on the light curve were excluded. We defined this as "averaged amplitude spectrum", and considered it to be the representative amplitude spectrum of the object in the sector. The reason why Lomb-Scargle is not simply applied to the entire light curve for each sector is to reduce the effect of several days of missing points on some light curves.', '4.3.3 Decomposition into peaks and red noise': "To extract the peaks from the spectrum, we estimated a red-noise component of the spectrum. First, we made a local 25th percentile \nFigure 4. Dynamic amplitude spectrum of V0332+53, sector 59. The top panel is a light curve, and the bottom is a dynamic spectrum. The shaded area is where the frequency is less than 1/3 d -1 or the number of extracted light curve points is less than 160. \n<!-- image --> \nFigure 5. Averaged amplitude spectrum of V0332+53, sector 59. The top panel shows a 2D histogram of dynamic amplitude spectrum (Figure 4) projected along the time axis, and the bottom panel shows an averaged amplitude spectrum. The averaged amplitude spectrum can be paraphrased as the centroid of the histogram for each frequency. 'Log density' is the logarithm of density (arbitrary units) and its base is 10. The left shaded area is where the frequency is less than 1/3 d -1 . See also Section 4.3.3 for the description of the red noise. \n<!-- image --> \n(first quartile) curve by extracting a portion of the spectrum with a window moving across all frequencies, and calculating the 25th percentile for each window. Then the 25th percentile curve is smoothed with a Gaussian filter. The obtained curve can be considered to be a rough estimate of the continuum. The reason for using the 25th percentile rather than the mean or median is the nature of the amplitude spectrum. Unlike random errors, which can be both positive and negative, the peaks in the amplitude spectrum are almost always positive. Therefore, the mean or median are expected to be systematically larger than the continuum. Finally, the red noise spectrum was obtained by model fitting to the roughly estimated continuum. We utilised the model used in Bowman et al. (2019) and Nazé et al. (2020) with some modifications. The model consists of a Lorentzian- \nFigure 6. Red noise estimation of V0332+53, sector 59. The smoothed local 25th percentile is the roughly estimated continuum, and white noise level is the constant component of the red noise model. 'Photometric Error' means a spectrum estimated from the photometric error. 'Normalized Amplitude' is the ratio of amplitude to the flux converted from Tmag. The left shaded area is where the frequency is less than 1/3 d -1 . \n<!-- image --> \nike term and a constant term: \n𝐴 RN ( 𝜈 ) = √︄ 𝑎 2 0 1 + ( 𝜈 / 𝜈 0 ) 𝛾 + 𝐶 2 , (8) \nwhere 𝜈 is a frequency, 𝐴 RN is a red noise amplitude spectrum, 𝑎 0 is a scaling factor, 𝜈 0 is a characteristic frequency, 𝛾 is an exponent, and 𝐶 is a white noise level. When 𝛾 = 2, the first term is the Lorentzian. If both sides of Equation (8) are squared, the equation agrees with Bowman et al. (2019) and Nazé et al. (2020). The square root is applied to make the dimension of both sides amplitude rather than power. Figure 6 shows the red noise estimation of V0332+53, sector 59. Incidentally, the error-derived amplitude (cf. Section 4.3.2) was smaller than the white noise level for all light curves as shown in Figure 6, even if putting into account the underestimation of the error. \nThe peaks were extracted by subtracting the red noise from the amplitude spectrum. We determined that the peaks whose amplitude were larger than twice the red noise were significant. For each source, we evaluated the amplitudes for all sectors at frequencies for which a significant peak was detected in at least one of the sectors. The amplitude of the detected peak was normalised by the flux calculated from the Tmag of the source. This normalisation is based on the assumption that Tmag is the typical magnitude of the source in the TESS pass-band, and ignores long-term optical variations.", '5.1 Long-term multi-wavelengths activities': 'Figures 7 to 11 are long-term multi-wavelength light curves of five sources. They exhibited significant X-ray outbursts in both 2-20 keV and 15-50 keV light curves, and OIR variability of ≳ 0.5 mag. The X-ray light curves of SAX J2103.5+4545 during outbursts show two components: a sharp peak with a duration of about 10 days and a relatively gentle bump of about 100 days. This feature was also seen in previous outbursts of this source (cf. Camero et al. 2014). \nThe Crab pulsar has a luminosity of ∼ 10 37 erg s -1 (Kirsch et al. \nFigure 7. Multi-wavelength light curves of 4U0115+634. The gray shaded regions correspond to TESS observed sectors. An X-ray outburst can be confirmedat ∼ 58900MJD,whichisconsistent with the Atel report (Nakajima et al. 2020). \n<!-- image --> \nFigure 10. A similar figure to Figure 7 for SAX J2103.5+4545. Recurrent X-ray/OIR outbursts can be confirmed at ∼ 58600, 59000, and 59500 MJD. \n<!-- image --> \nFigure 8. Asimilar figure to Figure 7 for GRO J2058+42. An X-ray outburst can be confirmed at ∼ 58500 MJD, which is consistent with the GCNC report (Barthelmy et al. 2019). \n<!-- image --> \n2005), a distance of ∼ 2 kpc (Lin et al. 2023), and is detected by the GSContheorder of 1 photons s -1 cm -2 (Morii et al. 2011). By comparing GSC count rates and distances of five targets with those of the Crab pulsar, the peak luminosities of these outbursts can be roughly estimated to be ≳ 10 37 erg s -1 . In addition, the orbital periodicity expected for normal outbursts was not confirmed, and the duration of outbursts was ∼ 100 days, which is longer than orbital periods other than RX J0440.9+4431. Outbursts of SAX J2103.5+4545 appear to be periodic, but their interval ( ∼ 200 days) is not consistent with its orbital period (12.67 days). Therefore, all detected X-ray outbursts can be classified as giant outbursts. \nThe time scales for all OIR light curve variations were about hundreds of days, which are consistent with the typical time scale of the circumstellar disk evolution. Figure 12 shows the OIR color variations of 4U0115+634. The spectral index shown in the figure represents the color, and is calculated for given two bands as follows: \n𝛼 ( 𝜈 0 , 𝜈 1 ) = log GLYPH<0> 𝑓 𝜈 0 / 𝑓 𝜈 1 GLYPH<1> log ( 𝜈 0 / 𝜈 1 ) , (9) \nwhere 𝜈 0 and 𝜈 1 , and 𝑓 𝜈 0 and 𝑓 𝜈 1 are photon frequencies and fluxes in the given two bands, respectively, and 𝛼 is a spectral index. This corresponds to the slope of the line connecting two points in the dou- \nFigure 9. A similar figure to Figure 7 for RX J0440.9+4431. Optical light curves exceeded the saturation level and are unavailable for analysis. An X-ray outburst can be confirmed at ∼ 60000 MJD, which is consistent with the Atel report (Nakajima et al. 2022). \n<!-- image --> \nFigure 11. Asimilar figure to Figure 7 for V0332+53. An X-ray outburst can be confirmed at ∼ 58700 MJD. \n<!-- image --> \nFigure 12. OIR color variations of 4U0115+634. Top and bottom panel show the J-band light curve and OIR color variations, respectively. The vertical axis in the bottom panel is the spectral index, which is redder at the bottom and bluer at the top. \n<!-- image --> \nand a larger value indicates a bluer spectrum. The figure shows a "redder when brighter" tendency that are common for OIR variations of Be stars. This tendency was confirmed for all targets except RX J0440.9+4431 whose optical light curves were unavailable. Therefore, all of the observed long-term OIR variations can be attributed to the disk evolution. Thus, observed 𝐽 -band flux, and hence 𝑚 𝐽 -𝑚 bb 𝐽 , can be used as a measure of the disk development. \nRecent studies on giant outbursts (Martin et al. 2011, 2014; Okazaki et al. 2013) suggested that such events should occur when a neutron star passes near a sufficiently developed disk that extends to the orbit of the neutron star. Since the orbital period of BeXBs is typically shorter than the time scale of the disk growth, giant outbursts are expected to occur at or shortly before the maximum IR excess. However, all detected giant outbursts were initiated months to hundreds of days after the peak of the 𝐽 -band light curve. The same was observed in the 2017 giant outburst of Swift J0243.6+6124 (AlfonsoGarzón et al. 2024). This discrepancy can be explained by assuming that the peak IR excess and the maximum outer disk radius are not \nTable 4. Detected peak frequencies of five sources. Peaks with a check mark in the \'Harm.\' column are those which have a possible integer multiple relationship to the frequency of the other peaks. \nsimultaneous. Since the mass is supplied to the disk from the Be star, i.e., from the innermost edge of the disk, there should be a time lag of about the viscous time scale between the mass supply and the outward extension of the disk. In other words, the total mass and outermost radius never reach the maximum at the same time, and the former does earlier. In addition, since the volume emissivity of thermal bremsstrahlung is proportional to the square of the electron density, IR excess should depend on both disk radius and mass, and the peak of IR excess should also be before the radius peak. Hence, the observed delay can be attributed to the time lag between mass supply and disk extension.', '5.2 Amplitude spectra': 'We summarised detected peak frequencies in Table 4, and showed red noise subtracted amplitude spectra of five sources in Figures 13 to 17. When the ratio of two peak frequencies matched an integer within the 1 𝜎 error range, we determined that the pair of peaks has possible harmonic relationship, except in cases where the error of ratio exceeded 0.5. The possible harmonic relationships were confirmed for only two of the five, 4U0115+634 and RX J0440.9+4431. These were the only two having peaks of 𝜈 ≲ 𝜈 b . In all five sources, the maximum peak was detected at 𝜈 = 2-4 d -1 with a normalized amplitude of ∼ 1 per cent. There was at least one peak per source where the normalized amplitude varied clearly between sectors. And the maximum peak was the amplitude varying peak for all sources.', '5.3 Peak amplitude and the IR excess correlations': "Figures 18 to 22 show the relationship between peak amplitudes and the IR excess for five sources. For all sources except GRO J2058+42, at least one peak exhibited significant anti-correlation between its amplitude and the IR excess. Other peaks did not show significant correlations. We refer these peaks as 'anti-correlated peaks' and 'non-correlated peaks', respectively. Except for GRO J2058+42, the maximum peak belonged to the anti-correlated peaks. The ratio of the maximum to minimum amplitudes of the anti-correlated peaks \nFigure 13. Red noise subtracted amplitude spectra of 4U0115+634. 'Normalized Amplitude' is the amplitude normalised by the flux calculated from the Tmag of the source. 'Red Noise × 2' line is the threshold of the peak detection. Vertical dashed lines indicate positions of detected peaks. \n<!-- image --> \nFigure 14. A similar figure to Figure 13 for GRO J2058+42. \n<!-- image --> \nwas typically 5-6. For RX J0440.9+4431, both the anti-correlated and non-correlated peaks coexist (e.g. peaks at 𝜈 = 2 . 2 , 2 . 6).", '6.1 Origin of periodic flux oscillations': 'The origin of the periodic flux oscillation, which appears as a peak in the amplitude spectrum, is the rotation or pulsation of the Be donor star. According to Balona (2019) and Balona & Ozuyar (2020), rotation-derived flux oscillations in massive stars are originated from gas clouds co-rotating with the star, and these appear in the power spectrum as clusters of peaks or broad humps, and integer multiplied harmonics. Furthermore, the frequency of the rotation-derived fundamental mode should be smaller than 𝜈 b , because the rotation of the Be star does not reach a critical speed in most cases. In light of this, peaks of 𝜈 < 2 d -1 in 4U0115+634 and RX J0440.9+4431 may be \nFigure 15. A similar figure to Figure 13 for RX J0440.9+4431. \n<!-- image --> \nFigure 16. A similar figure to Figure 13 for SAX J2103.5+4545. \n<!-- image --> \nFigure 17. A similar figure to Figure 13 for V0332+53. \n<!-- image --> \nFigure 18. IR excess and peak amplitude relationship in 4U0115+634. (top) Red noise subtracted normalised amplitude spectrum. Basically the same as Figure 13. (bottom) The further to the right, the larger the IR excess. Points with the same marker style and connected by dashed lines correspond to the peak of the same frequency and are plotted with values in multiple sectors. The color of the marker is the same as that of the vertical line in the top panel. In the upper right box are the correlation coefficients for each peak. Note that dashed lines simply connect adjacent points and ignore the time series. \n<!-- image --> \nFigure 19. A similar figure to Figure 18 for GRO J2058+42. \n<!-- image --> \nFigure 20. A similar figure to Figure 18 for RXJ 0440.9+4431. \n<!-- image --> \nFigure 21. A similar figure to Figure 18 for SAX J2103.5+4545. \n<!-- image --> \nrotation-derived fundamental modes, and their integer multiples may be harmonics. On the other hand, the other peaks do not have rotationderived features, and are considered to be pulsation-derived. However, the possibility that they are harmonics of the rotation-derived peaks cannot be completely ruled out. Furthermore, it should be noted that the ratio of the frequencies of the rotation-derived and pulsation-derived peaks could be close to an integer by chance, or that either the fundamental or harmonics could not be detected due to lack of signal-to-noise ratio. \nFigure 22. A similar figure to Figure 18 for V0332+53. \n<!-- image -->', '6.2 Interpretation of observed anti-correlations': 'The anti-correlation between flux oscillation amplitude and the IR excess is a notable result. According to the conventional idea of the pulsation-driven mass ejection, the pulsations should be active when the disk is growing, and thus the flux oscillation amplitude should be positively correlated with the IR excess, but the result was the opposite. In other words, if the anti-correlated peak amplitude variation we found is derived by the variation in the pulsation amplitude, it may disprove pulsation-driven mass ejection. However, if the pulsation amplitude did not change but only the flux oscillation amplitude changed, or if the anti-correlated peak is not pulsation-derived but rotation-derived, it is possible to explain the result without denying the pulsation-driven mass ejection. We examined following four scenarios to interpret the result: \n- 1st: Co-rotating gas cloud scenario,\n- 2nd: Fully covered photosphere scenario,\n- 3rd: Partially covered photosphere scenario,\n- 4th: Internal state transition scenario. \nWe have to mention the possibility that peak amplitude variations were caused by low-frequency beats ( 𝜈 ∼ 1 / 100 d -1 ), rather than the astrophysical mechanism. However, in this case, the anti-correlation can only be explained as being due to observations made at phases that happen to be so, which is inconsistent with the fact that anticorrelations were confirmed in four of the five sources. It is also possible that the anti-correlations are outliers. Our data set has too small a sample size to examine statistical significance.', '6.2.1 1st: Co-rotating gas cloud scenario': 'In the first scenario, we consider that changes in the density distribution of co-rotating gas clouds are the cause of amplitude variations. In other words, the origin of flux oscillations are not pulsations but the rotation. Since rotation-derived flux oscillations are caused by co-rotating gas clouds, the amplitude of flux oscillation should be \nFigure 23. Schematics of the first scenario. These illustrations view the Be star along its rotation axis. On the left is the case of a low mass ejection, which has a larger dynamic range of co-rotating gas cloud density along the rotational direction. On the other hand, the right side is when mass ejection is active, and the density distribution in the rotational direction is homogenised by the increased amount of co-rotating gas, resulting in a smaller dynamic range of density. \n<!-- image --> \ndetermined by the dynamic range of gas cloud density around the rotation axis. Therefore, the peak amplitude is expected to decrease in conjunction with the mass ejection of the Be star and the disk growth due to that the amount of co-rotating gas increased and homogenised around the rotation axis (Figure 23). However, unlike pulsation, which can have multiple modes, rotation is a single mode oscillation, so it is unlikely that rotation-derived anti-correlated and non-correlated peaks coexist. Furthermore, this scenario provides no explanation for amplitude variations of pulsation-derived flux oscillations. Therefore, in this scenario, all anti-correlated and noncorrelated peaks must be rotation-derived and pulsation-derived, respectively. Nevertheless, only 4U0115+634 and RX J0440.9+4431 have peaks with features suggesting a rotation origin, and to apply this scenario to other sources, it would have to be that the harmonics were detected but the fundamental mode was not.', '6.2.2 2nd: Fully covered photosphere scenario': 'The second scenario is based on the idea that the amplitude of the pulsation itself are constant, but the flux oscillation amplitude varies linked to the change in the amount of circumstellar materials (e.g. disk, stellar wind) and their reprocessing. We considers a situation in which most or all of the photosphere is covered by the circumstellar materials, as seen from our perspective. In this scenario, the photons coming from the Be star are scattered by the circumstellar materials, with at least 20 per cent of the photons passing through without being scattered even once (Figure 24). If the main scattering process is Thomson scattering, the reprocessing itself does not affect on the SED. However, because photons emitted from various regions on the photosphere are mixed, the scattered flux is considered to have lost periodicity. Since the mass ejected from the Be star should be supplied not only to the disk but also to the circumstellar materials, weexpect to observe the peak amplitude decrease in conjunction with the disk growth. This logic can also be applied to explain rotationderived oscillations. However, all peak amplitudes should decrease uniformly because almost all photons are reprocessed at the same rate in this mechanism. Thus, this scenario cannot explain non-correlated peaks. \nBecausethesituation in this scenario is simple, it is easy to estimate the density of the circumstellar materials. The following relationship \nFigure 24. Schematics of the second scenario. The green arrows are photon streams from the Be star, and the size indicates the number of photons pass through without scattering. In this scenario, the photons coming from the Be star are uniformly scattered by surrounding circumstellar materials at the same rate. \n<!-- image --> \nis expected to be satisfied when photons pass through a material: 𝜏 = e -𝑁𝜎 , (10) \nwhere 𝜏 is a transmittance, 𝑁 is a electron column density, and 𝜎 is a scattering cross section. Substituting 𝜏 = 0 . 2 and the Thomson scattering cross section 𝜎 𝑇 = 6 . 7 × 10 -25 cm 2 for 𝜎 , the column density is calculated as 𝑁 = 2 . 4 × 10 24 cm -2 . Assuming a spherical symmetric stellar wind as scattering materials, an electron number density 𝑛 is expected to follow an inverse square law with respect to a distance from the star 𝑟 due to the conservation of particle number, when ignoring particle acceleration: \n𝑛 = 𝑛 0 GLYPH<18> 𝑅 ∗ 𝑟 GLYPH<19> 2 , (11) \nwhere 𝑛 0 is a number density at the surface and 𝑅 ∗ is a radius of the star. 𝑁 is then equal to integral of 𝑛 from the stellar surface to infinity: \n𝑁 = ∫ 𝑛 d 𝑟 \n∞ 𝑅 ∗ (12) \n= 𝑛 0 𝑅 ∗ . (13) \nIf 𝑅 ∗ = 10 𝑅 ⊙ , then 𝑛 0 = 3 . 4 × 10 12 cm -3 . Assuming that the material is hydrogen, the numbers of electrons and nucleons are equal, thus a density at the surface is 𝜌 = 6 × 10 -12 g cm -3 . This is comparable to the observed density of the inner disk ( = 10 -13 -10 -10 g cm -3 , cf. Gies et al. 2007; Pott et al. 2010; Schaefer et al. 2010; Štefl et al. 2012; Meilland et al. 2012). Figure 25 shows the estimated density profile. The estimated density is ∼ 100 times larger than the numerical calculated wind density profile of the B1Ve star by Curé (2004). If the density distribution is not spherically symmetric, i.e., if the circumstellar material in our line of sight is locally dense, the less average density can be accounted for. In this case, there should be a clump at least 100 times denser than normal spherical symmetric winds. For example, the winds of Wolf-Rayet (WR) stars are clumpy \nFigure 25. Roughly estimated density profile. The unit of density is g cm -3 and the base of logarithm is 10. The flatness in 𝑢 ∼ -1 is due to the neglect of particle acceleration near the stellar surface. The range of the vertical axis is the same as in Figure 15 of Curé (2004). \n<!-- image --> \nas indicated by the stochastic variation in the line profiles (Moffat et al. 1988). The flux-limiting radiation hydrodynamic simulation of Moens et al. (2022) predicted a dynamic range of ∼ 100 in density of clumpy winds of WR stars. In addition, analysis of the UV and optical spectra of eight galactic O-type supergiants by Hawcroft et al. (2021) showed that half of the wind velocity field was covered by dense clumps and that their density was 25 times the average density. \nIncidentally, it is not obvious whether the effect of the temperature increase of the photosphere due to the return of some of the scattered photons to the Be star can be ignored in the situation where 80 per cent of the photons are scattered. So, we calculated the fraction of returning photons and obtained the value of about 10-20 per cent in the case considered here (Appendix B). Thus, at least the effect on the estimation of Be star properties in Section 4.1 can be considered negligible.', '6.2.3 3rd: Partially covered photosphere scenario': 'The third scenario shares the basic idea with the second one (Section 6.2.2) that the circumstellar materials obscure the Be star, but differs in that they obscure only a portion of the star, not the entire star (Figure 26). In this case, since some areas on the photosphere are not covered, the presence of non-correlated peaks can be tolerated. The question here is whether it is possible to reduce the peak amplitude by 80 per cent just by hiding part of the photosphere. For example, assuming the obscuration by the disk as shown in Figure 26, the obscured area is at most 50 per cent of the entire photosphere, and it seems difficult to achieve the amplitude reduction of 80 per cent. It can be satisfied in the case where the flux oscillations are generated in a limited region around the Be star. For the rotation-derived flux oscillations, this logic is applicable because they originate from co-rotating gas clouds. However, the probability that the co-rotating gas cloud is hidden by the disk is at most 50 per cent, so it is slightly difficult to explain why significant anti-correlations were found in four of the five sources. On the other hand, it is non-trivial whether the requirement is satisfied for the pulsation-derived flux oscillations. Non-radial pulsations cause the local expansion and contraction, and the geometry of the oscillation is characterised by the spherical har- \nFigure 26. Schematics of the third scenario. In this scenario, the flux oscillation amplitude is reduced because the region near the Be star or the photosphere contributing to the flux oscillation is covered by something around the Be star (in this figure, the disk) when viewed from our perspective. If the flux oscillations are caused by a larger region, this scenario cannot account for the amplitude variation of ∼ 80 percent. \n<!-- image --> \nmonics. The local brightness at the expanding region is expected to be smaller due to gravitational darkening. In low𝑙 modes, the number of nodal surfaces appearing on the photosphere is less, so each area of expansion or contraction is relatively large, and it is thought that the expanding and contracting regions do not appear equal when viewed from one direction. In this case, periodic flux oscillations can be generated by the brightness variations mentioned above, and the area contributing to the flux oscillations is relatively large. On the other hand, the same logic cannot simply be applied to cases where the expanding and contracting regions appear equal, such as in high𝑙 modes. In this case, flux oscillations can be generated by a local brightness difference (e.g. star spots) between the expanding and contracting regions. Therefore, some of the pulsation-derived flux oscillations may be caused by a limited region on the photosphere. However, the amplitude of the flux oscillations produced by this mechanism is considered to be smaller than that produced by the expansion and contraction of a larger region in low𝑙 modes. That makes difficult to interpret the anti-correlations of the maximum peaks in this scenario. In addition, local brightness differences on the photosphere are expected to be visible and hidden as the Be star rotates, which would cause rotation-linked features in light curves. Therefore, it is also difficult to apply this scenario to SAX J2103.5+4545 and V0332+53, which have no significant rotation-derived features in their amplitude spectra.', '6.2.4 4th: Internal state transition scenario': 'In the fourth scenario, we considered that the pulsation amplitude is indeed inversely correlated with the disk growth. The pulsations in early-type massive stars are driven by the 𝜅 -mechanism (Dziembowski & Pamiatnykh 1993; Dziembowski et al. 1993; Gautschy & Saio 1993; Pamyatnykh 1999; Miglio et al. 2007). In the temperature range where the Rosseland opacity 𝜅 is positively correlated with the temperature 𝑇 , 𝜅 variations provide positive feedback to 𝑇 variations, causing growth of thermodynamic fluctuations and thus pulsations. Since the amount of this feedback should depend on the \nFigure 27. Schematic 𝑇 -𝜅 curve and temperature dependence of d 𝜅 / d 𝑇 . Pulsations should be most active where d 𝜅 / d 𝑇 is largest. And it is qualitatively expected that the pulsation will be inactive at higher or lower temperature regions. \n<!-- image --> \nslope of the 𝑇 -𝜅 curve (i.e. d 𝜅 / d 𝑇 ), the intensity of 𝜅 -mechanism, and thus the pulsation amplitude, is expected to have dependence on the inner temperature distribution. Therefore, we first assume that some variation in the temperature inside the Be star has caused the pulsation to become inactive. The 𝜅 -mechanism is expected to be most active at the steepest point on the low-temperature side slope of the iron-group element derived bump (so-called Z-bump) in the 𝑇 -𝜅 curve. And therefore, d 𝜅 / d 𝑇 becomes smaller in the higher temperature as it is closer to the Z-bump peak. Conversely, moving to the low temperature side also reduces d 𝜅 / d 𝑇 as it approaches the bottom of valley between Z-bump and the hydrogen-derived bump. In other words, a decrease in pulsation amplitude is qualitatively expected for both temperature increases and decreases (Figure 27). If the star is hotter in the pulsation-inactive state than the pulsation-active state, relatively strong radiation pressure may promote mass ejection. On the other hand, considering the low temperature pulsation-inactive state, the lower temperature may be caused by the star expansion and the outer layer is more loosely bound than the active state, which is also expected to be relatively favourable for mass ejection. In any case, the pulsation-inactive state is qualitatively expected to be more active in mass ejection than the pulsation-active state. Note that these are only qualitative discussions, and a more quantitative discussion requires a detailed understanding of the structure and temperature distribution inside Be stars. \nSince this mechanism works for all pulsations, it cannot account for pulsation-derived non-correlated peaks. In addition, this scenario provides no explanation for the rotation-derived flux oscillations. Therefore, the origin of the anti-correlated and non-correlated peaks should be pulsation and the rotation, respectively. This is not so unreasonable since most of the anti-correlated peaks did not have the rotation-derived features, as discussed in Section 6.1.', '7 CONCLUSIONS': "The BeXBs exhibit characteristic and poorly understood phenomena such as two types of outbursts. To clarify them, we need to understand the nature of the circumstellar disk of the Be donor star. In particular, \nTable 5. Summary of four scenarios. It shows whether each scenario can explain the observed results. ' ⃝ ', ' × ', and ' △ ' indicate explainable, inexplicable, and conditionally explainable, respectively. \nthe mechanism driving the mass ejection of Be stars, which grows their circumstellar disks, is one of the critical unresolved issues. \nWe analysed long-term variations in the flux periodicity of five galactic BeXBs using TESS light curves, and compared them with the long-term multi-wavelength light curves, to constrain the behaviour of their circumstellar disk. We obtained X-ray, optical, and NIR light curves for Swift , MAXI, ZTF, ATLAS, and Gattini-IR. Then we confirmed that there was OIR variability on time scale of hundreds of days derived from the circumstellar disk and they exhibited giant outbursts. We extrapolated the 𝐽 -band magnitudes from the optical catalogued magnitudes using a simple blackbody model and subtracted them from the Gattini-IR 𝐽 -band light curve to evaluate the IR excess. In addition, we made amplitude spectra from TESS light curves, and extract peaks of periodic flux oscillations derived by the rotation or pulsations. \nAs a result, we found the anti-correlations between peak amplitudes and the IR excess. We examined the four scenarios to explain these results. Table 5 shows whether each scenario can explain the observed results. The first scenario, which assumes that all pulsationderived oscillations appears as non-correlated peaks, is unreasonable, and the second and third scenarios, which attempt to explain anticorrelations by reprocessing of the circumstellar materials, have severe conditions to explain the results. Therefore, we determined that it is most reasonable to interpret the observed anti-correlations as indicating an true anti-correlation between pulsation amplitudes and disk growth. In other words, the conventional idea of pulsation-driven mass ejection of Be stars may be incorrect. From this perspective, the most plausible scenario is the fourth one that assumed Be star internal state transition. \nThere were only two or three sectors with both TESS and Gattini-IR data available for each source, so it is difficult to examine the statistical significance of the confirmed anti-correlations. In addition, there were no TESS observations covering the near-infrared brightening phase when the mass ejection should be most active, for the target sources of this study. Therefore, further accumulation of data from future TESS observations is desirable. It is also important to check whether other BeXBs exhibit the same anti-correlation.", 'ACKNOWLEDGEMENTS': "M.Niwanowassupported by Grant-in-Aid for JSPS Fellows. This research was supported by JSPS KAKENHI Grant Number 23KJ0913, and 23K25910. This work includes data collected by the TESS mission, and funding for the TESS mission is provided by NASA's \nScience Mission Directorate. This research has made use of the MAXI data provided by RIKEN, JAXA, and the MAXI team. We acknowledge the Samuel Oschin 48-inch Telescope at the Palomar Observatory as part of the Zwicky Transient Facility project, supported by the National Science Foundation under Grant No. AST1440341 and a collaboration including Caltech, IPAC, the Weizmann Institute for Science, the Oskar Klein Center at Stockholm University, the University of Maryland, the University of Washington, Deutsches Elektronen-Synchrotron, and Humboldt University, Los Alamos National Laboratories, the TANGO Consortium of Taiwan, the University of Wisconsin at Milwaukee, and Lawrence Berkeley National Laboratories. Operations are conducted by COO, IPAC, and UW. This work has made use of data from the Asteroid Terrestrial-impact Last Alert System (ATLAS) project. The Asteroid Terrestrial-impact Last Alert System (ATLAS) project is primarily funded to search for near earth asteroids through NASA grants NN12AR55G, 80NSSC18K0284, and 80NSSC18K1575; byproducts of the NEO search include images and catalogs from the survey area. This work was partially funded by Kepler/K2 grant J1944/80NSSC19K0112 and HST GO-15889, and STFC grants ST/T000198/1 and ST/S006109/1. The ATLAS science products have been made possible through the contributions of the University of Hawaii Institute for Astronomy, the Queen's University Belfast, the Space Telescope Science Institute, the South African Astronomical Observatory, and The Millennium Institute of Astrophysics (MAS), Chile. This paper makes use of data from the AAVSO Photometric All Sky Survey, funded by the Robert Martin Ayers Sciences Fund and NSF (AST-1412587). The Pan-STARRS1 Surveys (PS1) and the PS1 public science archive have been made possible through contributions by the Institute for Astronomy, the University of Hawaii, the Pan-STARRS Project Office, the Max-Planck Society and its participating institutes, the Max Planck Institute for Astronomy, Heidelberg, and the Max Planck Institute for Extraterrestrial Physics, Garching, The Johns Hopkins University, Durham University, the University of Edinburgh, the Queen's University Belfast, the Harvard-Smithsonian Center for Astrophysics, the Las Cumbres Observatory Global Telescope Network Incorporated, the National Central University of Taiwan, the Space Telescope Science Institute, the National Aeronautics and Space Administration under Grant No. NNX08AR22G issued through the Planetary Science Division of the NASA Science Mission Directorate, the National Science Foundation Grant No. AST-1238877, the University of Maryland, Eotvos Lorand University (ELTE), the Los Alamos National Laboratory, and the Gordon and Betty Moore Foundation. This work has made use of data from the European Space Agency (ESA) mission Gaia ( https://www.cosmos.esa.int/gaia ), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https: //www.cosmos.esa.int/web/gaia/dpac/consortium ). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. This work utilized Python libraries for analysis and visualization: NumPy (Harris et al. 2020), SciPy (Virtanen et al. 2020), pandas (Wes McKinney 2010), Matplotlib (Hunter 2007). This work made use of Astropy: 3 a community-developed core Python package and an ecosystem of tools and resources for astronomy (Astropy Collaboration et al. 2013, 2018, 2022).", 'DATA AVAILABILITY': 'Swift /BATandZTFlightcurves used in this work are available in web pages of Swift /BAT Hard X-ray Transient Monitor 4 and NASA/IPAC Infrared Science Archive 5 , respectively. ZTF data we used are included in the ZTF public data release 15. The MAXI and ATLAS light curves in this work were generated by publicly available web services: MAXI on-demand 6 and ATLAS Forced Photometry 7 , respectively. As for the TESS light curves, the FFIs are available from MAST 8 . The Gattini-IR light curve is not publicly available.', 'REFERENCES': 'Stassun K. G., et al., 2018, AJ, 156, 102 \nTable A1. Typical properties of O-B type main sequence stars (Pecaut & Mamajek 2013). Extracted from https://www.pas.rochester.edu/ ~emamajek/EEM\\_dwarf\\_UBVIJHK\\_colors\\_Teff.txt . \nStassun K. G., et al., 2019, AJ, 158, 138 \nTamura K., Tsunemi H., Kitamoto S., Hayashida K., Nagase F., 1992, ApJ, 389, 676 \nTonry J. L., et al., 2018, PASP, 130, 064505 \nTorrejón J. M., Schulz N. S., Nowak M. A., 2012, ApJ, 750, 75 \nTorres G., Andersen J., Giménez A., 2010, A&ARv, 18, 67 \nTownsend L. J., Kennea J. A., Coe M. J., McBride V. A., Buckley D. A. H., Evans P. A., Udalski A., 2017, MNRAS, 471, 3878 \nTsygankov S. S., Doroshenko V., Lutovinov A. A., Mushtukov A. A., Poutanen \nJ., 2017, A&A, 605, A39 \nVasilopoulos G., et al., 2020, MNRAS, 494, 5350 \nVirtanen P., et al., 2020, Nature Methods, 17, 261 \nWenger M., et al., 2000, A&AS, 143, 9 \nWes McKinney 2010, in Stéfan van der Walt Jarrod Millman eds, Pro- ceedings of the 9th Python in Science Conference. pp 56 - 61, doi:10.25080/Majora-92bf1922-00a \nWest B. F., Becker P. A., Vasilopoulos G., 2024, ApJ, 966, L5 \nWilson-Hodge C. A., et al., 2018, ApJ, 863, 9 \nWilson C. A., Weisskopf M. C., Finger M. H., Coe M. J., Greiner J., Reig P., Papamastorakis G., 2005, ApJ, 622, 1024 \nŠtefl S., Le Bouquin J. B., Carciofi A. C., Rivinius T., Baade D., Rantakyrö \nF., 2012, A&A, 540, A76', 'APPENDIX A: TYPICAL PROPERTIES OF O-B TYPE MAIN-SEQUENCE STARS': 'Table A1 summaries typical properties of O-B type main sequence stars (Pecaut & Mamajek 2013).', 'APPENDIX B: A FRACTION OF PHOTONS RETURNING TO THE BE STAR DUE TO SCATTERING IN SPHERICALLY SYMMETRIC STELLAR WIND': 'A variation in the number of photons N as they pass through matter over a microscopic distance d 𝑟 can be written as follows: \nd N = -𝑛𝜎 N d 𝑟, (B1) \nwhere 𝑛 is a electron number density and 𝜎 is a scattering cross section. By solving Equation (B1) with substituting Equation (11), a \nnumber of pass-through photons N pass is obtained: \nN pass = N 0 exp GLYPH<20> -𝑛 0 𝜎𝑅 ∗ GLYPH<18> 1 -𝑅 ∗ 𝑟 GLYPH<19> GLYPH<21> , (B2) = N 0 exp h -𝑁𝜎 GLYPH<16> 1 -˜ 𝑟 -1 GLYPH<17> i , ( 𝑁 = 𝑛 0 𝑅 ∗ , ˜ 𝑟 = 𝑟 / 𝑅 ∗ ) (B3) = N 0 exp h -𝐴 GLYPH<16> 1 -˜ 𝑟 -1 GLYPH<17> i , ( 𝐴 ≡ 𝑁𝜎 ) \n(B4) \nwhere 𝑅 ∗ is a stellar radius, N 0 and 𝑛 0 are a number of photons and an electron number density at the stellar surface ( 𝑟 = 𝑅 ∗ ), 𝑁 is an electron column density, and ˜ 𝑟 is a normalised radius. Note that Equation (B3) is consistent with Equation (10) in the limit of ˜ 𝑟 → ∞ . In terms of the model in Section 6.2.2, N pass corresponds to the number of photons passing a sphere of radius ˜ 𝑟 that have never experienced scattering. Since scattering experienced photons also pass through the same plane, N is expected to be between N pass and N 0 . The differential number of scattered photons is equal to the absolute value of the right side of Equation (B1): \nd N scatt = 𝑛𝜎 N d 𝑟, (B5) \n= 𝐴 N d˜ 𝑟 ˜ 𝑟 2 , (B6) \nwhered N scatt is a number of photons scattered in a thin spherical shell of radius ˜ 𝑟 and thickness d˜ 𝑟 . If the scattering process is Thomson scattering and incident photons are unpolarized, the scattering is isotropic. In this case, the fraction of photons returning to the Be star is proportional to the solid angle of the star viewed from the scattering point. Assuming the spherical star, the solid angle of the star Ω can be calculated as follows: \nΩ = ∫ sin -1 1 ˜ 𝑟 2 𝜋 sin 𝜃 d 𝜃, \n0 (B7) = 2 𝜋 GLYPH<16> 1 -√︁ 1 -˜ 𝑟 -2 GLYPH<17> , (B8) = 2 𝜋𝐹 ( ˜ 𝑟 -1 ) . GLYPH<16> 𝐹 ( 𝑥 ) ≡ 1 -√︁ 1 -𝑥 2 GLYPH<17> (B9) \nThen, the differential number of returning photons d N ret can be calculated from d N scatt and Ω : \nd N ret = Ω 4 𝜋 d N scatt , (B10) = 𝐴 2 N 𝐹 ( ˜ 𝑟 -1 ) d˜ 𝑟 ˜ 𝑟 2 . (B11) \nWe can obtain the total number of returning photons N ret by integrating d N ret from the stellar surface to the infinity: \nN ret = ∫ ∞ 1 𝐴 2 N 𝐹 ( ˜ 𝑟 -1 ) d˜ 𝑟 ˜ 𝑟 2 . (B12) \nIf N = N 0 , the fraction of returning photons N ret /N 0 is calculated as follows: \nN ret N 0 = 𝐴 2 ∫ ∞ 1 𝐹 ( ˜ 𝑟 -1 ) d 𝑟 ˜ 𝑟 2 , (B13) = 𝐴 2 ∫ 1 0 𝐹 ( 𝑢 ) d 𝑢, GLYPH<16> 𝑢 = ˜ 𝑟 -1 GLYPH<17> (B14) = 𝐴 2 GLYPH<16> 1 -𝜋 4 GLYPH<17> = 0 . 11 𝐴. (B15) \nSimilarly, the case N = N pass is as follows: \nN ret N 0 = 𝐴 2 ∫ ∞ 1 exp h -𝐴 GLYPH<16> 1 -˜ 𝑟 -1 GLYPH<17> i 𝐹 ( ˜ 𝑟 -1 ) d˜ 𝑟 ˜ 𝑟 2 , (B16) \n= 𝐴 2 e -𝐴 ∫ 1 0 𝐹 ( 𝑢 ) e 𝐴𝑢 d 𝑢, (B17) \n= 𝐴 2 e -𝐴 GLYPH<18> e 𝐴 -1 𝐴 -∫ 1 0 √︁ 1 -𝑢 2 e 𝐴𝑢 d 𝑢 GLYPH<19> . (B18) \nHere the first order modified Bessel and Struve function of the first kind, 𝐼 1 and 𝐿 1 can be written in following integral representations: \n𝐼 1 ( 𝑥 ) = 2 𝜋 𝑥 ∫ 1 0 √︁ 1 -𝑢 2 cosh ( 𝑥𝑢 ) d 𝑢, (B19) \n𝐿 1 ( 𝑥 ) = 2 𝜋 𝑥 ∫ 1 0 √︁ 1 -𝑢 2 sinh ( 𝑥𝑢 ) d 𝑢. (B20) \nHence, \n𝜋 𝐼 1 ( 𝑥 ) + 𝐿 1 ( 𝑥 ) 2 𝑥 = ∫ 1 0 √︁ 1 -𝑢 2 [ cosh ( 𝑥𝑢 ) + sinh ( 𝑥𝑢 )] d 𝑢, (B21) = ∫ 1 0 √︁ 1 -𝑢 2 e 𝑥𝑢 d 𝑢. (B22) \nTherefore, \nRight side of (B18) = 𝐴 2 e -𝐴 GLYPH<20> e 𝐴 -1 𝐴 -𝜋 𝐼 1 ( 𝐴 ) + 𝐿 1 ( 𝐴 ) 2 𝐴 GLYPH<21> , (B23) \n= 1 2 GLYPH<26> 1 -e -𝐴 GLYPH<20> 1 + 𝜋 𝐼 1 ( 𝐴 ) + 𝐿 1 ( 𝐴 ) 2 GLYPH<21> GLYPH<27> . (B24) \nSubstituting 𝜎 = 𝜎 T = 6 . 7 × 10 -25 cm 2 and 𝑁 = 2 . 4 × 10 24 cm -2 to Equations (B15) and (B24), the fraction is calculated as N ret /N 0 = 0 . 13 , 0 . 17. In other words, about 10-20 % of the total photons return to the Be star in the model considered in Section 6.2.2. \nThis paper has been typeset from a T E X/L A T E X file prepared by the author.'} |
2024arXiv240907678R | A new inflationary scenario driven by a slowlyrolling homogeneous scalar field whose potential Vleftvarphiright is given by a generalized exponential function is investigated. Within the it slowroll approximation we obtain the main predictions of the model and compare them with current data from cosmic microwave background and largescale structure observations. We show that this single scalar field model admits a wider set of solutions than usual exponential scenarios and predicts acceptable values of the spectral index running of the spectral index and tensortoscalar ratio for the remaining number of it efolds lying in the interval N 55 pm 5 and an energy scale on which lambda geq sqrt2 in particular we observe that the value of the model parameter kappa depends on the analysis. Finally the primordial local nonGaussianity is briefly discussed where we conclude that kgtrsim 0.02 for ftextNLtextlocal ll 1. | 2024-09-01T00:00:00Z | ['10.48550/arXiv.2409.07678', '2024arXiv240907678R', 'arXiv:2409.07678'] | ['Astrophysics - Cosmology and Nongalactic Astrophysics'] | Generalized inflation in the context of kappadeformed theories | 2,024 | 175 | 0 | ['EPRINT_HTML', 'EPRINT_PDF'] | 0 | https://arxiv.org/pdf/2409.07678.pdf | {'B. W. Ribeiro, a,b, 2 I. M. Macêdo, a, 1 F. C. Carvalho. a, 3': '- a Departamento de Física, Universidade do Estado do Rio Grande do Norte, Av. Professor Antonio Campos - Pres. Costa e Silva, Mossoró-RN, 59610-210, Brasil.\n- b Departamento de Astronomia, Observatório Nacional, Rua General José Cristino 20921-400, Rio de Janeiro-RJ, Brasil. \nE-mail: [email protected], [email protected], [email protected] \nAbstract. A new inflationary scenario driven by a slowly-rolling homogeneous scalar field whose potential V ( φ ) is given by a generalized exponential function is investigated. Within the slow-roll approximation we obtain the main predictions of the model and compare them with current data from cosmic microwave background and large-scale structure observations. We show that this single scalar field model admits a wider set of solutions than usual exponential scenarios and predicts acceptable values of the spectral index, running of the spectral index and tensor-to-scalar ratio for the remaining number of e -folds lying in the interval N = 55 ± 5 and an energy scale on which λ ≥ √ 2 ; in particular, we observe that the value of the model parameter κ depends on the analysis. Finally, the primordial local non-Gaussianity is briefly discussed where we conclude that k ≳ 0 . 02 for f local NL ≪ 1 .', '1 Introduction': "In the standard Hot Big Bang (HBB) cosmology, the cosmic inflation [1-3] is assumed to set the initial conditions driving the universe towards homogeneity, flatness, and absence of primordial monopoles, and generate the primordial inhomogeneities that lead to the formation of cosmic structures [4, 5]. The cosmic inflation is a stage of accelerated expansion triggered by a homogeneous slowly-rolling scalar field φ , so-called inflaton , whose energy density is dominated by its potential V ( φ ) . Such an accelerated expansion took place in the very early universe and lasted just a split second, coming to the end once the kinetic and potential energy densities become comparable. Subsequently, the inflaton decays fully filling the universe with a hot plasma of relativistic particles, i.e, radiation, in a process known as reheating [6-8]. \nDuring such an inflationary period, quantum fluctuations in the inflaton were stretched by ultrafast expansion and converted into cosmic microwave background (CMB) fluctuations, which are the seeds from which the large-scale structure (LSS) of the universe has evolved. Furthermore, the cosmic inflation predicts a stochastic background of primordial gravitational waves (PGWs) [9-11] resulting from tensor perturbations in the metric, whose main observational signature is a curl-like pattern in the CMB polarization map, called B-modes [12, 13]. Since the universe before the last scattering is opaque to electromagnetic radiation, the PGWs represent a powerful window to probe the physics of the early universe and beyond. \nIn the framework of the simplest single field slow-roll inflation, both scalar and tensor perturbations are expected to have nearly scale-invariant spectra. A nearly scale-invariant spectrum is characterized by a specific pattern of fluctuations that are almost the same at all scales, i.e., they have compatible amplitudes and shapes regardless of the size of the region being observed. Such a spectrum can be completely described by a simple two-parameter power law. In addition, the initial spatial distributions of both scalar and tensor perturbations are nearly Gaussian, which means that all statistical information about them is contained in \ntheir two-point correlation functions or, equivalently, in their respective power spectra. These predictions have been confirmed by CMB (over the years by COBE [14], WMAP [15], and Planck [16] satellites) and LSS (2dFGRS [17], LSST [18] and DES [19]) observations. \nSince the temperature fluctuations measurements are close to the cosmic-variance limit for the tensor-to-scalar ratio, improvements are practically due to polarization. At present, a number of missions have been initiated to identify B-modes polarization in the CMB radiation and determine the tensor-to-scalar ratio. Among the most famous experiments, we highlight the ground-based AdvACT [20], CLASS [21], Keck/BICEP3 [22], Simons Array [23], and SPT-3G [24], balloons-borne EBEX 10k [25] and SPIDER [26], and satellites CMBPol [27], COrE [28] and LiteBIRD [29]. Detecting the PGWs signal will lead the cosmology into a new era of discovery where it will be possible to go back to ∼ 10 -32 s after the Big Bang. As a result, it will be possible (albeit indirectly) to examine phenomena related to a new physics at the grand unification scale and the quantum behavior of the spacetime metric [30]. \nDespite the fact that the cosmic inflation is the cosmologists' favorite paradigm for explaining the history of the early universe and the formation of LSS, numerous questions remain unanswered. These include the following [31]: \n- · although many toy models of inflation have been proposed, embedding inflation into a realistic particle physics model has proven to be a challenge;\n- · primordial inflation requires that the universe be dominated by the energy density of a purely hypothetical ingredient, referred to as inflaton, for which there is no direct evidence;\n- · if inflation is generated by a scalar matter field, a spacetime singularity must have preceded this period meaning that the inflationary cosmology is incomplete;\n- · the theory of cosmological fluctuations is based on the General Relativity theory, that is without doubt inapplicable at the sub-Planckian scales present during the early phases of inflation; and\n- · have not been found (to date) any signs of the PGWs background, predicted by the cosmic inflation, despite meticulous searches for them. \nThe above-mentioned conceptual difficulties are the main motivations for searching for some alternative theories to primordial inflation. The most recurrent in the literature are: varying light speed [32], topological defects [33], ekpyrotic and cyclic cosmology [34], string gas [35], and bouncing cosmologies [36]. As we can see, none of these alternatives theories is as well developed as inflationary cosmology at present, and none of them solves the standard HBB cosmology problems as elegantly as cosmic inflation does. Furthermore, as previously stated, the most recent and robust CMB and LSS observations are in fully agreement with the inflationary predictions. \nIt should be emphasized that the detailed particle physics mechanism responsible for the primordial inflation is still unknown, so any description of this era requires an appropriate extrapolation of the known physical laws. In consequence, the standard approach is strictly phenomenological with a considerable freedom in modeling the inflationary potential. In this sense, several shapes of the potential V ( φ ) have been proposed in the literature giving rise to a number of inflationary models, from conservative approaches like power law to alternative approaches like tachyon and ghost (see refs. [37-39] for a review on inflation). \nA simple and interesting class of inflationary models is that described through the usual exponential potential \nV ( φ ) ∝ exp( -λφ ) , (1.1) \nas originally investigated in refs. [40, 41]. It turns out that scalar fields with exponential potentials are very common in a number of particle physics theories. Indeed, they naturally occur in supergravity [42], superstrings [43], higer-order [44], higher-dimensional [45], and Kaluza-Klein theories [46], among others. In this paper we propose a generalization of the usual exponential scenario in eq. (1.1) in light of the κ -deformed functions [47]. Generalized cosmological scenarios of early and late universe have been previously investigated in ref. [48]. In particular, it has been shown that the β -exponential potential proposed in ref. [49] can be derived from the framework of the braneworld scenarios and that it provides a good description of the observational data [50]. On the other hand, the κ -generalized scenario studied in ref. [51] has been proposed in the context of the gravity-thermodynamics conjecture [52], such that a new cosmological scenario emerges based on modified Friedmann equations. In contrast, the κ -generalization of the ordinary exponential inflationary scenarios that we are proposing here occurs within the standard hypothesis, i.e., without modifying the usual Friedmann equations or introducing new thermodynamic conjectures concerning the spacetime structure. \nThis paper is organized as follow. In section 2 we introduce the formal basis of the inflationary dynamics in the framework of the slowly-rolling homogeneous scalar field. In section 3 we consider quantum fluctuations in the inflaton and develop mathematical tools in order to characterize them statistically, such as the primordial power spectrum, spectral index, running of the spectral index, and tensor-to-scalar ratio. In addition, we also provide a brief description of possible primordial non-Gaussianities (NGs) through the derivation of the non-linearity parameter. In section 4 we present our κ -generalized model, its main predictions and compare our results with the current Planck CMB and LSS data. Finally, we discuss our results in light of the current status of the cosmic inflation in section 5.", '2 Dynamics of Inflation': 'In this section we shall develop the main concepts regarding the physics of the scalar field, which is assumed to be the main driver of the primordial inflation. In this way we will show how such a scalar field produces an almost de Sitter expansion, in early times, required to explain the current features of the universe. Within the slow-roll approximation, we shall define some of the main inflationary parameters and discuss general aspects of this approach.', '2.1 Physics of the scalar field': "The simplest dynamical model of inflation involves a homogeneous single scalar field φ ( t ) , called inflaton, minimally coupled with the gravity and characterized by the potential V ( φ ) . Such a theory is described by the action [38] \nS = ∫ d 4 x √ -g [ 1 2 R + 1 2 g µν ∂ µ φ∂ ν φ -V ( φ ) ] , (2.1) \nin units for which M 2 Pl ≡ (8 πG ) -1 = ℏ = c = 1 . \nIn the present context, the Friedmann-Lemaître-Roberton-Walker (FLRW) metric ansatz fits the global properties about the statistical homogeneity and isotropy of the universe, i.e., \nds 2 = -dt 2 + a 2 ( t ) ( dr 2 1 -Kr 2 + r 2 d Ω 2 ) , K = 0 , ± 1 , (2.2) \nwhere K is the spatial curvature parameter and a ( t ) is the scale factor. Assuming a spatially flat ( K = 0 ) 3-section and a perfect fluid treatment, the stress-energy tensor corresponding to the extra scalar field φ is described by \nT ( φ ) µν = -2 √ -g δS φ δg µν = ∂ µ φ∂ ν φ -g µν [ 1 2 ∂ ρ φ∂ ρ φ + V ( φ ) ] . (2.3) \nFor a homogeneous field configuration ( ∇ φ = 0), this leads to the energy density and pressure \nρ φ = 1 2 ˙ φ 2 + V ( φ ) , (2.4) \nP φ = 1 2 ˙ φ 2 -V ( φ ) , (2.5) \nrespectively, where overdots indicate derivative with respect to the cosmic time t . Hence, the equation of state (EoS) corresponds to \nw φ = P φ ρ φ = 1 2 ˙ φ 2 -V ( φ ) 1 2 ˙ φ 2 + V ( φ ) . (2.6) \nThen varying the action in eq. (2.1) with respect to the scalar field φ , it is possible to arrive at the Klein-Gordon equation, \n¨ φ +3 H ˙ φ + V ' ( φ ) = 0 , (2.7) \nwhere primes indicate the derivative with respect to the field φ . On the other hand, varying this action with respect to the metric tensor g µν , we get the Friedmann equations, \nH 2 = 1 3 [ 1 2 ˙ φ 2 + V ( φ ) ] , (2.8) \na a = -1 3 [ ˙ φ 2 -V ( φ ) ] . (2.9) \nEqs. (2.8) and (2.9) govern the evolution of the universe filled by a homogeneous scalar field φ ( t ) whose dynamics is determined by eq. (2.7). In this way, a complete and suitable solution involves solving all three equations at the same time.", '2.2 Slow-roll approximation': "Notice then that a dynamical homogeneous scalar field φ ( t ) may induce an early inflationary epoch - without undermining the successes of the standard HBB cosmology - provided that the potential term V ( φ ) is dominant over the kinetic term, i.e., ˙ φ 2 ≪ V , and sufficiently flat, i.e., V ' , V '' ≪ V . In addition, inflation will only be sustained for a sufficiently long period of time if the second time derivative of φ is small enough, i.e., | ¨ φ | ≪ 3 H ˙ φ . This conditions are known as slow-roll conditions, leading to the approximation [38] \nV ' ≃ -3 H ˙ φ, (2.10) \nH 2 ≃ 1 3 V ( φ ) , (2.11) \n˙ H ≃ -1 2 ˙ φ 2 . (2.12) \nIn this case, the dynamics of inflation can be expressed in terms of the slow-roll parameters as defined by [39] \nϵ = 1 2 ( V ' V ) 2 , (2.13) \nη = V '' V -1 2 ( V ' V ) 2 , (2.14) \nξ 2 = V ' V ''' V 2 . (2.15) \nIn order to eqs. (2.7)-(2.9) are in agreement with the slow-roll approximation, the above slow-roll parameters must be extremely small, i.e., \nϵ ≪ 1 , | η | ≪ 1 , ξ 2 ≪ 1 . (2.16) \nHowever, an accelerated expansion phase requires only that a > 0 implying ϵ < 1 . Hence, an inflationary period ends when ϵ = 1 . \nWhen second-order contributions in the primordial power spectra are included, Planck TT, TE, EE+lowE+lensing(+BK15) data constrain the slow-roll parameters 1 as [16] \nϵ V < 0 . 0097 (0 . 0044) at 95 % CL , η V = -0 . 010 +0 . 007 -0 . 011 ( -0 . 015 ± 0 . 006) at 68 % CL , ξ 2 V = 0 . 0035 +0 . 0078 -0 . 0072 (0 . 0029 +0 . 0073 -0 . 0069 ) at 95 % CL . (2.17) \nAs we will see later, these constraints - especially those on ϵ - combined to the tensor-to-scalar ratio and the amplitude of the scalar fluctuations, provide an upper bound on the energy scale of inflation when the pivot scale k ∗ exits the Hubble radius.", '2.3 Number of e -folds': 'Within the slow-roll approach, it is simple to calculate the scale factor at any instant between the beginning and the end of inflation. Since the expansion is assumed to be very large in this stage, it is more convenient to compute it in terms of the number of e -folds remaining before the end of inflation, defined as [38] \nN ≡ ln ( a end a ) = ∫ φ φ end dφ √ 2 ϵ ( φ ) , (2.18) \nwhere φ end denotes the value of the scalar field at the end of inflation. This quantity measures then the amount of physical expansion during inflation. \nTo determine the number of e -folds corresponding to a particular scale k in terms of the present Hubble scale k 0 = a 0 H 0 , we need a model describing the complete history of the Universe. In the standard HBB scenario, inflation is followed by a period of reheating, then a period dominated by radiation, then one dominated by non-relativistic matter, and \nfinally the current one dominated by the cosmological constant. In this background, assuming instantaneous transitions between one era to another, then one has [16] \nN ( k ) ≃ 67 -ln ( k a 0 H 0 ) + 1 4 ln ( V 2 k ρ end ) -1 -3 w int 12(1 + w int ) ln ( ρ end ρ th ) -1 12 ln ( g th ) , (2.19) \nwhere V k is the potential energy when the scale k crosses the Hubble radius during inflation ( k = a k H k ), ρ end is the energy density at the end of inflation, w int corresponds to the effective EoS between the end of inflation and the thermalization energy scale ρ th , and g th is the number of effective bosonic degrees of freedom at the energy scale ρ th . Planck collaboration assumes g th = 10 3 , a pivot scale k ∗ = 0 . 002 Mpc -1 , and an uncertainty of 50 < N ∗ < 60 [16].', '3 Cosmological Perturbations': "Although inflation arose in a context in which the main problems to be solved were those of flatness, horizon and cosmic relics, the most robust inflationary prediction is the generation of density fluctuations as seeds for LSS in the Universe. Quantum fluctuations of the inflaton field are stretched on large scales by the accelerated expansion and frozen after the scale of perturbations leaves the Hubble radius during inflation. Long after inflation ends, the perturbations cross the Hubble radius again providing a natural explanation for the observed anisotropies in the CMB and the LSS observed today [53, 54]. On the other hand, if the energy scale of inflation is high enough, this mechanism is also expected to generate a subtle background of PGWs that can polarize the CMB photons, leading to a very distinctive signature in the B-modes spectrum on large angular scales. [37, 55]. \nFundamentally, we are interesting in only small fluctuations around the homogeneous single scalar field φ ( t, x i ) = ¯ φ ( t ) + δφ ( t, x i ) , where from now on we will use a bar in order to identify background quantities and neglect non-linear terms in the perturbations. The inflationary field has been assumed to be minimally coupled to the gravity. This means that any perturbation δφ will induce perturbations on the metric δg µν which are divided into scalar, vector and tensor modes according to the scalar-vector-tensor (SVT) decomposition. In this way, metric perturbations sourced by the fluctuations of the inflaton can be described by the perturbed FLRW line element [37], \nds 2 = -(1 + 2Φ) dt 2 +2 aB i dtdx i + a 2 [(1 -2Ψ) δ ij + E ij ] dx i dx j , (3.1) \nwhere B i = ∂ i B -S i and E ij = 2 ∂ ij E +2 ∂ ( i F j ) + h ij . Since the perturbations are decoupled at the linearized level, they evolve separately [56-58]. Scalar fluctuations are often described in terms of the comoving curvature perturbation R = Ψ + ( H/ ˙ ¯ φ ) δφ , obeying the following equation, \n1 a 3 ϵ ∂ ∂t ( a 3 ϵ ˙ R ) + k 2 a 2 R = 0 . (3.2) \nTensor fluctuations in turn can be described in terms of the decomposition h ij = h ( t ) e (+ , × ) ij , where e ij are the eigenmodes of the spatial Laplacian operator, and h ( t ) is some amplitude term obeying the following equation, \nh +3 H ˙ h + k 2 a 2 h = 0 . (3.3) \nThus, tensor fluctuations give rise to PGWs with two possible polarization states, + and × , and time-dependent amplitude h = h ( t ) . Vector fluctuations are expected to vanish in the presence of only scalar fields [37]. \nSince the primordial scalar and tensor perturbations are expected to be nearly Gaussian, they can be described in terms of their two-point correlation functions, \n⟨R k R k ' ⟩ = (2 π ) 3 δ 3 ( k + k ' ) P s ( k ) , (3.4) \n⟨ h k h k ' ⟩ = (2 π ) 3 δ 3 ( k + k ' ) P T ( k ) , (3.5) \nwhere P s ( k ) and P T ( k ) are the scalar and tensor power spectra, respectively. Nevertheless, throughout this paper we will work in terms of the dimensionless spectra 2 in the framework of the single field slow-roll approximation, which are respectively given by \nP s ( k ) = 4 πk 3 (2 π ) 3 ∣ ∣ R 2 ∣ ∣ = ( 1 12 π 2 )( V 3 V ' 2 ) , (3.6) \nP T ( k ) = 2 4 πk 3 (2 π ) 3 ∣ ∣ h 2 ∣ ∣ = ( 1 12 π 2 )( V 3 V ' 2 ) , (3.7) \nwhere the factor 2 in eq (3.7) encodes the two independent polarizations of the graviton, and R and h are the solutions of eqs. (3.2) and (3.3) respectively. Here, both power spectra are evaluated at the Hubble-radius crossing, k = aH . Since both scalar and tensor perturbations are sourced by the inflaton quantum fluctuations in an almost de Sitter background, we must expect the primordial power spectra to be nearly flat. \nFor single field models, the scale-dependence of the primordial power spectra can be approximated by a power-law form of the adiabatic scalar and tensor components, thus the scalar and tensor spectral indices are \nn s -1 = [ d ln P s ( k ) d ln k ] k = k ∗ (3.8) \nand \nn T = [ d ln P T ( k ) d ln k ] k = k ∗ , (3.9) \nrespectively. Here, k ∗ denotes an arbitrary pivot scale. Notice that the scale-invariant power spectra correspond to n s = 1 and n T = 0 [59]. \nPerturbations generated in a single field slow-roll regime are expected to be only weakly scale-dependent. Using eqs. (3.6) and (3.7) we can write the scalar and tensor spectral indices in terms of the slow-roll parameters as \nn s -1 = 2 η -4 ϵ (3.10) \nand \nn T = -2 ϵ . (3.11) \nIndeed, Planck TT, TE, and EE+lowE+lensing data give us n s = 0 . 9649 ± 0 . 0042 at 68 % CL in the baseΛ CDM model [16], confirming a nearly flat power spectrum just as expected in the single field slow-roll inflation. This scenario do not significantly changes when we consider Λ CDM extensions. For instance, for the baseΛ CDM + α s + β s model, Planck TT, TE, EE+lowE+lensing data give us n s = 0 . 9625 ± 0 . 0048 at 68 % CL [16]. Direct detection of tensor modes has not yet been achieved.", '3.1 Running of the spectral index': 'Because there is a weak scale-dependence on the spectral indices of both scalar and tensor perturbations, we shall seek a way to measure them. Considering the third terms proportional to (1 / 2) ln 2 ( k/k ∗ ) in the power spectra expansions 3 , the respective coefficients are called running of the scalar and tensor spectral indices, defined as [60] \nα s = [ dn s d ln k ] k = k ∗ , α T = [ dn T d ln k ] k = k ∗ . (3.12) \nSince ( d/d ln k ) = (1 /H )( d/dt ) , we can say that the running of the spectral index quantify the rate of change of { n s , n T } per Hubble time. In terms of the slow-roll parameters we have [37] \nα s = 16 ϵη -8 ϵ 2 -2 ξ 2 , (3.13) \nα T = -4 ϵ ( ϵ -η ) . (3.14) \nNotice then that, in this scenario, α s , α T ≪ 1 . Indeed, Planck 2018 data are consistent with the single field slow-roll inflation as it has constrained α s = 0 . 002 ± 0 . 010 at 68 % CL in the baseΛ CDM + α s + β s model [16].', '3.2 Tensor-to-scalar ratio': 'An important observational quantity is the called tensor-to-scalar ratio r , defined as the ratio between amplitudes of the tensor and scalar power spectra at pivot scale k ∗ , i.e., \nr = P T ( k ∗ ) P s ( k ∗ ) = A T A s . (3.15) \nIn the single field slow-roll framework, we obtain \nr = 16 ϵ = -8 n T , (3.16) \nwhere the second equality is called consistence relation, being a single field slow-roll inflation prediction [61]. Since ϵ ≪ 1 in the slow-roll regime, we expect that the amplitude of the tensor modes to be suppressed by the scalar modes. However, it must be possible to observe them. In a universe dominated by the energy density of a single slowly-rolling scalar field, the total field excursion is related to the tensor amplitude by ∆ φ/M Pl = √ r/ 8 N [62]. Thus, for N = 50 we get the lower bound [63] \n∆ φ M Pl = √ r 3 . 2 × 10 -3 . (3.17) \nNotice then that a large value for the tensor-to-scalar ratio, r ≥ 3 . 2 × 10 -3 , is associated with both significant amplitude of the tensor perturbations and a high scale for the inflationary energy, ∆ φ ≥ M Pl . Therefore, single field slow-roll inflation generates a PGWs background which may be observable in the low multipoles of the CMB anisotropies, but only if the scalar field variation is at least of the order of the Planck scale [62]. \nWithout observing the tensor modes, Planck was only able to infer the following upper limits considering the baseΛ CDM + r model [16]: \n- · r < 0 . 11 and r 0 . 002 < 0 . 10 for Planck TT, TE, EE+lowEB+lensing; and\n- · r < 0 . 061 and r 0 . 002 < 0 . 056 for Planck TT, TE, EE+lowE+lensing+BK15, \nOn the other hand, the following restrictions have been obtained by Forconi et al. [63] considering the baseΛ CDM + r + α s model: \n- · r < 0 . 159 for Planck TT, TE, EE+lowEB+lensing; and\n- · r < 0 . 0658 for Planck TT, TE, EE+lowE+lensing+BK15. \nIn any case, the constraints on this parameter are at 68 % CL, while the upper bounds are at 95 % CL. Therefore, the Planck 2018 baseline plus BK15 constraint on r implies in the upper bound on the energy scale of inflation [16], \nV ∗ = 3 π 2 M 4 Pl 2 A s r < ( 1 . 6 × 10 16 GeV ) 4 (95 % CL ) , (3.18) \nor, equivalently, \n̸ \nH ∗ M Pl < 2 . 5 × 10 -5 (95 % CL ) . (3.19) \nAdditionally, Forconi et al. [63] have obtained constraints on the PGWs considering nonvanishing curvature, Ω K = 0 , and ACTPol, SPT-3G and WMAP data combinations. Anyway, the results do not significantly change from those explained above.', '3.3 Primordial non-Gaussianity': 'Curvature perturbations generated by quantum fluctuations in an inflationary phase produce a quasi-Gaussian random density field. Nevertheless, detectable amounts of non-Gaussianity (NG) can be produced in scenarios such as inflaton self-interactions, additional light and/or heavy fields, and multi-fields. Thus even a small amount of primordial NG would be a sufficient argument to go beyond the simplest single field slow-roll scenario. In this way, primordial NG ends up being an important tool for discriminating between inflation theories [64, 65]. \nNon-Gaussianity is statistically characterized by a non-vanish three-correlation point function, or its Fourier transform, the bispectrum B s , defined as [38] \n⟨R k 1 R k 2 R k 3 ⟩ = (2 π ) 3 δ 3 ( k 1 + k 2 + k 3 ) B s ( k 1 , k 2 , k 3 ) . (3.20) \nThere are three main bispectrum templates most explored in the literature, the so-called local, equilateral and orthogonal shapes. In this work we shall explore the local template. \nThe simplest way in order to parameterize possible primordial NGs is through non-linear corrections to the Gaussian curvature perturbation R G , i.e., \nR = R G + 3 5 f local NL [ R 2 G -〈 R 2 G 〉] , (3.21) \nwhere f local NL quantify the degree of local non-linearity on the perturbation R 4 . In this way, it is easy to show that \nf local NL = 5 6 [ B s ( k 1 , k 2 , k 3 ) P s ( k 1 ) P s ( k 2 ) + P s ( k 2 ) P s ( k 3 ) + P s ( k 3 ) P s ( k 1 ) ] . (3.22) \nIn the single field slow-roll approach, both inflationary power spectrum and bispectrum can be obtained from the δN formalism [66, 67], so that the non-linearity parameter gives [37, 68] \nf local NL = -5 6 ( η -2 ϵ ) = -5 12 ( n s -1) . (3.23) \nThe second equality is the second consistence relation for the single field slow-roll inflation. As we can see, the local NG is completely suppressed by the slow-roll parameters, or equivalently n s -1 , in the slow-roll inflation. Therefore, any robust measurement revealing a large f local NL rules out the single field inflation [69, 70].', '4 κ -Exponential Inflation': 'In this section we propose a new inflationary model, the κ -exponential inflation. This model generalizes the conventional exponential scenarios originally studied in refs. [40, 41], described in eq. (1.1). In the scope of the slow-roll approximation, we derive the main predictions of the model, such as the slow-roll parameters, number of e -folds, scalar spectral index and its running, tension-to-scalar ratio, and primordial NGs, and its cosmological consequences. Finally, we confront our results with the current observations.', '4.1 The model': 'In the context of the κ -deformed theories, we propose the following generalization of the inflationary potential in eq. (1.1), given by \nV ( φ ) ∝ exp κ ( -λφ ) , (4.1) \nwhere the above generalized exponential and its inverse, called κ -logarithmic, are defined as \nexp κ ( x ) ≡ ( √ 1 + κ 2 x 2 + κx ) 1 κ , (4.2) \nln κ ( x ) ≡ x κ -x -κ 2 κ , (4.3) \nwhich reduce to ordinary exponential and logarithm functions, respectively, as the parameter κ ∈ [ -1 , 1] approaches zero. Notice, x ≡ -λφ for our purposes right here [47, 71, 72]. \nSome properties of the ordinary exponential and logarithmic functions are preserved in the above generalized approach. For instance, we have \nexp κ [ln κ ( x )] = x, (4.4) \nexp κ ( -x ) exp κ ( x ) = 1 , (4.5) \nexp -κ ( x ) = exp κ ( x ) . (4.6) \nThe first one reflects the fact that these functions are inverses of each other, while the second one allows us to construct a generalized statistical. Indeed, the κ -generalized distribution is defined as [72] \nf i ∼ exp κ ( -βε i + βµ ) , (4.7) \nwhose asymptotic limits give \nf i ∼ { exp( -βε i + βµ ) , βε i -βµ → 0; N ε -1 k i , ε i →∞ . (4.8) \nFigure 1 : The generalized potential, eq. (4.1), as a function of the field φ for some selected values of the deformation parameter κ and λ = 1 . 5 . \n<!-- image --> \nNotice that it reproduces the Boltzmann ordinary distribution at low energy limit, whereas a power law tail is achieved to describe high energy systems. Finally, the last one tell us that the generalized exponential is symmetric with respect to the parameter κ . This means that symmetric values of κ (e.g., κ = ± 0 . 5 ) give rise to the same solution. \n̸ \nWe display in figure 1 the behavior of the generalized potential, eq. (4.1), as a function of the field φ . In particular, it must be noted that for all κ = 0 the generalized potential presents a quasi-exponential (power law) behavior, whereas for κ = 0 the usual potential, eq. (1.1), is fully recovered. We also emphasize that because of the symmetry property highlighted in eq. (4.6), we have considered only some selected positive values of κ on this plot. However, in what follows, we will see that the straight relations between some inflationary parameters will result in functions that are not symmetric in κ . In these cases, we should also consider negative values of the parameter κ .', '4.2 Slow-roll results': "By considering the κ -generalized potential in eq. (4.1), the slow-roll parameters, eqs. (2.13)(2.15) give \nϵ = λ 2 2 1 [1 + κ 2 λ 2 φ 2 ] , (4.9) \nη = λ 2 2 1 [1 + κ 2 λ 2 φ 2 ] ( 1 + 2 κ 2 λφ √ 1 + κ 2 λ 2 φ 2 ) , (4.10) \nξ 2 = λ 4 [1 + κ 2 λ 2 φ 2 ] 2 [ (1 -κ 2 ) + 3 κ 2 λφ √ 1 + κ 2 λ 2 φ 2 ( 1 + κ 4 λφ √ 1 + κ 2 λ 2 φ 2 )] . (4.11) \nHence, the number of e -folds defined in eq. (2.18) gives \nN = 1 2 λ [ φ √ 1 + κ 2 λ 2 φ 2 + 1 κλ ln ( √ 1 + κ 2 λ 2 φ 2 + κλφ ) ] . (4.12) \nAs expected, the above expressions reduce to the usual exponential results ϵ = η = λ 2 / 2 , ξ 2 = λ 4 , and N = φ/λ in the limit κ → 0 . \n<!-- image --> \nFigure 2 : The first slow-roll parameter ϵ as a function of the field φ [eq. (4.10)] for selected values of λ and (a) κ = 0 . 1 and (b) κ = 0 . 3 . We see that increasing λ also increases the value of the field at the end of inflation, φ end , characterized by ϵ ( φ end ) = 1 . \n<!-- image --> \nIn addition, we can compute the value of the field at the end of inflation by setting ϵ ( φ end ) = 1 , i.e., \n̸ \nFor this value to be physically acceptable, i.e., φ end ∈ R , we must have | λ | ≥ √ 2 , where we can disregard negative values of λ as we are interested in only decreasing potentials. In this sense, we show in figure 2a the behavior of the slow-roll parameter ϵ as a function of the field φ for selected values of λ and κ = 0 . 1 ; figure 2b is the same analysis, but for κ = 0 . 3 . Notice that increasing λ also increases the value of the field at the end of inflation in both cases, as we can see from eq. (4.13). \nφ end = 1 κ ( 1 2 -1 λ 2 ) 1 2 , ∀ κ = 0 . (4.13) \nLet us now consider both the inflationary parameters, spectral index and tensor-to-scalar ratio. Replacing the above results for the slow-roll parameters into eq. (3.10) and eq. (3.16), we obtain \nn s -1 = -λ 2 [1 + κ 2 λ 2 φ 2 ] ( 1 -2 κ 2 λφ √ 1 + κ 2 λ 2 φ 2 ) , (4.14) \nr = 8 λ 2 1 [1 + κ 2 λ 2 φ 2 ] , (4.15) \nand the straight relation between n s and r , \nn s -1 = -r 8 [ 1 -2 κ ( 1 -r 8 λ 2 ) 1 2 ] . (4.16) \nNotice that the above eq. (4.16) is the straight generalization of the well-known expression r = 8(1 -n s ) , expected in the usual exponential scenario, as well as a scale invariant spectrum suggest vanishing primordial tensor modes. On the other side, the running of the spectral index gives \nα s = 2 λ 4 κ 2 [1 + κ 2 λ 2 φ 2 ] 2 [ 1 + λφ √ 1 + κ 2 λ 2 φ 2 ( 1 -3 κ 2 λφ √ 1 + κ 2 λ 2 φ 2 )] , (4.17) \n<!-- image --> \nFigure 3 : Trajectories for different values of κ on the planes (a) n s -r and (b) n s -α s . In (a) we consider the first order in the slow-roll approximation, eq. (4.16); whereas (b) is the numerical solution envolving eqs. (4.14) and (4.17). In both cases, the hatched area corresponds to the current Planck plus lensing observations at 68 % CL [73]. \n<!-- image --> \nor, equivalently, \nα s = 2 κ 2 ( r 8 ) 2 { 1 + 1 κ ( 1 -r 2 λ 2 ) 1 2 [ 1 -3 κ ( 1 -r 8 λ 2 )] } , (4.18) \nwhere we have replaced eq. (4.15) into eq. (4.17) in order to obtain eq. (4.18). Notice that κ = 0 implies α s = 0 . This means that the deformation parameter κ makes the primordial fluctuations acquire a very small dependence on the scale according to α s ∝ κ 2 . Next, we show in figures 3a and 3b the curves on the plane n s -r , originating from eq. (4.16), and on the plane n s -α s , corresponding to the numerical solution of eqs. (4.16) and (4.18), respectively, for different values of the parameter κ and λ = 1 . 5 . As we can see, for a sizable range of κ , the model's predictions in both cases are in fully agreement with current bounds from CMB plus lensing Planck data, r < 0 . 11 and n s = 0 . 9649 ± 0 . 0042 , at 68 % CL [74]. Notice that this time we also have used negative values of κ since they provide different solutions than those considering positive κ . Notice also that, regardless of the value of κ , n s = 1 implies r = 0 , which in turn implies α s = 0 . \nBy combining eqs. (4.12) and (4.14), we obtain the spectral index n s as a function of N , so that we can compare the model's predictions with the current observation bounds on these quantities. It is important to point out again that both the PGWs and structure formation observations constraint N to 50 -60 e -folds between the horizon exit and the end of inflation [16]. In figure 4a we then show the solutions n s ( κ, N ) for N = { 50 , 55 , 60 } . As we can see, only a small range of κ > 0 is in fully agreement with the Planck observations of n s (hatched area). \nLastly, replacing eqs. (4.9) and (4.10) into eq. (3.23) we get the generalized non-linearity parameter, \nf local NL = 5 λ 2 12 1 [1 + κ 2 λ 2 φ 2 ] ( 1 -2 κ 2 λφ √ 1 + κ 2 λ 2 φ 2 ) . (4.19) \n<!-- image --> \nFigure 4 : (a) Spectral index n s and (b) local non-linearity parameter f local NL as functions of the parameter κ for selected values of the number of e -folds ranging the interval N = 55 ± 5 and λ = 1 . 5 . \n<!-- image --> \nFirst, note the maximum value for the non-linearity parameter, f local NL = 5 λ 2 / 12 , corresponding to the usual exponential scenario κ = 0 . Then, in a similar way to n s ( κ, N ) - or equivalently using the second equality in eq. (3.23) -, the solutions f local NL ( κ, N ) for N = { 50 , 55 , 60 } are displayed in figure 4b. Since our κ -generalization has introduced the lower limit λ ≥ √ 2 , we can get large NGs (i.e., f local NL ∼ 1 ) from very small values of κ (i.e., κ ≲ 0 . 02 ). This means that if the single field slow-roll approximation, characterized by the consistence relation in eq. (3.23), is true, then we must have a κ not so close to zero.", '5 Discussion': "The inflationary paradigm represents one of the greatest successes of the fusion between the concepts of the quantum field theory and modern cosmology, bringing changes in the way we understand the primordial universe. It is the favorite hypothesis of most cosmologists to explain the origin of primordial fluctuations, being the basis for the emergence of structures, as well as for the flatness and statistical isotropy of the universe. In this way, a number of inflationary models whose the physical motivations are rooted in the modern particle physics have been proposed. \n̸ \nFrom eq. (4.13) we have found that | λ | ≥ 2 , where we have disregarded negative values as we are only interested in decreasing potentials. As it happens, the scalar field φ must be a real quantity after all. This constraint has straight influence on the energy scale of inflation, as well as on the slow-roll parameter ϵ , which signals the end of inflation. Indeed, increasing λ also increases the value of the field at the end of inflation, as we can see in eq. (4.13) and figures 2a and 2b. \nIn this paper we have investigated the main cosmological consequences of a new inflationary scenario characterized by the generalized potential in eq. (4.1). This model is the result of the improvement of the exponential models, which reduce to the form of eq. (1.1), in the context of the κ -deformed theories [47, 71, 72]. As discussed in section 4, this generalized potential behaves like a simple power law for all κ = 0 and exactly like an exponential function for κ = 0 , as we can see in figure 1. √ \nNext, we have analyzed the scalar spectral index, its running, and tensor-to-scalar ratio in light of the n s -r and n s -α s planes. The straight relation between n s and r is given in eq. (4.16), while for n s and α s we have needed to numerically solve eqs. (4.16)-(4.18). Figures 3a and 3b show us the solutions r ( n s ) and α s ( n s ) for different values of the parameter κ and λ = 1 . 5 , respectively. In both cases, the hatched area corresponds to the current Planck plus lensing observations at 68 % CL. As we can see, the model's prediction for these parameters are compatible with CMB and LSS observational constraints. \nWe also have analyzed the spectral index n s as a function of the parameter κ for three different values of the number of e -folds corresponding to the range N = 55 ± 5 and λ = 1 . 5 . Figure 4a shows us that only small range of κ , including only positive values, is compatible with the current bounds on n s from Planck plus lensing data, n s = 0 . 9649 ± 0 . 0042 at 68 % CL (hatched area). \nSince the single field slow-roll inflationary mechanism is expected to produce small (but detectable) primordial NGs, we also have investigated possible non-vanishing contributions to the three-point correlation function encoded in the local non-linearity parameter. In figure 4b we show the behavior of f local NL as a function of the parameter κ for N = { 50 , 55 , 60 } and λ = 1 . 5 . As we can see, f local NL has a maximum value when κ = 0 (usual scenario) and a minimum for κ → 1 . The maximum one depends only on λ , which obeys λ ≥ √ 2 . Hence, we can get large local NGs, i.e., f local NL ≳ 1 , for very small κ , tipically κ ≲ 10 -2 . This means that, if the single field slow-roll inflation is the true mechanism for generating the primordial fluctuations, we expect a κ not so close to zero. \nHowever, there is a tenacious observational nuisance when it comes to primordial NGs, since not even the most current and accurate measurements of f NL (be it local, equilateral or orthogonal) allow us to make any statements about it. It turns out that there are many sources of NGs in the CMB anisotropies beyond the primordial one, including systematics effects and astrophysical contamination [75]. Well-known examples of contamination are those produced by secondary non-linear anisotropies, generated since the epoch of matter-radiation decoupling at z ∼ 1100 , namely the integrated Sachs-Wolfe effect, gravitational lensing, Sunyaev-Zel'dovich effect, polarization, among others [76]. In addition, another important source of NG in the CMB anisotopies are the non-linearities in the evolution of the photonbaryon fluid at recombination [77]. \nAs a conclusion, we highlight the κ -exponential inflation as a promising model describing the origin of the primordial fluctuations, whose the main predictions such as the number of e -folds, tensor-to-scalar ratio, scalar spectral index and its running, and primordial NGs are in fully agreement with the most current and accurate Planck measurements. Our theoretical results tell us that the value of κ compatible with the observations depends on the analysis, e.g., n s -r plane analysis points out to κ < 0 , n s -κ to κ > 0 , and α s -n s to both. Meanwhile, the f local NL -κ analysis discards negatives and very close to zero values of κ . Finally, combining the n s -κ and f local NL -κ analyses we conclude that 0 . 02 ≲ κ ≲ 0 . 2 . 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2024arXiv240910623M | fR theories of modified gravity may be compatible with current observations if the deviations from general relativity are sufficiently well screened in dense environments. In recent work arXiv2310.19955 we have shown that approximations commonly used to assess whether galaxies are screened or unscreened fail to hold in observationally interesting parts of parameter space. One of the assumptions commonly made in these approximations and more broadly in the study of fR models is that the mass of the scalar mode can be neglected inside a galaxy. In this work we demonstrate that this approximation may fail spectacularly and discuss the implications of this for tests of the theory. | 2024-09-01T00:00:00Z | ['10.48550/arXiv.2409.10623', '2024arXiv240910623M', 'arXiv:2409.10623'] | ['Astrophysics - Cosmology and Nongalactic Astrophysics', 'Astrophysics - Astrophysics of Galaxies', 'General Relativity and Quantum Cosmology'] | Galactic Compton Wavelengths in fR Screening Theories | 2,024 | 175 | 0.26 | ['EPRINT_HTML', 'EPRINT_PDF'] | 0 | https://arxiv.org/pdf/2409.10623.pdf | {'Bradley March, a Clare Burrage a and Aneesh P. Naik b': 'a \nSchool of Physics and Astronomy, University of Nottingham, University Park, Nottingham NG7 2RD, UK b Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK \nE-mail: [email protected], [email protected], \[email protected] \nAbstract. f ( R ) theories of modified gravity may be compatible with current observations if the deviations from general relativity are sufficiently well screened in dense environments. In recent work [1] we have shown that approximations commonly used to assess whether galaxies are screened, or unscreened, fail to hold in observationally interesting parts of parameter space. One of the assumptions commonly made in these approximations, and more broadly in the study of f ( R ) models, is that the mass of the scalar mode can be neglected inside a galaxy. In this work we demonstrate that this approximation may fail spectacularly and discuss the implications of this for tests of the theory.', '1 Introduction': "In recent years, the continuing puzzle of the dark sector has motivated the study of scalartensor theories of gravity, in which the scalar field mediates gravitational-strength fifth forces [2-9]. For such a fifth force to evade detection in stringent Solar System tests of gravity, the theory must possess some screening mechanism , such that the force is suppressed in environments resembling the Solar System (i.e., environments of high ambient density or deep gravitational potential). Various screening mechanisms have been proposed along these lines, for example, the Vainshtein and symmetron mechanisms [10-13]. In this work, we are concerned with the class of theories exhibiting chameleon screening [14, 15], in which the mass of the scalar grows large in dense regions. In particular, we focus on the theory of Hu-Sawicki f ( R ) gravity [2, 16], which has received especially widespread attention in the literature. \nGalactic scales present a particularly interesting area of study in f ( R ) theories. Baker et al. [17] showed that galactic scales inhabit a 'desert' in parameter space with comparatively few observable probes. In the years since that work, various studies have sought to explore this desert, with considerable success [18-31]. Indeed, the strongest constraints to date on f ( R ) gravity were obtained in recent works by Desmond and Ferreira [32], and Landim et al [33] in a search for chameleon-induced equivalence principle violations in a large sample of observed galaxies. \nIn parallel with these analyses of observable probes, there has been work aiming to improve our theoretical understanding of the interplay between chameleon fifth forces and galaxies. Because galaxies inhabit the deeply non-linear regime of cosmological structure formation, such an understanding can only be obtained by numerical computation, typically with various simplifying approximations. For example, recent work has produced cosmological galaxy formation simulations incorporating a quasi-static chameleon fifth force [34]. Another example is our previous work [1] in which f ( R ) solvers were applied to static galactic models \nto solve for the scalar field and accompanying fifth forces. For a review of the approaches and challenges to numerically modelling theories with screened fifth forces see ref. [35]. \nWhen studying the behaviour of the scalar field within a galaxy, and the resulting screening of the fifth force, the various components of the galaxy (stars, gas and dark matter) are typically treated as continuous fluids of varying density. However, this approximation might not be a very good one in the case of the stellar component, or even the dark matter component if it is composed of compact objects and/or small-scale subhaloes. The goal of the present work is to test the approximation that the distribution of stars in a galaxy can be treated as a smooth density distribution. In particular, we calculate the Compton wavelength of the f ( R ) scalar field across a series of typical galaxies, assuming a continuous stellar density profile, and compare this length scale with the typical stellar separation. To anticipate our results, we find that in the screened regions of galaxies, the Compton wavelength is often far smaller than the stellar separation, suggesting that continuous fluid approximation is inadequate. \nThis work is structured as follows. In section 2 we present the Hu-Sawicki f ( R ) gravity theory, including expressions for the Compton wavelength. Section 3 describes our numerical solutions for the scalar field across our galactic model. Section 4 describes the results, before concluding remarks in section 5. Throughout this paper we work in natural units such that 8 πG = c = ℏ = 1 , except where units are explicitly stated. The code used to solve the field profiles and produce the figures contained within this paper is publicly available at github.com/Bradley-March/scalar-compton-wavelength.", '2.1 f ( R ) Gravity': 'In f ( R ) theories, modifications to GR are achieved by substituting the Ricci scalar, R , in the Einstein-Hilbert action with the generalised R + f ( R ) . This allows the action to incorporate higher-order curvature terms, \nS = 1 2 ∫ d 4 x √ -˜ g [ R + f ( R )] + S m [ ˜ g µν , ψ SM i ] , (2.1) \nhere R = R (˜ g ) represents the scalar-curvature of the Jordan frame metric. \nSuch an action can be reformulated into the general scalar-tensor framework via the field redefinition [16] \nϕ = -ln(1 + f R ) 2 β ( ϕ ) , V ( ϕ ) = Rf R -f ( R ) 2(1 + f R ) 2 and β ( ϕ ) = 1 √ 6 , (2.2) \nwhere \nf R ≡ d f ( R ) d R (2.3) \nserves as the scalar field in the theory. \nFor certain functional forms of f ( R ) , the theory can display the chameleon screening mechanism [36-38]. We adopt the well-known Hu-Sawicki model [2]. This model is characterised by the functional form \nf ( R ) = -am 2 1 + ( R/m 2 ) -b , (2.4) \nwhere a and b are positive dimensionless parameters, and m has dimensions of inverse length. For simplicity, and consistency with the majority of literature, we set b = 1 . The remaining parameters, a and m , can be related by ensuring that in the high curvature limit, R ≫ m 2 , gravity reverts to GR +Λ CDM, i.e. f ( R ) ≈ -2Λ where Λ is the cosmological constant, giving \nam 2 = 2Λ . (2.5) \nThis allows the theory to be characterised by a single model parameter, which we choose to be the present-day background field value f R 0 \na = -4Ω 2 Λ (Ω m +4Ω Λ ) 2 1 f R 0 and m 2 = -3 H 2 0 (Ω m +4Ω Λ ) 2 2Ω Λ f R 0 , (2.6) \nwhere Ω m and Ω Λ are the cosmological density parameters for matter and dark energy, respectively. Throughout this work, we use Ω m = 0 . 3 and Ω Λ = 0 . 7 , except where specifically stated. \nBy extremising the action, eq. (2.1), with respect to the metric we derive a set of modified Einstein field equations. Taking the trace and then applying the Newtonian limit, under the small field | f R | ≪ 1 and quasi-static approximations |∇ f R | ≫ ∂ t f R , we obtain the f ( R ) equation of motion, \n∇ 2 f R = 1 3 ( δR -δρ ) , (2.7) \nwhere δρ and δR represent the density perturbation and curvature perturbation, respectively. Within the Hu-Sawicki model the curvature perturbation is defined as \nδR = R 0 [√ f R 0 f R -1 ] , (2.8) \nwhere zero subscripts denote present-day background values. The acceleration due to the fifth force from our scalar field is \na 5 = 1 2 ∇ f R , (2.9) \nagain assuming | f R | ≪ 1 . \nAn examination of the fifth force equation, eq. (2.9), reveals that the effect of the scalar field is minimised when the field is constant and its gradient is approximately zero. A region in which the field is constant is referred to as the screened region. An analytical expression for the field profile can be derived in this limit, by noting that a flat field profile also implies a vanishing Laplacian in the f ( R ) equation of motion, eq. (2.7). For the right hand side of this equation to vanish we require \nf scr R f R 0 = ( 1 + δρ R 0 ) -2 , (2.10) \nwhere f scr R represents the analytic approximation to the screened field value. This screened solution corresponds to the curvature tracing the minimum of the effective scalar potential. This method has been utilised in the literature, e.g. ref. [2]. \nOutside of a few special simplified cases, for example the screened solution discussed in eq. (2.10), it is not possible to solve the highly non-linear equation of motion for the f R field analytically. In later sections of this work, where we wish to understand the behaviour of the \nfield in a galaxy, we utilise a two-dimensional, cylindrically symmetric numerical solver, based on MG-GADGET [39], to solve a discretised version of the equation of motion, eq. (2.7). For a given input f R 0 and δρ , the code performs an iterative Gauss-Seidel method, to generate the field profile f R . Section 3.2 of ref. [1] provides a comprehensive description of the solver.', '2.2 Compton Wavelength': 'The Compton wavelength of our theory is given by \nλ C ≡ m -1 ϕ , (2.11) \nwhere the scalar mass, m ϕ , is defined as \nm 2 ϕ ≡ d 2 V ( ϕ ) d ϕ 2 . (2.12) \nThe scalar field, ϕ , and potential, V ( ϕ ) , are defined in eq. (2.2). We compute the mass in the field picture ( ϕ , not f R ) because it is only in this picture that the scalar degree of freedom is canonically normalised, however, after applying the simplifications described later in this section both forms are equivalent to leading order. \nThe first derivative of the potential is \nd V d ϕ = d R d ϕ d V d R = -β R +2 f ( R ) -Rf R (1 + f R ) 2 , (2.13) \nwhere we have used \nd R d ϕ = ( d ϕ d R ) -1 = -2 β (1 + f R ) f RR , (2.14) \ngiven f RR ≡ d f R d R . The second derivative, resulting in the scalar mass, is \nm 2 ϕ = d 2 V ( ϕ ) d ϕ 2 = 2 β 2 (1 + f R ) 2 f RR [ (1 + f R ) 2 -f RR ((3 -f R ) R +4 f ( R )) ] . (2.15) \nWe can simplify this scalar mass using the functional form of f ( R ) , eq. (2.4), and its derivatives \nf ( R ) = -am 2 R R + m 2 , (2.16) \nf R = -am 4 ( R + m 2 ) 2 , (2.17) \nf RR = 2 am 4 ( R + m 2 ) 3 . (2.18) \nIn the high curvature regime, R ≫ m 2 , these expressions are well approximated as \nf ( R ) ≈ -am 2 , (2.19) \nf R ≈ -am 4 R 2 , (2.20) \nf RR ≈ 2 am 4 R 3 . (2.21) \nEliminating R between equations (2.20) and eq. (2.21) gives \nf RR ≈ 2 m 2 √ ( -f R ) 3 a . (2.22) \nSubstituting these expressions into the scalar mass, eq. (2.15), and assuming | f R | ≪ 1 , we obtain \nm 2 ϕ 2 β 2 m 2 ≈ √ a 2 ( -f R ) -3 / 2 -3 √ a ( -f R ) -1 / 2 -a. (2.23) \nEq. (2.6) shows that a ∼ | f R 0 | -1 . Therefore, given that | f R | ≪ 1 , the latter two terms in this equation are subdominant to the first, leaving us with the rather simple \nm 2 ϕ ≈ β 2 √ am 2 ( -f R ) -3 / 2 . (2.24) \nThis expression is equivalent to the previously obtained result, equation (48) in ref. [2], albeit we have expanded the f RR term. \nUsing this simplified scalar mass, we determine the Compton wavelength of our theory to be \nλ C ≈ ( ( -f R ) 3 am 4 β 4 ) 1 / 4 . (2.25) \nBy substituting the values for β and the model parameters a and m (eqs. (2.2) and (2.6) respectively), we arrive at the final expression \nλ C ≈ 1 H 0 √ 2 Ω m +4Ω Λ ( f 3 R f R 0 ) 1 / 4 . (2.26) \nIn the cosmological background, f R → f R 0 , the Compton wavelength reduces to \n¯ λ C ≈ 1 H 0 √ 2 | f R 0 | Ω m +4Ω Λ . (2.27) \nIn previous work, e.g. ref. [40], this equation is often quoted as ¯ λ C ≈ 32 √ | f R 0 | 10 -4 Mpc , where the prefactor of 32 requires adopting cosmological parameters of h = 0 . 73 , Ω m = 0 . 24 and Ω Λ = 0 . 76 .', '3.1 Galactic Density Model': 'To represent the galactic density profile we employ a two-component model, comprising a dark matter and stellar disc component. The dark matter is represented by the standard NFW profile [41], \nρ DM ( r ) = ρ NFW r r NFW ( 1 + r r NFW ) 2 , (3.1) \nwhere r is the spherical radial coordinate, and ρ NFW and r NFW are the characteristic density and length scales. The stellar disc is represented by a double exponential profile \nρ SD ( R,z ) = Σ disc 2 z disc e -R/R disc e -| z | /z disc , (3.2) \nwhere R represents the cylindrical radial coordinate, z represents the coordinate perpendicular to the galactic disc, and Σ disc , R disc and z disc are the characteristic density and length scales. \nIn order to maintain the simplicity of our model and reduce the number of input parameters, we utilise several empirical relations to relate the various density profile scales to a single input parameter, the virial mass of the dark matter halo, M vir . The methodology employed to achieve this is outlined in appendix B of ref. [1]. We cut off the central density singularity of the NFW profile by setting the density to be constant within 50 pc. We also add a sharp outer cutoff at 2.2 R vir to address the logarithmically divergent mass [42]. Here the virial radius, R vir , is the radius at which the enclosed mass equals the virial mass, M vir .', '3.2 Separation Between Stars': 'In section 4 we will discuss how the Compton wavelength of the f R field compares to the spatial separation between stars. Under the simplifying assumption of a single universal stellar mass M ∗ and neglecting binary or multiple systems, the separation between stars is approximately given by \nS = ( ρ SD M ∗ ) -1 3 . (3.3) \nwhere ρ SD is the stellar density profile. In the remainder of this work we assume M ∗ = M ⊙ in this estimate. For a Milky-Way-like galaxy, we would estimate the typical stellar separation at a galactic radius of 8 kpc is 2.5 pc, which is of similar order to the Solar Systemα Centauri separation of 1.3 pc.', '4.1 Compton Wavelength in Galaxies': 'Figure 1 provides an example of the field profile, with a background value of f R 0 = -10 -6 , and related quantities that we can compute in a Milky-Way-like galaxy; specifically: the numerically calculated field profile, treating the stars as a continuous fluid, f R , (top left); the Compton wavelength calculated from this field profile, eq. (2.26), λ C , (top right); the analytic screened field value, f scr R , eq. (2.10) (bottom left); and the galactic density profile, δρ , eqs. (3.1) and (3.2) (bottom right). The sharp transition in f R values shown in figure 1 corresponds to the position of the screening surface, r s . Inside this surface fifth forces are suppressed and the field is small; outside, the field increases towards the background value of f R 0 and fifth forces are unsuppressed. For the parameters chosen in this figure, the galaxy is partially screened, with a screening surface located within the stellar disc. \nFigure 1 also illustrates a number of relationships between parameters we have previously discussed. That λ C ∝ | f R | 3 / 4 , eq. (2.26), can be seen from the similarity of the profiles of the field (top left) and Compton wavelength (top right). We find that in the screened region the numerically calculated field (interior of the top left plot) is well approximated by the analytically derived screened field value (bottom left), i.e. f R ( r < r s ) ≈ f scr R . We also see the relationship of eq. (2.10) in the similarity between the density profile (bottom right) and the screened field (bottom left). \nConsidering the magnitude of the Compton wavelength in the top right panel of figure 1, we notice that outside the screened region the Compton wavelength is large, reaching values O (Mpc) . However, inside the screened region the value of the Compton wavelength plummets to sub-parsec scales, significantly smaller than the typical separation of stars, eq. (3.3). \nFigure 1 . An example solution for a Milky-Way-like galaxy with M vir = 1 . 5 × 10 12 M ⊙ , and f R 0 = -10 -6 , plotted to a maximum radius of 10 kpc. Top left shows the field solution obtained from the numerical solver. Top right is the Compton wavelength, eq. (2.26). Bottom left shows the screened field value, eq. (2.10). Bottom right is the galactic density profile. This example illustrates a partially screened case, where the scalar field (and Compton wavelength) change dramatically at the screening surface, which is located within the stellar disc. \n<!-- image --> \nFigure 2 . Comparison between average stellar separation (purple) and Compton wavelength (orange) as a function of galactocentric radius, in a Milky-Way-like galaxy with M vir = 1 . 5 × 10 12 M ⊙ and f R 0 = -10 -6 . Two curves are shown for the Compton wavelength: the numerical solution extracted from our solver (solid) and the analytic solution in the screened regime (eq. (2.10); dashed). The curve representing the stellar separation has been truncated at the point where the stellar density is subdominant to the cosmic mean density. The figure shows that the Compton wavelength is orders of magnitude smaller(larger) than the stellar separation in the screened(unscreened) regime. \n<!-- image --> \nWe expand on this visualisation in figure 2 by comparing the lengths of the Compton wavelength and the stellar separation, eq. (3.3), along the radial axis of the galactic disc, using the same parameters as in the previous figure. It is now evident that the Compton wavelength is orders of magnitude smaller than the stellar separation inside the screened portion of the galaxy, but rapidly grows to be much larger in the unscreened region of the galaxy. \nThe parameters in figures 1 and 2 were specifically chosen to simulate a galaxy with partial screening, where the screening radius lies within the stellar disc. In general, varying M vir and f R 0 , we find that most locations in parameter space result in a fully screened or fully unscreened stellar disc. Figure 3 shows the proportion of stellar mass located in a region of the galaxy where λ C > S , for a range of M vir and f R 0 parameters. In the top left of the plot, where the galaxies are fully unscreened, we find f R ≈ f R 0 everywhere, and thus λ C ≈ ¯ λ C ≫ S . In the lower right region, the galaxies have fully screened stellar discs, resulting in λ C = λ scr C ≪ S , as illustrated by the example in figure 2. In the following section, we will further discuss the implications of the partially screened solutions, represented by the coloured (not black and white) points in figure 3. \nFigure 3 . Proportion (in percent) of the stellar mass within the region where the Compton wavelength (calculated numerically, assuming a smooth stellar profile) is larger than the stellar separation, across a range of f R 0 and M vir values. The top-left white region corresponds to fully unscreened solutions. This figure shows that the majority of galaxies that are not fully screened have stellar profiles with separations less than the Compton wavelength. \n<!-- image -->', '4.2 Scalar Field Values in the Interstellar Medium': 'At a given point in space, the scalar field is sourced primarily by mass within a few Compton wavelengths; the contributions of more distant matter are exponentially suppressed. Where the Compton wavelength is larger than the typical separation between objects, the field profile evolves as if the density profile of the individual sources had been smoothed, with a smoothing scale of order the Compton wavelength. When the Compton wavelength is less than the typical separation between objects, there is no smoothing effect, and the field responds to the objects individually. When studying the behaviour of the f ( R ) field within a galaxy it is common to treat the stellar component as a fluid with a smooth density profile (for example refs. [32, 34]). As we have seen, this approximation is not valid in much of the interesting parameter space. In this section we consider how the field might behave in the absence of the smoothing of the stellar density profile. \nIn the small Compton wavelength regime, it becomes invalid to consider the stellar density profile as a continuous distribution. As a result, we expect the scalar field to take differing field values near to stars, and deep in the interstellar medium. This is impossible to study analytically and will be challenging to solve numerically given the dynamical range \nof the problem. However, we can suggest some behaviour of the field in this regime. We expect that, in between the stars, the field will relax to the dark-matter-only solution, f DM R , with pockets around the now-discrete stars where the field becomes orders of magnitude smaller. Therefore, except in the neighbourhood of the stars, the field evolves as if the stellar component of the galaxy was absent. \nThere is also a rare, more complex case where, if we include the smoothed stellar mass in the galactic density profile, the Compton wavelength is smaller than the stellar separation, but removing the stellar contribution from the density profile, the dark-matter-only solution results in a Compton wavelength larger than the stellar separation (i.e. λ DM+SD C < S < λ DM C ). In this highly non-linear regime, we suggest that the field in the interstellar medium relaxes to larger values causing an increase in the Compton wavelength until the Compton wavelength becomes large enough to be comparable to the stellar separation. A similar situation to this, although on very different scales, was considered in ref. [43]. In this situation, we suggest that the typical field value will be that which gives a Compton wavelength of order the stellar separation. However, further numerical study would be needed to understand this case in detail. \nIf it is the case that treating the stellar population as a discrete distribution would result in the field decreasing from f DM+SD R to f DM R , then there would be a knock-on effect on the position of the screening surface, r s . In figure 4 we compare the screening radii derived in the two cases: by assuming the stellar profile is a continuous distribution that contributes to the field profile, f DM+SD R ; and by neglecting the contribution of the stellar density profile, i.e. a dark matter only solution, f DM R . In both cases we calculate the screening radius using the curvature-to-density parameter derived in our previous work [1]. \nFigure 4 shows that if r s is situated within the stellar disc, and λ C < S , then the decrease in the field values would act to shift r s towards the centre of the galaxy. Overall this would lead to less of the galaxy being screened, in this narrow diagonal band of parameter space. Although this only affects a small part of the parameter space, it remains relevant for current constraints, since many contemporary tests rely on partially screened stellar populations as observable tracers of equivalence principle violations.', '4.3 Fifth Forces Around Stars': 'Compton wavelengths that are shorter than the interstellar separation, mean that there can be significant variation in the value of the field in the space between the stars. Here we provide an estimate of whether this can lead to observable fifth forces in the Solar System. \nWe approximate the Sun as a sphere of constant density, embedded in a background in which the field is screened, f R = f scr R . The star is screened as long as | f R | ≲ Φ N ( R ∗ ) where Φ N ( R ∗ ) is the Newtonian potential evaluated at the surface of the star. As | f scr R | ≪ | f R 0 | , this is a weaker condition than that normally applied, which states that a star is screened if | f R 0 | ≲ Φ N ( R ∗ ) . At the position of the Sun, f scr R ∼ 4 × 10 -11 f R 0 , and λ C ∼ 10 18 f 1 / 2 R 0 m . The fifth force, in this approximation of the Solar System, is \nF 5 F N ≈ ( | f R | 5 × 10 -6 )( R ⊙ r ) 2 e -r/λ C , (4.1) \nwhich is extremely small, and well beyond the sensitivity of current probes. \nFigure 4 . Comparisons of the screening radius assuming no stellar contribution to the galactic density ( r DM s ) and assuming a smooth stellar density profile ( r DM+SD s ). Left shows the absolute distance between the two r s solutions, and right shows the ratio between them. In both cases, the upper-left white region corresponds to fully unscreened solutions. In the lower-right regions, the screening radii are larger than the extent of the stellar profile and thus are unchanged by ignoring the stellar contribution. In the right plot these correspond to ratios of 1, in the left we have masked these values. Assuming no stellar contribution has the largest impact on solutions where the screening radius is within the stellar disc, corresponding to the thin diagonal highlighted in the left plot, or the ratios with a value less than one in the right plot. \n<!-- image -->', '5 Conclusions': 'Scalar fields react to matter on scales equivalent to the Compton wavelength of the field. When modelling populations of discrete objects, it is common practice to treat the profiles as continuous distributions, assuming that many discrete objects are contained within a Compton wavelength, acting to smooth out the scalar field profile. \nIn the context of chameleon f ( R ) gravity in galactic environments, surveys are beginning to probe the regime where the screening radius may lie within the galactic disc and where the Compton wavelength can become significantly smaller than the typical separation of stars within the galaxy. As such, it is important to exercise caution when assuming a continuous stellar density profile. \nA proper treatment of the stars as individual discrete objects may result in a heterogeneous field profile, in which the mass of the stars is inconsequential to the field in the interstellar medium, allowing the interstellar field to relax to a larger field value permitted by the dark matter density only solution. In regions closer to the high-density stars, the \nscalar field will become further screened, adopting smaller, more screened field values. Such a situation could cause galaxies with partially screened stellar discs to become less screened, as neglecting the stellar mass profile pushes the screening surfaces towards the centre of the galaxy. In this case, there is the possibility that modified gravity tracers that rely on the galaxy being unscreened could become observable once more. \nIn conclusion, we should always be careful to check that the assumptions we are making actually hold in the environments we study, and as the accuracy of observations improves we should also improve the accuracy of our theoretical predictions.', 'References': "- [1] C. Burrage, B. March and A.P. Naik, Accurate computation of the screening of scalar fifth forces in galaxies , JCAP 2024 (2024) 004 [ 2310.19955 ].\n- [2] W. Hu and I. 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2024arXiv240909182P | In the present work we obtain and analyze a new class of analytical solutions of magnetically charged black bounces in kessence theory spherically symmetric in 31dimensions coupled to nonlinear electrodynamics NED. We consider two metric models SimpsonVisser and Bardeen for the kessence configurations n 13 and n 15. We obtain in an analytical way which scalar field field potential and Lagrangian NED are necessary to support the metrics. We analyze the behavior of these quantities and the energy conditions due to the scalar field and the NED. | 2024-09-01T00:00:00Z | ['arXiv:2409.09182', '2024arXiv240909182P', '10.48550/arXiv.2409.09182'] | ['General Relativity and Quantum Cosmology', 'High Energy Physics - Theory'] | Magnetically charged blackbounce solution via nonlinear electrodynamics in a kessence theory | 2,024 | 175 | 0.16 | ['EPRINT_HTML', 'EPRINT_PDF'] | 5 | https://arxiv.org/pdf/2409.09182.pdf | {'Carlos F. S. Pereira ∗': 'Departamento de Física, Universidade Federal do Espírito Santo, Av. Fernando Ferrari, 514, Goiabeiras, 29060-900, Vitória, ES, Brazil.', 'Denis C. Rodrigues †': 'Núcleo Cosmo-ufes & Departamento de Física, Universidade Federal do Espírito Santo, Av. Fernando Ferrari, 514, Goiabeiras, 29060-900, Vitória, ES, Brazil.', 'Marcos V. de S. Silva ‡': 'Departamento de Física, Programa de Pós-Graduação em Física, Universidade Federal do Ceará, Campus Pici, 60440-900, Fortaleza, Ceará, Brazil', 'Júlio C. Fabris §': 'Núcleo Cosmo-ufes & Departamento de Física, Universidade Federal do Espírito Santo, Av. Fernando Ferrari, 514, Goiabeiras, 29060-900, Vitória, ES, Brazil and National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), 115409, Kashirskoe shosse 31, Moscow, Russia.', 'Manuel E. Rodrigues ¶': 'Faculdade de Ciências Exatas e Tecnologia, Universidade Federal do Pará Campus Universitário de Abaetetuba, 68440-000, Abaetetuba, Pará, Brazil and Faculdade de Física, Programa de Pós-Graduação em Física, Universidade Federal do Pará, 66075-110, Belém, Pará, Brazil.', 'H. Belich ∗∗': 'Departamento de Física e Química, Universidade Federal do Espírito Santo, Av.Fernando Ferrari, 514, Goiabeiras, Vitória, ES 29060-900, Brazil. \nIn the present work, we obtain and analyze a new class of analytical solutions of magnetically charged black bounces in k-essence theory, spherically symmetric in (3+1)-dimensions, coupled to nonlinear electrodynamics (NED). We consider two metric models, Simpson-Visser and Bardeen, for the k-essence configurations n = 1 / 3 and n = 1 / 5 . We obtain in an analytical way which scalar field, field potential, and Lagrangian NED are necessary to support the metrics. We analyze the behavior of these quantities and the energy conditions due to the scalar field and the NED. \nKeywords: Phantom fields, Black-bounce, k-essence theory, electrodynamics, energy conditions.', 'I. INTRODUCTION': "General relativity in its classical form is the simplest theory to describe gravitational and cosmological scenarios. Several objects can be predicted within this framework, including black holes, wormholes, neutron stars, and interesting phenomena such as gravitational waves and light deviation in the presence of a strong field, among others [1-6]. The best-known spherically symmetric black hole solution is the Schwarzschild solution. This solution is quite simple, as it is described solely by the mass of the black hole and has only one event horizon, which is a non-return surface, and one physical singularity, a point where geodesics are interrupted. Regular black holes emerge as alternatives that stand out compared to typical black holes due to the absence of singularities within their interiors, with the first being proposed by Bardeen in 1968 [7]. Later, Beato and Garcia showed that the matter content associated with the Bardeen metric that solved Einstein's equations was nonlinear electrodynamics [8]. Usually, static black holes with regularized centers can always have nonlinear electrodynamics as their source of matter, with the sources being either electric or magnetic [9-12]. The same is not necessarily true for rotating solutions [13]. \nRecently, Simpson and Visser used the Schwarzschild metric with a regularization procedure, r 2 → r 2 + a 2 , where a is the regularization parameter, to obtain the so-called black-bounce solutions [14]. This regularization method essentially removes the point where the singularity existed, along with its surrounding neighborhood, thereby creating a sort of wormhole within a black hole. These spacetimes can be classified as two-way traversable wormholes ( a > 2 m ), one-way wormholes ( a = 2 m ), and regular black holes with symmetric horizons ( a < 2 m ), where a represents the throat of the wormhole. In the case a = 2 m , the event horizon coincides with the position of the wormhole's throat, and in this situation, we have what is called a black throat. In works [15, 16], it was verified that the matter content associated with the Simpson-Visser metric is a combination of nonlinear electrodynamics and a phantom scalar field, where the electromagnetic source is a magnetic charge. It is important to note that these same solutions can also be obtained by considering an electrically charged source [17]. \nFollowing Simpson and Visser's work, a wide range of scenarios have been explored. These include modifications in the area functions [18], modifications in the mass of the object making it depend on the radial coordinate [19], and a regularization of the Reissner-Nordström spacetime [20, 21]. Furthermore, there are works in the context of modified gravity [22-24], non-conservative theory [25], Braneworld [26], and black strings [27-29]. This class of solutions has also been extended to spherically symmetric and stationary spacetimes [20, 28, 30, 31], as well as to the study of light deflection and gravitational lensing effects [32-36]. In the mass limit tending to zero, the Simpson-Visser spacetime transforms into the topologically charged Ellis-Bronnikov spacetime, and in this context, quantum systems were explored [37-42]. \nConventionally, in general relativity, solutions for regular black holes and traversable wormholes require that energy conditions be violated, thus requiring exotic matter. For the case of regular black holes with only nonlinear electrodynamics, the strong energy condition is always violated. However, there are also solutions that violate other energy conditions [43]. In the case of the scalar field, a canonical scalar field minimally coupled with gravitational theory is not sufficient to generate regular solutions, necessitating the presence of a phantom scalar field, which violates the null energy condition [44]. \nThe initial proposal of this work was to investigate possible modifications in the energy conditions, for example, arising from nonlinear electrodynamics incorporated into a k-essence theory. As we will see below, the electromagnetic sector is not modified by the presence of the k-essence function, however, the phantom and potential scalar field configurations are altered. \nThe paper is structured as follows: Section II establishes the theoretical basis of the k-essence model in conjunction with nonlinear electrodynamics, including the derivation of the equations of motion. In Sections III and IV, we apply the methodology to two specific models and derive the physical quantities for the k-essence configurations of n = 1 / 3 and n = 1 / 5 . In Section V, we derive the energy conditions generically for each configuration, n = 1 / 3 and n = 1 / 5 . In Section VI, we analyze the power conditions for each model. Finally, the conclusions are presented in Section VII.", 'II. GENERAL RELATIONS': "The k-essence theories are characterized by the presence of scalar fields whose kinetic terms are introduced in a noncanonical way and have been recently explored in the context of wormhole solutions [45, 46]. Now, we will introduce a term representing nonlinear electrodynamics. Therefore, consider the model described by the action below: \nS = ∫ d 4 x √ -g [ R -F ( X,ϕ ) + L ( f )] , (1) \nwhere R is the Ricci scalar, X = ηϕ ; ρ ϕ ; ρ denotes the kinetic term, and L ( f ) represents the contribution from electromagnetism, where f = H µν H µν 4 , with H µν = ∂ µ A ν -∂ ν A µ being the electromagnetic tensor, and A µ being the four-dimensional vector potential. Though k-essence models can include a potential term and non-trivial couplings, the scalar sector is typically minimally coupled to gravity. The parameter η = ± 1 is introduced to avoid imaginary terms in the kinetic expression X . By selecting different forms for the function F ( X,ϕ ) , k-essence theories can describe both phantom [47-50] and standard scalar fields. \nBy varying the above action 1 with respect to the fields and the metric tensor, we obtain the following equations of motion: \nG µν = T ϕ µν + T EM µν , (2) \nη ∇ α ( F X ϕ α ) -1 2 F ϕ = 0 , (3) \n∇ µ [ L f H µν ] = 0 , (4) \nwhere G µν is the Einstein tensor, T ϕ µν and T EM µν are the stress-energy tensors of the scalar field ϕ and the electromagnetic field, respectively, F X = ∂F ∂X , F ϕ = ∂F ∂ϕ , ϕ µ = ∂ µ ϕ , and L f = ∂L ∂f . \nThe energy-momentum tensor for each of the fields is defined by: \nT ϕ µν = -F 2 g µν + ηF X ∇ µ ϕ ∇ ν ϕ, (5) \nT EM µν = L ( f ) 2 g µν -L f 2 H µ α H να . (6) \nThe line element representing the most general spherically symmetric and static spacetime takes the form: \nds 2 = e 2 γ ( u ) dt 2 -e 2 α ( u ) du 2 -e 2 β ( u ) d Ω 2 , (7) \nwhere u is an arbitrary radial coordinate and d Ω 2 = dθ 2 + sin 2 θdφ 2 is the area element. Since our spacetime is spherically symmetric and static, we can assume that the scalar potential is a function only of the radial coordinate, ϕ = ϕ ( u ) . \nFor our purposes, we are only interested in possible magnetically charged solutions. Thus, the non-zero component of the Maxwell-Faraday tensor is given by \nH 23 = q m sin θ, (8) \nwhere the electromagnetic field is defined by \nf ( u ) = q 2 m 2 e 4 β ( u ) , (9) \nwith q m being the magnetic charge. \nThus, we write the general equations of motion, which are the same as those contained in Refs.[45, 46, 51]. However, they are now modified by nonlinear electrodynamics. It is assumed that the function X = -ηe -2 α ( ϕ ' ) 2 is positive, which implies that η = -1 . As a result, the equations of motion take the form: \n2 ( F X e -α +2 β + γ ϕ ' ) ' -e α +2 β + γ F ϕ = 0 , (10) \nγ '' + γ ' (2 β ' + γ ' -α ' ) -e 2 α 2 ( F -XF X ) + e 2 α 2 [ L ( f ) -q 2 m L f e 4 β ] = 0 , (11) \n-e 2 α -2 β + β '' + β ' (2 β ' + γ ' -α ' ) -e 2 α 2 [ F -XF X -L ( f )] = 0 , (12) \n-e -2 β + e -2 α β ' ( β ' +2 γ ' ) -F 2 + XF X + L ( f ) 2 = 0 . (13) \nThe notation used here follows that used in reference [51]. The following coordinate transformation is defined: u = x , and the quasi-global gauge α ( u ) + γ ( u ) = 0 is employed. As a result, the line element in Eq. (7) can be expressed in the following form: \nds 2 = A ( x ) dt 2 -dx 2 A ( x ) -Σ 2 ( x ) d Ω 2 , (14) \nwhere the metric functions are defined as A ( x ) = e 2 γ = e -2 α and e β = Σ( x ) . The equations of motion defined in Eqs. (10-13) can then be rewritten in the new coordinates. Combining Eqs. (11-13), we get the following expressions: \n2 A Σ '' Σ -XF X = 0 , (15) \nA '' Σ 2 -A ( Σ 2 ) '' +2 -q 2 m L f Σ 2 = 0 , (16) \nwhere the primes now represent derivatives with respect to x . \nThe two remaining equations, Eq. (10) and Eq. (13), are rewritten in the new coordinates as \n2 \n( \nF \nX \nA \nΣ \nϕ \n) \n- \nΣ \nF \nϕ \n= 0 \n, \n(17) \n1 Σ 2 ( -1 + A ' Σ ' Σ+ A Σ ' 2 ) -F 2 + XF X + L ( f ) 2 = 0 . (18) \nIt has been established in previous works [45, 46] that pursuing black-bounce solutions solely with the kinetic term of the k-essence function is not mathematically consistent. Therefore, when constructing these new solutions, we must incorporate a scalar potential given by F ( X ) = F 0 X n -2 V ( ϕ ) , where F 0 is a constant, n is a real number, and V ( ϕ ) is the potential function. Using Eq. (15), we can derive a general expression for the scalar field that depends on both the angular metric function Σ( x ) and the metric function A ( x ) , unlike most studies, where scalar field depends solely on the angular function and its derivatives [44, 51, 52].", 'III. FIRST MODEL': "In this section, we will consider the ingredients of the black-bounce solution proposed by Simpson-Visser in Ref. [14]. Specifically, we assume that the throat radius is exactly the magnetic charge, a = q m . Therefore, the metric functions are given by: \nA ( x ) = 1 -2 m √ x 2 + q 2 m , and Σ( x ) = √ x 2 + q 2 m . (19) \nIn the subsection below, we will calculate all quantities present in the equations of motion (15-18) using metric functions Eq. (19). \nA. case n = 1 3 \nConsidering the differential equation Eq. (15) for the metric functions Eq. (19), we can obtain the phantom scalar field for the configuration n = 1 3 and the parameter η = -1 . Therefore, its expression is given by \nϕ ( x ) = ( 6 q 2 m F 0 ) 3 / 2 x 4 q 2 m Σ 4 + 3 x 8 q 4 m Σ 2 -2 mx (15 q 4 m +8 x 4 +20 x 2 q 2 m ) 15 q 6 m Σ 5 + 3 arctan ( x q m ) 8 q 5 m , (20) \nIn the same way, using Eq. (16), we can find the first electromagnetic quantity \nL f ( x ) = 6 m √ x 2 + q 2 m . (21) \nFinally, using Eqs. (17)-(18), we can find the scalar potential V ( x ) as well as the electromagnetic quantity L ( f ) . Therefore, we have: \nV ( x ) = 2 q 2 m (5Σ -8 m ) 5Σ 5 , (22) \nL ( x ) = 12 mq 2 m 5Σ 5 . (23) \nElectromagnetic quantities in general must obey the following relationship: \n∂L ∂x = ( ∂f ∂x ) L f . (24) \nIn this way, we can use the expression in Eq. (9) for the Simpson-Visser area function and then write the electromagnetic quantity L ( x ) as a function of the invariant f = H µν H µν / 4 = q 2 m 2Σ 4 . Thus, we have: \n2 \n' \n' \n2 \nL ( f ) = 24 m 4 √ 2 | f | 5 / 4 5 √ | q m | . (25) \nTo better visualize the form of the scalar potential in Eq. (22), we can use an auxiliary variable defined by the transformation ψ = arctan ( x q m ) , where considering the asymptotic limits x → ±∞ is equivalent to considering ψ →± π 2 . Thus, the potential in the new coordinates is given by: \nV ( ψ ) = 2 cos 4 ( ψ ) q 2 m -16 m cos 5 ( ψ ) 5 q 3 m . (26) \nIt was verified in Ref. [46] that regardless of the choice for the phantom scalar field configuration n = 1 / 3 , 1 / 5 , 1 / 7 , . . . , the energy conditions remain unchanged, preserving the results obtained in Ref. [46]. The objective here is to examine in the following sections whether the energy conditions associated with the electromagnetic part are modified as we vary the power of the k-essence field. \nThe asymptotic form of the scalar field Eq. (20) is written as: \nϕ ( x →∞ ) = -ϕ ( x →-∞ ) = -√ 3 / 2 (45 πq m -256 m ) 20 q 3 m , (27) \nwhere F 0 = 1 . \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 1. The scalar field, given by Eq. (20), is represented in a) for different charge values and considering regions inside and outside any possible horizon. Similarly, we have the potential, given by Eq. (21), in b), and the electromagnetic functions, given by Eq. (22) in c) and Eq. (23) in d). All these are for n = 1 / 3 setting, and we are adopting F 0 = 1 . \n<!-- image --> \nIn panel 1, we have the graphical representation of the physical quantities obtained in the model above for the configuration of n = 1 / 3 . Regarding the scalar field (1(a)), looking at its asymptotic form Eq. (27), it is clear that there is a limit for charge values at which sign inversion occurs. For instance, for x → + ∞ , the field is positive for charge values around q m ≈ 1 . 81 m . Regarding the potential (1(b)), we can verify the same behavior presented in works [45, 46], where ,for certain charge values, already within the event horizon, the curves begin to create a minimum that grows as the charge becomes more internal. Likewise, we have the behavior of the electromagnetic functions in figures (1(c)) and (1(d)). \nB. case n = 1 5 \nConsidering the same procedure as in the previous section, we can explicitly obtain the same quantities, but now for a phantom scalar field configuration n = 1 / 5 . Therefore, we have: \nϕ ( x ) = ( 10 q 2 m F 0 ) 5 / 2 35 arctan ( x q m ) 128 q 9 m + 93 x 128 q 2 m Σ 8 + 35 x 7 128 q 8 m Σ 8 + 385 x 5 384 q 6 m Σ 8 + 511 x 3 384 q 4 m Σ 8 + 63 m 2 arctan ( x q m ) 64 q 11 m + + ( 10 q 2 m F 0 ) 5 / 2 [ 193 m 2 x 64 q 2 m Σ 10 + 63 m 2 x 9 64 q 10 m Σ 10 + 147 m 2 x 7 32 q 8 m Σ 10 + 42 m 2 x 5 5 q 6 m Σ 8 + 237 m 2 x 3 32 q 4 m Σ 10 ] + -( 10 q 2 m F 0 ) 5 / 2 [ 4 mx q 2 m Σ 9 + 512 mx 9 315 q 10 m Σ 9 + 256 mx 7 35 q 8 m Σ 9 + 64 mx 5 5 q 6 m Σ 9 + 32 mx 3 3 q 4 m Σ 9 ] , (28) \nV ( x ) = 4 q 2 m Σ 4 -36 mq 2 m 5Σ 5 , (29) \nL ( x ) = 12 mq 2 m 5Σ 5 . (30) \nAn observation to be made is that the electromagnetic quantity L f , obtained by Eq. (15), does not depend on the scalar field and is therefore the same as that obtained in the previous section III A. Likewise, we can write the electromagnetic quantity L ( f ) in terms of the invariant as done in the previous section. Therefore, we have \nL ( f ) = 24 m 4 √ 2 | f | 5 / 4 5 √ | q m | . (31) \nWe can also express the scalar potential for this field configuration using the most appropriate variables as done in the previous section. Thus, we have \nV ( ψ ) = 4 cos 4 ( ψ ) q 2 m -36 m cos 5 ( ψ ) 5 q 3 m . (32) \nIn panel 2, we have the graphical representation of the scalar field, Eq. (28), and potential, Eq. (29), for the configuration n = 1 / 5 . As illustrated in figure 2(a), in the limit of x → ∞ the scalar field grows for increasingly larger charge values and tends to zero for small values of the radial coordinate. The behavior of the scalar potential is qualitatively the same as the previous configuration (III A), changing only by a numerical shift (Figure 2(b)).", 'IV. SECOND MODEL': 'In this section, we will work with the metric functions corresponding to a Bardeen-type spacetime as investigated in Refs. [19, 52]. In this model, as in the previous case, the wormhole throat coincides with the magnetic charge q m = a . Therefore, we have: \nA ( x ) = 1 -2 mx 2 ( x 2 + q 2 m ) 3 / 2 , and Σ( x ) = √ x 2 + q 2 m . (33) \n<!-- image --> \nFigure 2. Scalar field, Eq. (28), in panel a) and potential, Eq. (29), in panel b) for different charge values and considering regions inside and outside any possible horizon, for the configuration n = 1 / 5 . Here, we define F 0 = 1 . \n<!-- image --> \nAs discussed in Refs. [19, 52], this space-time has four horizons, two event horizons and two Cauchy horizons. The extreme horizon is obtained for q m = q ext = 4 m/ 3 √ 3 and the others are symmetric and labeled by ( -x + , -x C , x C , x + ) . Where x + is the event horizon and x C is the Cauchy horizon. \nA. case n = 1 3 \nFollowing the same steps carried out in section III A, we can obtain all physical quantities of interest for this space-time. Therefore, we have \nϕ ( x ) = ( 6 q 2 m F 0 ) 3 / 2 5 x 8 q 2 m Σ 4 + 3 x 3 8 q 4 m Σ 4 -16 mx 7 105 q 6 m Σ 7 -2 mx 3 q 4 m 3 q 6 m Σ 7 -8 mx 5 15 q 4 m Σ 7 + 3 arctan ( x q m ) 8 q 5 m , (34) \nL f ( x ) = 22 mx 2 Σ 3 -4 mx 4 q 2 m Σ 3 -4 mq 2 m Σ 3 + 4 mx 2 q 2 m Σ , (35) \nV ( x ) = 2 q 2 m Σ 4 -32 mq 4 m 35Σ 7 -16 mx 2 q 2 m 5Σ 7 , (36) \nL ( x ) = 64 mq 4 m 35Σ 7 + 52 mx 2 q 2 m 5Σ 7 . (37) \nWriting the electromagnetic quantity L ( x ) in terms of the invariant f and considering the appropriate change of variables for the scalar potential, we can rewrite them as: \nL ( f ) = 52 m (2 | f | ) 5 / 4 5 √ | q m | -60 m √ | q m | (2 | f | ) 7 / 4 7 , (38) \nV ( ψ ) = 2 cos 4 ( ψ ) q 2 m -32 m cos 7 ( ψ ) 35 q 3 m -4 m cos 3 ( ψ ) sin 2 ( ψ ) 5 q 3 m . (39) \nThe asymptotic form for the scalar field Eq. (34) is written as: \nϕ ( x →∞ ) = -ϕ ( x →-∞ ) = √ 3 / 2 (315 πq m -256 m ) 140 q 3 m , (40) \nwhere F 0 = 1 . \nIn panel 3, we have a graphical representation of the physical quantities obtained above for the configuration n = 1 / 3 for Bardeen-type spacetime. In figure 3(a), we show the behavior of the scalar field, Eq. (34), in all regions of the \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 3. Scalar field Eq. (34) for some charge values and considering regions inside and outside any possible horizon is represented in a). Similarly, we have the potential Eq. (36) b) and the electromagnetic functions Eq. (35) c) and Eq. (37) d). All for n = 1 / 3 setting, adopting F 0 = 1 . \n<!-- image --> \nspacetime, considering different charge values. For some values of charge, q m > q ext = 4 m 3 √ 3 , there is no event horizon. Note in the asymptotic form of the scalar field Eq. (40) that for a certain limiting value of charge within the horizon, around q m ≈ 0 . 26 m , the scalar field tends to reverse sign. \nFor the scalar potential 3(b), we observe a behavior similar to the cases previously investigated. For regions outside the external horizon, the potential behaves like a barrier. For regions within the horizon, the potential starts to exhibit two symmetrical minima that tend to grow as the charge becomes more internal to extreme radius. The electromagnetic quantities Eq. (35) and Eq. (37) exhibit behavior similar to that presented in section III A and can be visualized in figures 3(c) and 3(d). \nB. case n = 1 5 \nFollowing the same steps as in the previous section IV A, we can obtain the same quantities, but now for the phantom field power n = 1 / 5 . In this way, we have: \nϕ ( x ) = ( 10 q 2 m F 0 ) 5 / 2 35 arctan ( x q m ) 128 q 9 m + 93 x 128 q 2 m Σ 8 + 35 x 7 128 q 8 m Σ 8 + 385 x 5 384 q 6 m Σ 8 + 511 x 3 384 q 4 m Σ 8 + 9 m 2 arctan ( x q m ) 512 q 11 m + + ( 10 q 2 m F 0 ) 5 / 2 [ -9 m 2 q 2 m x 512Σ 14 + 1199 m 2 x 5 2560 q 2 m Σ 14 -15 m 2 x 3 128Σ 14 + 9 m 2 x 13 512 q 10 m Σ 14 + 15 m 2 x 11 128 q 8 m Σ 14 + 849 m 2 x 9 2560 q 6 m Σ 14 + 18 m 2 x 7 35 q 4 m Σ 14 ] + \n-( 10 q 2 m F 0 ) 5 / 2 [ 4 mx 3 3 q 2 m Σ 11 + 512 mx 11 3465 q 10 m Σ 11 + 256 mx 9 315 q 8 m Σ 11 + 64 mx 7 35 q 6 m Σ 11 + 32 mx 5 15 q 4 m Σ 11 ] , (41) \nV ( x ) = 8 mq 2 m 5Σ 5 + 4 q 2 m Σ 4 -88 mq 4 m 35Σ 7 -44 mx 2 q 2 m 5Σ 7 , (42) \nL ( x ) = 68 mx 2 q 2 m 5Σ 7 + 176 mq 4 m 35Σ 7 -16 mq 2 m 5Σ 5 . (43) \nFinally, by expressing the potential Eq. (42) in the new coordinates, as well as the electromagnetic quantity Eq. (43) in terms of the invariant f , we have: \nV ( ψ ) = 8 m cos 5 ( ψ ) 5 q 3 m + 4 cos 4 ( ψ ) q 2 m -88 m cos 7 ( ψ ) 35 q 3 m -44 m cos 5 ( ψ ) sin 2 ( ψ ) 5 q 3 m , (44) \nL ( f ) = 52 m (2 | f | ) 5 / 4 5 √ | q m | -60 m √ | q m | (2 | f | ) 7 / 4 7 . (45) \n<!-- image --> \nFigure 4. Scalar field, Eq. (41), in panel a) and potential, Eq. (42), in panel b) with different charge values and considering regions inside and outside any possible horizon, for the configuration n = 1 / 5 . Here, we define F 0 = 1 . \n<!-- image --> \nIn panel 4, we have the graphical representation for the scalar field Eq. (41) and potential Eq. (42) with a configuration n = 1 / 5 . Qualitatively, the scalar field 4(a) and the potential 4(b) exhibit similar behavior to the previously investigated cases, with only minor numerical shifts. \nAn important observation is that the physical quantities associated with electromagnetism are the same regardless of the power of the k-essence field. Surprisingly, these quantities are the same as those obtained for the canonical case [52]. Logically, these electromagnetic functions, which remain unchanged about the k-essence field, are specific to each model of interest. As is the case with the functions Eqs. (25) and (31) which refer to the Simpson-Visser model and are the same expressions obtained for the canonical scalar field in [52]. The other electromagnetic quantity Eq. (21) is also the same for any power of the k-essence field and coincides with the canonical case. This fact occurs because it does not depend on the scalar field and is obtained directly through Eq. (16). \nThe same analysis can be extended to the second model (Bardeen type) when comparing the functions Eqs. (38) and (45) as well as the expression Eq. (35) which are the same found for the canonical scalar field [52]. \nThe fact that the electromagnetic quantities remain unchanged is compensated by changes in the phantom scalar field and scalar potential, as a kind of counterterms.', 'V. GENERAL ENERGY CONDITIONS': 'Weknow that black bounce solutions violate the null energy condition, which must violate all other energy conditions [14, 53]. In this section, we will make the generic construction for the energy conditions, that is, just considering the \nstatic and spherically symmetric metric Eq. (14), as well as the stress-energy tensor defined in Eqs. (5-6). The energy conditions are defined by the combination of energy density and radial and tangential pressures as 1 \nNEC ϕ,EM 1 , 2 = WEC ϕ,EM 1 , 2 = SEC ϕ,EM 1 , 2 ⇐⇒ ρ ϕ,EM + p ϕ,EM 1 , 2 ≥ 0 , (46) \nSEC ϕ,EM 3 ⇐⇒ ρ ϕ,EM + p ϕ,EM 1 +2 p ϕ,EM 2 ≥ 0 , (47) \nDEC ϕ,EM 1 , 2 ⇐⇒ ρ ϕ,EM + p ϕ,EM 1 , 2 ≥ 0 and ρ ϕ,EM -p ϕ,EM 1 , 2 ≥ 0 , (48) \nDEC ϕ,EM 3 = WEC ϕ,EM 3 ⇐⇒ ρ ϕ,EM ≥ 0 , (49) \nwith the subindices ϕ and EM in the energy conditions above, representing the scalar field and the electromagnetic field. \nT µ ν = diag [ ρ ϕ,EM , -p ϕ,EM 1 , -p ϕ,EM 2 , -p ϕ,EM 2 ] , (50) \nwhere the energy density is denoted by ρ ϕ,EM , while p ϕ,EM 1 and p ϕ,EM 2 represent the radial and tangential pressures, respectively. This expression for the stress-energy tensor above Eq. (50) refers to the part outside the event horizon A > 0 . Within the event A < 0 horizon we have: \nT µ ν = diag [ -p ϕ,EM 1 , ρ ϕ,EM , -p ϕ,EM 2 , -p ϕ,EM 2 ] . (51)', 'A. Energy conditions for n = 1 / 3': "As the energy conditions depend directly on the scalar field, these components of the stress-energy tensor are to the field configuration where n = 1 / 3 . For the region A > 0 , the components are defined as: \nρ ϕ = -3 A Σ '' Σ + V ( x ) , ρ EM = L ( x ) 2 , (52) \np ϕ 1 = A Σ '' Σ -V ( x ) , p EM 1 = -L ( x ) 2 , (53) \np ϕ 2 = p ϕ 3 = 3 A Σ '' Σ -V ( x ) , p EM 2 = p EM 3 = -L ( x ) 2 + q 2 m L f ( x ) 2Σ 4 . (54) \nThe components of the stress-energy tensor for the region A < 0 are defined by: \nρ ϕ = -A Σ '' Σ + V ( x ) , ρ EM = L ( x ) 2 , (55) \np ϕ 1 = 3 A Σ '' Σ -V ( x ) , p EM 1 = -L ( x ) 2 , (56) \np ϕ 2 = p ϕ 3 = 3 A Σ '' Σ -V ( x ) , p EM 2 = p EM 3 = -L ( x ) 2 + q 2 m L f ( x ) 2Σ 4 . (57) \nThe energy conditions for the region outside the event horizon A > 0 are: \nDEC \nϕ \n2 \n= \n- \nNEC ϕ 1 = WEC ϕ 1 = SEC ϕ 1 = -2 A Σ '' Σ ≥ 0 , (58) \nNEC ϕ 2 = WEC ϕ 2 = SEC ϕ 2 = 0 , (59) \nSEC ϕ 3 = 4 A Σ '' Σ -2 V ( x ) ≥ 0 , (60) \nDEC ϕ 1 = -4 A Σ '' Σ +2 V ( x ) ≥ 0 , (61) \nDEC ϕ 2 = -6 A Σ '' Σ +2 V ( x ) ≥ 0 , (62) \nDEC ϕ 3 = WEC ϕ 3 = -3 A Σ '' Σ + V ( x ) ≥ 0 . (63) \nLikewise, for the electromagnetic part \nNEC EM 1 = WEC EM 1 = SEC EM 1 = 0 , (64) \nNEC EM 2 = WEC EM 2 = SEC EM 2 = q 2 m L f ( x ) 2Σ 4 ≥ 0 , (65) \nSEC EM 3 = -L ( x ) + q 2 m L f ( x ) Σ 4 ≥ 0 , (66) \nDEC \nEM \n1 \n= \nL \n( \nx \n) \n≥ \n0 \n, \n(67) \nDEC EM 2 = L ( x ) -q 2 m L f ( x ) 2Σ 4 ≥ 0 , (68) \nDEC EM 3 = WEC EM 3 = L ( x ) 2 ≥ 0 . (69) \nPerforming the same analysis now within the horizon A < 0 , we have: \nNEC ϕ 1 = WEC ϕ 1 = SEC ϕ 1 = 2 A Σ '' Σ ≥ 0 , (70) \nNEC ϕ 2 = WEC ϕ 2 = SEC ϕ 2 = 2 A Σ '' Σ ≥ 0 , (71) \nSEC ϕ 3 = 8 A Σ '' Σ -2 V ( x ) ≥ 0 , (72) \nDEC ϕ 1 = -4 A Σ '' Σ +2 V ( x ) ≥ 0 , (73) \n4 \nA \nΣ \nΣ \n+2 \nV \n( \nx \n) \n≥ \n0 \n, \n(74) \nDEC ϕ 3 = WEC ϕ 3 = -A Σ '' Σ + V ( x ) ≥ 0 . (75) \nThe electromagnetic part within the horizon is the same expressions obtained above in Eqs. (64-69).", 'B. Energy conditions for n = 1 / 5': "The components of the stress-energy tensor for the region outside the event horizon A > 0 for the scalar field configuration in question are defined as \nρ ϕ = -5 A Σ '' Σ + V ( x ) , ρ EM = L ( x ) 2 , (76) \np ϕ 1 = 3 A Σ '' Σ -V ( x ) , p EM 1 = -L ( x ) 2 , (77) \np ϕ 2 = p ϕ 3 = 5 A Σ '' Σ -V ( x ) , p EM 2 = p EM 3 = -L ( x ) 2 + q 2 m L f ( x ) 2Σ 4 . (78) \n'' \nLikewise for the internal region A < 0 \nρ ϕ = -3 A Σ '' Σ + V ( x ) , ρ EM = L ( x ) 2 , (79) \np ϕ 1 = 5 A Σ '' Σ -V ( x ) , p EM 1 = -L ( x ) 2 , (80) \np ϕ 2 = p ϕ 3 = 5 A Σ '' Σ -V ( x ) , p EM 2 = p EM 3 = -L ( x ) 2 + q 2 m L f ( x ) 2Σ 4 . (81) \nThe energy conditions external to the horizon A > 0 for this field configuration are given by \nNEC ϕ 1 = WEC ϕ 1 = SEC ϕ 1 = -2 A Σ '' Σ ≥ 0 , (82) \nNEC ϕ 2 = WEC ϕ 2 = SEC ϕ 2 = 0 , (83) \nSEC ϕ 3 = 8 A Σ '' Σ -2 V ( x ) ≥ 0 , (84) \nDEC ϕ 1 = -8 A Σ '' Σ +2 V ( x ) ≥ 0 , (85) \nDEC ϕ 2 = -10 A Σ '' Σ +2 V ( x ) ≥ 0 , (86) \nDEC ϕ 3 = WEC ϕ 3 = -5 A Σ '' Σ + V ( x ) ≥ 0 . (87) \nLikewise, for the electromagnetic part \nNEC EM 1 = WEC EM 1 = SEC EM 1 = 0 , (88) \nNEC EM 2 = WEC EM 2 = SEC EM 2 = q 2 m L f ( x ) 2Σ 4 ≥ 0 , (89) \nSEC EM 3 = -L ( x ) + q 2 m L f ( x ) Σ 4 ≥ 0 , (90) \nDEC EM 1 = L ( x ) ≥ 0 , (91) \nDEC EM 2 = L ( x ) -q 2 m L f ( x ) 2Σ 4 ≥ 0 , (92) \nDEC EM 3 = WEC EM 3 = L ( x ) 2 ≥ 0 . (93) \nThe internal energy conditions at the horizon A < 0 for this field configuration are given by \nNEC ϕ 1 = WEC ϕ 1 = SEC ϕ 1 = 2 A Σ '' Σ ≥ 0 , (94) \nNEC ϕ 2 = WEC ϕ 2 = SEC ϕ 2 = 2 A Σ '' Σ ≥ 0 , (95) \nSEC ϕ 3 = 12 A Σ '' Σ -2 V ( x ) ≥ 0 , (96) \nDEC ϕ 1 = -8 A Σ '' Σ +2 V ( x ) ≥ 0 , (97) \nDEC ϕ 2 = -8 A Σ '' Σ +2 V ( x ) ≥ 0 , (98) \nDEC ϕ 3 = WEC ϕ 3 = -3 A Σ '' Σ + V ( x ) ≥ 0 . (99) \nThe electromagnetic part within the horizon is the same expressions obtained above in Eqs. (88-93). \nAnalyzing the energy conditions in a generic way, we can emphasize that the relations obtained above in relation to the phantom scalar field remain unchanged for higher powers of the k-essence field n = 1 / 5 , 1 / 7 , 1 / 9 , . . . , and it is sufficient to analyze the lowest power case n = 1 / 3 for the specific model of interest. This result corroborates the analyses carried out in [46].", 'A. Simpson-Visser model': 'The energy conditions for the scalar field and electromagnetism in the region external to the event horizon A > 0 and considering the n = 1 / 3 configuration, are set out explicitly below, where the quantities used will be used Eqs. (20-23) \nNEC ϕ 1 = -2 q 2 m ( √ q 2 m + x 2 -2 m ) ( q 2 m + x 2 ) 5 / 2 ≥ 0 , SEC ϕ 3 = -8 mq 2 m 5 ( q 2 m + x 2 ) 5 / 2 ≥ 0 , (100) \nDEC ϕ 1 = 8 mq 2 m 5 ( q 2 m + x 2 ) 5 / 2 ≥ 0 , DEC ϕ 2 = -2 q 2 m ( 5 √ x 2 + q 2 m -14 m ) 5( x 2 + q 2 m ) 5 / 2 ≥ 0 , (101) \nDEC ϕ 3 = WEC ϕ 3 = -q 2 m ( 5 √ x 2 + q 2 m -14 m ) 5( x 2 + q 2 m ) 5 / 2 ≥ 0 . (102) \nNEC EM 2 = 3 mq 2 m ( q 2 m + x 2 ) 5 / 2 ≥ 0 , SEC EM 3 = 18 mq 2 m 5 ( q 2 m + x 2 ) 5 / 2 ≥ 0 , (103) \nDEC EM 1 = 12 mq 2 m 5 ( q 2 m + x 2 ) 5 / 2 ≥ 0 , DEC EM 2 = -3 mq 2 m 5 ( q 2 m + x 2 ) 5 / 2 ≥ 0 , (104) \nDEC EM 3 = WEC EM 3 = 6 mq 2 m 5 ( q 2 m + x 2 ) 5 / 2 ≥ 0 . (105) \nAnalyzing the energy conditions for the scalar field outside the event horizon ( A > 0 ), we observe that the null energy condition NEC ϕ 1 Eq. (58) is violated, implying a violation of the dominant energy condition DEC ϕ 1 . All other energy conditions for the scalar field are violated except for NEC ϕ 2 Eq. (59). Regarding the energy conditions of the electromagnetic part, all of them are satisfied everywhere, except for the dominant energy condition DEC EM 2 Eq. (104). \nThe energy conditions for the region internal to the event horizon ( A < 0 ) explicitly for the model in question are defined by \nNEC ϕ 1 = NEC ϕ 2 = 2 q 2 m ( √ q 2 m + x 2 -2 m ) ( q 2 + x 2 ) 5 / 2 ≥ 0 , SEC ϕ 3 = 4 q 2 m ( 5 √ x 2 + q 2 m -12 m ) 5( x 2 + q 2 m ) 5 / 2 ≥ 0 , \nDEC ϕ 1 = DEC ϕ 2 = 8 mq 2 m 5 ( q 2 + x 2 ) 5 / 2 ≥ 0 , DEC ϕ 3 = WEC ϕ 3 = q 2 m ( 5 √ x 2 + q 2 m -6 m ) 5( x 2 + q 2 m ) 5 / 2 ≥ 0 . \nm (106) m (107) \nNote that the electromagnetic energy conditions within the event horizon are the same as those defined for the outer region Eqs. (103-105). \nReexamining the energy conditions, we observe that the null energy conditions NEC ϕ 1 = NEC ϕ 2 , Eq. (106), are violated within the event horizon. Specifically, focusing on the dominant energy conditions DEC ϕ 1 = DEC ϕ 2 , Eq. (107), they are not violated, however, since these conditions include the null energy conditions, they are also violated. The strong energy condition SEC ϕ 3 , Eq. (106), is always violated within the horizon, as well as the dominant energy condition DEC ϕ 3 , Eq. (107).', 'B. Bardeen type model': 'Replacing the metric functions in the energy conditions, in the region where A ( x ) > 0 , Eqs. (82-87), we have: \nNEC ϕ 1 = -2 q 2 m ( ( q 2 m + x 2 ) 3 / 2 -2 mx 2 ) ( q 2 m + x 2 ) 7 / 2 ≥ 0 , SEC ϕ 3 = 8 mq 2 m ( 8 q 2 m -7 x 2 ) 35 ( q 2 m + x 2 ) 7 / 2 ≥ 0 , (108) \nDEC ϕ 1 = 8 mq 2 m ( 7 x 2 -8 q 2 m ) 35 ( q 2 m + x 2 ) 7 / 2 ≥ 0 , DEC ϕ 2 = -2 q 2 m ( 32 mq 2 m -98 mx 2 +35 ( q 2 m + x 2 ) 3 / 2 ) 35 ( q 2 m + x 2 ) 7 / 2 ≥ 0 , (109) \nDEC ϕ 3 = WEC ϕ 3 = -q 2 m ( 32 mq 2 m -98 mx 2 +35 ( q 2 m + x 2 ) 3 / 2 ) 35 ( q 2 m + x 2 ) 7 / 2 ≥ 0 . (110) \nFor the electromagnetic part: \nNEC EM 2 = m ( 13 q 2 m x 2 -2 q 4 m ) ( q 2 m + x 2 ) 7 / 2 ≥ 0 , SEC EM 3 = 6 mq 2 m ( 91 x 2 -34 q 2 m ) 35 ( q 2 m + x 2 ) 7 / 2 ≥ 0 , (111) \nDEC EM 1 = 4 mq 2 m ( 16 q 2 m +91 x 2 ) 35 ( q 2 m + x 2 ) 7 / 2 ≥ 0 , DEC EM 2 = mq 2 m ( 134 q 2 m -91 x 2 ) 35 ( q 2 m + x 2 ) 7 / 2 ≥ 0 , (112) \nDEC EM 3 = WEC EM 3 = 2 mq 2 m ( 16 q 2 m +91 x 2 ) 35 ( q 2 m + x 2 ) 7 / 2 ≥ 0 . (113) \nThe null energy condition for the scalar field NEC ϕ 1 , Eq. (108), is clearly violated outside the horizon since A > 0 . Therefore, the associated dominant energy condition DEC ϕ 1 Eq. (109) is also violated. The secondary null energy condition NEC ϕ 2 , Eq. (59), is satisfied, while the dominant energy conditions DEC ϕ 2 , Eq. (109), and DEC ϕ 3 , Eq. (110), are violated. The strong energy condition SEC ϕ 3 , Eq. (108), is violated for throat radii between -√ 8 / 7 q m > x > √ 8 / 7 q m . \nFor the electromagnetic part, the main null energy condition NEC EM 1 Eq. (64) is satisfied within the horizon, as well as the dominant energy conditions DEC EM 1 Eq. (112) and DEC EM 3 Eq. (113). The secondary null energy condition NEC EM 2 Eq. (111) is violated for throat radii between -√ 2 / 13 q m < x < √ 2 / 13 q m . Likewise, the dominant energy condition DEC EM 2 Eq. (112) is violated for throat radii between -√ 134 / 91 q m > x > √ 134 / 91 q m . Finally, the strong energy condition SEC EM 3 Eq. (111) is violated for throat radii between -√ 34 / 91 q m < x < √ 34 / 91 q m . \nThe energy conditions for the region interior to the event horizon are given by \nNEC ϕ 1 = NEC ϕ 2 = 2 q 2 m ( ( x 2 + q 2 m ) 3 / 2 -2 mx 2 ) ( x 2 + q 2 m ) 7 / 2 ≥ 0 , (114) \nSEC ϕ 3 = 4 q 2 m ( 16 mq 2 m -84 mx 2 +35 ( q 2 m + x 2 ) 3 / 2 ) 35 ( q 2 m + x 2 ) 7 / 2 ≥ 0 , (115) \nDEC ϕ 1 = DEC ϕ 2 = 8 mq 2 m ( 7 x 2 -8 q 2 m ) 35 ( q 2 m + x 2 ) 7 / 2 ≥ 0 , (116) \nDEC ϕ 3 = WEC ϕ 3 = q 2 m ( -32 mq 2 m -42 mx 2 +35 ( q 2 m + x 2 ) 3 / 2 ) 35 ( q 2 m + x 2 ) 7 / 2 ≥ 0 . (117) \nIn the analysis of the energy conditions within the event horizon, A < 0 , clearly the null energy conditions NEC ϕ 1 = NEC ϕ 2 Eq. (114) are violated, therefore the dominant energy conditions are also violated DEC ϕ 1 = DEC ϕ 2 Eq. (116). Finally, the strong energy condition SEC ϕ 3 Eq. (115) is not violated within the horizon while the dominant energy condition DEC ϕ 3 Eq. (117) is violated.', 'VII. CONCLUSION': 'In the present work, we start from an action that describes the k-essence theory with a scalar field assuming a power form. This theory has been used to investigate black-bounce solutions [45] and generalizations for different \nscalar field configurations [46]. We extend this framework by introducing non-linear electrodynamics [52] to explore possible charged black-bounce solutions for scalar field strengths differing from the canonical case n = 1 . Specifically, we construct magnetic solutions such that the throat of the wormhole coincides with the magnetic charge a = q m . \nFrom the equations of motion, Eqs. (15-18), we can observe that the electromagnetic function L f , contained in the equation Eq. (16), does not depend on the form of the scalar field but rather only of the metric functions A ( x ) and Σ( x ) . Which leads to the conclusion that this function is the same as obtained for the canonical scalar field depending only on the chosen model. \nWe analytically derive all the functions involved for the Simpson-Visser models (Section III) and for a Bardeen-type solution (Section IV) for k-essence configurations with n = 1 / 3 and n = 1 / 5 . We can verify that for each model of specific form the electromagnetic functions L f ( x ) and L ( x ) do not change with the variation in the power of the k-essence field. In fact, these functions are the same as those obtained in the canonical case and investigated in [15, 52]. This behavior implies that the modifications in the scalar field due to k-essence are counterbalanced by the potential, keeping L f ( x ) and L ( x ) unchanged. This behavior can be extended to the other powers of the phantom scalar field n = 1 / 3 , 1 / 5 , 1 / 7 , . . . . \nWe graphically represent the scalar field, potential, and electromagnetic functions for each of the models studied. Qualitatively, the electromagnetic functions L f ( x ) and L ( x ) for both the Simpson-Visser model (Section 1) and the Bardeen model (Section 3) exhibit similar behavior. Regarding the behavior of the scalar field for both models in this configuration, it tends to invert its sign asymptotically x → ±∞ for some charge values, as illustrated in Eqs. (27) and (40). The potential for both models and configurations tends to behave similar to a potential barrier for regions outside the horizon and creates minima as it becomes more internal to the horizon. \nFinally, we analyzed the energy conditions for the scalar and electromagnetic fields for each of the models investigated. As previously observed in [46], the energy conditions for the scalar field do not change with the power of the k-essence, so it suffices to analyze only the case of the lower power n = 1 / 3 . Thus, in the Simpson-visser model, VIA, the null energy condition NEC ϕ 1 and its associated dominant energy condition DEC ϕ 1 are violated outside the horizon, A > 0 . As well as all other energy conditions, except for NEC ϕ 2 . For within the event horizon, A < 0 , all energy conditions are violated, Eqs. (106-107). Regarding the analysis of electromagnetic energy conditions for the Simpson-Visser model, all conditions are satisfied everywhere, except DEC EM 2 , Eq. (104). \nIn the analysis of the energy conditions of the second model VI B, the energy conditions NEC ϕ 1 , DEC ϕ 1 , DEC ϕ 2 , and DEC ϕ 3 are violated where A > 0 . The null energy condition NEC ϕ 2 is satisfied and SEC ϕ 3 , Eq. (108), is conditionally violated. 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2024arXiv240913391R | Prompt emission of GRB 230812B stands out as one of the most luminous events observed by both the FermiGBM and LAT. Prompt emission spectral analysis both timeintegrated and resolved of this burst supports an additional thermal component together with a nonthermal indicating the hybrid jet composition. The spectral parameters alpha Ep and kT of the bestfit BandBlackbody model show a tacking behaviour with the intensity. Further the low energy afterglow emission is consistent with the synchrotron emission from the external forward shock in the ISM medium. LAT detected very high energy emission VHE deviating from the synchrotron mechanism possibly originating from the Lorentz boosting of prompt emission photons by accelerated electrons in the external shock via Inverse Compton IC or Synchrotron Self Compton SSC emission mechanisms. The comparison of the prompt and afterglow emission properties of this burst revealed that unlike the bright prompt emission the afterglow of GRB 230812B is fainter than the other SNdetected bright bursts GRB 130427A and GRB 171010A at a similar redshift. | 2024-09-01T00:00:00Z | ['arXiv:2409.13391', '2024arXiv240913391R', '10.48550/arXiv.2409.13391'] | ['Astrophysics - High Energy Astrophysical Phenomena'] | Prompt and afterglow analysis of the FermiLAT detected GRB 230812B | 2,024 | 175 | 0.49 | ['EPRINT_HTML', 'EPRINT_PDF'] | 0 | https://arxiv.org/pdf/2409.13391.pdf | {'PROMPT AND AFTERGLOW ANALYSIS OF THE Fermi -LAT DETECTED GRB 230812B': 'Amit K. Ror 1,2 , S. B. Pandey 1 , A. Aryan 1,3 , Sudhir Kumar 2 , and A. J. Castro-Tirado 4', 'RESUMEN': "La emisi'on temprana de GRB 230812B se destaca como uno de los eventos m'as luminosos observados por instrumentos del sat'elite Fermi tales como GBM y LAT. El an'alisis espectral de emisi'on temprana (tanto integrado en el tiempo como en sus diversas fases temporales) de esta explosi'on respalda una componente t'ermica adicional y otra no t'ermica, lo que indica una composici'on h'ıbrida del jet (chorro). Los par'ametros espectrales 𝛼 , E 𝑝 y kT del modelo espectral de Band junto a cuerpo negro son los que proporcionan un mejor ajuste. Adem'as, las emisiones de la postluminiscencia a energ'ıas m'as bajas son consistentes con la emisi'on de sincrotr'on del choque externo que se va propagando hacia adelante barriendo el medio del medio interestelar. El instrumento LAT detect'o una emisi'on de muy alta energ'ıa (VHE) que se desv'ıa de los esperado del mecanismo de emisi'on sincrotr'on, posiblemente originada por el aumento del factor de Lorentz de los fotones de emisi'on temprana motivado por electrones acelerados en el choque externo a trav'es de mecanismos de emisi'on de Compton inverso (IC) o auto sincrotr'on Compton (SSC). La comparaci'on de las propiedades de emisi'on temprana y de la posterior de este estallido revel'o que, a diferencia de la emisi'on simult'anea (gamma) y m'as brillante, la postluminiscencia posterior de GRB 230812B es m'as d'ebil que la de los otros estallidos de rayos gamma tambi'en brillantes y asociados con una supernova (GRB 130427A y GRB 171010A) con desplazamientos al rojo similares.", 'ABSTRACT': 'Prompt emission of GRB 230812B stands out as one of the most luminous events observed by both the Fermi -GBM and LAT. Prompt emission spectral analysis (both time-integrated and resolved) of this burst supports an additional thermal component together with a non-thermal, indicating the hybrid jet composition. The spectral parameters 𝛼 , Ep, and kT of the best-fit Band+Blackbody model show a tacking behaviour with the intensity. Further, the low energy afterglow emission is consistent with the synchrotron emission from the external forward shock in the ISM medium. LAT detected very high energy emission (VHE) deviating from the synchrotron mechanism, possibly originating from the Lorentz boosting of prompt emission photons by accelerated electrons in the external shock via Inverse Compton (IC) or Synchrotron Self Compton (SSC) emission mechanisms. The comparison of the prompt and afterglow emission properties of this burst revealed that, unlike the bright prompt emission, the afterglow of GRB 230812B is fainter than the other SN-detected bright bursts (GRB 130427A and GRB 171010A) at a similar redshift. \nKey Words: Gamma ray burst - mechanism: Blackbody, Synchrotron, Synchrotron self Compton.', '1. INTRODUCTION': "Gamma-ray bursts (GRBs) are distant and powerful cosmic events characterized by two phases: A highly variable prompt emission in soft 𝛾 -rays and hard X-rays with a duration of (T90) seconds (s) to minutes, followed by a smoothly decaying afterglow emission spanning a broad temporal (hours-days) and energy ranges (radio-TeV energies; Kumar & Zhang 2015). GRBs are traditionally classified based on temporal and spectral properties into short/hard (SGRB) and long/soft (LGRBs; Kouveliotou et al. 1993). SGRBs (T90 < 2 s) come from compact binary mergers, and LGRBs (T90 > 2 s) are likely originating from a massive star collapse (Galama et al. 1998; Hjorth et al. 2003; Abbott et al. 2017). However, GRB 200826A (T90 ∼ 1.2 s from collapsar; Ahumada et al. 2021), GRB 211211A \n(T90 ∼ 50 s; Troja et al. 2022), and GRB 230307A (T90 ∼ 35 s; Levan et al. 2023) come from mergers are outlier to the traditional classification. Newly introduced classifications include intermediate bursts and ultra-long GRBs (ULGRBs, T90 > 1000 s; Mukherjee et al. 1998; Boer et al. 2015; Ror et al. 2024), broadening the classification beyond traditional LGRBs and SGRBs. \nOver the 50 years of study, the GRB prompt emission remains a subject of ongoing exploration (Pe'er 2015). Several theories describe its origin from internal shock or magnetic reconnection in the ultra-relativistic jet. The jet composition can be baryonic or magnetic, and the mechanism of prompt emission from these ultra-relativistic jets ( synchrotron , non-thermal, thermal, or hybrid) is still an open question (Zhang et al. 2018; Gupta et al. 2024). A widely employed Band function (Band et al. 1993), introduced during the BATSE era, characterizes the non-thermal spectral components of GRBs. It incorporates four parameters: (1) normalization constant, (2) low-energy spectral index ( 𝛼 ∼ -1), (3) high-energy spectral index ( 𝛽 ∼ -2), and (4) peak energy (Ep ∼ 300 keV; Preece et al. 2000). However, results of time-resolved spectral analysis from \nKaneko et al. (2006) have revealed variations, with a harder 𝛼 ∼ -0.72. Subsequent studies of 1500 time-resolved spectra from 78 bright bursts (Wang et al. 2024) revealed that the 𝛼 values from 80% of spectra crossing the limit (-3/2,-2/3) are due to the fast and slow cooling limits from synchrotron emission mechanism known as synchrotron line of death (Preece et al. 2000). Recently, new models have been used to fit complicated spectra like the physical synchrotron and blackbody to constrain the emission mechanism (Burgess et al. 2020). Studies have shown that in the case of GRBs, where 𝛼 crosses the limit (-3/2,-2/3), still the physical synchrotron model is found to best fit the spectra in some cases (Burgess et al. 2020). In addition, the study of the spectral parameter evolution during the prompt emission serves as a valuable tool to constrain emission mechanisms. The peak energy Ep shows four types of evolution: flux tracking, hard to soft, soft to hard, and a chaotic (Golenetskii et al. 1983; Norris et al. 1986). Time-resolved spectral analysis of bright GRBs from Fermi -GBM revealed that the Ep obtained for most of the cases shows flux tracking, but hard to soft evolution is common in single pulse GRBs (Wang et al. 2024; Li et al. 2021). The evolution of 𝛼 is, on the other hand, less predictable. However, recently Gupta et al. (2021, 2022); Gupta (2023); Ror et al. (2023) and Ror et al. (2024) determined the flux tracking evolution of 𝛼 in GRB 140201A and GRB 201216C, respectively. \nGRB afterglows from the cooling population of accelerated electrons from external shocks generally decay following a simple or broken PL. The late-afterglow light curve (hereafter LC) of some nearby GRBs exhibits a distinct bump, indicating the emergence of the underlying supernova (SN; Galama et al. 1998; Hjorth et al. 2003). Detailed spectroscopic examinations of these underlying SNe reveal that they are predominantly hydrogen-deficient broad-line type Ic-BL, often categorized as striped envelope supernovae (SESNe), for more detail about SESNe please refer to Pandey et al. (2021). Notably, SN-associated GRBs display a luminosity of three to five orders of magnitude less than typical LGRBs, so detectable only in the close proximity up to a redshift of ∼ 1. The energy source of GRB-associated SNe is believed to be thermal heating from radioactive Nickel trapped within the ejecta. Recently, the magnetar modelhasemergedasacompelling framework to explain the power source for GRB-associated SNe (Kumar et al. 2022). Another plausible energy source for GRBs associated with SNe involves the interaction between the ejected material and the circumstellar medium surrounding the burst. In the case of GRB 230812B, Srinivasaragavan et al. (2024) used the radioactive heating model to explain the power source of SN 2023pel to constrain the radioactive Nickel mass M 𝑁𝑖 = 0.38 ± 0.01 M ⊙ and the total ejecta mass of 1.0 ± 0.6 M ⊙ . Recently, Wang et al. (2024) demonstrated that GRB 230812B is one of the shortest long-duration GRBs (T90 ∼ 3 s) with a collapsar origin. This event is analogous to GRB 200826A (Ahumada et al. 2021), where a longer jet-bore time through the stellar envelope was proposed to account for the shorter burst duration. \nFig. 1. Represents the Fermi -GBM LCs of GRB 230812B, plotted from the CSPEC (green), CTIME (blue), and TTE (magenta) data files. The Upper panel shows the 1s binned LC of CSPEC and 64 ms of CTIME and TTE observation. Similarly, the lower panel shows the LC binned with 1s for all CSPEC, CTIME, and TTE. The vertical gray region corresponds to the pulse pileup, which we removed from our data analysis. \n<!-- image --> \nUntil 2018, LAT was the only very high-energy instrument that successfully detected GeV emissions from more than 100 GRBs (Fraija et al. 2019). LAT detected the maximum energy photon from GRB 130427A with an energy of ∼ 95 GeV (Ackermann et al. 2014). After 2018, six long GRBs (GRB 180720B, GRB 190114C, 190829A, GRB 201015A, GRB201216C,andGRB221009A)weredetected as having TeV emissions associated with their prompt and afterglow emissions (Gupta et al. 2021; Ror et al. 2023, 2024). A single synchrotron component could not explain the double bump observed in the broadband SED of VHE-detected GRBs, and an additional Synchrotron Self Compton (SSC) or inverse Compton (IC) component is required (Fraija et al. 2019; MAGIC Collaboration et al. 2019; Fraija et al. 2021; H. E. S. S. Collaboration et al. 2021; Ror et al. 2023). In our work, we have tried to constrain the possible origin of GeV detection associated with the GRB 230812B. \nThe content of the paper is as follows: § 2 is devoted to multi-band observation and analysis of GRB 230812B, § 3 to results, § 4 to discussion, and § 5 to summary and conclusion. In this paper, the chosen cosmological parameters are the Hubble constant H0 = 71 km s -1 Mpc -1 and the density parameters Ω Λ = 0.73 and Ω m = 0.27. Uncertainties are given at 1 𝜎 level if not stated otherwise. The representation of afterglow flux follows the convention expressed by F( 𝜈 , t) = t -𝛼 𝜈 -𝛽 . \nFig. 2. The multi-wavelength prompt emission LC of GRB 230812B is plotted from the GBM TTE observation. (a) GBM-BGO observation plotted in the 0.2-1 MeV (gray) and 1-30 MeV (green). (b) Similarly, the NaI LCs in the energy ranges 50-300 keV and 8-50 keV are plotted. (c) Evolution of hardness ratio (HR, number of photons detected in 50-300 keV / number of photons in the 8-50 keV) from the combined NaI-3, NaI-6, and NaI-7 scintillation detectors of Fermi -GBM. \n<!-- image -->", '2.1 . Prompt emission observation and analysis': "On August 12, 2023, at 18:58:12.05 UT (hereafter T0), the Gamma-Ray Burst Monitor (GBM, Meegan et al. 2009) on board Fermi gamma-ray space telescope ( Fermi ) discovered and localised GRB 230812B (Fermi GBM Team 2023; Lesage et al. 2023). During the prompt emission phase, the initial 3.3 ± 0.1 s exhibited 95% of the burst's fluence in 𝛾 -rays, hence considered the T90 duration. However, a softer tail persisted for approx 10 s post-detection (Lesage et al. 2023; Roberts et al. 2023). Prompt emission's preliminary analysis revealed that the time-integrated spectra within the 0-32 s time-interval were well-described by the band function, yielding parameters of Ep = 273 ± 3 keV, 𝛼 = -0.80 ± 0.01, and 𝛽 = -2.47 ± 0.02. The fluence recorded during this period was 2.5201 ± 0.0002 × 10 -4 erg cm -2 by Roberts et al. (2023). In addition to Fermi -GBM, GECAM -C (Xiong et al. 2023), Konus -Wind (Frederiks et al. 2023), and AGILE /MCAL (Casentini et al. 2023) also detected the intense prompt emission from GRB 230812B. \nGBM observations for this burst were acquired from the official web page of Fermi Science Support center /five.sup . Before analyzing the Fermi -GBM data, we carefully check the observed LC from the three types of spectral file (CTIME, CSPEC, and TTE) using the GBM-Tool python package (Goldstein et al. 2023). In the 64 ms bin LC, we found a glitch around [1.12-1.312] s in the TTE data. A thin \nmagenta spike in the TTE LC at around 1.2 s (see upper panel of Figure 1) corresponds to the artificially generated spike mentioned by Roberts et al. (2023). This bright spike is not present in the CSPEC and CTIME data files, which might be due to their coarse temporal or spectral resolution. However, the LC plotted with 1.024 s resolution from the TTE, CSPEC, and CTIME data files showed similar results without any discrepancy (see bottom panel of Figure 1). TTE data in the interval [1.12-1.312] s was lost due to its band-width limit, and also, there is observed pileup in all types of data files during the time-interval [0.5-1.4] s (Roberts et al. 2023). We have not utilized the data corresponding to this time range during the time-resolution spectral fitting of GBM data. As recommended by the Fermi -GBM team, we have used NaI-3,6,7 and BGO-0 detectors for the GBM data analysis. All other detectors are blocked by different parts of the spacecraft and cannot be considered reliable (Roberts et al. 2023). \nWe closely followed the methodology described in Ror et al. (2023), for the prompt emission spectral analysis. The time-integrated spectrum analysis in the temporal range of -0.1s to 5s was performed by combining the detectors NaI-3,6,7 and BGO-0, obtaining a spectrum spanning the energy range of ∼ 0.01 - 40 MeV. As usual, we discarded the 33-37 keV range to avoid the sodium K-edge. The time-integrated spectrum in the given temporal and energy range was extracted utilizing the GtBurst software (Caballero-Garc'ıa et al. 2023; Castro-Tirado et al. 2024). To employ the Bayesian analysis method for the temporal and spectral analysis of GBM data, we used a Python-based software package, Multi-Mission Maximum Likelihood ( 3ML , Vianello et al. 2015). For the spectral fitting within 3ML , we utilized a multi-nest sampler with 10,000 iteration steps, as described in Vianello et al. (2018). We have used several empirical models such as Power Law (PL), Band , and Cutoff PL ( CPL ) and physical Synchrotron and Blackbody ( BB ) models and their combinations ( PL+BB , Band+BB , CPL+BB ), to fit the time-integrated spectrum. Burgess et al. (2020) provides the complete details on the physical synchrotron model. To compare between the different models, we used the deviance information criteria (DIC). \nFurther, we performed the time-resolved spectral analysis of Fermi -GBM data to study the parameter's evolution during the prompt emission. For the time-resolved spectral analysis, the GBM LC from the brightest detector (NaI-6) was rebinned using the methodology known as Bayesian block binning by setting the false alarm probability at 0.01 (Scargle et al. 2013). The obtained time slices were then applied to all other detectors. The advantages of Bayesian block binning methods are given in Burgess (2014). From this method, we obtained 29 spectra, out of which only 24 are statistically significant with S > 20 (Vianello et al. 2018). After extracting the spectrum using GtBurst , we fit each spectrum with several empirical and physical models as discussed above for time-integrated analysis. The results of the spectral analysis for the prompt emission are presented \nin Figure 3 and elaborated upon in section 3.", '2.2 . Afterglow observation and analysis': 'index 𝛽 ox = 0.74 0 . 01 0 . 01 . The optical r-band to X-ray SED and corresponding PL fit is presented in the right panel of Figure 4. \nInitially, the burst was beyond the observational range of the Burst Alert Telescope onboard Neil Gehrel Swift Observatory ( Swift -BAT, Barthelmy et al. 2005). Following a sequence of tiled observations, the Swift X-ray Telescope (XRT) successfully localized the burst at T0 +25 ks (Evans & Swift Team 2023; Kennea & Swift Team 2023; Page & Swift-XRT Team 2023). All Swift -XRT observations were taken in photon count (PC) mode. The preliminary XRT LC and spectrum were modeled by a simple PL and absorbed PL with indices 𝛼 x = 1.80 ± 0.4 and Γ x ( 𝛽 x+1) = 1.82 ± 0.15 (Beardmore et al. 2023). \nTo delve into the afterglow characteristic of GRB 230812B, the X-ray temporal and spectral data were obtained from the online repository /six.sup of UK Swift Science Data Centre (Evans et al. 2007, 2009). For the optical data analysis, we obtained the data reported from GCN circulars /seven.sup . The redshift ( 𝑧 =0.36) of the burst was measured from the spectroscopic observations using the 10.4m Gran Telescopio de Canarias (de Ugarte Postigo et al. 2023). \nIn the case of GRB 230812B, the flux density X-ray LC at 10 keV showed no flare or plateau throughout the XRT observations from T0 +25 ks to T0 +1400 ks. We modeled the X-ray LC with a simple PL using the MCMC technique. We obtained a PL index of 𝛼 x = -1.22 + 0 . 05 -0 . 05 . The X-ray flux density LC and the PL model fitted to it are shown in the left panel of Figure 4. \nSimilarly, the optical LC also seems to decay without breaks. Following a similar methodology, we fitted the optical LC using a simple PL and obtained the decay index of 𝛼 o = 1.08 0 . 03 -0 . 03 . As shown in the left panel of Figure 4, the optical LC starts deviating from the PL decay at ∼ 4 days after T0. This is because the underlying supernova emission began to emerge about four days after the burst trigger (Srinivasaragavan et al. 2024; Hussenot-Desenonges et al. 2023). \nWe have retrieved the X-ray spectra in the time range of 25 ks to 38 ks and the energy range of 0.3 - 10 keV. For the XRT spectrum fitting, we utilized an absorbed PL model with a multiplicative Galactic absorption component ( phabs ) and the host absorption component ( zphabs ) in the X-ray spectral fitting package ( XSPEC ; Arnaud 1996; Gupta et al. 2022). The galactic hydrogen column density (NHGal) was fixed at 2.02 × 10 20 cm -2 during the spectral fitting. There is no flare or any other variation present in the XRT LC. We first calculated the host hydrogen column density (NHz = 7.20 × 10 20 cm -2 ) along the line of sight by fitting the absorbed PL to the average spectrum. Then, by fixing both NHGal and NHz, we again fit the XRT spectrum with the same absorbed PL model and obtained a spectral index of 𝛽 𝑥 = 0.68 + 0 . 08 -0 . 08 . After that, we created an optical r-band to X-ray spectral energy distribution (SED) in the time interval 25 ks - 38 ks. We fit the optical-X-ray SED with the PL and obtained a PL', '2.3 . Fermi -LAT observation and analysis': 'At the moment of the GBM detection, the LAT boresight angle was 29 · . Scotton et al. (2023) reported that Fermi -LAT detected the GeV photons from the GRB 230812B.Preliminary LAT spectral analysis above 100 MeV revealed the PL distribution of energy with the photon index Γ LAT = 2.16 ± 0.14. Fermi -LAT recorded a photon with a maximum energy of 72 GeV at T0 +32.2s (Scotton et al. 2023). \nWe employed the GtBurst software to download Fermi -LAT data spanning T0 to T0 +10 ks. Subsequently, we conducted an unbinned likelihood analysis on the time-integrated LAT data, covering the energy range of 0.1-100 GeV. In our analysis of LAT data, we selected a 10 × 10 deg 2 region centered around the GRB 230812B. The instrumental response function P8R3 SOURCE, a maximum zenith cut of 100 degrees, and a skybinning of 0.2 degrees were applied. Finally, we utilized the gtsrcprob tool to calculate the probability of observed photons associated with GRB 230812B. \nThe time-integrated spectral analysis of LAT data yielded a photon index Γ LAT = -1.99 ± 0.13, with an average flux = 3.79 ± 0.68 × 10 -9 erg s -1 cm -2 in the 0.1-10 GeV energy range. Additionally, we extracted the spectral file for XSPEC . We again fit the extracted spectra in XSPEC and obtained a photon index of Γ LAT = 2.03 + 0 . 27 -0 . 23 and a flux of 3.52 ± 0.98 × 10 -9 erg s -1 cm -2 , consistent within the error. We saved the QDP file from the XSPEC and plotted the LAT spectra in Figure 5. \nFollowing this, we performed the time-resolved analysis of LAT data. Initially, we binned the LAT data in 5 s bins, and subsequently, the bin width was increased, as illustrated in Figure 6. Up to the initial 100 s, the instrument response function P8R3 TRANSIENT020 was utilized, followed by a switch to the instrument response function P8R3 SOURCE . The outcomes of the time-resolved spectral analysis are presented in Figure 6. The flux obtained in the range 0.1-10 GeV decreased with time following a PL decay, with the decay index 𝛼 LAT = 1.04 ± 0.08.', '3.1 . Prompt emission': "The prompt emission multichannel LC of GRB 230812B is presented in the upper and middle panels of Figure 2. The evolution of hardness ratio (HR, i.e. no. of photons in 50-300 keV / no. of photons in 8-50 keV) is presented in the lower panel of Figure 2. The prompt emission LC is the fast rise and exponential decay (FRED) type with a single broad pulse. The HR remains > 1 for almost T0 to T0 + 2 s and seems to follow the intensity of the burst, indicating that GRB 230812B is a hard burst. \nThe time-integrated spectrum of GRB 230812B is best modeled by Band+BB with the minimum DIC value of \nFig. 3. Represents the evolution of the prompt emission parameter obtained from the time-resolution spectral analysis of GRB 230812B. In each plot, the left Y-axis represents the flux plotted in the background as a step function, where each step represents the width of the Bayesian block in which the spectral parameters are evaluated, and the X-axis ticks are midpoints of the Bayesian block. The orange-shaded region represents the time range where CPL+BB is the best fit, and the red-shaded region is the region where Band+BB is the best fit. The difference in the DIC Δ DIC= DIC Band+BB -DIC CPL+BB is shown in the upper-left panel. A dashed line at -10 is plotted to select the best-fit model. The upper-right panel represents the evolution of low energy spectral index 𝛼 obtained from CPL+BB (orange) and Band+BB (red). Two dotted lines at -3/2 and -2/3 represent the limits of synchrotron emission from external shock (Preece et al. 2000). Similarly, the middle-left and middle-right panels represent the evolution of Ep and kT, respectively. The lower two panels represent the magnetic field strength B and the energy distribution index 𝑝 of the shock-accelerated electrons. \n<!-- image --> \n0.5 \n9531 over synchrotron , Band, CPL , and CPL+BB with DIC values of 9570, 10356, 24725, and 13234, respectively. The best-fit parameter obtained from the best-fit model are 𝛼 = -1.02 + 0 . 01 -0 . 01 , Ep = 411.35 + 8 . 21 -8 . 18 keV, 𝛽 = -2.75 + 0 . 04 -0 . 04 and kT = 24.05 + 0 . 32 -0 . 32 keV. \nThe prompt emission time-resolved spectral fitting results are presented in Figure 3. The upper left panel of Figure 3 illustrates the difference between the DIC values of two best-fit models, Band+BB and CPL+BB , i.e., Δ DIC = DIC Band+BB -DIC CPL+BB . Δ DIC < -10 means that the \nFig. 4. Left panel: XRT (green) and r-band optical LCs (red) are plotted. The colored dotted line and the shaded area around them represent the PL fit and associated error bar for both LCs. The vertically shaded region corresponds to the time interval used to construct an optical r-band to X-ray SED. Right pane: optical to X-ray SED. Optical r-band data is plotted in a red square, and X-ray observation is shown in blue. The blue dotted line represents the PL fitted to X-ray spectra at 0.3-10 keV. The red dotted line represents the PL model fitted to the combined r-band to X-ray SED. Green squares and the dotted line represent the LAT spectra and corresponding power law fit, respectively. \n<!-- image --> \nBand+BB is best fitting; otherwise, CPL+BB is best fitting. The time-resolved spectral analysis revealed that during the rising phase of the LC's prompt emission, the time-resolved spectra are best fitted by the CPL+BB model. Conversely, during the decay phase of the LC, all the spectra are best fitted by the Band+BB model based on the lower DIC values. In Figure 3, the orange shade region represents the interval during which CPL+BB is best fitting, and the red shaded region represents the interval during which Band+BB best fits the time-resolved spectra. The total isotropic 𝛾 -ray energy release 𝐸 𝛾,𝑖𝑠𝑜 = 8.74 ± 1.61 × 10 52 erg along with the Ep value is found to be consistent with the Amati correlation for LGRBs (Amati 2006).", "3.2 . The prompt emission parameter's evolution": 'The evolution of the low energy spectral index 𝛼 obtained from the best-fit model is presented in the upper right panel of Figure 3. Throughout the prompt emission, 𝛼 seems to follow the intensity. The 𝛼 obtained from the CPL+BB in the 2-10 s time intervals deviates from the flux tracking. This might be due to the CPL+BB model not fitting well in the given time interval. Similar to the 𝛼 , the peak energy Ep of the spectrum is also found to be following the intensity. Hence, the GRB 230812B is a double-tracking burst. The time-integrated and time-resolved spectral analysis results showed that the single empirical Band or CPL do not fit well the spectra, but an addition of a thermal component is required. However, the evolution of the thermal energy component kT is somewhat complicated. It closely follows the flux up to T0 +3 s. After that, it shows almost a constant value of ∼ 7 keV, but with larger error bars. Nevertheless, the physical Synchrotron model did not fit the spectrum well. However, it is still reasonable to check for the evolution of the magnetic field strength (B) obtained from fitting \nwith the Synchrotron model. The evolution of B closely matches with the evolution of Ep, as shown in the lower panel of Figure 3. The electron energy distribution in the synchrotron model follows a PL with a PL index 𝑝 . The parameter 𝑝 is obtained from fitting the synchrotron model and is also showing a flux tracking evolution as shown in the lower right panel of Figure 3.', '3.3 . Afterglow emission and the closure relations': 'TheobservedX-rayandopticalr-bandLCsofGRB230812B do not show any flare, plateau, or bump (see Figure 4). Both X-ray and r-band LCs were fitted with a simple PL obtaining temporal decay indices of 𝛼 x = -1.22 + 0 . 05 -0 . 05 and 𝛼 o = 1.08 0 . 03 -0 . 03 . Both X-ray and optical decay indices are consistent with each other within 3 𝜎 . In addition, the spectral indices 𝛽 x = 0.68 + 0 . 08 -0 . 08 and 𝛽 ox = 0.74 0 . 01 0 . 01 obtained from the fitting of XRT spectra in 0.3-10 keV and combined optical r-band to X-ray SED are also consistent within 1 𝜎 . This indicates that both X-ray and optical emissions come from the same spectral regime. We checked the closure relation followed by the GRB external shock (Sari et al. 1998). We found that the optical and X-ray spectral and temporal indices satisfy the relation 𝛽 = 𝑝 -1 2 and 𝛼 = 3 ( 𝑝 -1 ) 4 in the spectral regime 𝜈 < 𝜈 < 𝜈 < 𝜈 \nm o x c in the ISM-like surrounding medium. \nFermi -LAT data was not included in the Spectral Energy Distribution (SED) due to the absence of optical and X-ray observations during the Fermi -LAT detection. From Fermi -LAT temporal and spectral analysis we obtained decay indices of 𝛼 LAT = 1.04 ± 0.08 and 𝛽 LAT ( Γ LAT-1) = 1.03 + 0 . 27 -0 . 23 . The observed 𝛽 LAT is slightly steeper than 𝛽 o and 𝛽 x. The difference Δ 𝛽 = 𝛽 LAT𝛽 x is 0.4 ± 0.3, which is consistent with the assumption that 𝜈 𝑐 might lie between the X-ray and LAT observation in the ISM medium. \nFig. 5. Represents the fitting results of time average LAT spectra in the energy range 0.1-100 GeV \n<!-- image --> \n6 \nFig. 6. Top panel: represents the Fermi -LAT LC of GRB 230812B in the energy range of 0.1-10 GeV. Squares denote the observed flux, and the dotted line represents the PL fitted to the LC. The shaded area represents the 1 𝜎 error associated with the fit. The middle panel represents the evolution of the photon index derived from the time-resolved fitting of LAT spectra. The bottom panel displays the temporal distribution of LAT-detected photons, with the color mapindicating the probability of the photons being associated with GRB230812B.Theredstar at 32 s represents the maximum energy photon of 72 GeV from Fermi -LAT. The red dotted line represents the maximum energy allowed by synchrotron emission for GRB 230812B. \n<!-- image -->', '3.5 . Afterglow LC comparison': 'We compared the X-ray and optical afterglow LCs of GRB 230812B with a set of 15 SN-detected GRB from Kumar et al. (2022) in Figure 7. We found the Swift -XRT and optical r-band afterglow LC of GRB 230812B is consistent \nwith other GRBs in the background and does not possess an exceptional bright afterglow or SN emissions. Even the optical and X-ray LC of GRB 230812B is relatively fainter than the bright bursts of GRB 130427A (Xu et al. 2013) and GRB 171010A (Kumar et al. 2022) at similar redshifts as shown in Figure 7. Our analysis concluded that the afterglow of GRB 230812B and its associated SN 2023pel (Hussenot-Desenonges et al. 2023; Srinivasaragavan et al. 2024) is an intermediate bright GRB/SN compared to other GRBs.', '3.6 . Possible origin of GeV emission from Fermi -LAT': 'The observed power-law decay of the LAT light curve and spectra indicate that the LAT emission originates from an external shock. However, unlike optical and X-ray observations, the emission mechanism of LAT-detected photons still needs to be completely clarified. According to Fraija et al. (2019), the maximum energy of photons from synchrotron external shock model can be 10 GeV × ( Γ 0/100) × (1+z) -1 . Where Γ 0 is the bulk Lorentz factor, and z is redshift. Considering the bulk Lorentz factor Γ 0 = 315, calculated it from the observed E 𝛾, iso using the relation given in Liang et al. (2010). Using the above relations, we calculated that, in the case of GRB 230812B, the Synchrotron emission mechanism could only generate a maximum energy photon of 23 GeV, as shown with a red dotted line in the lower panel of the Figure 6. From the lower panel of Figure 6, it is clear that all photons lie below this level except for 72 GeV photons. Thus, the origin of the 72 GeV photons cannot be from synchrotron emission. This has been observed earlier for the VHE bursts (GRB 180720B, 190114C, 190829B, GRB 201015A, GRB 201216C: Fraija et al. 2019; MAGIC Collaboration et al. 2019; Fraija et al. 2021; H. E. S. S. Collaboration et al. 2021; Ror et al. 2023), where the observed VHE photons are explained via synchrotron self Compton (SSC) or external Inverse Compton (IC) . It is possible that the non-thermal photons from the prompt emission were Lorentz boosted by accelerated electrons in the external shock resulting in the observed LAT emission. This suggests that the IC and SSC are the potential emission mechanisms that could be responsible for the VHE 72 GeV photons observed from GRB 230812B.', '3.7 . GRB 230812B compared with other S- and LGRBs': 'The characteristics of GRB 230812B compared to the other burst from the IceCube /eight.sup catalog are shown in the Figure 8. In the upper right panel of Figure 8, we compared the E 𝛾, iso of redshift detected bursts. The observed E 𝛾, iso of the bursts is found to correlate with the luminosity distance (D 𝐿 ). We fitted a PL and found the relation: log10 ( E 𝛾, iso ) = ( 45 . 42 ± 0 . 16 ) + ( 1 . 66 ± 0 . 04 ) × log10 ( DL ) . \nGRB 230812B, with the observed distance of 1.9 Gpc corresponding to the cosmological age of 9.75 Gyr, is inconsistent with the above relation. Along with GRB \n<!-- image --> \nFig. 7. A comparison of optical r-band (left) and Swift -XRT (right) luminosity LCs of GRB 130427A, GRB 171010A, and GRB 230812B (red) with other SN-detected bursts (grey) in the background, see Ror et al. (2023) for more details. \n<!-- image --> \n130427A and GRB 171010A, GRB 230812B lies outside of the 5 𝜎 range of relations, as shown in Figure 8. This is due to its exceptionally high prompt emission brightness at z=0.36. Most of the other GRBs at this redshift are fainter than GRB 230812B, except for GRB 130427A and GRB 171010A. In the lower-left panel of Figure 8, we compared the observed fluence of GRB 230812B (red star), GRB 130427A and GRB 171010A along with other bursts from the IceCube catalog. GRB 230812B is the brightest burst with T90 < 5 s, and only slightly fainter than GRB 130427A and GRB 171010A at the similar redshift. This huge flux detected in such sort time scale is the possible region for observed pileup in Fermi -GBM. The lower right panel compares the Ep (i.e., Hardness) of the Fermi -GBMdetected bursts. GRB 230812B is harder than the GRB 171010A and softer than the GRB 130427A at a similar redshift. GRB 230812B with T90 > 2 s lies around the dividing line of the population of SGRBs and LGRBs. However, based on the observed underlying SN and other observed properties, GRB 230812B is classified as a LGRB. Our analysis indicates that GRB 230812B is one of the brightest Fermi -GBM and LAT-detected LGRB with the hardest prompt spectra.', '4. SUMMARY AND CONCLUSION': "This work presents a detailed prompt and afterglow analysis of Fermi -LAT GRB 230812B. Due to its extreme brightness, the detecting instruments ( Fermi -GBM) suffer a pileup in the time range [0.5-1.4] s, which we exclude from our analysis. The time-integrated spectral analysis showed that the spectrum in the Fermi -GBM energy range is the best fit by the Band+BB function. In addition to this, all the time-resolved spectra favour an additional thermal component along with a dominant non-thermal component. All of these findings indicate the hybrid (baryonic + magnetic) jet composition of the burst. Spectral parameter evolution from our time-resolved analysis showed that the low energy spectral index 𝛼 crosses the slow and fast cooling limit imposed by the synchrotron emission \nmechanism. The evolution of 𝛼 poses a challenge to linking the non-thermal prompt emission with pure synchrotron emission. In addition, both 𝛼 and Ep were found to track the intensity of the burst throughout the prompt emission duration. The X-ray and optical LCs of this burst do not show any signature of late central engine activity (flare or plateau). Both X-ray and optical emissions are consistent with the closure relation in spectral regime 𝜈 𝑚 < 𝜈 𝑜 < 𝜈 𝑥 < 𝜈 𝑐 in the ISM-like surrounding medium. The Fermi -LAT analysis showed that in the early stage ( < 10 ks), the LAT spectral index 𝛽 LAT is harder than the 𝛽 x, indicating that the LAT emission is coming from a different spectral regime or entirely different origin. With the assumption of synchrotron origin of the LAT photons, we calculated that the break frequency 𝜈 𝑐 may lie or remain between the X-ray and LAT frequencies throughout the observations. However, the synchrotron emission mechanism is unable to explain the origin of the 72 GeV photons detected by LAT at T0 + 32s >> T90, i.e., during the afterglow phase. The most plausible explanation for the VHE-detected photons is IC or SSC boosting of prompt or early afterglow photons by the shock-accelerated electrons in the external medium. The comparison of X-ray and optical LCs of GRB 230812B with similar bursts (GRB 130427A and 171010A) revealed that, unlike the bright prompt emission, the afterglow of GRB 230812B was fainter than the other SN-detected GRBs at a similar Ep, E 𝛾, iso, and redshift. Moreover, previous studies have shown that low-redshift GeV-TeV detected GRBs, such as GRB 130427A and GRB 201015A, are associated with supernovae (Ror et al. 2023). This event further reinforces the idea that the association of low-redshift GeV-TeV detected GRBs with underlying supernovae is a common occurrence. \nAcknowledgments: AKR are thankful to Dr. Rahul Gupta for his continuous support during the analysis and writing of the paper. SBP acknowledges the financial support of ISRO under the AstroSat archival \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFig. 8. Prompt emission characteristic of GRB 230812B compared with SN-detected GRB 130427A and GRB 171010A at similar redshift. Upper left panel: represents GRB 230812B in Amati correlation space (Amati 2006). Upper right panel: represents the E 𝛾, iso distribution of GRBs plotted against the luminosity distance in Mpc and the universe's age in Gyr. Lower left panel: represents the fluence distribution of GRBs along with the T 90 duration on the observer frame. Lower right panel: Ep in the observed frames plotted with the T 90 duration. A vertical dashed line at 2s separates between SGRBs and LGRBs. \n<!-- image --> \ndata utilization program (DS 2B-13013(2)/1/2021-Sec.2). SBP also acknowledges support from DST grant no. DST/ICD/BRICS/Call-5/CoNMuTraMO/2023(G) for the present work. AA acknowledges funds and assistance provided by the Council of Scientific & Industrial Research (CSIR), India, under file no. 09/948(0003)/2020-EMR-I. AA also acknowledges the Yushan Young Fellow Program by the Ministry of Education, Taiwan, for financial support. AJCT acknowledges support from the Spanish Ministry project PID2020-118491GB-I00 and Junta de Andalucia grant P20 010168. This research has used data obtained through the HEASARC Online Service, provided by the NASA-GSFC, in support of NASA High Energy Astrophysics Programs.", 'REFERENCES': "Kumar, P. & Zhang, B. 2015, Phys. Rep., 561, 1. doi:10.1016/j.physrep.2014.09.008 \nSrinivasaragavan, G. P., Swain, V., O'Connor, B., et al. 2024, ApJ, 960, L18. doi:10.3847/2041-8213/ad16e7 \nHussenot-Desenonges, T., Wouters, T., Guessoum, N., et al. 2023, arXiv:2310.14310. doi:10.48550/arXiv.2310.14310 \nBeardmore, A. 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2024PDU....4601623R | In this paper we thoroughly explore two crucial aspects of a quantum Schwarzschild black solution within fourdimensional spacetime i the weak deflection angle ii the rigorous greybody factor and iii the Dirac quasinormal modes. Our investigation involves employing the GaussBonnet theorem to precisely compute the deflection angle and establishing its correlation with the Einstein ring. Additionally we derive the rigorous bounds for greybody factors through the utilization of general bounds for reflection and transmission coefficients in the context of Schrodingerlike onedimensional potential scattering. We also compute the corresponding Dirac quasinormal modes using the WKB approximation. We reduce the Dirac equation to a Schrodingerlike differential equation and solve it with appropriate boundary conditions to obtain the quasinormal frequencies. To visually underscore the quantum effect we present figures that illustrate the impact of varying the parameter mmlmath altimgsi1.svg displayinline idd1e1115mmlmsubmmlmrowmmlmirmmlmimmlmrowmmlmrowmmlmn0mmlmnmmlmrowmmlmsubmmlmath or more specifically in terms of the parameter mmlmath altimgsi108.svg displayinline idd1e1125mmlmimmlmimmlmath. This comprehensive examination enhances our understanding of the quantum characteristics inherent in the Schwarzschild black solution shedding light on both the deflection angle and greybody factors in a fourdimensional spacetime framework. | 2024-12-01T00:00:00Z | ['10.1016/j.dark.2024.101623', '2024PDU....4601623R', '2024arXiv240910930R', 'arXiv:2409.10930', '10.48550/arXiv.2409.10930'] | ['General relativity', 'Black holes', 'Weak deflection angle', 'Greybody factor', 'Quasinormal modes', 'General Relativity and Quantum Cosmology'] | An effective model for the quantum Schwarzschild black hole Weak deflection angle quasinormal modes and bounding of greybody factor | 2,024 | 175 | 0.34 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML'] | 1 | https://arxiv.org/pdf/2409.10930.pdf | {'An Effective Model for the Quantum Schwarzschild Black Hole: Weak Deflection Angle, Quasinormal Modes and Bounding of Greybody Factor': '´ Angel Rinc´on , 1, 2, ∗ Ali ¨ Ovg¨un , 3, † and Reggie C. Pantig 4, ‡ \n1 Departamento de Fsica Aplicada, Universidad de Alicante, Campus de San Vicente del Raspeig, E-03690 Alicante, Spain 2 Sede Esmeralda, Universidad de Tarapac´a, Avda. Luis Emilio Recabarren 2477, Iquique, Chile 3 Physics Department, Eastern Mediterranean University, Famagusta, 99628 North Cyprus, via Mersin 10, Turkey 4 Physics Department, Map´ua University, 658 Muralla St., Intramuros, Manila 1002, Philippines (Dated: September 18, 2024) \nIn this paper, we thoroughly explore two crucial aspects of a quantum Schwarzschild black solution within four-dimensional space-time: i) the weak deflection angle, ii) the rigorous greybody factor and, iii) the Dirac quasinormal modes. Our investigation involves employing the Gauss-Bonnet theorem to precisely compute the deflection angle and establishing its correlation with the Einstein ring. Additionally, we derive the rigorous bounds for greybody factors through the utilization of general bounds for reflection and transmission coefficients in the context of Schrodinger-like one-dimensional potential scattering. We also compute the corresponding Dirac quasinormal modes using the WKB approximation. We reduce the Dirac equation to a Schrodinger-like differential equation and solve it with appropriate boundary conditions to obtain the quasinormal frequencies. To visually underscore the quantum effect, we present figures that illustrate the impact of varying the parameter r 0 , or more specifically, in terms of the parameter α . This comprehensive examination enhances our understanding of the quantum characteristics inherent in the Schwarzschild black solution, shedding light on both the deflection angle and greybody factors in a four-dimensional space-time framework. \nPACS numbers: 95.30.Sf, 04.70.-s, 97.60.Lf, 04.50.+h Keywords: General relativity; Black holes; Weak deflection angle; Greybody factor; Quasinormal modes.', 'I. INTRODUCTION': "As predicted by Einstein's general theory of relativity and other theories of gravity, black holes are fundamental objects in the universe. They are of great importance in classical and quantum gravity. Black holes are the perfect arena to combine general relativity with quantum mechanics in light of several well-known concepts where classical and quantum features must coexist. In this sense, the concept of Hawking radiation [1, 2], although not yet observed, has captivated the scientific community. However, regardless of their relevance, black holes have several disadvantageous properties, for example, they contain a curvature singularity, which means that the black hole spacetime is geodesically incomplete. Also, unless their spin is exactly zero, black holes contain a Cauchy horizon, which makes it impossible to predict the future evolution of physical quantities [3]. To gain a deeper understanding of black hole physics, it is useful to study black holes in alternative theories of gravity, especially those in which quantum features can emerge naturally. Black holes and their properties can offer insights into the foundations of general relativity and beyond. They assume essential roles in both classical and quantum regimes, which is why they are regarded as ideal candidates for investigating alternative theories of gravity. \nThe study of black holes is well motivated from at least two complementary perspectives. The first motivation is the observational one. The direct detection of gravitational waves from black hole mergers [4-8] and the groundbreaking image of a supermassive black hole [9-14] at the center of galaxy Messier 87 by the Event Horizon Telescope project [15] have intensified research into various aspects of black hole physics. These recent advancements have sparked heightened interest and exploration in understanding black holes. Thus, the relevance of black holes is supported not only by observational evidence but also by theoretical arguments such as the 'no-hair theorem' or the 'no-hair conjecture' [16]. Such a conjecture states that, regardless of how the how the BH is formed, it can be parameterized by only three quantities and these are: i) the mass ii) the electric charge, and finally iii) the angular momentum [17]. However, in some special cases, black holes can have more than three quantities that are useful to describe their properties (see [18-20] and references therein). From the vast amount of black hole solutions in four-dimensional spacetime, we can highlight certain solutions in light of their importance. The first solutions in vacuum, with electric charge and rotation, are: i) The Schwarzschild solution [21], ii) \nthe Reissner-Nordstrom solution [22, 23], iii) the Kerr solution [24] and, finally, iv) the Kerr-Newman solution [25]. Such solutions are often denoted as the 'black-hole' of general relativity and include the three basic parameters, i.e., { M,Q,J } . From the last four works, the Schwarzschild black hole (a non-rotating, spherically symmetric solution to general relativity with a singularity at its centre, surrounded by an event horizon), albeit simple, is the most conventional 'toy model' or background, and it has been significantly used merge quantum features into a classical background. Be aware and notice that a consistent quantum theory of gravity remains an open task in modern theoretical physics. Thus, there are a significant number of approaches that attempt to obtain a well-defined theory of quantum gravity, for instance, we can mention [26-34] and references therein. In particular, we can highlight several approaches, as are, for instance, the asymptotic safety scenario which is based on the renormalization group flow that controls coupling constants [28]. The formalism has been implemented in several contexts (in particular, in black hole physics) producing good results consistent in some limits with GR (see [35-38] and references therein). An interesting application to black hole physics is the work of Bonanno and Reuter [39] (usually referred to as the 'quantum improved' black hole formalism), where they studied the implications of asymptotically safe gravity on the Schwarzschild metric, where the ordinary Newton constant in the classical metric component is replaced by the running gravitational constant obtained from the renormalization group equation. As in the case of black holes in asymptotically safe gravity, formalism has been used to analyze how quantum features perturb classical backgrounds (see [40-44] and references therein). In addition, a related approach is the scale-dependent scenario, where not only the Newton constant but also all other coupling constants appearing in the action are replaced by functions of a given energy scale. In such a formalism, the need to establish a direct link between the energy scale and the physical coordinate is avoided (in problems with a very high degree of symmetry). For further details see [45-61] and references therein. In both cases, the running Newton coupling acquires a certain scale dependence and evolves. In particular, for black holes, Newton's coupling evolves with the radial coordinate. As a consequence, the global structure of the quantum black hole solution differs significantly from its classical solution. In the case of the improved Schwarzschild black hole, such a solution admits an inner (or Cauchy) horizon in addition to the black hole event horizon. Alternatively, in the case of the scale-dependent Schwarzschild black hole, the event horizon was found to be slightly modified. These two examples suggest that the horizon, and hence its thermodynamic properties, may be perturbed in quantum black holes. \nBlack holes can be investigated in light of Loop quantum gravity [62], one of the best-known (and also accepted) theories of quantum gravity [63]. One of the main motivations for revisiting black holes within the framework of loop quantum gravity is the discretization of geometric properties at the Planck scale. This quantum discreteness arises from the canonical quantization procedure when applied to the action of general relativity, which is suitable for describing the coupling of gravity with gauge fields and fermions. More precisely, loop quantum gravity is a mathematically well-defined, background-independent, and non-perturbative quantization of general relativity. Some of the most important outcomes are: i) the calculation of the physical spectra of the geometric quantities such as area and volume, leading to quantitative predictions of Planck-scale physics, ii) a derivation of the black hole entropy method proposed by Bekenstein and Hawking, iii) a non-trivial scientific representation of quantum physical spaces micro-structure that is distinguished by Planck scale discreteness, among others. In light of previous remarks, the loop quantum gravity theory has provided a path to parameterize quantum properties of spacetime revealed by a black hole. \nLast but not least, the so-called quasinormal modes (QNMs) are important features of the late-time response of a black hole to external perturbations. The study of black hole perturbations has gained significant importance in the last decade, especially after the first detection of gravitational waves. Dominant QNMs are observed in the late-time gravitational wave signals of black holes (and other compact objects). In particular, collaborations such as LIGO and VIRGO have confirmed the detection of QNMs [64]. QNMs are also an essential tool for testing the stability of black holes. If the imaginary part of the quasinormal frequencies is positive, then the black hole is unstable to perturbations; otherwise, it remains stable. For a detailed explanation, see [65]. Let us focus on the Dirac quasinormal modes: they are particularly relevant because they provide crucial insights into the behavior of fermionic fields, such as spin-1/2 particles, near black holes. The study of these modes completes the topic of quasinormal modes, which usually includes only i) scalar, ii) electromagnetic, and iii) gravitational perturbations. \nThe structure of our work is as follows: In the next section, we summarize the fundamental aspects of a novel model for a quantum Schwarzschild black hole. Moving on to the third section, we calculate the deflection angle by comparing the solution with the Schwarzschild case. Subsequently, in the fourth section, we compute the Rigorous Bounds of Greybody factors for a massless scalar field. After that, in section five, we compute the Dirac QNMs and compare our results with the classical Schwarzschild case α = 0 . Finally, we conclude our work with a summary and concluding remarks in section six. Throughout the study, we adopt the mostly positive metric signature ( -, + , + , +) and employ geometrical units where the universal constants are set to unity ( c = 1 = G 0 ).", 'II. AN EFFECTIVE MODEL FOR THE QUANTUM SCHWARZSCHILD BLACK HOLE': "Let us start by considering a static region recently investigated in the context of LQG [66, 67]. To construct such a space, we first should take advantage of the Ashtekar-Barbero variables and then rewrite the corresponding Poisson algebra. Also, for an adequate effective descriptions a polymerization procedure is typically required. Doing so, an discrete parameter λ accounts the quantum spacetime. Given that the concrete construction of this spacetime have been recently shown in [66, 67] we will focus on the study of some properties of the static region (exterior), in particular: i) the deflection angle and ii) the rigorous bounds of Greybody Factors. Furthermore, we only considered specializing in static and spherically symmetric case for conservative reasons since (1) there was no consensus across different estimates of Sgr A*'s rotation, and (2) the effect of the spin parameter a on the shadow is small for the non-extremal case. Although there is a considerable effect relative to an observer's inclination position, it is still small (See the complete discussion in [68]). \nIn order to keep the paper self-contained, we will briefly summarize the more relevant equations needed to understand how this solution is found. The first step is to write a spherically symmetric spacetime using the new variables of Ashtekar (see for instance [69] and references therein). The set of triads, labeled E a i , are the canonical variables in LQG, and A i a are their corresponding SO(3) connections. In the spherically symmetric case, only three pairs of canonical variables remain, namely { η, P η , A ϕ , E ϕ , A x , E x } . As originally described in [70], a polar set of variables is chosen and that x is used because it is not necessarily described by the standard radial coordinate. Note that it is always possible to define the gauge invariant variables K i as a function of the connections A i and η , in other words, a more suitable variable can be defined. This means that E x , K x and E ϕ , K ϕ both represent the canonically conjugated pairs. In the context of General Relativity, diffeomorphism invariance is based on four constraints. The first one, the Hamiltonian H , and the remaining three come from the diffeomorphism constraint D . For this concrete case, we can write down the relevant quantities as follows: \nH = -˜ E ϕ 2 √ ˜ E x ( 1 + ˜ K 2 ϕ ) -2 √ ˜ E x ˜ K x ˜ K ϕ + 1 8 √ ˜ E x ˜ E ϕ ( ( ˜ E x ) ' ) 2 -√ ˜ E x 2 ( ˜ E ϕ ) 2 ( ˜ E x ) ' ( ˜ E ϕ ) ' + √ ˜ E x 2 ˜ E ϕ ( ˜ E x ) '' , (1) D = ( ˜ E x ) ' ˜ K x + ˜ E ϕ ( ˜ K ϕ ) ' . (2) \nHere the notation and symbols are clarified as follows: i) the prime denotes the derivative with respect to x , ii) ˜ E i are the symmetry-reduced triad components, and finally, iii) ˜ K i means the conjugated momenta with i = { x, ϕ } . Given that the holonomies of the connection have well-defined operators in loop quantum gravity, a polymerization procedure is needed (see [71] for clarifications). Thus, the basic idea is to replace the variables by an exponential form. In the case where real variables are used, a suitable replacement is of the form \n˜ x → sin ( λx ) λ , (3) \nwhere the last parameters are defined as follows: i) x is the variable and ii) λ is the polymerization parameter. Note that the classical theory is valid in the limit λ → 0 . Moreover, the polymerization parameter λ is directly related to the length of the loop along which the holonomy is computed, since it is responsible for the space-time discretization. It is important to note that there may be anomalies in the last replacement since the modified constraint algebra is typically not closed. Alternatively, both K ϕ and E ϕ can be replaced by ˜ K ϕ and ˜ E ϕ using the expressions: \n˜ K ϕ → sin ( λK ϕ ) λ , (4) \n˜ E ϕ → E ϕ cos ( λK ϕ ) , (5) \n̸ \nand thus the theory becomes free of anomalies [66]. The canonical transformation is bijective while cos ( λK ϕ ) = 0 and the dynamical content of the theory is equivalent to that obtained in GR. The case cos ( λK ϕ ) = 0 could be crucial because it could be a sign of new physics. Due to the fact that the Hamiltonian constraint diverges there, a regularization is required. As can be viewed in [66], using a simplified map { t, x } = { ˜ t, ˜ r } and choosing E x = ˜ r 2 and K ϕ = 0 we get the corresponding spacetime. Finally, taking into account the last ingredients, we can now introduce an effective quantum-corrected Schwarzschild space-time background [66, 72]. As the discussion is too involved from a mathematical point of view, we will not go into the technical details here and refer the interested reader to [66, 72]. In addition, this \nspacetime, along with similar ones, has been partially studied, and related calculations have been done independently during the submission of this paper. Please see [73-75] for other studies that have used similar ideas. \nThe spacetime metric for the quantum Schwarzschild black hole is: \nds 2 = -( 1 -2 M r ) dt 2 + ( 1 -r 0 r ) -1 ( 1 -2 M r ) -1 dr 2 + r 2 d Ω 2 , (6) \nwith r ∈ (2 M, ∞ ) . This region is asymptotically flat, and will describe one exterior domain. Also, notice that the parameter r 0 contains the quantum effect via the following relation \nr 0 = 2 M ( λ 2 1 + λ 2 ) ≡ αr H , (7) \nbeing r 0 smaller than the Schwarzschild horizon (which is r s = 2 M ). The event horizon, r H , obtained demanding g rr = 0 , is not modified and then, the horizon is the same that its classical counterpart (i.e. r s = 2 M = r H ). The black hole mass, M , and the auxiliary parameter, r 0 , can also be written with help of three well-known definitions of mass in spherical systems. First, notice that the black hole mass, is not represented by M , statement originally explained in [72] and reinforced in [76]. Commonly we have three well-known masses. They are i) the Komar mass M K ii) the ADM mass M ADM and finally iii) the Misner-Sharp mass M MS . These quantities are given by \nM K = M √ 1 -r 0 r = 1 2 r H ( 1 -α r H r ) 1 / 2 , (8) \nM ADM = M + r 0 2 = 1 2 r H ( 1 + α ) , (9) \nM MS = M + r 0 2 -Mr 0 r = 1 2 r H ( 1 + α ( 1 -r H r )) . (10) \nBe aware and notice that two of them, i.e., the Komar and Misner-Sharp masses, can be different each other, because of the modified spacetime is not a solution of the Einstein's equations [77]. In addition, for large values of r and for spherically symmetric and asymptotically flat spacetimes, the ADM and Misner-Sharp masses must be equal [78], as can be checked by simple inspection. Thus, the black hole mass, M , and the 'quantum' correction, r 0 , can be interpreted in term of the different masses according to: \nM = lim r →∞ M K , (11) \nr 0 = 2 lim r →∞ ( M MS -M K ) . (12) \nIn the next sections, we aim to study the deflection angle and the greybody bounds of the quantum Schwarzschild black hole described by Eq. (6). For instance, light deflection near a black hole has a role in gravitational lensing, which is a powerful tool for probing both the strong-field and weak-field regime of gravity. By studying how the deflection angle is modified in the quantum Schwarzschild scenario, we can gain a deeper understanding of how quantum corrections affect photon trajectories. Greybody factors, on the other hand, influence the spectrum of Hawking radiation. Rigorous bounds on these factors in the quantum Schwarzschild context can reveal how quantum effects modify the black hole evaporation process. This has implications for understanding the end state of black hole evaporation and the information loss paradox. The polymerization parameter λ arises from a specific quantization approach in Loop Quantum Gravity (LQG), and by studying the deflection angle and greybody factors, we can gain some insights into how quantum gravity might alter observable phenomena around black holes, as these effects could be observed with current or future astronomical instruments.", 'III. DEFLECTION ANGLE': "It was shown by [79] that in a static and spherically symmetric (SSS) spacetime with no asymptotic flatness, The Gauss-Bonnet theorem (GBT) can be written as \nˆ α = ∫∫ D KdS + ϕ RS , (13) \nsince \n[∫ K √ gdr ]∣ ∣ ∣ ∣ r = r ps = 0 . (21) \nThe prime denotes differentiation with respect to r . The weak deflection angle for non-asymptotic spacetime is then [79, 81], \nˆ α = ∫ ϕ R ϕ S [ -A ( r ) ( E 2 -A ( r ) ) C ' -E 2 C ( r ) A ( r ) ' 2 A ( r ) ( E 2 -A ( r )) √ B ( r ) C ( r ) ∣ ∣ ∣ ∣ r = r ( ϕ ) ] dϕ + ϕ RS . (22) \nwe find \n[∫ K √ gdr ]∣ ∣ ∣ ∣ r = r ϕ = -ϕ RS -(cos ϕ R -cos ϕ S ) ( v 2 +1 ) r H 2 v 2 b -(cos ϕ R -cos ϕ S ) αr H 2 b . (23) \nWith the above expression, Eq. (18) is needed. One should note that if ϕ S is given, ϕ RS = π -2 ϕ S . The cosine of ϕ is then \ncos ϕ = √ 1 -b 2 u 2 -r H u [ v 2 ( b 2 u 2 -1 ) -1 ] 2 √ v 2 (1 -b 2 u 2 ) , (24) \nK \ngdr \n= \n- \n, \n(20) \nwhere r ps is the radius of the particle's circular orbit, and S and R are the radial positions of the source and receiver respectively. These are the integration domains, and we note that the infinitesimal curve surface dS is given by \ndS = √ gdrdϕ. (14) \nFurthermore, ϕ RS is the coordinate position angle between the source and the receiver defined as ϕ RS = ϕ R -ϕ S , which can be found through the iterative solution of \nF ( u ) = ( du dϕ ) 2 = C ( u ) 2 u 4 A ( u ) B ( u ) [ ( E J ) 2 -A ( u ) ( 1 J 2 + 1 C ( u ) ) ] , (15) \nwhere we have used the substitution r = 1 /u and the corresponding angular momentum and energy of the massive particle given the impact parameter b as \nJ = µvb √ 1 -v 2 , E = µ √ 1 -v 2 . (16) \nwe find \nF ( u ) = 1 b 2 -u 2 -[ 1 + ( b 2 u 2 -1 ) v 2 ] r H u v 2 b 2 + au ( b 2 u 2 -1 ) r H b 2 . (17) \nThe above enables one to solve for the azimuthal separation angle ϕ as \nϕ = arcsin( bu ) + r H [ v 2 ( b 2 u 2 -1 ) -1 ] 2 bv 2 √ 1 -b 2 u 2 , (18) \nwhich is the also the direct expression for ϕ S as u is replaced by u S . Meanwhile, the expression for the receiver is ϕ R = π -ϕ S where u S should be replaced by u R . \nLeaving the angle ϕ for a while, the Gaussian curvature K in terms of connection coefficients can be calculated as [80] \nK = 1 √ g [ ∂ ∂ϕ ( √ g g rr Γ ϕ rr ) -∂ ∂r ( √ g g rr Γ ϕ rϕ )] = -1 √ g [ ∂ ∂r ( √ g g rr Γ ϕ rϕ )] , (19) \nsince Γ ϕ rr = 0 . If in a certain spacetime there is an analytical solution for the r ps , then we have the relation \n∫ \nr \nr \n( \nϕ \n) \nps \n√ \nA \n( \nr \n) \n2 \nA \n( \nr \n) ( \nE \n( \nE \n2 \n- \nA \n( \nr \n) \n2 \n- \nA \n( \nr \n)) \n) \nC \n' \n- \nE \n√ \n2 \nB \n( \nr \n) \nC \n( \nr \n) \nC \n( \nr \n) \nA \n( \nr \n) \n' \n∣ \n∣ \n∣ \n∣ \nr \n= \nr \n( \nϕ \n) \n<!-- image --> \nFIG. 1. Weak deflection angle (in µ as) using M87* parameters. The left panel shows the behavior of the deflection angle of timelike particles according to Eq. (25) as finite distance is considered while α varies. In contrast, the right panel shows how Θ behaves for photons. The case without finite distance is depicted by the dotted lines. We also added the Schwarzschild deflection angle for comparison as shown. The vertical dotted line represents our location from M87* SMBH, which is 16 . 8 Mpc. \n<!-- image --> \nwhich should be applied to the source and the receiver. Using the above expression to Eq. (23), we get the final analytic expression for the weak deflection angle that accommodates both time-like particles and finite distance as \nΘ = [ r H 2 b ( 1 + 1 v 2 + α )]( √ 1 -b 2 u 2 S + √ 1 -b 2 u 2 S ) . (25) \nAssuming that u R = u S , and these are distant from the black hole ( u → 0 ), \nΘ = r H b ( 1 + 1 v 2 + α ) . (26) \nFinally, when v = 1 , \nΘ null = r H b (2 + α ) . (27) \nWe plot the results in Fig. 1. Since the actual parameters are used for the finite distance 1 /u = 16 . 8 Mpc, we need to use a log-log plot to see the changes. The general observation is that the parameter α increases the deflection angle relative to the Schwarzschild case. Also, timelike particles provide a higher deflection angle than photons. The inset plot on the right panel shows the difference between the approximation r →∞ and not ignoring the finite distance. The difference only occurs when the impact parameter of photons is so large (comparable to 16 . 8 Mpc). \nThe weak deflection angle has direct application in a phenomenon called the Einstein ring. Here, we will determine the effects of the α parameter on the Einstein ring. To begin with, let us define the distance of the source and the receiver with respect to the lensing object as d S and, d R respectively. Through the thin lens equation, we then have d RS = d S + d R so that the position of the weak field images is given by [82] \nd \nRS \ntan \nβ \n= \nsin \nθ \n- \nd \nS \ncos(ˆ \nα \n- \nθ \n) \n. \n(28) \nIt is well known that an Einstein ring is formed when β = 0 , and the above equation leads then to the angular radius [83-92] \nθ E ≈ d S d RS ˆ α. (29) \nIn addition, since the Einstein ring is assumed to be small it is safe to take relation b = d R sin θ ∼ d R θ . Then, the weak deflection angle (29) takes the explicit form \nθ E = 1 v √ 2 r H d S d R √ 1 + v 2 (1 + α ) d S + d R . (30) \nd \nR \nsin(ˆ \nα \n- \nθ \n) \nFIG. 2. Einstein ring (in µ as) using M87* parameters. \n<!-- image --> \nThis angular radius is depicted in Figure 2 as a function of the impact parameter for different values of α This plot corresponds to the right panel in Figure 25, where we can see that it produces a greater value for the angular radius. Let us now consider M87*. Our location is d R ≈ 16 . 8 Mpc from the galactic center. Some additional works where the weak deflection angle is calculated are [93-97] and references therein.", 'IV. RIGOROUS BOUNDS OF GREYBODY FACTORS': "In what follows we will focus on bounds for the greybody factors [98-112] of a quantum Schwarzschild black hole inspired by loop quantum gravity as follows: \nds 2 = -| g tt | dt 2 + g rr dr 2 + r 2 ( dθ 2 +sin 2 θdϕ 2 ) (31) \nwhere lapse functions are \ng tt = ( 1 -2 M r ) , (32) \ng rr = 1 ( 1 -r 0 r )( 1 -2 M r ) . (33) \nHere, we consider the Klein-Gordon equation for the massless scalar field and Maxwell equation for electromagnetic field as follows: \n1 √ -g ∂ µ ( √ -gg µν ∂ ν Φ ) -m 2 Φ = 0 (34) \n1 √ -g ∂ µ ( √ -gg σµ g ρν F ρσ ) = 0 . (35) \nwhere F ρσ = ∂ ρ A σ -∂ σ A ρ and A µ is the vector potential. \nIn the case of a spherically symmetric background, with the metric functions given in (31). As usual, we proceed by considering the obeying the separated variable form \nΦ( t, r, θ, ϕ ) = ∑ l,m ψ l ( r ) r Y lm ( θ, ϕ ) e -iωt (36) \nwhere Y lm ( θ, ϕ ) are the spherical harmonic functions and l is the angular momentum quantum number [113-116]. \nIn order to simplify this equation, we can transform the radial coordinate r to the 'tortoise coordinate' r ∗ through the variable transformation \ndr ∗ dr = √ g rr ∣ ∣ g -1 tt ∣ ∣ , (37) \nwhere r ∗ is the tortoise coordinate defined in Schwarzschild metric [117]. We mention that this tortoise coordinate is defined only outside the event horizon. In this way, the radial part of the scalar-field equation of motion can be written as a Schr odinger-like equation \n∂ 2 r ∗ ψ l ( r ∗ ) + ω 2 ψ l ( r ∗ ) = V i ( r ) ψ l ( r ∗ ) (38) \nwith the effective potential V i in the form for scalar ( i = s ) and electromagnetic ( i = e ) \nV s = g rr g ' tt -g tt g ' rr 2 rg 2 rr + g tt l ( l +1) r 2 (39) \nV e = g tt l ( l +1) r 2 (40) \nFor the quantum Schwarzschild black hole, the effective potential for scalar field ( i = s )is calculated as follows: \nV s = (2 M -r ) ( -l ( l +1) r 2 +6 αM 2 -( α +2) Mr ) r 5 , (41) \nand for i = e is calculated as follows: \nV e = -l ( l +1)(2 M -r ) r 3 , (42) \nwhich is same with Schwarzschild case so for there is not any quantum effect from spacetime. \nNow, with help of the effective potential, V ( r ) , we study the lower rigorous bound for the greybody factor of the quantum black hole to probe the impact of α on the bound. The formula to derive the rigorous bound of greybody factor is given as follows [118, 119]: \nT ≥ sech 2 1 2 ω ∫ ∞ -∞ | V i | dr ( √ g rr ∣ ∣ g -1 tt ∣ ∣ , (43) \nand the boundary of the above formula is slightly modified when the cosmological constant is included [99] as follows: \nT ≥ T b = sech 2 1 2 ω ∫ ∞ r H | V i | ( √ g rr ∣ ∣ g -1 tt ∣ ∣ dr = sech 2 ( A ℓ 2 ω ) , (44) \nwhere the factor A ℓ is defined according to the following expression: \nA ℓ = ∫ ∞ r H | V i | ( √ g rr ∣ ∣ g -1 tt ∣ ∣ dr. (45) \nHence, the bounds of the greybody factor of bosons are numerically plotted in Figs. Fig.s 4 for ℓ = 0 and for ℓ = 1 . The graph shows that when the parameter of α increases, the bound of the greybody factor of bosons also increases. It is found that the quantum Schwarzschild black hole behave as good barriers.", 'V. QNMS: DIRAC PERTURBATIONS': "In the following section, we will outline the fundamental expression for calculating Quasinormal Modes (QNMs) for neutral Dirac particles. To maintain the discussion as general as we can, we will also consider a spherically symmetric background in \nFIG. 3. Effective potential for M = 1 , Left: l = 0 and Right: l = 1 \n<!-- image --> \nFIG. 4. Greybody transmission probability for M = 1 , Left: l = 0 and Right: l = 1 \n<!-- image --> \na d -dimensional space-time and finally, we will consider the particular case d = 4 at the end of equations (see [120] for details). The metric is then given by the following line element \nd s 2 = -| g tt | d t 2 + g rr dr 2 + r 2 dΩ 2 d -2 , (46) \nwhere d Ω 2 d -2 represents the metric for the ( d -2) -dimensional sphere and the two metric potentials are then given as usual in this case, i.e., \ng tt = ( 1 -2 M r ) ≡ f ( r ) , (47) \ng rr = 1 ( 1 -r 0 r )( 1 -2 M r ) ≡ ( 1 -r 0 r ) -1 f ( r ) -1 . (48) \nLet us perform a conformal transformation (see [121, 122] and references therein): \ng µν → g µν = Ω 2 g µν , (49) \nψ → ψ = Ω -( d -1) / 2 ψ, (50) \nγ µ ∇ µ ψ → Ω ( d +1) / 2 γ µ ∇ µ ψ, (51) \nNow, the metric becomes, after considering Ω = 1 /r \nd s 2 = -1 r 2 f ( r )d t 2 + 1 r 2 ( 1 -r 0 r ) -1 f ( r ) -1 d r 2 +dΩ 2 d -2 , (52) \nwhere we have considered ψ = r ( d -1) / 2 ψ . Now, let us take advantage of the fact that we can split the t -r part and the ( d -2) -sphere part which allow us to decouple the equation and solve the t -r . Thus, writing down the Dirac equation in the form (for massless particles): \nγ µ ∇ µ ψ = [ ( γ t ∇ t + γ r ∇ r ) ⊗ 1 + γ 5 ⊗ ( γ a ∇ a ) S d -2 ] ψ = 0 , (53) \nwhere as usual ( γ 5 ) 2 = 1 . In what follows, let us change our notation by omitting the bars for simplicity. We shall now let χ ( ± ) be the eigenspinors for the ( d -2) -sphere (see [123] for further details): \nℓ \n( γ a ∇ a ) S d -2 χ ( ± ) ℓ = ± i ( ℓ + d -2 2 ) χ ( ± ) ℓ , (54) \nHere ℓ = 0 , 1 , 2 , . . . . Since the eigenspinors are orthogonal, it is always possible to expand ψ as: \nψ = ∑ ℓ ( ϕ (+) ℓ χ (+) ℓ + ϕ ( -) ℓ χ ( -) ℓ ) . (55) \nThe Dirac equation acquires the simple form: \n{ γ t ∇ t + γ r ∇ r + γ 5 [ ± i ( ℓ + d -2 2 )]} ϕ ( ± ) ℓ = 0 , (56) \nwhich is precisely the 2 -dimensional Dirac equation. In order to solve the corresponding differential equations we make the explicit choice of the Dirac matrices, namely: \nγ t = r √ f ( r ) ( -iσ 3 ) , (57) \nγ r = √ 1 -r 0 r √ f ( r ) rσ 2 , (58) \nand σ i are the so-called Pauli matrices, defined as: \nσ 1 = ( 0 1 1 0 ) , σ 2 = ( 0 -i i 0 ) , σ 3 = ( 1 0 0 -1 ) . (59) \nAlso, γ 5 is written in term of the Pauli matrices as follows \nγ 5 = ( -iσ 3 )( σ 2 ) = -σ 1 . (60) \nThe spin connections are then found to be: \nΓ t = σ 1 ( 1 4 r 2 √ 1 -r 0 r ) d d r ( f ( r ) r 2 ) , (61) \nΓ r = 0 . (62) \nAt this point should be mentioned that the treatment for + sign solution is completely equivalent to the -sign case, the reason why we will focus on of them, the positive one. Also let us define the parameter ξ in term of the dimension d and the angular number ℓ : \nξ ≡ ℓ + 1 2 ( d -2) , (63) \nwith ξ being +1 , +2 , ... and d = 1 , ..., N. Taking the last facts into account, the Dirac equation can then be simplified to be: \n{ r √ f ( r ) ( -iσ 3 ) [ ∂ ∂t + σ 1 ( 1 4 r 2 √ 1 -r 0 r ) d d r ( f ( r ) r 2 )] + √ 1 -r 0 r √ f ( r ) rσ 2 ∂ ∂r +( -σ 1 )( i ) ( ξ ) } ϕ (+) ℓ = 0 (64) \nPutting spatial and temporal part separately, we finally have a first order partial differential equation for ϕ (+) ℓ , i.e., \nσ 2 (√ 1 -r 0 r √ f ( r ) r ) [ ∂ ∂r + r 2 √ f ( r ) d d r ( √ f ( r ) r )] ϕ (+) ℓ -iσ 1 ξϕ (+) ℓ = iσ 3 ( r √ f ( r ) ) ∂ϕ (+) ℓ ∂t . (65) \nNow, let us to obtain solutions of the form: \nϕ (+) ℓ = ( √ f ( r ) r ) -1 / 2 e -iωt ( iG ( r ) F ( r ) ) , (66) \nwhere the potentials are given by \nV ± = W 2 ± d W d r ∗ . (76) \nIt should be mentioned that the effective potentials V + and V -are supersymmetric to each other, the reason why the two functions F and G share the same spectra (i.e., they are isospectral), both for scattering and quasi-normal. In addition, for ϕ ( -) ℓ , we have these two potentials. In this respect, in what follows we will consider using the positive sign, having clarified that the potential is the same in this case. Now, utilizing the standard radial coordinate we have the concrete form of the effective potential according to: \nV + = f ( r ) [ ξ 2 r 2 + √ 1 -r 0 r ( ξf ' ( r ) 2 r √ f ( r ) -ξ √ f ( r ) r 2 )] (77) \nor, replacing the metric potential explicitly, we have \nV + = ( 1 -2 M r ) [ ξ 2 r 2 + √ 1 -r 0 r Mξ r 3 √ 1 -2 M r -ξ r 2 √ 1 -2 M r ] . (78) \nThe Dirac equation can then be reduced to: \nσ 2 (√ 1 -r 0 r √ f ( r ) r ) ( i d G ( r ) d r d F ( r ) d r ) -iσ 1 ξ ( iG ( r ) F ( r ) ) = σ 3 ω ( r √ f ( r ) ) ( iG ( r ) F ( r ) ) . (67) \nBy taking the components separately, we finally write a set of coupled first-order differential equations in terms of variables G ≡ G ( r ) and F ≡ F ( r ) as follows: \n[ √ 1 -r 0 r f ( r ) ] d G ( r ) d r -[ √ f ( r ) r ξ ] G ( r ) = + ωF ( r ) , (68) \n[ √ 1 -r 0 r f ( r ) ] d F ( r ) d r + [ √ f ( r ) r ξ ] F ( r ) = -ωG ( r ) . (69) \nTo calculate the quasinormal modes for neutral Dirac particles of this background, becomes convenient to introduce the well-known tortoise coordinate r ∗ , and the auxiliary function W ( r ) as follows \nr ∗ ( r ) ≡ ∫ r d¯ r [ f (¯ r ) √ 1 -r 0 ¯ r ] -1 , (70) \nW ( r ) = ξ √ f ( r ) r (71) \nTaking advantage of the last two equations, the first-order linear coupled equations (68) and (69) can be reduced to \n[ d d r ∗ -W ( r ) ] G ( r ) = + ωF ( r ) (72) \n[ d d r ∗ + W ( r ) ] F ( r ) = -ωG ( r ) (73) \nThe set of first-order differential equations for G ( r ) and F ( r ) can be trivially decoupled to obtain two Schrodinger-like differential equations, for some concrete effective potential, i.e., \nd 2 F d r ∗ 2 +[ ω 2 -V -] F = 0 , (74) \nd 2 G d r ∗ 2 +[ ω 2 -V + ] G = 0 , (75) \nFIG. 5. Effective potential for this quantum-inspired Schwarzschild black hole. For simplicity, we show V ( r ) varying ξ (proportional to angular number ℓ ) and M (the black hole mass), assuming the classical case, i.e., α = 0 . Left panel: Effective potential with a fixed mass and varying ξ , for α = 0 . Right panel: Effective potential with a fixed value ξ and varying the mass M , for α = 0 . \n<!-- image --> \nAt this point, we should choose appropriated outgoing boundary conditions at the horizon and spatial infinity i.e., nothing should come in from asymptotic infinity to disturb the system and nothing should come out of the horizon. In other words, the boundary conditions for Schrodinger-like equations (74) and (75) which are \n{ F, G } → exp(+ iωr ∗ ) , r ∗ →-∞ , (79) \n{ F, G } → exp( -iωr ∗ ) , r ∗ → + ∞ . (80) \nNote that, as ψ ∼ exp( -iωt ) , a frequency with a negative imaginary part implies a decaying (stable) mode. Conversely, a frequency with a positive imaginary part means an increasing (and therefore unstable) mode. The effective potential V ( r ) is shown for the Dirac case, assuming d = 4 (for simplicity) against the radial coordinate for different values of the set of parameters { ξ, M, α } . We include the Schwarzschild case with α = 0 for comparison purposes. To get some insights into how the effective potential V ( r ) looks like in the Dirac case, we show, in Figs. (5) and (6), its behavior against the radial coordinate for different values of the set of parameters { ξ, M, α } . For comparison reasons, we include the classical Schwarzschild case. We have considered in Figs. (5) and (6) the following cases: \n- · Fig. (5) shows the evolution of the effective potential for the classical case ( α = 0) varying the two free parameters ξ and M . Such figures are included for comparison. From the left panel we can see that as ξ increases, the maximum of the potential increases, at which point it shifts to the right. All the solutions converge at small radii because their associated horizons are equal in contract to the other cases shown. Similarly, the right panel shows that as M increases (for a fixing ξ ), the maximum of the potential decreases, now shifting to the right. \n̸ \n- · Fig. (6) shows the evolution of the effective potential for the quantum-inspired case ( α = 0) varying the two free parameters ξ and M . The case α = 0 is included for comparison. The left panel shows how the effective potential varies for fixed { ξ, M } and different values of the parameter α . The figure confirms that as α increases, the maximum of the potential decreases, at the same time as it shifts to the right. In addition, the effective potential tends to overlap quickly between small and large radii. The middle panel shows the potential V ( r ) for fixed { ξ } and different values of the BH mass M and the parameter α . The right panel shows the potential V ( r ) for fixed { M } and different values of the parameters ξ and α . Note that in the middle and right panels, the quantum corrections are compared with the quantum-inspired case and the modifications, although present, are small.", 'VI. QUASINORMAL SPECTRUM: WKB APPROXIMATION': "The characteristic oscillations of black holes, known as quasinormal modes (QNMs), reveal how the black hole responds to external perturbations. Analogous to the oscillations of a struck bell, these modes represent the 'ringing' of black holes. Quasinormal modes are defined by complex frequencies, where the real component is the oscillation frequency and the imaginary component is the damping rate induced by gravitational wave emission. The quasinormal spectra of BHs can only be obtained accurately in a few cases, e.g: i) when the induced differential equation (for the radial part of the wave \nFIG. 6. Effective potential for this quantum-inspired Schwarzschild black hole. We show V ( r ) varying ξ and M , for several different parameter values α . Left panel: Effective potential fixing ξ and M , varying α . Middle panel: Effective potential for a fixed parameter ξ , varying M and α . Right panel: Effective potential for a fixed mass M , varying ξ and α . All figures show that the peak of the potential is slightly shifted with respect to its classical counterpart ( α = 0) . \n<!-- image --> \nfunction) can be transformed into the Gauss' hypergeometric function (see [124-130] and references therein), or ii) when the potential barrier takes the form of the Poschl-Teller potential (see [131-136] and references therein). Given the complexity and non-trivial nature of the underlying differential equation, it is important to use numerical or semi-analytical methods to compute the relevant quasinormal frequencies. As a result, several strategies have been developed for this purpose, some of which are widely used. \nSpecifically: i) the method of continued fraction, together with its modifications, originally introduced in [137-139], ii) the Frobenius method and its generalization (see the references in [140-142]), iii) the asymptotic iteration method (see [143-145] and references therein). among other possibilities. Additional details and other methods commonly used to compute QNMs can be found in [146]. Since the effective potential has a well-behaved form, in the present work we will use the WKB semiclassical method to obtain the corresponding quasinormal frequencies (see [147-151] and references therein). Schutz and Will obtained the first-order calculation using the WKB method [147], followed by improvements made by Iyer and Will [148], who developed a semi-analytic expression including second- and third-order corrections. The method and its lower-order corrections are quite efficient for determining the lowest overtones among the complex frequencies of, for example, an oscillating Schwarzschild BH. The accuracy of the approximation improves as the angular harmonic index ℓ ( ℓ ∝ ξ ) increases but deteriorates as the overtone index increases. After all these works and improvements, R.A. Konoplya generalized the method up to the 6th order [152], while J. Matyjasek and M. Opala improved the formulas from the 7th to the 13th order [153]. \nBe aware and note that the higher-order equations of the WKB method have not been mathematically proven to consistently converge to the theoretical resolution. In this sense, Konoplya proposed a 'preliminary' criterion for identifying errors by simply subtracting frequencies at successive orders using the WKB method. The idea is reasonable, but it does not provide a strict criterion for selecting the order of the WKB approximation that minimizes the errors. It is important to note that the WKB approximation gives the best results when the sixth/seventh order is used. Note that it depends strongly on the background, the reasons why we cannot ensure that a certain order is the best regardless of the background. Since the metric potentials are not complicated and the effective potential has the appropriate form, the the WKB method is sufficient to obtain accurate results. The method takes advantage of the one-dimensional Schrodinger-like equation corresponding to a potential barrier. In short, the WKB formula uses the matching of asymptotic solutions, i.e. a combination of incoming and outgoing waves, together with a Taylor expansion centered around the peak of the potential barrier at x = x 0 . More precisely, the expansion covers the region between the two turning points (the roots of the effective potential U ( x, ω ) ≡ V ( x ) -ω 2 ). In this paper, we will consider the WKB method for computing 6th-order QN spectra, using the following generalized expression \nω 2 n = V 0 +( -2 V '' 0 ) 1 / 2 Λ( n ) -iν ( -2 V '' 0 ) 1 / 2 [1 + Ω( n )] , (81) \nwhere i) the second derivative of the potential at maximum is represented by V '' 0 , ii) ν = n +1 / 2 , iii) the maximum of the effective barrier is represented by V 0 , and finally iv) n = 0 , 1 , 2 ... is the overtone number. The functions Λ( n ) , Ω( n ) are well-known and also quite long, which is the reason why we bypass the deduction and write down the concrete expression for them. Instead, they can be consulted in [150]. Finally, we have used a Wolfram Mathematica [154] notebook using the WKB method on any order from one to six [155]. In our calculations, we will consider only values n < ξ . For higher order WKB corrections, the interested reader may consult [155, 156]. \nFor Dirac perturbations, we summarize our results in table (I) varying n , α and ξ . We used the QN frequencies using the WKB method of 6th order . Based on the frequencies obtained, all modes are found to be stable (given the negative value of the quasinormal frequencies). In order to guaranties the accuracy of the method, we consider the limited range n < ξ , \nTABLE I. Dirac Quasinormal frequencies (varying ξ , n and α ) with M = 1 for the model considered in this work. \n<!-- image --> \nFIG. 7. Dirac Quasinormal modes for this quantum-inspired Schwarzschild black hole. We show ω 0 varying ξ , M and n . Left panel: Real part of quasinormal modes against the parameter α , assuming a fixed mass and varying ξ , and n . Right panel: Imaginary part of quasinormal modes against the parameter α , assuming a fixed mass and varying ξ , and n . \n<!-- image --> \na sector in which the WKB method works perfectly and agrees with the other papers (we became aware of these papers during the revision process) [76, 157-159].", 'VII. CONCLUSION AND FUTURE PERSPECTIVES': 'In the present paper, we have mainly investigated i) the weak deflection angle and ii) the rigorous Greybody bounds of a recent black hole solution in four-dimensional spacetime inspired by loop quantum gravity. After a brief and compact review of the relevant expressions, such as the metric potentials and the alternative mass definitions, we have used the Gauss-Bonnet theorem to obtain a closed expression for computing the weak deflection angle. We have thus obtained an analytical formula to find ˆ α , using as a background the recently discovered quantum Schwarzschild black hole in terms of the black hole mass, or equivalently the Schwarzschild radius r H , plus the auxiliary parameter α , which is always less than one. It should be emphasised that the weak field deflection provides an alternative angle to study quantum effects when light rays are scattered with very large impact parameters. The weak deflection angle (which includes both time-like particles and finite distance) Θ (and also Θ null ) is obtained and plotted in Fig.(1) for different values of the parameter α and compared with the Schwarzschild black hole. We observe a qualitatively similar behaviour, although higher than the classical counterpart. A closed quantity related to the weak deflection angle is also computed with the manuscript. This is the case of the so-called Einstein ring, θ E , which is computed explicitly and shown in Fig.(2) for several values of the parameter α . We can conclude that the Einstein ring is significantly lower than the Schwarzschild case (using M87* parameters). In addition, we have obtained the rigorous Greybody bounds for this black hole solution by considering the Klein-Gordon equation for massless scalar fields, writing the corresponding Schrodinger-like equation, and identifying the effective potential. Thus we have well-defined cases: the first when s = 0 and the second when s = 1 , where s is the spin of the perturbation. Having obtained V l ( r ) (also plotted in Fig.(3)), we have obtained T b (i.e. the greybody transmission probability) for both cases. The exact expressions for T b are found and the numerical solution is plotted in Fig.(4). Our results show that as α increases, the transmission probability tends to the Schwarzschild bound. \nFinally, we compute the Dirac quasinormal modes for this quantum Schwarzschild black hole. We first study the effective potential (see Figs. (5) and (6)), as well as the real and imaginary parts of the quasinormal frequencies (see also Fig. (7)), for different values of the parameter α . In the light of our calculations, we conclude that this black hole is stable to perturbations (given the negative sign of the imaginary part of the quasinormal frequency).', 'VIII. 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2024arXiv240817419V | Context. To date three nuclear transients have been associated with highenergy neutrino events. These transients are generally thought to be powered by tidal disruptions of stars TDEs by massive black holes. However AT2019aalc hosted in a Seyfert1 galaxy was not yet classified due to a lack of multiwavelength observations. Interestingly the source has rebrightened 4 years after its discovery. Aims. We aim to classify the transient and explain the mechanism responsible for its second optical flare. Methods. We conducted a multiwavelength monitoring program from radio to Xrays of AT2019aalc during its rebrightening in 2023. Results. The observations revealed a uniquely bright UV counterpart and multiple Xray flares during the second optical flaring episode of the transient. The second flare similarly to the first one is also accompanied by IR dust echo emission. A longterm radio flare is found with an inverted spectrum. Optical spectroscopic observations reveal the presence of Bowen Fluorescence lines and strong highionization coronal lines indicating an extreme level of ionization in the system. Conclusions. The results suggest that the transient can be classified as a Bowen Fluorescence Flare BFF a relatively new subclass of flaring active galactic nuclei AGN. AT2019aalc can be also classified as an extreme coronal line emitter ECLE. We found that in addition to AT2019aalc another BFF AT2021loi is spatially coincident with a highenergy neutrino event. The multiwavelength properties of these transients suggest a possible connection between ECLEs BFFs and TDEs in AGN. | 2024-08-01T00:00:00Z | ['2024arXiv240817419M', 'arXiv:2408.17419', '2024arXiv240817419V', '10.48550/arXiv.2408.17419'] | ['Astrophysics - High Energy Astrophysical Phenomena', 'Astrophysics - Astrophysics of Galaxies'] | Back from the dead AT2019aalc as a candidate repeating TDE in an AGN | 2,024 | 175 | 0.62 | ['EPRINT_HTML', 'EPRINT_PDF'] | 3 | https://arxiv.org/pdf/2408.17419.pdf | {'Back from the dead: AT2019aalc as a candidate repeating TDE in an AGN': 'Patrik Milán Veres 1 , Anna Franckowiak 1 , 2 , Sjoert van Velzen 3 , Bjoern Adebahr 1 , Sam Taziaux 1 , Jannis Necker 4 , 5 , Robert Stein 6 , Alexander Kier 1 , Ancla Müller 1 , Dominik J. Bomans 1 , 2 , Nuria Jordana-Mitjans 1 , Marek Kowalski 4 , Erica Hammerstein 7 , 8 , 9 , Elena Marci-Boehncke 1 , Simeon Reusch 4 , 5 , Simone Garrappa 10 , Sam Rose 6 , and Kaustav Kashyap Das 11 \n- 1 Ruhr University Bochum, Faculty of Physics and Astronomy, Astronomical Institute (AIRUB), Universitätsstraße 150, 44801 Bochum, Germany e-mail: [email protected]\n- 2 Ruhr Astroparticle and Plasma Physics Center (RAPP Center)\n- 3 Leiden Observatory, Leiden University, Postbus 9513, 2300 RA Leiden, The Netherlands\n- 4 Deutsches Elektronen-Synchrotron (DESY), Platanenallee 6, D-15378 Zeuthen, Germany\n- 5 Institut fur Physik, Humboldt-Universität zu Berlin, D-12489 Berlin, Germany\n- 6 Division of Physics, Mathematics, and Astronomy, California Institute of Technology, Pasadena, CA 91125, USA\n- 7 University of Maryland, Department of Astronomy, 4296 Stadium Dr, College Park, MD 20742, USA\n- 8 NASA Goddard Space Flight Center, Astrophysics Science Division, 8800 Greenbelt Rd, Greenbelt, MD 20771, USA\n- 9 Center for Research and Exploration in Space Science and Technology, NASA / GSFC, Greenbelt, MD 20771, USA\n- 10 Department of Particle Physics and Astrophysics, Weizmann Institute of Science, 76100 Rehovot, Israel.\n- 11 Cahill Center for Astrophysics, California Institute of Technology, MC 249-17, 1200 E California Boulevard, Pasadena, CA, 91125, USA', 'ABSTRACT': 'Context. To date, three nuclear transients have been associated with high-energy neutrino events. These transients are generally thought to be powered by tidal disruptions of stars (TDEs) by massive black holes. However, AT2019aalc, hosted in a Seyfert-1 galaxy, was not yet classified due to a lack of multiwavelength observations. Interestingly, the source has re-brightened 4 years after its discovery. \nAims. We aim to classify the transient and explain the mechanism responsible for its second optical flare. \nMethods. We conducted a multi-wavelength monitoring program (from radio to X-rays) of AT2019aalc during its re-brightening in 2023. \nResults. The observations revealed a uniquely bright UV counterpart and multiple X-ray flares during the second optical flaring episode of the transient. The second flare, similarly to the first one, is also accompanied by IR dust echo emission. A long-term radio flare is found with an inverted spectrum. Optical spectroscopic observations reveal the presence of Bowen Fluorescence lines and strong high-ionization coronal lines indicating an extreme level of ionization in the system. \nConclusions. The results suggest that the transient can be classified as a Bowen Fluorescence Flare (BFF), a relatively new sub-class of flaring active galactic nuclei (AGN). AT2019aalc can be also classified as an extreme coronal line emitter (ECLE). We found that, in addition to AT2019aalc, another BFF AT2021loi is spatially coincident with a high-energy neutrino event. The multi-wavelength properties of these transients suggest a possible connection between ECLEs, BFFs and TDEs in AGN. \nKey words. galaxies: active - galaxies: Seyfert - quasars: emission lines - individual: AT2019aalc - Neutrinos', '1. Introduction': "Tidal disruption events (TDEs) are rare transient events, which occur when a star approaches a supermassive black hole (SMBH) and the tidal forces of the latter rip the star apart (Rees 1988). About half of the star's material is accreted around the black hole, generating a luminous outburst across the electromagnetic spectrum. Despite most of these transient sources have been discovered by optical telescopes, the origin of their optical / UV emission is poorly understood. One model is colliding stellar debris stream shocks (Piran et al. 2015). Another scenario is reprocessed soft X-ray / far UV emission (originating from the debris accretion onto the black hole) by an extended elliptical disk, or an envelope, or an outflow surrounding the central debris disk (for a recent review see Bu et al. 2022). TDE optical light curves are characterized by quick and colorless evolving but also \nfast (on monthly timescales) decaying optical emission. Several TDEs were detected in X-rays with a very soft X-ray spectrum typically explained by late-time accretion disk formation (Saxton et al. 2020). In addition, a handful of TDEs produced detectable radio emission, explained by delayed 1 non-relativistic outflows (Horesh et al. 2021; Cendes et al. 2023), or only in 4 instances by newly launched relativistic on-axis jets (Berger et al. 2012; Brown et al. 2015; Cenko et al. 2012; De Colle & Lu 2020; Yao et al. 2023a). Spectroscopically, TDEs are characterized by a strong blue continuum and broad emission lines ( > 10 4 kms -1 ) e.g., H α and H β . As the photosphere shrinks over time, the flux of the ionized lines such as He ii 4686Å tend to increase while the He ii / H α ratio increases (e.g., Hung et al. 2017; Charalam- \nulos et al. 2022a) and, in a few cases, Bowen Fluorescence (BF; Bowen 1934, 1935) lines e.g., N iii at 4640Å appear (e.g., Blagorodnova et al. 2019; Charalampopoulos et al. 2022a). \nTo date, roughly 120 promising candidate TDEs are known. However, since most of the above-mentioned observed properties do not exclusively characterize TDEs, spectroscopic coverage is required to distinguish TDEs from 'TDE impostors' such as conventionally accreting active galactic nuclei (AGN), changing-look AGN (CLAGN), supernovae (SNe) or even stellar collisions (Zabludo ff et al. 2021). \nTDEs generally result in one optical flare, however, in a few cases a re-brightening episode (or episodes) was detected even years after the initial flare. These might be explained in the following way. Under special circumstances, a star on a grazing orbit might be partially disrupted and only a fraction of the stellar mass will be deposited onto the SMBH. The secondary disruption of a surviving core after it reaches the pericenter again results in X-ray re-brightening explained by renewed accretion. The first systematically identified repeating partial TDE, AT2020vdq, has been classified after simultaneous multi-wavelength monitoring (Somalwar et al. 2023). Only a few other partial TDEs or candidates are known to date, these are ASASSN 14ko (Payne et al. 2021); eRASSt J045650.3-203750 (Liu et al. 2023); RX J133157.6-324319.7 (Hampel et al. 2022); AT2018fyk (Wevers et al. 2019a, 2023); AT2019avd (Chen et al. 2022) and AT2022dbl (Lin et al. 2024). \nBased on the observed properties of TDE host galaxies, French et al. (2020) estimated a TDE rate of 10 -5 per year per galaxy which is well borne out by recent data (Yao et al. 2023b). Importantly, a similar fraction of TDEs should take place in AGN hosts (e.g., Chan et al. 2019; Ryu et al. 2024). The multiwavelength characteristics of such a TDE should clearly di ff er from the ones taken place in quiescent galaxies, as the AGN is surrounded by a pre-existing accretion disk. Due to the extreme di ff erence in density between the two, neither the star nor the accretion disk are a ff ected after the first passage through the disk. However, the confined debris stream has such a low density after the star is tidally disturbed that when it interacts with the disk the collision between the stream and the disk can potentially dissipate much of the kinetic energy possessed by the stream and the disk gas near the impact point, and it can be radiated away (Chan et al. 2019). These transients are challenging to distinguish from AGN variability. Only a few candidate TDEs taken place in AGN (in the following: TDE-AGN) were studied earlier (e.g., PS16dtm - a TDE in a Narrow-Line Seyfert-1 (NLSy1) galaxy (Blanchard et al. 2017) or the multiple soft X-ray flares in IC 3599 (Campana et al. 2015) and GSN 069 (Shu et al. 2018)). In contrast with TDEs seen in earlier quiescent galaxies, these transients are characterized by slowly decaying optical emission and bumps tend to appear after the peak. \nThe family tree of AGN-related transients became much broader in the past years (e.g., Frederick et al. 2021). In addition to TDEs, flaring NLSy1s, CLAGN and di ff erent types of SNe were classified as typical nuclear transients. Recently, Trakhtenbrot et al. (2019) identified a new class of flares from accreting SMBHs. In 2019, the transient events AT2017bgt, F010042237 and OGLE17aaj have been identified as the first members of the class based on their unique optical spectroscopic features. Later, the peculiar X-ray loud transient AT2019avd (Trakhtenbrot et al. 2020) was found with remarkable common properties, however, its classification as an AT2017bgt-like transient is unclear. A more promising case is AT2021loi (Makrygianni et al. 2023), a UV-bright transient event with very similar multiwavelength properties to the 3 instances studied by Trakhtenbrot \net al. (2019). Those show unobscured AGN-like optical spectra with significant and persistent BF lines, extremely strong UV emission and slowly, on yearly scales, decaying optical emission. These AGN are characterized by significantly and longerterm intensified accretion and might be powered by a binary TDE or an interaction between an outflow and the broad-line region (BLR) or binary SMBHs (Trakhtenbrot et al. 2019). Notably, these transients take place in di ff erent types of AGN; AT2017bgt and OGLE17aaj in narrow-line AGN, AT2021loi in a broad-line AGN, F01004-2237 in an ultra-luminous infrared galaxy (ULIRG) while AT2019avd occurred in an earlier quiescent galaxy. Hereafter, we will refer to this new class of AGN flares as Bowen Fluorescence Flares (BFFs), following Makrygianni et al. (2023). Using the Zwicky Transient Facility (ZTF, Masci et al. 2019) Public Survey data, Dgany et al. (2023) found only one BFF candidate (AT2021seu) among 223 spectroscopically classified transients, suggesting that these flares are rare. Other candidates, namely VT J1548 (Somalwar et al. 2022) and AT2022fpx, (Koljonen et al. 2024) were found recently. \nIn recent years, two nuclear transients were found to be spatially coincident with neutrino events detected by the IceCube Neutrino Observatory based on ZTF follow-up campaigns of IceCube neutrino alerts (Stein et al. 2023). The TDE AT2019dsg is likely associated with the IceCube neutrino, IC191001A (Stein et al. 2021) and a further promising association, between the candidate TDE AT2019fdr and the IceCube neutrino IC200530A (Reusch et al. 2022) points to TDEs as a potential new class of sources of the extragalactic neutrinos. Both sources were found to have unusually luminous mid-infrared emission, and a search for similar flares (van Velzen et al. 2024) led to the identification of the transient event AT2019aalc (a.k.a. ZTF19aaejtoy) with the neutrino alert IC191119A (see Sect. 2. for explanation). \nIn this paper, we focus on the nuclear transient event and neutrino source candidate AT2019aalc especially on its significant optical re-brightening that has started 4 years after its initial flare (Veres et al. 2023). \nWe conducted multi-wavelength observations (from radio to X-rays) during the re-brightening and compare it to the initial flare associated with the high-energy neutrino, with the aim to characterize the multi-wavelength properties of this unusual transient event and to classify it. \nThe paper is organized as follows. We explain the discovery of the transient event and give an insight about its host galaxy in Sect. 2, we summarize our observing campaign and the data reduction in Sect. 3, we present the results of the observations in Sect. 4 and discuss them in Sect. 5 and finally give a summary and conclude in Sect. 6. Throughout the paper, we adopt a flat Λ CDMcosmological model with parameters H 0 = 70 km s -1 Mpc -1 , ΩΛ = 0 . 73, and Ω m = 0 . 27. In this model (Wright 2006), 1 mas angular size corresponds to 0 . 72 pc projected linear size at the source redshift z = 0 . 0356 (Ahn et al. 2012). This redshift corresponds to a luminosity distance of 158 . 3 Mpc. All magnitudes are given in AB system (Oke 1974).", '2. The transient event: AT2019aalc': 'Studying the infrared properties of AT2019dsg and AT2019fdr have revealed a strong reverberation signal i.e. the infrared dust echo, known as reprocessed emission of optical / UV photons in the infrared by hot dust around the SMBH, at a distance of 0 . 1 -1 pc (Dou et al. 2016; Lu et al. 2016; van Velzen et al. 2021b, 2024). In addition to the exceptionally high infrared luminosities (due to the infrared echo) of these transients, the neutrino detection times appear to be temporally consistent with the \ninfrared luminosity peaks (Stein et al. 2021; Reusch et al. 2022). This discovery motivated a systematic search for neutrino emission using an extended sample of black hole flares (van Velzen et al. 2024). This archival search revealed the nuclear transient AT2019aalc with a powerful optical flare and dust echo emission in spatial coincidence with the high-energy neutrino event, IC191119A (IceCube Collaboration 2019) detected by the IceCube Observatory. \nThe host galaxy of AT2019aalc (2MASX J15241665 + 0451192) is a barred spiral (SBa) galaxy hosting an active nucleus (Oh et al. 2013). Based on its Sloan Digital Sky Survey (SDSS; Abazajian et al. 2009) spectrum it was further classified as a broad-line Seyfert-1 galaxy (Trump et al. 2013; Liu et al. 2019). The galaxy was detected in archival radio sky surveys e.g., the Very Large Array Faint Images of the Radio Sky at Twenty-cm (VLA FIRST; Condon et al. 1998). Fig. 1 shows the Pan-STARRS (Chambers et al. 2016) i-band optical image of the host galaxy together with the contours of the FIRST radio data (at 1 . 4 GHz) and the position of the X-ray counterpart detected by Swift / XRT in 2023 / 2024. The host galaxy is not present in any X-ray catalogs prior to the optical discovery of the transient in 2019. \nIn addition to the unusually significant infrared echo, AT2019aalc shares multi-wavelength characteristics with the two other neutrino candidate transient events: a soft X-ray spectrum and transient radio emission (van Velzen et al. 2024). Moreover, similar to the other two instances, the infrared peak of the transient event is temporarily coincident with the detection time of the likely associated neutrino event. Notably, Winter & Lunardini (2023) estimated the highest neutrino fluence for AT2019aalc, due to its high estimated SMBH mass ( M BH = 10 7 . 2 M ⊙ , van Velzen et al. 2024) and low redshift ( z = 0 . 0356, Ahn et al. 2012). \nSince the proposed association with the high-energy neutrino event was announced roughly 2 years after the discovery of the transient event, the source was not at the focus of attention or being directly monitored at the time when the neutrino was emitted. Thus, the lack of spectroscopic monitoring during the first optical flare hindered a clear classification of the event. Nevertheless, out of the more than 10 4 AGN detected by the ZTF, less than 1% show similarly rapid and large outbursts (Reusch et al. 2022; van Velzen et al. 2024) suggesting a very unusual AGN case. The TDE-characteristic evolution of the blackbody light curve including a large and rapid optical flux increase is compatible with a TDE-related scenario. However, due to the host galaxy classification, extreme AGN variability provides an alternative explanation for the transient event. \nInterestingly, roughly 4 years after the first flare, a significant optical re-brightening started in mid-May 2023 (Veres et al. 2023), illustrated by the long-term ZTF di ff erence light curve shown in Fig. 2. Although double-peaked optical light curves could be a quasi-universal signature of TDEs around massive ( > 10 7 M ⊙ ) black holes (Wevers et al. 2019b), such a long double peaked light curve with two comparable peak magnitudes would be very untypical for a regular TDE. Throughout this paper we will discuss the results of our observations taken around and after the peak of the second optical flare in the light of di ff erent TDE,- and AGN-related scenarios. \nFig. 1. Pan-STARRS i-band image centered on the host galaxy position of the transient event AT2019aalc. The object located to the west of the galaxy core is a foreground star. The green contours represent the FIRST radio observations taken at 1 . 4 GHz. The lowest contour level is drawn at 3 σ image noise level corresponding to 0 . 4 mJy beam -1 . Additional positive contour levels increase by a factor of 2. The peak brightness is 5 . 8 mJy beam -1 . The red circle points to the Swift -XRT 0 . 3 -10 keV astrometrically-corrected position, determined using the Swift -XRT data products generator (Evans et al. 2014). Its size indicates the 90% confidence positional error. \n<!-- image --> \nFig. 2. Optical light curve of AT2019aalc based on ZTF di ff erence photometry. The transparent markers indicate negative flux i.e. the flux decreased below the mean flux of the reference images. Two distinct flares can be seen. The second flare peaked 4 years after the first one and is still ongoing. The blue vertical line indicates the arrival time of the neutrino event IC-191119 associated with the transient. \n<!-- image -->', '3. Observations and data reduction': '3.1. Optical/UV \n3.1.1. ZTF \nZTF forced photometry at the location of AT2019aalc was obtained following the recommendations outlined in Masci et al. \n(2023). We corrected for extinction with adopting a galactic extinction of E ( B -V ) = 0 . 039 (Schlafly & Finkbeiner 2011). As shown in Fig. 2, the forced photometry yields a negative flux in the period between the two flares (and also just before the first flare). This implies that during these epochs, the flux decreased below the mean flux of the reference images. For the r -band these were obtained in between March and May, 2018. While the baseline used to obtain the di ff erence flux of the g -band is based on images obtained between March and September, 2018.', '3.1.2. UVOT': 'The Neil Gehrels Swift Observatory (Arnaud 1996a) performed 25 Target of Opportunity (ToO) observations (project codes: 18972 and 19188, Proposers: Reusch, Veres& Sniegowska) on AT2019aalc from 2023-06-23 to 2024-05-27 with a total exposure time of 33130 ks. The Ultraviolet and Optical Telescope (UVOT, Roming et al. 2005) onboard the satellite observed the target field with the following four filters in each epoch: UW2 (central wavelength, 1928 Å), UVM2 (2246 Å), UW1 (2600 Å) and U (3465 Å). We analyzed the UVOT images using the Swift UVOT tools included in the heasoft 6.30 software package. We measured the flux using uvotsource , applying a circular aperature of 15 arcseconds (which capture nearly all the flux of the host galaxy). The background was measured using four 6 arcsecond regions in four quadrants away from the host galaxy. To estimate the UV flux before the flare, we use GALEX (Martin et al. 2005) observations obtained in 2007 and 2011. We use the same 15" circular aperture that was applied to the Swift / UVOT observations and use the g P hoton software (Million et al. 2016) to extract the flux. Over the 4 year baseline the GALEX observations, the NUV magnitudes change by about 0.3 mag. Following van Velzen et al. (2021a), we combine the mean GALEX flux with the SDSS (York et al. 2000) magnitudes and apply Prospector (Johnson & Leja 2017) to find the best-fit stellar population synthesis model (Conroy et al. 2009) that describes these data. From this model we estimate the baseline flux in the UVOT filters and this baseline is subtracted to obtain the di ff erence flux. Finally, we corrected for galactic dust extinction for each filter. \nThe baseline flux in the UVM2 band is 17.2, implying a flux increase of 1.8 mag relative to the peak of the UVM2 light curve. This large flux increase implies that, at the peak of the flare, we are not sensitive to the value of the baseline flux. This is useful because the baseline of the ZTF forced photometry was obtained around 2018 (at the start of the ZTF survey), while the baseline that is used to obtain the UVOT di ff erence flux is mainly determined by the GALEX observations that were obtained a decade earlier.', '3.2. Infrared': 'The Palomar 200-inch (P200) near-infrared telescope equipped with the Wide Field Infrared Camera (WIRC, Wilson et al. 2003) observed AT2019aalc on 2023-07-05. The imaging was obtained in the J-, H,- and Ks-bands. The data was processed with the S warp software (Bertin et al. 2002). We performed background subtraction using the photutils . background class and measured the flux using a circular aperture with a radius of 9 pixels. The corresponding zero-point magnitudes for this aperture size were calculated using the automatic pipeline for each filter. The error bars are the computed Poisson and background error of the \nphotometry 2 and we also took into account the uncertainty of the derived zero-point magnitudes. Finally, we corrected for galactic dust extinction. \nWe obtained mid-infrared photometry from archival observations of the Wide-Field Survey Explorer (WISE, Wright et al. 2010), which is continuously monitoring the sky at 3 . 4 µ m(W1) and, due to the NEOWISE project (Mainzer et al. 2011), at 4 . 6 µ m (W2). It visits any point in the sky roughly every six months for about ten times. Using timewise (Necker & Mechbal 2024; Necker et al. 2024), we downloaded all available multi-epoch photometry data from the AllWISE data release and single-epoch photometry from the NEOWISE-R data releases and stacked the data per visit to obtain robust measurements. All selected single-exposure photometry data points are within 1 arcsecond of the nucleus of the host. The resulting light curve spans almost 14 years, starting in February 2010 until July 2023.', '3.3. X-ray': "The Swift 's X-ray Telescope (XRT, Burrows et al. 2005) observations were obtained simultaneously with the UVOT observations. The observations were performed in photon counting (PC) mode. For each of the epochs, we performed the data reduction using the Space Science Data Center (SSDC) interactive archive in the following way. First, cleaned event-files were produced using the xrtpipeline task. The exposure map, used to correct for the loss of flux caused by some of the CCD pixels not being used to collect data, was created with the xrtexpomap tool. The ximage tool allowed us to determine the source detection level and the source position. In case of non-detections, 3 σ upper limits for the count rate were derived and converted to flux using the WebPIMMS 3 tool and assuming the mean power-law index of the observations yielded in detections. When our target source was detected above the 3 σ level, we ran the swxrtdas tool in order to extract the source and the background spectra. Source counts were retrieved from a circle with a radius of 20 pixel (47 arcseconds), while background counts were extracted using an annular region with 80 / 120 pixel inner / outer radius. In addition, Ancillary Response Files (ARFs) i.e. e ff ective response files for the extracted spectra and Response Matrices Files (RMF), that are used to link the instrumental channel scale with the physical wavelength scale, were automatically created. Finally, the grppha tool of the heasoft 6.32 software package was used to perform background subtraction using the source and background spectra and to create a readable spectrum for spectral fitting tools. The relevant ARF and RMF files were applied during this step. We rebinned each spectrum to have a minimum of 3 counts per spectral bin. Channels below 0 . 3 keV were ignored. \nWe fitted the individual X-ray spectra using the XSPEC fitting environment (Arnaud 1996b) in order to be able to study the X-ray flux evolution of AT2019aalc. First, we have determined the needed Galactic neutral hydrogen column density, N H, based on the HI4PI survey (HI4PI Collaboration et al. 2016). Then, we loaded the source spectrum and converted the instrumental Channel scale to 'physical' wavelength scale. The spectrum was then fitted using Cash-statistic (C-stat, Cash 1976). To find the best-fitting model, we first fitted the stacked spectrum with a ab- \nsorbed power-law (T babs * powerlaw in XSPEC) model as most of the AGN are dominated by a simple power-law and the last XMM data of AT2019aalc before our monitoring program was best-fit with a power-law model (Pasham 2023). We performed renormalization and ignored bad data points. When fitting the model, the galactic column density was fixed whilst the powerlaw index and the normalization were free to vary. The resultant fit quality is C-Stat = 161 . 49 for 168 degrees of freedom. We than added a blackbody model component, to take into account the soft excess. The blackbody temperature and the normalization of the new component were also free to vary and we fitted again which resulted in a better fit with C-Stat = 149 . 0 for 166 degrees of freedom. We implemented the likelihood ratio test (lrt in XSPEC) to estimate the statistical significance of the inclusion of the blackbody component. This test runs a given number of simulations of datasets based on the two models provided (which are the simple power-law and the more complex powerlaw + blackbody models in our case) and calculates the likelihood ratio for the more complex model relative to the simpler one. After running 100 simulations, the test resulted only in smaller likelihood di ff erences than the observed one ( ≈ 12) clearly implying that the di ff erence is unlikely to be due to random fluctuations. Therefore, accounting for the soft X-ray excess results in a significantly better fit than a normal power-law fit to model the stacked spectrum. The source luminosity of each dataset was determined with adding the clumin component in front of the model components. We fixed the normalization in the additive models (power-law and blackbody) and performed the fit again. Finally, we derived the 90% confidence level (2 . 706) error of the luminosity and the power-law index and, when it was relevant, the blackbody temperature. When fitting the individual datasets, a simple power-law model was enough to adequately describe the data and the soft excess becomes significant only in the stacked spectrum. Only one of the individual datasets is better fit with a blackbody model than with a power-law + blackbody or a simple power-law models.", '3.4.1. VLBI observations': "We observed AT2019aalc with the European VLBI Network (EVN) and the enhanced Multi-Element Remotely Linked Interferometer Network (e-MERLIN, Garrington et al. 2004). Our very long baseline interferomety (VLBI) maps enable us to study the pc-scale radio structure of AT2019aalc. These observations were carried out at 1 . 7 GHz (L-band) on 2023 June 09 (project code: EV027, PI: Veres). Each of the eight intermediatefrequency channels (IFs) were divided into thirty-two 500 kHz spectral channels, resulting in a total bandwidth of 128 MHz. The whole observation time was 10 hr. The experiment was performed in phase-referencing mode with 5-min duty cycles (with spending 3 . 5 min on the target during each cycle) which resulted in an on-source time of 6 . 7 h including fringe-finder scans and slewing. The following 15 antennas participated in the observations: Cm: Cambridge, UK; Da: Darnhall, UK; De: De ff ord, UK, Ef - E ff elsberg, Germany; Hh - Hartebeesthoek, South Africa; Jb - Jodrell Bank Mk2, UK; Kn - Knockin, UK; MC - Medicina, Italy; Pi - Pickmere, UK; O8 - Onsala, Sweden; T6 - Tianma, China; Tr - Toru'n, Poland; Ur - Urumqi, China; Wb - Westerbork, The Netherlands. The data were recorded in left and right circular polarizations with a data rate of 1 Gbit s -1 . The correlation was done at the EVN Data Processor (Keimpema et al. 2015) at the Joint Institute for VLBI European Research In- \nrastructure Consortium (Dwingeloo, The Netherlands) with an averaging time of 2 s. The Compact Symmetric Object (CSO) 1521 + 0430 (a.k.a. 4C 04.51) at an angular distance of 0 . 83 · was observed as phase-reference calibrator. Earlier EVN observations of this calibrator source taken at 1 . 7 GHz resolved two lobes, with a hint of a jet-like emission (Xiang et al. 2002). In addition, the following radio sources were observed as fringefinders: J1430 + 1043, J1550 + 0527 and J1751 + 0939. The bright CSO J1407 + 2827 (a.k.a. OQ208) was observed as amplitude calibrator and in order to calibrate the Merlin antennas properly. \nThe data were calibrated with the U.S. National Radio Astronomy Observatory Astronomical Image Processing System (AIPS, Greisen 1990) software package following the standard procedures. We performed time flagging using the .uvflg ancilliary table. The amplitude calibration has been done using the measured system temperatures 4 and gain curves of each telescope contained in the .antab ancilliary table. Since our observations were conducted at below 5 Ghz, we corrected for the dispersive ionospheric delays. Then, we corrected for the parallactic angle and performed global fringe-fitting for the calibrator sources. We imaged the phase-calibrator with the D ifmap software (Shepherd et al. 1994), using the hybrid mapping technique with clean deconvolution (Högbom 1974). It included several iterations of cleaning and phase self-calibration before we performed an overall amplitude self-calibration to obtain the antenna-specific gain correction factors. We then applied these amplitude corrections for telescopes with > 5% deviation from the gain solutions in AIPS. Using the clean component (CC) model we built up in D ifmap , we repeated the fringe-fitting in AIPS for the phase-reference calibrator source to correct for the source structure since it contributes to the measured interferometric phases. Finally, we interpolated the solutions of the last fringe-fitting to the data of our target source. As a final step, we exported the calibrated visibility file from AIPS to D ifmap for imaging purposes described in the following. \nAfter flagging the outlier data points, we imaged the target source using the hybrid mapping technique including the clean algorithm and phase-only self-calibration. Then, we obtained the antenna-specific gain correction factors. At the end of the procedure, several steps of amplitude and phase self-calibration were performed, starting from the total observation time, gradually for shorter and shorter solution time intervals. \nA simple elliptical Gaussian modelfit component (Pearson 1995) was fitted directly to the visibility data. This way we can describe the brightness distribution of the radio source for a quantitative analysis and to reveal any possible structure of it. Finally, to decrease the noise level, we performed 1000 clean iterations in the residual image with a small loop gain of 0 . 01.", '3.4.2. ATCA monitoring': 'The Australia Telescope Compact Array (ATCA) 5 , located at Narrabri, New South Wales, Australia, was used to monitor AT2019aalc. The observations were conducted between 202306-28 and 2024-05-04, utilizing the array in the 6km (6A and 6D) and (H168) configuration. The observations were conducted at frequencies of 2 . 1 GHz, 5 . 5 GHz, 9 GHz, 17 GHz and 19 GHz. The flux calibrator was the standard ATCA primary calibrator, \n5 \nhttps://csiropedia.csiro.au/ \naustralia-telescope-compact-array/ \n1934-638, and 1548 + 056 is used as phase calibrator. The observations are summarized in Table 1. \nThe data reduction procedures were based on the Multichannel Image Reconstruction, Imaging Analysis and Display ( miriad ; Sault et al. 1995). The data reduction procedure was conducted in accordance with standard protocol. In order to mitigate the impact of radio frequency interference (RFI) during flux and phase calibration, the interactive flagging tool bflag was utilised. Furthermore, automated flagging routines pgflag were applied to address interference for the source. Any corrupted data identified during calibration was manually flagged. \nDue to the limited ( u , v )-coverage and the low elevation of the source resulting in a very elongated beamshape, we were not able to perform self-calibration and cleaning. Fortunately, AT2019aalc was always dominating the flux within the resolution elements of our short observations. Therefore, we imaged the region around our source for each observation using an oversampling of the synthesised beam by a factor of at least 10. Under the assumption that AT2019aalc is always a point-source, we can take the highest pixel value in the image as our measured flux. Due to the uncertainties this method introduced we used a plain 10 % flux error.', '3.5. Spectroscopic observations': "The Low Resolution Imaging Spectrograph (LRIS; Oke et al. 1995) mounted on the Keck-I telescope at Maunakea measured the spectrum of AT2019aalc, on 2021-07-06 (PI: Kulkarni), almost exactly at half time between the two optical flares. At that time the source was not significantly detected with ZTF as indicated by the forced photometry results. Later, during the second optical flare, we obtained 5 more optical spectra of AT2019aalc using the following instruments: LRIS (PI: Kulkarni), DeVeny Spectrograph mounted on the Lowell Discovery Telescope (LDT) in Arizona (PI: Hammerstein) and the Double Beam Spectrograph (DBSP, Oke & Gunn 1982) mounted on the 200 inch telescope at Palomar Observatory, California (PIs: Kulkarni and Kasliwal). These optical spectra were reduced following the standard procedures using the automated reduction pipeline of the LRIS (Perley 2019), the software package P ype I t (Prochaska et al. 2020) for DeVeny, and the software packages DBSP DRT (Mandigo-Stoba et al. 2022) and P ype I t (Prochaska et al. 2020) for the DBSP. We normalized the flux of each spectrum and plotted it versus the rest-frame wavelength. Details of these observations of AT2019aalc are summarized in Table 2. The obtained spectra cover a timescale of a year and allow us to study the spectroscopic evolution of the transient during its optical re-brightening. We describe the features of each spectra in Subsect. 4.5 and discuss the remarkable lines in Subsect. 5.1. \nTo determine the type of AGN, we performed a stellar population synthesis fit of the host galaxy's SDSS spectrum with ( p PXF; Cappellari & Emsellem 2004; Cappellari 2017, 2023) and the newest GALAXEV (Bruzual & Charlot 2003) models (CB2019), assuming a Kroupa (2001) initial mass function (IMF) and an upper mass limit of 100, as templates. Beforehand, the spectrum was corrected for Milky Way extinction using the dust reddening maps by Schlafly & Finkbeiner (2011) and Cardelli et al. (1989) extinction curve. Furthermore, it was redshift corrected and logarithmically rebinned. We included several emission lines to the fit and assumed four kinematic components: One for all stellar templates, one for all forbidden lines, one for all allowed emission lines, and a last one for a second, broad component of all Balmer lines up to H δ . Additionally, we included two separate dust components, each one with the \nextinction curve of Cardelli et al. (1989): a first one for stellar, and a second one for nebular emission. We therefore fixed the Balmer line ratio and limited doublets of the preinstalled lines to their theoretical ratio using the keywords tie \\_ balmer and limit \\_ doublets of p PXF. However, we only fixed the ratio for the strongest lines of the Balmer series up to H δ to avoid possible contamination of H ϵ by the [Ne iii ] λ 3967 line. For the fit with p PXF we included additive and multiplicative polynomials up to tenth order ( degree= 10, mdegree= 10). In order to correct for intrinsic dispersion caused by the spectrograph, we assume a resolving power of 1500 at 3800 Å and 2500 at 9000 Å. 6 Linear interpolation is used to calculate values for wavelengths in between. \nThe SPS fit of the latest DBSP spectrum was conducted in a manner analogous to the one applied to the SDSS. However, additional kinematic components were required to account for the internal structure in order to achieve a satisfactory result. Each emission line with an excitation potential above 100 eV is treated as a separate component. The [S ii ] λ 6731 doublet is fitted as an additional component as well.", '4.1. Optical/UV': 'The optical and UV light curves of AT2019aalc are shown in the bottom panel of Fig. 5. The transient was first detected by the ZTF in January 2019 with a g-band magnitude of 18 . 44 mag. After the initial detection of the first optical flare in May 2019 ( g ≈ 18 . 1), AT2019aalc had brighten to a peak magnitude of r ≈ 16 . 7 ( M = -19 . 4) reached on 2019-06-18 on a timescale of roughly 60 days. After the quickly and monotonically evolving flare, a slow decay has started, and the source was last detected with ZTF on 2021-02-18 with a magnitude of g ≈ 18 . 3 before its second flare. The re-brightening started mid-May 2023, approximately 4 years after the initial flare (Veres et al. 2023). The second flare has evolved very similarly to the original one. From the beginning of the second flare (the source was detected anew on 2023-05-11 at r ≈ 19 . 3 mag), a 2 . 8 mag rise can be seen till the peak of the flare on 2023-07-13 with r ≈ 16 . 2 mag ( M = -19 . 8), on a timescale of only 60 days. Both flares evolved with constant color. However, the second flare peaked at roughly 1 . 5 times the peak luminosity of the first flare. The peak of the second flare was followed by a short plateau phase lasted for around 25 days. Both flares decay very slowly, on timescales of years. The first flare decayed with a power-law index of b ≈ -0 . 45 while the second has been decaying even slower with a power-law index of b ≈ -0 . 13 in r -band. The decaying phases are not monotonous as bumps are tend to appear. \nPrior to the discovery of AT2019aalc, the host galaxy of the transient has been monitored by several optical surveys. To study the optical variability of AT2019aalc prior to the first optical flare in 2019, we performed aperture photometry of the All-Sky Automated Survey for Supernovae (ASAS-SN; Shappee et al. 2014; Kochanek et al. 2017) using ATLAS forced photometry server (Tonry et al. 2018; Smith et al. 2020) and the Catalina Real-Time Transient Survey (CRTS; Djorgovski et al. 2011) data. In addition, Palomar Transient Factory (PTF Law et al. 2009) magnitudes were extracted from the PTF Lightcurve Table 7 . No significant optical variability can be seen between April 2005 and the discovery of the transient in January 2019 by ZTF (see Fig. 4). \nTable 1. Summary of the ATCA radio continuum emission observations of AT2019aalcTable 2. Summary of the optical spectroscopic observations of AT2019aalc. δ t f 2 is time relative to the optical peak of the second flare. \nBoth optical flares of AT2019aalc detected by ZTF rising with constant color g -r ≈ 0 and show reddening when decaying. The second flare has similar color evolution to the first one when evolving but with a color of g -r ≈ 0 . 2 around the peak. \nThe first optical flare was not monitored in UV. The UV monitoring of the source started on 2023-06-23 around 20 days before the optical peak of the second flare. The UV flare evolved quickly. The U-band peak is coincident with the optical peak while the shorter wavelength bands (UM2, UW1 and UW2) peaked 2 -4 weeks later 8 . The transient is most luminous at the shortest wavelength ranges covered by UVOT. The UV emission peaks around 10 44 erg / s in filter UW2. After the peak epoch, the UW2 -U color shows reddening which suggests increasing absorption or a drop of the blackbody temperature. Towards the end of the UVOT monitoring, the measured UM2 magnitudes decrease back to the level of the baseline GALEX-NUV magnitudes measured between 2007 and 2011. \nComparing the peak UVM2 magnitude of the UVOT monitoring to the GALEX-DR5 (Bianchi et al. 2011) NUV magnitude implies that the UV luminosity increased by a factor of ≈ 7 with respect to the archival flux. The amplitude of this flare is higher than typically seen for AGN. Only 8 out of the 305 AGN presented in the GALEX Time Domain Survey (Gezari et al. 2013) have larger amplitude NUV variability.', '4.2. Infrared': 'We detected AT2019aalc in all of the three bands observed with the WIRC. The observed magnitudes are plotted in the top panel \nof Fig. 5. The host galaxy of AT2019aalc is presented in the 2MASS All-Sky Catalog of Point Sources (Cutri et al. 2003) as 2MASS 15241666 + 0451191. The observed magnitudes in April 2000 are higher with ≈ 0 . 5 mag in J-, and Ks-bands and with ≈ 0 . 6 mag in H-band indicating the clear brightening of the IR counterpart with respect to archival data. \nThe top panel of Fig. 5 also shows the long-term light curve starting just before the first optical flare. There are clear signs of variability throughout the infrared light curve starting in February 2010 with the distinct first dust echo flare that peaks around six months after the first optical peak. The rise of the dust echo of the second flare becomes visible in the latest WISE data release. The light curve varies by around 0 . 3 mag in W1 and 0 . 4 mag in W2 before the dust echo flare, which is consistent with statistical AGN variability (Berk et al. 2004). In contrast, the dust echo presents an increase of 1 . 0 mag in W1 and 1 . 2 mag in W2. The first detection of the dust echo of the second flare shows a brightening of 0 . 8 mag in both bands and can be expected to increase further in upcoming data releases. Because the brightness is significantly above the limiting magnitude where Eddington bias becomes an issue for stacked single-exposure photometry (Necker et al. 2024), we can estimate the pre-flare mid-IR color using the median of the light curve prior to the dust echo flare with m W1 -m W2 ≈ 0 . 6. Although there is clear AGN activity prior to the dust echo flare, this is still below the widely used AGN identification cut of m W1 -m W2 ≥ 0 . 8 (Stern et al. 2012). However, according to the reliability of WISE selection as a function of a simple W1 -W2 color selection (Stern et al. 2012), the color of AT2019aalc suggests a reliability of only ≈ 70%. With the onset of the dust echo, the IR emission becomes dominated by the heated dust, which is shown in Fig. 6. The change from bluer to redder colors come from cooling of the dust. The re-brightening of the second dust echo is consequently accompanied by another color change towards the blue. \nFig. 3. Multi-wavelength (bottom plot: optical / UV and X-ray, middle plot: radio and upper plot: IR) light curves of AT2019aalc. Here we show only the positive ZTF fluxes based on the forced photometry results. The light blue vertical line indicates the IceCube detection of the high-energy neutrino event IC-191119. \n<!-- image --> \nFig. 4. Long-term optical light curves of the host galaxy of AT2019aalc spacing approximately 14 years before the discovery of the transient in January 2019 indicated with a blue vertical line. No obvious signs of variability can be seen before the first detection of the transient by the ZTF. \n<!-- image --> \nNotably, AT2019aalc is not part of the Flaires sample (Necker et al. 2024), a list of dust-echo-like infrared flares interpreted as extreme AGN accretion events or TDEs. This is because the broad application of the Flaires pipeline necessarily \nincluded a strict cut on extraneous variability that AT2019aalc did not pass. The authors did note however that a desirable relaxation of this criterion would include AT2019aalc. The source is, however, part of the dust echo sample of van Velzen et al. (2024) and, notably has the highest dust echo flux of all ZTF transients.', '4.3. X-ray': 'AT2019aalc was monitored with the Swift / XRT at 0 . 3 -10 keV energies for 11 weeks with weekly cadence around the peak of the optical flare and later for additional months with weekly and bi-weekly cadences. The luminosity evolution is shown in the bottom panel of Fig. 5 while the count rate variability is plotted in the top panel of Fig. 7. The first two visits show an increase of the X-ray emission. This flaring episode peaked at L X ≈ 6 × 10 42 erg s -1 on 2023-06-29 (MJD = 60124) (see Fig. 5.) before it experienced a very quick decreasing. This flare peaked 2 weeks before the optical peak. The count rate dropped by a factor of ≈ 10, from 0 . 05 to 0 . 005, within a timescale of ≈ 60 days. Later, the X-ray emission shows limited variability for several months, however, less significant flaring episodes appear that peaked in the end of September 2023 (MJD = 60215) and mid-February 2024 (MJD = 60356). The latter was followed by a bump in the optical and UV light curves around 2 weeks later while no optical / UV data is available around the another X-ray increasing. In the end of April 2024, a rapid and extreme flaring episode was \n<!-- image --> \n<!-- image --> \nFig. 5. The WIRC infrared images of AT2019aalc in J-, H-, and Ksbands taken on 2023-07-05 (roughly a week before the optical peak). The object located to the west from the target source is a foreground star. \n<!-- image --> \ndetected in X-rays that lasted less than a month but reached a luminosity of L X ≈ 4 × 10 43 erg / s which is almost a magnitude larger compared to the peak of the first X-ray flaring episode. Interestingly, this X-ray flare was also accompanied by an increasing phase in the optical 1 -2 weeks later, according to the most recent ZTF observations 9 . Notably, ≈ 90 days passed between both the first and second and the third and fourth X-ray flaring episodes. While no data is available for a longer period between the second and third flares when the source was too close to the Sun. \nIn Fig. 8 we show the source spectrum derived from the full observing campaign. The spectrum is dominated by a power-law \nFig. 6. The long-term IR variability of AT2019aalc based on WISE magnitudes in W1 and W2. The W1 -W2 color indicates heated dust as the dominant origin of the IR emission. The second optical flare seems to be accompanied by another IR flaring episode with color change towards the blue. \n<!-- image --> \n<!-- image --> \n60150 \n60200 \n60250 \n60300 \n60350 \n60400 \n60450 \nFig. 7. The Swift-XRT count rate and photon index variability of AT2019aalc during the observing campaign. In most cases the data is well described by a power-law spectral model (black), except during the most recent flaring episode, where a blackbody model is preferred (red). The blackbody temperature is indicated on the red y-axis. \n<!-- image --> \ncomponent but a blackbody component is also significantly presented. The spectral index of the stacked spectrum fitted with a power-law + blackbody model is Γ = 2 . 8 + 0 . 4 -0 . 4 while the blackbody temperature is kT = 89 + 10 -12 eV. These imply an overall unabsorbed flux of FX = 5 . 1 + 0 . 15 -0 . 15 × 10 -13 erg cm -2 s -1 corresponding to a luminosity value of LX = 1 . 55 + 0 . 15 -0 . 14 × 10 42 erg / s. The count \nFig. 8. The overall Swift -XRT 0 . 3 -10 keV spectrum of AT2019aalc. We binned the spectrum to have a minimum of 5 counts per spectral bin and fit with an absorbed blackbody + power-law model. \n<!-- image --> \nrate of the observations is 0 . 028 ± 0 . 001 cts s -1 . The photon index variability of AT2019aalc is presented in the bottom panel of Fig. 7. \nAT2019aalc ( = SRGe J152416.7 + 045118) was observed by the eROSITA (Sunyaev et al. 2021) soft X-ray telescope four times starting from 2020-02-02 (roughly half year after the peak of the first optical flare) with a 6 months cadence. The X-ray light curve reached a plateau between August 2020 and January 2021 with a flux of FX ≈ 4 . 6 × 10 -13 erg cm -2 s -1 in the energy range of 0 . 3 -2 keV. The source had a soft thermal spectrum described with a blackbody temperature of kT = 172 ± 10 eV (van Velzen et al. 2024). One observation with XMM-Newton Observatory (Jansen et al. 2001) has been performed in 2021-02-21 during a period of optical quiescence. The best-fit power-law index and the observed 0 . 3 -8 keV flux of this observation were Γ = 2 . 6 + 0 . 1 -0 . 1 and FX ≈ 5 . 1 + 0 . 15 -0 . 25 × 10 -13 erg cm -2 s -1 (68% confidence), respectively (Pasham 2023). These imply that the two X-ray flares observed with the Swift / XRT peaked at much higher luminosities than the plateau or long-term flaring episode between the optical flares, however, there is no X-ray data from around the peak of the first optical flare. Furthermore, the eROSITA results confirm the presence of a blackbody component in the X-ray spectrum, which is very soft and dominating the variability of the X-ray source.', '4.4.1. Temporal evolution': 'The host galaxy of AT2019aalc was detected in the NRAO VLA Sky Survey (NVSS, White et al. 1997) in February 1995 and in the FIRST survey in February 2000 (both at 1.4 GHz) with peak flux densities of 6 . 0 ± 0 . 5 mJy and 6 . 2 ± 0 . 1 mJy respectively, suggesting no substantial variability of the host galaxy prior to the discovery of the transient event. \nLater, the transient\'s sky location has been observed during the first and second epochs of the Very Large Array Sky Survey (VLASS). The VLASS covers the entire sky visible to the Karl G. Jansky Very Large Array (VLA) at 2 -4 GHz wide \nband o ff ering an angular resolution of 2 . 5 arcsec and an rms of 0 . 12 mJy beam -1 in each epoch (Lacy et al. 2020). One of the main goals of the VLASS multi-epoch survey is to detect various types of extragalactic radio transients, such as TDEs, accretion state changing AGNs and core-collapse supernova (Lacy et al. 2020; Zhang et al. 2022). To date, the first two epochs of the survey have been fully published. Our source has peak flux density values of 2 . 9 ± 0 . 2 mJy / beam (on 2019-03-14) and 5 . 7 ± 0 . 2 mJy / beam (on 2021-11-06), corresponding to 3 GHz radio luminosities of 3 × 10 38 and 5 × 10 38 erg s -1 in the Epoch 1 and Epoch 2 images, respectively. This factor of ≈ 2 increase in flux from the first VLASS epoch (three months before the optical peak) to the second VLASS epoch (2 years post-peak) is statistically significant (at the 8 σ -level, as estimated from the rms in the VLASS \'Quick Look" images, van Velzen et al. 2024). \nIn addition, we found a radio source at the position of AT2019aalc in the first data release (DR1) of the Rapid ASKAP Continuum Survey (RACS; McConnell et al. 2020). The RACS DR1 includes initial observations made for RACS-low (central frequency of 888 MHz) 10 . Moreover, the source has been observed in two di ff erent fields and, consequently, at two di ff erent epochs. The observed peak flux densities of 9 . 2 ± 0 . 5 mJy / beam (on 2019-04-24) and 9 . 5 ± 0 . 5 mJy / beam (on 2020-04-30) correspond to a 888 MHz radio luminosity of ≈ 2 . 5 × 10 38 erg s -1 and indicate that the radio flare seen in the VLASS data started at least 300 days after the peak of the first optical flare, sometime between April 2020 and November 2021. \nInterestingly, the radio luminosity in the beginning of our ATCA monitoring is at the same level within uncertainties ( L R ≈ 5 × 10 38 erg s -1 ) when the source was observed for the second time in the VLASS in April 2021. Since the radio brightness does not show rapid variability during the months of our ATCA monitoring, we can reasonably assume that the sharp increasing resulted in a long-lasting plateau. Our multi-frequency monitoring with the ATCA implies no substantial variability on monthly timescales at 9 GHz while it shows slow decreasing at 2 . 1 GHz and at 5 . 5 GHz over time. We observed the radio source at 17 -19 GHz at one epoch. We present the ATCA radio multifrequency light curves of AT2019aalc together with the archival observations in the middle panel of Fig. 5 and focusing on the ATCA monitoring in Fig. 9.', '4.4.2. The radio spectrum': 'To further characterize the radio emission of AT2019aalc, we derived the radio spectral index of the radio source in each epoch of our ATCA monitoring. The spectral index between 2 . 1 GHz and 9 . 0 GHz ( α , defined as S ∝ ν α ) does not show any obvious signs of variability on a timescale of 9 months. We give an average spectral index of α 9GHz 2.1GHz ≈ -1 . 1. We calculated a preflare spectral index using the VLASS epoch 1 and RACS epoch 1 flux densities (these flux values are explained in Sect. 4.4.1.). The spectral index we derived between 0 . 888 GHz and 3 . 0 GHz this way is α 3GHz 888MHz = -0 . 96 ± 0 . 07. \nOur high-frequency K-band observations centered at 18 GHz, however, do not follow the steep power-law and suggest a turnover between 9 GHz and 18 GHz. The spectrum of high frequency peaked radio sources are usually explained with significant absorption taken place in the system due to synchrotron self-absorption (although free-free absorption may also play a \nFig. 9. The radio flux density evolution of AT2019aalc during the time interval of our ATCA monitoring of the transient at 4 di ff erent frequencies. \n<!-- image --> \nFig. 10. Radio spectra of AT2019aalc between 3 GHz and 18 GHz based on our ATCA monitoring. We derived a prior radio flare spectrum between 888 MHz and 3 GHz from archival fluxes. The times in the legend are relative to the peak of the second optical flare. \n<!-- image --> \nrole, (e.g., Berton et al. 2020). We further discuss the features of the radio spectrum in Subsect. 5.3.', '4.4.3. VLBI detection': 'The final naturally-weighted VLBI map of AT2019aalc is shown in Fig. 11. We detected a single radio-emitting feature with a signal-to-noise ratio (SNR) of ≈ 19 σ with our 1 . 7 GHz EVN + eMERLIN VLBI observation. We obtained the following parameters of the fitted component; a total flux density of S = 21 . 0 ± 1 . 8 mJy and a size of d = 2 . 23 × 1 . 39 ± 0 . 3 × 0 . 3 mas at a position \nRelative Right Ascension (mas) \n<!-- image --> \nFig. 11. Naturally weighted 1 . 7 GHz EVN + e-MERLIN high-resolution VLBImapofAT2019aalc. The peak brightness is 17 . 3 mJy beam -1 . The grey ellipse in the lower-left corner represents the Gaussian restoring beam. Its parameters are 4 . 82 mas × 3 . 7 mas (FWHM) at a major axis position angle of PA = 60 · . 8. The lowest contour level is drawn at ± 3 σ image noise level corresponding to 0 . 08 mJy beam -1 . Further positive contour levels increase by a factor of 2. The red dashed contours represent the negative contours. The yellow ellipse is the fitted Gaussian modelfit component to describe the brightness distribution of the radio source. \nangle of ϕ = -73 . 3 · . We calculated the uncertainties of the flux density and size parameters following the formulas of Fomalont (1999). The minimum resolvable size ( θ lim) of a Gaussian component fitted to naturally weighted VLBI data was calculated following Kovalev et al. (2005). This yields θ lim = 1 . 05 × 0 . 81 mas implying that the fitted Gaussian component is resolved. We note that at high SNR, θ lim can be significantly smaller than the size of the resolving beam (Bertero & de Mol 1996; Kovalev et al. 2005) due to applying a specific a priori hypothesis about the shape of the emitting region which is in our case the two-dimensional Gaussian used to fit the observed brightness distribution of the radio source. \nThe positional uncertainty was estimated as follows. When calculating the VLBI positional uncertainty, we took into account both the positional uncertainty of the phase calibrator source and its angular separation from our target source. The position of the phase calibrator source is listed with an error of 0 . 18 mas in the third realization of the International Celestial Reference Frame (ICRF3; Charlot et al. 2020). The angular separation between the target and the calibrator source signifies a positional error of about 1 . 5 mas at 1 . 7 GHz (Chatterjee et al. 2004). The resolution of the observation depending on the thermal noise of the interferometric images (Reid et al. 1988) has been also considered. Following Reid et al. (1988), we estimated a thermal noise-limited positional accuracy of 0 . 1 mas. The resultant positional uncertainty of 1 . 5 mas is therefore dominated by the uncertainty arising from the angular separation between the calibrator and the target source. The radio position agrees with the source position in the Gaia DR3 (Gaia Collaboration et al. 2023) catalog within the positional uncertainties. \nWe calculated the brightness temperature of the radioemitting feature found at the position of AT2019aalc using the following equation (Condon et al. 1982; Ulvestad et al. 2005): \nT b = 1 . 22 × 10 12 (1 + z ) S θ maj × θ min ν 2 K, (1) \nwhere S is the flux density of the fitted Gaussian component measured in Jy, θ maj and θ min are the major and minor axes full width at half maximum (FWHM) of the fitted component in mas, and ν is the observing frequency in GHz. The obtained brightness temperature is T b = (3 . 0 ± 0 . 8) × 10 9 K at 1 . 7 GHz. The brightness temperature is an e ff ective parameter that is commonly used in radio astronomy to describe the physical properties of emitting material in astrophysical objects (e.g., Lobanov 2015). It can be determined by imaging and modeling the structure of a given emitting region. The derived value is crucial to characterize the detected radio emission, which can be either thermal or non-thermal.', '4.5. Spectroscopic results': 'Wepresent the archival and the newly obtained optical spectra of AT2019aalc in Fig. 12 with marking the notable emission lines. In the following, we highlight the remarkable line detection of the host spectrum observed before the first optical flare (pre-flare spectrum), the one observed between the two flares (post-flare-1 spectrum) and 5 spectra that were taken during the second flare (over-flare-2 spectra).', '4.5.1. Pre-flare spectrum': 'Aspectrum of the host galaxy of AT2019aalc was taken in April 2008 by the SDSS spectrograph. We present this spectrum with the best fits for the stellar and gas components in Fig. 13. \n- -We identify strong Balmer emission lines. These lines are generally thought to be originating from the broad-line region i.e. the vicinity of the SMBH. We estimate the FWHM of the Balmer lines to FWHM(Balmer) ≈ 2800 kms -1 for the broad component of the Balmer lines (see Fig. 13 subplots) in the host spectrum, which is typical for an unobscured, broad line AGN. The line widths (FWHM(Balmer) ≈ 2600 km s -1 ) presented in the SDSS DR7 broad-line AGN catalog (Liu et al. 2019) for our source are consistent with this classification. In addition to the Balmer lines, we detect the recombination line He i λ 5876 which is also produced in the BLR.\n- -The remarkable [O iii ] λ 4959 and [O iii ] λ 5007 forbidden lines exhibited already in the SDSS spectrum indicate that the host AGN was not completely dormant before the optical flares, which is further supported by the radio detections prior to the first flare (Subsect. 4.4.1). Other forbidden emission lines commonly seen in AGN such as [S ii ] λ 6716, [S ii ] λ 6731 and [N ii ] λ 6584 are detected as well. These lines are produced in the narrow-line region (NLR) in AGN. We note that the sulfur lines are also present in the spectra of the TDE AT2019dsg, which was hosted by a quiescent galaxy. We detect the [O ii ] λ 3726 + [O ii ] λ 3729 line doublet, which is a potential indicator of star formation or AGN-driven outflows (e.g., Santoro et al. 2020).', '4.5.2. Post-flare-1 spectrum': 'The first LRIS spectrum was taken almost 2 years after the first flare and 2 years before the second one. \n- -We detected an increase in the Balmer lines series and the forbidden lines [O iii ] λ 4959 and [O iii ] λ 5007. The Balmer lines are also remarkably broadened.\n- -In addition to the strong Balmer emission lines, we can identify the He ii λ 3203 He ii and λ 4686 transitions and the Bowen Fluorescence line O iii λ 3133. Wavelength coincidences between emission lines ("line fluorescence") can be an important source of radiative excitation. In the case of Bowen Fluorescence, the He ii -O iii and N iii coincidences lead the mechanism. The BF lines occur due to the excitation in certain states of O iii and N iii by the absorption of He ii Lyman α photons. These excited states lead to a cascade of transitions that can be observed as emission lines in the optical and NUV regimes (Netzer 1990; Trakhtenbrot et al. 2019). Interestingly, the existence of BF emission lines in astronomical objects has been predicted by Netzer et al. (1985) as the required gas densities are consistent with what is expected from the extremely dense environment in some AGN and gaseous nebulae.\n- -We detect the high ionization coronal lines [Fe vii ] λ 6089, [Fe x ] λ 6375 and [Fe xiv ] λ 5303. The very high ionozation potentials of 235 eV and 262 eV of [Fe xiv ] λ 5303 and [Fe x ] λ 6375, respectively, point to very energetic processes at work. First Oke & Sargent (1968) studied coronal lines in detail in a Seyfert galaxy (NGC 4151). These lines are commonly detected in AGN and commonly thought to be powered by photoionization of the NLR (Cerqueira-Campos et al. 2021). In some extreme cases, a shock model i.e. collosional ionization for the coronal line region is required to explain the origin of these lines as it is the case in the Seyfert galaxy NGC 1068 (Oliva 1997).\n- -We identify the [Ne V] emission lines at 3345Å and 3426Å in the spectrum. These lines are not strongly excited in the inner regions of AGNs, but are prominent emission lines in the lower density extended emission line regions (EELRs) excited by the AGN via photoionization or shocks. The forbidden line [O iii ] λ 4363Å also appears, however, bending from H γ is possible.', '4.5.3. Over-flare-2 spectra': 'We performed 5 optical spectroscopic observations of AT2019aalc during its second flare. These are characterized by a blue continuum with several lines appearing in comparison to the spectra discussed above. We compare the second LRIS spectrum of AT2019aalc (covering the broadest wavelength range) with a composite SDSS spectrum based on 10112 Seyfert 1 galaxies published in Pol & Wadadekar (2017). This comparison shows similarities but also significant di ff erences discussed below. \n- -The line widths and strengths of the earlier detected broad Balmer emission lines (H α , H β and H γ ) increased and both the line widths and strengths of the Balmer lines peaked 2 days before the optical peak. Two lines of the series (H δ and H η ) were detected only in the over-flare-2 spectra.\n- -Over the second optical flare of AT2019aalc we detected another BF line N iii λ 4640 together with the transition line He ii λ 4686. This doublet has not been detected in the \nFig. 12. The optical spectra of AT2019aalc normalized between 7400 and 7500Å. The most remarkable lines are indicated with vertical lines. Days relative to the second optical peak are given to the right from each spectra. The SDSS host spectrum was observed around 11 years before the first optical flare. \n<!-- image --> \nFig. 13. The SDSS spectrum of the host galaxy of AT2019aalc provided by p PXF supplemented by two subplots showing H α (top right corner) and H β (lower right corner). The input spectrum is shown as solid black line, the best stellar and stellar + gas fit are represented as solid red and orange lines, respectively (Cappellari 2023). Additionally, plotted with an arbitrarily chosen o ff set, one can see the individual gas components (solid blue), total gas emission (solid magenta) and fit residuals (green diamonds) (Cappellari 2023). \n<!-- image --> \narchival SDSS spectrum taken ≈ 11 years before the discovery of the transient or in the post-flare-I spectrum. We detected this doublet already in the first spectrum taken during the second flare. The strengths of these lines increased significantly around the second optical flare and peaked on 2023-07-11, very close to the optical continuum peak (2023-07-13). As we detected the Bowen line N iii λ 4640 even 315 days after the optical peak of the second flare, we can conclude the long-term persistence of at least one BF line after the second flare as well. In addition, the second LRIS spectrum shows a clear increase in the BF line O iii λ 3133 and in He ii λ 3203 when comparing it to the postflare-1 spectrum. Both lines increased by a factor of ≈ 2 with respect to the post-flare-1 LRIS spectrum. We might detected O iii λ 3760 which is also known as a Bowen line (Selvelli et al. 2007), however, due to a possible blending from Fe vii λ 3759, its detection is questionable. Nevertheless, this Bowen line is created through the same channel (O1) as the above discussed O iii λ 3133 which suggests its detection. O iii λ 3760 was detected in some metal-rich TDEs before (Leloudas et al. 2019) such as AT2019dsg (Cannizzaro et al. 2021). \n- -We detect the high-ionization coronal lines seen in the postflare-1 LRIS spectrum over the second flare in each spectrum. Furthermore, a new coronal line (Fe xi λ 7892, ionization potential: 361 eV) appears on 2023-08-17, around a month after the peak of the second optical flare. Interestingly, the line intensities of these lines are larger after the optical peak than around the peak in contrast with the BF lines and the above-mentioned ionized helium lines which experienced their maximums very close to the continuum peak. Moreover, [Fe xi ] λ 7892 and [Fe x ] λ 6375 are not just persistent but the strongest in the late-time DBSP spectrum, taken almost a year after the optical peak of the second flare. We detect the high-ionization coronal line [S xii ] λ 7611 only in the late-time spectrum. This line is an extreme ionization line with an ionization potential of 504 . 8 eV thus implies that unusually high level of ionization is taken place in AT2019aalc \n- almost a year after its second optical peak. For this coronal line we estimate FWHM(S xii λ 7611) ≈ 3400 km s -1 .\n- -We detected the [Ne V] emission lines at 3345Å and 3426Å in the second LRIS spectrum, similarly to the post-flare-1 spectrum and implying continuous ionization of the EELR. The [O iii ] λ 4363Å line appears in the late-time spectrum.', '5.1.1. Balmer lines': 'The line width of the Balmer lines and its limited variability are not common for regular TDEs and flaring AGN which tend to have decreasing and increasing Balmer line widths over time, respectively (Gezari 2021). However, this behavior is more compatible with the spectroscopic properties of the TDE-AGN PS16dtm (Blanchard et al. 2017). Still, the slight increasing and broadening indicate high-velocity gas moving in the BLR, close to the SMBH. In addition, a more complex structure of these lines seen in the late-time DBSP spectrum.', '5.1.2. Bowen lines': "The Bowen Fluorescence mechanism in astrophysical environments is a good indicator of absorbed EUV -soft X-ray flux (at wavelengths shorter than the He ii Lyman limit of h ν ≥ 54 . 4 eV) as these high-energy photons are converted into the He ii Lyman αλ 303 . 782Å emission required for the excitation of the O iii and N iii (Selvelli et al. 2007). This is consistent with the multiple soft X-ray flares and the extreme UV luminosities of AT2019aalc. The BF line O iii λ 3133 has been observed in the Bowen Fluorescence Flares AT2017bgt (Trakhtenbrot et al. 2019) and AT2021loi and only in a few Seyfert galaxies before (e.g., Malkan 1986; Schachter et al. 1990), however, we note that due to the atmospheric limit at ≈ 3100Å these lines are not easy to detect. This line is typically much weaker in normal Seyfert galaxies as in TDEs, BFFs or in our case (see Fig. 14). The BF line N iii λ 4640 is not generally seen in AGN (e.g., Vanden Berk et al. 2001), however, has been observed in several optical TDEs (Charalampopoulos et al. 2022b) and other SMBH-related transients; the BFFs AT2017bgt, F01004-2237, OGLE17aaj (Trakhtenbrot et al. 2019) and AT2021loi (Makrygianni et al. 2023), the peculiar transient event AT2019avd (Malyali et al. 2021) and a few flaring NLSy1s (Frederick et al. 2021). In a TDE case, the disrupted star's material forms an accretion disk and the photosphere absorbs the EUV -soft X-ray photons originating from accretion processes and this way provide a plausible explanation for the appearance of the BF lines. BFFs have been first identified by Trakhtenbrot et al. (2019) who revealed three flares (AT2017bgt, F01004-2237 and OGLE17aaj) in AGN with the appearance of persistent Bowen lines. Later, Makrygianni et al. (2023) studied a transient, AT2021loi, classified as a BFF. The detection of a BF line almost 2 years after the first optical flare of AT2019aalc suggests its long persistence, similar to the cases of the BFFs studied by Trakhtenbrot et al. (2019). In the case of AT2017bgt, this Bowen line was still detected 470 days after the discovery of the transient. These lines in BFFs identified by Trakhtenbrot et al. (2019) may originate from a pre-existing BLR (a region with dense line-emitting gas) which was suddenly exposed to the intense ionizing UV emission, which also indicates enhanced accretion onto the SMBH. Alternatively, the BF lines might be related to \nFig. 14. The over-flare-2 LRIS spectrum of AT2019aalc in comparison with the SDSS composite spectrum of 10112 Seyfert 1 galaxies published in Pol & Wadadekar (2017) and normalized between 7400 and 7500Å. The most remarkable lines are shown with vertical lines. The most significant di ff erences are the appearance of the Bowen lines (marked with blue) and the high-ionization coronal lines (marked with green) in the spectrum of AT2019aalc and the line strengths of the Balmer line series and the helium lines. \n<!-- image --> \na newly launched outflow driven by a sudden increase in accretion rate which interacts with the BLR. We compare the overflare-2 high-resolution LRIS spectrum of AT2019aalc with the spectra of the BFFs AT2017bgt and AT2021loi, the BFF candidate AT2019avd, the TDE-Bowen AT2018dyb and the repeating partial TDE AT2020vdq in Fig. 15 with focusing on the N iii λ 4640 + He ii λ 4686 emission line doublet. We further compare the second LRIS spectrum of AT2019aalc with the LRIS spectrum of AT2021loi in Fig. 16 zooming into the O iii λ 3133 and He ii λ 3203 region. \nThe increasing of these lines around the optical peak indicate that these lines were enhanced by the Bowen Fluorescence mechanism. The BF process is particularly strong in some He ii , O iii and N iii transitions. These lines are obviously detected during the second optical flare, while atypical for normal Seyfert 1 galaxies (see a comparison plot in Fig. 14). Moreover, the O iii lines at 3341 , 3429 and 3444 Å normally detected in AGN do not show up in our spectra, similarly to the case of AT2017bgt. These results suggest that the BF mechanism plays an important role and this mechanism traces significantly enhanced accretion onto the SMBH suggesting the unusual nuclear transient nature of AT2019aalc. \nThe He ii λ 4686 ionized helium line is presented in our spectra taken between the optical flares, and is also increased during the second optical flare. This emission line is commonly present in AGN and known as a recombination line, tracing ionized gas in the vicinity of the central SMBH responsible for the photoionization (e.g. Wang & Kron 2020). This line is known to be created by photoionization due to (soft) X-ray photons with a rough correspondence of 1 He ii photon emitted for each 0 . 3 -10 keV X-ray photon (Pakull & Angebault 1986; Schaerer et al. 2019; Cannizzaro et al. 2021). In the spectra of AT2019aalc, He ii λ 4686 is present together with strong N iii and O iii BF lines and the less commonly detected He ii λ 3203 tran- \nion, indicating its relation to the Bowen Fluorescence mechanism. Furthermore, we calculated a maximum line intensity ratio of F (He ii ) / F (H β ) ≈ 0 . 7 whose value significantly exceeds the typical ratios estimated for AGN (earlier Vanden Berk et al. (2001) estimated ratios of F (He ii ) / F (H β ) ≤ 0 . 05 while Shirazi & Brinchmann (2012) gives F (He ii ) / F (H β ) ≤ 0 . 1). Our value is, however, more comparable with the line ratio estimated for the BF transient AT2017bgt which is F (He ii ) / F (H β ) ≈ 0 . 5 (Trakhtenbrot et al. 2019). The line intensities of He ii λ 4686 and N iii λ 4640 peak 2 days before the continuum peak, suggesting extreme ionization by UV / soft X-ray photons around the peak. The He ii λ 4686 / He i λ 5876 ratio is sensitive to the change in the temperature and density as these lines belong to the same element but in two di ff erent ionization states (e.g., Ili'c et al. 2009). We found the highest ratio close to the continuum peak, implying increased temperature in the BLR. The smallest ratio can be estimated from the post-flare-1 and late-time spectrum taken 315 days after the continuum peak that is due to a drop in the temperature.", '5.1.3. Coronal lines': 'Wedetected several high-ionization coronal lines. Two of the detected coronal lines present already in the post-flare-1 spectrum taken 2 years after the first optical peak indicating a connection to the first optical flare. \nBased on a study of SDSS spectra, Wang et al. (2012) reported the discovery of 7 Extreme Coronal Line Emitter (ECLE) galaxies which show extremely strong coronal lines. Three TDEs (AT2017gge, AT2019qiz and AT2022upj) were also found to have coronal lines (Onori et al. 2022; Newsome et al. 2022; Short et al. 2023). The BFF AT2021loi (Makrygianni et al. 2023) and the spectacular transient event AT2019avd (Malyali et al. 2021) have significantly detected coronal lines, too. Based on the \nFig. 15. The second LRIS spectrum of AT2019aalc zoomed into the He ii λ 4686 + N iii λ 4640 doublet region compared to that of the BFFs AT2021loi (Makrygianni et al. 2023) and AT2017bgt (Trakhtenbrot et al. 2019), the peculiar transient event AT2019avd (Trakhtenbrot et al. 2020), the repeating TDE AT2020vdq (Somalwar et al. 2023) and the TDE-Bowen AT2018dyb (Pan et al. 2018). All of the plotted spectra were normalized at 5200Å. \n<!-- image --> \npublicly available spectra of AT2017bgt on the Transient Name Server 11 (TNS) we see enhanced coronal line emission in this BFF as well. Wang et al. (2012) states that the most possible origin of the ECLEs are remnants of an earlier TDE outburst which was later supported by Short et al. (2023) who revealed that the TDE AT2019qiz more likely follows the line ratio correlations of ECLEs instead of regular AGN. Interestingly, the coronal lines in AT2019qiz appeared around 400 days after the \noptical peak, further supporting this scenario. Moreover, the estimated rate of coronal line emitters is consistent with the estimated rate of TDEs (Wang et al. 2012). According to Wang et al. (2012), the ECLEs could be explained through the coronal lines arising from Fe liberated from dust grains that were destroyed by the TDE flare. The large dust echo of AT2019aalc is consistent with this picture. Furthermore, the TDEs with coronal line detection clearly have more luminous IR dust echo compared to other optical TDEs. AT2019qiz is one of the few TDEs in- \nFig. 16. The second LRIS spectrum of AT2019aalc zoomed into the O iii λ 3133 + He ii λ 3203 doublet region compared to that of the BFF AT2021loi published by Makrygianni et al. (2023). The spectra were normalized at 5200Å. \n<!-- image --> \ncluded in the accretion flare sample of van Velzen et al. (2024) due to its strong dust echo. Another coronal line-detected TDE AT2017gge is part of the mid-infrared (MIR) dust echo flare sample of Hinkle (2024) which contains 19 ambiguous nuclear transients with high dust covering factors. The extreme dust echo seems to characterize the BFFs as well. The BFFs AT2021loi, AT2017bgt, OGLE17aaj and the candidate BFF AT2019avd are present in the above-mentioned MIR dust echo sample (Hinkle 2024) and 2 of them are also part of the Flaires sample of dustecho-like IR flares (Necker et al. 2024). The Bowen line transient AT2019pev that shows strong high-ionization coronal lines (Frederick et al. 2021; Yu et al. 2022) is also part of the MIR flare sample of Hinkle (2024). These findings suggest a connection not only between the ECLEs and TDEs, but also the BFFs. Fig. 17 shows the [Fe x ] λ 6375 versus the [O iii ] λ 5007 luminosities of the EECLEs studied by Wang et al. (2012), AT2019qiz (Short et al. 2023), two BFFs and AT2019aalc together with the SDSS Seyfert sample of Gelbord et al. (2009). This clearly implies an o ff set of the ECLEs, BFFs and AT2019aalc from the regular AGN sample. \nThe [Fe x ] λ 6375 line evolution is shown in Fig. 18. The strength of this coronal line in AT2019aalc is further increased during the second flare and seem to be peaking lately with respect to the continuum peak. The coronal line TDE AT2019qiz shows a late-time increase in the this line as well (Short et al. 2023). Similarly to the case of AT2019qiz, we also found that [Fe x ] λ 6375 line is the strongest high-ionization coronal line presented in the spectra, while [Fe vii ] λ 6089 is the weakest one. The coronal line [Fe x ] λ 6375 has been detected in several AGN before, however, the maximum line intensity ratio we calculated ([Fe x ] λ 6375 / [O iii ] λ 5007 ≈ 0 . 65) clearly overshoots the maximum ratio ( ≈ 0 . 24, Nagao et al. (2000) found for AGN. For AT2021loi, Makrygianni et al. (2023) calculated even higher ratios, up to 2 . 06.The line width of [Fe x ] λ 6375 shows a clear evolution during the monitoring of AT2019aalc as we estimated values of FWHM([Fe x ] λ 6375) = 1200 -1800 kms -1 . Interestingly, the line widths we estimated for [Fe x ] λ 6375 are clearly larger than the maximum values found for coronal line emitters (around 1000 km s -1 , Wang et al. 2012), however, very consistent with the width estimated for AT2021loi. For AT2021loi, Makrygianni et al. (2023) gives a line width of ≈ 2000 km s -1 which implies that the line is consistent with a BLR instead of NLR origin since the latter would require the typical values estimated for coronal line emitters (200 -1000 km s -1 , Wang et al. 2012). The BLR \nFig. 17. [Fe x ] λ 6375 versus [O iii ] λ 5007 luminosities of ECLEs, the TDE AT2019qiz (Short et al. 2023), two BFFs and AT2019aalc. A sample of di ff erent types of Seyfert galaxies (Gelbord et al. 2009) is plotted as well. The black line represents the mean ratio calculated for the sample. The transient sources have clearly higher [Fe x ] / [O iii ] luminosity ratios than the Seyfert sample. Interestingly, the two BFFs, AT2019aalc and the TDE AT2019qiz all have significant dust echo emission and are presented in IR dust echo flare samples. This suggests that the coronal line emission can be explained via liberated Fe from the dust grains by TDE(-like) flares. \n<!-- image --> \norigin would naturally explain the increasing emission in these lines after the optical peak (due to the light travel time) but is also consistent with the distance between the dust responsible for the reprocessed IR emission and the SMBH (0 . 1 -1 pc, van Velzen et al. 2024). This suggests a connection between the unusually large dust echos and the extreme coronal lines in these transient sources. An alternative explanation is collisional excitation of these coronal lines. This is compatible with a detailed multi-wavelength study of AT2019qiz which suggests that outflows (powered by accretion or stream collision) are responsible for driving the optical flare of this coronal-line TDE (Nicholl et al. 2020). \nThe sulfur line [S xii ] λ 7611 has the second highest ionization potential among the optical lines detected in AGN before (the highest one is [Ar xiv ] λ 4412 which we do not detect in AT2019aalc). We zoom into the SPS fitted DBSP spectrum to show the detection of this line in Fig. 19. We can reasonably assume that the most recent and very strong X-ray flare, started less than a month before the appearance of the sulfur line, is responsible for the creation of this emission line. Apart from 5 ECLEs in the sample of Wang et al. (2012), [S xii ] λ 7611 has only been reported in 4 Seyfert galaxies (Kraemer & Crenshaw 2000; Mazzalay et al. 2010; Cerqueira-Campos et al. 2021; Oh et al. 2022) including the neutrino emitter (IceCube Collaboration et al. 2022) NGC 1068 and in the nuclear transients VT J1548 (Somalwar et al. 2022) and AT2022fpx (Koljonen et al. 2024). VT J1548 is a radio-detected transient that was classified as a promising BFF candidate and notably with remarkable TDE-like characteristics (Somalwar et al. 2022). Similarly, AT2022fpx shares properties both with TDEs and BFFs (Koljonen et al. 2024). To our knowledge, AT2019aalc is so far only the third nuclear transient detected with a line requiring such a high creation ionization potential. \nFig. 18. The [Fe x ] λ 6375 line evolution of AT2019aalc. The days in the legend are relative to the second optical peak. \n<!-- image --> \nFig. 19. The late-time DBSP spectrum taken almost a year after the optical peak of the second flare indicates the appearance of the highionozation coronal line [S xii ] λ 7611. The SPS fit (Subsect. 3.5.) of the spectrum allowed us to characterize this sulfur line. \n<!-- image -->', '5.1.4. Forbidden lines': 'We find evidence for the forbidden line [O iii ] λ 4363Å in the post-flare-1 LRIS spectrum and the late-time DBSP spectrum. Its detection, however, remains ambiguous because the Balmer line series are complex in our spectra, including blending from H γ . Nevertheless, the detections of [O iii ] λ 4363Å 2 years after the first flare and 1 year after the second flare is not unexpected. After all, this emission line requires densities much lower than the UV-line emitting clouds, and consistent with the outer NLR (or simple ISM) clouds (Ji et al. 2024) but is located inner than the emitting region of [O iii ] λ 5007Å (Nagao et al. 2001). The line \nratio of [O iii ] λ 4363Å / H γ ∼ 1 suggests extremely hard ionizing radiation from the ionizing source. Using James Webb Space Telescope (JWST) observations, this line is sometimes detected and studied in detail in highz AGN (used to distinguish AGN from star-forming regions), however, such a ratio was detected only in a very few cases (Übler et al. 2024; Ji et al. 2024; Mazzolari et al. 2024). The [O iii ] λ 4363Å / [O iii ] λ 5007Å ratio of ≈ 0 . 7 is clearly higher than calculated for AGN which are typically below 0 . 2 (Baskin & Laor 2005; Übler et al. 2024; Binette et al. 2024)). \nThe highly ionized neon lines typically originate from star formation activity or photoionization of the EELR by the AGN. In some extreme cases, high-velocity AGN-driven shocks that ionize extended emission regions are expected to be responsible for the presence of the lines [Ne V] at 3345Å and 3426Å (Maddox 2018). The strengths of these lines in AT2019aalc are higher than the [O ii ] λ 3726 + [O ii ] λ 3729 doublets which indicate their origin related to AGN activity since [Ne V] is not excited by star formation (Maddox 2018). These shocks are possibly, but not exclusively, related to radio jets as high shock velocities of 900 km s -1 are required to produce these lines (Maddox 2018). We estimated slightly lower velocities, which are therefore more consistent with the EELR origin and significant AGN photoionization in this outer region. From the increasing line intensities of [O iii ] and [Ne V] we can conclude that the NLR and EELR are also strongly ionized, however, in the late-time spectrum these decreased faster than the Bowen line N iii λ 4640. The ionization level of the BLR is therefore more persistent than in the outer regions.', '5.2. Optical re-brightening': 'The long-term ZTF light curve of AT2019aalc shows two well separated flares. Both flares evolved quickly and reached their peaks within 2 months, followed by a slow decay on yearly timescales. The first flare decayed for more than a year, while the second flare has been decaying for around a year. \nSeveral ZTF monitored extragalactic transients have been found with bumps seen in their optical light curves (see Soraisam et al. (2022) for a recent review) 12 . Most of them are SNe, however, a couple of them have been identified as TDEs (AT2020acka, AT2020nov and AT2021ehb). The latter two TDEs show prominent bumps in their light curves while decaying, while AT2020acka exhibits a long and monotonic bump. These TDEs are, however, experienced these bumps within a year from their peak. Moreover, the second flare of AT2019aalc peaked even at higher luminosities than the initial one. AT2020nov and AT2020acka experienced less significant second bumps, while AT2021ehb has two comparable peaks. The optical emission of these transients are poorly understood, and Soraisam et al. (2022) also states that their identifications as TDEs are ambiguous. Generally, TDEs are characterized by smoothly and rapidly declining light curves ( F ∝ t -5 / 3 as expected from mass fallback considerations (Rees 1988) without any prominent bumps in the case of full disruption. AT2019aalc in turn has a slowly decaying first flare with a power-law fit returning a power-law index of b ≈ -0 . 45 and a second flare that has started 4 years after the initial flare. The first systemically identified repeating partial TDE (AT2020vdq), however, shows not just a double-peaked long-term optical light curve, \nbut its second flare is ≈ 3 times more luminous compared to the first one (Somalwar et al. 2023). This implies an even more significant optical re-brightening than the one experienced by AT2019aalc. The second flares of partial TDEs are expected to follow an extremely steep t -9 / 4 power-law when declining. In the case of AT2020vdq, the second flare declined much slower (Somalwar et al. 2023) than this but still clearly faster than the second flare of AT2019aalc which decayed with a power-law index of b ≈ -0 . 13. Other candidate repeating TDEs have been found with periodic flares at di ff erent wavelengths; ASASSN 14ko with more than 20 observed optical / UV / X-ray flares (Payne et al. 2021), eRASSt J045650.3-20375 with 3 X-ray / UV flares (Liu et al. 2023), AT2022dbl with two optical flares (Lin et al. 2024), while AT2018fyk (Wevers et al. 2019a, 2023) and RX J133157.6-324319.7 (Hampel et al. 2022) showed a one-time Xray / UVand X-ray re-brightening episodes, respectively. In these cases, due to its grazing orbit, the star orbiting in the vicinity of the SMBH is thought to be only partially disrupted. After another pericenter passage, the surviving core thus feeds a new accretion flare, resulting in X-ray / UV re-brightening. For the Bowen-line and coronal-line emitter transient AT2019avd with two distinct optical flares, Chen et al. (2022) discusses a partial TDE scenario. \nInterestingly, most of the sources of the small sample of classified BFFs show optical re-brightening or bumps. Fig. 20 shows the optical evolution of the three BFFs with detected re-brightening episodes together with AT2019aalc. The optical light curve of AT2017bgt shows a bump roughly 400 days after its initial flare which is also clearly seen in the binned light curve published in Trakhtenbrot et al. (2019). One of the two other BFFs originally studied by (Trakhtenbrot et al. 2019), F01004-2237, experienced a significant optical re-brightening around 11 years after its initial flare detected in 2010. Originally, Catalina Sky Survey (CSS) light-curve data revealed that this ULIRG had undergone a luminous continuum flare (Tadhunter et al. 2021) and later Trakhtenbrot et al. (2019) classified it as a BFF similar to AT2017bgt. The binned light curve of this source published in Makrygianni et al. (2023) shows a bump around a year after its peak. We investigated the recent activity of the source and found that it significantly re-brightened in 2021 based on its ATLAS monitoring. The recent flare was more luminous than the first one and decayed faster. In addition to these two BFFs, AT2021loi shows a bump during its decaying, roughly a year after the first flare reached its peak. Makrygianni et al. (2023) suggests that optical re-brightening or bumps seen when decaying might be common features of BFFs. Only one transient classified as a BFF, OGLE17aaj, has not shown signs of optical re-brightening or bumps in its light curve to date. The first flares of the studied BFFs decay on yearly timescales, i.e. much slower than TDE flares but more similar to AT2019aalc. However, the BFFs with re-brightening episodes tend to have a faster decaying second flare / bump in comparison with their initial flare, while in the case of AT2019aalc the second flare decays slower. The BFFs classified so far tend to have remarkable variances in their optical light curve evolution in addition to the similarities we see. These might be explained by di ff erent subclasses of BFFs, similarly to regular TDEs, where di ff erent subclasses have been defined based on their light curve evolution (e.g., Charalampopoulos et al. 2023).', '5.3. Possible explanations of the long-term radio flare': 'In this subsection we discuss the long-term radio variability of AT2019aalc. The source experienced a brightening of its flux \ndensity by a factor of 2 on a timescale of less than 1 . 5 years followed by a long-term plateau ( ≥ 3 years) indicated by the archival radio survey data and our ATCA monitoring. Although radio variability on yearly timescales is a common phenomenon of Seyfert galaxies, the sharp, high-amplitude radio flare and especially the long-term plateau are generally not seen in intrinsically variable Seyfert galaxies (e.g., Mundell et al. 2009; Koay et al. 2016) and therefore require further investigation. The unusual radio spectrum with a high frequency turnover is also discussed below. \n- -The brightness temperature of T b = (3 . 0 ± 0 . 8) × 10 9 K at 1 . 7 GHz significantly exceeds 10 5 K, known as an upper limit for supernova remnants and He ii regions in star-forming galaxies (Condon 1992). Consequently, the radio emission we detected must be non-thermal and originate from physical processes associated to AGN-like (including TDEs) activity (see Bontempi et al. 2012 as an example how the estimated brightness temperatures help to characterize the radio emission of Seyfert galaxies). This way, we can also safely rule out thermal free-free emission from the accretion disk or the dusty torus (e.g., the case of NGC 1068, Gallimore et al. 1997) as the origin of the detected radio emission.\n- -Advection-dominated accretion flows have also been proposed to explain the origin of the radio emission of lowluminosity AGN (e.g., Seyfert galaxies, Mahadevan 1997). Assuming that accretion flows dominate the radio emission of AT2019aalc, following Yi & Boughn (1998) we estimate an expected 5 GHz radio luminosity of ≈ 4 x 10 35 erg s -1 , a magnitude of 3 lower than the observed one, clearly indicating that the advection-dominated accretion flows cannot explain the observed radio emission in our case.\n- -Magnetized coronal winds originating from the AGN accretion disk could produce the non-thermal radio emission of AT2019aalc. Laor & Behar (2008) found a correlation of log( LR / LX ) ≈ -5 studying the Palomar-Green (PG; Green et al. 1986) radio-quiet quasar sample (in good agreement with the correlation found for coronally active stars by Guedel & Benz (1993). Later, Behar et al. (2015) came to the same conclusion using a sample of seven radio-quiet Seyfert galaxies. Magnetized coronal winds were proposed to explain the observed radio emission of the Seyfert galaxy Mrk 590 by Koay et al. (2016), however, the authors noted that a jet contribution cannot be ruled out. These two types of outflows might even be equivalent and cannot be distinguished in Seyfert galaxies with unresolved compact core features seen in their VLBI maps (e.g., Kharb et al. 2015), like in the case of AT2019aalc. For our source, based on the mean ATCA C-band and the extrapolated Swift / XRT 2 -20 keV luminosities, we estimate a ratio of log( LR / LX ) ≈ -2 suggesting that magnetized coronal winds alone cannot fully explain the observed radio emission of AT2019aalc. Moreover, coronal emission should be rapidly variable at radio wavelengths and should evolve similarly to the X-ray flux, which is clearly not the case. Although, a contribution from the mechanisms discussed so far to the total radio flux remains a possibility, we need to look for another possible origin to explain the main origin.\n- -Out-flowing synchrotron emitting material induced by enhanced accretion are commonly detected in AGN and TDEs in the forms of non-relativistic outflows or relativistic jets which can be either on or o ff -axis relative to the observer. These features are often visible on VLBI radio maps of radio-detected Seyfert galaxies (e.g., Kozák et al. 2024) and \nFig. 20. Long-term optical light curves of AT2019aalc and three classified BFFs. The light curves indicate that optical re-brightening and bumps may be a common property of BFFs; yet, the various sources di ff er in when these episodes start with regard to the initial flares. \n<!-- image --> \na few TDEs (e.g., Paragi et al. 2017; Mohan et al. 2022). In our case, the high brightness temperature of the EVNdetected radio feature together with its position (in good agreement with the optical Gaia position) indicate its radio core identification, however, no other components were detected. Nevertheless, the steep ( α << -0 . 5) radio spectrum revealed by our ATCA observations indicates the presence of another, steep spectrum synchrotron-emitting component(s) in the system as radio cores are characterized by a flat radio spectrum. The steep radio spectrum may be the composition of the flat spectrum radio core and the additional, steep spectrum component(s). The ATCA radio spectrum indicates a high-frequency turnover above 9 GHz which is a possible hint of a newly ejected radio-emitting component. We further discuss the radio spectrum of AT2019aalc in the following. \nThe turnover suggests the multiple component nature of the spectrum. Radio spectra with high-frequency excess have been observed in a few radio-quiet narrow-line Seyfert 1 galaxies (Berton et al. 2020). These sources are thought to be kinematically young AGN with small linear sizes or heavily absorbed AGN. The high-frequency peak might be explained with the kinematic age of a newly ejected component. Since older sources are larger than young ones, the expanding relativistic jet or nonrelativistic outflow loses energy as the density decreases and the spectral peak moves to lower energies over time. With other words, as the projected linear size of the radio source increases, the turnover frequency decreases. These extremely young radio sources therefore peak at very high frequencies (e.g., Berton et al. 2020). The low-frequency peak is, however, more likely to be explained with star formation activity (as NLSy1 galaxies have a significantly higher star formation rate as other AGNs) instead of AGN emission. The latter is assumed to be synchrotron self-absorbed (SSA) to explain the faintness or the non-detection of these galaxies at lower frequencies. As another possibility to explain the high-frequency excess of, Berton et al. (2020) dis- \ncusses a scenario in which the AGN emission is absorbed below 10 GHz due to free-free absorption (FFA). In that case, the circumnuclear gas might be ionized both by the AGN and by hot stars and might act as a screen for the jet at low frequencies or the passage of the jet in the interstellar medium causes the formation of a cocoon of ionized gas and this way being responsible for the FFA. Finally, coronal emission could also be responsible for the inverted spectrum. \nIn the case of AT2019aalc, the coronal emission scenario can be safely excluded as we should see a rapidly variable radio light curve which should evolve similarly to the X-ray emission. On the other hand, the other two scenarios might also be challenging. The large-amplitude variability concluded from the VLASS observations and the EVN detection of a high brightness temperature ( Tb >> 10 5 K) feature at 1 . 7 GHz clearly imply AGN-like activity over a star formation scenario at these frequencies and is incompatible with significant absorption caused by SSA or FFA. Moreover, the presence of a relatively strong [O iii ] λ 5007 line in the SDSS host spectrum implies earlier AGN activity as being responsible for the radio emission before the long-term flare. Therefore, our results indicate that the radio spectrum of AT2019aalc is strongly dominated by AGN activity below 10 GHz as well, even if a contribution from star formation, especially at 2 . 1 GHz, is probable. \nNevertheless, the contribution of a newly ejected component to the overall radio spectrum remains a possibility. In the cases of the earlier discussed NLSy1 galaxies, the high frequency peak is thought to be suggesting a newly born radio source with no signs of earlier jet activity. Similarly to gigahertz-peaked sources (GPS), the high-frequency peak is thought to be shifting towards lower energies over time. This way, NLSy1 galaxies might be extremely young GPS which was first suggested by Oshlack et al. (2001). In our case, a newly ejected component can be responsible for the high-frequency excess whilst the low-frequency part is naturally explained with optically thin synchrotron emission of the AGN indicating that we do not see a newly born radio- \nAGN but an AGN with recently enhanced, intermittent activity. The result is a two-component radio spectrum with a relatively steep spectrum at lower frequencies and a high-frequency excess. The steepness of the spectrum between 888 MHz and 9 GHz might be explained with cooling of electrons originating from past AGN activity, and also a contribution from star formation can play a role here. Järvelä et al. (2021) states that restarted AGN activity can explain at least some of the radio spectra studied by Berton et al. (2020). The broad-line AGN classification of the host galaxy of AT2019aalc is more consistent with a restarted AGN picture as this type of AGN are thought to be older than NLSy1s making a restarted activity scenario over a newly born radio source one more probable. The restarted activity results in a peaked spectrum at higher frequencies, superimposed on a spectrum from a period (or periods) of earlier activity peaking at lower frequencies. Intermittent activity is further supported by theoretical models suggesting that sources with high accretion rates are more prone to this kind of behavior (Czerny et al. 2009). This is compatible with the accretion-induced lines seen in the post-flare-1 optical spectrum of AT2019aalc (see Subsubsect. 4.5.2 and van Velzen et al. 2024). \nWe note that the fitted ellipsoid model fit component to the EVN visibility data is aligned to the northwest-southeast direction (clearly di ff ers from the position angle of the restoring beam indicating the angle-dependent resolution of the VLBI array) which potentially indicates a newly ejected radio-emitting component. Higher frequency observations with improved angular resolution are required in order to resolve the radio structure we detected at 1 . 7 GHz. We observed AT2019aalc with the EVN and the Korean VLBI Network (KVN) jointly at 21 GHz in March 2024 and expect to reach sub-mas-scale angular resolution, roughly an order of magnitude better than with the currently discussed observation. These results will be presented in a follow-up publication (Veres et al. in prep.)', '5.4. X-ray properties': 'The X-ray spectra of AGN at energies above 2 keV have a powerlaw-like shape. These X-ray photons are believed to be triggered by Compton up-scattering of the accretion disk photons o ff hot electrons surrounding the disk, in a hot ( ≈ 10 9 K) optically thin corona above the disk (e.g., Kammoun et al. 2015). Below 2 keV energies many AGNs show an excess in their spectrum referred to as the soft X-ray excess. This soft excess can be well modeled by a blackbody model with a best-fit temperature in the range 0 . 1 -0 . 2 keV. The origin of the excess might be explained via Comptonized disk emission and the reflection of hard X-ray photons from the surface of the disk can account also for the excess (Done et al. 2012). \nThe stacked Swift / XRT X-ray spectrum of AT2019aalc can be adequately described with an absorbed blackbody + powerlaw model. We note that in the cases of strongly beamed jetted sources e.g. blazars, the X-ray spectrum can be well modeled with a simple or broken power-law without adding any additional model component (e.g., Kammoun et al. 2015; Acciari et al. 2022). Although AT2019aalc shows enhanced radio activity, its brightness temperature suggests a non-beamed nature of its radio emission in agreement with the X-ray spectrum. \nThe X-ray spectrum is dominated by a power-law component with a soft power-law index of Γ ≈ 2 . 8 while the thermal component is extremely soft with a blackbody temperature of kT ≈ 90 eV derived from the Swift / XRT observations. Most of the BLSy1s typically have a significantly harder X-ray spectrum with a power-law index Γ = 1 . 7 -2 (e.g., Pons & Watson 2014). \nOur value is more typical for sources characterized by enhanced accretion, e.g., flaring NLSy1s or TDEs which have typically soft X-ray spectra with Γ > 2 (Boller et al. 1996; Saxton et al. 2020). \nThe soft X-ray excess appears in the stacked spectrum and becomes more significant over time as we added a blackbody component in order to better describe the X-ray spectrum. Furthermore, the data of the peak of the second X-ray flare is clearly better-fit with a single blackbody model than with a power-law one (Subsect. 4.3) implying not just the need of the blackbody component to adequately fit the data but even its dominance over the power-law component at this epoch. A softening of the power-law index over time can be seen as well. These results suggest significantly enhanced accretion or disk instability. Delayed accretion flares have been observed in TDEs (Saxton et al. 2020). In our case, the post-flare-1 spectrum shows clear signs of accretion and the second flare is followed by enhanced accretion activity as well. \nThe soft X-ray flux of Seyfert galaxies was found to be well correlated with the observed fluxes in the coronal line [Fe x ]. The coronal lines to X-ray flux ratio is log( f [Fex] / fx ) = -3 . 43 ± 0 . 55 for both broad and narrow-line Seyfert 1 galaxies (Gelbord et al. 2009). For AT2019aalc we estimate values of log( f [Fex] / fx ) ≈ -3 . 45 when optical spectroscopic and X-ray observations were performed nearly simultaneously (due to the rapid X-ray variability we consider only observations taken on the same day). This implies that the expected soft X-ray luminosity is consistent with the observed values. The soft X-ray photons excite the coronal lines continuously and not years later originating from a past TDE as it was suggested for ECLEs by Wang et al. (2012). The photoionization or collisional excitation of the dust grains is rather due to a current event. This further supports that the reoccurring optical / UV outbursts seen during the decaying of the flares are driven by the soft X-ray flares which lead these outbursts only with a couple of weeks. These suggest an accretiondriven origin of the reoccurring outbursts in the optical / UV during the decaying of the flare.', '5.5. SED fitting': "Following Reusch et al. (2022), we fitted two blackbody models in order to describe the SED of the transient using lmfit . These fitted curves to the optical / UV ('blue' blackbody) and IR datapoints ('red' blackbody) are shown in Fig. 21, together with a combined double blackbody fit. The red blackbody fit was introduced to account for the IR dust echo. These curves fit the SED around the optical peak of the investigated flare. \nThe continuum emission of TDEs are well described by a thermal blackbody model (Gezari 2021). The case of AT2019aalc appears to be more complex. An excess in the UW2 band can be seen regardless the consideration of a simple or the double blackbody fit. The UV emission have a contribution from the Bowen Fluorescence mechanism and potentially also from Comptonization explaining the poor fit towards the blue. The red wing of the optical and the IR part cannot be well fitted with a simple blackbody model suggesting a complex, multicomponent dusty reprocessing region. The excess in the r -band might be explained with the early appearance of some Fe lines in this range (such as [Fe x ] λ 6375 and [Fe vii ] λ 6088) indicating the release of this element previously locked up by the dust grains. Since all of this seem to be started already around the optical peak, at least a fraction of the dust is located on ∼ subpc scales to the SMBH. This is further consistent with the extreme line widths of the coronal lines (see Subsect. 5.1.3). Therefore, not \nFig. 21. The extinction-corrected SED of the second flare of AT2019aalc as measured around the optical peak. Two blackbody models to the optical / UV (blue curve) and IR datapoints (red curve) were fitted separately and a combined fit (black dashed curve) is also shown. A clear excess in r -band and the WISE bands can be seen. \n<!-- image --> \nonly the dusty tori is responsible for the IR dust echo. It is possible that the significant amount of dust in the nucleus region is the consequence of enhanced star formation sometime in the past. The Baldwin, Phillips& Terlevich (BPT, Baldwin et al. 1981) emission-line ratio diagnostics of the host SDSS spectrum are log([O iii ] λ 5007 / H β ) ≈ 0 . 3 and log([N ii ] λ 6583 / H α ) ≈ -0 . 4. We plotted the BPT line-ratio of SDSS spectra with indicating the ratios of the host galaxy of AT2019aalc and the BFFs AT2021loi, AT2017bgt and F01004-2237 in Fig. 22 using as -troml (Vanderplas et al. 2012; Ivezi'c et al. 2014). All of the investigated galaxies are in the composite region, consequently, in addition to AGN activity, star formation activity might also have a remarkable contribution for producing the narrow lines seen in the spectra. Moreover, these ratios place these two galaxies solidly in the region of the diagram where the post-starburst sample of French et al. (2015) is located. We note that to derive the ratios for AT2019aalc and F01004-2237, we used pre-flare spectra. While as no host spectra are available for AT2017bgt and AT2021loi, we considered the spectra taken as early as possible in these cases.", '5.6. What type of transient is AT2019aalc?': "AT2019aalc shares similarities with TDEs such as the soft Xray spectrum, the dust echo and the optical flare(s) rising with constant color. The UV bright nature of the transient with a blue optical / UV peak color of NUV -r ≈ 0 . 2 is another TDE-like feature and not typical for AGN and SNe (Gezari 2021). Bowen lines appear in the spectra of severeal TDEs even if these are typically weaker than in our case. The most remarkable di ff erence is, however, the decay of the optical emission, which is significantly slower than for any TDE identified so far. The color evolution of the flares is also atypical for TDEs. \nThe unobscured AGN-like optical spectra, the presence of persistent and strong Bowen lines (O iii λ 3133 and N iii λ 4640), the extremely luminous UV flare, on yearly timescales decaying optical emission and the optical re-brightening altogether suggest a Bowen Fluorescence Flare classification of AT2019aalc. Furthermore, we classify AT2019aalc as an Extreme Coronal Line Emitter because of the detection of strong high-ionization coronal lines. \nFig. 22. The BPT narrow line ratio diagram (Baldwin et al. 1981) of SDSS spectra. The location of the dividing line is taken from Kewley et al. (2001). The dashed lines represent the composite region i.e. where both star formation and AGN activity play a role. The line ratios of the host galaxy of AT2019aalc derived from the fitted SDSS spectrum are indicated with a green star symbol. For the BFFs AT2021loi and AT2017bgt the ratios are taken from Makrygianni et al. (2023) and Trakhtenbrot et al. (2019) and indicated with red and blue star symbols, respectively. To derive the ratios for the BFF F01004-2237, we downloaded its 6df (Jones et al. 2009) spectrum which was taken years before the first flaring episode of the transient. The location of the BFFs in the diagram suggests remarkable current or past star formation activity which might explain the extreme dust echo emission of the transients. \n<!-- image --> \nThe mechanism that powers the long-term accretion enhancement in BFFs is still unclear. TDEs in AGN might play a role. In a single star case, the slowly decaying optical / UV emission might be explained by the dissipation of energy due to the interaction between the stellar debris stream and the preexisting accretion disk (similarly to the case of the TDE-AGN PS16dtm, Blanchard et al. 2017) or the AGN's radiation field. The bumps in optical and UV during the decay of the flares of AT2019aalc led by soft X-ray flares indicate the flare's accretiondriven origin consistently within this picture. Moreover, the flaring episodes in X-rays seem to be happen periodically which is expected from interaction between a circularizing stellar stream and the accretion disk. The origin through reprocessing is further consistent with the ionization of the BLR and so on the increase in the Bowen and the Balmer lines seen in the spectra during the second flare. The dusty region (expected from a source with IR dust echo) and the liberation of Fe and S from these dust grains due to photoionization by the soft X-ray flares provide an explanation for the high-ionization coronal lines. The increase of the W1 -W2 color implies significant cooling expected from cooling dust after the onset of the dust echo. This scenario provides explanation for one slowly decaying flare with recurring outbursts, however, AT2019aalc exhibits two distinct optical flares. The second flare is even more luminous than the first one, similarly to the case of the partial TDE AT2020vdq (Somalwar et al. 2023). A partial TDE scenario might explain the second optical flare. \nAlthough the slowly decaying optical emission and the rebrightening seem to be common features of BFFs, the long-term light curves still show di ff erences as well. The re-brightening \nepisodes started at di ff erent timescales for the individual sources, and more BFFs need to be classified to better understood their unique optical light curves. Nevertheless, (partial) TDEs occurring in AGN can be a possible origin of BFFs. This would explain the TDE-like properties of these sources. The most remarkable di ff erences i.e. the slowly decaying optical emission and the continuum emission not being well described by blackbody models might appear due to a more complex environment the TDE takes place with comparing to quiescent galaxies. Furthermore, a possible connection between the high-ionization coronal lines (which are most probably related to TDE activity, Wang et al. 2012) and the BFFs is consistent with this picture. A known potential partial TDE in an AGN is the case of the active galaxy IC 3599. This low-luminosity AGN exhibits periodic soft X-ray flares interpreted with a partially disrupted star, which results in reoccurring outbursts after each passage of the surviving core around the central black hole (Campana et al. 2015). Notably, IC 3599 has more prominent high-ionization coronal lines than most AGN (similarly to the coronal-line detected transients, see Fig. 17). Interestingly, IC 3599 is the only known AGN with fading coronal lines (Frederick et al. 2019). \nAlternatively, a double TDE i.e. the tidal disruptions of a pair of binary stars might explain the double-peaked optical light curve. Notably, double TDEs may represent nearly 10% of all stellar tidal disruptions (Mandel & Levin 2015). \nThe strength of the high-ionozation coronal lines that regular AGN activity cannot explain the unusual behaviour of this transient. It is not clear why we do not see as strong high-ionozation coronal lines in normal AGNs as in the transients with significant dust echo emission i.e. TDEs and BFFs. It might be explained with dust sublimation due to the UV radiation of the AGN (the sublimation radius is typically in the subpc range, van Velzen et al. 2021b) which clears the dust from the nucleus region. AGN-driven outflows play also a role here. The limited variability of the host galaxy of AT2019aalc prior the first flare and its location in the BPT diagram suggest weak AGN activity before the transient occurred. Therefore, the dust possibly originating from star formation activity in the nucleus could survive. Additionally, the first flare could not clear the nucleus region which suggests that it must have been way less luminous in UV than the second flare. Unfortunately, due to lack of UV data around the first flare this hypothesis cannot be further tested. Nevertheless, the detection of some Fe lines in the post-flare-1 spectrum implies that a certain fraction of the dust was destroyed following the first flare. \nInterestingly, the host AGN of other BFFs also showed weaker pre-transient activity than persistently accreting AGN (Trakhtenbrot et al. 2019; Makrygianni et al. 2023). The TDElike features suggest that TDEs might restarted these AGN. Beyond the TDE-related scenarios, it is possible that the restarted AGN activity is rather explained with the dusty environment itself. \nAs we studied AT2019aalc in radio in detail, we compare the radio properties with that of the BFFs available. Based on the radio observations explained in Makrygianni et al. (2023) AT2021loi has an inverted radio spectrum between 5 and 10 GHz while the source was not detected at 3 GHz which can be compatible with a newly ejected jet or outflow component, similarly to the case of AT2019aalc. Although no radio spectrum is available for AT2017bgt, we investigated the VLASS QL Images of this BFF to study its radio flux variability. We found a radio source at the optical position of AT2017bgt with flux densities of 3 . 6 mJy (on 2019-05-21), 6 . 6 mJy (on 2021-10-19) and 4 . 5 mJy (on 2024-06-11) at a central frequency of 3 GHz. \nThe host galaxy of AT2017bgt was detected in the FIRST (at 1 . 4 GHz) in July 1998 with a flux density of 0 . 9 mJy. In addition, the BFF candidate VT J1548 was also found to exhibit a radio spectrum peaking around 5 GHz (in contrast with non-jetted AGN that peak below 1 GHz, e.g., Padovani et al. 2017) which was explained with an ejection of a synchrotron-emitting outflow (Somalwar et al. 2022). These findings allow us to conclude that newly ejected radio outflow / jet components play a crucial role in BFFs. These outflows ejected due to enhanced accretion might be partly responsible for the liberation of Fe and S from the dust grains located relatively close to the SMBH. The collisional ionization in the case of AT2019aalc is supported by the extreme broadness of the coronal lines especially [S xii ] λ 7611.", '5.7. BFFs and high-energy neutrinos': "AT2019aalc has been associated with the high-energy neutrino event IC-191119A (most probable neutrino energy: 177 TeV) detected by the IceCube Observatory (IceCube Collaboration 2019; van Velzen et al. 2024). Motivated by the classification of AT2019aalc suggested by its multi-wavelength properties (Subsect. 5.6), we searched for spatial coincidences between the earlier classified BFFs and the neutrino alerts from the IceCube Event Catalog of Alert Tracks (ICECAT-1; Abbasi et al. 2023). Wefound that the BFF AT2021loi is located within the 90% rectangular uncertainty contour of the high-energy neutrino event IC-230511 (most probable neutrino energy: 167 TeV) detected in May 2023 (IceCube Collaboration 2023). In the full likelihood contours published in ICECAT-1, the source is just outside the 90% contour. We present the source positions together with the neutrino best-fit positions and both contours in Figs. 23 and 24. We also report a coincidence between the 'prototype' BFF AT2017bgt and the high-energy neutrino event IC-200410A. This event is, however, a poorly reconstructed cascade event, therefore we do not discuss it further here. \nThe neutrino associated with AT2021loi is 680 days delayed with respect to its first optical peak and 290 days with respect to its second optical peak. Note, that the 3 nuclear transients (including AT2019aalc) are associated with neutrinos that were delayed by 150 -290 days, which might be explained with delayed mass accretion, delayed jet formation or outflow ejection (van Velzen et al. 2024). In the case of AT2021loi, the inverted radio spectrum (Makrygianni et al. 2023) is compatible with a newly ejected jet or outflow component, similarly to the case of AT2019aalc. However, from the current data, we cannot tell if the radio-emitting region of AT2019aalc has been ejected before or after the neutrino. Although we do not see a clear increase in the 888 MHz RACS data 6 months after the neutrino arrival. However, if the radio flare is indeed related to a newly ejected, synchrotron self-absorbed component, it is not surprising that it does not result in any increasing at low frequencies at early times. Interactions between the out-flowing material and clouds in the surrounding of the SMBH (Wu et al. 2022) or between ultra-high energy protons (accelerated in a jet or outflow) and infrared photons from the dust echo (Winter & Lunardini 2023) were proposed to explain the observed neutrino emission of the 3 nuclear transients. These findings indicate that outflows play an important role in the neutrino production. Moreover, Winter & Lunardini (2023) estimate the highest neutrino fluence for AT2019aalc among the 3 studied nuclear transients associated with neutrinos. A similar case study for AT2021loi would help us to further constrain the neutrino production models of nuclear transient sources. \n<!-- image --> \ncreases, similar to the first flare when it peaked delayed with respect to the optical and was explained with a dust echo flare. We detected a very soft X-ray flare that peaked before the optical / UV emission peaks. Later, the Swift / XRT detected an almost 10 times more luminous X-ray flare which is clearly dominated by a blackbody component, suggesting enhanced accretion or disk instability. \nFig. 23. The blue and red contours represent the 50% and 90%CL containment detailed contours of the high-energy neutrino event IC191119A, respectively. The orange box is the 90%CL rectangular containment region. The neutrino best-fit position is shown with a blue point. The optical position of AT2019aalc is marked with a green star sign. This coincidence was originally published in van Velzen et al. (2024). The detailed contours are presented here for the first time.Fig. 24. The blue and red contours represent the 50% and 90%CL containment detailed contours of the high-energy neutrino event IC230511, respectively. The orange box is the 90%CL rectangular containment region. The neutrino best-fit position is shown with a blue point. The optical position of AT2021loi is marked with a green star sign. \n<!-- image --> \nIf the connection between the high-ionization coronal line detection and the unusually large dust echos of nuclear transients is real, this can can be used to reveal more sources with strong dust echo and therefore more candidate neutrino sources. Notably, both AT2019aalc and AT2021loi are characterized by significant dust echos (van Velzen et al. 2024; Hinkle 2024; see also Subsect. 4.2) and unusually strong high-ionization coronal lines (Makrygianni et al. 2023; see also Subsect. 5.1).", '6. Conclusions': 'We conducted a multi-wavelength campaign during the second optical flare of the nuclear transient AT2019aalc associated with a high-energy neutrino event. We found that the second optical flare is even more luminous and decays even slower than the first one. Both flares evolve rapidly and with constant color. The optical flare is accompanied by UV emission which peaks at extremely high luminosities (at ≈ 10 44 erg / s). The IR emission in- \nOur VLBI observation provides a hint of extension of the radio-emitting source and clearly favors an AGN-like origin over a star-formation scenario. The radio monitoring revealed a long-term flare and an inverted spectrum above 9 GHz, which is most probably explained by a newly ejected, self synchrotronabsorbed outflow or jet component. As we do not see obvious signs of variability well before the first optical flare, we suggest that the transient event in 2019 renewed the activity of the host AGN. Restarted AGN tend to have inverted radio spectrum. This also naturally explains the long-term flare. \nThe results from optical spectroscopy help us to better understand the event responsible for restarting the AGN. In the optical spectra, the presence of persistent Bowen lines are consistent with intensified accretion over a year after the second flare. The UV bright nature of the transient is consistent with the Bowen mechanism which pumps lines mostly in this range. The line intensities of the Bowen lines and the temperature of the BLR both peak around the peak of the optical flare indicating that the optical flare is driven by the ionization of the clouds of most probably the BLR. As we find the X-ray flaring episodes prior to the rise in optical / UV, the origin of the latter is most probably reprocessed X-ray photons which are also required for the extreme ionization implied by the Bowen lines. The X-ray flares are very soft which explains the unusually strong He ii lines. These findings altogether suggest the accretion-driven origin of the flaring episode we monitored. Note, that reprocessed soft X-ray emission i.e. by an outflow surrounding the disk is a possible explanation for the origin of the optical / UV emission of TDEs (Bu et al. 2022). \nWe detected several persistent or even late-time increasing high-ionization coronal lines in the spectra of AT2019aalc. The detection of [S xii ] λ 7611 requires an extremely high ionization potential. To our knowledge, AT2019aalc is only the third extragalactic transient event detected with this line and only 4 Seyfert galaxies (including the neutrino emitter NGC 1068) and 5 ECLEs were reported with a line with such a high ionization potential. The Fe / X-ray ratios disfavor a past TDE interpretation responsible for the coronal lines, which was proposed to explain these lines seen in most of the ECLEs studied earlier. The slowly decaying optical emission might instead be explained with continuous photoionization of the dust grains (located at distances consistent with the BLR instead of the NLR as seen for ECLEs before). This is consistent with the early observation of the dust echo emission with respect to the optical peak which suggests that the dusty region is located very close to the SMBH. Based on the narrow-line ratios, we found that enhanced star formation activity is a favorable explanation for the dusty environment. Moreover, the gas-rich environment might explain the restarted activity of the AGN. \nThe high [Fe x ] λ 6375 / [O iii ] λ 5007 luminosity ratios of the BFFs, coronal line-detected TDEs and ECLEs suggest a possible connection between these source types. These sources tend to have large dust echo-like flares, which implies that the liberation of Fe from the dust grains can be explained via their TDE(-like) flaring episodes. The significant photoionization is supported by the detection of strongly ionized neon lines in the EELR. A delayed outflow / jet component and collisional excita- \non provides an alternative explanation for the coronal lines and their late-time increasing or appearance. The BFF and coronalline emitter AT2021loi has an inverted radio spectrum and based on survey data we found a flaring radio source at the position of AT2017bgt. These findings suggest that newly ejected radioemitting components play an important role not only in BFFs, but might be connected to the coronal line emission. A detailed radio study of other BFFs and coronal line emitters is crucial to better understand the appearance of these highly ionized lines. \nAltogether, our results are consistent with the Bowen Fluorescence Flare classification of AT2019aalc. We also classify AT2019aalc as an Extreme Coronal Line Emitter. A connection between the ECLEs, some TDEs and BFFs suggested by our spectroscopy results makes a non-standard TDE-related scenario more probable to explain the optical flares and i.e. the restarted activity of the host AGN. Moreover, due to the pre-existing BLR and therefore the high densities, TDE-AGN are more capable for the Bowen Flurosence mechanism and the high-ionization coronal lines. Only a few TDEs were detected with coronal lines; these are also known as TDEs with extreme dust echo emission. \nContinuous interactions of a pair of binary star and the SMBH favored by Trakhtenbrot et al. (2019) might explain the re-brightened and slowly decaying optical emission of BFFs. An alternative scenario for the two distinct optical flares is a partial disruption of a single star in an AGN. Here, the slowly decaying reprocessed optical / UV emission originates from soft X-ray flares powered by continuous tidal interaction between the stellar debris and the pre-existing accretion disk. The double-peaked optical light curve is explained with a second disruption due to the surviving core after the first passage. However, more BFFs are needed to study the mechanism responsible for these transients. TDE-related scenarios nevertheless provide an explanation not only for the suddenly enhanced activity of the nucleus but also for the high metallicity of the ionized gas. Future observations will help us to better understand the nature of the transient; a binary TDE would not produce a third flare, a repeating TDE would lead to a roughly periodic flaring, while some other AGN activity would lead to erratic future flares. \nMotivated by the high-energy neutrino associated with AT2019aalc, we cross-matched the ICECAT-1 with the positions of the known 4 BFFs. We found that AT2021loi is coincident with the high-energy neutrino event IC-230511 confirming the flaring AGN-neutrino connection studied by van Velzen et al. (2024). Apart from the known multi-wavelength properties of BFFs, AT2021loi shares characteristics with AT2019aalc such as late-time IR flare (due to dust echo), high-ionozation coronal lines in its spectra (and also r -band excess), radio detection (and inverted radio spectrum) and re-brightening episode / bumps in optical. In both cases the host galaxies are classified as broadline AGN with remarkable star formation activity.', 'Acknowledgements': "We thank Benny Trakhtenbrot for useful discussions. PMV, AF, BA, ST, AK, DJB, EM-B acknowledge the support from the DFG via the Collaborative Research Center SFB1491 Cosmic Interacting Matters - From Source to Signal . EH acknowledges support by NASA under award number 80GSFC21M0002. \nThe European VLBI Network is a joint facility of independent European, African, Asian, and North American radio astronomy institutes. Scientific results from data presented in this publication are derived from the following EVN project code: EV027. e-MERLIN is a National Facility operated by the University of Manchester at Jodrell Bank Observatory on behalf of STFC. \nWe acknowledge the use of public data from the Swift data archive. This research has made use of the XRT Data Analysis Software (XRTDAS) developed under the responsibility of the ASI Science Data Center (ASDC), Italy. The Australia Telescope Compact Array is part of the Australia Telescope National Facility https: // ror.org / 05qajvd42 which is funded by the Australian Government for operation as a National Facility managed by CSIRO. We acknowledge the Gomeroi people as the Traditional Owners of the Observatory site. Funding for the Sloan Digital Sky Survey V has been provided by the Alfred P. Sloan Foundation, the Heising-Simons Foundation, the National Science Foundation, and the Participating Institutions. SDSS acknowledges support and resources from the Center for HighPerformance Computing at the University of Utah. SDSS telescopes are located at Apache Point Observatory, funded by the Astrophysical Research Consortium and operated by New Mexico State University, and at Las Campanas Observatory, operated by the Carnegie Institution for Science. The SDSS web site is www.sdss.org. SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration, including Caltech, The Carnegie Institution for Science, Chilean National Time Allocation Committee (CNTAC) ratified researchers, The Flatiron Institute, the Gotham Participation Group, Harvard University, Heidelberg University, The Johns Hopkins University, L'Ecole polytechnique fédérale de Lausanne (EPFL), Leibniz-Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Extraterrestrische Physik (MPE), Nanjing University, National Astronomical Observatories of China (NAOC), New Mexico State University, The Ohio State University, Pennsylvania State University, Smithsonian Astrophysical Observatory, Space Telescope Science Institute (STScI), the Stellar Astrophysics Participation Group, Universidad Nacional Autónoma de México, University of Arizona, University of Colorado Boulder, University of Illinois at Urbana-Champaign, University of Toronto, University of Utah, University of Virginia, Yale University, and Yunnan University. This publication makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory / California Institute of Technology, and NEOWISE, which is a project of the Jet Propulsion Laboratory / California Institute of Technology. WISE and NEOWISE are funded by the National Aeronautics and Space Administration. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. This work has made use of data from the Asteroid Terrestrial-impact Last Alert System (ATLAS) project. The Asteroid Terrestrial-impact Last Alert System (ATLAS) project is primarily funded to search for near earth asteroids through NASA grants NN12AR55G, 80NSSC18K0284, and 80NSSC18K1575; byproducts of the NEO search include images and catalogs from the survey area. This work was partially funded by Kepler / K2 grant J1944 / 80NSSC19K0112 and HST GO-15889, and STFC grants ST / T000198 / 1 and ST / S006109 / 1. The ATLAS science products have been made possible through the contributions of the University of Hawaii Institute for Astronomy, the Queen's University Belfast, the Space Telescope Science Institute, the South African Astronomical Observatory, and The Millennium Institute of Astrophysics (MAS), Chile. This work is based on data from eROSITA, the soft X-ray instrument aboard SRG, a joint Russian-German science mission supported by the Russian Space Agency (Roskosmos), in the interests of the Russian Academy of Sciences represented \nby its Space Research Institute (IKI), and the Deutsches Zentrum für Luft- und Raumfahrt (DLR). The SRG spacecraft was built by Lavochkin Association (NPOL) and its subcontractors, and is operated by NPOL with support from the Max Planck Institute for Extraterrestrial Physics (MPE). The development and construction of the eROSITA X-ray instrument was led by MPE, with contributions from the Dr. Karl Remeis Observatory Bamberg & ECAP (FAU Erlangen-Nuernberg), the University of Hamburg Observatory, the Leibniz Institute for Astrophysics Potsdam (AIP), and the Institute for Astronomy and Astrophysics of the University of Tübingen, with the support of DLR and the Max Planck Society. The Argelander Institute for Astronomy of the University of Bonn and the Ludwig Maximilians Universität Munich also participated in the science preparation for eROSITA. The Pan-STARRS1 Surveys (PS1) and the PS1 public science archive have been made possible through contributions by the Institute for Astronomy, the University of Hawaii, the Pan-STARRS Project O ffi ce, the Max-Planck Society and its participating institutes, the Max Planck Institute for Astronomy, Heidelberg and the Max Planck Institute for Extraterrestrial Physics, Garching, The Johns Hopkins University, Durham University, the University of Edinburgh, the Queen's University Belfast, the Harvard-Smithsonian Center for Astrophysics, the Las Cumbres Observatory Global Telescope Network Incorporated, the National Central University of Taiwan, the Space Telescope Science Institute, the National Aeronautics and Space Administration under Grant No. NNX08AR22G issued through the Planetary Science Division of the NASA Science Mission Directorate, the National Science Foundation Grant No. AST-1238877, the University of Maryland, Eotvos Lorand University (ELTE), the Los Alamos National Laboratory, and the Gordon and Betty Moore Foundation. These results made use of the Lowell Discovery Telescope (LDT) at Lowell Observatory. Lowell is a private, non-profit institution dedicated to astrophysical research and public appreciation of astronomy and operates the LDT in partnership with Boston University, the University of Maryland, the University of Toledo, Northern Arizona University and Yale University. The upgrade of the DeVeny optical spectrograph has been funded by a generous grant from John and Ginger Giovale and by a grant from the Mt. Cuba Astronomical Foundation. Some of the data presented herein were obtained at Keck Observatory, which is a private 501(c)3 non-profit organization operated as a scientific partnership among the California Institute of Technology, the University of California, and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W. M. Keck Foundation. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Maunakea has always had within the Native Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain. This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center / California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. Based on observations obtained with the Samuel Oschin Telescope 48-inch and the 60-inch Telescope at the Palomar Observatory as part of the Zwicky Transient Facility project. ZTF is supported by the National Science Foundation under Grant No. AST-2034437 and a collaboration including Caltech, IPAC, the Weizmann Institute for Science, the Oskar Klein Center at Stockholm University, the University of Maryland, Deutsches Elektronen-Synchrotron and Humboldt Univer- \ny, the TANGO Consortium of Taiwan, the University of Wisconsin at Milwaukee, Trinity College Dublin, Lawrence Livermore National Laboratories, and IN2P3, France. Operations are conducted by COO, IPAC, and UW.", 'References': "- Abazajian, K. N., Adelman-McCarthy, J. K., Agüeros, M. A., et al. 2009, ApJS,\n- 182, 543 Abbasi, R., Ackermann, M., Adams, J., et al. 2023, ApJS, 269, 25 Acciari, V. A., Aniello, T., Ansoldi, S., et al. 2022, ApJ, 927, 197 Ahn, C. P., Alexandro ff , R., Allende Prieto, C., et al. 2012, ApJS, 203, 21 Arnaud, K. A. 1996a, in Astronomical Society of the Pacific Conference Series, Vol. 101, Astronomical Data Analysis Software and Systems V, ed. G. H. Jacoby & J. 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2024SPIE13097E..2TM | In this contribution we report the ongoing progresses of the project FATE an operational automatic forecast system conceived to deliver forecasts of a set of astroclimatic and atmospheric parameters having the aim to support the science operations i.e. the Service Mode at the Very Large Telescope. The project has been selected at conclusion of an international open call for tender opened by ESO and it fits with precise technical specifications. In this contribution we will present the ultimate goals of this service once it will be integrated in the VLT operations the forecasts performances at present time and the state of the art of the project. FATE is supposed to draw the roadmap towards the optical turbulence forecast for the ELT. | 2024-08-01T00:00:00Z | ['arXiv:2409.05133', '2024SPIE13097E..2TM', '2024arXiv240905133M', '10.1117/12.3018041', '10.48550/arXiv.2409.05133'] | ['Astrophysics - Instrumentation and Methods for Astrophysics', 'Physics - Atmospheric and Oceanic Physics'] | FATE an operational automatic system for optical turbulence forecasting at the Very Large Telescope | 2,024 | 175 | 0.38 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML'] | 0 | https://arxiv.org/pdf/2409.05133.pdf | {'FATE - an operational automatic system for optical turbulence forecasting at the Very Large Telescope': "Elena Masciadri a , Alessio Turchi a , Luca Fini a , Alberto Ortolani b , Valerio Capecchi b , Francesco Pasi b , Angel Otarola c , and Steffen Mieske c \na INAF-Osservatorio Astrofisico di Arcetri, L.go Enrico Fermi 5, Firenze, Italy b LaMMA, Via Madonna del Piano 10, Sesto Fiorentino, Firenze, Italy c ESO, Alonso de C'ordova 3107, 763000, Santiago, Chile", 'ABSTRACT': 'In this contribution we report the on-going progresses of the project FATE, an operational automatic forecast system conceived to deliver forecasts of a set of astroclimatic and atmospheric parameters having the aim to support the science operations (i.e. the Service Mode) at the Very Large Telescope. The project has been selected at conclusion of an international open call for tender opened by ESO and it fits with precise technical specifications. In this contribution we will present the ultimate goals of this service once it will be integrated in the VLT operations, the forecasts performances at present time and the state of the art of the project. FATE is supposed to draw the roadmap towards the optical turbulence forecast for the ELT. \nKeywords: turbulence, turbulence forecast, numerical modelling, adaptive optics, machine learning', '1. INTRODUCTION': 'FATE is an acronym that states for Forecast system for Atmosphere and Turbulence for ESO. It is the winner project of an international Call for Tender (CfT) opened by ESO on 2020. The goal of the CfT was to supply automatic forecasts of optical turbulence and atmospheric parameters in operational mode for the VLT. The ESO request was to provide forecasts in operational mode for two years with the possible extension of up to further three years. The main goals/objectives of the ESO CfT were: \n- · to decreases the amount of out-of-constraints observations due to unforeseen changes in meteorological or optical turbulence conditions\n- · to assist the support astronomers in making decision at night\n- · to enable more aggressive short-term scheduling of the observations with well understood risks\n- · to best prepare the mode operations of the ELT at Cerro Armazones by gaining experience on the use of the forecast to maximize the science return of the VLT at Cerro Paranal. \nIn other words a two fold objective: to retrieve forecast of the main parameters (atmospheric and optical turbulence) to support the Service Mode at the VLT 1, 2 but also to prepare the roadmap towards the ELT. Since more than a decade ESO applies the Short-Term Scheduling (STS) whose aims is that to optimise the calculation of how many highly ranked observations are completed for unit time. Forecasts at short time scales are therefore useful to such an optimisation. \nThe roles and the actors inside FATE are as in the following: (1) ESO is the customer, (2) INAF is the contractor with the PI-ship, (3) the FATE team is composed by two institutes: INAF and LaMMA, both located in Florence, Italy. \nINAF have a robust and long-time expertise in the optical turbulence forecast field and atmospheric parameters relevant for the ground-based astronomy. The most relevant progresses in the context of the optical turbulence forecast have been reached in the proposition of parameterisation solutions, charaterisation of sites, proposition of model calibration methods hybrid forecast methods. 3-13 LaMMA has a long-time experience in managing operational meteo services and it is the meteo structure supporting the Civil Protection at an italian regional level that is in Tuscany. \nINAF has in FATE the following responsibilities: (1) scientific responsibility of the project, (2) development of the optical turbulence model Astro-Meso-Nh 3 and development of the automatic forecast system, 14 (3) development of algorithms having the aims to achieve the technical specifications and (4) R&D to further improve forecast performances. \nLaMMA has the responsibility to manage FATE forecasts in an operational mode on a daily base. \nWehighlight that INAF had carried out in the past a feasibility study (called MOSE - Phase A (2011-2013) and Phase B (2014-2015)) for ESO aiming to test the possibility to provide reliable forecasts of the optical turbulence above Cerro Paranal and Cerro Armazones. Such a study lead to the publication of a few papers. 8, 9, 15, 16', '2. MODEL CONFIGURATION': 'We refer readers to Masciadri et al. 2023 13 for a complete and extended description of the model configuration for FATE. We report here just a synthetic summary. In Fig.1 is shown the toy model of the forecast scheme related to the forecast at a time scale of 1 day (1d) for the night time. We are considering the forecasting of the night J. The mesoscale model Astro-Meso-Nh ∗ is initialiazed and forced with forecasts provided by the European Centre for Medium Range Weather Forecasts (ECMWF) HRES model and calculated at 00:00 UT of the day (J-1). Such a model has 137 vertical levels and a horizontal resolution of roughly 9 km. We start the simulation with the mesoscale model Astro-Meso-Nh six hours later (at 06:00 UT) when the initialisation data are accessible and we are supposed to deliver the forecasts to ESO two hours before the sunset. To obtain the forecast at two days (2d) and three days (3d) initialization data are calculated at 00:00 UT - 24h and 00:00 UT 48h. We highlight that the interval of time between 06:00 UT and the time of the delivery (i.e. two hours before the sunset) does not correspond to the computation time of a forecast. Actually different forecasts are executed in this interval of time. More precise and detailed infos in this respect are in Masciadri et al. 2023. 13 The toy model related to the day time is the same but it is shifted of 12 hours. That means that, for the forecast of the day J, initialization data from ECMWF are calculated at 12:00 UT and the simulations starts at 18:00 UT of the day (J-1). In that case we are supposed to deliver forecast two hours before the sun-rise. We use a grid-nesting technique 19 centred above Cerro Paranal (i.e. the point of interest). We have three imbricated domains and the highest horizontal resolution of the innermost domain is equal to 500 m (Fig.2). \nIt is important to highlight that the configuration of the automatic forecast system selected for FATE is the result of a trade-off between: \n- - the number of parameters that have to be forecasted\n- - technical specifications requested by ESO\n- - hardware cost\n- - the operational configuration that depends on the ESO requests therefore they depends on the Statement of Work (SoW). \nTwo different products are produced by FATE: \n- 1. Forecasts at long forecast time scale (FTS): at 1 day (1d), 2 days (2d) and 3 days (3d). These three forecasts correspond to the following time scales (23h - 33h), (47h - 57h) and (71h - 81h) respectively. For', 'TOY MODEL (NIGHT)': "Figure 1. Toy model representing the configuration of the automatic forecast system related to the forecast at long time scale for 1 day. As described more extensively in 13 the forecast refers to the interval [23h - 33h]. The mesoscale model Astro-Meso-nh is initialized with data coming from the ECMWF and calculated at 00:00 UT. \n<!-- image --> \nFigure 2. Meso-NH model grid-nesting configuration. The second column shows the number of horizontal grid points, the third column the domain extension, and the fourth column the horizontal resolution ∆X . \ncompleteness, the FTS for the 1d case is calculated as (23h - 33h) = T ini - T fc (see Masciadri et al. 2023 13 for details). For the forecasts at long time scale we use a pure hydrodynamical approach that is we use only the atmospherical Astro-Meso-Nh model. \n- 2. Forecasts at short forecast time scale (FTS) more precisely at 1h or 2h that are also the most relevant intervals for the Service Mode of the VLT (but also of most of Observatories). We refer to Masciadri et al. 2023 - Section 3 for the calculation of the FTS in the case of shirt FTS. We use here a hybrid approach based on the autoregression technics that has been proposed in Masciadri et al. 2020 10 and that we call 'AR technics'. Such an approach implies the use outputs of the forecast obtained with the Astro-Meso-Nh mesoscale atmospherical model plus real-time observations related to a finite number N of nights in the past. It has been proved by Masciadri et al. 2020 10 that the optimized configuration is obtained with N=5.", '3. TECHNICAL SPECIFICATIONS': "We summarise here the technical specifications defined by ESO in the SoW. Fig.3-5 show the categories within which the atmospherical parameters values for WS, RH and PWV should be distinguished. In the case of WS, when it flows with values between 12 ms -1 and 18 ms -1 there are pointing restrictions, when the the wind velocity is larger than 18 ms -1 the dome is closed. In the case of RH, the 50% value is related to a few humidity-sensitive instruments, 70% for all the other. Also ESO defined a subjective classification of the sky \nFigure 3. Categories, extracted from the SoW, identifying the values of the WS to be discriminated. \n<!-- image --> \nFigure 4. Same as Fig.3 but for the RH. \ntransparency (Fig.6). We have four categories: the photometric sky, the clear sky, the variable thin cirrus and the variable thick cirrus. Such a classification has been proven to be strictly correlated to the cloud cover † therefore we provide a forecast of the sky transparency passing by the forecast of the cloudiness fraction. In it planned by ESO in the future to introduce a classification of the sky transparency based on the fluctuation analysis of the infrared sky temperature measured by the radiometer LHATPRO. 21 \nIn Fig.7 and Fig.8 we have the categorisation for the seeing and the GLF. In Fig.9-Fig.10 is shown the joint categorization for seeing and τ 0 that are thresholds of seeing and τ 0 for which seeing and τ 0 are minor or equal to 10%, 20% ect. of the cumulative distribution calculated on a climatological scale. We highlight the fact that requirements of ESO are very, very strict. Indeed, looking at Fig.7 we can see that the requested accuracy can arrive for the seeing up to 0.1' in the range [0.5' - 0.8'] and looking at Fig.9 the accuracy of τ 0 can be smaller than 1 ms. This is because only one instrument is taken as a reference (the MASS-DIMM in this case). \nOn the other side, there is no evidence, at present, that these accuracies can be reached by observations. Indeed, for example, Masciadri et al. 2023 13 showed that, on a sample of 157 nights, the intrinsic dispersion SD obs (i.e. RMSE at which we rested the bias) between observations from a Stereo-SCIDAR and a MASS-DIMM is of the order of 0.24' for the seeing and 1.22 ms for the τ 0 . Both values are larger than the accuracy requested by ESO. Other authors showed even larger uncertainties. For example Griffiths et al. 2024 20 measured an uncertainty of 0.28' for the seeing and 2.21 ms for τ 0 between two other instruments (RINGSS and 24HSHIMM) located basically in the same place and at the same height above the ground. Considering that FATE forecasts are supposed to be used for the Service Mode of the four UTs (and annexed instruments) and the VLTI extended on a surface of one or two hundreds meters as shown in Fig.11 this element will have to be taken into account at a certain point.", '4. PARAMETERS/DELIVERABLES': 'Following the Statement of Work (SoW) of the CfT, the parameters for which ESO required forecasts and the characteristics of the deliverables are the following:', 'NIGHT TIME': '- - Forecasts at long time scale are at 1d, 2d and 3d.\n- - We are considering forecasts of eight parameters: the wind speed (WS) at 30 m above the ground level (a.g.l.), the wind direction (WD) at 30 m a.g.l., the relative humidity (RH) at 30 m a.g.l., the precipitable water vapour (PWV), the sky transparency plus three astro-climatic parameters i.e the seeing ( ε ), the wavefront coherence time ( τ 0 ) and the ground layer fraction (GLF). \n<!-- image --> \nFigure 5. Same as Fig.3 but for the PWV. \nFigure 6. Same as Fig.3 but for the sky transparency. \nFigure 7. Same as Fig.3 but for the seeing. \nFigure 8. Same as Fig.3 but for the GLF. \nFigure 9. Same as Fig.3 but for the seeing and the τ 0 . \nFigure 10. Table reflecting the joint categorisation for seeing and τ 0 shown in Fig.9. \n<!-- image --> \nFigure 11. Cerro Paranal plateau with the four UTs. \n<!-- image --> \nFigure 12. Automatic forecast system of FATE. \n<!-- image --> \n10 \n15 \nOBS-WS (ms) \nFigure 13. Density function maps associated to the scattering plots obtained with forecasts and observations of WS, WD, RH and PWV during the night (left) and day time (right). Sample of 4 months related to the commissioning data-set. \n<!-- image --> \n100 \n200 \n300 \nPWV \nRH \nOBS (dea) \n10 \n12 \n14 \n16 \n10 \n20 \n30 \n40 \n50 \nOBS (mm) \nOBS-RH x 100 (%) \nOBS (mm) \n- - Forecast at short time scale. ESO requires upgrades of the forecast at a frequency of 1h.\n- - The forecast temporal sampling is 10 minutes. We highlight that General Circulation Models (GCMs) can not provide such a kind of products.', 'DAY TIME': 'The specifications for the day time are the same of the night time but we are not supposed to provide forecasts of the three astroclimatic parameters. \nAmong the deliverables we mention also the probability that individual parameters fall in the categories defined by ESO in the SoW. In case the categories are joint categorisation (see for example Fig.9-Fig.10), we provide a joint probability.', '5. AUTOMATIC FORECAST SYSTEM': 'The automatic forecast system connects institutes located in different places in the world. Initialisation and forcing data are delivered by ECMWF to ESO. These data are retrieved from ESO and computations of the forecasts with the mesoscale model are performed by FATE in Florence, more precisely in the LaMMA buildings where are located the HPC facilities and the NAS servers for the data storage. Finally, forecasts are delivered to ESO-Santiago (main delivery server) and ESO-Garching (back-up). In Fig.12 is shown the complete flowchart/structure of the automatic forecast system. Such a system is divided in three sectors: (A) initialisation and forcing data, (B) input and output data and (C) automatic forecast system. The important thing to retain is that, for each step, we envisaged back-up servers and also back-up networks. At least in the part of the flowchart that depends on FATE (see Section 8 considerations for more infos).This to maximise the temporal coverage and minimise the possibility of failures. ESO indeed established a system of penalties in case forecasts \n16 \n50 \n14 \n12 \n10 \n40 \n2 \n8 30 \n20 \n10 \n10 \n15 \n20 \n100 \n200 \n300 \nOBS (dea) \nPW \n10 \n12 \n14 \n16 \n16 \n14 \n12 \n10 \n1 \nFigure 14. Density function maps of WS, WD, RH and PWV related to the night time. The red square report results of the forecasts obtained during the commissioning. In blue squares results as treated following Masciadri et al. 2023 13 study. Sample of 4 months related to the commissioning data-set. \n<!-- image --> \n<!-- image -->', 'commissioning': '<!-- image --> \nFigure 15. As in Fig.14 but for seeing, τ 0 and GLF. In this picture we are considering all the values of the seeing without any filtering. \n<!-- image --> \nscattering plots for the 4 months of data of the commissioning. We can notice that there is a very good correlation between forecasts and observations in both cases (night and day). However results and under our expectation as shown in Fig.14 in the night time case. In the latter figure we report the comparison of results obtained with the commissioning data-set and results as expected from the analysis/study performed in Masciadri et al. 2023. 13 Similarly, in Fig.15 are shown the density function maps of the seeing, τ 0 and GLF related to the commissioning data-set (see density function maps inside red squares). It is possible to observe that the correlation between forecasts and measurements is very good ‡ and the uncertainty between forecasts and observations is already smaller than that obtained with observations (see digression in Section 3). However performances are weaker than those expected therefore those obtained with the treatment presented in Masciadri et al. 2023 13 study applied to the same sample of nights. The positive thing is that the origin of the problem has been identified and we are working to overcome that.', 'Masciadri et al. 2023': 'are not delivered in duly time. The automatic system includes a continuous monitoring of the whole flowchart of information to identify the origin of possible failure. Finally, as shown in Fig.12, INAF can access both the LaMMA structure and ESO delivering servers.', '6. TIMELINE': "Here the complete timeline: \n- - On 2019/07/19: ESO publishes a Request for Information (RfI)\n- - On 2020/01/07: ESO publishes the Call for Tender (CfT) that implies 2 selection phases\n- - On 2020/02/26: deadline of the CfT. We have afterwards 7 months of stop due to COVID. The VLT was closed in that time.\n- - 2020/09/24: project FATE is pre-selected for phase 2 that implies the participation to a 'validation test'\n- - [2021/01/11 - 2021/03/11]: duration of the validation test aiming to prove the ability of the teams to carry on such a project in operational phase\n- - 2021/11/08: ESO designates INAF winner of the CfT\n- - 2022/10/26: FATE contract is signed\n- - 2022/11/01: FATE development phase starts\n- - 2023/07/31: FATE developments phase ends\n- - [2023/09/01 - 2023/12/31]: duration of the commissioning\n- - 2024/04/31: signature of an amendment requested by ESO\n- - 2024/06/01: operational phase starts", '7. PRELIMINARY RESULTS': 'We report here just a few results related to the forecasts at short time scale (AR) as these are the most interesting and critical one for the Service Mode. In Fig.13 are shown the density function maps related to the associated \n<!-- image -->', '8. LESSONS LEARNED': "On 2024 May 17th on the web page of ESO appeared a message that recited '...Due to an important system up-grade several of ESO communication systems are unavailable...' . This problem has been experienced because ESO had planned an upgrade of the network security system. As a result however ESO was basically not reachable for one week. The consequence on FATE was that it remained completely stuck for the whole duration of time. More precisely: \n- - initialisation and forcing data from ECMWF could not be delivered to ESO servers\n- - ESO archive observations useful for AR techniques related to the short FTS were unreachable\n- - FATE could not deliver forecasts to ESO servers \nOf course it is absolutely normal and understandable that ESO takes care to protect its network and that means that, even if they are rare, these events may occur during the time. Solutions to this problem can be envisaged and have been discussed even if it is not suitable to deal about that in this context for security reasons. \nIn terms of lessons learned suitable to be highlighted we also mention a modified way to store measurements of the PWV in the ESO archive during the night time that we suggested to implement as it would improve the performances of the forecasts at short time scale of that parameter. This has been already discussed with the ESO counterpart and approved. At present, indeed, the frequency of measurements of the PWV done at at zenith and stored in the ESO archive during the night time is lower that that during the day time. This is due to the necessity to perform measurements at different lines of sight as it is important the calibration of astronomical observations with respect to the telluric lines that requires very precise measurements of the PWV at different lines of sight. However it has been observed 22 that the PWV horizontal distribution is quite homogeneous. Spatial variability has been documented to be of the order of rms = 3% (implying that for a 3 mm PWV, the rms is below 0.1 mm). We therefore suggested to store in the ESO archive measurements done along whatever lines of sight but corrected by the airmass [using the elevation angle 1./cos(90-ElevationAngle)]. This should guarantee a high frequency of observations of 1-2 min that is necessary for us to provide performances as shown in. 11,12", '9. CONCLUSIONS': 'We can say that the FATE development phase has been concluded within the scheduled program. The commissioning lasted four months [2023/09/01 - 2023/12/31] showing no major problems and it led: \n- - to prove the strengthness of the automatic forecast system that revealed to be very robust (100% reliability i.e. no failures);\n- - to produce an automatic report to be delivered to ESO with monthly frequency containing the state of art of the success rate of the forecast system and the percentage of ESO real-time observations availability;\n- - performances of forecasts at long time scales revealed to be those expected;\n- - performances of forecasts at short time scales showed an uncertainty with respect to observations that is better than the experienced dispersion of observations from different instruments. However performances are lower than what expected. More precisely, the RMSE is larger than that obtained following the treatment presented in Masciadri et al. 2023. 13 The problem has been identified and we are working to overcome that;\n- - we identified a few lessons learned at conclusion of the commissioning phase;\n- - Operational Phase started on 2024 June 1st. \nFor what concerns the next steps, we can synthetically say that: \n- - the next milestone/verification is planned for June 2025\n- - we are engaged on R&D to improve forecasts performances and to investigate machine learning approaches\n- - we planned to up-grade soon the forecasts at short time scales (AR approach) by changing the frequency of the up-grade forecasts to 10 min instead of 1 hour.', 'ACKNOWLEDGMENTS': 'The study is funded by the contract FATE N. PO102958/ESO/20/95952/FLAB. We acknowledge the Tuscany Region for the contribution to this activity. Initialisation data of the Astro-Meso-Nh model come from the HRES model of ECMWF. We acknowledge a few members of ESO staff for useful discussions: Alain Smette, Miska Le Louarn, Michele Cirasuolo, Olivier Hainaut, Joseph Anderson.', 'REFERENCES': '- [1] Silva, D.R., SPIE Proceedings 4844 , 94\n- [2] Anderson, P.A., Sedaghati, E., Cikota, A., Behara, N., Otarola, A, Steffen, M., 2024, SPIE - Observatory Operations: strategies, processes, systems X , 13098-6\n- [3] Masciadri, E., Vernin, J., Bougeault, P., 1999, A&ASS , 137 , 185\n- [4] Masciadri, E. & Jabouille, P., 2001, A&A , 376 , 727\n- [5] Masciadri, E. & Egner, S., 2006, PASP , 118 , 1604\n- [6] Lascaux, F., Masciadri, E., Hagelin, S., 2011, MNRAS , 411 , 693\n- [7] Hagelin, S., Masciadri, E., Lascaux, F., 2011, MNRAS , 412 , 2695\n- [8] Masciadri, E., Lascaux, F., Fini, L., 2013, MNRAS , 436 , 1968\n- [9] Masciadri, E., Lascaux, F., Fini, L., 2017, MNRAS , 466 , 520\n- [10] Masciadri, E., Martelloni, G., Turchi, A., 2020, MNRAS , 492 , 140\n- [11] Turchi, A., Masciadri, E., Kerber, F., Martelloni, G., 2019, MNRAS , 482 , 206\n- [12] Turchi, A., Masciadri, Pathak, P, Kasper, M., 2020, MNRAS , 497 , 4910\n- [13] Masciadri, E.,Turchi, A., Fini, L., 2023, MNRAS , 523 , 3487\n- [14] Turchi, A., Masciadri, E., Fini, L., 2024, SPIE - Software and Cyberinfrastructure for Astronomy VIII , 13101\n- [15] Lascaux, F., Masciadri, E., Fini, L. , 2013, MNRAS , 436 , 3147\n- [16] Lascaux, F., Masciadri, E., Fini, L. , 2015, MNRAS , 449 , 1664\n- [17] Lafore J.-P., Stein, J., Asencio, N. et al., 1998, Annales Geophysicae, 16, 90\n- [18] Lac, C. et al., 2018, Annales Geosci. 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2024arXiv240910711R | Recent advances in machine learning algorithms have unlocked new insights in observational astronomy by allowing astronomers to probe new frontiers. In this article we present a methodology to disentangle the intrinsic Xray spectrum of galaxy clusters from the instrumental response function. Employing stateoftheart modeling software and data mining techniques of the Chandra data archive we construct a set of 100000 mock Chandra spectra. We train a recurrent inference machine RIM to take in the instrumental response and mock observation and output the intrinsic Xray spectrum. The RIM can recover the mock intrinsic spectrum below the 1sigma error threshold moreover the RIM reconstruction of the mock observations are indistinguishable from the observations themselves. To further test the algorithm we deconvolve extracted spectra from the central regions of the galaxy group NGC 1550 known to have a rich Xray spectrum and the massive galaxy clusters Abell 1795. Despite the RIM reconstructions consistently remaining below the 1sigma noise level the recovered intrinsic spectra did not align with modeled expectations. This discrepancy is likely attributable to the RIMs method of implicitly encoding prior information within the neural network. This approach holds promise for unlocking new possibilities in accurate spectral reconstructions and advancing our understanding of complex Xray cosmic phenomena. | 2024-09-01T00:00:00Z | ['2024arXiv240910711R', 'arXiv:2409.10711', '10.48550/arXiv.2409.10711'] | ['Astrophysics - Astrophysics of Galaxies'] | Deconvolving Xray Galaxy Cluster Spectra Using a Recurrent Inference Machine | 2,024 | 176 | 0.43 | ['EPRINT_HTML', 'EPRINT_PDF'] | 0 | https://arxiv.org/pdf/2409.10711.pdf | {'Deconvolving X-ray Galaxy Cluster Spectra Using a Recurrent Inference Machine': "Carter Lee Rhea , 1, 2, 3 Julie Hlavacek-Larrondo , 1 Alexandre Adam , 1, 4 Ralph Kraft , 5 ' Akos Bogd'an , 5 Laurence Perreault-Levasseur , 6, 4, 7 and Marine Prunier 1, 2, 8 \n1 D'epartement de Physique, Universit'e de Montr'eal, Succ. Centre-Ville, Montr'eal, Qu'ebec, H3C 3J7, Canada \n2 Centre de recherche en astrophysique du Qu'ebec (CRAQ) \n3 Iuvo-ai, Montr'eal, Qu'ebec, Canada \n4 Mila - Quebec Artificial Intelligence Institute, Montreal, Qu'ebec, Canada \n5 Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA 6 D'epartement de Physique, Universit'e de Montr'eal, Succ. Centre-Ville, Montr'eal, Qu'ebec, H3C 3J7, Canada 7 Center for Computational Astrophysics, Flatiron Institute, New York, USA \n8 Max-Planck-Institut fur Astronomie, Konigstuhl 17, D-69117 Heidelberg, Germany", 'ABSTRACT': "Recent advances in machine learning algorithms have unlocked new insights in observational astronomy by allowing astronomers to probe new frontiers. In this article, we present a methodology to disentangle the intrinsic X-ray spectrum of galaxy clusters from the instrumental response function. Employing state-of-the-art modeling software and data mining techniques of the Chandra data archive, we construct a set of 100,000 mock Chandra spectra. We train a recurrent inference machine (RIM) to take in the instrumental response and mock observation and output the intrinsic X-ray spectrum. The RIM can recover the mock intrinsic spectrum below the 1σ error threshold; moreover, the RIM reconstruction of the mock observations are indistinguishable from the observations themselves. To further test the algorithm, we deconvolve extracted spectra from the central regions of the galaxy group NGC 1550, known to have a rich X-ray spectrum, and the massive galaxy clusters Abell 1795. Despite the RIM reconstructions consistently remaining below the 1σ noise level, the recovered intrinsic spectra did not align with modeled expectations. This discrepancy is likely attributable to the RIM's method of implicitly encoding prior information within the neural network. This approach holds promise for unlocking new possibilities in accurate spectral reconstructions and advancing our understanding of complex X-ray cosmic phenomena.", '1. INTRODUCTION': "Galaxy clusters harbor large reservoirs of hot gas ( ∼ 10 7 -10 8 K), called the IntraCluster Medium (ICM), which account for the majority of baryonic matter in the cluster (e.g., Fabian & Allen 2003). This gas consists primarily of ionized hydrogen and helium but also contains numerous heavier elements (e.g., Mushotzky 1984; Mohr et al. 1999; Loewenstein 2003). Emission mechanisms such as thermal bremsstrahlung, boundfree atomic transitions, and the collisional excitation of hydrogen are responsible for the X-ray continuum (e.g., Markevitch & Vikhlinin 1997; Ettori & Fabian 1998; Sarazin et al. 1999; Markevitch et al. 1998). The Xray spectra of galaxy clusters also exhibit strong emis- \nCorresponding author: Carter Rhea \[email protected] \nsion lines from heavier atoms (e.g., Fabian 2012; Sarazin et al. 1999). In the soft X-ray regime (0 . 5 -2 keV), several of these emission lines are present such as several Iron species (Fe XIX, Fe XVII, and Fe XVIII), Oxygen species (O VII), Nitrogen species (N VII), and Silicon species (Si XIII and Si XIV). The rest-wavelengths of these lines are contained in the atomic database for collisional plasma tailored to high-energy X-ray astrophysics, AtomDB (Foster et al. 2012; this is primarily Chandra calibration data), and the calibration database, CalDB (Graessle et al. 2006). The underlying temperature of the ICM and the relative abundances of these metals have a substantial effect on the resulting X-ray spectrum (e.g., Fabian & Allen 2003; Markevitch & Vikhlinin 1997; Mushotzky 1984); this enables the study of galaxy clusters through fitting plasma physics models to observed X-ray spectra. \nThe ICM was first observed over five decades ago; the first targets were the brightest and most nearby galaxy clusters: Perseus, M87, and Coma (Bradt et al. 1967; Gursky et al. 1971; Gursky 1973). Since these first targets, X-ray observatories such as Uhuru (Sarazin 1986), EXOSAT (Giacconi et al. 1979), the EINSTEIN Observatory (Forman et al. 1978), XMM-Newton (Jansen et al. 2001; Wilman & Fabian 1999), and the Chandra X-ray Observatory ( CXO ) have revolutionized our understanding of the ICM (Forman et al. 2002a; Forman et al. 2002b; Vikhlinin et al. 2002; Mazzotta et al. 2001). \nIn particular, the CXO , with an unprecedented spectral and spatial resolution has changed the field of X-ray astronomy (Weisskopf et al. 2000; Weisskopf 2005; Weisskopf 2010). This observatory uses a charge-coupled device to capture high-energy photons between approximately 0 . 3 -10 keV. Not only does it collect the spatial information of the incident photons, but it also registers their energy on the device. Consequently, we can construct a spectrum for each pixel on the CCD. \nDue to its exquisite spatial resolution (0 . 492 arcsec per pixel), the CXO produces highly detailed, spatially resolved images of its targets. Precise spectroscopic measurements have allowed astronomers to map out temperature profiles and, in some cases, spatially resolved temperature maps (e.g., Bourdin et al. 2004; Pratt et al. 2007; Adam et al. 2017; Alden et al. 2019). These maps, in conjunction with metallicity maps, allow for studying the energetic history of the ICM by tracing gradients in the thermodynamic parameters. Moreover, these observations, used in synergy with other observations at different wavelengths, have shed light on the role of the central supermassive black hole in regulating the ICM (Fabian et al. 2009; Rafferty et al. 2006; Ruszkowski et al. 2019). \nTo extract these parameters, we fit theoretical models of the underlying astrophysics to the observed spectrum, considering any instrumental effects. One longstanding challenge in X-ray astronomy is that the observed X-ray spectrum, S ( E ' ), is the result of a convolution between the source's intrinsic spectrum, F ( E ), and the instrumental response, R ( E ' , E ) as shown in the equation: \nS ( E ' ) = ∫ ∞ 0 R ( E ' , E ) F ( E ) dE, (1) \nwhere E ' denotes the discrete photon energies captured by the detector and E is the measured energy. Despite the simplicity of equation 1, a direct deconvolution of the instrumental response function and model spectrum poses several issues (see Rhea et al. 2021a for an indepth discussion). Primarily, the instrumental response varies greatly as a function of position, and the detec- \nr energy space is limited and finite by design. Since the sampling of the detector space E ' is discrete, we can rewrite equation 1 as a matrix equation (Kaastra & Bleeker 2016): \nS i = ∑ j R ij F j . (2) \nWe have replaced the instrumental response function with its matrix counterpart, R ij . The first index, i , indicates the detector channel in E ' -space, while the second index, j , denotes the emitted photon energy in E -space. In this formulation, S i represents the observed photon count rate in units of counts s -1 for a given detector energy bin. F j is the model (also denoted as true) spectrum flux in units of counts m -2 s -1 in emitted energy bin j . We note that this discretization of equation 1 explicitly linearizes the equation. The ramifications of this choice are discussed in section 4.3. \nAs demonstrated in Rhea et al. (2021a), standard numerical methods, including different regularization methods, fail to deconvolve the true spectrum from the response matrix given the observed spectrum. Consequently, we began exploring a promising method known as the Recurrent Inference Machine (RIM) for deconvolution. This approach has shown remarkable success in deconvolving 2D radio images (Morningstar et al. 2018; Morningstar et al. 2019), separating gravitational lens sources from their foreground counterparts (Adam et al. 2022b), and reconstructing MRI images (Lønning et al. 2019). \nIn Rhea et al. (2023), we initially demonstrated the potential of this method for deconvolving X-ray spectra. Although astronomers have been able to fit the observed spectrum through a technique known as forward modeling, by which the theoretical model is convolved with the instrumental response and subsequently fitted, there are several benefits to having access to the intrinsic spectrum. Once deconvolved, these spectra can be fit directly, circumventing the need for forward modeling. Moreover, they can be used as inputs for machine learning algorithms such as a convolutional neural network to extract point estimates of the underlying thermodynamic parameters. This methodology can also study the line-by-line calibration of the CXO by extracting the intrinsic spectrum of a non-changing calibration target where the intrinsic spectrum is not expected to evolve. \nTherefore, we develop a new algorithm in this paper that can successfully disentangle the intrinsic Xray spectrum from the instrumental response function. Throughout this paper, we limit the case to the X-ray spectra of galaxy clusters but stress that the method we develop could be applied to all fields of X-ray astronomy. In § 2, we present the RIM and convolutional \nneural network used throughout this paper and the synthetic and natural X-ray observations used. We then discuss the results of using our RIM to deconvolve X-ray spectra in § 3. We compare fits using the deconvolved spectra with traditional methods, explore the reasoning behind the deconvolution, and explore failure modes in § 4. Finally, we conclude our paper in § 5. We assume a Hubble constant of H 0 =69.6, an energy density of matter, Ω M =0.286, an energy density of the vacuum, Ω vac =0.714, and a flat curvature.", '2. METHODS AND OBSERVATIONS': 'This paper aims to show that RIMs can successfully deconvolve the X-ray spectrum from the Chandra instrumental response function. Below, we outline in detail the machine and how it was trained.', '2.1. Recurrent Inference Machine': 'The RIM is an extension of a recurrent neural network (RNN) in which the RNN is used to provide updates to the solution mimicking a standard Newton-Raphson update; they have been used to solve 2D inverse problems (e.g., Putzky & Welling 2017; Morningstar et al. 2018; Morningstar et al. 2019). We adopt this architecture to solve our one-dimensional inverse problem described in equation 2. \nFigure 1. Schematic view of the RIM. Nodes in dark gray are treated as inputs to the RIM. Note that we separate the hidden layers before applying the RIM and combine them after applying the RIM. The teal box represents the RIM itself. \n<!-- image --> \nIn general, the RIM solves the generalized linear problem \ny = A x + n (3) \nby solving for x , the solution vector, recursively where y is the measurement vector, A is some operation which \nmanipulates the solution vector, and n is the noise vector. \nAt each step, the RIM uses the previous solution, x ( t ) , and the gradient of the log-likelihood function, ∇ log p ( y | x ( t ) ), to solve for an update to the solution ∆ x ( t ) . The solution is updated at each step as \nx ( t +1) = x ( t ) +∆ x ( t ) . (4) \nThe update of the solution can be viewed through the lense of a maximum a prosteriori (MAP) framework where \nˆ x MAP = argmax x [ log p ( y | x ) + log p ( x ) ] (5) \nFor this paper, we drop the prior term, log p ( x ), following Morningstar et al. (2018). \nMore specifically, we can express the iterative solution, x at time step t +1 as \nx ( t ) = x ( t ) + g θ ( x ( t ) , ∇ x log p ( y | x ) , h ( t ) ) h ( t +1) = g θ ( x ( t ) , ∇ x log p ( y | x ) , h ( t ) ) (6) \nwhere g θ is the neural network and θ represents the weights of the network, p ( y | x ) is the likelihood, and h is the hidden vector used in the neural network. \nFigure 1 shows the standard architecture of a RIM (e.g., Morningstar et al. 2018; Morningstar et al. 2019); the teal box represents the RIM itself. The methodology is as follows: \n- 1. Initialize the hidden vectors, h 1 and h 2 , the initial solution, x ( t =0) , and all weights of the gated recurrent units (GRUs) and convolutional layers of the RIM.\n- 2. Calculate the gradient of the likelihood function ∇ log p ( y | x ( t ) ).\n- 3. Pass the solution at the current time step, x ( t ) , the current value of the gradient of the likelihood function, ∇ log p ( y | x ( t ) ), and the state vectors, h 1 and h 2 to the RIM.\n- 4. Forward pass through the neural network.\n- 5. Update the current value of the solution, x ( t +1) with the output of the RIM, ∆ x , and the solution at the previous step, x ( t ) , following equation 4.\n- 6. Repeat steps 2-5 a predefined number of times. \nWe note that the constituent layers of the RIM, the convolutional layers and GRUs function as normal and \nare updated through standard back-propagation following the Adam optimization scale (Kingma & Ba 2017). Following Morningstar et al. (2019), we initialize x ( t =0) = 0 and each state vector, h ( t )=0 1 and h ( t =0) 2 , as the zero vector. We defined the likelihood as \nlog p ( y | x ( t ) ) = -1 2 ( y -A x ( t ) ) T C -1 ( y -A x ( t ) ) (7) \nwhere C ∈ A m × m is the covariance of the additive Gaussian noise of a particular observation. A Gaussian likelihood is ubiquitous in machine learning applications since the target distribution is often expected to be Gaussian. \nTherefore, the gradient is written as \n∇ x ( t ) log p ( y | x ( t ) ) = ( y -A x ( t ) ) T C -1 A (8) \nBy explicitly defining the physical model, we stress that our network is less of a black box since the physics of the problem is encoded in the likelihood. This allows us to incorporate the specific physics of the problem (i.e., the response matrix) in the updates. Furthermore, we apply an arcsinh transformation function on the gradient of the likelihood function before passing it to the RIM to avoid exploding gradients. We use a mean squared loss backpropagated through time function: \nL = 1 T T ∑ t =1 M ∑ i =1 (ˆ x ( t ) i -x i ) 2 (9) \nwhere ˆ x ( t ) i is the current best reconstruction at time t and M is the total number of spectral channels in the spectrum. Using this loss function, the RIM solution is considered at each time step. \nIn the case of the inverse problem investigated in this paper, we compute the loss function by comparing the result of our RIM with the actual spectrum. Thus, the RIM only sees the observed spectrum and the response matrix. The layers of the RIM are the following: \n- 1. Convolutional Layer 1: 64 filters with a tanh activation function, stride equal 1, and padding equal 1\n- 2. GRU 1: 800 nodes with sigmoid activation function\n- 3. Convolutional Layer 2: 128 filters with a tanh activation function, stride equal 1, and padding equal 1\n- 4. GRU 2: 800 nodes with sigmoid activation function\n- 5. Convolutional Layer 3: 128 filters with a linear activation function, stride equal 1, and padding equal 1 \nThe convolutional filters all have a kernel size of 3. We ran hyper-parameter tuning on the number of nodes in each GRU, filters and kernel size for each convolutional layer, and the batch size. We determined that eight convolutional filters and three kernel sizes were optimal.', '2.2. Synthetic X-ray Data': 'To train the RIM to deconvolve the intrinsic spectrum from the response matrix, we need groups of response matrices, true intrinsic spectra (referred to as ground truth ) and the corresponding convolved observed spectra. Since we cannot access ground truth spectra for real observations, we rely on synthetic spectra to train our algorithm. We create the synthetic spectra using the analysis software SOXS 1 , which was created to generate high spectral resolution mock spectra for the Lynx observatory (e.g., Gaskin & Swartz 2019). To create the synthetic spectra, we first generate 100,000 mock spectra of the ICM using the APEC model - a standard thermodynamic model that uses a combination of collisional ionization equilibrium (CIE) physics and nonequilibrium ionization (NEI) physics to model the emission of the ICM (Smith et al. 2001). CIE and NEI spectra are generated using AtomDB. We also trained the model on 200,000 mock spectra and found no difference in the reconstructions. To generate the spectra, users are required to input the temperature of the gas, the metal abundance in solar units, the redshift of the gas, z , and the normalization of the model defined as \nN = 10 -14 ∫ n e n p 4 π (1 + z ) 2 D 2 A dV (10) \nwhere n e and n p are the densities of the electrons and protons, respectively, D A is the angular diameter distance to the gas, and V is the volume of the gas. We randomly select the gas temperature from a uniform distribution between 0.4 and 8 keV. This range is chosen because it coincides with the majority of gas temperatures in nearby clusters as revealed by Chandra (e.g., Cavagnolo et al. 2009). The metallicity of the gas is randomly sampled from a uniform distribution between 0.2 Z ⊙ and 1.2 Z ⊙ . Similarly, these values are standard in nearby galaxy clusters. We restrict the redshift to only nearby galaxy clusters below z = 0 . 02 for this proof-ofconcept. We set the normalization by randomly selecting a value from a uniform distribution between 0.1 and \n10; we discuss the impact of this choice on the application to real spectra in section 3.2. We apply additional foreground absorption using the tbabs model (Wilms et al. 2000). We allow the column density, N H to vary between 0.001 × 10 20 cm -2 and 1 × 10 20 cm -2 . The SNR was uniformly sampled between 5 and 100. These SNR values run the gambit of typical observations (e.g., Cavagnolo et al. 2009) We provide the code to create the synthetic spectra in appendix A.1. \nA crucial free parameter governing synthetic data creation is the instrumental response. We limit our analysis to the ACIS instruments. Since the response matrix varies as a function of time and position on the chip, it is essential to sample the space of possible response matrices adequately. To do this, we mined the Chandra archive of observations of well-studied galaxy clusters designated as such in Mushotzky (1984). We created 1000 response matrices by randomly sampling 10 arcsec regions from the chip where the cluster center was located. \nWe create our training data on the fly using the forward model \nS = R ⊗ F + N (11) \nwhere N is a randomly sampled Gaussian distributon representing the noise. The sigma of the noisy distribution is randomly sampled, assuming a minimum and maximum signal to noise; we take the minimal value to be 10 and the maximum SNR to be 150. Below a SNR of 10, the observed X-ray data is dominated by the noise. By building our training data, we can explore a vast parameter space describing the intrinsic emission model, response matrices, and noise configurations. We show an example of mock ground truth spectrum, response matrix, and convolved observed spectrum in Figure 2.', '2.3. Chandra X-ray Observations': 'In this work, we apply our trained RIM to actual observations to validate its use. We select NGC 1550 as our first test since it is a galaxy group that has a rich X-ray spectrum with a well-constrained temperature ( kT = 1 . 37 ± 0 . 01 keV) and metallicity ( Z ≈ 0 . 27 Z ⊙ ; Sun et al. 2009; Kolokythas et al. 2020) at R 500 . NGC 1550 is at a redshift z = 0.0124 (Sun et al. 2009). The CXO has observed this target twice with the ACIS-I instrument. \nAs our second test, we select the galaxy cluster Abell 1795. Contrary to NGC 1550, Abell 1795 has a considerably hotter ICM (kT ≈ 6 keV; Ehlert et al. 2015); due to this higher temperature, the intrinsic X-ray spectrum is expected to have fewer emission lines than NGC 1550; therefore, this cluster is a good juxtaposition. Abell 1795 is located at a redshift of z = 0.062476 (Oegerle & \nHill 2001). In Table 1, we detail these observations. The data were downloaded directly from the data archive, chaser . All data were treated using CIAO (v.14.5.1) . \nThese objects were chosen since they represent two distinct points of the temperature scale in galaxy clusters and, thus, are represented by different spectral profiles. For NGC 1550, we use the two available observations to demonstrate how the deconvolution results change as a function of the observation. For Abell 1795, we select one random observation to highlight how the network responds to a different intrinsic spectrum. For each observation, the convolved observed spectrum is passed to the trained RIM, which, in turn, returns the deconvolved, intrinsic spectrum of the observation, referred to as the RIM Solution. \nAfter downloading the data, we reduce and clean the level 1 event files using an in-house reduction pipeline located at https://github.com/XtraAstronomy/AstronomyTools. \nThe pipeline first uses a CCD containing only background emission to construct a background light curves. We use the CIAO tool lc sigma clip to identify periods during the observation where the background differed from the 3σ level. These times were removed from the observations. We then destreak the event file and remove any bad pixels. The final processing is completed using acis process events with the vfaint option set to true since we are interested in the diffuse extended emission in the galaxy cluster. This entire process results in a level 2 event file for each ObsID. We extract the spectra within a radius of R 500 2 for the galaxy cluster using specextract , which creates the spectrum file and the two response files. These files are combined into a single response matrix using rmfimg with the argument product=true . We also extract a background spectrum from a region on the same CCD chip with no ICM emission.We subtract this background spectrum, S bkg , from the source spectrum, S src , using the following equation: \nS src -S bkg ∗ exp src ∗ bks src exp bkg ∗ bks bkg \nwhere exp src and exp bkg are the exposure times of the source and background spectra, respectively, while bks src and bks bkg are the source and background backscales.', '3. RESULTS': "<!-- image --> \n<!-- image --> \nFigure 2. Schematic of the convolution applied by the ACIS instrument on Chandra . This graphic demonstrates the response matrix's profound effect on the observed spectrum. The true intrinsic spectrum, the ground truth spectrum, is a typical emission spectrum from the ICM modeled using the apec model with a temperature of 2.0 keV and a metallicity of 0.3 Z ⊙ . The response matrix was taken from a randomly chosen ObsID (2707); the matrix displayed is the log of the response matrix. The convolved observed spectrum was created following the methodology outlined in § 2.2. \n<!-- image --> \nTable 1. Chandra ObsIDs used to test the RIM. We select two objects representing two extreme temperature scales in galaxy clusters, with NGC 1550 being a low temperature group and Abell 1795 being a massive hot galaxy cluster.", '3.1. Synthetic X-ray Data': "We run the RIM described in § 2.1 on 50,000 synthetic data created following the methodology outlined in § 2.2. We assign 90% The RIM was trained for 10 timesteps. We remind the reader that the network's output is the deconvolved spectrum (also denoted as the RIM Solution) for each time step. We applied a batch size of 64 and trained the network for 500 epochs. We tested different batch sizes and timesteps but found no effect on the final results. \nIn Figure 3, we show the true intrinsic spectrum, i.e. the ground truth, vs. the RIM solution (top left), the convolved observed spectrum vs. the RIM reconstruction (top right), the error in the RIM solution as compared to the ground truth (bottom left), and the error in the RIM reconstruction as compared to the convolved observed spectrum (bottom right) for a randomly selected test spectrum. The RIM solution is the direct output of the RIM, while the RIM reconstruction is the result of passing the RIM solution through the forward model (equation 1). The left-hand plot demonstrates that the RIM solution agrees with the literature model under 1σ . The most significant errors occur in the region between 0.5 and 1.0 keV where the strong sulfur, silicon, and lithium emission lines are present (e.g., Markevitch & Vikhlinin 1997; Mushotzky 1984; Sarazin et al. 1999). While the errors remain below the 1σ level, we note that the RIM misses small-scale features while glob- \nly reconstructing the spectrum; this has been noted for previous use cases of the RIM (e.g., Adam et al. 2022a). The right-hand plot demonstrates that the RIM reconstruction is indistinguishable from the mock observation to a 1σ level except for the same region between 0.5 and 1.0 keV. This indicates that the RIM is learning the noiseless intrinsic spectrum to an accuracy below 1σ except in this region. \nSince this region is physically essential, we attempted several changes to the hyper-parameters of the RIM, including a dynamic learning rate scheduler and training for more extended periods. However, none of these changes improved the results of the RIM in this critical section. We explore the ramifications of this in section 4.2.", '3.2. Chandra X-ray Observations': "We use both the extracted spectra and their associate response matrices to deconvolve the R 500 spectra created in section 2.3 for NGC 1550 and Abell 1795. \nTo normalize the data for NGC 1550, the spectra are multiplied by the observation's exposure times, 9,650 seconds and 9,990 seconds, respectively, for ObsID 3186 and ObsID 3187. In doing so, we ensure that the data fed to the network resembles the magnitude of the synthetic data. The noise level of the observation is estimated as the standard deviation of the intensity over the entire bandpass. In Figure 5, we show the results of \n<!-- image --> \nFigure 3. The RIM applied to synthetic X-ray spectra. In this figure, we show the results of the RIM deconvolution process on a randomly selected spectrum from the test set. We compare the ground truth in the upper left panel versus the RIM solution. In the upper right panel, we compare the convolved observed spectrum with the RIM reconstruction created by passing the RIM solution through the forward model. The bottom left panel shows the error between the ground truth and the RIM solution in units of σ . In the right bottom panel, we show the error between the convolved observed spectrum and the RIM reconstruction in units of σ . \n<!-- image --> \nthe RIM deconvolution for ObsID 3186 (top) and ObsID 3187 (bottom). In the left panel, we show the direct result of the RIM deconvolution (red), i.e. the RIM solution. In the center panel, we display the observed spectrum (black) and the RIM reconstruction (blue). We show the normalized residuals between the observed spectrum and the RIM reconstruction in the right panels. These results demonstrate that the RIM reconstruction matches the observed spectra below the 1σ level except for in the busiest area of the spectrum around 1 keV. We explore the limitations of this reconstruction in § 4.2. \nIn Figure 6, we show the results of the spectral deconvolution of ObsID 5289 (Abell 1795). Here, we used an exposure time of 14,950 seconds to normalize the data. Since Abell 1795 has a high temperature, the complex of strong emission lines around 1.0 keV is not as prominent as with NGC 1550. Figure 6 demonstrates similarly that the RIM reconstruction matches the observed spectrum below the 1σ level in all spectrum regions.", '4.1. Reasoning Behind Deconvolution': "Although the community has been working with the convolved X-ray spectra for the last several decades, deconvolution of X-ray spectra promises many benefits. Using deconvolved spectra, it will be possible to directly stack X-ray observations of the same target to improve the signal-to-noise. In doing so, we will no longer need to rely on simultaneous fitting of multiple observations, which is costly. \nMoreover, the deconvolved spectra can be passed to machine learning algorithms capable of extracting the \nunderlying model parameters (i.e. the temperature and metallicity) without worrying about the response matrices. The benefit of this methodology has been shown extensively in the literature (e.g., Fabbro et al. 2018; Rhea et al. 2020; Rhea et al. 2021b). \nX-ray time variability has been the focus of many studies (e.x. Mushotzky et al. 1993; Giannios et al. 2004; van der Klis 2006); these studies tend to focus on either photometric measurements, or, in the case when they are studying the spectra, model fits to convolved spectra. Our proposed methodology to deconvolve X-ray spectra could be used to study the variability of X-ray sources in time. Typically, changes to the spectrum due to time variability are nearly indistinguishable due to the convolution. However, if we have access to the deconvolved spectrum, the evolution of the spectrum will be clear without the use of additional modeling. \nThe RIM offers new possibilities for verifying the consistency of the CXO calibration over time. Indeed, contamination layers on Chandra's optical blocking filters cause the instrumental response of its detectors to evolve, leading to changes in the spectra of stable astrophysical sources observed across different cycles. These sources, which should have consistent spectra over time, are used as calibration sources to update the response matrix models, which are then used for data processing and spectral analysis. If the RMF calibration is accurate, the RIM-deconvolved spectra of these non-changing calibration targets across different cycles should be identical. If not, the RIM can be used to adjust the RMF by deconvolving recent spectra and comparing them to a reference spectrum from the initial cycle, which is assumed to be accurately calibrated. The \nFigure 4. Montage of 16 randomly selected mock observed spectra (black) and the RIM reconstructions (blue) for a variety of temperatures and metalicities. Included in each subfigure is the reduced chi squared value. The temperature increases from top to bottom and the metalicity increases from left to right. \n<!-- image --> \nFigure 5. Results of the RIM on the galaxy group NGC 1550. In the left panels, we show the RIM solution (i.e. the result of the deconvolution process) for ObsID 3186 and 3187, top and bottom, respectively (red). The middle panel shows the observed spectrum after background subtraction (black) and the RIM reconstruction (blue). The RIM reconstruction results from passing the RIM solution through the forward model developed in equation 2. The right panel shows the residual between the observed spectrum and the RIM reconstruction normalized to the noise level of the observation. \n<!-- image --> \nFigure 6. Results of the RIM on the massive galaxy cluster Abell 1795. In the left panels, we show the RIM solution (i.e. the result of the deconvolution process) for ObsID 5289 (red). The middle panel shows the observed spectrum after background subtraction (black) and the RIM reconstruction (blue). The RIM reconstruction results from passing the RIM solution through the forward model developed in equation 2. The right panel shows the residual between the observed spectrum and the RIM reconstruction normalized to the noise level of the observation. \n<!-- image --> \ncoefficients of the response matrix are iteratively modified until the deconvolved spectra match the reference, minimizing the residual and resulting in a recalibrated matrix.", '4.2. Limitations of the Model': 'In Section 3.1, we demonstrated that the RIM reconstructions match the synthetic ground truth spectra well below the noise level. However, we do not have access to the ground truth for the real data. Therefore, we instead compare the RIM solution of the spectrum predicted from the best-fit model in the literature. \nHere is the corrected version while keeping all LaTeX commands and symbols intact: \nMore specifically, in the case of NGC 1550, we use the best-fit model estimated in Kolokythas et al. (2020) and refer to this as kolokythas2020 from now on. Here, the authors fitted the convolved observed spectrum and estimated that the ICM can be modeled by an absorbed APEC model, where the metallicity is set to 0.27 Z ⊙ and the temperature is set 1.38 keV. The only parameter we modify is the normalization parameter to match the deconvolved spectra, i.e., the RIM solution. In Figure 7, we show the comparison between the model predicted from kolokythas2020 (dashed line) and the RIM solution from ObsID 3186 and ObsID 3187 (solid line). We stress that since we are deconvolving a real observation, there is no ground truth to compare the results to, but rather a fit that implicitly carries all the assumptions that go into the standard fitting procedure. \nFirst, Figure 7 shows that, while the deconvolution of ObsID 3186 and ObsID 3187 return similar intrinsic spectra, there are discrepencies between the two deconvolutions. Galaxy clusters change on the timescales of tens of millions of years, therefore it is unexpected for the intrinsic spectrum to change. We consider the driving force between the differences in deconvolution to be the considerable singularity of the response matrices which differ from one observation to the other even though the RIM has converged and shows excellent results on realistic synthetic data. This result highlights the intrinsic difficulties in deconvolving the X-ray spectrum from the response matrix. \nMore importantly, the deconvolved spectra do not match the expected model spectrum using the Kolokythas2020 model. The Kolokythas2020 model differs from the RIM solution in three main ways: the intensity of the powerlaw is slightly diminished, the intensity of the iron features around 0.8 - 1.2 keV are subdued, and the peak of the emission line around 2.2 keV is lower. This result can be interpreted as either the RIM solution, although relatively stable over different obser- \nFigure 7. In this figure we compare the RIM solution for both ObIDs (ObsID 3186 and ObsID 3187 in red) and the model taken from literature values (green) for NGC 1550. The later was obtained by estimating the thermodynamic properties of the cluster from the convolved observed spectrum in Kolokythas2020 and then creating a mock spectrum using in SOXS . \n<!-- image --> \nvations, is incorrectly capturing the intrinsic emission due to the inherent complexities of the inversion process, or the Kolokythas2020 model does not accurately represent the intrinsic emission. It is impossible to distinguish between the two scenarios without knowing the true result, which would require a higher resolution observation. \nFigure 8 shows the equivalent RIM solution for ObsID 5289 and the expected model spectrum of Abell 1795. We calculate the expected model spectrum by assuming a temperature of kT ≈ 5 keV and a metallicity of Z ≈ 0 . 5 Z ⊙ ; these values were approximated from Ettori et al. (2002) and Walker et al. (2014); we therefore refer to this as the Walker2014 model from now on. The galactic absorption was set to 1.2 × 10 20 cm -2 following measurements by Kalberla et al. (2005). While the RIM solution more closely matches the literature model, there are still discrepencies between the two in the location and amplitude of the emission lines. \nDespite these challenges with actual X-ray observations, the RIM performs remarkably well on synthetic data, consistently achieving highly accurate reconstructions that closely match the ground truth. This demonstrates the robustness of the method in controlled scenarios. However, further investigation is required to address the complexities of applying the RIM to actual observed data, which may involve refining the model and accounting for observational uncertainties. We discuss \nFigure 8. Same as in Fig. 7, but for Abell 1795. In this figure we compare the RIM solution for ObsID 5289 (red) and the model taken from literature values (green) taken from Walker2014. \n<!-- image --> \nthis further in the next section. Additionally, our results suggest that the thermodynamic parameters estimated from the convolved observed spectra may not fully capture the intrinsic properties of the cluster, potentially requiring a more nuanced approach.', '4.3. Future Improvements': "Given the inherent challenges of solving highly illposed inverse problems, here we investigate more stateof-the-art approaches that could improve the accuracy of the RIM in future applications. \nAdam et al. (2022b) applied the RIM to strongly gravitationally lensed systems, developing a method to simultaneously reconstruct undistorted images of the background source and the lens mass density distribution as pixelated maps. While successful, the authors showed that fine-tuning the RIM's objective function was necessary to improve reconstructions when fine structure was present in the data. Although this approach did not entirely eliminate the issue, it significantly improved the results. This methodology could potentially be adapted to X-ray spectra; however, since our current reconstructions are already at the noise level of the observations, this technique may have limited impact in our specific case. \nThe root of the problem lies in the way the RIM implicitly encodes the prior inside the neural network trained using the maximum likelihood estimate. Therefore, the RIM will produce blurred reconstructions because it returns an average spectrum corresponding to an observation. \nRecent advances in diffusion models have shown marked improvement over other techniques such as the RIM (e.g., Adam et al. 2022a; Dia et al. 2023; Adam et al. 2023). Diffusion-based techniques achieve better results by accurately encoding the priors using scorebased models that implicitly do not require the prior to be learned. \nTherefore, a future paper will use diffusion models to recover the intrinsic spectrum. Furthermore, since diffusion models are anchored in Bayesian statistics, this will also give us uncertainties on the reconstructions, which is crucial for comparing the deconvolution results with contemporary models bound by literature results. \nAnother potential improvement arises from our assumption when transitioning from equation 1 to equation 2, where we linearized each term in the integral equation. The linearization of equation 1 implictly assumes that the response matrix behaves in a linear manner from one bin in energy space to another. This is potentially true for high resolution spectra such as those obtained with the X-Ray Imaging and Spectroscopy Mission (XRISM), but it is likely not the case for Chandra ACIS-I or ACIS-S observations. If this is the case, then additional terms need to be added to equation 2 that capture the nonlinear nature of the initial spectral equation for the two equations to be truly equivalent. \nTherefore, in the case of new and upcoming high spectral resolution missions such as XRISM, new Athena, or LEM, we expect the linearization condition to hold since it requires a small bin size in energy space (Kaastra & Bleeker 2016). The RIM provides an accurate and efficient way in obtaining the actual intrinsic spectra of X-ray sources for these observatories. \nFurthermore, high spectral resolution missions will, by design, produce high resolution spectra where the response matrix does not cause strong degenerecies in the physical parameter space employing mixing emission lines. In future works, we will explore additional terms in the linearization of equation 1 and apply the RIM to future observatories. \nAnother potential challenge in recovering the intrinsic spectrum of real observations is that the ICM has a variety of temperatures but is rather fully characterized by a distribution of temperatures. While this poses a problem for the RIM which outputs a single expected spectrum, the diffusion-based model can capture this distribution in the posterior it will provide.", '5. CONCLUSIONS': "In this paper, we explored using a machine learning algorithm, the RIM, to deconvolve the intrinsic X-ray spectrum of an emitting source from the instrumental \nresponse function observed using data from the Chandra X-ray Observatory. \nWe demonstrate that the RIM effectively reconstructs the global features in modeled ICM spectra below a 1sigma noise level. Moreover, the RIMreconstructed spectra match the realistic synthetic observations closely. \nFurthermore, we apply our trained RIM to actual observations of the NGC 1550 and Abell 1795 galaxy clusters. The RIM, again, achieves a reconstruction error under 1σ . While these results represent a significant improvement over previous attempts to deconvolve X-ray spectra from the instrumental response, they highlight the method's inability to reconstruct local changes in the reconstructed spectra. In the case of using the deconvolved spectra for science, the local features are crucial since they encode physical information about the emitting gas such as the underlying temperature or metallicity. \nHowever, this assumes that the expected models are correct, which implicitly asserts that the literature values of thermodynamic parameters are accurate. While not unreasonable, these results present a different intrinsic spectrum of Abell 1795 and NGC 1550; this underscores the need to validate that the RIM results are physically consistent. \nFuture works will use carefully constructed priors to augment the RIM and score-based models for deconvolving X-ray spectra.", '6. SOFTWARE AND THIRD PARTY DATA REPOSITORY CITATIONS': 'Facilities: CXO \nSoftware: \nastropy', 'A. CHANDRA OBSERVATIONS FOR RESPONSE MATRICES': 'In this section, we enumerate the Chandra observations used to construct the response matrix catalog. \nA.1. SOXS Data Creation Commands \nThis section provides the code used to create the sythetic spectra. \n```\nfrom soxs import ApecGenerator , RedistributionMatrixFile rmf = RedistributionMatrixFile("region1.rmf") # THIS IS ONLY USED TO GET THE RANGE AND SPACING emin = rmf.elo[0] emax = rmf.ehi[-1] nbins = rmf.n\\_e binscale = "linear" agen = ApecGenerator(emin, emax, nbins, binscale=binscale) num\\_spec = 100000 true\\_ys = [] for i in tqdm(range(num\\_spec)): # Randomly select xspec parameters redshift = random.uniform(0.0, 0.8) # redshift temp = random.uniform(0.4, 8.0) # temperature in keV abundance = random.uniform(0.2, 1.2) # Metallicity abundance in Z\\_solar kT = (temp, "keV") norm = random.uniform(0.1, 1) spec = agen.get\\_spectrum(kT, abundance , redshift , norm) n\\_H = random.uniform(0.001, 0.01) # Foreground Galactic Absorption spec.apply\\_foreground\\_absorption(n\\_H, model="tbabs") true\\_ys.append(spec.flux)\n```', 'REFERENCES': '```\nAdam, A., Coogan, A., Malkin, N., et al. 2022a, E1, doi: 10.48550/arXiv.2211.03812\n``` \nAdam, A., Perreault-Levasseur, L., & Hezaveh, Y. 2022b, Pixelated Reconstruction of Gravitational Lenses using Recurrent Inference Machines, arXiv, \ndoi: 10.48550/arXiv.2207.01073'} |
2024EPJC...84.1182C | We investigate the radiative efficiency and jet power in the spacetime of a rotating black hole within the framework of loop quantum gravity LQG which includes an additional LQG parameter. The results show that as the LQG parameter increases the radiative efficiency decreases for slowly rotating black holes while it increases for rapidly rotating black holes. Furthermore the jet power is found to increase for different black hole spins. With the observed data from the wellknown sources A062000 H1743322 XTE J1550564 GRS1124683 GRO J165540 and GRS1915105 we make some constraints on the black hole spin parameter and the LQG parameter. The presence of the LQG parameter broadens the allowed range of the black hole spin parameter for sources A062000 H1743322 XTE J1550564 and GRO J165540. However for the source GRS 1915105 there is no overlap between the allowed parameter regions which implies that the rotating LQG black hole cannot simultaneously account for the observed jet power and the radiative efficiency as in other black hole spacetimes | 2024-11-01T00:00:00Z | ['arXiv:2409.06950', '10.48550/arXiv.2409.06950', '2024arXiv240906950C', '2024EPJC...84.1182C', '10.1140/epjc/s10052-024-13555-2'] | ['General Relativity and Quantum Cosmology'] | Signatures from the observed jet power and the radiative efficiency for rotating black holes in loop quantum gravity | 2,024 | 176 | 0.28 | ['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML'] | 0 | https://arxiv.org/pdf/2409.06950.pdf | {'Signatures from the observed jet power and the radiative efficiency for rotating black holes in loop quantum gravity': "Zhengwei Cheng 1 , Songbai Chen 1 , 2 ∗ , Jiliang Jing 1 , 2 † 1 \n1 1 Department of Physics, Institute of Interdisciplinary Studies, Key Laboratory of Low Dimensional Quantum Structures and Quantum Control of Ministry of Education, Synergetic Innovation Center for Quantum Effects and Applications, Hunan Normal University, Changsha, Hunan 410081, People's Republic of China 2 Center for Gravitation and Cosmology, College of Physical Science and Technology, Yangzhou University, Yangzhou 225009, People's Republic of China \n(Dated: October 29, 2024)", 'Abstract': 'We investigate the radiative efficiency and jet power in the spacetime of a rotating black hole within the framework of loop quantum gravity (LQG), which includes an additional LQG parameter. \nThe results show that as the LQG parameter increases, the radiative efficiency decreases for slowly rotating black holes while it increases for rapidly rotating black holes. Furthermore, the jet power is found to increase for different black hole spins. With the observed data from the well-known sources A0620-00, H1743-322, XTE J1550-564, GRS1124-683, GRO J1655-40, and GRS1915+105, we make some constraints on the black hole spin parameter and the LQG parameter. The presence of the LQG parameter broadens the allowed range of the black hole spin parameter for sources A0620-00, H1743-322, XTE J1550-564 and GRO J1655-40. However, for the source GRS 1915+105, there is no overlap between the allowed parameter regions, which implies that the rotating LQG black hole cannot simultaneously account for the observed jet power and the radiative efficiency as in other black hole spacetimes. \nPACS numbers: 04.70.Dy, 95.30.Sf, 97.60.Lf', 'I. INTRODUCTION': 'It is believed widely that general relativity could be replaced by quantum gravity at the Planck scale, even if general relativity is the most successful theory to describe gravity at present. One of interesting quantum theories of gravity is loop quantum gravity (LQG) [1-3] that preserves many features of general relativity while simultaneously employing quantization of both space and time at the Planck scale in the tradition of quantum mechanics. The effects originating from LQG could resolve the singularity problem appeared in cosmology and black hole physics [4-6]. Recently, based on the polymerization procedure in LQG, a regular and static black hole solution without any spacetime curvature singularity has been derived through the mini-superspace approach [7]. The effects of LQG in this solution are parameterized by the minimal area and the Barbero-Immirzi parameter. Additionally, the metric is self-dual in the context of T-duality because its form is invariant under the exchange r → a 0 /r [8], where a 0 is proportional to the minimum area A min in LQG and r is the standard Schwarzschild radial coordinate. The gravitational lensing [9] and the emission spectra [10] in this static self-dual black hole spacetime have been investigated. The effects of LQG on quasinormal modes have been researched for various perturbation fields, including scalar field [11-13], axial gravitational perturbations [14, 15] and polar [16] gravitational perturbations. Observational tests of the selfdual spacetime have been performed within the frame of the Solar system [17] as well as through the analysis of the orbit of S0-2 star orbiting Sgr A* in the central region of our Milky Way [18]. Since the real celestial bodies are inherently rotating, an effective rotating loop quantum black hole (LQBH) solution [19] has been obtained from the spherical symmetric LQBH using the modified Newman-Janis algorithm [20, 21]. The observable effects of LQG parameters, including black hole shadows and quasinormal modes, have been analyzed for this rotating spacetime [19]. Additionally, constraints on the parameters of this rotating LQG black hole have been studied using the observed data of Sgr A* from the Event Horizon Telescope [22]. \nEnergetic transient jets are commonly observed in astrophysical systems containing black holes [23, 24]. While numerous theoretical models have been proposed to explain these jets, the complete mechanism is still absent at present. The Blandford-Znajeck mechanism is certainly a popular scenario for explaining the formation of black hole jets. In this mechanism, black holes jets are assumed to be powered by the rotational energy of the black hole through the magnetic field whose field lines anchored in the horizon of black hole or in the accretion disk can be dragged and twisted by the spin of the black hole [25-27]. This means that the jet power could carry the information of the background black hole spacetime. Moreover, the continuum spectra released by the accretion disk near black holes contain the signals of the central celestial bodies [28-30]. And then, a novel method was firstly proposed [30, 31] to \ntest the metric of black holes by combing the estimates of the jet powers with the estimates of the radiative efficiency of the disks. Utilizing this method along with relevant observational data, one can constrain the metric parameters and further evaluate corresponding theories of gravity. Based on the relation between jet power and spin identified in [31-33], parameter constraints on the Kerr-Sen black hole were analyzed in [34], suggesting that Kerr black holes are favored over Kerr-Sen black holes with dilaton charges. Additionally, using data on radiative efficiencies and jet powers, the suitability of the Kerr-Taub-NUT spacetime to describe the geometry around known sources [35-40] was examined [41]. Similar analyses were extended to the case of a rotating black hole within a perfect fluid dark matter model [42]. In this paper, we will investigate effects of the LQG parameter on the radiative efficiency and the power of relativistic jets from the Blandford-Znajeck mechanism in the rotating LQG black hole spacetime, and further probe the possibility of indirect observational evidence of LQG by combining with the observed data from the known sources [35-40]. \nThe paper is organized as follows: In Sect. II, we briefly review the rotating LQG black hole and probe effects of LQG parameter on the radiative efficiency and the power of relativistic jets in this background spacetime. In Sect. III, we compare the theoretical model with observations from the known sources [35-40] and constrain the parameters of the black hole. Finally, we present a summary.', 'II. EFFECTS OF THE LQG PARAMETER ON RADIATIVE EFFICIENCY AND POWER OF RELATIVISTIC JETS AROUND A ROTATING BLACK HOLE': "Starting from the effective LQG-corrected Schwarzschild metric [7], a rotating black hole in LQG has been obtained by using the modified Newman-Janis algorithm [20, 21]. In the Boyer-Lindquist coordinates, its metric can be expressed as [19], \nds 2 = H Σ { ∆ Σ ( dt -a sin 2 θdϕ ) 2 -Σ ∆ dr 2 -Σ dθ 2 -sin 2 θ Σ [ adt -( k 2 + a 2 ) dϕ ] 2 } , (1) \nwith \n∆( r ) = r 2 [ 1 -2 GMr ( r + r ∗ ) 2 ] + a 2 , Σ( r ) = k 2 ( r ) + a 2 cos 2 θ, k 2 ( r ) = r 4 + a 2 0 ( r + r ∗ ) 2 , (2) \nwhere M and a denote the mass and the spin parameter of the black hole, respectively. The quantity r ∗ is related to the LQP parameter P by the equation r ∗ = 2 GMP/ (1 + P ) 2 , where P is defined as \nP ≡ √ 1 + ϵ 2 -1 √ 1 + ϵ 2 +1 . (3) \nThe quantity ϵ is defined as the product of the polymeric parameter δ and the Immirzi parameter γ , expressed as ϵ = γδ . Its value is strictly constrained within the range ϵ = γδ ≪ 1, which ensures the validity of the effective metric \nderived from the polymerization procedure in LQG [7]. This means that the possible physical range of P is given by 0 ≤ P ≤ ( √ 2 -1) 2 ≈ 0 . 1716. The parameter a 0 is related to the minimum area gap of LQG, denoted as A min , by the equation a 0 = A min / 8 π , which implies that a 0 is proportional to l Pl and is expected to be negligible. Thus, we here consider only the case a 0 = 0. The quantity H is a undefined regular function in the metric (1), which must satisfy lim a → 0 H = r 2 + a 2 0 /r 2 [17] and lim P → 0 H = r 2 + a 2 cos 2 θ because the rotating LQG black hole (1) is expected to recover to the non-rotating self-dual black hole [17] as a = 0 and to the Kerr black hole as P = 0. Combining with the previous discussion that a 0 can be negligible, we can set a 0 = 0 without loss of generality. Thus, we take the form H = r 2 + a 2 cos 2 θ as presented in [19]. In this case, the event horizon radius of rotating LQG black hole (1) can be obtained by solving a simplified equation \n(1 + P ) 4 r 4 -2 M (1 + P ) 2 (1 + P 2 ) r 3 +[4 M 2 P 2 + a 2 (1 + P ) 4 ] r 2 +4 a 2 MP (1 + P ) 2 r +4 a 2 M 2 P 2 = 0 , (4) \nwhich implies that the event horizon radius depends on the values of M , a , and P . The explicit expressions of the outer and inner horizon radii r ± are given in [19], which explore that the presence of the parameter P results in a decrease in the value of r + , while increasing the value of r -. Thus, the dependence of horizon radii r ± on the parameter P is analogous to their dependence on the spin parameter a . In Fig.(1), we present the physical parameter \nFIG. 1: Regions of existing two event horizons, one degenerate horizon and no horizon in the physical parameter space of P and a . The dashed line corresponds to the case a = 0 . 5413. \n<!-- image --> \nspace of P and a , highlighting regions corresponding to two event horizons, one degenerate horizon, and no horizon. It is shown that as 0 ≤ a < 0 . 5413, the black hole always has two horizons and it is non-extremal in the physical region of P ∈ [0 , ( √ 2 -1) 2 ]. As a ≥ 0 . 5413, the range of P existing black hole solution decreases with the spin parameter a , which is consistent with that obtained in [19]. In the high spin case with a = 1 . 0, there exists only an extremal black hole with P = 0. \nGenerally, the accretion process occurred near black holes are very complex. Here, we focus on only a simple Novikov-Thorne accretion disk model [28] . The Novikov-Thorne accretion disk is geometrically thin, allowing the heat produced by viscous stresses and dynamic friction to be efficiently dissipated through radiation across its surface. This cooling mechanism ensures that the disk remains in hydrodynamical equilibrium. In this geometrically thin disk model, it is assumed that gravitational forces dominate gas motion over gas pressure, causing the particles in the disk to follow circular geodesic orbits in the equatorial plane. The Novikov-Thorne disk exhibits a black body spectrum [28, 43-45], which is highly sensitive to the position of the inner edge of the accretion disk. Generally, it is assumed that the inner edge of this thin disk is located at the Innermost Stable Circular Orbit (ISCO) radius. With the observed mass of the black hole, the inclination angle of the disk, and the distance between the black hole and the observer, one can infer the corresponding location of the ISCO radius [46]. Theoretically, the ISCO radius can be inferred from the effective potential of a timelike particle orbiting a black hole. For an axially symmetric and stationary spacetime (1), with the time-like condition of the particle 4-velocity u µ u µ = -1, one can obtain \ng rr u 2 r + g θθ u 2 θ = V eff , (5) \nwith the effective potential V eff \nV eff = E 2 g ϕϕ +2 ELg tϕ + L 2 g tt g 2 tϕ -g tt g ϕϕ -1 , (6) \nwhere E and L are respectively the conserved energy and the angular momentum of the orbiting particle. In term of the conditions for particles moving along circular orbits V eff ( r ) = 0 and V ' eff ( r ) = 0 ( where the apostrophe ' denotes a derivative with respect to the radial coordinate r ), one can obtain \nE = -g tt -Ω g tϕ √ -g tt -2Ω g tϕ -Ω 2 g ϕϕ , (7) \nL = Ω g ϕϕ + g tϕ √ -g tt -2Ω g tϕ -Ω 2 g ϕϕ . (8) \nHere Ω = dϕ/dt is the angular velocity of the particle orbiting around the black hole [44] \nΩ = dϕ dt = -g tϕ,r ± √ {-g tϕ,r } 2 -{ g ϕϕ,r }{ g tt,r } g ϕϕ,r (9) \nTogether with the condition V '' eff ( r ) = 0, we can calculate the ISCO radius in the spacetime of a rotating black hole in LQG (1). Fig. (2) illustrates the dependence of the ISCO radius on the parameters a and P of the rotating black hole in LQG spacetime, indicating that the ISCO radius decreases as both black hole parameters increase. \nIn the Novikov-Thorne accretion disk, all emitted photons can escape from the disk's surface to infinity. Consequently, the radiative efficiency η is determined by the specific energy of a particle at the marginally stable orbit r isco , \ni.e, \nη = 1 -E isco , (10) \nThe dependence of the conversion efficiency η on the parameters a and P is illustrated in Fig. (3), which shows that the conversion efficiency η decreases with the LQG parameter P for slowly rotating black holes, while it increases for \nFIG. 3: Dependence of the conversion efficiency η on the parameters a and P of the rotating black hole in LQG spacetime. \n<!-- image --> \nrapidly rotating black holes. Additionally, as the black hole spin increases, the conversion efficiency η increases for different values of P , which is similar to that in the Kerr case. It is well known that black holes surrounded by the Novikov-Thorne accretion disk with the same radiative efficiency must own the same thermal spectrum [47], which can be utilized to estimate the parameters of the background black hole. Generally, to constrain the parameters a and P in the black hole metric (1), we must firstly construct a theoretical thermal spectrum model of the disk surrounding the black hole, and compute the statistical quantity χ 2 to find the best-fit values by comparing the observed spectra of the \n<!-- image --> \nFIG. 2: Dependence of the ISCO radius on the parameters a and P of the rotating black hole in LQG spacetime. \n<!-- image --> \nsources with the theoretical spectrum derived from the standard accretion disk model. This fitting method requires six free parameters: the black hole mass M and its spin a , the LQG parameter P , the mass accretion rate ˙ M , the black hole distance d , and the inclination angle i of the disk with respect to the line of sight of the distant observer, which implies that this procedure is probably long and time-consuming [46]. A simply approximate way to constraining black hole parameters is to estimate the radiative efficiency [29, 30]. The main reason is that the radiative efficiency depends on the spacetime metric and it directly connects the measured luminosity to the key physical parameter governing accretion dynamics. This method of applying radiative efficiency is extensively employed to constrain the black hole parameters in various scenarios [29-34, 41, 42]. \nBlack hole jets are one of the most spectacular astronomical sights in the sky. Generally, jets near the central compact body are classified as the steady non-relativistic jets and the transiently relativistic jets. The latter are believed to originate from close to the event horizon [26], making them valuable tools for extracting information about the central black hole. However, the mechanisms that generate relativistic transient jets are so complicated that they are not fully understood at present [48, 49]. The Blandford-Znajeck process is such a mechanism in which the relativistic jets are powered through extracting energy from the magnetic fields around the accretion disk, which are dragged and twisted by the spin of the black hole. Here we employ the Blandford-Znajeck mechanism to estimate the power of transiently relativistic jets around the rotating LQG black hole (1). Assuming that the electromagnetic field dominates and other contributions can be neglected, the total energy-momentum tensor can be approximated as \nT tot µν ≃ T EM µν = F µα F α ν -1 4 g µν F αβ F αβ , (11) \nwhere F µν = A ν,µ -A µ,ν is the electromagnetic field tensor related to the four potential A µ . The corresponding covariant conservation equation can be simplified as \n∇ µ T EM µν = 0 . (12) \nFor a force-free magnetosphere, the electromagnetic tensor satisfies \nF µν J ν = 0 , (13) \nwhere J ν is the current 4-vector. From Eq.(13), one can get the following relationship \nA t,r A ϕ , r = A t,θ A ϕ,θ = -ω ( r, θ ) , (14) \nwhere ω ( r, θ ) is defined as the electromagnetic angular velocity [25]. With the force-free condition (14), and the assumption that the four potential A µ is stationary and axisymmetric, one can find that the electromagnetic tensor \ncan be further expressed as \nF µν = √ -g 0 -ωB θ ωB r 0 ωB θ 0 B ϕ -B θ -ωB r -B ϕ 0 B r 0 B θ -B r 0 . (15) \nIn the context of the Blandford-Znajeck model, the power of the relativistic jets can be given by [25] \nP BZ = 4 π ∫ π/ 2 0 √ -gT r t dθ , (16) \nwhere T r t is the radial component of the Poynting flux \nT r t = g rr g θθ F rθ F θt -g rt g θθ F 2 tθ + g rϕ g θθ F ϕθ F θt . (17) \nThus, the spacetime affects the power of the relativistic jets through its metric determinant and the radial Poynting flux T r t . Assuming that the jet launching radius corresponds to the event horizon, the radial component of the Poynting flux can be further expressed as \nT r t = 2 r H M sin 2 θ ( B r ) 2 ω [Ω H -ω ] | r = r H , (18) \nwhere Ω H is the angular velocity evaluated at the event horizon r H with a form \nΩ H = -g tϕ g ϕϕ ∣ ∣ ∣ ∣ r H = a ( r ∗ + r H ) 2 a 2 ( r ∗ + r H ) 2 + r 4 H . (19) \nFrom Fig. (4), we find that the absolute value of the angular velocity Ω H at the event horizon increases with P . For \nFIG. 4: Dependence of angular velocity at the event horizon Ω H on the parameters a and P of the rotating black hole in LQG spacetime. \n<!-- image --> \nthe Blandford-Znajek model, the leading-order contribution to the jet power [25, 50, 51] is directly proportional to the square of Ω H \nP BZ = k Φ 2 tot Ω 2 H , (20) \nwhere k = 1 / 6 π for a split monopole field profile and k = 0 . 044 for a paraboloidal one [25]. Φ tot is the magnetic flux threading the event horizon, which is given by \nΦ tot = 2 π ∫ π 0 √ -g | B r | dθ . (21) \nAlthough this result is obtained in the Kerr metric [50, 51], the analyses of the Blandford-Znajeck mechanism in other theories of gravity show that the specific gravity model affects only higher-order corrections, while the jet power at leading order retains the same form as given in Eq. (20) [52]. Since the jet power P BZ depends on the metric through Ω H , we also present the change of angular velocity with the parameters a and P which characterize the gravitational field of the rotating black hole in loop quantum gravity (1).", 'III. COMPARISON OF THEORETICAL MODELS WITH OBSERVATIONS': 'The previous discussions show that both the radiative efficiency and the jet power are sensitive to the spacetime metric. Therefore, through comparing theoretical models with the corresponding observations, one can obtain some signatures for rotating black holes in LQG and gain insights into the observationally favored range of the LQG parameter P . There are observational samples from six X-ray binaries including GRS1915+105, GROJ1655-40, XTEJ1550-564, A0620-00, H1743-322 and GRS1124-683, whose jet power and radiative efficiency are known from observations [31, 32, 53]. With these data, the estimates of the spin parameter a for the Kerr black hole and the corresponding radiative efficiency η are listed in Table I. \nTABLE I: Parameters of the transient black hole binaries. The radiative efficiency η is obtained from the spin measurement for the Kerr metric. \nFollwing the procedure in [31, 32], one can calculate the jet power for the above six microquasars. Using the natural units, the proxy for the jet power is given by \nP jet = ( ν 5 GHz )( S tot ν, 0 Jy )( D kpc ) 2 ( M M ⊙ ) -1 , (22) \nwhere S tot ν, 0 is the beaming corresponding to the approaching and receding jets. Here the entire power in the transient jet is assumed to be proportional to the peak 5 GHz radio flux density. To correct for the beaming, the Lorentz factor \nrewritten as \nlog P BZ = log K +2logΩ H , (23) \nwhere K = k Φ 2 tot . The magnitude of K has been estimated by fitting above equation to the observed jet power P BZ as well as the angular velocity Ω H [31, 56]. It is shown that log K = 2 . 94 ± 0 . 22 for the Lorentz factor Γ = 2 and log K = 4 . 19 ± 0 . 22 for Γ = 5 at 90% confidence level [56]. Generally, the value of K is not a constant for all the sources. However, for the above six sources, each has a mass of approximately 10 solar masses and the corresponding Eddington-scaled mass accretion rates are also similar because transient jets happen during the transition from the hard to the soft state. This implies that for the above six sources the value of K can be treated as a constant and is independent of the spacetime geometry. Therefore, making use of these determined values of K together with the observed jet power of the sources in Table II, we can make certain constraints on the spin parameter a and the LQG parameter P of the rotating black hole (1). \nIn Figs.(5) and (6), we present the allowed regions in the parameter space P -a derived from the above six sources. In each panel, the red region and the blue region respectively correspond to the radiative efficiency and the jet power. The shade region indicates the scenario in which the compact object described by the rotating LQG metric (1) is a naked singularity rather than a black hole. With increasing of the LQG parameter P , the allowed value of the spin parameter a for the source system A0620-00 increases for different radiative efficiency η . For the sources H1743-322 and XTE J1550-564, the allowed value of the spin parameter a decreases for the cases with higher radiative efficiency η and increases for the cases with lower efficiency. However, for the sources of GRO J1655-40, GRS 1124-683 and GRS 1915+105, the allowed value of the spin parameter a decrease across all radiative efficiencies. The allowed value of a from the jet power decreases with P for all six sources. Moreover, the width of the allowed ranges of a narrows \nΓ associated with the jet is anticipated to fall within the range 2 ≤ Γ ≤ 5 [54, 55], which commensurates with the mildly relativistic jets in microquasars. The Doppler corrected jet powers with the Lorenz factor Γ = 2 and Γ = 5 for the six black hole sources are respectively listed in Table II [53, 56]. From Eq.(20), the power of the jet can be \nTABLE II: Proxy jet power values in units of kpc 2 GHz Jy M -1 ⊙ . \nFIG. 5: The allowed regions in the parameter space P -a for different sources. Panels from the top to the bottom respectively correspond to the sources A0620-00, H1743-322, XTE J1550-564 and GRO J1655-40. The red region and the blue region respectively correspond to the radiative efficiency and the jet power. Left panel is for the Lorentz factor Γ = 2 and the right one for Γ = 5. \n<!-- image --> \nwith increasing P from both the radiative efficiency and the jet power for these sources. The jet power provides the stronger constraint on a for the source A0620-00, while the radiative efficiency offers a stronger constraint on a for the source GRS 1915+105. \nCombining with the two constrain regions from the radiative efficiency and the jet power, we find that there exist intersection regions in both cases the Lorentz factor Γ = 2 and Γ = 5 for the sources A0620-00, H1743-322, XTE \nJ1550-564 and GRO J1655-40, which implies that the gravitational fields of these four sources could be described by the rotating LQG black hole (1). However, for the source GRS 1124-683, one can find the intersection of the constrain regions exists in the case Γ = 5 but disappears in the case Γ = 2. Especially, for the source GRS1915+105, there is no intersection for the two constrain regions from the radiative efficiency and the jet power in either cases Γ = 2 or Γ = 5. The results of the source GRS 1124-683 hint that the intersection regions could exist in the case with the more high Lorentz factor Γ for the source GRS1915+105. The largest possible allowed ranges of the black \nFIG. 6: The allowed area in the parameter space P -a for the sources GRS 1124-683 and GRS1915+105. The top panels are for the source GRS 1124-683 and the bottom one are for GRS1915+105. The red region and the blue region respectively from the radiative efficiency and the jet power. Left panel is for the Lorentz factor Γ = 2 and the right one for Γ = 5. \n<!-- image --> \nhole parameters a and P , derived from the intersection region between the radiative efficiency and the jet power, are listed in Table (III) for the six sources. Comparing with the Kerr black hole, the presence of the LQG parameter P \nTABLE III: The largest possible allowed ranges of the black hole parameters a and P from the intersection region between the radiative efficiency and the jet power of the six different sources. \nbroadens the allowed range of the black hole spin parameter a for the sources A0620-00, H1743-322, XTEJ1550-564 and GROJ1655-40. However, for the source GRS1124-683, one can find from the case Γ = 5 that the largest allowed \nrange of a for the rotating LQG black hole (1) is the same as that in the Kerr case. Moreover, the absence of an overlapping region corresponding to these two observational constraints for the source GRS1915+105 is also found in other spacetimes [41, 42]. This leads to some arguments [58] on the validity of the correlation between jet powers and black hole spins proposed by Narayan and McClintock in [31] and [32]. However, it is too early to assess these arguments due to the limited number of available sources. With the increasing high precision observational data, it is expected that the validity of such kind of correlations could be confirmed in the future.', 'IV. CONCLUSION': 'We have studied the radiative efficiency and the jet power in the spacetime of the rotating LQG black hole (1). Our results show that with the increase of the LQG parameter P , the conversion efficiency decreases for the slowly rotating black hole, but increases for the rapidly rotating black hole. The absolute value of the angular velocity Ω H at the event horizon, associated with the jet power, increases with the LQG parameter P . With the observed data from the well-known sources A0620-00, H1743-322, XTE J1550-564, GRS1124-683, GRO J1655-40, and GRS1915+105, we also make some constraints on the black hole spin parameter a and the LQG parameter P . It is found that the allowed ranges of a decreases with P based only on the radiative efficiency or from the jet power for these sources. The jet power provides the stronger constraint on a for the source A0620-00, while the radiative efficiency provides the stronger constraint on a for the source GRS 1915+105. \nWith the data from both the radiative efficiency and the jet power, we find that there exist intersection regions in both cases the Lorentz factor Γ = 2 and Γ = 5 for the sources A0620-00, H1743-322, XTE J1550-564 and GRO J1655-40, which implies that the gravity of these four sources could be described by the rotating LQG black hole (1). The presence of the LQG parameter P broadens the allowed range of the black hole spin parameter a for these four sources. However, for the source GRS 1124-683, one can find the intersection of the constrain regions exists in the case Γ = 5 but disappears in the case Γ = 2. Especially, for the source GRS1915+105, there is no intersection for the two constrain regions respectively derived from the radiative efficiency and the jet power in both cases Γ = 2 and Γ = 5. The phenomenon of no overlapped region corresponding to these two observational constraints for the source GRS1915+105 is also found in other spacetimes [41, 42], which yields some arguments [58] on the validity of the correlation between jet power and the black hole spin proposed by Narayan and McClintock in [31] and [32]. It is anticipated that the validity of such kind of correlations could be confirmed in the future with the increasing high precision observational data.', 'Acknowledgments': "This work was supported by the National Natural Science Foundation of China under Grant No.12275078, 11875026, 12035005, 2020YFC2201400, and the innovative research group of Hunan Province under Grant No. 2024JJ1006. \n- [1] C. Rovelli, Living Rev. Rel. 1 , 1 (1998) 1.\n- [2] T. Thiemann, Lect. 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